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| − | == Wind Power == | + | <h2> Wind Power </h2> |
| | + | <p>The power <i>P </i>of a wind-stream, crossing an area <i>A </i>with velocity <i>v </i>is given by |
| | + | </p><p> <img _fckfakelement="true" _fck_mw_math="P=\frac{1}{2}\rho A v^3" src="/images/math/a/f/f/aff47ae2d0c794edc79c72dc23327697.png" /><br /> |
| | + | </p><p>It varies proportional to air density <span class="texhtml">ρ</span>, to the crossed area <i>A </i>and to the cube of wind velocity <i>v</i>.  |
| | + | </p><p>The Power <i>P </i>is the kinetic energy |
| | + | </p><p><img _fckfakelement="true" _fck_mw_math="E=\frac{1}{2}mv^2" src="/images/math/d/0/9/d09bd4120bbded4606433d6eb4539e7c.png" /> |
| | + | </p><p>of the air-mass <i>m </i>crossing the area <i>A </i>during a time interval <br /> |
| | + | </p><p><img _fckfakelement="true" _fck_mw_math="\dot{m}=A \rho \frac{dx}{dt}=A\rho v" src="/images/math/b/3/e/b3ef61411620083fa8c4d12f2df4d414.png" />. |
| | + | </p><p>Because power is energy per time unit, combining the two equations leads back to the primary mentioned basic relationship of wind energy utilisation |
| | + | </p><p><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" /> |
| | + | </p><p>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 (<span class="texhtml"><i>v</i><sub>1</sub></span>) and behind the rotor area (<span class="texhtml"><i>v</i><sub>2</sub></span>) is <span class="texhtml"><i>v</i><sub>1</sub> / <i>v</i><sub>2</sub> = 1 / 3</span>. The maximum power extracted is then given by<br /> |
| | + | </p><p><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" /> |
| | + | </p><p>where <span class="texhtml"><i>c</i><sub><i>p</i>.<i>B</i><i>e</i><i>t</i><i>z</i></sub> = 0,59</span> 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 <span class="texhtml"><i>c</i><sub><i>p</i>.<i>B</i><i>e</i><i>t</i><i>z</i></sub> = 0,5</span>. |
| | + | </p> |
| | + | <h2> Unit abbreviations </h2> |
| | + | <table width="399" cellspacing="1" cellpadding="1" border="0" align="left" style=""> |
| | | | |
| − | The power ''P ''of a wind-stream, crossing an area ''A ''with velocity ''v ''is given by
| + | <tr> |
| − | | + | <td> m = metre = 3.28 ft.<br /> |
| − | [[File:]]
| + | </td> |
| − | | + | <td> HP = horsepower<br /> |
| − | It varies proportional to air density <span class="texhtml">ρ</span>, to the crossed area ''A ''and to the cube of wind velocity ''v''.
| + | </td></tr> |
| − | | + | <tr> |
| − | The Power ''P ''is the kinetic energy
| + | <td> s = second<br /> |
| − | | + | </td> |
| − | [[File:]]
| + | <td> J = Joule<br /> |
| − | | + | </td></tr> |
| − | of the air-mass ''m ''crossing the area ''A ''during a time interval
| + | <tr> |
| − | | + | <td> h = hour<br /> |
| − | [[File:]].
| + | </td> |
| − | | + | <td> cal = calorie<br /> |
| − | Because power is energy per time unit, combining the two equations leads back to the primary mentioned basic relationship of wind energy utilisation
| + | </td></tr> |
| − | | + | <tr> |
| − | [[File:]]
| + | <td> N = Newton<br /> |
| − | | + | </td> |
| − | 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 (<span class="texhtml">''v''<sub>1</sub></span>) and behind the rotor area (<span class="texhtml">''v''<sub>2</sub></span>) is <span class="texhtml">''v''<sub>1</sub> / ''v''<sub>2</sub> = 1 / 3</span>. The maximum power extracted is then given by
| + | <td> toe = tonnes of oil equivalent<br /> |
| − | | + | </td></tr> |
| − | [[File:]]
| + | <tr> |
| − | | + | <td> W = Watt<br /> |
| − | where <span class="texhtml">''c''<sub>''p''.''B''''e''''t''''z''</sub> = 0,59</span> 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 <span class="texhtml">''c''<sub>''p''.''B''''e''''t''''z''</sub> = 0,5</span>.
| + | </td> |
| − | | + | <td> Hz = Hertz (cycles per second)<br /> |
| − | == Unit abbreviations == | + | </td></tr></table> |
| − | | + | <p><br /> |
| − | {| style="" align="left" width="399" border="0" cellpadding="1" cellspacing="1"
| + | </p><p><br /> |
| − | |-
| + | </p><p><br /> |
| − | | m = metre = 3.28 ft.<br/>
| + | </p><p><br /> |
| − | | HP = horsepower<br/>
| + | </p><p><span class="texhtml">10<sup> − 12</sup></span> = p pico = 1/1000,000,000,000 |
| − | |-
| + | </p><p><span class="texhtml">10<sup> − 9</sup></span> = n nano = 1/1000,000,000 |
| − | | s = second<br/>
| + | </p><p><span class="texhtml">10<sup> − 6</sup></span> = µ micro = 1/1000,000 |
| − | | J = Joule<br/>
| + | </p><p><span class="texhtml">10<sup> − 3</sup></span> = m milli = 1/1000 |
| − | |-
| + | </p><p><span class="texhtml">10<sup>3</sup></span> = k kilo = 1,000 = thousands |
| − | | h = hour<br/>
| + | </p><p><span class="texhtml">10<sup>6</sup></span> = M mega = 1,000,000 = millions |
| − | | cal = calorie<br/>
| + | </p><p><span class="texhtml">10<sup>9</sup></span> = G giga = 1,000,000,000 |
| − | |-
| + | </p><p><span class="texhtml">10<sup>12</sup></span> = T tera = 1,000,000,000,000 |
| − | | N = Newton<br/>
| + | </p><p><span class="texhtml">10<sup>15</sup></span> = P peta = 1,000,000,000,000,000 |
| − | | toe = tonnes of oil equivalent<br/>
| + | </p><p><br /> |
| − | |-
| + | </p><p><a _fcknotitle="true" href="Portal:Wind">Portal:Wind</a> |
| − | | W = Watt<br/>
| + | </p> |
| − | | Hz = Hertz (cycles per second)<br/>
| + | |
| − | |}
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| − | <span class="texhtml">10<sup>− 12</sup></span> = p pico = 1/1000,000,000,000 | + | |
| − | | + | |
| − | <span class="texhtml">10<sup>− 9</sup></span> = n nano = 1/1000,000,000 | + | |
| − | | + | |
| − | <span class="texhtml">10<sup>− 6</sup></span> = µ micro = 1/1000,000 | + | |
| − | | + | |
| − | <span class="texhtml">10<sup>− 3</sup></span> = m milli = 1/1000 | + | |
| − | | + | |
| − | <span class="texhtml">10<sup>3</sup></span> = k kilo = 1,000 = thousands | + | |
| − | | + | |
| − | <span class="texhtml">10<sup>6</sup></span> = M mega = 1,000,000 = millions | + | |
| − | | + | |
| − | <span class="texhtml">10<sup>9</sup></span> = G giga = 1,000,000,000 | + | |
| − | | + | |
| − | <span class="texhtml">10<sup>12</sup></span> = T tera = 1,000,000,000,000 | + | |
| − | | + | |
| − | <span class="texhtml">10<sup>15</sup></span> = P peta = 1,000,000,000,000,000 | + | |
| − | | + | |
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| − | | + | |
| − | [[Portal:Wind]]
| + | |
| | | | |
| | [[Category:Wind]] | | [[Category:Wind]] |
Revision as of 10:09, 16 May 2012
Wind Power
The power P of a wind-stream, crossing an area A with velocity v is given by
![]()
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
![]()
of the air-mass m crossing the area A during a time interval
![]()
.
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 (v1) and behind the rotor area (v2) is v1 / v2 = 1 / 3. The maximum power extracted is then given by
![]()
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.
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HP = horsepower
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s = second
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J = Joule
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h = hour
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cal = calorie
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N = Newton
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toe = tonnes of oil equivalent
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W = Watt
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Hz = Hertz (cycles per second)
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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
<a _fcknotitle="true" href="Portal:Wind">Portal:Wind</a>
