# Incompressible Flow

An anlysis of Water Hammer based on the assumption that water is incompressible

**Contents**

### Key Facts

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

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

## Overview

Using**The Rigid Column Theory**the compressibility of the fluid is ignored and it is assumed that pressure changes caused by opening or closing a valve are felt instantaneously through out the pipe. In effect the water column is a solid column, which can accelerate or decelerated as an entity.

## The Gradual Valve Closure

A

**valve**is a device that regulates, directs or controls the flow of a fluid by opening, closing, or partially obstructing various passageways.
As the valve closes the pressure at the valve rises decelerating the water column.
Let be the pressure rise above the initial steady flow pressure, the deceleration of the water column, the rise in pressure, the cross sectional area and the length of the pipe.
The increase in Pressure Head is given by:

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{quote}## Valve Closed To Produce Constant Acceleration

**Acceleration**is the rate of change of velocity over time.

Let the time taken to close the valve be seconds.

Then: So for any given pipeline, if the maximum pressure rise is specified, the time taken to close the valve is directly proportional to the initial water velocity.

Then: So for any given pipeline, if the maximum pressure rise is specified, the time taken to close the valve is directly proportional to the initial water velocity.

## Valve Closed To Produce Uniform Rate Of Pressure Increase

**Pressure**is the force per unit area applied in a direction perpendicular to the surface of an object.

The maximum pressure rise
Where is a constant and is the total time of valve closure.

At any instant during closure
where is Constant.

The values of and can be found from the known initial and final conditions at and when . then: And the maximum pressure rise

The values of and can be found from the known initial and final conditions at and when . then: And the maximum pressure rise

## The Valve Closes So That The Area Decreases Uniformly With Time

This equation again relates the maximum pressure rise directly to the time of valve closure. The difference between this and the first case is that the rate of increase in has been controlled. The equation gives for a known pipe, the maximum pressure rise given any closure time. Note: The suffix "o" refers to the initial conditions when the head behind the valve and is equal to the gross supply head if the friction in the pipe is neglected and is the Inertia head at a time .**For Valve opening**

A

**nozzle**is a device designed to control the direction or characteristics of a fluid flow as it exits an enclosed chamber or pipe via an orifice.Consider the valve as a nozzle at the end of the pipe with a which is constant.

**Initial flow**At a time the discharge But: When is a maximum: The Maximum head Putting: Solving the quadratic: This equation again relates the maximum pressure rise directly to the time of valve closure. The difference between this and the first case is that the rate of increase in has been controlled. The equation gives for a known pipe, the maximum pressure rise given any closure time. Which must be the positive root for valve closure.**For valve opening.****For pipe 1**

Note: The above expression is the same as equation (2) with an "equivalent" pipe length of: And for a number of pipe sections of different cross sectional areas: These equations allow the calculation of an "overall length" which can be used to calculate pressure rises under differing valve opening regimes.

Applying Newton's Second Law for each pipe:
By continuity:
Putting equation (10) into equation (8)
Equation (11) can now be inserted into equation (9)

Note: The above expression is the same as equation (2) with an "equivalent" pipe length of: And for a number of pipe sections of different cross sectional areas: These equations allow the calculation of an "overall length" which can be used to calculate pressure rises under differing valve opening regimes.

N.B. The following two examples are based on the assumption that water is not compressible.

Note: The above expression is the same as equation (2) with an "equivalent" pipe length of: And for a number of pipe sections of different cross sectional areas: These equations allow the calculation of an "overall length" which can be used to calculate pressure rises under differing valve opening regimes.

N.B. The following two examples are based on the assumption that water is not compressible.

Example:

[imperial]

##### Example - Example 1

Problem

Find the

**maximum increase**in pressure head if the valve closes in 12 seconds.- a) So that the velocity decreases uniformly with time.
- b) So that the pressure rises uniformly with time.
- c) So that the valve discharge area decreases uniformly with time.
- d) The last 1000 ft. of pipe is 8 ft. diam. and the velocity decreases uniformly with time.

Workings

Initial velocity

- a) Velocity decreases uniformly with time.

- b) Pressure rises uniformly with time.

- c) Valve Discharge Area decreases Uniformly with time

- d) The last 1000 ft. of pipe is 8 ft. diam. and the velocity decreases uniformly with time.

- 1. Pipe friction is small compared to the pressure rise and has been neglected
- 2. The time of closure is greater than

Solution

- a) When Velocity decreases uniformly with time the Inertia Head

- b) When Pressure rises uniformly with time,
- c) When Valve Discharge Area decreases Uniformly with time,
- d) When the last of pipe is diam. and the velocity decreases uniformly with time,