An Analytical Model for the Prediction of Emptying Processes in Single Water Pipelines

Air pockets in water distribution networks can cause various operational issues, as their expansion during drainage operations leads to sub-atmospheric conditions that may result in pipeline collapse depending on soil conditions and pipe stiffness. This study presents an analytical solution for calculating air pocket pressure, water column length, and water velocity during drainage operations in a pipeline with an entrapped air pocket and a closed upstream end. The existing system of three differential equations is reduced to two first-order nonlinear differential equations, enabling a rigorous analysis of the existence and uniqueness of solutions. The system is then further reduced to a single secondorder nonlinear ordinary differential equation (ODE), providing an intuitive framework for examining the physical behaviour of the hydraulic and thermodynamic variables. Furthermore, through a change of variables, the second-order ODE is transformed into a first-order linear ODE, facilitating the derivation of an analytical solution. The analytical solution is validated by comparing it with a numerical solution. Additionally, a practical application demonstrates the effectiveness of the developed tool in predicting the extreme pressure values in the air pocket during the water drainage process in a pipe, within a controlled environment.