Issue #161, June / July 2014

Intermediate

- Intermediate

The third (and most important for siting wind turbines) variable is *wind speed*. You can see that it’s cubed in the equation, meaning small changes in wind speed yield larger changes in available power. A 26% increase in wind speed (from 10 to 12.6 mph) doubles the available energy, while a 20% drop (from 10 to 8 mph) cuts it in half. This is why it is critical to put a wind turbine on a tall tower where it can intercept strong wind.

As a wind turbine extracts kinetic energy from the wind, it does not consume air mass (only nuclear reactions consume mass), so it must be “consuming” the wind speed. In other words, the wind approaches the turbine at one speed and leaves at a lesser speed. This is how any wind turbine extracts energy from the wind—by slowing it down. The difference between the wind speed before and after it passes through the turbine defines the energy the turbine has extracted from the wind. This is the fundamental function of the wind turbine, and some turbines do it better than others.

It is not feasible to extract all of the power available in the wind—and no wind turbine can harness more energy than is available in the wind. Avoid any wind turbine that claims it can—no wind turbine can slow wind down to a speed of less than zero.

If a wind turbine were to extract all the available power in the wind—that is, slow the wind to a stop and capture all its power—we would say that turbine is 100% efficient. However, any wind turbine that did this would stop the wind, and then there would be no air movement from which to extract more power! An effective wind turbine must find a balance, slowing the wind enough to maximize power capture, yet still allow enough wind to pass through so it can keep capturing more.

Efficiency is defined as the ratio between the output power and the input power. For wind turbines, aerodynamic efficiency is referred to as the power coefficient, C_{p}, so the governing equation for wind turbine power output is:

**P _{output }= 1/2 × air density × area × wind speed^{3} × C_{p}**

We know C_{p} cannot be 100%, but what is the upper limit? In 1919, German scientist Albert Betz took the above governing equations for wind power, and used them to determine how much a theoretically “perfect” wind turbine could extract from the wind. His answer, which is referred to as Betz’ Law or the Betz limit, states that when the wind is slowed by two-thirds (wind speed out = 1/3 wind speed in), the wind turbine reaches its theoretical maximum possible efficiency, C_{P}-max., of 59.3%.

It is critical to note that Betz’ Law is derived from the governing equations, and not from any assumptions about wind turbine type. Therefore, the Betz limit is not restricted to any particular type or style of wind turbine, as some people mistakenly believe. The Betz limit is a physical limit that applies to *all* wind turbines.

In the real world, no wind turbine can ever even reach—let alone exceed—the Betz limit. But which turbines do the best, and how well do they do?

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Kenslow is actually correct. Betz totally ignores rotation in his calculations, and Glauert admits the calculations are wrong if rotation occurs. See page 194 of his most famous work "Airplane propellers", he proved nothing about wind turbines and was primarily focused on propeller design, only 17 out of 191 pages cover wind turbine design and they contain errors. Glauert died in 1934 before it was published and without proof reading it.

The differences between Betz' and Glauert's approaches explain the absence or inclusion of wake rotation in the calculations. Betz took the simplest approach of treating the entire turbine rotor as an ideal disc, often referred to as actuator disc theory (ADT). ADT considers the flow through the plane of the rotor as a whole. As such, it is both highly simplified and non-conservative since it treats the entire rotor area as having the same efficiency without considering real effects such as tip and hub losses, not to mention differences along the span of the blades.

Glauert uses the more realistic approach of dividing the blades into spanwise sections, often referred to as Blade Element Momentum (BEM) theory. This approach attempts to calculate the contribution of each spanwise position of each blade, omitting the hub (which has no airfoil) and including changes along the span due to different angular velocities.

Because ADT only considers flow through a plane, any rotational velocity would be irrelevant because the component of velocity that is not perpendicular to the plane would not pass through the plane. BEM can account for rotational velocity because the rotational component changes the inflow velocity and angle of attack for individual blade elements. BEM is certainly the more accurate method to determine propeller performance and is commonly used in wind turbine aero-elastic design codes today.

Having said all that, Betz' approach still represents an ideal case where all the flow through the disc (note that Betz' theorem is not limited to propellers, nor even circular discs or in fact even to air) is converted to useful power. If such an ideal disc could be created (it can't, but the propeller has proven to come closest), it could approach the Betz Limit for efficiency. A propeller may be influenced by rotational velocity components that Betz ignores, but any gains can never make up for the real-world losses of a real propeller (tip, hub and frictional losses), let alone exceed the ideal theoretical limit proven by Betz.

In summary, BEM is the more accurate approach for wind turbine design because it more accurately reflects reality. But reality does not solely incur benefits; the Betz limit still applies.

I'd be happy to write more on this topic Ben, though it sounds like you are asking for a summary of this very article. Please clarify if there is something more or different that you feel was not covered and maybe HP will consider it.

Yes, more precisely, Betz, Glauert, and modern Blade Element Momentum Theory all rely on the flawed 19th century Froude's actuator disc theory. This is part of the problem, Froude's ADT equations are in fact invalid and diverge from that of real airflow as axial induction factors approach 0.5. Also most all methods including Glauert's are based on simplifying assumptions which either remove or improperly account for rotational terms within the energy equation. That said, my only point is to emphasize that these are strictly very outdated theorems, not laws nor physical limits, and to accept otherwise serves only to stifle innovation.

It is important to remember that the theorems are just models, varying in complexity, and must be viewed as such. All engineering uses models to some extent, with the critical aspect being that the assumptions on which the model is based are applicable to the real situation being modeled. Therefore I would not categorize the ADT as "flawed" or even "outdated." It is a very good tool for a generalized performance model of energy extraction from a fluid flow. It is not so good for wind turbine blade design. I also would not blame the theorems for stifling innovation. If one chooses to use use the wrong tools for design, it is not the tools fault.

Finally, the ADT is in fact quite valid for what it is intended as it is based on fundamental fluid physics. As the article states, the Betz proof is technology agnostic and simply defines the amount that a fluid must be slowed (by 2/3rds) to maximize power extracted from the flow. It is quite valid for this purpose and is applicable to wind turbines regardless of any other - better - tools used to design them.

From your refernced source, Martin Hansen's Aerodynamics of Wind Turbines, 2nd ed. page 41 "It is possible to exceed the Betz limit by placing the wind turbine in a diffuser."

Hansen makes the mistake (in my view) of ignoring the area intercepted by the diffuser. By ignoring the diffuser area one can claim that the turbine could exceed the Betz limit, but this is technically incorrect, and - as Hansen notes - the diffuser has to be accounted for both structurally and economically (so why not aerodynamically?). Furthermore, as Hansen also notes, this concept of an augmented rotor "beating Betz" by (erroneously) ignoring the diffuser area has never been demonstrated on a full-size turbine despite many attempts costing many millions of dollars.

Betz limit is not a true limit. It doesn't take rotational velocities into account. That's why it was immediately discounted by the engineering community when it was claimed.

Michael,

I am not aware of Betz' theory being discounted by the engineering community when it was proposed in 1919, so please share any evidence you have in that regard. As for it not being a "true limit" you are correct in so far as it is only a theoretical limit and can never be truly achieved with a real wind turbine.

I am unsure of what you refer to as "rotational velocities," but the effect of wake rotation has been considered in regards to wind turbine rotor efficiency. In 1935, British aerodynamicist Hermann Glauert investigated the effect of wake rotation on theoretical rotor efficiency. His conclusion reinforced Betz' result, proving that the theoretical efficiency approaches 16/27 as the tip speed ratio of the rotor approaches infinity. [ref: Hansen M. O. L. (2015). Aerodynamics of Wind Turbines]

I don't think your question was directed to me Michael, but you raise an interesting point. A propeller rotor induces a rotational velocity as well as an axial velocity on the inflow. In general this provides no benefit but rather a loss because that induced rotation is energy imparted by the rotor into the flow (assuming no rotation originally exists in the inflow). There is potential that inflow with rotation could provide benefit to a rotor that induces opposite rotation, but that would require a specific condition that is unlikely in reality, and the benefit would not be great since the rotational induced velocity is a much smaller effect than axial induced velocity from which the vast majority of power is extracted from the flow. And in the end, ideal rotational velocity on a real rotor cannot provide enough benefit to exceed the ideal condition described by Betz in the theoretical world prescribed by the laws of physics.