# Enhancing Friction Models for Starting Up Water Installations Containing Trapped Air
## 5 Conclusions  
This research presented a proposed model to study the starting up of water installations, maneuvers that involve the presence of water and air. A differential-algebraic equations system was developed using the rigid water column approach, the polytropic law, and the piston-flow method. The rigid water column method was presented, considering an unsteady friction model, which is the most robust equation to analyze friction losses using actual mathematical models. The mathematical model underwent validation in an experimental facility consisting of a 7.6 m long PVC pipe, where the filling operation under analysis was conducted slowly, resulting in an isothermal evolution. The results affirm the suitability of the mathematical model for simulating experimental measurements of air pocket pressure. Based on the results, the Swamee-Jain equation was the best approach to compute the friction factor, when compared to the other formulations ([PERSON], [PERSON], Hazen-Williams, Blaissus, and von Karman-Prandtl).  
To demonstrate the practical application of this method, a water installation scenario (total length of 460 m, and internal pipe diameter of 150 mm) was presented to observe the responses of the main variables, such as air pocket pressure, water velocity, and water column length. The equation was used to study a pipeline with an undulating profile, which is of utmost importance, since actual water installations present this type of profile.  
Based on the results, the following recommendations are set forth:  
* It is of paramount importance for engineers and designers to carefully monitor the maximum pressure surges experienced during such transient events. In this context, during the experimental measurements (Run No. 2), a maximum value of 50.42 m was reached, significantly exceeding the initial absolute pressure head of the hydro-pneumatic tank (125,000 Pa or 12.74 m) and the air pocket pressure of 10.33 m. This occurred due to the compression of the air pocket during operation.
* Water utilities can assess existing water installations using the proposed model and the results presented for practical application involving three pipe branches, which depict the evolution of the gravity term component in a typical scenario. As the number of longitudinal slopes increases, more equations must be considered for evaluating the gravity term. This scenario becomes particularly relevant when air valves are either absent or damaged due to maintenance issues.
* In situations where air valves have not been installed, air pockets may persist within water installations, as observed in the experimental measurements and practical applications. This condition poses significant risks, as it can lead to additional pressure surges when a new filling operation is initiated to expel the air phase.  
The authors suggest that future research could focus on assessing filling operations which involves air valves, considering unsteady friction models.  
Conceptualization, A.A.-P. and O.E.C.-H.; methodology, A.A.-P. and O.E.C.-H.; formal analysis, O.E.C.-H.; validation, A.A.-P. and V.S.F.-M.; writing--original draft preparation, A.A.-P. and O.E.C.-H.; writing--review and editing, V.S.F.-M. All authors have read and agreed to the published version of the manuscript.  
This research did not receive any outside funding.  
Not applicable.  
Databases are available from the corresponding author.