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• 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 squaredso 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 forfinal 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 thatwindmill 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 videobuilt by and for a collection of educational video

• creators including Real Engineering, Mike BoydUp and Atom, Jordan Harrod - and me. And Nebula

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• 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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The paradox of windmills is this: extract kinetic  energy from the wind while still letting the

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# How Much Wind Should A Windmill Mill?

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Summer posted on 2021/04/01
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