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# Introduction to Turbines

An Introduction to Water Turbines including an analysis of the impact of a jet on a turbine vane or blade
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### Key Facts

Gyroscopic Couple: The rate of change of angular momentum ($\inline&space;\tau$) = $\inline&space;I\omega\Omega$ (In the limit).
• $\inline&space;I$ = Moment of Inertia.
• $\inline&space;\omega$ = Angular velocity
• $\inline&space;\Omega$ = Angular velocity of precession.

## Introduction

The British Government and the EU demands that the quantity electricity generated using fossil fuels be greatly reduced. The "green" alternatives such as wind wave and solar power are Dependant on climatic conditions and tidal power has great difficulty in generating continually over a 24 hour period. This is a real problem to the electricity supply companies who need to ensure that the demand for electricity can always be met This is no difficulty at present, since the quantity of "Green " electricity produced is not a significant percentage of the total, but as the number of wind farms increase, this will change. In many countries the majority of the power generation relies on steam turbines. These are highly efficient BUT inflexible. Basically they have to be kept spinning and they can not be quickly shut down or started.

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

Hydro electricity a reliable form of renewable energy. Water turbines are highly efficient and easily controlled to provide power as and when it is needed. In addition, currently the only system available to store large quantities of electrical power, is pumped storage. This involves pumping water into a high level reservoir. This can happen when the demand for electricity is low, at night for-instance. When the demand is high the supply can be rapidly increased by running the stored water through Turbines.

Turbines can be divided into two basic basic types. These are Impulse Turbines and Reaction Turbines.

### Impulse Turbines

In these the whole of the available energy of the water is converted to Kinetic Energy before the water acts on the moving parts of the turbine. In this type of turbine the cups or wheel passages are never entirely filled with water. To achieve this the turbine must be mounted slightly above the tail race.

The most popular example is the Pelton Wheel. These can be of great size

##### MISSING IMAGE!

The picture shows the arrangements of the cups on the wheel. Water is blasted at these cups by one or more jets mounted in the surrounding casing. This type of turbine is highly efficient. Power output can be increased by adding more jets around the circumference but there is a limit of three or maybe four. Above this number "spent" water tends to interfere with the jets and losses occur.

### Reaction Turbines

##### MISSING IMAGE!

In Reaction Turbines the rotation is mainly achieved by the reaction forces created by the acceleration of the water in the runner. The basic principle is the same as a rotating lawn sprinkler in which water enters the arms of the sprinkler at low velocity and leaves through the jets at high velocity. The exact manner in which this acceleration is achieved in Turbines depends upon the whether the runner is of the Propeller, Kaplin, Francis or Deriaz type.

The great variety possible in the geometry of the runner, makes Reaction Turbines suitable for a wide range of applications. In all types it is normal for a fraction of the hydraulic pressure to be converted into velocity as it passes through the inlet structure which consists of a spiral casing and a gate apparatus leading to the runner. The power from the water (Pressure and Velocity) is then converted into mechanical power in the runner. It is usual for the exit pressure from the runner to be below atmospheric. This is achieved by using a Draft Tube.
##### Axial Flow Turbines
Propeller and Kaplan have axial flow runners and are used for the lower heads. They are particular suitable for large installations. At low heads (Below 25 ft.) the running speed can be twice that of a Francis Turbine but even so the running speed of high output propeller turbines is less than 100 r.p.m. The fixed vane design of the propeller turbine does not lead to flexibility and the machines need to be run at or near their optimum output. At below 75% load the efficiency falls rapidly. To overcome this problem the

was developed. It is essentially a propeller turbine with variable blade angles. An interesting variation on the traditional design uses a horizontal shaft and an electric generator mounted in a metal shell which sits in the water-flow. Such machines are particularly suitable for very low heads and were developed for a French Tidal Power scheme.
##### Mixed Flow Turbines
The Francis Turbine is probably the most commonly used type of Turbine. It can operate from very low heads of about 10 ft. up to about 2000 ft. Turbines operating at these heads must have a large output since the low water quantities and the size of the water passages within the runner make construction difficult. At very low heads Propeller Turbines are usually a more economic solution.

A development of the Kaplan Turbine with a variable pitch design that improves the efficiency under less than full load at medium heads has been developed and is called the Deriaz Turbine after its inventor.

In these there is a pressure which in some cases amounts to half the head in the clearance space between the guide vanes and the wheel vanes.

The velocity with which the water enters the wheel is due to the difference between the pressure due to the head and the pressure in the clearance space.

## The Impact Of Water On A Vane

Before considering Turbines it is necessary examine the impact of water on the vanes in a turbine.

### General

To find the Force, work done, the efficiency of transfer of power from water to the vanes, it is necessary to know:

• The mass of liquid/sec. striking the vanes ( slugs/sec or kilograms/sec)
• The change in absolute (or relative) velocity of the liquid in the required
direction. (ft/sec. or meters/sec.)
Based on these the following can be calculated:

• The Force on the vane in the direction of water flow which can be found from Newton's Second Law.

Hence:
Force = Mass of water per second $\inline&space;X$ The change in Velocity
Note that the vane exerts an equal and opposite reaction on the water and the supply nozzle.

• The Work done per second by the water on the vane.

The Vane Velocity $\inline&space;X$ The force on the vane in the direction of the jet velocity

• Horse Power output = Work done per second $\inline&space;/&space;550$.
• Efficiency $\inline&space;\eta$ = Output Power/ Input power from jet.

### A Series Of Moving Plates

##### MISSING IMAGE!

The weight of water hitting the plates per second$\inline&space;&space;=&space;W&space;=&space;waV$ and it is assume that the water leaves the plates tangentially. N.B. For a single plate the weight of water per second striking the plate is $\inline&space;wa(V-v)$.

The change of velocity $\inline&space;=&space;(V&space;-&space;v)$

Force on plate = mass/sec. $\inline&space;X$ Change of velocity

$=&space;\frac{W}{g}\left(V&space;-&space;v&space;\right)$

Work done/second = Force $\inline&space;X$ Velocity =
$=&space;\frac{W}{g}\left(V&space;-&space;v&space;\right)v$

The input = the Kinetic energy of the jet $\inline&space;=&space;\displaystyle\frac{WV^2}{2g}$

The efficiency of the system,
$\eta&space;&space;=&space;\frac{2\,v(V&space;-&space;v)}{V^2}$

For a given jet velocity
$\frac{d\eta&space;}{dv}&space;=&space;\frac{2}{V^2}\left(V&space;-&space;v&space;\right)$

For maximum,
$\eta&space;\;\;\;\;V&space;=&space;2v$

and the value of the maximum efficiency $\inline&space;\eta&space;&space;=&space;50\%$

### A Single Plate Inclined To The Jet

##### MISSING IMAGE!

Neglecting friction and assuming that the water leaves the plate tangentially, the resulting force will be normal to the plate.

The weight of water striking the plate per second will be, as before, given by:

$W&space;=&space;w\,a(V&space;-&space;v)$

Change of velocity normal to the plate = $\inline&space;(V&space;-&space;v)\,sin\,\theta$.

Normal force on plate $\inline&space;F$ = $\inline&space;\displaystyle\frac{w\,a\,(v&space;-&space;v)\;sin\,\theta&space;}{g}$.

Component of $\inline&space;F$ in the direction of $\inline&space;v$ = $\inline&space;F\;sin\,\theta&space;&space;=&space;\displaystyle\frac{wa\;(v&space;-&space;v)^2\;sin^2\theta}{g}$.

Work done on plate
$=&space;\;&space;sin\,\theta&space;\times&space;v=&space;\frac{w\,a\,(V&space;-&space;v)^2\;vsin^2\theta&space;}{g}$

### Stationary Curved Vanes

##### MISSING IMAGE!

Force in the direction of $\inline&space;x$ =$\inline&space;\frac{W}{g}(Vcos\,\alpha&space;&space;-&space;V_1cos\,\beta&space;)$

$=&space;\displaystyle\frac{W}{g}(V_w&space;-&space;V_{w1})$

N.B. $\inline&space;\;W_{w1}$ is negative.

For a semi-circular blade both $\inline&space;\alpha$ and $\inline&space;\beta$ are zero and neglecting friction the force is twice that of a corresponding flat plate.

Force in direction $\inline&space;y$ = $\inline&space;\;\displaystyle\frac{W}{g}(V_f&space;-&space;V_{f1})$