# 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

### Key Facts

**Gyroscopic Couple**: The rate of change of angular momentum () = (In the limit).

- = Moment of Inertia.
- = Angular velocity
- = Angular velocity of precession.

## 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

The 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 .
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 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**: 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 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 Pipe

Notes: 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
Also:
Or:
where is the thickness of the pipe wall
Consider the situation in the pipe at a time sec.after an instantaneous valve closure. The pressure wave will have reached a point at a distance from the valve where The water in the length is now at rest and at a pressure above normal.
During time after closure of the valve a volume has passed point and this volume must be accommodated in addition to the initial volume . This is possible due to the compression of the water and the expansion of the pipe.
Assuming that and substituting from equation 29
Then:
Combining the above equation with
From the above two equations we can :
Eliminate by putting
And the increase in pressure head
or:
To eliminate :
By ignoring Pipe Expansion Equations (35 ) (38 ) (41) and (46 ) can be simplified so that:
And The head rise
And:
Thus the equivalent bulk modulus allowing for pipe expansion is given by:

## Pipe With Change Of Section

At 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 . The above diagram shows some instant when the pressure wave is between from equation (46)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: Similarly for the pressure wave in the small pipe: Note: is the velocity of sound in the small pipe. Adding equations ( 59 ) and(60 ) Equate to equation ( 56 ) substitute from ( ) and ( ) for V_0 & V_1 Using the value for found from equations ( 14 ) ( 27 ) and( ) this equation can be solved for and hence and . Notes:

- 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)

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 putting Thus: Solving the quadratic in The increase in the pressure headSolution

The

**increase in the pressure head**is