Question

A wind turbine with a rotor diameter of 40 m produces 90 kW of electrical power when the wind speed is 8 m/s. The density of air impinging on the turbine is 1.2 kg/m3. What fraction of the kinetic energy of the wind impinging on the turbine is converted to electrical energy

Answer 1

0.233

Step-by-step explanation:

Given that

Diameter of rotor, d = 40 m

Power of rotor, P = 90 kW

Speed of the wind, v = 8 m/s

Density of air, p = 1.2 kg/m³

It is a known fact that

KE = ½mv², where mass flow rate, m

m = p.A.v, where Area, A

A = πd²/4

A = (3.142 * 40²)/4

A = 3.142 * 1600/4

A = 3.142 * 400

A = 1256.8 m², substitute for A in the mass flow rate equation

m = p.A.v

m = 1.2 * 1256.8 * 8

m = 12065.28, substitute for m in the KE equation

KE = ½mv²

KE = ½ * 12065.28 * 8²

KE = 12065.28 * 32

KE = 386088.96 W or

KE = 386.1 kW

Fraction of kinetic energy converted to electric energy is

Fraction = Electric Power / Total KE

Fraction = 90 / 386.1

Fraction = 0.233