## 2 Materials and Methods
### Simulation Conditions and Process  
Urea fertilizers are widely utilized due to their high nitrogen content, cost-effectiveness, and rapid nutrient supply to crops upon application [25]. Consequently, urea fertilizers were selected as the test materials for this study. To obtain accurate measurements, a micrometer with a precision of 0.01 mm was employed to measure the length, width, and height of 100 fertilizer particles. The recorded results were 2.55 mm, 2.50 mm, and 2.16 mm, respectively. Using Equations (3) and (4), the equivalent sphere diameter and sphere rate of the fertilizer particles were calculated as 2.40 mm and 94.54%, respectively. \[\mathrm{D}=\sqrt[3]{\mathrm{LWT}} \tag{3}\]  
\[\varphi=\frac{\mathrm{P}}{\mathrm{L}}\times 100\% \tag{4}\]  
where \(D\) is the equivalent diameter (mm). \(L\) is the length of fertilizer particles (mm). \(W\) is the width of fertilizer particles (mm). \(T\) is thickness of fertilizer particles (mm). \(\varphi\) expresses the sphere rate (%). A previous study established that the particle model could be represented as a sphere when the exceeded sphere rate of fertilizer particles 90% [26]. Consequently, for this study, the fertilizer particle model was simplified as spheres with a diameter of 2.40 mm (Figure 1). It is essential to highlight that, in this project, fertilizer particles were simplified as breakage-free, wear-free, liquid bridge-free, and monodispersed. \begin{table}
\begin{tabular}{c c c} \hline \hline
**Force and Torque** & **Symbol** & **Equation** \\ \hline Normal force & \(\mathrm{F}_{\mathrm{c,ij}}^{\rightarrow\mathrm{n}}\) & \(\frac{3}{4}\mathrm{E}^{\mathrm{eq}}\sqrt{\mathrm{r}_{\mathrm{\bar{\eta}}}^{ \mathrm{eq}}\mathrm{s}_{\mathrm{\bar{\eta}}}^{3}}\) \\  & \(\mathrm{\overset{\rightarrow}{\rightarrow}}\) & \(-2\sqrt{\frac{5}{6}}\varphi\sqrt{\mathrm{S}^{\mathrm{m}}\mathrm{r}_{\mathrm{ \bar{\eta}}}^{\mathrm{eq}}\mathrm{\vec{v}}_{\mathrm{\bar{\eta}}}^{\mathrm{m}}}\) \\  & \(\mathrm{\overset{\rightarrow}{\rightarrow}}\) & \(-\mathrm{St}_{\mathrm{\bar{\eta}}}^{\mathrm{st}}\) \\ Tangential force & \(\mathrm{F}_{\mathrm{c,ij}}\) & \(-\mathrm{St}_{\mathrm{\bar{\eta}}}^{\mathrm{st}}\) \\  & \(\mathrm{F}_{\mathrm{d,ii}}\) & \(-2\sqrt{\frac{5}{6}}\varphi\sqrt{\mathrm{S}^{\mathrm{m}}\mathrm{r}_{\mathrm{ \bar{\eta}}}^{\mathrm{eq}}\mathrm{\vec{v}}_{\mathrm{\bar{\eta}}}^{\mathrm{st}} }\) \\  & \(\mathrm{\overset{\rightarrow}{\mathrm{T}}}^{\mathrm{st}}\) & \(\mathrm{\overset{\rightarrow}{\mathrm{r}}}_{\mathrm{i}}\times\left(\mathrm{ \overset{\rightarrow}{\mathrm{F}}}_{\mathrm{c,ij}}^{\mathrm{st}}+\mathrm{ \overset{\rightarrow}{\mathrm{F}}}_{\mathrm{d,ij}}\right)\) \\ Torque & \(\mathrm{\overset{\rightarrow}{\mathrm{T}}}^{\mathrm{sn}}\) & \(\mathrm{\overset{\rightarrow}{\mathrm{T}}}_{\mathrm{i}}\times\left(\mathrm{ \overset{\rightarrow}{\mathrm{T}}}_{\mathrm{i}}\right)+\mathrm{\overset{ \rightarrow}{\mathrm{T}}}_{\mathrm{i},\mathrm{\bar{\eta}}}^{\mathrm{sn}}\) \\  & \(\mathrm{\overset{\rightarrow}{\mathrm{T}}}_{\mathrm{r}}\) & \(-\mathrm{\overset{\rightarrow}{\mathrm{\mu}}}_{\mathrm{r}}\mathrm{F}_{\mathrm{c, ij}}\overset{\rightarrow}{\mathrm{\overset{\rightarrow}{\mathrm{\mu}}}}_{\mathrm{i}} \overset{\rightarrow}{\mathrm{\overset{\rightarrow}{\mathrm{\mu}}}}_{\mathrm{i}} \overset{\rightarrow}{\mathrm{\overset{\rightarrow}{\mathrm{\mu}}}}_{\mathrm{i}}\) \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Force and torque equations used in contact model. In this study, the focus was on the conveying process of the worm-type distributor. Building upon the research findings of [PERSON] [27], the worm-type distributor was segmented into the conveying zone, barrier zone, and distribution zone, which includes the hopper, helix, outlet, and shell. The specific geometric structural parameters are detailed in Figure 2a, and essential parameters were measured and calibrated during the simulation, with the remaining parameters sourced from relevant studies [28; 29]. Refer to Table 2 for more details.