Uniform internal heat generation at q = 5 X 107 W/m3 is occurring in a cylindrical nuclear reactor fuel rod of 50-mm diameter, and under steady-state conditions the temperature distribution is of the form T(r) = a + br2, where T is in degrees Celsius and r is in meters, while a = 800°C and b = -4.167 x 105°C/m2. The fuel rod properties are k = 30 W/m ∙ K, p = 1100 kg/m3, and c p = 800 J/kg ∙ K.

(a) What is the rate of heat transfer per unit length of the rod at r = 0 (the centerline) and at r = 25 mm (the surface)?

(b) If the reactor power level is suddenly increased to q2 = 108 W/m3, what is the initial time rate of temperature change at r = 0 and r = 25 mm?