Assume a wind turbine with a hub 50.0 m above the ground, a rotor diameter of 50.0 m, and a wind-power conversion efficiency of 25%. The turbine operates in an area with an average wind-power density of 500 watts/m2 at 50 m altitude. How much electrical energy (in kWh) can the wind turbine generate in a year?
The question is written to allow for a rather simple solution, unlike the challenges of real wind turbine power and energy estimation.
Energy = Power x Time
Electric power produced = conversion efficiency x gross wind power
Gross wind power = power density x turbine swept area
So, with all the equations at hand, we can solve the problem.
Turbine swept area = 2 x pi x (25 m)^2 = 3927 m^2
Wind power density = 500 W/m^2
Thus, Gross wind power = 500 W/m^2 x 3927 m^2 = 1.9635 x10^6 W = 1.9635 MW
Electric power produced = 0.25 * 1.9635 MW = 0.49087 MW
Since the problem identifies the wind power as long-term average, we can calculate the average energy produced in a year.
Energy (Joules) = Watts x seconds
Energy = 0.49087 MW x 31536000 s = 15.487 x10^6 MJ
But the question asked for energy measured in kilowatt-hours (kWh).
So, Energy = 490.87 kW * 8760 h = 4300100 kWh produced in 1 year.
Unfortunately, a real wind turbine would produce this energy in an unscheduleable manner, typically peaking when the demand (load) is lowest. That's the challenge of wind energy.
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