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| − | == Wind Power ==
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| − | The power ''P ''of a wind-stream, crossing an area ''A ''with velocity ''v ''is given by <br>
| + | [[Portal:Wind|► Back to Wind Portal]] |
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| − | <math>P=\frac{1}{2}\rho A v^3</math> <br>
| + | = Overview - Wind Power = |
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| − | <br>
| + | The power ''P ''of a wind-stream, crossing an area ''A ''with velocity ''v ''is given by |
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| − | It varies proportional to air density, to the crossed area ''A ''and to the cube of the wind velocity ''v''. | + | <math>P=\frac{1}{2}\rho A v^3</math><br/> |
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| | + | It varies proportional to air density <math>\rho</math>, to the crossed area ''A ''and to the cube of wind velocity ''v''. |
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| | + | The Power ''P ''is the kinetic energy |
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| | + | <math>E=\frac{1}{2}mv^2</math> |
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| | + | of the air-mass ''m ''crossing the area ''A ''during a time interval<br/> |
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| | + | <math>\dot{m}=A \rho \frac{dx}{dt}=A\rho v</math>. |
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| | + | Because power is energy per time unit, combining the two equations leads back to the primary mentioned basic relationship of wind energy utilisation |
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| | + | <math>P=\dot{E}=\frac{1}{2}\dot{m}v^2=\frac{1}{2}\rho A v^3</math> |
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| | + | 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 (<math>v_1</math>) and behind the rotor area (<math>v_2</math>) is <math>v_1/v_2=1/3</math>. The maximum power extracted is then given by<br/> |
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| | + | <math>P_{Betz}=\frac{1}{2} \rho A v^3 c_{P.Betz}</math> |
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| | + | where <math>c_{p.Betz}=0,59</math> 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 <math>c_{p.Betz}=0,5</math>. |
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| | + | <br/> |
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| | + | = Unit Abbreviations = |
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| | + | {| border="0" align="left" cellspacing="1" cellpadding="1" style="width: 399px" |
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| | + | | m = metre = 3.28 ft.<br/> |
| | + | | HP = horsepower<br/> |
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| | + | | s = second<br/> |
| | + | | J = Joule<br/> |
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| | + | | h = hour<br/> |
| | + | | cal = calorie<br/> |
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| | + | | N = Newton<br/> |
| | + | | toe = tonnes of oil equivalent<br/> |
| | + | |- |
| | + | | W = Watt<br/> |
| | + | | Hz = Hertz (cycles per second)<br/> |
| | + | |} |
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| | + | <br/> |
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| | + | <br/> |
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| | + | <br/> |
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| | + | <br/> |
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| | + | <math>10^{-12}</math> = p pico = 1/1000,000,000,000 |
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| | + | <math>10^{-9}</math> = n nano = 1/1000,000,000 |
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| | + | <math>10^{-6}</math> = µ micro = 1/1000,000 |
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| | + | <math>10^{-3}</math> = m milli = 1/1000 |
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| | + | <math>10^{3}</math> = k kilo = 1,000 = thousands |
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| | + | <math>10^{6}</math> = M mega = 1,000,000 = millions |
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| | + | <math>10^{9}</math> = G giga = 1,000,000,000 |
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| | + | <math>10^{12}</math> = T tera = 1,000,000,000,000 |
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| | + | <math>10^{15}</math> = P peta = 1,000,000,000,000,000 |
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| | + | <br/> |
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| | + | = Further Information = |
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| | + | *[[Wind Energy - Introduction|Wind Energy - Introduction]] |
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| | + | <br/> |
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| | + | = References = |
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| | + | <references /> |
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| | + | [[Category:Wind]] |
Latest revision as of 09:38, 12 August 2014
► Back to Wind Portal
[edit] Overview - Wind Power
The power P of a wind-stream, crossing an area A with velocity v is given by
Failed to parse (PNG conversion failed; check for correct installation of latex and dvipng (or dvips + gs + convert)): P=\frac{1}{2}\rho A v^3
It varies proportional to air density Failed to parse (PNG conversion failed; check for correct installation of latex and dvipng (or dvips + gs + convert)): \rho
, to the crossed area A and to the cube of wind velocity v.
The Power P is the kinetic energy
Failed to parse (PNG conversion failed; check for correct installation of latex and dvipng (or dvips + gs + convert)): E=\frac{1}{2}mv^2
of the air-mass m crossing the area A during a time interval
Failed to parse (PNG conversion failed; check for correct installation of latex and dvipng (or dvips + gs + convert)): \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
Failed to parse (PNG conversion failed; check for correct installation of latex and dvipng (or dvips + gs + convert)): P=\dot{E}=\frac{1}{2}\dot{m}v^2=\frac{1}{2}\rho A v^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 (Failed to parse (PNG conversion failed; check for correct installation of latex and dvipng (or dvips + gs + convert)): v_1
) and behind the rotor area (Failed to parse (PNG conversion failed; check for correct installation of latex and dvipng (or dvips + gs + convert)): v_2
) is Failed to parse (PNG conversion failed; check for correct installation of latex and dvipng (or dvips + gs + convert)): v_1/v_2=1/3
. The maximum power extracted is then given by
Failed to parse (PNG conversion failed; check for correct installation of latex and dvipng (or dvips + gs + convert)): P_{Betz}=\frac{1}{2} \rho A v^3 c_{P.Betz}
where Failed to parse (PNG conversion failed; check for correct installation of latex and dvipng (or dvips + gs + convert)): 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 Failed to parse (PNG conversion failed; check for correct installation of latex and dvipng (or dvips + gs + convert)): c_{p.Betz}=0,5
.
[edit] 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)
|
Failed to parse (PNG conversion failed; check for correct installation of latex and dvipng (or dvips + gs + convert)): 10^{-12}
= p pico = 1/1000,000,000,000
Failed to parse (PNG conversion failed; check for correct installation of latex and dvipng (or dvips + gs + convert)): 10^{-9}
= n nano = 1/1000,000,000
Failed to parse (PNG conversion failed; check for correct installation of latex and dvipng (or dvips + gs + convert)): 10^{-6}
= µ micro = 1/1000,000
Failed to parse (PNG conversion failed; check for correct installation of latex and dvipng (or dvips + gs + convert)): 10^{-3}
= m milli = 1/1000
Failed to parse (PNG conversion failed; check for correct installation of latex and dvipng (or dvips + gs + convert)): 10^{3}
= k kilo = 1,000 = thousands
Failed to parse (PNG conversion failed; check for correct installation of latex and dvipng (or dvips + gs + convert)): 10^{6}
= M mega = 1,000,000 = millions
Failed to parse (PNG conversion failed; check for correct installation of latex and dvipng (or dvips + gs + convert)): 10^{9}
= G giga = 1,000,000,000
Failed to parse (PNG conversion failed; check for correct installation of latex and dvipng (or dvips + gs + convert)): 10^{12}
= T tera = 1,000,000,000,000
Failed to parse (PNG conversion failed; check for correct installation of latex and dvipng (or dvips + gs + convert)): 10^{15}
= P peta = 1,000,000,000,000,000
[edit] Further Information
[edit] References
