Physics For Scientists And Engineers 6E - part 168

S E C T I O N 2 2 . 1 • Heat Engines and the Second Law of Thermodynamics

669

22.1 Heat Engines and the Second Law

of Thermodynamics

heat engine is a device that takes in energy by heat

and, operating in a cyclic

process, expels a fraction of that energy by means of work. For instance, in a typical
process by which a power plant produces electricity, coal or some other fuel is burned,
and the high-temperature gases produced are used to convert liquid water to steam.
This steam is directed at the blades of a turbine, setting it into rotation. The mechani-
cal energy associated with this rotation is used to drive an electric generator. Another
device that can be modeled as a heat engine—the internal combustion engine in an
automobile—uses energy from a burning fuel to perform work on pistons that results
in the motion of the automobile.

A heat engine carries some working substance through a cyclic process during

which (1) the working substance absorbs energy by heat from a high-temperature en-
ergy reservoir, (2) work is done by the engine, and (3) energy is expelled by heat to a
lower-temperature reservoir. As an example, consider the operation of a steam engine
(Fig. 22.1), which uses water as the working substance. The water in a boiler absorbs
energy from burning fuel and evaporates to steam, which then does work by expand-
ing against a piston. After the steam cools and condenses, the liquid water produced
returns to the boiler and the cycle repeats.

It is useful to represent a heat engine schematically as in Figure 22.2. The engine

absorbs a quantity of energy

! from the hot reservoir. For this discussion of heat en-

gines, we will use absolute values to make all energy transfers positive and will indicate
the direction of transfer with an explicit positive or negative sign. The engine does
work W

eng

(so that negative work W ! " W

eng

is done on the engine), and then gives up

a quantity of energy

! to the cold reservoir. Because the working substance goes

We will use heat as our model for energy transfer into a heat engine. Other methods of energy

transfer are also possible in the model of a heat engine, however. For example, the Earth’s atmosphere
can be modeled as a heat engine, in which the input energy transfer is by means of electromagnetic
radiation from the Sun. The output of the atmospheric heat engine causes the wind structure in the
atmosphere.

Figure 22.1 This steam-driven

locomotive runs from Durango to

Silverton, Colorado. It obtains its

energy by burning wood or coal.

The generated energy vaporizes

water into steam, which powers the

locomotive. (This locomotive must

take on water from tanks located

along the route to replace steam

lost through the funnel.) Modern

locomotives use diesel fuel instead

of wood or coal. Whether old-

fashioned or modern, such

locomotives can be modeled as

heat engines, which extract energy

from a burning fuel and convert a

fraction of it to mechanical energy.

Phil

Degginger/Stone/Getty

Hot reservoir at T

eng

Cold reservoir at T

Engine

Active Figure 22.2 Schematic

representation of a heat engine.

The engine does work W

eng

. The

arrow at the top represents energy

0 entering the engine. At the

bottom, Q

0 represents energy

leaving the engine.

At the Active Figures link

at http://www.pse6.com, you

can select the efficiency of the

engine and observe the

transfer of energy.

through a cycle, its initial and final internal energies are equal, and so %E

int

Hence, from the first law of thermodynamics, %E

int

Q & W ! Q " W

eng

, and with

no change in internal energy,

the net work W

eng

done by a heat engine is equal

to the net energy Q

net

transferred to it. As we can see from Figure 22.2,

net

| " |Q

|; therefore,

(22.1)

If the working substance is a gas,

the net work done in a cyclic process is the

area enclosed by the curve representing the process on a PV diagram. This is
shown for an arbitrary cyclic process in Figure 22.3.

The

thermal efficiency e of a heat engine is defined as the ratio of the net work

done by the engine during one cycle to the energy input at the higher temperature
during the cycle:

(22.2)

We can think of the efficiency as the ratio of what you gain (work) to what you give
(energy transfer at the higher temperature). In practice, all heat engines expel only a
fraction of the input energy Q

by mechanical work and consequently their efficiency

is always less than 100%. For example, a good automobile engine has an efficiency of
about 20%, and diesel engines have efficiencies ranging from 35% to 40%.

Equation 22.2 shows that a heat engine has 100% efficiency (e ! 1) only if

! ! 0—that is, if no energy is expelled to the cold reservoir. In other words, a heat

engine with perfect efficiency would have to expel all of the input energy by work. On
the basis of the fact that efficiencies of real engines are well below 100%, the
Kelvin–Planck form of the second law of thermodynamics states the following:

It is impossible to construct a heat engine that, operating in a cycle, produces no
effect other than the input of energy by heat from a reservoir and the performance
of an equal amount of work.

This statement of the second law means that, during the operation of a heat engine,
W

eng

can never be equal to

!, or, alternatively, that some energy !Q

! must be

rejected to the environment. Figure 22.4 is a schematic diagram of the impossible
“perfect” heat engine.

e !

eng

! Q

! " ! Q

! Q

1 "

! Q

eng

! " !Q

670

C H A P T E R 2 2 • Heat Engines, Entropy, and the Second Law of Thermodynamics

Area = W

eng

Figure 22.3 PV diagram for an

arbitrary cyclic process taking place

in an engine. The value of the net

work done by the engine in one

cycle equals the area enclosed by

the curve.

Thermal efficiency of a heat

engine

Quick Quiz 22.1

The energy input to an engine is 3.00 times greater than

the work it performs. What is its thermal efficiency? (a) 3.00 (b) 1.00 (c) 0.333
(d) impossible to determine

Quick Quiz 22.2

For the engine of Quick Quiz 22.1, what fraction of the en-

ergy input is expelled to the cold reservoir? (a) 0.333 (b) 0.667 (c) 1.00 (d) impossible
to determine

Example 22.1 The Efficiency of an Engine

An engine transfers 2.00 ' 10

J of energy from a hot reser-

voir during a cycle and transfers 1.50 ' 10

J as exhaust to a

cold reservoir.

(A)

Find the efficiency of the engine.

Solution The efficiency of the engine is given by Equation
22.2 as

0.250, or 25.0%

e ! 1 "

! Q

1 "

1.50 ' 10

2.00 ' 10

The impossible engine

Cold reservoir at T

Engine

Hot reservoir at T

eng

Figure 22.4 Schematic diagram of

a heat engine that takes in energy

from a hot reservoir and does an

equivalent amount of work. It is

impossible to construct such a

perfect engine.

S E C T I O N 2 2 . 2 • Heat Pumps and Refrigerators

671

22.2 Heat Pumps and Refrigerators

In a heat engine, the direction of energy transfer is from the hot reservoir to the cold
reservoir, which is the natural direction. The role of the heat engine is to process the
energy from the hot reservoir so as to do useful work. What if we wanted to transfer en-
ergy from the cold reservoir to the hot reservoir? Because this is not the natural direc-
tion of energy transfer, we must put some energy into a device in order to accomplish
this. Devices that perform this task are called

heat pumps or refrigerators. For exam-

ple, we cool homes in summer using heat pumps called air conditioners. The air condi-
tioner transfers energy from the cool room in the home to the warm air outside.

In a refrigerator or heat pump, the engine takes in energy

! from a cold reser-

voir and expels energy

! to a hot reservoir (Fig. 22.5). This can be accomplished

only if work is done on the engine. From the first law, we know that the energy given up
to the hot reservoir must equal the sum of the work done and the energy taken in from
the cold reservoir. Therefore, the refrigerator or heat pump transfers energy from a
colder body (for example, the contents of a kitchen refrigerator or the winter air out-
side a building) to a hotter body (the air in the kitchen or a room in the building). In
practice, it is desirable to carry out this process with a minimum of work. If it could be
accomplished without doing any work, then the refrigerator or heat pump would be
“perfect” (Fig. 22.6). Again, the existence of such a device would be in violation of the
second law of thermodynamics, which in the form of the

Clausius statement

states:

(B)

How much work does this engine do in one cycle?

Solution The work done is the difference between the
input and output energies:

What If?

Suppose you were asked for the power output of

this engine? Do you have sufficient information to answer
this question?

5.0 ' 10

eng

! Q

! " ! Q

! ! 2.00 ' 10

J " 1.50 ' 10

Answer No, you do not have enough information. The
power of an engine is the rate at which work is done by the
engine. You know how much work is done per cycle but you
have no information about the time interval associated with
one cycle. However, if you were told that the engine oper-
ates at 2 000 rpm (revolutions per minute), you could relate
this rate to the period of rotation T of the mechanism of the
engine. If we assume that there is one thermodynamic cycle
per revolution, then the power is

! !

eng

5.0 ' 10

000

min

1 min

60 s

1.7 ' 10

First expressed by Rudolf Clausius (1822–1888).

Cold reservoir at T

Heat pump

Hot reservoir at T

Active Figure 22.5 Schematic diagram of a heat pump,

which takes in energy Q

0 from a cold reservoir and

expels energy Q

0 to a hot reservoir. Work W is done

on the heat pump. A refrigerator works the same way.

▲

PITFALL PREVENTION

22.1 The First and Second

Laws

Notice  the  distinction  between
the  first  and  second  laws  of
thermodynamics.  If  a  gas  under-
goes  a  one-time  isothermal  process
%

int

Q & W ! 0. Therefore,

the  first  law  allows  all energy  in-
put  by  heat  to  be  expelled  by
work.  In  a  heat  engine,  however,
in which a substance undergoes a
cyclic  process,  only  a  portion of
the  energy  input  by  heat  can  be
expelled  by  work  according  to
the second law.

At the Active Figures link

at http://www.pse6.com, you

can select the COP of the heat

pump and observe the transfer

of energy.

It is impossible to construct a cyclical machine whose sole effect is to transfer energy
continuously by heat from one object to another object at a higher temperature
without the input of energy by work.

In simpler terms,

energy does not transfer spontaneously by heat from a cold

object to a hot object. This direction of energy transfer requires an input of energy to
a heat pump, which is often supplied by means of electricity.

The Clausius and Kelvin–Planck statements of the second law of thermodynamics

appear, at first sight, to be unrelated, but in fact they are equivalent in all respects. Al-
though we do not prove so here, if either statement is false, then so is the other.

Heat pumps have long been used for cooling homes and buildings, and they are

now becoming increasingly popular for heating them as well. The heat pump contains
two sets of metal coils that can exchange energy by heat with the surroundings: one set
on the outside of the building, in contact with the air or buried in the ground, and the
other set in the interior of the building. In the heating mode, a circulating fluid flow-
ing through the coils absorbs energy from the outside and releases it to the interior of
the building from the interior coils. The fluid is cold and at low pressure when it is in
the external coils, where it absorbs energy by heat from either the air or the ground.
The resulting warm fluid is then compressed and enters the interior coils as a hot,
high-pressure fluid, where it releases its stored energy to the interior air.

An air conditioner is simply a heat pump with its exterior and interior coils inter-

changed, so that it operates in the cooling mode. Energy is absorbed into the circulat-
ing  fluid  in  the  interior  coils;  then,  after  the  fluid  is  compressed,  energy  leaves  the
fluid through the external coils. The air conditioner must have a way to release energy
to the outside. Otherwise, the work done on the air conditioner would represent en-
ergy  added  to  the  air  inside  the  house,  and  the  temperature  would  increase.  In  the
same  manner,  a  refrigerator  cannot  cool  the  kitchen  if  the  refrigerator  door  is  left
open.  The  amount  of  energy  leaving  the  external  coils  (Fig.  22.7)  behind  or  under-
neath the refrigerator is greater than the amount of energy removed from the food.
The difference between the energy out and the energy in is the work done by the elec-
tricity supplied to the refrigerator.

The effectiveness of a heat pump is described in terms of a number called the

coeffi-

cient of performance (COP). In the heating mode, the COP is defined as the ratio of
the energy transferred to the hot reservoir to the work required to transfer that energy:

(22.3)

Note that the COP is similar to the thermal efficiency for a heat engine in that it is a
ratio of what you gain (energy delivered to the interior of the building) to what you
give (work input). Because |Q

| is generally greater than W, typical values for the COP

are greater than unity. It is desirable for the COP to be as high as possible, just as it is
desirable for the thermal efficiency of an engine to be as high as possible.

If the outside temperature is 25°F ("4°C) or higher, a typical value of the COP for a

heat pump is about 4. That is, the amount of energy transferred to the building is about
four times greater than the work done by the motor in the heat pump. However, as the
outside temperature decreases, it becomes more difficult for the heat pump to extract
sufficient energy from the air, and so the COP decreases. In fact, the COP can fall below
unity for temperatures below about 15°F ("9°C). Thus, the use of heat pumps that
extract energy from the air, while satisfactory in moderate climates, is not appropriate in
areas where winter temperatures are very low. It is possible to use heat pumps in colder

COP (heating mode)

energy transferred at high temperature

work done by heat pump

! Q

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C H A P T E R 2 2 • Heat Engines, Entropy, and the Second Law of Thermodynamics

Hot reservoir at T

Q

 = Q

Cold reservoir at T

Heat pump

Impossible heat pump

Figure 22.6 Schematic diagram

of an impossible heat pump or

refrigerator—that is, one that takes

in energy from a cold reservoir and

expels an equivalent amount of

energy to a hot reservoir without

the input of energy by work.

See, for example, R. P. Bauman, Modern Thermodynamics and Statistical Mechanics, New York,

Macmillan Publishing Co., 1992.

Figure 22.7 The coils on the back

of a refrigerator transfer energy by

heat to the air. The second law of

thermodynamics states that this

amount of energy must be greater

than the amount of energy

removed from the contents of the

refrigerator, due to the input of

energy by work.

Charles D. Winters

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