Finding the best height for windmill towers

[Imolopa]

To approximate windspeeds we normally use the natural logarithm relation:
u_air = V ln [ (z-d) / z_0 ]Where:
V - Characteristic speed, constant
z - height from the ground
z_0 - roughness length, constant
d - zero plane displacement, constant

The Betz limit tells us that in theory that the max possible fraction of power that can be extracted from the wind is:

C_P = 16/27 = 0.59

Since this is equivalent to a 1/3 reduction of the kinetic energy of the wind we get that:

v_windturbine = (2/3) v_air = v_airthroughthewindturbine

Now the extracted power would be given by:

P = 0.5 * C_p * rho * A * [v_windturbine]3Where:

rho - density of air at given conditions (assume constant in this ideal case)

A = pi * R2 - area covered by the windmill blades of radius R

So both of the top expressions into the expression for power returns:

P = 0.5 * C_p * rho * A * [(2/3) V ln [ (z-d) / z_0 ] ]3So assuming that:
R = 3 m
rho = 1.2 kg/m3
V = 5 m/s
d = 0.0001 m
z_0 = 30 m

Also we assume this is an ideal case and that the above holds for the site in question.

Now if divide P/z, we find that the peak values for P/z are achieved between z = 500 and 600 m of tower/hub height. The same results for a windmill that is double that size (R = 6m). However I'm quite sure I've seen claims of small windturbines decreasing in performance gain after around hundred metres of hub/tower height.

This last claim could be due to material costs and disadvantages of building stuff that is that tall, also the tower for the windmill would have to be wider for increasing heights (which in itself increases the material need considerably since the shape is cylindrical). Also the windmills normally don't operate when the windspeeds go above a certain limit due to safety/limitations, which I guess should be between 30 and 40 m/s.

However if we disregard this last paragraph, should I get results that would indicate peak values for P/z at around 100 m or between 100 and 200 m, iow. are my above results in the ballpark/realistic at all?

 

[mfb]

Multiple units are wrong.

With your model, the ideal height would be hundreds of meters (z₀*e³ neglecting d), but building such a large tower is impractical.

 

[CWatters]

https://en.wikipedia.org/wiki/Wind_gradient

The height above ground where surface friction has a negligible effect on wind speed is called the "gradient height" and the wind speed above this height is assumed to be a constant called the "gradient wind speed".[11][15][16] For example, typical values for the predicted gradient height are 457 m for large cities, 366 m for suburbs, 274 m for open terrain, and 213 m for open sea.[17]

 

[mfb]

274 m for open terrain, and 213 m for open sea.

The largest wind turbines are approaching those values as tip height (but not with the hub).
230 m onshore, 220m offshore, and they are working on R=100m blades, which would probably be installed on wind turbines with a height exceeding 250 meters.

 

[Imolopa]

Multiple units are wrong.

With your model, the ideal height would be hundreds of meters (z₀*e³ neglecting d), but building such a large tower is impractical.

Thanks corrected the units.

The largest wind turbines are approaching those values as tip height (but not with the hub).
230 m onshore, 220m offshore, and they are working on R=100m blades, which would probably be installed on wind turbines with a height exceeding 250 meters.

Interesting, so the critical factor here is the roughness length z_0. As we can see the results are highly sensitive to its value:
z₀= 1 => z₀e³ = 20 m
z₀= 2.5 => z₀e³ = 50 m
z₀ = 5 => z₀e³ = 100 m
z₀ = 10 => z₀e³ = 200 m
z₀ = 20 => z₀e³ = 402 m

And it seems like they don't even bother going above gradient height so that the entire "windmill-area" is above it since it's not even done on the tallest windmill in the world.
This might imply that the final gain does not win over the costs for building taller towers, even with much larger blades than the ones initially used in the problem here.

And yes it makes perfect sense to increase bladeradius R, since it increases the power P quite significantly since P is a function of R²

 

[mfb]

A larger blade radius is more challenging in terms of blade weight (square/cube law), and you need larger distances between larger windmills. You need taller cranes and so on. It is not immediately obvious that larger windmills win, but overall they do with recent advances in their design.