# Numerical Simulation of Particle Distribution in Capillary Membrane during Backwash

^{1}

^{2}

^{3}

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Methods

#### 2.1. Experimental Section

^{−1}) and soft (baker's yeast cells) particles (50 mg L

^{−1}) are chosen as additives to the feed in order to cause inorganic and organic particulate membrane fouling. Furthermore, spherical polystyrene particles (100 mg L

^{−1}) with well-defined size distribution are applied. The filtration device illustrated in Figure 1 (lower part) can generate simultaneously fouling layers in up to three single commercial UF capillary membranes under the same dead-end conditions. The device is operated for 1 liter of the particle suspension at constant flux of 180 L m

^{−2}h

^{−1}recording temperature (T) and pressure (P). The flow rate and additionally the density of the suspensions are recorded by high sensitive coriolis flowmeters (FL).

#### 2.2. Numerical Simulation

#### 2.2.1. Model Description

^{−3}and a kinematic viscosity of 1×10

^{−6}m

^{2}s

^{−1}. The particles are considered as non-deformable spheres with a constant density of 1050 kg m

^{−3}. Reynolds number Re of 200–700 related to the capillary diameter, maximum velocity and kinematic viscosity of the fluid indicates a laminar flow inside the capillary.

#### 2.2.2. Governing Equations

- Continuity equation$$\frac{\partial \left({\alpha}_{k}\right)}{\partial t}+\nabla \left({\alpha}_{k}{\overline{u}}_{k}\right)=0$$
- Momentum equation$$\frac{\partial \left({\alpha}_{k}{\overline{u}}_{k}\right)}{\partial t}+\nabla ({\alpha}_{k}{\overline{u}}_{k}{\overline{u}}_{k})+\nabla ({\alpha}_{k}{\overline{R}}_{k}^{\mathit{\text{eff}}})=-\frac{{\alpha}_{k}}{{\rho}_{k}}\nabla p+{\alpha}_{k}g+\frac{{\overline{M}}_{k}}{{\rho}_{k}}$$$${\alpha}_{k}=\frac{{V}_{k}}{V}$$$${\overline{R}}_{k}^{\mathit{\text{eff}}}=-{v}_{k}^{\mathit{\text{eff}}}(\nabla {\overline{u}}_{k}+\nabla {\overline{u}}_{k}^{T}-\frac{2}{3}I\nabla {\overline{u}}_{k})+\frac{2}{3}I{k}_{k}$$

_{k}[24].

_{r}with C

_{L,α}= 0.5. It is given by [26] as follows:

**Figure 2.**Positions of control lines (A, B, C) for the evaluation of velocity and particle distribution during backwash.

**Figure 3.**Positions of control lines (D = 500 μm, E = −500 μm) for the evaluation of velocity and particle distribution during backwash.

#### 2.2.3. Initial and Boundary Conditions

^{−2}h

^{−1}and applied at inlet. Dead-end is defined as closed end of the capillary with wall properties and no-slip condition. The ambient pressure is given at outlet. During numerical calculations a convergence criterion of 10

^{−6}is set to guarantee a converged solution. The time step control is realized with an adaptive time stepping procedure assumed a maximum Courant number of 0.8. The stable calculations run with the time step in the range of 10

^{−5}s. Gaussian linear scheme is implied for the approximation and interpolation. First order, bounded and implicit time discretisation scheme is specified for the solution.

## 3. Numerical Results

**Figure 5.**Particle distributions along control line A with backwashing flux of 300 L m

^{−2}h

^{−1}during 5 s (homogeneous initial distribution, particle diameter of 20 μm).

**Figure 6.**Particle distributions along control line B with backwashing flux of 300 L m

^{−2}h

^{−1}during 5 s (homogeneous initial distribution, particle diameter of 20 μm).

**Figure 7.**Particle distributions along control line C with backwashing flux of 300 L m

^{−2}h

^{−1}during 5 s (homogeneous initial distribution, particle diameter of 20 μm).

**Figure 8.**Particle distributions along control line C with not considered lift force and backwashing flux 300 L m

^{−2}h

^{−1}during 5 s (cf. Figure 7).

**Figure 9.**Particle distribution at control line D with backwashing flux of 300 L m

^{−2}h

^{−1}, (deposited layer as initial distribution, particle diameter of 7 μm).

**Figure 10.**Particle distribution at control line E with backwashing flux of 300 L m

^{−2}h

^{−1}, (deposited layer as initial distribution, particle diameter of 7 μm).

#### 3.1. Effect of the Particle Size

^{−2}h

^{−1}was constant along the inlet. Three different particle sizes were simulated 5 μm, 10 μm and 20 μm which had the same material properties.

**Figure 11.**Particle distribution along control line A with different particle size after 5 s (homogeneous initial distribution, backwashing flux of 300 L m

^{−2}h

^{−1}.

**Figure 12.**Particle distribution along control line B with different particle size after 5 s (homogeneous initial distribution, backwashing flux of 300 L m

^{−2}h

^{−1}.

**Figure 13.**Particle distribution along control line C with different particle size after 5 s (homogeneous initial distribution, backwashing flux of 300 L m

^{−2}h

^{−1}.

#### 3.2. Effect of the Backwash Flux

^{−2}h

^{−1}the particles are driven to the outlet without considerable lateral migration. The motion across the streamlines and the corresponding behavior of the suspended particles are caused by the relative velocity which at low Reynolds number points parallel to capillary axis. Thus, the lift force tends to drift away accumulated particles to the capillary outlet.

**Figure 14.**Particle distribution along control line A with various backwashing fluxes after 8 s (homogeneous initial distribution, particle diameter of 10 μm).

**Figure 15.**Particle distribution along control line B with various backwashing fluxes after 8 s (homogeneous initial distribution, particle diameter of 10 μm).

**Figure 16.**Particle distribution along control line C with various backwashing fluxes after 8 s (homogeneous initial distribution, particle diameter of 10 μm).

^{−2}h

^{−1}and 400 L m

^{−2}h

^{−1}.

**Figure 17.**Particle distribution at control line D with backwashing flux of 100 L m

^{−2}h

^{−1}, (deposited layer as initial distribution).

**Figure 18.**Particle distribution at control line E with backwashing flux of 100 L m

^{−2}h

^{−1}, (deposited layer as initial distribution).

## 4. Discussion and Conclusion

## Acknowledgments

## Conflicts of Interest

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## Nomenclature

$\frac{D}{Dt}$ | substantive derivative |

${\overline{R}}_{k}^{\mathit{\text{eff}}}$ | Reynolds stress tensor (m ^{2} s^{−2}) |

C_{D} | coefficient of interphase drag force |

C_{υ} | coefficient of interphase virtual mass force |

C_{L,α} | coefficient of interphase lift force |

d | particle diameter (m) |

g | gravitational acceleration (m s ^{−2}) |

I | identity tensor |

k | turbulent kinetic energy (J kg ^{−1}) |

M^{D} | drag momentum rate (kg m ^{−2} s^{−2}) |

M^{L} | lift momentum rate (kg m ^{−2} s^{−2}) |

M^{V} | virtual mass momentum rate (kg m ^{−2} s^{−2}) |

M_{k} | averaged interphase momentum rate acting on phase k (kg m ^{−2} s^{−2}) |

Re | Reynolds number |

u | velocity (m s ^{−1}) |

u_{r} | relative velocity (m s ^{−1}) |

V | volume (m ^{3}) |

α | volume fraction |

ν_{eff} | effective kinematic viscosity (m ^{2} s^{−1}) |

ρ | density (kg m ^{−3}) |

c | subscript refers to continuous phase |

k | subscript refers to the phase |

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**MDPI and ACS Style**

Mansour, H.; Keller, A.; Gimbel, R.; Kowalczyk, W. Numerical Simulation of Particle Distribution in Capillary Membrane during Backwash. *Membranes* **2013**, *3*, 249-265.
https://doi.org/10.3390/membranes3040249

**AMA Style**

Mansour H, Keller A, Gimbel R, Kowalczyk W. Numerical Simulation of Particle Distribution in Capillary Membrane during Backwash. *Membranes*. 2013; 3(4):249-265.
https://doi.org/10.3390/membranes3040249

**Chicago/Turabian Style**

Mansour, Hussam, Anik Keller, Rolf Gimbel, and Wojciech Kowalczyk. 2013. "Numerical Simulation of Particle Distribution in Capillary Membrane during Backwash" *Membranes* 3, no. 4: 249-265.
https://doi.org/10.3390/membranes3040249