## 4 Pressure Loss Correction Model for Bend Section
### Correction Model  
This paper uses the one-dimensional flow model of a horizontal straight pipe section for concrete pumping as a reference. It considers the effects of the bend radius, horizontal inclination angle, and pumping flow rate on the pressure loss in the bend. A correction model for the pressure loss of the bend is then established. The pressure loss value \(\Delta p_{w}\) of the bend is given by:  
\[\Delta p_{w}=\lambda\Delta p_{m}=\lambda 0.0055\Bigg{[}1+\left(20000K_{d}+ \frac{10^{6}}{\text{Re}}\right)^{\frac{1}{3}}\Bigg{]}\frac{l\rho_{s}v^{2}}{2d _{e}} \tag{17}\]  
where, \(\lambda\) is the correction factor. The bend radius, horizontal inclination angle, and flow rate directly affect the pressure loss in concrete pumping. To further explore the impact of these parameters, a three-factor, five-level orthogonal experimental design is used. The inclination angles of the pumping pipeline relative to the horizontal are 90\({}^{\circ}\), 45\({}^{\circ}\), 0\({}^{\circ}\), \(-\)45\({}^{\circ}\), and \(-\)90\({}^{\circ}\). The concrete pumping speeds are 40 m\({}^{3}\)/h, 55 m\({}^{3}\)/h, 70 m\({}^{3}\)/h, 100 m\({}^{3}\)/h, and 120 m\({}^{3}\)/h. The selected curved pipe radii of curvature are 195 mm, 235 mm, 275 mm, 315 mm, and 355 mm. To enhance the simulation of concrete flow, a straight pipe section of 500 mm is added before the elbow, and a straight pipe section of 400 mm is added after the elbow. The aggregate particle shape and physical parameters are consistent with those in the validation model. A total of 125 particle flow simulation analyses are conducted, with the results detailed in Table 4.  
Figure 8: Pressure distribution cloud at different pump flow rates. The relationship between the horizontal inclination of the bend and the pressure loss value is shown in Figure 9. For a flow rate of 40 m\({}^{3}\)/h and a bend radius of 195 mm, the pressure loss increases with a larger horizontal inclination. The correlation between the horizontal inclination angle and the pressure loss is linear, with a goodness-of-fit (R\({}^{2}\)) value of 0.9865. When the bend radius is 235 mm, 275 mm, 315 mm, or 355 mm, the pressure loss value relative to the horizontal inclination angle remains essentially consistent, following a linear distribution. The R\({}^{2}\) values exceed 0.97, indicating a good fit. At the same time, a larger curvature radius of the pumping pipeline results in lower pressure loss during concrete transportation. However, the spacing between each curve changes, indicating an interactive relationship between the pipeline curvature radius and the horizontal inclination angle. The relationship between the bend radius and concrete pumping pressure loss is illustrated in Figure 10. As the bend radius increases, the pumping pressure loss decreases when the horizontal inclination of the bend is 90\({}^{\circ}\) and the flow rate is 40 m\({}^{3}\)/h. At different flow rates, the relationship between pressure loss and bend radius exhibits similar characteristics. At the same time, as the flow rate increases, the corresponding pressure loss value also increases. Notably, the spacing between the curves changes, indicating a significant interaction between the bend radius and the flow rate. \begin{table}
\begin{tabular}{c c c c c c c} \hline \hline \multicolumn{2}{c}{**Horizontal**} & \multicolumn{2}{c}{**Pumping Flow**} & \multicolumn{5}{c}{**Radii of Curvature (mm)**} \\ \cline{3-7}
**Inclination Angle** & **Rate (m\({}^{3}\)/h)** & **195** & **235** & **275** & **315** & **355** \\ \hline \multirow{6}{*}{0\  
Figure 11 primarily illustrates the pressure loss associated with varying flow rates at different horizontal inclination angles. The figure shows that the flow rate and the concrete pumping pressure loss exhibit a clear linear relationship, with the R\({}^{2}\) exceeding 0.99. As the pumping flow rate increases, the pressure loss also rises. This linear trend remains consistent even when the layout of the pumping pipeline changes.