# A Stochastic Model Approach for Copper Heap Leaching through Bayesian Networks

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## Abstract

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## 1. Introduction

_{2}), which together with NO

_{x}and CO

_{2}, can cause major problems, such as acid rain and increased local pollution [5]. So, hydrometallurgy is a good alternative to process both oxidized minerals and sulfurized minerals, since it is more environmentally friendly [5,7,8].

^{2}) [9]. Finally, cupric ions (Cu

^{2+}) are obtained together with other ions of elements dissolved in the pregnant liquid solution (PLS), which ions all advance to the subsequent stage of solvent extraction [12].

## 2. Materials and Methods

#### 2.1. Machine Learning

#### 2.2. Data-Based Modeling in Mineral Processing

#### 2.3. Bayesian Networks

#### 2.4. Uncertainty Analysis

## 3. Results and Discussion

#### 3.1. Analysis of Uncertainty

#### 3.2. Bayesian Network Modeling

#### 3.3. Bayesian Network Validation

## 4. Conclusions and Future Works

#### 4.1. Conclusions

- Identifying the dependency relationships between independent variables and the response variable, in addition to dependency relationships between independent variables.
- Determining the variables that contribute most to explain the variability of the response.
- Assimilating quantitative knowledge in terms of the frequency of the occurrence of a given event (or level of recovery), using the parameters obtained by the BN, which will allow the identification of recurrent scenarios.
- The generation of copper recovery estimates based on partial knowledge of the operational variables considered in the study.

#### 4.2. Future Works

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## References

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**Figure 1.**Uncertainly analysis at 30 days of leaching, copper recovery distribution (

**a**) and normal probability plot (

**b**).

**Figure 2.**Uncertainly analysis at 60 days of leaching, copper recovery distribution (

**a**) and normal probability plot (

**b**).

**Figure 3.**Uncertainly analysis at 90 days of leaching, copper recovery distribution (

**a**) and normal probability plot (

**b**).

**Figure 6.**Distributions of independent variables high pile (

**a**), particle size (

**b**), surface velocity of the leaching flow through the bed (

**c**), effective diffusivity of the solute within the pores of the particle (

**d**) y porosity of the particle (

**e**).

**Figure 7.**Copper recovery based on leaching time (

**a**), particle size (

**b**), pile high (

**c**), superficial velocity of lixiviant flow (

**d**), effective diffusivity of the solute within particle pores (

**e**) and porosity of particle (

**f**).

**Figure 8.**Operational fit curve versus Bayesian networks outputs (

**a**) and Normal Probability Graph $E\left[Y\left(t\right)|{X}_{n}\right]$ (

**b**).

**Figure 9.**Normal Probability Graph $E\left[Y\left(t\right)|{X}_{n-1}\bigwedge E\left[x\right]\right]$, x: particle porosity (

**a**), velocity of lixiviant flow (

**b**), particle size (

**c**), effective diffusivity of solute within particle pores (

**d**) and porosity of particle (

**e**).

Variable/Value | Minimum | Average | Maximum |
---|---|---|---|

Leaching time (days) | 30 | - | 90 |

Pile height (cm) | 300 | 600 | 900 |

Particle size (mm) | 14 | 20 | 34 |

Surface velocity of the leaching flow through the bed ($c{m}^{3}/c{m}^{2}\xb7d$) | 10 | 30 | 50 |

Effective diffusivity of the solute within the pores of the particle ($c{m}^{3}/cm\xb7d$) | 0.05 | 0.10 | 0.15 |

Porosity of the particle (%) | 1.0 | 3.5 | 6.0 |

Model/Statistic | MAD | MSE | MAPE |
---|---|---|---|

BN | $1.32\times {10}^{-3}$ | $2.94\times {10}^{-6}$ | $2.49\times {10}^{-4}$ |

Model/Statistic | MAD | MSE | MAPE |
---|---|---|---|

$\mathrm{E}\left[\mathrm{Y}\left(\mathrm{t}\right)|{\mathrm{X}}_{\mathrm{n}-1}\bigwedge \mathrm{E}\left[\mathrm{x}:\mathrm{Pile}\text{}\mathrm{high}\right]\right]$ | $0.237$ | 0.073 | $0.234$ |

$\mathrm{E}\left[\mathrm{Y}\left(\mathrm{t}\right)|{\mathrm{X}}_{\mathrm{n}-1}\bigwedge \mathrm{E}\left[\mathrm{x}:\mathrm{Velocity}\text{}\mathrm{of}\text{}\mathrm{lixiviant}\text{}\mathrm{flow}\right]\right]$ | $0.241$ | 0.092 | $0.252$ |

$\mathrm{E}\left[\mathrm{Y}\left(\mathrm{t}\right)|{\mathrm{X}}_{\mathrm{n}-1}\bigwedge \mathrm{E}\left[\mathrm{x}:\mathrm{particle}\text{}\mathrm{size}\right]\right]$ | $1.32\times {10}^{-3}$ | $8.64\times {10}^{-3}$ | $0.109$ |

$\mathrm{E}\left[\mathrm{Y}\left(\mathrm{t}\right)|{\mathrm{X}}_{\mathrm{n}-1}\bigwedge \mathrm{E}\left[\mathrm{x}:\mathrm{effective}\text{}\mathrm{diffusivity}\right]\right]$ | $3.92\times {10}^{-2}$ | $9.85\times {10}^{-4}$ | $4.19\times {10}^{-3}$ |

$\mathrm{E}\left[\mathrm{Y}\left(\mathrm{t}\right)|{\mathrm{X}}_{\mathrm{n}-1}\bigwedge \mathrm{E}\left[\mathrm{x}:\mathrm{porosity}\right]\right]$ | $4.31\times {10}^{-2}$ | $3.78\times {10}^{-5}$ | $2.93\times {10}^{-3}$ |

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**MDPI and ACS Style**

Saldaña, M.; González, J.; Jeldres, R.I.; Villegas, Á.; Castillo, J.; Quezada, G.; Toro, N.
A Stochastic Model Approach for Copper Heap Leaching through Bayesian Networks. *Metals* **2019**, *9*, 1198.
https://doi.org/10.3390/met9111198

**AMA Style**

Saldaña M, González J, Jeldres RI, Villegas Á, Castillo J, Quezada G, Toro N.
A Stochastic Model Approach for Copper Heap Leaching through Bayesian Networks. *Metals*. 2019; 9(11):1198.
https://doi.org/10.3390/met9111198

**Chicago/Turabian Style**

Saldaña, Manuel, Javier González, Ricardo I. Jeldres, Ángelo Villegas, Jonathan Castillo, Gonzalo Quezada, and Norman Toro.
2019. "A Stochastic Model Approach for Copper Heap Leaching through Bayesian Networks" *Metals* 9, no. 11: 1198.
https://doi.org/10.3390/met9111198