Scholarly article on topic 'Predicting the Leakage Exponents of Elastically Deforming Cracks in Pipes'

Predicting the Leakage Exponents of Elastically Deforming Cracks in Pipes Academic research paper on "Environmental engineering"

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Abstract of research paper on Environmental engineering, author of scientific article — A.M. Cassa, J.E. van Zyl

Abstract In this study, the relationship between the conventional power equation and the FAVAD (Fixed and Variable Discharges) equation for modelling leakage as a function of pressure is investigated. It is shown that different leakage exponent (or N1) values are obtained for the same leak when measured at different pressures. Analytical exploration of the two equations shows that N1 tends to 0.5 when the system pressure tends to zero and 1.5 when the system pressure tends to infinity. A new term called the dimensionless leakage number, NL, is defined as the ratio between the variable and fixed portions of a leak, and it is shown that a single function can be used to describe the relationship between NL and N1. This relationship is combined with previous work to predict the head-area slope for cracks in pipes to predict the leakage exponent for a range of crack widths and lengths in different pipe materials.

Academic research paper on topic "Predicting the Leakage Exponents of Elastically Deforming Cracks in Pipes"

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ScienceDirect Procedia

Engineering

ELSEVIER Procedia Engineering 70 (2014) 302 - 310 ;

www.elsevier.com/locate/proeedia

12th International Conference on Computing and Control for the Water Industry, CCWI2013

Predicting the leakage exponents of elastically deforming

cracks in pipes

A. M. Cassaa*, J. E. van Zylb

aDepartment of Water Resources, Golder Associates Africa (Pty) Ltd, P. O. Box 6001, Halway House 1685, Gauteng, South Africa bDepartment of Civil Engineering, University of Cape Town, Private Bag X3, Rondebosch 7701, Cape Town, South Africa

Abstract

In this study, the relationship between the conventional power equation and the FAVAD (Fixed and Variable Discharges) equation for modelling leakage as a function of pressure is investigated. It is shown that different leakage exponent (or N1) values are obtained for the same leak when measured at different pressures. Analytical exploration of the two equations shows that N1 tends to 0.5 when the system pressure tends to zero and 1.5 when the system pressure tends to infinity. A new term called the dimensionless leakage number, NL, is defined as the ratio between the variable and fixed portions of a leak, and it is shown that a single function can be used to describe the relationship between NL and N1. This relationship is combined with previous work to predict the head-area slope for cracks in pipes to predict the leakage exponent for a range of crack widths and lengths in different pipe materials.

© 2013 The Authors.PublishedbyElsevierLtd.

Selection and peer-reviewunderresponsibilityoftheCCWI2013 Committee Keywords: Leakage; pressure; FAVAD; leakage exponent; mathimatical models

1. Introduction

Pressure management is commonly used as an effective technique to reduce the leakage rate from and prolong the service life of pipes in distribution systems (Lambert, 2001). Pressure management has proven to be more

* Corresponding author. Tel.: +27 11 313 1122; fax: +27 11 315 0317. E-mail address: acassa@golder.co.za

CrossMar]

1877-7058 © 2013 The Authors. Published by Elsevier Ltd.

Selection and peer-review under responsibility of the CCWI2013 Committee

doi: 10. 1016/j .proeng .2014.02.034

effective for leakage management than predicted by the Torricelli orifice equation, which describes the flow through an orifice to be proportional to the square root of the pressure head.

This paper aims to propose a method for predicting the leakage exponent of individual longitudinal, circumferential or spiral cracks in different pipe materials. This is based on the dimensionless leakage number used in combination with previous work by Cassa and Van Zyl (In press).

Nomenclature

A area of orifice (m2)

A0 initial leak area (under zero pressure conditions)

C coefficient of discharge

C' coefficient

Cd coefficient of discharge

d diameter of pipe (m)

E modulus of elasticity (N/m2)

g acceleration due to gravity (m/s2)

h pressure head at the orifice (m)

Lc length of crack (m)

m head-area slope

N1 leakage exponent

Nl leakage number

Q flow rate (m3/s)

t thickness of pipe (m)

Wc width of crack (m)

p density of water (kg/m3)

al longitudinal stress (№№)

2. Background

The hydraulics of orifices is well understood and a great deal of research has been conducted on different orifice shapes and conditions (van Zyl & Cassa, 2011). Orifice hydraulics is based on the Torricelli equation:

Where Q is flow rate, Cd the discharge coefficient, g acceleration due to gravity, A the area of the orifice and h the pressure head at the orifice. Leak openings in pipes are basically orifices and thus should adhere to Equation 1. However, in practice a more general leakage equation is used, even as far back as 1881 (Lambert, 2001):

Where N1 is the leakage exponent and C the leakage coefficient. Efforts to characterise the behaviour of system leaks with pressure have focused mainly on the leakage exponent, and field tests have found system leakage exponents substantially higher than 0.5 (Gebhardt, 1975; Lambert, 2001; Al-Ghamdi, 2011), with some values even as high as 2.79 (Farley & Trow, 2003). Van Zyl and Clayton (2007) suggested four factors that may be responsible for the higher than expected leakage exponents in individual leaks, i.e. leak hydraulics, pipe material behaviour, soil hydraulics and water demand. Of these factors, pipe material behaviour is considered the most important, meaning that the leak area increases with increasing pressure.

Q = Chm

Laboratory-based tests on individual leaks in pipes showed that for round holes the leakage exponent is close to 0.5 irrespective of the pipe material or hole size (Hiki, 1981; Grevenstein & van Zyl, 2007). However, the leakage exponent can be substantially higher for cracks (Avila, 2003; Grevenstein & van Zyl, 2007; Ferrante, 2012). The relationship is complicated for plastic pipe by hysteresis and plastic deformation as documented by Ferrante et al. (2011) and Massari et al. (2012).

Cassa et al. (2010) and Cassa and Van Zyl (2011) used finite element modelling under the assumption of linear elastic behaviour to show that the areas of all types of leak openings vary linearly with pressure. Thus the area of any leak undergoing elastic deformation can be described as:

Where A0 is the initial leak area (under zero pressure conditions) and m the head-area slope. Replacing this relationship into Equation 1 results in:

While this equation is identical in form to the FAVAD equation proposed by May (1994), it has an important interpretive difference in that leaks are not considered either fixed or variable, but all leaks are considered variable. Equation 4 can explain leakage exponents between 0.5 and 1.5, and shows that under elastic conditions, the pressure response of a leak can be fully characterized by knowing its initial area A0 and head-area slope m.

3. Power and FAVAD equations

The power leakage equation (Equation 2) is commonly used to model leakage, and is likely to remain in use. Thus it was deemed necessary to investigate the accuracy with which the power equation can model an elastically deforming leak. In addition, the link between the power and FAVAD (Equation 4) equations was investigated and a model is proposed for converting between them.

3.1. Modelling elastic leaks with the Power Equations

To investigate the performance of the power equation in modelling elastic leaks, it was tested on a 60 mm long leak in a 110 mm class 6 uPVC pipe. The crack was modelled using finite elements as described in Cassa and Van Zyl (2011). The leak area was determined at different pressures and plotted against the pressure head as shown in Figure 1. It can be seen from the figure that straight lines fit the behaviour very well, and the head-area slopes of the cracks is 1.195x10-6.

A = A0 + mh

2.0E-04 1.8E-04 1.6E-04

^ 1.4E-04 £

^ 1.2E-04 I 1.0E-04 ä 8.0E-05 ° 6.0E-05 4.0E-05 2.0E-05 0.0E+00

0 10 20 30 40 50 60 70 80 90 100 110

Pressure head (m)

_♦Longitudinal Crack_

Figure 1. The areas of a 60 mm long longitudinal crack in a class 6 uPVC pipe as a function of the pressure head as determined by finite element analysis

From these areas, the leakage flow rate can be estimated by applying the FAVAD equation (Equation 4) to obtain the behaviour of the leak flow rates as a function of pressure as shows in Figure 2. The conventional approach is to fit the power leakage equation to the flow to obtain the leakage exponents as shown. When this is done to the data in Figure 2, a leakage exponent of 0.91 is found. While the power curve fit reasonably well, it clearly doesn't follow the trend of the data for the full pressure range.

Pressure head (m) ♦Longitudinal crack

Figure 2. The flow through a 60 mm longitudinal crack in a class 6 uPVC pipe as a function of the pressure head

The limitation of the power equation is exposed by fitting it not through all the points, but at each point by

calculating the change in leakage flow for a small pressure increase. These results are shown in Figures 3 for the leakage exponent. The results show that the leakage exponent of a given leak is not fixed, but is higher at higher pressures and at lower pressures.

1.3 1.2 1.1 1

Z ., S 0.9 1

a 0.7 |

es" 0.

1 0.4 ! J

0.3 0.2 0.1

10 20 30 40 50 60 70 80 90 100 110

Pressure head (m)

♦Longitudinal crack

Figure 3. Leakage exponents obtained at different pressures for the h-Q curve in Figure 2

3.2. Analytical exploration

To investigate the relationship between the power and FAVAD equations further, an analytical exploration may be done by first equating Equations 2 and 4:

Chm = CdJ2g ((h05 + mh15 )

Dividing both sides by the orifice equation results in:

C'hN1~0-5 = 1 + — ; With C

CdA0j2g

The term mh/A on the very right of the equation represents the ratio between the variable and fixed portions of the leakage, and is now defined as the dimensionless leakage number NL:

Through further manipulation, an expression can be found for N1 in the form:

= \n( +1)-in(c')| 1

\n(h) 2

This equation confirms that the leakage exponent is a function of h, but is complicated by the fact that C' is also a function of h. However, by exploring the limits of h in Equation 9, it is found that:

■ In the limit as h reduces to zero, the leakage exponent N1 = 0.5.

■ In the limit as h increases to infinity, the leakage exponent N1 = 1.5.

4. Leakage number

Further exploration of the relationship between the power and FAVAD equations revealed a plot of the leakage exponent against the leakage number always falls on the same line, irrespective of the values that A0, m and h. This relationship is shown in Figure 5.

Based on this curve, it may be concluded that the relationship between N1 and NL display the following characteristics:

N1 = 1 when Nl = 1 N1 > 1 when Nl > 1 N1 < 1 when Nl < 1 N1 is practically 0.5 for all NL < 0.01 N1 is practically 1.5 for all NL > 100

0 ------

0.001 0.01 0.1 1 10 100 1000

Leakage number NL

Figure 4. Leakage exponents obtained at different pressures for the h-Q curve in Figure 2

It was possible to find the following expression that provides an exact formula for the conversion between N1 and Nl:

N1 - 0.5

Nl =--(10)

L 1.5 - N1

Finally, Equation 10 may be used to convert the pressure-leakage response of a leak between the conventional and FAVAD leakage models. At the same time, the leakage number allows the range of leakage exponents to be calculated for any pressure range.

5. Predicting leak pressure to changes in Pressure

The model above can be used to predict the leakage exponent of any leak at different pressures if the initial area A0 and head-area slope m of the leak are known. The initial area of a crack is simple to determine if its length and width are known. The head-area slopes of cracks were investigated by Cassa & Van Zyl (In press) using finite element modelling. They proposed the following equations to predict the head-area slopes as a function of the crack and pipe properties:

2.93157• d03379 • L480.100-5997(logL)2-p-g

miong =---(11}

3 . 7714 • d0 178569 • L6 051 • a00928 • 101'05(logL )2 • p • g m =-5----(12)

spiral e • t1- 6795 U '

1 64802xlO-5 • L487992662 av09182555 -10a 82763163(los)2 . D. g m =—-5-l-cLJk. (13)

circ E t0-33824224 186376316

While the crack width was found to have negligible impact on the head-area slope, the width of the crack has a major influence on its initial area, and thus from Equation 8 on its leakage number. This means that narrow cracks will have higher leakage exponents than wider cracks of the same length. In addition, cracks that close up under zero pressure conditions will have leakage exponents of 1.5.

Based on the above, it is now possible to plot the combinations of crack lengths and widths that will produce different ranges of leakage exponents for a given pipe diameter and pipe material. For example, Figure 6 gives the ranges of the leakage exponent for longitudinal cracks in 100 mm nominal diameter uPVC, asbestos cement and cast iron pipes. The graph is drawn for a pressure of 50 m, and will vary slightly if the pressure is changed.

It can be seen from the graphs that the leakage exponent will tend to 1.5 as the width of the crack reduces to zero. Figure 6 also shows that the same crack will have a significantly higher leakage exponent in uPVC than in the other two materials, and that asbestos cement will have slightly higher leakage exponents than cast iron. For instance, a crack with a length of 300 mm and an initial width of 1 mm will display leakage exponents of 1.5, 0.9 and 0.6 respectively in uPVC, asbestos cement and cast iron pipes.

m < 0.01; N1 = 0.5

m between 0.01 and 1; N1 between 0.5 and 1

^m between 1 and 100; N1 between 1 and 1.5

m > 100; N1 = 1.5

Figure 5. Leakage exponent diagrams for longitudinal cracks in Class 9 (or equivalent) a) PVC, b) asbestos cement and c) cast iron pipes

6. Conclusion

This study investigated the link between the conventional and FAVAD leakage models. It is shown that the leakage exponent N1 does not provide a good characterization of the pressure response of a leak, and different leakage exponents result for the same leak when measured at different pressures. A dimensionless leakage number Nl = mhjA is defined as a more consistent way to characterise the pressure response of leaks, and a formula

proposed for converting between the leakage number and leakage exponent. In combination with equations for the head-area slope developed by Cassa & van Zyl (In press), this allows the prediction of the range of leakage exponent for cracks in different pipe materials.

7. References

Al-Ghamdi, A. S., 2011. Leakage-pressure relationship and leakage detection in intermittent water distribution

systems. Journal of Water Supply: Research and Technology - Aqua, 60(3), pp. 178-183. Ávila, H., 2003. Determination of parameters for longitudinal faults and leaks in residential connections in PVC pipes, Columbia: Master's Thesis, Department of Civil and Environmental Engineering, Universidad de Los Andes.

Cassa, A. M. & van Zyl, J. E., 2011. Predicting the head-area slopes and leakage exponents of cracks in pipes.

Exeter, Centre of Water Systems, University of Exeter, College of Engineering Mathematics and Physical Science, pp. 485-490.

Cassa, A. M. & van Zyl, J. E., In press. Predicting the head-leakage slope of cracks in pipes subjects to elastic

deformations. Journal of Water Supply: Research and Technology - Aqua. Cassa, A. M., van Zyl, J. E. & Laubscher, R. F., 2010. A numerical investigation into the effect of pressure on

holes and cracks in water supply pipes. Urban Water Journal, 7(2), pp. 109-120. Farley, M. & Trow, S., 2003. Losses in Water Distribution Networks: A Practioner's Guide to Assessment,

Monitoring and Control. London: IWA Publishing. Ferrante, M., 2012. Experimental investigation of the effects of pipe material on the leak head discharge

relationship. Journal of Hydraulic Engineering, ASCE, 138(8), pp. 736-743. Ferrante, M., Massari, C. B. B. & Maniconi, S., 2011. Experimental evidence of hysteresis on the head-discharge relationship for a leak in a polyethylene pipe. Journal of Hydraulic Engineering, ASCE, 137(7), pp. 775780.

Gebhardt, D. S., 1975. The effects of pressure on domestic water supply including observations on the effect of

limited garden-watering restrictions during a period of high demand. Water SA, 1(1), pp. 3-8. Grevenstein, B. & van Zyl, J. E., 2007. An experimental investigation into the pressure-leakage relationship of

some failed water pipes. Journal of Water Supply: Research and Technology - Aqua, 56(2), pp. 117-124. Hiki, S., 1981. Relationship between leakage and pressure. Journal of Japan Waterworks Association, pp. 50-54. Lambert, A., 2001. What do we know about pressure/leakage relationships in distribution systems. Brno, Czech

Republic, IWA Publishing, pp. 89-96. Massari, C., Ferrante, M., Brunone, B. & Meniconi, S., 2012. Is the leak head-discharge relationship in

polyethylene pipes a bijective function?. Journal of Hydraulic Research, IAHR, 50(4), pp. 409-417. May, J., 1994. Leakage, Pressure and Control, BICS International Conference on Leakage Control Investigation

in underground Assets. London, BICS. van Zyl, J. E. & Cassa, A. M., 2011. Linking the power and FAVAD equations for modelling the effect of pressure on leakage. Exeter, Centre of Water Systems, University of Exeter, College of Engineering Mathematics and Physical Science, pp. 551-556. van Zyl, J. E. & Clayton, C. R. I., 2007. The effect of pressure on leakage in water distribution systems. Water Management, 160(WM2), pp. 109-114.