,

with

at ,

at ,

and

,

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:

and

.

Thus the equations become

,

at ,

at ,

and

,

with

and

.

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 and 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].