numerical modeling of ion transport and adsorption in

Liu et al., manuscript submitted to Desalination

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Numerical Modeling of Ion Transport and Adsorption in Porous Media: A Two-scale Study for Capacitive Deionization Desalination

Min Liu1, John Waugh2,3, Siddharth Komini Babu2, Jacob S. Spendelow2, Qinjun Kang1*

1 Earth and Environmental Science Division, Los Alamos National Laboratory, Los Alamos, NM 87545, USA

2 Materials Physics and Applications Division, Los Alamos National Laboratory, Los Alamos, NM 87545, USA

3 Material Science Department, Vanderbilt University, Nashville, TN 37235

* Corresponding author


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Abstract A two-scale model is presented to simulate the dynamic ion transport and adsorption processes in porous electrodes used for capacitive deionization (CDI). At the pore scale, the Stokes equation governing water flow in porous CDI electrodes is solved using the lattice Boltzmann method and Nernst-Planck equations describing ion transport is solved using the finite volume method. The ion adsorption process is considered at the surface of carbon electrodes. At the continuum scale, Darcy equation and advection-diffusion equation governing water flow and solute transport through CDI cells are solved using OpenFOAM®. After validation against analytical solutions and previously published results, the model is used to study the coupled water flow, ion transport and adsorption at the pore and continuum scales. In the pore-scale modeling, the effect of electrode microstructure, electrical potential and flow velocity on the adsorption processes is quantitatively investigated, and the relative importance of various parameters is determined. In addition, the average adsorption rate derived from the pore-scale simulations is used in the continuum model to simulate water desalination in a flow-through CDI cell. The evolution of salt concentration during desalination has been quantified. It is found that the electrode adsorption capability has a direct influence on overall ion adsorption. Results also demonstrate that a narrower spacer between the two electrodes allows for faster ion adsorption. The presented model provides a numerical tool to quantitatively analyze the ion transport and adsorption in porous electrodes. It can help improve the fundamental understanding of the adsorption processes in porous electrodes for capacitive deionization.

Keywords Pore-scale modeling; Multiscale simulation; Reactive transport; Adsorption; Porous media


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1. Introduction The increasing need for clean and drinkable water, in the face of a changing climate and growing global population, has become a critical challenge. Limited freshwater resources are insufficient to meet this growing demand (AlMarzooqi et al., 2014; Kang et al., 2017; Salamat and Hidrovo, 2020). Due to the abundance of seawater and brackish water resources, desalination becomes an attractive approach of great potential to produce clean and potable water.

Desalination techniques of removing inorganic ions mainly include thermal (distillation), pressure (reverse osmosis) and electrical driven (capacitive deionization and electrodialysis) methods. In distillation, the freshwater is obtained by firstly vaporizing the seawater or brackish water and then condensing the steam. The application of this technique is limited due to high energy consumption. In contrast, reverse osmosis (RO) is a pressure-driven and membrane-based method which employs semipermeable membranes to selectively remove the salt from seawater. However, the requirement of high pressure increases the operational cost while the transport through membrane limits the production efficiency of potable water (Qasim et al., 2019). Unlike distillation and RO, the capacitive deionization (CDI) method enables removal of the minor constituent (salt) from the major constituent (water) based on electrochemical adsorption in the active carbon electrodes, and is therefore particularly attractive for purification of dilute saltwater resources (i.e. brackish water) (Liu et al., 2020). During water desalination using CDI, seawater flows through/by the porous electrodes of the CDI cell. Salt ions such as Na+ and Cl- are driven by the electrostatic force and adsorbed at the surface of the porous active carbon materials in the electrodes and thus removed from the water (AlMarzooqi et al., 2014; Oren, 2008).

In the past ten years, there have been an increasing number of studies on CDI to improve performance of desalination (AlMarzooqi et al., 2014; Humplik et al., 2011; Qu et al., 2015; Qu et al., 2016; Salamat and Hidrovo, 2020). Different CDI cell architectures have been developed, aiming to increase the ion adsorption capability and reduce energy requirement. Among these, two of the most popular CDI cell architectures are flow-by and flow-through CDI (Remillard et al., 2018). These two systems both consist of two porous electrodes made of adsorptive carbon materials. In a flow-through cell, as shown in Figure 1, the feed water flow is perpendicular to the porous electrodes. An electrical potential is applied between the porous electrodes, driving charge on the surface of the carbon materials. The ion transport and adsorption in CDI cells occur in the pore space within the electrodes. The movement of ions is affected by advection, diffusion, and electrostatic force.

Figure 1 Schematic representation of the architecture of a flow-through CDI cell

Feed water Fresh water

Separator

Porous electrodes

Electrode Structure (Li et al. 2019)


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Apart from the architectures of the CDI cell, the high ion transport resistance and low ion adsorption capacity are currently the main limitations of CDI performance (Porada et al., 2012). Extensive research on improving ion adsorption capacity and decreasing ion transport resistance has been conducted (Huang et al., 2013; Kondrat et al., 2014; Liu et al., 2015; Porada et al., 2012; Satterfield et al., 1973; Wang et al., 2014). For instance, Porada et al. (2012) constructed CDI cells with different activated carbon materials and compared their salt adsorption capacity. They found that carbon materials with well-defined sub-nanometer pore sizes show significantly higher salt adsorption capacity at the same voltage, indicating the potential of carbide-derived carbon materials for energy-efficient water desalination. Qu et al. (2015) characterized the electric resistance in the CDI system by proposing measurable figures of merit. They measured resistive components in circuit models and found contact pressure between porous electrodes and current collectors can lead to low contact resistance. Nevertheless, these studies were mostly focused on experimental investigations, which are often time consuming and expensive. Alternatively, physics-based modelling can be performed under different operating conditions to find critical parameters more quickly and less expensively than experiments. Thus, the physics-based model that accounts for transport and adsorption of ions in the CDI electrodes has become a promising method to improve the desalination performance.

Recently, several numerical models have been developed to quantify and analyze ion transport and adsorption in porous CDI electrodes (Chen et al., 2018; Khan, 2003; Mugele et al., 2015; Qu et al., 2016; Salamat and Hidrovo, 2018). For example, Liu et al. (2019) developed a pore-scale model accounting for ion transport and adsorption in porous electrodes of a CDI cell. They used this model to investigate the transient ion adsorption at the pore scale and studied the effects of porous structures on the ion adsorption process. They found that the inlet velocity does not affect the final ion adsorption amount, but changes the time to reach the final steady state. Their results also showed that porous structures with a moderate particle aggregation degree have the best capacitive deionization performance. Salamat and Hidrovo (2020) studied the impact of micropores on adsorption/desorption of ions in a CDI cell. They included the sorption resistance and non-electrostatic attractive forces into their model and compared the simulation results with experiments using a circular CDI cell operating under various conditions. Their results showed that neglecting the electro-sorption resistance can result in up to 50% overestimation of the energy efficiency and overall desalination performance.

However, very few models have considered multiscale features by including pore-scale models representing explicit pore structures, and adsorption-transport behaviours at the scale of a CDI cell. Besides, no previous numerical study has been performed directly on the real geometries of carbon electrodes obtained from scanning electron microscope (SEM) images. Herein, a multiscale model based on direct numerical simulations is presented to examine ion transport and adsorption in CDI systems with different parameters. The effect of pore structures and electrical potential on ion adsorption in porous electrodes are quantitatively determined. The results are interpreted using ion concentrations and pore structures. The pore-scale results are also incorporated into the continuum model to analyze water flow, ion transport and adsorption at continuum scale.


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2. Numerical model setup Fluid flow, ion transport and adsorption in the porous electrodes are considered in the numerical models. In this section, the mathematical formulations for the pore and continuum-scale models are presented.

2.1.Pore-scale model

Laminar and incompressible fluid flow at low Reynolds number is assumed in the porous

electrodes. Thus, the continuity and the Stokes equations are considered as (Sahimi, 2011)

∇ ∙ 𝒖 = 0, (1)

∇p = µ∇!𝒖, (2)

where 𝒖 is fluid velocity vector, p is pressure, and µ is viscosity of the fluid.

In the current study, adsorption of Na+ and Cl- ions from saline water is considered. The transport

of ions is governed by the Nernst-Planck equation,

𝜕𝐶!𝜕𝑡

+ (𝒖 ∙ ∇)𝐶! = ∇ ∙ (𝐷!∇𝐶!) +𝑒𝑧!𝐷!𝐾"𝑇

∇ ∙ (𝐶!∇𝜓), (3)

where 𝐶", 𝐷" and 𝑧" represent the local concentration, diffusion coefficient, and ion valence of the i-th ion, respectively. 𝑒, 𝐾#, 𝜓, T represent the value of one proton charge, Boltzmann constant, electrical potential and temperature, respectively.

The distribution of electrical potential obeys the following equation:

∇#𝜓 = −𝜌$𝜀%𝜀&

(4)

where 𝜀$ is the vacuum electric permittivity and 𝜀% is the dimensionless dielectric constant. 𝜌& denotes the net charge density and is related to the ion concentration as follows:

𝜌$ =5𝐹𝑧!𝐶!!

(5)

where 𝐹 is the Faraday constant.

The adsorption on the surface of carbon materials in the porous electrodes is described by

𝐷!𝜕𝐶!𝜕𝑛

= 𝐾'𝐶! , (6)

with the adsorption constant 𝐾'.

The governing equations for fluid flow are solved via lattice Boltzmann method. An implicit finite volume method is employed to discretize the governing equations for ion transport (Liu et al., 2020a; Liu and Mostaghimi; Mostaghimi et al., 2013). The details of the numerical methods can be found in Mostaghimi et al. (2016) and Liu and Mostaghimi (2017b).


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2.2. Continuum model

In the continuum model, the fluid properties are averaged over a control volume. For an incompressible fluid and constant porosity, the continuity equation for the fluid phase at the continuum scale is described as

∇ ∙ 𝒖8 = 0, (7)

where 𝒖2 is the average velocity in each control volume. The velocity is solved by Darcy’s law,

𝒖8 = −𝑘µ∇p=, (8)

where k is the permeability and p3 is the pressure in the control volume. The conservation equations for transport of ions in a multicomponent aqueous fluid can be written as

𝜕(𝜑𝐶!̅)𝜕𝑡

+ ∇ ∙ 𝐽! = 𝑅! , (9)

where 𝜑 is the porosity, 𝐶2" is the concentration in each control volume, 𝑅" is adsorption term determined from pore-scale simulations and 𝐽" is the solute flux of the 𝑖 th species, which is expressed by

𝐽! = −𝜑𝐷!∇𝐶!̅ + 𝒖8𝐶!̅ , (10)

In the continuum model, the Salt Adsorption Capacity (SAC) is included, which is the amount of ion taken up by the adsorbent per unit mass, and can be expressed as

𝑆𝐴𝐶 =𝑞 ∫ ∆𝑐(𝑡)𝑑𝑡(

&𝑚

(11)

where 𝑞 is the volumetric flow rate, ∆𝑐(𝑡) is the difference between the initial concentration and current concentration (C0-C(t)), and 𝑚 is the mass of the porous electrodes. When the adsorption capacity is reached in the simulations, the adsorption stops and the adsorption rate will be automatically assigned to zero value in the model. The presented continuum model is implemented in the open-source simulation platform OpenFOAM® (http://www.openfoam.org), which is a C++ library that solves problems described by partial differential equations using finite-volume methods.

3. Model validation For the model validation of ion transport and adsorption due to advection and diffusion developed in Liu and Mostaghimi (2017b), we compare our simulation results with the previous studies by Zhou et al. (2015). An identical system is used in both Zhou et al. (2015) and our validation simulations. The model simulates the fluid flow through a channel with ion adsorption at the bottom surface (as shown in Figure 2a). The size (𝐿 × 𝐿) of the domain is 320×320 in lattice units.


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A constant concentration 1.0 is enforced at the inlet and surface adsorption obeying the first-order kinetics is considered at the bottom wall of the channel. No-slip boundary condition is applied at the upper wall. The adsorption rate constant is set as 1.0, while the diffusion coefficient is 1/6 in lattice units. A laminar flow regime is solved in the whole domain with a fully developed flow condition enforced at the outlet. The concentration distribution is obtained by solving the advection-diffusion equation. Details about the solution of the model are available in Zhou et al. (2015). Figure 2b shows ion mass flux along the bottom wall of the channel obtained from both Zhou et al. (2015) and our present study. It is clear that both results are in excellent agreement.

(a) (b)

Figure 2 (a) The schematic of the channel with advection, diffusion and adsorption at the bottom; (b) comparison with the results from Zhou et al. (2015).

We validate our simulation of ion migration caused by electrical potential by comparing with the analytical solutions in a half channel shown in Figure 3(a). The width of the domain is 1 µm. Zero potential and constant concentration of 1×10-2 mol/m3 are applied at the centerline. The constant electrical potential of −10mV is enforced at the bottom wall. The analytical solution of ion concentration is (Politzer and Truhlar, 2013)

𝐶) = 𝐶)*exp(−𝑒𝑧)𝜓𝐾"𝑇

), (12)

where the ion valence 𝑒 is 1.6 × 10()*𝐶. The value of Boltzmann constant 𝐾# is 1.38 × 10(!+𝐽/𝐾.

The analytical electrical potential distribution is computed as (Liu et al., 2019)

𝜓(𝑦) = 4𝐾"𝑇𝑒𝑡𝑎𝑛ℎ+,(tanh(

𝑒𝜓-4𝐾"𝑇

exp(−𝑦

U𝜀%𝜀&𝐾"𝑇2𝐶)*𝑒𝐹

))). (13)

The comparison of ion concentrations between our model and analytical solutions are demonstrated in Figure 3b. The simulation results of Na+ and Cl- concentrations present good agreement with analytical solutions.

L

L

Flow

No adsorption

Surface adsorption


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(a) (b)

Figure 3 (a) Half channel geometry for validation of electro-migration;(b) Comparisons with analytical solutions for ion concentration

4. Ion transport and adsorption in porous electrodes at pore scale In this section, pore-scale simulations of ion transport and adsorption are carried out in porous carbon-based electrodes. The impact of pore structures, specific surface area, flow velocity, and electrical potential on adsorption is quantitatively analyzed and investigated. In the models, only the charging process of CDI is considered. The electrical potential is assumed to be zero for inlet, outlet, upper and bottom boundaries. The ions are assumed to be adsorbed only on the surface of the carbon.

4.1.The effect of pore structures

The pore structure is a key factor affecting solute transport in porous media (Kang et al., 2014; Liu and Mostaghimi, 2017a; Liu et al., 2020b). In this section, six porous electrodes with different pore structures are constructed and shown in Figure 4. The porosity and surface area are the same in these porous structures. The size of the simplified microstructure domain used in pore-scale study is 2 µm ´ 2 µm (Figure 4). Notably, to avoid the boundary effect on the concentration distribution in Geometry_c and Geometry_f, we perform simulations in the domain with 3.3 µm ´ 2 µm for these two cases, and only compare the results in the selected section of 2 µm ´ 2 µm. The diameter of the solid materials is 0.2 µm, with a constant electrode potential of -20 mV. Feedwater with NaCl flows in from the inlet on the left.

(a) (b) (c)

Zero potentialConstant concentration

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(d) (e) (f)

Figure 4 Six constructed porous geometries used for simulations: a-Geometry_a; b-Geometry_b; c-Geometry_c; d-Geometry_d; e-Geometry_e; f-Geometry_f.

Adsorption simulations have been performed in the six porous structures at two different velocities: 2.1´10-4 m/s and 1.26´10-3 m/s. The concentration distributions of Na+ at steady state are shown in Figure 5 and 6. The different arrangements of adsorbent materials result in great differences in the ion concentration distributions. For example, for Geometry_b in Figure 5b, the adsorbent materials are placed in the middle of the domain. This results in low concentrations in the middle area due to strong adsorption. However, in Geometry_d (Figure 5d), the adsorbent materials are placed near the top and bottom. Most ions are transported through the middle area of porous electrodes, causing high concentrations in this area due to lack of adsorption. In Figure 5f, the adsorbent materials are placed at the back of the domain. There is no adsorbent in the upstream, causing very high local Na+ concentration.

(a) (b) (c)


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(d) (e) (f) Figure 5 Na+ concentration distributions in: a-Geometry_a; b-Geometry_b; c-Geometry_c; d-

Geometry_d; e-Geometry_e; f-Geometry_f, at velocity 2.1´10-4 m/s.

(a) (b) (c)

(d) (e) (f)

Figure 6 Na+ concentration distributions in: a-Geometry_a; b-Geometry_b; c-Geometry_c; d-Geometry_d; e-Geometry_e; f-Geometry_f, at velocity 1.26´10-3 m/s

In comparison with concentrations of Figure 5, ion concentrations are higher in the same geometries in Figure 6, due to faster flow and mass transfer. For example, in Geometry_a, the concentrations near the top and bottom boundary of the domain are much higher than those in Figure 5a. In addition, the concentrations in the downstream of Geometry_e are much higher than those in Figure 5e. This is because higher velocities cause faster transport of ions. With same adsorption rate constant, more Na+ is transported through the media during the same time period. This leads to higher concentration in the domain.

The salt removal percentage in these six structures are calculated and compared in Figure 7. The highest percentage at velocity 2.1´10-4 m/s (Figure 7a) and 1.26´10-3 m/s (Figure 7b) are 96% and 71%, respectively, resulting from Geometry_a for both flow rates. This means Geometry_a is most efficient for removing the salt ions, and is preferred for CDI cells. By contrast, Geometry_b has the lowest salt removal percentage for both cases, and hence the worst desalination performance.


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(a) (b)

Figure 7 Comparison between the salt removal percentage in different porous electrodes at velocity (a) 2.1´10-4 m/s and 1.26´10-3 m/s.

4.2.The significance of specific surface area for adsorption

Interactions between fluid and solid are closely related to the surface area, which is important for the physicochemical processes in porous media (Chen et al., 2013; Liu and Mostaghimi, 2018a). In this section, we investigate the impact of specific surface area (SA) on the ion adsorption processes. We generate four different structures of porous electrodes with various specific surface area but same porosity. The porous electrodes are demonstrated in Figure 8, with increasing specific surface area of 5 µm-1, 10 µm-1, 20 µm-1 and 30 µm-1. In the simulations, identical average flow velocity and adsorption rate constant are applied. The only difference in the simulations is the specific surface area.

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Geom_a Geom_b Geom_c Geom_d Geom_e Geom_f

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Geom_a Geom_b Geom_c Geom_d Geom_e Geom_f

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(c) (d)

Figure 8 Porous electrodes with different surface area: a-SA1=5 µm-1; b-SA2=10 µm-1; c-SA3=20 µm-1; d-SA4=30 µm-1

The steady-state concentration distributions of Na+ in porous electrodes with different specific surface areas are shown in Figure 9. When the specific surface area is 5 µm-1, the ion concentration in the pore space is high, excluding the area close to the surface of the porous electrode. With the increase of specific surface area, the concentration gradient between the inlet and outlet becomes larger, as a larger specific surface area leads to stronger adsorption.

(a) (b)

(c) (d)

Figure 9 Concentration distributions of Na+ ion in porous electrodes: a-SA1=5 µm-1; b-SA2=10 µm-1; c-

SA3=20 µm-1; d-SA4=30 µm-1


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The salt removal percentage for different cases is shown in Figure 10. This percentage increases from 61% for the case of lowest specific surface to 99% for the case of largest specific surface area. The results indicate a higher surface area of carbon electrode is more desirable for ion adsorption. However, to maintain the same velocity in these porous electrodes, different pressure gradients are applied. Pressure gradient (normalized by that for case SA1=5 µm-1) for different cases is also shown in Figure 10. The results show the higher the specific surface area, the larger the pressure gradient is required to obtain the same velocity. Though ion adsorption is stronger in electrodes with a larger specific area, the requirement of higher-pressure gradients would result in additional operating cost. Thus, it is suggested an optimized design should be made to achieve a balance between the adsorption ability and pressure requirements.

Figure 10 Comparison between normalized pressure gradient and salt removal percentage in porous

electrodes with different specific surface area

4.3. The effect of flow velocity

To represent the real structures of the porous electrodes, a SEM image is digitalized for simulations (Figure 11). Ion transport and adsorption are simulated in the geometry of porous electrodes shown in Figure 11b. The digitized image for modelling has a size of 43.3 × 32.5 µm. Initially the system is saturated with Na+ and Cl- and a Dirichlet boundary condition is considered at the inlet for the concentration equation. The diffusion coefficient of Na+ and Cl- is 1.0 × 10(*𝑚!/𝑠 (Parkhurst and Appelo, 1999). The adsorption constant is 𝐾' = 1.0 × 10(,𝑚/𝑠 (Liu et al., 2019). Constant concentration (C0=1.0) is enforced at the inlet (left). The electrical potential difference is set as 40mV, which is in the range of normal values in pore-scale studies by Liu et al. (2019). Fluid flows from left to right. Simulations are carried out at different flow velocities but the same adsorption constant and electrical potential.


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(a) (b)

Figure 11 (a) SEM images (43.3 × 32.5 µm) of mesoporous carbon reproduced from Li et al. (2019) and (b) digitalized microstructure from SEM for modeling.

The steady-state concentration distributions of Na+ at different flow velocities are demonstrated in Figure 12. Different flow velocities result in great variations in Na+ ion concentration distribution in porous electrodes. The Na+ concentration decreases along the flow direction as a result of combined transport and adsorption. At low velocity, for example, in Figure 12a, most Na+ is observed in fast flow paths. The concentration of Na+ ions in small pore spaces or narrow channels are obviously lower due to low local velocity and electrical attraction. With the increase of velocity, at velocity 2.4 × 10(,𝑚/𝑠 (Figure 12b), a higher concentration of Na+ in the bottom right corner is observed. This is due to the faster mass transfer at higher velocity. For a fixed adsorption rate constant, a faster mass transfer can lead to more ions transported through the pore space in electrodes. At the highest velocity in Figure 12c, Na+ ions are transported to more narrow pore throats.

(a) (b) (c)

Figure 12 The concentration of Na+ in the porous electrodes with velocity of: a-6 × 10+.𝑚/𝑠;c-2.4 × 10+/𝑚/𝑠; d-9.6 × 10+/𝑚/𝑠;

The vertically averaged concentration of Na+ ions as a function of distance from the inlet is shown in Figure 13. As the pore space is initially assumed to be saturated with salt water, the Na+ concentration at the inlet is 1.0. With the increase of distance from the inlet, the concentration decreases due to mass transfer and adsorption. The outlet Na+ concentration is higher at higher flow velocities. This indicates the faster flow is not preferred for stronger adsorption. At higher flow velocity, the mass transfer is faster so that more ions are flushed out the domain during the


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same period of time before getting adsorbed on the electrode surface. Therefore, the concentration remains high with the same electrical potential and adsorption rate constant.

Figure 13 The vertically averaged concentration of Na+ ions as a function of distance from the inlet

4.4. The impact of electrical potential difference

The impact of electrical potential on the ion adsorption in porous electrodes is also investigated. Simulations are performed on the geometry of the porous electrodes in Figure 11b at three values of electrical potential at the solid surface. Identical flow velocity is applied in all the cases. The Na+ concentration distributions are shown in Figure 14. At the lowest potential difference of 20mV (Figure 14a), the electro-migration of ions is quite weak. As a result, the Na+ ion concentration in the pore space is higher due to the weaker electrical attraction. When the potential at the solid surface is−40𝑚𝑉, Na+ concentration in many narrow flow paths or pore throats becomes much lower and even approaches zero. This indicates stronger electrical attraction accelerates the transport of Na+ ions to the solid surface, which are subsequently removed by adsorption. At the highest potential difference in Figure 14(c), there are more pore space and flow paths with low Na+ concentration because of the stronger adsorption.

(a) (b) (c)

Figure 14 Na+ concentration distribution with wall electrical potential (a) −20𝑚𝑉 (b) −40𝑚V (c) −60𝑚𝑉

The vertically averaged concentration of Na+ ions as a function of distance from the inlet at different electrical potential differences are compared in Figure 15. In the porous electrodes, the Na+ concentration is always higher at the lower potential difference. At a high potential difference (60𝑚𝑉), the outlet Na+ concentration is reduced to a quite low value 0.04. In the simulations at the

0

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0.6

0.8

1

1.2

0 10 20 30 40 50

Norm

alize

d co

ncen

tratio

n

Distance (!m)

Low velocityMedium velocityHigh velocity


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medium potential difference (40𝑚𝑉), the ion concentration decreases from 1 to 0.1. The highest outlet concentration is 0.15 when the potential difference is the lowest at 20𝑚𝑉 . The results indicate larger potential difference can effectively enhance the ion adsorption and reduce the Na+ ion concentration in electrodes. It also shows our model can be used to quantitatively describe the Na+ concentration variations during adsorption at different electrical potential differences. This allows predicting optimal electrode parameters under different operating conditions.

Figure 15 The vertically averaged concentration of Na+ ions as a function of distance from the inlet at

different potential difference

5. 3D modelling of ion adsorption in fibrous electrodes Fibers are popular adsorbent used in capacitive deionization electrodes (Huang et al., 2017; Liu et al., 2020). Figure 16 shows a SEM image of polyacrylonitrile (PAN) fibers in a porous electrode. In this section, we create fibrous structures and perform simulations of ion transport and adsorption in 3D porous electrodes.

Figure 16 SEM image of PAN fibers in porous electrodes


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Cylindrical fibers with given diameters are generated and randomly arranged in a 3D computational domain. The size of the domain is 150×150×150 voxels while the resolution of each voxel is 100nm. Figure 17 shows four samples of the reconstructed fibrous medium with same fiber diameter of 400nm. The black color refers to the carbon fibers and void area denotes the pore space. The corresponding porosity of the samples from (a) to (d) is 50%, 60%, 70% and 80%, respectively.

(a) (b)

(c) (d)

Figure 17 3D fibrous media with fiber diameter 400nm and porosity of (a) 50%; (b) 60%; (c) 70% and (d) 80%

We investigate the effect of porosity and fiber diameters on the ion adsorption behaviors in fibrous media. The potential at the material surface is set as -1.2V and the average flow velocity is 1.0´10-

6 m/s. Figure 18-20 show the Na+ concentration distribution in fibrous media with different fiber diameters and porosities. Figure 21 presents quantified salt removal percentage in fibrous media at different fiber diameters and porosities. For the same fiber diameter, the higher the porosity, the lower the salt removal percentage. The reason is that for a given fiber diameter, a lower porosity means denser fibers and higher specific surface area of the whole porous electrode, which leads to stronger adsorption for a fixed flow rate. For example, in the simulations with fiber diameter of 600nm (black squares), the salt removal percentage decreases from 84% to 28% when the porosity increases from 50% to 80%. This indicates higher porosity results in weaker salt adsorption in porous electrodes.

For the simulations with same porosity but different fiber diameters, the results show salt removal percentage increases as the fiber diameter decreases. Again, the reason is that for a given porosity, a smaller fiber diameter means higher specific surface, causing stronger adsorption in the porous

5µm


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electrodes when the same flow velocity is enforced. At porosity 70%, the salt removal percentage is calculated as 40% in fibrous media with a diameter of 800nm, but it can be as high as 62% when the fiber diameter is 400nm. It is indicated the finer fibers can greatly improve the salt removal percentage, causing better salt removal in porous electrodes.

(a) (b)

(c) (d)

Figure 18 Na+ concentration distribution in fibrous media with fiber diameter of 400nm at porosity (a) 50%;(b) 60%;(c) 70% and (d) 80%

(a) (b)


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(c) (d)


(a) (b)

(c) (d)



20

Figure 21 Salt removal percentage in fibrous media at different fiber diameters and porosities.

6. Macroscale transport in porous electrodes with ion adsorption Ion transport and adsorptions in porous electrodes occur at different scales. Studies of ion transport and adsorption across scales can help enhance the understanding of the adsorption process (Liu and Mostaghimi, 2018b; Raoof et al., 2010). In this section, we apply a continuum model to simulate the ion transport and adsorption in a CDI cell. The width of the computational domain in the continuum model is 1200 µm and the length is 680 µm (Figure 22). In this model, each grid node is considered as a control volume. We first calculate the average adsorption rate from pore-scale simulations using 𝑟 = -.

-/= ∫(.!(."#)-/

∆/. Then, we apply this rate to each control volume of

continuum model. The salt adsorption capacity is included into the model to study its effect on adsorption processes.

Figure 22 The computational domain of a flow-through CDI cell.

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In the simulations, a constant concentration is enforced at the inlet. The average flow velocity is set as 2.0 × 10(4𝑚/𝑠 (Qu et al., 2016). The computed adsorption rate from pore-scale model is 9.2´10-3 molm-3s-1. The adsorption capacity is set as 8.65 mg/g (Bryjak et al., 2015). During the simulations, the accumulative amounts of adsorbed ions is calculated at each time step. Once the adsorption capacity is reached, the adsorption module will be turned off automatically in the simulations and there will be no more adsorption in the cell.

Figure 23 shows the evolution of average Na+ concentration at the outlet of the CDI cell. The ion adsorption process consists of three stages. In the first stage, the concentration in the porous electrodes decreases with time due to adsorption. For example, initially the concentration is 1.0. With an increase in time, the concentration decreases to 0.55 at tD=8. Afterwards, the concentration distribution in the CDI cell reaches a steady state due to the limitation of adsorption rate, where the outlet concentration remains constant. This is the second stage with a steady adsorption rate. Over time, the adsorbed mass in the materials accumulates until it reaches its limited adsorption capacity. After the capacity is reached at tD=11, there is no more adsorption in the CDI cell, and the concentration starts to increase due to salt ions transport from inlet. This is the third stage. Finally, the concentration will recover to the initial status. In real applications, once the ion adsorption capability is reached, reversing the polarity is required to regenerate and release the concentrated ions into the discharge water. Energy recovery is possible during the regeneration process, which is similar to discharging a capacitor. In current study, the regeneration process is not considered.

Figure 23 The variations of Na+ concentration at the outlet with time

6.1.The effect of adsorption capacity

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The amount of desalination through the CDI systems is dependent on the ion adsorption capability. which is controlled by the physical properties of the carbon materials in electrodes. In this section, we performed simulations to investigate the effect of salt adsorbent capability on the ion adsorption. The values of salt adsorption capacity used in the simulations are 8.65 mg/g (low), 13.1 mg/g (medium) and 17.53 mg/g (high), respectively. Figure 24 demonstrates the evolution of outlet Na+ concentration in CDI cell with different SAC. Simulation results with three SAC present different adsorption time. At high SAC (black triangles), the adsorption process continues until tD=28, which lasts for the longest due to the high adsorption capacity. At the medium SAC, the Na+ keeps being adsorbed until tD=19. At the lowest SAC=8.65 mg/g, the adsorption stops only at tD=12 because of the low adsorption capacity. The results show the adsorption capability can greatly affect the adsorption time of the CDI cell. It is indicated that the charging and regeneration cycle has a great dependency on the adsorption capability. Meanwhile, the adsorption capability can also be estimated based on the adsorption time or charging cycle. This can be utilized to select the material with desirable adsorption capability for CDI cell fabrication.

Figure 24 The evolution of outlet concentration with different adsorbent capability

6.2.The impact of spacer thickness

The CDI cell consists of two electrodes and the spacer. The spacer is used to electronically isolate these electrodes. In the construction of CDI cell, spacers with different thickness are also used. However, the effect of spacer thickness on the adsorption has not been clearly studied. In this section, we explore the impact of spacer thickness by performing simulations in CDI cell with different spacer thickness: 40 µm, 80 µm and 160 µm.

Figure 25 shows the evolution of ion concentration in CDI cells with different spacer thickness. The results show that the spacer thickness will not affect the total salt adsorption amount of the CDI cell. However, it will affect the desalination time. When the spacer thickness is the smallest (40 µm), the adsorption reaches salt adsorption limitation at tD=19 and the Na+ concentration goes back to the initial saturation at tD=32. In the CDI cell with spacer thickness of 80 µm, adsorption stops at tD=22. At the spacer thickness of 160 µm, the adsorption continues until tD=24, and the

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concentrations take the longest time recovering to initial value at tD=45. Thicker spacer causes longer recovery time back to the initial concentration after adsorption capability limitation is reached. The quantitative comparison suggests thinner spacers are preferred in CDI cells to improve the desalination efficiency and water throughput.

Figure 25 Concentration evolution in CDI cells with different spacer thickness

7. Conclusions In this paper, we presented a numerical study of the electrical-chemical process involving ion transport and adsorption in porous electrodes at pore and continuum scales. Advection, diffusion, electrical migration and adsorption are simulated in the simulations. The model is validated by comparing results with previous results and analytical solutions. The impact of pore structures on adsorption is investigated. Adsorption simulations have been performed in six porous electrodes. A uniform pore structure increases ion adsorption, which is favorable for applications such as CDI that depend on high ion removal. The significance of the specific surface area of adsorbent materials is also quantitatively determined. The ion transport and adsorption demonstrate great dependency on flow velocity and electrical potential. The impact of velocity was studied by performing simulations on images of real porous electrodes. Simulations were also conducted in porous electrodes by enforcing different potential difference. The results indicate larger potential difference can effectively enhance the ion adsorption and reduce the Na+ ion concentration in electrodes. It also shows our model can be used to quantitatively describe the Na+ concentration variations during adsorption at different electrical potential differences. This allows predicting optimal electrode parameters under different operating conditions.

We also performed simulations in 3D fibrous media. For the simulations in fibrous media with same fiber diameter, the higher porosity is, the lower salt removal percentage is calculated, which indicates higher porosity results in weaker adsorption in porous electrodes. For the simulations with same porosity but different fiber diameters, the salt removal percentage in fibrous media with finer fiber is higher than those with larger fiber diameters. It is indicated that the finer fibers enhance the adsorption in porous electrodes.

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A continuum model was used to simulate the adsorption at the scale of CDI cell. The adsorption capacity was included in the model. Three stages of outlet salt concentration variations are captured. As the adsorption occurs, the adsorbed mass in the materials accumulates until it reaches the adsorption limitation. After the limitation is reached, the adsorption stops and the concentration starts to increase due to mass transfer from inlet. Finally, the concentration recovers to the initial status. Simulations were also carried out in CDI cells with different salt adsorption capabilities. It is indicated that the charging cycle has great dependency on the adsorption capability. Meanwhile, the adsorption capability can also be estimated based on the adsorption time or charging cycle. This can be utilized to select the materials with desirable adsorption capability for CDI cell fabrication. Besides, the effect of spacer thickness on Na+ ion adsorption is also investigated. The results show that the spacer thickness will not affect the total amount of adsorbed ions in the CDI cell, but it will reduce the desalination time efficiency. Thinner spacers are preferred in CDI cells to improve the desalination efficiency and water throughput.

The current model provides a numerical approach to better understand and quantitatively characterize adsorption in porous electrodes. This study investigates the strong dependency of reactive transport on pore structures and surface area and helps improve the understanding of ion transport and adsorption in porous electrodes with multiscale features.

Acknowledgments

Research presented in this article was supported by the Laboratory Directed Research and

Development (LDRD-ER) program of Los Alamos National Laboratory (LANL).


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numerical modeling of ion transport and adsorption in

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