Transient Heat Conduction with a Nuclear Heat Source

Consider a spherical nuclear fuel element consisting of a sphere of fissionable material of radius , surrounded by a spherical shell of alloy cladding with outer radius , initially at temperature , that is suddenly immersed at in a cooling bath of temperature . The heat equation describing this system is:
at ,
at ,
where is the radial coordinate (), is the thermal diffusivity (), is the thermal conductivity (), is the heat transfer coefficient between the sphere and the surrounding fluid (), and is the volume source of thermal energy () generated by the fissionable material. This source is approximated by a simple parabolic function:
where is a dimensionless positive parameter.
To solve this problem, it is convenient to introduce the following dimensionless variables:
Thus the equations become
at ,
at ,
Here is the Biot number, the ratio of internal resistance to conductive heat transfer in the sphere to the external resistance of convective heat transfer from the sphere to the surrounding fluid. The subscripts 1 and 2 refer to thermal properties of the spheres and , respectively.
We show the temperature distribution and the maximum temperature within the system, quantities that are significant when making estimates of thermal deterioration [1].


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A similar steady-state problem is solved in [1].
[1] R. B. Bird, W. E. Stewart and E. N. Lightfoot, Transport Phenomena, rev. 2nd ed., New York: John Wiley & Sons, Inc., 2007.
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