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December 2007
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Outer space: Blowin' in the wind

by John D. Barrow



Finding the maximal efficiency

We want to calculate the maximum value of windmill efficiency $P/P_0$. Write $y$ for $P/P_0$ and $x$ for $V/U$. We get

\[ y = \frac{1}{2}(1-x^2)(1+x). \]

Differentiating we get

\[ dy/dx = \frac{1}{2}(1 - 2x - 3x^2). \]

The maximum occurs when $dy/dx = 0$, in other words when

\[ 3x^2 + 2x - 1 = 0. \]

This happens for $x_0=1/3$ and $x_1=-1$. Discounting the negative solution we get a maximal efficiency of

\[ 1/2(1-x_0^2)(1+x_0) = 16/27. \]

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