Theory of Wind Turbine and Betz CoefficientPublished on 24/2/2012 and last updated on 12/8/2018
We define power as the change of energy per second. Hence, this extracted power can be written as, As mass m of the air passes in one second, we refer the quantity m as the mass flow rate of the wind. If we think of that carefully, we can easily understand that mass flow rate will be the same at the inlet, at the outlet and as well as at every cross-section of the air duct. Since, whatever quantity of air is entering the duct, the same is coming out from the outlet. If Va, A and ρ are the velocity of the air, the cross-sectional area of the duct and density of air at the turbine blades respectively, then the mass flow rate of the wind can be represented as Now, replacing m by ρVaA in equation (1), we get, Now, as the turbine is assumed to be placed at the middle of the duct, the wind velocity at turbine blades can be considered as average velocity of inlet and outlet velocities. To obtain maximum power from wind, we have to differentiate equation (3) in respect of V2 and equate it to zero. That is,
Betz CoefficientFrom, the above equation it is found that the theoretical maximum power extracted from the wind is in the fraction of 0.5925 of its total kinetic power. This fraction is known as the Betz Coefficient. This calculated power is according to theory of wind turbine but actual mechanical power received by the generator is lesser than that and it is due to losses for friction rotor bearing and inefficiencies of aerodynamic design of the turbine.
From equation (4) it is clear that the extracted power is
- Directly proportional to air density ρ. As air density increases, the power of the turbine increases.
- Directly proportional to the swept area of the turbine blades. If the length of the blade increases, the radius of the swept area increases accordingly, so turbine power increases.
- Turbine power also varies with velocity3 of the wind. That indicates if the velocity of wind doubles and the turbine power will increase to eight folds.