Betz limit
The theoretical maximum fraction of kinetic energy an open rotor can extract from the wind: 16/27, or about 59.3%.
Derived by Albert Betz in 1919 (and independently by Lanchester and Joukowsky), the limit follows from one-dimensional momentum theory applied to an ideal actuator disc. If a rotor slowed the air to zero it would block the flow; if it did not slow it at all it would extract nothing. The optimum lies in between: the wind downstream slows to one third of the free-stream speed, and the speed at the rotor plane is two thirds.
Here ρ is air density (about 1.225 kg/m³ at sea level, 15°C), A the swept area and v the wind speed. Cp is the power coefficient.
Real turbines fall short because of blade tip losses, wake rotation, profile drag and a finite number of blades. Modern three-bladed rotors reach aerodynamic Cp of roughly 0.48–0.52 at their design tip-speed ratio; after gearbox, generator and converter losses, the electrical Cp is typically 0.42–0.47.
The Betz limit applies to open (unducted) rotors in free flow. Ducted or augmented turbines can exceed 59% when referenced to rotor area, but not when referenced to the full duct exit area, so they have not proved economic. Analogous limits apply to tidal stream turbines, where blockage in narrow channels can raise the effective ceiling.
Related terms
All terms →- Power curve
The relationship between hub-height wind speed and a turbine's electrical output, used with a wind speed distribution to estimate annual energy production.
- Swept area
The area of the circle traced by a turbine's rotating blades; wind turbine power output is directly proportional to it, and it grows with the square of rotor diameter.
- Tip-speed ratio
The ratio of the linear speed of a turbine blade tip to the undisturbed wind speed; three-bladed utility turbines are designed for values of about 7 to 10.
- Wind power density
The mean power available in the wind per square metre of area perpendicular to the flow, in W/m², the standard measure of wind resource quality.
- Tidal stream power
Electricity generated by underwater turbines placed in fast tidal currents, analogous to wind turbines but driven by water about 800 times denser than air.