The Betz limit and the Maximum Efficiency for Horizontal Axis Wind Turbines

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In physics, the Betz limit is the maximum power that can be extracted from a wind turbine. This limit is based on the Betz equation, which states that no more than 59.3% of the kinetic energy in a flow of wind can be converted into electrical energy.

The Betz limit is important because it helps us understand the maximum amount of power we can extract from a domestic wind generator. While there are ways to improve on this limit, it serves as a good starting point for designing turbines that produce renewable energy.

(Original article and sketches by Dr. Les Bardbury / PelaFlow Consulting)

Can the Betz Limit be exceeded and does it apply to vertical axis wind turbines?

The simplest model of a wind turbine is the so-called actuator disc model where the turbine is replaced by a circular disc through which the airstream flows with a velocity Ut and across which there is a pressure drop from P1 to P2 as shown in the sketch.

At the outset, it is important to stress that the actuator disc theory is useful (as will be shown) in discussing the overall efficiencies of turbines but it does not help at all with how to design the turbine blades to achieve the desired performance.

turbine disc graph

The power developed by the wind turbine is:

power equation

where At is the turbine disc area. Volume flow continuity gives:

continuity equation

From momentum conservation, the force exerted on the turbine is equal to the momentum change between the flow far upstream of the disc to the flow far downstream of the disc. Thus:

momentum equation

The final basic equations are Bernoulli's equation applied upstream and downstream of the actuator disc:

Bernoulii upstream

Bernoulli downstream

where P∞ is the ambient pressure in the flow both far upstream and far downstream of the actuator disc.

From equations (4a),(4b), (3) and (2)

Equation 5

whence

Equation 6

i.e. the velocity through the actuator disc is the mean of the upstream and downstream velocities in the stream tube.

Finally, from equations (1), (5), and (3), the efficiency is given by:

Efficiency formula

The figure below shows the variation of efficiency (often referred to as the power coefficient, cp) with the ratio of downstream to upstream velocity. By differentiating equation (7), it is easy to show that the maximum turbine efficiency occurs when Ud/Uu=1/3 (i.e. when Ad/Au=3).

The efficiency is then η=16/27 ≈ 59%. This is the maximum achievable efficiency of a wind turbine and is known as the Betz limit - after Albert Betz who published this result in 1920. There are assumptions in the above analysis such as the neglect of radial flow at the actuator disc but these have only a small effect on the final limiting result.

Efficiency curve

The point to note here is that as you reduce the downstream velocity in the expectation of increasing the power extracted from the wind, the area of the upstream stream tube that passes through the turbine reduces in size.

In the limit, as the downstream velocity is reduced to zero, the area of the upstream stream tube that passes through the turbine is just half the turbine area and the efficiency is thus 50%.

Can the Betz limit be exceeded for horizontal axis wind turbines?

It is important to note that the equations leading up to the Betz limit represent an overall momentum balance argument and therefore the argument still applies to any horizontal axis 'device' that replaces the actuator disc in the above derivation.

The only question is what is the effective diameter of the stream tube that is influenced by the device?

There have been numerous devices that claim to improve the efficiency of a wind turbine and the shrouded turbine shown on the right is rather typical of these designs.

In these 'shrouded' turbines, the general idea seems to be to use the shroud to create a low-pressure region downstream of the turbine and thus draw more air through the turbine.

Generally, with these designs, there is little in the way of experimental data to support the efficiency claims but an exception to this seems to be some experimental and theoretical work carried out mainly at Kyushu University in Japan. The reference is given below. The figure on the right shows this design with a layout sketch.

In their report, the authors measure the turbine efficiency and, from a graph, they show a peak value of about 29% for the turbine on its own and a figure of about 110% with the shroud in place. In both cases, the efficiency or power factor is based on the swept rotor area.

However, as can be seen from the sketch, the ratio of the shroud diameter to the rotor diameter is about 2.53 (i.e. 1013/400) and, if we base the efficiency of the shrouded turbine on the shroud cross-sectional area, the peak efficiency falls to 17%.

In other words, a straightforward turbine with the diameter of the shroud would perform better in terms of efficiency than the shrouded turbine.

The point here is that the Betz derivation still applies, but the stream tube's diameter influenced by the shrouded turbine is closer to the shroud diameter than the turbine diameter.

This seems a fairly obvious conclusion and emphasizes the point that there is no way of getting around the overall momentum balance between far upstream and far downstream in the derivation of the Betz limit.

Moreover, in the case of the shrouded turbine, the drag on the shroud contributes nothing to the turbine power.

Reference

  • Abea K., Nishidab M., Sakuraia A. et al (2005) Experimental and numerical investigations of flow fields behind a small wind turbine with a flanged diffuser.
  • Journal of Wind Engineering and Industrial Aerodynamics Volume 93, Issue 12, December 2005, Pages 951–970.

FloDesign-Ogin shrouded wind turbine

An example of a shrouded wind turbine that is being put forward as a practical design is one that was originally designed by a company called FloDesign based in Massachusetts.

A quite large demonstrator unit was put up in 2011 at Deer Island in Massachusetts as shown in the figure. It was a design rated at 100 kilowatts.

The astonishing thing about this design is that it seems to have attracted multi-million dollar investments and yet nowhere in the web pages or downloadable literature is there any apparent awareness of the intrinsic limitations imposed by the arguments behind the Betz limit.

It is almost certainly the case that the design will be less efficient than a conventional design whose rotor diameter is the same as the shroud diameter.

It is difficult to see any advantages in designs like this and it is significant that no test results on the turbine have been published which meet the IEC 61400-12 standards.

Does the Betz limit apply to vertical axis wind turbines?

The short answer to this question is 'No' although it is not obvious how to produce an equivalent theorem for a VAWT. The arguments that are used to derive the Betz limit for a HAWT do not apply directly to a vertical-axis wind turbine.

It is possible that an equivalent theorem can be produced by splitting the approaching stream tube into two parts; one passing through the advancing blades and the other passing through the retreating blades.

The torque exerted on the VAWT will have to be matched by an equal and opposite angular momentum in the stream far downstream. It is much less obvious how to set up all the conservation relationships for a VAWT than for a HAWT but it would make a good student or even a good post-graduate project.

From an experimental point of view, the efficiencies of VAWTs based on their frontal area seem always to be lower than a HAWT of equivalent frontal area and no VAWT has yet been tested to IEC61400-12 standards that have efficiencies in the upper range of large HAWTs - which can be in the region of 45%.

In spite of their lower efficiency, there are situations where a VAWT might be preferable to a HAWT (i.e. a gusty urban environment or some location with severe space constraints).

Summary

Although this is a fairly weighty article, and the maths may be too involved for some users. If you arrived at this site looking for sustainable alternatives to everyday products, please don't feel put off, we don't normally do such a deep dive into physics!

Nevertheless, I felt it was important to preserve the work from the original Wind Power Program website, as it does provide one of the most comprehensive and detailed explanations of the Betz limit and its application in calculating the efficiency of horizontal axis turbines. Despite the fact that the original article is now a few years old, these formulae do not go out of date, and remain useful and applicable today.

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With her Master of Science in Renewable Energy Engineering (MSREE) from Oregon Tech, Steph is supremely well qualified to write on all aspects of renewable energy. She has already achieved a zero carbon footprint and her goal is to help as many other people as possible do the same. Her other hobbies include music, yoga, swimming and horror movies.
Stephanie Cole
With her Master of Science in Renewable Energy Engineering (MSREE) from Oregon Tech, Steph is supremely well qualified to write on all aspects of renewable energy. She has already achieved a zero carbon footprint and her goal is to help as many other people as possible do the same. Her other hobbies include music, yoga, swimming and horror movies.

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