Problem — Efficiency of a wind turbine blade

Challenge: efficiency of a wind turbine. A wind turbine has a rotor of radius $50\;\mathrm{m}$ and produces $1\;\mathrm{MW}$ in a wind of $10\;\mathrm{m/s}$ ($\rho_\text{air} = 1{,}2\;\mathrm{kg/m^3}$). Estimate its efficiency and compare it with the Betz limit ($\eta_\text{max} = 0{,}593$).

Solution

Data: $r = 50\;\mathrm{m}$, $P_\text{out} = 1\;\mathrm{MW} = 10^6\;\mathrm{W}$, $v = 10\;\mathrm{m/s}$, $\rho = 1{,}2\;\mathrm{kg/m^3}$.

Area swept by the rotor:

$A = \pi r^2 = \pi\cdot 50^2 = \pi\cdot 2500 \approx 7854\;\mathrm{m^2}$

Power of the wind passing through the rotor. The mass of air passing per second is $\dot m = \rho A v$, and its kinetic energy per unit time is $P_\text{wind} = \tfrac{1}{2}\dot m\,v^2 = \tfrac{1}{2}\rho A v^3$ (depends on the cube of the speed):

$P_\text{wind} = \tfrac{1}{2}\rho A v^3 = \tfrac{1}{2}(1{,}2)(7854)(10)^3 = 0{,}6\cdot 7854\cdot 1000 \approx 4{,}7\cdot 10^6\;\mathrm{W}$

$\ev{P_\text{wind} \approx 4{,}7\;\mathrm{MW}}$

Blade efficiency:

$\eta = \frac{P_\text{out}}{P_\text{wind}} = \frac{1{,}0\cdot 10^6}{4{,}7\cdot 10^6} \approx 0{,}21 = 21\%$

$\ev{\eta \approx 21\%}$

Comparison with the Betz limit. No turbine can extract more than $59{,}3\%$ of the wind’s energy (otherwise the air would come to a complete stop behind the rotor, blocking the flow). The ratio between actual efficiency and theoretical limit:

$\frac{\eta}{\eta_\text{Betz}} = \frac{0{,}21}{0{,}593} \approx 0{,}36 = 36\%$

$\ev{\eta/\eta_\text{Betz} \approx 36\%}$

Physical reading. The turbine captures about $21\%$ of the wind’s energy, a little over a third of the maximum physically allowed by Betz. The rest escapes because the air must keep flowing (it cannot stop) and because of unavoidable aerodynamic and mechanical losses. The strong dependence $P_\text{wind}\propto v^3$ explains why windy sites are so valuable: doubling the wind speed multiplies the available power by eight.

Collegamenti

Argomenti: Work and energy Concetti: Power · Kinetic energy Competenze: Fermi estimates