Wind Power - Betz's law

Description

Wind power is the conversion of wind energy into a useful form of energy, such as using wind turbines to produce electrical power, windmills for mechanical power, windpumps for water pumping or drainage, or sails to propel ships.

This equation calculates the power produced by a wind turbine.

Wind power in an open air stream is thus proportional to the third power of the wind speed; the available power increases eightfold when the wind speed doubles. Wind turbines for grid electricity therefore need to be especially efficient at greater wind speeds.

Betz’s law indicates the maximum power that can be extracted from the wind, independent of the design of a wind turbine in open flow. It was published in 1919, by the German physicist Albert Betz. The law is derived from the principles of conservation of mass and momentum of the air stream flowing through an idealized “actuator disk” that extracts energy from the wind stream. According to Betz’s law, no turbine can capture more than 16/27 (59.3%) of the kinetic energy in wind. The factor 16/27 (0.593) is known as Betz’s coefficient. Practical utility-scale wind turbines achieve at peak 75% to 80% of the Betz limit.

The Betz limit is based on an open disk actuator. If a diffuser is used to collect additional wind flow and direct it through the turbine, more energy can be extracted, but the limit still applies to the cross-section of the entire structure.

Modern large wind turbines achieve peak values for Cp in the range of 0.45 to 0.50, about 75% to 85% of the theoretically possible maximum. In high wind speed where the turbine is operating at its rated power the turbine rotates (pitches) its blades to lower Cp to protect itself from damage. The power in the wind increases by a factor of 8 from 12.5 to 25 m/s, so Cp must fall accordingly, getting as low as 0.06 for winds of 25 m/s.

Related formulas

Variables

Ppower (W)
Cppower coefficient (Betz limit - max value : 0.593) (dimensionless)
Arotor area (m2)
ρdensity of air (kg/m3)
vwind speed (m/s)