Question

The blades of a windmill sweep out a circle of area A. (a) If the wind flows at a velocity v perpendicular to the circle, what is the mass of the air passing through it in time t?(b) What is the kinetic energy of the air? (c) Assume that the windmill converts 25% of the wind’s energy into electrical energy, and that $A=30m^2$, v = 36 km/h and the density of air is $1.2kg{m}^{-3}$. What is the electrical power produced?

Answer 1

(a) Area of the circle swept by the windmill = A
Velocity of the wind = v
Density of air =$\rho$
Volume of the wind flowing through the windmill per sec = Av
Mass of the wind flowing through the windmill per sec $=\rho Av$
Mass m, of the wind flowing through the windmill in time $t=\rho Avt$

(b) Kinetic energy of air $=\frac{1}{2}m{v}^2$
$=\frac{1}{2}\left(\rho Avt\right)v^2=\frac{1}{2}\rho A{v}^{3}t$

(c) Area of the circle swept by the windmill $=A=30m^2$
Velocity of the wind = v = 36 km/h = 10 m/s
Density of air, $\rho =1.2kg{m}^{-3}$
Electric energy produced = 25% of the wind energy
$=\frac{25}{100}\times$ Kinetic energy of air
$=\frac{1}{8}\rho A{v}^{3}t$
Electrical power $=\frac{Electricalenergy}{Time}=\frac{1}{8}\frac{\rho A{v}^{3}t}{t}=\frac{1}{8}\rho A{v}^3=\frac{1}{8}\times 1.2\times 30\times (10{)}^3=4.5\times {10}^{3}W=4.5kW$