Difference between revisions of "Wind Energy - Physics"

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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>
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[[Portal:Wind|► Back to Wind Portal]]
  
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&nbsp;<br>
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= Overview - Wind Power =
  
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<br>
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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''.&nbsp;
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<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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|-
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| m = metre = 3.28 ft.<br/>
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| HP = horsepower<br/>
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|-
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| s = second<br/>
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| J = Joule<br/>
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|-
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| h = hour<br/>
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| cal = calorie<br/>
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|-
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| N = Newton<br/>
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| toe = tonnes of oil equivalent<br/>
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|-
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| W = Watt<br/>
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| Hz = Hertz (cycles per second)<br/>
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|}
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<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

Overview - Wind Power

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

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle P=\frac{1}{2}\rho A v^3}

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

The Power P is the kinetic energy

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

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\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

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 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 (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle v_1} ) and behind the rotor area (Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle v_2} ) is Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle v_1/v_2=1/3} . The maximum power extracted is then given by

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle P_{Betz}=\frac{1}{2} \rho A v^3 c_{P.Betz}}

where Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\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 Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\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)






Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 10^{-12}} = p pico = 1/1000,000,000,000

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 10^{-9}} = n nano = 1/1000,000,000

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 10^{-6}} = µ micro = 1/1000,000

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 10^{-3}} = m milli = 1/1000

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 10^{3}} = k kilo = 1,000 = thousands

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 10^{6}} = M mega = 1,000,000 = millions

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 10^{9}} = G giga = 1,000,000,000

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 10^{12}} = T tera = 1,000,000,000,000

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 10^{15}} = P peta = 1,000,000,000,000,000


Further Information


References

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