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11 December 2006
1
THE EFFECT OF PRESSURE ON LEAKAGE IN WATER
DISTRIBUTION SYSTEMS
JE van Zyl 1 (Ph.D.) and CRI Clayton 2 (Ph.D.)
1 Associate Professor,
Department of Civil Engineering Science,
University of Johannesburg,
PO Box 524, 2006 Auckland Park, South Africa,
Ph +2711 489 2345, Fax +2711 489 2148,
Email jevz@ing.rau.ac.za
2 Professor,
School of Civil Engineering and the Environment,
University of Southampton,
Highfield,
Southampton SO17 1BJ
Email cric@soton.ac.uk
Number of words: 4000
Number of tables and figures: 2
Key words: Hydraulics & Hydrodynamics, Pipes & pipelines, Water supply
Submitted to: Proceedings of the Institution of Civil Engineers, Water Management
11 December 2006
2
Abstract
The results of pressure management field studies have shown that the leakage exponent is
often considerably higher than the theoretical orifice value of 0.5. The purpose of this paper is
to identify and analyse factors that may be responsible for the higher leakage exponents. Four
factors are considered: leak hydraulics, pipe material behaviour, soil hydraulics and water
demand. It is concluded that a significant proportion of background leakage can consist of
transitional flow, and thus have a leakage coefficient value above 0.5 (although not above 1).
An important factor is pipe material behaviour: laboratory test results are presented to show
that pipe material behaviour can explain the range of leakage exponents observed in the field.
The complexity of the interaction between a leaking pipe and its surrounding soil is discussed
and it is concluded that the relationship between pressure and leakage is unlikely to be linear.
Finally, it is noted that if water demands are present in minimum night flows, the resulting
leakage exponent is probably underestimating the true value.
NOTATION
A orifice or hole area
c leakage coefficient
c' stress factor
C constant
Cd discharge coefficient
d hole diameter
d0 original hole diameter
Δd change in hole diameter
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3
D pipe diameter
E elasticity modulus
F shape factor for soil flow region
g acceleration due to gravity
h pressure head
k soil coefficient of permeability
n aspect ratio of a rectangle
P wetted perimeter
q flow rate
Qdem demand flow rate
R hydraulic radius
Re Reynolds number
t pipe wall thickness
v velocity
α leakage exponent
β water demand elasticity
ε material strain
ρ fluid density
σ material stress
ψ kinematic viscosity
Note to the Editor: The accepted symbol for kinematic viscosity is not ψ as used in this
manuscript, but the Greek letter nu (ν). The change was made to avoid confusion with the
symbol v, which looks very similar to the Greek nu in the font used. We will appreciate it if
the correct symbol ν (Greek nu) can be used in the printed paper, provided that it can be
distinguished clearly from the letter v.
11 December 2006
4
1. INTRODUCTION
Water distribution systems world-wide are aging and deteriorating, while the demands on
these systems, and thus on natural water resources, are ever increasing. Losses from water
distribution systems are reaching alarming levels in many towns and cities throughout the
world. Water losses are made up of various components including physical losses (leaks),
illegitimate use, unmetered use and under-registration of water meters. Leakage makes up a
large part, sometimes more than 70 % of the total water losses1.
One of the major factors influencing leakage is the pressure in the distribution system. In the
past the conventional view was that leakage from water distribution systems is relatively
insensitive to pressure, as described by the orifice equation:
ghACqd2=
... (1)
Where q the flow rate, Cd a discharge coefficient, A the orifice area, g acceleration due to
gravity and h the pressure head differential over the orifice. To apply this equation to leaks in
pipes it can be written in more general form as:
q ch
=
... (2)
Where c is defined as the leakage coefficient and α as the leakage exponent (α is sometimes
referred to as N1). A number of field studies have shown that α can be considerably larger
than 0.5, and typically varies between 0.5 and 2.79 with a median of 1.15 2. This means that
11 December 2006
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leakage in water distribution systems is much more sensitive to pressure than conventionally
believed. The range of exponents observed reflects substantial differences in the impact of
pressure on rate of leakage. For example, halving the pressure in a pipe will result in
reductions in flow rate of 29 %, 50 % and 82 % respectively for exponents of 0.5, 1.0 and 2.5.
The reasons for the high leakage exponents are not well understood, but an important cause is
believed to be the expansion of the hole opening with increasing pressure 2.
The large influence of the leakage exponent when estimating the potential impact of pressure
management on leakage rate means that it is essential to develop an understanding of the
mechanisms responsible for the observed behaviour. The purpose of this paper is to identify
possible causative factors and, where possible, quantify the effect of these factors on the
leakage exponent. The possible causative factors are discussed under four headings: leak
hydraulics, pipe material behaviour, soil hydraulics and water demand.
2. LEAK HYDRAULICS
The Orifice equation (equation 1) is derived for an orifice in the side of a tank and describes
the conversion of all the potential energy, in the form of pressure, to kinetic energy. The
discharge coefficient is added to incorporate energy losses and the reduction of jet diameter
downstream of the orifice. The pressure in the jet downstream of the orifice is assumed to
equal that of the surrounding fluid.
The hydraulic behaviour of orifices has been researched extensively and can be predicted with
some degree of certainty. The exponent of 0.5 is generally only true for large Reynolds
11 December 2006
6
numbers (Re). For smaller Reynolds numbers, equation 1 is typically modified by writing the
coefficient c as a function of the Reynolds number. This variable coefficient can also be
expressed as a fixed coefficient with an exponent that is not 0.5. For example, substituting the
expression for laminar flow through an orifice (from 3) into equation 1 results in an equation
with constant coefficient and an exponent of 1. For transitional flow, the equivalent exponent
will vary between 0.5 at the transitional-turbulent flow boundary to 1.0 at the laminar-
transitional flow boundary.
As noted above, the flow regime is determined by the Reynolds number (Re). Flow through
orifices is typically laminar at Re below 10 and turbulent at Re above 4000 to 5000 3. The
Reynolds number for a general leak opening or orifice can be written as:
P
q
vR
4
4
Re
=
=
... (3)
Where v velocity and ψ kinematic viscosity of the fluid, and R the hydraulic radius of the
orifice (defined as flow area A divided by the wetted perimeter P).
Since the kinematic viscosity of a fluid is a function of temperature, it follows from the
equation that the leakage flow rate for a fixed Reynolds number (e.g. for maximum laminar or
transitional flow) and fluid is only affected by two variables: the temperature of the fluid and
the wetted perimeter of the orifice. The viscosity of water approximately halves when its
temperature increases from 0 to 30 ˚C, meaning that the maximum laminar or turbulent flow
11 December 2006
7
will approximately double. Leak openings with large wetted perimeters (such as cracks) will
be able to sustain much larger laminar or transitional flow rates than circular openings with
the same areas.
It is possible to find an expression for the maximum laminar and transitional flow rates
through different types of leak openings for the typical pressure range in a water distribution
system. First, the flow rate is written as the product of the velocity and area of an opening.
For a circular opening, this is given by:
vDQ 2
25.0
=
…(4)
Where D the diameter of the leak opening. Writing equation 3 in terms of the hole diameter
and replacing it and equation 1 into equation 4 results in the expression:
ghC
q
d24
Re22
=
... (5)
For a rectangular leak opening with an aspect ratio of n, the expression is given by:
( )
ghnC
n
q
d24
Re122
2
+
=
... (6)
If a constant discharge coefficient (say Cd = 0.6) and kinematic viscosity (say ψ = 1.14 x 10-6
for water at 15 ˚C) are assumed, the equations can be used to estimate the maximum laminar
and transitional flow rates that are possible in water distribution networks. Cracks can be
11 December 2006
8
viewed as rectangular leak openings with high aspect ratios. The maximum laminar and
transitional flow rates for different types of leak openings are shown in Figure 1 for the 10 to
100 m pressure range, which covers the pressures found in most water distribution systems.
The figure shows that cracks can have much higher laminar or transitional flow rates than
round or square holes. This is due to the role of their much larger wetted perimeters. Theory
predicts that the maximum possible flow rates that are fully laminar are typically very small
(e.g. less than 3 l/day even for a crack with an aspect ratio of 10 000) and it is thus unlikely
that substantial losses from water distribution systems will occur in the fully laminar zone.
A distinction is often made between bursts and background leakage. Bursts are large
individual leaks that come to the surface or are found through active leakage control
initiatives. Background leakage comprises numerous small leaks that are very difficult or
impossible to detect without excavating the pipe. In a well-run system, much of a network’s
water loss that we seek to reduce through pressure control may thus result from background
leakage. This view is supported by water leakage figures for England and Wales (about 25
million connected properties) that has been estimated by OFWAT 4 in 2004 to be on average
of the order of 10 m3/km of main/day, or 360 l/h/km of main. Comparing this figure with the
maximum transitional flow rates above indicate that it is possible that much of the
background leakage can occur in this range, especially in systems that are likely to have pipes
that develop crack failures. Transitional flow can thus be an important cause of a leakage
exponent above 0.5 (although not above 1.0) when background leakage is a large contributor
to leakage from a system.
11 December 2006
9
3. PIPE MATERIAL BEHAVIOUR
Pipe material plays an important role in the leakage behaviour of pipes. Water pressure in a
pipe is taken up by stresses in the pipe wall, and thus may be a factor in failure and leakage
behaviour. The following effects can be linked to an increase in the internal pressure of a
pipe:
• Small cracks or fractures that do not leak at low pressures open up to create new leaks.
• The area of existing leak openings in a pipe increase due to increased stresses in the
pipe wall.
• The frequency of pipe bursts increases 2, 5 with a corresponding increase in
maintenance costs.
Greyvenstein and Van Zyl 6 used an experimental setup to measure the leakage exponents of
failed pipes taken from the field and pipes with artificially induced leaks. The study included
round holes, and longitudinal and circumferential cracks in uPVC, steel and asbestos cement
pipes. All flows were turbulent and leaks were exposed to the atmosphere. The resulting
leakage exponents varied between 0.42 and 2.4 as detailed in Table 1. The main findings of
the study were:
• The results confirm that the leakage exponents found in field studies are not
unrealistic.
• The highest leakage exponents occurred in corroded steel pipes, probably due to
corrosion reducing the support material around the hole. This is contrary to the
11 December 2006
10
perception that plastic pipes will have higher leakage exponents due to their lower
modulus of elasticity.
• Round holes had leakage exponents close to the theoretical value of 0.5 and no
significant difference was observed between steel and uPVC pipes.
• Besides corrosion holes, the largest exponents were found in longitudinal cracks. This
is due to the fact that circumferential stresses in pipes are normally significantly
higher than longitudinal stresses.
• The leakage exponents for circumferential cracks in uPVC pipes were sometimes less
than 0.5, suggesting that the leak opening might be contracting with increasing
pressure. This is explained by the fact that the experimental setup did not allow
substantial longitudinal stresses to develop in the pipe. It is thought that the
circumferential stresses caused the cracks to elongate, and at the same time reduce in
area. These results have subsequently been verified through finite element analysis 7.
Theoretical expressions for the longitudinal and circumferential stresses in a pipe under
pressure are given in many textbooks (for example, see 8). These expressions show that the
stresses in the circumferential direction are double those in the longitudinal direction. When a
discontinuity such as a hole is present, the pipe wall stresses are increased in the vicinity of
the hole. The circumferential stress in the vicinity of the hole is now written as:
t
gDhc 2
'
=
... (7)
Where σ the pipe wall stress, c' a stress factor, D the pipe diameter, h the pressure head and t
the pipe wall thickness. The stress factor incorporates both the variation in stress around the
11 December 2006
11
circumference of the hole and the stress concentration factor. Assuming linear elastic
behaviour, the wall stress can also be written in terms of the strain ε and elasticity modulus E:
0
d
EE
d
==
... (8)
Where d0 the original hole diameter and Δd the change in diameter due to the pressure in the
pipe. Using equations 7 and 8, the hole diameter
0
d d d= +
can now be expressed as:
( )
Chd
tE
gDhc
dd
+=
+=
1
2
'
1
0
0
... (9)
Where C is a constant. Substituting the equation for the area of a hole (based on equation 9)
into equation 1 results in the following expression for the leakage flow rate from a circular
hole in a pipe:
1 3 5
22
2 2 2
0
0.125 2
d
q g C d h Ch C h
= + +
... (10)
The relationship shows that the processes involved in the expanding leak opening are more
complex than the simple power relationship normally used to describe leakage. The equation
contains the sum of three terms with leakage exponents of 0.5, 1.5 and 2.5 respectively, which
seem to tie in well with field and experimental observations. However, when calculating the
11 December 2006
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leakage from typical pipes using equation 10 it is found that the terms with exponents 1.5 and
2.5 contribute little to the leak under normal pressure conditions.
Due to the material properties, pipes of different materials will fail in certain characteristic
ways. For instance, longitudinal cracks are common in asbestos cement pipes, while steel and
cast iron pipes often leak through corrosion holes. Small-diameter cast iron pipes typically fail
in bending leading to circumferential cracks which, because of the relatively high coefficient
of thermal expansion of the pipe material, may open and close as the temperature of the water
in the system changes. Understanding the failure behaviour of pipes and the associated
leakage exponents can assist with modelling the response of a given distribution system to a
change in pressure and in better managing leakage reduction programmes.
4. SOIL HYDRAULICS
A simplistic application of geotechnical seepage theory would suggest, in contrast to equation
(1), that if head losses through the pipe orifice are neglected, the flow rate should be linearly
proportional to the head of the water in the pipe, h, since following Darcy’s Law, the flow rate
(q) in the soil for a given head on the orifice water/soil boundary will be 13
q = F . k . h ... (11)
Where F is the form factor for the soil flow region, and k is the coefficient of permeability of
the soil. However this equation is underpinned by a number of assumptions, and these are not
generally valid for seepage around a water pipe.
11 December 2006
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Firstly, in soil seepage analysis it is generally safe to assume that the velocity component of
total head is very small, and can be ignored. The velocity of flow through soil under a
hydraulic gradient of unity varies from 10-2 m/s for a clean coarse sand to 10-8 m/s and
smaller for clays. Yet in contrast the hydraulics orifice equation (equation 1) predicts very
high velocities at the soil/water interface. There is clearly an incompatibility here, and an
equation for the combined system cannot be properly defined by the straightforward coupling
the orifice equation with the soil seepage equation, as has been previously done 9. Soil
outside the pipe will modify downstream jet behaviour, whilst perhaps also obscuring part of
the orifice itself. The simple Darcy soil seepage equation assumes that there is a fixed
upstream boundary geometry with constant head applied to it but, because of the high orifice
outlet velocity, both the boundary geometry and the head applied to it are likely to be
modified by scour of the soil boundary and fluidisation of the soil. A number of studies have
shown the complexity of these processes 10, 11, 12
Secondly, the downstream boundary conditions in the ground surrounding the pipe are not
generally constant regardless of flow rate. In many geotechnical seepage problems both
upstream and downstream boundaries can reasonably be assumed to have fixed geometries
and head conditions, and the position of any phreatic surface can be assumed fixed. Since
water pipes are generally laid above the ground water level the seepage flow net (and thus F
in equation 10 above) varies as a function of the rate of outflow from the pipe and the
coefficient of permeability of the soil. For any given soil permeability, increasing flow leads
to progressive build-up of pore pressure in the soil around the pipe, and eventual “mounding”
of water above it. For low flow rates relative to the permeability of the soil, seepage will not
11 December 2006
14
reach the ground surface, and leakage will go unnoticed. At higher rates of leakage water will
emerge at ground surface and a burst will be detected.
Thirdly there are limits to validity of Darcy’s law. A linear relationship between head and
flow in soil is, as observed by Osborne Reynolds in 1883, only valid for laminar flow. The
critical value of Re (expressed in soil mechanics as
/vDR=
, where v is the discharge
velocity (flow per unit cross section of soil), and D is the average soil particle diameter) at
which flow in soil changes from laminar to turbulent has been found to range between about 1
and 10 (for example, see 13) . Discharge velocity depends upon both hydraulic gradient and
permeability (which is itself a function of particle size). Under the low hydraulic gradients
(Δh/Δl << 1) typical of many soil seepage situations laminar flow can be expected in sands
and finer materials, but not in gravels. However the hydraulic gradient around a leaking pipe
will be much larger - water distribution pipes are generally buried at a depth of less than 1m,
and have supply heads of the order of 30 m. Non-laminar flow can therefore be expected in
most coarse granular soils and loose backfills.
Finally, the stress conditions in the ground contribute to the way in which flow takes place.
Calculations of Darcy flow generally assume permeability to be constant, with flow
distributed across the entire region of permeable soil. Considerations of force equilibrium
make it clear, however, that for a particulate material such as soil the maximum water
pressure in the pores between the particles, on any given plane, cannot exceed the (total
i
)
stress on that plane. Once the water pressure at any point in the ground rises above the minor
total principal stress (which may be in the horizontal or vertical direction, but is unlikely to
i
The total stress on a plane in a soil mass is the stress on that plane that arises as a result of external loading and
of the self weight of the soil. This is distinct from the effective stress, which governs the strength,
compressibility and to some extent permeability of soil, and is the numerical difference between total stress and
pore pressure on any given plane.
11 December 2006
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exceed 20-30kPa for typical pipe burial depths) then hydraulic fracture takes place. The soil
cracks along planes of weakness, flow occurs preferentially along these cracks, flow rates rise
through orders of magnitude, and conventional seepage analysis is no longer applicable (for
an example see 14). Because of their size, and for the reasons discussed above, flow along
these cracks is unlikely to be laminar. As heads increase, the move from Darcy flow to
hydraulic fracturing can be expected to produce flow increases that contribute to leakage
exponents greater than unity.
Even if the water pressure is not sufficiently high to cause hydraulic fracture, if upward flow
takes place in unbonded granular soil and its velocity become sufficiently great then
fluidisation may occur. “Piping”, as this is known, results when the upward force on the soil
particles resulting from seepage exceeds its buoyant self-weight, and occurs at a hydraulic
gradient approximately equal to unity. Since the particles in the fluidised zone move as an
integral part of the fluid, the overall permeability of the flow region is greatly reduced.
On average, leakage figures for a well-maintained system (such as the England and Wales
estimate of about 0.1 l/s/km of main 4) probably represent a few larger-volume infrequent
bursts combined with a much greater number of continuous but undetected losses from
smaller defects in the network. For example, vertical downward flow (gravitational flow, i.e.
without any development of excess pore water pressure in the soil) from a single 0.1 litre/s
leak would occupy a plan area of only about 10cm x 10cm in gravel, and 1m x 1m in sand,
suggesting that in coarse granular soil leaks will be absorbed without trace by the ground
around the pipe.
11 December 2006
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In summary, it can be concluded that the interaction between a leaking pipe and its
surrounding soil is complex, and requires further investigation. The relationship between head
loss and flow is unlikely to be linear, as a result of interaction of soil particles with the orifice,
turbulent flow in the soil, the changing geometry of the unconfined flow regime, hydraulic
fracturing and piping. Theoretical considerations suggest that small continuous leaks from
pipes will drain away without trace into underlying granular soil. This cannot be expected to
occur in lower permeability clays and silts, where hydraulic fracture is more likely, with leaks
rapidly becoming visible as wet patches and bursts at the ground surface.
5. WATER DEMAND
While water demand is not classified as leakage, it is often impossible to separate legitimate
water consumption from leakage measurements in the field. It is thus important to understand
the behaviour of water demand as a function of pressure. The effect of pressure on demand
Qdem can be expressed as 15:
dem
Q Ch
=
... (12)
With C a constant coefficient and β the elasticity of demand with respect to pressure. There is
a clear resemblance between equations for leakage (equation 2) and demand elasticity
(equation 12). The elasticity includes the effects of human behaviour, such as reacting to an
increased pressure by opening taps less to obtain the same flow rate. In a study of water
consumption patterns at a student village on the campus of the University of Johannesburg,
Bartlett 16 found the indoor demand elasticity for pressure to be approximately 0.2. Outdoor
water consumption such as garden irrigation is typically time-based rather than volume-based,
meaning that a higher exponent can be expected for outdoor use. The typical exponent for
11 December 2006
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outdoor irrigation equipment is around 0.5 5, 16 although soaker hoses were found to have
values as high as 0.75 17.
In large systems it becomes likely that even minimum measured night flows will include
some legitimate consumption. Since the combined ‘leakage exponent’ for outdoor and indoor
consumption is likely to be less than 0.5, it may be concluded that measured leakage
exponents in systems with demand are likely to underestimate the true leakage exponent of
the system, provided that the level of demand in the measured night flows do not differ
significantly.
6. CONCLUSIONS
The leakage exponent determined from field studies differ significantly from the theoretical
orifice exponent of 0.5. The purpose of this paper has been to identify and analyse factors that
may be responsible for the range of leakage exponents observed in the field. Leak hydraulics,
pipe material behaviour, soil hydraulics and water demand were considered as possible
causative factors. It is concluded that a significant proportion of background leakage can
consist of transitional flow, and thus have a leakage coefficient value above 0.5 (although not
above 1). Both experimental and theoretical investigations indicate that pipe material
behaviour can provide one explanation for the observed range of leakage exponents.
The interaction between a leaking pipe and its surrounding soil is complex, and flow rates are
unlikely to be a linear function of pressure, as a result of interaction of soil particles with the
11 December 2006
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jet and the orifice, turbulent flow in the soil, the changing geometry of the unconfined flow
regime, hydraulic fracturing and piping. Finally, if water demands are present in minimum
night flows, the resulting leakage exponent is probably an underestimate of the true value.
References
1. WHO - WORLD HEALTH ORGANISATION. Leakage management and control - a best
practice manual, 2001,WHO, Geneva.
2. FARLEY M. AND TROW S. Losses in water distribution networks, 2003, IWA
Publishing, London.
3. IDELCHIK I. E. Handbook of hydraulic resistance (3rd edition), 1994, Begell House,
New York.
4. OFWAT - OFFICE OF WATER SERVICES. Security of supply, leakage and the efficient
use of water 2003-2004 Report, 2004, OFWAT, Birmingham.
5. THORNTON J. and LAMBERT A. Progress in practical prediction of pressure: leakage,
pressure: burst frequency and pressure: consumption relationships, Proceedings of
Leakage 2005, 2005, IWA, London.
6. Greyvenstein, 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, Accepted for publication.
7. CASSA A.M. A numerical investigation into the behaviour of leak openings in pipes
under pressure, 2006, Master’s degree thesis, Department of Civil Engineering Science,
University of Johannesburg.
8. POPOV E.P. Mechanics of materials (second edition), 1978, Prentice Hall, London.
11 December 2006
19
9. WALSKI, T., BEZIS, W., POSLUSZNY, E.T., WEIR, M. AND WHITMAN, B.
Understanding the hydraulics of water distribution system leaks. Proc. 6th ASCE/EWRI
Annual Symp. on Water Distribution System Analysis, 2004, 27 June – 1 July, Salt Lake
City.
10. NIVEN, R.K, KHALILI, N. In situ fluidisation by a single internal vertical jet, Journal of
Hydraulic Research, 1998, 36, No2, 199-229.
11. RAJARATNAM, N. (1982), Erosion by submerged circular jets, Journal of Hydraulic
Div. Am. Soc. Civ. Eng., 1982, 108, No. 2, 262-267.
12. HANSON, G.J. Development of a jet index to characterize erosion resistance of soils in
earthen spillways, Trans ASAE, 1991, 34, No. 5, 2015-2020.
13. HARR M.E. Groundwater and seepage, 1962, McGraw Hill, New York.
14. BJERRUM L.J., NASH J.K.T.L., KENNARD R.M. and GIBSON R.E. Hydraulic
fracturing in field permeability testing. Geotechnique, 1972, 22, No.2, 319-332.
15. VAN ZYL J.E., HAARHOFF J. and HUSSELMANN M.L. Potential application of end-
use demand modelling in South Africa, Journal of the South African Institute of Civil
Engineering, 2003, 45 No. 2, 9 - 19.
16. BARTLETT L. Pressure dependant demands in Student Town Phase 3, 2004, Final Year
Civil Engineering Project Report, Department of Civil and Urban Engineering, Rand
Afrikaans University (now University of Johannesburg), South Africa.
17. CULLEN R. Pressure vs consumption relationships in domestic irrigation systems, 2004,
Research thesis, Department of Civil Engineering, University of Queensland, Australia.
11 December 2006
20
Table 1. Leakage exponents found in an experimental study by Greyvenstein 6
Failure type
Leakage exponent for pipe material
uPVC
Asbestos cement
Mild steel
Round hole
0.52
-
0.52
Longitudinal crack
1.38 – 1.85
0.79 – 1.04
-
Circumferential crack
0.41 – 0.53
-
-
Corrosion cluster
-
-
0.67 – 2.30
11 December 2006
21
1.E-05
1.E-04
1.E-03
1.E-02
1.E-01
1.E+00
1.E+01
1.E+02
1.E+03
1.E+04
1.E+05
10 20 30 40 50 60 70 80 90 100
Pressure Head (m)
Flow rate at the limit (l/h)
Round Square
Rectangular (aspect ratio 100) Rectangular (aspect ratio 10 000)
Transitional flow limits
Laminar flow limits
Fig 1. Maximum laminar and transitional flow rates for different types of leak openings
