An iron rod $$\left( {K = 41.5W/mK} \right)$$ of $$15mm$$ diameter and $$160mm$$ long extends out of a hot surface of temp $${150^ \circ }C$$ into environment at $${36^ \circ }C.$$The free end of the rod is insulated. If the film heat transfer coefficient is $$25W/{m^2}K,$$ calculate the rate of heat flowing out of the hot surface through the rod and the temp at the insulated end of the rod.

The total efficiency, $${\eta _t}$$ for a finned surface may be defined as the ratio of the total heat transferred by the combined area of the fins, $${A_r},$$ and the exposed surface to that by the total area, $$A$$, if it were maintained at the base temp, $${T_0}.$$ Assuming uniform heat transfer coefficient over the entire surface, derive an expression for the relationship between efficiency and effectiveness of the fin.