# Ballast Flow Characteristics of Discharging Pipeline in Shield Slurry System
## 2 Simulation Model of Ballast Motion in Slurry Discharging Pipeline
### Ballast Motion Model  
In the model, the ballast shape is a non-spherical particle in which the large type is flat chip, and the small type tends to be spherical. A random sample was used for the ballast within the tunneling distance from 120 to 1800 m (Figure 3a). Figure 3b shows the grading diagram of the ballast size that was obtained by the screening method. The figure shows that about 80% of the ballast is within 10-40 mm and the density of the ballast is between 2500 and 2750 kg/m\({}^{3}\). We used the Wadell [20] shape coefficient to define the shape of ballast, which is defined, as follows:  
\[\varphi_{b}=A_{S}/A_{b} \tag{4}\]  
where \(\varphi_{b}\) is the ballast shape coefficient, \(A_{S}\) is the surface area of a sphere of the same volume of the ballast, and \(A_{b}\) is the surface area of the ballast. Based on the field investigation, the shape coefficient of the ballast transported in the slurry system was mainly in the range of 0.7 to 0.9. Table 2 provides the three-dimensional size, volume, and surface area of ballast. \begin{table}
\begin{tabular}{c c} \hline \hline
**Parameter** & **Value** \\ \hline \(k\) & 2.10 \\ \(n\) & 0.22 \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Slurry rheological parameters. \begin{table}
\begin{tabular}{c c c c c c c c} \hline \hline
**Sample** & **L (mm)** & **W (mm)** & **H (mm)** & \(d_{e}\) (mm)** & \(V_{b}\) (m\({}^{3}\)) & \(A_{b}\) (m\({}^{2}\)) & \(\varphi_{b}\) \\ \hline a & 19 & 14 & 11.5 & 17 & \(2.51\times 10^{-6}\) & \(4.58\times 10^{-4}\) & 0.8 \\ b & 30 & 24.5 & 11.6 & 22 & \(5.57\times 10^{-6}\) & \(1.12\times 10^{-3}\) & 0.78 \\ c & 33.5 & 25 & 13 & 27 & \(1.04\times 10^{-5}\) & \(3.0\times 10^{-3}\) & 0.77 \\ d & 37 & 37 & 15 & 32 & \(1.68\times 10^{-5}\) & \(4.29\times 10^{3}\) & 0.75 \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Ballast shape parameters. Figure 3: Ballast size distribution: (**a**) the ballast sample collected on site and (**b**) the ballast size grading diagram. We divided the non-spherical ballast into four types on the basis of particle size, which is formed by the accumulation of 4 mm spherical particles. Figure 4 depicts the particle size of the ballast and the number of accumulated spherical particles. During the numerical solution process, the ballast is regarded as a discrete phase and the ballast stress follows [PERSON]'s second law. The ballast movement in the slurry system consists of the translation motion and rotation motion. The stress model of the \(j\) th ballast in the slurry is [21]:  
\[m_{j}\frac{d}{dt}(\mathbf{u}_{j})=m_{j}\Big{(}1-\rho_{f}/\rho_{j}\Big{)}g+ \sum_{i=1}^{n}\left(\mathbf{F}_{c,i}^{j}+\mathbf{F}_{d,i}^{j}\right)+\left( \mathbf{F}_{c,i}^{w}+\mathbf{F}_{d,i}^{w}\right)+\mathbf{F}_{j}^{f} \tag{5}\]  
The moment balance model of the \(j\) th ballast during rotation is:  
\[\frac{d}{dt}\Big{(}\mathbf{I}_{j}\omega_{j}\Big{)}=\sum_{i=1}^{n}(\mathbf{T}_ {i}^{j})+\mathbf{T}_{w}^{j}+\mathbf{T}_{DT}^{j} \tag{6}\]  
where \(m_{j}\) represents the mass of the \(j\) th ballast; \(\mathbf{u}_{j}\) and \(\omega_{j}\) represent the translational and rotational velocity of the \(j\) th ballast, respectively; \(\mathbf{I}_{j}\) represents the inertial mass of the \(j\) th ballast; \(\mathbf{F}_{c,i^{\prime}}^{j}\), \(\mathbf{F}_{d,i^{\prime}}^{j}\) and \(\mathbf{T}_{i}^{j}\) represent the contact force, viscoelastic force, and torque generated by \(j\) th ballast and the surrounding ballast \(i\) (\(i=1\), \(2...\), \(n\)), respectively; \(\mathbf{F}_{c,i^{\prime}}^{w}\), \(\mathbf{F}_{d,i^{\prime}}^{w}\) and \(\mathbf{T}_{w}^{j}\) represent the contact force, viscoelastic force, and torque generated by the interaction between the \(j\) th ballast and the inner wall of the pipe, respectively; and, \(\mathbf{T}_{DT}^{j}\) represents the moment of motion of the \(j\) th ballast. The contact force, viscoelastic force, and stress moment between the ballast and pipeline wall and ballast follow the Hertz-Mindlin no-slip model [22].