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How It Works
Wind turbine power depends on air density, rotor swept area, the coefficient of performance (Cp), wind speed, and the efficiencies of the generator and gearbox. Wind speed is cubed in the equation, so doubling it yields eight times the power. The Betz limit caps Cp at 0.593, though real turbines typically achieve 0.30–0.45.
Example Problem
A turbine has 40-m blades (ρ = 1.225, Cp = 0.35, Ng = 0.80, Nb = 0.95) in 12 m/s wind. What power does it generate?
- A = π × 40² = 5,027 m²
- P = 0.5 × 1.225 × 5,027 × 0.35 × 12³ × 0.80 × 0.95
- P ≈ 1,462,000 W (1.46 MW)
Frequently Asked Questions
What is the Betz limit?
The Betz limit (59.3%) is the theoretical maximum fraction of wind energy any turbine can capture. It exists because the turbine must slow the wind to extract energy, but cannot stop it entirely or air would stop flowing through the rotor.
Why does wind speed matter so much?
Power scales with V³. A site with 8 m/s average wind produces 3.4 times more energy than one with 6 m/s. This is why hub height and site selection are the most important design decisions.
What is a typical capacity factor?
Onshore turbines achieve 25–35% capacity factor; offshore turbines reach 35–50%. This means a 2 MW turbine at 30% capacity factor produces an average of 600 kW over a year.
What are cut-in and cut-out wind speeds?
Cut-in speed (typically 3–4 m/s) is the minimum wind needed to generate power. Cut-out speed (typically 25 m/s) is the maximum before the turbine shuts down to prevent structural damage.
Related Calculators
- Horsepower Calculator — convert between power units.
- Engine Equations Calculator — volumetric efficiency and displacement.
- Density Calculator — find air density at different altitudes for power output calculations.
- Wing Lift Calculator — aerodynamic lift principles shared with turbine blade design.
- Power Unit Converter — convert turbine output between watts, kilowatts, and horsepower.
Reference: Betz, A. 1919. “Das Maximum der theoretisch möglichen Ausnützung des Windes durch Windmotoren.” Zeitschrift für das gesamte Turbinenwesen.