# Calibration and Validation of a Cone Crusher Model with Industrial Data

^{1}

^{2}

^{3}

^{4}

^{*}

## Abstract

**:**

## 1. Introduction

^{®}—Andritz Automation, Mimic

^{®}—Emerson, and DynSim

^{®}—Aveva) are based on Whiten’s model. Such dynamic simulation software is in high demand in digital twin applications, control system design, operator training, and flowsheet evaluation.

## 2. Literature Review

^{®}. Anticoi et al. [26] analyzed the influence of the type of material on the parameters of the breakage function in a milling circuit. The parameters were calibrated against data from piston press tests and single compression strength tests; the optimal parameters were obtained by minimizing the root mean square error between the measured and calculated values. Ambiguous results led the authors to conclude that numerical algorithms may not be the best option for calibration in the presented scenario. In a fashion similar to their previous study, Anticoi et al. [33] also studied the influence of the operating conditions of a pilot-scale HPGR on the breakage function parameters.

## 3. Description of the Cone Crusher Model

#### 3.1. Classification Function

#### 3.2. Breakage Function

## 4. Proposed Calibration Strategy

- $DS=\{1,\dots ,\phantom{\rule{4pt}{0ex}}{n}_{d}\}$: set of ${n}_{d}$ datasets;
- $CL=\{1,\dots ,\phantom{\rule{4pt}{0ex}}{n}_{f}\}$: set of ${n}_{f}$ classes used to describe the PSD of the material;
- $CS{S}_{l}$: closed side setting (CSS; mm) of the cone crusher for dataset l, $l=1,\dots ,{n}_{d}$;
- $TP{H}_{l}$: throughput (dry t/h) for dataset l, $l=1,\dots ,{n}_{d}$;
- $F{80}_{l}$: 80% passing size (mm) of the feed for dataset l, $l=1,\dots ,{n}_{d}$;
- $LLE{N}_{l}$: length of the face of the mantle liner (mm) for dataset l, $l=1,\dots ,{n}_{d}$;
- $LH{R}_{l}$: liner age (hours) for dataset l, $l=1,\dots ,{n}_{d}$;
- $E{T}_{l}$: eccentric throw (mm) for dataset l, $l=1,\dots ,{n}_{d}$;
- ${\widehat{P}}_{i,l}\left({\dots}_{1}\right)$, ${P}_{i,l}$: predicted and measured product PSDs, respectively, for $i=1,\dots ,{n}_{f}$ and $l=1,\dots ,{n}_{d}$;
- $\theta $: vector of decision variables;
- $LB$: lower bound on the decision variables;
- $UB$: upper bound on the decision variables.

**B**is equal to one. Constraints (1515) and (16) define bounds on ${K}_{1}$ and ${K}_{2}$ based on empirical recommendations [20]. Finally, constraint (17) defines bounds on ${\theta}_{1}$.

## 5. Results and Discussion

#### 5.1. Evaluation of the FBS and CBS

#### 5.2. Calibration/Validation Using the FBS

#### 5.2.1. Scenario 1

#### 5.2.2. Scenario 2

#### 5.2.3. Scenario 3

## 6. Conclusions

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## References

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**Figure 1.**Schematic representation of a cone crusher, extracted from Yamashita et al. [37].

**Figure 4.**Measured and predicted PSD curves to compare the effect of the number of degrees of freedom of the breakage function on the model accuracy for dataset 1.

**Figure 5.**Measured and predicted PSD curves in Scenario 1 for the Serra Leste datasets. (

**a**) PSD predicted using the model calibrated on datasets 1, 2 and 3. Validation with Dataset 1. (

**b**) PSD predicted using the model calibrated on datasets 1, 2 and 3. Validation with Dataset 2. (

**c**) PSD predicted using the model calibrated on datasets 1, 2 and 3. Validation with Dataset 3.

**Figure 6.**Measured and predicted PSD curves in Scenario 1 for the S11D datasets. (

**a**) PSD predicted using the model calibrated on datasets 4 to 7. Dataset 4 is used for validation. (

**b**) PSD predicted using the model calibrated on datasets 4 to 7. Dataset 5 is used for validation. (

**c**) PSD predicted using the model calibrated on datasets 4 to 7. Dataset 6 is used for validation. (

**d**) PSD predicted using the model calibrated on datasets 4 to 7. Dataset 7 is used for validation.

**Figure 7.**Measured and predicted PSD curves in Scenario 2 for the Serra Leste datasets. (

**a**) PSD predicted using the model calibrated on datasets 2 and 3. Validation with Dataset 1. (

**b**) PSD predicted using the model calibrated on datasets 1 and 3. Validation with Dataset 2. (

**c**) PSD predicted using the model calibrated on datasets 1 and 2. Validation with Dataset 3.

**Figure 8.**Measured and predicted PSD curves in Scenario 2 for the S11D datasets. (

**a**) PSD predicted using the model calibrated on datasets 4 and 7. Validation with Dataset 5. (

**b**) PSD predicted using the model calibrated on datasets 4 and 5. Validation with Dataset 6. (

**c**) PSD predicted using the model calibrated on datasets 5 and 6. Validation with Dataset 7.

**Figure 9.**Measured PSDs for CSS = 35 mm, 38 mm and 41 mm. Predicted PSDs for CSS = 37 mm, 39 mm and 40 mm.

Iron Mine | Cone Crusher | Dataset | $\mathbf{CSS}$ (mm) | $\mathbf{TPH}$ (t/h) | $\mathit{F}80$ (mm) |
---|---|---|---|---|---|

1 | 35 | 883 | 102.36 | ||

Serra Leste | Metso HP400 | 2 | 38 | 986 | 102.36 |

3 | 41 | 998 | 102.36 | ||

4 | 28 | 368 | 45.35 | ||

S11D | Sandvik CH660 | 5 | 28 | 368 | 73.03 |

6 | 28 | 368 | 30.89 | ||

7 | 28 | 368 | 27.79 |

FBS | CBS | |||
---|---|---|---|---|

Dataset(s) Used for Calibration | SSE | Time [s] | SSE | Time [s] |

1 | $6.04\times {10}^{-5}$ | 20.72 | 98.66 | 1.43 |

2 | $3.01\times {10}^{-6}$ | 13.32 | 72.16 | 2.90 |

3 | $2.21\times {10}^{-6}$ | 12.92 | 76.10 | 1.53 |

1 and 2 | 22.51 | 180.39 | 407.95 | 1.94 |

1 and 3 | 1.26 | 79.04 | 308.75 | 1.51 |

2 and 3 | $1.28\times {10}^{-2}$ | 97.63 | 155.85 | 1.60 |

1, 2 and 3 | 21.23 | 58.07 | 755.82 | 3.76 |

Mean (StDev) | 6.43 (±10.56) | 66.01 (±60.50) | 267.90 (±250.22) | 2.10 (±0.89) |

FBS | CBS | |||
---|---|---|---|---|

Dataset(s) Used for Calibration | SSE | Time [s] | SSE | Time [s] |

5 and 7 | 10.15 | 30.94 | 244.83 | 10.45 |

4 and 7 | 7.67 | 22.29 | 258.50 | 3.23 |

4 and 5 | $4.94\times {10}^{-2}$ | 56.52 | 70.15 | 2.99 |

4, 5, 6 and 7 | 479.51 | 42.11 | 1064.00 | 2.50 |

Mean (StDev) | 124.34 (±236.82) | 37.96 (±14.79) | 409.37 (±444.76) | 4.79 (±3.78) |

Iron Mine | ${\mathit{\alpha}}_{0}$ | ${\mathit{\alpha}}_{1}$ | ${\mathit{\alpha}}_{2}$ | ${\mathit{\alpha}}_{3}$ | ${\mathit{\beta}}_{0}$ | ${\mathit{\beta}}_{1}$ | ${\mathit{\beta}}_{2}$ | ${\mathit{\beta}}_{3}$ | ${\mathit{\gamma}}_{0}$ |
---|---|---|---|---|---|---|---|---|---|

Serra Leste (1 to 3) | 0 | 0.905 | 0 | 0 | 0 | 0.096 | 0.095 | 0 | 3.000 |

S11D (4 to 7) | 0.155 | 0.441 | 0 | 0.053 | 0.047 | 1.698 | 0 | 0 | 3.000 |

Iron Mine | ${\mathit{\alpha}}_{0}$ | ${\mathit{\alpha}}_{1}$ | ${\mathit{\alpha}}_{2}$ | ${\mathit{\alpha}}_{3}$ | ${\mathit{\beta}}_{0}$ | ${\mathit{\beta}}_{1}$ | ${\mathit{\beta}}_{2}$ | ${\mathit{\beta}}_{3}$ | ${\mathit{\gamma}}_{0}$ | $\Phi $ | $\mathit{\delta}$ | $\mathit{\sigma}$ |
---|---|---|---|---|---|---|---|---|---|---|---|---|

Serra Leste (1 to 3) | 0 | 1.375 | 0.018 | 0 | 0 | 3.060 | 0 | 0 | 3.000 | 1.000 | 0.506 | 211.441 |

S11D (4 to 7) | 0.001 | 0.425 | 0 | 0.076 | 0.004 | 0.704 | 0.063 | 0.177 | 3.000 | 1.000 | 0.400 | 875.255 |

Datasets | ${\mathit{\alpha}}_{0}$ | ${\mathit{\alpha}}_{1}$ | ${\mathit{\alpha}}_{2}$ | ${\mathit{\alpha}}_{3}$ | ${\mathit{\beta}}_{0}$ | ${\mathit{\beta}}_{1}$ | ${\mathit{\beta}}_{2}$ | ${\mathit{\beta}}_{3}$ | ${\mathit{\gamma}}_{0}$ |
---|---|---|---|---|---|---|---|---|---|

2 and 3 | 0.431 | 0.249 | 0 | 0.096 | 0 | 2.507 | 0 | 0 | 1.695 |

1 and 3 | 0 | 7.481 | 0.277 | 0 | 0 | 2.451 | 0 | 0 | 2.636 |

1 and 2 | 0 | 0.950 | 0 | 0 | 0 | 0 | 0.082 | 0 | 3.000 |

4 and 7 | 0.021 | 0.391 | 0 | 0.109 | 0.017 | 1.309 | 0.030 | 0 | 3.000 |

4 and 5 | 0.193 | 0.486 | 0 | 0.005 | 0.033 | 1.703 | 0 | 0 | 3.000 |

5 and 6 | 0.387 | 0.454 | 0 | 0.029 | 0.206 | 1.693 | 0 | 0 | 3.000 |

Datasets | ${\mathit{\alpha}}_{0}$ | ${\mathit{\alpha}}_{1}$ | ${\mathit{\alpha}}_{2}$ | ${\mathit{\alpha}}_{3}$ | ${\mathit{\beta}}_{0}$ | ${\mathit{\beta}}_{1}$ | ${\mathit{\beta}}_{2}$ | ${\mathit{\beta}}_{3}$ | ${\mathit{\gamma}}_{0}$ | $\Phi $ | $\mathit{\delta}$ | $\mathit{\sigma}$ |
---|---|---|---|---|---|---|---|---|---|---|---|---|

2 and 3 | 0.024 | 2.127 | 0.303 | 2.380 | 0 | 3.500 | 0 | 0 | 2.275 | 1.000 | 0.534 | 75.929 |

1 and 3 | 0.022 | 29.596 | 1.410 | 2.276 | 0 | 3.218 | 0 | 0 | 3.000 | 1.000 | 0.514 | 237.607 |

1 and 2 | 0 | 0.500 | 0 | 0 | 0 | 0 | 0.067 | 0 | 2.688 | 0.016 | 0.487 | 242.818 |

4 and 7 | 0 | 0.642 | 0.011 | 0 | 0.047 | 0.096 | 0.093 | 0.379 | 3.000 | 1.000 | 0.343 | 577.232 |

4 and 5 | 0 | 0.858 | 0.051 | 0.192 | 0 | 0.397 | 0.094 | 0.076 | 3.000 | 1.000 | 0.408 | 764.263 |

5 and 6 | 0 | 0.903 | 0.035 | 0.053 | 0.001 | 0.813 | 0.109 | 0. | 3.000 | 1.000 | 0.417 | 644.004 |

CSS [mm] | *35 | 37 | *38 | 39 | 40 | *41 |
---|---|---|---|---|---|---|

$\widehat{P}80$ [mm] | 34.66 | 40.84 | 46.95 | 48.51 | 50.00 | 52.39 |

$P80$ [mm] | 33.78 | - | 47.16 | - | - | 53.90 |

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**MDPI and ACS Style**

Duarte, R.A.; Yamashita, A.S.; da Silva, M.T.; Cota, L.P.; Euzébio, T.A.M.
Calibration and Validation of a Cone Crusher Model with Industrial Data. *Minerals* **2021**, *11*, 1256.
https://doi.org/10.3390/min11111256

**AMA Style**

Duarte RA, Yamashita AS, da Silva MT, Cota LP, Euzébio TAM.
Calibration and Validation of a Cone Crusher Model with Industrial Data. *Minerals*. 2021; 11(11):1256.
https://doi.org/10.3390/min11111256

**Chicago/Turabian Style**

Duarte, Robson A., André S. Yamashita, Moisés T. da Silva, Luciano P. Cota, and Thiago A. M. Euzébio.
2021. "Calibration and Validation of a Cone Crusher Model with Industrial Data" *Minerals* 11, no. 11: 1256.
https://doi.org/10.3390/min11111256