# Fouling Mitigation via Chaotic Advection in a Flat Membrane Module with a Patterned Surface

^{1}

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## Abstract

**:**

## 1. Introduction

_{2}) and reducing the emission of CO

_{2}[3]. Depending on the relative direction between the feed flow and the permeate flow in a membrane filtration process, filtration is classified into dead-end filtration and crossflow filtration. The feed flow is perpendicular to the membrane surface in a dead-end filtration system, while the feed flow is tangential to the flat or tubular membrane in a crossflow filtration (CFF) system. The permeate flux is produced by the transmembrane pressure difference across the membrane. The accumulation of foulants on a membrane surface (membrane fouling) increases the resistance to filtration, leading to a decline in the permeate flux and lifespan of the membrane. Membrane fouling is caused by particle intrusion into membrane pores, narrowing or blocking flow passages through the pores, or by an adsorption of molecules on the membrane surface, forming a boundary layer with higher concentrations than those in the bulk flow [4,5,6,7,8].

## 2. Problem Definition

#### 2.1. Channel Geometry

#### 2.2. Governing Equations and Boundary Conditions

**.**At the inlet (${\Gamma}_{i}$), a uniform normal velocity ($\overline{u})$ is imposed, while at the outlet (${\Gamma}_{o}$), a constant static pressure (in this study, $p=0$) is specified. At the non-permeable patterned surface (${\Gamma}_{s}$), the velocity is zero, i.e., $\mathit{u}=\mathbf{0}$ on ${\Gamma}_{s}$. On the membrane surface (${\Gamma}_{w}$), a constant permeate flux is imposed, i.e., $\mathit{u}\xb7\mathit{n}={u}_{p}$, where $\mathit{n}$ is the unit outward normal vector at ${\Gamma}_{w}$ and ${u}_{p}$ the constant permeate velocity on the membrane surface. The Reynolds number (Re) of the bulk flow is defined as $\mathrm{Re}=\rho \overline{u}{h}_{c}/\mu $, where the uniform inlet velocity $\overline{u}$ is used as the characteristic velocity, and $\mathrm{Re}$ ranges from 50 to 500. In addition, we define another Reynolds number, called the wall Reynolds number (${\mathrm{Re}}_{w}$), defined by ${\mathrm{Re}}_{w}=\rho {u}_{p}{h}_{c}/\mu $, employing ${u}_{p}$ as the characteristic velocity. As for the permeate velocity, a constant ${u}_{p}$ is used, regardless of the inlet velocity, such that ${\mathrm{Re}}_{w}=0.01$, corresponding to a constant flux mode of operation.

#### 2.3. Simulation Details

^{®}Xeon

^{®}Silver 4210R 2.4 GHz) and 98 GB memory is used, and all computations are carried out using parallel computing with 10 processors.

## 3. Results and Discussion

#### 3.1. Convergence with Mesh Refinement

#### 3.2. Flow Characteristics

#### 3.3. Evolution of the Foulant Concentration

#### 3.4. Growth Rate of the Wall Concentration

#### 3.5. Pressure Loss in the Constant Permeate Flux Mode

## 4. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

## List of Symbols

$c$ | Concentration of foulant, mol/m^{3} |

${c}_{b}$ | Concentration in the bulk, mol/m^{3} |

${c}_{0}$ | Concentration at the inlet, mol/m^{3} |

${c}_{p}$ | Concentration in the permeate, mol/m^{3} |

${c}_{w}$ | Concentration on the wall (membrane surface), mol/m^{3} |

${c}_{w}^{*}$ | Dimensionless wall concentration |

${\overline{c}}_{w}^{*}$ | Dimensionless line-averaged wall concentration |

$\mathcal{D}$ | Diffusivity of foulants, m^{2}/s |

δ | Thickness of the concentration boundary layer (film layer), m |

$h$ | Half-height of a thin slit channel, m |

${h}_{c}$ | Channel height, m |

${h}_{g}$ | Groove depth, m |

${h}_{g}^{*}$ | Dimensionless groove depth |

${l}_{p}$ | Length of a periodic unit of the channel, m |

$\mathit{n}$ | Unit outward normal vector |

n | Exponent of the growth rate of the wall concentration |

$p$ | Pressure, Pa |

$\mathrm{Pe}$ | Péclet number |

$\mathrm{Re}$ | Reynolds number |

${\mathrm{Re}}_{c}$ | Critical Reynolds number |

${\mathrm{Re}}_{w}$ | Wall Reynolds number |

$\mathit{u}$ | Velocity vector, m/s |

$\overline{u}$ | Uniform normal velocity at the inlet, m/s |

${u}_{max}$ | Maximum velocity in the fully developed laminar flow through a thin slit, m/s |

${u}_{p}$ | Permeate velocity, m/s |

${v}_{dw}^{*}$ | Dimensionless downwelling velocity magnitude |

${w}_{c}$ | Channel width, m |

${w}_{g}$ | Groove width, m |

${z}^{*}$ | Dimensionless $z$–coordinate |

Greek Letters | |

${\Gamma}_{i}$ | Inlet boundary |

${\Gamma}_{o}$ | Outlet boundary |

${\Gamma}_{s}$ | Non-permeable solid boundary |

${\Gamma}_{w}$ | Membrane surface |

$\delta $ | Concentration boundary layer thickness, m |

$\mu $ | Viscosity, Pa∙s |

$\theta $ | Groove angle, ° |

## Appendix A. Back Transport of Tracer Particles

**Figure A1.**Back transport of 100 tracer particles for values of ${h}_{g}^{*}$ ($0$, $0.05$, $0.15$, and $0.3$) when $\mathrm{Re}=100$ and ${\mathrm{Re}}_{w}=0.01$. At the inlet (${z}^{*}=0$), the particles are initially located $0.05{h}_{c}$ away from the membrane surface (see the first row). Here ${N}_{p}$ is the number of tracer particles found at each cross section.

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**Figure 1.**The schematic representation of the flat membrane module consisting of a patterned solid surface (bottom surface) and a flat membrane (top surface). (

**a**) The non-permeable patterned surface with staggered herringbone-shaped grooves; (

**b**) The channel geometry with symmetry in the lateral direction ($x$‒direction) used in numerical simulations and an enlarged image around the inlet. In this figure, the top surface is the membrane (the area shaded in red); (

**c**) Top view of the periodic unit of the channel, where groove regions are shaded. The channel is geometrically periodic in the $z$‒direction with the period ${l}_{p}$. In each half cycle, there are six grooves. Here, ${h}_{c}$, ${h}_{g}$, ${w}_{c}$, ${w}_{g}$, and $\theta $ are the channel height, the groove depth, the channel width, the groove width, and the groove angle, respectively. The groove angle is fixed to $\theta =45\xb0$.

**Figure 2.**The change in the dimensionless average wall concentration (${\overline{c}}_{w}^{*}$) in the dimensionless $z$‒direction (${z}^{*}$) showing convergence with mesh refinement. Basic information on the four meshes used to check mesh convergence can be found in Table 1.

**Figure 3.**The cross-sectional velocity vectors and the magnitude of the cross-sectional velocity at the cross section located at ${z}^{*}=0.184$(shown in (

**a**)), scaled by the average inlet velocity, $\sqrt{\left({u}^{2}+{v}^{2}\right)}/\overline{u}$, representing the strength of the cross-sectional flow when $\mathrm{Re}=100$: (

**a**) the location of the cross section where the velocity vectors and velocity contours are plotted for the three groove depths; (

**b**) ${h}_{g}^{*}=0.05$; (

**c**) ${h}_{g}^{*}=0.15$; (

**d**) ${h}_{g}^{*}=0.3$. The maximum values at the three values of ${h}_{g}^{*}$, 0.05, 0.15, and 0.3, are 0.059, 0.115, and 0.152, respectively. The arrows in each contour plot represent the velocity vector projected onto the cross section (viewed from the outlet).

**Figure 4.**Cross-sectional flows at the cross section located at ${z}^{*}=0.184$ visualized by streaklines and contours of the downwelling velocity magnitude affected by the dimensionless groove depth (${h}_{g}^{*}$) and the Reynolds number ($\mathrm{Re}$). In contour plots, the dimensionless downwelling velocity magnitude, defined by ${v}_{dw}^{*}=-v/\overline{u}$, is plotted. While positive contours indicate downwelling flows, negative contours indicate upwelling flows. In each cross-sectional plot, the upper edge corresponds to the membrane.

**Figure 5.**Poincaré sections affected by the dimensionless groove depth (${h}_{g}^{*}$) at the three Reynolds numbers, $\mathrm{Re}=100$, $200$, and $500$.

**Figure 6.**The dimensionless concentration ${c}^{*}\left(=c/{c}_{0}\right)$ near the membrane surface at a cross section located at ${z}^{*}=8$ for the four values of ${h}_{g}^{*}$, $0$, $0.05$, $0.15$, and $0.3$, when $\mathrm{Re}=100$.

**Figure 7.**The change in the dimensionless average wall concentration (${\overline{c}}_{w}^{*}$) in the dimensionless $z$‒direction (${z}^{*}$), affected by the dimensionless groove depth, ${h}_{g}^{*}$: (

**a**) $\mathrm{Re}=50$; (

**b**) $\mathrm{Re}=100$; (

**c**) $\mathrm{Re}=200$; (

**d**) $\mathrm{Re}=500$.

**Figure 8.**The change in the dimensionless average concentration (${\overline{c}}_{w}^{*}$) at the end of the filtration channel ($\mathrm{at}{z}^{*}=10$) as a function of the dimensionless groove depth (${h}_{g}^{*}$) when $\mathrm{Re}=50$, $100$, $200$, and $500$.

**Figure 9.**The change in the dimensionless pressure loss $\Delta {p}^{*}$ in one periodic unit as a function of the dimensionless groove depth (${h}_{g}^{*}$) when $\mathrm{Re}=50$, $100$, $200$, and $500$.

**Table 1.**The number of elements and the minimum element size of the meshes used to check convergence with mesh refinement. Here, ${h}_{c}$ is the channel height.

Mesh | Number of Elements | Minimum Element Size |
---|---|---|

M1 | 8,608,896 | 0.01${h}_{c}$ |

M2 | 16,685,136 | 0.005${h}_{c}$ |

M3 | 30,380,832 | 0.0025${h}_{c}$ |

M4 | 41,685,120 | 0.00125${h}_{c}$ |

**Table 2.**The fitted values of $n$ at the three dimensionless groove depths when $\mathrm{Re}=50$, $100$, $200$, and $500$. In each row, numbers in bold indicate the values of ${h}_{g}^{*}$ and $n$ when the growth rate is minimum at a specific Reynolds number.

$Re$ | ${\mathit{h}}_{\mathit{g}}^{*}$ | $\mathit{n}$ |
---|---|---|

50 | 0.05 | 0.2347 |

0.10 | 0.2163 | |

0.15 | 0.2005 | |

0.20 | 0.1866 | |

0.25 | 0.1773 | |

0.30 | 0.1712 | |

100 | 0.05 | 0.2419 |

0.10 | 0.2191 | |

0.15 | 0.1916 | |

0.20 | 0.1749 | |

0.25 | 0.1668 | |

0.30 | 0.1633 | |

200 | 0.05 | 0.2524 |

0.10 | 0.2084 | |

0.15 | 0.1745 | |

0.20 | 0.1627 | |

0.25 | 0.1625 | |

0.30 | 0.1666 | |

500 | 0.05 | 0.2580 |

0.10 | 0.1577 | |

0.15 | 0.1342 | |

0.20 | 0.1411 | |

0.25 | 0.1559 | |

0.30 | 0.1771 |

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**MDPI and ACS Style**

Kim, K.T.; Park, J.E.; Jung, S.Y.; Kang, T.G.
Fouling Mitigation via Chaotic Advection in a Flat Membrane Module with a Patterned Surface. *Membranes* **2021**, *11*, 724.
https://doi.org/10.3390/membranes11100724

**AMA Style**

Kim KT, Park JE, Jung SY, Kang TG.
Fouling Mitigation via Chaotic Advection in a Flat Membrane Module with a Patterned Surface. *Membranes*. 2021; 11(10):724.
https://doi.org/10.3390/membranes11100724

**Chicago/Turabian Style**

Kim, Kyung Tae, Jo Eun Park, Seon Yeop Jung, and Tae Gon Kang.
2021. "Fouling Mitigation via Chaotic Advection in a Flat Membrane Module with a Patterned Surface" *Membranes* 11, no. 10: 724.
https://doi.org/10.3390/membranes11100724