Research paperModeling transport of charged species in pore networks: Solution of the Nernst–Planck equations coupled with fluid flow and charge conservation equations
Introduction
The Nernst–Planck equations are widely used in the literature to describe the transport of ionic species in electrochemical systems (Meng et al., 2014, Metti et al., 2016). With respect to porous media, the equations describe ion transport in a wide variety of applications such as electrochemical cells (van Soestbergen et al., 2010) and certain redox flow batteries (Sadeghi et al., 2019b). They are also used to analyze ion conduction in biological structures of pores (Bolintineanu et al., 2009), but probably the most common applications are for the study of ion transport mechanisms in clay soils and concrete. Smith et al. (2004) applied the NP equations to the analysis of transport through platy-clay soils and Pivonka et al. (2004) analyzed chloride diffusion in concrete for the estimation of structural degradation due to corrosion. Moreover, it has been shown that simulations based on the NP equations accurately predict ionic diffusion coefficients experimentally estimated on concrete (Narsilio et al., 2007). In a more recent work (Azad et al., 2016), the transport processes in a system including a concrete plug surrounded by clay stone were modeled using the NP equations. The developed numerical interface (Azad et al., 2016) accurately predicted the results on several complex geochemical transport problems studied by different authors (van der Lee et al., 2003, Xu et al., 2006, Xu et al., 2011, Nardi et al., 2014, Marty et al., 2015).
Another important field where the NP equations are used is modeling transport in capacitive charging and deionization (Biesheuvel and Bazant, 2010, Gabitto and Tsouris, 2015). Comparisons between simulation results and experimental data (Sharma et al., 2015) highlighted the capabilities of the NP based simulations to help in the design of capacitive deionization devices. While the transport of ionic species in the bulk of a solution flowing through a porous medium is generally described using the NP equations, a charge conservation equation is required to close the system. One option, perhaps the most accurate, uses the well-known Poisson equation for the electrostatic potential (Newman and Thomas-Alyea, 2012). The Poisson equation relates the electric charge density to the Laplacian of the potential and describes the movement of the charged species in solution. This yields the famous Poisson Nernst–Planck system of equations. Charge conservation can also be enforced through a Laplace equation for the potential which allows for further mathematical simplifications under certain assumptions (Newman and Thomas-Alyea, 2012). In the presence of fluid flow, the solution of the flow problem based on the mass and momentum conservation equations (Stokes or Navier–Stokes) enables the calculation of the advective term in the NP equations.
Solving electrochemical problems in porous media at the pore-scale based on the NP equations is generally carried-out using computational mesh that conforms to the real geometry of the system being analyzed. Different methods have been used to numerically solve the transport equations such as the finite difference (Bolintineanu et al., 2009, Meng et al., 2014, Sharma et al., 2015) and finite element (Samson and Marchand, 1999, Narsilio et al., 2007, Lu et al., 2010, Metti et al., 2016, Azad et al., 2016). However, it is well-known that direct numerical simulations (DNS) require significant computational resources. The same logic applies to many other transport problems such as pure diffusion or dispersion in porous media. PNM, as an alternative pore-scale modeling approach, requires substantially lower computing resources (compared to pore-scale DNS) and have been successfully applied to study physics such as diffusion reaction (Gostick et al., 2007) and dispersion (Sadeghi et al., 2019a) in porous media. However, the use of PNM to study transport of charged species is in its infancy. For instance, in a study of electrokinetic transport through charged porous media (Obliger et al., 2014), a steady-state PNM approach was used. This work (Obliger et al., 2014) is one of the first modeling electrochemical systems based on PNM. The used pore-scale microscopic transport coefficients were simple analytical relations obtained by solving the NP equations in a cylinder. Recently (Lombardo et al., 2019), a pore network model based on the NP equations was used to study porous electrodes in electrochemical devices. However, their approach (Lombardo et al., 2019) was based on the upwind scheme, which was recently shown to have high errors when Péclet number is above unity (Sadeghi et al., 2019a).
In this work, a more accurate method was developed and validated to solve the charge conservation NP system in pore networks. This new method will ultimately allow for accurate pore-scale simulation of transport in electrochemical systems with substantially lower computational cost compared to DNS approaches such as FEM. One aim of the present work is to identify the best approach among various options and to establish a numerically accurate and robust algorithm. Future work can then build on this solid foundation.
Although the simplifications related to PNM may induce additional errors into the numerical solution, it has been shown through comparisons between results of advection diffusion simulations, that the PNM approach provides reasonably accurate solutions (Yang et al., 2016) compared to those obtained from DNS using lattice Boltzmann and finite volume methods. This work presents a novel PNM framework for the simulation of charged species transport. The framework is based on highly accurate discretization schemes in addition to several charge conservation options. It also supports transient simulations and handles non-linear source terms.
Section snippets
Background
This work considers single-phase, isothermal, incompressible flow of a dilute electrolytic solution, treated as a Newtonian fluid, in a non-deformable porous medium. Assuming flow in the viscous-dominated regime (Agnaou et al., 2016, Agnaou et al., 2017), the movement of the electrolytic solution can be described using the following steady-state momentum and mass conservation (Stokes) equations and where is the velocity of the solution, its pressure, and its dynamic
Pore network modeling formulation
The pore network is a simplified representation of a real porous medium geometry, consisting of pore bodies interconnected by throats. Fig. 1 shows a pore-throat-pore conduit of a pore network. For the sake of simplicity regarding the conservation equations to be considered, idealized shapes are assigned to pores and throats. In this sense, and for a three-dimensional (3D) medium, pores and throats are generally represented by spheres and circular cylinders, respectively. For a two-dimensional
Comparisons with reference solutions
Ion transport problems over arbitrary disordered porous media were considered here. It is worth recalling that the structural disorder refers to the randomness in pores and throats sizes and in the coordination number of pores. The considered problems were solved numerically based on the PNM approach and, for the sake of comparison, based on the FEM. To assess the accuracy of different NMEs presented in Section 3, PNM simulations were performed using three different NMEs. The NMEs consist of
Conclusions
Ion transport problems in pore networks with random pore sizes and coordination numbers were considered and solved numerically using PNM and FEM solvers. The transport was modeled based on the NP equations for each charged species present in the electrolytic solution in addition to a charge conservation equation which relates the concentration of different species one to each other. In the presence of a fluid flow, the momentum and mass conservation equations, were adopted to describe the fluid
CRediT authorship contribution statement
Mehrez Agnaou: Conceptualization, Data curation, Formal analysis, Investigation, Methodology, Software, Validation, Visualization, Writing-original draft, Writing-review & editing. Mohammad Amin Sadeghi: Formal analysis, Data curation, Methodology, Writing-original draft, Writing-review & editing. Thomas G. Tranter: Software, Writing-original draft. Jeff T. Gostick: Conceptualization, Funding acquisition, Methodology, Software, Project administration, Resources, Supervision, Writing-original
Declaration of Competing Interest
The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.
Acknowledgments
This research was supported by CANARIE Canada .
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Computer code available on public repository https://github.com/PMEAL/OpenPNM.