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 22). 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 = 1/2 ρAv³₁. 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 Betz’s formula for F is not meaningful for r close to zero. How much wind would pass through the turbine if v₂ were zero? Is this realistic?

Figure 22

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1 P : = = ਸੁ - 1 2 watts