The Wings of Society – Theme 6 Version 2.2 Chapter 33 - Is That Top Turbine Yours?

Wind Turbine Maths Task Sheet

33.1Related subjects• Technology • Physics• Mathematics

How much energy will you be able to generate from the air, if you use a wind turbine? It cannot be avoided, you will have to use maths.

A wind turbine converts the kinetic energy of moving air into electric power.The following factors play a role:

• the size of the rotor, which determines the area that is swept by the rotor;

• the wind velocity;• the air density;• the shape of the turbine blades.

33.1.1 a) Explain whether the captured energy will

increase or decrease if the first three factors increase.

b) Explain in your own words why the shape of the turbine blades is so important.

Wind power is proportional to velocity CUBED!The standard formula for calculating the electric power to be expected from a wind turbine expressed in metric terms is:

kW = ½ ρ.V3.A .E

whereρ is the density of air;V is the wind velocity;A is the turbine rotor area;E is the Power Coefficient, an efficiency factor, that, theoretically, cannot exceed the 60%.

If the turbine works h hours at a constant wind speed, the total amount of converted energy will be:

kWh = ½ ρ.V3.A .E.h

As the wind speed changes constantly, these rather simple calculations will not give the actual yield of energy, but an average.

33.1.2 a) The length of a rotor blade is 20 m.

Now calculate the turbine rotor area, use the formula for a circle: A = лr2.

b) If the length of the rotor blades are increased by 20%, what percentage extra power can be expected?

Wind Turbine Maths

At some distance above the earth, wind is usually stronger than near the surface, while the air density almost remains the same.As wind power strongly benefits from wind velocity, it is worthwhile to erect high towers with wind turbines. However, high towers are very expensive for many reasons. That is why engineers are looking for alternative ways of positioning wind turbines at higher altitudes.

This example, designed by the Sky Windpower Company, is a giant kite-like turbine that consists of four tilted rotors. It is harnessed and controlled by a cable that also conducts the generated electricity to earth. The problem is, that it will come down if the wind is not strong enough.

Another design, from the Magenn Company, is actually a balloon. It will stay at its working altitude, even if there is no wind. The entire balloon spins around a central shaft with generators at both sides. With this system, too, the generated electricity is transported by the cable that secures it. The form of the turbine balloon guarantees its stability while working.

33.1.3 A wind turbine has blades of 15 m length.The air density on a normal day at sea level is 1.225 kg/m3. The Power Coefficient of this tower is calculated to be 30%.a) Calculate the electric power this turbine

provides at a wind speed of 4 m/s.b) Do the same for a wind speed of 8 m/s.c) Also calculate the average power in a strong

breeze.Do not forget to convert the wind speed into m/s!

Beauforts Scale & Wind Speeds

Bf km/h

1 1-5 Light air.

2 6-11 Light breeze.

3 12-19 Gentle breeze.

4 20-29 Moderate breeze.

5 30-39 Fresh breeze.

6 40-50 Strong breeze.

7 51-61 Near gale.

8 62-74 Gale.