Tree-shaped networks with loops

Dendritic flow architectures are being contemplated for thermal designs that provide high heat transfer densities for the cooling of electronics. Optimized tree networks maximize the flow access between one point (source, sink) and an infinity of points (line, area, volume). This paper is a fundamental study of a new class of dendritic flow architectures for thermal design: trees combined with closed-loop structures, as in the venation of leaves. The loops provide robustness to the design: the network continues to serve its assigned area even if one or more ducts are damaged. The study documents the achievement of performance and robustness systematically, by starting from the simplest architectures and proceeding toward the more complex, namely, point-circle networks with one loop size and two loop sizes, and networks with loops without and with branching levels. It is shown that the use of loops increases the global flow resistance relative to the dendritic design without loops. Damage, or removal of a duct from the network, also leads to an increase in global flow resistance. These effects become less important as complexity increases, provided that the network is optimized. A damaged peripheral duct induces a smaller penalty than a damaged duct that is situated close to the center of the network. In summary, optimized complex flow structures are robust. Loops are an attractive design feature for maintaining a high level of global performance when the structure experiences local damage.