# Friday Math Movie - New wind turbine design

By Murray Bourne, 14 May 2010

This makes a lot of sense to me. If their claims are right, it's more likely individual homes could generate their own power.

Very small changes in propellers can make huge differences in efficiency, and of course, math is at the core of the analysis.

When can I order mine?

### 6 Comments on “Friday Math Movie - New wind turbine design”

1. Justin Moretti says:

I can't actually see the video from where I'm typing. If the still snapshot is right, it closely resembles the compressor turbine of a jet engine. What's not clear is the scale. Power generated would be a function of torque and subsequent rate of spin, which in turn depends on the amount of power the blades can accept from the moving air. This in turn depends on mass flow past the blades, and mass flow is proportional to blade area, wind velocity and air density. The first is fixed by the design; the second is at the mercy of the prevailing winds; the third is to all intents and purposes invariant (unless you live at very high altitude, and then it only varies to the detriment of wind power). All of this depends on whether a light breeze is enough to start the blades of your precious wind turbine turning against starting friction, ongoing friction losses and back-EMF at the generator shaft, and even an impossibly 100% efficient multi-blade, controllable-pitch turbine can't change that when you haven't got enough blade diameter to give you adequate mass flow, torque etc.

On a whole-city scale, I think you'll have an additional problem in that air coming off the first wave of turbines isn't going to be moving in the appropriate manner to deliver energy efficiently to the next wave. That's why multi-stage compressors (or steam power turbines) are carefully crafted so that each successive stage is specifically shaped to accept the gases coming off the previous stage, depending on whether that stage is spinning under power or being spun. Add the aerodynamic disruption from houses, fences, trees, etc. and the downwind area of town might not derive all that much benefit. This is why all the wind farms I've seen are stretched out in a straight line. Everyone will rush to live on the leading edge of the prevailing winds, and those with the most money will win.

2. Murray says:

Thanks for the analysis, Justin. In the post, I did cast some doubt by saying "If their claims are right..." The old adage could hold here: if it seems too good to be true, it probably is.

3. Justin Moretti says:

You're welcome. Assuming perfect conversion of wind kinetic energy into power, a rough back of the envelope calculation neglecting inefficiencies showed wattage is about d-squared times v-cubed divided by about 50, with velocity in kph, blade disc diameter in metres, and pi plus conversion factors and constants subsumed into the denominator. I'll have to go hunt up specs of actual turbines in the field and see how close I got.

What this all boils down to if I'm right is that a seven metre diameter turbine with perfect conversion gives an output in kW equal to the cube of one-tenth of the wind speed - a brutal simplification, but it gives an equation that any reasonable and interested person could do in their head. Echoes of the story (? urban myth) of the fellow at the Trinity A-bomb test (it may have been Fermi) who grabbed a piece of paper, tore up the strips and flung them in the air as the blast wave reached the observation post, and correctly determined the order of magnitude of the blast from where they fell. Mathematics in action!

4. Justin Moretti says:

I went back and checked using this - [page no longer available] - for my source. Based on measurements made from the screen based on their description of turbine tower height, and taking their 10kW output at 5.5m/s as a check, I get 9950 to 10200W depending on whether I round the wind velocity conversion up or not. Not bad for an order-of-magnitude calculation made from first principles, but I still worry about the fact that I haven't allowed for losses and I'm still getting their promised output to within 50 watts out of ten thousand. That's TOO accurate.

5. Phil Randall says:

Question. For many years we have seen typical wind turbines operating on farms pumping water. Not vastly different from modern turbines except for a simple observance.
In viewing the swept area of a "farmers windmill" the blade surface facing the wind is some 80 % of swept area.
In viewing the swept area of a "commercial wind turbine" the blade area facing the wind is only some 10 % of the swept area.
I would like to understand the reasons for this obvious difference.
Phil Randall

6. Murray says:

Great question, Phil. I don't claim any expertise in this area - perhaps a reader can help?

Aircraft propellers also show some variance in blade area.

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