Wind Power Density, the Betz Limit, and Rotor Output

Wind power goes as the CUBE of speed, which is why siting beats equipment, and the Betz limit then caps what any rotor can take at 59.3%. A 380 ft rotor sweeps 113,411 sq ft, and in a 20 mph wind at sea-level density that disc has 4,614 kW passing through it -- 40.7 W per square foot. Betz caps extraction at 2,734 kW and a real coefficient of 0.45 gives 2,076 kW, which is 76% of the Betz ceiling and about as good as machines get. THE CUBE LAW IS THE FACT MOST OFTEN UNDERWEIGHTED. The same rotor at 25 mph makes 1.953 times the power, and at 10 mph one eighth of it. A site averaging 20% more wind has 73% more energy in it, which is why met tower data and hub height matter enormously and why moving a machine a short distance to better exposure can beat any equipment choice. BETZ IS NOT A DESIGN TARGET, IT IS A CEILING. A rotor that extracted all the wind energy would have to stop the air completely, and stopped air cannot get out of the way of the air behind it; the optimum slows the flow to a third of its upstream speed and captures 16/27. No rotor of any design beats it, so a machine advertised as extracting 65% is not a breakthrough, it is an error or a fraud -- and being able to say that in one line is worth having in front of a small-wind buyer. THIS IS NOT AN ENERGY ESTIMATE. Annual output depends on the distribution of speeds over the year, not on any single speed, and because power is cubic the average of the cubes is much larger than the cube of the average, so a mean wind speed put into this relation understates energy substantially. It does not account for cut-in and cut-out, rated power and clipping, availability, wake losses in an array, blade soiling and icing, or electrical losses, and air density must be corrected for site elevation and temperature. The manufacturer warranted power curve and a site-specific energy assessment govern any production estimate.

P = 0.5 x rho x A x v cubed, with the mass density taken as the entered weight density divided by g = 32.174 ft/s squared and A the swept area pi/4 x diameter squared; the Betz limit caps the extractable fraction at 16/27 = 0.593.

The free-stream power relation and the Betz limit of 16/27 by name. Real machines reach a power coefficient of 0.35 to 0.48 including drivetrain losses. The manufacturer's warranted power curve and a site-specific energy assessment govern any production estimate.

Arithmetic on the user's own rotor diameter, wind speed, and density; the Betz coefficient is a first-principles result, not a proprietary figure.

Estimate. AHJ and licensed professional govern.

Field names used by the API: wind_speed_mph, air_density_pcf, rotor_diameter_ft, power_coefficient, alt_wind_speed_mph, swept_area_ft2, power_in_wind_kw, power_density_w_ft2, betz_power_kw, extracted_power_kw, alt_ratio

• Power is cubic in speed 20% more wind is 73% more energy; siting beats equipmentfirst principles
• Betz is a ceiling, not a target 16/27 follows from conservation of mass and momentum aloneBetz's law
• A single speed is not an energy estimate the distribution is what determines annual outputIEC 61400-12