Heat conduction in fractal tree-like branched networks

The geometric structures and fractal dimensions of fractal tree-like branched networks have the significant influence on the efficiency of physiological, communication and transport processes. We analyze the heat conduction through symmetric fractal tree-like branched networks. We obtain the expression of thermal conductivity in the networks and analyze the relationship between the effective thermal conductivity (ETC) and the geometric structures of the networks. We have found that the ETC of the networks is always less than that of a single channel, and the value of the thermal conductivity of the network can tend to zero under certain conditions; as long as the branching number N is fixed, the heat conduction reaches the fastest rate at the same diameter ratio βm which is corresponding to the fractal dimension Dd = 2.0. We have also found that the heat conduction in the networks is rather different from Murray’s law both for laminar regime (2−1/3) and for turbulent flow regime (2−3/7).