# Enhancing Friction Models for Starting Up Water Installations Containing Trapped Air
## 4 Practical Application  
In this instance, a scenario featuring three pipe branches, as depicted in Figure 9, is suggested. More intricate systems can be tackled in an identical manner, given that the model is utterly universal. The analysis to explore the dynamics of a filling operation within a undulating pipeline installation is characterized as follows: a total length (\(L_{T}\)) of 460 m; an internal diameter (\(D\)) of 150 mm; three pipe branches measuring 200, 70, and 200 m, and longitudinal slopes of \(10^{\circ}\), \(5^{\circ}\), and \(15^{\circ}\), respectively; an absolute roughness (\(k_{s}\)) of 0.0015 mm; a polytropic coefficient (\(k\)) of 1.2; an initial hydro-pneumatic absolute pressure (\(p_{0}^{*}\)) of 265,000 Pa; an initial air pocket (\(x_{0}\)) of 40 m; and a resistance coefficient (\(R_{v}\)) of 15 ms\({}^{2}\)/m\({}^{6}\) for a fully open condition. The gravity term varies along the analyzed pipeline, considering its undulating profile. Table 3 presents the evolution of the gravity term and the application range for the three pipe branches for practical applications. Figure 8: Sensitivity analysis for different friction factor equations. Figure 9: A scheme including three pipe branches. The ODE 23 s solver method employing the Simulink tool in Matlab was used for the numerical resolution of differential-algebraic formulations included in Equations (1), (2), (4a), or (4b)-(6) in all simulations, in which the air pocket pressure is changing over time (\(p_{1}^{*}\)). Equation (2) must be computed considering the three equations presented in Table 3. The unsteady friction model (UFM) was employed, with calculations based on the Swamee-Jain equation, and the results are visualized in Figure 10. These data reveal an initial sharp increase in the onset of the hydraulic event, as shown in Figure 10a. At 61.35 s, the air pocket absolute pressure head reaches its maximum value of 115.60 m, representing an 11-fold increase compared to the initial air pocket pressure of 10.33 m (at atmospheric conditions). Subsequently, oscillations occur, with the air pocket pressure stabilizing around 9  
Regarding the water velocity (Figure 10b), it rapidly rises, peaking at 10.24 m/s at 4.03 s. When the air pocket pressure reaches its maximum (at 61.35 s), the water velocity drops to zero, followed by intermittent pulses at a velocity of 0 m/s, indicating a resting water column. The initial water column length was 40 m, but after the regulating valve was opened, the pipe quickly filled with water, displacing the air phase (Figure 10b). The final position of the water column length trends to a value of 364 m, which indicates that the hydraulic system could not be drained completely, since there are no air valves along the system. Therefore, an air pocket size of 66 m is found at the end of the transient event. The time of the occurrence of the maximum water column length (403.9 m) coincides with the peak air pocket pressure value (at 61.35 s). Figure 10c shows the gravity term evolution over time. The minimum and maximum values of the gravity term of 0.129 and 0.175 m/m are found at 27.6 and 61.35 s, respectively. The water column stays in pipe branch 1 (\(L_{1}\)) trough from 0 to 18.96 s. It occupies pipe branch 2 (\(L_{2}\)) from 18.96 to 27.62 s, and finally, the water column completes its movement in pipe branch 3. To study this kind of transient event, the effect of the gravity term should be considered in order to obtain suitable results [30]. Figure 10: Hydraulic and thermodynamic variable evolution: (**a**) air pocket pressure head; (**b**) water velocity and water column length; and (**c**) gravity term. The analysis presented in this section can be used by designers and engineers to evaluate the water distribution system, considering the scenario in which air valves have not been installed or have failed. The proposed model can be utilized not only by water utilities, but also by other industries that require the analysis of filling operations involving various liquids.