## 3 Results and Discussion
### The Circumferential Distribution of Force Chains  
To visualize the circumferential distribution of force chains in the barrier zone, four layers (sizes indicated in Figure 2b) were divided within the circumferential range after the particles reached dynamic stability. The dotted circle, from the outer layer to the inner layer, represents the boundary of the screw shell and the boundary of the screw axis (Figure 16). From Figure 16, it is evident that the force chains above the screw shell exhibit smaller sizes compared to those below (Figure 16a). Although the circumferential distribution of force chains under other conditions is not presented here, the described situations are representative. This suggests a clear distribution gradient of force chains along the circumferential direction in the screw drainer. At a rotational speed of 20 rpm, the force chains are generally small and exhibit a uniform distribution. When the rotational speed is increased to 50 rpm, the strength of the force chains significantly increases, and the distribution becomes uneven. There are no obvious changes at 80 rpm, and the circumferential distribution of the force chains remains consistent with that at 50 rpm. To quantify the uniformity of the circumferential distribution of force chains, the uniformity index (UI) was introduced. The data extraction method is depicted in Figure 2b. Initially, the average normal contact force was extracted and normalized. Subsequently, the standard deviation of the normalized average contact force in the Y-Z plane and its mean  
Figure 16: Circumferential distribution of force chains: spherical particles, speed of (a) 20 rpm, (b) 50 rpm, and (c) 80 rpm. Figure 15: The average coordination number of different rotational screw speeds. were calculated. Finally, UI was obtained. It is important to note that a lower UI indicates a more uniform contact force distribution. The calculation formula for the UI is:  
\[\mathrm{F_{ij}^{n^{\prime}}}=\frac{\mathrm{F_{ij}^{n}-F_{min}^{n}}}{\mathrm{F_{ max}^{n}-F_{min}^{n}}} \tag{9}\]  
\[\overline{\mathrm{F_{i}^{n^{\prime}}}}=\frac{1}{\sum_{i=1}^{m}n}\sum_{i=1}^{m }\sum_{j=1}^{n}\mathrm{F_{ij}^{n^{\prime}}} \tag{10}\]  
\[\mathrm{SD_{i}}=\sqrt{\frac{\sum_{j=1}^{n}\left(\mathrm{F_{ij}^{n^{\prime}}- \overline{F_{i}^{n^{\prime}}}}\right)^{2}}{n-1}} \tag{11}\]  
\[\overline{\mathrm{SD}}=\frac{1}{m}\sum_{m=1}^{m}\mathrm{SD_{i}} \tag{12}\]  
\[\mathrm{UI}=\frac{\overline{\mathrm{SD}}}{\overline{\mathrm{F_{i}^{n^{\prime} }}}} \tag{13}\]  
where \(F_{ij}^{n}\) is the average normal force of sample \(j\) in the layer \(i\). \(F_{max}^{n}\) and \(F_{min}^{n}\) are the maximum average normal force and minimum average normal force of all particles in each layer, respectively. \(F_{ij}^{n^{\prime}}\) is the normalized normal force of sample \(j\) in the layer \(i\). \(m\) and \(n\) are the number of divided sample layers and the number of samples in each layer. \(\overline{F_{i}^{n^{\prime}}}\) is the normalized average normal force of layer \(i\). \(SD_{i}\) is the standard deviation of all particles in layer \(i\). \(\hat{SD}\) is the average standard deviation of all layer numbers. \(\hat{UI}\) is the uniformity index of particles. The relationship between the rotational speed and UI is illustrated in Figure 17. With the increase in rotational speed, the UI initially increases and then declines. At a rotational speed of 20 rpm, the force chains exhibit a relatively uniform circumferential distribution, indicating a small difference in the normal contact force within the same layer and a small difference in the normal contact force between adjacent layers. It is crucial to note that the variation in the UI aligns with the results in Figure 16.  
The circumferential distribution of force chains is directly related to the energy transmission among particles. Therefore, particles with an axial length of 4 mm were tracked over 25 s, and their kinetic energy was extracted (Figure 18).