First published in the Science and Engineering forum of the For Freedom Forums

Posted: Sunday 25th March 2012 05:10 am

"Dow" Equation for the power and energy output of a wind farm
"The power and energy of a wind farm is proportional to (the square root of the wind farm area) times the rotor diameter". In his book which was mentioned to me on another forum and so I had a look, David MacKay wrote that the power / energy of a wind farm was independent of rotor size which didn't seem right to me considering the trend to increasing wind turbine size. Now I think the commercial wind-turbine manufacturing companies know better and very possibly someone else has derived this equation independently of me and long ago - in which case by all means step in and tell me whose equation this is. Derivation Assume various simplifications like all turbine rotors are the same size and height, flat ground and a rotationally symmetrical wind turbine formation so that it doesn't matter what direction the wind is coming from. Consider that an efficient wind farm will have taken a significant proportion of the theoretically usable power (at most the Betz Limit, 59.3%, apparently, but anyway assume a certain percent) of all the wind flowing at rotor height out by the time the wind passes the last turbine. So assume the wind farm is efficient or at least that the power extracted is proportional to the energy of all the wind flowing through the wind farm at rotor height. This defines a horizontal layer of wind which passes through the wind farm of depth the same as the rotor diameter. The width of this layer which flows through the wind farm is simply the width of the wind farm which is proportional to the square root of the wind farm area. Wind farm turbine formations Therefore the width or diameter of a rotationally symmetrical wind farm is a critically important factor and arranging the formation of wind turbines to maximise the diameter of the wind farm is important. Consider two different rotationally symmetrical wind turbine formations, I have called the "Ring formation" and the "Compact formation". Let n be the number of wind turbines in the wind farm Let s be the spacing between the wind turbines Ring formation Larger image also hosted here The circumference of the ring formation is simply n times s. Circumference = n x s The diameter of the ring formation is simply n times s divided by PI. Diameter = n x s / PI Compact formation Larger image also hosted here The area of the compact formation, for large n, is n times s squared. This is slightly too big an area for small n. Area = n x s^2 (for large n) The diameter of the compact formation, for large n, is 2 times s times the square root of n divided by PI. This is slightly too big a diameter for small n. Diameter = 2 x s x SQRT(n/PI) This is easily corrected for small n greater than 3 by adding a "compact area trim constant" (CATC) (which is a negative value so really it is a subtraction) to the s-multiplier factor. The CATC is 4 divided by PI minus 2 times the square root of 4 divided by PI. CATC = 4/PI - 2 x SQRT(4/PI) = - 0.9835 This CATC correction was selected to ensure that the compact formation diameter equation for n=4 evaluates to the same value as does the ring formation equation for n = 4, that being the largest n for which the ring and compact formations are indistinguishable. The CATC works out to be minus 0.9835 which gives Diameter = s x ( 2 x SQRT(n/PI) - 0.9835) (for n > 3) Ratio of diameters Larger image also hosted here It is of interest to compare the two formations of wind farm for the same n and s. The diameter of the ring formation is larger by the ratio of diameter formulas in which the spacing s drops out. Ring formation diameter : Compact formation diameter n/PI : 2 x SQRT (n/PI) - 0.9835 This ratio can be evaluated for any n > 3 and here are some ratios with the compact value of the ratio normalised to 100% so that the ring value of the ratio will give the ring formation diameter as a percentage of the equivalent compact formation diameter. Here are some examples, n = 4, 100 : 100 n = 10, 123 : 100 n = 18, 151 : 100 n = 40, 207 : 100 n =100, 309 : 100 n =180, 405 : 100 n =300, 514 : 100 n =500, 656 : 100 As we can see that for big wind farms, with more turbines, the ratio of diameters increases. Since the Dow equation for the power and energy of a wind farm is proportional to the diameter of the wind farm then it predicts that the power and energy of the ring formation wind farms will be increased compared to the compact formation wind farms by the same ratio. In other words, the Dow equation predicts, for example, that a 100 turbine wind farm in the ring formation generates 3 times more power and energy than they would in the compact formation, assuming the spacing is the same in each case. Practical application when designing a wind farm My recommendation would be to prefer to deploy wind turbines in a wind farm in the ring formation in preference to the compact formation all other things being equal. The compact formation can be improved up to the performance of a ring formation by increasing the turbine spacing so that the circumference is as big as the ring but then if a greater turbine spacing is permitted then the ring formation may be allowed to get proportionally bigger as well keeping its advantage, assuming more area for a larger wind farm is available. The ring formation may be best if there is a large obstacle which can be encircled by the ring, such as a town or lake where it would not be possible or cost effective to build turbines in the middle of it and so a compact formation with larger spacing may not be possible there. Where it is not possible to install a complete ring formation then a partial ring formation shaped as an arc of a circle would do well also. _________________ Peter Dow, |