Wind Energy - Physics

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)

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

$10^{-9}$ = n nano = 1/1000,000,000

$10^{-6}$ = µ micro = 1/1000,000

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

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

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

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

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

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