Question

69, Eiciency of wind turbines A wind turbine into electrical power. Let vi equal the upstream vel wind before it encounters the wind turb downstream velocity of the wind after it passes thequal swept out by the turbine blades. A wind turbine converts win stream velocity of the App f(x) sin r cos.r con fo)--2 In( 1) f(x) (sin1)(COs ) the rig fx)- a Minin volume where etermine the location and What an n the given interval, if they 73. Traject from a height a a. Assuming that v, >0. give a physical explanation to show reach it that 0 ss1 74. Maxim from sa b. The amount of power extracted from the wind depends on the where a -, the ratio of the downstream velocity to upstream ratio a. Find b. Dete velocity. Let R(r) equal the fraction of power that is extracted from the total available power in the wind stream, for a given value of r. In about 1920, the German physicist Albert Betz tion 75. Maximi showed that R()r)(1 maximu -2), where 0 srsI (a derivation of R is outlined in Exercise 70). Calculate R(I and explain how you could have arrived at this value without using the formula for R. Give a physical explanation of why is unlikely or impossible for it to be the case that r-I c. Calculate R(0) and give a physical explanation of why it is likely or impossible for it to be the case that r = 0. d. The maximum value of R is called the Betz limit. It represc the theoretical maximum amount of power that can be from the wind. Find this value and explain its physical n people Althou continu a. How the p h. Supp peop 76. Minimiz 64 have

maxima minima

Answer 1

neh'c ener net velo al龙wind velo citt equ andlV is cuinol net decrease in kinehc ener

ulen no coversion V)

2 4 2. 2 2- om ven

passing he nbne hich isnot possibo 2- di 3

the total power nea ns toward sind tanbine with veouty the to Cal avalable power -Ge :/ん订 can a Doner_ enn m 叼 maximum pe y l 6 = 59.26.