Specific Speed and Unit Conditions
- 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 FactsGyroscopic Couple: The rate of change of angular momentum () = (In the limit).
- = Moment of Inertia.
- = Angular velocity
- = Angular velocity of precession.
IntroductionIn 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.
- 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 TurbineThe 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 .
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
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.
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)
Therefore, (which is constant)
Therefore, (which is constant)
Specific Speeds For Differing Types Of Turbine
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 SpeedWhat 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.
- If is the Unit speed and the speed under a head And . Therefore . Or
(Where represents unit conditions and represents a Constant)
- 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)
- 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)
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The Performance Curves Of A Turbine
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Characteristic Curves And Iso-efficiency Curves For A Turbine Under All Operating Conditions
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An Example Of The Use Of Unit ConditionsA 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:
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