A Three-Scale Model of pH-Dependent Flows and Ion Transport with Equilibrium Adsorption in Kaolinite Clays: I. Homogenization Analysis

Lima, Simone Pedrosa; Murad, Márcio A.; Moyne, Christian; Stemmelen, Didier

doi:10.1007/s11242-010-9545-4

Cited by 24 publications

(30 citation statements)

References 27 publications

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“…onde V Dé a velocidade de Darcy e os parâmetro efetivos dados pelos problemas de fechamento na escala microscópica [3]. Para uma geometria de placas planas paralelas [2][6] a solução dos parâmetros efetivos acima são dadas na forma…”

Section: Modelo Macroscópicounclassified

“…Para o sistema hidrodinâmico consideramos uma condição de deslizamento do campo de velocidade dada pela equação de Helmholtz-Smoluchowsky dada por [3] [6]…”

Section: Introductionunclassified

“…onde o termo Γ i com (i = N a + , H + , Cl − , OH − )é a adsorção elétrica dosíons na camada dupla e γ H + representa a adsorção química derivado na modelagem nanoscópica do problema na seguinte forma [3] …”

Section: Introductionunclassified

“…onde C b = C N ab + + C Hb +é a concentração do bulk, ϕ o potencial elétrico com ζ = ϕ| Γ f s [2,3].…”

Section: Introductionunclassified

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Modelagem Computacional Multi-Escala de Fenômenos Eletrocinéticos em Meios Porosos Argilosos Carregados Eletricamente

Mariano¹,

Lima

Gomes³

2015

Proceeding Series of the Brazilian Society of Computational and Applied Mathematics

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Resumo: Neste trabalho de pesquisa propomos uma modelagem matemática e computacional em três escalas de fenômenos eletrocinéticos de solos argilosos carregados eletricamente. O sistema de equações microscópico que governa o acoplamento hidro-eletroquímicoé derivado na escala macroscópica utilizando a técnica de homogeneização de estruturas periódicas. O sistema de equações na escala de Darcy não linearé discretizado utilizando o método de elementos finitos. Diferentemente dos resultados obtidos por (Cirilo, 2013), neste trabalho propomos a utilização do método de Newton-Raphson para a discretização das equação e a utilização do método self-consistent para a derivação dos parâmetros efetivos em geometrias mais complexas. As soluções numéricas permitem descrever o processo de eletroremediação de solos argilosos. IntroduçãoNeste trabalho de pesquisa propomos uma modelagem computacional multiescala com o intuito de simular o acoplamento hidro-eletroquímico em meios porosos argilosos utilizando a técnica multiescala de homogeneização de estruturas periódicas e o método self-consistentA fase sólida consiste de partículas da argila e os poros preenchidos por uma solução contendo quatro solutos iônicos (N a + , H + , Cl − , OH − ). Na escala microscópica, o movimento da solução eletrolíticaé governado pelo problema de Stokes enquanto que o transporte dos solutos iônicos modelados utilizando equações do tipo Nernst-Planck. O modelo matemático microscópico/nanoscópicoé derivado na escala de Darcy utilizando a técnica de homogeneização de estruturas periódicas [4] [5]. O sistema de equações macroscópicoé suplementado por condições de contorno do tipo Dirichlet e as soluções numéricas do modelo não linear são obtidas utilizando o método de elementos finitos [6]. O modelo matemático/numérico permite simular o processo de eletroremediação de solos argilosos. Modelo MicroscópicoConsideramos o domínio microscópico um meio bifásico Ω = Ω s ∪Ω f ⊂ R 3 onde Ω sé a fase sólida carregada eletricamente e Ω f os poros preenchidos por uma solução aquosa com Γ f s a interface

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Section: Modelo Macroscópicounclassified

“…Para o sistema hidrodinâmico consideramos uma condição de deslizamento do campo de velocidade dada pela equação de Helmholtz-Smoluchowsky dada por [3] [6]…”

Section: Introductionunclassified

“…onde C b = C N ab + + C Hb +é a concentração do bulk, ϕ o potencial elétrico com ζ = ϕ| Γ f s [2,3].…”

Section: Introductionunclassified

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Modelagem Computacional Multi-Escala de Fenômenos Eletrocinéticos em Meios Porosos Argilosos Carregados Eletricamente

Mariano¹,

Lima

Gomes³

2015

Proceeding Series of the Brazilian Society of Computational and Applied Mathematics

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“…In a companion article (Lima et al 2009), the authors developed a three-scale theory to model reactive transport of four ionic species (Na + , Cl − , H + , OH − ) coupled with the movement of an aqueous solution in kaolinite clays. The macroscopic model was derived by double averaging the nanoscopic and microscopic descriptions of the medium (Fig.…”

Section: Introductionmentioning

confidence: 99%

A Three-Scale Model of pH-Dependent Flows and Ion Transport with Equilibrium Adsorption in Kaolinite Clays: II Effective-Medium Behavior

et al. 2010

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In part I (Lima et al., Transp Porous Media, 2009), a three-scale model governing the movement of an aqueous saline solution containing four monovalent species (Na + , H + , Cl − , OH − ) in kaolinite clays was derived. Unlike purely macroscopic approaches, the novelty of the formulation relied on the double averaging of the nanoscopic electrochemistry of particle/electrolyte solution interface ruled by the electrical double layer coupled with protonation/deprotonation reactions. The passage from the nano to the micro (pore)-scale gave rise to ion-sorbed concentrations and slip velocity at the solid/fluid interface which are coupled with the microscopic Stokes problem and Nernst-Planck equations governing the hydrodynamics and ion transport in the micropores. Application of a formal homogenization procedure led to macroscopic governing equations with effective electro-chemical parameters, such as retardation coefficients, electro-osmotic permeability, and electric conductivity. In this study, we reconstruct the constitutive laws of the macroscopic coefficients by solving the nano and microscopic closure problems. New generalized isotherms for Na + and H + − OH − sorption are build-up based on a perturbation approach and the limitations 123 46 S. A. de Lima et al. of classical Freundlich isotherm for modeling ion sorption at the solid/fluid interface are discussed. The macroscopic governing equations are discretized by the finite volume method and numerical simulations of a transient electroosmosis experiment for desalination of a clay sample by electrokinetics are presented. List of Symbolsa fs Surface area [m −1 ] f Characteristic function m, n, p Exponents of the adsorption isotherms k e Isoelectric point p Thermodynamic pressure [Pa] t Time [s] x, y Parallel and orthogonal coordinates [m] u Auxiliary perturbation parameter C ib Ionic bulk concentration with i = Na,H [mol/m 3 ] C H + 0 H + concentration at the solid surface [mol/l] C b Bulk concentration [mol/m 3 ] F Faraday constant [C/mol] H Half distance between the particles [m] K Equilibrium constant [l/mol] K W Ionic product of water [(mol/l) 2 ] L D Debye's length [m] L Sample length [m] M Metallic ions at the surface R ideal gas constant [J/(mol K)] R N , R H Ionic retardation coefficients T Temperature [K] Y Cell domain Y f , Y s Fluid and solid subdomains ∂Y fs Fluid/solid interface x Macroscopic coordinate [m] y Microscopic coordinate [m] A eff First Onsager coefficient [(C·m 2 )/(mol·s)] B eff Second Onsager coefficient [(C·m 2 )/(mol·s)] C eff Effective electric conductivity [C/(m·s)] D eff Net effective diffusivity H + -OH + [m 2 /s] D eff Na Effective sodium diffusivity [m 2 /s] K eff P Hydraulic conductivity [m 2 /(Pa·s)] K eff E Electroosmotic permeability [m/(V·s)] I eff f Effective electric current [C/(m 2 ·s)] J eff Effective ionic flux [mol/(m 2 ·s)] 123 A Three-Scale Model of Ion Transport with Equilibrium Adsorption in 1:1 Clays 47

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Textural versus electrostatic exclusion‐enrichment effects in the effective chemical transport within the cortical bone: A numerical investigation

Lemaire

Kaiser

Naı̈li

et al. 2013

Numer Methods Biomed Eng

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Interstitial fluid within bone tissue is known to govern the remodelling signals' expression. Bone fluid flow is generated by skeleton deformation during the daily activities. Due to the presence of charged surfaces in the bone porous matrix, the electrochemical phenomena occurring in the vicinity of mechanosensitive bone cells, the osteocytes, are key elements in the cellular communication. In this study, a multiscale model of interstitial fluid transport within bone tissues is proposed. Based on an asymptotic homogenization method, our modelling takes into account the physicochemical properties of bone tissue. Thanks to this multiphysical approach, the transport of nutrients and waste between the blood vessels and the bone cells can be quantified to better understand the mechanotransduction of bone remodelling. In particular, it is shown that the electrochemical tortuosity may have stronger implications in the mass transport within the bone than the purely morphological one.

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A Three-Scale Model of pH-Dependent Flows and Ion Transport with Equilibrium Adsorption in Kaolinite Clays: I. Homogenization Analysis

Cited by 24 publications

References 27 publications

Modelagem Computacional Multi-Escala de Fenômenos Eletrocinéticos em Meios Porosos Argilosos Carregados Eletricamente

Modelagem Computacional Multi-Escala de Fenômenos Eletrocinéticos em Meios Porosos Argilosos Carregados Eletricamente

A Three-Scale Model of pH-Dependent Flows and Ion Transport with Equilibrium Adsorption in Kaolinite Clays: II Effective-Medium Behavior

Textural versus electrostatic exclusion‐enrichment effects in the effective chemical transport within the cortical bone: A numerical investigation

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