# Enhancing Friction Models for Starting Up Water Installations Containing Trapped Air
## 3 Validation Model  
A specialized experimental setup was established in the hydraulic lab at the University of Lisbon in Lisbon, Portugal, with the primary aim of validating the governing equations outlined in Section 2. These experimental measurements have been previously reported [26]. This setup comprises several essential components: a hydropneumatic tank capable of providing various initial pressure heads, a 7.6-m long PVC pipe, a pressure transducer situated at the highest point within the setup, a manual valve (referred to as MV1) designed to isolate the pipeline from the hydropneumatic tank, and four electro-pneumatic valves (from BV1 and BV4). It should be noted that BV1 and BV2 are kept open during the entire filling operation. Air pockets are introduced by injecting them into the highest point in the system. For all experimental tests, a consistent initial air pocket size (denoted as \(x_{0}\)) of 0.517 m is configured. Throughout the experiments, the right water column is maintained at a constant level, serving as a blocking water column (see Figure 3). To facilitate the filling operation, a total left branch pipe measuring 3.867 m in length is employed, encompassing both the inclined and vertical sections (refer to Figure 3). The commencement of the filling process is initiated by opening BV4. A synthetic maneuvering of BV4 is incorporated, adhering to the manufacturer's specified opening time of 0.2 s, and accounting for a resistance coefficient (\(R_{v}\)) of \(1.7\times 10^{5}\) ms\({}^{2}\)/m\({}^{6}\) for full opening. As BV4 opens, the left water column begins to fill the hydraulic system, concurrently causing the compression of the injected air pocket. Simultaneously, BV3 is promptly closed to induce rapid compression of the trapped air. The blocking water column serves as a crucial boundary condition, effectively replicating the behavior of a single pipeline. Considering this setup and the observed processes, the filling operation within this configuration is simulated using Equations (1) to (6). The hydropneumatic tank is meticulously configured to comprise two distinct initial absolute pressure heads (referred to as \(p_{0}^{*}\)), measuring values between 175,000 and 125,000 Pa. A summary of the initial conditions for the experimental runs is provided in Table 2. \begin{table}
\begin{tabular}{c c} \hline
**Run** & **Initial Condition** \\ \hline No. 1 & \(p_{0}^{*}\) = 175,000 Pa; \(x_{0}\) = 0.517 m \\ No. 2 & \(p_{0}^{*}\) = 125,000 Pa; \(x_{0}\) = 0.517 m \\ \hline \end{tabular}
\end{table} 
Table 2: Initial conditions for experimental runs. Figure 3: Experimental facility. The air pocket pulses were conducted twice to ensure the accuracy and reliability of the measurements for the two separate experimental runs. The pressure head pattern of the air pocket for Run No. 1 is illustrated in Figure 4. Remarkably, both experimental repetitions exhibited a consistent and similar trend across both experimental runs. Therefore, an average of the experimental data was computed to further validate the mathematical model, as depicted in Figure 5. Analysis of the experimental results revealed that the maximum air pocket pressure head reached 32.02 m at 0.36 s for Run No. 1 and 50.42 m at 0.33 s for Run No. 2. For all experimental runs, a total time of 1.2 s was considered, since the air pocket pressure peaks are found during this interval. The initial values of air pocket pressure patterns increase rapidly until the maximum value is achieved; after that, these oscillations start to dissipate. It is of utmost importance that the pipe class can resist the maximum value in order to avoid a pipe rupture, as occurred during experimental analysis. The gravity term can be computed as \(\Delta z/L_{f}=\sin 30^{\circ}\), considering the analyzed water installation. Unsteady friction models prove to be more suitable than their steady friction  
Figure 4: Experimental repetition for Run No. 1. Figure 5: Average of experimental air pocket pressure head fluctuations for Runs No. 1 and No.