# The Development and Application of a TFM for Dense Particle Flow and Mixing in Rotating Drums

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## Abstract

**:**

## 1. Introduction

## 2. Model Development

#### 2.1. TFM Coupled with KTGF

#### 2.2. Frictional Solids Stress Model

_{2D}is the second invariant of the deviator of the strain rate tensor.

_{min}= 0.5.

#### 2.3. Boundary Condition Model

## 3. Model Application

#### 3.1. The Validation of the Model

#### 3.2. Study on the Flow of Dense Uniform Particles

#### 3.3. Study on Mixing and Segregation of Dense Binary Particles

## 4. Concluding Remarks

- (1)
- TFM coupled with KTGF is generally used to study the dilute granular flow. In order to apply it to the dense granular flow in rotating drums, the frictional viscosity model is supplemented to consider the friction between particles due to long-term contact. Several frictional viscosity models are proposed for the dense particles flow and mixing in specific rotating drums, but a general model needs to be developed;
- (2)
- The research on the flow and mixing of dense granular flow in the rotating drum began in 2007; thus, the development and application of the model are still in the exploratory stage. By properly adjusting the model parameters, the model can be used to study the uniform particle flow and binary particle mixing in the rotating drum with and without flights and achieve valuable results;
- (3)
- The application of the model is flexible. The rotation of the drum can be performed by moving the wall or moving mesh. The validation of the model is easily completed by comparison with the experimental results of particle velocity distribution, particle volume fraction and solids hold up in the flight et al.
- (4)
- Although the advantages of TFM compared with DEM include low computing resources and a suitability for industrial-scale simulation, the application of the TFM model is mainly focused on a laboratory-scale rotating drum (diameter less than 0.5 m), and has not been applied to the prediction or analysis of a industrial-scale rotating drum.

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

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**Figure 1.**(

**a**) Volume fraction of the granular solid phase of 3.68 mm and fill level of 31.40% for drum rotating at 1.45, 4.08, 8.91 and 16.4 rad/s from the left to the right [46]; (

**b**) axial distribution of the tracer particle volume fraction after an injection time of 60 s for different operating conditions of the drum [42]; (

**c**) solid volume fraction distributions (passive phase) without and with the k-ε-turbulence model [52]. (Note: all the legends indicate volume fraction of particles).

**Figure 2.**(

**a**) Simulated drum transverse plane for the radial segregation analysis [53]; (

**b**) the surface of the 0.775 mm solid phase (top row) and 0.385 mm solid phase (bottom row) [23]; (

**c**) the mixing process of binary particles in the drum with “+” baffle [21]. (Note: all the legends indicate volume fraction of particles).

The continuity equations for the gas phase and solid phase | |

$\frac{\partial}{\partial t}\left({\alpha}_{g}{\rho}_{g}\right)+\nabla \xb7\left({\alpha}_{g}{\rho}_{g}{\overrightarrow{v}}_{g}\right)=0$ | (1) |

$\frac{\partial}{\partial t}\left({\alpha}_{s}{\rho}_{s}\right)+\nabla \xb7\left({\alpha}_{s}{\rho}_{s}{\overrightarrow{v}}_{s}\right)=0$ | (2) |

${\alpha}_{s}+{\alpha}_{g}=1$ | (3) |

The conservation equations of momentum for the gas phase and solid phase | |

$\frac{\partial}{\partial t}\left({\alpha}_{g}{\rho}_{g}{\overrightarrow{v}}_{g}\right)+\nabla \xb7\left({\alpha}_{g}{\rho}_{g}{\overrightarrow{v}}_{g}{\overrightarrow{v}}_{g}\right)=-{\alpha}_{g}\nabla p+\nabla \xb7{\stackrel{=}{\tau}}_{g}+{\alpha}_{g}{\rho}_{g}\overrightarrow{g}+{{\rm K}}_{sg}\left({\overrightarrow{v}}_{s}-{\overrightarrow{v}}_{g}\right)$ | (4) |

$\frac{\partial}{\partial t}\left({\alpha}_{s}{\rho}_{s}{\overrightarrow{v}}_{s}\right)+\nabla \xb7\left({\alpha}_{s}{\rho}_{s}{\overrightarrow{v}}_{s}{\overrightarrow{v}}_{s}\right)=-{\alpha}_{s}\nabla p-\nabla {p}_{s}+\nabla \xb7{\stackrel{=}{\tau}}_{s}+{\alpha}_{s}{\rho}_{s}\overrightarrow{g}+{{\rm K}}_{sg}\left({\overrightarrow{v}}_{g}-{\overrightarrow{v}}_{s}\right)$ | (5) |

Solid pressure | |

${p}_{s}={\alpha}_{s}{\rho}_{s}{\mathsf{\Theta}}_{s}+2{\alpha}_{s}{}^{2}{\rho}_{s}{g}_{0}{\mathsf{\Theta}}_{s}\left(1+{e}_{s}\right)$ | (6) |

Radial distribution function [22] | |

${g}_{0}=\frac{1}{1-{\left(\frac{{\alpha}_{s}}{{\alpha}_{s,max}}\right)}^{\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$3$}\right.}}$ | (7) |

Stress–strain tensor for gas phase and solid phase | |

${\stackrel{=}{\tau}}_{g}={\alpha}_{g}{\mu}_{g}\left[\nabla {\overrightarrow{v}}_{g}+\nabla {\overrightarrow{v}}_{g}{}^{T}\right]-\frac{2}{3}{\alpha}_{g}{\mu}_{g}\nabla \xb7{\overrightarrow{v}}_{g}\stackrel{=}{I}$ | (8) |

${\stackrel{=}{\tau}}_{s}={\alpha}_{s}{\mu}_{s}\left[\nabla {\overrightarrow{v}}_{s}+\nabla {\overrightarrow{v}}_{s}{}^{T}\right]+{\alpha}_{s}\left({\lambda}_{s}-\frac{2}{3}{\mu}_{s}\right)\nabla \xb7{\overrightarrow{v}}_{s}\stackrel{=}{I}$ | (9) |

Shear viscosity of solid phase [23] | |

${\mu}_{s}={\mu}_{s,col}+{\mu}_{s,kin}$ | (10) |

${\mu}_{s,col}=\frac{4}{5}{\alpha}_{s}{\rho}_{s}{d}_{s}{g}_{0}\left(1+{e}_{s}\right){\left(\frac{{\mathsf{\Theta}}_{s}}{\pi}\right)}^{\frac{1}{2}}$ | (11) |

${\mu}_{s,kin}=\frac{10{\rho}_{s}{d}_{s}\sqrt{{\mathsf{\Theta}}_{s}\pi}}{96{g}_{0}\left(1+{e}_{s}\right)}{\left[1+\frac{4}{5}{\alpha}_{s}{g}_{0}\left(1+{e}_{s}\right)\right]}^{2}$ | (12) |

Bulk viscosity of solid phase [24] | |

${\lambda}_{s}=\frac{4}{3}{\alpha}_{s}{}^{2}{\rho}_{s}{d}_{s}{g}_{0}\left(1+{e}_{s}\right){\left(\frac{{\mathsf{\Theta}}_{s}}{\pi}\right)}^{\frac{1}{2}}$ | (13) |

The transport equation of the granular temperature [25] | |

$\frac{3}{2}\left[\frac{\partial}{\partial t}\left({\alpha}_{s}{\rho}_{s}{\mathsf{\Theta}}_{s}\right)+\nabla \xb7\left({\alpha}_{s}{\rho}_{s}{\overrightarrow{v}}_{s}{\mathsf{\Theta}}_{s}\right)\right]=\left(-{p}_{s}\stackrel{=}{I}+{\stackrel{=}{\tau}}_{s}\right):\nabla {\overrightarrow{v}}_{s}+\nabla \xb7\left({k}_{{\mathsf{\Theta}}_{s}}\nabla {\mathsf{\Theta}}_{s}\right)-{\gamma}_{{\mathsf{\Theta}}_{s}}+{\phi}_{gs}$ | (14) |

${k}_{{\mathsf{\Theta}}_{s}}=\frac{150{\rho}_{s}{d}_{s}\sqrt{{\mathsf{\Theta}}_{s}\pi}}{384{g}_{0}\left(1+{e}_{s}\right)}{\left[1+\frac{64}{5}{\alpha}_{s}{g}_{0}\left(1+{e}_{s}\right)\right]}^{2}+2{\alpha}_{s}{}^{2}{\rho}_{s}{d}_{s}{g}_{0}\left(1+{e}_{s}\right){\left(\frac{{\mathsf{\Theta}}_{s}}{\pi}\right)}^{\frac{1}{2}}$ | (15) |

${\gamma}_{{\mathsf{\Theta}}_{s}}=3\left(1-{e}_{s}{}^{2}\right){\alpha}_{s}{}^{2}{\rho}_{s}{g}_{0}{\mathsf{\Theta}}_{s}\left(\frac{4}{{d}_{s}}\sqrt{\frac{{\mathsf{\Theta}}_{s}}{\pi}}-\nabla \xb7{\overrightarrow{v}}_{s}\right)$ | (16) |

${\phi}_{gs}=-3{{\rm K}}_{sg}{\mathsf{\Theta}}_{s}$ | (17) |

The interphase momentum exchange coefficient of gas and solid [23] | |

${{\rm K}}_{sg}={{\rm K}}_{gs}=\{\begin{array}{c}\frac{150{\mu}_{g}\left(1-{\alpha}_{g}\right){\alpha}_{s}}{{\alpha}_{g}{d}_{s}^{2}}+\frac{1.75{\alpha}_{s}{\rho}_{g}\left|{\overrightarrow{v}}_{g}-{\overrightarrow{v}}_{s}\right|}{{d}_{s}}{\alpha}_{g}\le 0.8\\ \frac{3{C}_{D}{\rho}_{g}{\alpha}_{s}\left|{\overrightarrow{v}}_{g}-{\overrightarrow{v}}_{s}\right|}{4{d}_{s}}{\alpha}_{g}^{-1.65}{\alpha}_{g}0.8\end{array}$ | (18) |

${C}_{D}=\{\begin{array}{c}\frac{24}{Re}\left[1+0.15R{e}^{0.687}\right]Re1000\\ 0.44Re\ge 1000\end{array}$ | (19) |

$Re=\frac{{\alpha}_{g}{\rho}_{g}{d}_{s}\left|{\overrightarrow{v}}_{g}-{\overrightarrow{v}}_{s}\right|}{{\mu}_{g}}$ | (20) |

Solid pressure | |

${p}_{si}={\alpha}_{si}{\rho}_{si}{\theta}_{si}+2\frac{{d}_{sjsi}^{3}}{{d}_{si}^{3}}\left(1+{e}_{si}\right){\alpha}_{si}{\alpha}_{sj}{\rho}_{si}{g}_{0,sisj}{\theta}_{si}$${d}_{sisj}>\frac{{d}_{si}{g}_{0,sjsj}+{d}_{sj}{g}_{0,sisi}}{{d}_{si}+{d}_{sj}}$ | (21) |

Radial distribution function [22] | |

${g}_{0,sisi}=\frac{1}{1-{\left(\frac{{\alpha}_{s}}{{\alpha}_{s,max}}\right)}^{1/3}}+\frac{1}{2}{d}_{si}{\displaystyle {\displaystyle \sum}_{i=1}^{2}}\frac{{\alpha}_{si}}{{\alpha}_{sj}}$ | (22) |

${g}_{0,sisj}\frac{{d}_{si}{g}_{0,sjsj}+{d}_{sj}{g}_{0,sisi}}{{d}_{si}+{d}_{sj}}$ | (23) |

Packing limit [26] | |

${\alpha}_{s,max}=\left({\alpha}_{si,max}-{\alpha}_{sj,max}+\left[1-\sqrt{{d}_{sj}/{d}_{si}}\right]\left(1-{\alpha}_{si,max}\right){\alpha}_{sj,max}\right)\left({\alpha}_{si,max}+\left(1-{\alpha}_{si,max}\right){\alpha}_{sj,max}\right)\frac{W}{{\alpha}_{si,max}}+{\alpha}_{sj,max}$$\text{}W\le {\alpha}_{w}$ | (24) |

${\alpha}_{s,max}=\left[1-\sqrt{{d}_{sj}/{d}_{si}}\right]\left({\alpha}_{si,max}+\left(1-{\alpha}_{si,max}\right){\alpha}_{sj,max}\right)\left(1-W\right)+{\alpha}_{sI,max}$$\text{}W{\alpha}_{w}$ | (25) |

$W=\frac{{\alpha}_{si}}{{\alpha}_{si}+{\alpha}_{sj}}$$\text{}\mathrm{for}\text{}{d}_{si}{d}_{sj}$ | (26) |

${\alpha}_{w}=\frac{{\alpha}_{s,max}}{{\alpha}_{si,max}+\left(1-{\alpha}_{si,max}\right){\alpha}_{sj,max}}$ | (27) |

The solid–solid momentum exchange model | |

${K}_{sjsi}=\frac{3\left(1+{e}_{ij}\right)\left(\frac{\pi}{2}+{C}_{fr}{\pi}^{2}/8\right){\epsilon}_{si}{\rho}_{si}{\epsilon}_{sj}{\rho}_{sj}{\left({d}_{si}+{d}_{si}\right)}^{2}{g}_{0,sjsi}}{2\pi \left({\rho}_{si}{d}_{si}^{3}+{\rho}_{sj}{d}_{sj}^{3}\right)}\times \left|{v}_{si}-{v}_{sj}\right|$ | (28) |

Year of Publication | Focus of the Study | Validation Basis | Rotation Method | D (mm) | L (mm) | Flight or Not | Particle Type | d (mm) | ρ_{s}(kg/m ^{3}) | Particle Shape |
---|---|---|---|---|---|---|---|---|---|---|

2012 [45] | Dynamic characteristics and the rheology of a granular viscous flow scale up | Particle velocity and dimensionless active layer thickness | - | 400 | - | No | Uniform | 1.5 | 2900 | Spherical |

2013 [46] | Particle dynamic behavior | Solid flow regime and velocity distribution | - | 195 | 500 | No | Uniform | 1.09/3.68 | 2460 | Spherical |

2015 [17] | The effect of operating conditions on solids flow | Solids hold up in the flight | moving mesh | 108 | 500 | Yes | Uniform | 1.09/1.84/2.56 2.56 | 2455 2090 | Spherical |

2016 [42] | Predict the transverse and axial solid-flow patterns, the fluid-flow profile, and particle residence time | Particle and fluid velocities and residence time | moving wall | 390 | 450 | No | Uniform | 4.25 | 1370 | Spherical |

2016 [47] | Heat transfer and mixing characteristics | Velocity and temperature of particles | - | 203 | - | No | Uniform | 2.5 | 2627 | Spherical |

2017 [27] | Boundary condition effects on the particle dynamic flow | Solids hold up in the flight, the bed height and solid volume fraction distribution | moving mesh | 108 | 500 | Yes | Uniform | 1.09 | 2455 | Spherical |

2017 [48] | The effects of specularity and restitution coefficients under different solid-flow regimes | Solid volume fraction distribution | moving mesh | 300 | 450 | Yes | Uniform | 25 | 7890 | Spherical |

2017 [49] | The effects of parameters on heat transfer characteristics | Average temperature of granular materials | moving wall | 300 | 350 | Yes | Uniform | 1 | 3900 | Spherical |

2018 [50] | The effects of parameters on the hydrodynamic and granular temperature of particles | Particle velocity | moving wall | 215 | - | No | Uniform | 6.2 | 1164 | Spherical |

2018 [51] | Irregular particle (non-spherical) dynamics | Rice grains velocities and drum transverse plane | moving wall and moving mesh | 390 | 20/30/40 | No | Uniform | 3.44 * | 1465 | Non-spherical |

2019 [28] | The effects of parameters on the charge of solid in the flight | Solids hold up in the flight and solid volume fraction distribution | moving mesh | 108 | 500 | Yes | Uniform | 1.09 1.02 | 1551 963 | Spherical |

2020 [10] | Solid frictional viscosity and wall friction | Particle velocity and flow pattern | moving mesh | 100 | - | No | Uniform | 3 | 2500 | Spherical |

2021 [52] | The comparison between the Eulerian (CFD) and the Lagrangian (DEM) approaches | Solids hold up in the flight and solid volume fraction distribution | moving mesh | 108 | 500 | Yes | Uniform | 1.09 | 2455 | Spherical |

2007 [15] | Main features of solids motion and segregation | Particle velocity and concentration | - | 240 | 1000 | No | Binary | 1.5/3 | 2600 | Spherical |

2013 [32] | Particle segregation and model of granular viscosity | End-view bed profile | - | 45 | 50 | No | Binary | 0.385/0.775 | 2500 | Spherical |

2016 [53] | Quantitatively and qualitatively evaluates the mixture and segregation processes | Drum transverse plane | - | 220 | 500 | No | Binary | 6.35/1.13 | 2460 | Spherical |

2017 [30] | Particle segregation and model of granular viscosity | End-view bed profile | - | 500 | 500 | No | Binary | 0.385/0.545/0.775 | 2500 | Spherical |

2017 [31] | Effects of specularity coefficient on particle segregation | End-view bed profile | - | 500 | 500 | No | Binary | 0.385/0.545/0.775 | 2500 | Spherical |

2020 [21] | Mixing and segregation of particles | The evolution of the degree of mixing and mixing process | - | 150 | 10 | No | Binary | 3/1.5 | 2600 | Spherical |

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**MDPI and ACS Style**

Rong, W.; Li, B.; Feng, Y.
The Development and Application of a TFM for Dense Particle Flow and Mixing in Rotating Drums. *Processes* **2022**, *10*, 234.
https://doi.org/10.3390/pr10020234

**AMA Style**

Rong W, Li B, Feng Y.
The Development and Application of a TFM for Dense Particle Flow and Mixing in Rotating Drums. *Processes*. 2022; 10(2):234.
https://doi.org/10.3390/pr10020234

**Chicago/Turabian Style**

Rong, Wenjie, Baokuan Li, and Yuqing Feng.
2022. "The Development and Application of a TFM for Dense Particle Flow and Mixing in Rotating Drums" *Processes* 10, no. 2: 234.
https://doi.org/10.3390/pr10020234