# Multidimensional Optimization of the Copper Flotation in a Jameson Cell by Means of Taxonomic Methods

^{1}

^{2}

^{3}

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Experiment

#### 2.1. Laboratory Investigation

- separation tank diameter and height 200 mm and 900 mm, respectively;
- downcomer diameter and length: 0.020 m and 1.8 m, respectively;
- nozzle diameter 0.005 m;
- conditioning tank volume: 0.1 m
^{3}; - downcomer plunging length that is the depth to which the end of the downcomer is immersed in the separation tank: 0.5 m;
- feed rate and air rate: 100 cm
^{3}/s.

#### 2.2. Methodology of Taxonomic Methods

#### 2.2.1. Theoretical Background

_{j}was used, whose general formula is presented by Equation (3).

- i—number of the row;
- j—number of the column;
- n—number of investigated variables (flotation tests);
- l—number of variables (process evaluation factors);$${\mathrm{z}}_{\mathrm{ij}}=\frac{{\mathrm{x}}_{\mathrm{ij}}}{{\mathrm{x}}_{\mathrm{jmax}}},\text{}{\mathrm{x}}_{\mathrm{jmax}}=\underset{\mathrm{i}}{\mathrm{max}}\left({\mathrm{x}}_{\mathrm{ij}}\right)\u2014\mathrm{standardized}\text{}\mathrm{value}.$$

_{ij}the values e

_{1}, e

_{2}, …, e

_{n}were calculated by means of Equation (3). The smallest value allowed us to determine the optimal values of the considered factors.

#### 2.2.2. Application

## 3. Results and Discussion

## 4. Conclusions

_{1}, w

_{2}, w

_{3}, can be entered into the optimization function. In such a case, particular components of the F function should be multiplied by w

_{1}, w

_{2}, w

_{3}, respectively, where 0 < w

_{1}< 1, 0 < w

_{2}< 1, 0 < w

_{3}< 1 and w

_{1}+ w

_{2}+ w

_{3}= 1. The presented methodology can be used efficiently in the evaluation of all kinds of processes and when combined with modeling methods, it can be used as an algorithm of process quality monitoring.

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## References

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**Figure 1.**Scheme of a Jameson cell [39].

**Table 1.**Mineralogical composition of lithological types of Polish copper ores [6].

Lithological Type of Copper Ore | Content of Selected Metals in Lithological Types | Prevalent Copper-Bearing Minerals | |
---|---|---|---|

carbonates | Cu (%) | 1.69 | chalcocite in combination with digenite, bornite, covellite and chalcopyrite |

Ag (g/t) | 54 | ||

shales | Cu (%) | 6.02 | chalcocite-bornite and bornite-chalcopyrite minerals |

Ag (g/t) | 188 | ||

sandstones | Cu (%) | 1.29 | bornite-chalcopyrite and chalcocite-bornite minerals |

Ag (g/t) | 30 |

Particle Size Fraction d (μm) | Collector Type k | Collector Dosage s (g/t) | Time t (min) |
---|---|---|---|

0–20 20–40 40–71 | Aqueous solution of ethyl sodium xanthate—E Aqueous solution of isobutyl sodium xanthate—I | 100 150 | 1, 2, 4 6, 9, 12 17, 22, 30 |

Time t (min) | E | I | ||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|

100 (g/t) | 150 (g/t) | 100 (g/t) | 150 (g/t) | |||||||||

β | ϑ | ε | β | ϑ | ε | β | ϑ | ε | β | ϑ | ε | |

1 | 0.145 | 0.024 | 0.068 | 0.113 | 0.018 | 0.152 | 0.166 | 0.026 | 0.152 | 0.171 | 0.023 | 0.122 |

2 | 0.171 | 0.017 | 0.356 | 0.121 | 0.013 | 0.402 | 0.184 | 0.019 | 0.416 | 0.157 | 0.018 | 0.359 |

4 | 0.160 | 0.012 | 0.572 | 0.113 | 0.010 | 0.594 | 0.175 | 0.013 | 0.613 | 0.150 | 0.014 | 0.562 |

6 | 0.141 | 0.008 | 0.709 | 0.102 | 0.007 | 0.706 | 0.156 | 0.009 | 0.737 | 0.137 | 0.009 | 0.706 |

9 | 0.126 | 0.006 | 0.782 | 0.092 | 0.006 | 0.772 | 0.142 | 0.007 | 0.811 | 0.124 | 0.007 | 0.785 |

12 | 0.115 | 0.006 | 0.802 | 0.086 | 0.005 | 0.798 | 0.131 | 0.006 | 0.833 | 0.115 | 0.006 | 0.809 |

17 | 0.108 | 0.005 | 0.832 | 0.081 | 0.004 | 0.830 | 0.123 | 0.006 | 0.856 | 0.107 | 0.005 | 0.835 |

22 | 0.104 | 0.005 | 0.858 | 0.077 | 0.004 | 0.839 | 0.119 | 0.005 | 0.882 | 0.103 | 0.005 | 0.856 |

30 | 0.097 | 0.004 | 0.876 | 0.072 | 0.004 | 0.856 | 0.112 | 0.004 | 0.895 | 0.097 | 0.004 | 0.870 |

Time t (min) | E | I | ||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|

100 (g/t) | 150 (g/t) | 100 (g/t) | 150 (g/t) | |||||||||

β | ϑ | ε | β | ϑ | ε | β | ϑ | ε | β | ϑ | ε | |

1 | 0.076 | 0.024 | 0.065 | 0.065 | 0.026 | 0.017 | 0.063 | 0.018 | 0.049 | 0.167 | 0.021 | 0.137 |

2 | 0.076 | 0.021 | 0.188 | 0.081 | 0.023 | 0.148 | 0.075 | 0.016 | 0.220 | 0.135 | 0.013 | 0.336 |

4 | 0.077 | 0.019 | 0.301 | 0.078 | 0.021 | 0.256 | 0.082 | 0.013 | 0.403 | 0.121 | 0.013 | 0.505 |

6 | 0.075 | 0.017 | 0.414 | 0.079 | 0.019 | 0.372 | 0.074 | 0.011 | 0.509 | 0.114 | 0.010 | 0.633 |

9 | 0.076 | 0.015 | 0.506 | 0.077 | 0.017 | 0.453 | 0.068 | 0.010 | 0.569 | 0.105 | 0.009 | 0.706 |

12 | 0.073 | 0.014 | 0.545 | 0.076 | 0.016 | 0.501 | 0.063 | 0.010 | 0.589 | 0.097 | 0.008 | 0.733 |

17 | 0.073 | 0.012 | 0.602 | 0.073 | 0.015 | 0.538 | 0.059 | 0.009 | 0.615 | 0.091 | 0.008 | 0.754 |

22 | 0.074 | 0.011 | 0.645 | 0.073 | 0.014 | 0.577 | 0.058 | 0.009 | 0.639 | 0.088 | 0.007 | 0.777 |

30 | 0.071 | 0.011 | 0.677 | 0.071 | 0.013 | 0.610 | 0.055 | 0.009 | 0.658 | 0.083 | 0.007 | 0.796 |

Time t (min) | E | I | ||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|

100 (g/t) | 150 (g/t) | 100 (g/t) | 150 (g/t) | |||||||||

β | ϑ | ε | β | ϑ | ε | β | ϑ | ε | β | ϑ | ε | |

1 | 0.075 | 0.019 | 0.032 | 0.064 | 0.020 | 0.065 | 0.092 | 0.024 | 0.024 | 0.075 | 0.025 | 0.041 |

2 | 0.057 | 0.017 | 0.142 | 0.065 | 0.018 | 0.193 | 0.068 | 0.022 | 0.132 | 0.068 | 0.023 | 0.144 |

4 | 0.057 | 0.016 | 0.253 | 0.068 | 0.016 | 0.326 | 0.068 | 0.020 | 0.238 | 0.063 | 0.022 | 0.226 |

6 | 0.057 | 0.014 | 0.372 | 0.072 | 0.013 | 0.467 | 0.065 | 0.019 | 0.328 | 0.060 | 0.021 | 0.299 |

9 | 0.058 | 0.013 | 0.462 | 0.069 | 0.011 | 0.548 | 0.062 | 0.018 | 0.386 | 0.056 | 0.020 | 0.348 |

12 | 0.058 | 0.012 | 0.515 | 0.067 | 0.010 | 0.592 | 0.059 | 0.017 | 0.413 | 0.055 | 0.020 | 0.382 |

17 | 0.056 | 0.011 | 0.546 | 0.065 | 0.010 | 0.630 | 0.057 | 0.017 | 0.446 | 0.055 | 0.019 | 0.419 |

22 | 0.055 | 0.011 | 0.580 | 0.064 | 0.009 | 0.667 | 0.056 | 0.016 | 0.468 | 0.055 | 0.018 | 0.443 |

30 | 0.054 | 0.010 | 0.613 | 0.062 | 0.008 | 0.700 | 0.053 | 0.016 | 0.488 | 0.052 | 0.018 | 0.463 |

**Table 6.**The optimal values obtained by assumed particle size fraction, collector type, and its dosage.

Assumed Values | Optimal Values | |||||
---|---|---|---|---|---|---|

Particle Size d (µm) | Type of Collector k | Dosage of Collector s (g/t) | t (min) | β | ϑ | ε |

0–20 | E | 100 | 17 | 0.108 | 0.005 | 0.832 |

0–20 | E | 150 | 17 | 0.081 | 0.004 | 0.839 |

0–20 | I | 100 | 22 | 0.119 | 0.005 | 0.882 |

0–20 | I | 150 | 22 | 0.103 | 0.005 | 0.870 |

20–40 | E | 100 | 3 | 0.071 | 0.001 | 0.677 |

20–40 | E | 150 | 30 | 0.071 | 0.013 | 0.610 |

20–40 | I | 100 | 12 | 0.063 | 0.010 | 0.589 |

20–40 | I | 150 | 12 | 0.010 | 0.009 | 0.706 |

40–71 | E | 100 | 22 | 0.055 | 0.011 | 0.580 |

40–71 | E | 150 | 30 | 0.062 | 0.008 | 0.700 |

40–71 | I | 100 | 17 | 0.057 | 0.017 | 0.446 |

40–71 | I | 150 | 22 | 0.055 | 0.019 | 0.473 |

Assumed Values | Optimal Values | |||||
---|---|---|---|---|---|---|

Particle Size d (µm) | Type of Collector k | Dosage of Collector s (g/T) | t (min) | β | ϑ | ε |

0–20 | E | 100 | 22 | 0.104 | 0.005 | 0.858 |

0–20 | I | 150 | 22 | 0.119 | 0.005 | 0.882 |

20–40 | E | 100 | 30 | 0.071 | 0.011 | 0.677 |

20–40 | I | 150 | 12 | 0.105 | 0.009 | 0.709 |

40–71 | E | 100 | 22 | 0.064 | 0.009 | 0.667 |

40–71 | E | 150 | 17 | 0.057 | 0.017 | 0.446 |

Assumed Values | Optimal Values | |||||
---|---|---|---|---|---|---|

Particle Size d (µm) | Type of Collector k | Dosage of Collector s (g/T) | t (min) | β | ϑ | ε |

0–20 | I | 100 | 22 | 0.119 | 0.005 | 0.882 |

20–40 | I | 150 | 12 | 0.105 | 0.009 | 0.706 |

40–71 | E | 150 | 22 | 0.064 | 0.009 | 0.667 |

Assumed Values | Optimal Values | |||||
---|---|---|---|---|---|---|

t (min) | Particle Size d (µm) | Type of Collector k | Dosage of Collector s (g/t) | β | ϑ | ε |

1 | 20–40 | I | 150 | 0.167 | 0.021 | 0.137 |

2 | 0–20 | E | 150 | 0.121 | 0.013 | 0.402 |

4 | 0–20 | I | 100 | 0.175 | 0.013 | 0.613 |

6 | 0–20 | I | 100 | 0.156 | 0.009 | 0.737 |

9 | 0–20 | I | 100 | 0.142 | 0.007 | 0.833 |

12 | 0–20 | I | 100 | 0.123 | 0.006 | 0.856 |

22 | 0–20 | I | 100 | 0.119 | 0.005 | 0.882 |

30 | 0–20 | I | 100 | 0.112 | 0.004 | 0.895 |

**Table 10.**The optimal values by values of adjustable variables presented in Table 2.

Particle Size d (µm) | Type of Collector k | Collector Dosage s (g/T) | t (min) | β | ϑ | ε | F_{opt} |
---|---|---|---|---|---|---|---|

0–20 | I | 100 | 22 | 0.119 | 0.005 | 0.882 | 0.376 |

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**MDPI and ACS Style**

Niedoba, T.; Pięta, P.; Surowiak, A.; Şahbaz, O.
Multidimensional Optimization of the Copper Flotation in a Jameson Cell by Means of Taxonomic Methods. *Minerals* **2021**, *11*, 385.
https://doi.org/10.3390/min11040385

**AMA Style**

Niedoba T, Pięta P, Surowiak A, Şahbaz O.
Multidimensional Optimization of the Copper Flotation in a Jameson Cell by Means of Taxonomic Methods. *Minerals*. 2021; 11(4):385.
https://doi.org/10.3390/min11040385

**Chicago/Turabian Style**

Niedoba, Tomasz, Paulina Pięta, Agnieszka Surowiak, and Oktay Şahbaz.
2021. "Multidimensional Optimization of the Copper Flotation in a Jameson Cell by Means of Taxonomic Methods" *Minerals* 11, no. 4: 385.
https://doi.org/10.3390/min11040385