Betz's law
Practical utility-scale wind turbines achieve at peak 75–80% of the Betz limit.
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.
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.
[4] It is thus an example of Stigler's law, which posits that no scientific discovery is named after its actual discoverer.
The Betz Limit is the maximum possible energy that can be extracted by an infinitely thin rotor from a fluid flowing at a certain speed.
[5] In order to calculate the maximum theoretical efficiency of a thin rotor (of, for example, a wind turbine), one imagines it to be replaced by a disc that removes energy from the fluid passing through it.
The force exerted on the wind by the rotor is the mass of air multiplied by its acceleration:
Both of these expressions for power are valid; one was derived by examining the incremental work, and the other by the conservation of energy.
Thus no energy can be extracted other than at the front and back of the interaction region, fixing the airspeed of the actuator disk to be the average.
[6] This constant velocity effect is distinct from the radial kinetic energy loss that is also ignored.
[citation needed] It has a maximum value CP max = 16/27 = 0.593 (or 59.3%; however, coefficients of performance are usually expressed as a decimal, not a percentage).
Modern large wind turbines achieve peak values for CP in the range of 0.45 to 0.50,[2][full citation needed] about 75–85% of the theoretically possible maximum.
This is a correct calculation, but it only considers the incoming air which eventually travels through the rotor.
Although it is often touted (e.g.[10][11]) as the definitive upper bound on energy extraction by any possible wind turbine, it is not.
Despite the misleading title of his article,[12] Betz (nor Lanchester) never made such an unconditional claim.
[13][14] The second actuator disk could be, but need not be, in the far field wind zone (parallel streamline) for this consideration to hold.
[6] The reason for this surprising exception to a law based solely on energy and flux conservation laws lurks in the seemingly modest assumption of transverse uniformity of the axial wind profile within the stream lines.
[6] Mathematically, the derivation for a single actuator disk implicitly embeds the assumption that the wind does not change velocity as it transits the "infinitely thin" actuator; in contrast, in the dual actuator hybrid, the wind does change velocity as it transits, invalidating the derivation's key step requiring constant velocity.
[16] Most real wind turbines are aerodynamically "thin" making them approximate the assumptions of Betz law.
An increase in system efficiency is required to bring down the cost of electrical power production.
The assumptions of the Betz derivation[17] impose some physical restrictions on the nature of wind turbines it applies to (identical inlet/outlet velocity for example).
But beyond those assumptions, the Betz limit has no dependence on the internal mechanics of the wind extraction system, therefore S may take any form provided that the flow travels from the entrance to the control volume to the exit, and the control volume has uniform entry and exit velocities.
Any extraneous effects can only decrease the performance of the system (usually a turbine) since this analysis was idealized to disregard friction.
Any non-ideal effects would detract from the energy available in the incoming fluid, lowering the overall efficiency.
Some manufacturers and inventors have made claims of exceeding the limit by using nozzles and other wind diversion devices, usually by misrepresenting the Betz limit and calculating only the rotor area and not the total input of air contributing to the wind energy extracted from the system.
The Betz limit has no relevance when calculating turbine efficiency in a mobile application such as a wind-powered vehicle, as here the efficiency could theoretically approach 100% minus blade losses if the fluid flow through the turbine disc (or equivalent) were only retarded imperceptibly.
As this would require an infinitely large structure, practical devices rarely achieve 90% or over.
[citation needed] In 1934 H. Glauert derived the expression for turbine efficiency, when the angular component of velocity is taken into account, by applying an energy balance across the rotor plane.
In 2001, Gorban, Gorlov and Silantyev introduced an exactly solvable model (GGS), that considers non-uniform pressure distribution and curvilinear flow across the turbine plane (issues not included in the Betz approach).
[19] Computed optimal efficiency is, typically, between the Betz limit and the GGS solution.
