Wind Turbine Power: Calculating Max Output

[Donald.]

Hello everyone, I've found these forums extremely helpful in the past and I've decided to create an account in order to answer a question that has been going in my mind and I can't seem to find the answer.

The power a wind turbine can generate can be derived from the equation:
$Power = \frac{1}{2}\rho A v^3$ and the area ($A$) of the turbine can be calculated using the area of a circle equation: $A = \pi r^2$.

This equation would give you the theoretical maximum power obtained by the turbine, however all turbine have a center from which the blades are attached. In a 'real life' situation would one use the whole circle or the 'doughnut like' shape area to calculate the maximum power that can be obtained?

The doughnut area could be calculated using the formula:
$A = \pi r_{total}^2 - \pi r_{center}^2$


Thank you,
Donald.
$A = \pi (r_{total}^2 - r_{center}^2)$

 

[sophiecentaur]

Your formula is giving the total Kinetic Energy in the volume of air that passes each second. You could never get that much out at one location because that would imply that the wind has to have zero speed afterwards - and how do you get rid of it all?

Any turbine (air / gas / water) can only get a certain amount of the power out (finite efficiency) and I think the efficiency is, in fact, so low that the small adjustments you suggest, although correct, are not really relevant in the overall scheme of things.


This link suggests that you can't do better than 60% in any case. The Betz Limit, apparently. God that man was such a pessimist.