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Water Leak from Hole in Copper Pipe (photograph)
Leak Rate Equations
Leak Rate
Leaks are common occurrences in pipes, tubes, and tanks. The leak rate is based on the pressure difference
between the inside and outside of the pipe, the crack or hole area, and specific gravity of the liquid.
The pipe leak rate calculator computes the leak rate of a liquid out of a circular or noncircular crack or hole in a pipe or tank.
The leak rate calculation has builtin specific gravity values for water, seawater, diesel fuel, SAE 30 oil, and gasoline.
Specific gravity can be entered for other liquids. If the leak rate is known, the calculator can compute
pressure or crack area (and diameter).
Leak Rate Equations and Derivation
Liquid inside a pipe, tank, or other confined space flows out through a hole or crack.
The liquid in the pipe is at an elevation Z_{1}, has pressure P_{1}, and has velocity V_{1}.
As the liquid exits the hole, it increases to velocity V_{2} through the hole, pressure P_{2} at the hole exit, and
an elevation Z_{2} of the hole exit.
Diagram of liquid flowing out of pipe through crack:
The Bernoulli equation (Munson et al., 1998, p. 114; bernoulli.php) describes flow of an inviscid fluid flowing from
location 1 to location 2.
Specific gravity S = ρ/ρ_{w,std}, the ratio of density of the flowing liquid to density of water
at standard conditions (4^{o}C), which is 1000 kg/m^{3}.
The Bernoulli equation has many applications. Leak rate is one application where it is widely used. Since a pipe
is usually much larger diameter than the hole, V_{2}>>V_{1}. Since the elevation change Z_{2}Z_{1} is typically very small
compared to other terms in the Bernoulli equations for a pipe leak application, Z_{2}Z_{1}=0. Thus:
Solve for V_{2}:
Then with Q=V_{2}A_{2}, A=πd^{2} / 4, and ΔP=P_{1}P_{2}:
Due to the viscosity of liquids and the tortuous path of the liquid through the hole,
the measured leak rate from a hole is less that that computed by the above equation.
To account for these effects, a discharge coefficient C is commonly used in the equation (e.g. Munson et al., 1998, p. 531; McAllister, 2005, p. 356):
C=0.61 is a typical value determined from experiments (McAllister, 2005, p. 356).
Velocity out of the hole is computed approximately as:
A more accurate computation of velocity can be found using the methods at
our Discharge from a Tank calculation which splits C into a
velocity coefficient and a contraction coefficient. Our Bernoulli equation calculator applies the Bernoulli equation
to many situations including pipe leak, but does not include a discharge coefficient.
Variables
The units shown for the variables are SI (International System of Units); however, the
equations above are valid for any consistent set of units. Our calculation allows a
variety of units; all unit conversions are accomplished internally.
A = Hole or crack area (m^{2}).
C = Discharge coefficient. Need 0 < C ≤ 1. Suggested value is C=0.61 (McAllister, 2005, p. 356). For highly viscous liquids, C is lower.
d = Hole diameter (m).
g = Acceleration due to gravity, 9.8066 m/s^{2}.
Q = Leak rate (m^{3}/s).
S = Liquid specific gravity. In calculation, S for water is from Munson et al. (1998, p. B.2); seawater, SAE 30 oil, and gasoline are from Munson Table 1.6; and diesel is from Chevron (2007, p. 48).
V = Leak velocity (m/s).
ΔP = Pressure drop from inside of pipe to outside of pipe (or tank) (N/m^{2}).
ρ_{w,std} = Mass density of water at standard conditions (4^{o}C), 1000 kg/m^{3}.
π = 3.1415926...
Error Messages given by calculation
"Need A > 0",
"Need 0 < C ≤ 1",
"Need d > 0",
"Need S > 0",
"Need Q > 0",
"Need ΔP > 0".
References
Chevron Corporation. 2007. Diesel Fuels Technical Review. Retrieved from
https://www.chevron.com//media/chevron/operations/documents/dieselfueltechreview.pdf
McAllister, E. W. (editor). 2005. Pipeline Rules of Thumb Handbook. Elsevier  Gulf Professional Publishing.
Munson, B.R., Young, D.F, and Okiishi, T. H. 1998. Fundamentals of Fluid Mechanics. John Wiley and Sons, Inc. 3ed.
© 2017 LMNO Engineering, Research, and
Software, Ltd. All rights reserved.
Please contact us for consulting or other questions about pipe leak rate.
LMNO Engineering, Research, and Software, Ltd.
7860 Angel Ridge Rd. Athens, Ohio 45701 USA Phone: (740) 7072614
LMNO@LMNOeng.com https://www.LMNOeng.com

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