## 3 Results and Discussion
### Correlation between Material Properties and Breakage Rate Parameters  
#### 3.5.1 Simple Regression Analysis  
In the first part of this study simple regression analysis involving one dependent and one independent variable was carried out, while Excel software was used to establish potential correlations between material properties and breakage rate parameters. During this part, linear or non-linear functions were used and the equation with the highest correlation coefficient R\({}^{2}\) was considered for determining the breakage rate parameters (\(S_{i}\), \(\alpha_{T}\), \(\alpha\), \(S_{m}\) and \(x_{m}\)). It is mentioned that no similar functions are given so far in other studies in literature. The correlations between \(S_{i}\) values and \(V_{p}\), \(R_{L}\), \(UCS\) and \(E_{t}\) are presented in Figure 10. It is seen that very strong correlation between \(S_{i}\) and \(R_{L}\) (R\({}^{2}\) = 0.98) and strong correlations between \(S_{i}\) and \(V_{p}\) (R\({}^{2}\) = 0.85) and \(E_{t}\) (R\({}^{2}\) = 0.85) are obtained with the use of inverse linear functions. Furthermore, \(S_{i}\) is very well correlated with \(UCS\) with the use of an inverse exponential function (R\({}^{2}\) = 0.99). Similar results are obtained for the correlations between \(\alpha_{T}\) and \(V_{p}\), \(R_{L}\), \(UCS\) and \(E_{t}\) (Figure 11). Parameter \(\alpha_{T}\), which defines the breakage rate for a feed size of 1 mm, is correlated very well with \(V_{p}\), \(R_{L}\) and \(E_{t}\) using an inverse linear relationship. The very strong correlation (R\({}^{2}\) = 1) between \(\alpha_{T}\) and \(R_{L}\) is considered very important for the design of grinding circuits. \(R_{L}\) values are obtained from the application of the convenient and low-cost Schmidt hammer test, which can be used both in laboratory and in industrial mills, thus enabling the determination of breakage rate for a specific material in a grinding circuit. Figure 8: First order plots showing the mass % remaining for each size fraction of each material vs. grinding time; (**a**) quartz, (**b**) marble, (**c**) quartzite and (**d**) metasandstone. Figure 9: Variation of breakage rate \(S_{i}\) vs. upper feed particle size. Figure 11: Correlations between parameter \(\alpha_{T}\) and (**a**) P-wave velocity \(V_{p}\), (**b**) Schmidt rebound values \(R_{L}\), (**c**) uniaxial compressive strength _UCS_ and (**d**) tangent modulus of elasticity \(E_{t}\); horizontal error bars show standard deviation of the measurements. Figure 10: Correlations between breakage rate \(S_{i}\) and (**a**) P-wave velocity \(V_{p}\), (**b**) Schmidt rebound values \(R_{L}\), (**c**) uniaxial compressive strength _UCS_ and (**d**) tangent modulus of elasticity \(E_{t}\); horizontal error bars show standard deviation of the measurements. Figure 12 shows the correlations between parameter \(\alpha\) and \(V_{p}\), \(R_{L}\), \(UCS\) and \(E_{t}\). It is seen that parameter \(\alpha\) is very well correlated with \(V_{p}\), \(R_{L}\) and \(E_{t}\) with the use of inverse exponential functions. Parameter \(\alpha\) is constant for the same material and therefore the functions obtained can be used independently of grinding conditions. However, a moderate correlation (R\({}^{2}\) = 0.67) between \(\alpha\) and \(UCS\) is indicated. The correlations between maximum breakage rate \(S_{m}\) and \(V_{p}\), \(R_{L}\), \(UCS\) and \(E_{t}\) are presented in Figure 13. Very strong correlation (R\({}^{2}\) = 0.96) is shown between \(S_{m}\) and \(R_{L}\) with the use of an inverse linear function. Furthermore, \(S_{m}\) seems to follow an inverse exponential relationship with \(UCS\), with correlation coefficient R\({}^{2}\) = 0.95. However, the prediction of \(S_{m}\) from \(V_{p}\) and \(E_{t}\) values is somehow risky due to the moderate correlation between these variables. Figure 14 shows the correlations between the optimum size \(x_{m}\) and \(V_{p}\), \(R_{L}\), \(UCS\) and \(E_{t}\).