# Copper Mineral Leaching Mathematical Models—A Review

^{1}

^{2}

^{3}

^{4}

^{5}

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## Abstract

**:**

## 1. Introduction

_{2}emissions in the atmosphere, causing serious environmental problems [6]. However, it is hoped that the implications of the technological revolution in mining will contribute to mitigating the negative effects of mining on the environment in which it operates [7].

## 2. Leaching Process

#### 2.1. Overview and Industrial Applications

_{2}SO

_{4}), known as the curing process, aiming to begin the copper sulfation process in the oxidized minerals or sulfated minerals (cured with mixed sulfuric acid and chlorides solutions [44,45]). The mineral is discharged by means of a spreader machine, depositing it in a very organized manner and forming a continuous embankment from 6 to 8 m high: The leaching heap. Above this heap, a drip irrigation system is installed, and sprinklers cover all the exposed area. Under the heaps of materials to be leached, a waterproof membrane is installed in order to provide a system of drains (grooved pipes) that allow to collect the Pregnant Leach Solution (PLS) that can infiltrate through the heap [46].

#### 2.2. Bibliometric Analysis

## 3. Mineral Leaching Modeling

_{e}). It is supposed that B homogeneously diffuses all over the particle or mineral cluster, this way all the solid zone or stalled can be modeled as just one reagent. The reactive, as well as the reagent, concentrations, C

_{A}and C

_{B}, possess units of molar quantity per mineral volume unit.

#### 3.1. Generic Mineral Leaching Modeling

#### 3.2. Copper Leaching Modeling

_{2}SO

_{4}as leaching agent. The mathematical model consists of a system of differential equations: two diffusion–convection–reaction equations with Neumann boundary conditions, and an ordinary differential equation (see Equation (62)) where ${u}_{1}$, ${u}_{2}$ and ${u}_{3}$ are the H

_{2}SO

_{4}concentration and copper concentration in liquid and solid phase, respectively, D is the diffusion–dispersion tensor and the vector V is the fluid flow velocity. The system is complemented with non-homogeneous flow contour conditions, which correspond to the physical behavior of the irrigation and infiltration processes in leaching piles. The system of heap leach transport equations used is very similar to that of Cariaga et al. [99] which are derived from the compositional flow model considered by Kacur and Van Keer [100]. The results of the model generated by Cariaga et al. [98] show that the model satisfactorily predicts that main trends exhibited by the phenomenon studied, i.e. the time evolution of acid and copper concentration in the liquid solution extracted from the tailings.

_{2}SO

_{4}on copper extraction from chalcopyrite (CuFeS

_{2}). Similarly, and previously, Liu et al. [108] optimized copper leaching from a low-grade flotation middling through RSM, studying the effect of key parameters, i.e., sulfuric acid concentration, nitric acid concentration and leaching time, on the leaching efficiency.

_{2}SO

_{4}and chloride concentration) on the leaching of pure chalcocite to extract copper, fitting a quadratic model that allows to predict extraction. Saldaña et al. [123] develop an experimental design both to evaluate the impact of dependent variables on the response, and to generate analytical models (through multiple regressions) that represent the copper and manganese extractions. Pérez et al. [20] applied the surface optimization methodology using a central composite face design to evaluate the effect of leaching time, chloride concentration and sulfuric acid concentration on the level of copper extraction from covellite. The ANOVA developed by Pérez et al. [20] indicated that leaching time and chloride concentration have the most significant influence, while copper extraction was independent of sulfuric acid concentration. The experimental data was described using a quadratic model.

_{2}SO

_{4}concentration, leaching temperature and leaching time, on leaching efficiency are examined. In Sabzezari et al. [126], the RSM and CCD were employed to study the effect of leaching parameters (acid concentration, pulp density, oxidant concentration, microwave power and leaching time) on copper and zinc dissolution. While in Quezada et al. [127], non-linear regression was modeled to represent the dissolution of black copper oxides from residue leaching, as a function of Eh and time.

_{p}represent the reagent concentration, solid-to-liquid ratio, stirring rate and particle size, respectively. Trinh et al. [131], model the selective recovery of copper by acid leaching from waste sludge. Ambo et al. [132] model the selective leaching of copper from preconcentrated copper ores based on near-infrared sensors, revealing that the rate of leaching increases with increasing ammonium chloride concentration, temperature, decreasing particle size of the ore, the speed of agitation and the solid-liquid ratio. Lee et al. [133] use the SCM model to study the effect of mechanical activation on copper leaching from copper sulfide, CuS, by analyzing the leachability and apparent activation energy. Shi et al. [134], study the kinetics of copper extraction from foundry slag by pressure oxidative leaching with sulfuric acid, adjusting a kinetic equation of leaching. Zhang et al. [135], applied the SCM model to study the leaching behavior of copper and iron recovery from reduction roasting pyrite cinder. It was shown that the leaching process was controlled by mixed diffusion and chemical reaction, which indicated that the leaching rate was controlled by the lixiviant diffusion and surface reaction simultaneously, while that residues characterization indicated that free copper oxide, combined copper oxide and secondary copper sulfide almost completely dissolved in the H

_{2}SO

_{4}solution; however, chalcopyrite only partially dissolved due to the difficulty for H

_{2}SO

_{4}to leach copper (in the form of primary sulfides) at atmospheric pressure. Finally, Apua and Madiba [136] carry out an experimental investigation on the study of the leaching kinetics of copper oxide minerals, investigating the effect of time, pH, stirring speed and temperature on the extent of dissolution, fitting a potential function that explains the recovery of copper (and other metals) over time.

## 4. Conclusions and Future Perspectives

_{2}SO

_{4}+ chlorides. The mineral leaching process was developed by many authors, modeling the process mainly applied on an industrial scale of metallic mining of copper. Nevertheless, the trend in leaching processes points to the leaching of copper sulfide minerals (mainly using industrial applications) and to leaching processes that are environmentally friendly. These include the efficient use of water resources, an issue of special interest when considering novel paradigms such as smart industry, circular economy or green economy and the impact of production processes on the carbon footprint.

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

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Metallic Mineral | Publications |
---|---|

Copper, Gold, Silver, Uranium | Padilla et al. [29] |

Zinc | Qin et al. [30,31]; Petersen and Dixon [32] |

Nickel | McDonald and Whittington [33]; Oxley et al. [34]; Khalezov et al. [35] |

Platinum | Mwase et al. [36,37,38]; Schoeman et al. [39] |

Manganese | Krebs and Milligan [40]; Baumgartner and Groot [41] |

**Table 2.**Kinetics models suggested for the leaching process (X = fraction reacted, k = kinetic constant).

Model | Mechanism | Equation | Reference |
---|---|---|---|

$k=1-{\left(1-X\right)}^{1/3}$ | Chemical reaction control | (37) | [59] |

$k=1-\frac{2}{2}X-{\left(1-X\right)}^{2/3}$ | Diffusion control | (38) | [59] |

$k=1-{\left(1-0.45X\right)}^{1/3}$ | Surface chemical reaction by shrinking core model | (39) | [62] |

$k={\left[1-{\left(1-X\right)}^{1/3}\right]}^{2}$ | Diffusion through product layer | (40) | [63] |

$k=1-\frac{2}{2}X-{\left(1-X\right)}^{1/3}$ | Diffusion through a porous product layer by shrinking core model | (41) | [64] |

$k=\frac{1}{3}ln\left(1-X\right)+{\left(1-X\right)}^{1/3}-1$ | Interfacial transfer and diffusion across the product layer | (42) | [65] |

$k=1-3{\left(1-X\right)}^{2/3}+2\left(1-X\right)$ | Diffusion of hydrogen ions through a product layer by shrinking core model | (43) | [66] |

$k=1-{\left(1-X\right)}^{2/3}$ | Mixed control model by shrinking core model (diffusion control; chemical reaction control) | (44) | [67] |

$k=-ln\left(1-X\right)$ | Mixed control model (surface reaction control; and diffusion through sulfur layer) | (45) | [68] |

$k=\frac{1}{5}{\left(1-X\right)}^{-5/3}-\frac{1}{4}{\left(1-X\right)}^{-4/3}+\frac{1}{20}$ | Mixed control model based on reactant concentrations | (46) | [69] |

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**MDPI and ACS Style**

Saldaña, M.; Gálvez, E.; Robles, P.; Castillo, J.; Toro, N.
Copper Mineral Leaching Mathematical Models—A Review. *Materials* **2022**, *15*, 1757.
https://doi.org/10.3390/ma15051757

**AMA Style**

Saldaña M, Gálvez E, Robles P, Castillo J, Toro N.
Copper Mineral Leaching Mathematical Models—A Review. *Materials*. 2022; 15(5):1757.
https://doi.org/10.3390/ma15051757

**Chicago/Turabian Style**

Saldaña, Manuel, Edelmira Gálvez, Pedro Robles, Jonathan Castillo, and Norman Toro.
2022. "Copper Mineral Leaching Mathematical Models—A Review" *Materials* 15, no. 5: 1757.
https://doi.org/10.3390/ma15051757