*3.2. Discussion*

#### 3.2.1. Effect of Size on Particle Strength

To appreciate the significance of threshold energy, size normalised *Ex0* was plotted in Figure 11 and it was observed that generally the materials get weaker with increasing particle size. Most researchers attribute this to an increase in flaw density with increase in particle size. Apparently, it is not the case with the silica, which has the same threshold energy for the entire size range.

**Figure 11.** Comparison of specific threshold energy (J/kg).

It was observed for the coal that the increase in relative strength with size was gradual. Extrapolation of the model suggests that this starts to increase exponentially as particles become smaller than 10 g. As for the gold waste rock and the dolomite, the relative increase in toughness with decreasing particle size is at a much higher rate. To show the implication of this phenomenon, specific probability breakage results for both gold waste rock and silica were compared.

In Figure 12, breakage probabilities at similar specific energy input are compared for the gold waste rock for −13.2 + 11.2 mm and −19 + 16 mm particles; it is seen clearly that the breakage probabilities for similar specific energy input are higher for the coarser particles. In other words, the bigger particles are relatively weaker for this material [26]. In Figure 13, a similar comparison is made for silica material for the −26.5 + 22.4 mm and −19 + 16 mm particles, and it is seen that the breakage probability differences for similar specific energy input are not as drastic as those seen for the gold waste rock in Figure 12. Apparently, the smaller particles offered stronger resistance to breakage at low energy input while at higher energy input, they were broken relatively more easily.

**Figure 12.** Comparison of gold waste rock breakage probabilities for two particle sizes at specific impact energy input.

**Figure 13.** Comparison of silica breakage probabilities at similar specific impact energy input.

3.2.2. Difference in Damage Accumulation in Materials

The rate of deterioration with each impact can also be compared by normalising the part comprising energy terms in Equation (3) to get the following equation:

$$P\_b = 1 - e^{(-an^b)} \tag{5}$$

when parameters *a* and *b* are applied to this equation for the different materials, the result is what is seen in Figure 14. This reveals some interesting aspects about the materials; with the exception of silica, the other materials appear to become more difficult to break. Here, the silica stands out due to its brittle nature; it is seen that as long as the threshold energy is exceeded, each impact causes significant cumulative damage. However, it was noticed that, for both the coal and gold waste rock, the relative damage is minimal with each impact. This difference may probably be due to non-brittle deformation

behaviour or the tendency of some materials to plastically deform, thus requiring further breakage attempts before yielding.

**Figure 14.** Comparison of the rate of deterioration with each impact for different materials.

It is natural to assume that a high *Ex*<sup>0</sup> value would usually indicate that a particular material is difficult to break, but as demonstrated above, *Ex*<sup>0</sup> varies differently for some materials, and one material which is stronger at a smaller size can end up being stronger at a different size. It was also observed that a material with a low *Ex*<sup>0</sup> can still be difficult to break if it has a low rate of deterioration with successive impacts. As such, it is recommended to test particles contained in various size ranges, to capture the observed size effect.

The model described by Equation (3) was also tested against published data from Morrison et al. [14]; using the Excel solver to estimate model parameters, the model successfully described the data. Thus, there is a good indication that this model could be of wide applicability, and future tests on more different types of material are planned.

In Appendix B is included Figure A2, which presents an algorithm that has been proposed by Bwalya [15] that applies breakage probability data to DEM simulation to predict the comminution rate in size reduction equipment.

#### **4. Conclusions**

The fracture response of particles to energy input is dependent on the material types and flaw size and density, which varies with particle size. This has been the basis of most of the probability fracture models that have been developed recently. The new model has demonstrated that it is possible to characterize particle breakage properties from the size related threshold energy point of view. The model has revealed that materials exhibit unique trends in terms of how their threshold energy and rate of deterioration vary with particle size and each impact, respectively. Its ability to predict particle fracture probability has also been successfully demonstrated on four different materials. Among the materials tested, gold waste rock proved to be the toughest, and its relative increase in toughness with decreasing particle size was also higher than the other three materials. The difference in cumulative damage beyond the threshold energy may be attributed to non-brittle deformation behaviour or plastic deformation of particles. The purely brittle material will require a few impacts to disintegrate, while those with plastic deformation tendencies will endure several more impacts before complete failure. As for the differences in threshold energies, the existence of flaws which are relatively bigger and more frequent in larger particles was considered to be a key factor. The damage accumulation

*Minerals* **2020**, *10*, 710

as a result of repeated impacts is likely a function of properties such as the shape, composition, flaw distribution, and the presence of mineral grains, whose effects will be the object of future research by the authors.

**Author Contributions:** Conceptualization, M.M.B. and N.C.; methodology, M.M.B. and N.C.; software, M.M.B.; validation, M.M.B. and N.C.; formal analysis, M.M.B. and N.C.; investigation, M.M.B. and N.C.; resources, M.M.B. and N.C.; data curation, M.M.B. and N.C.; writing—original draft preparation, M.M.B. and N.C.; writing—review and editing, M.M.B. and N.C.; visualization, M.M.B. and N.C; supervision, M.M.B. and N.C.; project administration, M.M.B. and N.C.; funding acquisition, Y.Y. All authors have read and agreed to the published version of the manuscript.

**Funding:** This research received no external funding.

**Acknowledgments:** MINTEK is greatly acknowledged for allowing the authors to use the drop weight tester and their laboratory facility. The authors particularly mention Portai Mudau for ensuring that all their needs were met during the time of experiments.

**Conflicts of Interest:** The authors declare no conflict of interest.
