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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 wind turbine 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 can be extracted by 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 wind turbine), one imagines it to be replaced by a disc that removes energy from the fluid passing through it. At a certain distance behind this disc, the fluid that has passed through the disc has a reduced, but nonzero, velocity. Assumptions The disk is ideal, with no hub (so that 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 lower the efficiency. 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; the control volume must contain all incoming and outgoing flow in order to use the conservation equations. The flow is non-compressible. Density is constant, and there is no heat transfer. Uniform pressure is applied to the disk. (No radial dependence on pressure in this 1-D model.) Application of conservation of mass (continuity equation) Applying conservation of mass to the control volume, the mass flow rate (the mass of fluid flowing per unit time) is 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, S {\displaystyle S} is the area of the turbine, and A 1 {\displaystyle A_{1}} and A 2 {\displaystyle A_{2}} are the areas of the fluid before and after the turbine (the inlet and outlet of the control volume). The density times the area and speed must be equal in each of the three regions: before the turbine, while going through the turbine, and past the turbine. The force exerted on the wind by the rotor is the mass of air multiplied by its acceleration: Power and work The incremental work done by the force may be written and the power (rate of work done) of the wind is Substituting the force F computed above into the power equation yields 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 Substituting the mass flow rate from the continuity equation yields Both of these expressions for power are valid; one was derived by examining the incremental work, and the other by the conservation of energy. Equating these two expressions yields The density can't be zero for any v and S, so or The constant wind velocity across the rotor may be taken as the average of the upstream and downstream velocities. This is arguably the most counter-intuitive stage of the derivation of Betz's law. It is a direct consequence of the "axial flow" assumption, which disallows any radial mass flow in the actuator disk region. With no mass escape and a constant diameter in the actuator region, the air speed cannot change in the interaction region. 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. (Removing that restriction may allow higher performance than Betz law allows, but other radial effects must also be considered. This constant velocity effect is distinct from the radial kinetic energy loss that is also ignored.) Betz's law and coefficient of performance Returning to the previous expression for power based on kinetic energy: P = 1 2 m ˙ ( v 1 2 − v 2 2 ) = 1 2 ρ S v .... Discover the G H Betz popular books. Find the top 100 most popular G H Betz books.