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Betz' law- no ifs, ands, or buts

As part of my ongoing survey of all things related to sustainable design (or renewable energy or simply “green” ) I have started to take a look at wind energy.   Almost immediately I was confronted with Betz’ law which sets a limit on the efficiency of a wind turbine to about 59%.   In other  words,  I won’t be able to collect and convert the kinetic energy of the wind at a rate better than 59%.   When pondering the Betz’ law a few things struck me:

  1. Elegant- The derivation is  fairly straightforward.  The further removed I get from my academic skills the more my head starts to throb when I look at derivations but this one didn’t hurt!  It even makes me think if ever there was something I could derive this might have been it.  Too late, oh well….
  2. Dependencies:  After a quick scan of the derivation it started to sink in that this efficiency was dependent on nothing.  NOTHING!  Come on… Carnot gave us the wiggle room of source and sink temperatures allowing us to reach for higher efficiencies, but with Betz there is nothing.  Speaking of Carnot efficiency and extreme temperatures reminds me of an awful Keanu Reeves movie  that still haunts me today.  I can’t remember what it is called but after looking at IMDB I think it might have been Chain Reaction, which ironically is about a green alternative energy project.  They were no doubt working on a revolutionary energy supply, but based on my recollection it looked like a large cylindrical propane tank with clear vinyl tubes coming in and out all over the place.  I recall them discussing the temperatures they were pumping through this contraption and thinking that their real claim to fame was developing the tube that could withstand this temperature.
  3. What about fans?-After the shock of the Betz’ limit wore off I started to think about axial fans.  I expected to find an elegant derivation for a fan but so far haven’t seen one.  I wonder if fans are covered by the same limit?

Wind Energy, Sustainable Design

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About John Wilson

John WilsonJohn Wilson joined Mentor Graphics Corporation, Mechanical Analysis Division (formerly Flomerics Ltd) in 1999. John has worked on or managed more than 100 thermal and airflow design projects. His modeling and design knowledge range from Electronics Cooling IC packaging level to Data Centers and Clean Rooms. He has extensive experience in IC package level test and analysis correlation through his work at Mentor Graphics' San Jose based Thermal Test Facility. He is currently the Consulting Engineering Manager, WRO in the Mechanical Analysis Division. Visit John R Wilson's Blog

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Comments 6

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The derivation may not have dependencies, but it probably has underlying assumptions. By wind turbine, you probably mean a free-standing propeller. What is the most efficient means of blowing air through an enclosure, a free standing propeller? Certainly not. An axial fan with a tubular duct to direct the air is much more efficient. Perhaps you should look at ducted fans in reverse as a form of wind turbine. They are more expensive, but they may be more efficient over a certain range of speeds.

Simon Favre
11:29 PM Aug 11, 2009

The derivation as described in seems to be symmetric in that the power and energy can be considered to be supplied to drive the rotors as opposed to the power and energy being extracted from the air. So I'd say that yep, the max efficiency probably holds for axial fans as well (unless in a long duct)

Robin Bornoff
8:19 AM Aug 13, 2009

Betz's law claims a 59% maximum efficiency for any wind turbine. This is based on the assumption that wind turbines extract energy by slowing down the wind. If we start at 100% and the turbine is placed in the freestream, the down stream velocity of the wind would be reduced by no more than 16/27. But if the wind turbine device's design accelerates the wind to greater than 100% the freestream velocity it should follow that the efficiency is greater than 59%. This does not invalidate Betz's law as it only applies to devices that slow the wind.

12:15 PM Sep 12, 2009

I have just looked at the derivation in Wikipedia. I can follow the calculations but I find the definition of C_p, the coefficient of performance, to be somewhat subtle. It is not really an efficiency ratio, as I shall attempt to explain. Betz's calculation shows that the maximum power is P_max = (16/27)*(1/2)*rho*S*(v_1)^3 where S is the area swept by the rotors and v_1 is the far-field incident wind velocity. As far as I can tell this expression is correct. However, this quantity is then compared to the total power obtainable from a cylinder of air of cross-sectional area S and velocity v_1, which is simply P_cyl = (1/2)*rho*S*(v_1)^3, hence the ratio C_p = P_max/P_cyl = 16/27 = 59.3%. However, this is not the the true total power available. In the upstream far field we have a cylinder of air at v_1 but with a larger swept area A_1 > S. At the turbine we have a cylinder of air of swept area S but with a smaller velocity v < v_1. The cylinder of air used by Betz is entirely fictitious, so the ratio it is not really a measure of efficiency. That is not to say that it doesn't correlate with efficiency in some way and is not useful. However, we should take care in describing C_p as the theoretical efficiency limit, as is often done. Incidentally, this is why the the law may not apply to more complex turbines, such as the currently fashionable ducted versions, where not all of the air from the far-field area A_1 passes through the rotors, as assumed in the derivation.

5:44 AM Dec 18, 2009

Sorry, smaller swept area A_1 < S in the penultimate paragraph.

6:10 AM Dec 18, 2009

I've been looking for a satisfying yet rigorous derivation of this limit without success. Would it be facetious to note that... "For any attempts to obtain efficiencies greater than 60%...... all Betz are off"?

Pat OLeary
10:35 AM Feb 20, 2010

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