A cylindrical stainless steel vessel (k = 15 W/mK) with an inside diameter of 1 m and 0.1 m thick walls is full of radioactive material which generates energy at a rate of q W/m3 . The vessel is submerged in a bath of water, which is maintained at 25 C. The convective heat transfer coefficient between the vessel and water is 1000 W/m2 K. The ends of the cylindrical vessel are capped and very well insulated so that that heat transfer through the ends is negligible. a) Starting with an energy balance around a suitable control volume, derive the differential equation for steady-state one-dimensional heat conduction through the vessel wall and express the boundary conditions. b) Solve the differential equation to obtain a relation for the variation of temperature in the vessel wall. Sketch the temperature distribution.

c) Due to safety reason, the outside surface of the vessel should not exceed 50C. Determine the rate of allowable heat generation rate per unit volume in vessel.