# One-Dimensional Modeling of the Pressure Loss in Concrete Pumping and Experimental Verification
## 3 Concrete Pumping One-Dimensional Flow Model Construction
### Pressure Loss Analysis with the One-Dimensional Flow Model for Concrete Pumping  
After concrete pumping for a period of time, the concrete pumping flow and pumping pressure no longer change with time, and the flow of concrete in the pumping pipe is a constant flow. The average flow velocity in each cross-flow section of different sizes and directions remains unchanged in the straight pumping section, so the concrete pumping is a uniform flow. Along the concrete pumping direction, the pressure distribution in each section of the pumping pipe satisfies [PERSON]'s equation:  
\[\frac{p_{1}}{\gamma}+\frac{v_{1}^{2}}{2g}+z_{1}=\frac{p_{2}}{\gamma}+\frac{v_{ 2}^{2}}{2g}+z_{2}+h_{f} \tag{11}\]  
In the above, \(h_{f}\) is the head loss of concrete pumping from cross-flow section 1 to cross-flow section 2, \(\gamma\) is the volume weight of the mortar, \(v_{1}\) is the flow velocity in cross-flow section 1, and \(v_{2}\) is the flow velocity in cross-flow section 2. The straight pumping pipe is assumed to be a uniform pipe, and according to the continuity equation,  
\[v_{1}=v_{2}=v \tag{12}\]  
Then, the pressure difference \(\Delta p_{m}\) between cross-flow sections 1 and 2 is  
\[\Delta p_{m}=p_{1}-p_{2}=h_{f}\gamma+(z_{2}-z_{1})\gamma \tag{13}\]  
In a typical straight pumping pipe, the pumping pressure loss uniformly increases along the pumping length, and the pumping pressure loss is friction loss. According to the  
Figure 2: Schematic diagram of the one-dimensional flow model for concrete pumping.  
[PERSON] formula of fluid dynamics, the concrete pumping pressure loss \(p_{f}\) through cross-flow section 1 to cross-flow section 2 is  
\[p_{f}=h_{f}\gamma=\lambda\frac{l}{d_{e}}\frac{\rho_{s}v^{2}}{2} \tag{14}\]  
Here, \(\lambda\) is the resistance coefficient, which is to be found, \(d_{e}\) is the equivalent diameter of the lubrication layer, and \(\rho_{s}\) is the mortar density.  
The lubrication layer is an annular area formed by the wall of the pumping pipe and the plunger layer, and the equivalent diameter \(d_{e}\) is  
\[d_{e}=4\times\frac{\pi d^{2}-\pi(d-2\delta)^{2}}{\pi d+\pi(d-2\delta)}\times \frac{1}{4}=2\delta \tag{15}\]  
The friction loss coefficient \(\lambda\) and Reynolds number are related to the relative roughness of the coarse aggregate particles. The Reynolds number affects the lubrication layer movement in the flow pattern, and the relative roughness of the coarse aggregate particles directly affects the flow state of the region around the coarse aggregate particles, usually due to the abrupt change in the coarse aggregate structure around the coarse aggregate particles. When the coarse aggregate equivalent particle size is greater than the thickness of the lubrication layer, the mortar is detached from the particles and flows into the area around the particles, so vortices of different sizes are easily formed around the particles, which affect the lubrication layer movement in the flow pattern. The Reynolds number \(R_{e}\) and coarse aggregate particle relative roughness \(K_{d}\) are  
\[R_{e}=\
icefrac{{vd_{e}}}{{v}} \tag{16}\]  
\[v=\
icefrac{{u}}{{\rho_{S}}} \tag{17}\]  
\[K_{d}=\
icefrac{{K}}{{d_{e}}} \tag{18}\]  
where \(v\) is the mortar kinematic viscosity coefficient and \(u\) is the mortar dynamic viscosity coefficient. The mortar kinematic viscosity coefficient \(u\) can be directly calculated using the slump [28], which is expressed as  
\[u=\rho_{s}g\times T_{100}\times(0.117S_{1}-20.86)\times 10^{-3} \tag{19}\]  
Here, \(\rho_{s}\) is the density of the mortar, \(T_{100}\) is the time required for the cement paste to slump 100 mm, and \(S_{1}\) is the slump of the mortar.