Coupled Transport and Adsorption in Graded Filters: A Multi-Scale Analysis of Non-Solenoidal Effects

Abstract

We investigate the transport and adsorption of solutes within graded porous filters characterised by a spatially varying microstructure. While classical homogenisation theory typically assumes periodic media, we employ the method of multiple scales to derive an effective macroscopic model for ``near-periodic'' geometries where the porosity varies slowly over the longitudinal coordinate. A key novelty of this work is the departure from the standard solenoidal constraint; instead, we introduce a modified incompressibility condition derived from non-equilibrium thermodynamics that accounts for the coupling between the solute concentration and the solvent velocity. This leads to a generalised Darcy-scale description where the fluid velocity field is non-solenoidal within the porous domain. Through asymptotic analysis, we determine the leading-order concentration profiles and quantify first-order corrections that capture the interplay between the porosity gradient and the mixture composition. We evaluate filter performance across several metrics, including outflux concentration and total adsorption rate, under both fixed-flow and fixed-pressure-drop operating conditions. Our results demonstrate that the porosity gradient and the coupling parameter significantly influence the filtration efficiency, particularly as the medium approaches the clogging limit. The analysis reveals that the optimal filter design is highly sensitive to the chosen performance metric, highlighting the necessity of physically consistent boundary conditions and mixture dynamics in the design of high-efficiency graded filters.

Figures (10)

• Figure 1: Diagram of the filtration model for a two-dimensional square lattice, corresponding to the model description in Section \ref{['sec3.1']}. (a) A macroscopic view of the entire filter, showing decreasing porosity relative to the fluid flow. (b) A microscopic close-up of an individual cell $\omega(\hbox{\boldmath $x$})$. Source: dalwadi2015
• Figure 2: Evaluation of the effective macroscopic transport coefficients as a function of porosity for a two-dimensional square lattice. (left panel) Diffusion coefficient. The numerical evaluation (dashed line) is obtained by solving the cell problem \ref{['eq2.40']} for various porosity values within the range $\phi\in[1-\pi/4,1]$ using the FlexPDE software. The results are then plugged into \ref{['eq2.45real']} to obtain the plotted values. The closed-form approximation (solid line) represents evaluated expression \ref{['eq3.37']}. (right panel) Numerical solution of the cell problem \ref{['eq2.44']} yielding the effective permeability coefficient $\mathsfbi{K}=K\mathsfbi{I}_d$ as a function of porosity. The numerical solution requires care and is described in Appendix \ref{['sec.App-K']}.
• Figure 3: Numerically obtained fluid-volume averaged solute concentration $C^{f}=C/\phi$ (a), fluid velocity $U=\phi U^{f}$ (b) for three gradients $m=-0.3,0,0.3$. The evaluation is performed for a two-dimensional square lattice with the parameters $\phi_0=0.75,\, A=0.6,\, \mathrm{Pe}=3$ and $k=1$. Note that for the special case of divergence free flow field, $A=0$, the solution is $U=1$ and hence it can be viewed as a measure of the effect of the novel term, being a result of treating the filtration problem as a true mixture.
• Figure 4: Numerically obtained concentration profiles for linear porosity \ref{['eq3.32']} for $A=0$ and $A=0.6$, hence highlighting the effect of the new term. The evaluation is performed for a two-dimensional square lattice with the parameters ${\phi_0=0.75,\, \mathrm{Pe}=3,\, k=1}$ and $m=-0.3,0.3$. The lower two curves (solid and dashed) correspond to positive porosity gradient $m=0.3$, while the upper two to a negative porosity gradient $m=-0.3$. Note that the inlet solute concentration is affected (being larger for $A\neq 0$) as well as the amount of the particles being filtered out (again, being larger for $A\neq 0$).
• Figure 5: Numerically obtained concentration profiles for linear porosity \ref{['eq3.32']} for square and hexagonal lattices in two dimensions. The parameters used are $\phi_0=0.75$, $A=0.6,\, \mathrm{Pe}=3,\, k=1$ and $m=-0.3,0.3$. The effect of microscale geometry and the capability of the asymptotic method to reflect these effects in the derived macroscale model is apparent.