Moment analysis for predicting effective transport properties in hierarchical retentive porous media

We report on a new homogenization approach to solve, with drastically improved speed and accuracy, the general advection-diffusion equation in hierarchical porous media with localized diffusion and adsorption/desorption processes, thus opening the way to a much deeper understanding of the band broadening process in chromatographic systems. The proposed robust and efficient moment-based approach allows us to compute the exact local and integral concentration moments and hence provides exact solutions for the effective velocity and dispersion coefficients of migrating solute particles. Innovative to the proposed method is also that it not only produces the exact effective transport parameters of the long-time asymptotic solution, but also their entire transient. The analysis of the transient behaviour can be used, for example, to properly identify the time and length scales needed to achieve the macro-transport conditions. If the hierarchical porous media can be represented as the periodic repetition of a unit lattice cell, the method only requires the solution of the time-dependent advection-diffusion equations for the zeroth order and first-order exact local moments, exclusively on the unit cell. This implies an enormous reduction of the computational efforts and a significant improvement of the accuracy of the results when compared to the direct numerical simulation (DNS) approaches which require flow domains that are long enough to achieve steady-state conditions, and hence often cover tens to hundreds of unit cells. The reliability of the proposed method is verified by comparing its predictions with DNS results, in one, two and three dimensions, in both transient and asymptotic conditions. The influence of top and bottom no-slip walls on the separation performance of chromatographic columns with micromachined porous and nonporous pillars is discussed in detail.