It has been proposed that highly concentrated solar energy can be used to economically process materials when it is desirable to rapidly heat the material surface without significantly heating the bulk. In one such process for case hardening low-cost carbon steel, the surface of a thin disk is to be exposed to concentrated solar flux. The distribution of absorbed solar flux on the disk is given by

where r is the distance from the disk axis and q" max and Ro are parameters that describe the flux distribution. The disk diameter is 2Rs, its thickness is Zs, its thermal conductivity is k and its thermal diffusivity is α. The disk is initially at temperature Tinit and at time t = 0 it is suddenly exposed to the concentrated flux. Derive the set of explicit difference equations needed to predict how the disk temperature distribution evolves with time. The edge and bottom surface of the disk are insulated and reradiation from the disk may be neglected.GIVENSteel disk exposed to concentrated solarflux

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1-0.9 R, q" () = q" max