## 2 Materials and Methods
### Analysis of the Movement Status of Fertilizer Particles  
To analyze why the spiral cavity cannot be filled when the speed of fertilizer particles is low, the movement status of fertilizer particles is analyzed as shown in Figure 3.  
In Figure 3, \(D\) is the spiral diameter, mm; \(d\) is the spiral inner diameter, mm; \(d_{1}\) is the distance between the particles at point \(O\) and the axis, mm; \(b\) is the thickness of the spiral blade, mm; \(S\) is the spiral pitch, mm; \(F_{\rm t}\) is the axial force acting on the fertilizer particles, N; \(F_{\rm r}\) is the normal phase thrust on the particle, N; \(F\) is the combined force on the particles, N; \(F_{\rm w}\) is the tangential friction force, N; \(F_{\rm q}\) is the circumferential component force on the particle, N; \(\beta\) is the spiral rise angle at the position of the particle, \(\circ\); \(\gamma\) is the friction angle of particles on the spiral surface, \(\circ\); \(V_{\rm t}\) is the axial velocity of particles; \(V_{\rm r}\) is the absolute velocity of particles (without considering friction); \(V\) is the absolute velocity of particles (considering frictional force); \(V_{\rm q}\) is the tangential velocity of particles; \(V_{\rm w}\) is the circumferential velocity of the particles.  
Figure 1: Trend of filling state changes when rotational speed changes: (**a**) Low rotational speed state; (**b**) Medium rotational speed state; (**c**) High rotational speed state: 1. Fertilizer particle volume, 2. Remaining cavity volume, 3. Spiral blade.  
Figure 2: Filling coefficient variation curve.  
The circumferential and axial forces acting on the particles at point \(O\) are:  
\[F_{q}=F\sin(\beta+\gamma) \tag{1}\]  
\[F_{t}=F\cos(\beta+\gamma) \tag{2}\]  
The spiral rise angle of particles at point \(O\) is:  
\[\tan\beta=S/\pi d_{1} \tag{3}\]  
The friction angle between the particle and the spiral surface at point \(O\) is:  
\[\gamma=\arctan\mu_{1} \tag{4}\]  
In the formula, \(\mu_{1}\) is the friction coefficient between the spiral surface and the particles.  
Using Equations (1) to (4), it can be determined the magnitude of the spiral rise angle at the particle's position varies with its distance from the axis. When the friction angle on the spiral blade is fixed, the axial and circumferential forces on the particles vary with the size of the spiral rise angle [13; 14; 15]. The particle motion velocity at point \(O\) was analyzed to determine the particles' circumferential and axial motion velocities.  
The circumferential velocity of particles at point \(O\) is:  
\[V_{w}=n\pi r \tag{5}\]  
The absolute velocity of particles without considering friction is:  
\[V_{r}=V_{w}\sin\beta \tag{6}\]  
The absolute velocity of particles when considering the frictional force is:  
\[V=\frac{V_{r}}{\cos\gamma} \tag{7}\]  
The higher the axial velocity of particles, the faster the axial conveying speed in the spiral cavity and the stronger the fertilizer discharge capacity. When the circumferential velocity is high, the rotation of the spiral blade mainly plays a starring role in the particles. Therefore, analyzing the axial and circumferential velocities of particles reveals that:  
\[V_{t}=V\cos(\beta+\gamma)=\frac{n\pi r\sin\beta}{\cos\gamma}\cos(\beta+\gamma) \tag{8}\]  
Figure 3: Characteristics of particle motion in the spiral fertilizer applicator cavity.  
\[V_{q}=V\sin(\beta+\gamma)=\frac{n\pi r\pi\sin\beta}{\cos\gamma}\sin(\beta+\gamma) \tag{9}\]  
Combining Equations (8) to (9), it can be seen that the axial and circumferential forces on particles are mainly affected by the rotational speed \(n\), spiral pitch \(S\), and spiral diameter \(D\). Therefore, the structural size and rotational speed of the spiral blade are calculated.