This project uses the steady-state heat equation to model the temperature distribution in an industrial furnace used for metal production, for example, a blast furnace.
The heat flow is assumed to be steady-state, so that only an elementary ordinary differential equation (ODE) is needed, and is easily solved explicitly. In particular, we examine the issue of how one can determine thickness of the protective ``refractory'' lining on the furnace inner wall, and when it is wearing too thin, a safety issue.
This introduces the idea of an inverse problem, in which one wants to determine one or more important physical parameters for a system when these parameters cannot be measured directly. Students should have some exposure to ODE's, but no specific solution technique is needed (beyond simple integration). It is helpful if they have seen the idea behind Newton's Law of Cooling (heat flows from hot to cold, in proportion to temperature difference).
A computer algebra system can be helpful for performing computations, but is not essential.
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