OPTIMIZATION OF HEAT CONDUCTION FOR TREELIKE NETWORK WITH ARBITRARY CROSS-SECTIONAL SHAPE

To minimize the thermal resistance of the fractal treelike heat conduction network and develop an optimization principle applicable for the network with an arbitrary cross-sectional shape, this paper first establishes a theoretical model regarding the total thermal resistance of the symmetric treelike network with arbitrary cross-sectional shapes and then studies the effects of the geometric and structural parameters of the network on its total thermal resistance. The numerical simulations are also performed to analyze the influences of the geometric and structural parameters of symmetric treelike networks with circular, rectangular and triangular cross-sectional shapes on the total thermal resistance. Both the theoretical model and the numerical simulation show that the total thermal resistance of the network with an arbitrary cross-sectional shape first decreases and then increases with increasing cross-sectional area ratio but always increases with increasing length ratio of branches at two successive branching levels when the total branch volume is constant. When the cross-sectional area ratio is equal to the reciprocal of the branching number, the treelike network has the minimum total thermal resistance. This scaling law is applicable for the treelike network with an arbitrary cross-sectional shape to achieve the minimum total thermal resistance.