## 2 Materials and Methods
### Structural Size Analysis  
The amount of fertilizer discharged per unit of time via the spiral fertilizer applicator is the fertilizer discharge flow rate, mainly determined by the amount of single-circle fertilizer discharged and the rotational speed of the spiral fertilizer applicator [16]. The fertilizer discharge flow rate needs to be adjusted based on the fertilizer demand of crops. Taking the demand for corn mid-tillage fertilization as an example, after consulting relevant literature and agronomic requirements [17; 18], it is determined that the forward speed \(v_{\rm x}\) is 3 m/s, the width \(T\) of the operation ridge is 0.6 m, and the fertilizer application rate \(R\) per hectare is 375 kg/hm\({}^{2}\). According to Equation (10), the required limit fertilizer discharge \(Q\) is 67.5 g/s. \[Q=\frac{RTv_{x}}{10} \tag{10}\]  
In the formula: \(Q\) is the limit fertilizer discharge, g/s; \(R\) is the amount of fertilizer applied per hectare, kg/hm\({}^{2}\); \(T\) is the width of the homework ridge, m; \(v_{\rm x}\) is the forward speed, m/s. From Equations (11) to (12), it can be seen that when the filling coefficient \(\mu\) and particle volume mass \(\rho\) are fixed, the single-circle fertilizer discharge is mainly influenced by parameters such as the thickness \(b\) of the spiral blade, spiral diameter \(D\), spiral inner diameter \(d\), and spiral pitch \(S\). We determine the thickness \(b\) of the spiral blade as 2 mm, and calculate the spiral diameter \(D\), spiral inner diameter \(d\), and screw pitch \(S\) using Equations (13) to (15) [13; 19]. \[q=\Bigg{[}\frac{\pi(D^{2}-d^{2})S}{4}-b\frac{D-d}{2}L_{p}\Bigg{]}\rho\mu \tag{11}\]  
\[L_{p}=\sqrt{\left[\pi(D+d)/2\right]^{2}+S^{2}} \tag{12}\]  
\[D\geq K2.5\bigg{(}\frac{Q}{\mu\rho C}\bigg{)} \tag{13}\]  
\[d=(0.2\,\sim 0.35D) \tag{14}\]  
\[S=(0.5\,\sim 0.9D) \tag{15}\]  
In the formula: \(q\) is the single-circle fertilizer discharge amount of the spiral fertilizer applicator, g; \(L_{\rm p}\) is the average length of the screw teeth, mm; \(\mu\) is the filling coefficient of the spiral fertilizer applicator, with a value of 1; \(\rho\) is the volumetric mass of the material, in t/m\({}^{3}\), taken as 0.948 t/m\({}^{3}\)[20]; \(Q\) is the required maximum fertilizer discharge, t/h; \(C\) is the inclination coefficient of the spiral fertilizer applicator, as the spiral fertilizer applicator is horizontally placed, the inclination coefficient is taken as 1; \(K\) is the comprehensive coefficient of the material, with a value of 0.045 [21]. According to Equation (13), the minimum required spiral diameter \(D\) is 26.11 mm. Referring to the existing spiral fertilizer applicator, the range of spiral diameter \(D\) is determined to be 40-50 mm, and the inner diameter \(d\) is calculated to be 10 mm through Equations (14) to (15), with the spiral pitch \(S\) range of 20-40 mm. The rotational speed is the decisive factor affecting the fertilizer discharge flow rate, and the fertilizer discharge flow rate, fertilizer discharge uniformity, and filling coefficient all increase with the increase in rotational speed [22]. However, excessive rotational speed can cause the fertilizer particles to roll in the spiral cavity, causing the spiral blades to lose their axial propulsion function. Therefore, it is necessary to calculate the limit rotational speed of the spiral fertilizer feeder to ensure normal fertilizer discharge.