A wind turbine generates electricity by removing some part of the energy in the wind and converting it to electrical power. What kind of energy is associated with the wind? It is the kinetic energy due to the motion of the air. Within the hub of the moving turbine blades is an electrical generator. The action of the wind on the blades creates a torque on the armature of the generator, and this torque, acting across a magnetic field, induces electrical power.

To see if we can quantify all of this, imagine the air ahead of the blades is moving at a known wind speed V. Also, let’s assume the length of the blade, from tip to the hub center, is L. Then we have a horizontal, circular column of air of radius L ahead of the blades. All of the air in this air column moves through the turbine blades. The kinetic energy flowing through this column, per unit time, must be 1/2x(V^2)x(mass of air flowing through the column per unit time). The last term is just (RhoxAreaxV). The quantity Rho is the density of the air. It is the mass of air per unit volume, a constant equal to 1.225 kilogram/cubic meter. The area is just the area of the air column: A = PI*(L^2). We arrive at the following result:

K.E. flowing through the blades per unit time = 1/2xRhox(PIxL^2)x(V^3).

If you check the units on the right side of this expression, you will see that it has the units of energy per unit time, which is power. Now, we know that the efficiency is just the energy extracted as electrical power (output power) divided by the input power. Using the above expression we can write the efficiency as follows.

Eff = (Electrical Power Out)/[1/2xRhox(PIxL^2)x(V^3)]

There is a well known theorem due to Betz that states that the maximum theoretical efficiency that can be achieved by any wind turbine is 59.6% (approximately). This is based on the idea that we can never convert all of the incoming kinetic energy to electricity. There must be some remaining wind energy to push the air through the blades and beyond. To find out more about Betz’s theorem see the following web site:

http://www.ndsu.nodak.edu/ndsu/klemen/Perfect_Turbine.htm

It is important to note that even highly efficient turbines never achieve the Betz limit. A well designed, modern turbine might achieve an efficiency of 35 per cent. Finally, notice that we can rearrange the expression for efficiency to give us an estimate for electrical power, assuming the efficiency is known:

(Electrical Power Out) = Effx[1/2xRhox(PIxL^2)x(V^3)].

So, for example, we can use the last formula to estimate the power output of a ‘good’ wind turbine. Just insert the value 0.35 for the efficiency, along with the other known values L, Rho and V that characterize the wind turbine.

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