# A Posteriori Analysis of Analytical Models for Heap Leaching Using Uncertainty and Global Sensitivity Analyses

^{1}

^{2}

^{3}

^{4}

^{*}

^{†}

## Abstract

**:**

## 1. Introduction

_{3}) could be represented by four shrinking core models (spherical particles under reaction control, spherical particles under layer diffusion control, cylindrical particles under reaction control and cylindrical particles under layer diffusion control).

## 2. Analytical Models for Heap Leaching

#### 2.1. Mellado Model

^{3}/cm·s), ${\epsilon}_{o}$ is the porosity of the particle, $t$ is the time (s), $r$ the particle radius (cm). At the bulk level, Dixon and Hendrix considered that the heap is represented by the porous bulk formed of particles through which the leaching solution flows at a constant rate. Then, they defined the dimensionless time of the bulk [30,31]

^{3}/cm·s), ${\epsilon}_{b}$ is the volumetric fraction of the bulk solution in the bed, and $Z$ is the height of the heap (cm). Then, Equation (2) can be written for both dimensionless times as [27,28]:

#### 2.2. Uncertainty Analysis

#### 2.3. Sensitivity Analysis

#### 2.4. Regionalization of Input Variables

## 3. Results and Discussion

#### 3.1. Uncertainty Analysis

^{3}/cm·d, 0.03, 2.5 cm, 28 cm

^{3}/cm

^{2}·d and 600 cm, respectively.

#### 3.2. Global Sensitivity Analysis

^{3}/cm

^{2}·d and 57.0 cm

^{3}/cm

^{2}·d. The fifth case considered an effective diffusivity 0.06 cm

^{3}/cm

^{2}·d and 0.112 cm

^{3}/cm

^{2}·d. The sixth case considered a porosity of the particle of 0.01 and 0.06.

^{3}/cm

^{2}·s. Figure 3, Figure 11 and Figure 12 show the $S{T}_{i}$ index when the ${D}_{Ae}$ is 0.0864, 0.06 and 0.112 cm

^{3}/cm

^{2}·d. Figure 3, Figure 13 and Figure 14 show the $S{T}_{i}$ index when the ${\epsilon}_{o}$ is 0.03, 0.01 and 0.06. The effect of the nominal value and the uncertainty is important in the results observed.

#### 3.3. Regionalization

## 4. Conclusions

## Acknowledgments

## Author Contributions

## Conflicts of Interest

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**Figure 15.**Regionalization of the influential input variables $Z$, ${u}_{s}$ and ${\u03f5}_{b}$, $B$ (

**left**) and $\overline{B}$ (

**right**).

**Figure 16.**Plane $Z=300\mathrm{cm}$ including both groups $B$ (desired outcome, recovery >73.5%) and $\overline{B}$ (undesired outcome, recovery <73.5%)

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**MDPI and ACS Style**

Mellado, M.E.; Cisternas, L.A.; Lucay, F.A.; Gálvez, E.D.; Sepúlveda, F.D.
A Posteriori Analysis of Analytical Models for Heap Leaching Using Uncertainty and Global Sensitivity Analyses. *Minerals* **2018**, *8*, 44.
https://doi.org/10.3390/min8020044

**AMA Style**

Mellado ME, Cisternas LA, Lucay FA, Gálvez ED, Sepúlveda FD.
A Posteriori Analysis of Analytical Models for Heap Leaching Using Uncertainty and Global Sensitivity Analyses. *Minerals*. 2018; 8(2):44.
https://doi.org/10.3390/min8020044

**Chicago/Turabian Style**

Mellado, Mario E., Luis A. Cisternas, Freddy A. Lucay, Edelmira D. Gálvez, and Felipe D. Sepúlveda.
2018. "A Posteriori Analysis of Analytical Models for Heap Leaching Using Uncertainty and Global Sensitivity Analyses" *Minerals* 8, no. 2: 44.
https://doi.org/10.3390/min8020044