## 3 Discharge Simulation
### Numerical Analysis of Discharge Process Using Coupled Eulerian-Lagrangian (CEL) Method  
From the numerical analysis point of view, there are several Lagrangian analysis and Eulerian analysis methods to simulate the behavior of an element with a volume over time. The Lagrangian analysis method defines the behavior of the continuum as a function of the physical coordinates and the time of each element. Furthermore, because the nodes of each mesh move with the properties of the elements, the nodes must be accurately shared on the contact surface between the two objects (Figure 7a).  
\begin{table}
\begin{tabular}{c c} \hline \hline
**Parameters** & **Basic Values** \\ \hline Silo height, H (m) & 23.515 \\ Silo diameter, d\({}_{\mathrm{c}}\) (m) & 8.7 \\ Orifice diameter, d\({}_{0}\) (m) & 0.786 \\ Silo hotspot detector diameter, d\({}_{\mathrm{t}}\) (m) & 0.035 \\ Half angle of hopper, \(\beta\) (deg) & 20.65 \\ Bulk density, \(\rho\) (kg/m\({}^{3}\)) & 1019 \\ Young’s modulus, E (MPa) & 16 \\ Poisson ratio, v & 0.3 \\ Internal friction angle, \(\varphi\) (deg) & 35.96 \\ Dilatancy angle, \(\psi\) (deg) & 2.5 \\ Cohesion, c (Pa) & 0 \\ Wall friction, \(\mu\) & 0.49 \\ \hline \hline \end{tabular}
\end{table}
Table 3: Parameters used in finite element method (FEM) simulation.  
Figure 7: Deformation of a continuum in **(a)** Lagrangian and **(b)** Eulerian analyses [31].  
The Eulerian analysis method defines the behavior of a continuum as a function of spatial coordinates and time. Here, the behavior of an object is defined as the amount of movement of a substance within a specific region (Figure 7b). Therefore, there is an advantage in that the distortion or warping of the mesh does not occur during Eulerian analysis. Therefore, in this study, the silo discharge process was analyzed using the CEL analysis method.  
Figure 8 shows the calculation procedure of the CEL analysis method. In the CEL analysis method, elements that generate a large strain are processed by the Eulerian technique, and the remaining elements are processed by the Lagrangian technique. As shown in Figure 8, the volume fraction tool created a discrete field based on the Eulerian part. The Eulerian volume fraction (EVF) was calculated between the Eulerian part and the reference part. The EVF had a value between \"0\" and \"1,\" which represented the percentage of the area occupied by the specified material. Materials were thus assigned to predefined fields, and material boundaries were determined based on the EVF of all elements.  
Areas that were not defined in the Eulerian part were designated as voids. These voids had no material properties at first; however, as the analysis proceeded further, the material flowed or passed through the void region. During each calculation step of the CEL method, the calculation results were stored in each node and mapped to the computational mesh (step 2, Figure 8b). Subsequently, unknown variables were calculated using the Lagrangian formula; the velocity, acceleration, and displacement of each node were updated using these values (step 3, Figure 8b), and the stress and strain were calculated by the applied structural model. After mesh adjustment, the position of the node was updated (step 4, Figure 8b). To solve the Eulerian step, the deformed mesh was remapped to the original fixed mesh, and the material volume between adjacent elements was calculated by the EVF (Figure 8c). Finally, a Lagrangian step process was performed to explain phenomena such as energy, stress, and strain between adjacent elements, as shown in Figure 8b.