Solve the following:
Q: A nuclear reactor fuel element consists of a solid cylindrical pin of radius r1 and thermal conductivity kf. The fuel pin is in good contact with a cladding material of outer radius r2 and thermal conductivity kc. Consider steady-state conditions for which uniform heat generation occurs within the fuel at a volumetric rate q and the outer surface of the cladding is exposed to a coolant that is characterized by a temperature T8 and a convection coefficient h.
(a) Obtain equations for the temperature distributions Tf(r) and Tc(r) in the fuel and cladding, respectively. Express your results exclusively in terms of the foregoing variables.
(b) Consider a uranium oxide fuel pin for which kf = 2 W/m · K and r1 = 6 mm and cladding for which kc = 25 W/m · K and r2 = 9 mm. If q = 2 X 108 W/m3, h = 2000 W/m2 · K, and T8 = 300 K, what is the maximum temperature in the fuel element?
(c) Compute and plot the temperature distribution, T(r), for values of h = 2000, 5000, and 10,000 W/m2 · K. If the operator wishes to maintain the centerline temperature of the fuel element below 1000 K, can she do so by adjusting the coolant flow and hence the value of h?