The Minimum Weight One-Dimensional Straight Cooling Fin

Abstract

The Minimum Principle is used to obtain the minimum weight one-dimensional cooling fin without the length-of-arc assumption. This approach enlarges the class of optimum fin designs available for consideration. The heat-transfer coefficient and the, thermal conductivity can be temperature dependent. It is shown that the tip dimension vanishes for typical profiles. The results are specialized to make a comparison with results obtained with the length-of-arc assumption. With initial conditions obtained from the approximate method for kθ0 /q0 = 10, the fin is 15 percent shorter than the classical parabolic profile and contains about 1.6 percent less material. The shorter fin length in this example and the flexibility in specifying the heat flux per unit fin base area for any fin are practical design considerations. The optimum fin is described by a nonlinear two-point boundary-value problem whose solution is obtained numerically. Computational guides are provided.