Quick design of truncated pin fins of hyperbolic profile for heat-sink applications by using shortened power series

A heat sink normally consists of a substrate on which a bundle of pin fins made of a highly conducting metal like aluminum or copper is attached. Its performance can be qualified by the space they occupy in relation to the thermal resistance they provide, which requires the solution of the quasi one-dimensional heat conduction equation in the geometry of interest. This technical brief reports exact and approximate analytical procedures for the truncated pin fin of hyperbolic profile. The exact solution involves rather intricate Bessel functions, whereas the approximation is based on easily computable power series. Compared to the exact figures, simple, truncated power series having between five and eight terms yield results of excellent quality for practical combinations of the two controlling parameters: the thermal length mL and the coordinate ratio W. Results are presented for the two variables of interest in thermal design: the fin tip temperature (a local quantity) and the fin thermal efficiency (a global quality ratio).