# Numerical Optimization Simulation of Synchronous Four-Wing Rotor

^{1}

^{2}

^{3}

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Numerical Simulation and Optimization Model of Mixing Flow Field

#### 2.1. Physical Model

#### 2.2. Basic Assumptions

- The compound is fully filled in the mixing chamber;
- The compound is isothermal, and the temperature of each point in the flow field is consistent;
- Rubber flow is laminar, with a small Reynolds number;
- The gravity and inertia of the rubber are much smaller than the viscosity, and can be ignored;
- No slippage on the fluid wall;
- The compound is a power-law fluid, which meets the characteristics of a non-Newtonian fluid.

#### 2.3. Mathematical Equation

#### 2.3.1. Time-Varying Governing Equation

#### 2.3.2. Basic Equation

- ρ—the density of the melt;
- t—time;
- v
_{x}, v_{y}, v_{z}—the component of the velocity vector in the x, y, z direction;

- P—the pressure on a fluid microelement;
- ${\tau}_{ij}$—the component of the adhesive stress τ acting on the surface of the microelement due to molecular viscosity;
- gx, gy, gz—the mass force of the microelement along the x, y, z axis.

- T—temperature;
- C
_{v}—constant-volume specific heat; - qx, qy, qz—the heat flux density of fluid flowing in unit time and unit area along the x, y, z axis direction.

#### 2.4. Characteristics of Material Rheological Parameters

_{0}is the viscosity at zero shear rate (Pa·s), η

_{∞}is the infinite shear viscosity (Pa·s), λ is the rubber viscoelastic characteristic time (s), and n is the Power-law index.

_{0}= 1,423,133; η

_{∞}= 0.0138; λ = 14.6707; n = 0.2590189. The rheological curve of the BC model material is shown in Figure 2.

#### 2.5. Finite Element Meshing

## 3. Rotor Optimization Process and Results

#### 3.1. Standard PSO Algorithm

_{1}is a self-learning factor. When c

_{1}= 0, it is called a selfless particle swarm algorithm, which will cause particles to quickly lose group diversity and fall into an endless loop;

_{2}is a group learning factor. If c

_{2}= 0, it is called a self-cognitive particle swarm algorithm. Because there is no information transferred and shared between groups, it will directly lead to a slower convergence of the algorithm. Therefore, selecting an appropriate learning factor can avoid falling into an infinite loop while ensuring the convergence speed. Generally, c

_{1}= c

_{2}= 2 is selected;

- Determine the optimization interval, generate a certain number of particles, and initialize the speed and position of each particle in the particle swarm;
- Calculate the target vector of each particle and add the non-dominated solution to the external memory;
- Determine the initial best global position of the particles and the best position of each particle;
- Update the speed and position of the particles, and take certain measures to ensure the movement of particles in the optimization interval;
- Calculate the target vector value of each particle and adjust their individual best position;
- Update the external memory and select the global best position for each particle at the same time;
- Determine whether the target meets convergence conditions. If it does, the algorithm stops and enters the optimal value. If it does not, it returns to step 3 to recalculate.

#### 3.2. Optimization Process and Results

## 4. Analysis of Synchronous Rotor before and after Optimization

#### 4.1. Comparison of Rotor Configuration

#### 4.2. Comparative Study of Contours

#### 4.2.1. Pressure Cloud Analysis

#### 4.2.2. Shear Rate Cloud Analysis

^{−1}, and the maximum shear rate of the rotor before optimization is 291.5 s

^{−1}. This is because the pressure difference between the front and rear peak surfaces in the optimized rotor is greater during rotation, which increases the volume flow rate and linear flow velocity of the rubber at the crest gap. The increase of the shear rate has an important effect on the improvement of the dispersion and mixing ability of the internal mixer.

#### 4.2.3. Mixing Index Cloud Analysis

#### 4.3. Comparative Numerical Analysis

#### 4.3.1. Analysis of the Stretch Length

#### 4.3.2. Analysis of Average Mixing Efficiency

#### 4.3.3. Analysis of the Global Distribution Index

#### 4.3.4. Separation Scale and Visual Analysis

#### 4.4. Comparative Experimental Study

#### 4.4.1. Experimental Raw Materials and Formulas

#### 4.4.2. Experiment and Test Process

#### 4.4.3. Experimental Results and Analysis

## 5. Conclusions

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## References

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**Figure 7.**Comparison of internal mixer rotor expansion before and after optimization (photograph of specimens).

**Figure 11.**Mean logarithmic stretch length of the optimized rotor (black) and rotor before optimization (red) as a function of time.

**Figure 12.**Average mixing efficiency of the optimized rotor (black) and rotor before optimization (red) as a function of time.

**Figure 13.**Global distribution index for the optimized rotor (black) and rotor before optimization (red) as a function of time.

**Figure 14.**Separation scale as a function of time curve of distributed mixing between two internal mixer chambers for the optimized rotor (black) and rotor before optimization (red).

**Figure 16.**Particle distribution at 3, 10, and 40 s between the two chambers for the rotor (

**a**) before and (

**b**) after optimization.

Geometric Parameters | Parameter Values | Geometric Parameters | Parameter Values |
---|---|---|---|

Rotor wing width (mm) | 3 | Helix angle of short-wing (°) | Variable (Parametric control) |

Gap between rotors (mm) | 1 | Maximum swivel circle diameter (mm) | 62 |

Front surface radius (mm) | 31 | Ratio of long and short wings | Variable (Parametric control) |

Back surface radius (mm) | 100 | Center distance of rotors(mm) | 63 |

Base circle diameter (mm) | 35 | Axial length of rotor(mm) | 93 |

Helix angle of long-wing (°) | Variable (Parametric control) |

Group | Serial Number | Helix Angle of Long Wing (°) | Helix Angle of Short Wing (°) | Length of Long Wing (mm) | Length of Short Wing (mm) |
---|---|---|---|---|---|

1 | 1 | 41 | 46 | 68 | 25 |

2 | 40 | 53 | 72 | 21 | |

3 | 34 | 47 | 64 | 29 | |

4 | 41 | 46 | 72 | 21 | |

5 | 35 | 39 | 64 | 29 | |

6 | 37 | 46 | 68 | 25 | |

2 | 1 | 36 | 46 | 66 | 27 |

2 | 36 | 52 | 71 | 22 | |

3 | 34 | 47 | 64 | 29 | |

4 | 35 | 47 | 68 | 25 | |

5 | 34 | 44 | 64 | 29 | |

6 | 37 | 47 | 67 | 26 | |

3 | 1 | 37 | 47 | 66 | 27 |

2 | 33 | 47 | 70 | 23 | |

3 | 34 | 47 | 64 | 29 | |

4 | 33 | 47 | 66 | 27 | |

5 | 34 | 46 | 64 | 29 | |

6 | 35 | 48 | 67 | 26 | |

4 | 1 | 36 | 47 | 66 | 27 |

2 | 33 | 45 | 67 | 26 | |

3 | 34 | 47 | 64 | 29 | |

4 | 34 | 47 | 66 | 27 | |

5 | 34 | 47 | 64 | 29 | |

6 | 34 | 48 | 66 | 27 | |

5 | 1 | 34 | 46 | 65 | 28 |

2 | 35 | 46 | 65 | 28 | |

3 | 34 | 47 | 64 | 29 | |

4 | 34 | 47 | 65 | 28 | |

5 | 34 | 47 | 64 | 29 | |

6 | 34 | 47 | 64 | 29 |

Group | 1 | 2 | 3 | 4 | |
---|---|---|---|---|---|

Serial Number | |||||

1 | 12.8196 | 10.7240 | 11.2609 | 10.8838 | |

2 | 13.4446 | 12.5034 | 12.8554 | 10.9776 | |

3 | 10.5061 | 10.5061 | 10.5061 | 10.5061 | |

4 | 12.7168 | 12.3610 | 10.5796 | 10.7224 | |

5 | 11.5913 | 11.0641 | 10.7576 | 10.5061 | |

6 | 12.8548 | 12.3343 | 12.7344 | 10.7512 |

Rotor Type | Synchronous Rotor before Optimization | Synchronous Rotor after Optimization | |
---|---|---|---|

Ridge Parameters | |||

Helix angle of long wing (°) | 35 | 34 | |

Helix angle of short wing (°) | 35 | 47 | |

Length of long wing (mm) | 72 | 64 | |

Length of short wing (mm) | 21 | 29 |

Material Name | Phr | Material Name | Phr |
---|---|---|---|

RC2557S | 110 | Silical115MP | 65 |

SBR1723 | 30 | Si69 mix | 8.4 |

ZnO | 2 | N234 | 25 |

SAD | 2 | Antilux111 | 1.8 |

4020 | 2 | V700 | 4 |

Serial Number | Mooney of Optimized Rotor (ML_{1+4}) | Mooney of the Rotor Before Optimization (ML_{1+4}) | Density of the Optimized Rotor (g/cm^{3}) | The Density of the Rotor Before Optimization (g/cm^{3}) | Carbon Black Dispersion of the Optimized Rotor | Carbon Black Dispersion of the Rotor Before Optimization |
---|---|---|---|---|---|---|

1 | 87.16 | 92.16 | 1.109 | 1.134 | 8.2 | 7.4 |

2 | 81.46 | 88.12 | 1.135 | 1.185 | 8.4 | 7.2 |

3 | 84.15 | 91.67 | 1.104 | 1.125 | 7.9 | 7.9 |

4 | 84.29 | 89.64 | 1.149 | 1.158 | 8.3 | 7.1 |

5 | 81.49 | 87.4 | 1.143 | 1.194 | 8.4 | 8.1 |

6 | 86.78 | 88.87 | 1.156 | 1.105 | 7.9 | 7.2 |

7 | 86.65 | 91.23 | 1.130 | 1.105 | 8 | 7.3 |

8 | 83.75 | 84.69 | 1.139 | 1.102 | 8.1 | 7.8 |

9 | 85.98 | 86.56 | 1.153 | 1.168 | 8.4 | 7.2 |

10 | 81.87 | 92.23 | 1.132 | 1.160 | 8.2 | 7.6 |

11 | 84.93 | 86.62 | 1.117 | 1.111 | 8 | 7.9 |

12 | 82.59 | 89.06 | 1.159 | 1.180 | 8.2 | 8.1 |

13 | 83.8 | 91.36 | 1.102 | 1.162 | 8.2 | 7.9 |

14 | 86.39 | 92.26 | 1.120 | 1.107 | 8.1 | 7.9 |

15 | 82.18 | 89.01 | 1.158 | 1.107 | 7.8 | 7.5 |

16 | 82.88 | 83.83 | 1.122 | 1.114 | 8.3 | 7.3 |

17 | 84.04 | 83.14 | 1.119 | 1.179 | 8.2 | 7.6 |

18 | 83.1 | 88.67 | 1.107 | 1.109 | 8.1 | 7.3 |

19 | 86.09 | 88.19 | 1.155 | 1.124 | 7.8 | 8 |

20 | 87.32 | 91.94 | 1.108 | 1.124 | 8.1 | 7.8 |

21 | 81.93 | 89.89 | 1.120 | 1.111 | 8 | 8 |

22 | 82.44 | 89.09 | 1.154 | 1.186 | 7.9 | 8 |

23 | 85.46 | 87.77 | 1.130 | 1.170 | 8.4 | 7.6 |

24 | 83.34 | 85.27 | 1.137 | 1.173 | 8.1 | 7.4 |

25 | 87.23 | 91.94 | 1.135 | 1.165 | 7.8 | 7.8 |

26 | 87.22 | 87.93 | 1.142 | 1.152 | 8.1 | 8.2 |

27 | 85.08 | 83.09 | 1.102 | 1.133 | 8.3 | 7.9 |

28 | 86.49 | 89.41 | 1.132 | 1.166 | 7.9 | 8 |

29 | 83.51 | 87.74 | 1.102 | 1.112 | 8.4 | 7.8 |

30 | 85.01 | 83.36 | 1.150 | 1.115 | 8 | 7.7 |

μ | 84.49 | 88.4 | 1.131 | 1.141 | 8.1 | 7.7 |

σ | 1.92 | 2.86 | 0.0189 | 0.0306 | 0.19 | 0.33 |

C.V | 2.28% | 3.23% | 1.67% | 2.68% | 2.31% | 4.24% |

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**MDPI and ACS Style**

Wang, K.; Liu, H.; Chang, T.; Han, D.; Pan, Y.; Wang, C.; Bian, H.
Numerical Optimization Simulation of Synchronous Four-Wing Rotor. *Materials* **2020**, *13*, 5353.
https://doi.org/10.3390/ma13235353

**AMA Style**

Wang K, Liu H, Chang T, Han D, Pan Y, Wang C, Bian H.
Numerical Optimization Simulation of Synchronous Four-Wing Rotor. *Materials*. 2020; 13(23):5353.
https://doi.org/10.3390/ma13235353

**Chicago/Turabian Style**

Wang, Kongshuo, Haichao Liu, Tianhao Chang, Deshang Han, Yiren Pan, Chuansheng Wang, and Huiguang Bian.
2020. "Numerical Optimization Simulation of Synchronous Four-Wing Rotor" *Materials* 13, no. 23: 5353.
https://doi.org/10.3390/ma13235353