Optimization of tree-shaped flow distribution structures over a disc-shaped area

In this paper, we review the fundamental problem of how to design a flow path with minimum overall resistance between one point (O) and many points situated equidistantly on a circle centred at O. This is a fundamental problem in energy engineering: the distribution of fluid, energy, electric power, etc., from points to surrounding areas. This problem is also fundamental in heat transfer and electronics cooling: how to bathe and cool with a single stream of coolant a disc-shaped area or volume that generates heat at every point. This paper outlines, first, a direct route to the construction of effective tree-shaped flow structures. The starting point is the optimization of the shape of each elemental area, such that the length of the flow path housed by the element is minimized. Proceeding towards larger and more complex structures—from elements to first constructs, second constructs, etc.—the paper develops tree-shaped flow structures between one point and a straight line, as an elemental problem, and a circle and its centre. We also consider the equivalent tree-shaped networks obtained by minimizing the pressure drop at every step of the construction, in accordance with geometric constraints. The construction method is applied to a fluid flow configuration with laminar fully developed flow. It is shown that there is little difference between the two methods. The minimal-length structures perform very close to the fully optimized designs. These results emphasize the robustness of optimized tree-shaped flows. Copyright © 2003 John Wiley & Sons, Ltd.