Wind Speed and Power
Before venturing into the dynamics of wind flow, we should first differentiate between wind speed and wind power.
Wind Speed
Wind speed is the rate at which air flows past a point above the earth’s surface. Wind speed can be quite variable and is determined by a number of factors which will be discussed in this section.
Wind Power
Wind power is a measure of the energy available in the wind. It is a function of the cube (third power) of the wind speed. If the wind speed is doubled, power in the wind increases by a factor of eight (23). This relationship means that small differences in wind speed lead to large differences in power. For example, assume that one person takes a speed measurement of 10 mph and another person, at the same time but at a neighboring site, gets a reading of 12.6 mph. For this difference of 2.6 mph, there is a 100% difference in the available wind power (103 = 1000 vs. 12.63 = 2000)! This example points out that minor differences in wind speed due either to site selection or measurement errors can have a major bearing on a decision to invest in a wind turbine. For this reason, a thorough and accurate wind study is imperative before buying a wind turbine.
The amount of power available in the wind is determined by the equation w = 1/2 r A v3 where w is power, r is air density, A is the rotor area, and v is the wind speed. This equation states that the power is equal to one-half, times the air density, times the rotor area, times the cube of the wind speed. Air density varies according to elevation, temperature and weather fronts. For the purposes of calculating wind power, the variations in weather fronts are too small to significantly affect electric power output, so the formula for air density is: p = (1.325 x P) / T where T is the temperature in Fahrenheit + 459.69 and P is the pressure in inches of Mercury adjusted for elevation.
A standard value for air density can be used to reflect the typical average air temperature (59°F) adjusted to sea level. The standard value may be used as an approximation for Iowa. The power equation can then be simplified to the following, depending on whether you use English or metric units.
Simplified Power Equation
English units | Metric units | |
w = 0.0052 A v ^{ 3 } | w = 0.625 A v ^{ 3 } | |
where w is power in watts, and A is the cross-sectional area in square feet swept out by the wind turbine blades, and v is the wind speed in miles per hour. | where w is power in watts, and A is the cross-sectional area in square meters swept out by the wind turbine blades, and v is the wind speed in meters per second. |
The metric version is shown because the metric system is frequently used by the wind energy industry. English to metric conversion tables are posted here. For example, a wind speed of one meter per second is the same as 2.24 miles per hour. One square meter is the same as 10.76 square feet.
Although this power equation shows an exponential increase in wind power as wind speed increases, in practice, the actual power increase in a wind turbine is more linear than is predicted by the equation. This is because a wind turbine is not perfectly efficient. The power curve of a wind turbine is actually more significant. Nevertheless, an increase of 2 mph in average wind speed can mean a 50% rise in the electricity produced by a turbine.
The area swept by most wind turbine blades is circular. The area can be calculated byusing the equation for the area of a circle: Swept Rotor Area = A = (pi) r2 where r is the rotor radius (half the diameter, or the distance from the hub to a blade tip). The equation states that the swept rotor area, A, is equal to (pi) or 3.14 times the square of the rotor radius. Below is a sample power calculation.
Assume a wind speed of 15 mph, standard air density, and a rotor radius of 10 feet. The rotor area (A) is equal to 3.14 times (10 ft.)2, or 314 square ft. Using the appropriate simplified power equation (English units), the wind power is:
w = 0.0052 A v3
w = 0.0052 (314 sq. ft.) (15 mph)3
w = 0.0052 (314) (3375)
w = 5,511 watts
w = 5.511 kilowatts
Wind power density (not to be confused with air density) is a term commonly used to describe the wind power available per unit area swept by the blades, or w/A. Wind power density is normally expressed in metric units which are watts per square meter (abbreviated W/m2). If using English units, square feet must first be converted to square meters, and miles per hour to meters per second.
To determine the wind power density in metric units (W/m2) from the sample power calculation given above, it is necessary to divide the calculated power (5,511 watts) by the rotor area (314 sq. ft.), which must first be converted to square meters. Dividing 314 sq. ft. by 10.76 equals 29.18 square meters. The wind power density is:
A short-hand method for calculating wind power density in metric units (W/m2) when the wind speed is known is:
Wind Power Density
= 0.056 v3 where v is in miles per hour
Wind Power Density
= 0.625 v3 where v is in meters per second
Using the previous example for a 15 mph wind speed: Wind Power Density = 0.056 (15 mph)3 = 189 W/m2
Because of the wind’s normal variability, and the effect of this variability on the cube of the wind speed, the power equation should only be used for instantaneous or hourly wind speeds and not for long-term averages. To illustrate why, consider two places where the average speed is 15 mph. At the first location the wind always blows at 15 mph, giving a power density of 189 W/m2. At the second site the speed fluctuates. The wind is at 10 mph half the time (power density = 56 W/m2), and at 20 mph the other half (power density = 448 W/m2). The mean power density here is (56 W/m2 x 1/2) + (448 W/m2 x 1/2) or 252 W/m2. At both locations the mean speed is exactly 15 mph, but there is 33% more power at the site with varying speeds.
In the real world, the wind varies constantly. Actual wind power density at most sites can range from 1.7 times to 3 times greater than that calculated from only the mean wind speed. For a typical 15 mph site in Iowa, the actual average wind power density is about 400 W/m2.