This study considers the transient heating of an object in a process furnace or in an enclosure with radiating walls from a specified initial condition to a specified steady state following a prescribed temperature history. The necessary power input distributions during the whole heating process for the heaters to satisfy the design specifications are sought. The problem is a transient inverse boundary condition estimation problem, where the geometry and the properties of the surfaces are specified and the boundary condition on the heater wall is to be found by making use of the information provided at the design surface for each time step. The boundary condition estimation problems require the solution of Fredholm equations of the first kind, which are known to be highly ill-posed. The introduction of the transient nature makes the problem even more interesting, challenging and also realistic. In order to model radiative heat transmission, the Monte Carlo method is used as it is aimed to develop a generic solution technique for straight-sided, two-dimensional enclosures of arbitrary shape. The use of the Monte Carlo method also enables us to handle the specularly reflecting surfaces or blockage effects with ease. The inverse problem is solved by the conjugate gradient method, a well-known N-step solution technique, which provides smooth and accurate results after first few steps.
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