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We investigate band broadening in the most widely adopted configuration of micro-pillar array columns (μ PACs)—specifically, a cylindrical pillar array where both the pillar walls and the channel bottom are coated with a thin layer of mesoporous material. The two-zone moment analysis method is adopted to investigate the dispersion properties of μ PACs in a broad range of shell thicknesses, reduced fluid velocities, and retention factors. Three different models of the unit cell, of increasing complexity, have been implemented, namely a two-dimensional model and two different three-dimensional models with and without the retentive bottom layer, the presence of which seems to have a very significant effect on the plate height curves. Model predictions are compared with experimental van Deemter curves for uncoated and coated porous layers, and a robust relationship between the intra-particle (porous-zone) diffusion coefficient D pz and the retention factor k ′ is established. [ABSTRACT FROM AUTHOR]

The paper develops a robust and computationally efficient homogenization approach, grounded on exact local and integral moments, to investigate the temporal evolution of effective dispersion properties of solute particles in periodic media possessing absorbing/desorbing walls. Adsorption onto and desorption from active walls allow linear and reversible mass transfer between the solid surface and the fluid phase. The transient analysis reveals some important features of the dispersion process that cannot be captured by asymptotic approaches aimed at determining exclusively the long-range/large-distance dispersion properties. Two case studies are considered: the dispersion of an analyte in a sinusoidal channel with adsorbing/desorbing walls and the retentive pillar array column for liquid chromatography. For both systems, the transient analysis shows how the tortuous fluid motion induced by the sinusoidal walls or by the presence of pillars induces wide and persistent temporal oscillations of the effective velocity and dispersion coefficient even for a steady (non-pulsating) Stokes flow. The adsorption/desorption process strongly amplifies the phenomenon of the overshoot for the effective dispersion coefficient that, on short/intermediate time scales, reaches values significantly larger than the asymptotic one. Moreover, the method proposed allows a detailed analysis of the temporal evolution of the skewness of the marginal distribution of the analyte along the main stream direction. It clearly shows that the time scale for achieving the macro-transport regime, which implies a Gaussian (symmetric) marginal pdf, is largely underestimated if one bases the analysis on the attainment of constant asymptotic values for the effective velocity and for the dispersion coefficient. [ABSTRACT FROM AUTHOR]

We report on the possibility to extend to bi-continuous packings the two models for the effective longitudinal diffusion D eff , or B-term band broadening, recently proposed for discontinuous chromatographic beds. In bi-continuous packings, like monolithic columns, solutes experience a connected end-to-end pathway in both the mobile and stationary zones, as opposed to discontinuous packings, wherein the stationary adsorptive zone is distributed over a set of isolated elements. Since it is unclear whether a densely packed bed of spherical particles should be treated as a continuous or a bi-continuous medium, this extension is also crucial to fully understand the behaviour of packed particle beds. The proposed models for the effective longitudinal diffusion D eff originate from the adoption of the Two Zone Moment Analysis (TZMA) method by which D eff can be expressed as a linear combination of two essential quantities γ m and γ s , referred to as effective zone-diffusion factors. In the present work we propose two analytical models for γ m and γ s that now cover both the discontinuous and the bi-continuous case. To validate the theory, several bi-continuous packings are investigated, including the tetrahedral skeleton model (TSM), six different Triple Periodic Minimal Surface (TPMS) monoliths and randomly packed beds of spheres. For all of these, the models provide highly accurate results for D eff over a wide range of porosities and zone retention factors k ″. The comparison with literature experimental data for both monolithic silica columns and columns packed with fully porous and porous-shell particles is also presented. • Bi-continuous chromatographic beds are investigated. • The two-zone moment-analysis method (TZMA) has been adopted. • Two models are proposed for D eff , an approximate and a general one. • The TSM and six TPMS monoliths are considered to model monolithic silica columns. • Three randomly packed beds of spheres are analyzed. [ABSTRACT FROM AUTHOR]

This paper develops a homogenization approach, based on the introduction of exact local and integral moments, to investigate the temporal evolution of effective dispersion properties of point-sized and finite-sized particles in periodic media. The proposed method represents a robust and computationally efficient continuous approach, alternative to stochastic dynamic simulations. As a case study, the exact moment method is applied to analyze transient dispersion properties of point-sized and finite-sized particles in sinusoidal tubes under the action of a pressure-driven Stokes flow. The sinusoidal structure of the tube wall induces a significant variation of the axial velocity component along the axial coordinate. This strongly influences the transient behavior of the effective axial velocity V z (t) and of the dispersivity D z (t) , both exhibiting wide and persistent temporal oscillations, even for a steady (not-pulsating) Stokes flow. For a pointwise injection of solute particles on the symmetry axis, many interesting features appear: negative values of the dispersion coefficient D z (t) , values of D z (t) larger than the asymptotic value D z (∞) , and anomalous temporal scaling of the axial variance of the particle distribution. All these peculiar features found a physical and theoretical explanation by adopting simple transport models accounting for the axial and radial variation of the axial velocity field and its interaction with molecular diffusion. [ABSTRACT FROM AUTHOR]

• A moment approach is proposed for the estimate of effective transport parameters. • Transient and asymptotic conditions can be analyzed with ease and great accuracy. • Direct Numerical Simulation results are used to validate the moment-based approach • Porous pillar array columns are investigated. • The role of top and bottom walls on the separation of porous pillar array columns is quantified. 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. [ABSTRACT FROM AUTHOR]