Overview - Wind Power

The power P of a wind-stream, crossing an area A with velocity v is given by

$P={\frac {1}{2}}\rho Av^{3}$

It varies proportional to air density $\rho$ , to the crossed area A and to the cube of wind velocity v.

The Power P is the kinetic energy

$E={\frac {1}{2}}mv^{2}$

of the air-mass m crossing the area A during a time interval

${\dot {m}}=A\rho {\frac {dx}{dt}}=A\rho v$ .

Because power is energy per time unit, combining the two equations leads back to the primary mentioned basic relationship of wind energy utilisation

$P={\dot {E}}={\frac {1}{2}}{\dot {m}}v^{2}={\frac {1}{2}}\rho Av^{3}$

The power of a wind-stream is transformed into mechanical energy by a wind turbine through slowing down the moving air-mass which is crossing the rotor area. For a complete extraction of power, the air-mass would have to be stopped completely, leaving no space for the following air-masses. Betz and Lanchester found, that the maximum energy can be extracted from a wind-stream by a wind turbine, if the relation of wind velocities in front of ( $v_{1}$ ) and behind the rotor area ( $v_{2}$ ) is $v_{1}/v_{2}=1/3$ . The maximum power extracted is then given by

$P_{Betz}={\frac {1}{2}}\rho Av^{3}c_{P.Betz}$

where $c_{p.Betz}=0,59$ is the power coefficient giving the ratio of the total amount of wind energy which can be extracted theoretically, if no losses occur. Even for this ideal case only 59% of wind energy can be used. In practice power coefficients are smaller: todays wind turbines with good blade profiles reach values of $c_{p.Betz}=0,5$ .

Unit Abbreviations

m = metre = 3.28 ft.	HP = horsepower
s = second	J = Joule
h = hour	cal = calorie
N = Newton	toe = tonnes of oil equivalent
W = Watt	Hz = Hertz (cycles per second)