Chemical Engineering July 2021 - 35

Feature Report
Part 2
Can You Trust the
Joukowsky Equation for
Waterhammer?
The Joukowsky equation is often relied upon to determine the maximum possible fluid pressure
inside a pipe, but there are certain scenarios where this equation does not return the expected
conservative result for overpressurization
W
e engineers
love our formulas.
Especially
when
we know they can quickly
give us a conservative,
worst-case answer.
As a young engineer in
250
- Simulation
Martin
200
150
100
the aerospace and power
industries, I participated
in multiple meetings and
read several reports that
used the Joukowsky
equation to determine a
maximum possible fluid
pressure inside a pipe.
Since everyone " knew "
that the Joukowsky
equation predicted the maximum possible
pressure, this was a quick and reasonable
thing to do. Unbeknownst to my managers,
colleagues and myself at the time, this was a
potentially dangerous thing to do. Because
while the Joukowsky equation often returns
a conservative result, it cannot be relied upon
to do so in every situation.
This
50
Pressure rise Eq. 1
= 104 m (340 ft)
0.1
0.2
0.3 0.4 0.5 0.6
Time, s
0.7 0.8 0.9
article summarizes
the
important
points of a recent journal article [1] that discusses
three different situations for which the
Joukowsky equation does not give a worstcase,
conservative answer.
The Joukowsky equation
The Joukowsky equation [2]
relates the
instant change in piezometric (hydraulic)
head, H, to an instant change in velocity, V,
often conceptualized as an instantaneous
valve closure. The Joukowsky equation is
sometimes referred to by other names, such
as the " Basic Waterhammer Equation, " the
" Instantaneous Waterhammer Equation "
or the " Maximum Theoretical Waterhammer
Equation. " The relationship is shown in
800
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600
500
400
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200
100
1
FIGURE 1. The graph shown here, from Example 2, illustrates the experimental and numerical
predictions of pressures during transient cavitation, compared to the maximum
predicted pressure from the Joukowsky equation (from Ref. 1)
Equation (1):
∆HJ = -a∆V⁄g
Trey Walters
Applied Flow
Technology
IN BRIEF
THE JOUKOWSKY
EQUATION
EQUATION
ASSUMPTIONS
CASES WHERE THE
JOUKOWSKY EQUATION
IS NOT CONSERVATIVE
TRANSIENT CAVITATION
AND LIQUID COLUMN
SEPARATION
CONCLUDING REMARKS
(1)
where a is the wavespeed (also known as
the celerity) and g is the acceleration due
to body forces (32.2 ft/s2 or 9.8 m/s2 for
stationary systems at the earth's surface)
due to gravity. The wavespeed is related to
the speed of sound in the liquid, but also
includes pipe structural interaction. Note
that the negative sign in Equation (1) means
that a reduction in velocity leads to an increase
in piezometric head. Equation (1) is
more typically found in civil engineering applications,
which frequently use piezometric
head and hydraulic gradeline concepts.
The relationship between the change in piezometric
head and pressure, P, is given by
Equation (2):
∆P = ρg∆H
(2)
where ρ is the liquid density.
Combining Equations (1) and (2) results
CHEMICAL ENGINEERING WWW.CHEMENGONLINE.COM
JULY 2021
35
Pressure, ft of water abs.
Pressure, m of water abs.

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Chemical Engineering July 2021

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