Heat conduction in a spherical nuclear fuel element. Consider a spherical nuclear fuel ele- ment as shown in Fig. 10B.3. It consists of a sphere of fissionable material with radius R, surrounded by a spherical shell of aluminum "cladding" with outer radius R. Inside the fuel element, fission fragments are produced that have very high kinetic energies. Collisions between these fragments and the atoms of the fissionable material provide the major source of thermal energy in the reactor. Such a volume source of thermal energy resulting from nuclear fission we call S, ([=] energy/volume time). This source will not be uniform throughout the sphere of fissionable material; it will be the smallest at the center of the sphere. For the pur- pose of this problem, we assume that the source can be approximated by a simple parabolic function (10B.3-1) Here S, is the volume rate of heat production at the center of the sphere, b is a dimensionless positive constant, and r is the radial distance from the center of the sphere. The known tem- perature at the outside cladding surface (atr=Rc) is T. The thermal conductivities of the fuel element and cladding are ke and ke. Determine the temperature profiles in the fuel element and the cladding. Answers: TF(r) = Te(r) = SR 3kc SR {[- ( + ) ] + [ ( ; ) ] } + (1 + b) (1 R ) + SR 3kc To 6k +To Coolant Tr(r) -RE Rc + Aluminum cladding Sphere of fissionable material [1 +(+;)] To S = 5,0 Fig. 108.3 A spherical nuclear fuel assembly, showing the temperature distribution within the system.