# An Analytical Model for the Prediction of Emptying Processes in Single Water Pipelines
## 4 Analytical solution
### Validation of the Analytical Solution  
The analytical solution is now checked by first numerically solving the original system (4)-(6) and comparing its solution with the analytical solution (20)-(22) and (32)-(34) on the respective time intervals. During the test, the following parameters were considered: \(L_{T}=600\) m, \(f=0.018\), \(D=0.35\) m, \(R_{v}=0.06\) s\({}^{2}\)/m\({}^{-6}\), \(x_{0}=200\) m, \(k=1.2\), \(p_{atm}^{*}=101325\) Pa, \(\theta=\sin^{-1}(3/120)\) rad.  
Figures 3-5 display graphical representations of these solutions with the integrals (22) and (34) evaluated numerically. The analytical and numerical solutions show excellent concordance, validating the use of the analytical solution as a predictive tool for emptying processes.  
Figure 6 presents the solution considering the first seven terms of the series, expressed explicitly through Equations (29) and (35) for the initial two intervals corresponding to \(\frac{dL}{dt}\geq 0\) and \(\frac{dL}{dt}\leq 0\), respectively. The results demonstrate rapid convergence of the analytical solution to the numerical solution with relatively few terms.  
Furthermore, the analytical model was validated using experimental measurements conducted by the authors, as reported in Reference [5], where a single pipeline with an internal diameter of 42 mm and a total length of 4.36 m was employed. In this case, conditions similar to those of the practical application were considered, implying that the upstream end remained closed during the simulations. For the analysis, the initial air pocket size, \(x_{0}\), ranged from 0.205 to 0.450 m; longitudinal pipe slopes varied between 0.457 and 0.515 rad; and resistance coefficients at the regulating valves ranged from 11.89 to 138.41 \(\times\) 10\({}^{6}\) ms\({}^{2}\)/m\({}^{6}\). The analytical solution was applied under the assumption of instantaneous valve opening to provide conservative (safety) estimates, while the experiments were conducted considering a gradual valve opening. Figure 7 presents a comparison between the computed and measured evolution of the air pocket pressure over time, for an initial air pocket volume of 0.450 m, a longitudinal slope of 0.457 rad, and a resistancecoefficient of 30.86 \(\times\) 10\({}^{6}\) ms\({}^{2}\)/m\({}^{6}\). The discrepancy observed during the initial seconds is attributed to the experimental setup, in which the regulating valve was gradually opened using an opening manoeuvre time of 0.3 s.  
Figure 8 compares the maximum air pocket pressures obtained from 12 measurements, demonstrating that the analytical model can accurately capture these values, which are crucial to detect if a pipeline can have a risk of collapse.  
Figure 4: Analytical solution for the water column length \(L(t)\).  
Figure 3: Analytical solution for the water velocity \(v(t)\).  
Figure 5: Analytical solution for the air pocket pressure \(p_{1}^{*}(t)\).  
Figure 6: Convergence of the series in the analytical solution for the water velocity \(v(t)\). The red line corresponds to the numerical solution.  
Figure 7: Comparison of the air pocket pressure pulses for \(x_{0}\) = 0.45 m, \(\theta\) = 0.457 rad, and \(R_{p}\) = 30.86 \(\times\) 10\({}^{6}\) ms\({}^{2}\)/m\({}^{6}\).