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 winds energy into electrical energy, and that A = 30 $m^2$, v = 36 km/h and the density of air is 1.2 kg $m^3$. What is the electrical power produced?

Answer 1

(a)

Density of air $=\rho kg/m^3$
Time taken = $t$ sec
Velocity of air = $v$ m/s
Mass of the air = density $\times$ volume $=\rho Avt$

(b)

$KE=\frac{1}{2}m{v}^2$
$=\frac{1}{2}\left(\rho Avt\right)v^2$
$=\frac{1}{2}\left(\rho A{v}^{3}t\right)$

(c)

Velocity, v= 36 km/hr = 10 m/s

$\rho =1.2kg/m^3$

$A=30m^2$

Work done = change in kinetic energy

$=\frac{1}{2}\times 1.2\times 30\times {10}^{3}tNm$

$=18000tJ$

Power = $work/t=18000W$

Efficiency = 25%

Actual power$=18000\times 25/100=4500W$