Analysis of the smallest detectable leakage flow rate of negative pressure wave-based leak detection systems for liquid pipelines

doi:10.1016/j.compchemeng.2007.08.011

Computers & Chemical Engineering

Volume 32, Issue 8, 22 August 2008, Pages 1669-1680

https://doi.org/10.1016/j.compchemeng.2007.08.011 Get rights and content

Abstract

One of the most concerned performance indices of leak detection and location systems is the smallest detectable leakage flow rate (SDLFR). Based on the physical model of pipeline, mathematical description for the amplitude change of negative pressure wave (NPW) and its attenuation traveling along the pipeline have been deduced. On the basis of mathematical description, the methods for evaluating SDLFR and determining the sensitive point of the pipeline have been proposed. Further study shows that factors such as leakage position, pressures at inlet and outlet, flow rate, density, wave speed in the pipeline, precision of the instruments and noise will have influences on SDLFR. And the results are of great significance to evaluate the performance of the systems based on NPW method.

Introduction

Pipelines are of great importance to chemical and petrochemical industries. For safety and environmental reasons, pipeline leakage surveillance is becoming more and more important in the management of pipeline and many kinds of leak detection and location systems have been developed and widely used in this area (Zhang, Wang, Liu, & Chang, 2001). In these systems the most prevalent method is based on negative pressure wave (NPW) for its relatively low cost, easy implementation, acceptable detection sensitivity and location precision. Nowadays more and more attention has been attracted to improve the performance of the systems and new demands have been proposed concerning the performance indices such as the smallest detectable leakage flow rate (SDLFR), false alarm rate and miss alarm rate.

In this paper mathematical description of the amplitude change and its attenuation of the NPW traveling along the pipeline has been deduced based on the physical model of the pipeline. On the basis of the mathematical description, the calculation method of SDLFR and sensitive point of pipeline have been proposed. Further study has been carried on the influences of factors on the SDLFR. And simulation results have been presented for illustration.

Section snippets

Mathematical model of the pipeline

The physical model of pipeline consists of two equations, namely momentum equation and continuity equation (Wylie, Streeter, & Suo, 1993; Yan, 1986): $\frac{P_{x}}{ρ} + V V_{x} + V_{t} + g sin a + \frac{f V | V |}{2 D} = 0$ $V P_{x} + P_{t} + ρ a^{2} V_{x} = 0$ where the subscript x and t denote partial differentiation, for example V_x denotes the partial differentiation of V to x. Let P denote the pressure, ρ the density of the fluid, V the velocity of the fluid, g the acceleration due to gravity, α the angle of the pipeline inclined with the horizontal, f the

Calculation of pressure change and leakage flow rate at leakage site

Let ${\bar{P}}_{0}$ and ${\bar{P}}_{L_{P}}$ denote the steady pressures at inlet and outlet of the pipeline, L_p denotes the length of the pipeline, l denotes the distance of the leakage site from the inlet, ${\bar{P}}_{l}$ denotes the pressure at distance l before leaking. According to the pressure gradient of the pipeline we have: $\frac{{\bar{P}}_{0} - {\bar{P}}_{l}}{l} = \frac{{\bar{P}}_{0} - {\bar{P}}_{L_{P}}}{L_{P}}$ so, ${\bar{P}}_{l} = {\bar{P}}_{0} - \frac{l}{L_{P}} ({\bar{P}}_{0} - {\bar{P}}_{L_{P}})$ After a leakage happened at l, the leakage flow rate Q_l satisfies the orifice equation: $Q_{l} = C_{k} A_{k} \sqrt{2 ρ (P_{l} - P_{g})}$ where C_k and A_k are the discharge coefficient and area of

Attenuation of the NPW traveling along the pipeline

In Section 3 the mathematical descriptions for the amplitude change of the NPW and the leakage flow rate are given. In order to determine the SDLFR, the mathematical description for the attenuation of the NPW traveling to the inlet and outlet must be deduced.

In pipelines convective terms are ignorable, thus Eqs. (1) and (2) can be written as $P_{x} + \frac{Q_{t}}{A} + ρ g sin a + \frac{f Q | Q |}{2 D ρ A^{2}} = 0$ $P_{t} + \frac{a^{2} Q_{x}}{A} = 0$ where Q is the mass flow rate.

The pressure P and mass flow rate Q of the pipeline can be expressed as $P = \bar{P} + p, Q = \bar{Q} + q$

Calculation of the SDLFR

Assume that pressure sensors are mounted only at the inlet and outlet of the pipeline. The necessary condition for a leakage detectable is that the pressure change caused by the leaking exceeds the threshold (the least valid pressure change) determined by the precision of the pressure sensor and the noises. So the SDLFR problem is converted to detect the smallest valid pressure changes of inlet |p₀| and outlet $| p_{L_{P}} |$ , i.e. $| p_{0} | > h_{u} = λ_{0} \sqrt{h_{0}^{2} + σ_{0}^{2}}$ $| p_{L_{p}} | > h_{d} = λ_{L_{p}} \sqrt{h_{L_{P}}^{2} + σ_{L_{P}}^{2}}$ where h₀ and $h_{L_{p}}$ denote the errors

Conclusion

Based on the physical model of the pipeline, mathematical description of the amplitude change of the NPW and its attenuation traveling along the pipeline has been funded and on the basis of these description, methods for determining the SDLFR and sensitive point of a pipeline are proposed. The results are of significant for the performance evaluation of the leak detection and location systems.

Acknowledgements

This work is supported by the National 863 Program of China (program No. 2006AA04Z428) and Honeywell (China) Ltd.

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Modeling of the pipeline as a lumped parameter system
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Analysis of the smallest detectable leakage flow rate of negative pressure wave-based leak detection systems for liquid pipelines

Abstract

Introduction

Section snippets

Mathematical model of the pipeline

Calculation of pressure change and leakage flow rate at leakage site

Attenuation of the NPW traveling along the pipeline

Calculation of the SDLFR

Conclusion

Acknowledgements

Models of pipelines in transient mode

Mathematical and Computer Modeling of Dynamical Systems

Pipeline simulation technique

Mathematics and Computers in Simulation

Verification of various pipeline models

Mathematics and Computers in Simulation

Modeling of the pipeline as a lumped parameter system

Atutomatika