Compressible Flow
This analysis of water hammer allows for the compressibility of water and the expansion of the pipe
Contents
- Overview
- 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
Overview
Although 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 Closure

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
.
By Newton's second law, the pressure force equals the change of momentum per second.
i.e.
Pressure rise at valve
where
Notes:
- 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 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:
Pressure
Which is commonly written as:
Pressure rise
Where
is the change of velocity.
And:
The increase in Pressure head

Instantaneous Partial Closure Of Valve
If 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
The Relationship Between Pressure Rise, Speed Of Sound And Bulk Modulus Allowing For Lateral Expansion Of The The Pipe
Notes: Bulk modulus- Assume a thin cylinder constrained longitudinally and subject to an internal


Pipe With Change Of Section


Flow through valve

- If the velocity of sound is the same for both pipes then
- Putting
= infinity i.e.the pipe entering the reservoir
Example:
[imperial]
Example - Example 1
Problem
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.
Workings
a) Using equation (39)
Thus:
Solving the quadratic in
The increase in the pressure head

b) When the valve is fully open using equation (42)
And when the valve is closed by 1/5 th Combining the above two equations and puttingSolution
The increase in the pressure head is 
Last Modified: 10 Jan 12 @ 17:02 Page Rendered: 2022-03-14 15:56:28