Wind Turbine Question

Question

Suppose you own land which is on a mountain. You notice that at the top elevation of your property there is light and steady wind. You measure the wind with anemometer over the course of weeks and find the time average wind speeds is 4 m/s. You can afford to purchase a small wind turbine that has rotors of 5 m in length. You research the average overall efficiency (capturing the wind, mechanical, electrical) and estimate it to be 25%. What is the average power (kW) that could be generated from installing this wind turbine? Assume the average temperature is 25 °C.

Answer 1

I don't know what the exact meaning of "... turbine that has rotors of 5 m in length." I guess it gives you the size of the swept area by the turbine blades. But is the 5-m the diameter or the radius? I am not sure. So let's just assume that the 5-m be the radius. Therefore the swept area A = π·r² = 3.14×5×5 = 78.5 (m²)^. With the wind speed of 4 m/s, the volume of the air moving across the turbine is 78.5×4 = 314 m³/s. Under 25°C condifition, the density of the air is 1.184 kg/m³ (from wiki), so the mass of the moving air is 314×1.184 = 371.8 kg/s. The kinetic energy of the air in one second is 1/2 × m × v² = 0.5×371.8×4² = 2974 kg·(m/s)². Consider the overall efficiency of 25%, the power generated from the wind will be 2974×0.25 = 743 w = 0.743 kw.