# Why is it claimed that Betz's limit is a law?

• I
• normandajc
In summary: The force induced on the profiles of the blades is due to the relative speed created by the rotation speed and the speed of the fluid.
normandajc
TL;DR Summary
The mathematician Betz defined from the calculation of the kinetic energy of the wind a maximum power coefficient. His theory is correct, but it is not a law.
Large wind turbines have become very efficient and have a power coefficient close to that defined by Betz.
However, large wind turbines are stopped when the wind is too strong, not because they produce too much, but because their blades are subject to bending stresses which may break their blades.
The power coefficient of betz is Cp_Betz = 16/27 (~119%).
If piezoelectric systems could fully convert mechanical stress into electricity, the power coefficient would be double 2 x Cp_Betz = 2x16/27 (~119%).
These wind turbines use the principle of aircraft wings. The force induced on the profiles of the blades is due to the relative speed created by the rotation speed and the speed of the fluid.
The induced force can be broken down into two forces, one force combined with the radius that creates engine torque and another much larger force that creates bending stresses on the blades.
If we consider the case of the sailboat, the force induced on the sail can be broken down into a force that serves to make the sailboat move forward and another much greater force that seeks to make the sailboat heel and creates stresses on the keel.
By using hydrofoils, the stresses on the keel are transformed into a flow around a profile.
The force induced on this profile allows the yacht to move faster and prevents the yacht to heel.
If we consider Darrieus type turbines, the force induced on the blade profile creates a compressive stress on the arm supporting the blade during one half turn and an extension stress during the other half turn.
It is possible to transform this alternative constraint into recovery of additional energy as for the sailboat.
Betz's limit theory is correct, but the Betz limit has never been a law. Yet many books, articles, wikipedia assert that it is a law that has never been demonstrated and that can never be demo
see
https://hal.archives-ouvertes.fr/hal-01300531/
https://hal.archives-ouvertes.fr/hal-01982516v8

Delta2
normandajc said:
If piezoelectric systems could fully convert mechanical stress into electricity,
To get energy out them the internal forces have to do work, so the geometry of the system must change. How do you define the efficiency (specifically the available wind energy) for a turbine with varying geometry and thus varying cross sectional area?

normandajc said:
If we consider the case of the sailboat...
The Betz limit applies to stationary wind turbines in the reference frame of the ground. For wind-powered vehicles, it still applies in the rest frame of the vehicle, but not in the ground frame. Unless you redefine the available wind energy based on the increased volume of air that a vehicle can encounter (compared to a stationary turbine)

See section 2.4 here:
https://orbit.dtu.dk/files/3748519/2009_28.pdf

Last edited:
The advantage of a Darrieus type turbine is that the stresses in the arms vary alternately from compression to extension during each revolution. The conversion system is a connecting rod and crank system. Instead of the piston moving, the crankshaft moves. The profile almost describes the path of a circle. The section swept by the blades almost doesn't vary. look at this page of my website
http://www.cyberquebec.ca/_layout/?uri=http://www.cyberquebec.ca/normandajc/

If you put a wind turbine on a moving platform, you must consider the speed of the fluid the turbine receives. The relative velocity, not the velocity relative to the ground, should be considered.

normandajc said:
Large wind turbines have become very efficient and have a power coefficient close to that defined by Betz.
However, large wind turbines are stopped when the wind is too strong, not because they produce too much, but because their blades are subject to bending stresses which may break their blades.

The power coefficient is not dependent on wind speed. You can have a low power coefficient with a high wind speed and a high power coefficient with a low wind speed. So it is not true that a turbine is stopped because it reaches Betz limit or something like that.

normandajc said:
The power coefficient of betz is Cp_Betz = 16/27 (~119%).
If piezoelectric systems could fully convert mechanical stress into electricity, the power coefficient would be double 2 x Cp_Betz = 2x16/27 (~119%).

How would this work? Because this is simply not true:
• The efficiency is defined as the power extracted by the turbine (i.e. converted to rotational speed and torque) divided by the kinetic energy flowing through the turbine disk (the area swept by the blades) when the turbine would not be there. Therefore a 119% efficiency would mean that you recover 19% more energy than there is kinetic energy flowing through the turbine disk. I don't see how this would work.
• A stress is a force, not a power source. You cannot convert a stress to energy without letting this stress do some work, like bending the blades. But since you can only bend the blades so far, this cannot be a continuous source of energy.

normandajc said:
These wind turbines use the principle of aircraft wings. The force induced on the profiles of the blades is due to the relative speed created by the rotation speed and the speed of the fluid.
The induced force can be broken down into two forces, one force combined with the radius that creates engine torque and another much larger force that creates bending stresses on the blades.

In fact, the turbine 'converts', if you will, this axial force in a torque, so the bending stress is directly related to the torque and rpm of the blade.

normandajc said:
If we consider the case of the sailboat, the force induced on the sail can be broken down into a force that serves to make the sailboat move forward and another much greater force that seeks to make the sailboat heel and creates stresses on the keel.
By using hydrofoils, the stresses on the keel are transformed into a flow around a profile.
The force induced on this profile allows the yacht to move faster and prevents the yacht to heel.

Not true. You are convoluting two functions of the keel. One is to provide stability by use of its weight. The keel lowers the center of gravity which together with the center of buoyancy provides a righting moment (arm + force). The force on the sails plus the hydrodynamic resistance of the ship provides a heeling moment, and these two need to be in balance. So a yacht MUST heel to be able to move forward.

A keel also provides lateral surface area which makes the resistance of moving sideways much larger than the resistance moving forward. This resistance to sideways motion largely reduces (but not completely cancels) the sideways (leeway) velocity caused by the sideway component of the force in the sail. This make the yacht go mostly forwards, even when the largest component of the force in the sail is sideways. The keel does not however magically make the yacht move faster.

normandajc said:
Betz's limit theory is correct, but the Betz limit has never been a law. Yet many books, articles, wikipedia assert that it is a law that has never been demonstrated and that can never be demo

Not every law is created equal. You have the physical laws of which there is no way around (conservation of energy / momentum etc.). And you have the more practical laws like Betz limit and Hook's law and many more.
But there is a solid theory for why the Betz limit holds:
A turbine converts kinetic energy to rotational velocity and torque. Reduction of kinetic energy means lowering the velocity. Not reducing the kinetic energy means obviously that you cannot recover any energy, reducing the kinetic energy to zero however also means that you are not recovering energy because now the flow velocity through your turbine disk is zero. Thus somewhere between not reducing the kinetic energy and reducing the kinetic energy to zero there is a maximum of energy that can be extracted. This is the Betz limit.

How would this work? Because this is simply not true:
If I understand a sailboat equipped with hydrofoil can not go faster. A glider that goes against the wind cannot go forward. Yet this is possible because it is the relative speed that is the key to the mystery. For a Darrieus turbine, the relative speed is the combination of the rotation speed and the speed of the fluid.

"In fact, the turbine 'converts', if you will, this axial force in a torque, so the bending stress is directly related to the torque and rpm of the blade."
For a horizontal axis turbine, the bending stress is constant over time if the fluid velocity is constant. This is not the case for a Darrieus turbine.

I never said that the keel allows you to speed up. On the other hand, replacing the keel with hydrofoils, allows for better performance on the sailboat.

normandajc said:
On the other hand, replacing the keel with hydrofoils, allows for better performance on the sailboat.
And inflating your tires gives you better gas mileage in your car. But neither is relevant to the Betz limit.

TeethWhitener
normandajc said:
If I understand a sailboat equipped with hydrofoil can not go faster. A glider that goes against the wind cannot go forward.

If you mean by 'a sailboat equipped with hydrofoil' a sailboat that can 'fly' on its hydrofoil, then yes, a sailboat on hydrofoils is faster, but this is only due to the massive reduction in resistance and has nothing to do with the Betz limit.

A glider also goes downward. And wind direction is not relevant anymore when you are in the air.

normandajc said:
Yet this is possible because it is the relative speed that is the key to the mystery.

There is nothing mysterious about relative speed. But this has also nothing to do with Betz limit, flying sailboats or gliders...

TeethWhitener
"Not every law is created equal"
If we consider a trolley with a vertical plate exposed to the wind. If the trolley is free and without any resistance, it will move at wind speed and the vertical plate is not stressed.
If the wheels of the trolley are welded, the trolley speed is zero and the vertical plate is under maximum stress.
In the case of a wing profile, we add a displacement speed U which changes everything.
Normal force to the profil of the blade creates stresses in the arms of a Darrieus type turbine.
When calculating a mast for a wind turbine, we consider the force defined by Betz associated with the swept area.
In theory, the Betz force associated with the swept area is used entirely to produce energy, and is used entirely to calculate the stresses.
it is possible to transform these constraints into kinetic energy with darrieus turbine

#### Attachments

• 2.jpg
26.2 KB · Views: 166
• 1.jpg
28.1 KB · Views: 135
Arjan82 said:
If you mean by 'a sailboat equipped with hydrofoil' a sailboat that can 'fly' on its hydrofoil, then yes, a sailboat on hydrofoils is faster, but this is only due to the massive reduction in resistance and has nothing to do with the Betz limit.

A glider also goes downward. And wind direction is not relevant anymore when you are in the air.

The normal force on the hydrofoils allows the sailboat to be lifted and makes it possible to counter the tilt. By this trick, we transform potential energy (stress on the keel) into kinetic energy.

I've completely and utterly lost you. Don't have a clue what you mean.

normandajc said:
Betz's limit theory is correct, but the Betz limit has never been a law. Yet many books, articles, wikipedia assert that it is a law that has never been demonstrated and that can never be demo...
I wouldn't get too hung up on the term "law" here. It isn't used consistently in science, and when it is, is usually describing a mathematical relation, irrespective of how accurate/applicable it is. E.G., we still use the name "Newton's Law of Gravity."

I don't understand why it is impossible to transform constraints into additional energy recovery. It is potential energy that we transform into kinetic energy. Large wind turbines are stopped when the constraints are too high. That is, when the potential energy is too high. It is not the kinetic energy that is too great. For a sailboat, it is possible to transform potential energy into kinetic energy by replacing the keel with hydrofoils. For a Darrieus-type turbine, I transform potential energy into kinetic energy, by replacing the arms with crank and connecting rod systems. This allows me to obtain a much higher efficiency than the one defined by Betz.

normandajc said:
Betz's limit theory is correct, but the Betz limit has never been a law.
normandajc said:
This allows me to obtain a much higher efficiency than the one defined by Betz.
Oh...so you don't really think the Betz limit is correct, and are using the "it's not a law" thing as an argument against it. That's really not a good look.

As @A.T. indicated, the problem for wind turbines is one of geometry. The cross sectional area of a moving parcel of wind has to get larger as the speed drops in order to satisfy continuity/conservation of mass.

This really bears no relation to what is happening with a sailboat or glider, and this idea of a multiplier effect is a misunderstanding of what sailboats do, and strongly implies a violation of conservation of energy/perpetual motion.

Arjan82 and jbriggs444
normandajc said:
I don't understand why it is impossible to transform constraints into additional energy recovery. It is potential energy that we transform into kinetic energy. Large wind turbines are stopped when the constraints are too high. That is, when the potential energy is too high. It is not the kinetic energy that is too great.

Well... the forces (dynamic and/or static) on the structure get too high, then they stop. The forces on the structure get too high because the wind velocity is too high, the wind velocity is a measure for kinetic energy, therefore the turbine indeed stops because the kinetic energy gets too high.

But also note that Betz limit does not require high wind velocities, in fact it is completely unrelated to wind velocity. Also with very mild winds (way before any structural failure is imminent) the turbine efficiency is limited by the Betz limit. The Betz limit just tells you how much, in theory, of the incident kinetic energy can be converted to useful energy (RPM + torque).

normandajc said:
For a sailboat, it is possible to transform potential energy into kinetic energy by replacing the keel with hydrofoils.

I don't see what type of potential energy is converted to kinetic energy here. What potential energy do you mean here? The gravitational potential energy actually increases rather than decreases since the boat is lifted out of the water.

Note that a foiling sailboat is not converting more wind energy to forward velocity than a comparable non-foiling ship. The only reason the velocity is increased is because the resistance is decreased. But the amount of captured energy doesn't necessarily change (it in fact is more likely to become less since the relative wind angle is more forwardly directed which is generally less capable of generating useful energy, i.e. forward force + velocity)

normandajc said:
For a Darrieus-type turbine, I transform potential energy into kinetic energy, by replacing the arms with crank and connecting rod systems. This allows me to obtain a much higher efficiency than the one defined by Betz.

Ok, what I think you mean here is that the stresses in the arms are alternating between compressing and stretching forces. This could be used to drive some kind of converter producing energy. This however will not generate any extra energy. Since the energy that is used to drive this converter cannot also be used to generate a torque on the turbine and vice versa (is this why you think there is a factor 2 increase in efficiency?). This means that now you've just got two parallel conversions, each converting a part of the incoming kinetic energy.

But it doesn't matter how you convert the kinetic energy to whatever other kind of energy, or combination of other types of energy, or combination of converters, or combination of types of energy and converters or whatever... This input amount of kinetic energy is your only input of energy to the system.

But a reduction of kinetic energy here means simply a lower flow velocity. But then Betz still applies, no reduction of flow velocity means no energy capturing, reduction to zero means also no energy capturing, but somewhere in between there is a maximum amount of energy conversion. This is the Betz limit, and there is no way around it.

russ_watters
Betz's theory is absolutely correct. It is calculated in relation to kinetic energy.
Using a relative velocity which is the combination of the rotational speed and the fluid velocity creates an induced force on the profile. Part of this force associated with the radius creates a driving torque. This is consistent with Betz's theory. Large wind turbines have an efficiency close to that defined by Betz. The other part of this force creates stresses. It is potential energy that Betz's theory does not take into account. But by transforming these constraints into energy recovery. It is potential energy that is transformed into kinetic energy. Betz did not take this possibility into account.

You now mean the internal stresses of the structure that keep the turbine upright? That is not a source of energy! As I said in my first post, these stresses are forces (stress is force per unit area). To convert a force to energy you need to let it do some work. Do work means let the force act some distance in the direction of application. In this case that means you can only do work by letting the turbine deform. But energy is work per unit time. So to produce a continuous flow of energy you need a constant rate of work. In this case that means continuous deformation in the direction of that force! (the moment you deform in the other direction you require energy rather than produce it). A turbine can deform only so far before it breaks, so this is not a continuous supply of energy!

russ_watters
normandajc said:
The other part of this force creates stresses. It is potential energy that Betz's theory does not take into account. But by transforming these constraints into energy recovery. It is potential energy that is transformed into kinetic energy. Betz did not take this possibility into account.
What potential energy? It looks to me like you are confusing force and energy here.

Are the papers linked in the OP written by you? Are they published in reputable scientific journals? One thing I noticed is that while you go through a lot of math to create a new efficiency limit, you never state what that limit actually is (at least in the first paper). What, exactly, is the efficiency limit you think is possible?

To calculate the structure of the mast, this force is taken into account
Coef 1/2 rho S V^2
The coefficient is close to the one defined by Betz.
60% of the kinetic energy of the wind can provide energy, but 60% of the kinetic energy of the wind creates stress. Why is it not possible to transform these stresses into additional energy recovery?

normandajc said:
Why is it not possible to transform these stresses into additional energy recovery?
Because stress isn't energy. A book sitting on a table puts a stress on the table, but no energy is being expended.

Because stress isn't energy. A book sitting on a table puts a stress on the table, but no energy is being expended.
for your example, I agree since the stress is constant over time.

normandajc said:
To calculate the structure of the mast, this force is taken into account
Coef 1/2 rho S V^2
The coefficient is close to the one defined by Betz.

That this coefficient is close to Betz (numerically) is entirely coincidental (if that is true, I didn't check it). What you call 'Coef' here is the drag coefficient and is entirely unrelated to the Betz limit. It so happens to be the case that the force on an object is generally correlated to the amount of incoming kinetic energy (not strange I would say, but note that this relation is not exact!). So determining a drag coefficient means you can easily estimate the force on a system for a particular wind speed, it is an engineering tool. But, as noted by many, force and energy are two entirely different things.

russ_watters
normandajc said:
For Darrieus type turbines, the stresses vary during each turn. You do have a notion of time. It is possible to transform constraints into additional energy recovery.
http://www.cyberquebec.ca/_layout/?uri=http://www.cyberquebec.ca/normandajc/

Although you could convert these stresses to energy, it will not give you extra energy! It means that the torque and/or rpm of the turbine is reduced!

russ_watters
I'm not questioning the Stirling engine. I am also not questioning the fact that you indeed can convert an alternating force into energy by a Stirling engine with a decent efficiency.

What I am pointing out however, is that the inflow of kinetic energy to the turbine can be used once, and once only. So if you build a Darrieus type turbine with a Stirling engine, than that is absolutely fine, great! But what will NOT happen, is that this arrangement in some way will convert any energy that wasn't there in the first place. The total efficiency (rotating turbine + Stirling engine) of this system will not surpass Betz.

russ_watters
normandajc said:
For Darrieus type turbines, the stresses vary during each turn. You do have a notion of time. It is possible to transform constraints into additional energy recovery.
Time varying stress still isn't energy.

I agree with @Arjan82 ; at best, you are trying to use the same energy twice, and glossing-over it.

Since evidently this is your own unpublished "personal theory", it is not appropriate for discussion on PF. If you get it published in a reputable journal we can discuss it further, but I don't think publication is likely since it appears to suggest a violation of conservation of energy.

Last edited:
davenn, vanhees71 and Arjan82

## 1. What is Betz's limit?

Betz's limit, also known as the Betz law, is a theoretical limit on the maximum amount of energy that can be extracted from the wind by a wind turbine. It states that no turbine can capture more than 59.3% of the kinetic energy of the wind.

## 2. Why is it called a "law"?

Betz's limit is considered a law because it is based on mathematical and physical principles and has been extensively tested and validated through experiments and observations. It is a fundamental concept in the field of wind energy and is widely accepted by the scientific community.

## 3. How was Betz's limit derived?

Betz's limit was first proposed by German physicist Albert Betz in 1919. It was derived from the laws of conservation of mass and energy, as well as the principles of fluid dynamics. Betz used mathematical equations to calculate the maximum amount of energy that could be extracted from the wind without causing a significant decrease in wind speed.

## 4. Is Betz's limit still relevant today?

Yes, Betz's limit is still widely used and relevant in the field of wind energy. While modern wind turbines have become more efficient, they still cannot exceed the theoretical limit set by Betz's law. This limit serves as a benchmark for evaluating the performance of wind turbines and improving their design.

## 5. Are there any exceptions to Betz's limit?

There are some rare cases where the efficiency of a wind turbine can exceed Betz's limit, such as when the turbine is placed in a wind tunnel or in conditions with very high wind speeds. However, these exceptions do not invalidate the law and are not representative of real-world scenarios.

• General Engineering
Replies
8
Views
3K
• General Engineering
Replies
16
Views
6K