Transient Wave Scattering and Its Influence on Transient Analysis and Leak Detection in Urban Water Supply Systems: Theoretical Analysis and Numerical Validation
Abstract
:1. Introduction
2. Problem Statement and Study Framework
3. Models and Methods
3.1. One-Dimentional (1D) Transient Model
3.2. Multi-Scale Perturbation Method
4. Analytical Results and Analysis
4.1. Results of Regular Non-Uniformities
- Subcritical detuning: ,
- Supercritical detuning: ,
- Bragg resonance: ,
4.2. Results of Random Non-Uniformities
5. Numerical Validation
5.1. Settings of Numerical Tests
5.2. Validation for Regular Case
5.3. Validation for the Random Case
6. Results Discussion and Implications
6.1. Energy Analysis of Transient Wave Scattering
6.2. Impacts on Transient Modelling and Analysis
6.3. Impacts on Transient-Based Leak Detection
7. Conclusions
Acknowledgments
Conflicts of Interest
References
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Type | Case No. | λw/2λb | A0 (m2) | δA | Distribution Function | Correlation Function |
---|---|---|---|---|---|---|
Regular | 1 | >1 | 1.0 | σA/A0 = 0.20 | Degenerate (deterministic) | 0 for ζ ≠ 0 1 for ζ = 0 |
2 | =1 | |||||
3 | <1 | |||||
Random | 4 | >1 | 1.0 | σA/A0 = 0.23 | Uniform | e−α|ζ| |
5 | =1 | |||||
6 | <1 | |||||
7 | >1 | 1.0 | σA/A0 = 0.23 | Upper triangular | e−α|ζ| | |
8 | =1 | |||||
9 | <1 |
Case No. | Uniform Distribution | Upper Triangular Distribution | |||||
---|---|---|---|---|---|---|---|
4 | 5 | 6 | 7 | 8 | 9 | ||
Wave scattering factor (φ) | Analytical | 125.0 | 75.0 | 159.4 | 125.0 | 75.0 | 159.4 |
Numerical | 127.5 | 74.5 | 157.0 | 125.8 | 76.3 | 152.5 | |
Relative error (%) | 2.0 | 0.7 | 1.5 | 0.7 | 1.7 | 4.5 |
Case | Real Leak Information, xL* & AL* | Predicted Leak Information, xp* & Ap* | Max. Error, |xL* − xp*| & |AL* − Ap*| | |||
---|---|---|---|---|---|---|
TRM | TDM | SRFM | ITM | |||
T1 | No Leak | 0.50 & 0.01 | 0.50 & 0.038 | 0.25 & 0.024 | 0.46 & 0.031 | --- & --- |
T2 | 0.1 & 0.002 | 0.49 & 0.012 | 0.44 & 0.042 | 0.17 & 0.019 | 0.34 & 0.035 | 39% & 40% |
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Duan, H.-F. Transient Wave Scattering and Its Influence on Transient Analysis and Leak Detection in Urban Water Supply Systems: Theoretical Analysis and Numerical Validation. Water 2017, 9, 789. https://doi.org/10.3390/w9100789
Duan H-F. Transient Wave Scattering and Its Influence on Transient Analysis and Leak Detection in Urban Water Supply Systems: Theoretical Analysis and Numerical Validation. Water. 2017; 9(10):789. https://doi.org/10.3390/w9100789
Chicago/Turabian StyleDuan, Huan-Feng. 2017. "Transient Wave Scattering and Its Influence on Transient Analysis and Leak Detection in Urban Water Supply Systems: Theoretical Analysis and Numerical Validation" Water 9, no. 10: 789. https://doi.org/10.3390/w9100789