## 1 Introduction  
Soybeans are mostly stored in bulk after harvesting, cleaning, and drying. Temperature and moisture in the grain storage ecosystem are principal factors affecting the security of grain storage. To ensure grain quality, grain bulk must be kept below a safe moisture level and temperature [1, 2]. Effective aeration, temperature control, and humidity control measures are required promptly after monitoring abnormally high local grain temperatures. Aeration, the most commonly utilized method in grain storage technology, maintains the temperature in the grain bulk and avoids the temperature fluctuation caused by convection, grain metabolism, and insect and microbial growth [3, 4]. In the grain bulk, the vertical pressure distribution is different due to the self-weight of grain. The grain is subjected to different levels of compression. The bulk density and porosity change with vertical pressure [5, 6, 7]. The process of aeration is to force external air into the grain bulk and thus reduce the grain temperature. However, airflow through the grain is hindered. Airflow resistance is the pressure drop per unit of grain depth due to friction and turbulence. Therefore, the fan power in the aeration system needs to overcome the airflow resistance through the grain bulk. Fans that are selected incorrectly lead to uneven cooling or increased aeration costs [8]. Specifying the airflow resistance through grain bulk is the basis of aeration system design. Granular materials have creeping compression characteristics, which are more obvious under vertical pressure. As a typical porous medium, grain bulk has an irregular pore structure, making airflow characteristics more complex and variable [9]. [PERSON] et al. [10] found that the long axis of grain particles is more inclined in the horizontal direction after they enter the bin, which results in a higher porosity in the horizontal direction as compared to the vertical direction, leading to a difference in airflow resistance in the horizontal and vertical direction. [PERSON] [11] showed that the bulk density and porosity change with the spatial position of the grain bulk, and the pressure drop considering the change in porosity will be higher than the value calculated by the Shedd equation. The pressure drop through grain bulk is greatly affected by the pore characteristics. For grain aeration technology, it is critical to determine the pressure drop through grain bulk under vertical pressure. Pressure drop in rice [12; 13; 14], wheat [15; 16], soybeans [17], corn [18], rape [19], and other crops has been studied using a variety of variables such as airflow velocity, bed depth, moisture content, filling mode, and theoretical models of pressure drop calculation [20; 21]. [PERSON] et al. [22] found that since the grain is compacted with the increase in depth, the airflow velocity and pressure drop through the grain bulk change significantly with the heterogeneous pore structure. [PERSON] et al. [23] showed that the airflow resistance through oat grains increased with the external load and loading time. [PERSON] et al. [24] and [PERSON] et al. [25] showed that the porosity decreased under vertical pressure, which increased the airflow resistance of the grain bulk and hindered the aeration operation. [PERSON] et al. [26] verified that the pore structure in grain bulk is more complex and tortuous after compacting, resulting in higher airflow resistance. Experimental and theoretical formula calculation methods are often used to study the pressure drop of the grain bulk, but there are few accurate and efficient numerical calculation models. [PERSON] et al. [27] obtained a nonlinear model by the finite element method to study the airflow distribution in a three-dimensional grain-packed bed. [PERSON] et al. [28] simulated the airflow distribution in the bin, after considering the inhomogeneity of porosity at different depths. But the above numerical calculation model cannot reflect the real pore structure distribution of packed grain particles.