Outer space: Blowin' in the wind

Submitted by plusadmin on December 1, 2007

December 2007

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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