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  

has partnered with CuriosityStream, which offers  thousands of documentaries and nonfiction titles,  

to give you access to both platforms in one go...  for just a few dollars a month! I really enjoyed  

watching the European Inventor Award episode about  Henrik Stiesdal, who has been intimately involved  

in the invention and development of modern wind  turbines, building the first offshore wind farm,  

and more. Sign up for CuriosityStream using the  link in the description and you'll also get access  

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