Optimal structure of damaged tree-like branching networks for the equivalent thermal conductivity

Damage of tree-like branching networks has being widely received considerable attention due to their wide existence in nature, engineering, cardiovascular system, lungs and cooling system of micro channels for electronic components. In the present work, we suppose the damaged number of channels of at some branching level of a tree-like branching network, and then the network becomes asymmetric. The transport properties of fluid flow and heat transfer properties in the damaged network will significantly differ from those in an undamaged network. The equivalent thermal conductivity of the damaged network is derived in this work. We find that the damaged channel number and branching levels have significant effect on optimal diameter ratio and the optimal thermal conductivity. It is found that the larger number of damaged channels results in the lower optimal thermal conductivity and the higher optimal diameter ratio. While the higher damaged branching levels lead to the lower optimal diameter ratio and higher optimal thermal conductivity. It is also found that the length ratios of the channels and total number of branching levels have no effect on optimal diameter ratio and the optimal thermal conductivity of damaged network. The present results may provide the important theoretical base for research and design of thermal transport systems.