Calculation of Wind Power
To understand how important it is to mount a wind turbine on a tall tower, consider a simple mathematical equation. It’s called the power equation and is used to calculate the power available from the wind. This equation shows us that three factors influence the output of a wind energy system: (1) air density, (2) swept area, and (3) wind speed — all explained shortly.
The power equation is: P = d . A . V³/2
P stands for the power available in the wind (not the power a wind generator will extract — that’s influenced by efficiency and other factors). Density of the air is d. Swept area is A. Wind speed is V.
Air Density
Air density is the weight of air per unit volume, which varies with elevation. As a general rule, anticipate a decrease in the air density of about 3 percent per 1,000 feet (300 meters) increase in elevation. As a result, air density doesn’t affect the power available from the wind until elevation reaches 2,500 feet (760 meters) above sea level.
At 3,000 feet (about 910 meters) above sea level, the air density is 9 percent lower than at sea level. At 5,000 feet (about 1,525 meters) air density declines by about 15 percent.
Air density is also a function of relative humidity, although the difference between a dry and humid area is usually negligible.
Temperature also affects the density of air. Warmer air is less dense than colder air. Consequently, a wind turbine operating in cold (denser) winter winds would produce slightly more electricity than the same wind turbine in warmer winds blowing at the same speed.
Although temperature and humidity affect air density, they are not factors we can change. Installers must be aware of the reduced energy available at higher altitudes, however, so they don’t create unrealistic expectations for a wind system.
Although density is not a factor we can control, wind installers do have control over a couple of other key factors, notably, swept area (A) and wind speed — both of which have a much greater impact on the amount of power available to a wind turbine and the electrical output of the machine than air density.
Swept Area
Swept area is the area of the circle that the blades of a wind machine create when spinning. It is a wind machine’s collector surface. The larger the swept area, the more energy a wind turbine can capture from the wind. Swept area is determined by blade length. The longer the blades, the greater the swept area. The greater the swept area, the greater the electrical output of a turbine.
As the equation suggests, the relationship between swept area and power output is linear. Theoretically, a ten percent increase in swept area will result in a ten percent increase in electrical production. Doubling the swept area doubles the output.
When shopping for a wind turbine, always convert blade length to swept area, if the manufacturer has not done so for you (they usually do). Swept area can be calculated using the equation A = π · r².
In this equation, A is the area of the circle, the swept area of the wind turbine The Greek symbol is π, which is a constant: 3.14. The letter r stands for the radius of a circle, the distance from the center of the circle to its outer edge. For a wind turbine, radius is usually about the same as the length of the blade.
Because swept area is a function of the radius squared, a small increase in radius, or blade length, results in a large increase in swept area.
As an example, a wind generator with an 8-foot blade has a swept area of 200 square feet. A wind generator with a 25 percent longer blade, that is, a 10-foot blade, has a 314 square-foot swept area. Thus, a 25 percent increase in blade length results in a 57 percent increase in swept area and, theoretically, a 57 percent increase in electrical production.
Wind Speed
Although swept area is more important than density, wind speed is even more important in determining the output of a wind turbine. That’s because the power available from the wind increases with the cube of wind speed. This relationship is expressed in the power equation as V³.
Consider an example: Suppose that you mount a wind machine 18 feet (5.5 meters) above the ground surface on the grasslands of Nebraska. Suppose that the wind is blowing at eight miles per hour (3.6 meters per second). A friend, who knows how important it is to mount a wind machine on a tall tower, installs an identical wind turbine on a 90-foot (27-meter) tower. When the wind is blowing at 8 mph where your turbine flies at 18 feet, an anemometer on your friend’s 90-foot tower indicates that the wind is blowing at 10 miles per hour. Wind speed is 25 percent higher. What’s the difference in available power?
The power available in the wind can be approximated by multiplying the wind speeds by themselves three times. (Units aren’t important for this comparison.) For the lower turbine the result is 8 x 8 x 8 or 512. The power available to the wind turbine mounted on a 90-foot tower is 10 cubed or 10 x 10 x 10 which is 1,000.
Thus, a two-mile-per-hour increase in wind speed, a paltry 25 percent increase, doubles the available power. Put another way, a 25 percent increase in wind speed yields an increase of nearly 100 percent.
The important lesson is that because power is function of V³, a small increase in wind speed results in a very large increase in the power available to a wind turbine. This can result in a very large increase in the electrical output of a wind turbine.
Although winds are out of our control, homeowners can affect the wind speed at their wind turbines by choosing the best possible site and by installing their machines on the tallest towers.
Shopping Tip
Because swept area is such an important determinant of the output of a wind turbine, we strongly recommend focusing more on the swept area of a wind turbine than on its rated power — at least until the industry can come up with a standardized way of measuring and reporting rated power.