Energy and Human Ambitions on a Finite Planet, 2021a

6 Putting Thermal <strong>Energy</strong> to Work 96
Example 6.5.1 What is the limit to efficiency of maintaining a freezer
at −10 ◦ C in a room of 20 ◦ C?
First, we express the temperatures in Kelvin: T c ≈ 263 K <strong>and</strong> ΔT
30 K. 43 The maximum efficiency, by Eq. 6.10, computes to ε 43: Note that ΔT 30 in either K or ◦ cool ≤ 8.8
C.
(880%).
Example 6.5.2 What is the limit to efficiency of keeping a home
interior at 20 ◦ C when it is −10 ◦ C outside?
First, we express the temperatures in Kelvin: T h ≈ 293 K <strong>and</strong> ΔT
30 K. 44 The maximum efficiency, by Eq. 6.11, computes to ε 44: Note that ΔT 30 in either K or ◦ heat ≤ 9.8
C.
(980%).
Box 6.4: Is >100% Really Possible?
At first, it seems to be spooky <strong>and</strong> impossible that efficiencies can
be greater than 100%. Example 6.5.1 essentially says that as many as
8.8 J of thermal energy can be moved for a mere 1 J input of work! The
situation bears analogy to the martial art of Jiu Jitsu, whereby the
opponent’s momentum is used to their detriment, requiring little
work to direct its flow. In this case, we convince a bundle of thermal
energy sitting in the freezer to move outside where it is hotter (uphill;
against natural flow) <strong>and</strong> in the process use less energy than the
amount of thermal energy residing in the bundle.
The fact that our “efficiency” metrics come out to be greater than
100% is an illusion: an artifact of how we defined ε cool <strong>and</strong> ε heat .
Conservation of energy is not violated; we’re just putting the small
piece (ΔW) in the denominator to form the efficiency metric. 45 In
this sense, it’s not the usual sort of efficiency measure, which puts
the largest quantity (total budget) in the denominator.
Maybe the situation can be compared more
underst<strong>and</strong>ably to money transfers, where
one might pay a $20 fee to wire $1,000 from
account A to account B. It doesn’t mean
that $1,000 was created out of $20—just that
$20 was spent (like ΔW) to move a much
larger sum into account B. But if account A
belonged to somebody else, it would seem
like you just turned $20 of your own money
into $1,000 at a gain of 5,000%, even though
it really came from elsewhere.
45: Following the example numbers in Figure
6.5, we would say that ε cool , defined as
ΔQ c /ΔW, is 2.0, <strong>and</strong> ε heat is 3.0.
In the case of heating, it is worth comparing the output of a heat pump
to the application of direct heat. Let’s revisit the scenarios explored in
Section 6.3.
Example 6.5.3 If a house’s thermal performance is 150 W/ ◦ C <strong>and</strong> we
want to maintain 20 ◦ C inside while the outside temperature is a frigid
−20 ◦ C, we would need to supply 6,000 W of energy 46 to the home in 46: 150 W/ ◦ C times 40 ◦ C.
the form of burning fuel (natural gas, propane, firewood) or electricity
for direct-heating application. 47
47: . . . e.g., four space heaters each expending
1,500 W
But according to Eq. 6.11, a heat pump could theoretically move
6,000 W of thermal energy by only applying 820 W without violating
the Second Law, since ε heat ≤ 293/40 7.3 <strong>and</strong> 6,000 J (ΔQ h ) divided
by 7.3 (to get ΔW) is 820 J. 48 48: We are solving for ΔW ΔQ h/ε heat ,
<strong>and</strong> consider the energy moved in one second
in order to go from W to J.
Engineering realities will prevent operating right up to the thermody-
© 2021 T. W. Murphy, Jr.; Creative Commons Attribution-NonCommercial 4.0 International Lic.;
Freely available at: https://escholarship.org/uc/energy_ambitions.