# Performance Analysis and Testing of Spiral Quantitative Fertiliser Distributors in Orchards
## 2 Materials and Methods
### Design of Critical Parts
#### 2.2.3 Fertilizing Apparatus  
The fertilizing apparatus comprised a steady-flow screw, a conveyor screw and fertilizer tanks (Figure 3). The stead-flow screw was able to stabilize fertilizer flows and deliver fertilizer to the lower tank quantitatively. Next, the conveyor screw uniformly discharged fertilizers into the soil. The single-ring fertilizer quantity of the conveyor screw is an essential index for measuring the fertilizer distribution performances of two-level spiral fertilizer distributors, and it reflects the stability and uniformity of the fertilizer distribution [17]. Without consideration of axial resistance and rotation speed, the calculation formula for single-ring fertilizer quantity is expressed as follows:  
\[q=\Bigg{[}\frac{\pi\big{(}D^{2}-d^{2}\big{)}S}{4}-bhL\Bigg{]}\rho\varphi \tag{1}\]  
\[L=\sqrt{[\pi(D+d)/2]^{2}+S^{2}} \tag{2}\]  
\[h=(D-d)/2 \tag{3}\]  
where \(D\) is the outer diameter of the conveyor screw (mm), d is the inner diameter of the conveyor screw (mm), \(S\) is the pitch of the spiral fertilizer distributor (mm), \(b\) is the average thickness of the screw thread of the spiral fertilizer distributor (mm), \(h\) is the depth of the screw thread of the spiral fertilizer distributor (mm), \(L\) refers to the average length of the screw thread of the spiral fertilizer distributor (mm), \(\rho\) is the unit weight of fertilizers (g/mm\({}^{3}\)), and \(\varphi\) is the filling coefficient of the spiral fertilizer distributor. Equation (1) shows that the volume of discharged fertilizers in a single ring (\(q\)) is determined by \(D\), \(d\), \(S\), \(P\) and \(\varphi\). Hence, it changes with \(D\), \(d\) and \(S\). The relationship between \(D\) and fertilizing quantity is defined as follows:  
\[D=K\bigg{(}\frac{Q}{\varphi\lambda\varepsilon}\bigg{)}^{\frac{3}{\delta}} \tag{4}\]  
\[K=\bigg{(}\frac{1}{47 cA}\bigg{)}^{\frac{2}{\delta}} \tag{5}\]  
where \(Q\) refers to the spiral fertilizing quantity (t/h); A is the comprehensive characteristic coefficient of the materials; \(K\) is the comprehensive coefficient of the materials; \(c\) is the proportional coefficient of the pitch and diameter; \(\lambda\) is the unit volume mass of the materials (t/m\({}^{3}\)); and \(\varepsilon\) is the conveying coefficient. Figure 3: Fertilizing apparatus: 1. fertilizer tanks; 2. steady-flow screw; 3. conveyor screw. The fertilizing quantity of the fertilizer distributor under continuous operation is expressed as follows:  
\[Q_{S}=v_{m}g/s \tag{6}\]  
where \(Q_{S}\) is the fertilizing quantity, \(g\) is the fertilizing quantity after advancing for some distance and \(s\) is the advancing distance. During the fertilizer transmission of the fertilizer distribution device, affected by the rotation of the conveyor screw, fertilizers move in a compound motion along the spiral axis rather than making a simple straight motion along the axis line [21]. To avoid the rolling of fertilizers perpendicular to the conveying direction rather than in an axial conveying motion because of the excessively high rotation speed, the relationships of the spiral rotation speed of the conveyor with the outer diameter and fertilizing quantity while meeting the fertilizing quantity are expressed as follows:  
\[n_{max}\leq A/\sqrt{D} \tag{7}\]  
\[Q_{c}=47D^{2}S\varphi\lambda\varphi n_{min} \tag{8}\]  
First of all, consider the fertilizer required for each square meter of fruit trees and the forward speed to find out the amount of fertilizer required, combined with the physical characteristics of the fertilizer itself (fertilize filling coefficient \(\varphi\), the integrated material properties of the coefficient \(A\), integrated coefficients \(K\) and the mass of fertilizer per unit of volume \(\lambda\)) to deduce the design parameters of the conveying screw. Secondly, the conveying screw parameters are determined by the amount of fertilizer discharged through a single circle and the amount of single-circle discharge from the fertilizer, to arrive at the design parameters of the steady-flow screw.