# Ballast Flow Characteristics of Discharging Pipeline in Shield Slurry System
## 2 Simulation Model of Ballast Motion in Slurry Discharging Pipeline
### Numerical Model of Slurry System Pipe  
The pipeline section being studied starts at the outlet of the P2.3 pump, ends at the underlying surface, and consists of three parts: inclined pipeline, horizontal or nearly horizontal pipeline, and vertical pipeline. The P2.3 pump is located at the 259 segment rings away from the tunnel entrance, and the width of each segment ring is 1.2 m. The length of the inclined pipeline is 2.5 m, the vertical pipeline is 21 m, and the horizontal pipeline is over 300 m, which account for over 90% of the total length of the slurry pipeline system. Figure 5 marks the corresponding structures. In line with the calculation capacity of the workstation and research requirements, when considering the similarity of the ballast flow characteristics in the extension part of the pipeline, the size of horizontal pipeline was reduced, while the sizes of the vertical and the inclined pipelines were retained, and Table 4 shows the structural parameters of each pipeline section.  
The numerical model of the slurry pipeline system is divided into three parts: inlet, outlet, and wall. Figure 5 indicates the slurry flow direction and gravity direction. The inlet surface of the pipeline is set as the velocity inlet. We assumed that the velocity of this surface is evenly distributed along the section of the pipe, and the inlet velocity was determined in line with the actual condition. The outlet surface was set as the free stream. The non-slip wall model was adopted. The numerical model adopts the hexahedral structured mesh division. For calculation accuracy, the minimum mesh size must be larger than the particle size. The smallest mesh division size was 10 mm and the number of mesh divisions was 66,056.  
Figure 6 shows the flow chart of the CFD-DEM coupling procedure. Firstly, the pressure and velocity in the fluid were solved using the CFD model. Subsequently, the CFD results were transferred to the two-way coupling method (CFD-DEM) to calculate the forces on the particles, such as the drag force F\({}_{D}\), rotation lift force F\({}_{M}\), and pressure gradient force F\({}_{P}\). The position and velocity of particles were solved and updated based on [PERSON]'s second law. Finally, the particle volume information and  
\begin{table}
\begin{tabular}{c c c} \hline \hline
**Parameter** & **Value** & **Unit** \\ \hline \(D_{p}\) & 0.3 & m \\ \(L_{1}\) & 2.5 & m \\ \(L_{2}\) & 30 & m \\ \(L_{3}\) & 21 & m \\ \(\alpha\) & 45 & \({}^{\circ}\) \\ \hline \hline \end{tabular}
\end{table}
Table 4: Slurry system pipeline structure parameters.  
Figure 5: The numerical model of pipeline in slurry system.  
forces on the fluid from particles were transferred to the CFD solver. This is a complete calculation process and subsequent calculations are performed in this cycle.  
The slurry flow model was solved while using the semi-implicit method for pressure linked equations (SIMPLEC) algorithm, and the ballast motion model was solved using the explicit time integral method (central difference method). The time steps were set to 0.005 s and 0.0001 s in CFD (Fluent12.0, ANSYS Inc., USA and 2013) and DEM software (EDEM2.2, DEM Solutions Ltd., Britain and 2014), respectively, to ensure the model convergence. The simulation time of each working condition was 50 s, with a total of 64 groups. Each group required nearly 14 h on average for the data processing. The calculation was conducted by a 32-core Intel Gold 6103 (2.2 GHz) and 128 GB RAM workstation computer, which required approximately 45 days in total.