# Development of a Simulator for Random and Non-Random Breakage of Particles and Liberation of Grains Based on Voronoi Tessellation

^{*}

## Abstract

**:**

^{®}. Results showed that for the samples used in this study, the proportion of the intergranular breakage changes as the grade of the ore changes, with an agreement between simulations and experiments, independently from the energy level of comminution.

## 1. Introduction

_{1}+ l

_{2}and its Free Surface (FS) is equal to l

_{3}. Each of these amounts transform to a phase-specific parameter by dividing by phase area [25].

_{1}is the Area of Phase 1. After the breakage, phase-specific variables for the product particles should be calculated. Suppose that a fracture line which leads to pure transgranular breakage divides phase 1 into two bodies attached to other phases (as seen in Figure 1b with areas of A

_{12}and A

_{13}, where A

_{12}+ A

_{13}= A

_{1}). It can be seen that:

_{12}and SA

_{13}are the Surface Areas (not Phase-Specific) of new generated particles with areas of A

_{12}and A

_{13}. Therefore, the change in phase-specific variables after the breakage is calculated as:

_{8}):

## 2. Materials and Methods

#### 2.1. Simulation

^{®}codes. The ore before the breakage was simulated as a space (matrix or first phase) in which some squares (mineral grains or second phase) were emplaced. Then the Voronoi diagrams were generated and the Voronoi polygons were considered as the propagated cracks (as shown in Figure 2a). The seeds (centers of Voronoi polygons) were randomly distributed in random regions of a two-dimensional space. Different Particle Size Distributions (PSDs) were produced by changing the number of seeds and it mimicked different levels of grinding energy. After the breakage, the relation between the cracks and all produced particles (phase one and two; mineral and the gangue) was investigated and the contribution of each type of breakage was measured based on the method developed by Leiβner [25]. The simulator was able to produce different proportions of each type of breakage in a single run, by changing a manipulated geometrical criterion in the simulator. However, the criterion itself did not replicate the targeted proportion of inter/transgranular breakage, as the criterion defined when a phase boundary was considered as a fracture and how the density of cracks changed in different regions. Then the criterion was manually changed (it could be done by an optimization algorithm) in order to approximate the average value of the simulated proportion of inter/transgranular breakage to such a value obtained from images of crushed particles. Therefore, the calculation of such a proportion was based on the same methodology and in 2D for both simulations and images of crushed artificial stones. Another criterion was also considered in these simulations; i.e., the d80 of simulated particles falls in the range of ±10% of d80 of laboratory samples. Therefore, it was tried to reproduce the particle size distribution too. If random breakage was targeted, all the cracks were considered to propagate independently from the composition (no matter whether it was in phase one or two), but when preferential breakage was considered, some cracks inside a phase were removed to produce different crack densities in different phases. For pure detachment, cracks were obliged to change their path through the phase boundaries, i.e., the boundary between two phases was assumed as a crack. Therefore, in the first step of simulation, the crack propagation was random (Figure 2b), and then by using the preferential algorithm, the crack propagation pattern changed (Figure 2c). Such algorithms led to different liberation spectrums and PSDs based on the targeted type of breakage, as shown in Figure 3 and Figure 4 [34]. In Figure 2b,c, the colored (other than red) particles represent mixed particles (in this text, mixed particle refers to a two-phase particle that some part of the surface of its valuable phase is free and exposed, while the locked particles have no free surfaces).

#### 2.2. Validation

## 3. Results

## 4. Discussion

## 5. Conclusions

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## References

- Leißner, T.M.; Mütze, T.; Bachmann, K.; Rode, S.; Gutzmer, J.; Peuker, U.A. Evaluation of mineral processing by assessment of liberation and upgrading. Miner. Eng.
**2013**, 53, 171–173. [Google Scholar] [CrossRef] - Bole, J.; Lin, C.L.; Miller, J.D. Experimental verification of the PARGEN simulator for liberation analysis. Int. J. Miner. Process.
**1993**, 37, 209–221. [Google Scholar] [CrossRef] - Herbst, J.A.; Rajamani, K.; Lin, C.L.; Miller, J.D. Development of a multicomponent-multisize liberation model. Miner. Eng.
**1988**, 1, 97–111. [Google Scholar] [CrossRef] - Gay, S. A liberation model for comminution based on probability theory. Miner. Eng.
**2004**, 17, 525–534. [Google Scholar] [CrossRef] - Wills, B.; Atkinson, K. Some observations on the fracture and liberation of mineral assemblies. Miner. Eng.
**1993**, 6, 697–706. [Google Scholar] [CrossRef] - Hsih, C.S.; Wen, S.B. An extension of Gaudin’s liberation model for quantitatively representing the effect of detachment in liberation. Int. J. Miner. Process.
**1994**, 42, 15–35. [Google Scholar] [CrossRef] - King, R.P. Modeling and Simulation of Mineral Processing Systems; Butterworth-Heinemann: Oxford, UK, 2001. [Google Scholar]
- Gaudin, A. Principles of Mineral Dressing; McGraw-Hill: New York, NY, USA, 1939. [Google Scholar]
- Klimpel, R.R.; Austin, L.G. A Preliminary Model of Liberation from a Binary System. Powder Technol.
**1983**, 34, 121–130. [Google Scholar] [CrossRef] - Wiegel, R.L. A random model for mineral liberation by size reduction. Trans. AIME
**1967**, 238, 179–189. [Google Scholar] - King, R.P.; Schneider, C.L. Mineral liberation and the batch communition equation. Miner. Eng.
**1998**, 11, 1143–1160. [Google Scholar] [CrossRef] - Mariano, R.; Evans, C.; Manlapig, E. Definition of random and non-random breakage in mineral liberation-A review. Miner. Eng.
**2016**, 94, 51–60. [Google Scholar] [CrossRef] - Sutherland, D.; Fandrich, R. Selective fracture and liberation of minerals. In Proceedings of the Chemeca 96: Excellence in Chemical Engineering: 24th Australian and New Zealand Chemical Engineering Conference and Exhibition, Sydney, Australia, 30 September–2 October 1996; pp. 83–88. [Google Scholar]
- Wills, B.A.; Finch, J. Wills’ Mineral Processing Technology: An Introduction to the Practical Aspects of Ore Treatment and Mineral Recovery; Butterworth-Heinemann: Oxford, UK, 2015. [Google Scholar]
- Bérubé, M.A.; Marchand, J.C. Evolution of the mineral liberation characteristics of an iron ore undergoing grinding. Int. J. Miner. Process.
**1984**, 13, 223–237. [Google Scholar] [CrossRef] - Revnivtsev, V.K.; Ye, P.; Finkelstein, G.A.; Zarogatsky, L.P.; Ivanov, N.A.; Blekhman, I.I.; Ivanov, B.G. Selective liberation of minerals in inertial cone crushers. Powder Technol.
**1984**, 38, 195–203. [Google Scholar] [CrossRef] - Austin, L.G.; Luckie, P.T. The problems of quantifying mineral liberation: A review. Part. Part. Syst. Charact.
**1988**, 5, 122–129. [Google Scholar] [CrossRef] - Laslett, G.; Sutherland, D.; Gottlieb, P.; Allen, N. Graphical assessment of a random breakage model for mineral liberation. Powder Technol.
**1990**, 60, 83–97. [Google Scholar] [CrossRef] - Choi, W.; Adel, G.; Yoon, R. Liberation modeling using automated image analysis. Int. J. Miner. Process.
**1988**, 22, 59–73. [Google Scholar] [CrossRef] - Fandrich, R.G.; Bearman, R.A.; Boland, J.; Lim, W. Mineral liberation by particle bed breakage. Miner. Eng.
**1997**, 10, 175–187. [Google Scholar] [CrossRef] - Garcia, D.; Lin, C.L.; Miller, J.D. Quantitative analysis of grain boundary fracture in the breakage of single multiphase particles using X-ray microtomography procedures. Miner. Eng.
**2009**, 22, 236–243. [Google Scholar] [CrossRef] - Xu, W.; Dhawan, N.; Lin, C.L.; Miller, J.D. Further study of grain boundary fracture in the breakage of single multiphase particles using X-ray microtomography procedures. Miner. Eng.
**2013**, 46, 89–94. [Google Scholar] [CrossRef] - King, R.P. Calculation of the liberation spectrum in products produced in continuous milling circuits. In Proceedings of the 7th European Symposium on Comminution, Ljubljana, Slovenia, 12–14 June 1990; pp. 429–444. [Google Scholar]
- Little, L.; Mainza, A.N.; Becker, M.; Wiese, J.G. Using mineralogical and particle shape analysis to investigate enhanced mineral liberation through phase boundary fracture. Powder Technol.
**2016**, 301, 794–804. [Google Scholar] [CrossRef] - Leißner, T.; Hoang, D.; Rudolph, M.; Heinig, T.; Bachmann, K.; Gutzmer, J.; Schubert, H.; Peuker, U. A mineral liberation study of grain boundary fracture based on measurements of the surface exposure after milling. Int. J. Miner. Process.
**2016**, 156, 3–13. [Google Scholar] [CrossRef] - Tanemura, M.; Ogawa, T.; Ogita, N. A new algorithm for three-dimensional voronoi tessellation. J. Comput. Phys.
**1983**, 51, 191–207. [Google Scholar] [CrossRef] - Machado Leite, M.R. Liberation by size reduction. Consequences and improvements on flotation kinetics. In Innovations in Flotation Technology; Springer: Dordrecht, The Netherlands, 1992; pp. 149–170. [Google Scholar]
- Vassiliev, P.V.; Ledoux, H.; Gold, C. Modeling Ore Texture and Mineral Liberation Using 3D Voronoi Diagrams. In Proceedings of the International Conference “Numerical Geometry, Grid Generation and High Performance Computing”, Moscow, Russia, 10–13 June 2008; pp. 10–13. [Google Scholar]
- Khalesi, M.R.; Bazin, C.; Hodouin, D.; Bellec, S. A grinding-liberation model for the size reduction of gold ores. In Proceedings of the World Gold Conference, Johannesburg, South Africa, 26–30 October 2009. [Google Scholar]
- Khalesi, M.R.; Bazin, C.; Hodouin, D.; Bellec, S. Simulation of gold grain exposure of ground ore using Voronoi tessellation. IFAC Proc. Vol.
**2009**, 42, 43–48. [Google Scholar] [CrossRef] - Rozenbaum, O.; Machault, J.; Le Trong, E.; Tankeu, Y.G.N.; Barbanson, L. Ore fragmentation modelling for the evaluation of the liberation mesh size. In Proceedings of the 13th SGA Biennal Meeting, Nancy, France, 24–27 August 2015; pp. 1447–1450. [Google Scholar]
- Van der Wielen, K.P.; Rollinson, G. Texture-based analysis of liberation behaviour using Voronoi tessellations. Miner. Eng.
**2016**, 89, 93–107. [Google Scholar] [CrossRef] - Ueda, T.; Oki, T.; Koyanaka, S. A general quantification method for addressing stereological bias in mineral liberation assessment in terms of volume fraction and size of mineral phase. Miner. Eng.
**2018**, 119, 156–165. [Google Scholar] [CrossRef] - Mirzaei, Z.S.; Khalesi, M.R. Liberation Analysis Using a New Simulator of Random and non-Random Breakage. In Proceedings of the 16th International Mineral Processing Symposium (IMPS 2018), Antalya, Turkey, 23–25 October 2018. [Google Scholar]
- Johnston, I.W.; Choi, S.K. Synthetic soft rock for laboratory model studies. Int. J. Rock Mech. Min. Sci. Geomech. Abstr.
**1986**, 23, 251–263. [Google Scholar] [CrossRef] - Kiss, L.; Schönert, K. Liberation of two-component material by single particle compression and impact crushing. Aufbereitungs-Technick
**1980**, 5, 223–230. [Google Scholar] - Woollacott, L.C.; Valenta, M. Use of synthetic ore particles to test a transformation function in liberation analysis. Miner. Eng.
**1996**, 9, 1017–1032. [Google Scholar] [CrossRef] - Ueda, T.; Oki, T.; Koyanaka, S. Experimental analysis of mineral liberation and stereological bias based on X-ray computed tomography and artificial binary particles. Adv. Powder Technol.
**2018**, 29, 462–470. [Google Scholar] [CrossRef] - Rezvani, A.; Khalesi, M.R.; Mirzaei, Z.S.; Albijanic, B. A simple method for determination of liberation spectrum of high-grade ores by images analysis of crushed particles. Adv. Powder Technol.
**2019**. submitted. [Google Scholar] - Ueda, T.; Oki, T.; Koyanaka, S. Statistical effect of sampling particle number on mineral liberation assessment. Miner. Eng.
**2016**, 98, 204–212. [Google Scholar] [CrossRef] - Ueda, T.; Oki, T.; Koyanaka, S. Numerical analysis of the general characteristics of stereological bias in surface liberation assessment of ore particles. Adv. Powder Technol.
**2018**, 29, 3327–3335. [Google Scholar] [CrossRef]

**Figure 1.**(

**a**) A particle containing three phases with phase 1 as the valuable mineral; (

**b**) transgranular breakage occurs, dividing the particle and phase 1 into two bodies; (

**c**) intergranular breakage occurs, detaching phase 1.

**Figure 2.**(

**a**) Generation of Voronoi diagrams (as propagated cracks) on a virtual two-phase ore (red phase as matrix and yellow phase as grains); (

**b**) the shape of mixed particles after random breakage, (

**c**) the shape of mixed particles after preferential breakage.

**Figure 8.**Dependency of contribution of intergranular breakage on the grade of the ore with plaster grains; (

**a**) energy level of 117.2 (kJ), (

**b**) energy level of 102 (kJ).

**Figure 9.**Dependency of contribution of intergranular breakage on the grade of the ore with cement grains; (

**a**) energy level of 117.2 (kJ), (

**b**) energy level of 102 (kJ).

**Figure 10.**Increasing the proportion of intergranular breakage by increasing the grain size to particle size ratio (dg/dp).

© 2019 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (http://creativecommons.org/licenses/by/4.0/).

## Share and Cite

**MDPI and ACS Style**

Mirzaei, Z.S.; Khalesi, M.R.
Development of a Simulator for Random and Non-Random Breakage of Particles and Liberation of Grains Based on Voronoi Tessellation. *Minerals* **2019**, *9*, 341.
https://doi.org/10.3390/min9060341

**AMA Style**

Mirzaei ZS, Khalesi MR.
Development of a Simulator for Random and Non-Random Breakage of Particles and Liberation of Grains Based on Voronoi Tessellation. *Minerals*. 2019; 9(6):341.
https://doi.org/10.3390/min9060341

**Chicago/Turabian Style**

Mirzaei, Zeinab Sadat, and Mohammad Reza Khalesi.
2019. "Development of a Simulator for Random and Non-Random Breakage of Particles and Liberation of Grains Based on Voronoi Tessellation" *Minerals* 9, no. 6: 341.
https://doi.org/10.3390/min9060341