“Disc-point” mass transfer Constructal optimizations with Darcy and Hagen–Poiseuille flows in porous media

Analogizing with Rocha et al.’s [L.A.O. Rocha, S. Lorente, A. Bejan, Int. J. Heat Mass Transfer 45 (8) (2002) 1643–1652] heat conduction model of a disc-shaped body, a “disc-point” mass transfer model with porous media is investigated by using Constructal theory. The heat flow with Fourier law is considered in Rocha et al.’s heat conduction model, while the mass flow with both Darcy law and Hagen–Poiseuille law are considered in the mass transfer model, respectively. The optimal constructs of radial- and branched-pattern discs are obtained by taking maximum pressure drop minimization as optimization objective. The results show that there exist optimal aspect ratios which lead to the minimization of maximum pressure drop of the radial-pattern disc with Darcy flow and Hagen–Poiseuille flow in the channels, respectively. For the first order branched-pattern disc, different definitions of the first order channel fraction lead to different optimization results. When the first order open space fraction is defined as the ratio of the total open space surface area of the whole disc to the whole disc area, there exists an optimal elemental open space fraction which leads to the minimum dimensionless maximum pressure drop of the first order branched-pattern disc. When the product of the cubic of first order open space fraction and the reciprocal of the dimensionless permeability of low permeability porous media is 1000, the critical dimensionless radius, which determines whether the radial- or branched-pattern design is adopted, is 1.78. That is, when the radius of the branched-pattern disc is higher than the critical radius, branched-pattern design should be adopted, otherwise the radial-pattern design should be adopted. The comparison between the optimal constructs of the mass transfer model in this paper and Rocha et al.’s heat conduction model are carried out.