19
Heat Engines and Refrigerators
Recommended class days: 3 minimum, 4 preferred
Background Information
Thermodynamics, properly presented, is a grand subject. It is a sweeping, “big picture” view
of energy. Unfortunately, this big picture is barely glimpsed in the standard presentation, which
becomes focused on myriad details and tedious calculations.
Indeed, thermodynamics really does require attention to details and equations. There are
many different processes, many different parameters, and many different relationships between
the parameters—some general, others applicable only to specific processes. There are 20 or so
distinct and important equations in this chapter (many in Table 19.1), depending on how you define
“important,” plus many other minor equations. It takes practice and experience—and the guidance
of the instructor—for students to have any hope of seeing the big picture without drowning in this
sea of details.
However, there is a systematic procedure to follow when analyzing heat engines, a procedure
given in Problem-Solving Strategy 19.1. By following this strategy in your examples, and emphasizing it, you can help students distinguish what thermodynamics is all about from the details of
the calculations.
Student Learning Objectives
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To understand the thermodynamics of the four basic processes of an ideal gas.
To understand the physics of simple heat engines and refrigerators.
To recognize that thermodynamics has practical applications to real devices.
To learn that there is a limit to the efficiency of a heat engine.
Pedagogical Approach
In order to avoid the complications of phase changes, this chapter considers only heat engines
and refrigerators using an ideal gas as the working substance. Students are now familiar with the
four basic gas processes, and they should be able to visualize how these come about. Table 19.l
summarizes the information. Even so, students will need additional practice with ideal-gas
calculations before becoming adept.
The presentation of information is straightforward. Care has been taken to:
• Emphasize that thermodynamics is about the transfer and transformation of energy.
• Use a variety of gas processes so that students can recognize the similarities and differences.
• Use realistic examples.
This chapter finally introduces the work done by the system, distinguishing it with the symbol
Ws. The net work done by a heat engine during one full cycle is called Wout and the work done on a
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Instructor’s Guide
refrigerator during one full cycle is Win. It is important to distinguish between these because work
really is used in these three different contexts. Experienced physicists can get away with the single
symbol W because they recognize, from the context, how W is being used. Students, however, will
get these three ideas hopelessly confused unless the notation is clear.
The heat engine embodies the central concepts of thermodynamics—transferring and transforming energy. A heat engine is a complex idea, a model of many different kinds of real engines.
Many texts don’t give a clear explanation of what a heat engine is or of the assumptions behind it.
It’s particularly difficult for students to understand:
• Why a heat engine must use a cyclical process.
• That a heat engine requires both a heat input and a heat output.
• That the common statement W  Q means Wout  QH  QC.
• Why Wout is the area inside the closed pV curve.
• Why (Eth)net  0.
Sections 19.1 and 19.2 give an overview of these important issues before diving into the details.
There’s a tacit assumption in the text that all processes take place sufficiently slowly and remain
sufficiently close to thermal equilibrium that they can be represented by a pV trajectory and that
dQ  TdS. The text does not distinguish between reversible and irreversible processes. The usual
analysis of heat engines assumes that the working substance follows a reversible cycle, with
Sgas  0, but the entire process, including the heat reservoirs, is irreversible (except for a Carnot
engine) because Stotal  0. I’ve found that the subtleties of these issues are lost on students, and the
time needed to explicate them is better used for more important issues. Hence the decision to sidestep the issue and focus just on reversible processes for which the standard analysis is valid.
Irreversible processes, such as the free expansion of a gas, are not considered. Students who take a
more advanced course in thermodynamics will encounter these ideas at a later time.
This textbook does not use the term “heat reservoir.” This term suggests that heat is a substance
“in” the reservoir—such as water in a lake. After endeavoring to convince students that heat is a
transfer of energy, not a kind of energy, it’s not good to use antiquated terminology that suggests
otherwise. Thus this textbook refers to the “hot reservoir” and “cold reservoir” of the heat engine as
being the source and sink for the transfer of heat energy. Perhaps not yet ideal terminology, but at
least an improvement.
As a word of caution, you should not use heat-engine cycles with a
downward-sloping straight line section such as that shown in the figure. As
Dickerson and Mottman (Amer. J. Phys. 62, 558 (1994)) have pointed out,
the analysis of such as process is not as straightforward as is tacitly assumed
by many texts. In such a process with T  0, which appears to be a cooling
process, the heat Q is not purely QC. A more careful analysis, beyond what is
appropriate for the introductory course, shows that heat is transferred to the
system until the straight line is tangent to an adiabat, then from the system for the remainder of the
process. The Q computed as Eth  Ws is the combined QH  QC, but a simple analysis doesn’t yield
QC and QH separately.
This textbook gives more attention to ideal-gas refrigerators than do other textbooks. Refrigerators present some interesting and instructive ideas. Although it’s often said that a refrigerator is a
heat engine running backward, it turns out that not every heat engine run backward is a refrigerator.
In order for heat energy QC to be transferred from the cold reservoir to the gas in the refrigerator, the
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gas has to be at a colder temperature than the cold reservoir. Similarly, the gas must later be hotter
than the hot reservoir in order to exhaust energy QH. A refrigerator cycle must use adiabatic
processes to bring about these conditions. Attention to these rather subtle points focuses students’
attention on some of the key ideas of thermodynamics.
Using Class Time
There is a lot of information in this chapter, and students need to see (and practice) a wide variety of
examples. Consequently, a minimum of three days are needed to do justice to Chapter 19. A fourth
day will allow more time for examples.
DAY 1: There aren’t many demonstrations that can be used with heat engines, so the instructor
needs to resist the temptation to fall back into a purely lecture mode. There are still many
opportunities to pose qualitative questions to students, and students should be involved at different
stages of working example problems.
The first day is used to introduce heat engines and the idea of the efficiency of an engine. The
energy-transfer diagrams commonly used to portray heat engines and refrigerators are new to
students, and many do not understand these diagrams without some explanation. That’s why I start
by showing the diagram for a simple situation in which heat energy is transferred from hot to cold
through a copper bar. This introduces the idea of an energy reservoir and the terms QH and QC in a
familiar context, before adding the complication of the system doing work. Simple examples show
that work can be transformed entirely into heat energy, and you can show how this looks on an
energy-transfer diagram. Now you’ve set the stage for asking whether—and how—heat energy can
be transformed into work. A critical idea is that any practical device for transforming heat into work
must be cyclical, and thus (Eth)net  0.
Once the basic idea is established, the first law (i.e., energy conservation) allows you to relate
Wout to QH and QC and to make the standard definitions of efficiency and coefficient of performance.
The “pipes” of the energy-transfer diagram implicitly conserve energy, but you want to explicitly
note that “what goes in must come out.” The fact that there are no perfect heat engines—
asserted for now, proven later—leads to the important conclusion that a heat engine has both a heat
input QH and a heat output QC.
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Instructor’s Guide
DAY 2: The energy-transfer diagrams show what we want heat engines and refrigerators to do, but
not how to do it. Day 2 can now be devoted to ideal-gas heat engines. Although examples quickly
get into details, the “big picture” to emphasize at the beginning is:
• Any closed pV trajectory has some processes in which heat QH is transferred into the system and
some processes in which QC is transferred out of the system. Heat transfer requires a
temperature difference, so these processes require a hot reservoir and a cold reservoir.
• For a clockwise trajectory, the system always does work Wout on the environment.
• Thus, regardless of the details, heat energy is being partially transformed into work and the
remaining heat energy is exhausted to the cold reservoir.
Of course, the devil is in the details.
To make the idea of a heat engine “real,” you need to associate the pV diagram with real
physical processes. Textbook Figure 19.8 shows a simple heat engine, Figure 19.9 represents the
heat engine as a pV diagram, and Example 19.1 then does a calculation of the efficiency of such an
engine. Class examples need not be this detailed, but it’s certainly good to give some idea about
how the processes might be carried out.
A good class example uses a heat engine similar to Example
19.1, but with the isothermal process replaced by an adiabatic
process. Have students analyze the engine qualitatively before
getting into the mathematics. In particular, ask them to:
• Determine whether Ws is positive, negative, or zero for each
process.
• Determine whether Q is positive, negative, or zero for each
process.
• Describe in words and pictures how the cycle is accomplished.
Then they can follow the steps of Problem-Solving Strategy 19.1 to determine:
• p, V, and T at each corner.
• Ws, Q, and Eth for each process.
• Wout, QH, QC, and (Eth)net.
• The efficiency .
Encourage them to put the values in a table, since otherwise they’ll lose track of information. As a
follow-up, ask for the power output of the engine if it runs at 1200 rpm.
Another interesting heat-engine example is shown in the
diagram here. Students will begin by seeking an equation to
calculate Q along the diagonal, and it’s worth letting them find that
there is no formula. They must find Ws as the area under the curve,
the temperatures from the ideal gas law, Eth from nCVT, and
finally use the first law Q  Ws  Eth. Nearly every important idea
is used in this example, and it helps convince students that—despite
the numerous equations—there really is a logic to thermodynamics.
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DAY 3: Day 3 can turn to refrigerators and then the Carnot cycle, ending with the interesting and
profound conclusion that heat engines have a maximum efficiency. This will likely be a day spent
mostly in lecture mode. The logical argument that the particular cycle called “the Carnot cycle” is
the maximum-efficiency cycle is subtle and probably not worth spending much time on. The more
important idea for students to remember is that there is such a cycle and what the maximum
efficiency is.
No one is surprised by now that an engine can’t violate conservation of energy—you can’t get
something for nothing. But it is truly surprising that not only is there a limit to the efficiency of a
heat engine, but you can state quite clearly what the limit is. And for most practical devices, the
limiting efficiency is not very high.
Simple exercises about engine efficiency are easy to devise. The
figure shows an example in which you can ask, “Is there some heat
engine for which this is a physically possible behavior?” In this case, the
efficiency of 30%, although low, exceeds the Carnot efficiency of 27%.
So no engine can work this efficiently between temperatures of 100°C
and 0°C.
Sample Reading Quiz Questions
1. What is the generic name for a cyclical device that transforms heat energy into work?
2. The area enclosed within a pV curve is
a. the work done by the system during one complete cycle.
b. the work done on the system during one complete cycle.
c. the heat transferred to the system during one complete cycle.
d. the heat transferred out of the system during one complete cycle.
3. The maximum possible efficiency of a heat engine is determined by
a. its design.
d. the amount of heat that flows.
b. the maximum and minimum pressure.
e. the compression ratio.
c. the maximum and minimum temperature.
4. The engine with the largest possible efficiency uses a
a. Brayton cycle.
d. Joule cycle.
b. Carnot cycle.
e. Otto cycle.
c. Diesel cycle.
f. Stirling cycle.
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Instructor’s Guide
Sample Exam Questions
These questions cover the material of Chapters 18–19.
1. a. Why is CP larger than CV? Write a short answer in which you make explicit use of a pV
diagram.
b. Why is CV for a diatomic gas larger than CV for a monatomic gas?
2. A monatomic gas is adiabatically compressed to 1/8 of its original volume. Do each of the
following quantities change? If so, does the quantity increase or decrease, and by what factor?
If not, why not?
a. The rms speed vrms.
c. The temperature.
b. The mean free path .
d. The molar heat capacity CV.
3. Two gas cylinders, A and B, each hold 100 cm3 of nitrogen gas at a temperature of 30°C and a
pressure of 1 atm. Cylinder A undergoes an isobaric expansion to 300 cm3. Cylinder B
undergoes an isothermal expansion that does the same amount of work as cylinder A.
a. What are the final volume, temperature, and pressure of each cylinder?
b. Show both processes on a pV diagram. Provide an appropriate numerical scale on each axis.
(Note: It’s good to provide an empty set of pV axes.)
4. A heat engine uses 120 mg of helium (A  4) as its working substance. The gas follows the
thermodynamic cycle shown.
a. What is the thermal efficiency of this engine?
b. What is the maximum possible thermal efficiency of an engine that operates between Tmax
and Tmin?