# Enhancing Friction Models for Starting Up Water Installations Containing Trapped Air
## 3 Validation Model
2.  
counterparts when it comes to accurately portraying filling operations involving entrapped air, primarily due to the involvement of a variable friction factor during transient events. To assess this evolution, the Brunone friction coefficient was employed. Additionally, the Swamee-Jain equation (as outlined in Equation (4b)) was utilized to calculate the fluctuation in the friction factor during steady flow conditions. This choice was made based on the similarity of results compared to those obtained using the Colebrook-White formula, which is founded on a physical formulation. Reynolds numbers ranging from 0 to 2000, indicative of laminar flow, were effectively modeled using Equation (4a). For the remaining Reynolds number conditions, the Swamee-Jain formulation was employed, since it is well-suited for values situated within the critical and turbulent flow zones. The calibration of the polytropic coefficient (\(k\)) was performed across three distinct types of evolution: (i) an isothermal process characterized by \(k=1.0\), (ii) an intermediate process with \(k=1.2\), and (iii) an adiabatic process denoted by \(k=1.4\). Figure 6 offers a comparative analysis of these polytropic coefficients for both Run No. 1 and No. 2. Notably, the most favorable outcome was achieved when employing an isothermal evolution, as it allowed the mathematical model to closely track fluctuations observed in the average experimental results. For Run No. 2, the utilization of a polytropic coefficient of \(k=1.0\) (representing an isothermal process) yielded a peak air pocket pressure head of 50.00 m. In contrast, the employment of intermediate and adiabatic processes resulted in values of 48.14 and 47.19 m, respectively. Consequently, a polytropic coefficient of \(k=1.0\) was selected for the analysis, aligning closely with the measured peak air pocket pressure head of 50.42 m. Additionally, the mathematical model for unsteady friction models (UFM) demonstrated its capacity to accurately depict oscillations within the measured air pocket pressure patterns. The root mean square error (RSME) was computed for Run No. 1 and No. 2, yielding values of 1.36 and 1.34 m, respectively. The Brunone friction coefficient and the Vardy's shear decay coefficient depend on the Reynolds number. Figure 7 shows the evolution of these variables. The Reynolds number varies from 0 to 110,625 (for Run No. 1) and from 0 to 149,910 (for Run No. 2). The maximum value of the Reynolds number is found at 0.25 s and 0.24 s for Runs No. 1 and No. 2, respectively. The Brunone friction coefficient (\(k_{\delta}\)) and Vardy's shear decay coefficient (\(C^{*}\)) presented a cycle behavior, with an increasing trend for both numerical runs, with maximum values of 0.034 and 0.0047, respectively. Figure 6: _Cont._  
To analyze the behavior of other friction models during the filling operations, the formulas of [PERSON], [PERSON], Hazen-Williams, [PERSON], and [PERSON] were considered. These formulas have been used instead of Equation (4b) (Swamee-Jain). The application of the [PERSON] equation [27] is justified for a relationship of \(k_{s}/D>0.01\); the Wood equation [28] for Re > 10,000 and \(10^{-5}<k_{s}/D<0.04\); and the Hazen-Williams [29] equation is employed for a \(D>75\) mm and \(v_{f}<3\) m/s. The Moody equation is described as shown:  
\[f=0.0055\Bigg{[}1+\left(20000\frac{k_{s}}{D}+\frac{10^{6}}{\text{Re}}\right)^{ 1/3}\Bigg{]} \tag{7}\]  
The Wood formulation is:  
\[f=0.094\bigg{(}\frac{k_{s}}{D}\bigg{)}^{0.225}+0.53\bigg{(}\frac{k_{s}}{D} \bigg{)}+88\bigg{(}\frac{k_{s}}{D}\bigg{)}^{0.44}\text{Re}^{-1.62(\frac{k_{s} }{D})^{0.134}} \tag{8}\]  
and, the Hazen-Williams equation is given by:  
\[f=\frac{133.89}{C_{HW}^{1.851}D^{0.017}\
u_{f}^{0.15}\text{Re}^{0.15}} \tag{9}\]  
where, \(C_{HW}\) = the Hazen-Williams coefficient. The Blaissus equation can be used for smooth pipes (\(k_{s}<0.305\delta^{\prime}\)), where \(\delta^{\prime}\) = the boundary layer thickness.