The Maths of Wind Power
When a wind turbine is operating, if all the wind energy were converted into electrical power, the turbine would be 100% efficient.
The calculations would go as follows, illustrated in the diagram.
The area swept out by the blades iswhereis the length of the blades.
In one second the volume of air that passes the turbine iswhereis the wind speed perpendicular to the blades.
The density of the air passing through the blades of the turbine is
The mass of the air passing through the turbine blades per second is then
Then the kinetic energy of the air passing the turbine is
If the wind turbine were 100% efficient, all the kinetic energy of the wind would be converted into electrical energy, because if it were, the speed of the air having passed through the turbine would have to be zero. And the air has to move away from the turbine to make room for air constantly passing through. The above derivation does imply, correctly, that doubling the wind speed means the available energy increases by a factor of eight, so if the efficiency of the wind turbine stays the same, the power output of the turbine will also increase by a factor of eight.