RESUMO
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.
RESUMO
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+ transportis postulated to capture its retention in the electrical double layer. Thetwo-scalenano/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.
Neste artigo propomos um modelo em três escalas para descrever o acoplamento entre o fluxo eletroosmótico e o transporte de íons incluindo fenômenos de adsorção em uma caulinita. A argila é caracterizada por três escalas nano/micro e macroscópica. A escala microscópica é constituída por micro-poros saturados por uma solução aquosa contendo quatro íons monovalentes e partículas sólidas carregadas eletricamente circundadas por uma dupla camada elétrica fina. O movimento dos íons é governado pelas equações de Nernst-Planck e a influência da dupla camada sobre o fluxo aquoso é modelada por uma condição de contorno de deslizamento da componente tangencial do campo de velocidade (condição de Helmholtz-Smoluchowski). Além disso, uma condição de adsorção na interface fluido-sólido para os íons Na+ é postulada capturando a retenção do sódio na dupla camada elétrica. O modelo em duas escalas nano/micro incluindo a adsorção do sal e a condição de deslizamento da velocidade é homogeneizado levando a derivação das equações macroscópicas na escala de Darcy. Um dos aspectos inovadores do modelo em três escalas é a reconstrução da lei constitutiva para o coeficiente de partição que governa a adsorção do Na+ na dupla camada elétrica. Para ilustrar as potencialidades do modelo em três escalas na simulação da eletroremediação de solos argilosos, o modelo macroscópico é discretizado utilizando o método de volumes finitos no intuito de simular a dessalinização de uma amostra de caulinita por técnica de eletrocinética.
