# A Water/Ion Separation Device: Theoretical and Numerical Investigation

## Abstract

**:**

## Featured Application

**The proposed application could be exploited for the design of a desalination device at various scales, depending on the desired flow rate of clean water.**

## Abstract

## 1. Introduction

## 2. Materials and Methods

#### 2.1. System Model

^{+}and Cl

^{−}ions flow between two planar, periodic, carbon surfaces, theoretically analogous to Poiseuille flow (see the inset of Figure 1). In a previous work, we have investigated the effect of the electric field strength, E, and the distance between the two wall surfaces, h, on desalination performance in a stand-alone nanochannel [30]. In this work, we consider a parallel network of such nanochannels and extend the investigation over the whole network.

^{+}near the upper wall, Cl

^{−}ions near the lower wall, and clean water in the region around the cell centerline. During the simulations, the solution density is constant, ρ = 1078 kg/m

^{3}, and temperature remains constant at T = 300 K with the application of Nose–Hoover thermostats.

**r**

_{c}= 9 Å is the cut-off radius. Moreover, coulombic interaction between hydrogen, oxygen, and ions is given by ${V}_{c}=\frac{C{q}_{i}{q}_{j}}{{\epsilon}_{0}{\mathit{r}}_{ij}}$, where C the energy conversion constant, q

_{i}and q

_{j}the charges of interacting atoms, and ε

_{0}is the dielectric constant. For two different species i and j we consider ${\sigma}_{ij}=\left({\sigma}_{i}+{\sigma}_{j}\right)/2$, ${\epsilon}_{ij}=\sqrt{\left({\epsilon}_{i}{\epsilon}_{j}\right)}$. When i and j correspond to wall (w) and fluid (f) atoms, the ratio ${\epsilon}_{wf}/{\epsilon}_{ff}$ represents whether the wall is treated as hydrophobic (${\epsilon}_{wf}/{\epsilon}_{ff}$ = 0.1 and 0.2) or hydrophilic (${\epsilon}_{wf}/{\epsilon}_{ff}$ = 0.5 and 1.0). Water molecules comply to the SPC/E (extended simple point charge) pair potential, which has been found to reproduce adequately the structural and dynamic properties of water [32], especially under confinement [33].

#### 2.2. Mathematical Formulation

_{ch}, is obtained from the time-averaging of the mean square displacement of N fluid particles [39] as

_{P}, and a perpendicular to the wall direction, D

_{T}, component, as

_{P}, or perpendicular, μ

_{T}, we, respectively, substitute in Equation (2) the diffusion coefficients D

_{P}and D

_{T}.

_{ch}is directly influenced by the size effect of the shear viscosity μ. As D

_{ch}is decreasing with the channel width, μ is increasing. Shear viscosity is connected to friction, i.e., a macroscopic property, and it becomes clear that it also affects the ion and clean water flow rate for the proposed device.

_{y}is the channel y-dimension. To disjoin calculations from L

_{y}, since the simulation system is periodic in the y-direction, we consider the more general volumetric flow rate per unit length to be

## 3. Results and Discussion

#### 3.1. Shear Viscosity Calculations

_{P}and μ

_{T}, respectively, we obtain a close-up view of the internal mechanism that affects ion movement inside each cell. For the narrowest cell under investigation (h = 3 nm), in Figure 2a, we observe that there exists increased shear viscosity perpendicular to the wall, μ

_{T}, significantly higher that the parallel component, μ

_{P}. As the strength of E increases from E = 0.0 to 1.0 V/Å, μ

_{T}decreases monotonically. In contrast, μ

_{P}remains unaffected by the electric field value. The channel shear viscosity is, though, unaffected, remaining around the water bulk value for the range of E investigated here. In a weaker E range, the viscosity component parallel to the electric field has been found to increase monotonically with the electric field strength, while the perpendicular component first decreases and then increases with E [41]. We also note that experimental results do not always adhere to simulation, and this is attributed to the lower E strength applied, compared to simulated nanoscale systems [42].

#### 3.2. Volumetric Flow Rate

#### 3.3. Configuration Issues

_{x}for a cell of height h that could yield satisfying ion separation results has been estimated in a previous work and is given by ${L}_{x}=320h-27.46$ [30]. For the device dimension on z-direction equal to ${L}_{z}=1\mathrm{cm}$, the number of cells, N, in the array can be calculated, as well as the resulting volumetric flow rate q (per unit length). Depending on the configuration capability, one may choose to construct an array of 395,257 cells in a (${L}_{x}$, ${L}_{z}$) = (1 cm, 7.33$\times $10

^{−4}cm) where $q$ = 1.54$\times $10

^{−7}lt/day or an array of only 80 cells with macroscale dimensions (${L}_{x}$, ${L}_{z}$) = (1 cm, 3.64 cm) where $q=3.76\text{}\mathrm{lt}/\mathrm{day}$.

## 4. Conclusions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

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**Figure 1.**Τhe proposed desalination device, consisted by an array of N cells. The inset figure presents a detailed view of each cell. Green and red circles are Na

^{+}and Cl

^{−}ions, respectively. The external electric field E

_{z}is applied perpendicular to the piston-driven flow. Clean water is gathered from each cell to the common outlet, while the remaining water/ion solution (brine) is recirculated to further separate water from ions.

**Figure 2.**Shear viscosity, μ, in parallel and perpendicular to the wall direction, and as total channel value vs. the strength of the electric field, E, for (

**a**) h = 3 nm and (

**b**) h = 6 nm.

**Figure 3.**Shear viscosity vs. the wall wettability strength, ${\epsilon}_{wf}/{\epsilon}_{ff}$, for (a) h = 3 nm, (b) h = 6 nm, (c) h = 9 nm, and (d) h = 15 nm. E = 0.1 V/Å.

**Figure 4.**Volumetric flow rate per unit length calculations for various cell heights vs. the wall wettability strength. E = 0.1 V/Å.

**Table 1.**Calculated volumetric flow rate per unit length, for various cases of cell heights and number of cells incorporated in a device.

$\mathit{N}$ | $\mathit{h}$ | ${\mathit{L}}_{\mathit{z}}\left(\mathbf{cm}\right)$ | ${\mathit{L}}_{\mathit{x}}\left(\mathbf{cm}\right)$ | $\mathit{q}$ (lt/day) |
---|---|---|---|---|

395,257 | 2.53 × 10^{−6} | 1 | 7.33 × 10^{−4} | 1.54 × 10^{−7} |

293,255 | 3.41 × 10^{−6} | 1 | 9.89 × 10^{−4} | 2.79 × 10^{−7} |

227,273 | 4.40 × 10^{−6} | 1 | 1.28 × 10^{−3} | 4.66 × 10^{−7} |

180,733 | 5.53 × 10^{−6} | 1 | 1.61 × 10^{−3} | 7.37 × 10^{−7} |

80,000 | 1.25 × 10^{−5} | 1 | 3.63 × 10^{−3} | 3.76 × 10^{−6} |

8000 | 1.25 × 10^{−4} | 1 | 3.64 × 10^{−2} | 3.76 × 10^{−4} |

800 | 1.25 × 10^{−3} | 1 | 3.64 × 10^{−1} | 3.76 × 10^{−2} |

80 | 1.25 × 10^{−2} | 1 | 3.64 | 3.76 |

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**MDPI and ACS Style**

Sofos, F.
A Water/Ion Separation Device: Theoretical and Numerical Investigation. *Appl. Sci.* **2021**, *11*, 8548.
https://doi.org/10.3390/app11188548

**AMA Style**

Sofos F.
A Water/Ion Separation Device: Theoretical and Numerical Investigation. *Applied Sciences*. 2021; 11(18):8548.
https://doi.org/10.3390/app11188548

**Chicago/Turabian Style**

Sofos, Filippos.
2021. "A Water/Ion Separation Device: Theoretical and Numerical Investigation" *Applied Sciences* 11, no. 18: 8548.
https://doi.org/10.3390/app11188548