Betz Limit
Maximum fraction of wind energy any turbine can capture.
Formula
Variables
Example
No turbine can exceed ~59.3% capture
How it works
The Betz limit gives the theoretical upper bound on how much kinetic power in a steady wind stream can be extracted by an ideal turbine. It works because a turbine cannot stop the air completely: if the wind behind the rotor were reduced to zero, no more air could flow through the blades, so the optimal design leaves the air moving while still slowing it down as much as possible. A momentum-and-energy analysis of the air passing through an ideal actuator disk shows the maximum occurs when the downstream wind speed is one-third of the upstream speed, leading to a maximum power coefficient of 16/27, or about 59.3%.
When to use it
Use this limit when analyzing wind turbine performance, comparing measured power coefficients to the theoretical maximum, or checking whether a design claim is physically plausible. It is especially useful in wind-energy engineering and fluid dynamics problems involving idealized energy extraction from a moving air stream.
Worked example, step by step
Suppose a wind turbine has a swept area of 120 m^2 and the wind speed is 10 m/s. Step 1: Compute the wind power available in the flow: P_wind = 1/2 ρ A v^3 P_wind = 1/2 × 1.2 × 120 × 10^3 P_wind = 0.6 × 120 × 1000 P_wind = 72,000 W Step 2: Apply the Betz limit to find the maximum extractable power: P_max = C_p,max × P_wind P_max = (16/27) × 72,000 P_max ≈ 0.5926 × 72,000 P_max ≈ 42,667 W Step 3: State the result: The largest possible ideal output is about 4.27 × 10^4 W, or 42.7 kW.
Common mistakes to avoid
- Treating 59.3% as the efficiency of every real turbine instead of a theoretical upper bound
- Confusing the power coefficient C_p with the overall electrical efficiency of the turbine and generator
- Using the Betz limit without first calculating the available wind power from air density, rotor area, and wind speed
Did You Know?
Albert Betz proved in 1919 that slowing the wind too much blocks the flow — capping efficiency at 59.3%.
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