Energy and Human Ambitions on a Finite Planet, 2021a

11 Hydroelectric <strong>Energy</strong> 175
a small finger. We want to know how much mass must be lifted to
yield the same amount of gravitational potential energy as is contained
in a battery or equivalent volume of gasoline. In the comparison, we
will imagine having a hoist that can lift a large mass 7 4 m high—about
house-height.
A st<strong>and</strong>ard AA battery cell has a charge rating of 2.5 Ah 8 <strong>and</strong> operates
at about 1.5 V. Following the development in Sec. 5.8 (p. 76), we multiply
these two numbers to get 3.75 Wh, translating to 13.5 kJ. Equating this to
mgh, where we know g ≈ 10 m/s 2 <strong>and</strong> h 4 m, we find that m ≈ 340 kg.
That’s really heavy—about the mass of 4–5 people. 9 Meanwhile, the AA
battery is a puny 0.023 kg. Reflect for a moment on this comparison,
visualizing 340 kg lifted 4 m above the ground providing the same
amount of energy as a AA battery held in your h<strong>and</strong>.
7: ...arock, for instance
8: The number is usually given as, e.g.,
2,500 mAh (milli-amp-hours).
9: Amuse yourself by picturing 4–5 people
slung haphazardly into a net <strong>and</strong> hoisted
to roof height—a very odd (<strong>and</strong> grumpy?)
replacement for a AA battery.
Gasoline is even more extreme. At an energy density around 34 kJ per
mL of volume, filling a AA-sized cup 10
10: . . . just over 7 mL
with gasoline yields about 250 kJ
of energy. 11 Performing the same computation, we would need to lift over
6,000 kg (6 metric tons) to a height of4mtogetthesame energy content.
Typical cars have masses in the 1,000–2,000 kg range, so we’re talking
about something like 4 cars! One caveat is that we are not typically able
to convert the thermal energy in gasoline 12 into useful work at much
12: . . . via combustion; see Sec. 6.4 (p. 88)
better than 25%, while gravitational potential energy can be converted
at nearly 100%. Still, being able to lift 1,500 kg 13 to a height of 4 m using
the energy in 7 mL of gasoline is rather impressive, again emphasizing
that gravitational potential energy is pretty weak. It only amounts to
significance when the masses (volumes) of water are rather large.
11: Thus, gasoline is nearly 20 times as
energy-dense as a AA battery by volume.
Usually, we will discuss energy density by
mass, in which case the ∼5× denser battery
provides nearly 100× less energy per gram
than does gasoline.
13: ...now just one car, rather than four;
it means this small volume of gasoline can
propel a car up a4mhill
11.2 Hydroelectric <strong>Energy</strong>
The basic idea behind hydroelectricity is that water in a reservoir behind
a dam (Figure 11.2) creates pressure at the base of the dam that can
force water to flow through a turbine that drives a generator to make
electricity—sharing elements of Fig. 6.2 (p. 90) but spinning the turbine
by water flow instead. The amount of energy available works out to be
the gravitational potential energy corresponding to the height of water
at the lake’s surface relative to the water level on the other side. It’s as
if dropping the water from the surface to the turbine <strong>and</strong> asking how
much potential energy it gave up in the process. In reality, water is not
dropping from the lake surface, but the force on the water at the turbine
is determined by the height of water above it: the “pressure head,” as
it is called. The process is highly efficient, approaching 90% capture of
the potential energy in the water delivered as electrical power from the
generator.
reservoir
high pressure
turbine blades
penstock
h
river
Figure 11.2: Cross section of a dam, holding
back a reservoir of water at height, h,over
the downstream river.
© 2021 T. W. Murphy, Jr.; Creative Commons Attribution-NonCommercial 4.0 International Lic.;
Freely available at: https://escholarship.org/uc/energy_ambitions.