Radioactive wastes are packed in a long, thin-walled cylindrical container. The wastes generate thermal energy non-uniformly according to the relation q = q_v [1 – (r/r_o) ²], where q is the local rate of energy generation per unit volume, q_o is a constant, and r₀ is the radius of the container. Steady-state conditions are maintained by submerging the container in a liquid that is at T_∞, and provides a uniform convection coefficient h.

Obtain an expression for the total rate at which energy is generated in a unit length of the container. Use this result to obtain an expression for the temperature Ts of the container wall.

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T h