Energy and Human Ambitions on a Finite Planet, 2021a

12 Wind <strong>Energy</strong> 189
have never met something so fast before. For a modern derivation of the
Betz limit <strong>and</strong> how efficiency depends on tip speed, see [71]. The largest
turbines—having 150 m diameter rotors—are rated for up to 10 MW of
electrical power production.
[71]: Ragheb et al. (2011), “Wind Turbines
Theory - The Betz Equation <strong>and</strong> Optimal
Rotor Tip Speed Ratio”
Example 12.2.1 How much power could you expect a small (4 m
diameter) 3-blade wind turbine situated atop your house to deliver in
a respectable 5 m/s breeze?
The radius is 2 m <strong>and</strong> we’ll pick a middle-of-the-road efficiency of
45%: P 1 2 · 0.45 · (1.25 kg/m3 ) · π · (2m 2 ) · (5m/s) 2 comes to about
14
450 W.
14: Not too impressive: hard to get much
wind power on a household scale, although
10 m/s would give 3.6 kW.
Besides the limit on how much power can be pulled out of the air by a
single turbine, we also find limits on how densely they may be populated
in a given area: how much space is required between turbines so that
one does not disrupt the other. Obviously, it would not serve to put
one turbine directly behind another, as they would at best split the
available power arriving as wind. Even side by side, it is best to leave
room between windmills so that additional rows are not deprived of
wind power. A rule of thumb is to separate turbines by at least 5–8
diameters side-to-side, <strong>and</strong> 7–15 diameters 15 along the (prevailing) wind 15: An older “rule of thumb” was 5 side-toside
<strong>and</strong> 7–8 deep, but newer work suggests
direction. For the sake of illustration, Figure 12.5 shows a spacing on
as much as 8 diameters side-to-side <strong>and</strong> 15
the denser side of the range, but otherwise we adopt the more recent
diameters deep.
recommendations <strong>and</strong> use 8 diameters side-to-side <strong>and</strong> 15 diameters
deep [75]. This works out to a 0.65% “fill factor,” meaning that 0.65% of 16: . . . one πR 2 rotor area for every 8D ×
the l<strong>and</strong> area contains an associated rotor cross section. 16 15D 120D 2 120 × (2R) 2 480R 2 of
l<strong>and</strong> area
D
rotor (viewed from above)
D
prevailing wind direction
10D
5D
Figure 12.5: Overhead view of wind farm
turbine locations, for the case where separations
are 10 rotor-diameters along the
wind direction, <strong>and</strong> 5 rotor diameters in
the cross-wind direction—a geometry that
yields 1.6% area “fill factor.” Current recommendations
are for 15 <strong>and</strong> 8 rotor diameters,
which is significantly more sparse than
even this depiction, leading to 0.65% area
fill. Note that most wind turbines can turn
to face the wind direction, for times when
its direction is not the prevailing one.
In order to compare to other forms of renewable energy, we can evaluate
a power per unit l<strong>and</strong> area (in W/m 2 ) by the following approach:
power
area 1
2 ερ airπR 2 v 3
480R 2 π
960 ερ airv 3 , (12.3)
employing the rule-of-thumb 8 × 15 turbine placement scheme. Using
an efficiency of 40% <strong>and</strong> v 5 m/s, 17 we get 0.2 W/m 2 —which is 1,000
times smaller than solar’s ∼200 W/m 2 insolation (Ex. 10.3.1; p. 167).
A final general note about wind generation is somewhat obvious: the
17: Recall that this choice gave sensible
global wind power estimates lining up with
Table 10.2 (p. 168).
© 2021 T. W. Murphy, Jr.; Creative Commons Attribution-NonCommercial 4.0 International Lic.;
Freely available at: https://escholarship.org/uc/energy_ambitions.