Kinetic Energy

From Wind wiki

The kinetic energy of an object is the energy which it possesses due to its motion. It is directly proportional to the mass of the object and the square of its velocity. Because air has mass and it moves to form wind, it has kinetic energy. For the translational kinetic energy (E_t ) ,that is, the kinetic energy associated with rectilinear (straight line) motion, of a body with constant mass m, whose center of mass is moving in a straight line with speed v, is equal to E_t=1/2 mv^2

Now, for reasons of finding the kinetic energy of moving air molecules (wind), we must calculate the mass of the wind in a specified volume. To do this we assume, say, an arbitrary cylindrical air mass (to correspond with the geometry of a wind turbine) with cross-sectional area, A, and depth, d. The volume, , of this cylindrical air mass is equal to the cross sectional area multiplied by the depth:

V=Ad

Now that we have the volume we can extract the mass, m, of the cylinder of air from the density (ρ) equation:

ρ=m/V

m=ρV

To find the velocity (which is perpendicular to the cross-sectional area) of the cylinder of air, we recall that the cylinder’s depth is d. The velocity is then a question of how fast this depth takes to pass through a given plane parallel to the cross-sectional area (which is analogous to it passing through the blades of a turbine). If it takes a time t to pass through this plane then the velocity would be the depth divided by the time:

v=d/t

d=vt

Now we can make the necessary substitutions in order to find an equation for the kinetic energy, E_t:

E_t=1/2 mv^2 =1/2 (ρV) v^2 (substituting: m=pV) =1/2 ρ(Ad) v^2 (substituting: V=Ad) =1/2 ρA(vt) v^2 (substituting: d=vt) =1/2 ρv^3 At ∎

So now we can see that the kinetic energy of the air mass is proportional to the cube of the velocity. In the section Wind Power we will derive the equation for wind power.