# Enhancing Friction Models for Starting Up Water Installations Containing Trapped Air
## 1 Introduction  
Reliability is of paramount concern in the design of hydraulic systems, largely due to the substantial costs associated with such systems. Therefore, engineers are tasked with a critical responsibility: the careful selection of pipe resistance classes. This selection is pivotal in safeguarding hydraulic installations against the adverse effects of pressure surges, as outlined by [PERSON] et al. (2019) [1]. Throughout the last decades, a considerable amount of attention has been focused on the examination of water hammer effects, primarily driven by incidents of pipe collapses, concentrating on monophasic conditions involving water [2, 3]. Initially, the focus of analysis centered on the behavior of the water phase and encompassed operations such as valve closures and pump stoppages. These hydraulic phenomena have been subjected to extensive scrutiny through a combination of numerical simulations and experimental studies, often employing commercial software packages such as HAMMER (OpenFlows Hammer Edition Update 3) and Allievi (Version 3.0.0), among others. The global landscape has responded to the challenges posed by transient flow within hydraulic installations by adopting regulations that govern its management, a practice observed across numerous countries. As a result, engineers and designers are capitalizing on the wealth of knowledge accumulated in this field to craft robust and effective designs, as articulated by [PERSON] et al. (1999) [4]. In recent decades, there has been a surge in interest regarding the analysis of transient flow phenomena involving entrapped air. This surge is driven by the recognition that air pockets can induce more extreme absolute pressure patterns compared to transient flow scenarios in the absence of air, as discussed in various works [1; 5; 6]. This peculiarity arises from the inherent greater elasticity of air compared to that of both water and the pipes themselves. When an air pocket undergoes a volume reduction due to water compression, its pressure increases, and vice versa. Modeling the transient flow with entrapped air demands the utilization of complex formulations to faithfully represent the intricate interplay between the water and air phases. Numerous approaches have been employed to simulate filling operations within hydraulic installations. For instance, [PERSON] et al. (2002) [6] harnessed the method of characteristics, in conjunction with polytropic laws and a steady friction factor, to model the temporal evolution of water during a filling operation. Advanced mathematical models have been used to analyze pipe operations involving entrapped air in water distribution systems [1], in which the utilization of the rigid water column model was considered, in combination with steady friction models. Computational fluid dynamics (CFD) techniques have been utilized to periodically study maneuvers involving entrapped air, offering insights into the evolution of various variables, such as air pocket temperature, air pocket pressure, water velocity, air pocket density, and air pocket size patterns [7]. CFD techniques are also utilized to examine water phenomena when one-dimensional mathematical models fall short, particularly in scenarios in which the water column obstructs the internal orifice of an air valve without fully expelling the injected air pocket [8]. There is no specialized software designed for performing one-dimensional simulations related to the analysis of transient flows with entrapped air pockets in water distribution systems, which is a current challenge for the water sector. Water utilities have suffered the negative effects of transient events with air entrapped in their infrastructures over recent decades; however, they only can act considering practical recommendations presented by the American Water Works Association (AWWA) [9]. For pipeline filling, the manual recommends that the filling velocity should not surpass 0.3 m/s, which can be achieved using a potable water pump or a throttled flow rate for the filling operation.