Analysis of a transient in a pipeline with a leak using Laplace transforms

Abstract

A transient or water hammer event is initiated whenever a steady-state condition in a pipeline is disturbed either by a planned event or accidentally. When the transient reaches a leak, the transient will be reflected and transmitted, which results in a different transient event compared to the transient from the same pipeline without a leak. In a previous study, an analytical solution for the variation in hydraulic grade line (HGL)expressed in terms of a Fourier series has been obtained based on a pipeline with known initial conditions and constant boundary conditions. Based on the analytical solution, a leak detection method was developed previously using leak-induced transient damping. However, The Fourier series approach is not well suited to the case where the boundary conditions vary during the transient event. A Laplace transform solution approach overcomes this difficulty and is the focus of this paper. Normalized hyperbolic governing equations for a pressure transient in a pipeline with a leak are derived, where the discontinuity induced by a leak is considered by using a delta function. In addition the orifice-leak equation is linearized. The accuracy of the analytical solution has been verified by nonlinear numerical analyses using the method of characteristics. The effects of a leak on pipeline transients induced by a pulse boundary perturbation and a continuously changing boundary perturbation are investigated in detail.