turbine headscratcher

[dvoss]

I was pondering about how much windpower could be extracted from a given volume of the atmosphere if we were able to situate turbines wherever we wanted at any altitude within the volume. Imagine they are on towers so thin that they don't contribute significantly to turbulence.

Consider a space with volume V through which air is flowing at a steady speed v in a constant direction.

I put a wind turbine in there (much smaller than the space) and it produces power at P = 0.5 rho A v^3 where rho is air density and A is the area swept out, so let's say P = 0.5 rho pi r^2 v^3, r being the rotor radius.

Now there's a limit to how many turbines you can jam in there before you start to see significant drops in speed due to turbulence. In ordinary terrestrial wind farms the rule of thumb appears to be that the area you need per turbine is about 40 times the square of the rotor diameter (ie, 160 times the square of the radius).

So I was initially thinking you could just think of it as a set of layers, each 2r high, each layer with turbines spaced as above with one turbine per (160 times the square of the radius).

But this gives the perverse result P/V = (0.5 rho pi r^2 v^3) / (160 r^2) / (2r) = pi/720 * rho v^3/r, implying that the amount of power you can extract is inversely proportional to radius, which can't be right.

So what do you reckon? Imagine this is just a rectangular prism of the atmosphere, somewhere high enough above the ground that we can ignore any effects from the solid surface. It should be within current with constant wind speed and direction, and much smaller than the space where that current applies, but larger than the wind turbines. Is there a maximum amount of power that we can derive by placing wind turbines in that space, and what would the configuration be?

[Cragyon]

I'm trying to wrap my head around what you're writing...

I think that I agree that optimal location of a wind turbine should be explored and I'm aware that wind turbines are often places on top of hills and other high wind locations.

As far as calculating the aerodynamics of a wind turbine that math is beyond me.

[Kelly_Bramble]

Here's an over view of the power available from wind

https://www.engineersedge.com/power_...wind_15848.htm

[Cragyon]

Also, there is...

Wind Turbine Power and Torque Equation and Calculator