Abstract
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 electro- chemistry 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 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.
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Abbreviations
- a 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/m3]
- \({C_{\rm H^{+}_{0}}}\) :
-
H+ concentration at the solid surface [mol/l]
- C b :
-
Bulk concentration [mol/m3]
- 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 , Ys:
-
Fluid and solid subdomains
- ∂Y fs :
-
Fluid/solid interface
- x :
-
Macroscopic coordinate [m]
- y :
-
Microscopic coordinate [m]
- A eff :
-
First Onsager coefficient [(C·m2)/(mol·s)]
- B eff :
-
Second Onsager coefficient [(C·m2)/(mol·s)]
- C eff :
-
Effective electric conductivity [C/(m·s)]
- \({{\bf \widehat{D}^{\rm eff}}}\) :
-
Net effective diffusivity H+–OH+ [m2/s]
- \({{\bf D^{\rm eff}_{\rm Na}}}\) :
-
Effective sodium diffusivity [m2/s]
- \({{\bf K_{P}^{\rm eff}}}\) :
-
Hydraulic conductivity [m2/(Pa·s)]
- \({{\bf K_{E}^{\rm eff}}}\) :
-
Electroosmotic permeability [m/(V·s)]
- \({{\bf I_f^{\rm eff}}}\) :
-
Effective electric current [C/(m2·s)]
- J eff :
-
Effective ionic flux [mol/(m2·s)]
- V D :
-
Darcy’s velocity [m/s]
- \({{\bf I_f^{\rm eff}}}\) :
-
Spatial gradients
- Δx :
-
Mesh size
- Δt :
-
Time step
- \({< \cdot >}\) :
-
Volumetric average operator
- \({< \cdot >_{\rm fs}}\) :
-
Interfacial average operator
- Na+, H+, Cl−, OH− :
-
Ionic species
- α, β:
-
Electro-chemical coefficients
- δ :
-
Half particle thickness [m]
- \({\epsilon}\) :
-
Perturbation parameter
- \({\tilde{\epsilon}_{0}}\) :
-
Permittivity of the free space [C/(V m)]
- \({\tilde{\epsilon}_{r}}\) :
-
Dielectric constant
- \({\phi}\) :
-
Macroscopic electric potential [V]
- \({\overline{\phi}}\) :
-
Dimensionless macroscopic electric potential
- \({\gamma_{\rm H^{+}}}\) :
-
Protonic concentration at the surface [mol/m2]
- η :
-
Porosity
- κ P :
-
Characteristic tensorial function
- μ f :
-
Viscosity of the water [Pa·s]
- π :
-
Microscopic pressure
- σ :
-
Surface charge density [C/m2]
- ζ :
-
Zeta potential [V]
- \({{\overline \zeta}}\) :
-
Dimensionless zeta potential
- \({\Theta}\) :
-
Nonlinear parameter in the advection term of the protons
- Γ:
-
Sorbed ionic concentration at the surface [mol/m2]
- ΓMAX :
-
Density of available sites [sites/nm2]
- Ω:
-
Macroscopic domain
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de Lima, S.A., Murad, M.A., Moyne, C. et al. A Three-Scale Model of pH-Dependent Flows and Ion Transport with Equilibrium Adsorption in Kaolinite Clays: II Effective-Medium Behavior. Transp Porous Med 85, 45–78 (2010). https://doi.org/10.1007/s11242-010-9546-3
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DOI: https://doi.org/10.1007/s11242-010-9546-3