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Propeller Dynamics

Essential reading for model aircraft contest fliers. This is the only book on the market explaining propeller theory in non-mathematical terms. A rattling good read, I know, I wrote it.


Wind Turbine Design


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 start.

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.

Figure 1.

Oops, what's this? Where did the downwash and interference vectors spring from, we don't know those!!! Alarm, Alarm, DALEK attack!

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 first place.

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!

Figure 2.

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!


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