# Specific Speed and Unit Conditions

**Contents**

- Introduction
- Principles Of Similarity Applied To Turbines.
- Specific Speed Of A Turbine
- Specific Speeds For Differing Types Of Turbine
- An Example Of The Use Of Specific Speed
- Unit Conditions
- The Performance Curves Of A Turbine
- Characteristic Curves And Iso-efficiency Curves For A Turbine Under All Operating Conditions
- An Example Of The Use Of Unit Conditions
- Fundamental Similarity Conditions And Model Testing
- 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.

## Introduction

In the selection and/or design of turbines for a particular application it is common to rely on model testing. The results are then scaled up using the following principles:## Principles Of Similarity Applied To Turbines.

**similar model**means :

- Geometrically similar - made from the same drawings but to a different scale.
- Dynamically similar - Operating conditions and equal efficiencies.

- All the linear dimensions will be in the same ratio.
- All angles will be the same, the velocity triangles will be geometrically similar and all velocities will be in the same ratio.

## Specific Speed Of A Turbine

The**Specific Speed**of a turbine is the speed in rotations per minute (r.p.m.) at which a similar model of the turbine would run under a head of 1ft. when of such size as to develop 1 H.P.

(

**Note:**The suffix "s" is used to denote the values associated with the Specific Turbine) Each type of Turbine (Pelton Wheel, Francis etc.) has it's own characteristic limits of .

**Revolutions per minute**(

**R**.

**P**.

**M**.) is a measure of the frequency of a rotation. It annotates the number of full rotations completed in one minute around a fixed axis.

And . Therefore,

But, = The Area of flow The Velocity of flow

and Or, But the weight of water per second is . Which is,

H.P.output of the Turbine . Which is

**Note:**The efficiencies are equal (where Constant)

But for the specific Turbine, and are 1.

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### Notes On The Use Of The Specific Speed Of A Turbine

**Horse power**(

**H**.

**P**.) is the name of several units of measurement of power.

The mechanical horsepower (imperial horsepower), of exactly 550 foot-pounds per second is approximately equivalent to 745.7 watts.

Horse power was originally defined to compare the output of steam engines with the power of draft horses.

- is based on the values of , and used at the design point. i.e. at maximum efficiency.
- is NOT dimensionless and there are different values in each of the measurement systems.

**The Dimensions of Specific Speed**

can be made dimensionless and still be a constant by dividing by and this is called the

**The Speed Number**.

##### The Specific Speed Of A Particular Form Of Turbine

- For a particular type of Turbine is constant.

and . Therefore . Or (which is constant)

But

Therefore, (which is constant)

Therefore, (which is constant)

## Specific Speeds For Differing Types Of Turbine

**Specific speed**is a non-dimensional number used to classify pump impellers as to their type and proportions.

In

**Imperial**units it is defined as the speed in revolutions per minute at which a geometrically similar impeller would operate if it were of such a size as to deliver one gallon per minute against one foot of hydraulic head.

In

**metric**units flow may be in or and head in , and care must be taken to state the units used.

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## An Example Of The Use Of Specific Speed

What type of turbine would be used if the supply head is of 10 cu.ft/sec with a head of 225 ft. ? Assume an efficiency of 80%.It would therefore be necessary to use a Turgot Turbine. However it might be possible to use a Pelton Wheel with two jets.

Power per jet . Therefore per Jet

From the above table it can be seen that the value of is too high. It is therefore worth considering a Pelton Wheel with four jets.

Now:

## Unit Conditions

**unit operating conditions**for a turbine are those under which that particular turbine would run when working under a head of 1 ft. (or unit head in any other system) assuming there no change in efficiency.

### Unit Speed

- If is the Unit speed and the speed under a head And . Therefore . Or

(Where represents unit conditions and represents a Constant)

Unit speed:

### Unit Quantity

- The Unit quantity of a Turbine is the flow through the turbine when operating under a head of 1 ft. assuming similar conditions.Let,
- be the flow under a head .
- Therefore is the area of flow velocity.

(where is a Constant)

### Unit Power

- The Unit Power of a given turbine is the power output of the turbine when operating under a head of 1 ft. assuming no change in efficiency .If is the output under a head

Then: If is unchanged And: (where is a Constant)

But

Unit PowerNote:##### MISSING IMAGE!

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## The Performance Curves Of A Turbine

**Performance Curves**are plotted for a constant head and a constant Gate opening (or needle valve setting) and are on the basis of speed in r.p.m.

### Francis Turbine

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### Pelton Wheel

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## Characteristic Curves And Iso-efficiency Curves For A Turbine Under All Operating Conditions

**turbine**is a rotary engine that extracts energy from a fluid flow and converts it into useful work.

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## An Example Of The Use Of Unit Conditions

A**Francis Turbine**develops 3240 h.p. at 120 rpm when under a head of 36 ft. What would be the speed and output under a head of 25 ft. assuming no loss in efficiency.

Unit Power, . Therefore

## Fundamental Similarity Conditions And Model Testing

And, , . Therefore, . Or .

Substitute for in the above equation: . Or

Or substitute for : Or

Power, if is unchanged

From Equation (146), . Or

From Equation (148), . Or

i.e. (where is a Constant)

From equations (144) and (150) . Or

These seven expressions allow the performance of the prototype turbine to be estimated from tests on the model. Note that there are, in fact, only three independent equations.

The efficiency predicted for a large Turbine from test carried out on a model are usual lower than that obtained from the actual prototype. This is because of the relatively greater frictional losses in the smaller passages of the model.

Strictly speaking, the surface finish of the model should be geometrically similar to that of the prototype. The reduction in efficiency is said to be due to scale effects and is correcter for in practice by the use of empirical equations such as:

**The Moody equation**

##### Example - Example 1

At what

**speed**must the model be run and if it develops 135 h.p. and uses 38cu.ft.of water per second at this speed, what

**power**will be obtained from the full scale Turbine, assuming that it's efficiency is 3% better than that of the model?

With a value for of 138 the Turbine must be a Propeller Turbine.

- The
**specific speed**is - The
**power**is