Thermal analysis of an annular fin with (a) simultaneously imposed base temperature and base heat flux and (b) fixed base and tip temperatures

The performance of an annular fin with uniform thickness losing heat by convection to its surroundings is studied analytically for two sets of boundary conditions: (a) simultaneously prescribed base temperature and base heat flux, (b) specified base and tip temperatures. For (a) analytical expressions are derived for the temperature distribution, the tip heat flow, and the Biot number at the tip. For the fin tip to be insulated, a relationship between the base heat flow, the radii of the fin, and the fin parameter must be satisfied. Based on the base heat flow for the insulated tip, an expression is derived for the tip temperature. For part (b), analytical solutions are presented for the temperature distribution, base heat flow, and tip heat flow. For the adiabatic tip condition, a relation between the tip temperature, the fin parameter, and the radii of the fin must be satisfied. The condition under which the fin tip can be freely exposed to the environment is established. The graphical results provide a comprehensive picture of the thermal performance of the annular fin under both cooling and heating conditions.