Power wind calculation and blade area

[Grinch]

Dear all,
I'm just doing some thinking around the maximum theoretical power that can be extracted from wind by using a wind plant.
Derivation of the power inside a block of wind is not difficult then I'll simply report some consideration regarding this subject (look at the attached paper for details).
At first we've to assume some air density, speed, and area of the air parcel that is going to the plant, put all together will give (look at the Pwr relation on the page 2 of attached PDF):

Pwr = 1/2 * (rho * V^3 * A)

As defined A stand for the area of the air parcel.
Now I've a question about the role of the blade area into the output power that I can generate from the plant. Looking at the formulae seems that the blade area have no influence because the area A are simply calculated as pi*R^2 where R is the blade length.
Of course there is some consideration about the blade speed, if too slow only a little part of the air will be collected, if too high turbolence may also decrease the output power and then there is a compromise from the air speed and the blade speed but the concept is there is a way to show a dipendence from the blade area and the total air parcel cross section area or really this is not important and the final resul will be independent from the geometrical blade area?
In other word, if I suppose a blade area equal to S and n°3 blade my total useful area will be equal to 3S then, if we've a unity ideal conversion ratio the energy that I can collect from the air parcel should be proportional to 3S then also the final power sould be proportional to the 3S area... or not... (take also care that my question is different than the Bets law http://en.wikipedia.org/wiki/Betz'_law, is just a matter to know if really the blade area isn't important or have some role into the final energy and then power computation).

Thanks in advance

Grinch

 

[Grinch]

I'm sorry to boring again, could someone take a look at my question?

Tnx
Grinch

 

[Crosson]

I looked at your question and I think that the analysis of the WPD you have presented is reasonable.

is there is a way to show a dipendence from the blade area and the total air parcel cross section area or really this is not important and the final resul will be independent from the geometrical blade area?

The total power collected will very much depend on the blade area and the design of the blades. To calculate this dependence you would need to analyze the Bernoulli force that spins the blades:

http://hyperphysics.phy-astr.gsu.edu/Hbase/pber.html

The problem is that the WPD is just a theoretical maximum. To determine the output in a specific case we would need need to analyze the shape of the blades in a detailed way.

 

[Grinch]

Dear Crosson,
first we like to thank you for your time, after that your phrase give me some comfort:

The problem is that the WPD is just a theoretical maximum. To determine the output in a specific case we would need need to analyze the shape of the blades in a detailed way

I'm totally agree with you, the power extracted from the wind is just a theoretical calculation, in other word I'll expect to extract a maximum power equal to the calculated one if I'm able to collect every energy from the air parcel, but a simple things show that is not possible for many reason, most simply when blade are rotating a time t only a part of the air parcel will be intercepted and then some energy is lost, at t+dt blade rotate but some other part of the air parcel has gone away trought the area not covered by the blade and so on... focusing on this simple reasoning I've figured out that this formula isn't really applicabile and give us a maximum theoretical value that can't be absolutely reached. Other than there is also a consideration concerning the Betz Law that set another limit to the maximum power achievable from wind.
Then from this point of view the maximum power value should be considered as a indicator of the maximum final value of the power that I'll can extract from wind...

Well Crosson, again I like to say thank you for your time.
If you have some other ideas or calculation or some other infos please feel free to let me know.

Regards
Grinch

 

[RTW69]

Maximum power is just the kinetic energy in the wind. There are numerious wind turbine websites that have performance curves for various wind machines. You can deduce system efficiencies from these curves. You can Compare different efficiencies with different blade diamenters, configurations etc.