Energy and Human Ambitions on a Finite Planet, 2021a

12 Wind <strong>Energy</strong> 187
able to replace the energy currently derived from fossil fuels.
12.2.1 Wind Turbines
To underst<strong>and</strong> practically-available energy, we back up <strong>and</strong> consider
how much air hits a wind turbine whose rotor diameter is R. Figure 12.3
illustrates the concept. If the wind speed is v, the air travels a distance
vΔt in time interval Δt. 9
9: We can pick any value for Δt: a long
interval makes a very long cylinder, while a
small Δt results in a short, stubby cylinder.
In the end, the value we chose for Δt will
cancel out, so as not to matter.
v
R
vt
Cylinder volume: R 2 vt
Figure 12.3: Wind power concept. In time
interval Δt at wind speed v, a volume of
air encounters the rotor having the shape
of a cylinder of radius R <strong>and</strong> length vΔt.
Note that most wind turbines are designed
to pivot about a vertical axis to face into the
wind, whatever the direction.
The cross-sectional area of the wind turbine (rotor) is defined as the
area swept out by the blades, so πR 2 . Thus the volume of the cylinder
of air interacting with the turbine over time interval Δt is the “base”
(circular) area of the cylinder times its “height” (straight length, vΔt),
or V πR 2 vΔt. We know the density of the air, 10 so the mass of the 10: ρ air ≈ 1.25 kg/m 3
cylinder is m ρ air V ρ air πR 2 vΔt. The kinetic energy contained in this
cylinder of air is therefore K.E. 1 2 mv2 1 2 ρ airπR 2 v 3 Δt. Now let’s get
rid of that pesky Δt. Think for a moment what happens if we divide both
sides by Δt: it will definitely get rid of the Δt on the right-h<strong>and</strong>-side, but
what does the left-h<strong>and</strong> side mean: energy over time? Hopefully, this is
familiar by now as the concept of power.
Table 12.1: Wind power scales as the cube
of wind speed.
Definition 12.2.1 The power delivered by a wind turbine of radius R in
wind speed v <strong>and</strong> operating at efficiency ε is
P wind 1 2 ερ airπR 2 v 3 . (12.2)
where ρ air ≈ 1.25 kg/m 3 at sea level. Efficiency has been inserted as ε <strong>and</strong>
tends to be 40–50% for modern turbines.
Notice that the delivered power scales, sensibly, with the area of the wind
turbine’s blade path, but more importantly <strong>and</strong> perhaps surprisingly
as the velocity cubed (Table 12.1). The cubed part should make you sit
Speed
Power
0 0
1 1
2 8
3 27
4 64
5 125
.
.
10 1,000
© 2021 T. W. Murphy, Jr.; Creative Commons Attribution-NonCommercial 4.0 International Lic.;
Freely available at: https://escholarship.org/uc/energy_ambitions.