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Analytical Solutions of Branching Fins

Ahmad Fakheri
Bradley University

DOI: 10.1615/TFESC1.fnd.013226
pages 895-907

KEY WORDS: Branching fins, Constructal law, Murray’s Law, Tree shaped structures, Bifurcating systems


This paper presents a closed form analytical solution for the conductance of branching fins, which is then used to obtain solutions for several specific cases. It is shown that in the absence of imposed constraints, branching fins do not have a global optimum, and the system's conductance asymptotes to a constant value for large number of branches and/or children. Furthermore, a bush pattern, where there are fewer branching generations, but a higher number of children at each generation provides higher conductance compared to a tree type shape. Imposing constraints on the problem limits the range of acceptable solutions to a subset of all solutions that satisfy the imposed constraints. The reported optimums are either the largest or the smallest value in the subset, and therefore, depending on the imposed constraints, different maxima may be obtained. The requirements like maximum conductance, minimum work, minimum resistance, constant volume, and constant area are not mandated by nature, and require better justification. Even for constrained problem, the tree shape does not provide any benefits over a simple fin array, which is able to provide a much higher conductance. The Constructal law does not appear to be applicable to this inanimate system.

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