# Improving the Flocculation Performance of Clay-Based Tailings in Seawater: A Population Balance Modelling Approach

^{1}

^{2}

^{3}

^{4}

^{5}

^{6}

^{*}

## Abstract

**:**

^{2}) greater than 0.9 and Goodness of Fit (GoF) greater than 95% in most cases, wherein the fitting parameters allowed for analysing the impact of magnesium and clays on the collision efficiency, collision frequency, and fragmentation rate. The model is predictive with few parameters, and it is potentially a powerful tool for water management optimisation.

## 1. Introduction

^{−1}. Under these conditions, the fractal dimension may be considered to be constant. This finding is essential for the purpose of the present study, which recognises an experimental system that is similar to that worked by Quezada et al. [31].

## 2. Materials and Methods

#### 2.1. Materials

^{+}10.8 g/L, Ca

^{2+}0.42 g/L, Mg

^{2+}1.41 g/L, and K

^{+}0.4 g/L was obtained by atomic absorption spectrometry. The composition of Cl

^{−}was determined using the argentometric method, which was 19.6 g/L, while the HCO

_{3}

^{−}concentration was determined by acid-base volumetry, whose value was 0.15 g/L.

^{6}Da. High purity lime was used to form the magnesium precipitates in the seawater treatment.

#### 2.2. Seawater Treatment

#### 2.3. Flocculation Tests

^{−1}and, therefore, this can be constant over the process. At the same time, a suspension was flocculated for two minutes, to later allow it to settle. This allowed for obtaining the fractal dimension by applying the methodology that was proposed in Heath et al. [20]. The authors used the hindered settling rate from the sedimentation tests and the size of the aggregates from the FBRM probe.

#### 2.4. PBM Modelling

- The first and second terms describe the aggregate formation of size $i$ from smaller aggregates.
- The third and fourth terms describe the aggregation death of size $i$ to higher aggregates.
- The fifth term represents the breakage formation of size $i$ from the rupture of a higher aggregate.
- The sixth term represents the breakage death of size $i$ by creating smaller aggregates.
- The superscript max
_{1}is the maximum number of intervals used to represent the complete aggregate size spectrum; max_{2}corresponds to the largest interval from which the aggregates in the current range are produced.

#### 2.4.1. Aggregation Kernel

#### 2.4.2. Breakage Kernel

#### 2.4.3. Model Expressions

#### 2.4.4. Model Optimisation

## 3. Results

#### 3.1. Fractal Dimension and Initial Volume Distribution

^{2+}10 mg/L), but when 0% is reported, it corresponds to raw seawater (Mg

^{2+}1420 mg/L). The fractal dimension increases by increasing the proportion of treated water, which is directly related to a lower presence of solid magnesium complexes that affect the selectivity of the flocculant [14]. Interestingly, the presence of kaolin also leads to more open and porous aggregates, lowering the fractal dimension. This can be accounted for the affinity that flocculant has toward solid magnesium complexes and the fine nature of kaolin. When both are present, the polymer adsorption is stable on their surfaces, making branched aggregates. When magnesium and/or kaolin diminished, the aggregates are mainly composed by quartz, where the polymer affinity is lower, creating spherical-like structures.

#### 3.2. Flocculation Kinetics

#### 3.3. Optimized PBM Model Parameters

^{2}are higher than 0.95, so the model provides little deviation concerning the experimental results (Table 3).

_{1}and s

_{2}. The higher these parameters, the structure presents a greater willingness to fragment. There is a particular trend for the studied systems, showing that the aggregation kernel dominates the kinetics and, therefore, there is more noise in the patterns of s

_{1}. Even so, it is inferred that the higher the magnesium content, the greater the fragmentation. The result is satisfying, since the structures are considerably smaller; however, the low fractal dimension suggests open and porous structures, which are more prone to fragmentation.

#### 3.4. Physical Parameters Calculations

#### 3.5. Modelling Validation

^{2}are listed in Table 4.

## 4. Conclusions

^{2}values that are greater than 0.9 and GoF greater than 95% in most cases, wherein the analysis of the fitting parameters suggests that, by reducing the magnesium content, there is greater efficiency of particle collision and a greater collision frequency, as a result of the greater radius of gyration of the aggregates. However, fragmentation rates are slightly higher, especially since larger structures are more prone to breakage.

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## References

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**Figure 1.**(

**a**) Fractal dimension of flocculated aggregates and (

**b**) initial volume particle distribution of qz/kao 75/25. Solid concentration 13 wt%, flocculant dose 15 g/t (for fractal dimension), mixing intensity 188 s

^{−1}.

**Figure 2.**Experimental (symbol) vs. calculated (lines) evolution of the aggregates at varied proportions of quartz/kaolin (Qz/Kao) and treated seawater (Tsw). Solid concentration 13 wt%, flocculant dose 15 g/t, mixing intensity 188 s

^{−1}. Quartz/Kaolinite ratio (

**a**) 50/50; (

**b**) 75/25; (

**c**) 100/0.

**Figure 5.**Collision efficiency for the mean chord length over time. Quartz/Kaolinite ratio (

**a**) 50/50; (

**b**) 75/25; (

**c**) 100/0.

**Figure 6.**Collision frequency for the mean chord length over time. Quartz/Kaolinite ratio (

**a**) 50/50; (

**b**) 75/25; (

**c**) 100/0.

**Figure 7.**Breakage rate for the mean chord length over time. Quartz/Kaolinite ratio (

**a**) 50/50; (

**b**) 75/25; (

**c**) 100/0.

**Figure 8.**Permeability of aggregates for the mean chord length over time. Quartz/Kaolinite ratio (

**a**) 50/50; (

**b**) 75/25; (

**c**) 100/0.

**Figure 9.**Experimental (symbol) vs. calculated (lines) flocculation kinetic. Solid concentration 13 wt%, flocculant dose 15 g/t, mixing intensity 188 s

^{−1}. Three conditions are evaluated according Table 4.

Variable | Expression | Input Parameters | Reference |
---|---|---|---|

Collision efficiency | ${\alpha}_{i,j}=\left({\alpha}_{max}-{\alpha}_{min}\right){e}^{-{k}_{d}t}+{\alpha}_{min}$ | Vajihinejad and Soares [30] | |

Collision frequency | ${\beta}_{i,j}=\frac{1}{6}{\left({f}_{i}{d}_{i}+{f}_{j}{d}_{j}\right)}^{3}G$ | Veerapaeni and Wiesner [38] | |

Breakage rate | ${S}_{i}={s}_{1}{G}^{{s}_{2}}{d}_{i}$ | Pandya and Spielman [36] | |

Distribution breakage | ${\Gamma}_{i,j}=\{\begin{array}{c}{V}_{j}/{V}_{i}forj=i+1\\ 0forj\ne i+1\end{array}$ | ||

Dimensionless frequency function | ${f}_{i}=\mathrm{exp}\left(-\frac{{\left(i-1\right)}^{2}}{{K}_{k}{d}_{f}^{4}}\right)$ | This work | |

Initial number concentration | ${N}_{0,i}=\varphi \frac{v\left({d}_{i}\right)}{{V}_{i}}$ | $\varphi =0.054$ | |

Particle diameter | ${d}_{i}={d}_{0}{\left(\frac{{2}^{i-1}}{C}\right)}^{\frac{1}{{d}_{f}}}$ | ${d}_{0}\left({d}_{f}1.7\right)=1\mathsf{\mu}\mathrm{m}$ ${d}_{0}\left({d}_{f}1.7\right)=0.1\mathsf{\mu}\mathrm{m}$ | Mandelbrot [39] |

Particle volume | ${V}_{i}={V}_{0}{2}^{i-1}=\frac{4}{3}\pi {d}_{0}^{3}{2}^{i-1}$ | ||

Shear rate | $G={\left(\frac{{N}_{p}{N}^{3}{D}^{5}}{V}\frac{{\rho}_{sus}}{{\mu}_{sus}}\right)}^{\frac{1}{2}}$ | ${\mu}_{sus}=5{10}^{-3}\mathrm{kg}/\left(\mathrm{ms}\right)$ ${N}_{p}=0.6$ $D=8\mathrm{cm}$ $V=0.25\mathrm{L}$ | |

Suspension density | ${\rho}_{sus}={\left(\frac{w}{{\rho}_{s}}+\frac{1-w}{{\rho}_{w}}\right)}^{-1}$ | $w=0.08$ ${\rho}_{s}=2600\mathrm{kg}/{\mathrm{m}}^{3}$ ${\rho}_{w}=1000\mathrm{kg}/{\mathrm{m}}^{3}$ | |

Porosity | ${\varphi}_{i}=1-C{\left(\frac{{d}_{i}}{{d}_{0}}\right)}^{{d}_{f}-3}$ | $C=0.65$ | Vainshtein et al. [40] |

Permeability | ${K}_{i}=\frac{{d}_{i}^{2}}{72}\left(3+\frac{4}{1-{\varphi}_{i}}-3\sqrt{\frac{8}{1-{\varphi}_{i}}-3}\right)$ | Li and Logan [35] |

Variable | Expression |
---|---|

Objective function | $OF\left({\alpha}_{max},{\alpha}_{min},{k}_{d},{s}_{1},{s}_{2}\right)={{\displaystyle \sum}}_{{t}_{i}}^{{t}_{f}}{\left({d}_{exp}-{d}_{mod}\right)}^{2}$ |

Model diameter | ${d}_{mod}=\frac{{{\displaystyle \sum}}_{i=1}^{max}{N}_{i}{d}_{i}^{4}}{{{\displaystyle \sum}}_{i=1}^{max}{N}_{i}{d}_{i}^{3}}$ |

Coefficient of determination | ${R}^{2}=1-\frac{{{\displaystyle \sum}}_{i=1}^{max}{\left({d}_{agg,exp,i}-{d}_{agg,mod,i}\right)}^{2}}{{{\displaystyle \sum}}_{i=1}^{max}{({d}_{agg,exp,i}-\langle {d}_{agg,exp}\rangle )}^{2}}$ |

Goodness of fit | $GoF(\%)=100\frac{\langle {d}_{agg,exp}\rangle -\mathrm{std}}{\langle {d}_{agg,exp}\rangle}$ |

Standard error | $\mathrm{std}={\left(\frac{1}{n-f}{{\displaystyle \sum}}_{{t}_{i}}^{{t}_{f}}{\left(\langle {d}_{agg,exp}\rangle -\langle {d}_{agg,mod}\rangle \right)}^{2}\right)}^{\frac{1}{2}}$ |

System | Treated Seawater, % | R^{2} | GoF, % |
---|---|---|---|

50/50 | 0 | 0.760 | 94.9 |

40 | 0.949 | 96.5 | |

60 | 0.918 | 95.8 | |

75 | 0.954 | 96.3 | |

90 | 0.899 | 93.9 | |

100 | 0.976 | 97.5 | |

75/25 | 0 | 0.885 | 95.8 |

40 | 0.900 | 95.6 | |

60 | 0.964 | 96.3 | |

75 | 0.957 | 96.0 | |

90 | 0.963 | 96.4 | |

100 | 0.984 | 97.2 | |

100/0 | 0 | 0.946 | 95.8 |

40 | 0.939 | 95.6 | |

60 | 0.945 | 96.0 | |

75 | 0.984 | 97.7 | |

90 | 0.957 | 95.7 | |

100 | 0.966 | 95.7 |

System Qz/Kao, Tsw/Rsw | ${\mathit{\alpha}}_{\mathit{m}\mathit{a}\mathit{x}}$ | ${\mathit{\alpha}}_{\mathit{m}\mathit{i}\mathit{n}}$ | ${\mathit{k}}_{\mathit{d}}$ | ${\mathit{s}}_{1}$ | ${\mathit{s}}_{2}$ | GoF | R^{2} |
---|---|---|---|---|---|---|---|

85/25, 50/50 | 3.07 × 10^{−4} | 2.64 × 10^{−4} | 6.55 × 10^{−2} | 1.81 × 10^{−6} | 3.69 | 95.4% | 0.933 |

95/5, 75/25 | 8.07 × 10^{−4} | 5.23 × 10^{−4} | 3.01 × 10^{−2} | 1.37 × 10^{−6} | 3.49 | 96.9% | 0.978 |

85/5, 100/0 | 1.32 × 10^{−4} | 9.41 × 10^{−4} | 2.21 × 10^{−2} | 2.63 × 10^{−3} | 2.22 | 93.6% | 0.890 |

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**MDPI and ACS Style**

Quezada, G.R.; Jeldres, M.; Robles, P.; Toro, N.; Torres, D.; Jeldres, R.I.
Improving the Flocculation Performance of Clay-Based Tailings in Seawater: A Population Balance Modelling Approach. *Minerals* **2020**, *10*, 782.
https://doi.org/10.3390/min10090782

**AMA Style**

Quezada GR, Jeldres M, Robles P, Toro N, Torres D, Jeldres RI.
Improving the Flocculation Performance of Clay-Based Tailings in Seawater: A Population Balance Modelling Approach. *Minerals*. 2020; 10(9):782.
https://doi.org/10.3390/min10090782

**Chicago/Turabian Style**

Quezada, Gonzalo R., Matías Jeldres, Pedro Robles, Norman Toro, David Torres, and Ricardo I. Jeldres.
2020. "Improving the Flocculation Performance of Clay-Based Tailings in Seawater: A Population Balance Modelling Approach" *Minerals* 10, no. 9: 782.
https://doi.org/10.3390/min10090782