Consider the configuration of Example 3.8, where uniform volumetric heating within a stainless steel tube is induced by an electric current and heat is transferred by convection to air flowing through the tube. The tube wall has inner and outer radii of r1 = 25 mm and r2 = 35 mm, a thermal conductivity of k = 15 W/m ∙ K, an electrical resistivity of Pe = 0.7 X 10-6 fl. m, and a maximum allowable operating temperature of 1400 K.

(a) Assuming the outer tube surface to be perfectly insulated and the air flow to be characterized by a temperature and convection coefficient of T∞,1 = 400 K and hi = 100 W/m2 ∙ K, determine the maximum allowable electric current I.

(b) Compute and plot the radial temperature distribution in the tube wall for the electric current of part (a) and three values of hi (100,500, and 1000 W/m2 ∙ K). For each value of hi' determine the rate of heat transfer to the air per unit length of tube.

(c) In practice, even the best of insulating materials would be unable to maintain adiabatic conditions at the outer tube surface. Consider use of a refractory insulating material of thermal conductivity k = 1.0 W/m ∙ K and neglect radiation exchange at its outer surface. For hi = 100 W/m2 ∙ K and the maximum allowable current determined in part (a), compute and plot the temperature distribution in the composite wall for two values of the insulation thickness (δ = 25 and 50 mm). The outer surface of the insulation is exposed to room air for which T∞.2 = 300 K and h2 = 25 W/m2 ∙ K. For each insulation thickness, determine the rate of heat transfer per unit tube length to the inner air flow and the ambient air.