Optimum shapes of convective pin fins with variable heat transfer coefficient

Abstract

Principles of variational calculus are used to determine the shapes of convective pin fins that maximize heat dissipation, given the amount of fin material. The analysis considers the convective heat transfer coefficient h to depend on the fin diameter D according to the relationship h ∝ 1/D n, where n takes on values from 0.2 to 0.5 depending on the type of pin fin configuration and flow condition ( 1 : AIChE J , Vol. 29, p. 1043, 1983). The Euler equations, which are nonlinear and coupled, are formulated and solved for the cases of both length and weight constraints as well as only weight constraint. The resulting quadrature formulae are represented in the form of a convinient design plot, from which the optimum design parameters may be obtained and used to determine the fin and temperature profiles as well as the cooling performance. The solutions under both constraints yield considerably simple results for the case of only weight constraint, which corresponds to the diameter and excess temperature of the fin tip being zero. An important result is that the Schmidt criterion ( 2 : Z. Verein. Deutsch Ing., Vol. 70, pp. 885, 947, 1926) of a linear temperature profile also holds for pin fins of specified weight with a variable heat transfer coefficient. Finally, by using Pontryagin's minimum principle ( 3 : The Mathematical theory of Optimal Processes, Wiley, New York, 1962), it is demonstrated that the problems of maximizing cooling (for a given fin weight) and minimizing weight (for a given fin cooling) are identical as both are governed by the same optimum design equations.