Maximum Cp of traditional multi bladed rotors

[Adriaan Kragten]

In another post, there was confusion about the maximum Cp which can be expected for the multi bladed rotor of a traditional water pumping windmill. I thought that no reliable measurements were availble but I have researched my library and at my own surprice, I found some valuable information. This information was given in the German magazine Windräder WKJ 1/83, page 10 up to 16. The article is named: "Die Göttinger Lieferungen von 1932" commented by Ulrich Stampa.

Wind tunnel measurements have been performed ordered by L. Prandtel and A. Betz already in 1927 and 1932 on eight different scale models with diameters D in between 0.75 and 1.1 m. No 1, no 2 and no 5 were models of multiblade rotors. I can read German but the given information about the characteristics is a bit confusing because one uses German symbols for the dimensionless coefficients. The thrust coefficient Ct is called Cw. The torque coefficient Cq is called Cd. The power coefficient Cp is called Cl. On the last page of the article there is an overview of the coefficients of all eight scale models. However, one has made serious mistakes by copying the values from the given characteristics of each model into this table. So I have read the values of Cpmax, Ct and Cqopt. for the optimum tip speed ratio lambda opt. and Cqstart for lambda = 0 directly from the given characteristics. I have mentioned only model no 1, 2 and 5. Model no 2 has been measured for beta = 0° and beta is 10°. Beta is the yaw angle but I have only used the values for beta = 0°.

Model no 1. D = 0.81 m. Number of blades B = 18. Optimum tip speed ratio lambda opt. = 1.35. Maximum power coefficient Cpmax = 0.27. Thrust coefficient Ct = 0.94. Optimum torque coefficient Cqopt. = 0.20.
Starting torque coefficient Cqstart = 0.23.
Model no 2. D = 0.8 m. Number of blades B = 27. Optimum tip speed ratio lambda opt. = 0.9. Maximum power coefficient Cpmax = 0.298. Thrust coefficient Ct = 0.9. Optimum torque coefficient Cqopt. = 0.33.
Starting torque coefficient Cqstart = 0.48.
Model no 5. D = 0.9 m. Number of blades B = 20. Optimum tip speed ratio lambda opt. = 1.1. Maximum power coefficient Cpmax = 0.36. Thrust coefficient Ct = 0.85. Optimum torque coefficient Cqopt. = 0.33.
Starting torque coefficient Cqstart = 0.6.

So model no 5 has the highest maximum Cp of 0.36. It also has the highest ratio Cqstart / Cqopt. So this model has the best rotor geometry for a water pumping windmill driving a single acting piston pump. All three models differ strongly in blade geometry and blade angles and so there isn't a standard multi bladed rotor which can be used as a reference for what can be expected from a traditional multi bladed rotor. One has to realise that these rotors were probably designed by try and error and than one will find a different geometry if one designs a multibladed rotor with a design tip speed ratio of about 1 using the aerodynamic theory as given in my report KD 35.

In chapter 2 of my report KD 490: "Water pumping with a windmill", I describe traditional water pumping windmills which make use of a traditional multi bladed rotor. Such rotors are very heavy and the use of a single acting piston pump results in a starting wind speed which is always higher than the design wind speed for which the rotor runs at its design tip speed ratio. Development of new water pumping windmills of this type is therefore not advised. If a piston pump is wanted, I advise the use of a floating piston valve (see chapter 3) as this allows the use of a rotor with a much lower starting torque coefficient. Therefore a much lighter rotor with a design tip speed ratio of 2 can be used (see report KD 704 fot the VIRYA-3.6L2).

[bigrockcandymountain]

In chapter 2 of my report KD 490: "Water pumping with a windmill", I describe traditional water pumping windmills which make use of a traditional multi bladed rotor. Such rotors are very heavy and the use of a single acting piston pump results in a starting wind speed which is always higher than the design wind speed for which the rotor runs at its design tip speed ratio. Development of new water pumping windmills of this type is therefore not advised. If a piston pump is wanted, I advise the use of a floating piston valve (see chapter 3) as this allows the use of a rotor with a much lower starting torque coefficient. Therefore a much lighter rotor with a design tip speed ratio of 2 can be used (see report KD 704 fot the VIRYA-3.6L2).

9
Those cp measurements are higher than i have seen mentioned for multiblade mills. 

I wonder what their blades looked like.  The ones i have are widest chordwise at the tip and have a curve rolled in with maybe a 0.5m radius approximately. 

They are 2.4m diameter rotors and i think they have 15 blades. 

I know at really low wind speeds they aren't making much power, but it is still impressive to see them pump water in a wind that you almost don't notice.  Our water table is only about 6m down, so they aren't working very hard. 

Efficiency in a remote water pumping situation is usually not a very high priority.  Reliability, ease of maintenance, low installed cost, durability, etc are all generally more important. 

The old water pumper that I use every day can't be much less than 100 years old.  I had it down about 15 years ago because the well collapsed and it got moved to the new well.  It got new bronze plain bearings at that time, so it should be good for another 50 years or so.

[Adriaan Kragten]

The article of Ulrich Stampa gives a picture of each tested rotor and a picture with the measured characteristics. My scanner isn't functioning at this moment but I will scan the pictures of model no 5 when it works again. The scale model rotor no 5 has 20 blades and a diameter of 0.9 m. The blade lenght is about 2/3 * R and so there is a circle in the centre of the rotor with no blades. The airfoil is cambered only a little (5 %) and this makes that the airfoil has a low minimum Cd/Cl ratio (see report KD 398). The blade angle at the tip is 27° and the chord at the tip is 109 mm. Half way the blade lenght, the blade angle is 37° and the chord is 126 mm. At the blade root, the blade angle is 47° and the chord is 134 mm. So the chord is increasing at decreasing radius what is opposite to most traditional multi bladed rotors but probably better in accordance with the aerodynamic theory. The supporting construction of the blades isn't specified in detail but has four spokes and two rings.

The geometry of the used wind tunnel isn't specified and it is also not specified if the wind tunnel is open, closed or has only an open measuring section. So it can't be guaranted that the measurements have not been influenced by tunnel blockage. But even at 1932, tunnel blockage was a known fenomenon and wind tunnel measurements of Göttingen are famous for its accuracy.

[MattM]

I want to know what Cp is of non-traditional multi bladed rotors.  If nature uses constant ratio curves and s-curves, I'd think those would be better than what we a have mostly seen with wind turbines.  What is in nature has taken millions of years of trial and error.

[Adriaan Kragten]

I want to know what Cp is of non-traditional multi bladed rotors.  If nature uses constant ratio curves and s-curves, I'd think those would be better than what we a have mostly seen with wind turbines.  What is in nature has taken millions of years of trial and error.

I don't quite understand what you mean. Nature does't produce windmill rotors. Nature produces wings of birds and there is a strong analogy in between wings of birds and blades of wind turbines. The aerodynamic theory is based on how nature functions around aerodynamical shaped airfoils and how the generated power can be maximised. So if you design a multi bladed rotor using the aerodynamic theory, you get a result which is best supported by nature. Traditional multi bladed rotors with blades for which the chord becomes larger at increasing radius are not supported by the aerodynamic theory and therefore they have a lower maximum Cp. The rotor of model no 2 had such blades and the maximum Cp is a lot lower than that of rotor of model no 5. However, the increase of the chord at decreasing radius for rotor of model no 5 is only limited and it is certainly possible to design a multi bladed rotor with constant chord blades which has a high maximum Cp. However, the blade angles of such a rotor will be much smaller than that of the tested rotors, even if cambered plate airfoils would be used.

The main reason for the low maximum Cp of rotors with a low design tip speed ratio is wake rotation. If you look at figure 4.2 of report KD 35, you can see that the maximum Cp (if there are no other losses) is 0.416 for lambda = 1, that it is 0.481 for lambda = 1.5 and that it is 0.513 for lambda = 2. Most traditional multi bladed rotors have an optimum lambda of about 1. As you have also reduction of the maximum Cp because of tip losses, the drag of the airfoil and the supporting structure and the fact that the blade length is much shorter than R, a maximum Cp of 0.3 for a rotor with an optimum lambda of 1 is already rather good. However, if the optimum lambda is increased from 1 up to 1.5 or 2, it is much easier to get a high maximum Cp because then the losses due to wake rotation are much lower. This is one of the reasons why we designed water pumping windmills with rotors with lambda design = 2 in stead of 1 when I worked at the Wind Energy Group of the University of Technology Eindhoven. Another reason is that the required solidity for a rotor with lambda design = 2 is much lower than for a rotor with lambda design = 1 and therefore you can use much lighter rotors with only six or eight blades.

[MagnetJuice]

Nature does't produce windmill rotors.

Man produces windmill rotors using the laws of nature.

Nature produces curves to do its work as efficiently as possible.

Fibonacci, Golden Ratio, Golden Ellipse, Constant Ratio curves, Humpback whale fins, Parabolas, Orbits of celestial bodies, and many more.

Man produces windmills focusing on economics to maximize profits.

If man would closely examine the natural curves and applied them to windmills, they could build them in a way that would minimize the losses due to wake rotation.

Some focus on the past. Others focus of the future and the possibilities that seem to be impossible now.

Ed

[MattM]

I agree, MagnetJuice.

A traditional blade can disrupt the air going over and around the blades in proximity to it.  Nature has a way of avoiding those limitations using simple geometry. 

The curves in nature tend to bend the disturbance away from proximal surfaces.  Its there to see but people over simplify what they focus upon.  While aerofoils sort of resemble natural occuring shapes, you'll  be hard pressed to find them.  Aerofoils in nature are cross sectionally not aerofoils, but rather some level of s-curve that are shape changing.

Man loves symmetry and consistency, which is why blade shapes are made from rigid materials and have elements to look elegant.  Pretty, sure, but unnatural.  So it probably isn't correct at any level to suggest dwe borrowed aerofoils from nature.

[joestue]

I agree, MagnetJuice.

A traditional blade can disrupt the air going over and around the blades in proximity to it.  Nature has a way of avoiding those limitations using simple geometry. 

The curves in nature tend to bend the disturbance away from proximal surfaces.  Its there to see but people over simplify what they focus upon.  While aerofoils sort of resemble natural occuring shapes, you'll  be hard pressed to find them.  Aerofoils in nature are cross sectionally not aerofoils, but rather some level of s-curve that are shape changing.

Man loves symmetry and consistency, which is why blade shapes are made from rigid materials and have elements to look elegant.  Pretty, sure, but unnatural.  So it probably isn't correct at any level to suggest dwe borrowed aerofoils from nature.

I dont think you understand that the airfoils in nature are optimized for the bird or fish doing work on the water or air. Not the fluid doing work on the animal.

This is why airplane props and windmills are somewhat inverse profiles.

[JW]

When I lived in Miami Fl we used to go out to the Everglades. Lots of Airboats out there. I remember looking at a guys boat he had indexable carbon fiber blades they were both smooth and curved. Very nice looking. Got a real good look at the system.

[Adriaan Kragten]

If man would closely examine the natural curves and applied them to windmills, they could build them in a way that would minimize the losses due to wake rotation.

Ed

Wake rotation is the result of the law of Newton. If the wind exerts a torque on the rotor, the rotor will exert a torque on the wind in the opposite direction. If this torque is large because of the low tip speed ratio, the wake behind the rotor will rotate strongly against the direction of rotation of the rotor and so a lot of energy will be lost in the wake. The torque level of rotors with a high tip speed ratio is much lower for the same power as for rotors with a low tip speed ratio and therefore much less power will be lost because of wake rotation.

[MattM]

joestue,

In all politeness, your summation is incorrect.  You can shoehorn circumstances to appear as such.  But most of nature harvests the energy in the environment rather than fights against it.  Quite contrary to your theory, anything that struggled within its environment would be replaced with life that benefits from it.

Adrian,

The wake is typically from disorderly airflow, not really a direct product of torque.  If anything it may be an inverse relationship to the torque imparted by wind on the blade.  Birds use tip feathers to create linear flow at the end of their wingspan.  Without them, sustained gliding becomes difficult.  Birds rely heavily on the glide phase.  The better gliders have better tip controls.

[Adriaan Kragten]

joestue,

In all politeness, your summation is incorrect.  You can shoehorn circumstances to appear as such.  But most of nature harvests the energy in the environment rather than fights against it.  Quite contrary to your theory, anything that struggled within its environment would be replaced with life that benefits from it.

Adrian,

The wake is typically from disorderly airflow, not really a direct product of torque.  If anything it may be an inverse relationship to the torque imparted by wind on the blade.  Birds use tip feathers to create linear flow at the end of their wingspan.  Without them, sustained gliding becomes difficult.  Birds rely heavily on the glide phase.  The better gliders have better tip controls.

No, I am sure that I am right. You have different types of wakes. One is the rotation of the whole wake behind the rotor. So this is the wake of the stream-tube behind the rotor as given in figure 4.1 of KD 35. As A2 = 2 * Ar, this wake expands from the rotor diameter D up to a diameter of square root of 2 times D. This is called wake rotation and the losses due to wake rotation are presented in figure 4.2 of report KD 35. The losses due to wake rotation are higher as the torque level is higher. So these losses are higher as the tip speed ratio is lower.

The second type of wake is coming from the blade tips because there is a pressure difference in between the front and the back side of the airfoil. This kind of losses are called tip losses and these losses depend on the number of blades B. These losses, including the losses of wake rotation, are presented in figure 4.3 of report KD 35. Tip losses create a lot of turbulence and noice.
The third losses are caused by airfoil drag. If the boundary layer isn't laminar but turbulent, like you have with airfoils with a sharp nose, airfoil drag can also result in turbulence behind the rotor plane and so in noice production. The influence of airfoil drag on the maximum Cp is presented by figure 4.5 up to 4.11 of KD 35. These figures include the losses due to wake rotation and the tip losses depending on the number of blades. So losses due to wake rotation are always there even if there is no turbulence.
The fourth losses are caused by the fact that the blade length k is generally shorter than R. These losses are calculated by formula 6.3 of KD 35.

The aerodynamic theory as presented in my report KD 35 is not my invention but is developed by Betz and Glauert already in 1935 and I have used the same terminology which was used in many earlier reports of the University of Technology Eindhoven and other technical studies about this subject. My report was checked by my boss Ir. Paul Smulders and by Dr. G. Stacey who has a degree in aerodynamics and you can believe that what I have written is right.

[JW]

Ahhh here it comes the Betz Limit

[Adriaan Kragten]

Ahhh here it comes the Betz Limit

As you can see in figure 4.2 of KD 35, the line for only wake rotation is approaching the horizontal dotted line of the Betz limit (Cp = 16/27 = 0.593) at very high tip speed ratio's. So if there would be no other losses than wake rotation, the tip speed ratio must be chosen very high to get a high value of Cp. But in figure 4.7 for a 3-bladed rotor, it can be seen that a really high maximum value of Cp th is only possible for a very low Cd/Cl ratio.

If we could obtain a Cd/Cl ratio of 0.03, a maximum Cp th of about 0.455 is realised for a lambda of about 4.5. If we could obtain a Cd/Cl ratio of 0.02, a maximum Cp th of about 0.48 is realised for a lambda of about 6. If we could obtain a Cd/Cl ratio of 0.01, a maximum Cp th of about 0.515 is realised for a lambda of about 8. The value of Cp th found this way, then has to be corrected to find the real maximum Cp with formula 6.3 because the blade lenght k is generally shorter than R or because the airfoil is stalling at low values of r for constant chord blades.

Very big wind turbines have large chords and therefore very high Reynolds numbers resulting in very low Cd/Cl ratios. If big wind turbines have blades which are effecitive up to the hub as used for the wind turbines of Enercon, they therefore can have a maximum Cp of about 0.5 at a lambda of about 8. But the rather small 3-bladed wind turbines with an effective blade length k considerably shorter than R as made on this forum, perform very well if the maximum Cp is about 0.4 for lambda is about 6.         

[MattM]

JW-

Yep, the Betz limit won't be broken AFAICT.  Distribution of the disturbance, however, is directly related to shaping.  So its not about breaking Betz, its about how simple construction can be to approach what people are getting out of elaborate aerofoils.  I would think people would ditch complicated aerofoils if they had a simpler path available.

[Adriaan Kragten]

JW-

Yep, the Betz limit won't be broken AFAICT.  Distribution of the disturbance, however, is directly related to shaping.  So its not about breaking Betz, its about how simple construction can be to approach what people are getting out of elaborate aerofoils.  I would think people would ditch complicated aerofoils if they had a simpler path available.

Yes, chosing the correct airfoil is important. There are modern airfoils which have a very low Cd/Cl ratio for a certain lift coefficient Cl and so for a certain angle of attack alfa. But at higher or lower angles alfa than the optimum value, the Cd/Cl ratio becomes much higher. These airfoils are also sensible for small deformation of the airfoil due to wear, insects or dust. Traditional airfoils with a flat lower side like the NACA 4412, the Gö 623 or the Gö 711 have a higher minumum Cd/Cl value but the Cd/Cl value is rather low for a large alfa range. One must chose such airfoils for a constant chord blade because the lift coefficient is low at the blade tip and high at the blade root resulting in a small angle of attack at the blade tip and a large angle of attack at the blade root. But is is possible to use cambered sheet airfoils with a low camber of 7.14 % (see KD 398) if the design tip speed ratio isn't very high. Figure 4.7 of KD 35 shows that a maximum Cp th of about 0.41 is possible for a 3-bladed rotor with a lambda = 3.5 if the Cd/Cl ratio is 0.05.

[Adriaan Kragten]

My scanner is working again and so I have made a scan of the rotor drawing and the characteristics of rotor no 5. I have added this scan as an attachment. In the figure with the characteristics it can be seen that the maximum Cp is about 0.365 for an optimum lambda of about 1.1. The measured thrust coefficient is about 0.85 for lambda = 1.1 but much higher for lower values of lambda.

Betz has found that the wind speed in the rotor plane has to be reduced to 2/3 V to get the maximum Cp of 16/27. The thrust coefficient Ct for this condition is 8/9 = 0.889. The real Ct is always lower for at least two reasons. The first reason is that the blade length is shorter than R. The blade length of the tested rotor is about 2/3 R which means that 1/9 of the swept rotor area isn't effective and so only 8/9 of the swept rotor area is effective. This results in a Ct = 8/9 * 8/9 = 0.79. The second reason is that the rotor has tip losses which results in a decrease of the pressure difference at the blade tip and so in decrease of the thrust. So a reasonable Ct of this rotor is about 0.75 for the optimum lambda. The fact that one has measured 0.85 for the optimum lambda is an indication that probably there has been some tunnel blockage and that therefore a maximum Cp of 0.365 is a bit too optimistic. I think that a maximum Cp of about 0.32 is possible for such a rotor.

 Rotor no 5, Göttinger Lieferungen 1932.pdf (15.46 kB - downloaded 47 times.)

[Adriaan Kragten]

I have also made a scan of the pictures of rotor no 2 and have added this scan as attachment. This rotor has blades for which the chord increases at increasing radius. The blade angles are much larger than that of rotor no 5, especially at the blade root. This rotor has been measured for a yaw angle (called beta) of 0° and a yaw angle of 10°. The difference in between the curves is very large for only a difference of 10° in yaw angle. This rotor has a maximum Cp of about 0.3 at a lambda of about 0.9. The Cq for lambda = 0 is about 0.48 which is much lower than the value of 0.6 which is realised for rotor no 5.

 Rotor no 2, Göttinger Liefrungen 1932.pdf (146.43 kB - downloaded 43 times.)

[topspeed]

Ahhh here it comes the Betz Limit

According to Betz...2 dimensional propeller creates a slowed down airmass behind it...and thus no more than 59,3 % efficiency can never be achieved....with a 2 dimesional propeller disc  (disc shaped actuator).

https://en.wikipedia.org/wiki/Betz%27s_law

[MattM]

Just an FYI, topspeed, but the Betz limit is kind of a monthly topic rotation item around these parts.

[topspeed]

double

[topspeed]

Just an FYI, topspeed, but the Betz limit is kind of a monthly topic rotation item around these parts.

Yes it is good to remember..since there is at least one company that actually measures more than the 59,3% of their products.

I have't even been close yet.

I wish this conmpany would somehow verify the results.

[topspeed]

triple

[Adriaan Kragten]

Ahhh here it comes the Betz Limit

According to Betz...2 dimensional propeller creates a slowed down airmass behind it...and thus no more than 59,3 % efficiency can never be achieved....with a 2 dimesional propeller disc  (disc shaped actuator).

https://en.wikipedia.org/wiki/Betz%27s_law

The Betz coefficient is not the same as the aerodynamic efficiency. A Betz coefficient of 16/27 corresponds to an aerodynamic efficiency of 8/9. The difference is explained at page 15 of my report KD 35. For the aerodynamic efficiency, one has to compare the power at the beginning and at the end of the stream-tube as given by figure 4.1.

[topspeed]

Ahhh here it comes the Betz Limit

According to Betz...2 dimensional propeller creates a slowed down airmass behind it...and thus no more than 59,3 % efficiency can never be achieved....with a 2 dimesional propeller disc  (disc shaped actuator).

https://en.wikipedia.org/wiki/Betz%27s_law

The Betz coefficient is not the same as the aerodynamic efficiency. A Betz coefficient of 16/27 corresponds to an aerodynamic efficiency of 8/9. The difference is explained at page 15 of my report KD 35. For the aerodynamic efficiency, one has to compare the power at the beginning and at the end of the stream-tube as given by figure 4.1.

 

So within that ...can VAWT actually "legally" exceed the 59.3 %

[MattM]

100% efficieny of 59.3% is your theoretical maximum.

[Adriaan Kragten]

100% efficieny of 59.3% is your theoretical maximum.

This is again the wrong use of the word efficiency. The maximum Cp is 16/27 = 0.593 if there are no losses. The Cp is defined as the mechanical power supplied by the rotor devided by the wind power which if flowing through an area equal to the swept area of the rotor. But this is not the aerodynamic efficiency! For any efficiency we always take the power which comes out of the system devided by the power which goes in the system. The system for a wind turbine, is the expanding stream-tube which starts far before the rotor and ends far behind the rotor. According to Betz, the cross sectional area far before the rotor is 2/3 of the swept area of the rotor. The aerodynamic efficiency is therefore a factor 3/2 larger than the maximum Cp and so 3/2 * 16/27 = 8/9. So one should not use the word efficiency if one means the Cp.

As there are always losses due to wake rotation, tip losses and aerodynamic drag of the airfoil, the real maximum Cp will always be lower than 16/27. The practical maximum is about 0.5 for very big rotors. The only way to come higher than 16/27 is by changing the shape of the stream-tube by using a duct around the rotor. But an effective duct must be much larger than the rotor diameter and an effective duct is therefore more expensive than a bigger rotor which supplies the same power as a smaller rotor with a duct.

[MattM]

Theoretical maximum is not 8/9ths, but a non-existent perfection.  That 8/9th would be a practical maximum.  And that is based on information gleamed from a wind tunnel, which you give reasoning for the limitations involved.  Nobody is trying to hit that level of precision here.  Nobody is aiming at 2/3 or .5 efficiency, which requires substantial budgets.  This is just an exercise in language.  There are no legally defined terms that restrict an advertisement from claiming the ability to harvest higher than 59.3% of available energy by area or volume.

[DamonHD]

@MattM: Fraudlent claims are widely legally contestable, especially if made in bad faith.  But maybe I'm misunderstanding you!

Rgds

Damon

[MattM]

I certainly do not advocate for lying.

Here in the states you can file complaints but they usually go nowhere.  And to sue generally requires you to prove considerable damages incurred.  A true circus nowadays.

[topspeed]

@MattM: Fraudlent claims are widely legally contestable, especially if made in bad faith.  But maybe I'm misunderstanding you!

Rgds

Damon

What I have seen are claims of small turbines being exaggerated.

[Adriaan Kragten]

Theoretical maximum is not 8/9ths, but a non-existent perfection.  That 8/9th would be a practical maximum.  And that is based on information gleamed from a wind tunnel, which you give reasoning for the limitations involved.  Nobody is trying to hit that level of precision here.  Nobody is aiming at 2/3 or .5 efficiency, which requires substantial budgets.  This is just an exercise in language.  There are no legally defined terms that restrict an advertisement from claiming the ability to harvest higher than 59.3% of available energy by area or volume.

You have clearly not understood what I have written. I don't say that the maximum Cp can be higher than 16/27. I only say that a maximum Cp of 16/27 corresponds to an aerodynamic efficiency of 8/9 and that you must not talk about efficiency if you mean Cp. So the maximum efficiency is 8/9 if there are no losses but that is because, if we talk about efficiency, we relate the generated mechanical power of the rotor to the area of the stream-tube far before the rotor and not to the swept area of the rotor which is used for the definition of the Cp. Believe me, this is way how the aerodynamic efficiency and the Cp are defined in aerodynamics.

The fact that a Cp of 16/27 corresponds to an aerodynamic efficiency of 8/9 demonstrates that the real conversion of the kinetic power in the wind to the mechanical power of the rotor is much better than if you take only the ratio 16/27 of the maximum Cp into consideration. But we normally use the Cp because it is more practical to use the swept area of the rotor as reference area for the undisturbed wind speed than to take a 2/3 smaller area far before the rotor. But if we calculate the generated mechanical power P (using formula 4.1 out of report KD 35) we have to use the undisturbed wind speed V and this is the wind speed far before the rotor or the wind speed in the rotor plane if no rotor is placed. That is the problem with the Cp; one compares two different situations which can't happen at the same time. One divides the generated mechanical power of the rotor by the power flowing through the rotor plane if no rotor is placed.

Many people believe that Cpmax = 16/27 is the maximum fraction of the kinetic power in the wind which can be conversed into mechanical power of the rotor. But then my question is: What has happened with the remaining 11/27. Nobody can answer this question because the believe that 16/27 is the maximum fraction which can be conversed into mechanical power is incorrect. Cpmax = 16/27 corresponds to an aerodynamic efficiency of 8/9. If now it is asked: What has happened with the remaining 1/9? The answer is that 1/9 is just the kinetic power which is left in the wake far behind the rotor.

The derivation of the Betz coefficient is given in chapter 4.2 of  my report KD 35. I have used the same method as used by Betz in 1926. So what is done is that the kinetic power is determined in the area A1 far before the rotor and in the area A2 far behind the rotor. The difference is the mechanical power which can be extracted by the rotor if there are no losses. If you compare this mechanical power to the power available in the area A1, you get the aerodynamic efficiency. If you compare this power to the kinetic power which is flowing through an area equal to the swept rotor area, for no rotor in place, you get the Cp. So if you use the Cp in formula 4.1 of KD 35, the maximum Cp is 16/27 if there are no losses. But in this formula you have to use the swept area of the rotor and the undisturbed wind speed far before the rotor.

The real maximum Cp is much lower than 16/27 because of the four reasons given in chapter 4.3 of KD 35. So anyone who claimes to have measured a Cp higher than 0.5 is a liar or he has measured the wind speed at the wrong place.

[topspeed]

How many percent does the ANEW company lie about their produced electricity coeffiecient when they are at 70% ?