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
Thermodynamics relates heat and motion
thermo = heat
dynamics = motion
Stirling engine
Please watch this video
https://www.youtube.com/watch?v=wGRmcvxB_dk&list=PLZbRNoceG6UmydboILKclQv7Seqy4waCE&index=
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Carnot’s question
How much work can be produced from a given quantity of heat?
“…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...”
Modern translations
Motive power: work
Motive power of heat: work produced by heat
Limit: Carnot limit, Carnot efficiency
Carnot, Reflections on the Motive Power of Fire (1824)
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Device runs in cycle
So they can run steadily over many, many cycles
Heat, Q
Work, W
Device
DU device = 0
DSdevice = 0
5
Isolated system
When confused, isolate.
Isolated
system
IS
Isolated system conserves mass over time:
Isolated system conserves energy over time:
Isolated system generates entropy over time:
Define more words:
dmIS
dt
dEIS
dt
dSIS
dt
=0
=0
³0
ì > 0, irreversible process
dSIS ïï
í =0, reversible process
dt ï
ïî <0, impossible process
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Reservoir of energy. Reservoir of entropy
A (purely) thermal system with a fixed temperature
QR
reservoir of energy and entropy
Fixed TR
Changing UR, SR
The reservoir has a fixed temperature:
TR
The reservoir receives energy by heat
Conservation of energy:
DU R = QR
The reservoir increases entropy
DU R
DS
=
(Reversible process. Clausius-Gibbs equation):
R
T
R
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Thermodynamics permits heater
A device runs in cycle to convert work to heat
Isolated system
Reservoir of energy, TR
Heat, Q
Heat, Q
Device
Work, W
Weight goes down.
Device runs in cycle:
DU device = 0, DSdevice = 0
Isolated system conserves energy:
Q =W
Isolated system generates entropy:
Q
³0
TR
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Thermodynamics forbids
perpetual motion of the second kind
a device runs in cycle to produce work by receiving heat from a single reservoir
Isolated system
Reservoir of energy, TR
Heat, Q
Weight
goes up
Heat, Q
Device
Work, W
Device runs in cycle:
Isolated system conserves energy:
Isolated system generates entropy:
DU device = 0, DSdevice = 0
Q =W
-
Q
³0
TR
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Carnot’s remarks
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”
High-temperature source, TH
Q
Low-temperature sink, TL
Thermal contact of reservoirs of different temperatures generates entropy, and does no work.
Sgen =
Q Q
TL TH
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Two reservoirs
For an engine running in cycle to convert heat to work, a single reservoir
will not do; we need reservoirs of different temperatures.
Isolated system of 4 parts
Q
DU H = -QH , DSH = - H
TH
High-temperature source, TH
QH
DU engine = 0, DSengine = 0
Engine
QL
DU weight =Wout , DSweight = 0
Wout
Low-temperature sink, TL
Q
DU L = QL , DSL = L
T
L
Isolated system conserves energy
Wout -QH +QL +0 = 0
isolated system generates entropy
Q
Q
0- H + L +0 ³ 0
TH TL
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Carnot cycle
Clapeyron (1834)
Carnot (1824)
Gibbs (1873)
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Steam power plant
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Thermal efficiency
desired output )
(
(efficiency ) = required input
(
)
( theraml efficiency ) =
(net work out)
( heat from high-temperature source)
htheraml =
Wnet out
QH
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Carnot efficiency
Isolated system
Isolated system conserves energy:
Wnet out = QH -QL
Isolated system generates entropy:
Q
Q
- H + L ³0
TH TL
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
Q
T
= L,
=1- L
TH TL
QH
TH
QH
Wnet out
Q
T
< L,
<1- L
TH TL
QH
TH
QH
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All real processes are irreversible
So many ways to generate entropy (i.e., to be irreversible)
Friction
Heat transfer through a
temperature difference
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Carnot (1824): Two reservoirs
Reflections on the Motive Power of Fire.
…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.
Modern translation
Caloric: entropy
Reservoir of caloric: Thermal reservoir
Carnot (1796-1832)
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Carnot: “The steam is here only a means
of transporting the caloric.”
High-temperature source, TH
Q
DSin = H
TH
High-temperature source, TH
Q
Engine
Low-temperature sink, TL
Q
DSout = L
TL
Low-temperature sink, TL
Thermal contact generates entropy
Sgen =
Q Q
TL TH
Reversible engine transports entropy
QH
Q
= L
TH TL
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Carnot’s analogy in his own words
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)
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Carnot’s analogy in pictures
high-height source
Wout
Low-height sink
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Carnot’s analogy in modern terms
Fall of caloric (entropy)
Fall of water
zH
zL
1
2
TH
1
2
4
3
TL
4
m1g
3
m2g
S1
S2
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
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Carnot efficiency
reversible engine running between two reservoirs of fixed temperatures TH and TL
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
Wnet out
QH
T
=1- L
TH
TL = 300K
TH = 600K
T
300K
1- L =1= 0.5
TH
600K
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https://flowcharts.llnl.gov/
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What you need to know about energy, The National Academies.
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Wasted energy
Yang, Stabler, Journal of Electronic Materials. 38, 1245 (2009)
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Refrigerator
Isolated system
desired output )
(
(efficiency ) = required input
(
)
QL
(coefficient of performance, COP) = W
net in
Isolated system conserves energy:
Isolated system generates entropy:
Carnot limit:
Wnet in = QH -QL
QH
Q
- L ³0
TH TL
COPR £
TL
TH -TL
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Isolated system
Heat pump
desired output )
(
(efficiency ) = required input
(
)
QH
(coefficient of performance, COP) = W
net in
Isolated system conserves energy:
Isolated system generates entropy:
Carnot limit:
Wnet in = QH -QL
QH
Q
- L ³0
TH TL
COPHP £
TH
TH -TL
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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):
Wnet out
QH
•
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): W
T
net out
•
T
=1- L
TH
QH
<1- L
TH
Carnot cycle also limits the coefficients of performance of refrigerators and heat pumps.
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