# A Coupled CFD-DEM Model for Resolved Simulation of Filter Cake Formation during Solid-Liquid Separation

^{1}

^{2}

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Materials and Methods

#### 2.1. Filtration Equation

#### 2.2. Filtration Experiments

#### 2.3. Numerical Simulations

#### 2.3.1. Fluid Flow Calculation with CFD

#### 2.3.2. Particle Movement Calculation with DEM

#### 2.3.3. CFD-DEM Coupling

#### 2.4. Particle Sedimentation

#### 2.4.1. Theory

^{3}and the turbulent regime has $R{e}_{P}$ > 10

^{3}[50].

#### 2.4.2. 3D Models

#### 2.5. Separation Process

^{11}m

^{−3}. Table 3 summarizes the relevant parameters of the filtration simulation.

_{3}) of the glass beads was measured in water with the Horiba Laser Diffraction Particle Size Analyzer (Retsch GmbH, Haan, Germany) in the experiments and then transformed into the number-based distribution (Q

_{0}) which can be used as input parameter in the simulations.

## 3. Results

#### 3.1. Filtration Experiments

^{10}m

^{−2}, and the y-intercept of the graph yields the filter medium resistance with Equation (2) to ${R}_{M,Exp}$ = 6.3·10

^{7}m

^{−1}. Because of the slightly varying pressure difference in the experiments, the pure filter medium resistance ${R}_{M0,Exp}$ is obtained in the range between 2.1·10

^{6}m

^{−1}and 3.9·10

^{6}m

^{−1}.

#### 3.2. Simulation of the Particle Sedimentation

^{−1}and the Reynolds number $R{e}_{P,t,theor}$ = 0.3324 are calculated. This indicates that the particle settles at the beginning of the transition regime between the laminar and turbulent flow. This analytical solution is compared to the results of the simulations concerning the single particle settling in models A1, B1, C1 and D1 in the next section.

#### 3.3. Simulation of the Filtration Process

- In (a) the filter medium model is shown before the particle generation.
- In (b) the calculated flow velocities and the initial particle positions are shown at the beginning of the filtration when the first particles are reaching the filter medium and the fluid passes through the filter medium. The local flow velocity maxima in the area of the pores can be seen. Accordingly, the time-dependent filtrate velocity (Figure 19a) shows a maximum before this point of time because of the minimum pressure loss that is determined only by the filter medium.
- In (c) it can be seen that some particles are already clogging the pores of the filter medium, which leads to the increase of the pressure drop and the resulting decrease of the filtrate volume flow.
- In (d) the further filter cake formation process at the real time of 98 ms is shown. The filter cake height reaches 0.5 mm. It can be seen in Figure 20 that at this time point 145 particles have already been deposited in the filter cake and the other 16 generated particles are still settling. The particles in the filter cake do not remain fixed, some of them move slowly resulting in a reorganization and packing of the filter cake. These cake formation micro processes take place due to fluid flow, gravitation and contact forces between the particles.

**Figure 18.**CFD-DEM simulation of the filtration process with the woven wire cloth filter medium (

**a**) filter medium model, before the process starts at $t$ = 0 ms, (

**b**) inflowing suspension at $t$ = 3.9 ms, (

**c**) particles are retained by the filter medium $t$ = 40 ms, (

**d**) filter cake formation at $t$ = 98 ms.

^{6}m

^{−1}. After the particle-free fluid flow through the filter, the slope of the curve increases and after 6 mL of filtrate volume becomes linear. When the linear section of the graph begins, the linear fit represents the specific filter cake resistance. The slope of it according to Equation (3) leads to the specific filter cake resistance of ${r}_{K,Sim}$ = 7.8·10

^{10}m

^{−2}(Figure 19b).

#### 3.4. Comparison of Simulation and Experiment

^{10}m

^{−2}matches the experimental value of ${r}_{K,Exp}$ = 7.7·10

^{10}m

^{−2}. The experimentally obtained values of ${R}_{M0,Exp}$ were in the range between 2.1·10

^{6}m

^{−1}and 3.9·10

^{6}m

^{−1}, which is in good agreement with the simulation results of ${R}_{M0,Sim}$ = 3.4·10

^{6}m

^{−1}.

## 4. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

## References

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**Figure 3.**LBM coupling scheme with particle (orange), particle boundary (red) and fluid (white) nodes. (

**a**) New fluid nodes (blue) after particle movement, (

**b**) surrounding layer of fluid nodes (brown) used to compute viscous and pressure forces which act on the particle.

**Figure 5.**Particle impact on a solid surface and rebound in a closed container with liquid: (

**a**) particle sedimentation down to the bottom wall and counter flow of liquid, (

**b**) particle approaches the wall and squeezes liquid from the gap, (

**c**) first contact with wall, (

**d**) particle-wall contact deformation δ, (

**e**) rebound of particle, fluid streams back into gap, (

**f**) rebound of particle, counter flow of fluid.

**Figure 6.**Different model arrangements to investigate particle settling behavior. The particle is 80 µm in diameter for each case. (

**A1**) Basic model with periodic boundaries, (

**B1**) basic model with coarser grid, (

**C1**) basic model with solid wall boundaries, (

**D1**) basic model with filter medium.

**Figure 7.**(

**A2**) from model A1 which is reduced in width, with periodic boundaries, (

**C2**) from model C1 which is reduced in width, with solid wall boundaries, (

**A3**) from model A1 which is enlarged in width, with periodic boundaries, (

**C3**) from model C1 which enlarged in width, with solid wall boundaries, (

**A4**) from model A1 which is enlarged in height and one added particle, with periodic boundaries, (

**A5**) from model A1 which is enlarged in height and two added particles, with periodic boundaries.

**Figure 8.**Computed wire cloth, (

**a**) side view in a sectional plane through a wire, (

**b**) top view of the complete model wire cloth structure.

**Figure 10.**Typical filtration diagram obtained in the pressurized housing experiments with the filter medium (no spacer) and the filter cake formed from glass particles.

**Figure 11.**Particle settling velocity with the corresponding images at characteristic time points of sedimentation obtained in the simulation case A1.

**Figure 12.**Particle settling behavior at time $t=$ 2 ms after it was dropped in different model arrangements, particle diameter ${d}_{P}$ = 80 µm, model height $H$ = 800 µm, voxel length: (

**A1**) ${\lambda}_{A}=$ 2 µm, (

**B1**) ${\lambda}_{B}=$ 4 µm, (

**C1**) ${\lambda}_{C}=$ 2 µm, (

**D1**) ${\lambda}_{D}=$ 2 µm.

**Figure 13.**Particle Reynolds number $R{e}_{P}$ from simulations A1, B1, C1 and D1 compared to the theoretical sedimentation curve according to Ladenburg.

**Figure 14.**Variation of the model width and influence of the periodic boundary conditions illustrated as neighboring particles in the grayscale images in A2: (

**A1**) basic model, (

**A2**) narrow channel (half the width of A1), (

**A3**) wider channel (double width of A1), (t = 2 ms).

**Figure 15.**(

**a**) Influence of model width on the settling behavior for different models: with periodic boundaries (A1, A2, A3) and solid wall boundaries (C1, C2, C3), as well as theoretical approach according to Ladenburg. (

**b**) Ratio of the particle velocity (${\mathrm{v}}_{\mathrm{p}}$) to the maximum velocity (terminal velocity ${\mathrm{v}}_{\mathrm{p},\mathrm{t}}$) depending on the ratio of the distance to wall $\mathrm{x}$ and particle diameter d

_{p}.

**Figure 16.**Variation of number and distance between particles settling in a row. Particle 1 has the same initial position in all simulations. Particle 2 has the same position in A4 and A5. (

**A1**) basic model, (

**A4**) basic model enlarged in height with one additional particle, (

**A5**) basic model enlarged in height with two additional particles, (t = 2 ms).

**Figure 17.**Influence of the neighboring particles following one another on the settling behavior: (

**a**) vertical distance of the particle to the bottom wall over settling time, (

**b**) particle vertical velocity component (in z-direction) over time.

**Figure 19.**(

**a**) Time-dependent filtrate velocity and (

**b**) filter curve obtained with CFD-DEM simulation and described by filter equation in Equation (3).

Property | Unit | Value |
---|---|---|

mesh size ${d}_{Pore}$ | µm | 32 |

wire diameter ${d}_{F}$ | µm | 28 |

filter area $A$ | cm^{2} | 20 |

pressure difference $\mathsf{\Delta}p$ | Pa | 1000 |

particle density ${\rho}_{P}$ | kg∙m^{−3} | 2200 |

particle size median value ${\mathrm{x}}_{50,3}$ | µm | 82.2 |

particle size modal value ${\mathrm{x}}_{\mathrm{mod},3}$ | µm | 82.7 |

x_{10,3} ^{1} | µm | 69.6 |

x_{90,3} ^{2} | µm | 95.3 |

^{1}10% of particle volume has a diameter smaller than the value of x

_{10,3},

^{2}90% of particle volume has a diameter smaller than the value of x

_{90,3.}

Property | Unit | Model | Value |
---|---|---|---|

Voxel length $\lambda $ | µm | A1, A2, A3, A4, A5, C1, C2, C3, D1 | 2 |

B1 | 4 | ||

Model size (L × W × H) | µm × µm × µm | A1, C1, D1 | 320 × 320 × 802 |

A2, C2 | 160 × 160 × 802 | ||

A3, C3 | 640 × 640 × 802 | ||

A4, A5 | 320 × 320 × 1128 | ||

B1 | 320 × 320 × 804 | ||

Wire diameter ${d}_{W}$ | µm | D1 | 28 |

Pore size ${d}_{Pore}$ | µm | D1 | 32 |

Particle density ${\rho}_{P}$ | kg∙m^{−3} | every model | 2200 |

Particle diameter ${d}_{P}$ | µm | every model | 80 |

Fluid density $\rho $ | kg∙m^{−3} | every model | 1000 |

Fluid viscosity $\mu $ | Pa∙s | every model | 0.001 |

Number of particles | - | A1, A2, A3, B1, C1, C2, C3, D1 | 1 |

A4 | 2 | ||

A5 | 3 | ||

Ratio ${d}_{P}/L$ | - | A1, B1, C1, D1, A4, A5 | 0.25 |

A2, C2 | 0.5 | ||

A3, C3 | 0.125 |

Property | Unit | Value |
---|---|---|

Voxel length $\lambda $ | µm | 2.5 |

Wire diameter ${d}_{W}$ | µm | 28 |

Pore size ${d}_{Pore}$ | µm | 32 |

Particle density ${\rho}_{P}$ | kg∙m^{−3} | 2200 |

Particle diameter ${d}_{P}$ of the generated fractions | µm | 55 |

63 | ||

72 | ||

83 | ||

95 | ||

109 | ||

Particle concentration ${c}_{n}$ | m^{−3} | 10^{11} |

Pressure difference $\mathsf{\Delta}p$ | Pa | 1000 |

Fluid density $\rho $ | kg∙m^{−3} | 1000 |

Fluid viscosity $\mu $ | Pa∙s | 0.001 |

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**MDPI and ACS Style**

Puderbach, V.; Schmidt, K.; Antonyuk, S.
A Coupled CFD-DEM Model for Resolved Simulation of Filter Cake Formation during Solid-Liquid Separation. *Processes* **2021**, *9*, 826.
https://doi.org/10.3390/pr9050826

**AMA Style**

Puderbach V, Schmidt K, Antonyuk S.
A Coupled CFD-DEM Model for Resolved Simulation of Filter Cake Formation during Solid-Liquid Separation. *Processes*. 2021; 9(5):826.
https://doi.org/10.3390/pr9050826

**Chicago/Turabian Style**

Puderbach, Vanessa, Kilian Schmidt, and Sergiy Antonyuk.
2021. "A Coupled CFD-DEM Model for Resolved Simulation of Filter Cake Formation during Solid-Liquid Separation" *Processes* 9, no. 5: 826.
https://doi.org/10.3390/pr9050826