Betz's law

Simple cartoon of two air molecules shows why wind turbines cannot actually run at 100% efficiency.

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%.

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 did.[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 places an upper bound on the annual energy that can be extracted at a site. Even if a hypothetical wind blew consistently for a full year, no more than the Betz limit of the energy contained in that year's wind could be extracted.

Essentially increasing system economic efficiency results from increased production per unit, measured per square meter of vane exposure. An increase in system efficiency is required to bring down the cost of electrical power production. Efficiency increases may be the result of engineering of the wind capture devices, such as the configuration and dynamics of wind turbines, that may increase the power generation from these systems within the Betz limit. System efficiency increases in power application, transmission or storage may also contribute to a lower cost of power per unit.

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.[5]

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.[5]

Returning to the previous expression for power based on kinetic energy:

${\displaystyle {\dot {E}}={\tfrac {1}{2}}{\dot {m}}(v_{1}^{2}-v_{2}^{2})}$

${\displaystyle ={\tfrac {1}{2}}\rho Sv(v_{1}^{2}-v_{2}^{2})}$

${\displaystyle ={\tfrac {1}{4}}\rho S(v_{1}+v_{2})(v_{1}^{2}-v_{2}^{2})}$

${\displaystyle ={\tfrac {1}{4}}\rho Sv_{1}^{3}\left(1-\left({\frac {v_{2}}{v_{1}}}\right)^{2}+\left({\frac {v_{2}}{v_{1}}}\right)-\left({\frac {v_{2}}{v_{1}}}\right)^{3}\right).}$

The horizontal axis reflects the ratio v₂/v₁, the vertical axis is the "power coefficient " C_p.

By differentiating ${\displaystyle {\dot {E}}}$ with respect to ${\displaystyle {\tfrac {v_{2}}{v_{1}}}}$ for a given fluid speed v₁ and a given area S, one finds the maximum or minimum value for ${\displaystyle {\dot {E}}}$. The result is that ${\displaystyle {\dot {E}}}$ reaches maximum value when ${\displaystyle {\tfrac {v_{2}}{v_{1}}}={\tfrac {1}{3}}}$.

Substituting this value results in

${\displaystyle P_{\text{max}}={\tfrac {16}{27}}\cdot {\tfrac {1}{2}}\rho Sv_{1}^{3}.}$

The power obtainable from a cylinder of fluid with cross-sectional area S and velocity v₁ is

${\displaystyle P=C_{\text{p}}\cdot {\tfrac {1}{2}}\rho Sv_{1}^{3}.}$

The reference power for the Betz efficiency calculation is the power in a moving fluid in a cylinder with cross-sectional area S and velocity v₁:

${\displaystyle P_{\text{wind}}={\tfrac {1}{2}}\rho Sv_{1}^{3}.}$

The "power coefficient"[6] C_p (= P/P_wind) is dimensionless ratio of the extractable power P to the kinetic power P_wind available in the undistributed stream. It has a maximum value C_p 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 C_p in the range of 0.45 to 0.50,[2] about 75–85% 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 C_p to protect itself from damage. The power in the wind increases by a factor of 8 from 12.5 to 25 m/s, so C_p must fall accordingly, getting as low as 0.06 for winds of 25 m/s.

Intuitively, the speed ratio of [V₂/V₁ = 0.333] between outgoing and incoming wind, leaving at about a third of the speed it came in, would imply higher losses of kinetic energy. But since a larger area is needed for slower-moving air, energy is conserved.

All energy entering the system is taken into consideration, and local "radial" kinetic energy can have no effect on the outcome, which is the final energy state of the air leaving the system, at a slower speed, larger area and accordingly its lower energy can be calculated.

The last step in calculating the Betz efficiency C_p is to divide the calculated power extracted from the flow by a reference power value. The Betz analysis uses for its power reference, reasonably, the power of air upstream moving at V₁ contained in a cylinder with the cross-sectional area S of the rotor.

The Betz limit has no dependence on the geometry 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.

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.[7] Due to the Glauert model, efficiency is below the Betz limit, and asymptotically approaches this limit when the tip speed ratio goes to infinity.

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).[8] They utilized and modified the Kirchhoff model,[9] which describes the turbulent wake behind the actuator as the "degenerated" flow and uses the Euler equation outside the degenerate area. The GGS model predicts that peak efficiency is achieved when the flow through the turbine is approximately 61% of the total flow which is very similar to the Betz result of 2/3 for a flow resulting in peak efficiency, but the GGS predicted that the peak efficiency itself is much smaller: 30.1%.

Recently, viscous computations based on computational fluid dynamics (CFD) were applied to wind turbine modeling and demonstrated satisfactory agreement with experiment.[10] Computed optimal efficiency is, typically, between the Betz limit and the GGS solution.

• The Betz limit - and the maximum efficiency for horizontal axis wind turbines
• Pierre Lecanu, Joel Breard, Dominique Mouazé. Betz limit applied to vertical axis wind turbine theory