Wind turbines work by converting the kinetic energy in the wind first into rotational kinetic energy in the turbine and then electrical energy that can be supplied, via the national grid.
In order to make wind farm economically viable, we need to know the expected power and energy output of each wind turbine.
Turbine blades capture the wind which forces the rotation of the rotor. It has direct relation between longer blade and higher rotational power.
The standard formula can be used to apply the aerodynamic performance of the blade to the wind speed and the size of the blades, as per LM Wind Power
Energy = (1/2) x Mass x (Velocity)2
The air density changes slightly with air temperature and with elevation. The ratings for wind turbines are based on standard conditions of 59° F (15° C) at sea level.
A density correction should be made for higher elevations as shown in the Air Density Change with Elevation graph. A correction for temperature is typically not needed for predicting the long-term performance of a wind turbine. For calculating the power, let’s consider air density of 1.23 kg/m3
To find out the mass hitting the blades per second, we can use the formula.
Mass/sec = velocity (meters/sec) x area (square meters) x density (kg/cubic meter)
Mass/sec = V x pi x blade length in meters square x 1.23 kg/m3
Hence, Mass/sec = V x (L)2 x 3.86
Energy/Sec (Watts) = (1/2) x [V x (L)2 x 3.86] x V2 = (1.93) x (L)2 x (V)3
Now, to better understand the formula; we will find the total power that would theoretically be available in the swept area of a (e.g.) 12 kw turbine with (e.g.) 4.5 meter wind turbine blade length and wind at 10 m/s will be:
Watts = 1.93 x (4.5)2 x (10)3 = 1.93 x 20.25 x 1000 = 39,082 W
A German physicist Albert Betz concluded in 1919 that no wind turbine can convert more than 16/27 (59.3%) of the kinetic energy of the wind into mechanical energy turning a rotor.
This is known as the Betz Limit or Betz’ Law. The theoretical maximum power efficiency of any design of wind turbine is 0.59 (i.e. no more than 59% of the energy carried by the wind can be extracted by a wind turbine). This is called the “power coefficient” and is defined as: C=0.59
Betz limited power = 39,082 x 0.59 = 23,058 W
There are other factors which we need to consider like friction losses, generator efficiency, the gearbox, bearings, blade efficiency and electrical losses.
Considering above mentioned factors, the actual power now we get around 12,000 watts of power out of a 4.5 m long blade wind turbine system.