New generations of safer nuclear reactors including the pebble bed reactor use spherical nuclear fuel pellets that consist of a fissionable core of radius, \(r_{\text {core }}\), and thermal conductivity, \(k_{\text {core }}\), that generates an energy per volume \(\dot{q}\). To handle the material and prevent problems if coolant is lost, the spheres are encased in an inert graphite coating of thermal conductivity, \(k_{\text {coat }}\), and outer radius, \(r_{\text {coat }}\). The fuel pellets are immersed in a fluid of temperature, \(T_{\infty}\), and heat transfer coefficient, \(h\).

a. Derive the energy balances for the core and coating of the fuel pellet.

b. What are the boundary conditions at the core of the pellet? The surface of the pellet in contact with the fluid? The interface between the core and cladding of the pellet?

c. Solve for the temperature distribution for the core and the shell. What is the heat flow rate from a pellet into the fluid?

d. If we are interested in maximizing the heat transfer to the fluid and in turn maximizing reactor safety, how thick should the cladding layer be?