Water Hammer

Water hammer. Instant valve closure. Joukowski impulse equation. Pressure wave travel time.

 
 

 

Water Hammer. Maximum Pressure Change due to Sudden Valve Closure using Joukowski Equation
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© 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 water hammer calculation into the calculation below to compute wave travel time (may need to use keyboard control-c and control-v instead of right-clicking):

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

Units in 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.


INTRODUCTION TO WATER HAMMER RAPID VALVE CLOSURE CALCULATIONS
Prior to water hammer, 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 also called water hammer.

Water hammer sudden valve closure Joukowski equation for water hammer

 

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

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

    tw = 2L/c

EQUATIONS FOR WATER HAMMER PRESSURE DUE TO RAPID VALVE CLOSURE AND FOR WAVE TRAVEL TIME
The water hammer instantaneous valve closure calculation predicts the maximum increase in pressure that will occur due to a sudden valve closure. The valve closure time in water hammer 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 water hammer instantaneous valve closure calculation provides the engineer with the expected maximum pressure increase. The water hammer calculation can also be used in reverse - to compute the pipe velocity - if a maximum pressure rise due to water hammer is input. For the "Click to Calculate" button to function, the water hammer 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 for water hammer. 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 in a water hammer event. The water hammer 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 causing water hammer, 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 the water hammer event. In either case, the pressure increase due to water hammer 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, during water hammer 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):

Water hammer wave speed and modulus equations

The ΔP equation for water hammer 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.

Water hammer Wave Travel Time Equation
Instantaneous valve closure due to water hammer 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

In water hammer, 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.


VARIABLES IN WATER HAMMER PRESSURE AND WAVE EQUATIONS
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 in water hammer event due to sudden valve closure [F/L2].
ΔV = Change in velocity in water hammer [L/T].
ρ = Fluid density [M/L3].

WATER HAMMER FLUID PROPERTIES, PIPE PROPERTIES, and PIPE WALL THICKNESS

Fluid Properties
Fluid density, viscosity, and elastic modulus provided by the drop-down menus in the water hammer 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 water hammer calculation have been compiled from the references shown below.


WATER HAMMER ERROR MESSAGES GIVEN BY PRESSURE AND WAVE TRAVEL TIME CALCULATIONS
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 in water hammer calculation.

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

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

WATER HAMMER REFERENCES
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.

 

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Please contact us for consulting or other questions about water hammer.

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

 

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