wind turbine Blade Aerodynamics
I’m writing a special section here because I’ve made a little breakthrough. I’ve created a tool that should make the task of matching wind turbine blades with their generators much easier. Please let me know what you think, especially if you still don’t follow, or if you want more!
The Constant-Wind-Velocity Curve
Having drowned myself in math to make sense of rotor aerodynamic performance and I’ve finally found a way to present the performance of a rotor on a simple graph. I hope that with a little explanation, anyone will be able to understand. Hobbyists who like to build their own wind turbine blades may put this “mental picture” to use when selecting sizes and twist for their wind turbine rotor blades, trying to pick parameters that match a given alternator. It also works vice-versa, if choosing an alternator that will work well with a given set of blades.
Starting with the basic chart, the curve is a fairly simple bump, but one side is steep, like a sand dune. The bottom scale is the speed the rotor turns, in RPM, and the vertical scale is power. Be sure to think about mechanical power when looking at the graph! I am not talking about any electrical machines – we could be pumping water with the wind rotor for all it matters. There is simply a driving power at the rotor blades’ shaft and it going to be “used” somehow.
This chart is a plot of a wind rotor’s performance at a constant wind speed. I’ve never seen a graph like this before, but it is useful enough that I want to explain how to use it. Most charts involving wind turbines put the wind speed on the bottom axis, so that the reader can quickly appreciate a whole wind turbine’s performance. Here, I’m breaking the problem down into smaller pieces. By focusing on only the blades, at one wind speed at a time, I can show how the rotor behaves when being forced to work at different RPM’s.
There is an important thing going on in the background when I plot the graph this way. By keeping the wind speed constant, and varying the RPM, it is the ANGLE OF ATTACK of the blades that is being changed. For example, when a rotor is allowed to turn faster in the same speed of oncoming wind, the “total wind” each blade can intercept is changed in both strength and direction because the proportion of the airflow coming from turning increases. Increasing one component and keeping the other constant changes the angle between them. The angle of attack changes – in this case becomes lower.
There is a big step in this curve. The step down on the left side is the point of stall. Any slower than that and there is much less lift generated by the airfoil of the blade, leaving you with diminishing coefficient of lift and diminishing tangential speed. We can refer back to the lift-curve of a typical airfoil to check.
At an angle of attack = 0, this airfoil (which is slightly curved or “cambered”) generates a small amount of lift. If it was the airfoil of a wind turbine blade, this case represents high speed rotation, also known as high Tip Speed Ratio. This is useful to keep in mind: High TSR = Low Angle of Attack and vice-versa.
Forcing the wind turbine to rotate more and more slowly at a given wind speed is like forcing the blades to work at a higher angle of attack. At some point, you probably guessed by now, the angle of attack gets so high that the blade stalls. The lift coefficient drops off drastically, and only in certain cases does it gradually recover. It’s worth knowing, however, that the lift does not drop to zero, just to some lower unsteady value that is much lower than it was before.
The bump-shaped curve looks a lot like the lift-curve of the airfoil, doesn’t it?! The stall is obvious in the sudden drop on the left, just like the sudden drop of lift on the airfoil curve. Also, zero lift would be the maximum speed of the rotor, and you can find that on the constant-wind-velocity chart, too. The peak of the curve on the CwV curve corresponds to the airfoil’s point of peak Lift-over-Drag ratio.
The armchair aerodynamicists out there are going “Ahhha!” So did I.
Back to Constant-Wind-Velocity graph. What happens when the blades get larger? Or a lower TSR is used? I can answer these questions by showing how the graph moves as we tweak the variables that were used to create it (long rambling calculus, don’t ask).
Now this curve is a bit cluttered, but I’ll try to help make sense of it. The red solid line is the same as before. If we were to try a larger diameter, keeping all other values the same, we will see the graph move up to the upper black dashed line. The lower dashed line is what you get if you make the blades a bit smaller. That’s probably intuitive of course: more blades = more power.
Look carefully, and you can also notice that the black dashed lines moved a bit left and right. This is showing what is often learned by the experienced wind turbine builder: larger blades need to turn more slowly, smaller blades can turn fast. All three have the same design TSR – it’s the radius to the tip of the blade that is affecting things.
And what about the infamous TSR? How does that affect the turbine? Of course I had to graph that one, too. That’s the pair of blue dashed lines (above), and maybe it’s easy to guess that by increasing the TSR, the graph moves to the right (higher RPM), and to the left (lower RPM) at lower TSR. You may not expect to see that higher TSR blades seem to have a little bit more power, but that really is the case. The reason for that is much more difficult to explain. I will say that in aerodynamics, things that move faster work more efficiently than things that move more slowly. To learn more about why that happens, you’ll have to read up about “Reynold’s Number”. Google search or Wikipedia that term (proceed at your own risk).
There is still lots more to do with this graph! How about plotting multiple wind speeds on it? This is what that looks like; the progression is a gradually growing hump that rises and spreads out across the graph. All of the bumps are similar; just scaled differently. The units don’t matter much, either. I could be plotting lines at 20, 30 + 40 meters per second, or they could be in miles per hour.
With a bunch of these CWV curves plotted, now it’s easy to find the “sweet spot”. Here is the same chart, this time with peak Power Coefficient (Cp) marked on it. Here at last is the tool that we can use for designing, selecting and matching alternators. I’m giving you a peek at what I have worked out for my own 8-foot diameter rotor, which I am currently using to drive my motor conversion alternators.
Let’s consider what happens when the alternator doesn’t match these blades. The alternator could be so large that it demands more power than the blades can deliver, at any RPM. For example, if the alternator requires 1300 Watts of shaft power to be turned at 300 RPM, it would definitely stall the blades! A smaller alternator, or perhaps we should say a “better matched alternator” would do well if it required that 1300 Watts only when turning at about 500 RPM. We can say it’s matched for this speed, now. Hopefully the match holds well for other speeds, though that isn’t always the case.
Here are 3 different alternators, compared on the same set of prop curves. The difference is clear. There is one alternator (marked with “X”) that is skirting the stall of the blades throughout the entire operating range. The performance of thiw wind turbine will be, shall we say, lackluster. Especially as the blades get bugs and dirt on them!
The right-hand generator power curve has its own problems (marked with diamonds). Now we are looking at a generator that does not put enough load on the blades to keep their speed down. The blades of this wind turbine will turn 100’s of RPM faster than necessary. Furling will be more energetic, too, becoming more difficult to control. And just in case that wasn’t bad enough, this combination will probably make more noise.
Yes, Goldilocks, the middle one is just right! The middle curve, marked with squares, runs right up the middle of the prop curves, hitting the points of maximum Cp all the way.
If only things were that perfect in the real world…
I am currently testing out these charts on my own wind turbine. The generator that I have right now has a power curve much like the X’s on the graph above. It is clearly stalling the blades, too! I am waiting for a storm or some strong winds greater than 40 kph or so to see if the blades can break out of stall at any time. In the light breezes of the past 2 weeks, I don’t think it’s spun more than 300 RPM.
I hope this was informative. Or at the very least helped you with your college homework.
– Steven Fahey, July 27, 2010
2009 Entry (Other thoughts about propellor/wind rotor aerodynamics)
Despite many setbacks, I’ve never given up on making a computer program that can predict the performance of a set of prop blades for any given set of dimensions, wind speeds, and rotational speeds. I suppose it’s not important to most, because building your own windmill blades isn’t really a mystery any more. Certainly Hugh Piggott has done a lot for hobbyists, including myself, to make it easy to find a set of parameters that work and are easy to make.
For me, using a “plug and chug” spreadsheet isn’t enough, but trying to make my own analysis has always proved too daunting and I would give up. Last year, my interest was re-ignited by Jim Overington in New Zealand, with whom I had a stimulating exchange of e-mails and spreadsheets that came close to making the proverbial better mousetrap. The project wasn’t finished, however, partly due to being away for holidays and also due to a disagreement on how to use airfoil data. While Jim preferred using numerically generated airfoil lift/drag plots, I insisted on using data from the old NACA wind tunnel tests. Due to these exchanges, I suddenly realized that my previous explorations of aerodynamic analysis failed because I didn’t take into account the Reynolds Number that scales airfoil lift and drag up and down with size. A horrible error and it was making my previous results worthless. Small scale models of things simply cannot get the same coefficient of lift as their full-size counterparts, and by using airfoils designed for airplanes, there is a huge penalty to pay.
Cracking open a book called “Airplane Aerodynamics” by Domasch, Sherby, and Connolly, I found an excellent and clear analysis of an aircraft propeller using a method called “blade element analysis”. They took into account all sorts of important details in their analysis, all of which are also important considerations in wind turbine props, too. The biggest blockade to using their analysis at face value was the fact that aircraft propellers ADD power into the wind, accelerating the air, while wind turbines SUBTRACT power from the wind, slowing it down. The directions of some arrows on the diagram are reversed, and of course, the lift on the aircraft prop is on the front, while it’s on the back of a wind turbine blade. I re-drew the diagram for the wind turbine case. Once that was done I could step through the algebra and discovered than any differences in the direction of the arrows would eventually be cancelled out and only a few “-” signs were necessary here and there to make it work correctly.
When the wind passes through the disk of a wind turbine, some of the kinetic energy in the wind is taken out. With less kinetic energy, the air streaming out the back of the wind turbine is slower. At the disk itself, then, the speed of the wind is half of that difference. Before I understood this, any time I did TSR calculations, I just assumed that the prop would see the same wind speed. This isn’t true! The wind is slowed down slightly, making the TSR of the prop slightly higher than you would think. Often a prop turning at a seeming TSR=6 will actually be experiencing TSR=7 or even =8, right at the blades.
Hugh Piggott takes this fact carefully into account in his “blade design” spreadsheet. That must be why his blades work so well, so often, even for novice builders. Hugh’s spreadsheet goes even further and makes sure that there is enough chord for the diameter and speed that the user desires. Without enough blade chord there would not be enough surface area for the lift to act upon to produce the torque. It is calculated in a quick and efficient way, too. Now that I’ve done the work myself I commend him for the thought and effort that was put into it.
Going through this process, I discovered to my horror that my 8-ft diameter blades were given the wrong twist! They were given too much twist and have the wrong angle of attack. I’d intended to have TSR=7, but actually they are more like TSR=4! I did not follow the Hugh Piggott guidelines and so I get what I deserve. I am glad that I did not repeat the error on the next blades, where matching the generator will be much more difficult.
I’ve finally reached the point that I can do just a bit more than Hugh’s spreadsheet. With one program, I can match the generator’s power curve to a prop for any TSR, diameter, chord, twist, and airfoil, in any wind speed. Here is an example:
Diameter = 8 feet
Chord = 7 inches (root)
Taper = 50% (tip)
TSR = 7.0
Airfoil = NACA3415
Angle of Attack = 5 deg
Material = wood
Blades = 3
Altitude = 3000 feet
This graph has several lines plotted. The red lines are power. The lower red line is the power that can be developed in 30 kph wind when the prop is allowed to turn at different RPM’s. The peak power of 330 Watts in 30 kph wind happens at 315 RPM. This means it has a peak Cp of 0.22, at TSR=4.8
I repeated the same analysis at 45 and 60 kph winds and plotted graphs for each. Then I went hunting for the peak values at many other wind speeds and plotted them as the blue line.
Any time that I find the peak Cp for a real prop, then remove the factors accounting for Reynolds number, the peak Cp shoots back up to nearly 59%. Now I can see that the small scale and slow speed of small wind turbines is the key factor preventing high performance. Larger wind turbines gain an advantage simply through their larger geometry.
The way this is plotted, it can be adjusted until it agrees with the plot of the generator’s load and speed. In this example below, on the left you can see that the generator’s load curve is just a bit less than that of the blades, making it a good match to the blades. It is very hard to make a generator this perfect (it’s only a made-up example in this case). Beside it are generator and blade curves that do NOT match. That set-up will run too slow in low winds, and run away in strong ones.
There is one very important conclusion to be drawn from this: The selection of the blade’s airfoil can become important, if the matching of the blade and generator are to be close. There are several classes of airfoils that are designed to sustain their good lift and low drag properties even at low Reynolds Numbers. After all this work it is now clear how valuable they can be. I have now come full circle. At the top of this article, I mentioned a disagreement between myself and Jim Overington about the true value of the airfoil selection. At the time, I downplayed it. It now should rise in importance in my mind, and I also owe Mr. Overington a debt for pointing it out.