Abstract
A new three-scale model to describe the coupling between pH-dependent flows and transient ion transport, including adsorption phenomena in kaolinite clays, is proposed. The kaolinite is characterized by three separate nano/micro and macroscopic length scales. The pore (micro)-scale is characterized by micro-pores saturated by an aqueous solution containing four monovalent ions and charged solid particles surrounded by thin electrical double layers. The movement of the ions is governed by the Nernst-Planck equations, and the influence of the double layers upon the flow is dictated by the Helmholtz-Smoluchowski slip boundary condition on the tangential velocity. In addition, an adsorption interface condition for the Na+ transport is postulated to capture its retention in the electrical double layer. The two-scale nano/micro model including salt adsorption and slip boundary condition is homogenized to the Darcy scale and leads to the derivation of macroscopic governing equations. One of the notable features of the three-scale model is there construction of the constitutive law of effective partition coefficient that governs the sodium adsorption in the double layer. To illustrate the feasibility of the three-scale model in simulating soil decontamination by electrokinetics, the macroscopic model is discretized by the finite volume method and the desalination of a kaolinite sample by electrokinetics is simulated.
Highlights
The quality of groundwater in clayey soils is strongly influenced by the electrochemistry, which is dictated by the electrochemical interactions that occur between the pore fluid and the minerals
A number of laboratory-scale and field-scale studies have shown the technical feasibility of electrokinetic processes in removing various contaminants from fine-grained soils
In addition to the ionic concentrations in the electrical double layer, we incorporate adsorption/desorption phenomena owing to protonation/deprotonation reactions involving the H + ions at the surface of the particles
Summary
In addition to the ionic concentrations in the electrical double layer, we incorporate adsorption/desorption phenomena owing to protonation/deprotonation reactions involving the H + ions at the surface of the particles. By subtracting the Nernst-Planck relations (18) for the Cl− from the N a+, using the above definition together with the electroneutrality condition (2) and the ionic product of water (3) by recalling our time scale assumption, where CHb+ and CO Hb− are assumed constant, we obtain conservation of charge. It remains to establish the interface conditions for the Nernst-Planck equations To this end, we invoke the conservation of mass at the interface, which states that the time derivative of the EDL adsorption quantities i (i = N a+) balances the diffusive flux normal to the particle surface (Auriault and Lewandowska 1996).
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