# A Population Balance Model for Shear-Induced Polymer-Bridging Flocculation of Total Tailings

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## Abstract

**:**

## 1. Introduction

## 2. Materials and Methods

#### 2.1. Flocculation Data of Total Tailings Obtained by FBRM

^{−3}. The total tailings sample and the particle size distribution (PSD) of the total tailings obtained in the previous work are shown in Figure 1 [25].

^{−1}to 25 g t

^{−1}, 0.005% to 0.15%, and 51.60 s

^{−1}to 412.90 s

^{−1}, respectively. About 29 experimental runs were designed by response surface methodology (RSM). The square-weighted mean chord length (SWMCL) of floc obtained by FBRM during the flocculation process was used to represent the floc size. The dynamic variation of the SWMCL of floc with flocculation time was shown in Figure 2 [25].

#### 2.2. Population Balance Model (PBM)

_{1}is the number concentration of primary particles in channel 1. The quantities $\beta $ and $\alpha $ are collision frequency and collision efficiency, respectively. By $S$, we denote breakage rate coefficient. Moreover, $\Gamma $ means a breakage distribution function. The superscripts $\mathrm{max}1$ and $\mathrm{max}2$ represent the maximum number of size channels used to describe the full-size range of particles and flocs, respectively, by aggregation and breakage. Values of $\mathrm{max}1$ and $\mathrm{max}2$ were determined in Section 3.1.

#### 2.2.1. Aggregation Kernel

#### 2.2.2. Breakage Kernel

#### 2.3. PBM Solution and Parameter Fitting Methodology

## 3. Results and Discussion

#### 3.1. Initial Population of Total Tailings Particles

^{−20}m

^{3}. At the same time, the largest volume of flocs is 5.23 × 10

^{−10}m

^{3}, which is about 2

^{36.37−1}times of ${V}_{0}$. Accordingly, we divided the size range of particles and flocs into 37 numerical channels. That is to say, both $\mathrm{max}1$ and $\mathrm{max}2$ in Equation (1) are 37. In the 29 experimental runs, there are five solid fractions (SFs), 5 wt%, 10 wt%, 15 wt%, 20 wt%, and 25 wt%. Based on the volume-based PSD (Figure 1b) and the SFs, the corresponding initial population of total tailings particles, that is, the number concentration distribution (m

^{−3}) of primary total tailings particles, can be calculated as shown in Figure 3. Because the PSDs of total tailings are the same under deferent SFs, the proportion of the same channel under deferent SFs is the same. At the same time, high SF means a high number concentration.

#### 3.2. Fitting the Parameters of PBM

#### 3.3. Analysis of the Parameters

#### 3.4. Validation of PBM

^{−1}. Correspondingly, the fitting parameters ${f}_{1}$, …, ${f}_{6}$ were 0.93340, 0.83532, 0.04749, 0.11752, 0.76734, and 1.65428, respectively. The corresponding collision efficiency and breakage rate coefficient estimated are shown in Figure 8. It can be found from Figure 8b that the breakage rate coefficient is much lower than that of run 16 and run 20 (Figure 6), indicating that the flocs are not easy to break under the optimal conditions.

## 4. Conclusions

## Author Contributions

## Funding

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## References

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**Figure 2.**Dynamic evolution of the square-weighted mean chord length of floc with flocculation time [25].

**Figure 3.**Number concentration distribution of primary total tailings particles assigned in 37 channels.

**Figure 4.**Dynamic evolution of the square-weighted mean chord length of floc obtained by experiment (symbols) and modeling (continuous lines) in (

**a**) run 1 to 10; (

**b**) run 11 to 20, and (

**c**) run 21 to 29. In the legend, E and M represent experiment and modeling, respectively.

**Figure 5.**Collision efficiency in PBM of (

**a**) run 16; (

**b**) run 20. The x-axis and y-axis represent the numerical channel I and j, respectively. The z-axis denotes collision efficiency ${\alpha}_{ij}$.

**Figure 6.**Breakage rate coefficient in PBM of (

**a**) run 16; (

**b**) run 20. The x-axis and y-axis represent the shear rate $G$ and gyration radius ${r}_{gi}$, respectively. The z-axis denotes the breakage rate coefficient ${S}_{i}$.

**Figure 7.**Predicted vs. actual values plot for responses: (

**a**) ${f}_{1}$; (

**b**) ${f}_{2}$; (

**c**) ${f}_{3}$; (

**d**) ${f}_{4}$; (

**e**) ${f}_{5}$; and (

**f**) ${f}_{6}$.

**Figure 8.**(

**a**) Collision efficiency and (

**b**) breakage rate coefficient in PBM under the optimal conditions.

**Figure 9.**Dynamic evolution of the square-weighted mean chord length of floc obtained by experiment (symbols) and modeling (continuous lines) under the optimal conditions.

Run. | Factors (Uncoded) | Responses | R^{2} | ||||||||
---|---|---|---|---|---|---|---|---|---|---|---|

SF, x_{1} (wt%) | FD, x_{2} (g t^{−1}) | FC, x_{3} (%) | G, x_{4} (s^{−1}) | f_{1} | f_{2} | f_{3} | f_{4} | f_{5} | f_{6} | ||

1 | 15 | 15 | 0.1500 | 51.60 | 0.906760 | 0.981830 | 0.045583 | 0.071916 | 1.238100 | 1.949340 | 0.8799 |

2 | 25 | 15 | 0.1500 | 232.25 | 0.884400 | 0.906411 | 0.024310 | 0.019595 | 1.042457 | 1.945652 | 0.9121 |

3 | 5 | 25 | 0.0775 | 232.25 | 0.881730 | 0.922640 | 0.021378 | 0.007813 | 1.106460 | 1.894710 | 0.8615 |

4 | 15 | 15 | 0.0050 | 412.90 | 0.887640 | 0.928360 | 0.024536 | 0.016628 | 1.123560 | 1.913460 | 0.9492 |

5 | 15 | 15 | 0.0775 | 232.25 | 0.862600 | 0.959000 | 0.015087 | 0.056445 | 1.100100 | 1.989600 | 0.9677 |

6 | 15 | 15 | 0.0775 | 232.25 | 0.863360 | 0.935460 | 0.002770 | 0.052764 | 1.090710 | 1.962360 | 0.9504 |

7 | 15 | 15 | 0.0775 | 232.25 | 0.867900 | 0.942720 | 0.003796 | 0.046269 | 1.086990 | 1.952670 | 0.8714 |

8 | 15 | 5 | 0.0775 | 51.60 | 0.986006 | 0.954247 | 0.017934 | 0.269841 | 1.074363 | 1.918388 | 0.6896 |

9 | 25 | 5 | 0.0775 | 232.25 | 0.868640 | 0.911100 | 0.023525 | 0.015922 | 1.087500 | 1.883550 | 0.8287 |

10 | 15 | 25 | 0.1500 | 232.25 | 0.994610 | 0.780870 | 0.039762 | 0.174078 | 0.224532 | 1.353810 | 0.9463 |

11 | 25 | 15 | 0.0775 | 51.60 | 0.913510 | 0.953530 | 0.028176 | 0.075294 | 1.177080 | 1.875150 | 0.8578 |

12 | 15 | 15 | 0.0050 | 51.60 | 0.899170 | 0.962320 | 0.036402 | 0.090126 | 1.167180 | 1.967460 | 0.8402 |

13 | 5 | 15 | 0.1500 | 232.25 | 0.865380 | 0.937700 | 0.000980 | 0.037248 | 1.073790 | 1.940640 | 0.8627 |

14 | 15 | 5 | 0.0775 | 412.90 | 0.876230 | 0.898450 | 0.026484 | 0.005345 | 1.037370 | 1.919790 | 0.8567 |

15 | 15 | 5 | 0.0050 | 232.25 | 0.879900 | 0.925820 | 0.020141 | 0.008921 | 1.115490 | 1.891080 | 0.8934 |

16 | 15 | 15 | 0.0775 | 232.25 | 0.864620 | 0.945730 | 0.007218 | 0.051825 | 1.092600 | 1.968210 | 0.9721 |

17 | 5 | 15 | 0.0775 | 51.60 | 0.866723 | 0.742204 | 0.029235 | 0.104340 | 0.571991 | 1.404482 | 0.8773 |

18 | 25 | 25 | 0.0775 | 232.25 | 0.894745 | 0.922552 | 0.039456 | 0.049053 | 1.049720 | 1.766263 | 0.7917 |

19 | 15 | 15 | 0.0775 | 232.25 | 0.913690 | 0.985850 | 0.023613 | 0.020428 | 1.172490 | 2.003190 | 0.8818 |

20 | 15 | 25 | 0.0775 | 51.60 | 0.901440 | 0.962160 | 0.034187 | 0.087948 | 1.116930 | 2.028450 | 0.9678 |

21 | 25 | 15 | 0.0050 | 232.25 | 0.800013 | 0.608530 | 0.008808 | 0.012363 | 0.657667 | 1.079004 | 0.9215 |

22 | 15 | 15 | 0.1500 | 412.90 | 0.885230 | 0.926670 | 0.023514 | 0.006677 | 1.123440 | 1.908900 | 0.8874 |

23 | 5 | 15 | 0.0050 | 232.25 | 0.840267 | 0.689191 | 0.018029 | 0.065081 | 0.434195 | 1.275404 | 0.7366 |

24 | 15 | 25 | 0.0775 | 412.90 | 0.982242 | 0.870247 | 0.009020 | 0.165464 | 0.709983 | 1.764251 | 0.8417 |

25 | 25 | 15 | 0.0775 | 412.90 | 0.887640 | 0.928360 | 0.024536 | 0.016628 | 1.123560 | 1.913460 | 0.6573 |

26 | 5 | 15 | 0.0775 | 412.90 | 0.831532 | 0.973551 | 0.030851 | 0.165089 | 0.577301 | 1.717226 | 0.8328 |

27 | 15 | 5 | 0.1500 | 232.25 | 0.895850 | 0.934230 | 0.031360 | 0.010204 | 1.143030 | 1.920150 | 0.7514 |

28 | 5 | 5 | 0.0775 | 232.25 | 0.884920 | 0.920920 | 0.017997 | 0.007352 | 1.115130 | 1.894710 | 0.6050 |

29 | 15 | 25 | 0.0050 | 232.25 | 0.887650 | 0.928390 | 0.024556 | 0.016685 | 1.123560 | 1.913490 | 0.9515 |

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**MDPI and ACS Style**

Ruan, Z.; Wu, A.; Bürger, R.; Betancourt, F.; Ordoñez, R.; Wang, J.; Wang, S.; Wang, Y.
A Population Balance Model for Shear-Induced Polymer-Bridging Flocculation of Total Tailings. *Minerals* **2022**, *12*, 40.
https://doi.org/10.3390/min12010040

**AMA Style**

Ruan Z, Wu A, Bürger R, Betancourt F, Ordoñez R, Wang J, Wang S, Wang Y.
A Population Balance Model for Shear-Induced Polymer-Bridging Flocculation of Total Tailings. *Minerals*. 2022; 12(1):40.
https://doi.org/10.3390/min12010040

**Chicago/Turabian Style**

Ruan, Zhuen, Aixiang Wu, Raimund Bürger, Fernando Betancourt, Rafael Ordoñez, Jiandong Wang, Shaoyong Wang, and Yong Wang.
2022. "A Population Balance Model for Shear-Induced Polymer-Bridging Flocculation of Total Tailings" *Minerals* 12, no. 1: 40.
https://doi.org/10.3390/min12010040