Mass Transfer Simulation In An Inclined Two-Layer Porous Channel By The Lattice Boltzmann Method

Abstract

The study numerically investigates the mass transfer in an inclined two-layer porous channel in the gravitational field. The lower region of the channel is occupied by a porous medium, while the upper region consists of a pure fluid. The initial concentration distribution is such that the impurity is localized in the central part of the porous domain. The upper and lower walls of the channel are solid and no-flux boundary condition for the concentration is applied. Periodic boundary conditions are applied for the velocity field on the side walls, and the flow is driven by the longitudinal component of the velocity induced by the gravitational field and channel inclination. For the concentration field two boundary condition types are examined on the side walls: periodic boundary conditions, and a non-periodic characterized by a vanishing concentration at the left wall and a constant flux condition at the right wall. The problem is solved for constant porosity and permeability coefficients, with the Schmidt number fixed at \(\varvec{Sc = 10^3 }\). The study focuses on the diffusion of an impurity into a viscous pure fluid for various Darcy numbers \(\varvec{Da}\). The simulations are conducted using the Lattice Boltzmann Method (LBM) on a D2Q9 lattice. A modified multiple relaxation-time (MRT) LBM scheme was introduced for the mass transfer simulation in porous media. The effectiveness and applicability of the proposed scheme for such classes of problems are substantiated through the presented results. For the periodic boundary conditions, it is shown that the integral concentration within the domain is conserved, and the concentration profiles both inside and outside the porous layer converge toward the average value. In contrast, under non-periodic boundary conditions, the impurity is gradually washed out of the domain. The obtained numerical results also demonstrate that the type of boundary condition imposed on the concentration field at the side walls has a negligible effect on the velocity field. At a higher Darcy number \(\varvec{Da = 10^{-2} }\), the evolution of the impurity is more pronounced, and the system reaches a steady state more rapidly. For lower Darcy numbers (\(\varvec{ Da = 10^{-3} }\) and \(\varvec{Da = 10^{-4}}\)), the impurity evolution rate outside the porous matrix is approximately the same, whereas within the porous matrix, the evolution is more intense for larger Darcy numbers.