CP The maximum power that can be extracted by a wind

Chapter 20, Problem 20.58

(choose chapter or problem)

The maximum power that can be extracted by a wind turbine from an air stream is approximately

               \(P=k d^{2} v^{3}\)

where d is the blade diameter, v is the wind speed, and the constant \(k=0.5\mathrm{\ W}\cdot\mathrm{s}^3/\mathrm{m}^5\). (a) Explain the dependence of P on d and on v by considering a cylinder of air that passes over the turbine blades in time t (Fig. P20.58). This cylinder has diameter d, length L = vt and density \(\rho\). (b) The Mod-5B wind turbine at Kahaku on the Hawaiian island of Oahu has a blade diameter of 97 m (slightly longer than a football field) and sits atop a 58-m tower. It can produce 3.2 MW of electric power. Assuming 25% efficiency, what wind speed is required to produce this amount of power? Give your answer in m/s and in km/h. (c) Commercial wind turbines are commonly located in or downwind of mountain passes. Why?