In 1919 physicist Alfred Betz argued that the maximum efficiency of a wind turbine is

Chapter 4, Problem 77

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In 1919 physicist Alfred Betz argued that the maximum efficiency of a wind turbine is around 59%. If wind enters a turbine with speed v1 and exits with speed v2, then the power extracted is the difference in kinetic energy per unit time: P = 1 2 mv2 1 1 2 mv2 2 watts where m is the mass of wind flowing through the rotor per unit time (Figure 20). Betz assumed that m = A(v1 + v2)/2, where is the density of air and A is the area swept out by the rotor. Wind flowing undisturbed through the same area A would have mass per unit time Av1 and power P0 = 1 2Av3 1. The fraction of power extracted by the turbine is F = P /P0.(a) Show that F depends only on the ratio r = v2/v1 and is equal to F (r) = 1 2 (1 r2)(1 + r), where 0 r 1. (b) Show that the maximum value of F, called the Betz Limit, is 16/27 0.59. (c) Explain why Betzs formula for F is not meaningful for r close to zero. Hint: How much wind would pass through the turbine if v2 were zero? Is this realistic? 1 0.1 0.2 0.3 0.5 0.4 0.6 0.5 F (A) Wind flowing through a turb