- The document discusses the Carnot cycle, which is a reversible thermodynamic cycle consisting of two isothermal processes and two isentropic processes between a high-temperature reservoir at TH and a low-temperature reservoir at TL. - All reversible engines operating between these two temperature reservoirs have a maximum thermal efficiency known as the Carnot efficiency of 1 - TL/TH. Real irreversible engines have efficiencies below this Carnot limit. - The Carnot cycle established fundamental limits on heat engines, refrigerators, and heat pumps based on the maximum temperature difference available from the reservoirs. It laid the foundation for the modern science of thermodynamics.

1. CARNOT CYCLE
I am teaching Engineering Thermodynamics using the textbook by Cengel and Boles. This
set of slides overlap somewhat with Chapter 6. But here I assume that we have established
the concept of entropy, and use the concept to analyze the Carnot cycle in the same way as
we analyze any other thermodynamic process. An isolated system conserves energy and
generates entropy.
I did add a few slides to show how Carnot motivated his idea of entropy using the analogy of
waterfall. I used the Dover edition of his book.
I went through these slides in one 90-minute lecture.
Zhigang Suo, Harvard University

2. Thermodynamics relates heat and motion
thermo = heat
dynamics = motion

3. Stirling engine
3
https://www.youtube.com/watch?v=wGRmcvxB_dk&list=PLZbRNoceG6UmydboILKclQv7Seqy4waCE&index=
22
Please watch this video

4. Carnot’s question
How much work can be produced from a given quantity of heat?
4
“…whether the motive power of heat is unbounded, whether the possible
improvements in steam-engines have an assignable limit, a limit which the
nature of things will not allow to be passed by any means whatever...”
Carnot, Reflections on the Motive Power of Fire (1824)
Modern translations
Motive power: work
Motive power of heat: work produced by heat
Limit: Carnot limit, Carnot efficiency

5. Device runs in cycle
So they can run steadily over many, many cycles
5
Heat, Q
Device
Work, W
DSdevice =0
DUdevice =0

6. Isolated system
When confused, isolate.
6
Isolated system conserves mass over time:
Isolated system conserves energy over time:
Isolated system generates entropy over time:
Define more words:
Isolated
system
IS
dmIS
dt
= 0
dEIS
dt
= 0
dSIS
dt
³ 0
dSIS
dt
> 0, irreversible process
=0, reversible process
<0, impossible process
ì
í
ï
ï
î
ï
ï

7. 7
Reservoir of energy. Reservoir of entropy
A (purely) thermal system with a fixed temperature
The reservoir has a fixed temperature:
The reservoir receives energy by heat
Conservation of energy:
The reservoir increases entropy
(Reversible process. Clausius-Gibbs equation):
TR
DUR =QR
DSR =
DUR
TR
reservoir of energy and entropy
Fixed TR
Changing UR, SR
QR

8. 8
Heat, Q
Weight goes down.
Thermodynamics permits heater
A device runs in cycle to convert work to heat
Device
Work, W
Heat, Q
Reservoir of energy, TR
Isolated system
Q
TR
³ 0
Device runs in cycle:
Isolated system conserves energy:
Isolated system generates entropy:
DUdevice =0, DSdevice =0
Q =W

9. Thermodynamics forbids
perpetual motion of the second kind
a device runs in cycle to produce work by receiving heat from a single reservoir
9
Device
Work, W
Heat, Q
Reservoir of energy, TR
Isolated system
Heat, Q
Weight
goes up
-
Q
TR
³ 0
Device runs in cycle:
Isolated system conserves energy:
Isolated system generates entropy:
DUdevice =0, DSdevice =0
Q =W

10. Carnot’s remarks
10
1. “Wherever there exists a difference of temperature, motive power can be
produced.”
1. To maximize motive power, “contact (between bodies of different
temperatures) should be avoided as much as possible”
Low-temperature sink, TL
High-temperature source, TH
Q
Thermal contact of reservoirs of different temperatures generates entropy, and does no work.
Sgen =
Q
TL
-
Q
TH

11. Two reservoirs
11
For an engine running in cycle to convert heat to work, a single reservoir
will not do; we need reservoirs of different temperatures.
DUH = -QH , DSH = -
QH
TH
Engine
High-temperature source, TH
Low-temperature sink, TL
QH
QL
Isolated system conserves energy
isolated system generates entropy 0-
QH
TH
+
QL
TL
+0 ³ 0
Isolated system of 4 parts
DUL =QL, DSL =
QL
TL
DUengine =0, DSengine =0
DUweight =Wout, DSweight =0
Wout -QH +QL +0=0
Wout

12. 12
Carnot cycle
Carnot (1824)
Clapeyron (1834)
Gibbs (1873)

13. 13
Steam power plant

14. 14
Thermal efficiency
htheraml =
Wnet out
QH
theraml efficiency
( )=
net work out
( )
heat from high-temperature source
( )
efficiency
( )=
desired output
( )
required input
( )

15. 15
Carnot efficiency
-
QH
TH
+
QL
TL
³ 0
Isolated system conserves energy:
Isolated system generates entropy:
All reversible engines running in cycle
between reservoirs of two fixed temperatures
TH and TL have the same thermal efficiency
(Carnot efficiency):
All real engines are irreversible. For an
irreversible (i.e. real) engine running in cycle
between reservoirs of two fixed temperatures
TH and TL, the thermal efficiency is below the
Carnot efficiency:
Wnet out =QH -QL
Isolated system
QH
TH
<
QL
TL
,
Wnet out
QH
<1-
TL
TH
QH
TH
=
QL
TL
,
Wnet out
QH
=1-
TL
TH

16. 16
All real processes are irreversible
So many ways to generate entropy (i.e., to be irreversible)
Friction Heat transfer through a
temperature difference

17. Carnot (1824): Two reservoirs
Reflections on the Motive Power of Fire.
17
…the re-establishing of equilibrium in the caloric; that is, its passage from a body in
which the temperature is more or less elevated, to another in which it is lower. What
happens in fact in a steam-engine actually in motion? The caloric developed in the
furnace by the effect of the combustion traverses the walls of the boiler, produces
steam, and in some way incorporates itself with it. The latter carrying it away, takes it
first into the cylinder, where it performs some function, and from thence into the
condenser, where it is liquefied by contact with the cold water which it encounters
there. Then, as a final result, the cold water of the condenser takes possession of
the caloric developed by the combustion... The steam is here only a means of
transporting the caloric.
These two bodies, to which we can give or from which we can remove the heat
without causing their temperatures to vary, exercise the functions of two unlimited
reservoirs of caloric.
Carnot (1796-1832)
Modern translation
Caloric: entropy
Reservoir of caloric: Thermal reservoir

18. Carnot: “The steam is here only a means
of transporting the caloric.”
18
Engine
High-temperature source, TH
Low-temperature sink, TL
DSin =
QH
TH
Thermal contact generates entropy Reversible engine transports entropy
QH
TH
=
QL
TL
DSout =
QL
TL
Sgen =
Q
TL
-
Q
TH
Low-temperature sink, TL
High-temperature source, TH
Q

19. Carnot’s analogy in his own words
19
The motive power of a waterfall depends on its height and on the quantity of
the liquid; the motive power of heat depends also on the quantity of caloric
used, and on what may be termed, on what in fact we will call, the height of its
fall, that is to say, the difference of temperature of the bodies between which
the exchange of caloric is made. In the waterfall the motive power is exactly
proportional to the difference of level between the higher and lower reservoirs.
In the fall of caloric the motive power undoubtedly increases with the difference
of temperature between the warm and the cold bodies; but we do not know
whether it is proportional to this difference. We do not know, for example,
whether the fall of caloric from 100 to 50 degrees furnishes more or less motive
power than the fall of this same caloric from 50 to zero. It is a question which
we propose to examine hereafter.
Modern translations
Motive power: work
Caloric: entropy
Carnot, Reflections on the Motive Power of Fire (1824)

20. Carnot’s analogy in pictures
20
Low-height sink
high-height source
Wout

21. Carnot’s analogy in modern terms
21
Fall of water Fall of caloric (entropy)
Reservoirs Two reservoirs of water Two reservoirs of caloric (entropy)
Height of fall zH - zL TH - TL
What is falling? Quantity of water, (m2g – m1g) Quantity of entropy, (S2 – S1)
Work produced by the fall (zH – zL)(m2g – m1g) (TH – TL)(S2 – S1)
12 Gain water from source Gain entropy from source
23 Drop elevation at constant quantity of water m2g Drop temperature at constant entropy S2
34 Lose water to sink Lose entropy to sink
41 Raise elevation at constant quantity of water m1g Raise temperature at constant entropy S1
TH
TL
zH
zL
Fall of water Fall of caloric (entropy)
S2
m2g
1 2
3
4
1 2
3
4
S1
m1g

22. Carnot efficiency
reversible engine running between two reservoirs of fixed temperatures TH and TL
22
Wnet out
QH
=1-
TL
TH
Carnot efficiency:
Low-temperature reservoir is the atmosphere:
High-temperature reservoir is limited by materials
(Melting point of iron is 1811 K. Metals creep at
temperatures much below the melting point.)
Carnot efficiency in numbers 1-
TL
TH
=1-
300K
600K
=0.5
TL = 300K
TH =600K

23. 23
https://flowcharts.llnl.gov/ 23

24. 24
What you need to know about energy, The National Academies.

25. 25
Yang, Stabler, Journal of Electronic Materials. 38, 1245 (2009)
Wasted energy

26. 26
Refrigerator
efficiency
( )=
desired output
( )
required input
( )
coefficient of performance, COP
( )=
QL
Wnet in
COPR £
TL
TH -TL
QH
TH
-
QL
TL
³ 0
Isolated system conserves energy:
Isolated system generates entropy:
Carnot limit:
Wnet in =QH -QL
Isolated system

27. 27
Heat pump
efficiency
( )=
desired output
( )
required input
( )
coefficient of performance, COP
( )=
QH
Wnet in
COPHP £
TH
TH -TL
Isolated system conserves energy:
Isolated system generates entropy:
Carnot limit:
Wnet in =QH -QL
QH
TH
-
QL
TL
³ 0
Isolated system

28. Summary
• Thermodynamics permits heater (a device running in cycle to convert work to heat).
• Thermodynamics forbids perpetual motion of the second kind (a device running in cycle
to produce work by receiving heat from a single reservoir of a fixed temperature).
• Carnot cycle: A reversible cycle consisting of isothermal processes at two temperatures
TH and TL, and two isentropic processes.
• All reversible engines running in cycle between reservoirs of two fixed temperatures TH
and TL have the same thermal efficiency (Carnot efficiency):
• All real engines are irreversible. For an irreversible (i.e. real) engine running in cycle
between reservoirs of two fixed temperatures TH and TL, the thermal efficiency is below
the Carnot efficiency (Carnot limit):
• Carnot cycle also limits the coefficients of performance of refrigerators and heat pumps.
Wnet out
QH
=1-
TL
TH
Wnet out
QH
<1-
TL
TH
28