The Efficiency of Turbines
Key FactsGyroscopic Couple: The rate of change of angular momentum () = (In the limit).
- = Moment of Inertia.
- = Angular velocity
- = Angular velocity of precession.
IntroductionThis section considers the efficiency of both Impulse and Reaction Turbines.
The Efficiency Of Turbines
Efficiency is used with the specific purpose of relaying the capability of a specific application of effort to produce a specific outcome effectively with a minimum amount or quantity of waste, expense, or unnecessary effort.
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- be the total head behind the nozzle i.e. the sum of the pressure and velocity heads. This equals - The Pipe line losses of .
- = Jet Velocity
- = the exhaust velocity of water leaving the vanes.
- The Work done per second on the vanes ( per lb of water per second)
- The theoretical hydraulic efficiency of the Turbine is equal to:
It is a vector physical quantity (both magnitude and direction are required to define it).
Impulse is defined as a force multiplied by the amount of time it acts over.
The impulse can be calculated as the integral of force with respect to time. Alternately, impulse can be calculated as the difference in momentum between two given instances.
Impulse Turbine Allowing For Friction
- The frictional losses are in the nozzle, , the runner and various mechanical losses, . The effect of these is to reduce i.e. to reduce the hydraulic efficiency
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Example - Example 1
( one Horse Power (HP) is 550 ft.lb/sec. One cubic ft. of water weighs 62.4 lb., 32.2 ft/second squared)
Neglecting windage losses, find:
- a) The efficiency of the runner.
- b) The diameter of each jet.
- a) The efficiency of the runner is
- b) The diameter of each jet is or approx. inches.
Reaction TurbinesThis section covers Inward Radial flow, Mixed flow or Axial Turbines with a propeller shaft. These may be sited below the tail race or above it with a draft tube.
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- be the absolute velocity at the entry to the runner.
- be the absolute velocity at the exit of the runner.
- be the head.
Where the Supply is low it is usual to have an open flume supply with a short Pinstock e.g. as found in a Propeller Turbine. In this arrangement the head is the vertical height from the water level in the "fore bay" to the level in the ail Race.
For larger heads the Pinstock is longer and there is a smaller throughput of water e.g A Radial Flow Francis Turbine. is now the Total Head in the Supply Pipe just before entering the Turbine casing i.e. Pressure Velocity and Datum . Where appropriate measurements are relative to the Tail Stock Datum.
The Gross Head is measured from the the supply reservoir and includes Pinstock Losses.
The Draft Tube is designed to convert Kinetic Energy of the water being discharged into a reduced Pressure. This gives increases the suction through the Turbine and enables the height Z ( see diagram) to be included in the Supply head .
- 1. For a Reaction Turbine
- 2. For Axial Flow
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- be the number of blades.
- be the blade thickness.
- be the width of the runner.
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Example - Example 1
The runner is 24 inches outside diameter and 16 inches inside diameter and the widths at the entrance and exit are 2 and 3 inches respectively.
The pressure at entry to the guides is + 87 ft.head and the kinetic energy there can be neglected. The pressure at discharge is - 6 ft.head.
If the losses in the guides and moving vanes are taken as where f is the radial component of flow calculate:
- a)The speed of the runner in r.p.m. for tangential flow on to the running vanes
- b) The horse-power given to the runner by the water.
- a)The speed of the runner is
- b) The horse-power given to the runner by the water is Kilo watts