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:

where m is the mass of wind flowing through the rotor per unit time (Figure 19). Betz assumed that m = ρA(v1 + v2)/2, where ρ 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 ρAv1 and power P0 = 1/2ρAv31. 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 ˆ’ r2)(1 + r), where 0 ‰¤ r ‰¤ 1.
(b) Show that the maximum value of F(r), called the Betz Limit, is 16/27 ‰ˆ 0.59.
(c) Explain why Betz's formula for F(r) is not meaningful for r close to zero. How much wind would pass through the turbine if v2 were zero? Is this realistic?

Ps2mv_1mf watts muwatts 0.5 0.5 0.4 0.3 02 0.1 0.5 (A) Wind flowing through a turbine B) Fis the fraction of energy extracted by the turbine as a function ofr FIGURE 19