Wind Energy - Physics

Wind Power

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

 <img _fckfakelement="true" _fck_mw_math="P=\frac{1}{2}\rho A v^3" src="/images/math/a/f/f/aff47ae2d0c794edc79c72dc23327697.png" />

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

The Power P is the kinetic energy

<img _fckfakelement="true" _fck_mw_math="E=\frac{1}{2}mv^2" src="/images/math/d/0/9/d09bd4120bbded4606433d6eb4539e7c.png" />

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

<img _fckfakelement="true" _fck_mw_math="\dot{m}=A \rho \frac{dx}{dt}=A\rho v" src="/images/math/b/3/e/b3ef61411620083fa8c4d12f2df4d414.png" />.

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

<img _fckfakelement="true" _fck_mw_math="P=\dot{E}=\frac{1}{2}\dot{m}v^2=\frac{1}{2}\rho A v^3" src="/images/math/e/d/d/eddae381c857bc2114fd643b32111bf9.png" />

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 (v₁) and behind the rotor area (v₂) is v₁ / v₂ = 1 / 3. The maximum power extracted is then given by

<img _fckfakelement="true" _fck_mw_math="P_{Betz}=\frac{1}{2} \rho A v^3 c_{P.Betz}" src="/images/math/0/8/3/08377cd8c23f47f4ce336b72d8422baf.png" />

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

10³ = k kilo = 1,000 = thousands

10⁶ = M mega = 1,000,000 = millions

10⁹ = G giga = 1,000,000,000

10¹² = T tera = 1,000,000,000,000

10¹⁵ = P peta = 1,000,000,000,000,000

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