# Enhancing Friction Models for Starting Up Water Installations Containing Trapped Air
## 2 Mathematical Model  
This section presents the development of a rigid water column model considering an unsteady friction model for performing filling operations in pressurized pipelines. A filling process begins when a regulating valve is opened; then, the initial water column starts to fill a pipeline installation, and the air volume exhibits a reduction in size. Initially, the hydraulic system is at rest, but the reduction of the injected air volume produces an increase in the air pocket pressure head. The pipeline should be sized to resist the maximum pressure during a transient flow event. The proposed model considers the following assumptions: (i) the water movement is simulated using the rigid water column model, (ii) an unsteady friction model, and (iii) a polytropic law to represent the air phase; and (iv) the air-water interface is considered perpendicular to the main direction of the flow. The proposed model can be applied to water pipelines with small internal diameters. Figure 1 presents a scheme of water pipeline filling. A pipeline has several branches along its chainage, which are described by the length (\(L_{i}\)) and a longitudinal slope (\(\theta_{i}\)). The total number of pipe branches is represented by \(n\). Longitudinal slopes (\(\theta_{i}\)) can have positive or negative values for ascending or descending pipe branches, respectively. A complete modeling of a filling process can be achieved using the formulations described as follows:  
1. Mass oscillation equation: the Euler equation describes the water movement along a pipeline installation, which considers the inertia system with the term \(dv_{f}/dt\). The equation must be applied between the upstream end of the water installation and the pipe branch where the tail water column is located. In addition, the Euler equation \begin{table}
\begin{tabular}{c c c c} \hline \hline
**Author** & **RWCM** & **EWCM** & **CFD** \\ \hline [PERSON] and [PERSON] (1996) [15] & ✓ & & \\ [PERSON] et al. (1999) [16] & ✓ & & \\ [PERSON] et al. (2016) [17] & ✓ & & \\ [PERSON] et al. (2013) [18] & & ✓ & \\ [PERSON] et al. (2019) [19] & & ✓ & \\ [PERSON] et al. (2011) [7] & & & ✓ \\ [PERSON] et al. (2021) [20] & & & ✓ \\ [PERSON] et al. (2021) [21] & & & ✓ \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Summary of mathematical models used for the water phase. Figure 1: Scheme of a water pipeline filling. contains both steady and unsteady terms for computing the friction factor. The equation is applied as shown:  
\[\frac{dv_{f}}{dt}=\frac{p_{0}^{*}-p_{1}^{*}}{\rho_{w}L_{f}}+g\frac{\Delta z}{L_{f} }-g\bigg{(}f\frac{v_{f}|v_{f}|}{2 gD}+\frac{k_{\delta}}{g}\frac{dv_{f}}{dt} \bigg{)}-\frac{R_{v}gA^{2}v_{f}|v_{f}|}{L_{f}} \tag{1}\]  
where \(p_{0}^{*}\) = upstream pressure supplied by a tank or pump, \(v_{f}\) = water filling velocity, \(\rho_{w}\) = water density, \(p_{1}^{*}\) = air pocket pressure, \(g\) = gravitational acceleration, \(f\) = friction factor, \(\Delta z\) = difference elevation between two points of a pipeline, \(D\) = internal pipe diameter, \(L_{f}\) = filling column length, \(R_{v}\) = resistance coefficient, \(A\) = cross-sectional area, and \(k_{\delta}\) = Brunone friction coefficient. The Brunone friction coefficient (\(k_{\delta}\)) [22] depends on the Vardy's shear decay coefficient (\(C^{*}\)), which is a function of a laminar or turbulent flow condition. For a laminar flow condition, it takes a value of \(C^{*}\) = 0.00476, and for a turbulent regime, it can be computed by the expression \(C^{*}=\frac{7.41}{\text{Re}^{\log\left(143/\text{Re}^{\log\left(15\right)} \right)}}\). The Reynold number is calculated as \(Re=v_{f}D/v\), where \(v\) = kinematic viscosity. 2. Gravity term: for making calculations, it of utmost important to determine the gravity term (\(\Delta z/L_{f}\)), which varies depending on where the water filling is located in regards to the pipe branch \(i\).