Optimal tree-shaped networks for fluid flow in a disc-shaped body

Abstract

In this paper we consider 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 centered at O. The flow may proceed in either direction, from the center to the perimeter, or from the perimeter to the center. This problem is an integral component of the electronics cooling problem of how to bathe and cool with a single stream of coolant a disc-shaped area or volume that generates heat at every point. The smallest length scale of the flow structure is fixed ( d), and represents the distance between two flow ports on the circular perimeter. The paper documents a large number of optimized dendritic flow structures that occupy a disc-shaped area of radius R. The flow is laminar and fully developed in every tube. The complexity of each structure is indicated by the number of ducts ( n 0) that reach the central point, the number of levels of confluence or branching between the center and the perimeter, and the number of branches or tributaries (e.g., doubling vs. tripling) at each level. The results show that as R/ d increases and the overall size of the structure grows, the best performance is provided by increasingly more complex structures. The transition from one level of complexity to the next, higher one is abrupt. Generally, the use of fewer channels is better, e.g., using two branches at one point is better than using three branches. As the best designs become more complex, the difference between optimized competitors becomes small. These results emphasize the robustness of optimized tree-shaped networks for fluid flow.