Can we motivate mathematically why wind turbines almost always have 3 flappers and aeroplane propellers can have any number of flappers?

Question

Firstly I know some might frown upon a question so very broad and applied as this one. It really may not be a well defined mathematical question as some people would prefer on the site. I am okay with moving this to physics or some other more suitable SE if enough people want to do that.

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Supposedly both air propellers as well as wind turbines want to utilize the displacement of air in order to either turn it into motion or energy.

Is there some mathematical justification for the choice in difference of number of blades of an aircraft or a wind turbine?

How could we set up some kind of calculation for this? Which equations would be useful?

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Own work What I suspect would be very useful is to use differential equations to model how flow of air effects acceleration of flappers (and other way around) and how this affects generation of electricity.

Answer 1

Let's take for granted that for wind turbines, fluid dynamics and economics dictate that we want as few blades as possible, for efficiency. Let's also take for granted that for any rotating apparatus, mechanics dictates that we want the center of mass and the moment of inertia to be invariant under rotations, for stability. The "why" of these dictates probably belongs on other sites.

For the mathematics: We want to place the blades at the roots of unity, $z^n-1=0$. The center of mass is the sum of the solutions, which is the coefficient of $z^1$, which is $0$ if and only if $n>1$.

The moment of inertia requirement means that we also want a sum of the form $\sum_k\cos^2(k\alpha+\theta)$ to be constant. Using the double angle formula, this is equivalent to something like $\sum_k\cos(2k\alpha+\theta)=0$. If $n$ is odd, this reduces to the same constraint in the previous paragraph, so we just need $n>1$, which means $n\geq 3$. If $n$ is even, this reduces to the previous constraint on $n/2$, so we need $n/2>1$, which means $n\geq 4$. Combining the even and odd cases, we have a constant moment of inertia iff $n\geq 3$.

So for both wind turbines and aircraft propellers, we need $n\geq 3$.

For wind turbines in particular, the least $n$ that solves $n\geq 3$ is, of course, $n=3$.

Answer 2

I conjecture that this is not a mathematical problem at all, but rather an engineering and legal one.

You want to extract lots of energy from the wind with your turbine. If you put tons of blades close together, each will interfere with the airflow over the next one, so there's clearly an upper limit. The assembly needs to be balanced to avoid huge non-axial torques, so you need more than 1 blade (although Hagen suggests that in some cases this may not be true...or maybe his "one blade" is a diameter rather than a radius, i.e., what I'd call a 2-blade setup).

The towers (often) have maximum height restrictions, and you can only fit so many of them in a given space, so cramming in more blades makes sense, hence 3-blade rather than 2-blade. But why not 4 or 6 or 12 blade ones?

I'm guessing that each additional blade gives a bump in output, but perhaps not in proportion to the number of blades: the 5th blade increases the output less than the 4th one did, and so on. And blades are expensive. And more blades requires a more substantial axle, gearbox, etc., not to mention a stronger tower. So there's a point where the additional cost for the $n$th blade exceeds the value of the incremental power generated.

And looking at common windmills suggests that this point is at about $n = 3$. :)