Difference between revisions of "Wind Energy - Physics"
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| + | [[Portal:Wind|► Back to Wind Portal]] | ||
| + | |||
| + | = Overview - Wind Power = | ||
The power ''P ''of a wind-stream, crossing an area ''A ''with velocity ''v ''is given by | The power ''P ''of a wind-stream, crossing an area ''A ''with velocity ''v ''is given by | ||
| − | + | <math>P=\frac{1}{2}\rho A v^3</math><br/> | |
| − | It varies proportional to air density < | + | It varies proportional to air density <math>\rho</math>, to the crossed area ''A ''and to the cube of wind velocity ''v''. |
The Power ''P ''is the kinetic energy | The Power ''P ''is the kinetic energy | ||
| − | + | <math>E=\frac{1}{2}mv^2</math> | |
| − | of the air-mass ''m ''crossing the area ''A | + | of the air-mass ''m ''crossing the area ''A ''during a time interval<br/> |
| − | + | <math>\dot{m}=A \rho \frac{dx}{dt}=A\rho v</math>. | |
Because power is energy per time unit, combining the two equations leads back to the primary mentioned basic relationship of wind energy utilisation | Because power is energy per time unit, combining the two equations leads back to the primary mentioned basic relationship of wind energy utilisation | ||
| − | + | <math>P=\dot{E}=\frac{1}{2}\dot{m}v^2=\frac{1}{2}\rho A v^3</math> | |
| + | |||
| + | 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/> | ||
| − | + | <math>P_{Betz}=\frac{1}{2} \rho A v^3 c_{P.Betz}</math> | |
| − | + | 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>. | |
| − | + | <br/> | |
| − | + | = Unit Abbreviations = | |
| − | {| | + | {| border="0" align="left" cellspacing="1" cellpadding="1" style="width: 399px" |
|- | |- | ||
| m = metre = 3.28 ft.<br/> | | m = metre = 3.28 ft.<br/> | ||
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| + | <br/> | ||
| + | <br/> | ||
| + | <br/> | ||
| + | <br/> | ||
| + | <br/> | ||
| + | <math>10^{-12}</math> = p pico = 1/1000,000,000,000 | ||
| + | <math>10^{-9}</math> = n nano = 1/1000,000,000 | ||
| + | <math>10^{-6}</math> = µ micro = 1/1000,000 | ||
| − | < | + | <math>10^{-3}</math> = m milli = 1/1000 |
| − | < | + | <math>10^{3}</math> = k kilo = 1,000 = thousands |
| − | < | + | <math>10^{6}</math> = M mega = 1,000,000 = millions |
| − | < | + | <math>10^{9}</math> = G giga = 1,000,000,000 |
| − | < | + | <math>10^{12}</math> = T tera = 1,000,000,000,000 |
| − | < | + | <math>10^{15}</math> = P peta = 1,000,000,000,000,000 |
| − | < | + | <br/> |
| − | + | = Further Information = | |
| − | + | *[[Wind Energy - Introduction|Wind Energy - Introduction]] | |
| + | <br/> | ||
| + | = References = | ||
| − | + | <references /> | |
[[Category:Wind]] | [[Category:Wind]] | ||
Latest revision as of 09:38, 12 August 2014
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 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 \rho} , to the crossed area A and to the cube of wind velocity v.
The Power P is the kinetic energy
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 E=\frac{1}{2}mv^2}
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
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
