The paradox of windmills is this: extract kinetic energy from the wind while still letting the
wind past. Unlike a dam, where you extract the potential energy from the height of the water
and so you can lower the water as much as you want without slowing the flow, with a windmill,
you extract kinetic energy from the *motion* of the wind, which causes the wind to slow down.
So how much wind should a windmill mill?
Well, the maximum possible amount of energy you can extract from the
wind is simply the difference between its initial and final kinetic energy.
Kinetic energy depends on speed squared, so if the outgoing wind speed is, say,
half of the incoming speed, then the final wind energy will be a quarter of the initial energy,
meaning that three quarters of the energy of the wind was extracted when it passed the windmill.
But then there's the question of how much of the wind you can actually extract energy
from. It turns out that the speed of the wind at a windmill is halfway between the
incoming and outgoing wind speeds. So for a final wind speed of half the incoming speed,
the speed at the windmill will be three fourths the incoming speed,
which means only three fourths as much wind can actually pass the windmill every second.
Combining these two facts means that a windmill that slows the wind to half its
initial speed allows three fourths as much wind to pass and extracts three fourths of
the energy from that wind, for a total efficiency of three fourths times three fourths, or 56.25%.
Doing the same math for other possible wind speed reductions,
you find that the most efficient windmill possible slows the wind to a third of its incoming speed.
Reducing to 1/3 speed achieves a kinetic energy "efficiency" of v^2-(v/3)^2 equals 88.88% of
the energy of wind passing the windmill is extracted. And reducing the wind to
1/3 speed after the windmill means the wind passing the windmill is going at 2/3 speed,
so 2/3 as much wind passes through the windmill. And 88.88% energy extraction
from 2/3 of the wind results in an overall efficiency of 88.88*2/3 = 59.259 percent.
A windmill that slows the wind down below 1/3 speed will get a higher
percentage of the energy but from even less wind,
for a lower overall efficiency. And a windmill that slows the wind down less will get a lower
percentage of the energy from somewhat more wind, again for a lower overall efficiency.
So how much wind should a windmill mill? 2/3 of it, for 59% efficiency in extracting the energy.
Obviously, real world windmills are far more complicated and have way
more engineering considerations, so the 59% number may not apply exactly.
Ok, Henry here, I've put a bonus video on Nebula that covers some details I had
to leave out of the main video. Nebula is the Streamy-award-nominated independent streaming
service that's the co-sponsor of this video, built by and for a collection of educational video
creators including Real Engineering, Mike Boyd, Up and Atom, Jordan Harrod - and me. And Nebula
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