Page 32 - Fluid Power Engineering

10 Chapter Tw o

FIGURE 2-1 Cylinder r
of air in front of the
rotor.
m ˙ = ρAν
=

ν Wind
Cylinder of air
Turbine

The mass (m) from which energy is extracted is the mass contained
in the volume of air that will flow through the rotor. For a horizontal
axis wind turbine (HAWT), the volume of air is cylindrical, as shown
in Fig. 2-1. Approximation of a uniform cylinder will be relaxed later
in the chapter.
Most are familiar with kinetic energy of a solid object of fixed mass.
With air flow, it is convenient to think of mass in a cylinder of air of
radius r. Since v m/s is the wind speed, the mass contained in cylinder
of length v meters and radius r is the amount of mass that will pass
through the rotor of turbine per second. It is, therefore, convenient to
use mass per second ( ˙m) in Eq. (2-1).

1
˙ E = ˙ mv 2 (2-2)
2
˙ m = ρ Av (2-3)

where ρ is air density and A is the cross-section area. ˙m is the amount
of matter contained in a cylinder of air of length v. ˙ E is energy per
second, which is the same as power P

1 1
2
˙ E = P = ρ Avv = ρ Av 3 (2-4)
2 2
3
3
3
2
3
2
Units of power are (kg/m )m m /s = kg m /s = J/s = Watts. Other
units of power are kiloWatts (kW), megaWatts (MW), gigaWatts (GW),
and horsepower (HP). Units of energy are Watt-seconds (=1 J), Watt-
hours, kiloWatt-hours (kWh), megaWatt-hours (MWh), etc.
2
For a HAWT, A = πr , where r is the radius of the rotor, therefore:
1 2 3
P = ˙ E = ρπr v (2-5)
2
The distinction between power and energy is important. If a wind
turbine operates at a constant power of 10 kW for 2 h, then it will
produce 20 kWh of energy, which is 72 million J (or Watt-seconds).