## 4 Results and Discussion
### Pressure Drop
#### 4.4.2 Numerical Simulation of Pressure Drop  
The Ergun [44] equations and Forchheimer [45] equations (as Equations (7) and (8), respectively), are the most widely used methods for calculating the flow resistance coefficient and pressure drop in porous media. A PathFinder code was developed by [PERSON] [46] to calculate pore-scale features based on DEM simulations of particle-packed structures. The center coordinates of each particle were calculated by a C program with an accuracy of 10  
Figure 8: Radial mean dimensionless velocity under different vertical pressures. Figure 9: Pressure drop test values under different vertical pressures. and converted to a format readable by PathFinder code. In this study, the changes in airflow paths during soybean packing and compression were recreated using the PathFinder code. The flow resistance coefficient term in the Forchheimer equation was determined based on the DEM particle information at different vertical pressures. The pressure drop results from numerical simulations and equations were compared with the experimental results for verification to verify model accuracy. The results of the PathFinder code calculation are shown in Table 2.  
Ergun Equation:  
\[-\frac{\Delta P}{L}\ =\ 150\mu\frac{(1-\epsilon)^{2}}{\epsilon^{3}}\frac{u}{{d_{p}} ^{2}}+1.75\rho\frac{(1-\epsilon)}{\epsilon^{3}}\frac{u^{2}}{d_{p}} \tag{7}\]  
Forchheimer Equation:  
\[-\frac{\Delta P}{L}\ =\ A\cdot\mu\cdot u+B\cdot\rho\cdot u^{2} \tag{8}\]  
where \(\mu\) is the dynamic viscosity of the air, \(\mu\) = 1.79 \(\times\) 10\({}^{-5}\) kg/(m-s); \(\rho\) is the density of the air, \(\rho\) = 1.025 kg/m\({}^{3}\); \(u\) is the inlet velocity; \(A\) is the expression for the permeability coefficient in the Kozeny-Carman equation, \(A\ =\ \frac{1}{L}\ =\ C_{kz}\cdot\tau\cdot S_{0}^{2}\cdot\frac{(1-\epsilon)^{2}}{ \epsilon^{3}}\), \(C_{kz}\) is the Kozeny-Carman pore shape factor, \(C_{kz}\) = 5.0; \(\tau\) is the tortuosity; \(S_{0}\) is the the specific surface of the packed bed. A total of nine inlet points in the center and periphery of the packed bed were selected for airflow path calculation. The airflow path distribution calculated by PathFinder software is shown in Figure 10. The shape and length of the airflow path change with the vertical pressure. The deviation from the bottom inlet point to the top outlet increases with the vertical pressure, and the airflow path becomes more tortuous, with continuous bends in the airflow path at 250 kPa. As the vertical pressure increases, the particles move toward the pore area, making the packed bed more compact and the pore connection path curved and narrow. Compared to the central and peripheral airflow paths, the curvature of the path near the peripheral region was more intense as the vertical pressure increased. The path curvature due to vertical compression can visually reflect the phenomenon of increased pressure drop in the packed bed. The pressure drop results simulated by the DEM-CFD method for different inlet velocities and vertical pressure conditions are shown in Figure 11, and are being compared with the empirical equations and experimental results. The calculated results of the DEM-CFD simulation and the Forchheimer equation under different vertical pressures were in close agreement with the experimental results, and the average difference was less than 10%. The Ergun equation underestimates the pressure drop values clearly, with an average difference of more than 24% in the uncompressed state (0 kPa)