Abstract
A finite element solution for a mass transport model for porous materials accounting for sorption hysteresis is presented in this paper. The model is prepared for modeling of concrete durability, but the general presentation makes it suitable for other porous materials like soil and tissues. The model is an extended version of the Poisson–Nernst–Planck (PNP) system of equations. The PNP extension includes a two-phase vapor and liquid model coupled by a sorption hysteresis function and a chemical equilibrium term. The strong and weak solutions for the equation system are shown, and a finite element formulation is established by Galerkin’s method. A single-parameter implicit time integration scheme is used for solving the transient response, and the out-of-balance solution is minimized by using a modified Newton–Raphson scheme in which the tangential stiffness is not computed exactly. The sorption hysteresis is added to the solution procedure by a rate function. The hysteresis effect is described by scanning curves defined between two boundary sorption isotherms. A numerical example was constructed to show the applicability and compare a simple approach and a extended approach within the sorption hysteresis model. The examples illustrate the impact of changing relative humidity at the mass transport boundary on the adsorption and desorption stages of a cement-based material. Changes in the pore solution ion concentrations are a result of the changing moisture content, which are shown by the example. Comparing the two approaches showed significant deviations in the liquid content and ion concentrations, in parts of the domain considered.
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Abbreviations
- \(c_{i}^{l}\) :
-
Ionic concentration of species \(i\) in liquid phase \(l\) (mol/l)
- \(\varepsilon ^{l}\) :
-
Liquid-phase volume fraction (\(\hbox {m}^{3}/\hbox {m}^{3}\))
- \(\varepsilon _{v}\) :
-
Vapor-phase volume fraction (\(\hbox {m}^{3}/\hbox {m}^{3}\))
- \(\rho _{i}^{l}\) :
-
Density of species \(i\) in the liquid phase \(l\) (\(\hbox {kg}/\hbox {m}^{3}\))
- \(D_{i}^{l}\) :
-
Diffusion coefficient for constituent \(i\) (\(\hbox {m}^{2}/\hbox {s}\))
- \(A_{i}^{l}\) :
-
Ion mobility (m\(^{2}\)/s V)
- \(z_{i}\) :
-
Valence (–)
- \(\Phi \) :
-
Electrical potential (V)
- \(q_{i}\) :
-
Chemical interactions (1/s)
- \(\xi _{d}\) :
-
Relative dielectricity coefficient (V/m)
- \(\xi _{0}\) :
-
Dielectricity coefficient of vacuum (V/m)
- \(F\) :
-
Faraday’s constant (C/mol)
- \(\rho _{w}\) :
-
Water density (kg/m\(^{3}\))
- \(D_{\varepsilon ^{l}}\) :
-
Diffusion coefficient for the liquid phase (m\(^{2}\)/s)
- \(\hat{m}_{l}\) :
-
Mass exchange of liquid phase
- \(\mathbf{v}^{l,s}\) :
-
Liquid-phase \(l\) velocity with respect to the solid \(s\) (m/s)
- \(\hat{m}_{l}\) :
-
Mass exchange of liquid phase
- \(\rho _{vs}\) :
-
Vapor-saturation density (kg/m\(^{3}\))
- \(\varepsilon _{p}\) :
-
Porosity (m\(^{3}\)/m\(^{3}\))
- \(\phi _{v}\) :
-
Relative humidity (–)
- \(D_{\phi }\) :
-
Diffusion coefficient for the vapor phase (m\(^{2}\)/s)
- \(\hat{m}_{v}\) :
-
Mass exchange of vapor phase
- \(\varepsilon ^{l,eq}\) :
-
Sorption hysteresis function
- \(\tilde{\square }\) :
-
Non-linear variable notation
- \(R\) :
-
Rate constant for mass exchange between liquid and vapor (1/s)
- \(w\) :
-
Weight function spatial domain \(w\left( x,y,z\right) \)
- \(W\) :
-
Weight function time domain \(W\left( t\right) \)
- \(V\) :
-
Volume (m\(^{3}\))
- \(S\) :
-
Surface (m\(^{2}\))
- \(\mathbf{j}_{i}^{l}\) :
-
Boundary flux of species \(i\)
- \(\mathbf{n}\) :
-
Normal vector
- \(\mathbf{j}^{\Phi }\) :
-
Electrical displacement
- \(\mathbf{j}^{l}\) :
-
Boundary flux of liquid
- \(\mathbf{j}^{\phi }\) :
-
Boundary flux of vapor
- \(\mathbf{N}\) :
-
Global shape function
- \(\mathbf{a}\) :
-
State variables vector
- \(\mathbf{C}\) :
-
Mass matrix
- \(\mathbf{K}\) :
-
Stiffness matrix
- \(\mathbf{f}\) :
-
Force vector
- \(R_g\) :
-
Universal gas constant (V C/K mol)
- \(t\) :
-
Time (h)
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Jensen, M.M., Johannesson, B. & Geiker, M.R. A Numerical Comparison of Ionic Multi-Species Diffusion with and without Sorption Hysteresis for Cement-Based Materials. Transp Porous Med 107, 27–47 (2015). https://doi.org/10.1007/s11242-014-0423-3
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DOI: https://doi.org/10.1007/s11242-014-0423-3