Question

A wind-powered generator converts wind energy into electric energy, Assume that the generator converts a fixed fraction of wind energy intercepted by its blades into electrical energy. For wind speed $v$, the electric power output will be proportional to:

• A. $v$
• B. $v^2$
• C. $v^3$
• D. $v^4$

Answer 1

The correct option is C $v^3$

Let ρρ be the density of the air and AA be the cross-section area of the blades. Consider a cylindrical volume VV with cross-sectional area AA and length xx (see figure). The mass of air contained in volume VV is m=ρAxm=ρAx and its kinetic energy is,.

$K=\frac{1}{2}m{v}^2=\frac{1}{2}\rho Ax{v}^2$

The time taken by this volume to intersect the blades is Δt=x/vΔt=x/v. The air transfers its entire kinetic energy to the blades (assuming 100%100% efficiency). Thus, the power generated is,

$P=\frac{K}{\Delta t}=\frac{\frac{1}{2}\rho Ax{v}^2}{\Delta t}=\frac{1}{2}\rho A{v}^3$