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Finding the best height for windmill towers

  1. Nov 19, 2016 #1
    To approximate windspeeds we normally use the natural logarithm relation:
    u_air = V ln [ (z-d) / z_0 ]

    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


    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?
    Last edited: Nov 19, 2016
  2. jcsd
  3. Nov 19, 2016 #2


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    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.
  4. Nov 19, 2016 #3


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  5. Nov 19, 2016 #4


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    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.
  6. Nov 20, 2016 #5
    Thanks corrected the units.
    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
  7. Nov 20, 2016 #6


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    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.
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