Constructal design of porous-filled tree-shaped networks for optimal fluid transport: An analytical approach

In this article, a general mathematical framework is developed for the flow distribution in a porous-filled tree-network with rigid and impermeable walls using the Darcy–Brinkman model. The governing model equations, incorporating heterogeneous permeability, are solved by the wavelet collocation method for flow in a single tube. Further, the local study has been extended to a tree-shaped network by electrical circuit analogy. Mathematical analysis is performed for various fluids in single and two-layer fluid flow with different permeabilities to evaluate the flow variables responsible for the flow resistance. The developed theory is used to establish the mathematical relationship associated with the optimal flow in a self-symmetric, porous-filled, tree-shaped network. The explicit expression of the optimal geometry for flow transport is obtained in terms of the tube aspect ratio, tube length, and branching angle between the parent and daughter branches, using rational function approximation. It is observed that the optimal flow has strong dependencies on the permeability of the porous matrix, offering various applications in natural and engineered fluid transport phenomena.