Optimizing permeability in fractal tree-like branched networks

The overall permeability of composites with self-similar fractal tree-like networks is studied. Under the constraint of total volume, we derived a dimensionless expression of effective permeability and discussed in detail the relationship between the dimensionless effective permeability and the geometrical parameters of the tree-like network including diameter ratio, length ratio, branching number and fractal dimension. From the study, it was shown that, the dimensionless effective permeability of the tree-like network decreases with the increase of bifurcation number (N), branching length ratio (γ), branching levels (m) or fractal dimensions of channel length (D) when other parameters are kept constant. It was also found that, the dimensionless effective permeability of the tree-like networks reaches maximum when the diameter ratio β satisfies β^- =N^-1/Δ, where Δ=3, N is the bifurcation number N=2, 3, 4,......This optimal diameter ratio for maximum effective permeability of the fractal tree-like networks obeys Murry's law, but does not obey the WBE model of plants.