Long metallic rods of circular cross section are heat treated by passing an electric current through the rods to provide uniform volumetric heat generation at a rate q (W/m3). The rods are of diameter D and are placed in a large chamber whose walls are maintained at the same temperature T∞ , as the enclosed air. Convection from the surface of the rods to the air is characterized by coefficient h.
a) Assuming negligible internal resistance, derive a differential equation that could be solved to determine the rod temperature as a function of time during the heating process. Express your result in terms of the foregoing variables and appropriate physical properties of the rod material. Do not attempt to solve.
b) Obtain an expression that could be used to determine the steady-state temperature of the rod.
c) Neglecting radiation effects and prescribing an initial (t = 0) rod temperature of = T ∞, solve the equation in part a to obtain the transient temperature response of the rod.