Problem 4.16: Consider a nuclear fuel element of spherical form, consisting of a sphere of...

Problem 4.16: Consider a nuclear fuel element of spherical form, consisting of a sphere of "fissionable" material surrounded by a spherical shell of aluminum "cladding" as shown in Fig. P4.16. Nuclear fission is a source of thermal energy, which varies non-uniformly from the center of the sphere to the interface of the fuel element and the cladding. We wish to determine the temperature distribution in the nuclear fuel element and the aluminum cladding.

The governing equations for the two regions are the same, with the exception that there is no heat source term for the aluminum cladding. We have

where subscripts 1 and 2 refer to the nuclear fuel element and cladding, respectively. The heat generation in the nuclear fuel element is assumed to be of the form

where q0 and c are constants depending on the nuclear material. The boundary conditions are

(a) Develop the finite element model, (b) give the form of the assembled equations, and (c) indicate the specified primary and secondary variables at the nodes. Use two linear elements to determine the finite element solution for the temperature distribution, and compare the nodal temperatures with the exact solution