# Study of CFD-DEM on the Impact of the Rolling Friction Coefficient on Deposition of Lignin Particles in a Single Ceramic Membrane Pore

^{1}

^{2}

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Governing Equation of Fluid–Solid Two-Phase Flow

#### 2.1. Fluid Phase Control Equation

^{3}, respectively; S is the sum of the fluid resistance ${F}_{\mathrm{D}}$ acting on the volume of a grid cell, N; and $\u2206V$ is the volume of the mesh unit, m

^{3}.

#### 2.2. Discrete Model

^{2}, rad/s, and N·m, respectively. ${\mathit{M}}_{\mathbf{J}\mathbf{K}\mathbf{R},\mathit{i}}$ is the adhesive torque in collisions with other particles, N·m; ${\mathit{F}}_{\mathbf{J}\mathbf{K}\mathbf{R}}$ refers to the Van der Waals adhesion force, N; ${\mathit{F}}_{\mathbf{f}\mathbf{p}}$ is the particle-fluid force, N; and ${\mathit{F}}_{\mathbf{c}}$ denotes the particle–particle contact force, N.

^{2}; ${E}^{*}$ refers to the relative Young’s Modulus, Pa; $\alpha $ is the normal overlap, m; and ${R}^{\mathrm{*}}$ represents the relative radius, m.

^{2}; ${C}_{\mathrm{D}}$ refers to the drag coefficient, which depends on the Reynolds number $Re$, and can be calculated by Equation (11) [37].

#### 2.3. The Directional Constant Torque Model

## 3. Computational Set-Up

#### 3.1. Geometry and Computational Domain

#### 3.2. Boundary Conditions and Parameter Settings

## 4. Results and Discussion

#### 4.1. Particle Deposition Process

#### 4.2. Impacts of the Rolling Friction Coefficient on Deposition Morphology of Lignin Particles

#### 4.3. Impacts of the Rolling Friction Coefficient on the Deposition Structure of Lignin Particles

## 5. Conclusions

- (1)
- The deposition of lignin particles on ceramic membranes was dynamic, which mainly included capturing ceramic membranes in the initial filtration and deposited lignin particles. Formation of a dendritic structure not only made the deposition morphology of lignin particles look like a “forest,” but also greatly improved the efficiency in capturing the lignin particles.
- (2)
- The rolling friction coefficient among the lignin particles crucially affected the deposition morphology, average coordination number, coordination number distribution, and porosity of the particles; the average coordination number decreased from 3.96 to 2.73, and the porosity increased from 0.65 to 0.73, when it increased from 0.1 to 3.0.
- (3)
- Reasonably providing a rolling friction coefficient among the lignin particles could replace spherical lignin particles with non-spherical particles. Impacts of the rolling friction coefficient on the deposition morphology, coordination number, and porosity of lignin particles enabled the simulation to be closer to the real lignin filtration by setting the rolling friction coefficient among the lignin particles as 0.6–2.4.

## Author Contributions

## Funding

## Institutional Review Board Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## References

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**Figure 2.**Pore structure of the ceramic membrane and the size of the computational domain. (

**a**) Isometric view; (

**b**) YZ plan view.

**Figure 4.**Change in the deposition morphology of the lignin particles with time. (

**a**) A (T = 0.1 ms); (

**b**) B (T = 0.5 ms); (

**c**) C (T = 1.0 ms); (

**d**) D (T = 1.5 ms); (

**e**) E (T = 2.0 ms); (

**f**) F (T = 2.2 ms).

**Figure 6.**Curves of the cumulative number of penetrated particles and efficiency in capturing lignin particles with time.

**Figure 7.**Impacts of the rolling friction coefficient on the deposition morphology of lignin particles. (

**a**) ${\mu}_{\mathrm{p}-\mathrm{m}}=0.1$; (

**b**) ${\mu}_{\mathrm{p}-\mathrm{m}}=1.0$; (

**c**) ${\mu}_{\mathrm{p}-\mathrm{m}}=2.0$; (

**d**) ${\mu}_{\mathrm{p}-\mathrm{m}}=3.0$.

**Figure 9.**Impacts of the rolling friction coefficient among the lignin particles on coordination number distribution. (

**a**) ${\mu}_{\mathrm{p}-\mathrm{m}}=0.1$; (

**b**) ${\mu}_{\mathrm{p}-\mathrm{m}}=1.0$; (

**c**) ${\mu}_{\mathrm{p}-\mathrm{m}}=2.0$; (

**d**) ${\mu}_{\mathrm{p}-\mathrm{m}}=3.0$.

**Figure 10.**Impacts of the rolling friction coefficient between the lignin particles and membranes on the coordination number distribution (

**a**) ${\mu}_{\mathrm{p}-\mathrm{p}}=0.6$; (

**b**) ${\mu}_{\mathrm{p}-\mathrm{p}}=1.8$; (

**c**) ${\mu}_{\mathrm{p}-\mathrm{p}}=3.0$.

Material | Particle | Membrane | Black Liquor |
---|---|---|---|

Diameter (μm) | 1 | 10 | - |

Density (kg/m^{3}) | 1451 | 3100 | 1004 |

Shear modulus (Pa) | 2 × 10^{7} | 7 × 10^{10} | - |

Poisson’ ratio | 0.25 | 0.2 | - |

Viscosity (Pa·s) | - | - | 1.467 |

Velocity (m/s) | 0.5 | - | 0.5 |

Collision Parameters | Coefficient of Restitution | Coefficient of Static Friction | Coefficient of Rolling Friction | Surface Energy (J/m^{2}) |

Particle–particle | 0.1 | 2.0 | 0.1–3 | 0.6 |

Particle–membrane | 0.1 | 2.0 | 0.1–3 | 1 |

Simulation Parameters | Particle Generation Rate/s | Total Number of Particles | Time Step of DEM/s | Time Step of CFD/s |

5 × 10^{5} | 1000 | 1 × 10^{−10} | 1 × 10^{−8} |

Group | Mesh Quantity | Pressure Drop (Pa) |
---|---|---|

1 | 28,900 | 527.34924 |

2 | 37,544 | 527.9762 |

3 | 46,400 | 528.85345 |

4 | 50,270 | 529.08069 |

5 | 59,048 | 529.62501 |

**Table 4.**Proportion of the number of contacts between the lignin particles and membranes to the total.

${\mathsf{\mu}}_{\mathbf{p}-\mathbf{m}}$ | 0.1 | 1.0 | 2.0 | 3.0 | |
---|---|---|---|---|---|

${\mathsf{\mu}}_{\mathbf{p}-\mathbf{p}}$ | |||||

0.1 | 13.8% | 17.3% | 13.7% | 17.3% | |

0.6 | 12.1% | 11.1% | 10.4% | 11.3% | |

1.2 | 16.7% | 11.2% | 14.6% | 15.5% | |

1.8 | 18.4% | 12.8% | 13.0% | 14.2% | |

2.4 | 17.5% | 15.1% | 14.4% | 16.3% | |

3.0 | 17.2% | 14.2% | 16.4% | 13.1% |

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**MDPI and ACS Style**

Wang, H.; Wang, X.; Wu, Y.; Wang, S.; Wu, J.; Fu, P.; Li, Y. Study of CFD-DEM on the Impact of the Rolling Friction Coefficient on Deposition of Lignin Particles in a Single Ceramic Membrane Pore. *Membranes* **2023**, *13*, 382.
https://doi.org/10.3390/membranes13040382

**AMA Style**

Wang H, Wang X, Wu Y, Wang S, Wu J, Fu P, Li Y. Study of CFD-DEM on the Impact of the Rolling Friction Coefficient on Deposition of Lignin Particles in a Single Ceramic Membrane Pore. *Membranes*. 2023; 13(4):382.
https://doi.org/10.3390/membranes13040382

**Chicago/Turabian Style**

Wang, Hao, Xinyuanrui Wang, Yongping Wu, Song Wang, Junfei Wu, Ping Fu, and Yang Li. 2023. "Study of CFD-DEM on the Impact of the Rolling Friction Coefficient on Deposition of Lignin Particles in a Single Ceramic Membrane Pore" *Membranes* 13, no. 4: 382.
https://doi.org/10.3390/membranes13040382