# Trends in Modeling, Design, and Optimization of Multiphase Systems in Minerals Processing

^{1}

^{2}

^{*}

## Abstract

**:**

## 1. Introduction

^{−13}m with time scales of 10

^{−16}s, are of significant interest in understanding the interaction of minerals with reagents. Large scales, e.g., plants length scales of 10

^{3}m with time scales of 10

^{6}s, are important in terms of plant integration and environmental impact.

## 2. Molecular Dynamic Modeling

#### 2.1. Collector/Depressor Adsorption on Different Mineral Surfaces in the Flotation Process

#### 2.2. Interaction of Clay Minerals, Water, and Interlayer Structures

_{2}emissions and energy consumption. For this reason, understanding cement hydration mechanisms was the main motivation of this study to provide an academic basis for the design of new environmentally friendly cement. Finally, Kubicki et al. [13] studied the vibrational spectra on clays by DFT approaches. Herein, they presented an overview of quantum mechanical calculations to predict vibrational frequencies of molecules and materials such as clays and silicates. For creating a realistic model, the vibrational frequencies were calculated by two analytical methods, Raman and infrared intensities.

## 3. Computational Fluid Dynamics (CFD) in Multiphase Systems

## 4. Design and Optimization

## 5. Artificial Intelligence (AI) Applied to Multiphase Systems

## 6. Response Surface Methodology (RSM)

^{2}= 0.9989) was obtained, which indicates a good agreement with experimental values. Similar good results were observed using a Box–Bhenken design in copper sulphide ore grinding in a ball mill [150]. However, several processes do not follow a second-order polynomial behavior and, consequently, a poor adjustment of the model is obtained (see Figure 10). The immediate consequence is incorrect optimization. The related literature proposes different approaches in the modeling of surface response instead of polynomial models. For example, regression of Gaussian processes has been proposed, since these models can model complex functions [141,154]. Also, the use of SVM regression as a prediction model has been proposed [155]. However, the most popular alternative has been ANNs [156].

## 7. Uncertainty and Sensitivity Analyses

## 8. Discussion and Conclusions

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

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**Figure 1.**Levels of length and time alongside the modeling and optimization tools analyzed in this manuscript (CFD—computational fluid dynamics; RSM—response surface methodology; AI—artificial intelligence; GSA—global sensitivity analysis).

**Figure 3.**Example of an equilibrium snapshot from a molecular dynamics (MD) simulation of water adsorption on the side surfaces of (

**a**) kaolinite and (

**b**) montmorillonite at 298 K and 1 bar [11].

**Figure 4.**Examples of computational fluid dynamics (CFD) multiphase modeling in mineral processing. (

**a**) CFD-predicted net attachment rates after flotation time in the stirred cell [31]; (

**b**) Bubble volume fraction (unit in vol %) distribution in a pipe for a backfill material [32]; (

**c**) Predicted contours of (

**c1**) pressure and (

**c2**) tangential velocities in Renner’s cyclone [33].

**Figure 5.**Example of the physical interactions, the model components, and the data flows in discrete element simulation–smoothed particle hydrodynamics (DEM–SPH) modeling of wet grinding [53].

**Figure 6.**Relationship between scales in bubble behavior in flotation (

**a**) macro-scale: cell level, (

**b**) meso-scale: bubble coalescence and breakup studies, and (

**c**) micro/nano-scale: molecular interactions [57].

**Figure 7.**Linear circuit analysis (LCA) formulation example, including uncertainty analysis: (

**a**) individual unit, (

**b**) two-stage circuit [71].

**Figure 9.**Comparison of flotation circuit design: (

**a**) design and ash contents of the initial circuit, (

**b**) design and ash contents of the new circuit, adapted from Pirouzan et al. [95].

**Figure 10.**(

**a**) Quartz recovery using a second-order polynomial as a prediction model (R

^{2}= 0.931), (

**b**) quartz recovery using an artificial neural network as a prediction model (R

^{2}= 0.982) [156].

**Figure 11.**(

**a**) Bubble–particle collision probability, P

_{c}, versus bubble diameter, (

**b**) prediction absolute error versus number of interpolation points [158].

**Figure 12.**(

**a**) The comminution-specific energy histogram of a SAG (semi-autogenous grinding) mill under three uncertainty magnitudes. (

**b**) The regionalization of fresh ore flux fed (F), percentage of mill volume occupied by steel balls (J

_{b}), and percentage of critical speed (ϕ

_{c}) [162].

**Figure 13.**(

**a**) Designed circuit using the methodology. (

**b**) Sobol total index for each stage and for chalcopyrite (Cp), chalcopyrite–pyrite (CpPy), pyrite–arsenopyrite, and silica (Sc) [169].

**Table 1.**Flotation circuit design methodologies (adapted from Reference [84]) (LP linear programming; NLP nonlinear programming; MILP, mixed-integer linear programming; MINLP, mixed-integer nonlinear programming).

Reference | Model Type | Cell or Bank Model | Entrainment Model | Froth Recovery Model | Algorithm Used | Maximum Number of Species | Maximum Number of Cell or Bank |
---|---|---|---|---|---|---|---|

Mehrotra and Kapur [85] | NLP | Bank | no | no | Mathematical programming | 3 | 4 |

Reuter et al. [86] | LP | Bank | no | no | Mathematical programming | 3 | 4 |

Reuter and Van Deventer [87] | LP | Bank | no | no | Mathematical programming | 3 | 5 |

Schena et al. [88] | MINLP | Bank | no | no | Mathematical programming | 2 | 4 |

Schena et al. [83] | MINLP | Bank | no | no | Mathematical programming | 2 | 6 |

Guria et al. [89] | NLP | Cell | no | no | Genetic Algorithm | 3 | 4 |

Guria et al. [90] | NLP | Cell | no | no | Genetic Algorithm | 2 | 2 |

Cisternas et al. [81] | MINLP | Bank | no | no | Mathematical programming | 3 | 4 |

Méndez et al. [82] | MINLP | Bank | no | no | Mathematical programming | 3 | 3 |

Ghobadi et al. [91] | MINLP | Bank | yes | no | Genetic Algorithm | 3 | 2 |

Maldonado et al. [92] | NLP | Bank | no | no | Mathematical programming | 2 | 6 |

Hu et al. [93] | MINLP | Cell | yes | yes | Genetic Algorithm | 2 | 8 |

Cisternas et al. [94] | MINLP | Bank | no | no | Mathematical programming | 3 | 5 |

Pirouzan et al. [95] | NLP | Bank | no | no | Genetic Algorithm | 2 | 4 |

Calisaya et al. [96] | MILP MINLP | Bank | no | no | Mathematical programming | 5 | 7 |

Acosta-Flores et al. [84] | MILP MINLP | Bank Cell | no | yes | Mathematical programming | 15 | 3 8 |

Lucay et al. [97] | MINLP | Bank | no | no | Tabu-search | 7 | 5 |

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**MDPI and ACS Style**

Cisternas, L.A.; Lucay, F.A.; Botero, Y.L.
Trends in Modeling, Design, and Optimization of Multiphase Systems in Minerals Processing. *Minerals* **2020**, *10*, 22.
https://doi.org/10.3390/min10010022

**AMA Style**

Cisternas LA, Lucay FA, Botero YL.
Trends in Modeling, Design, and Optimization of Multiphase Systems in Minerals Processing. *Minerals*. 2020; 10(1):22.
https://doi.org/10.3390/min10010022

**Chicago/Turabian Style**

Cisternas, Luis A., Freddy A. Lucay, and Yesica L. Botero.
2020. "Trends in Modeling, Design, and Optimization of Multiphase Systems in Minerals Processing" *Minerals* 10, no. 1: 22.
https://doi.org/10.3390/min10010022