## 4 Pressure Loss Correction Model for Bend Section  
### One-Dimensional Flow Model for Horizontal Straight Pipe Section  
In the pumping pipe, the concrete is divided into a plunger layer and a lubrication layer. In the plunger layer, aggregate particles and mortar move together as a cohesive unit. This results in minimal friction between the particles and the mortar, which corresponds to a lower pressure loss. In the lubrication layer, the mortar flows in layers, experiencing friction between the mortar and the pumping pipe wall, friction due to the relative slip between different mortar layers, and friction from the relative movement between the lubrication layer and the plunger layer. This friction results in significant energy consumption and corresponds to considerable pressure loss during the pumping process. The factors affecting pumping pressure loss include pipe wall roughness, lubrication layer thickness, aggregate particle shape and size, viscosity of the lubrication layer, Reynolds number, and concrete pumping speed. The one-dimensional flow model assumes that the lubrication layer and the critical point of the plunger layer are filled with coarse aggregate particles. Coarse aggregate particles of different shapes are approximated as spherical particles with equivalent sizes, as illustrated in Figure 4, where \(v\) represents the average concrete velocity, \(l\) is the distance between the selected control body overflow sections 1 and 2, \(z_{1}\) is the height of the axis of overflow section 1 from the reference datum, \(z_{2}\) is the height of the axis of overflow section 2 from the reference datum, \(p_{1}\) is the average pressure at overflow section 1, \(p_{2}\) is the average pressure at overflow section 2, \(d\) is the inner diameter of the pumping pipeline, \(\delta\) is the thickness of the lubrication layer, \(K\) is the equivalent particle size of the coarse aggregate particles, and \(\tau_{0}\) is the viscous force between the lubrication layer and the plunger layer. To calculate the pressure loss along the pumping pipeline, the Darcy formula is used, with the along-range loss coefficient determined by Moody's chart. This coefficient is primarily dependent on the Reynolds number and relative roughness during concrete conveying. The pressure difference \(\Delta p\) between the two cross-sectional streams of concrete is calculated as follows:  
\[\Delta p_{m}=0.0055\left[1+\left(20000K_{d}+\frac{10^{6}}{\text{Re}}\right)^{ \frac{1}{3}}\right]\frac{l\rho_{s}v^{2}}{2d_{e}} \tag{14}\]  
Figure 4: Schematic diagram of one-dimensional flow model for concrete pumping [10]. \begin{table}
\begin{tabular}{c c c c} \hline \hline
**Average Flow** & **Experimental Data** & **Simulated Calculation of** & **Relative Error** \\
**Velocity** & **Value** & **Pressure Loss Value** & \\ \hline
0.63 m/s & 90.1 kPa/m & 83.6 KPa/m & \(-6.59\%\) \\
0.89 m/s & 96.8 KPa/m & 89.6 KPa/m & \(-7.43\%\) \\
1.13 m/s & 102.3 KPa/m & 93.8 KPa/m & \(-8.28\%\) \\ \hline \hline \end{tabular}
\end{table} 
Table 3: Comparison of experimental data and simulation results. where, \(d_{e}\) is the equivalent diameter of the lubrication layer, \(K_{d}\) is the relative roughness of the lubrication layer in contact with the plunger layer, and Re is the Reynolds number of the mortar in the lubrication layer. The lubrication layer is the annular region formed by the pumping pipe wall and the plunger layer, and its equivalent diameter \(d_{e}\) is:  
\[d_{e}=2\delta \tag{15}\]  
Relative roughness \(K_{d}\) is defined as the ratio of absolute roughness \(K\) to the equivalent diameter \(d_{e}\). According to the one-dimensional flow model assumption, the area where the lubrication layer contacts the plunger layer consists of coarse aggregate particles with uniform particle size. Thus, the absolute roughness is considered to be the average particle size of the aggregate particles. Aggregate particles have complex shapes and different sizes.