[brandmeyer]

The single-blade model reduction is just a rhetorical device used to guide the reader on the principle of energy generation, not on viability.

As you point out, single-blade turbines have been done but they have their own suite of mechanical problems. A more practical way to save capital cost is to under-size the electrical system relative to the blade area. You may still only get 30% capacity factor of the blades and tower, but you could get 40% or better on the electrical system for a small win.

[rwallace]

Thanks! That actually reminds me of something else I was wondering: why only three blades, leaving so much empty space for wind to pass through unimpeded? I understand adding a fourth or fifth would give you slightly less energy per blade, but wouldn't it give you more energy per windmill, for greater overall cost efficiency? Or is it the case that for some reason I'm not aware of, most of the cost is in the blades themselves, such that energy per blade is the most important factor?

[brandmeyer]

The error in logic is the part where you see so much empty space for wind to pass through. The horizontal velocity of the wind is quite a bit lower than the circumferential velocity of the turbine blade. So only a relatively short distance passes before another blade comes along.

I had a great animation in mind that shows it clearly, but I can't seem to find it right now. To get a static idea, See the corkscrew graphics in figures 4.18 and 4.19 of https://scholarworks.umass.edu/cgi/viewcontent.cgi?article=1...

Basically, there is a relationship between number of blades, blade tip speed, and flow speed that is optimized. For the same flow speed, power, and area, a two-blade turbine has to spin faster to reach optimum and makes more drag. A four-blade turbine has to spin slower to reach optimum, but it is only barely closer to the Betz limit than the corresponding optimized three-blade turbine. The difference is small enough that its difficult to measure or simulate - some other inter-blade losses rise in importance as well.

edit to add: Here's a deep dive into tip speed ratio as it applies to wind turbine design: https://cdn.intechopen.com/pdfs/16242/InTechWind_turbines_th...

[jacquesm]

It would give you more torque, so you will see this in windmills that require a lot of starting power such as water pumpers. For other purposes efficiency of extraction is more important than starting torque. Wind does not 'pass through unimpeded', the speed of the mill is carefully calibrated to slow down the maximum amount of air without causing it to pool behind the machine. That is precisely why there is a maximum amount of energy that can be extracted to begin with.

[brandmeyer]

> It would give you more torque, so you will see this in windmills that require a lot of starting power such as water pumpers.

This is the classical rationale, but having worked on tons of fluid systems, I don't buy it. Its true that you will get a lower tip speed ratio, and therefore a lower RPM for the same power output.

But in practice, that just changes the gearing required for the fluid pump.

IMO, the real reason that fluid applications used four blades and more was that humanity didn't know any better. The theory behind wind turbine optimization wasn't fully developed until the early 1980's.

[jacquesm]

You are making a very common error.

A windmill that isn't rotating yet is an entirely different device than one that is already turning. Once moving, indeed, the three bladed device with a gearing has the same torque as one that has many more blades. But when it isn't turning yet the device with more blades has far more torque than the one with only three blades and that is why it can get started at all.

Pumps are particularly unforgiving, most of the designs out there will have an uneven load distribution on a single rotation of the outgoing shaft to the pump and so will get stopped at the point where the required torque is at its maximum. Precisely because of the distinction made by the questioner: why does so much of the surface allow air to 'slip through'. Which it will in fact do when the mill is standing still or still moving very slowly.

So on a three bladed device the start-up torque will be a small fraction of what it is on a device with many blades, which prevents the three bladed device to start rotating effectively stalling the rotor until the amount of wind is high enough to overcome the initial resistance of the mechanism, which can be quite high. In low wind situations this will allow the multi-blade device to produce far more yield, in moderate wind situations it will produce somewhat less yield and in very high wind situations it may cause the three-blader to overspeed and destroy itself unless it has a very good furling mechanism (such as variable pitch). On a three blader the only parts active to generate this start-up torque are the three blade roots out to about 1/3rd of the rotor diameter.

This limitation applies to purely mechanical direct drives without any trickery such as start-up relays or clutches that only engage when the mill is spinning at a sufficiently high rate, in which case the rotor can stay unloaded long enough for it to get into a high enough rate of rotation that a torque convertor such as an electric motor or gearbox can be engaged. But that is a far more complex setup, one that waterpumpers that run for decades without maintenance can do without.

For electricity generating windmills the situation is a bit different in that most of these have an output voltage below which the energy the mill produces isn't used at all, no current will flow until the output voltage exceeds this lower limit. This effectively acts as a clutch that only engages when the mill is already in an efficient domain of operation. MPP tracking can help extract some power below that speed, something that may be useful in domains where there are long periods of low speed wind.

Just because there wasn't a good computational model for a long time doesn't mean there wasn't a wealth of practical knowledge about windmills empirically gained over three centuries. We already knew what worked and what didn't, we just didn't know why it worked, and we weren't able to scale up the smaller machines to the sizes we see today. The latter was also heavily affected by the development of stronger materials such as carbon fiber.

At a guess you have a highly developed theoretical knowledge about windmills, but not a very well developed practical knowledge, as in actually building some of these models to verify your theoretical knowledge. If you had you would not make such a simple error because you'd know from actual observation that this difference exists in practice and why it exists, rather than just to make a claim from what you know from your - in this case faulty - theoretical knowledge.