Much progress has been made over the last decade in understanding the behaviour of flow through leak openings with changes in water mains pressure. In particular it has been established that variations in leak areas with pressure is the main factor responsible for the range of leakage exponents observed in practice, and several numerical and experimental studies have investigated this behaviour. This paper provides an overview of the advances in leakage modelling over the last decade and then presents the results of a new experimental study of various leak types (round holes and longitudinal, spiral and circumferential cracks) in different pipe materials (unplasticised polyvinylchloride, modified polyvinylchloride, high density polyethylene and steel). The experimental results are evaluated in light of the latest theoretical advances and recommendations are made for further experimental studies.
INTRODUCTION
The International Water Association (IWA) has been at the forefront of water loss management for several decades. Their efforts were formalised in 1995 with the formation of the Water Loss Task Force, which has made extensive contributions to the field, including the internationally used IWA Water Balance, Infrastructure Leakage Index and several other benchmarks and guidelines (Lambert 2015).
One of the important activities of the IWA Water Loss Task Force (now the Water Loss Specialist Group) has been to investigate why leak flow rates are often significantly more sensitive to pressure than predicted by the orifice equation (Lambert 1997, 2000; Lambert et al. 2013).
While the value of N1 should be 0.5 to comply with the orifice equation, in field studies it has been found to range between 0.36 and 2.95 (see Schwaller & Van Zyl (2014) for a summary of the ranges for N1 found in different countries).
It is important to understand the causes of the observed high leakage exponent values as this may allow engineers to better predict the response of systems to changes in pressure, and ultimately improve the management of leakage. Van Zyl & Clayton (2007) proposed a number of potential causes of the observed range of N1 values, including leak hydraulics, soil hydraulics and variations in leak area with pressure. Other factors have also been shown to play a role, including errors in field test assessment of average pressure and leakage estimates, the spatial distribution of leaks (Schwaller & Van Zyl 2014; Schwaller et al. 2015), soils surrounding the pipe (Walski et al. 2006; Van Zyl et al. 2013) and changes in axial momentum (Ferrante et al. 2013a, 2013b). It is now widely accepted that changes in leak area are the most important causative factor of the observed behaviour.
Greyvenstein & van Zyl (2007) published the results of an exploratory experimental study in Journal of Water Supply: Research and Technology – AQUA, showing that the leakage exponents measured in the field are not unrealistic, and can be reproduced in the laboratory. They found leakage exponents between 0.41 and 2.30 and agreed that variation in leak area is the most likely cause for the deviation from the orifice equation.
Significant progress on understanding the behaviour of leaks areas in pipes has been made in the last decade. Thus the aims of this paper are two-fold: to provide a review and synthesis of developments in understanding leakage behaviour over the last decade and to present the results of a recent experimental study in the light of these findings. It is hoped that the proposed experimental method and data analysis will provide a uniform basis for the further experimental work required to establish the characteristics of the various leak types in different pipe materials and sections, including service connections and leaks at joints and fittings.
The next section provides an overview of the current understanding of leak area behaviour and its implications for leakage modelling and management. This is followed by the description of a proposed standard experimental procedure and data analysis of different leak types (round holes and longitudinal, spiral and circumferential cracks) in different pipe materials (unplasticised polyvinylchloride (uPVC), modified PVC (mPVC), high density polyethylene (HDPE) and steel).
LEAK AREA VARIATION AND ITS IMPLICATIONS FOR LEAKAGE MODELLING
Pressure and pipe wall stresses
Pressure in a water pipe causes stresses to develop in the pipe walls, resulting in material strain and subsequently changes in the areas of leak openings in pipes. It is possible to derive theoretical equations for the circumferential and longitudinal stresses in a closed cylindrical container resulting from water pressure (for instance, see Cassa et al. (2010)). These equations show that both circumferential and longitudinal stresses are linear functions of the pressure, but that the circumferential stresses are double the size of the longitudinal stresses.
However, unlike pressure vessels, pipes are not closed at their ends and are generally supported by thrust blocks at bends and junctions that transfer the longitudinal forces to the soil. Thus it is the circumferential rather than longitudinal pipe wall stresses that will vary as a result of changes in pressure.
It should be noted that several external factors, such as the weight of soil, external loads, soil movements and thermal expansions also influence the pipe wall stresses. However, unlike the circumferential stresses induced by fluid pressure, these stresses are independent of water pressure and thus are likely to have a constant impact on leak area variations.
Linearity of the pressure-area relationship
The mechanisms through which pipe material can deform in response to changes in pressure are elastic, viscoelastic and plastic deformation and fracture.
Over the last decade, several experimental and modelling studies have been carried out under elastic deformation conditions on a large range of pipe materials, section properties, leak types and loading conditions. These studies concluded that all leak areas vary as a linear function of pressure (Buckley 2007; Cassa & Van Zyl 2013; Van Zyl & Cassa 2014).
Plastics such as PVC and PE are viscoelastic materials, which means that they display both elastic and viscous deformation behaviour. Several researchers have investigated the resultant time-dependent response of leak areas to changes in pressure (Ferrante et al. 2011, 2013a, 2013b; Ferrante 2012; Massari et al. 2012; De Marchis et al. 2016; Fox et al. 2016a, 2016b). Ssozi et al. (2016) showed that at any given time after loading, plastic pipes also have a linear pressure-area relationship, but that the slope of the relationship increases with time until it stabilises at the relaxation time of the material.
Unlike the elastic and viscoelastic cases, leak areas in pipes undergoing plastic (permanent) deformation or fracture cannot be assumed to have linear pressure-area relationships. However, both these processes are non-reversible and can thus only continue for a limited time before they will either stabilise (i.e. become elastic or viscoelastic) or result in catastrophic fracture (Buckley 2007). In addition, they can only occur when the water pressure is increased (i.e. pipe wall stresses are increased) and not when the pressure is decreased. Since zonal pressure management virtually always involves lowering of pressures, the effects of plastic deformation and fracture may be ignored in analysing pressure management zone data.
The linearity of the pressure-leakage relationship means that it is only necessary to know a leak opening's initial area and head-area slope to fully characterise its area and thus its hydraulic behaviour using Equation (1). The following points summarise the main findings on the head-area slope of pipe leaks (Greyvenstein & van Zyl 2007; Cassa et al. 2010; Cassa & van Zyl 2013; Van Zyl & Cassa 2014):
The areas of round holes in all materials are stable and vary very little with pressure. In practice this means that their head-area slopes may be assumed to be zero.
The head-area slopes of all types of leaks in steel pipes are very small and may also be assumed to be zero. However, corrosion failures in metal pipes have not been studied in detail and further work is required to determine whether reduced wall thicknesses due to corrosion will have a significant effect on the head-area slope.
The head-area slopes of circumferential cracks are generally small and often negative, meaning that the crack area reduces with increasing pressure (a result of the circumferential stresses elongating the crack, pulling it closed due to Poisson's ratio effect).
- Longitudinal cracks have the largest head-area slopes of all leak types. Cassa & Van Zyl (2013) found that the crack width and Poisson's ratio of the pipe material had a negligible effect on the head-area slope. They proposed an equation for predicting the head-area slope of longitudinal cracks as a function of the pipe diameter d, crack length Lc, elasticity modulus of the pipe material E and pipe wall thickness t (ρ is the density of water and g acceleration due to gravity):4
Van Zyl & Cassa (2014) showed that this equation provides good results based on data from several different laboratory studies.
Implications for leak hydraulics
The form of this equation was earlier proposed by Ledochowski (1956) and particularly May (1994) whose paper became highly influential as the FAVAD (Fixed and Variable Area Discharges) equation.
The first term of Equation (5) is the orifice equation and describes the flow through a fixed initial area of the leak. The second term in the equation describes the flow through the expanded area of the leak.
While Equation (5) or (7) seems to predict a leakage exponent between 0.5 and 1.5, it can be seen from Equation (9) that the leakage exponent can adopt a wider range. In particular, the leakage exponent will approach infinity when the leakage number approaches minus one.
The implications for leakage modelling can be summarised as follows (Van Zyl & Cassa 2014; Van Zyl et al. 2017):
The N1 power equation is an empirical equation not based on fundamental fluid mechanics principles. It can provide reliable results when used within its calibration pressure range, but may result in significant errors if used to extrapolate beyond this range.
The leakage exponent of a system (with a given set of leaks) is not constant, but varies with system pressure. Higher pressures in the same system will result in higher N1 values, while lower pressures will result in lower N1 values.
While the head-area slope is not affected by the width of a crack, the leakage number (Equation (8)) is significantly affected due to the change in the initial area. From Equation (9) it can now be shown that the same crack will have substantially higher leakage exponents at smaller crack widths. The implication of this observation is that the leakage exponents determined in laboratory tests, where a slit is normally machined into a pipe, will tend to underestimate the leakage exponent of the same crack length that forms in the field without removal of pipe material (and thus with a smaller width).
Finally, it can be shown from Equations (6)–(9) that leak openings with negligibly small head area slopes (i.e. fixed leak openings) will result in N1 = 0.5 and leak openings with initial areas close to zero (such as cracks closing under zero pressure conditions) will result in N1 = 1.5.
Implications for DMA leakage modelling
Leakage in water distribution systems is normally assessed at DMA (district metered area) level through minimum night flow and water balance analyses. While larger leaks can mostly be detected and repaired, little is known about the number and distribution of smaller background leaks.
The behaviour of systems with many leaks adhering to the power leakage equation was studied by Ferrante et al. (2014), who found that the DMA leakage exponent is larger than the mean individual leakage exponent due to the spatial variability of leaks. Schwaller & Van Zyl (2014) found that individual leaks adhering to the FAVAD equation can result in a wide range of DMA leakage exponents, similar to that observed in practice.
The linearity of the leak area-pressure relationship has important implications for the understanding of DMA leakage behaviour. In particular it means that the total DMA leakage, which is the sum of the individual leak behaviours, will also be a linear function of average zone pressure and that Equations (3) and (5)–(8) can thus be applied to DMAs with many leaks.
In practice this means that if the DMA leakage at the time of steady minimum night flow is known at two different Average Zone Night Pressures, which are currently used to determine N1 using Equation (2), the same data may be used in Equation (7) to estimate the sum of effective initial areas and head area slopes for all the leaks – both detectable and non-detectable – in the DMA.
The resulting DMA initial area provides an estimate for the total leak area under zero pressure conditions and the DMA effective head-area slope can be used in combination with the known values for different leak types to estimate the dominant leak type in the DMA. In addition the N1 for the DMA can be estimated at different pressures from its initial area and head-area slope using Equations (8) and (9).
Schwaller et al. (2015) used a spreadsheet model with stochastic leak distributions to confirm that the initial area and head-area slope of a DMA provide good estimates of the sum of the initial areas and head-area slopes of individual leaks in the system respectively. Unrealistic values of the system leakage parameters may be used to diagnose potential problems, such as measurement errors or leaking boundary valves (this is already the case in the expert software PresCalcs (Lambert 2016)).
Finally, Kabaasha et al. (2016) incorporated the FAVAD approach into hydraulic network models and showed that significant errors in the leakage demand at nodes can be made when the current power equation approach is used.
The next section of the paper reports on the results of an experimental study of the pressure-leakage behaviour of pipe leaks, and analyses these results in light of the theory discussed in this section.
EXPERIMENTAL STUDY
Introduction
An experimental study was conducted to investigate the behaviour of various types of leaks (round holes and longitudinal, spiral and circumferential cracks) in 100 mm nominal diameter pipes of different materials predominantly used for distribution mains. The pipe materials that were included in the study were uPVC, mPVC, HDPE and steel. mPVC is a pipe material designed to have additional ductility and more stable long-term characteristics than uPVC (DPI 2010).
The properties and dimensions of the leaks and pipes used in this study are summarised in Table 1. The elasticity modulus of the materials were not measured, but estimated from literature. All cracks had a width of 1 mm.
Properties and dimensions of the leaks and pipes investigated in this study
| Pipe material | Internal diameter (mm) | Wall thickness (mm) | Elasticity modulus E (MPa) | Main dimensions of leaks tested (mm) | |||
|---|---|---|---|---|---|---|---|
| Round holes | Cracks | ||||||
| Longitudinal | Spiral | Circumferential | |||||
| uPVC | 105.5 | 4.54 | 3 | 12 | 50, 100 | 50, 100 | 50, 100 |
| mPVC | 106.6 | 3.44 | 3 | 12 | 50 | 50 | 50 |
| HDPE | 103.6 | 6.38 | 1 | 12 | 73, 100 | 50, 78 | 54, 80 |
| Steel | 105.1 | 4.92 | 203 | 12 | 50, 100 | 50, 105 | 53 |
| Pipe material | Internal diameter (mm) | Wall thickness (mm) | Elasticity modulus E (MPa) | Main dimensions of leaks tested (mm) | |||
|---|---|---|---|---|---|---|---|
| Round holes | Cracks | ||||||
| Longitudinal | Spiral | Circumferential | |||||
| uPVC | 105.5 | 4.54 | 3 | 12 | 50, 100 | 50, 100 | 50, 100 |
| mPVC | 106.6 | 3.44 | 3 | 12 | 50 | 50 | 50 |
| HDPE | 103.6 | 6.38 | 1 | 12 | 73, 100 | 50, 78 | 54, 80 |
| Steel | 105.1 | 4.92 | 203 | 12 | 50, 100 | 50, 105 | 53 |
Experimental setup and procedure
Schematic layout and photograph of the main components of the experimental setup.
The system was held together with three 20 mm diameter stainless steel rods secured to the end sections. The steel rods took up the longitudinal forces exerted by the water pressure and thus prevented longitudinal stresses being induced in the sample pipe walls. Test sections were all 800 mm long, leaving a minimum distance of 350 mm between the leak opening and pipe section end.
One end section was connected to a pumped water supply from an underground sump through a 25 mm calibrated magnetic flow meter with an accuracy of ±0.5%. The downstream end was fitted with a calibrated pressure transducer with a working range of 0–20 bar and accuracy of ±0.5%. Water was supplied by a variable speed submersible centrifugal pump capable of supplying a flow of 5 L/s at a head of 100 m.
At the start of an experimental run, a small flow was introduced and the setup tilted towards an opening on the downstream end to remove all trapped air. The setup was then placed horizontally with the pressure transducer and leak on the same level. The leak discharged into the atmosphere and no flow existed in the system apart from the leakage.
Logged flow and pressure data for a typical experimental run.
Experimental data points were obtained by identifying stable sections of the flow and pressure graphs and then taking the average values over each of the ranges. These (typically 25) average values were then plotted and analysed to determine the leakage characteristics for each experiment.
It was observed that the first raising leg of tests on cracks in plastic pipes were often distinctly different from the rest of the data, possibly due to initial plastic deformation before the leak opening stabilised. In such cases the first few (between one and six) data points were omitted from the results.
RESULTS
The results of the experiments are summarised in Table 2. The analyses conducted to obtain these results are first explained, followed by a discussion of significant observations and their implications for leakage modelling.
Summary of experimental results
| Pipe material | Length/diameter (mm) | N1 | 95% SCI for | 95% SCI for | p for | Cd | ||
|---|---|---|---|---|---|---|---|---|
| Round holes | ||||||||
| uPVC | 12 | 0.499 | 68.3 | ±0.11 | −0.00141 | ±0.00230 | 12.5 | 0.603 |
| mPVC | 12 | 0.500 | 68.3 | ±0.17 | 0.00021 | ±0.00360 | 87.9 | 0.604 |
| HDPE | 12 | 0.501 | 70.3 | ±0.16 | 0.00180 | ±0.00345 | 4.5 | 0.621 |
| Steel | 12 | 0.497 | 67.4 | ±0.18 | −0.00486 | ±0.00365 | 0.2 | 0.596 |
| Longitudinal slits (width = 1 mm) | ||||||||
| uPVC | 50 | 0.886 | 27.4 | ±2.08 | 0.28658 | ±0.0193 | <0.1 | 0.547 |
| uPVC | 100 | 1.041 | 52.9 | ±2.59 | 2.51200 | ±0.129 | <0.1 | 0.529 |
| mPVC | 50 | 0.989 | 37.5 | ±6.00 | 0.86711 | ±0.139 | <0.1 | 0.749 |
| HDPE | 73 | 0.835 | 62.8 | ±11.7 | 0.97499 | ±0.348 | <0.1 | 0.860 |
| HDPE | 100 | 0.798 | 94.5 | ±12.0 | 2.03031 | ±0.578 | <0.1 | 0.945 |
| Steel | 50 | 0.500 | 22.0 | ±0.22 | 0.00011 | ±0.00275 | 10.3 | 0.440 |
| Steel | 100 | 0.529 | 44.7 | ±0.13 | 0.02166 | ±0.00205 | <0.1 | 0.447 |
| Spiral slits (width = 1 mm) | ||||||||
| uPVC | 50 | 0.657 | 36.6 | ±0.53 | 0.10802 | ±0.00800 | <0.1 | 0.732 |
| uPVC | 100 | 0.797 | 78.4 | ±1.31 | 1.24352 | ±0.0476 | <0.1 | 0.784 |
| mPVC | 50 | 0.798 | 43.2 | ±1.47 | 0.37375 | ±0.0288 | <0.1 | 0.865 |
| HDPE | 50 | 0.656 | 33.0 | ±3.08 | 0.09377 | ±0.0451 | <0.1 | 0.660 |
| HDPE | 78 | 0.703 | 69.0 | ±4.68 | 0.48148 | ±0.123 | <0.1 | 0.884 |
| Steel | 50 | 0.490 | 22.8 | ±0.08 | −0.00267 | ±0.00095 | 6.4 | 0.457 |
| Steel | 105 | 0.523 | 44.2 | ±0.17 | 0.01615 | ±0.00265 | 1.6 | 0.421 |
| Circumferential slits (width = 1 mm) | ||||||||
| uPVC | 50 | 0.455 | 34.8 | ±0.26 | −0.02170 | ±0.00380 | <0.1 | 0.696 |
| uPVC | 100 | 0.327 | 50.2 | ±1.28 | −0.11423 | ±0.0189 | <0.1 | 0.502 |
| mPVC | 50 | 0.433 | 37.2 | ±0.27 | −0.03403 | ±0.00380 | <0.1 | 0.743 |
| HDPE | 54 | 0.185 | 20.6 | ±1.74 | −0.06212 | ±0.0213 | <0.1 | 0.382 |
| HDPE | 80 | −0.262 | 37.4 | ±5.14 | −0.20706 | ±0.0646 | <0.1 | 0.467 |
| Steel | 53 | 0.499 | 22.6 | ±0.37 | −0.00008 | ±0.00485 | 33.3 | 0.427 |
| Pipe material | Length/diameter (mm) | N1 | 95% SCI for | 95% SCI for | p for | Cd | ||
|---|---|---|---|---|---|---|---|---|
| Round holes | ||||||||
| uPVC | 12 | 0.499 | 68.3 | ±0.11 | −0.00141 | ±0.00230 | 12.5 | 0.603 |
| mPVC | 12 | 0.500 | 68.3 | ±0.17 | 0.00021 | ±0.00360 | 87.9 | 0.604 |
| HDPE | 12 | 0.501 | 70.3 | ±0.16 | 0.00180 | ±0.00345 | 4.5 | 0.621 |
| Steel | 12 | 0.497 | 67.4 | ±0.18 | −0.00486 | ±0.00365 | 0.2 | 0.596 |
| Longitudinal slits (width = 1 mm) | ||||||||
| uPVC | 50 | 0.886 | 27.4 | ±2.08 | 0.28658 | ±0.0193 | <0.1 | 0.547 |
| uPVC | 100 | 1.041 | 52.9 | ±2.59 | 2.51200 | ±0.129 | <0.1 | 0.529 |
| mPVC | 50 | 0.989 | 37.5 | ±6.00 | 0.86711 | ±0.139 | <0.1 | 0.749 |
| HDPE | 73 | 0.835 | 62.8 | ±11.7 | 0.97499 | ±0.348 | <0.1 | 0.860 |
| HDPE | 100 | 0.798 | 94.5 | ±12.0 | 2.03031 | ±0.578 | <0.1 | 0.945 |
| Steel | 50 | 0.500 | 22.0 | ±0.22 | 0.00011 | ±0.00275 | 10.3 | 0.440 |
| Steel | 100 | 0.529 | 44.7 | ±0.13 | 0.02166 | ±0.00205 | <0.1 | 0.447 |
| Spiral slits (width = 1 mm) | ||||||||
| uPVC | 50 | 0.657 | 36.6 | ±0.53 | 0.10802 | ±0.00800 | <0.1 | 0.732 |
| uPVC | 100 | 0.797 | 78.4 | ±1.31 | 1.24352 | ±0.0476 | <0.1 | 0.784 |
| mPVC | 50 | 0.798 | 43.2 | ±1.47 | 0.37375 | ±0.0288 | <0.1 | 0.865 |
| HDPE | 50 | 0.656 | 33.0 | ±3.08 | 0.09377 | ±0.0451 | <0.1 | 0.660 |
| HDPE | 78 | 0.703 | 69.0 | ±4.68 | 0.48148 | ±0.123 | <0.1 | 0.884 |
| Steel | 50 | 0.490 | 22.8 | ±0.08 | −0.00267 | ±0.00095 | 6.4 | 0.457 |
| Steel | 105 | 0.523 | 44.2 | ±0.17 | 0.01615 | ±0.00265 | 1.6 | 0.421 |
| Circumferential slits (width = 1 mm) | ||||||||
| uPVC | 50 | 0.455 | 34.8 | ±0.26 | −0.02170 | ±0.00380 | <0.1 | 0.696 |
| uPVC | 100 | 0.327 | 50.2 | ±1.28 | −0.11423 | ±0.0189 | <0.1 | 0.502 |
| mPVC | 50 | 0.433 | 37.2 | ±0.27 | −0.03403 | ±0.00380 | <0.1 | 0.743 |
| HDPE | 54 | 0.185 | 20.6 | ±1.74 | −0.06212 | ±0.0213 | <0.1 | 0.382 |
| HDPE | 80 | −0.262 | 37.4 | ±5.14 | −0.20706 | ±0.0646 | <0.1 | 0.467 |
| Steel | 53 | 0.499 | 22.6 | ±0.37 | −0.00008 | ±0.00485 | 33.3 | 0.427 |
SCI, simultaneous confidence interval.
A linear model was fitted to the data points for each experiment to determine the model parameters and the 95% simultaneous confidence intervals for each parameter.
and are the pressure head and effective area respectively for observation i (i = 1,2,…, n), and , which captures noise, follows a normal distribution with a mean of zero and variance , and is independently drawn for each observation.The intercept and slope of the linear relationship, contained in , are estimated as
so as to maximise the likelihood function.
is the upper 2.5th percentile of a t-distribution with n − 2 degrees of freedom.
may be determined using the equation: is the upper 5th percentile of an F-distribution with 2 and n − 2 degrees of freedom; X is the n-by-2 design matrix (first column contains 1 s, second column contains the values); and is the estimated value of , and equals the sum of squared residuals (differences between observed values and fitted expected values) divided by n−2.The simultaneous confidence interval widths for the experiments were found to be on average 27% (varying between 24.6 and 29.3%) larger than the single parameter confidence intervals.
The effective initial area and its 95% simultaneous confidence interval are given in columns 4 and 5 of Table 2 respectively. Similarly, the effective head-area slope and its 95% simultaneous confidence interval are given in columns 6 and 7.
The p-value in column 8 of Table 2 is the probability of observing the effective head-area slope or a more extreme estimate, where the true value of the head-area slope is equal to zero (null hypothesis: ). A small p-value provides evidence against this null.
The final column of Table 2 gives an estimate of the discharge coefficient of the leak openings. Since the initial area of the leak openings can be estimated from the pipe samples, it is possible to estimate the discharge coefficient by dividing the effective initial area by the actual initial area.
DISCUSSION
The results in Table 2 and Figure 1 show that the areas of 12 mm round holes varied very little with pressure for all pipe materials tested. Even at the largest absolute 95% interval, the effective head-area slopes were all below 0.01 mm2/m. All the measured leakage exponent values were equal to the theoretical value of 0.50 at two significant digits.
This means that in practice 12 mm round holes (and likely a wider range of diameters) can be assumed fixed and thus to adhere to the orifice equation. For experimental studies, 12 mm round holes can be used as a benchmark for the accuracy and consistency of the results.
Steel pipes also displayed small head-area slopes with leakage exponents at or close to 0.50 for the range of leaks tested. Longer longitudinal and spiral slits displayed larger head-area slopes than shorter slit lengths (the same trend was observed for other pipe materials tested). These results support the practice of assuming N1 values of 0.5 for networks with steel or cast iron pipes. However, it should be noted that pipes with extensive corrosion damage may have larger N1 values as shown in a limited study using very low pressures by Greyvenstein & van Zyl (2007).
All circumferential slits tested were found to display negative head-area slopes, meaning that the areas of these slits decreased with increasing pressure resulting in leakage exponents below 0.5. Longer slits displayed more negative head-area slopes than shorter slits. The 80 mm long circumferential slit in HDPE had the most negative head-area slope and a negative N1 value, which means that not only the slit area, but also the flow rate through the slit decreased with increasing pressure.
Comparison of the effective head area slope predicted using Equation (4) and a discharge coefficient of 0.6 with the experimentally determined values.
It has been shown that the leakage exponents of most leaks are not fixed, but vary with fluid pressure (Van Zyl & Cassa 2014). While the experimental study only estimates a single overall N1 for each experiment, it is possible to calculate the actual range of N1 values for each experiment from the effective initial area, head area slope and the range of experimental test pressures. This is done by first calculating the leakage number for each data point using Equation (8) and then the leakage exponent at this pressure using Equation (9).
The range of N1 values for each experiment against the single N1 determined experimentally.
Comparison of the N1 power and FAVAD equation flow predictions for 100 mm slits in uPVC, HDPE and steel pipes.
The FAVAD equation is based on a fundamental fluid mechanics theory, incorporating the linear head-area slope demonstrated in Figure 4. Thus it can be assumed to describe the true behaviour, while the empirical N1 equation results in significant modelling errors outside the measured pressure range. The capacity of the pumping system used in the experiment did not allow data to be collected at higher pressures, and thus further work is required to verify this assumption.
Both equations performed well on the steel pipe due to the practically zero head-area slope, and thus pure orifice flow that is equally well described with the FAVAD and N1 power equations.
CONCLUSIONS
The aims of this paper were to provide a review of the developments in understanding the behaviour of leaks over the last decade and use these to interpret the results of a recent experimental study on the pressure-leakage behaviour of various leak types in different pipe materials.
Recent research has established that the areas of all leaks types vary as linear functions of pressure under both elastic and viscoelastic conditions. This behaviour can be characterised by determining the initial area and head-area slope of a leak opening.
It has been established and confirmed by this study that round holes have very small head-area slopes that can normally be assumed as zero, circumferential slits have small head-area slopes that are often negative and longitudinal slits have the largest head-area slopes. An equation for predicting the head-area slope of longitudinal slits (Cassa & Van Zyl 2013) was shown to perform reasonably well in this study.
Finally, the study showed that the N1 power equation produces good results when used within its calibrated range, but can result in significant errors when used outside this range.
ACKNOWLEDGEMENTS
The authors would like to acknowledge the assistance of Reshma Kassanjee of the Statistical Consulting Service at the Department of Statistical Sciences, University of Cape Town for the statistical analysis, as well as Allan Lambert for constructive comments on the manuscript.
