Given the technological sophistication of today's wind turbines, it's quite humbling to think that their theoretical maximum efficiency was derived by wind turbine pioneer Albert Betz in 1920. Betz' Law, as it is now known, is a relatively simple proof that the maximum efficiency of a wind turbine, irrespective of its design, cannot exceed 59%. Still, some believe laws are there to be broken - at least in the virtual simulation world.
Betz' Law
The efficiency of a wind turbine is measured as the ratio between the energy extracted from the wind to perform useful work (e.g., electricity) and the total kinetic energy of the wind without the presence of a wind turbine. To understand the reasoning behind Betz' Law, consider a 100% efficiency, i.e., extracting all the kinetic energy from the wind and thus bringing the air to a standstill. The paradox of bringing the air to a stop means that there's no way for the air to drive a rotating machine, so no useful work can be extracted. Now consider the other extreme, i.e., the wind turbine doesn't reduce the wind speed at all. Again conservation of energy dictates that no useful work will be accomplished by the wind turbine. Clearly the maximum theoretical efficiency lies somewhere between these two extremes. Betz' Law simply and elegantly proves the maximum efficiency of a wind turbine can't exceed 59%.
Horizontal-Axis Wind Turbine
The latest horizontal-axis wind turbines typically have efficiencies in the range of 35-40%, so clearly there's no conflict there between theory and practice. If electricity generators and distribution are taken into consideration, then efficiency drops to the 10-30% range.
Virtual Wind Turbine Simulation
A recent article entitled "CFD Modeling for Wind Turbines" cites efficiencies for a shrouded wind turbine way in excess of the Betz Law limit (shown by Figure 4 in the article). Straightaway this should be a cause for the kind of skepticism usually reserved for perpetual motion devices. Now, maybe Betz' Law is flawed and the researchers have found a loophole, or more likely, they have made a mistake in their calculations. To make such an extraordinary claim against a well-regarded theory requires extraordinary evidence. Their study is based on Computational Fluid Dynamics (CFD) calculations. I believe this is an excellent example of how not to use advanced Computer-Aided Engineering (CAE) tools, such as CFD.
Reality Check
CAE tools are only as good as the engineers and researchers that know when and where to apply them. Good practitioners know that when a simulation contradicts a well-proven law, such as conservation of energy, or in this case Betz' Law, there is likely a problem with their simulation and not vice-versa. It is relatively easy to use CAE tools to perform virtual simulations that have no real world equivalent. There are a number of reasons why a good simulation can turn bad:
- Underlying assumptions of the tool and the physical models it supports are not met, for instance a linear stress analysis model is no use if the stresses are likely to exceed the elastic limit and make a material deform plastically
- Specification of inappropriate boundary conditions and material properties can produce subtle inaccuracies in simulation results
- Low fidelity (coarse mesh) model representation
- Mistyped numbers or incorrect unit conversions can cause a simulation to produce results that are of by orders of magnitude
To at least recognize these problems it is essential that a practitioner has a good understanding of their physical problem domain, such that they know when a simulation is approaching the improbable. Oftentimes this would take the form of a low fidelity hand calculation (ballpark figure) to cross correlate a simulation result. The best CAE practitioners validate their CAE tools against similar, known test cases prior to applying it to a new case.
Lesson
So the lesson here is that you need to know your domain prior to picking up your CAE tool and not vice-versa. Then, when you find a simulation result that contradicts a physical law, you'll know it for what it is: a revolution in physics, or, most likely, a simple modeling mistake.
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I'm not sure there is anything wrong with the claims...
Why do you say the performance is "way in excess of the Betz Law limit"?
All they have done is created a venturi effect to accelerate a larger volume of air through the turbine.
The power output of any turbine, shrouded or not, would quadrupal if you increased the wind speed by 50% (5 m/s to 7.5 m/s) so this is correct.
I assume then that the Benz law if calculated for the larger parcel of air (not just the rotor diameter size) that is effected by the shroud still holds for the higher power output.
If this is correct then the only value of the shroud is it allows the use of smaller blades for the same amount of power. However the trade off is you need to build this giant shroud which would be totally impractical since it would not be able to withstand high/storm wind loads. Blades feather in high winds to prevent damage. A shroud can't.
It is just not a practicle tradeoff. But the results aren't necesarily flawed or unrealistic.
Betz' Law isn't Broken
I guess I was a little hasty in calling this shrouded wind turbine study flawed. I think you are right that Betz' Law isn't broken if the inlet diameter of the shroud is used instead of the blade diameter to calculate the maximum Betz power.
Thanks for taking the time to point out my mistake.
Betz Law
Betz "Law" is derived from momentum theory, which uses conservation of mass and momentum across a "rotor disc". The "59% efficiency" result follows from the Momentum Theory result that only half of the velocity change associated with work extraction (or addition in the case of a helicopter rotor) occurs at the disc, while the rest occurs downstream, somewhere in the "far wake".
I don't much of a problem with this for a low disc-loading rotor (I do for a counter-rotating pair of coaxial rotors), where the flow is dominantly axial. However, I don't see this being so evident (or unbeatable) in the case of a vertical axis turbine, where the work extraction clearly occurs with turning of the flow.
As an example, I wonder if you can reconcile the Betz Law with the case of a Pelton Wheel, where the capture area is only the area of the blades exposed to the flow. Just a thought. Let me confess that I don't have any results to advertise, showing greater than Betz efficiency. More to the point, is there any derivation out there specifically considering, say, a centrifugal device (or a high-disc-loaded vertical axis wind turbine) and arriving at the same result as the axial-flow Betz law?
One aspect to consider is that a typical helicopter jet engine turbine shaft extracts much more than 59% of the energy in the air blowing into it. This is because it uses rotor and stator stages. In a 50% reaction stage, 50% of the static pressure change occurs in the stator (nozzle), and the rest right away in the rotor. Of course only the rotor part produces useful work, but that is where most of the stagnation pressure drop occurs as well.
The user above proved why Betz law should not be taken as religion even for the purely axial case. If you use a nozzle upstream of the rotor, you can achieve part of the desired engineering result (small rotor, large work extraction).
If you now take the exhaust from this, and say, turn it in a Pelton turbine, you may do even better.
Best regards
naray9