# Discrete Element Method (DEM) and Experimental Studies of the Angle of Repose and Porosity Distribution of Pellet Pile

^{1}

^{2}

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Experimental Work

_{1}and V denote the void volume (mL) of beakers full of particles and the volume of the empty beaker, respectively. In order to reduce the errors, the experiment was repeated nine times, and the average result was reported.

## 3. Simulation Method and Conditions

#### 3.1. Discrete Element Method (DEM)

_{i}g, contact force and viscous contact damping force, where K

_{n}, K

_{t}, γ

_{n}and γ

_{t}express the normal elastic constant, tangential elastic constant, normal damping constant and tangential damping constant, respectively. A particle with the mass of m

_{i}contacts with K particles, and the contact force between them depends on the deformation between particles, δ

_{n}. In the equation, u

_{i}, v

_{n}and v

_{t}represent the translational velocity, and the component of relatively velocity for the normal and tangential directions.

_{i}and ω

_{i}denote the moment of inertia and rotational velocity, respectively.

#### 3.2. Simulation Conditions

_{p}is the volume of a single pellet, and n is the number of particles in the box.

## 4. Results and Discussion

#### 4.1. Simulation and Experimental Study of Angle of Repose

#### 4.1.1. Angle of Repose by the Discharging Method

#### 4.1.2. The angle of Repose by the Lifting Method

#### 4.1.3. Simulated vs. Experimental Angles of Repose

#### 4.2. Porosity Distribution of Pellet Pile

#### 4.2.1. Simulated vs. Experimental BPD

#### 4.2.2. Effects of Rolling and Static Friction on BPD

## 5. Conclusions

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## References

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**Figure 1.**(

**a**) Experimental apparatus and schematic of the measurement of pellet pile and (

**b**) the geometric models of different packing method in simulation.

**Figure 3.**Vertical cross-sections of the pellet pile simulated under different (

**a**) rolling and (

**b**) static friction (

**b**) coefficients.

**Figure 4.**Extracted contours of the pellet pile for different (

**a**) rolling and (

**b**) static friction coefficients.

**Figure 5.**Relationship of the angle of repose of pellet piles and (

**a**) rolling friction coefficient with ${\mu}_{s}=0.15$, and (

**b**) static friction coefficient with ${\mu}_{r}=0.12$. Error bars indicate the deviation of the angle of repose in different directions of the pile.

**Figure 6.**(

**a**) Angle of repose and (

**b**) and normalized effective diameter of the heap (top view) with different coefficient of static and rolling friction for different drop heights in the Discharging Method. (The numbers in the two figures represent rolling and static friction coefficients, respectively) The inserted subfigure in (

**b**) is a top view of the simulated pile.

**Figure 7.**The angle of repose for different lifting velocities. Pink dotted and solid lines represent the angle of repose with different packing methods, but the same DEM parameters.

**Figure 9.**Comparisons of simulated (with and without containers) and experimental bottom porosity distribution (BPD) of the pellet pile.

**Figure 11.**BPD of the pellet pile and the inserted graph is the average porosity (the average value of seven points on a curve) with different (

**a**) static and (

**b**) rolling friction coefficients.

**Figure 12.**Frequency distribution of coordination number with different (

**a**) static and (

**b**) rolling friction coefficients.

**Figure 13.**The average coordination number of the whole heap for different static (${\mu}_{r}=0.12$) and rolling friction (${\mu}_{s}=0.15$) coefficients.

Parameters | Equations |
---|---|

K_{n}, K_{t} | ${K}_{n}=\frac{4}{3}{\gamma}^{*}\sqrt{{R}^{*}{\delta}_{n}}$,${K}_{t}=8{G}^{*}\sqrt{{R}^{*}{\delta}_{n}}$ |

γ_{n}, γ_{t} | ${\gamma}_{n}=-2\sqrt{\frac{5}{6}}\beta \sqrt{{s}_{n}{m}^{*}}\gg 0,{\gamma}_{t}=-2\sqrt{\frac{5}{6}}\beta \sqrt{{s}_{t}{m}^{*}}\gg 0$ |

Sn, St | ${S}_{n}=2{\gamma}^{*}\sqrt{{R}^{*}{\delta}_{n}},{S}_{t}=8{G}^{*}\sqrt{{R}^{*}{\delta}_{n}}$ |

β | $\beta =\frac{\mathrm{ln}\left(e\right)}{\sqrt{{\mathrm{ln}}^{2}\left(e\right)+{\pi}^{2}}}$ |

$\frac{1}{{\gamma}^{*}}$ | $\frac{1}{{\gamma}^{*}}=\frac{\left(1-{v}_{1}{}^{2}\right)}{{Y}_{1}}+\frac{\left(1-{v}_{2}{}^{2}\right)}{{Y}_{2}}$ |

$\frac{1}{{G}^{*}}$ | $\frac{1}{{G}^{*}}=\frac{2\left(2-{v}_{1}\right)\left(1+{v}_{1}\right)}{{Y}_{1}}+\frac{2\left(2-{v}_{2}\right)\left(1+{v}_{2}\right)}{{Y}_{2}}$ |

$\frac{1}{{R}^{*}},\frac{1}{{m}^{*}}$ | $\frac{1}{{R}^{*}}=\frac{1}{{R}_{1}}+\frac{1}{{R}_{2}},\frac{1}{{m}^{*}}=\frac{1}{{m}_{1}}+\frac{1}{{m}_{2}}$ |

$\Delta {M}_{r}^{k}$ | $\Delta {M}_{r}^{k}=-{k}_{r}\Delta {\theta}_{r},{k}_{r}={k}_{t}\xb7{R}^{*2}$ |

$\left|{M}_{r,t+\Delta t}^{k}\right|\ll {M}_{r}^{m}$ | |

${M}_{r,t+\Delta t}^{k}={M}_{r,t}^{k}+\Delta {M}_{r}^{k},{M}_{r}^{m}={\mu}_{r}{R}^{*}{F}_{n}$ | |

${M}_{r,t+\Delta t}^{d}$ | ${M}_{r,t+\Delta t}^{d}=-{C}_{r}\dot{{\theta}_{r}}(|{M}_{r,t+\Delta t}^{k}|<{M}_{r}^{m})$ ${M}_{r,t+\Delta t}^{d}=-{C}_{r}\dot{{\theta}_{r}}(|{M}_{r,t+\Delta t}^{k}|={M}_{r}^{m})$ ${C}_{r}={\eta}_{r}{C}_{r}^{crit},{C}_{r}^{crit}=2\sqrt{{I}_{r}{k}_{r}}$ ${I}_{r}={\left(\frac{1}{{I}_{i}+{m}_{i}{r}_{i}^{2}}+\frac{1}{{I}_{j}+{m}_{j}{r}_{j}^{2}}\right)}^{-1}$ |

**Table 2.**Physical and contact parameters used in DEM simulation, including pellet particle and walls.

Parameters | Values |
---|---|

Particle number | 100,000 |

Particle density | 4837 kg/m^{3} |

Time step | 10^{−5} s |

Young’s modulus | 2.5 × 10^{11} Pa (pellet), 2 × 10^{11} Pa (steel plane), 7.2 × 10^{10} (plexiglass) |

Poisson ratio (p-p; p-w; p-g) | 0.25, 0.3, 0.2 |

Coefficient of restitution (p-p; p-w; p-g) | 0.4, 0.35, 0.2 |

Coefficient of friction (p-w; p-g) | 0.4, 0.25 |

Rolling friction coefficient (p-w; p-g) | 0.4, 0.15 |

Size of pellet | 8 mm, 14 mm, 20 mm |

Cases | Rolling Friction Coefficient | Static Friction Coefficient | Barrel Size (Diameter: m) |
---|---|---|---|

Case1 | 0.05 | 0.15 | 0.1 |

Case2 | 0.15 | ||

Case3 | 0.12 | 0.15 | |

Case4 | 0.12 | 0.45 |

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**MDPI and ACS Style**

Wei, H.; Li, M.; Li, Y.; Ge, Y.; Saxén, H.; Yu, Y.
Discrete Element Method (DEM) and Experimental Studies of the Angle of Repose and Porosity Distribution of Pellet Pile. *Processes* **2019**, *7*, 561.
https://doi.org/10.3390/pr7090561

**AMA Style**

Wei H, Li M, Li Y, Ge Y, Saxén H, Yu Y.
Discrete Element Method (DEM) and Experimental Studies of the Angle of Repose and Porosity Distribution of Pellet Pile. *Processes*. 2019; 7(9):561.
https://doi.org/10.3390/pr7090561

**Chicago/Turabian Style**

Wei, Han, Meng Li, Ying Li, Yao Ge, Henrik Saxén, and Yaowei Yu.
2019. "Discrete Element Method (DEM) and Experimental Studies of the Angle of Repose and Porosity Distribution of Pellet Pile" *Processes* 7, no. 9: 561.
https://doi.org/10.3390/pr7090561