Step 6: Blade design
Lift – based wind turbines are by far the most efficient type, so we decided to use an aerofoil (wing) shape used in wind turbines already, the imaginatively named FX-83-W-108. See http://worldofkrauss.com/foils/52
This aerofoil was chosen because it has a good Lift/Drag ratio of 68.785. This means that for every force it creates in drag, it creates 68.785 times more force in lift. The aerofoil also has a broad range of angles of attack in which it works, from -5 to +8 degrees. Basically this just gives us a little margin for error when we make the blades.
The first step in optimising the blade design is really to calculate how much power there is in the wind. Since our project involved a wind tunnel, we had a more or less constant wind speed. The formula is:
Wind Power = 0.5 * (air density) * (area) * (wind speed)^3
This gives power in Watts – make sure you use S.I units (i.e. metres, kilograms, seconds, etc.)
-The air density at sea level at 20 degrees C is about 1.204 kgm -3
-The area refers to the area that the turbine will occupy. For our design, this was the area of the end of our duct, i.e. pi * 0.14*0.14 = 0.0616 square metres.
-The wind speed is the speed of the air through the area the turbine will occupy. As you can see, a small increase in wind speed makes a large increase in power.
We had a wind speed of about 11 metres per second and an area of 0.0616 square metres, so this gave us the power in the wind as about 50 Watts.
Due to something called the “Betz Limit,” the maximum possible power that can be extracted from the wind by a turbine is 59.3% of this wind power. I won’t go in to the reasons here, but you can look it up if you’re really interested…
So now we’ve got our maximum possible power output as 59.3% of 50 Watts, which gives about 29 Watts.
This number assumes that the turbine is 100% efficient, which is impossible. The large white turbines you see all over the place these days manage about 75 – 85% efficiency, which is quite impressive. We’re not that good, so 50% efficiency sounds reasonable. This gives us the theoretical power output from our turbine as about 14 Watts.
The next bit is some more maths unfortunately – but this is the last bit!
What we need to do now is work out how big the blades need to be to achieve our calculated power output. This also depends on the speed we want the turbine to spin at.
The aerofoil we chose works best with an airspeed of about 22-30 metres per second (50-70 mph), so we need to make sure that the turbine will spin fast enough to allow this.
To work out the speed of the blade at a certain point, we use:
U = ω*r
- U is the speed of the blade
- ω is the rotational speed in radians per second
- r is the radius in metres.
We chose a rotational speed of 1500 rpm. To convert this to radians per second, multiply by 2*pi, and then divide by 60;
(1500 * 2 * pi)/60 = 157 radians per second
The blade tips will have a radius of 140mm from their centre of rotation (because of the size of the duct), so the tip speed will be:
U = ω*r = 157 * 0.14 = 22 metres per second
So this is how fast the blade is moving through the air perpendicular to the wind. To find the total airspeed experienced by the blade at the tip, we use Pythagoras:
Total speed = √((U^2 )+V^2)
U is the tip speed, measured earlier as 22 metres per second
V is the wind speed, calculated before as 11 metres per second
So we get a total airspeed of 24.6 metres per second at the blade tip, which is nicely in the middle of the range of optimum speeds for our aerofoil.
OK, next the big equation to get our blade area:
Blade area = Power/[ 0.5*ρ*√(U^2+V^2 )*(Cl UV-CdU^2)]
-Power is the wind turbine power we calculated before, 14 Watts
- ρ is the density of air, again about 1.204 kg per cubic metre
-V is the wind speed in metres per second – in this case 11m/s
-U is the tip speed of the blades in metres per second – in this case 22m/s
-Cl is the coefficient of lift for our aerofoil, found on the data sheet. Our aerofoil has a coefficient of lift of 1.138
-Cd is the coefficient of drag, which is 0.01654
So from the equation, we get the optimum blade area for our turbine’s speed and power output to be 0.003536 square metres.
We decided to have two blades (any more and they would be very small and fragile) so this gave us each blade area as 0.001768 square metres. Using a blade width of 2.5cm gives a blade length of about 7cm.
So now we have our theoretical power output, our turbine’s rotational speed, the number of blades we need, and the dimensions that the blades need to be. We’re almost ready to do a CAD model of the blades now – there’s just a tiny bit more maths first…
The final thing we need to work out is the angle of the blades at various points along the blade radius. This is for a couple of reasons – firstly, the aerofoil works best at an “angle of attack” of 5 degrees. This means that the blades will work best if they are tilted up by 5 degrees to the direction of air flow. The second reason is that the blades will experience airflow at different angles along the radius of the blade, as the blade is moving faster through the air at its tip than it is at the root.
To calculate the angle “α” that the blades need to be turned into wind from their direction of travel, we use:
α = 95 - tan^(-1)(U/V)
-U is the speed of the blade at a specific radius (U = ω*r)
-V is the wind speed, always 11m/s in this case
Since our blades will be 7cm long, and have a maximum radius of 14cm, the root of the blade will be 7cm from the centre of rotation. So from root to tip, the angles are:
Radius(m) V(m/s) U(m/s) α(degrees)
0.07 11 10.99 50.0
0.08 11 12.56 46.2
0.09 11 14.13 42.9
0.10 11 15.70 40.0
0.11 11 17.27 37.5
0.12 11 18.84 35.3
0.13 11 20.41 33.3
0.14 11 21.98 31.6
OK, the maths is finally done, and now we can go on to the next step – modelling the blade in CAD software.
You can use the aerofoil coordinates from the website, save them as a .txt file, and then import them in to Solidworks to give the aerofoil shape. Once the coordinates are saved as a .txt file, go to insert > curve > curve through xyz points in Solidworks, and insert your aerofoil file on to one of the basic planes. Then select this plane, click on the sketch of the aerofoil, and select “convert entities.” This can then be scaled and rotated to a certain angle using the “move entities” toolbar.
Then, go to insert > reference geometry > insert planes, and insert 7 planes, each at a distance of 10mm from each other. Select each plane in turn, click on the aerofoil shape, and select “convert entities.” This will project the aerofoil on to each plane. As before, this can then be scaled (we used a scale of 2.5, to make the blade 2.5cm from leading to trailing edge) and you can also rotate the blade to the angles calculated before.
Then select “lofted boss/base,” and select all the angled aerofoil profiles. This will give you the main part of the blade!
All that is left to do now is make a “key” to allow the blade to slot in to the hub, and also a piece at the end to slot in to the outer ring. These can both be done by sketching on the appropriate planes, and using the “extrude” tool to make them 3D.
The blade is now ready for Rapid Prototyping!