# Study on Critical Velocity of Sand Transport in V-Inclined Pipe Based on Numerical Simulation

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## Abstract

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## 1. Introduction

## 2. Numerical Method

#### 2.1. Governing Equations

**g**is the acceleration of gravity, ${\mu}_{q}$ and ${\lambda}_{q}$ are the shear and bulk viscosity of phase q, $\overline{\overline{I}}$ is the unit tensor, ${\mathit{F}}_{q}$ is an external body force, ${\mathit{F}}_{lift,q}$ is a lift force, ${\mathit{F}}_{wl,q}$ is a wall lubrication force, ${\mathit{F}}_{vm,q}$ is a virtual mass force, ${\mathit{F}}_{td,q}$ is a turbulent dispersion force, and p is the pressure shared by all phases. ${\mathit{R}}_{pq}$ is an interaction force between phases. ${\mathit{v}}_{pq}$ and ${\mathit{v}}_{qp}$ are the interphase velocities.

#### 2.2. Turbulence Model

#### 2.3. Computational Domain and Grid-Independent Analysis

#### 2.4. Solution Strategies and Boundary Conditions

## 3. Results and Discussion

#### 3.1. Comparison between Empirical Formula and Numerical Simulation

#### 3.1.1. Empirical Formula for Critical Velocity

#### 3.1.2. Numerical Simulation for Critical Velocity

#### 3.2. Effect of Simulated Pipeline Length

#### 3.3. Effect of Pipe Inclination

#### 3.4. Effect of Particle Size

## 4. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## Abbreviations

CFD | computational fluid dynamics |

ASM | algebraic slip mixture |

DEM | discrete element method |

## References

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**Figure 3.**Net mass flow rate changes with flow time (horizontal pipe $L\phantom{\rule{4.pt}{0ex}}=\phantom{\rule{4.pt}{0ex}}80\phantom{\rule{3.33333pt}{0ex}}\mathrm{m}\phantom{\rule{4.pt}{0ex}}d\phantom{\rule{4.pt}{0ex}}=\phantom{\rule{4.pt}{0ex}}0.02\phantom{\rule{4.pt}{0ex}}\mathrm{mm}$).

**Figure 4.**Sand volume concentration changes with flow time (horizontal pipe $L\phantom{\rule{4.pt}{0ex}}=\phantom{\rule{4.pt}{0ex}}80\phantom{\rule{3.33333pt}{0ex}}\mathrm{m}\phantom{\rule{4.pt}{0ex}}d\phantom{\rule{4.pt}{0ex}}=\phantom{\rule{4.pt}{0ex}}0.02\phantom{\rule{4.pt}{0ex}}\mathrm{mm}$).

**Figure 5.**Net mass flow rate changes with flow time (V-inclined pipe $L\phantom{\rule{4.pt}{0ex}}=\phantom{\rule{4.pt}{0ex}}80\phantom{\rule{3.33333pt}{0ex}}\mathrm{m}\phantom{\rule{4.pt}{0ex}}d\phantom{\rule{4.pt}{0ex}}=\phantom{\rule{4.pt}{0ex}}0.02\phantom{\rule{4.pt}{0ex}}\mathrm{mm}$).

**Figure 6.**Net mass flow rate changes with flow time (V-inclined pipe $L\phantom{\rule{4.pt}{0ex}}=\phantom{\rule{4.pt}{0ex}}150\phantom{\rule{3.33333pt}{0ex}}\mathrm{m}\phantom{\rule{4.pt}{0ex}}d\phantom{\rule{4.pt}{0ex}}=\phantom{\rule{4.pt}{0ex}}0.02\phantom{\rule{4.pt}{0ex}}\mathrm{mm}$).

**Figure 7.**Net mass flow rate changes with flow time (V-inclined pipe $L\phantom{\rule{4.pt}{0ex}}=\phantom{\rule{4.pt}{0ex}}200\phantom{\rule{3.33333pt}{0ex}}\mathrm{m}\phantom{\rule{4.pt}{0ex}}d\phantom{\rule{4.pt}{0ex}}=\phantom{\rule{4.pt}{0ex}}0.02\phantom{\rule{4.pt}{0ex}}\mathrm{mm}$).

**Figure 8.**Sand volume concentration changes with flow time (V-inclined pipe $L\phantom{\rule{4.pt}{0ex}}=\phantom{\rule{4.pt}{0ex}}80\phantom{\rule{3.33333pt}{0ex}}\mathrm{m}\phantom{\rule{4.pt}{0ex}}d\phantom{\rule{4.pt}{0ex}}=\phantom{\rule{4.pt}{0ex}}0.02\phantom{\rule{4.pt}{0ex}}\mathrm{mm}$).

**Figure 9.**Sand volume concentration changes with flow time (V-inclined pipe $L\phantom{\rule{4.pt}{0ex}}=\phantom{\rule{4.pt}{0ex}}150\phantom{\rule{3.33333pt}{0ex}}\mathrm{m}\phantom{\rule{4.pt}{0ex}}d\phantom{\rule{4.pt}{0ex}}=\phantom{\rule{4.pt}{0ex}}0.02\phantom{\rule{4.pt}{0ex}}\mathrm{mm}$).

**Figure 10.**Sand volume concentration changes with flow time (V-inclined pipe $L\phantom{\rule{4.pt}{0ex}}=\phantom{\rule{4.pt}{0ex}}200\phantom{\rule{3.33333pt}{0ex}}\mathrm{m}\phantom{\rule{4.pt}{0ex}}d\phantom{\rule{4.pt}{0ex}}=\phantom{\rule{4.pt}{0ex}}0.02\phantom{\rule{4.pt}{0ex}}\mathrm{mm}$).

**Figure 12.**Sand volume concentration contours at different cross-sections of horizontal pipe (inflow velocity = 0.4 m/s).

**Figure 13.**Sand volume concentration contours at different cross-sections of V-inclined pipe (inflow velocity = 0.4 m/s).

**Figure 14.**Net mass flow rate changes with flow time (V-inclined pipe $L\phantom{\rule{4.pt}{0ex}}=\phantom{\rule{4.pt}{0ex}}80\phantom{\rule{3.33333pt}{0ex}}\mathrm{m}\phantom{\rule{4.pt}{0ex}}d\phantom{\rule{4.pt}{0ex}}=\phantom{\rule{4.pt}{0ex}}0.05\phantom{\rule{4.pt}{0ex}}\mathrm{mm}$).

**Figure 15.**Net mass flow rate changes with flow time (V-inclined pipe $L\phantom{\rule{4.pt}{0ex}}=\phantom{\rule{4.pt}{0ex}}80\phantom{\rule{3.33333pt}{0ex}}\mathrm{m}\phantom{\rule{4.pt}{0ex}}d\phantom{\rule{4.pt}{0ex}}=\phantom{\rule{4.pt}{0ex}}0.1\phantom{\rule{4.pt}{0ex}}\mathrm{mm}$).

**Figure 16.**Sand volume concentration changes with flow time (V-inclined pipe $L\phantom{\rule{4.pt}{0ex}}=\phantom{\rule{4.pt}{0ex}}80\phantom{\rule{3.33333pt}{0ex}}\mathrm{m}\phantom{\rule{4.pt}{0ex}}d\phantom{\rule{4.pt}{0ex}}=\phantom{\rule{4.pt}{0ex}}0.05\phantom{\rule{4.pt}{0ex}}\mathrm{mm}$).

**Figure 17.**Sand volume concentration changes with flow time (V-inclined pipe $L\phantom{\rule{4.pt}{0ex}}=\phantom{\rule{4.pt}{0ex}}80\phantom{\rule{3.33333pt}{0ex}}\mathrm{m}\phantom{\rule{4.pt}{0ex}}d\phantom{\rule{4.pt}{0ex}}=\phantom{\rule{4.pt}{0ex}}0.1\phantom{\rule{4.pt}{0ex}}\mathrm{mm}$).

Scheme Number | Grid Number | Slurry Velocity (m/s) | Relative Error (%) |
---|---|---|---|

1 | 764,000 | 0.3029 | 0.66 |

2 | 1,487,600 | 0.3010 | 0.03 |

3 | 1,855,600 | 0.3009 | 0 |

Parameters | Horizontal Pipe | V-Inclined Pipe |
---|---|---|

±2° | ||

Grid cell size (m) | 0.05 | 0.05 |

Pipe length L (m) | 80 | 80/150/200 |

Pipe diameter D (mm) | 2600 | 2600 |

Liquid density $\rho $ ($\mathrm{kg}\xb7{\mathrm{m}}^{-3}$ ) | 998.2 | 998.2 |

Sand density ${\rho}_{s}$ ($\mathrm{kg}\xb7{\mathrm{m}}^{-3}$) | 2300 | 2300 |

Particle size d (mm) | 0.02 | 0.02\0.05\0.1 |

Sand content ${C}_{s}$ ($\mathrm{kg}\xb7{\mathrm{m}}^{-3}$) | 9.71 | 9.71 |

Sand volume concentration ${C}_{V}$ (%) | 0.42 | 0.42 |

Inflow velocities (m/s) | 0.3–0.7 | 0.3–1.6 |

Reference | Empirical Formula |
---|---|

Durand [16] | ${v}_{c}={F}_{L}{\left(2gD\frac{{\rho}_{s}-\rho}{\rho}\right)}^{\frac{1}{2}}$ |

Wasp [12] | ${v}_{c}=3.28{{C}_{V}}^{0.243}{\left(2gD\frac{{\rho}_{s}-\rho}{\rho}\right)}^{\frac{1}{2}}{\left(\frac{d}{D}\right)}^{\frac{1}{6}}$ |

Shook [13] | ${v}_{c}=2.43\frac{{{C}_{V}}^{\frac{1}{3}}}{{{C}_{D}}^{\frac{1}{4}}}{\left(2gD\frac{{\rho}_{s}-\rho}{\rho}\right)}^{\frac{1}{2}}$ |

He Wuquan [19] | ${v}_{c}=1.8644K{{C}_{W}}^{0.2341}{\left(gD{\omega}^{2}\frac{{\rho}_{s}-\rho}{\rho}\right)}^{\frac{1}{4}}$ |

**Table 4.**Comparison between empirical formula and numerical simulation under different particle sizes.

Particle Size | Emprical Formula | Numerical Simulation | |
---|---|---|---|

Wasp | He Wuquan | ||

0.02 mm | 0.99 m/s | 0.43 m/s | 0.4 m/s |

0.05 mm | 1.16 m/s | 1.10 m/s | 1.1 m/s |

0.1 mm | 1.30 m/s | 2.05 m/s | 1.5 m/s |

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**MDPI and ACS Style**

Yao, R.; Qi, D.; Zeng, H.; Huang, X.; Li, B.; Wang, Y.; Bai, W.; Wang, Z. Study on Critical Velocity of Sand Transport in V-Inclined Pipe Based on Numerical Simulation. *Water* **2022**, *14*, 2627.
https://doi.org/10.3390/w14172627

**AMA Style**

Yao R, Qi D, Zeng H, Huang X, Li B, Wang Y, Bai W, Wang Z. Study on Critical Velocity of Sand Transport in V-Inclined Pipe Based on Numerical Simulation. *Water*. 2022; 14(17):2627.
https://doi.org/10.3390/w14172627

**Chicago/Turabian Style**

Yao, Rao, Dunzhe Qi, Haiyan Zeng, Xingxing Huang, Bo Li, Yi Wang, Wenqiang Bai, and Zhengwei Wang. 2022. "Study on Critical Velocity of Sand Transport in V-Inclined Pipe Based on Numerical Simulation" *Water* 14, no. 17: 2627.
https://doi.org/10.3390/w14172627