Theoretical Investigation of Leak’s Impact on Normal Modes of a Water–Filled Pipe: Small to Large Leak Impedance

Abstract

Recent research shows the potential of resonant frequency-based leakage detection methods. However, there is a disagreement in whether a leak shifts the normal modes (often called natural or resonant modes) and whether a leak introduces additional peaks to the frequency response function (FRF) of the pipeline. In this paper, the impact of a leak on the normal modes is investigated. The trajectories of normal modes in the frequency complex plane with varying leak size are studied. The key parameter that represents the leak size and controls the trajectories of the normal modes is the ratio of the acoustic impedance of the pipe to the resistance impedance of the leak. It is found that, as the impedance ratio increases from zero (i.e., no leak), each normal mode shifts toward the upper-half complex plane of frequency by a leak, where the imaginary part is a measure of the leak-induced damping of the wave. When the impedance ratio is less than the order of one, the leak-induced normal-mode frequency shift is negligible, which supports the theory put forward by proponents of the no-shift and no-additional-peak hypothesis. When the impedance ratio is of the order of one or larger, not only is the shift of the FRF's peak significant, but also new peaks appear, which supports the theory raised by proponents of the leak-induced additional peaks hypothesis.

On sound propagation along an infinite rectangular duct-like structure with a finite slot opening and its modelling

“…The square openings act like dampers. Similar observation has been made by Lin et al8 though they focussed on normal duct modes.Strong sound leakage is found at the duct eigen-frequencies of strong sound transmission because of the strong excitation of acoustic modes [Fig.3(b)]. Sharp narrowband drops of sound power, hereinafter referred as 'dip', which are not found in the non-leaking duct and the transmitted power spectra [Fig.3(a)], are also observed at ka/p ~ 1.12, 1.50 and 2.29.…”

“…There are also investigations which look into the change in the duct frequency responses due to duct leakage of different sizes 7 and the mechanisms behind the change. 8,9 However, only normal duct modes are considered.…”

Multi‐Sensor Fusion for Transient‐Based Pipeline Leak Localization in the Dempster‐Shafer Evidence Framework

“…Let and denote the amplitudes of incident and reflected wave at the leak. They have the relationship (Contractor, 1965): where is the impedance of the pipe and is the impedance of the leak, is the gravitational acceleration, is the internal cross‐sectional area of pipeline, is the steady‐state head at the leak, and “” is due to the fact that in general (Lin et al., 2019). With Equation , it is able to obtain a rough estimate of the amplitude of reflected signal, once the leak size is known.…”

“…is the impedance of the leak, g is the gravitational acceleration, A is the internal cross-sectional area of pipeline, 0 L H is the steady-state head at the leak, and "" is due to the fact that in general Z Z Lin et al, 2019). With Equation 1, it is able to obtain a rough estimate of the amplitude of reflected signal, once the leak size is known.…”