# Overview - Wind Power

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

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

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

The Power P is the kinetic energy

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

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

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

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

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

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 (${\displaystyle v_{1}}$) and behind the rotor area (${\displaystyle v_{2}}$) is ${\displaystyle v_{1}/v_{2}=1/3}$. The maximum power extracted is then given by

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

where ${\displaystyle c_{p.Betz}=0,59}$ 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 ${\displaystyle c_{p.Betz}=0,5}$.

# 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)

${\displaystyle 10^{-12}}$ = p pico = 1/1000,000,000,000

${\displaystyle 10^{-9}}$ = n nano = 1/1000,000,000

${\displaystyle 10^{-6}}$ = µ micro = 1/1000,000

${\displaystyle 10^{-3}}$ = m milli = 1/1000

${\displaystyle 10^{3}}$ = k kilo = 1,000 = thousands

${\displaystyle 10^{6}}$ = M mega = 1,000,000 = millions

${\displaystyle 10^{9}}$ = G giga = 1,000,000,000

${\displaystyle 10^{12}}$ = T tera = 1,000,000,000,000

${\displaystyle 10^{15}}$ = P peta = 1,000,000,000,000,000