Published by Australian Academy of Science
Wind power gathers speed Box 4 | The power of the winds

The potential power offered by wind turbines can be demonstrated by a series of equations. The kinetic energy (KE) of a mass is expressed by the equation:

where m is the mass of the air (or other object) and v is its velocity or speed.

For a unit volume (1 cubic metre) of material

where d is the density, or mass per unit volume. (For air, d is about 1 kilogram per cubic metre.)

The mass of the air that actually supplies energy to a wind turbine is related to the area (A) covered by the sweep of the blades. For the standard wind turbine with blades that revolve on a horizontal axis (that is, the blades themselves are perpendicular to the ground), this area is expressed in the following equation:

where p = 3.1416 and D is the diameter swept by the rotor blades.

The kinetic energy available to the wind turbine therefore becomes:

The power available to the wind turbine in a given unit of time is a factor of kinetic energy and the distance that the wind travels in that time, which is determined by the wind velocity. Thus:

Simplifying the equation:

and

This last equation is the one of most interest. Since the density (d) of air varies only slowly with temperature or height above the Earth's surface, the only variables that can influence the power available for electricity generation are the diameter (D) of the turbine blades and the velocity (v) of the wind. (If d and D are in metres and v is in metres per second then the calculated power is in watts.)

The equation shows us that power is directly proportional to the cube of velocity: this means that a small increase in wind speed can produce a large increase in power. For example, if windspeed is 10 kilometres per hour, the value of v³ in the equation would be 1000. If windspeed doubled to 20 kilometres per hour, the value of v³ would be 8000. A doubling of windspeed therefore leads to an eight-fold increase in power.

Similarly, since power is proportional to the square of turbine diameter, rotor length is also an important determinant of the amount of electricity produced. For example, if the turbine diameter is 10 metres, then the value of D² in the equation would be 100. If D = 20, then D² = 400. Thus, a doubling of diameter produces a four-fold increase in power.

These two factors have important implications for the design and location of wind turbines. First, the stronger the wind, the more effective the turbine (up to a point – if the wind is too strong, the turbine will be damaged). Second, the longer the rotor blades, the better (again, up to a point – if they are too long they become unwieldy and more susceptible to damage).

There are other considerations affecting the potential power of wind turbines, such as the efficiency at which they convert the kinetic energy of the wind into electricity. This can be expressed as the power coefficient, which is the power produced by the turbine as a percentage of the power of the undisturbed wind passing through an area equal to that swept by the rotor.

For windmills built before 1900, the power coefficient was usually less than 5 per cent. Technological innovation has led to considerable improvements in modern wind turbines which can now achieve power coefficients of about 35 per cent. This means that they can convert more than a third of the wind's power into electricity. The theoretical maximum in most situations is 59 per cent.

Related site

• The power of the wind: Cube of wind speed (Danish Wind Industry Association)

Other boxes

Box 1. What causes the wind?

Box 2. The environmental credentials of wind power

Box 3. Energy basics