In aerodynamics, 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 windmill of any mechanism 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 75–80% 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.
Concepts
Betz's law applies to all Newtonian fluids, including wind. If all of the energy coming from wind movement through a turbine were extracted as useful energy, the wind speed afterward would drop to zero. If the wind stopped moving at the exit of the turbine, then no more fresh wind could get in; it would be blocked. In order to keep the wind moving through the turbine, there has to be some wind movement, however small, on the other side with some wind speed greater than zero. Betz's law shows that as air flows through a certain area, and as wind speed slows from losing energy to extraction from a turbine, the airflow must distribute to a wider area. As a result, geometry limits any turbine efficiency to a maximum of 59.3%.
Independent discoveries
British scientist Frederick W. Lanchester derived the same maximum in 1915. The leader of the Russian aerodynamic school, Nikolay Zhukowsky, also published the same result for an ideal wind turbine in 1920, the same year as Betz. It is thus an example of Stigler's law, which posits that no scientific discovery is named after its actual discoverer.
Proof
The Betz Limit is the maximum possible energy that may be derived by means of an infinitely thin rotor from a fluid flowing at a certain speed.
In order to calculate the maximum theoretical efficiency of a thin rotor (of, for example, a windmill) one imagines it to be replaced by a disc that withdraws energy from the fluid passing through it. At a certain distance behind this disc the fluid that has passed through flows with a reduced velocity.
Assumptions
The disk does not possess a hub and is ideal (so if it were conceptualized as a rotor it might have an infinite number of blades, which have no drag). Any non-idealities are assumed to only lower this idealized value making it an upper bound. As an effectively 1-D model, the flow into and out of the disk is axial, and all velocities are transversely uniform. This is a control-volume analysis, and to construct a solution, the control volume must contain all flow going in and out, failure to account for that flow would violate the conservation equations. The flow is non-compressible. Density remains constant, and there is no heat transfer. Uniform pressure is applied to the disk (again a 1-D model, so no radial dependence on pressure)
Application of conservation of mass (continuity equation)
Applying conservation of mass to this control volume, the mass flow rate (the mass of fluid flowing per unit time) is given by
where v1 is the speed in the front of the rotor, v2 is the speed downstream of the rotor, v is the speed at the fluid power device, ρ is the fluid density, the area of the turbine is given by , and and are the areas of the fluid before and after reaching the turbine.
So the density times the area and speed should be equal in each of the three regions: before, while going through the turbine and afterward.
The force exerted on the wind by the rotor is the mass of air multiplied by its acceleration. In terms of the density, surface area and velocities, this can be written as
Power and work
The work done by the force may be written incrementally as
and the power (rate of work done) of the wind is
Now substituting the force F computed above into the power equation will yield the power extracted from the wind:
However, power can be computed another way, by using the kinetic energy. Applying the conservation of energy equation to the control volume yields
Looking back at the continuity equation, a substitution for the mass flow rate yields