Energy and Human Ambitions on a Finite Planet, 2021a

6 Putting Thermal <strong>Energy</strong> to Work 95
6.5 Heat Pumps
We can flip a heat engine around <strong>and</strong> call it a heat pump. In this case,
we apply some external work to drive a heat flow opposite its natural
direction—like pushing heat uphill. This is how a refrigerator 38 works,
for instance. Figure 6.5 sets the stage.
...to hot
...from cold...
drives heat flow...
large hot reservoir at T h
Q h
30
Q c
20
10
W
Q h
= Q c
+ W
large cold reservoir at T c
example quantities (J)
external work
supplied
38: ...orafreezer, or air conditioner
Figure 6.5: Heat pump energy balance. The
application of work (ΔW; from an electrical
source, for instance) can drive heat to flow—
counterintuitively—from a cold reservoir
(like the interior of a freezer) to a hotter environment.
Example T c → T h pairs might
include freezer-interior → room-air; cooledinside
→ summer-outside; winter-outside
→ warmed-inside. We still must satisfy conservation
of energy (ΔQ h ΔQ c + ΔW),
where ΔQ is a heat flow. Entropy constraints
limit how large ΔQ c can be for a given ΔW
input. Arrow widths are proportional to energy,
<strong>and</strong> red numbers are example energy
amounts, for use in the text.
A very similar chain of logic can be applied to this configuration,
invoking the Second Law to guarantee no entropy decrease. We define
the efficiency according to the application <strong>and</strong> what we care about,
giving rise to two different figures of merit.
Definition 6.5.1 ε cool : For cooling applications, 39 we care about how much
heat can be removed from the cooler environment (ΔQ c ) for a given input
of work (ΔW). The efficiency is then characterized by the ratio ε cool
ΔQ c /ΔW.
Definition 6.5.2 ε heat : For heating applications, 40 we care about the heat
delivered to the hot bath (ΔQ h ) for a given amount of input work (ΔW). The
efficiency is then characterized by the ratio ε heat ΔQ h /ΔW.
The derivation goes similarly to the one above, but now we require that
the entropy added to the hot bath must not be smaller than the entropy
removed from the cold bath so that the total change in entropy is not
negative. 41 In this case, the maximum allowed efficiencies for cooling
<strong>and</strong> heating via heat pumps are:
39: . . . freezer, refrigerator, air conditioner
40: . . . home heating via heat pump
41: Imposing this condition has the result
that ΔS h ≥ ΔS c ; opposite Eq. 6.4 since the
direction of flow changed.
<strong>and</strong>
ε cool ≤
ε heat ≤
T c
T h − T c
T c
ΔT , (6.10)
T h
T h − T c
T h
ΔT . (6.11)
These look a lot like Eq. 6.9, but turned upside down. The maximum
42
efficiencies can be larger than unity!
42: See Box 6.4.
Temperature must be in Kelvin for these
relations.
© 2021 T. W. Murphy, Jr.; Creative Commons Attribution-NonCommercial 4.0 International Lic.;
Freely available at: https://escholarship.org/uc/energy_ambitions.