Energy and Human Ambitions on a Finite Planet, 2021a

12 Wind <strong>Energy</strong> 195
6. A typical house may have a floor area around 150 m 2 (1,600 square
feet). If the floor–to-ceiling distance is typically 2.5 m, how much
mass is in the air within the house? Could you lift this much mass
if h<strong>and</strong>ed to you as bags of rocks?
7. Atmospheric pressure is about 10 5 N/m 2 , meaning that a 100,000 N i This is called the scale height of the atmosphere,
h 0 , which you may wish to compare
weight of air—corresponding to a mass of 10,000 kg—sits atop
to the tallest mountains on Earth or the
very square meter of the ground (at or near sea level). If the air
height at which airplanes fly. The actual
density were constant at 1.25 kg/m 3 —rather than decreasing with density of the atmosphere decreases exponentially
as a function of height, with h 0
height as it actually does—how high would the atmosphere extend
to result in this weight (mass)?
being the characteristic scale.
8. Comparing the pale green region in the southeastern U.S. to the
purple region of the plains states in Figure 12.2, how much more
power would we expect out of the same rotor placed in the plains
than in the southeast (by what factor is it bigger in the plains)?
9. How much more powerful is a hurricane-strength wind of 50 m/s
hitting your house than is a light breeze of 5 m/s?
10. How much power would a moderate-sized 50%–efficient wind
turbine produce whose radius is 10 m at wind speeds of 5 m/s,
10 m/s, 15 m/s, <strong>and</strong> 20 m/s? Express the answers in kW or MW,
depending on what is most natural.
11. The Betz limit says that we get to keep no more than 59% of the
available wind power. If 59% of the kinetic energy in a lump of air
moving at speed 38 v is removed, how fast is it going afterwards, 39
as a fraction of the original speed?
38: Make up your own velocity or solve in
symbols/variables: same either way.
39: This relates to wind speed just behind
a wind turbine
12. The largest wind turbines have rotor diameters 40 around 150 m. 40: not radius
Using a sensible efficiency of 50%, what power does such a jumbo
turbine deliver at a maximum design wind speed of 13 m/s?
13. A recent news article announces the largest wind turbine yet,
measuring 220 m in diameter <strong>and</strong> having a maximum power output
of 13 MW. Using a reasonable efficiency, calculate the velocity of
the wind at which maximum power is reached.
14. Compare the tip speed of a three-blade turbine operating at its
optimal efficiency (as per Figure 12.4) in a moderate wind of 7 m/s
to typical freeway driving speeds in the same units.
15. Traveling down the road, you carefully watch a three-bladed wind
turbine, determining that it takes two seconds to make a full
revolution. Assuming it’s operating near the peak of its efficiency
curve 41 according to Figure 12.4, how fast do you infer the wind
speed to be if the blade length 42 appears to be 15 m long?
16. Building from the result in Problem 15, how much power is this
windmill delivering if its efficiency is about 50%?
41: Hint: focus on tip speed.
42: . . . corresponding to radius of the rotor
Reflect on the fact that just estimating the
rotor blade length <strong>and</strong> timing its revolution
is enough for you to produce an estimate of
power being generated!
© 2021 T. W. Murphy, Jr.; Creative Commons Attribution-NonCommercial 4.0 International Lic.;
Freely available at: https://escholarship.org/uc/energy_ambitions.