It's taken about 20 years, but I think I've finally worked out why wind power varies with the square of the change in wind speed.

For some reason it jumped into my mind this morning with my wake up cup of tea.

I realised that when you double wind speed you also double the amount of air over the sail.

ie. if you have a 5m2 sail and a wind speed of 10m/s you have 50 cubic metres of air per sec flowing over the sail.

if you half sail area to 2.5m2 then you have 25m3 per sec flowing over the sail and half the power.
If you then double the wind speed you're back to 50m3 per sec over the sail but at twice the speed so the power is also double.
If you then increase sail size back to original 5m2 you have twice the volume of air (100m3 per sec) and twice the wind speed so 4 times the power.

Does this make sense to anybody else, or is my old brain befuddling me?

Another simple approach is work out how much energy is available to harvest from the wind. Air has 'mass', so just like any other object that is moving, the kinetic energy contained within the moving air mass is given by the formula: KE = 1/2 m.v^2

The reason Kinetic Energy has a v^2 term in it is that to accelerate an object from 0 to 1 m/s, you can be standing still and push it with a certain amount of force for a certain amount of time, but to then accelerate it from 1 to 2 m/s, you also need to push it with the same amount of force for the same amount of time, but that force be moving alongside the object. What that means is you end up applying the same force, but over a larger distance, which takes increasingly more energy to do. (Work = force x distance)

Coming back to your reasoning, and what you have described is correct. You are looking at the reverse of the above paragraph, and are talking about taking the energy from the wind. Think of it like this....you might first think that wind going twice the speed has twice the energy...but you get twice as much of it coming past. In other words "twice x twice", or "twice squared". That is not really the precise way to describe it, but it is a corollary of the correct explanation, and illustrates the general concept.

You're on the right track Decrepit, you've allowed for, but forgotten to count one of the velocity effects. As Harrow said in a round about way, the power in the wind goes up with the cube of the wind speed, at least as far as power-generating wind turbines are concerned. The kinetic energy in a unit parcel of air goes up as the square of velocity, but then the number of parcels of air passing through the turbine in a unit of time, (or over the sail) also goes up again with velocity. That's a power of 3 all up. But maybe you've already allowed for the fact that you have to use a smaller sail in stronger wind?

Ian K said..

You're on the right track Decrepit, you've allowed for, but forgotten to count one of the velocity effects. As Harrow said in a round about way, the power in the wind goes up with the cube of the wind speed, at least as far as power-generating wind turbines are concerned. The kinetic energy in a unit parcel of air goes up as the square of velocity, but then the number of parcels of air passing through the turbine in a unit of time, (or over the sail) also goes up again with velocity. That's a power of 3 all up. But maybe you've already allowed for the fact that you have to use a smaller sail in stronger wind?

OH NO!!!!

Cube of wind speed? really?
Well I don't think I can wait another 20years for the next light bulb moment, can somebody explain why the kinetic energy is the square of velocity?

decrepit said...






OH NO!!!!

Cube of wind speed? really?
Well I don't think I can wait another 20years for the next light bulb moment, can somebody explain why the kinetic energy is the square of velocity?

You have to go to the original definition of energy or work, and that is
work = force times the distance over which it is applied.

So you can push on a brick wall as hard as you like but if you don't move it you haven't done any work and the wall hasn't increased its energy. If you manage to move a pile of bricks with the same force a bit of distance then you have done work, but the bricks will have gained heat energy with friction rather than kinetic energy.

If you push a wheeled mass a given distance with a given force it will not get hotter, it will gain kinetic energy. The kinetic energy will also equal the force applied times the distance the mass moves. That's not a useful way of expressing kinetic energy though, so we have to do a bit of maths to express the energy in terms of the velocity it has gained after being pushed for this distance.

The velocity is increasing as it is pushed according to v = at (acceleration times time since the push started)

The distance the mass is moved is the integration of its velocity over time = 1/2 at^2 (Just a little bit of calculus)

f = ma

so the force times the distance = ma * 1/2 a t^2

but at = v

So force times the distance = 1/2mv^2 = the kinetic energy

Thanks Ian, I'm not good at thinking mathematically, but I think I'm getting my head around it.

decrepit said..

Ian K said..



You're on the right track Decrepit, you've allowed for, but forgotten to count one of the velocity effects. As Harrow said in a round about way, the power in the wind goes up with the cube of the wind speed, at least as far as power-generating wind turbines are concerned. The kinetic energy in a unit parcel of air goes up as the square of velocity, but then the number of parcels of air passing through the turbine in a unit of time, (or over the sail) also goes up again with velocity. That's a power of 3 all up. But maybe you've already allowed for the fact that you have to use a smaller sail in stronger wind?

OH NO!!!!Cube of wind speed? really?

Well I don't think I can wait another 20years for the next light bulb moment, can somebody explain why the kinetic energy is the square of velocity?

Decrepit, your thinking is sort of correct but you need to make the distinction between FORCE,.. which does increase with the square of the wind speed, and POWER which increases with the cube of the wind speed.
So the FORCE you feel on the sail does increase with the square of the speed as you said, but because power has a velocity component multiplier in the equation, this gives it another wind speed dependent multiplier and thus is proporrtional to the cube of the wind speed.

Don't worry about what Newton said. He has caused nothing but trouble since he came up with all those explanations of why he was accosted by an apple falling from a tree. He should have just shut up and eaten the apple.
Also,.. just be happy that the light is still burning.
In some people it seems to have gone out. (even though they still claim to be architects )
Ohhh that was a low blow. Sorry aboominator.

Thanks Pweedas,
I had a feeling it was something like that, it's a very long time since I did physics at school.