Energy Analysis of the Resonant Frequency Shift Pattern Induced by Nonuniform Blockages in Pressurized Water Pipes

Abstract

Blockages in urban water supply systems are commonly formed from various complicated physical, chemical, and biological processes; thus, they usually constrict randomly and nonuniformly along their lengths. Although the transient-based method has been developed for blockage detection, the applications of this method are mainly limited to blockages of uniform constriction along their lengths (termed as uniform blockages). The influence of blockages with linearly varying diameters (termed as linear nonuniform blockages) on transient frequency responses was studied by the authors in a previous study. It was found that the resonant frequency shifts induced by linear nonuniform blockages have totally different patterns from that of uniform blockages. Specifically, the resonant frequency shifts induced by linear nonuniform blockages become less evident for higher harmonics. But the physical mechanism of this pattern is still unclear. This study intends to clarify this phenomenon from an energy perspective. For this purpose, the energy transmission coefficient of an unbounded pipeline containing various blockages is analytically derived, which is numerically validated by the method of characteristics. Afterward, the influence of nonuniform blockage properties on the energy transmission is investigated systematically based on the validated result. The results indicate that the impedance of nonuniform blockages is frequency dependent, which becomes smaller for higher frequency waves. This means that nonuniform blockages have a less blocking effect on the propagation of higher frequency waves; thus, the resonant frequency shifts induced by nonuniform blockages become less evident.