# 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

- Santamarina, J.C.; Torres-Cruz, L.A.; Bachus, R.C. Why coal ash and tailings dam disasters occur. Science
**2019**, 364, 526–528. [Google Scholar] [CrossRef] - Silva Rotta, L.H.; Alcântara, E.; Park, E.; Negri, R.G.; Lin, Y.N.; Bernardo, N.; Mendes, T.S.G.; Souza Filho, C.R. The 2019 Brumadinho tailings dam collapse: Possible cause and impacts of the worst human and environmental disaster in Brazil. Int. J. Appl. Earth Obs. Geoinf.
**2020**, 90, 102119. [Google Scholar] [CrossRef] - Wang, C.; Harbottle, D.; Liu, Q.; Xu, Z. Current state of fine mineral tailings treatment: A critical review on theory and practice. Miner. Eng.
**2014**, 58, 113–131. [Google Scholar] [CrossRef] - Qi, C.; Fourie, A. Cemented paste backfill for mineral tailings management: Review and future perspectives. Miner. Eng.
**2019**, 144, 106025. [Google Scholar] [CrossRef] - Chen, X.; Jin, X.; Jiao, H.; Yang, Y.; Liu, J. Pore connectivity and dewatering mechanism of tailings bed in raking deep-cone thickener process. Minerals
**2020**, 10, 375. [Google Scholar] [CrossRef] - Grabsch, A.F.; Fawell, P.D.; Adkins, S.J.; Beveridge, A. The impact of achieving a higher aggregate density on polymer-bridging flocculation. Int. J. Miner. Process.
**2013**, 124, 83–94. [Google Scholar] [CrossRef] - Fawell, P.D.; Nguyen, T.V.; Solnordal, C.B.; Stephens, D.W. Enhancing Gravity Thickener Feedwell Design and Operation for Optimal Flocculation through the Application of Computational Fluid Dynamics. Miner. Process. Extr. Metall. Rev.
**2019**, 42, 496–510. [Google Scholar] [CrossRef] - Ruan, Z.; Wu, A.; Bürger, R.; Betancourt, F.; Wang, Y.; Wang, Y.; Jiao, H.; Wang, S. Effect of interparticle interactions on the yield stress of thickened flocculated copper mineral tailings slurry. Powder Technol.
**2021**, 392, 278–285. [Google Scholar] [CrossRef] - Qi, C.; Fourie, A.; Chen, Q.; Tang, X.; Zhang, Q.; Gao, R. Data-driven modelling of the flocculation process on mineral processing tailings treatment. J. Clean. Prod.
**2018**, 196, 505–516. [Google Scholar] [CrossRef] - Jiao, H.; Wang, S.; Yang, Y.; Chen, X. Water recovery improvement by shearing of gravity-thickened tailings for cemented paste backfill. J. Clean. Prod.
**2020**, 245, 118882. [Google Scholar] [CrossRef] - Jiao, H.; Wu, Y.; Wang, H.; Chen, X.; Li, Z.; Wang, Y.; Zhang, B.; Liu, J. Micro-scale mechanism of sealed water seepage and thickening from tailings bed in rake shearing thickener. Miner. Eng.
**2021**, 173, 107043. [Google Scholar] [CrossRef] - Betancourt, F.; Celi, D.; Cornejo, P.; del Río, M.; Macera, L.; Pereira, A.; Rulyov, N. Comparison of ultra-flocculation reactors applied to fine quartz slurries. Miner. Eng.
**2020**, 148, 106074. [Google Scholar] [CrossRef] - Hornn, V.; Park, I.; Ito, M.; Shimada, H.; Suto, T.; Tabelin, C.B.; Jeon, S.; Hiroyoshi, N. Agglomeration-flotation of finely ground chalcopyrite using surfactant-stabilized oil emulsions: Effects of co-existing minerals and ions. Miner. Eng.
**2021**, 171, 107076. [Google Scholar] [CrossRef] - Hornn, V.; Ito, M.; Shimada, H.; Tabelin, C.B.; Jeon, S.; Park, I.; Hiroyoshi, N. Agglomeration–Flotation of Finely Ground Chalcopyrite Using Emulsified Oil Stabilized by Emulsifiers: Implications for Porphyry Copper Ore Flotation. Metals
**2020**, 10, 912. [Google Scholar] [CrossRef] - Hornn, V.; Ito, M.; Yamazawa, R.; Shimada, H.; Tabelin, C.B.; Jeon, S.; Park, I.; Hiroyoshi, N. Kinetic Analysis for Agglomeration-Flotation of Finely Ground Chalcopyrite: Comparison of First Order Kinetic Model and Experimental Results. Mater. Trans.
**2020**, 61, 1940–1948. [Google Scholar] [CrossRef] - Hornn, V.; Ito, M.; Shimada, H.; Tabelin, C.B.; Jeon, S.; Park, I.; Hiroyoshi, N. Agglomeration-Flotation of Finely Ground Chalcopyrite and Quartz: Effects of Agitation Strength during Agglomeration Using Emulsified Oil on Chalcopyrite. Minerals
**2020**, 10, 380. [Google Scholar] [CrossRef] - Concha, F.; Rulyov, N.N.; Laskowski, J.S. Settling velocities of particulate systems 18: Solid flux density determination by ultra-flocculation. Int. J. Miner. Process.
**2012**, 104–105, 53–57. [Google Scholar] [CrossRef] - Senaputra, A.; Jones, F.; Fawell, P.D.; Smith, P.G. Focused beam reflectance measurement for monitoring the extent and efficiency of flocculation in mineral systems. AIChE J.
**2014**, 60, 251–265. [Google Scholar] [CrossRef] - Sharma, S.; Lin, C.L.; Miller, J.D. Multi-scale features including water content of polymer induced kaolinite floc structures. Miner. Eng.
**2017**, 101, 20–29. [Google Scholar] [CrossRef][Green Version] - Elfarissi, F.; Pefferkorn, E. Fragmentation of Kaolinite Aggregates Induced by Ion-Exchange Reactions within Adsorbed Humic Acid Layers. J. Colloid Interface Sci.
**2000**, 221, 64–74. [Google Scholar] [CrossRef] - Odriozola, G.; Schmitt, A.; Moncho-Jordá, A.; Callejas-Fernández, J.; Martínez-García, R.; Leone, R.; Hidalgo-Álvarez, R. Constant bond breakup probability model for reversible aggregation processes. Phys. Rev. E
**2002**, 65, 031405. [Google Scholar] [CrossRef] [PubMed][Green Version] - Jeldres, R.I.; Fawell, P.D.; Florio, B.J. Population balance modelling to describe the particle aggregation process: A review. Powder Technol.
**2018**, 326, 190–207. [Google Scholar] [CrossRef] - 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. [Google Scholar] [CrossRef] - Tanguay, M.; Fawell, P.; Adkins, S. Modelling the impact of two different flocculants on the performance of a thickener feedwell. Appl. Math. Model.
**2014**, 38, 4262–4276. [Google Scholar] [CrossRef] - Wu, A.; Ruan, Z.; Bürger, R.; Yin, S.; Wang, J.; Wang, Y. Optimization of flocculation and settling parameters of tailings slurry by response surface methodology. Miner. Eng.
**2020**, 156, 106488. [Google Scholar] [CrossRef] - Hounslow, M.J.; Ryall, R.L.; Marshall, V.R. A discretized population balance for nucleation, growth, and aggregation. AIChE J.
**1988**, 34, 1821–1832. [Google Scholar] [CrossRef] - Spicer, P.T.; Pratsinis, S.E. Coagulation and Fragmentation: Universal Steady-State Particle-Size Distribution. AIChE J.
**1996**, 42, 1612–1620. [Google Scholar] [CrossRef] - Oyegbile, B.; Ay, P.; Narra, S. Flocculation kinetics and hydrodynamic interactions in natural and engineered flow systems: A review. Environ. Eng. Res.
**2016**, 21, 1–14. [Google Scholar] [CrossRef][Green Version] - Mandelbrot, B.B. Self-affine fractals and fractal dimension. Phys. Scr.
**1985**, 32, 257–260. [Google Scholar] [CrossRef] - Kusters, K.A.; Wijers, J.G.; Thoenes, D. Aggregation kinetics of small particles in agitated vessels. Chem. Eng. Sci.
**1997**, 52, 107–121. [Google Scholar] [CrossRef][Green Version] - Quezada, G.R.; Ramos, J.; Jeldres, R.I.; Robles, P.; Toledo, P.G. Analysis of the flocculation process of fine tailings particles in saltwater through a population balance model. Sep. Purif. Technol.
**2020**, 237, 116319. [Google Scholar] [CrossRef] - Heath, A.R.; Bahri, P.A.; Fawell, P.D.; Farrow, J.B. Polymer flocculation of calcite: Population balance model. AIChE J.
**2006**, 52, 1641–1653. [Google Scholar] [CrossRef] - Veerapaneni, S.; Wiesner, M.R. Hydrodynamics of fractal aggregates with radially varying permeability. J. Colloid Interface Sci.
**1996**, 177, 45–57. [Google Scholar] [CrossRef] - Neale, G.; Epstein, N.; Nader, W. Creeping flow relative to permeable spheres. Chem. Eng. Sci.
**1973**, 28, 1865–1874. [Google Scholar] [CrossRef] - Li, X.Y.; Logan, B.E. Permeability of fractal aggregates. Water Res.
**2001**, 35, 3373–3380. [Google Scholar] [CrossRef] - Vainshtein, P.; Shapiro, M.; Gutfinger, C. Mobility of permeable aggregates: Effects of shape and porosity. J. Aerosol Sci.
**2004**, 35, 383–404. [Google Scholar] [CrossRef] - Camp, T.R.; Stein, P.C. Velocity Gradients and Internal Work in Fluid Motion. J. Bost. Soc. Civ. Eng.
**1943**, 30, 219–237. [Google Scholar] - Smoluchowski, M.V. Versuch einer mathematischen Theorie der Koagulationskinetik kolloider Lösungen. Z. Phys. Chem.
**1918**, 92U, 129–168. [Google Scholar] [CrossRef][Green Version] - Adler, P. Heterocoagulation in shear flow. J. Colloid Interface Sci.
**1981**, 83, 106–115. [Google Scholar] [CrossRef] - Soos, M.; Sefcik, J.; Morbidelli, M. Investigation of aggregation, breakage and restructuring kinetics of colloidal dispersions in turbulent flows by population balance modeling and static light scattering. Chem. Eng. Sci.
**2006**, 61, 2349–2363. [Google Scholar] [CrossRef] - Selomulya, C.; Bushell, G.; Amal, R.; Waite, T.D. Understanding the role of restructuring in flocculation: The application of a population balance model. Chem. Eng. Sci.
**2003**, 58, 327–338. [Google Scholar] [CrossRef] - Antunes, E.; Garcia, F.A.P.; Ferreira, P.; Blanco, A.; Negro, C.; Rasteiro, M.G. Modelling PCC flocculation by bridging mechanism using population balances: Effect of polymer characteristics on flocculation. Chem. Eng. Sci.
**2010**, 65, 3798–3807. [Google Scholar] [CrossRef] - Pandya, J.D.; Spielman, L.A. Floc breakage in agitated suspensions: Theory and data processing strategy. J. Colloid Interface Sci.
**1982**, 90, 517–531. [Google Scholar] [CrossRef] - Pandya, J.D.; Spielman, L.A. Floc breakage in agitated suspensions: Effect of agitation rate. Chem. Eng. Sci.
**1983**, 38, 1983–1992. [Google Scholar] [CrossRef] - Chen, W.; Fischer, R.R.; Berg, J.C. Simulation of particle size distribution in an aggregation-breakup process. Chem. Eng. Sci.
**1990**, 45, 3003–3006. [Google Scholar] [CrossRef] - Asuero, A.G.; Sayago, A.; González, A.G. The Correlation Coefficient: An Overview. Crit. Rev. Anal. Chem.
**2006**, 36, 41–59. [Google Scholar] [CrossRef] - Ruan, Z.; Wu, A.; Wang, J.; Yin, S.; Wang, Y. Flocculation and settling behavior of unclassified tailings based on measurement of floc chord length. Chin. J. Eng.
**2020**, 42, 980–987. [Google Scholar] [CrossRef]

**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