Exact,analytic temperature distributions of pin fins with constant thermal conductivity and power‐law type heat transfer coefficient

In this Technical Note, the problem of determining the temperature distribution in a pin fin with power‐law heat transfer coefficients is addressed. It is demonstrated that the governing fin equation, a nonlinear second‐order differential equation, is exactly solvable for the entire range of the exponent n in the power‐law heat transfer coefficients. The exact, closed‐form analytical solutions in implicit form are convenient for physical interpretation and optimization for maximum heat transfer. Furthermore, it is proved that the exact solutions have three different structures: (1) dual in the range of , (2) unique or dual in the range of , and (3) unique in the range of . Additionally, exact analytical expressions for the fin efficiency and the fin effectiveness are provided, both as a function of the dimensionless fin parameter for the gamma of n under study.