A 9'- 6" diameter turbine at a 15 mph wind can not exceed 719 watts output and at 30 mph wind it cannot exceed 5882 watts well my wind turbine produces 1117.291 watts at a 15 mph wind and 6984.554 watts at a 30 mph wind the winds input is 1213 watts at a 15 mph wind and 9926 watts at a 30 mph wind before I designed the turbine I had no idea that Betz Law even existed but now i do but law's were made to be broken

If you can get the same readings outside in the open then it is better as the wind is more random and gusts and swirls are hardly ever in a straight line as you can get in a tunnel but you can test stress resistance in high winds in a tunnel as high winds travel straighter.

regards

Ok, as an aerodynamicist by trade and multiple degrees, I feel the need to chime in here. I came across this while trying to verify another of Betz' claims, and I would like to explain so people stop making the mistakes with it.

Betz' Law is based on a control volume analysis, and to do that correctly, you must choose correct control volume. Betz' original assumptions for his control volume contains ALL the air that flows into the fluid power device, and ALL that flows out. So, if you have a nozzle, your control volume must extend from the beginning of the nozzle, to the farfield exit. The same can be said for a venturi type design, even though you have no nozzle, you are drawing air from a larger area than the projected frontal area of the rotor.

Consider a 747 engine. That engine swallows on the order of 100k cubic feet per second of air, but the nozzle diameter is only on the order of 15ft (numbers not exact, just making the point), so to properly analyze the efficiency of a jet engine using a control volume analysis, your initial entrance to the control volume must contain a massive area in front of the nozzle (227 sq feet at 300 mph, for 100k cubic feet).

So, consider a wind tunnel with your VAWT in it, where the freestream air is moving at a constant speed. If you want to know your efficiency, you must measure all the air in front of the turbine with a anemometer of some sort, and any air that is moving slower than the freestream is being effected by your turbine. Therefore, your control volume area must extend at least out to there.

If you are doing a venturi-type design, chances are good you are drawing air from in front and the sides of the control volume, all of this must be taken into account, otherwise you are effectively falsifying your efficiency number.

The responder who talked about rotation in the flow must remember as well that the amount of rotation in a flow is conserved as well, and a conservation of vorticity equation can be written for that flow. By no means does this enably you to violate Betz' law. If you find you are getting a higher efficiency, you must re-examine your control volume, and you will find that your turbine is consuming air from outside it's projected frontal area.

Hope this makes sense to everyone.

As a guy in business development, not physics or engineering, I would love if someone would give a straight up answer to this question... Does Betz Law apply to VAWTs? I have read reasonable arguments saying it does, and some saying it might not. From the evidence I find online, the majority of people seem to think it does. When I speak to an engineer on one of my teams (whose wk is not related to anything on this topic board), he says it absolutely is not.

I would love it if someone smarter than myself and who works in this area of expertise would say YES or NO. I understand there are ways to bend your efficiency numbers, stators, shrouds etc. But thats not what I am looking for, yes or no.

Anyone??

Betz’s Law is applicable to non-shrouded, transverse-flow turbines having a flow velocity through the turbine equal to the average of the incoming and exiting flow velocities. The assumption regarding the average flow velocities is part of the proof, so it is incorrect to apply Betz’s Law to applications where this initial assumption is not the case. Also, the theoretical efficiency of 50.26 percent is the result of theoretically removing 88.89 percent of the energy from the two-thirds of the incoming flow that passes through the turbine and removing zero energy from the remaining one-third of the incoming flow that passes around the turbine. According, the theoretical efficiency of the turbine is dependent on what is defined to be the power of the incoming flow. If the power of the incoming flow is defined in terms of the cross-sectional area of the flow that eventually passes through the turbine, then the maximum theoretical efficiency is 88.89 percent. However, if incoming power is defined in terms of a cross-sectional area equal to the cross-sectional area of the turbine, as in the case of Betz’s Law, then the maximum theoretical efficiency is 50.26 percent.

The maximum theoretical efficiency for turbines that do not correlate to either of the above cases must accordingly be calculated for the specific design. For example, the inclusion of shrouding around the turbine can alter the velocity through the turbine to be other than the average of the incoming and exiting flow velocities (the main assumption used in Betz’s Law) and thereby changes the maximum theoretical efficiency. Accordingly, modifying a design to provide a different maximum theoretical efficiency does not violate Betz’s Law, because Betz’s Law is specific to a set of conditions where the velocity through the turbine must be equal to the average of the incoming and exiting flow velocities. Likewise, altering the definition of what is considered to be incoming power also does not violate Betz’s Law, because incoming power in Betz’s Law is predefined in terms of the incoming cross-sectional area that is equal to that of the turbine.

If you have reservations about the above, then perform the following calculations at the point of maximum theoretical efficiency as determined by Betz's Law. This occurs when the incoming velocity is three times the exiting velocity. Accordingly: Velocity v(in) is equal to 3 v(ex). The velocity through the turbine v(t) is equal to [v(in) + v(ex)]/2 by definition. v(t) is thereby also equal to 2 v(ex). The outlet cross-sectional flow area A(ex) is equal to 3 A(in). And, the turbine area A(t) is equal 1.5 A(in).

Accordingly: The incoming energy power P(in) is equal to ½ rho A(in) v(in)^3. The exiting P(ex) is equal to ½ rho A(ex) v(ex)^3. And, the output power (power removed) P(out), as typically stated in proofs of Betz’s Law, is equal to ½ rho A(t) v(t) [v(in)^2 - v(ex)^2], which should also be equal to the difference of the incoming power and the exiting power.

Performing the calculations and normalizing for A(in) equal to 1, v(ex) equal to 1, and rho equal to 1, results in the following: P(in) equal to 13.5, P(ex) equal to 1.5, and P(out) equal to 12. The efficiency P(out)/P(in) is therefore equal to 12/13.5 or 0.8889. If we modify the definition of incoming power to use the cross-sectional area equal to that of the turbine (as defined by Betz’s Law), P(in) becomes equal to rho A(t) V(in)^3, which is equal to 20.25, and the efficiency thereby decreases to 12/20.25, which equals 59.26 percent, in agreement with Betz’s Law.

It seems a priori that a device placed in a free stream should extract energy as the difference between the upstream and the downstream airflow. However, it appears that in the case of a vortex this does not apply.

I'm not a meteorologist but a tornado seems to operate at 100% efficiency within a defined perimeter. Get close enough and you will be sucked in and exit somewhere at the top of the vortex. If a device were constructed to function as a tornado would not the same results be expected?

It is my understanding that with a Pelton wheel you already have the fluid(water) at its static pressure, the nozzle then converts this to kinetic energy in the form of velocity and it impacts the "blade" losses its connect energy(velocity) and falls by the wayside. with wind the wind is still in the way. I would wonder what the best impact energy transfer would be if the air would get out of the way?(change betz assumptions -so to speak)
L8R

A Venturi Effect fluid Turbine of a Utility Design, that is fashioned in such a way as to maximize energy production from fluids (namely air or water) in motion, by mechanically creating a vortex that reduces fluid pressure at the axis of rotation of said art device, thereby causing the free stream velocity to increase as a function of said vortex.

Upon a mass of fluid impacting the airfoil(s) or hydrofoil(s) leading edge, a reduction in fluid pressure begins on the curved outside surface of the airfoil(s) or hydrofoils and continues to decrease as the trailing edge is approached by said fluid flow. The difference in fluid flow velocity between the curve side and the concave side of the airfoil(s) or hydrofoil(s) creates a venturi effect at the boundary layer where the low pressure fluid flow of the curve side impacts the higher pressure fluid flow of the concave side.

This venturi effect causes the lower pressure curved side fluid flow to accelerate. The fluid flows intersect at the central axis causing the turbine to rotate as the fluid masses merge and accelerate through the central axis, exiting the top of the fluid turbine in a vortex.

Airfoil or hydrofoil pitching moment is located ahead of the center of pressure (closer to the leading edge) at the boundary layer between the two fluid streams. Approximately Thirty-five percent of the rotational force of said art device is a result of the pitching moment of the airfoil(s) or hydrofoil(s), due to the extreme camber of said airfoil(s) or hydrofoil(s). Fifty-five percent of the rotational force is from lift and ten percent results from dynamic drag.

A Venturi Effect fluid Turbine that is functional in a wide range of fluid velocities, from 2 meters/sec. to 20 meters/sec. This is accomplished by sizing and increasing the number of airfoil(s) or hydrofoil(s) as a function of average (peak) fluid velocity (in the area of operation) and attaching said art device to a continuously variable transmission. The increase in rotational force, from sizing, is described by the logarithmic function f(x)= ax.

A Venturi Effect fluid Turbine offers several advantages over prior art devices; namely greater efficiency extracting energy, from a fluid, per volume of installation space, silent operation, more efficient energy production, environmentally friendly to birds, can be sized to fit the operating environment without sacrificing energy output and appears as a solid object to Doppler radar.

A Venturi Effect fluid Turbine of an Ornamental Design that forms the shape of a helix. The helix design is aesthetically pleasing and represents the essence of creation. The design allows for “artwork” adornments on airfoils to suit the installation needs of the user's environment.

Proponents of horizontal wind turbine (HAWT) technology often deride vertical wind apparatus as inefficient. As promulgated, vertical axis wind turbines (VAWT) fall under one of two categories, Savonius or Darrieus or some variation thereof. Because of this design iteration factor, all but a few of these devices allow all rotors to generate a rotational force simultaneously. This is due to the fact that fifty percent of the time the rotor is on the downwind side of the free stream and the collection device is either blocked from the airflow or the orientation of the rotors are such that they provide zero, or in some cases, negative lift. Darrieus wind turbines also suffer from blade stall at low wind velocities and need a starter device or Savonius turbine to initiate rotation.

The Venturi Effect Fluid Turbine can be considered a hybrid of Savonius and Darrieus types. In the case of the Savonius type, the collection buckets are replaced by a lifting device, namely an airfoil or hydrofoil, that extends the length of the central axis. However, unlike a typical Darrieus device, the airfoils (rotors) trailing edge extend to the central axis, preventing airflow from passing through the central axis without affecting (all) four rotors simultaneously. This is accomplished by twisting the airfoil 180 degrees, in a helical fashion, around the central axis of rotation.

A fundamental principal of wind collection devices states that potential energy is a function of the swept area (a) = pi x r2. In other words, the larger the area the propellers move through the more potential energy available.

Swept area for a Venturi Effect Fluid Turbine is calculated using the equation for the area of a cylinder
A = (2 x pi x r2) + (2 x pi x rh)

Wind Turbine Power:1
P = 0.5 x rho x A x Cp x V3 x Ng x Nb
where:
P = power in watts (746 watts = 1 hp) (1,000 watts = 1 kilowatt)
rho = air density (about 1.225 kg/m3 at sea level, less higher up)
A = rotor swept area, exposed to the wind (m2)
Cp = Coefficient of performance (.59 {Betz limit} is the maximum theoretically possible, .35 for a good design)
V = wind speed in meters/sec (20 mph = 9 m/s)
Ng = generator efficiency (50% for car alternator, 80% or possibly more for a permanent magnet generator or grid-connected induction generator)
Nb = gearbox/bearings efficiency (depends, could be as high as 95% if good)

Using the above equation the following power output for a 9m2 and 38m2 at wind velocity 5.5 m/sec. is 432w and 1800w respectively. It is my premise that the operating principle of this device type should, in practice, produce a significantly higher power output. However this is dependent on Betz's law at greater than 59% Cp. If for instance the Cp were 120% due to aforementioned venturi effect then the output would increase to 880w and 3700w. With the further addition of airfoils, the Cp over the theoretical Betz maximum should double each time the number of airfoils is doubled.

The above claims do not exceed Betz's law as defined: What is occurring here is the acceleration of the free stream, localized to the volume and flow of fluid through the VEFT. Betz's law is 59% upstream of the VEFT and is 59% flowing through the VEFT. However the velocity of the air through the VEFT is greater than that of the free stream velocity due to the venturi effect.

So the power output, of the VEFT, would quadruple if you increased the flow velocity by 50% (5.5 m/s to 8.0-+ m/s) as it flows through the VEFT. So this comparison is more like the comparison in radio broadcasting to gain over an isotropic radiator. In other words a directional antenna can be measured to produce more power from the same input over an omni directional antenna. This seemingly magic power increase only exists in one direction, so no laws of physics are broken but the gain in power output is real.

This question is a result of a lack of clarification on the betz limit. either of these two solutions will show that the betz limit remains unbroken:-

1. Use the actual velocity of the wind within the shroud.
2. Use an effective swept area which includes the influence of shroud.

The betz limit applies to ALL machines which intend to harness the kinetic enery of air. It is based on a dimensionless tube of flowing air regardless of the mechanism used.

If you are measuring higher efficiencies in the wind tunnel i would be very skeptical of results. typically the models used in wind tunels are small and have very low reynolds numbers resulting in less than optimum efficiency (unless you have taken this into account during the design?). I found a mono-bladed turbine to be more efficient in the wind tunnel just because the optimisation theory allows a longer chord length.

I'm not 100% certain about this but there is also the posibility that if your turbine is close to the size of your working section in the wind tunnel, then air is forced through it rather than being allowed to flow around it which is how i personally understand the betz limit. i.e. create too much resistance by trying to take too much power (or a total blockage) and the air upstream will be diverted around the turbine. this maybe the reason for artificially high efficiency readings.

if the air lost all its forward motion could it not then fall off the propeller due to potential energy or possibly sink or rise due to thermal changes? It would not necessarily have to pass beyond the blade.

The Betz limit only applies to wind turbines built in Germany before 1926. Just like Newton's laws of motion only apply to objects in England prior to 1687. Everyone knows that!

Betz law applies to a single isolated turbine. The model Betz uses is that the upstream wind flows in in a cylinder of first radius and first speed, while the wind flowing out is at a lower speed, and so must be flowing in a cylinder of larger radius due to conservation of volume (constant density assumed).

But how can this be? How did the downstream wind at the output of the turbine get to expand into the space taken by the air on either side of it without changing the density? What if there is a line of turbines, and the downstream wind from the turbines on either side are also outputting wind that is attempting to expand into the same space as their neighbors? Even better, consider a semi-infinite array of turbines being blown by a semi-infinite swath of wind. There is no more cross-sectional area on the downstream side of the turbines for the exhaust air to occupy than there is on the upstream side. Something's gotta give here!

I don't know the final result, but it seems clear that, while Betz law may predict the maximum efficiency of a single, isolated turbine, the model does not work for a "wind farm".

AMEN! Finally someone points out the simple and obvious fact that the Lanchester Betz Zhukosky limit does not apply to ducted / concentrators because the machines perturb airflow upstream, in the slipstream and downstream. Not all ducts do that, but not all ducts are created equal. Flow is not constant, It is altered in a ducted machines. Pls read the theorem in its entirety. It applies to uniform flow, constant mdot, in an incompressible fluid, single actuator disk propeller. It does not account for altered back-pressure (pressure drops). It is only a reference. But Euler still applies and that is the misconception. Confusing viscous effects with CPmax.

"ignorance is not the hurdle to progress, the illusion of knowledge is"

Betz postulate can be treated as a sort of misunderstanding from the start.

The theoretical model that Betz used for determination of the maximum effectiveness coefficient of an axial wind turbine has practically nothing to do with the turbines (apart from the name).

This theoretical model applies to the ideal fluid break.

The whole matter can be found described in greater detail in a research paper by T. Koronowicz and J. Szantyr entitled Comparative analysis of the theoretical models of ideal propulsor, ideal fluid brake, ideal screw propeller and ideal axial wind turbine

The paper can be found in PDF here: http://www.bg.pg.gda.pl/pmr/pdf/PMRes_2013_2.pdf

Naturally, a 100% efficiency is impossible - however the Betz limit is not valid in the context of wind turbines.

(More in the article under the link in my previous post).

Why do you believe that ? And ignore complete blocked wind that is released into a pulley or gears or both with capacity of light speed. Eienstiens modifications of Newtonians phyics law.

Hi Richard, you started this discussion on Betz Law and the intelligent comments on your site so far have been genuine with few crack-pots.

Betz Law is correct in defining, in physics, how a propeller driven windmill operates- nothing more and nothing less. Indeed, professor Betz used a 1 square foot test plate with a scale attached to measure 100% of the wind energy potential available to a PROPELLER driven wind mill.

Professor Betz used the single dimension, 1 square flat test plate to represent measurements for the propeller because in geometry the propeller rotates on a single dimension, flat plane and operates side-ways to the energy source. Professor Betz didn’t test a 1 square foot test plate composed of four 3 inch, half round turbine blades that would yield more surface area (that contains volume) and test much higher for wind energy potential.

From a single point of reference, the wind flows 90 degrees before the point and 90 degrees beyond the point. Professor Betz did not test a multidimensional turbo machinery rotor operating directly into the wind energy source because it did not pertain to the operation of a propeller driven wind mill which he was defining in physics.

Professor Betz would roll over in his grave if he knew that his propeller driven wind mill dynamic has been re-branded as turbine machinery dynamic.

Betz Law is not the Holy Grail in wind energy science; it correctly defines how a propeller driven wind mill operates. If you change the rotor dynamic to a multidimensional turbo machinery rotor dynamic that operates directly into the wind energy force and has flow through; you have to change over to the physics governing the new dynamic. Betz Law does not apply to the new dynamic because the physics of the new dynamic are different than that of a propeller driven wind mill.

In real science, working models of the concept are proof of the concept and trumps conjecture. I have 3 working models of the concept of “Low Pressure Turbine Dynamics” that prove that there is more wind energy potential than Professor Betz demonstrated with his propeller driven wind mill test. I have demonstrated, with my models, that there can be advancement in wind energy technology without disagreeing with Betz Law. The disagreement is by brainwashed people who think that a propeller is turbine machinery dynamic, science is a religion, and Betz Law is the Holy Grail in wind energy technology.

Check out my video Richard, on YouTube demonstrating the new multidimensional turbine dynamic with ducted flow through and be sure to read the description.
Baker Turbo-Vortex Wind Turbine Turbina

https://www.youtube.com/watch?v=wTeiSHpbFt4

bombuli; its the law of wind, Petot rules

Thanks Taff for your interest in wind energy SCIENCE. Here is the geometry SCIENCE explanation of wind energy potential and efficiency.
Link:
Home of the 6 and 8 helix wind turbomachinery rotor dynamic.
Wind Energy Basics: Wind Energy Potential.
Turbine Dynamic vs Propeller Dynamic: Wind Energy Efficiency.
The two wind mill machine rotor dynamics have only one thing in common; they both have a center line of operation into the wind flow. The propeller rotor dynamic operates from the center line in a single dimension on a flat plane, rotating side-ways to the wind flow.
A 6 and 8 helix turbomachinery rotor dynamic operates from the center line in a multidimensional plane, rotating 90 degrees in front of the center line to 90 degrees past the center line. This 180 degree arc is the full WIND ENERGY POTENTIAL from a center line, not the 1 degree that the propeller dynamic WIND ENERGY POTENTIAL operates in.
A propeller dynamic operates in a ratio of only 1/180 of the WIND ENERGY POTENTIAL from a center line so how can a propeller dynamic be about 50% efficient to the WIND ENERGY POTENTIAL? It’s not, it is only about 50% efficient in it’s rotation in the 1 degree intersect with the wind flow and about 50% of the wind flow must pass through it for it to turn.
Oh wait, it gets better: The TOTAL WIND ENERGY POTENTIAL for a 6 and 8 helix turbomachinery rotor is not only the 180 degree arc on top of the turbine dynamic but the bottom 170 degree arc as well, via ducting, before releasing the column of captive wind flow from Larry’s Windcatcher. Now, the propeller dynamic is operating in only a ratio of 1/350 of the TOTAL WIND ENERGY POTENTIAL.
The Turbo Vortex Wind Turbine dynamic is hundreds of times more EFFICIENT in the TOTAL WIND ENERGY POTENTIAL available to it than a propeller dynamic in the same amount of wind flow. I challenge the National Renewable Energy Laboratory to prove any different, but then, they are more political propaganda and corporate Status Quo oriented than science oriented. Instead of inventing a 6 or 8 helix turbomachinery rotor dynamic in their laboratory to operate in the TOTAL WIND ENERGY POTENTIAL, the NREL re- branded the propeller dynamic as a turbine machinery dynamic; that is political propaganda, (first the propagandist steal your words, then they steal their meaning) not science. Without government corporate oligarchy welfare tit subsidy and the NREL, the corporate propeller wind farms would not exist because modern propeller dynamic windmills have never been economically viable.
I hope that explanation demonstrates to you Taff the uphill battle it is for original inventors when SCIENCE is replaced with political corruption by monopoly corporate/ government agenda and propaganda. You see, until the NREL recognizes the SCIENCE, it doesn’t exist; brainwashed Americans will not accept the concept until they are told by someone in authority to accept the concept. I have made it this far on my own and I do not expect any different. Do you know of any honest investors, I don’t. E-mail me until I reply. My g-mail is all geeked up and I did not receive your email. I answer all e-mails received.
Open source? Excuse me, but do you realize how difficult it is to construct an aluminum 6 or 8 helix rotor? The lay-out is all about geometry and most people do not have the skill level to deal with aluminum which is essential (weight to surface area ratio, balance and flexibility).
Here is a link to my 12 ft. in diameter, 6 ft. wide wind turbine that I finished ducting and will soon be hooked up to the Continental Divide Coop electrical grid. The turbine dynamic has more surface area than a 3 bladed, 90’ diameter variable pitch propeller dynamic.
Link: https://www.youtube.com/watch?v=kyQB48QJLWc

Is it with H-Darrieus that you have observed this phenomena ?

Actually it's a trick there in the technology. Betz law considered the velocity at which the wind speed struck the blades of the turbine. In case of windmill kept in open there is no difference between wind speed and the wind speed with which the blades are struck. But, when put inside the throat of a venturi, a difference arises out it. Inside the venturi, the speed increases due to the internal property of the venturi (that I don't want to discuss here) and the speed with which the blades were struck by air will be higher than the wind velocity in open. If we consider that velocity with which the blades were struck, we can see that the Betz law isn't violated at all. So, in short, it's overcoming Betz law without violating it.