# 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