Flat Plate Key Problem

Let the rate of energy per unit volume u x) be generated in a flat plate and let the thickness and the thermal conductivity of the plate be 2i and k(T), respectively. Under steady conditions, the total energy generated in the plate is transferred, with a heat transfer coefficient h, to an ambient at temperature Too This plate could be one of the fuel plates of a nuclear reactor core or one of the elements of an electric heater.4 

Following the five steps of formulation, first we consider the differential system (Step 1) shown in Fig. 2.12(a). The first law of thermodynamics (Step 2), Eq. (1.16) interpreted in terms of Fig. 2.12(b), yields 

4 For electrical applications, PR = PRnO- + T) and the energy generation is more conveniently represented by u (T). 

In the last step of our formulation, we need the origin of the coordinate axis. Because of the geometric as well as the thermal symmetry of the problem, this origin is assumed to be on the midplane. Then, the first boundary condition is (Step 5) 

For a distributed energy generation and a variable thermal conductivity, multiplying Eq. (2.47) by dx and integrating the result gives 

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This is the same result as the one found for the flat-plate key problem, Eq. (2.57) Adding the dads has no effect on the outside surface temperature for flat-plate fuel elements (what about, cylindrical elements ). Let us now look at the inside temperatures. Since the same total energy balances the heat flux at any cross section of clads,... 

Also, Eq. (2.58) of the flat plate key problem gives the temperature of the midplane relative to the temperature of the interface,... 

For an application of the foregoing general considerations, reconsider the flat plate (key problem) of Section 2.3, Let the uniform internal energy u " be suddenly generated and thereafter held constant in the plate which has a uniform initial temperature Foe-The governing equation for a variable conductivity, obtained. from the one-dimensional form of Eq. (3.73), is... 

The second corresponds to our key problem for the flat plate (of thickness l). Because of the symmetry with respect to the middle plane, each surface of the plate transfers one-half of the energy generated within the plate, that is,... 

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