# 3 Theoretical Power of Wind

Kinetic Energy

• KE= ½ mv2, where m = mass & v = velocity

Air’s Mass

• m = ρAvt, where ρ= air density  A = area through which air passes  v = velocity & t= time

Wind Energy

• substituting m = ρAvt  into KE= ½ mv^2 results in  KE = ½ ρAvtv^2 or wind energy = ½ ρAtv^3

Power

• Energy = Power * time
• Power = Energy/time
• wind energy = ½ ρAtv^3
• wind power = ½ ρAv^3

wind power = ½ ρAv^3

• wind power is directly proportional to the swept area
• wind power is directly proportional to ρ, air density.
• wind power is directly proportional to v^3, air velocity cubed.

Clipper Wind: wind power ∝ swept area

• Swept area = πr^2 or π(d/2)^2 where d is the diameter
• The blade length or radius of the Clipper Wind Liberty 2.5 MW Wind Turbine (C100) is 48.8 meters and a rotor diameter of 100meters
• The swept area = π(d/2)^2 = π(100meters/2)^2 = 7854m^2  (industry uses this method) however,
• With blade length only swept area = π(r/2)^2 = π(48.7m/2)^2  = 7,451m^2

Acciona Energy: wind power ∝ swept area

• swept area = πr^2 or π(d/2)^2 where d is the diameter
• The blade length or radius of the AW-82/1500 Wind Turbine is 40.3 meters and the diameter is 82m
• The swept area = π(d/2)^2 = π(82meters/2)^2 = 5281m^2  (industry uses this method) however,
• With blade length only swept area = πr^2 = π(40.3m)^2 = 5,102 m^2

Wind power ∝ ρ (air density)

• air density decreases with increases in altitude (for the same wind velocity a turbine is more efficient in Iowa than in the mountains)
• air density decreases with increases in temperature (wind turbines are more efficient in the winter than summer)
• Try this air density calculator

Wind power ∝ v^3

• Velocity is the most important contributor to wind power
• Example:
• If when v = 5.25 m/s,
the wind power is 187.5 kW, then
• When v = 10.5 m/s,
the wind power is 1500 kW

This is an 8x increase in power for a 2x increase in velocity 