Rapid Valve Closure Pressure Calculation

Maximum water hammer pressure. Joukowski impulse equation. Pressure wave travel time.


Maximum pressure and wave travel time calculations are mobile-device-friendly as of February 26, 2015

Maximum Pressure Change due to Sudden Valve Closure using Joukowski Equation
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Make selections: 
© 2015 LMNO Engineering,
 Research, and Software, Ltd. www.LMNOeng.com
Fluid Density: 
Fluid Elastic Modulus, Ef
Pipe Elastic Modulus, Ep  psi
Pipe Inside Diameter, D: 
Pipe Wall Thickness, w:   inch
Celerity, c: 
Velocity Change, ΔV:   ft/s
Pressure Change, ΔP: 

Pressure Wave Travel Time Calculator (free)
Copy and paste the celerity from the above calculation into the calculation below to compute wave travel time (may need to use keyboard control-c and control-v instead of right-clicking):

Pipe Length, L: 
Celerity, c: 
Wave travel time, tw
© 2015 LMNO Engineering,  Research, and Software, Ltd.

Units in rapid valve closure water hammer calculators: cm=centimeter, ft=foot, g=gram, G=Giga, k=kilo, kg=kilogram, lb=pound, m=meter, M=Mega, mm=millimeter, N=Newton, Pa=Pascal, psf=pound per square foot, psi=pound per square inch, s=second.

Liquid initially flows at a constant velocity through a pipe. A downstream valve closes instantaneously, and the liquid slams against the closed valve causing a pressure spike ΔP.

Sudden valve closure Joukowski equation


Liquid flows from reservoir through pipe at steady state, then valve closes

If a valve is closed faster than the wave travel time, then it is considered an instantaneous valve closure.

    tw = 2L/c

The instantaneous valve closure calculation predicts the maximum increase in pressure that will occur due to a sudden valve closure. The valve closure time is considered to be instantaneous if the valve closes faster than (or equal to) the time required for a pressure wave to travel two pipe lengths (i.e. the time for the wave to travel upstream from the valve, reflect off the upstream boundary and return to the valve). The pressure predicted by the instantaneous valve closure calculation provides the engineer with the expected maximum pressure increase. The calculation can also be used in reverse - to compute the pipe velocity - if a maximum pressure rise is input. For the "Click to Calculate" button to function, the pressure calculation requires registration, but the wave travel time calculation does not.

Joukowski Equation
One-dimensional momentum conservation for frictionless flow is used to derive the Joukowski equation. The equation was developed for a liquid flowing steadily through a pipe and then instantly the velocity drops to zero due to a sudden valve closure. The equation assumes that liquid compression and pipe friction are negligible. Though the Joukowski equation's primary applicability is for a liquid velocity that drops to zero upon contacting a closed valve, the equation is valid for any instantaneous drop in velocity, not necessarily a drop to zero velocity. The Joukowski equation is seen with and without a negative sign on the right hand side depending on whether the pressure wave is traveling upstream or downstream. In either case, the pressure increase is a positive number. The Joukowski equation is (Wylie, 1993, p. 4; Chaudhry, 1987, p. 8; Hwang and Houghtalen, 1996, p. 118):

ΔP =  ρ c ΔV

The equation for wave speed, c, is based on mass conservation and allows the pipe wall material to expand (Wylie, 1993, p. 6; Chaudhry, 1987, p. 34; Hwang and Houghtalen, 1996, p. 115):

Wave speed and modulus equations

The ΔP equation was derived for liquid upstream of the valve and does not include effects downstream of the valve. The D/(wEp) portion was derived using a thin-walled pipe approximation.

Wave Travel Time Equation
Instantaneous valve closure is defined to occur if the valve is closed faster than the wave travel time. The wave travel time is (Hwang and Houghtalen, 1996, p. 119):

tw = 2L/c

The wave travel time, tw, is the time for a pressure wave to propagate from the valve, upstream to the reservoir, and back down to the valve. If a valve takes longer than tw to close, then our other water hammer calculation should be used, where the user can enter the valve closure time.

Dimensions: F=Force, L=Length, M=Mass, T=Time

c = Celerity (wave speed) [L/T].
D = Inside diameter of pipe [L].
E = Composite elastic modulus [F/L2].
Ef = Elastic modulus of fluid [F/L2].
Ep = Elastic modulus of pipe material [F/L2].
L = Pipe length [L].
tw = Wave travel time [T].
w = Pipe wall thickness [L].
ΔP = Maximum pipe pressure increase due to sudden valve closure [F/L2].
ΔV = Change in velocity [L/T].
ρ = Fluid density [M/L3].


Fluid Properties
Fluid density, viscosity, and elastic modulus provided by the drop-down menus in the calculation have been compiled from the closed conduit pipe flow references shown on our literature web page.

Pipe Properties
The pipe material elastic moduli built into our calculation have been compiled from the references shown below.

Pressure calculation
"Need Density > 0". "Need Ef > 0". "Need Ep > 0". "Need Diameter > 0". "Need Thickness > 0".
Density, elastic modulus of fluid and pipe, diameter, and wall thickness must be entered as positive numbers.

"Need ΔV > 0". "Need ΔP > 0". If ΔV or ΔP is selected as an input, it must be positive.

Wave travel time calculation
"Need Length>0". "Need Celerity>0". Length and celerity (wave speed) must be entered as positive numbers.

Chaudhry, M. Hanif. 1987. Applied Hydraulic Transients. Van Nostrand Reinhold Co. 2ed.

Hwang, Ned H .C. and Robert J. Houghtalen. 1996. Fundamentals of Hydraulic Engineering Systems. Prentice Hall, Inc. 3ed.

LMNO Engineering, Research, and Software. 2009. Newsletter comparing water hammer calculations.

Mays, Larry W. 1999. Hydraulic Design Handbook. McGraw-Hill.

Wylie, E. Benjamin and Victor L. Streeter. 1993. Fluid Transients in Systems. Prentice-Hall, Inc.


© 2009-2015 LMNO Engineering, Research, and Software, Ltd.  All rights reserved.

Please contact us for consulting or other questions.

LMNO Engineering, Research, and Software, Ltd.
7860 Angel Ridge Rd.   Athens, Ohio 45701  USA   Phone and fax: (740) 592-1890
LMNO@LMNOeng.com    http://www.LMNOeng.com

August 25, 2015: Made text fields show 8 significant figures rather than 16. Calculation still uses double precision internally.


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Related calculation:

Water Hammer (user enters valve closing/opening time)


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