# One-Dimensional Modeling of the Pressure Loss in Concrete Pumping and Experimental Verification
## 2 Empirical Formula for the Concrete Pumping Pressure Loss  
Assuming that concrete is an incompressible fluid, the flow of concrete in a pipeline is a plunger flow. The performance of the concrete in the pipe flow is unchanged, and the frictional resistance between the concrete and the pipe wall and the internal bond strength of the concrete are the main sources of pressure loss during the transportation process. According to the empirical formula of [PERSON] [27], the drag of flow \(f\) is  
\[f=k_{1}+k_{2}\Bigg{(}1+\sideset{}{{}^{t_{2}}}{\Big{/}}t_{1}\Bigg{)}v \tag{1}\]  
Here, \(k_{1}\) is the coefficient of adhesion (unit: Pa), which is the resistance coefficient caused by the concrete sticking to the pipeline, \(k_{2}\) is the speed coefficient (unit: Pa\(\cdot\)s/m), which is the resistance coefficient caused by the speed of the concrete flow in the pipeline, \(v\) is the average flow velocity of the concrete, \(t_{1}\) is the time for the pumping piston to push the concrete to flow, and \(t_{2}\) is the switching time of the distribution valve; \(t_{2}/t_{1}\) is generally taken as 0.3. The larger \(k_{1}\) is, the greater the thrust required for transformation of the concrete state of motion, and the larger \(k_{2}\) is, the more difficult it is to improve the concrete delivery; \(k_{1}\) and \(k_{2}\)[27] are related to the concrete match ratio, the smoothness of the pipe wall, etc. In engineering, the concrete slump \(S\) is commonly used for calculation:  
\[k_{1}=300-S \tag{2}\]  
\[k_{2}=400-S \tag{3}\]  
The axial pressure difference in the concrete unit pipe is  
Figure 1: Schematic diagram of stress analysis of the concrete unit in a straight pipe section.  
\[p\pi R^{2}-(p+dp)\pi R^{2}=-\pi R^{2}dp \tag{4}\]  
The plunger flow resistance is  
\[2\pi Rdxf=2\pi R\bigg{[}k_{1}+k_{2}(1+\sfrac{t_{2}}{t_{1}})v\bigg{]}dx \tag{5}\]  
The plunger flow inertia force is  
\[\pi R^{2}dx\rho dv/dt \tag{6}\]  
The concrete gravity is  
\[\pi R^{2}\rho g\sin\theta dx \tag{7}\]  
According to the force analysis of the concrete unit of the straight pipe section, the combined force \(\sum F\) in the axial direction of the concrete unit is given by Equations (4)-(7):  
\[\sum F=\pi R^{2}dp-2\pi R\bigg{[}k_{1}+k_{2}\bigg{(}1+\sfrac{t_{2}}{t_{1}} \bigg{)}v\bigg{]}dx-\pi R^{2}dx\rho dv/dt-\pi R^{2}\rho g\sin\theta dx \tag{8}\]  
In the above, \(\rho\) is the density of the concrete, \(\theta\) is the angle between the pumping pipe and the reference datum, and \(dv/dt\) is the concrete acceleration.  
For the plunger pump form, it is generally accepted that the flow rate of concrete in one stroke is basically constant, i.e., \(dv/dt=0\), and that the pumped concrete is in uniform motion, i.e., \(\sum F=0\); thus, the following is obtained:  
\[dp=\sfrac{2}{R}[k_{1}+k_{2}(1+t_{2}/t_{1})v]dx+\rho g\sin\theta dx \tag{9}\]  
Considering the pulse periodicity in the pumping system and the concrete flow properties inside the pipeline, the above equation is integrated over \(x\) between the two ends. Assuming that the axial length of the two cross-flow sections is \(l\), the expression for the pumping pressure difference between the two cross-flow sections is  
\[\Delta p_{c}=\sfrac{2}{R}[k_{1}+k_{2}(1+t_{2}/t_{1})v]l+\rho g(z_{2}-z_{1}) \tag{10}\]