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 = 30 m², v = 36 km/h and the density of air is 1.2 kg m⁻³. What is the electrical power produced?

Answer 1

it is given that,
Area swept by the blades = A
Velocity of wind flow = v
Density of air = ρ

(a) volume of the air passing through windmill per second , V = Av

we know, mass = density × volume
so, mass of the air passing through windmill per second , m = ρAv

hence, mass of the air passing through windmill in time t sec , M = ρAvt


(b) kinetic energy of the air = 1/2 Mv²

= 1/2 (ρAvt)v²

= 1/2 ρAv³t


(c) a/c to question,
windmill converts 25% of the wind's energy into electrical energy

so, electrical energy = 25 % of kinetic energy or wind's energy
= 25/100 × 1/2 ρAv³t
= 1/8 ρAv³t

given, ρ = 1.2 kg/m³ , A = 30m², v = 36km/h = 10m/s

so, electrical energy = 1/8 × 1.2 × 30 × (10)³ × t
= 1/8 × 12 × 3 × 1000t
= 36000t/8
= 4500t J

we know, power = energy/time
so, electrical power = electrical energy/time
= 4500t/t = 4500 W or 4.5kW