This analysis of water hammer allows for the compressibility of water and the expansion of the pipe
- Pressure Rise Following Instantaneous Closure
- Pressure Rise For Instantaneous Partial Reduction Of Flow.
- Instantaneous Partial Closure Of Valve
- The Relationship Between Pressure Rise, Speed Of Sound And Bulk Modulus Allowing For Lateral Expansion Of The The Pipe
- Pipe With Change Of Section
- Page Comments
Key FactsGyroscopic Couple: The rate of change of angular momentum () = (In the limit).
- = Moment of Inertia.
- = Angular velocity
- = Angular velocity of precession.
OverviewAlthough it is usual to consider water as incompressible and of a uniform density, this is clearly not true. It would, for example, be impossible to control the depth of a submersible vessel unless the density of water increased with depth. Here we consider the impact of compressibility for the purpose of computing water hammer and the subsequent pipe expansion.
Pressure Rise Following Instantaneous ClosureThe pressure rise at the valve
Where is the density and the velocity of Sound for the fluid (Water)
Let the initial pipe velocity be At the instant that the valve is closed a pressure wave, moving at the speed of sound sets off up the pipe bringing the water to rest. Consider the instant shown above, secs after valve closure. At this moment the length of the column brought to rest is .
Pressure rise at valve
- The pressure at the valve remains at above normal from the instant the valve is closed until the pressure wave is reflected from the open end of the pipe as a wave of normal pressure and velocity reaching the valve at time .
- A wave of reduced pressure, below normal will then set off up the pipe.
- The time is the period of the pipe.
- The above proof still applies if the valve is closed in a time of closure i.e. the valve is closed before the pressure wave returns to the valve.
Pressure Rise For Instantaneous Partial Reduction Of Flow.The Pressure rise
Where is the change of velocity.
The increase in Pressure head
The initial velocity is reduced to by the sudden partial valve closure. seconds later the pressure wave will have traveled a distance up the pipe as before Applying Newton's second Law:
Which is commonly written as: Pressure rise
Where is the change of velocity. And: The increase in Pressure head
Instantaneous Partial Closure Of ValveIf the reduction in area of the valve is known rather than the reduction of flow or pipe velocity,then the pipe may be treated as a nozzle with a coefficient of discharge of From the Initial Conditions and from the change in Valve Area: The change of pressure head
The Relationship Between Pressure Rise, Speed Of Sound And Bulk Modulus Allowing For Lateral Expansion Of The The PipeNotes: Bulk modulus = Increase of Pressure/Volumetric strain
- Assume a thin cylinder constrained longitudinally and subject to an internal
Hoop stress/Hoop strain = Young's Modulus Hoop strain
where is the thickness of the pipe wall
To eliminate :
And The head rise
Thus the equivalent bulk modulus allowing for pipe expansion is given by:
Pipe With Change Of SectionAt the initial steady flow conditions it is assumed that friction; velocity head and contraction losses can be neglected. Therefore the pressure throughout both pipes is .
Flow through valve
And from equation (16) also:
Equations ( 54 ) ( 55 ) and ( 56 ) can be solved for and hence and and hence . Now consider an instant in time after the pressure wave has passed through the junction and a reflected wave has set off up the pipe from to . By continuity:
Using equation ( 16 ) for the pressure wave in the large pipe:
- If the velocity of sound is the same for both pipes then
- Putting = infinity i.e.the pipe entering the reservoir
Example - Example 1
A pipe carrying water has a valve at the discharge end. The bulk modulus,, is
- a) If the initial velocity in a pipeline is show that an instantaneous closure of the valve will result in a pressure rise at the valve of
- b) Given that the initial velocity is 12 ft/sec. and the pressure head in the pipe is 600 ft, find the increase in pressure head which results when the valve is instantaneously closed by 1/5 th.
a) Using equation (39)
b) When the valve is fully open using equation (42)
The increase in the pressure head is