# Evolution and Physical Characteristics of a Raceway Based on a Transient Eulerian Multiphase Flow Model

^{*}

## Abstract

**:**

^{2}to 1.644 m

^{2}. The depth and height of the raceway increased considerably with velocity, while the width slightly increased.

## 1. Introduction

## 2. Model Description

#### 2.1. Conservation Equations

#### 2.2. Constitutive Relations

#### 2.3. Turbulence Equations

#### 2.4. Geometry and Operating Conditions

#### 2.5. Grid and Time Step Independence

## 3. Results and Discussion

#### 3.1. Raceway Evolution Characteristics

#### 3.2. Raceway Size Characteristics

^{2}to 1.644 m

^{2}, and the depth increased from 0.386 m to 1.109 m. This was due to the increased gas kinetic energy because of the increased blast volume and velocity. Therefore, increasing the blast velocity is very effective for increasing the depth of the raceway in order to develop the central gas flow in an actual BF.

#### 3.3. Pressure Distribution

#### 3.4. Flow Pattern

## 4. Conclusions

- (1)
- The evolution process of the raceway can be divided into three stages: rapid expansion, slow contraction, and gradual stabilization. The shape of the raceway was that of an upturned bag at high blast velocity.
- (2)
- The blast velocity had a significant effect on the size of the raceway. As the velocity increased, the depth, height, and surface area of the raceway considerably increased, while the width slightly increased.
- (3)
- The gas pressure in the raceway was higher than that of the particle bed, while the solid granular pressure was lower. The raceway did not exhibit a single-cycle flow pattern, but exhibited a complex multiphase and multi-cycle flow pattern.

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## Notation

Symbol | Meaning |

${\alpha}_{i}$ | $i$ phase volume fraction |

${\rho}_{i}$ | $i$ phase density, kg/m^{3} |

${U}_{i}$ | $i$ phase velocity, m/s |

${\tau}_{i}$ | $i$ phase stress–strain tensor, Pa |

${P}_{i}$ | $i$ phase pressure, Pa |

$g$ | Gravity acceleration, m/s^{2} |

$S$ | Source term |

$\beta $ | Momentum exchange coefficient |

${C}_{D}$ | Drag coefficient |

${d}_{s}$ | Solid-phase diameter, m |

${\mu}_{g}$ | Gas-phase viscosity, Pa$\xb7$s |

$Re$ | Reynolds number |

$I$ | Unit stress tensor |

${\mu}_{eff,g}$ | Gas effective viscosity, Pa$\xb7$s |

${k}_{g}$ | Gas turbulent kinetic energy, Pa$\xb7$s |

${\mu}_{t,g}$ | Gas turbulent viscosity, Pa$\xb7$s |

${\epsilon}_{g}$ | Gas turbulent dissipation rate |

${g}_{0}$ | Solid radial distribution function |

$e$ | Coefficient of restitution for particle collisions |

$\Theta $ | Granular pseudo-temperature |

${k}_{s}$ | Diffusion coefficient |

${\mathsf{\gamma}}_{s}$ | Particle collisional dissipation of energy |

${\alpha}_{s,min}$ | Friction packing limit |

${\alpha}_{s,\text{}\mathrm{max}\text{}}$ | Packing limit |

${\xi}_{s}$ | Solid bulk viscosity, Pa$\xb7$s |

${\mu}_{s}$ | Solids shear viscosity, Pa$\xb7$s |

${\mu}_{s,kin}$ | Solid kinetic viscosity, Pa$\xb7$s |

${\mu}_{s,col}$ | Solid collisional viscosity, Pa$\xb7$s |

${\mu}_{s,fr}$ | Solid frictional viscosity, Pa$\xb7$s |

${P}_{friction}$ | Frictional pressure, Pa |

$\varphi $ | Angle of internal friction |

${I}_{2D}$ | Second invariant of the deviatoric stress tensor |

$Fr$ | Froude number |

${G}_{k,g}$ | Gas-phase turbulent kinetic energy |

${\Pi}_{{k}_{g}}$ | Turbulent kinetic energy source term |

${\Pi}_{{\epsilon}_{g}}$ | Turbulent dissipation rate source term |

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**Figure 4.**Raceway shapes after they stabilize at different blast velocities: (

**a**) 150 m/s; (

**b**) 200 m/s; (

**c**) 250 m/s; (

**d**) 300 m/s.

**Figure 6.**(

**a**) Gas pressure and (

**b**) solid granular pressure for Case 3 in the symmetry plane at 40 s.

**Figure 8.**The symmetry plane of Case 3 at 40 s: (

**a**) gas velocity streamline; (

**b**) solid velocity streamline.

**Figure 9.**The tuyere level plane of Case 3 at 40 s: (

**a**) gas velocity streamline; (

**b**) solid velocity streamline.

Item | Formula |
---|---|

Gas stress | ${\tau}_{g}=-{P}_{g}I+{\mu}_{eff,g}(\nabla {U}_{g}+\left(\nabla {U}_{g}{)}^{T}\right)-\frac{2}{3}\left({\mu}_{eff,g}\left(\nabla \xb7{U}_{g}\right)I+{\rho}_{g}{k}_{g}\right)$ |

Gas effective viscosity | ${\mu}_{eff,g}={\mu}_{g}+{\mu}_{t,g}$ |

Gas turbulent viscosity | ${\mu}_{t,g}={\rho}_{g}{C}_{\mu}\frac{{k}_{g}^{2}}{{\epsilon}_{g}}$ (${C}_{\mu}=0.09$) |

Item | Formula |
---|---|

Solid stress | ${\tau}_{s}=\left(-{P}_{s}+{\xi}_{s}\nabla \xb7{U}_{s}\right)I+{\mu}_{s}\left\{\left(\nabla {U}_{s}+\nabla {U}_{s}^{T}\right)-\frac{2}{3}\nabla {U}_{s}I\right\}$ |

Solid pressure | ${P}_{s}={\alpha}_{s}{\rho}_{s}\Theta +2{\rho}_{s}\left(1+e\right){\alpha}_{s}^{2}{g}_{0}\Theta $ |

Diffusion coefficient | ${k}_{s}=\frac{150{\rho}_{s}{d}_{s}\sqrt{\Theta \pi}}{384\left(1+e\right){g}_{0}}{[1+\frac{6}{5}{g}_{0}{\alpha}_{s}\left(1+e\right)]}^{2}+2{\alpha}_{s}^{2}{\rho}_{s}{d}_{s}{g}_{0}\left(1+e\right){(\frac{\Theta}{\pi})}^{1/2}$ |

Particle collisional dissipation of energy | ${\mathsf{\gamma}}_{s}=3\left(1-{e}^{2}\right){g}_{0}{\rho}_{s}{\alpha}_{s}^{2}\Theta (\frac{4}{{d}_{s}}(\frac{\Theta}{\pi}{)}^{1/2}-\nabla \xb7{U}_{s})$ |

Solid radial distribution function | ${g}_{0}=\frac{3}{5}\left[1-\right[\frac{{\alpha}_{s}}{{\alpha}_{s,\text{}\mathrm{max}\text{}}}{]}^{1/3}{]}^{-1}$ |

Solid bulk viscosity | ${\xi}_{s}=\frac{4}{3}{\alpha}_{s}^{2}{\rho}_{s}{d}_{s}{g}_{0}\left(1+e\right){(\frac{\Theta}{\pi})}^{1/2}$ |

Solids shear viscosity | ${\mu}_{s}={\mu}_{s,kin}+{\mu}_{s,col}+{\mu}_{s,fr}$ |

Solid kinetic viscosity | ${\mu}_{s,kin}=\frac{10{\rho}_{s}{d}_{s}\sqrt{\Theta \pi}}{96\left(1+e\right){g}_{0}}{[1+\frac{4}{5}{g}_{0}{\alpha}_{s}\left(1+e\right)]}^{2}$ |

Solid collisional viscosity | ${\mu}_{s,col}=\frac{4}{5}{\alpha}_{s}^{2}{\rho}_{s}{d}_{s}{g}_{0}\left(1+e\right){(\frac{\Theta}{\pi})}^{1/2}$ |

Solid frictional viscosity | ${\mu}_{s,fr}=\frac{{P}_{friction}\text{}\mathrm{sin}\text{}\varphi}{2\sqrt{{I}_{2D}}}$ |

Frictional pressure | ${P}_{friction}=\{\begin{array}{c}Fr\frac{{\left({\alpha}_{s}-{\alpha}_{s,min}\right)}^{2}}{{\left({\alpha}_{s,max}-{\alpha}_{s}\right)}^{5}},Fr=0.1{\alpha}_{s},{\alpha}_{s}\ge 0.5\\ 0{\alpha}_{s}0.5\end{array}$ |

Parameters | Value |
---|---|

Number of calculation units | 109,516 |

Time Step | 0.0001 s |

Particle density | 1000 kg/m^{3} |

Angle of internal friction | 30° |

Tuyere equivalent diameter | 0.113 m |

Initial solid volume fraction | 0.6 |

Solid packing limit | 0.7 |

Friction packing limit | 0.61 |

Initial bed particle height | 4 m |

Outlet pressure | 303,000 Pa |

Case | Blast Velocity (m/s) | Injection Angle | Particle Diameter (m) |
---|---|---|---|

1 | 150 | 5° | 0.01 |

2 | 200 | 5° | 0.01 |

3 | 250 | 5° | 0.01 |

4 | 300 | 5° | 0.01 |

Number of Grid Cells | Time Step (s) | Depth (mm) | Height (mm) | Width (mm) | Deviation |
---|---|---|---|---|---|

109,516 | 0.0001 | 631 | 458 | 264 | - |

300,672 | 0.0001 | 640 | 461 | 267 | <2% |

109,516 | 0.00005 | 637 | 462 | 266 | <1% |

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## Share and Cite

**MDPI and ACS Style**

Peng, X.; Wang, J.; Zuo, H.; Xue, Q.
Evolution and Physical Characteristics of a Raceway Based on a Transient Eulerian Multiphase Flow Model. *Processes* **2020**, *8*, 1315.
https://doi.org/10.3390/pr8101315

**AMA Style**

Peng X, Wang J, Zuo H, Xue Q.
Evolution and Physical Characteristics of a Raceway Based on a Transient Eulerian Multiphase Flow Model. *Processes*. 2020; 8(10):1315.
https://doi.org/10.3390/pr8101315

**Chicago/Turabian Style**

Peng, Xing, Jingsong Wang, Haibin Zuo, and Qingguo Xue.
2020. "Evolution and Physical Characteristics of a Raceway Based on a Transient Eulerian Multiphase Flow Model" *Processes* 8, no. 10: 1315.
https://doi.org/10.3390/pr8101315