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Wind Turbines Swept Area, Solidity Page 2

<< Continuation from Previous Page

Solidity Continuation

There is a limit on how big the solidity ratio can be before the air no longer can flow across the rotor, I other words the number of blades can not be such that no air will pass through the rotor.

The German Scientist Albert Betz developed a mathematical calculation in which he established that the maximum theoretical energy that a wind turbine can capture from the wind is 59.3%.

So far no known machine has achieved theoretical Betz limit mainly due to aerodynamics

imperfections on rotors and blades causing drags and turbulence loses.

Tip Speed Ratio

Another important parameter in the selection or design of wind turbine is the tip speed ratio.

The Tip Speed Ratio is defined as the ratio of the wind speed to the rotational speed of the blades at the tip. The bigger the wind turbine blades the higher the blade tip speed or tangential speed at the tip on very large blades the tips speed can be very high in the order of 75 meters per seconds or more, this high speed imposes additional design considerations to assure the structural integrity of the blades and the turbines as a whole. Sometimes birds are killed when intercepted by these high speed rotating blades, bird fatalities are raising concern among environmentalists which are demanding measures to reduce such occurrences.

Large and very large turbines can cause noise and wind turbulence, something to keep in mind when designing or selecting a location for the installation of large wind turbines.

The tip speed ratio Tsr is expressed mathematically by this simple equation

Tsr = (Tip speed in m/s)/(Given wind speed in m/s) or:

Tsr = Ts/Ws

Ts = Blade tip speed in m/s

Ws =Wind speed in m/s

Example:

How the Tsr for a small wind turbine will if it has a rotor with a diameter of 3.5 m with a tip speed of 13 m/s when hit by a wind speeding at 7 m/s?

Tsr = (13m/s)/(7m/s) = 1.85

The table below shows typical turbine rotors tip speed ratios in accordance withe their size,

for a given wind speed.

The tip speed ratio data is normally part of the specifications for a wind turbine and it something derived from measurements and testing, although it could be calculated but with not so much precision, always will be necessary the testing and verification to render the final performance specs.

Tip Speed to RPM Relationship

Some manufacturers give the RPM (revolutions per minute) for a given wind speed instead of the tip speed ratio, but do not worry it can be easily calculated by the tangential speed formula.

Tangential speed = Tip speed

Tangential speed = (rad/s) x (r) = angular speed x (r)

Example:

Calculate the wind turbine tip speed with a rotor with the parameters given.

Diameter = 7 m

RPM = 300

The radio r of the rotor will be = 7/2 = 3.5m

We need to convert RPM to RPS (revolutions per second)

300 RPM =300/60 = 5 RPS

Now we need to convert these 5RPS into Radians per second;

Each RPS = 360 degree, therefore 5 RPM will be = 5 x (360) = 1800 degrees in other words, in one second the five revolution will sweep the circle of 360 degree five times.

Now because the formula for tangential speed is given in radians, we need to convert these degrees into radians (rad):

1 rad = 57.2957 degrees

Then 1800 degrees/s = 1800/57.2957 = 31.415 rad/s; this is the angular speed of the rotor now the tip or tangential speed of the rotor blades will be,

(rad/s)(r) = 31.415 x (3.5) = 109.9 m/s

Number of blades selection:

Turbine type |
Rotor Diameter m |
Given Wind Speed m/s |
Tip Speed m/s |
Tip Speed Ratio |

Farm Windmill |
3 |
7 |
13 |
1.85 |

Small Turbines |
2 |
10 |
35 |
3.5 |

Medium Size Turbines |
40 |
12 |
66 |
5.5 |

Very Large Turbines |
60- |
12 |
75- |
6.25- |

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