Mathematical Modeling of Leak Detection Pipelines Conveying … · Liu et al [12]presented particle filters are sequential Monte Carlo methods based on point mass (or “particle”)

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Mathematical Modeling of Leak Detection

Pipelines Conveying Fluids

M. S. El Masri1 H. A. El Gamal2 E. M. Attia3

Student , ME, Professor Associate Prof.

Arab Academy for Science Alexandria University Arab Academy for Science

Department of Mech. Eng.

Alexandria Governorate, Egypt Alexandria Governorate, Egypt Alexandria Governorate, Egypt

Abstract: The leakage from pipelines in various industrial

applications is the cause of environmental problems and the loss

of conveyed fluid, It is therefore essential to detect the location

of unintended leakage resulting from cracks, holes and pores.

This requires the knowledge of pipe geometrical configuration

and nature of fluid inside it. Mathematical models to simulate

the leakage problem are needed to provide suitable predictions

necessary to help developing technical methods for the detection

of any leakage problem. Pipeline leak detection techniques and

systems are based on computational leak methods implemented

on several pipeline systems. The leak detection includes volume

mass balance, pressure monitoring, transient flow detection

methods. The transient leakage detection has become one of the

major applications of transient simulation techniques aiming to

detect and locate pipeline leakage efficiently. In the present

work a method of leakage from pipelines having cracks is

introduced. It is based on the continuous introduction of

sinusoidal pressure waves of small amplitudes at the entrance of

the pipeline. The variation of the amplitude of the pressure

gradient and the shear stress on the inside surface of the pipeline

in the streamwise flow direction are to be numerically evaluated.

The effect of the geometrical parameters concerning the pipe

and crack along with the flow parameters namely the Strouhal

and Reynolds numbers are examined to provide information

necessary for the technique to be implemented. Since this

transient simulation is of a numerical kind accurate solution of

the unsteady partial differential equations of the flow system is

very much required. It is the aim of the present work to

undertake this task.

Keywords Modeling- leakage- detection- fluids- incompressible

flow-pipelines

1. INTRODUCTION

Baghdadi and Mansy [1] presented a novel method

for leak detection in pipelines. The method is based on a

unidimensional flow analysis. The theoretical findings have

been verified experimentally for two different hole

geometries (circular and rectangular). The comparison set

between theory and experiment confirms the physical realism

of the mathematical model.

Turner [ 2 ] considered a pipeline network in which gas

pressure and temperature were measured at every

measurement location, and mass flows were measured at

some of these locations including all inlets and outlets to the

network. Leak detection is the process of determining

whether a leak is present in a particular section. Once a leak

is detected, the time it appeared, its size and its location are

determined.

Turner and Mudford [ 3 ] described a method of detecting

leaks in gas pipelines by use of flow, pressure and

temperature measurements that are usually made in long-

distance natural gas pipelines for other reasons.

Sharp and Campbell [4]applied a technique which involves

the injection of a sound pulse into the object under

investigation and recording of the resultant reflections.

Analysis of the reflections gives information about the bore

profile and input impedance of the object.

Silva et al [ 5 ] presented on-line computational technique

used in the analysis of hydraulic transients caused by leakage,

in order to detect and locate pipeline ruptures. Pressure

transients were obtained in two PVC pipelines 3/4"(19mm)

diameter by using four pressure transducers connected to a

PC computer equipped with A/D and D/A converters.

Miller et al [6] constructed a reference standard for setting up

and evaluating Acoustic Emission (AE) equipment to be used

in pipeline leak detection.

Fukushima et al [ 7 ] put forward a leak detection method

based on a dynamic simulation with the wave equations is

presented. An industrial application to one of the longest gas

pipeline is also presented with its performance information.

Verde [ 8 ] solved the multi-leak detection problem using

only sensors of flow and pressure at the extremes of the duct,

and using the analytical redundancy given of these

measurements.

Covas and Ramos [9] focused on leakage detection and

location in pipe networks based on a recent and novel

approach, known as inverse transient analysis. The main idea

behind this methodology is the identification of leaks location

in pipe networks using observed pressure data, collected

during the occurrence of transient events, and the

minimization of the difference between observed and

calculated parameters.

Wang et al [10] made attempts to detect leaks in pipelines

which contribute to damping of transient events. That fact

leads to a method of finding location and magnitude of leaks.

Because the problem of transient flow in pipes is nearly

linear, the solution of the governing equations can be

expressed in terms of a Fourier series.

Mpesha et al [11] presented a new procedure utilizing

transient state pressures to detect leakage in piping systems.

Transient flow, produced by opening or closing a valve, is

analyzed in the time domain by the method of characteristics

and the results are transformed into the frequency domain by

the fast Fourier transform.

International Journal of Engineering Research & Technology (IJERT)

ISSN: 2278-0181

www.ijert.orgIJERTV4IS110042

(This work is licensed under a Creative Commons Attribution 4.0 International License.)

Vol. 4 Issue 11, November-2015

149

Department of Mech. Eng. Department of Mech. Eng.

Liu et al [12]presented particle filters are sequential Monte

Carlo methods based on point mass (or “particle”)

representations of probability densities, which can be applied

to estimate states in nonlinear and non-Gaussian systems

without linearization.

Xu et al [13]described how the belief rule based expert

systems can be trained and used for pipeline leak detection.

Pipeline operations under different conditions are modeled by

a belief rule base using expert knowledge, which is then

trained and fine-tuned using pipeline operating data, and

validated by testing data.

Covas, Ramos and Almeida [ 14 ]Focused on leakage

detection in pipe systems by means of the standing wave

difference method ( SWDM ) used for cable fault location in

electrical engineering.

Huang et al [15] proposed a hybrid configuration of Mach-

Zehnder and Sagnac interferometer as sensing frame. In this

interferometer, there are two light paths that have the same

optical length but travel different sequence paths. Because the

propagation lights of the two light paths pass through the

leaking point at different times, the resulting phase signals

differ respectively.

Paivar, Salahshoor and Hourfar [16] proposed a neural

decision making approach to oil pipeline leak localization.

The one main methods, model based fault detection is used (

to find leaks quantity and location-making ) to form a novel

fault diagnosis scheme.

Koppel, Ainola and Puust [17] Proposed a mathematical

model for the determination of unregistered consumption and

leakage using the heads and flows at the inlet and at the outlet

of the main or at some nodes of the network.

Abhulimen and Susu [ 18 ] developed a novel model for

detecting leaks in complex pipeline network systems. The

model was derived from the theory of Liapunov stability

criteria. A leak is detected if the resulting eigenvalues from

the deviation flow matrix have values less than a

predetermined value.

Pedro et al [19] proposed the impulse response function as a

means of leak detection in pressurized fluid pipelines. The

experimental extraction of the impulse response function

indicates that frequency content of the injected transient

signal must be taken into account to minimize distortion.

Chuanhu, Guizeng and Hao [ 20 ]proposed 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

Ekuakille, Vendramin and Trotta [ 21 ] studied a spectral

analysis response, used for leak detection, the generally based

on fast Fourier transform (FFT).

Zhou et al [ 22 ]A belief rule base inference methodology

using the evidential reasoning approach (RIMER) has been

developed recently, where a new belief rule base (BRB) they

proposed to extend traditional if-then rules and can capture

more complicated causal relationships using different types

of information with uncertainties, but these models are

trained off-line and it is very expensive to train and re-train

them.

Olunloyo and Ajofoyinbo [ 23 ]proposed a model for real

time leakage detection in pipelines based on integration of a

detection subsystem and Global Positioning System (GPS)

receiver.

Shuqing et al [24] Proposed a new time-frequency analysis

method of Hilbert- Huang Transform applied in Pipeline leak

detection. Choosing the negative pressure wave signal from

dynamic pressure transmitter as research object, the dynamic

pressure signal is decomposed by the empirical mode

decomposition, and we obtain the proper mode functions that

satisfy the condition.

González et al [25] focused on the modeling and simulation

of a gas distribution pipeline network with a special emphasis

on gas ducts. Gas ducts are the most important components of

such kind of systems since they define the major dynamic

characteristics.

Wan and Qiu [26] introduced an optical fiber early-warning

system based on Mach-Zehnder in order to monitor the

normal operation of pipelines. Three single-mode fiber in the

cable which is buried along the pipeline probe vibration

signal on the ground, in which the two was not in the same

casing as a sensing fiber, the formation of intervention arm,

the other one as the transmission fiber, the return of the

interference signal.

Zhou et al [ 27 ]put forward a recursive algorithm based on

the Bayesian reasoning approach is proposed to update a

belief rule based (BRB) expert system for pipeline leak

detection and leak size estimation.

Reddy et al [ 28 ] Proposed dynamic simulation models that

can be used along with flow and pressure measurements, for

on-line leak detection and identification in gas pipeline

networks. They proposed method uses the available pressure

and flow rate measurements, sampled at regular intervals and

is based on an efficient state estimation technique, developed

by the authors (Reddy.

Reddy et al [29] evaluated the performance of the a proposed

leak detection and identification methodology, using

experiments with compressed air on a laboratory scale

network.

Liang and Zhang [30] presented a novel pipeline leak

detection scheme based on gradient and slope turns rejection

(GSTR). Instead of monitoring the pipeline under constant

working pressure, GSTR introduces a new testing method

which obtains data during the transient periods of different

working pressures.

Meng et al [ 31 ]summarized in detail the technologies of

recognizing and extracting wave characteristics, which is to

distinguish leaking and disturbing signals from time and

frequency domain.

Mandal, Chan and Tiwari [32] proposed a novel leak

detection scheme based on rough set theory and support

vector machine (SVM) was to overcome the problem of false

leak detection. In their approach, ‘rough set theory’ is

explored to reduce the length of experimental data as well as

generate rules further;

Jing and Zhi-Hong [33] studied pipelines leakage factors,

based on Grey relational analysis (GRA) to analyze and


ISSN: 2278-0181




150

evaluate all the factors and draw an order of factors

influencing on pipeline leakage.

II. THEARETICAL ANALYSIS

Fig. 1 shows the cracked pipe geometry and the coordinate

system. It is to be noted that the crack is modeled here as

circumferential slot in order to facilitate the theoretical

analysis as well as the numerical treatment using the finite

difference approximation

Fig.(1) The cracked pipe geometry and the coordinate system

Where,

Vm is the Mean flow velocity, vr is the Radial velocity

vz is the Axial velocity , W is the Crack size,

L- The length of the pipeline, R-pipe's radius.

A. Governing equations

2 2 2

2 2 2 3 2

1 1 1( ) ( ) ( )r r z r r r

r i

V

r r r z z r r z

(1)

2 2 2

2 2 2 3 2

1 1 1( ) ( ) ( )i i z i i r

i r

V

r r zr z r r z

(2)

2 2

2 2

1r r rrr

r r r z

(3)

2 2

2 2

1i i iir

r r r z

(4)

B – Dimensionless form

Let:

* rr =

R , * z

z =L

, * WW = ,

L

*t = ωt

* r

r

m

vv =

V , * z

z

m

vv =

V, r

r

m

vv

V

2

1 1 1 1( ) ( )z

z

m m m

vv

V V r r V r R r

,

m

R

V

,2

1 1( )r

m

vr r V R

, 2

mV R

Equation (1), (2), (3) and (4) in dimensionless form may be

written as,

2

2 2 22 2

2 2 3 2

1 1 1 1( ) ( ) ( )

( ) 0

r r r rr

e

rz i

R R

r r r L z r z L Rr

RV S

z L

(5)

And,

2 2 22 * 2

2 2 2 3 2

1 1 1 1( ) ( ) ( )

( ) 0

i i i ii

e

iz r

R R

r r r L z r r z L R

RV S

z L

(6)

2 22

2 2

1( )r r r

r

Rr

r r r L z

(7)

2 22

2 2

1( )i i i

i

Rr

r r r L z

(8)

III. RESULTS AND DISCUSSION

Fig.2 shows the amplitude of the stream function (*

r ) of the

perturbed sinusoidal flow in the pipe in the absence of a

crack. There exists a small value of phase shift from the input

sinusoidal signal applied at the pipe entrance as seen from Fig

3. It is clear from the results that the signal has rapidly

developed from the entrance of the pipe as can be inferred

from the parallel streamlines along the axis of the pipe. This

rapid development is true from the pipe centre to its wall.

This obviously shows that there is no attenuation of the signal

along the pipe.

Fig.(2) Amplitude of the flow stream function in the pipe without crack

0.450.425

0.4

0.35

0.30.275

0.20.175

0.125

0.1

0.075

0.05

0.025

amp of stream function Iso Contours

z*

r*

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

r

L

Vm

Vz

R vz

L

vr

W

Crack ( slot )

z


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151

Fig.(3) Phase shift of the flow stream function in the pipe without crack

On the other hand Fig. 4 Shows the amplitude of the stream

function ( *

i ) in the presence of a crack positioned axially at

the middle of the pipe length. It is clear that the presence of

the crack has affected the development of the signal not only

close to crack but even at the entrance of the pipe. This is

attributed to the loss of momentum from the crack out

flowing from the pipe wall which to be substituted at the pipe

entrance.

Fig.(4) Amplitude of the flow stream function in the pipe

with crack.

It can also be seen from Fig. 5 that the phase shift has also

been affected by the presence of the crack despite its small

value. It is be noted in this respect that the case just

considered are for Reynolds number (Re=1) and Strouhal

number (S=1). The effect of (Re) and(S) on the behavior of

the signal is more or less qualitatively the same.

Fig. 5 phase shift of the flow stream function in the pipe with

crack.

Is this respect that the axial position of the crack does not

qualitatively affect the behavior of the signal as can be seen

from Figs. 6 and 7 for positions (z*=0.1) and (z*=0.8) close

the pipe entrance and exit respectively. This is true for all

values of Re and S considered in this work.

0.00028

0.00084

0.0014

0.00196

0.00252

0.00308

0.00364

0.0042

0.00476

0.00532

phase shift of stream function Iso Contours

z*

r*

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

0.450.425

0.3750.35

0.30.275

0.25

0.20.175

0.15

0.125

0.075

0.05

0.025

amp of stream function Iso Contours

z*

r*

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

0.0002980.000674

0.001050.00143

0.00180.00218

0.002930.0033

0.00406

0.00481

0.00556

0.00593

0.00669

0.00706

0.00669

phase shift of stream function Iso Contours

z*

r*

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

10

20

30

40

50

60

70

80

90

z-direction

am

p.

pre

ssure

(Ap)

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

z-direction

am

p.

pre

ssure

(Ap)

Crack zone

a-Re=1

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

0

2

4

6

8

10

12

14

16

18

z-direction

am

p.

pre

ssure

(Ap)

b-Re=50


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152

Fig .6 a,b,c Variation of amplitude of pressure gradient with Reynolds

number (R=0.1m , R/L=0.01 , Vm=0.1m/s , U*=1 , S=1 , W=1mm ,

q=0.001 Q)

For the pressure gradient phase shift it is seen that the

increases in Re increase its absolute value considerably

although it is relatively small. This can be seen from Figs (7).

a- 1eR

b- 50eR

C- 100eR

Fig,.7 a,b,c Variation of phase shift of pressure gradient with Reynolds

number (R=0.1m , R/L=0.01 , Vm=0.1m/s , U*=1 , S=1 , W=1mm ,

q=0.001 Q)

A -Pressure gradient and shear stress on the pipe wall in the

presence of a crack:

Fig. 8 shows the effect of Re on the distribution of the wall

shear stress amplitude along the pipe. Similar to the behavior

observed for the pressure gradient it is the case for wall shear

stress as far as the form of the distribution and the indication

of the presence of the crack are concerned. It is quite clear

from the figures that the value of the shear stresses amplitude

varies slightly, that is less than 10%, as Re increase from 1 to

100 with S=1.

a- 1eR

b- 50eR

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

z-direction

am

p.

pre

ssure

(Ap)

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1-7

-6

-5

-4

-3

-2

-1

0x 10

-3

z-direction

phase s

hift

pre

ssure

(Cp)

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1-0.09

-0.08

-0.07

-0.06

-0.05

-0.04

-0.03

-0.02

-0.01

0

z-direction

phase s

hift

pre

ssure

(Cp)

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1-0.12

-0.1

-0.08

-0.06

-0.04

-0.02

0

z-direction

phase s

hift

pre

ssure

(Cp)

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

z-direction

am

p.

shear

str

ess(B

s)

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

z-direction

am

p.

shear

str

ess(B

s)

c- Re=100


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C - 100eR

Fig. 8 a, b ,c Variation of amplitude of shear stress with Reynolds number

(R=0.1m , R/L=0.01 , Vm=0.1m/s , U*=1 , S=1 , W=1mm , q=0.001 Q)

From Fig. 9 it is seen that the absolute value of the phase shift

of the shear stress increases considerably with the increase in

Re.

a- 1eR

b- 50eR

c-Re=100

Fig.9 a, b c Variation of phase shift of shear stress with

Reynolds number(R=0.1m , R/L=0.01 , Vm=0.1m/s , U*=1 , S=1 ,

W=1mm , q=0.001 Q)

Figs. (10) Show the effect of Strouhal number S on the

amplitude of the pressure gradient along the pipe wall while

keeping Re=1. The results forms are indicative of the pressure

of the crack irrespective of the value of S.

On the other hand the increase in S from 1 to 100 shows no

appreciable change in the pressure gradient (of order about

10%).

a-S=1

b- S=50

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

z-direction

am

p.

shear

str

ess(B

s)

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1-7

-6

-5

-4

-3

-2

-1

0x 10

-3

z-direction

phase s

hift

shear

str

ess(D

s)

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1-0.09

-0.08

-0.07

-0.06

-0.05

-0.04

-0.03

-0.02

-0.01

0

z-direction

phase s

hift

shear

str

ess(D

s)

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1-0.12

-0.1

-0.08

-0.06

-0.04

-0.02

0

z-direction

phase s

hift

shear

str

ess(D

s)

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

10

20

30

40

50

60

70

80

90

z-direction

am

p.

pre

ssure

(Ap)

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

10

20

30

40

50

60

70

80

90

z-direction

am

p.

pre

ssure

(Ap)


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c- S=100

Fig. 10 a,b,c Variation of amplitude of pressure gradient with Strouhal

number(R=0.1m , L/R=0.01 , Vm=0.1m/s , U*=1 , Re=1 , W=1mm ,

q=0.001 Q)

In a similar fashion the absolute value of the phase shift of

the pressure gradient increases considerably with the increase

of Strouhal number S although the value is relatively small

(see Fig.11).

(a) a-S=1

b-S=50

c- S=100

Fig.11 a,b,c Variation of phase shift of pressure gradient with Strouhal

number (R=0.1m , R/L=0.01 , Vm=0.1m/s , U*=1 , Re=1 ,

W=1mm , q=0.001 Q)

Figs.12 show the variation of the amplitude of the wall shear

stress along the pipe with the Strouhal number (S). The

behavior is quite similar to the effect of (Re) in that the shear

stress varies by the order of only (10%).

a- S=1

b- S=50

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

10

20

30

40

50

60

70

80

90

z-direction

am

p.

pre

ssure

(Ap)

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1-7

-6

-5

-4

-3

-2

-1

0x 10

-3

z-direction

phase s

hift

pre

ssure

(Cp)

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1-0.09

-0.08

-0.07

-0.06

-0.05

-0.04

-0.03

-0.02

-0.01

0

z-direction

phase s

hift

pre

ssure

(Cp)

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1-0.12

-0.1

-0.08

-0.06

-0.04

-0.02

0

z-direction

phase s

hift

pre

ssure

(Cp)

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

z-direction

am

p.

shear

str

ess(B

s)

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

z-direction

am

p.

shear

str

ess(B

s)


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c- S=100

Fig. 12 a,b,c Variation of amplitude of shear stress with

Strouhal number(R=0.1m , R/L=0.01 , Vm=0.1m/s , U*=1 , Re=1 ,

W=1mm , q=0.001 Q)

Exactly the same observation as for the effect of Re applies

to the phase shift of the shear stress and its relation to

Strouhal number as can be seen from Figs.(13).

This behavior is very well expected by the inspection of the

equations governing the perturbed flow originating from the

velocity sinusoidal variations at the pipe entrance. As is clear

from Eqns.(5) and (6) after multiplying both equations by Re

that the dependence of the solution can be put effectively in

terms of ReS.

a- S=1

b- S=50

c- S=100

Fig. 13 a,b,c Variation of phase shift of shear stress with

Strouhal number(R=0.1m , R/L=0.01 , Vm=0.1m/s , U*=1 , Re=1 ,

W=1mm , q=0.001 Q)

The results have also been tested for the effect of the crack

size (W) and the percentage amount of leak on the behavior

and magnitude of the flow resulting from the velocity input

signal.

It is found that the of the size of the crack and the amount of

leak both have no sensible effect on the flow signal as can be

seen figs.14. The aspect ratio (R/L) is also found to have no

appreciable effect on the behavior and magnitude of the

disturbing signal. Although he was the effect of this ratio

clearly shows the value of the pressure at the entrance to the

pipe as in Fig.15 where we note increasing pressure to

increase the value of (R/L)

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

z-direction

am

p.

shear

str

ess(B

s)

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1-7

-6

-5

-4

-3

-2

-1

0x 10

-3

z-direction

phase s

hift

shear

str

ess(D

s)

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1-0.09

-0.08

-0.07

-0.06

-0.05

-0.04

-0.03

-0.02

-0.01

0

z-direction

phase s

hift

shear

str

ess(D

s)

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1-0.12

-0.1

-0.08

-0.06

-0.04

-0.02

0

z-direction

phase s

hift

shear

str

ess(D

s)


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156

a- 21 10W x

b- 41 10W x

c- 51 10W x

Fig. 14 a,b,c Variation of amplitude of pressure gradient with crack size (W)

=1 , S=1 , q=0.001 Q )*, Vm=0.1m/s , U.010(R=0.1m , Re=1 , L/R=

a- R/L=100

b- R/L=2000

C - R/L=5000

Fig. 15 a,b,c Variation of amplitude of pressure gradient with Ratio (R/L)(R=0.1m , Vm=0.1m/s , U*=1 , Re=1 , S=1 , W=1mm , q=0.001

Q)

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

10

20

30

40

50

60

70

80

90

z-direction

am

p.

pre

ssure

(Ap)

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

10

20

30

40

50

60

70

80

90

z-direction

am

p.

pre

ssure

(Ap)

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

10

20

30

40

50

60

70

80

90

z-direction

am

p.

pre

ssure

(Ap)

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

10

20

30

40

50

60

70

80

90

z-direction

am

p.

pre

ssure

(Ap)

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

200

400

600

800

1000

1200

1400

1600

1800

z-direction

am

p.

pre

ssure

(Ap)

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

500

1000

1500

2000

2500

3000

3500

4000

4500

z-direction

am

p.

pre

ssure

(Ap)


ISSN: 2278-0181




157

III- CONCLUSIONS

The following conclusions may be drawn from the foregoing

analysis and discussions:

The presence of a crack in the pipe affects the

development of the disturbing signal irrespective of the

axial position of the crack. This shows how effective the

signal is in detecting the presence of a crack.

Reynolds number is seen to have no sensible effect on

the detection of the crack whether on the pressure

gradient and the shear stress along the pipe wall.

Strouhal number has also no sensible effect on the

detection of the crack as far as the pressure gradient and

wall shear are concerned.

The aspect ratio of the pipe, the size of the crack and the

percentage amount of leak from it are not to affect the

disturbing velocity signal used.

The velocity sinusoidal signal applied at the entrance of

the pipe is shown to be successful in detecting the

presence of pipe crack.

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