## 4 Results and Discussion
### Radial Voidage  
The distribution of the center coordinates of all particles in the X-Y plane is shown in Figure 5. It is found that the distribution of particle center coordinates is similar under different vertical pressures, and the center coordinates tend to be organized in circles in the vicinity of the cylindrical wall surface. The distance from the circle formed by the periphery particle coordinates to the cylindrical wall is approximately the same as the particle radius. The degree of random distribution of coordinates increased with distance from the cylindrical wall. The particle center coordinates are more dispersed in the X-Y plane as the vertical pressure increase. Figure 4: Compression characteristics in soybean bulk. Figure 5: Particle center coordinates under different vertical pressures: (**a**) 0 kPa (**b**) 50 kPa (**c**) 150 kPa (**d**) 250 kPa. The radial porosity distribution is an important indicator to verify the packing quality of the DEM. The pore distribution inside the packing structure links with the radial position under the constraint effect of the wall. The soybean packed bed is dense at the location of small radial porosity values. To verify the accuracy of the particle packing structure generated by DEM, the radial porosity distribution under different vertical pressures was compared with the results calculated by [PERSON]. The empirical equation widely used to predict the radial porosity distribution in a cylindrical packed bed was derived by [PERSON] [42] experimentally, with the porosity being expressed as:  
\[\varepsilon(r)\ =\ \left\{ \begin{array}{l}2.14n^{2}-2.53n+1,n\leq 0.637\\ \overline{\varepsilon}+0.29e^{-0.6n}\times\cos[2.3\pi(n-0.16)]+0.15e^{-0.9n},n>0.637\end{array} \right. \tag{5}\]  
where \(\varepsilon(r)\) is the radial porosity of the packed bed; \(n\) is the wall dimensionless distance; \(n\ =\ (R-r)/d_{p}\); \(r\) is the distance from the radial position of the packed bed to the central axis; \(d_{p}\) is the particle radius; \(\overline{\varepsilon}\) is the central porosity of the packed bed. The particle-packed region is divided into multiple coaxial cylindrical surfaces along the radial direction. The porosity is the ratio of the fluid area intercepted by the cylindrical surface to the cylindrical area, and the calculation formula can be expressed as [43]:  
\[\varepsilon(r)\ =\ \frac{A_{\text{void}}(r)}{2\pi rH} \tag{6}\]  
where \(A_{\text{void}}(r)\) is the fluid area, m\({}^{2}\); \(\varepsilon(r)\) is the radial porosity of the packed bed. Figure 6 shows that the radial porosity distribution of the packed bed calculated using DEM and [PERSON]'s formula is in close agreement. The radial porosity distribution of the packed bed at different vertical pressures was similar, and the maximum radial porosity was about 1, near the cylindrical wall (\(n=0\)), owning to the point contact between the cylindrical wall and the particle. The fluid area intercepted by the coaxial cylindrical surface was approximately the entire cylindrical area. With increasing dimensionless distance, the minimum radial porosity was 0.25 at about the distance of the particle radius from the cylindrical wall, which is the same location as the center coordinate distribution near the cylindrical wall. The radial porosity oscillates and decreases toward the center of the packed bed, with a smaller amplitude toward the center, eventually approaching the average porosity of the packed bed. From Figure 6, the porosity of the center of the packed bed in the uncompressed state (0 kPa) was \(\overline{\varepsilon}\approx 0.43\). The central porosity of the packed bed was 0.39, 0.36, and 0.35, respectively, in 50, 150, and 250 kPa compression states. The amplitude of radial porosity oscillation decreases with the increase in vertical pressure.