## 3 Results and Discussion
### The Axial Distribution of Force Chains  
To achieve the visualization of the axial distribution of force chains in the barrier zone, samples were selected within the axial range of 102 mm \(\leq\) X \(\leq\) 182 mm after particles reached dynamic stability (within the red dotted line range in Figure 2b). The cloud chart illustrating the axial distribution of force chains under different rotational speeds is presented in Figure 13. Clearly, the relatively strong force chain distribution in the barrier zone significantly decreased at 20 rpm (Figure 13a). As the rotational speed increased, there was an extensive distribution of relatively strong force chains in the barrier zone at 80 rpm, approaching the outlet gradually (Figure 13c). These results indicate that the increase in rotational speed strengthens the disturbance of the helical blades to the particles, promoting the axial distribution of force chains simultaneously. The axial coefficient of variation (CV) was employed for the quantitative characterization of the axial distribution uniformity of force chains. The CV, also known as the relative standard deviation, is defined as the ratio of the standard deviation to the mean of the axial distribution of force chains. A higher CV value indicates lower uniformity in the axial distribution. \[\rm F_{i}^{n}=\frac{\sum_{r=1}^{N_{r}}F_{i,r}^{n}}{N_{r}} \tag{5}\]  
Figure 12: Relationship between average normal contact force and average mass flow rate. The screw rotational speed is 50 rpm. Figure 13: The axial distribution of the force chain: speed (a) 20 rpm, (b)50 rpm, (c) 80 rpm. \[\overline{\mathrm{F}_{\mathrm{i}}^{\mathrm{n}}}=\frac{\sum_{\mathrm{i=1}}^{ \mathrm{N}}F_{\mathrm{i}}^{\mathrm{n}}}{\mathrm{N}} \tag{6}\]  
\[\sigma=\sqrt{\frac{\sum_{\mathrm{i=1}}^{\mathrm{N}}\left(\overline{\mathrm{F}_{ \mathrm{i}}^{\mathrm{n}}}-\mathrm{F}_{\mathrm{i}}^{\mathrm{n}}\right)^{2}}{ \mathrm{N}-1}} \tag{7}\]  
\[\mathrm{CV}=\frac{\sigma}{\overline{\mathrm{F}_{\mathrm{i}}^{\mathrm{n}}}} \tag{8}\]  
where \(F_{\mathrm{i},r}^{n}\) is the average normal force of sample \(i\) at time \(r\). \(N_{r}\) is the number of time, and it is set 0.1 s in this study. \(F_{i}^{n}\) is the average normal force of sample \(i\) in time, and its unit time is 19 s-25 s. \(N\) is number of samples, and it is set 20 in this study. \(\overline{F_{i}^{n}}\) is the average normal force of samples. \(\sigma\) is the standard deviation of samples. \(CV\) is the coefficient of variation. The CV of force chains under different rotational speeds is illustrated in Figure 14. As the rotational speed increases, the CV exhibits a declining trend. The qualitative results of the axial distribution of force chains in Figure 13 and the quantitative CV results in Figure 14 are generally consistent. Given that force and energy transmission among particle swarms occur through contact, the coordination number is a crucial index for studying the contact characteristics of particle systems. It directly reflects the contact number [36]. The average coordination number of all particles in the barrier zone under different rotational speeds was used to effectively investigate the force transmission among particles. As depicted in Figure 15, the average coordination number decreases with the increase in rotational speed. This decline is attributed to the increase in centrifugal force and the strengthened disturbance of helical blades to the particle swarm, which is associated with the higher rotational speed. The axial distribution diagram of force chains clearly shows a significant increase in force chains and a more uniform distribution with the rise in rotational speed. Although the average coordination number decreases, the residual particles are more tightly connected with the increased force chains, leading to more consistent movements. This observation aligns with the axial distribution of force chains. In summary, rotational speed influences the axial distribution of force chains, as evidenced by differences in CV.