Analysis of thermal conductivity in tree-like branched networks

Asymmetric tree-like branched networks are explored by geometric algorithms. Based on the network, an analysis of the thermal conductivity is presented. The relationship between effective thermal conductivity and geometric structures is obtained by using the thermal-electrical analogy technique. In all studied cases, a clear behaviour is observed, where angle (δ, θ) among parent branching extended lines, branches and parameter of the geometric structures have stronger effects on the effective thermal conductivity. When the angle δ is fixed, the optical diameter ratio β* is dependent on angle θ. Moreover, γ and m are not related to β*. The longer the branch is, the smaller the effective thermal conductivity will be. It is also found that when the angle θ < δ/2, the higher the iteration m is, the lower the thermal conductivity will be and it tends to zero, otherwise, it is bigger than zero. When the diameter ratio β1 < 0.707 and angle δ is bigger, the optimal k of the perfect ratio increases with the increase of the angle δ; when β1 > 0.707, the optimal k decreases. In addition, the effective thermal conductivity is always less than that of single channel material. The present results also show that the effective thermal conductivity of the asymmetric tree-like branched networks does not obey Murray's law. (classical areas of phenomenology)