## 2 Materials and Methods
### Crushing Experiment
#### 2.2.2 Quantitative Evaluation Indicators for the Degree of Fragmentation  
The fragmentation degree of shells and kernels are important for the quantitative evaluation of crushing quality. Therefore, in order to quantitatively analyze the effect of MC on walnut crushing, the degree of fragmentation of shells and kernels were characterized quantitatively using relevant indicators, respectively. (1) Quantitative evaluation indicators for the degree of shell fragmentation  
The size distribution of the shell fragments is closely related to the crushing characteristics and mechanism [20]. To determine the size distribution of the shell fragments, and considering the shell fragments with complex morphology, the projected area (PA) was obtained using the image processing function embedded in MATLAB R2018b software. The image processing process of the shell fragments is presented in Figure 3. Note that the threshold value, which is used to distinguish the boundaries of the shell fragments, plays an important role in determining shell fragment PA. Research indicates that, when \begin{table}
\begin{tabular}{c c} \hline \hline
**Parameters** & **Size** \\ \hline Grading accuracy/(\%) & \(\geq\)96 \\ Power of cracker/kw & 0.75 \\ Cracking angle/(\({}^{\circ}\)) & 0–1 (0.47) \\ Overall weight/kg & 165 \\ Roller speed/(r/min) & 0–210 (108) \\ Length \(\times\) width \(\times\) height/(mm \(\times\) mm \(\times\) mm) & 1200 \(\times\) 800 \(\times\) 1200 \\ Productivity/(kg/h) & 1080 \\ Inlet size/(mm \(\times\) mm) & 94 \(\times\) 45 \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Parameters of the crushing machine. Figure 2: (**A**) Three-dimensional structure and (**B**) pictorial diagram of the walnut-crushing apparatus. compared to alternative methods like k-means clustering and the watershed algorithm, the adaptive thresholding technique is extensively utilized in boundary extraction due to its heightened robustness and adaptability [3; 21]. Thus, the adaptive threshold was used to determine shell fragments' PAs. A fractal dimension (\(D_{\text{b}}\)) that can reflect the statistical characteristics of material fragmentation has been commonly used to analyze chaotic fragment distributions [22]. Therefore, to quantitatively characterize the fragmentation degree of the shell, the \(D_{\text{b}}\) is adopted. In this study, a method proposed by [PERSON] et al. was used to determine \(D_{\text{b}}\) from the fragment size distribution [23]. The number of fragments whose equivalent length is greater than or equal to \(L_{\text{eq}}\) is:  
\[N=N_{0}\big{(}L_{\text{eq}}/L_{\text{eqmax}}\big{)}^{-D_{\text{b}}} \tag{1}\]  
\[L_{\text{eq}}=\sqrt{PA} \tag{2}\]  
where \(N_{0}\) is the number of fragments with the largest equivalent length \(L_{\text{eqmax}}\). Note that the larger \(D_{\text{b}}\) is, the larger the fragmentation degree is. The conversion of Equation (1) can be obtained as follows:  
\[D_{\text{b}}=\frac{\log\left(N/N_{0}\right)}{\log\big{(}L_{\text{eqmax}}/L_{ \text{eq}}\big{)}} \tag{3}\]  
For statistical generality and accuracy, \(L_{\text{eqs}}>22\) mm were excluded in this study. This is due to the small percentage and poor continuity of distribution in this range, as shown in Figure 4. It should be noted that a similar method was used in Refs. [24; 25]. Based on this, the \(\lg(L_{\text{eqmax}}/L_{\text{eq}})\) vs. \(\lg N\) curve of the shell fragments is shown in Figure 5. As shown, the curve has a good correlation with the fitted straight line (R\({}^{2}\) = 0.95), which indicates that the \(L_{\text{eq}}\) distribution of fragments follows the fractal distribution very well. From the fitted straight line, the \(D_{\text{b}}\) also can be obtained: \(D_{\text{b}}\) = 1.61. In addition, the fitting curve slightly deviates from the initial data points, which is likely due to the decrease in the number of large fragments. Figure 3: The image processing process of the shell fragments. 2.