g The electric power needs of a community are to be met by windmills with 40-m-diameter rotors. The windmills are to be located where the wind is blowing steadily at an average velocity of 6 m/s. Determine the minimum number of windmills that need to be installed if the required power output is 2100 kW. The gas constant of air is 0.287 kPa·m3/kg·K.

Given Data

The diameter of the wind mills is d = 40m

Velocity of the air is V = 6 m / s

Required power output is: P ₀ = 2100 k W

Expression to calculate the exergy of the air is

E = V ² / 2

Substitute the value in above expression

E = (6 m / s) ² / 2

E = 18 m ² / s ² x (1kJ/kg / 1000m²/s²)

E = 0.018 k J / k g

Expression to calculate the density of the air is

P v = m R T

m / v = P / RT ⋯ ⋯ (I)

Here

m is the mass of the air,

v is the volume of the air,

P is the atmospheric pressure,

T is the standard temperature at the atmospheric pressure and

R is the gas constant

As the density is

ρ = m / V

Substitute the value in expression (I)

ρ = 101 kP a / (0.287 k J / k g ⋅ K) (298 K)

ρ = 1.180 k g / m ²

Expression to calculate the mass flow rate is

m = ρ A V ⋯ ⋯ (I I)

Here A is the area of the windmill

Expression to calculate the A is

A = π / 4 d ²

Substitute the value in above expression

A = π / 4 (40 m ²)

A = 1256.63 m ²

Substitute the value in expression (II)

m = (1.180 k g / m ³) (1256.63 m ²) (6 m / s)

m = 8896.94 k g / s

Expression to calculate the maximum power available to the windmill is

P w = m (V ² / 2)

Substitute the value in above expression

P w = 8896.94 k g / s ((6m/s) ²/2)

P w = 160144.92 W * (1 W / 1000 k W)

P w = 160.144 k W

Expression to calculate the number of windmills required is

n = P o / P w

Substitute the value in above expression

n=2100kw/160.144kw

n=13.11