Factors to consider when evalutating small wind generators.

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 speed3(cubed) in mph = watts/meter2(squared)



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

10mph

15mph

20mph

25mph

30mph

35mph

40mph

54 watts/m2

184 watts/m2

437 watts/m2

855 watts/m2

1477 watts/m2

2146 watts/m2

3502 watts/m2

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/m2

108 watts/m2

257 watts/m2

504 watts/m2

871 watts/m2

1233 watts/m2

2066 watts/m2

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)radius2

Area for a 1 meter diameter (3.3 foot) rotor = (3.14) .52 = 3.15 * .25 = .785 sq meters
Area for a 2 meter diameter (6.6 foot) rotor = (3.14) 12 = 3.14 * 1 = 3.14 sq meters
Area for a 3 meter diameter (9.9 foot) rotor = (3.14) 1.52 =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/m2

84 watts/m2

201 watts/m2

395 watts/m2

683 watts/m2

967 watts/m2

1621 watts/m2

2 meters

97 watts/m2

339 watts/m2

807 watts/m2

1582 watts/m2

2735 watts/m2

3871 watts/m2

6487 watts/m2

3 meters

228 watts/m2

770 watts/m2

1825 watts/m2

3564 watts/m2

6158 watts/m2

9779 watts/m2

14.6 kw/m2



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.

Mechanical Inefficiencies

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.