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

[John Denker <jsd@av8n.com>]

On 04/03/2008 10:02 AM, Rick Tarara wrote:

> The available energy may increase as the cube of the velocity (see below), 

I hope that's a typo.

Available _power_ (not "energy") goes like the cube, under these 
conditions, more or less.

> but the transfer to the turbine is via momentum transfer and the air 
> momentum at the blades goes as the square of the velocity (mass per second 
> increases along with the actual momentum).  

That's not the right physics.

The force on the blades depends on momentum /transferred/ which
does not scale like the total momentum of the air.  You're not
deflecting the air through a fixed angle in the lab frame.  It
helps to analyze the system in a frame comoving with the blade,
whereupon the primacy of /angle of attack/ becomes apparent.

You want to fly the blade at the optimal angle of attack, as
nearly as you can.  The optimal angle of attack depends on 
the lift-to-drag ratio, including contributions from parasite
drag and induced drag.  This is nontrivial because induced 
drag depends strongly on angle of attack.

> But then we have a torque 
> problem as well,  as the wind is hitting different distances from the axis 
> of rotation (effective lever arm then 1/2 of the blade length?).  

It's not that simple.  There is a lot of twist in the blade,
because you want both the tip and the root (and everything in
between) to fly at more-or-less the optimal angle of attack.

Twist might be something you can discuss with students.

Most gen-ed students -- and indeed most of the professional 
physicists I know -- are not very skilled at three-dimensional 
geometry.  Feynman said "maybe if we were birds" we would have 
more of a feel for maneuvering in 3D.

You can pick it apart into a collection of N problems in 2D
rather than one big problem in 3D.  Consider the blade at the
moment when it is horizontal, and do two of these 2D cases:
 a) A section near the hub is moving upward at a speed rω while 
  the wind is moving horizontally at a speed |V|.
 b) Meanwhile a section near the tip is moving vertically upward
  at a speed Rω while the wind is moving horizontally at the
  aforementioned speed |V|.  R is the radius, and R >> r.

In each case, draw the two vectors and their resultant.  
  http://www.av8n.com/how/htm/yaw.html#sec-p-factor-angle
The resultant is the /relative wind/ seen by the blade section.
The two relative wind vectors (a) and (b) are not parallel!
Therefore if you want the sections are each to fly anywhere
near the optimal angle of attack (which you do!), the
blade angle must be different at the two locations.  Lots
different.  That is to say, there needs to be lots of twist.

To make this work, for a fixed blade, you need the rotational
speed to scale in proportion to the airspeed.

> At the 
> generator itself, doubling the rotational speed would double the rate of 
> flux change and therefore double the induced emf, but that should quadruple 
> the power through a fixed load V^2/R.  

In the immortal words of Richard Nixon, we could do that, but
it would be wrong.  Applying a load like that to the rotor
would not allow the rotor to spin at the optimal speed.

In the real world, they control the excitation of the field
coils, so that the torque of the generator matches the torque
of the rotor at an appropriate speed.  This costs some efficiency
in the generator.

At the next level of detail, they trade off a little bit of rotor
aerodynamic efficiency to improve generator efficiency at the 
lowest airspeeds.  Most systems have hubs that can change the
blade angle ... for just this purpose.  Of course changing the
blade angle screws up the calculation of optimal twist that we
did a few paragraphs ago ... which explains why the overall
efficiency goes to zero at a non-zero airspeed:  At low airspeed,
if you coarsen the pitch too much, the roots will go to /negative/
angle of attack while the tips are still at a too-large angle 
of attack.


>  .... Published curves still look (to me) pretty 
> linear over an appreciable range of wind speeds.

See previous note, including:
  http://www.av8n.com/physics/img48/wind-power.png