Hi there folks!!
Now for another great act of plagiarism! Yes its time to steal
some more of Larrabee's work, and even a little of Liebeck
and Adkins. The latter fellows were engineers at Douglas,
but of late work with Boeing. I was only a few miles from
Boeing Seattle last year, but never got to meet them. (They
didn't know I was there) (they don't know I exist!). I shall
have some more words to say on this neglect of Supercool later.
Today's lesson is on wind turbines. Pens ready?
Backs straight? Chewing gum stuck under the table? Then let's
Several years ago my son Jim figured he would like to retire
at age 30. He would have made it too, but the stock exchange
proved fickle and now he has a new plan. Yes, he is going
to sell power to our electricity utility here in sunny (oops,
I mean windy) Western Australia. All he has to do is build
50 windmills, put them up on high towers, connect them to
the grid, and watch the money roll in. So far, he has built
one small unit (see mpeg)
and is now ripping Ford Laser cars apart for parts to build
a 10 kilowatt unit. No problem for you, dear reader, but I
drive a 1982 Ford Laser and it gets very agitated when it
sees my son coming in his 4WD!
Click to enlarge
Naturally, he turned to his dear old Dad for
the design of his turbine blades, since Dad is smart but mostly
cheap. Dad, of course, knows that a wind turbine is just a
propeller running backwards, so this should be a piece of
cake. At least, that's what Liebeck and Adkins reckon; just
change a few angle definitions and Bob's yer uncle.
In yer dreams! No wonder Douglas went belly
up. But Larrabee said the same, just change the sign of power
to negative (thats right, suddenly power is a vector!) and
carry on using the propeller equations. Do I have to tell
you this doesn't work? Or have you guessed already from the
tone of my text?
I went to some wind turbine text-books, and what a dog's breakfast
they are. In not one of them did I find a proper vector diagram
to illustrate the theory. Wrong diagrams, yes. Right but confusing
diagrams, yes. There was nothing to it but to do it all myself.
So don't be too surprised if I have it wrong as well.
So lets look at this from what we already know. A propeller
rips around and blows air out the back, the more the better.
A turbine rips around, and sucks air in from the front. Sucks?
That's right, if you do it right, the air zooms in the front,
and comes out the back slowly. Different from propellers,
you see. Also the airfoil on the turbine is upside down, which
is where I started to lose the plot. So you see, this is where
the problems start.
I ran my prop code, and told my son, look at this great design,
it's 80% efficient. Naturally, he was delighted. Good old
Dad! Just one thing Dad: why do the text books say a wind
turbine can't be more efficient than 60%? Why, they are all
idiots, son, can't you see that, where is your faith? So after
he had gone home, I snuck back to the text books.
It seems that the result of the slipstream being slowed down
after going through the turbine, is that all the air piles
up there and prevents a fresh lot coming in from the front.
Not much chance of getting energy out of it then. I guess
you could call this a choking effect. The more blades, the
wider the blades, the more the choking, which eventually shuts
you down at 60% energy conversion.
Now one for the theorists. A key feature of propeller theory
is a hang over from actuator disc theory, with a key finding:
the slipstream velocity a long way downstream is double the
increase in airspeed at the prop disc. Well, something like
that. So what about a wind turbine? Can we say that the slipsteam
velocity behind the turbine is a half that at the turbine?
Sounds reasonable to me, but I can't see where it is in the
turbine theory of our propeller Gods.
Time to move on. How can we design a wind turbine blade? First
off, check out Figure 1. We've got the wind whistling in from
left to right, so that's one thing we know. Also we know what
design RPM to shoot for, so the speed of the blade elements
are known. Just to make it easy on ourselves, we'll guess
the shape of the blades as well, so we know the radial distribution
of airfoil chord. Hell, we've nearly finished, all we need
to find out are the blade angles and the amount of power we
can suck out of the wind. The figure has all these vectors
marked on it, plus a few extras.
Oops, what's this? Where did the downwash and interference
vectors spring from, we don't know those!!! Alarm, Alarm,
Well, I guess this thing is going to work by generating lift
from the airfoil. That's going to make it rotate. And you
get lift as a result of the air downwash from the airfoil,
so that better be somewhere on the diagram. The interference
factors are just the components of the downwash vector, so
that's no mystery. The torque component of the interference
velocity "a'" is pointed in the same in the same
direction as the rotation, so that adds to the inflow speed
in the plane of rotation. Which is good, because its equal
and opposite reaction force is what spins the turbine in the
The thrust interference velocity "a" is in the opposite
direction to the wind, which is why the windspeed down wind
is slowed up. Also, its reaction force tries to push the tower
over, so we might like to know what that is.
Read on, we are almost finished.
Check out Figure 2. Its just the same as Figure 1, but this
time we have some blade angles marked. Here is the trick.
If we knew "phi", then we could quickly calculate
the downwash and interference velocities. Then if we knew
"alpha", the section angle-of-attack, we would have
the blade angle. Vunderbar!
This is easily done. First we choose a lift
coefficient for our airfoil which we know works well, say
0.5. Most airfoils work well for that value. Then we start
with a small guess for the blade angle, and then run over
a range of "alpha's" for each of which we calculate
the achieved lift coefficient using a formula from Lieback
and Adkins, or even the "inexact" formula of Larrabee.
If we get a value that matches our value of 0.5, we are done.
If not, we must try a new, larger value of "beta"
and start again. Eventually we'll get one, and then we know
the blade angle. From some other formulae, we also get the
power absorbed and thrust on the tower for each blade element.
Add them up and we are done
Pretty cool, huh? But read on , we are almost finished. Have
we got the best blade shape? Remember, we guessed that. Perhaps
we need to modify it, but how?
The answer lies in the slipstream. The best shape is found
when the value of the stream velocity behind the turbine is
radially uniform. With a few more equations, we can calculate
those values. If the value is too great at one point, then
the chord is too great. Reduce it by simple proportion compared
to a reference chord on the blade, say at the 75% radius.
Then repeat all of the above until the the stream velocity
is reasonably uniform. Job done.
But read on, we are almost done!
Just to help out, I include below the source code in Quick
Basic. If you run this you will see the graphics in action
and hopefully that will add to your understanding. Please
note this is not freeware. If you want to run it, you must
send me AUD$10000. If you don't send me the money, I won't
show you the deliberate mistake I made in the code. If you
find the mistake(s), tell me and I will let you run the code
100 times before you must pay again.
The source code is reasonably commented, so have fun (download
source code). Also attached an EXE file (download
exe, 60k) in case you don't have QB. It will run under
DOS and Win 98, but it won't run under Windows XP (expletive
deleted). Probably not Win 2000 either. Thank you Bill.
Now you can't read on, because you have been done!