Question 2.68: A wind turbine is rotating at 15 rpm under steady winds flow......
A wind turbine is rotating at 15 rpm under steady winds flowing through the turbine at a rate of 42,000 kg/s. The tip velocity of the turbine blade is measured to be 250 km/h. If 180 kW power is produced by the turbine, determine (a) the average velocity of the air and (b) the conversion efficiency of the turbine. Take the density of air to be 1.31 kg/m³.
A wind turbine produces 180 kW of power. The average velocity of the air and the conversion efficiency of the turbine are to be determined.
Assumptions The wind turbine operates steadily.
Properties The density of air is given to be 1.31 \mathrm{~kg} / \mathrm{m}^3.
Analysis (a) The blade diameter and the blade span area are
Then the average velocity of air through the wind turbine becomes
\displaystyle V=\frac{\dot{m}}{\rho A}=\frac{42,000 \mathrm{~kg} / \mathrm{s}}{\left(1.31 \mathrm{~kg} / \mathrm{m}^3\right)\left(6140 \mathrm{~m}^2\right)}=\mathbf{5 . 2 3} \mathbf{~m/s}(b) The kinetic energy of the air flowing through the turbine is
Then the conversion efficiency of the turbine becomes
\displaystyle \eta=\frac{\dot{W}}{\mathrm{K\dot E}}=\frac{180 \mathrm{~kW}}{574.3 \mathrm{~kW}}=\mathbf{0 . 3 1 3}=\mathbf{3 1 . 3\%}Discussion Note that about one-third of the kinetic energy of the wind is converted to power by the wind turbine, which is typical of actual turbines.