Calculation of the power produced by a windmill

The formula below gives the power generated by a windmill.

Windmills are not a new source of energy : they have been used to produce mechanical power or to pump water since several centuries ago, and they are now being developed to generate electricity, but not in very significant amounts, as can be seen below.

The most efficient windmills are not more than 50% efficient in the most even in the most windy places of the world. The power produced by a windmill is given by the following formula: 

- D is the diameter of the propeller in meters

- V is the velocity of the wind in meters/second

This formula is proportional to the square of the diameter of the propellers, which helps us to understand why modern windmills have grown bigger and bigger, with ever larger diameters, in their attempt to produce significant amounts of energy.

This formula also demonstrates how variable the power output can be depending on the meteorological conditions (speed of the wind).

For example, the latest generation of windmills, using the most modern technologies and construction materials (see photo below), are almost 100 meters in diameter, and stretch to the sky, higher than a 40 stories high building (about half the size of the Eiffel tower). Such a windmill will produce about 1 MW of power when the wind is blowing at a high speed of 10 m/s.

It would require 1500 such giant windmills, assuming that the wind is always blowing at full capacity, to produce as much energy as one nuclear reactor of 1500 MW (only a few meters wide and high).

But on an ordinary day, and in most locations, the wind rarely blows at 10 m/s. So what happens in a more moderate wind, let's say for example 1 m/s ?

If the speed of the wind is 1 m/s then these huge modern windmills, produce only about 1 kW of power (1000 times less than when the wind is blowing at 10 m/s), not even enough power for a single clothes-washing or drying machine.

These huge windmills cost about a million dollars each to build, that's a lot of money for an installation that doesn't produce enough energy to run a single washing machine in a moderate wind...

Nevertheless, windmills can be useful, to produce small amounts of power far from the electrical grid, for example to produce power for a radio and navigation instruments on a sailing boat.

This is possible because the navigation instruments consume a lot less energy than a clothes-washing machine.

Windmills are not to be ruled out entirely, they can be very useful in local applications tu produce SMALL amounts of energy, but, quite unfortunately, they cannot replace the nuclear or fossile fuel plants to produce electricity in industrialized countries.

Photo of a modern windmill, about 150 m high and almost 100m in diameter, delivers about 2MW of power when the wind blows hard (10m/s) but about 1000 times less - it then hardly produces anything - when the wind blows more moderately (1m/s).