## 4 Analysis of CFD Simulation Results  
### Analysis of Simulation Results of Non-Clogging Conditions  
According to the Bernoulli equation for fluid transport in pipelines, the energy equation for slurry transport in filling systems is given by the following:  
\[P+\rho gH=\frac{\rho v^{2}}{2}+I, \tag{8}\]  
In the equation, \(P\) represents the pressure provided by the filling pump at the pipeline inlet (Pa); \(P\) denotes the slurry density (kg/m\({}^{3}\)); \(g\) is the gravitational acceleration (9.81 m/s\({}^{2}\)); \(H\) refers to the height difference between the pipeline inlet and outlet (m); \(v\) is the slurry flow velocity at the outlet (m/s); and \(I\) is the total resistance loss in the pipeline (Pa).  
Generally, the pipeline resistance loss consists of two parts [25,26]: the frictional loss along the straight pipe section, \(i_{s}\), and the local resistance loss, \(i_{b}\), which is primarily associated with 90\({}^{\circ}\) elbows [27, 28, 29].  
The frictional resistance loss, \(i_{s}\), along the straight pipe section can be expressed as the product of the total length of the straight section, Ls, and the unit length frictional resistance loss, \(i_{s}\), as shown in the following equation:  
\[i_{s}=\frac{16}{3D}\tau_{0}+\eta\frac{32v}{D^{2}}, \tag{9}\]  
\begin{table}
\begin{tabular}{c c c c} \hline \hline \multirow{2}{*}{**Case Number**} & \multicolumn{2}{c}{**The Degree of**} & \multirow{2}{*}{**Blockage Pattern**} & \multirow{2}{*}{**Blockage Location/m**} \\  & & & \\ \hline
1 & 0\% & not & 0 \\ \hline
2–7 & 25\% & A, B, and C blockages &
\begin{tabular}{c} 13.0 \\ The center of the elbow \\ \end{tabular} \\ \hline
8–13 & 50\% & A, B, and C blockages &
\begin{tabular}{c} 13.0 \\ The center of the elbow \\ \end{tabular} \\ \hline
14–19 & 75\% & A, B, and C blockages &
\begin{tabular}{c} 13.0 \\ The center of the elbow \\ \end{tabular} \\ \hline \hline \end{tabular}
\end{table}
Table 1: Slurry pipeline simulation scheme.  
where \(\tau_{0}\) is the initial yield stress of the fluid, \(D\) is the pipeline diameter, \(\eta\) is the fluid viscosity, and \(\
u\) is the fluid's kinematic viscosity (dynamic viscosity divided by density).  
The flow behavior of slurry in pipe bends is more complex, and in the literature, the local losses in the pipe-bend section are generally considered in conjunction with the overall frictional resistance losses along the pipe. The local resistance is typically estimated as 10% to 20% of the total resistance loss in the pipeline. As part of the pipeline system, the pipe bend shares the same inner diameter and slurry parameters as the straight sections, and its resistance loss can also be calculated based on the unit length. The unit length local resistance loss, \(i_{b}\), of the pipe bend is typically expressed as a multiple of the unit length frictional resistance loss, \(i_{s}\), as follows:  
\[i_{b}=ks_{i}=k\bigg{(}\frac{16}{3D}\tau_{0}+\eta\frac{32\
u}{D^{2}}\bigg{)}, \tag{10}\]  
Under the condition of no blockage, the pressure distribution of the pipeline during normal operation is shown in Figure 10, which includes both the simulation results and theoretical calculations. As shown in Figure 10, the pressure distribution obtained from the software simulation aligns closely with the theoretical calculation results, and the pressure decreases linearly with increasing pipeline length. Overall, the calculated pressure values and the simulation results are in good agreement, with a deviation of 9.12%. This error primarily arises from the selection of the coefficient k at the pipe bend. Due to the complex flow conditions at the bend, it is difficult to accurately determine the k value. Nevertheless, it can be concluded that this model design closely reflects the actual operating conditions of the pipeline.  
Figure 10: Distribution of pressure along the non-clogging pipe.