Optimization of Finned Arrays

Abstract

There are many occasions when the optimization of a finned array can be profitable from the standpoint of either minimum weight for a prescribed energy transfer or for enhanced energy transfer from a given weight of material. Kraus et al. [1], following the assumptions posed Murray [2] and Gardner [3], showed that for individual longitudinal fins of rectangular profile (and others) and in finned arrays composed of these fins, conditions of heat flow and temperature excess at any point on a fin are induced by similar conditions at the fin base. In particular, it has been shown that, for a single fin, there is a linear transformation which maps conditions from the fin tip to conditions at the fin base

$$ \left[ {\begin{array}{*{20}{c}}{{\theta _b}} \\{{q_b}}\end{array}} \right] = {\text{ T}}\left[ {\begin{array}{*{20}{c}}{{\theta _a}} \\{{q_a}}\end{array}} \right] = \left[ {\begin{array}{*{20}{c}}{{\tau _{11}}}&{{\tau _{12}}} \\ {{\tau _{21}}}&{{\tau _{22}}}\end{array}} \right]\left[ {\begin{array}{*{20}{c}}{{\theta _a}} \\{{q_a}} \end{array}} \right] $$

((1))

and it was demonstrated that this T matrix could be used to obtain a new parametization, called the input admittance that would completely describe the performance of a fin. The conventional fin efficiency was abandoned and it was proposed that single fins and finned arrays should be characterized by this single, yet important, parameter. The input admittance was further shown by Kraus and Snider [4] to be the ratio of the heat entering the base of a single fin or a finned array to the temperature excess at the base of the fin or the finned array.