- Wind Turbine Theory: Wind turbines extract power from the wind by converting kinetic energy as air passes through an imaginary duct.
- Power Definition: Power is defined as the change in kinetic energy per second as wind flows through the turbine.
- Mass Flow Rate: Mass flow rate is the quantity of air passing through the duct per second, calculated as ρVaA.
- Betz Coefficient: The Betz Coefficient is a theoretical limit indicating that only 59.25% of the wind’s kinetic energy can be converted into usable power.
- Betz Limit Derivation: The Betz Limit derivation shows that extracted power depends on air density, the swept area of the turbine blades, and the cube of wind velocity.
An ideal actuator-disc model represents a wind turbine as a rotor inside an imaginary streamtube. The undisturbed upstream wind speed is V1 and the downstream wake speed is V2. The rotor extracts energy by slowing the air, so the downstream speed is lower than the upstream speed. If mass m passes through the streamtube each second,
its inlet kinetic energy is:
Its outlet kinetic energy is:
The difference between these two energy rates is:
Because m is the mass passing each second, this kinetic-energy difference is the ideal power extracted from the wind.
Power is energy transferred per unit time, so the extracted power is:
The mass flow rate is constant through a steady streamtube. As the air slows, the streamtube cross-section changes so that the same mass passes each section per second.
At the rotor, let Va be air speed, A be swept area and ρ be air density. The mass flow rate is ρVaA.
Replacing m with ρVaA in equation (1) gives:
For the ideal one-dimensional actuator disc, the speed at the rotor is the average of the upstream and downstream speeds.
To find maximum extracted power, differentiate equation (3) with respect to V2 and set the result to zero. The optimum downstream speed is one-third of the upstream speed.
Betz Coefficient
The Betz Coefficient, also called the Betz limit, is the ideal maximum power coefficient of 16/27, which is about 0.5926 or 59.3%. It applies to an ideal, unconfined single actuator disc. The theory of wind turbine therefore does not predict that all available wind power can become shaft power. A real turbine captures less because of rotor aerodynamics plus bearing, drivetrain and generator losses.
Equation (4) shows that the available wind power follows these relationships when the other quantities remain fixed:
- Power is directly proportional to air density ρ. Denser air carries more kinetic energy through the same swept area at the same speed.
- Power is directly proportional to swept area. Since A = πR², longer blades increase the area intercepted by the rotor.
- Available wind power is proportional to wind speed3. Doubling wind speed gives eight times the available power at unchanged density and swept area, although a real turbine limits output above its rated speed.
