Wind Turbine Calculator

The power in a moving airstream is the kinetic energy of the mass flowing through the rotor disc each second, which works out to one half times air density times swept area times velocity cubed. Two terms in that expression dominate design. Swept area grows with the square of rotor diameter, so a 120 metre rotor intercepts four times the airflow of a 60 metre one. Velocity enters as a cube, so a site averaging 8 metres per second holds roughly 2.4 times the power of one averaging 6. Air density is the quiet third term: the standard 1.225 kilograms per cubic metre applies at sea level and 15 degrees C, but a site at 1,500 metres elevation sees about 1.06, roughly a 13 percent power penalty that must be accounted for in turbine selection. No rotor captures all of that power. The power coefficient Cp is capped at 16 over 27, about 0.593, because extracting all kinetic energy would require the downstream air to stop and block the flow. Modern three-bladed machines reach Cp values of 0.45 to 0.50 near their optimal tip-speed ratio of roughly 6 to 8, which is 75 to 85 percent of the theoretical ceiling. The cubic law also explains why this instantaneous calculation cannot be scaled directly to annual energy: because power varies as the cube of a fluctuating speed, using the mean speed underestimates true annual output at most sites. Real turbines also cut in near 3 to 4 metres per second, hold constant rated power above roughly 12 to 15, and shut down near 25 to protect the drivetrain.

Windmills ground grain and pumped water for centuries, but electricity generation began with Charles Brush, who built a 17 metre, 144-blade turbine in Cleveland, Ohio in 1888 to charge batteries in his basement. Poul la Cour, a Danish scientist, began systematic experiments at Askov in 1891 and recognized that fewer, faster blades with proper aerodynamic profiles outperformed the traditional many-bladed rotor, establishing the research tradition that would later define Danish turbine design. The theoretical limit was derived independently by three people. Frederick Lanchester published relevant work in Britain in 1915, Albert Betz published the result in Germany in 1919 and again in his 1926 book on wind energy, and Nikolai Zhukovsky presented equivalent findings in Russia around 1920, which is why it is sometimes called the Lanchester-Betz-Joukowsky limit. Scaling to utility size took decades: the 1.25 megawatt Smith-Putnam turbine on Grandpa Knob in Vermont ran intermittently from 1941 until a blade failed in 1945, and Johannes Juul built the 200 kilowatt Gedser turbine in Denmark in 1957, whose three-bladed upwind layout with stall regulation and an induction generator became the template for the modern industry. Commercial development accelerated after the 1970s oil shocks, and rotor diameters have since grown from about 15 metres to more than 220 metres on the largest offshore machines.