## 2 Materials and Methods
### Key Component Parameter Design and Analysis
#### 2.3.2 Effect of the Spike Teeth Parameters on Material Impacting through the Screener  
Four rows of spike teeth were designed on the rotating axis of the differential tremmel screen, and the spike teeth were installed in a staggered manner at 90\({}^{\circ}\) to prevent the BSF sand materials from being affected by the axial side-by-side obstruction of the adjacent spike teeth (Figure 8). Above, \(a\), with a value of 1.0 m, expresses the length of the rotating axis of the tremmel screener. Given the body parameters of the BSFL, the teeth trace distance \(l_{1}\) is 15 mm, which is less than the average body length of the BSFL, which is beneficial for loosening the BSFL and sand materials. The radius \(R_{2}\) of the rotation axis, which was a solid round tube in our experiments, was 12.5 mm. The number of spike teeth increased, and the frequency with which the spike teeth impacted the BSFL increased with the excessively small spike teeth spacing \(b\), thus reducing the contact time with the screener surface. Since the separation object was the BSF sand, a staggered design of 90\({}^{\circ}\) was selected for the spike teeth, i.e., \(b\) was 19.625 mm to reduce the repeated impact on the material in the radial direction of the screener and to increase the drop flow frequency. Simplifying the collision between the spike teeth and the BSF sand particles, the analytical model of the collision between the BSF sand particles and the spike teeth is illustrated in Figure 9.  
As shown in Figure 9, a coordinate system was built with the rotation center O of the tremmel screener as the origin. The horizontal direction was the \(x\)-axis, and the vertical direction was the \(y\)-axis. OB was simplified as the spike teeth of the tremmel screener. The spike teeth rotated clockwise around the point O at an angular velocity \(w_{1}\). The tremmel screener rotated counterclockwise at an angular velocity \(w\). The circle O\({}_{1}\) was the simplified BSF sand particles. After the agglutinate BSF sand particles collided with the spikes, they moved obliquely at the speed \(v\). The spikes were fixed on the built-in rotating shaft of the tremmel screener. Driven by the motor, the motion state of the spike teeth remained unchanged after the collision. Above, OC is the position of the spike teeth when the spike teeth collide twice with the BSF sand, while (\(x_{o}\), \(y_{o}\)) denote the coordinates at the collision  
Figure 8: Arrangement of the nail teeth. Note: \(a\) denotes the length of the rotating axis of the tremmel screener, m; \(l_{1}\) represents the teeth trace distance, m; \(R_{2}\) is the radius of the rotation axis, m; and \(b\) expresses the spike teeth spacing, m.  
point A. After the collision, the agglutinate BSF sand particles move obliquely at the speed \(v\). At this time, the motion trajectory equation of the BSF sand particles is expressed as follows:  
\[ \begin{cases}\begin{aligned} x=x_{o}-v\sin\gamma t\\ y=y_{o}+v\cos\gamma-\frac{1}{2}gt^{2}\end{aligned}\end{cases}  \tag{13}\]  
where \(t\) is time, \(s\). The function equation of the angle between the line connecting the BSF sand particles and the center of rotation and the horizontal \(x\)-axis is written as follows:  
\[ \begin{aligned} \varphi_{(t)}=\tan^{-1}\frac{y_{o}+v_{1}\cos \gamma-\frac{1}{2}gt^{2}}{x_{o}-v\sin\gamma t}\end{aligned}  \tag{14}\]  
The angle equation between the spike teeth and the horizontal \(x\)-axis direction is written as follows:  
\[ \begin{aligned} \varphi_{t}^{\prime}=\varphi_{(t=0)}-wt \end{aligned}  \tag{15}\]  
When \(t=0\), i.e., \(\varphi_{(t)}=\varphi_{t}^{\prime}\), we can obtain the position at which the spike teeth of the tremmel screener and the BSF sand particles collided for the first time. The BSF sand particles moved in the direction perpendicular to the spike teeth when the BSF sand particles collided with the spike teeth. The speed was slightly greater than the linear velocity in the circumferential direction of the spike teeth.