To approximate windspeeds we normally use the natural logarithm relation:(adsbygoogle = window.adsbygoogle || []).push({});

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]3

Where:

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 ] ]3

So 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?

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# Finding the best height for windmill towers

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