www.lesswrong.com/posts/rQxrSRPJGmksGjK8z/it-s-time-for-a-self-reproducing-machi...

This claim conflicts with the standard wind-power relationship used by the U.S. Department of Energy.

DOE educational material gives the basic wind-power equation as P = 1/2 ρ A V^3 (equivalently P = 1/2 ρ π r^2 V^3 for a circular rotor), where power depends on air density, swept area, and wind speed cubed. That means if you make a similar turbine larger while holding wind speed constant, the captured power scales with the rotor's swept area—i.e. with the square of a linear dimension such as blade length—not with the fifth power of height.

DOE's small-wind guide likewise says the amount of power a turbine produces is determined by the diameter of its rotor, and gives annual energy output as proportional to D²V³. DOE's overview of modern turbines also explains that taller towers help because winds generally increase with altitude (wind shear), while larger rotor diameters help because they sweep more area. Those are two separate effects; DOE does not describe turbine output as obeying a universal fifth-power rule in tower height.

So the article's statement is incorrect as written: there is no general law that windmill power scales as height^5. At most, taller turbines can gain some additional output from higher wind speeds at higher altitude, but that is site-dependent and is not the same as a fixed fifth-power scaling law.