1. Introduction

2. Analytical Model

2.2. Displacement fields

The NL equation is the fundamental equation governing wave motion in elastic and homogeneous media. If the media is subjected to a non-equilibrium local force f, the NL equation can be written in vector form [24] as

$$\left(\lambda +2\mu \right)\nabla \left(\nabla \bullet \mathit{u}\right)-\mu \nabla \times \nabla \times \mathit{u}+\mathit{f}=\rho \frac{{\partial }^2\mathit{u}}{{\partial t}^2},$$

(10)

where u is the displacement vector, λ and μ are Lamé constants, and ρ is the density of the media. The displacement field in cylindrical coordinates is specified by three potentials: the scalar potential Φ for P wave, and two vector potentials, ${\rm X}{\widehat{\mathit{e}}}_z$ for the SH wave and $\Psi {\widehat{\mathit{e}}}_z$ for the SV wave. Previously, we adapted the model proposed by Morse and Freshbach [19], expressed by

$$\mathit{u}=\nabla \Phi +\nabla \times \left({\rm X}{\widehat{\mathit{e}}}_z\right)+a\nabla \times \nabla \times \left(\Psi {\widehat{\mathit{e}}}_z\right),$$

(11)

because the three components were easily separated. The displacement vector can be described by the displacement components in the $\left(r,\theta ,z\right)$ coordinates

$$\mathit{u}=u_r\widehat{\mathit{r}}+u_{\theta }\widehat{\mathit{\theta }}+u_z\widehat{\mathit{z}},$$

(12)

where

Turbulence is generated through the pinhole, but it is maximized at the leaking orifice due to edge discontinuities. Since the hole cross-sectional area is very small compared to the cylinder surface, the force due to turbulent outflow at the orifice surface can be treated as PS. As the PS, the force f due to FRS is expressed as follows

$$\mathit{f}={\mathit{P}}_w\delta (\mathit{x}-{\mathit{x}}_0)e^{-i\omega t}.$$

(16)

In Eq. (16), ω is the angular frequency that transmits FRS energy from the PS to the cylinder and resonates with the energy of cylinder materials. As a solution to the delta function, Green’s function $g(\mathit{x};{\mathit{x}}_0)$ is defined as

$${\nabla }^{2}g\left(\mathit{x};{\mathit{x}}_0\right)=\delta (\mathit{x}-{\mathit{x}}_0).$$

(17)

In cylindrical coordinates, Eq. (17) is expressed as

$${\nabla }^{2}g\left(r,\theta ,z;r_0,{\theta }_0,z_0\right)=\frac{\delta \left(r-r_0\right)\delta \left(\theta -{\theta }_0\right)\delta \left(z-z_0\right)}{r},$$

(18)

and the solution of Eq. (18) is given as {18}

$$g\left(r,\theta ,z;r_0,{\theta }_0,z_0\right)=A_{v1}{g}_r\left(r;r_0\right)g_{\theta }\left(\theta ;{\theta }_0\right)g_z\left(z;z_0\right),$$

(19)

where $A_{v1}$ is the coupling constant, expressed as the first root of the Bessel function in $g_r$ and the integer v of the aperiodicity in $g_{\theta }$. To make the position of the PS the new origin, as shown in Figure 1(b), the coordinates $\left(r,\theta ,z;r_0,{\theta }_0,z_0\right)$ are replaced by $\left(\xi ,\vartheta ,\eta ;0\right)$, defined as

where the PS is located on the $x_i$ axis, $r_0=x_{0i}$. Previously, we derived the Green’s function responsible for the PS located inside the cylinder. Assuming that the Green’s function is periotic (v = 0)

where l is the length of the cylinder, and a and b are the outer and inner diameters, respectively. (Note. Corrects erratum in Eq. (16) for the η range in Ref. [18].) For gas leakage, the Green’s function is non-zero on the outer surface of the cylindrical shell because the PS is located on the outer surface of the cylindrical shell. In this study, the value of ${\kappa }_z$ was determined empirically.

In Eq. (20), the value of ξ is the shortest distance between the PS and the point where the detecting sensor is projected onto the equatorial layer containing the PS. However, as shown in Figure 1(b), there is no linear distance between the two points across the hollow interior. Since the thickness is much shorter than the diameter of the outer circle, we simply use the value of ξ as the length of the arc around the outer circle,

$$\xi \approx {\xi }_p=\pi a\theta /180°,$$

(23)

where θ is angle between the PS and the projected point.

From Eqs. (16) and (17), the force vector can be rewritten as

$$\mathit{f}=\mathit{P}{\nabla }^{2}g\left(\xi ,\vartheta ,\eta \right)e^{-i\omega t}=\nabla \left[\nabla \bullet \mathit{P}g\left(\xi ,\vartheta ,\eta \right)\right]-\nabla \times \left[\nabla \times \mathit{P}g\left(\xi ,\vartheta ,\eta \right)\right]e^{-i\omega t},$$

(24)

where ${\nabla }^2=\frac{{\partial }^2}{{\partial \xi }^2}+\frac{1}{\xi }\frac{\partial }{\partial \vartheta }+\frac{1}{{\xi }^2}\frac{{\partial }^2}{{\partial \vartheta }^2}+\frac{{\partial }^2}{{\partial \eta }^2}$.

Previously [18], the three potential functions were specified as CFIPs generated by the PS.

where φ, χ and ψ are scalar functions. These scalar functions were determined by solving Eq. (10) combined with Eqs. (11) and (25)–(27) as follows

where ${\alpha }^2=k_p^2-k_{\eta }^2>0$, and ${{\beta }^2=k}_s^2{-k}_{\eta }^2>0$. In Eqs. (28)–(30), $A_m$, $B_m$ and $C_m$ are the coupling constants.

As shown in Figure 1(a), the CF vectors generated by the FRS act in the axial ($P_z$) and the tangential ($P_{\theta }$) directions. There is no the CF in the radial ($P_r$) direction because the surface boundary is perpendicular to the radial direction. Substituting Eq. (28) into Eq. (25), the CFIPs for the P wave is given by

$${\Phi }_z=P_{\mathrm{w}}\left[\left(-i{k}_{\eta }\right)A_m{J}_m\left(\alpha \xi \right)\mathrm{c}\mathrm{o}\mathrm{s}\left(m\vartheta \right)e^{-i{k}_{\eta }\eta }-\frac{k_p^2}{\rho {\omega }^2}\frac{J_m\left({\kappa }_z\xi \right)}{{\alpha }^2-{\kappa }_z^2}\frac{\partial G_{01}}{\partial \eta }\right],$$

(31)

$${\Phi }_{\theta }=-P_{\mathrm{w}}\left[A_{m}m\frac{1}{\xi }{J}_m\left(\alpha \xi \right)\mathrm{s}\mathrm{i}\mathrm{n}\left(m\vartheta \right)e^{-i{k}_{\eta }\eta }\right],$$

(32)

Similarly, substituting Eq. (29) into Eq. (26), we get the CFIPs for the SH wave as

$${\mathsf{{\rm X}}}_z=0,$$

(33)

$${{\rm X}}_{\theta }=-P_{\mathrm{w}}\left\{B_{m}m{J}_m\left(\beta \xi \right)\mathrm{c}\mathrm{o}\mathrm{s}\left(m\vartheta \right)e^{-i{k}_{\eta }\eta }-\frac{k_p^2}{\rho {\omega }^2}\frac{G_{01}}{{\beta }^2-k^2}\left[J_m\left({\kappa }_z\xi \right)+\xi \frac{\partial J_m\left({\kappa }_z\xi \right)}{\partial \xi }\right]\right\}.$$

(34)

Substituting Eq. (30) into Eq. (27), the CFIPs for the SV wave is given by

$${\Psi }_z=0,$$

(35)

$${\Psi }_{\theta }=-P_{\mathrm{w}}{C}_m\left[\frac{1}{\xi }{J}_m\left(\beta \xi \right)+\frac{\partial J_m\left(\beta \xi \right)}{\partial \xi }\right]\cos \left(\mathit{m\vartheta }\right)e^{-i{k}_{\eta }\eta },$$

(36)

Since all CFIPs for the P, SH, and SV waves at a given CF direction have been fully derived due to pinhole leakage, the displacement components in the $\left(\xi ,\vartheta ,\eta \right)$ coordinates generated by the CFIPs can be obtained using the component form in Eqs. (13)–(15). Notes. In the same way as in Ref. [18], the component of the displacement d due to P_f is expressed as

$$u_{df}=P_f\left(A_{mf}{F}_{df}^1+B_{mf}{F}_{df}^2+C_{mf}{F}_{df}^3+F_{df}^4\right),$$

(37)

where $P_f$ is $P_z$ or $P_{\theta }$.

For $P_z$, the radial components $u_{rz}$,

the tangential components $u_{\theta z}$,

and the axial component $u_{zz}$,

(Note. Corrects the erratum in Eq. (62) for $F_{zz}^4$ in Ref. [18].)

For $P_{\theta }$, the radial components $u_{r\theta }$,

the tangential components $u_{\theta \theta }$,

and the axial component $u_{z\theta }$,

The remaining task is to evaluate the coupling constants $A_m$, $B_m$ and $C_m$. Let’s apply boundary conditions to the outer surface in the same way as in Ref. [18]. There is no stress on the outer surface of the cylinder because the effect of the atmosphere pressure on the displacement field is negligible

$${\sigma }_{rr}={\sigma }_{r\theta }={\sigma }_{rz}=0.$$

(58)

Introducing Eq. (58) to the stress-strain displacement relations for the circular cylindrical shell studied in this work yields a system of three linear algebraic equations given as follows

$$\left[\begin{array}{ccc}{a}_{11f}& a_{12f}& a_{13f}\\ a_{21f}& a_{22f}& a_{23f}\\ a_{31f}& a_{32f}& a_{33f}\end{array}\right]\left[\begin{array}{c}{A}_{mf}\\ B_{mf}\\ C_{mf}\end{array}\right]=\left[\begin{array}{c}{b}_{1f}\\ b_{2f}\\ b_{3f}\end{array}\right].$$

(59)

Elements $a_{ijf}$ and $b_{if}$ of the matrix in Eq. (59) are given in the Appendix A. For the $P_z$ CF, Eq. (59) becomes very simple because all elements except $a_{i1z}$, $b_{1z}$ and $b_{3z}$ are zero. We get

$$B_{mz}=0,$$

(60)

$$A_{mz}=\frac{b_{1z}}{a_{11z}}=\frac{b_{3z}}{a_{31}}.$$

(61)

The values of $k_{\eta }$ at a given location of the PS can be obtained by solving the roots of the function

$$f\left(k_{\eta }\right)=\frac{b_{1z}}{a_{11z}}-\frac{b_{3z}}{a_{31z}}=0.$$

(62)

To evaluate the displacements at position $\left(\xi ,\eta \right)$ on the outer surface, the arrival times $\left(\tau \mathrm{s}\right)$ of the AE signal generated by the PS force must be introduced into Eq. (37). The arrival times of the P and S waves propagating with velocities $c_P$ and $c_S$ are given as

$${\tau }_P=\frac{\sqrt{{\xi }^2+{\eta }^2}}{c_P},{\tau }_P=\frac{\sqrt{{\xi }^2+{\eta }^2}}{c_S},$$

(63)

respectively, where ξ is given by Eq. (23). Finally, the displacement fields generated by the gas leakage can be summarized as follows

$$u_{df}=P_f\left[\left(t-{\tau }_P\right)\left(A_{mf}{F}_{df}^1+F_{df}^4{\delta }_{dr}\right)+\left(t-{\tau }_S\right)\left(B_{mf}{F}_{df}^2+C_{mf}{F}_{df}^3+F_{df}^4{\delta }_{d\theta }\right)\right]\times e^{-i\omega t},$$

(64)

where $d=r,\theta$and z, and $f=z$ and θ. Based on the wave characteristics, Eq. (64) can be divided into the P, SH, and SV waves as

3. Experimental

4. Results and Discussion

4.2. Verification of CFIP model

Stainless steel cylindrical structure (ρ = 7.80 × 10³ kg/m3, c_p = 5.98 km/s, c_s = 3.30 km/s) was used for the simulation.

Before simulating the displacement field, the relation between $k_{\eta }$ and ${\kappa }_z$ was determined from Eq. (61). As shown in Figure B3, as the value of ${\kappa }_z$ increased, the value of k_η gradually decreased until ${\kappa }_z$= 15, while above ${\kappa }_z$ = 15, the value of k_η is almost independent of the ${\kappa }_z$ value. The angular-dependence of the simulated displacement field is very effective in determining the ${\kappa }_z$ value. The angular dependence of the observed AE amplitudes appears to be related to the Bessel function $J_0\left({\kappa }_z\xi \right)$, which is involved in the Green’s function in Eq. (20). The Bessel function goes to zero at a certain value of ξ depending on the value of ${\kappa }_z$. As shown in Figure B4, no minimum appears until ${\kappa }_z=2$. Above this value, a single minimum appears at $\vartheta =176°$ for ${\kappa }_z=3$, $\vartheta =141°$ for ${\kappa }_z=5$, and $\vartheta =87°$ for ${\kappa }_z=8$. Above ${\kappa }_z=8$, a well of multiple minima appears: for ${\kappa }_z=9$, there are two minima position at $\vartheta =78°$ and 164°. In the simulation, we set ${\kappa }_z=9$, and set FRS = 91 N/m² for p₀ = 2 bar and 250 N/m² for p₀ = 4 bar. These results led us to conclude that the displacement fields generated by a gas leakage in the cylindrical geometry is 2π-periodic: m = 0. Therefore, Eq. (64) is rewritten as

$$\begin{gathered} u_{df}=P_f\left[\left(t-{\tau }_P\right)\left(A_{0f}{F}_{df}^1+F_{df}^4{\delta }_{dr}\right)\right. \\ \left.+\left(t-{\tau }_S\right)\left(B_{0f}{F}_{df}^2+C_{0f}{F}_{df}^3+F_{df}^4{\delta }_{d\theta }\right)\right]e^{-i\omega t}, \end{gathered}$$

(68)

The displacement field, expressed by Eq. (68), indicates that the AE wave is active if $F_{df}^4$ is nonzero. From Eqs. (40), (42), (45), (49), (53) and (57), it can be found that $F_{rz}^4$, $F_{zz}^4$ and $F_{\theta \theta }^4$ are nonzero, but $F_{\theta z}^4$, $F_{r\theta }^4$ and $F_{z\theta }^4$ are zero. Considering that the nonzero $F_{df}^4$ terms are proportional to $1/{\omega }^2$, Eq. (68) can be rewritten for multi-frequency waves as

$$u_{df}\left({\omega }_1,\cdots ,{\omega }_n\right)=\sum _{i=1}^n{W}_i{u}_{df}\left({\omega }_i\right),$$

(69)

$$W_i=\frac{{\omega }_i^2}{\sum _i^{}{\omega }_i^2},$$

(70)

Eq. (70) gets contribution with equal weights at the configured frequencies. Therefore, the observed weight of the i component $\left(w_{\mathrm{e}\mathrm{m}\mathrm{p},i}\right)$ is compensated to Eq. (70). The final formula of the AE wave is

$$u_{df}\left({\omega }_1,\cdots ,{\omega }_n\right)=\sum _{i=1}^n{w}_{\mathrm{e}\mathrm{m}\mathrm{p},i}\frac{{\omega }_i^2}{\sum _i^{}{\omega }_i^2}{u}_{df}\left({\omega }_i\right),$$

(71)

where $u_{rz},u_{zz}$ and $u_{\theta \theta }$ are active and the others are inactive. Among these active displacements, $u_{zz}$ is dominant (hereafter, referred to as $u_z$).

The AE wave shown in Figure 3(a) was chosen as the model for the simulation. The frequency components and their observed weights are listed in Table 1. In the simulation, these data were introduced into Eq. (71). Figure 6(a) shows the simulated $u_z$ displacement and corresponding FFT spectrum for D0.2P4η1$\mathsf{\vartheta }$0.001 (FRS = 250 N/m²) using the eight frequencies listed in Table 1. Since $g\left(\xi ,\vartheta ,\eta \right)$ diverges to infinity at $\mathsf{\vartheta }=0$, we set the minimum angle to 0.001° in the simulation. As shown in Figure 6(b), the FFT spectrum of the simulated AE wave is in good agreement with the observed spectrum (Table 1).

Table 1.

${\nu }_i$/kHz	70.2	102.3	120.8	145.6	173.4	244.5	270.3	284.5
$w_{\mathrm{e}\mathrm{m}\mathrm{p},i}$	0.091	0.045	0.069	0.18	0.45	0.097	0.015	0.058

Here, we demonstrated the validity of our approach by evaluating the multi-frequency $u_z$ as a function of $\vartheta$ for angular dependence and a function of η for axial dependence. The maximum value was taken from the simulated $u_z$ displacements. For the angular dependency both experimental values and calculated values (after baseline correction) were converted to relative values for a value of $\vartheta =120°$. A baseline correction to the experimental values was necessary to match the two values, because the calculated minimum was nearly zero (Figure B4). As shown in Figure 7, the results for the angular dependency show good agreement between the observed and simulated values.

For the axial dependence, the simulation was also performed at $\vartheta$ = 0.001 instead of $\vartheta$ = 0. Simulated values given in meter were converted to those given in mV by multiplying the appropriate factor. The results for D0.2 and D1.2 are shown in Figure 8 and the others are shown in Figure B5. When η approaches zero, there is a sudden increase in the observed AE amplitude. The experimental results show that the AE signal increases abnormally when approaching the leakage point.

5. Conclusions

References

1. Korlapatia, N.V.S.; Khana, F.; Noora, Q.; Mirza, S.; Vaddiraju, S. Review and Analysis of Pipeline Leak Detection Methods. J. Pipeline Sci. Eng. 2022, 2, 100074. [Google Scholar] [CrossRef]
2. Reddy, R.S.; Payal, G.; Karkulali, P.; Himanshu, M.; Ukil, A.; Dauwels, J. Pressure and flow variation in gas distribution pipeline for leak detection. 2016. [Google Scholar] [CrossRef]
3. Tian, X.; Jiao, W.; Liu, T. Intelligent leak detection method for low-pressure gas pipeline inside buildings based on pressure fluctuation identification. J. Civ. Struct. Heal. Monit. 2022, 12, 1191–1208. [Google Scholar] [CrossRef]
4. Huang, S.-C.; Lin, W.-W.; Tsai, M.-T.; Chen, M.-H. Fiber optic in-line distributed sensor for detection and localization of the pipeline leaks. Sensors Actuators A: Phys. 2007, 135, 570–579. [Google Scholar] [CrossRef]
5. Quy, T.B.; Kim, J.-M. Pipeline Leak Detection Using Acoustic Emission and State Estimate in Feature Space. IEEE Trans. Instrum. Meas. 2022, 71, 1–9. [Google Scholar] [CrossRef]
6. Brunner, A.J.; Barbezat, M. Acoustic Emission Leak Testing of Pipes for Pressurized Gas Using Active Fiber Composite Elements as Sensors. J. Acoust. Emiss. 2007, 25, 42–50. [Google Scholar]
7. Xiao, R.; Joseph, P.; Li, J. The leak noise spectrum in gas pipeline systems: Theoretical and experimental investigation. J. Sound Vib. 2020, 488, 115646. [Google Scholar] [CrossRef]
8. Putra, I.A.; Prasetiyo, I.; Sari, D.P. The Use of Acoustic Emission for Leak Detection in Steel Pipe. Appl. Mech. Mater. 2015, 771, 88–91. [Google Scholar] [CrossRef]
9. Meribout, M. Gas Leak-Detection and Measurement Systems: Prospects and Future Trends. IEEE Trans. Instrum. Meas. 2021, 70, 1–13. [Google Scholar] [CrossRef]
10. Meng, L.; Yuxing, L.; Wuchang, W.; Juntao, F. Experimental study on leak detection and location for gas pipeline based on acoustic method. J. Loss Prev. Process. Ind. 2012, 25, 90–102. [Google Scholar] [CrossRef]
11. Ullah, N.; Ahmed, Z.; Kim, J.-M. Pipeline Leakage Detection Using Acoustic Emission and Machine Learning Algorithms. Sensors 2023, 23, 3226. [Google Scholar] [CrossRef] [PubMed]
12. Quy, T.B.; Kim, J.-M. Real-Time Leak Detection for a Gas Pipeline Using a k-NN Classifier and Hybrid AE Features. Sensors 2021, 21, 367. [Google Scholar] [CrossRef] [PubMed]
13. Lighthill, M.J. On sound generated aerodynamically. Part I: General theory. Proc. R. Soc. Lond. 1952, 211, 564–587. [Google Scholar] [CrossRef]
14. Pollock, A.A.; Pepper, C.E. Quantitative Analysis of Acoustic Emission from Gas Leaks in a Model Piping System. 1999. [Google Scholar] [CrossRef]
15. Yoshida, K.; Akematsu, Y.; Sakamaki, K.; Horikawa, K. Effect of Pinhole Shape with Divergent Exit on AE Characteristics during Gas Leak. J. Acoust. Emiss. 2003, 21, 223–229. [Google Scholar]
16. Laodeno, R.N.; Nishino, H.; Yoshida, K. Characterization of AE Signals Generated by Gas Leak on Pipe with Artificial Defect at Different Wall Thickness. Mater. Trans. 2008, 49, 2341–2346. [Google Scholar] [CrossRef]
17. Mostafapour, A.; Davoudi, S. Analysis of leakage in high pressure pipe using acoustic emission method. Appl. Acoust. 2013, 74, 335–342. [Google Scholar] [CrossRef]
18. Kim, K.B.; Kim, B.K.; Kang, J.-G. Modeling Acoustic Emission Due to an Internal Point Source in Circular Cylindrical Structures. Appl. Sci. 2022, 12, 12032. [Google Scholar] [CrossRef]
19. Morse, P.M.; Feshbach, H. Methods of Theoretical Physics; McGraw-Hill: New York, NY, USA, 1953; pp. 1764–1767. [Google Scholar]
20. http://brennen.caltech.edu/fluidbook/basicfluiddynamics/turbulence.htm.
21. Bomelburg, H.J. Estimation of Gas Leak Rates Through Very Small Orifices and Channels. BNWL-2223-77; Pacific Northwest Laboratory, 1977.
22. Bariha, N.; Mishra, I.M.; Srivastava, V.C. Harzard Analysis of Failure of Natural Gas and Petroleum Gas Pipelines. J. Loss Prev. Process Ind. 2016, 40, 217–226. [Google Scholar] [CrossRef]
23. Li, Y.; Zhou, P.; Zhuang, Y.; Wu, X.; Liu, Y.; Han, X.; Chen, G. An Improved Gas Leakage Model and Research on the Leakage Field Strength Characteristics of R290 in Limited Space. Appl. Sci. 2022, 12, 5657. [Google Scholar] [CrossRef]
24. Eringen, A.C.; Suhubi, E.S. Fundamentals of Linear Elastodynamics. In Elastodynamics; Academic Press: New York, NY, USA, 1975; Volume II, pp. 343–366. [Google Scholar]