Chapter 4.1, Problem 69E

Efficiency of wind turbines A wind Turbine converts wind energy into electrical power. Let v₁ equal the upstream velocity of the wind before it encounters the wind turbine, and let v₂ equal the downstream velocity of the wind after it passes through the area swept out by the turbine blades.

a.    Assuming that v₁ > 0, give a physical explanation to show that $0\le \frac{v_2}{v_1}\le 1$.

b.    The amount of power extracted from the wind depends on the ratio $r=\frac{v_2}{v_1}$, the ratio of the downstream velocity to upstream velocity. Let R(r) equal the fraction of power that is extracted from the total available power in the wind stream, for a given value of r. In about 1920, the German physicist Albert Betz showed that $R\left(r\right)=\frac{1}{2}\left(1+r\right)\left(1-r^2\right)$, where $0\le r\le 1$ (a derivation of R is outlined in Exercise 70).

Calculate R(1) and explain how you could have arrived at this value without using the formula for R. Give a physical explanation of why it is unlikely or impossible for it to be the case that r = 1.

c.    Calculate R(0) and give a physical explanation of why it is unlikely or impossible for it to be the case that r = 0.

d.    The maximum value of R is called the Betz limit. It represents the theoretical maximum amount of power that can be extracted from the wind. Find this value and explain its physical meaning.