On the optimum thermal design of individual longitudinal fins with rectangular profile

The optimum design of single longitudinal fins with constant thickness, considering different uniform heat transfer coefficients on the fin faces and on the tip, has been approached by meals of an accurate mathematical method yielding the solution of constrained minimization (maximization) problems. Starting from the classical one-dimensional (1-D) model of the fin, the optimum design problem is reduced to a constrained optimization one by considering the limitations of the fin thermal convenience criterion, of the I-D accuracy criterion, or of the geometric constraint on the primary surface of the fin array. The analysis, developed in dimensionless form, shows that the existence and the uniqueness of the solution are nor ensured in any case, and the condition of the solution existence is often a consequence of the imposed constraints. A comparison between the results obtained and those achieved by applying to the fin optimization the half-thickness rule (HTR), based on the Harper and Brown approximation, has been carried out for some meaningful cases. Even if the HTR is usually satisfactory to design optimum fins, its uncritical use is very questionable.