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Energy & Renewable Energy Engineering DISCUSSION
I installed a small horizontal-axis wind turbine with a 2 m rotor diameter, sold as 1 kW. With an anemometer next to it reading a steady 6 m/s, the output is only 100 to 150 W. The seller says this is normal, but it is barely a tenth of the rating.
How is the power of a wind turbine related to wind speed and rotor diameter, what is the maximum I could possibly get at 6 m/s, and what does the 1 kW rating actually refer to?
The power carried by wind through the rotor area is P = 0.5 × ρ × A × v³, with air density ρ about 1.225 kg/m³ at sea level. A 2 m rotor sweeps A = π × 1² = 3.14 m². At 6 m/s: 0.5 × 1.225 × 3.14 × 216 = 416 W in the wind.
No turbine can take all of it, because the air has to keep moving to get out of the way. The theoretical maximum fraction is the Betz limit, 16/27 or 59.3 percent, which here is 246 W. Real small turbines, after blade, generator and electronics losses, achieve an overall coefficient of perhaps 0.25 to 0.35, giving 104 to 145 W. Your measurement is what the physics allows.
The 1 kW figure is the output at the rated wind speed chosen by the manufacturer, commonly 11 to 12 m/s or more for small machines. Because power goes with the cube of speed, doubling the wind from 6 to 12 m/s multiplies the available power by 8: 416 × 8 = about 3,330 W in the wind, and about 1,000 W out at an overall coefficient of 0.3.
So the rating is plausible for this rotor, but only in a wind that most sites see for a small fraction of the year. Rated power without the rated wind speed and the power curve tells you very little. When comparing turbines, compare rotor area and the annual energy at your average wind speed instead.
For energy estimates, do not just put the average wind speed into the cube law. Because of the cube, the windy hours contribute far more than the calm hours take away, so the mean power is higher than the power at the mean speed. For a typical (Rayleigh) wind distribution the factor is about 1.9.
Example: at a site averaging 5 m/s, the power in the wind at exactly 5 m/s is 0.5 × 1.225 × 3.14 × 125 = 240 W, and the long-term mean is about 1.9 × 240 = 460 W in the wind. At an overall coefficient of 0.3 that is an average output near 140 W, or roughly 1,200 kWh per year, an average of 14 percent of the nameplate. Expect somewhat less in practice, because nothing is produced below the cut-in speed.
Height and siting usually matter more than the choice of turbine. Wind speed increases with height above ground, approximately as v2 = v1 × (h2 / h1)^α, with α about 0.14 over open flat terrain and larger over rough ground. Going from a 10 m to a 20 m tower in open terrain raises the speed by 2^0.14 = 1.10, and the power by 1.10³ = 1.34, a third more energy from the same turbine.
Turbulence is just as important. A rotor on a roof or close behind trees and buildings sees slow, swirling air that cuts output and wears the machine. The usual guideline is to have the rotor well above any obstacle within a wide radius, and to measure the wind at the proposed hub position before buying.
