Ion Transport Equations Research Articles

Harvesting Enhanced Blue Energy in Charged Nanochannels Using Semidiluted Polyelectrolyte Solution.

Blue energy generation in nanochannels based on salinity gradients is currently the most promising method in the area of nonconventional energy production. We used a semidiluted pure sodium carboxymethylcellulose (NaCMC)-KCl aqueous solution to study the characteristics of blue energy generation within a charged nanochannel. We solve the corresponding equations for ionic transport using a numerical technique based on the finite element method. Our analysis focused on the electric double layer (EDL) potential field, open circuit current, diffuse potential, electric conductance, maximum generated pore power, and maximum energy conversion efficiency by varying concentrations of the salt in the left-side reservoir and the bulk polyelectrolyte. The results indicate that as the polyelectrolyte concentration increases, the extent of EDL overlap considerably reduces. With an increase in polyelectrolyte concentration, the open circuit current increases, while the diffuse potential reduces. It was observed that both electrical conductance and maximal pore power improve considerably with higher polyelectrolyte concentrations. Interestingly, our modeling framework demonstrates a power density substantially higher (up to 16.31 W/m2) than earlier configurations and surpasses the established commercial limit (5 W/m2). Furthermore, our findings reveal that the reservoir salt concentration significantly affects the rate of decline in the maximum energy conversion efficiency as the polyelectrolyte concentration increases. The research paves the way for the development of high-power-density devices with several practical applications.

Intrinsic rotation modulation by diffusive neutral particles in tokamaks

The modulated transport model, a model kinetic ion transport equation for the pedestal and scrape-off layer (SOL), is generalized to self-consistently include effects of a single neutral particle species. The neutrals contribute additional transport terms, modifying the v∥ -dependent orbit-averaged ion diffusivities of the original work and the resulting predicted intrinsic rotation of the ions. After making simplifying assumptions of the neutral transport, in particular taking the continuous transport limit via a short charge-exchange step expansion, we derive relatively simple analytic expressions that capture the diffusive neutral physics. Within the scope of the model’s validity, the neutral-driven intrinsic rotation can compete with the turbulence-driven intrinsic rotation. However, for physically motivated parameters, the neutral-driven intrinsic rotation appears negligible, either on a term-by-term basis or due to a strong cancellation between the neutral-driven momentum diffusion and pinch terms. It appears that a treatment containing finite charge-exchange steps is necessary to capture neutral transport of strong flow momentum into the confined region from the SOL.

Influence of non-linear slip dependent zeta potential on the bi-layered electrohydrodynamic flow in an electrically actuated microsystem

In this paper, the electro-osmotic flow (EOF) of two immiscible fluids through an electrically actuated micro-slit with a non-linear slip-dependent zeta potential is considered. The main objective of this work is to enhance the EOF velocity of non-polar fluid with an intensive flow variation in micro-confinements. We present the comparison between the Poisson–Nernst–Planck (P–N–P) and the Poisson–Boltzmann model and illustrate the impact of different parameters, such as Debye–Hückel parameter, slip coefficient, and interfacial potential drop, on the ionic concentration and the induced potential in detail. In this study, the interface between two immiscible fluids is considered to be planar, and a potential drop is observed close to the interface due to the development of back-to-back diffuse layers. At the interface between two layers, the continuity of the velocity and total stresses (Maxwell stress and shear stress) are taken into account to investigate the flow field in the system of immiscible electrolytes. The P–N–P model, linked with the ion transport equation and the Poisson equation, is employed to describe the motion of electrolyte solutions. The non-linear governing equations are numerically computed using a pressure-correction-based finite volume technique based on a staggered grid algorithm. Closed-form analytical solutions are derived for both steady and unsteady EOF field of two immiscible fluids for the low wall zeta potential and non-overlapping electrical double layer. The analytical solutions are well validated with our numerical results under suitable assumptions. It is also observed that the slip-dependent zeta potential successfully provides a considerable enhancement in EOF velocity over a broad range of parameters such as zeta potential, Debye–Hückel parameter, as well as boundary slip parameter. The results for both layers are highly affected by unequal wall potential, and the ion concentrations are highly actuated by the external electric field.

Diffusiophoresis of a highly charged rigid colloid in a hydrogel incorporating ion steric interactions

Diffusiophoresis, mediated by an imposed ionic concentration gradient, is a useful technique in directed migration of colloids. Theoretical studies on diffusiophoresis in a polymer hydrogel medium are rather limited owing to the presence of nonlinear terms in the fluid transport equations. The hydrogel creates a hydrodynamic interaction and short-range steric interaction to the dispersing particle and macromolecules. The volume exclusion due to a finite ion size consideration is important in the context of a highly charged particle. In this study, we consider the finite ion size effects on diffusiophoresis of a charged particle with ζ≥1 in a hydrogel medium. The modified ion transport equations coupled with the Brinkman equation for fluid flow and the Poisson equation for electric field are solved numerically in a coupled manner through a control volume approach. Our results based on the Brinkman model incorporating only the hydrodynamic interactions deviate from the existing experimental results. A close comparison with this experimental data is established when the computed mobility is modified to incorporate the short-range steric factor. We have considered the impact of the hindered diffusivity of ions in the gel medium. Our results show that the modification in the ionic diffusivity is significant for low porosity and high charge density situations. The impact of the finite ion size effect is found to be significant when the bulk ionic concentration is large enough to create an electric double layer thinner than the particle size. We have analyzed the size dependency of the mobility in the gel medium. A simplified expression for the mobility in a hydrogel medium is proposed.

Developing a phenomenological model to simulate single and mixed scale formation during flow in porous media: Coupling a salt precipitation model with an ion transport equation under dynamic conditions

Water flooding and pressure maintenance are recommended to improve oil recovery practices after low recovery of petroleum reservoirs occurs during primary production. Salt crystal formation is a frequent occurrence when using these techniques. Several experimental, numerical, and theoretical studies have been done on the mechanisms underlying scaling and permeability reduction in porous media; however, there has not been a satisfactory model developed. This study developed a phenomenological model to predict formation damage caused by salt deposition. Compared with existing models, which provide a scaling tendency, the proposed model predicts the profile of scale deposition. The salt precipitation model simulates reactive fluid flow through porous media. A thermodynamic, kinetic, and flow hydrodynamic model was developed and coupled with the ion transport equation to describe the movement of ions. Further, a set of carefully designed dynamic experiments were conducted and the data were compared with the model predictions. Model forecasts and experimental data were observed to have an average absolute error (AAE) ranging from 0.68% to 5.94%, which indicates the model's suitability.

An Open-Source Continuum Model for Anion-Exchange Membrane Water Electrolysis

Anion-exchange membrane (AEM) electrolysis has the potential to produce green hydrogen at low cost by combining the advantages of conventional alkaline electrolysis and proton-exchange membrane electrolysis. The alkaline environment in AEM electrolysis enables the use of less expensive catalysts such as nickel, whereas the use of a solid polymer electrolyte enables differential pressure operation. Recent advancements in AEM performance and lifetime have spurred interest in AEM electrolysis, but many open research areas remain, such as understanding the impacts of water transport in the membrane and salt content in the electrolyte on cell performance and degradation. Furthermore, integrating electrolyser systems into renewable energy grids necessitates dynamic operation of the electrolyser cell, which introduces additional challenges. Computational modelling of AEM electrolysis is ideally suited to tackle many of these open questions by providing insight into the transport processes and electrochemical reactions occurring in the cell under dynamic conditions.In this work, an open-source, transient continuum modelling framework for anion-exchange membrane (AEM) electrolysis is presented and applied to study electrolyzer cell dynamic performance. The one-dimensional cell model contains coupled equations for multiphase flow in the porous transport layers, a parameterized solution property model for potassium hydroxide electrolytes, and coupled ion and water transport equations to account for water activity gradients within the AEM. The model is validated with experimental results from an AEM electrolyser cell. We find that pH gradients develop within the electrolyte due to the production and consumption of hydroxide, which can lead to voltage losses and cell degradation. The influence of these pH gradients on potential catalyst dissolution mechanisms is explored and discussed. Finally, initial studies of transient operation will be presented.This work has been performed in the frame of the CHANNEL project. This project has received funding from the Fuel Cells and Hydrogen 2 Joint Undertaking (now Clean Hydrogen Partnership) under grant agreement No 875088. This Joint undertaking receives support from the European Union's Horizon 2020 Research and Innovation program, Hydrogen Europe and Hydrogen Europe Research. Some of this work has been performed within the MODELYS project "Electrolyzer 2030 – Cell and stack designs" financially supported by the Research Council of Norway under project number 326809.

A physicochemical model-based digital twin of Li–S batteries to elucidate the effects of cathode microstructure and evaluate different microstructures

We are presenting a physicochemical model-based digital twin of Li–S batteries which involves continuum, volume-averaged ion transport equations including species migration, dissolution and precipitation, and electrochemical reactions, where the cathode microstructure is taken into account in terms of pore size distribution, hierarchy and tortuosity. The digital twin is applied in simulations of galvanostatic charge-discharge of a Li–S cell, exploring three types of cathode microstructure, activated carbon, graphene and hollow particles, two different amounts of sulphur and two different cathode coating thicknesses. Very good agreement is observed between predictions and experimental data in all case-studies. The best cathode is 70 wt% sulphur in host with porous hollow particles which trap the polysulphides from migrating to the anode. The simulations reveal that other reasons for low sulphur utilization, preventing the Li–S cell from reaching its theoretical capacity, are undissolved sulphur in micropores and sulphur leaching to macropores, especially in the cathode region near the separator, and reprecipitating.

Homogenization of the linearized ionic transport equations in random porous media

In this paper we obtain the homogenization results for a system of partial differential equations describing the transport of a N-component electrolyte in a dilute Newtonian solvent through a rigid random disperse porous medium. We present a study of the nonlinear Poisson–Boltzmann equation in a random medium, establish convergence of the stochastic homogenization procedure and prove well-posedness of the two-scale homogenized equations. In addition, after separating scales, we prove that the effective tensor satisfies the so-called Onsager properties, that is the tensor is symmetric and positive definite. This result shows that the Onsager theory applies to random porous media. The strong convergence of the fluxes is also established. In the periodic case homogenization results for the mentioned system have been obtained in Allaire et al (2010 J. Math. Phys. 51 123103).

A numerical solver for investigating the space charge effect on theelectric field in liquid argon time projection chambers

This paper reports the development of a numerical solver aimed to simulate the interaction between the space charge (i.e. ions) distribution and the electric field in liquid argon time projection chamber (LArTPC) detectors. The ion transport equation is solved by a time-accurate, cell-centered finite volume method and the electric potential equation by a continuous finite element method. The electric potential equation updates the electric field which provides the drift velocity to the ion transport equation. The ion transport equation updates the space charge density distribution which appears as the source term in the electric potential equation. The interaction between the space charge distribution and the electric field is numerically simulated within each physical time step. The convective velocity in the ion transport equation can include the background flow velocity in addition to the electric drift velocity. The numerical solver has been parallelized using the Message Passing Interface (MPI) library. Numerical tests show and verify the capability and accuracy of the current numerical solver. It is planned that the developed numerical solver, together with a Computational Fluid Dynamics (CFD) package which provides the flow velocity field, can be used to investigate the space charge effect on the electric field in large-scale particle detectors.

Global Electric Fields at Mars Inferred from Multifluid Hall-MHD Simulations

In the Martian induced magnetosphere, the motion of planetary ions is significantly controlled by the ambient electric fields, which can be decomposed into three components: the motional, Hall, and ambipolar electric fields. Each of them is dominant in different regions and provides the ion acceleration with a particular effectiveness. Therefore, it is necessary to characterize the global distribution of these electric field components. In this study, a global multifluid Hall-MHD model is applied, which considers the motional, Hall, and ambipolar electric fields in ion transport and magnetic induction equations to self-consistently investigate the morphology of the electric fields in the Martian space environment. Numerical results suggest that the motional electric field is dominant in the upstream of the bow shock and in the magnetosheath along the Z MSE direction, leading to the formation of the ion plume escape channel. At the bow shock, the ambipolar electric field points outward, to decelerate and deflect the solar wind plasma flow. In the magnetosheath region, the ambipolar and motional electric fields with inward direction tend to reaccelerate the solar wind ions. However, along the magnetic pileup boundary, the Hall electric field pointing outward prevents the solar wind ions from penetrating the Martian induced magnetosphere, which also prevails in the Martian magnetotail region, to accelerate the ions’ tailward escape. This is the first systematic investigation of the global distribution of electric fields, which is helpful to understand the processes of ion acceleration/deceleration and escape within the Mars–solar wind interaction.

Diffusiophoresis of a soft particle incorporating ion partitioning and hydrophobic core

The diffusiophoresis of a soft particle governed by an imposed ionic concentration gradient is studied based on a modified ion transport equation. The surface of the rigid core of the soft particle is considered to be hydrophobic, which is coated with a charged polyelectrolyte layer (PEL) whose dielectric constant is different from the dispersion medium. This step-change in the dielectric constant of the PEL and the dispersion medium creates an ion partitioning effect. The governing equations based on the coupled Navier–Stokes–Nernst–Planck–Poisson equations are modified to incorporate the ion partitioning effect. A simplified model based on the linear perturbation technique under a small applied concentration gradient in the scale of the particle radius is developed. The simplified model agrees well with the exact numerical solutions for a Debye length smaller than the particle size. We have analyzed the impact of the core slip length and PEL permittivity on the mobility of the soft particle. The interfacial slip enhances the impact of the core charge density on the diffusiophoresis. In this case, a reversal of mobility occurs at a thinner Debye length compared to a hydrophilic core. Ion partitioning attenuates the concentration of mobile counterions in PEL, leading to a reduction in screening of PEL fixed charge. We find that the diffusiophoresis of the soft particle in a non-z:z asymmetric electrolyte strongly depends on the valence of the counterions. Diffusiophoresis of the soft particle in a KCl-NaCl mixture is considered to achieve positive mobility.

Diffusiophoresis of a Charged Droplet in Asymmetric as Well as Mixed Electrolytes through Numerical and Semianalytic Models

A numerical study on the diffusiophoresis of a droplet in an electrolyte medium is carried out by solving the full set of coupled governing equations, which are based on the conservation principle. Diffusiophoresis is considered for monovalent as well as non-z:z electrolytes and mixed electrolytes. The numerical model is supplemented with a semianalytic simplified model based on first-order perturbation analysis, which agrees with the numerical model for a low to moderate range of surface potential. The mobility for a low-viscosity fluid at a thinner Debye length is dominated by the chemiphoresis part, which creates the mobility to become an even function of the surface charge density for a monovalent electrolyte. Such a pattern in mobility does not appear in a non-z:z asymmetric electrolyte. At a thinner Debye length, diffusiophoresis becomes independent of the diffusion field, hence the mobility is independent of the composition of electrolytes in a mixed monovalent electrolyte solution. Our results show that the size-based sorting of droplets is efficient when a mixed electrolyte is considered. We have also addressed the finite ion size effects by considering a modified ion transport equation. One of the key features of the present study is the simplified semianalytical model for the diffusiophoresis of a droplet in a z:z electrolyte as well as in non-z:z and mixed electrolytes, which is shown to be valid up to a moderate range of surface potential for a finite Debye length.

Nonlinearity of ion temperature gradient mode with entropy drift

We investigated the role of entropy in linear and nonlinear structures and obtained the linear dispersion relation from the set of Braginskii’s ion transport equations, demonstrating that entropy has an effect on the mode’s linear features. We also discovered that the system’s entropy is responsible for nonlinear structure changes such as compression to refractive types, as well as other features influenced by it and other plasma parameters such as dust density, charge number and mass, and so on. Our observations are very useful for space plasma observations such as interstellar space, planetary rings, and laboratory plasma because in the mentioned plasmas there is a lot of entropy due to the temperature differences.

Separation of ion component from solid hydrocarbon materials by laser ablation

Plasma usually consists of multiple ion component. Ion-component separation occurs in various conditions, and profoundly affects the plasma dynamic evolution. In this work, ion-component separation in two-ion-component plasma is investigated in the hydrodynamic condition. Starting from the Landau-Fokker-Planck equations of two-ion-component plasma, the ion transport equations are reduced through the Chapman-Enskog approach. The transport equations are then transformed into a set of linear algebraic equations and solved by expanding the perturbed ion distribution functions into the series of Sonine polynomials. The diffusive ion mass flows with inclusion of baro-diffusion, thermo-diffusion and electro-diffusion are thus obtained. With these efforts, the complete ion fluid equations are presented, which can be used to describe the processes of ion-component separation. We evaluate ion-component separation in the case of a solid CH plate target ablated with a laser pulse, by solving the ion diffusion equation with the hydro states output from the one-dimensional radiative hydro code Multi-1D. The simulation results show that ion-component separation mainly occurs around ablation front and under-dense region, and that the effect of ion-species separation on plasma hydrodynamic evolution is minor and can be neglected. For those physical processes sensitive to ion concentration such as Thomson scattering, however, the effect of ion-component separation is significant, which means that ion-component separation should be included in the study of laser plasma interaction.

Simulation of a strong steady-state plasma shock in a warm dense matter regime

The structure of collisional plasma shocks has been subject to an extensive, multi-decadal investigation—in the hydrodynamic, hybrid kinetic ion/electron fluid, and fully kinetic ion/electron limits. Despite this thoroughness, all of these studies apply exclusively to classical, weakly coupled plasmas. Here, we report the first results for a planar hydrodynamic simulation of a strong, steady-state shock in a subspace of the warm dense matter (WDM) regime. Specifically, we consider a plasma of fully degenerate electrons with moderate-to-strongly coupled ions. Since the WDM ion and electron transport coefficients and equation of state differ markedly from their non-degenerate, weak-coupling equivalents, we find that the structure of a WDM plasma shock notably deviates from the ideal plasma picture.

Upscaling the Poisson–Nernst–Planck equations for ion transport in weakly heterogeneous charged porous media

The Poisson–Nernst–Planck (PNP) equations govern the continuum level description of ions in electrolytes and especially the impact of charged surfaces. In numerous applications such surfaces are complex, varying on a small lengthscale compared to the overall scale of the system, often prohibiting the direct prediction of the osmotic swelling pressures induced by ion behaviours in Debye layers near surfaces. With periodicity, upscaling techniques can be readily used to determine the behaviour of the swelling pressure on large lengthscales without solving the PNP equations on the complex domain, though generalising to cases where the periodicity is only approximate is more challenging. Here, we generalise a method by Bruna and Chapman (2015) for upscaling a non-periodic diffusion equation to the PNP equations. After upscaling, we find a rational derivation of the swelling pressure closely resembling the classical, though phenomenological, use of Donnan membrane theory predictions for the swelling pressure in cartilage, together with a novel contribution driven by heterogeneous fixed (surface) charges. The resulting macroscale model is also shown to be thermodynamically consistent, though its comparison with a recent upscaled models for swelling pressure in cartilage mechanics emphasises the need to understand how macroscale models depend on differing upscaling techniques, especially in the absence of perfect periodicity.

Energetic variational approach for prediction of thermal electrokinetics in charging and discharging processes of electrical double layer capacitors

This work proposes a new variational, thermodynamically consistent model to predict thermal electrokinetics in electric double layer capacitors (EDLCs) by using an energetic variational approach. The least action principle and maximum dissipation principle from the non-equilibrium thermodynamics are employed to develop modified Nernst–Planck equations for non-isothermal ion transport with temperature inhomogeneity. Laws of thermodynamics are employed to derive a temperature evolution equation with heat sources due to thermal pressure and electrostatic interactions. Numerical simulations successfully predict temperature oscillation in the charging–discharging processes of EDLCs, indicating that the developed model is able to capture reversible and irreversible heat generations. The impact of ionic sizes and scan rate of surface potential on ion transport, heat generation, and charge current is systematically assessed in cyclic voltammetry simulations. It is found that the thermal electrokinetics in EDLCs cannot follow the surface potential with fast scan rates, showing delayed dynamics with hysteresis diagrams. Our work thus provides a useful tool for physics-based prediction of thermal electrokinetics in EDLCs.

The electrical impedance of carbon xerogel hierarchical electrodes

Many electrodes used in energy storage devices are produced by grinding a micro- and meso- or macro-porous material into a powder and depositing it on a flat substrate, which leads to a hierarchical structure. At the largest scale the structure is that of a flat layer; the intermediate structure is that of porous grains; the smallest scale is that of the microporous skeleton that makes up the grains.The paper uses electrical impedance spectroscopy to investigate both experimentally and through mathematical modelling how electric charges are stored and transported in such structure. The paper focuses on electrodes produced with a variety of carbon xerogels soaked with KCl solutions. Nonane adsorption is used during the electrode preparation to selectively block the micropores, and discriminate their contribution from that of larger pores. The impedance data are analysed with a mathematical model built on the Poisson–Boltzmann equation for charge storage and on the Nernst–Planck equation for ion transport, which accounts for the hierarchical structure of the electrodes.The equilibrium charge storage in large pores is found to be quantitatively described by the classical electric double layer. In micropores, however, non-ideality of the electrolyte has to be invoked to account for the measured capacitance, hinting at ion desolvation. As for the charge transport, each structural level is found to have a characteristic frequency, above which the storage of electric charges is transport-limited at the considered scale. The hierarchical structure of the electrode is responsible for salient characteristics of the impedance. In particular the Warburg regime, with impedance proportional to (iω)−1/2, is observed when transport is rate-limiting at one scale only. In hierarchical electrodes, an additional regime is observed with impedance proportional to (iω)−1/4 when transport limitations are present at two scales simultaneously.

Surface Charge Characteristics of DC-GIL Insulator Under Multiphysics Fields: 3-D Modeling

Surface charge accumulation on the insulator of direct current gas-insulated transmission lines (DC-GILs) under thermal-electric multiphysics fields is investigated using the finite element method (FEM). To obtain the accurate temperature and charge distributions in the horizontally installed gas-insulated transmission lines (GILs) with a basin-type insulator, a three-dimensional (3-D) geometric model is constructed. First, the equations and parameters are optimized to improve the computing efficiency. The thermal gradient (TG) negligibly affects the surface electrical conductivity. The recombination and diffusion terms in the ion transport equation can also be removed. After omitting these terms, the calculation efficiency is improved by approximately 70% without affecting the calculation accuracy. The surface charge density increases under TG but varies slightly in different radial directions along the insulator surface. In contrast, the electric field distribution obviously depends on the radial direction. The calculations confirm that the improved method properly analyzes the surface charge characteristics under thermal-electric fields and can thus evaluate the insulation performance of DC-GILs.

Experimental and simulation study on chloride diffusion in unsaturated concrete under the coupled effect of carbonation and loading

In this paper, the transport of chloride ions in marine concrete was investigated under coupled environment of carbonization and various static sustained loads. Firstly, the influences of concrete strength, exposed ages of concrete, flexural and compressive loads on chloride ion diffusion in concrete were studied by the salt spray erosion and carbonation erosion alternate test. The result showed that the content of chloride ions decreased with the increasing concrete strength, and increased with age. And the chloride ion content increased with the increase of the flexural stress and they first decreased then increased with the increase of the compressive stress level. Carbonization accelerates the chloride ion transmission and increases the chloride ion content to a certain extent under the experimental conditions. Then, the chloride ion transport equation in partially saturated concrete was formulated considering such multiple factors as carbonization, load, saturation, relative humidity and temperature. On the basis of the established chloride transport equation, the chloride diffusion process of the samples was simulated by using COMSOL software at the macro level, and the calculated results display fine consistency with experimental data. Finally, the multi-phase mesoscopic numerical models were proposed by two-dimensional meso-structure of samples with randomly distributed convex polygon aggregates. It is found that aggregate has a significant impact on the transport path and distribution uniformity of chloride ions.