How this calculator works
The kinetic energy passing through a rotor each second is one half times the air density times the swept area times the cube of the wind speed. Nothing can capture all of it: Albert Betz showed in 1920 that no rotor can extract more than 16/27, or 0.593, of the power in the wind, because slowing the air completely would stop the flow through the rotor. Real machines reach a coefficient of performance of about 0.45 to 0.50, and mechanical, electrical and control losses take another 10 to 20 percent. That is why the calculator asks for both a coefficient and a loss figure. A horizontal-axis rotor sweeps a circle of the rotor diameter, while a vertical-axis machine sweeps a rectangle of diameter times height, which is why the two need different inputs. The wind speed is usually the annual average at hub height; because power rises with the cube of speed, a site with 8 m/s average produces far more than twice the energy of a site with 4 m/s.
Formula
swept area (m2) = pi x (diameter / 2)^2 for a horizontal axis, or diameter x height for a vertical axis. power in the wind (W) = 0.5 x air density x swept area x wind speed^3. electrical power (W) = power in the wind x coefficient of performance x (1 - losses). Wind speed for a target power = cube root of (2 x power / (density x area x coefficient x (1 - losses))). Annual energy (kWh) = electrical power / 1000 x hours at that speed.
Worked example
A 50 m diameter rotor sweeps 1,963.5 m2. In 8 m/s wind with an air density of 1.225 kg/m3 the wind power is 0.5 x 1.225 x 1,963.5 x 512 = 615.8 kW. At a coefficient of 0.45 and 15 percent losses the electrical output is 615.8 x 0.45 x 0.85 = 235.5 kW. For 2,000 hours a year at that speed the energy is 471,000 kWh, worth $37,700 at $0.08 per kWh. To reach 2 MW with this rotor and coefficient the wind would have to be the cube root of (2,000,000 / (0.5 x 1.225 x 1,963.5 x 0.45 x 0.85)) = 16.3 m/s.
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Frequently asked questions
What is the Betz limit?
The maximum fraction of the kinetic energy in the wind that any rotor can capture, 16/27 or about 0.593. It comes from the need to leave the air moving behind the rotor: if you stopped it completely, new air could not flow through. No turbine can beat it.
Why is the output so sensitive to wind speed?
The power in the wind rises with the cube of speed. Going from 6 to 8 m/s is a 33 percent rise in speed but a 137 percent rise in power. That is also why a wind farm sited at a slightly better location can double its output.
What does the hourly figure mean?
Wind does not blow at one speed all year. The calculator multiplies the power at your chosen speed by the number of hours a year you expect that speed. For an annual estimate, use the capacity factor instead, or run the calculation across a wind speed distribution.
Are small turbines worth it?
In a built-up area, usually not. Rooftop wind speeds are low and turbulent, and the output falls with the cube of speed. A small turbine in an open site can work, but it rarely matches the cost of the same money spent on solar panels.