# Analysis and Optimization of Grinding Performance of Vertical Roller Mill Based on Experimental Method

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## Abstract

**:**

## 1. Introduction

## 2. Material and Methods

#### 2.1. Material Properties

#### 2.2. Experimental Setup

#### 2.3. Grinding Performance Indexes

## 3. Response Surface Models and Parametric Study

#### 3.1. Design of Experimental (DOE)

#### 3.2. Response Surface Models

#### 3.3. Analysis of Variance (ANOVA)

#### 3.3.1. Analysis of Variance of Energy Consumption (${E}_{cs}$) Responses

^{2}is 0.934, which is consistent with the adjusted R

^{2}of 0.9774. The difference between predicted R-squared and the adjusted R-squared was less than 0.2. According to the analysis results in Table 3, order of parameter effect on energy consumption $\left({E}_{cs}\right)$ can be obtained by comparing the F-values and p-values magnitudes as follows: P > P

^{2}> n > ω > nω > ω

^{2}> n

^{2}> Pn > Pω.

#### 3.3.2. Analysis of Variance of Grinding Energy Efficiency (η) Responses

^{2}is 0.854, which is consistent with the adjusted R

^{2}of 0.9213. It can be seen that the difference between predicted R-squared and the adjusted R-squared is less than 0.2. According to the analysis results in Table 4, order of parameter effect on grinding energy efficiency (η) can be obtained by comparing the F-values and P-values magnitudes as follows: n > ω > P > P

^{2}> ω

^{2}> n

^{2}> nω > Pn > Pω.

#### 3.4. Validation of the RS Models

#### 3.5. Parametric Study

#### 3.5.1. Effect of Operating Parameters on ${E}_{cs}$ Response

#### 3.5.2. Effect of Operating Parameters on $\eta $ Response

## 4. Multi-Objective Optimization Design (MOD)

#### 4.1. Description of the Optimization Problem

#### 4.2. Desirability Function Approach

#### 4.3. Design Optimization Results

## 5. Conclusions

## Author Contributions

## Funding

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## References

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**Figure 1.**(

**a**) Particle size distribution of limestone after crushing; (

**b**) X-ray spectrum of raw limestone analyzed by XRD.

**Figure 2.**(

**a**) Structure of the laboratory-scale VRM used in experiments. (

**b**) Two-dimensional diagram of VRM.

**Figure 4.**(

**a**) Variation of ${E}_{cs}$ with n & P; (

**b**) variation of ${E}_{cs}$ with n & ω; (

**c**) perturbation plot of ${E}_{cs}$.

**Figure 5.**(

**a**) Variation of $\eta $ with ω and n; (

**b**) variation of $\eta $ with P and n; (

**c**) perturbation plot of $\eta $.

Parameters | Code | −1 | 0 | 1 |
---|---|---|---|---|

Loading pressure (MPa) | P | 6 | 7 | 8 |

Rotation speed (rpm) | n | 350 | 450 | 550 |

Moisture content (%) | ω | 0 | 1 | 2 |

No. | P (MPa) | n (rpm) | ω (%) | ${\mathit{E}}_{\mathit{c}\mathit{s}}$ (kWh/kg) | $\mathit{\eta}$ (kg/kWh) |
---|---|---|---|---|---|

1 | 7.00 | 450.00 | 1.00 | 0.03 | 0.834 |

2 | 6.00 | 550.00 | 1.00 | 0.028 | 0.856 |

3 | 8.00 | 450.00 | 2.00 | 0.04 | 0.756 |

4 | 8.00 | 450.00 | 0.00 | 0.038 | 0.962 |

5 | 7.00 | 350.00 | 2.00 | 0.028 | 0.756 |

6 | 8.00 | 550.00 | 1.00 | 0.043 | 0.751 |

7 | 6.00 | 450.00 | 0.00 | 0.024 | 1.021 |

8 | 7.00 | 450.00 | 1.00 | 0.03 | 0.834 |

9 | 6.00 | 350.00 | 1.00 | 0.025 | 1.042 |

10 | 7.00 | 550.00 | 2.00 | 0.035 | 0.426 |

11 | 7.00 | 450.00 | 1.00 | 0.03 | 0.834 |

12 | 6.00 | 450.00 | 2.00 | 0.027 | 0.812 |

13 | 7.00 | 450.00 | 1.00 | 0.03 | 0.834 |

14 | 7.00 | 550.00 | 0.00 | 0.029 | 0.654 |

15 | 7.00 | 350.00 | 0.00 | 0.026 | 0.994 |

16 | 7.00 | 450.00 | 1.00 | 0.028 | 0.834 |

17 | 8.00 | 350.00 | 1.00 | 0.038 | 0.942 |

Source | Sum of Squares | Df | Mean Square | F-Value | p-Value |
---|---|---|---|---|---|

Model | 4.948 × 10^{−4} | 9 | 5.498 × 10^{−5} | 77.75 | <0.0001 |

P | 3.781 × 10^{−4} | 1 | 3.781 × 10^{−4} | 534.72 | <0.0001 |

n | 4.050 × 10^{−5} | 1 | 4.050 × 10^{−5} | 57.27 | 0.0001 |

ω | 2.113 × 10^{−5} | 1 | 2.113 × 10^{−5} | 29.87 | 0.0009 |

Pn | 1.000 × 10^{−6} | 1 | 1.000 × 10^{−6} | 1.41 | 0.2731 |

Pω | 2.500 × 10^{−7} | 1 | 2.500 × 10^{−7} | 0.35 | 0.5708 |

nω | 4.000 × 10^{−6} | 1 | 4.000 × 10^{−6} | 5.66 | 0.0490 |

P^{2} | 4.655 × 10^{−5} | 1 | 4.655 × 10^{−5} | 65.83 | <0.0001 |

n^{2} | 1.392 × 10^{−6} | 1 | 1.392 × 10^{−6} | 1.97 | 0.2034 |

ω^{2} | 1.918 × 10^{−6} | 1 | 1.918 × 10^{−6} | 2.71 | 0.1435 |

Residual | 4.950 × 10^{−6} | 7 | 7.071 × 10^{−7} | ||

Lack of Fit | 1.750 × 10^{−6} | 3 | 5.833 × 10^{−7} | 0.73 | 0.5860 |

Pure Error | 3.200 × 10^{−6} | 4 | 8.000 × 10^{−7} | ||

Cor Total | 4.998 × 10^{−4} | 16 | |||

R^{2} | Adjusted R^{2} | Predicted R^{2} | Adequate precision | ||

0.9901 | 0.9774 | 0.934 | 30.429 |

Source | Sum of Squares | Df | Mean Square | F-Value | p-Value |
---|---|---|---|---|---|

Model | 0.34 | 9 | 0.038 | 21.82 | 0.0003 |

P | 0.013 | 1 | 0.013 | 7.42 | 0.0296 |

n | 0.14 | 1 | 0.14 | 79.48 | <0.0001 |

ω | 0.097 | 1 | 0.097 | 56.27 | 0.0001 |

Pn | 6.250 × 10^{−6} | 1 | 6.250 × 10^{−6} | 3.625 × 10^{−3} | 0.9537 |

Pω | 2.250 × 10^{−6} | 1 | 2.250 × 10^{−6} | 1.305 × 10^{−3} | 0.9722 |

nω | 2.500 × 10^{−5} | 1 | 2.500 × 10^{−5} | 0.015 | 0.9075 |

P^{2} | 0.063 | 1 | 0.063 | 36.35 | 0.0005 |

n^{2} | 0.014 | 1 | 0.014 | 8.29 | 0.0237 |

ω^{2} | 0.020 | 1 | 0.020 | 11.38 | 0.0119 |

Residual | 0.012 | 7 | 1.724 × 10^{−3} | ||

Lack of Fit | 0.012 | 3 | 4.023 × 10^{−3} | ||

Pure Error | 0.000 | 4 | 0.000 | ||

Cor Total | 0.35 | 16 | |||

R^{2} | Adjusted R^{2} | Predicted R^{2} | Adequate precision | ||

0.9656 | 0.9213 | 0.854 | 18.79 |

Parameter | $\mathit{P}$ (MPa) | $\mathit{n}$ (rpm) | $\mathit{\omega}$ (%) | ${\mathit{E}}_{\mathit{c}\mathit{s}}\text{}(\mathbf{kWh}/\mathbf{kg})$ | $\mathit{\eta}\text{}(\mathbf{kg}/\mathbf{kWh})$ | Desirability |
---|---|---|---|---|---|---|

Example | 7 | 450 | 1 | 0.03 | 0.834 | / |

Optimzed | 6 | 350 | 2 | 0.02314 | 1.11343 | 0.449 |

Parameter | ${E}_{cs}$ (kWh/kg) | $\eta $ (kg/kWh) |
---|---|---|

Predicted | 0.02314 | 1.11343 |

Experimental | 0.02135 | 1.02981 |

Error | 7.7% | 9.2% |

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**MDPI and ACS Style**

Liu, C.; Chen, Z.; Mao, Y.; Yao, Z.; Zhang, W.; Ye, W.; Duan, Y.; Xie, Q.
Analysis and Optimization of Grinding Performance of Vertical Roller Mill Based on Experimental Method. *Minerals* **2022**, *12*, 133.
https://doi.org/10.3390/min12020133

**AMA Style**

Liu C, Chen Z, Mao Y, Yao Z, Zhang W, Ye W, Duan Y, Xie Q.
Analysis and Optimization of Grinding Performance of Vertical Roller Mill Based on Experimental Method. *Minerals*. 2022; 12(2):133.
https://doi.org/10.3390/min12020133

**Chicago/Turabian Style**

Liu, Chang, Zuobing Chen, Ya Mao, Zhiming Yao, Weili Zhang, Weidong Ye, Yuanyuan Duan, and Qiang Xie.
2022. "Analysis and Optimization of Grinding Performance of Vertical Roller Mill Based on Experimental Method" *Minerals* 12, no. 2: 133.
https://doi.org/10.3390/min12020133