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 cp.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 cp.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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