The Model for Temperature Calculation

L vov et al. [4, 5] used the following procedure to model the temperature distribution in a powder sample in the presence of residual air (or foreign gas) and water vapour. The specimen was considered to be composed of horizontal layers of material with the layer thickness equal to a powder grain diameter, so that the modelling could be reduced to a reconstruction of a vertical temperature distribution between the sample layers. (A single crystal sample corresponds to a one-layer specimen.) The furnace temperature above and below the specimen was assumed to be the same so the analysis was limited to consideration of half of this multilayer sample, from a central, zeroth or first layer, up to the nth, outer layer. 

If the thermal conductivity at point contacts between grains is neglected and decomposition conditions are assumed to be stationary, a heat balance equation for any fth layer of the sample can be formulated, which connects the amounts of heat expended for decomposition, radiation, and heat conduction of water vapours and residual air, with the heat accumulated from radiation and heat transfer from the neighbouring i — l)th and i + l)th layers  

The following additional boundary conditions can be introduced for the central ( = 0 or i = 1) coldest layer  

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