Question

At a certain location, wind is blowing steadily at 5 mph. Suppose that the mass density of air is 0.0796 lbm/ft3 and determine the radius of a wind turbine with a power generation potential of 1000 kW. Express your answer in feet. Also, determine how the power generation potential (PGP) scales with wind speed, that is, find a such that PGP = Cva, where C is a constant.

Answer 1

The radius of a wind turbine is 691.1 ft

The power generation potential (PGP) scales with speed at the rate of 7.73 kW.s/m

Step-by-step explanation:

Given;

power generation potential (PGP) = 1000 kW

Wind speed = 5 mph = 2.2352 m/s

Density of air = 0.0796 lbm/ft³ = 1.275 kg/m³

Radius of the wind turbine r = ?

Wind energy per unit mass of air, e = E/m = 0.5 v² = (0.5)(2.2352)²

Wind energy per unit mass of air = 2.517 J/kg

PGP = mass flow rate * energy per unit mass

PGP = ρ*A*V*e

`PGP = \rho *(\pi r^2)/(2) *V*e \\\\r^2 = (2*PGP)/(\rho*\pi *V*e) , r=\sqrt{ (2*PGP)/(\rho*\pi *V*e)} = \sqrt{ (2*10^6)/(1.275*\pi *2.235*2.517)}`

r = 210.64 m = 691.1 ft

Thus, the radius of a wind turbine is 691.1 ft

PGP = CVᵃ

For best design of wind turbine Betz limit (c) is taken between (0.35 - 0.45)

Let C = 0.4

PGP = Cvᵃ

take log of both sides

ln(PGP) = a*ln(CV)

a = ln(PGP)/ln(CV)

a = ln(1000)/ln(0.4 *2.2352) = 7.73

The power generation potential (PGP) scales with speed at the rate of 7.73 kW.s/m