{"id":737,"date":"2010-05-18T17:28:35","date_gmt":"2010-05-19T00:28:35","guid":{"rendered":"https:\/\/c21-wp.phas.ubc.ca\/index.php\/wind-turbines-betz-law-explained"},"modified":"2023-03-31T14:21:39","modified_gmt":"2023-03-31T21:21:39","slug":"wind-turbines-betz-law-explained","status":"publish","type":"article","link":"https:\/\/c21.phas.ubc.ca\/article\/wind-turbines-betz-law-explained\/","title":{"rendered":"PRIVATE: Wind Turbines – Betz Law Explained"},"content":{"rendered":"

How did Betz come up with this?<\/b><\/p>\n

The work done on the turbine = change in kinetic energy of the wind: W<\/i> =\u0394K<\/i>. The speed v<\/i>2<\/sub> behind the turbine is slower than the speed in front of the turbine v<\/i>1<\/sub>, and the average speed at the location of the turbine is<\/p>\n

$v_{av} = \\dfrac{1}{2}(v_1 + v_2)$<\/p>\n

The mass streaming through the turbine, found as above, is<\/p>\n

\\begin{eqnarray}
\n\\dfrac {\\Delta m}{\\Delta t} &=& \\rho A v_{av} \\nonumber \\\\
\n&=& \\rho A \\dfrac{1}{2}(v_1 + v_2) \\nonumber
\n\\end{eqnarray}<\/p>\n

while the available wind power due to this is<\/p>\n

\\begin{eqnarray}
\nP &=& \\dfrac{\\Delta K}{\\Delta t} \\nonumber \\\\
\n&=& \\dfrac{\\dfrac{1}{2} \\Delta m(v_1^2 – v_2^2)}{\\Delta t} \\nonumber \\\\
\n&=& \\dfrac{1}{4} \\rho A(v_1 + v_2)(v_1^2 – v_2^2) \\nonumber
\n\\end{eqnarray}<\/p>\n

and the undisturbed wind power (the power of the wind if it did not pass through the turbine) is<\/p>\n

\\begin{eqnarray}
\nP_o &=& \\dfrac{K}{\\Delta t} \\nonumber \\\\
\n&=& \\dfrac{1}{2} \\rho Av_1^3 \\nonumber
\n\\end{eqnarray}<\/p>\n

When graphing P<\/i>\/P<\/i>o<\/sub>, the maximum power output is found at v<\/i>2<\/sub>\/v<\/i>1 <\/sub> = 0.33 (Fig. 3).<\/p>\n

\"\"<\/p>\n

Figure 3.<\/b>The plot agrees with Betz’s conclusions that the maximum power output (of 59.3%) occurs when v2<\/sub><\/i> is 1\/3 of v1<\/sub><\/i>. To view the spreadsheet used to produce this graph, see Betz_Law_Spreadsheet_Data.xls<\/a>.<\/p>\n","protected":false},"author":14,"featured_media":1217,"template":"","tags":[],"date_post_made_public":"2010-11-05","post_authored_by":"","hook":"An explanation of Betz' Law","big_ideas":"","thumbnail_for_post":"\"\"","series":false,"number_in_series":"0","supporting_classroom_materials":false,"supporting_experiment":false,"related_articles":false,"related_experiments":false,"related_classroom_materials":false,"_links":{"self":[{"href":"https:\/\/c21.phas.ubc.ca\/wp-json\/wp\/v2\/article\/737"}],"collection":[{"href":"https:\/\/c21.phas.ubc.ca\/wp-json\/wp\/v2\/article"}],"about":[{"href":"https:\/\/c21.phas.ubc.ca\/wp-json\/wp\/v2\/types\/article"}],"author":[{"embeddable":true,"href":"https:\/\/c21.phas.ubc.ca\/wp-json\/wp\/v2\/users\/14"}],"version-history":[{"count":3,"href":"https:\/\/c21.phas.ubc.ca\/wp-json\/wp\/v2\/article\/737\/revisions"}],"predecessor-version":[{"id":3109,"href":"https:\/\/c21.phas.ubc.ca\/wp-json\/wp\/v2\/article\/737\/revisions\/3109"}],"wp:featuredmedia":[{"embeddable":true,"href":"https:\/\/c21.phas.ubc.ca\/wp-json\/wp\/v2\/media\/1217"}],"wp:attachment":[{"href":"https:\/\/c21.phas.ubc.ca\/wp-json\/wp\/v2\/media?parent=737"}],"wp:term":[{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/c21.phas.ubc.ca\/wp-json\/wp\/v2\/tags?post=737"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}