# Describing Mining Tailing Flocculation in Seawater by Population Balance Models: Effect of Mixing Intensity

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## Abstract

**:**

## 1. Introduction

## 2. Model

#### 2.1. Aggregation Kernel

#### 2.2. Breakage Kernel

#### 2.3. Shear Rate

#### 2.4. PBM Solution

## 3. Materials and Methods

#### 3.1. Materials

^{3}acquired from Donde Capo (Chilean Store, Santiago, Chile) were used, which were pulverised and screened with -#325 mesh. The SiO

_{2}content, estimated by XRD analysis, was higher than 99 wt%. Kaolin particles were acquired from Ward’s Science (Rochester, NY 14586, Rochester, United States), and the XRD analysis indicated 84% kaolinite and 16% halloysite. Seawater from the coast of Antofagasta (Chile) was provided by the Department of Marine Resources from the University of Antofagasta and subsequently filtered by UV for the removal of microorganisms. The ionic composition was determined with varied chemical methods depending on the ion type. By atomic absorption spectrometry: Na

^{+}10.5 g/L, Mg

^{2+}1.42 g/L, Ca

^{2+}0.38 g/L, K

^{+}0.40 g/L. The composition of Cl

^{−}was determined by argentometric method and was 19.1 g/L. The concentration of HCO

^{3−}was determined by acid–base volumetry to be 0.17 mg/L. NaOH was used as a pH modifier; anionic SNF704 and high molecular weight were used as a flocculant.

#### 3.2. Flocculation Kinetics

#### 3.3. Viscosity

^{−1}.

#### 3.4. Sedimentation

^{3}(35 mm internal diameter) and then slowly inverted twice by hand (the whole cylinder rotation process took 4 s). After 10 min of settling, the supernatant fluid was rescued and stirred to homogenize the suspended solids. Then, a 50 mL aliquot was used for turbidity measurements in a Hanna HI98713 turbidimeter, which performed ten readings in 20 s, delivering the average of the point readings.

#### 3.5. Conditions

## 4. Results

#### 4.1. Initial Particle Size

#### 4.2. Fractal Dimension

#### 4.3. Flocculation Kinetics Modelling

^{−1}. The PBM captures all the complex stages of particle flocculation, that is, initial growth of aggregates by particle bridging and subsequent size reduction as a result of fragmentation.

^{2}summarized in Table 2 show that the quality of the PBM is very good in all the cases analyzed and that the effect of representing the fractal dimension of the aggregates as a constant independent of the flocculation time is a correct decision, at least to represent the flocculation kinetics.

#### 4.4. Aggregation, Breakage, and Permeability Modelling

^{−1}) that leads to the larger aggregates (225 µm) the results of Figure 4a,b for collision frequency and collision efficiency, respectively, and the results of Figure 4c for aggregate permeability, suggest that there are no large differences if one or the other fractal dimension is used.

^{−1}), the results of Figure 4c suggest that the fragmentation rate is better represented by the variable fractal dimension, the use of a constant fractal dimension can lead to significant deviations.

#### 4.5. Optimized Parameters

^{−1}), the three parameters of the collision efficiency can be estimated considering a constant fractal dimension. Finally, the two parameters of the breakage rate (${s}_{1}$, ${s}_{2}$) change erratically with the shear rate but are not affected by the choice of the fractal dimension, whether constant or variable. For the optimal mixing conditions, the breakage rate parameters can be determined assuming a constant fractal dimension.

#### 4.6. Prediction Capability

^{–1}.

## 5. Conclusions

^{–1}, the fractal dimension of the aggregates decreases monotonously over time, indicating that fragmentation of aggregates leads to lower-dimension Euclidean structures. This study revealed that compensations between aggregation and breakage rates lead to a very good representation of the flocculation kinetics of the particle tailings in seawater, and in addition, that the representation of flocculation kinetics under optimal conditions is equally good with a constant or variable fractal dimension. The mixing intensity of 163 s

^{–1}was found to maximize the size of the tailing aggregates, that is, ca. 225 µm at ca. 20 s. The aggregation and breakage functions and corresponding parameters are sensitive to the choice of the fractal dimension of the aggregates, whether constant or time-dependent; however, under optimal conditions, a constant average of the fractal dimension is sufficient. The predictive capacity of the model can be used to find the optimal flocculation conditions based on a few experimental flocculation data at different mixing intensities, and the optimal flocculation time can be used to make decisions regarding the effectiveness of the flocculant used.

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## References

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**Figure 1.**Normalized initial volume distribution of particles of synthetic tailings in seawater at pH 8 for different mixing intensities (shear rate G).

**Figure 2.**Temporal evolution of the fractal dimension during flocculation of synthetic tailing particles in seawater at pH 8 as a function of shear stress (G).

**Figure 3.**Flocculation kinetics of synthetic tailing particles in seawater at pH 8 as a function of shear stress (G). Open circles correspond to experimental data and solid lines to the best fit with the population balance model (PBM). (

**A**): Shear rate range from 89–251 s

^{−1}. (

**B**): Shear rate range from 300–462 s

^{−1}.

**Figure 4.**PBM representation of collision frequency (

**a**), collision efficiency (

**b**), aggregate permeability (

**c**) and fragmentation rate (

**d**) for the largest aggregates considering constant fractal dimension of the aggregates (dashed lines) and time-dependent fractal dimension (solid lines) for different mixing intensities. Aggregate size range is 200–600 µm (bin i = 15).

**Figure 5.**Experimental (symbols) vs. calculated (continuous lines) time evolution of the mean diameter of aggregates of tailing particles as a function of pH in seawater. Constant pH condition at 8.

**Figure 6.**Optimum aggregation parameters vs. shear rate for constant and variable fractal dimension, (

**a**) maximum and minimum collision efficiencies and (

**b**) collision efficiency decay constant ${k}_{d}$.

**Figure 7.**Optimum breakage parameters vs. shear rate for constant and variable fractal dimension, (

**a**) ${s}_{1}$ and (

**b**) ${s}_{2}$.

${\mathit{i}}_{\mathit{m}\mathit{a}\mathit{x}}$ | $30$ | $-$ |
---|---|---|

$\varphi $ | $0.054$ | - |

$c$ | $0.65$ | - |

${N}_{p}$ | $0.6$ | - |

$D$ | $8.0$ | cm |

$V$ | $0.25$ | $\mathrm{L}$ |

${\rho}_{s}$ | $2600$ | kg/m^{3} |

${\rho}_{w}$ | $1000$ | kg/m^{3} |

${\mu}_{sus}$ | $0.005$ | $\mathrm{kg}/\mathrm{ms}$ |

$w$ | $0.08$ | - |

${d}_{0}$ | $0.0005$ | cm |

**Table 2.**Quantitative results of GoF and R

^{2}when the PBM is used with constant fractal dimension (${d}_{f}$ mean) or variable (${d}_{f}$ var) fractal dimension dependent on flocculation time.

Mixing | GoF, % | R^{2} | ||
---|---|---|---|---|

Shear Rate (s^{−1}), Mixing Rate | ${\mathit{d}}_{\mathit{f}}\mathbf{var}$ | ${\mathit{d}}_{\mathit{f}}\mathbf{mean}$ | ${\mathit{d}}_{\mathit{f}}\mathbf{var}$ | ${\mathit{d}}_{\mathit{f}}\mathbf{mean}$ |

89 (100 rpm) | 90.1 | 89.2 | 0.8895 | 0.8698 |

131 (130 rpm) | 92.8 | 93.6 | 0.9449 | 0.9565 |

163 (150 rpm) | 91.6 | 91.6 | 0.9177 | 0.9174 |

214 (180 rpm) | 90.6 | 90.2 | 0.8963 | 0.8881 |

251 (200 rpm) | 91.6 | 91.3 | 0.8956 | 0.8889 |

300 (225 rpm) | 91.1 | 90.4 | 0.9267 | 0.9151 |

330 (240 rpm) | 89.5 | 88.2 | 0.8898 | 0.8611 |

372 (260 rpm) | 92.5 | 92.3 | 0.9055 | 0.9024 |

462 (300 rpm) | 93.5 | 94.5 | 0.9038 | 0.9319 |

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## Share and Cite

**MDPI and ACS Style**

Quezada, G.R.; Ayala, L.; Leiva, W.H.; Toro, N.; Toledo, P.G.; Robles, P.; I. Jeldres, R.
Describing Mining Tailing Flocculation in Seawater by Population Balance Models: Effect of Mixing Intensity. *Metals* **2020**, *10*, 240.
https://doi.org/10.3390/met10020240

**AMA Style**

Quezada GR, Ayala L, Leiva WH, Toro N, Toledo PG, Robles P, I. Jeldres R.
Describing Mining Tailing Flocculation in Seawater by Population Balance Models: Effect of Mixing Intensity. *Metals*. 2020; 10(2):240.
https://doi.org/10.3390/met10020240

**Chicago/Turabian Style**

Quezada, Gonzalo R., Luís Ayala, Williams H. Leiva, Norman Toro, Pedro G. Toledo, Pedro Robles, and Ricardo I. Jeldres.
2020. "Describing Mining Tailing Flocculation in Seawater by Population Balance Models: Effect of Mixing Intensity" *Metals* 10, no. 2: 240.
https://doi.org/10.3390/met10020240