Chapter 4.1, Problem 70E

Derivation of wind turbine formula A derivation of toe function R in Exercise 69, based on three equations from physics, is outlined here. Consider again the figure given in Exercise 69, where v_1 equals the upstream velocity of the wind just before the wind stream encounters the wind turbine, and equals the downstream velocity of the wind just after the wind stream passes through the area swept out by the turbine blades. An equation for the power extracted by the rotor blades, based on conservation of momentum, is $P=v^2\rho A\left(v_1-v_2\right)$, where v is the velocity of the wind (in m/s) as it passes through the turbine blades, p is the density of air (in kg/m³), and A is the area (in m²) of the circular region swept out by the rotor blades.

  a.    Another expression for the power extracted by the rotor blades, based or conservation of energy is $P=\frac{1}{2}\rho vA\left(v_1{}^2{v}_2{}^2\right)$. Equate the two power equations and solve for v.

  b.    Show that $P=\frac{\rho A}{4}\rho vA\left(v_1+v_2\right)\left(v_1{}^2-v_2{}^2\right)$.

  c.    If the wind were to pass through the same area A without being disturbed by rotor blades, the amount of available power would be $P_0=\frac{\rho A{v}_1{}^3}{2}$. Let $r=\frac{v_2}{v_1}$ and simplify the ratio $\frac{P}{P_0}$ to obtain the function R(r) given in Exercise 69.