🌬️Wind Power Calculator - Betz Limit, Turbine Power Curve, Annual Energy

How to use Wind Power Calculator - Betz Limit & Turbine Energy

1. 1Set altitude and temperature for the air density, then adjust wind speed, rotor diameter and induction factor a to watch the streamtube and the Betz derivation
2. 2On the turbine tab choose a preset or edit rated power, diameter, cut-in, rated and cut-out speeds and tip-speed ratio
3. 3On the annual energy tab enter the mean wind speed at the measurement height, the terrain and the Weibull shape factor k
4. 4Read the hub-height wind, annual energy, capacity factor and the wind histogram, or look up a speed in the Beaufort table

FAQ

Where does the 59.3% Betz limit come from?

Treat the rotor as a thin disk that creates a pressure jump. Mass and momentum conservation give the thrust T = ṁ(v − v_w), and Bernoulli applied upstream and downstream gives the same thrust, so the speed at the disk is exactly the average of the upstream and wake speeds. With a the fractional slowdown at the disk, the power coefficient is Cp = 4a(1−a)², which peaks at a = 1/3 with Cp = 16/27 ≈ 59.3%. Stopping the air completely would block the flow, so you would extract nothing.

If the wind speed doubles, do I really get 8 times the power?

The power in the wind scales with v³, so between cut-in and rated speed a turbine's output rises roughly with the cube of the speed. At rated speed the blades pitch to cap the output, and above the cut-out speed (often 25 m/s) the turbine shuts down. So annual energy is very sensitive to the mean wind but far from 8×: raising the annual mean from 6 to 7 m/s typically adds 30–40% for the same turbine.

What is a capacity factor, and what do wind farms achieve?

It is the actual annual energy divided by what the turbine would make running at rated power for all 8760 hours of the year. Onshore wind typically reaches 20–35% and good offshore sites 40–50%. It depends on the wind resource and the design: low-wind turbines with a large rotor for their rating have higher capacity factors. The tool's estimate excludes wake and downtime losses unless you enter them under other losses.

Why are wind turbines getting taller with longer blades?

Ground friction slows the wind near the surface, and speed rises with height roughly as v ∝ h^α, with α ≈ 0.14 over open land and up to 0.3 in suburbs. Raising the hub from 10 m to 100 m can increase the mean speed by over 40% and nearly double the power density. The swept area grows with the square of the diameter, so doubling the rotor quadruples the area, which is why offshore rotors now exceed 200 m on 150 m hubs.

How should I read the Weibull k and c parameters?

A Weibull distribution describes how often each wind speed occurs over a year. The scale c is proportional to the mean speed (v̄ = cΓ(1+1/k)), and the shape k describes how steady the wind is: k ≈ 2 is the common Rayleigh case, steady trade winds and offshore sites reach 2.5–3, and gusty mountain sites can fall below 1.8. For the same mean speed a smaller k means more strong-wind hours and a higher power density.