Instantaneous Valve Closure Pressure Calculation. |
Maximum water hammer pressure. Joukowski impulse equation. |
To: LMNO Engineering home page (more calculations) Water Hammer (user enters valve closing/opening time) |
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Liquid initially flows at a constant velocity through a pipe. A downstream valve closes, and the liquid slams against the closed valve causing a pressure spike ΔP.
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Definition of instantaneous valve closure: An instantaneous valve closure occurs if the closure time is less than or equal to 2L/c. 2L/c is the time required for a pressure wave to reflect off the valve at the closure, then travel upstream to the reservoir, and return to the valve. |
Register to enable "Calculate" button.
Copy and paste the Celerity from the above calculation into the calculation below to compute 2L/c (may need to use keyboard control-c and control-v instead of right-clicking) :
INTRODUCTION
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
wave travels two pipe lengths in a time of 2L/c. If the valve takes longer then 2L/c to
close, then the water hammer pressures can be computed by our other
water hammer calculation which predicts pressures for finite valve opening or closing
times. 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 2L/c calculation does not.
EQUATIONS
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):
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.
Instantaneous valve closure occurs if the valve is closed in time < 2L/c (Hwang and Houghtalen, 1996, p. 119). The product 2L/c 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 2L/c to close, then our other water hammer calculation should be used, where the user can enter the valve closure time.
VARIABLES
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].
ΔP = Maximum pipe pressure increase due to sudden valve closure [F/L2].
ΔV = Change in velocity [L/T].
w = Pipe wall thickness [L].
ρ = Fluid density [M/L3].
FLUID PROPERTIES, PIPE PROPERTIES, and PIPE WALL THICKNESS
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.
ERROR MESSAGES given by 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.
"Need Velocity > 0". "Need Pressure > 0". If ΔV or ΔP is selected as an input, it must be positive.
2L/c calculation
"Need Length>0". "Need Celerity>0". Length and celerity
(wave speed) must be entered as positive numbers.
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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Software, Ltd. (All Rights Reserved)
LMNO Engineering, Research, and Software, Ltd.
7860 Angel Ridge Rd. Athens, Ohio USA (740) 592-1890
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