## 2 Material and Methods
### Characterization of Fragment Size and Shape Distributions
#### 2.4.1 Generalized Extreme Value (GEV) Distribution  
Some methods have been proposed to describe the particle size distributions, such as GEV [38], Rayleigh [39], Weibull [23] and log-normal distributions [40]. This paper uses the three-parameter GEV distribution to fit the cumulative probability distribution of coal fragment sizes, similar to the analysis method of experimental results conducted by [PERSON] [26] and [PERSON] [21]. Because the GEV distribution is a complete set of statistical distributions developed within extreme value theory to combine the Gumbel, Frechet and Weibull distributions, also known as type I, II and III extreme value distributions, solving the limitation of one type of extreme value distribution function [38]. As shown in Figure 6, taking the coal fragments under an impact velocity of 253 m/s as an example, the three-parameter GEV distribution and Weibull distribution models are used to fit the cumulative probability of coal fragmentation. The fitting results illustrate that both distribution models can describe the cumulative probability distribution of coal fragments for most particle size ranges, such as 0.6-5.0 mm; and the fitting correlation coefficients of both curves are more than 0.99. However, the size scope of the application for GEV distribution function is wider than that for the Weibull distribution, especially for coal fragments with small particle sizes. It can be seen that from the partial magnification in Figure 6b,c that the Weibull distribution model cannot describe the cumulative distribution of coal fragments smaller than 0.55 mm, and the fitted cumulative probability will reach a stead value of 1 when the fragment size is larger than 5.5 mm. Moreover, as shown in Figure 6 b, the GEV distribution model is more suitable for characterizing the size distribution of the small-sized particles. The same phenomenon also appears for other coal fragments in WJD under different jet impact velocities and coal strengths. Additionally, the added parameters (\(\mu\), \(\sigma\), and \(\xi\)) can characterize the location, scale and shape of distribution curves of fragments under different hydraulic parameters and coal seam properties. Therefore, the GEV distribution model can accurately characterize the size distribution of coal fragments.  
The cumulative distribution model of the GEV distribution can be described as follows [39]:  
\[F(x;\mu,\sigma,\xi)=\exp\left\{-\left[1+\xi\left(\frac{x-\mu}{\sigma}\right) \right]^{-1/\xi}\right\} \tag{3}\]  
\begin{table}
\begin{tabular}{c c c c c c} \hline
**No.** & **7** & **8-1** & **8-2** & **8-3** & **8-4** \\ \hline Impact velocity (m/s) & 268 & 219 & 237 & 253 & 268 \\ Mass percentage & 16\% & 21\% & 25\% & 29\% & 32\% \\ \hline \end{tabular}
\end{table}
Table 2: Mass percentage of the particles with the sizes less than 0.3 mm.  
where \(x\) is the equivalent diameter of fragment, \(\mu\) is the location parameter, \(\sigma\) is the scale parameter, and \(\xi\) is the shape parameter. These parameters must satisfy \(1+\xi(x-\mu)/\sigma>0\).  
Next, the derivative of Equation (3) is taken, which is the probability density function as shown in Equation (4). The peak values of the distribution curves correspond to the dominant fragment sizes, i.e., the modes of the probability density distribution curves. To a certain extent, the dominant fragment size indicates the centralized distribution of the experimental data. Therefore, it can be used to characterize the distribution of coal fragments subjected to high-pressure water jets in WJD.  
\[f(x;\mu,\sigma,\xi)=\frac{1}{\sigma}\bigg{[}1+\xi\bigg{(}\frac{x-\mu}{\sigma} \bigg{)}\bigg{]}^{-1/\xi-1}\exp\Bigg{\{}-\bigg{[}1+\xi\bigg{(}\frac{x-\mu}{ \sigma}\bigg{)}\bigg{]}^{-1/\xi}\Bigg{\}} \tag{4}\]