The power available in the wind is a function of the density of the air and the speed of the wind. The calculations follow. Using formulas taken from Paul Gipe's book "Wind Power for home and business", we find the following:

** Power density** is the power (in watts)
contained in air at sea level, at 60 degrees F. at a given speed:

**Power density = .05472
x speed ^{3}(cubed) in mph = watts/meter^{2}(squared)**

Inserting the appropriate numbers for a few common wind
speeds we get:

10mph |
15mph |
20mph |
25mph |
30mph |
35mph |
40mph |

54 watts/m |
184 watts/m |
437 watts/m |
855 watts/m |
1477 watts/m |
2146 watts/m |
3502 watts/m |

*Table
1 Power density (watts/sq. meter) at different wind speeds:*

This is the raw energy of the wind. Unfortunately, it
isn't what we can extract from the wind. There are two limitations we
need to consider. ** The Betz limit** and the inefficiencies
of any machine we build. There are others as well, but we won't deal
with those right now.

**The Betz limit of Energy from the wind**

Given the nature of the wind, there is a finite amount
of energy that we can get from it. This has been derived and is known
as the Betz Limit. This works out to 59%.

Applying the Betz
limit of 59% - The most power physically able to be extracted from
the wind, assuming a device efficiency of 100%.

10mph |
15mph |
20mph |
25mph |
30mph |
35mph |
40mph |

31 watts/m |
108 watts/m |
257 watts/m |
504 watts/m |
871 watts/m |
1233 watts/m |
2066 watts/m |

*Table
2 Betz limit of 59% of the power per sq. meter at various wind speeds*

Of course, wind turbines are not 100% efficient, neither are generators or electric transmission techniques. The challenge is to come as close to the Betz limit as possible by optimizing the rotor design and reducing loss due to friction and generation. The following are the Betz numbers for various size rotors. First we'll determine the swept area for the rotor sizes. For horizontal axis wind generators, this is equal to the area of a circle with the rotor length being the diameter.

**Area of a circle =
(p)radius ^{2}**

Area for a 1 meter diameter (3.3 foot) rotor = (3.14)
.5^{2} = 3.15 * .25 = .785 sq meters

Area for a 2 meter
diameter (6.6 foot) rotor = (3.14) 1^{2} = 3.14 * 1 = 3.14 sq
meters

Area for a 3 meter diameter (9.9 foot) rotor = (3.14) 1.5^{2}
=3.14 * 2.25 = 7.065 sq meters

Applying Betz -- **(.59)Power Density * area**
for various rotor sizes

Rotor diameter |
10 mph |
15 mph |
20 mph |
25 mph |
30 mph |
35 mph |
40 mph |
---|---|---|---|---|---|---|---|

1 meter |
24 watts/m |
84 watts/m |
201 watts/m |
395 watts/m |
683 watts/m |
967 watts/m |
1621 watts/m |

2 meters |
97 watts/m |
339 watts/m |
807 watts/m |
1582 watts/m |
2735 watts/m |
3871 watts/m |
6487 watts/m |

3 meters |
228 watts/m |
770 watts/m |
1825 watts/m |
3564 watts/m |
6158 watts/m |
9779 watts/m |
14.6 kw/m |

Some
wind turbine manufacturers use 27 or 28 mph as a standard wind speed
for rating the performance of their turbines. If we apply the same
windspeed to the Betz limit for the diameters shown above, we get the
following:

1 meter diameter |
2 meter diameter |
3 meter diameter |

498 watts |
1993.9 watts |
4.486 kilowatts |

*Table
4 Power density of the raw wind @ 27mph = 1077 watts/sq meter. The
Betz limit (59%) applied to this ---> 635 watts/sq meter*

The numbers shown above should serve as a sanity
check since there is no machine made that is 100% efficient (see
below). If you think you've beaten these numbers, check your figures.

In addition to the limitations imposed by the Betz limit and air density, there is the fact that all mechanical devices suffer from energy loss due to friction. Electrical devices, like generators and alternators, suffer from energy loss due to inefficiencies of electrical conductivity in copper wire. In addition to this, there are speed ranges in which generators/alternators are not efficient and the fact that at high speeds wind turbines tend to self destruct, requiring that they furl away from the wind or be manually shut down during high wind events.

These all combine to result in wind generators that are extracting 12 to 30% of the power contained in the wind as usable electrical energy. There are some wind generators that are proported to extract as much as 40% that are verifiable. If we calculate 30% of the total power in the wind for the turbine sizes and wind speeds shown above, we get the table below, units are watts.

Rotor diameter |
10 mph |
15 mph |
20 mph |
25 mph |
30 mph |
35 mph |
40 mph |
---|---|---|---|---|---|---|---|

1 meter |
12.89 |
43.49 |
103.09 |
201.35 |
347.94 |
552.51 |
824.74 |

2 meters |
51.55 |
173.97 |
412.37 |
805.41 |
1391.75 |
2210.05 |
3298.96 |

3 meters |
115.98 |
391.43 |
927.83 |
1812.17 |
3131.43 |
4972.6 |
7422.66 |

It's not that wind turbines of this size can't extract
more energy from the wind than this, some do. What is shown in the
table above is the ballpark figure for what wind turbines of this
size are likely to extract at these wind speeds. Those who advertise
figures drastically higher than these, and especially higher than the
Betz numbers (above), should be questioned.