$(30$ pts$)$ A thin but very long circular fin of radius $R$ and length $L($i$.$e$.,LR)$ is mounted on a wall $($as shown in the figure below$).$ The outside wall temperature of the pipe is $T_w$ and the ambient air temperature is $T_a.$ Heat is transferred to the ambient air by surface convection and neglect the heat loss from the edge of the fin. Assume that away from the wall, the temperature of the rod decreases along the rod and reaches the air temperature at another end. Assume that the heat transfer coefficient $h$ and the thermal conductivity $k$ of the fin are constants.

Air temperature: $T_a$

Work out the governing equation, identify the boundary conditions, and derive the temperature distribution along the fin and the cooling efficiency.