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Re: [Phys-l] Wind generator output versus wind speed.



On 04/03/2008 08:44 AM, Rick Tarara wrote:

From various texts, the theoretical maximum power transfer from wind to
electrical power is cubic in wind speed

OK, roughly speaking.

and also depends on the area swept
by the rotors.

And a bunch of other things.

The cubic part comes from the KE which is of course
quadratic in speed but with the added linear factor of wind mass.

That's not the real physics. It's dimensionally correct, but
that's about all.

-- The KE of the wind is due to motion in a direction perpendicular
to the rotor disk.
-- In contrast, the power comes from forces (i.e. torque) in
the plane of the disk, tending to rotate the rotor.

We agree that the one scales like the other over a certain range
... but that is not to say the one "is" the other. KE is always
KE and always scales like airspeed squared, whereas the force on
the blade scales like airspeed squared only over a limited range
of Reynolds numbers.

.... http://www.clemson.edu/scies/wind/Poster-Schmidt.pdf

Slide 11 above shows some typical power curves

Is that real data? It looks like an "artist's impression" ...
to put it politely.

--many others can be found on-line.

I see lots of "artists' impressions". They are not helpful.

This Danish one looks real to me:
http://www.windpower.org/en/tour/wres/pwr.htm

The part that seems "linear" at first glance is not wildly different
from a cubic. I have added a cubic (in red) to the Danish data
here:
http://www.av8n.com/physics/img48/wind-power.png
Remember we are talking about a cubic that starts at the /origin/,
not at the cut-in point.

We see that the Danish data is a little worse than cubic at very
low airspeeds, and also at very high speeds. This seems entirely
plausible to me.

In any case, the Danish web page calls itself a "tour" ... and is
a long way from being a scientific paper. In particular, we have
no way of knowing what part of the result is due to the aerodynamic
efficiency of the rotor and what part is due to the electromechanical
efficiency of the back-end of the system.

Note that at high airspeed, you *want* the wind turbine to have lousy
efficiency! Otherwise
a) you would break something, or
b) you would incur an absurd capital cost for overdesigning the thing
so as to operate efficiently in exceptional conditions. There's a
big lesson here: Some of what you see can be explained in terms of
physics, but a lot of it has to be explained in terms of engineering
and economics. Just because something is physically possible doesn't
mean it is worth doing.


BTW ... some airplane propellers are optimized for low airspeed,
to give good takeoff performance at the expense of lousy cruise
performance ... and conversely some are optimized for high airspeed,
to give good cruise performance at the expense of lousy takeoff
performance. The power curves are radically different in the two
cases. I would be very surprised if similar words did not apply to
wind turbines.

------------

To summarize:

1) There is a difference between the efficiency of the rotor and the
efficiency of the system as a whole.

2) Physics says "airspeed cubed" for the rotor alone, in this range
of Reynolds numbers ... but that's for an absolutely ideal rotor.
Such a rotor is entirely imaginary.

We expect the rotor -- and the other parts of the system -- to be
less than 100% efficient, especially when operating at very low or
very high airspeeds.

3) Don't rely on the unaided eyeball when deciding whether something
is cubic or not.