Energy and Human Ambitions on a Finite Planet, 2021a

1 Exponential Growth 12
which we can rearrange to isolate temperature, satisfying
T 4
0.707 × 1360 W/m2
. (1.11)
4σ
Solving for T yields T ≈ 255 K, or −18 ◦ C (about 0 ◦ F). This is cold—too
cold. We observe the average temperature of Earth to be about 288 K, or
15 ◦ C (59 ◦ F). The difference of 33 ◦ C is due to greenhouse gases—mostly
H 2 O—impacting the thermal balance by preventing most radiation from
escaping directly to space. We’ll cover this more extensively in Chapter
9.
Armed with Eq. 1.11, we can now estimate the impact of waste heat on
Earth’s equilibrium temperature. Using the solar input as a baseline,
we can add increasing input using the exponential scheme from the
previous section: starting today at 18 TW <strong>and</strong> increasing at 2.3% per
year (a factor of 10 each century). It is useful to express the human
input in the same terms as the solar input so that we can just add to
the numerator in Eq. 1.11. In this context, our current 18 TW into the
projected area πR 2 ⊕ adds 0.14 W/m2 to the solar input (a trivial amount,
today), but then increases by a factor of ten each century. Taking this
in one-century chunks, the resulting temperatures—adding in the 33 K
from greenhouse gases—follow the evolution shown in Table 1.4. At
first, the effect is unimportant, but in 300 years far outstrips global
warming, <strong>and</strong> reaches boiling temperature in a little over 400 years! If we
kept going (not possible), Earth’s temperature would exceed the surface
temperature of the sun inside of 1,000 years!
Years Power Density (W/m 2 ) T (K) ΔT (C)
100 1.4 288.1 0.1
200 14 288.9 ∼1
300 140 296.9 ∼9
400 1,400 344 56
417 2,070 373 100
1,000 1.4 × 10 9 8,600 8,300
A potential inconsistency in our treatment
is that we based our exploration of energy
scale on solar energy as a prelude to stellar
energy capture. But in the thermodynamic
treatment, we implicitly added our power
source to the existing solar input. If the sun
is the source, we should not double-count
its contribution. Nonetheless, continued, relentless
growth would eventually dem<strong>and</strong> a
departure from solar capture on Earth <strong>and</strong>
drive the same thermodynamic challenges
regardless. Synthesizing the messages: we
can’t continue 2.3% growth for more than a
few centuries using sunlight on Earth. And
if we invent something new <strong>and</strong> different to
replace the fully-tapped solar potential, it
too will reach thermodynamic limits within
a few centuries.
Table 1.4: At a constant energy growth rate
of 2.3% per year, the temperature climb
from waste heat (not CO 2 emissions) is slow
at first, but becomes preposterous within a
few-hundred years. Water boils in just over
400 years, <strong>and</strong> by 900 years Earth is hotter
than the sun! The scenario of continued
growth is obviously absurd.
Connecting some ideas, we found in the previous section that we would
be consuming the sun’s entire output in 1,400 years at the 2.3% growth
rate. It st<strong>and</strong>s to reason that if we used a sun’s worth of energy confined
to the surface of the earth, the (smaller) surface would necessarily be
hotter than the sun (in 1,400 years), just like a light bulb filament is hotter
than human skin despite putting out the same power—owing to the
difference in area. 19
One key aspect of this thermal radiation scenario is that it does not depend
on the form of power source. It could in principle be fossil fuels, nuclear
fission, nuclear fusion, or some form of energy we have not yet realized
<strong>and</strong> may not even have named! Whatever it is, it will have to obey
thermodynamics. Thus, thermodynamics puts a time limit on energy
growth on this planet.
19: This can be gleaned from Eq. 1.8 or Eq.
1.9.
© 2021 T. W. Murphy, Jr.; Creative Commons Attribution-NonCommercial 4.0 International Lic.;
Freely available at: https://escholarship.org/uc/energy_ambitions.