Wind Turbine Power – Betz Limit and a Day of Wind
PhysicsPhysics & the EnvironmentAges 15–16
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Sign in to playA wind turbine captures P = ½ρAv³·Cp through its swept area A = πR². Adjust the blade length, the day's mean wind speed, the power coefficient Cp (never above the Betz limit of 59.3 %), the cut-in, rated and cut-out speeds and the air density, then follow the power curve, the wind over 24 hours and the day's energy compared with a household's daily use.
Lesson: Renewable energy: wind power, turbine output and the Betz limit
What it shows
The wind passing through a turbine's swept area A = πR² each second carries kinetic energy, giving a wind power of ½ρAv³: doubling the blade length gives four times the power, and doubling the wind speed gives eight times. A rotor can extract only a fraction Cp of it; Betz showed that Cp can never exceed 16/27 ≈ 59.3 %. Real turbines start at a cut-in speed, hold their rated power above the rated speed and shut down at the cut-out speed to protect themselves. The model keeps Cp constant and uses simulated wind data for the day.
How to use
Press Play to run the clock through the day and watch the red dot move along the power curve. Change Blade R and Mean wind to see how strongly the output depends on each, and compare the day's energy with the value calculated from the mean speed. Then adjust Cp, Rated v, Cut-in v and Cut-out v, choose a higher altitude under Air, and press New day for different weather.
Parameters you can change
- Blade length (rotor radius) 1–80 m
- Mean wind speed of the day 2–14 m/s
- Power coefficient Cp 0.1–0.59
- Rated wind speed 8–16 m/s
- Cut-in wind speed 2–5 m/s
- Cut-out wind speed 18–30 m/s
- Air density Sea level, 15 °C (1.225 kg/m³), 1000 m altitude (1.112 kg/m³), 2000 m altitude (1.007 kg/m³)
- Household electricity use per day 2–40 kWh
Questions to explore
- If the blade length doubles, by what factor does the power increase? What if the wind speed doubles?
- Why does using the mean wind speed give the wrong energy for the day?
- Why must a turbine stop in very strong winds, and why can no turbine exceed 59.3 %?
