Chapter 19

Heat Engines

Important Concepts

Heat Engines

Refrigerators

Important Concepts

2nd Law of Thermo

• Heat flows spontaneously from a substance

at a higher temperature to a substance at a

lower temperature and does not flow

spontaneously in the reverse direction.

• Heat flows from hot to cold.

• Alternative: Irreversible processes must

have an increase in Entropy; Reversible

processes have no change in Entropy.

• Entropy is a measure of disorder in a system

2nd Law: Perfect Heat Engine Can NOT exist!

No energy is expelled to the cold reservoir

It takes in some amount of energy and does an equal amount of work

e = 100%

It is an impossible engine

No Free Lunch!

Limit of efficiency is a Carnot Engine

Second Law – Clausius Form

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.

Or – energy does not transfer

spontaneously by heat from a cold

object to a hot object.

Section 22.2

Heat Pumps and Refrigerators Heat engines can run in reverse

This is not a natural direction of energy transfer

Must put some energy into a device to do this

Devices that do this are called heat pumps or refrigerators

energy transferred at high tempCOP =

work done by heat pump

h

heating

Q

W

energy transferred at low tempCOP =

work done by heat pump

C

cooling

Q

W

A heat pump, is essentially an air conditioner installed backward. It extracts energy from colder air outside and deposits it in a warmer room. Suppose that the ratio of the actual energy entering the room to the work done by the device’s motor is 10.0% of the theoretical maximum ratio. Determine the energy entering the room per joule of work done by the motor, given that the inside temperature is 20.0°C and the outside temperature is –5.00°C.

Carnot cycle

0.100h hQ Q

W W

293 K0.100 0.100 1.17

293 K 268 K

h h

h c

Q T

W T T

1.17 joules of energy enter the room by heat for each joule of work done.

energy transferred at high tempCOP =

work done by heat pump

h

heating

Q

W

Heat Pumps

Refrigerators

Reversible and Irreversible Processes The Arrow of Time! Play the Movie Backwards!

Entropy

The Maximum Efficiency

H

h h

h c

Q TCOP

W T T

c cC

h c

Q TCOP

W T T

and 1c c c

c

h h h

Q T Te

Q T T

Reversible and Irreversible Processes The reversible process is an idealization.

All real processes on Earth are irreversible.

Example of an approximate reversible process:

The gas is compressed isothermally

The gas is in contact with an energy reservoir

Continually transfer just enough energy to keep the temperature constant

Section 22.3

The change in entropy is equal to zero for a reversible process and increases for irreversible processes.

Heat Engine A 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

A heat engine carries some working substance through a cyclical process

The working substance absorbs energy by heat from a high temperature energy reservoir (Qh)

Work is done by the engine (Weng) Energy is expelled as heat to a

lower temperature reservoir (Qc)

Eint = 0 for the entire cycle

Eint = 0 for the entire cycle

eng h cW Q Q

eng1h c c

h h h

W Q Q Qe

Q Q Q

Thermal efficiency is defined as the ratio of the net work done by the engine during one cycle to the energy input at the higher temperature

Thermal Efficiency of a Heat Engine

Rank in order, from largest to smallest, the work Wout

performed by these four heat engines.

A. Wb > Wa > Wc > Wd

B. Wb > Wa > Wb > Wc

C. Wb > Wa > Wb = Wc

D. Wd > Wa = Wb > Wc

E. Wd > Wa > Wb > Wc

eng1h c c

h h h

W Q Q Qe

Q Q Q

Rank in order, from largest to smallest, the work Wout

performed by these four heat engines.

A. Wb > Wa > Wc > Wd

B. Wb > Wa > Wb > Wc

C. Wb > Wa > Wb = Wc

D. Wd > Wa = Wb > Wc

E. Wd > Wa > Wb > Wc

eng1h c c

h h h

W Q Q Qe

Q Q Q

Analyze this engine to determine (a) the net work done per cycle, (b) the engine’s thermal efficiency and (c) the engine’s power output if it runs at 600 rpm. Assume the gas is monatomic and follows the ideal-gas process above.

2nd Law: Carnot’s Theorem

No real heat engine operating between two energy reservoirs can be more efficient than a Carnot engine operating between the same two reservoirs

All real engines are less efficient than a Carnot engine because they do not operate through a reversible cycle

The efficiency of a real engine is further reduced by friction, energy losses through conduction, etc.

1796 – 1832 French engineer

Section 22.4

The Carnot cycle is an ideal-gas cycle that consists of the two

adiabatic processes (Q = 0) and the two isothermal processes

(ΔEth = 0) shown. These are the two types of processes allowed in

a perfectly reversible gas engine.

Section 22.4

The Carnot cycle is an ideal-gas cycle that consists of the two

adiabatic processes (Q = 0) and the two isothermal processes

(ΔEth = 0) shown. These are the two types of processes allowed in

a perfectly reversible gas engine.

Carnot Cycle, A to B

A → B is an isothermal expansion.

The gas is placed in contact with the

high temperature reservoir, Th.

The gas absorbs heat |Qh|.

The gas does work WAB in raising the

piston.

Section 22.4

Carnot Cycle, B to C

B → C is an adiabatic expansion.

The base of the cylinder is replaced by a thermally nonconducting wall.

No energy enters or leaves the system by heat.

The temperature falls from Th to Tc.

The gas does work WBC.

Section 22.4

Carnot Cycle, C to D

The gas is placed in thermal contact

with the cold temperature reservoir.

C → D is an isothermal compression.

The gas expels energy |Qc|.

Work WCD is done on the gas.

Section 22.4

Carnot Cycle, D to A

D → A is an adiabatic compression.

The base is replaced by a thermally

nonconducting wall.

So no heat is exchanged with the

surroundings.

The temperature of the gas increases

from Tc to Th.

The work done on the gas is WDA.

Section 22.4

Carnot Engine – Carnot Cycle A heat engine operating in an ideal, reversible cycle (now called a Carnot cycle) between two reservoirs is the most efficient engine possible. This sets an upper limit on the efficiencies of all other engines

and 1c c c

c

h h h

Q T Te

Q T T

Temperatures must be in Kelvins

Carnot Cycle Problem An ideal gas is taken through a Carnot

cycle. The isothermal expansion occurs at 250°C, and the isothermal compression takes place at 50.0°C. The gas takes in 1 200 J of energy from the hot reservoir during the isothermal expansion. Find

(a) the energy expelled to the cold reservoir in each cycle and

(b) (b) the net work done by the gas in each cycle.

Refrigerators

• Understanding a refrigerator is a little harder

than understanding a heat engine.

• Heat is always transferred from a hotter

object to a colder object.

• The gas in a refrigerator can extract heat

QC from the cold reservoir only if the gas

temperature is lower than the cold-reservoir

temperature TC. Heat energy is then

transferred from the cold reservoir into the

colder gas.

• The gas in a refrigerator can exhaust heat

QH to the hot reservoir only if the gas

temperature is higher than the hot-reservoir

temperature TH. Heat energy is

then transferred from the warmer gas into

the hot reservoir.

Coefficient of Performance

The effectiveness of a heat pump is described by a number called the coefficient of performance (COP)

In heating mode, the COP is the ratio of the heat transferred in to the work required

energy transferred at high tempCOP =

work done by heat pump

hQ

W

COP, Heating Mode

COP is similar to efficiency

Qh is typically higher than W

Values of COP are generally greater than 1

It is possible for them to be less than 1

We would like the COP to be as high as possible

COP, Cooling Mode

In cooling mode, you “gain” energy from a cold temperature reservoir

A good refrigerator should have a high COP

Typical values are 5 or 6

COP cQ

W

Carnot Cycle in Reverse

Theoretically, a Carnot-cycle heat engine can run in reverse

This would constitute the most effective heat pump available

This would determine the maximum possible COPs for a given combination of hot and cold reservoirs

Can this refrigerator be built?

c cC

h c

Q TCOP

W T T

COP

H C

c

W Q Q

Q

W

Free Expansion

Consider an adiabatic free expansion.

This process is irreversible since the

gas would not spontaneously crowd

into half the volume after filling the

entire volume .

Section 22.7

The change in entropy is greater than zero for a irreversible processes.

The Limits of Efficiency

Everyone knows that heat can produce motion. That it

possesses vast motive power no one can doubt, in

these days when the steam engine is everywhere so well

known. . . . Notwithstanding the satisfactory condition to

which they have been brought today, their theory is very

little understood. The question has often been raised

whether the motive power of heat is unbounded, or

whether the possible improvements in steam engines

have an assignable limit.

Sadi Carnot

A perfectly reversible engine must use only two types of

processes:

1. Frictionless mechanical interactions with no

heat transfer (Q = 0)

2. Thermal interactions in which heat is transferred in

an isothermal process (ΔEth = 0).

Any engine that uses only these two types of processes is

called a Carnot engine.

A Carnot engine is a perfectly reversible engine; it has the

maximum possible thermal efficiency and, if operated as

a refrigerator, the maximum possible coefficient of

performance.

The Limits of Efficiency

Entropy and Heat

The original formulation of entropy dealt with the transfer of energy by heat in a

reversible process.

Let dQr be the amount of energy transferred by heat when a system follows a

reversible path.

The change in entropy, dS is

The change in entropy depends only on the endpoints and is independent of the

actual path followed.

The entropy change for an irreversible process can be determined by calculating

the change in entropy for a reversible process that connects the same initial and

final points.

rdQdS

T

Section 22.6

S in a Free Expansion

Consider an adiabatic free expansion.

This process is irreversible since the

gas would not spontaneously crowd

into half the volume after filling the

entire volume . Q = 0 but we need to find Qr

Choose an isothermal, reversible

expansion in which the gas pushes

slowly against the piston while energy

enters from a reservoir to keep T

constant.

1f fr

ri i

dQS dQ

T T

Section 22.7

Since Vf > Vi , S is positive

This indicates that both the entropy and the disorder of the gas increase as a result of the irreversible adiabatic expansion . ln f

i

VS nr

V

1f fr

ri i

dQS dQ

T T

Entropy increases when: • Temperature increases • Q flows into the system • Volume increases • Pressure decreases??

Entropy on a Microscopic Scale

We can treat entropy from a

microscopic viewpoint through

statistical analysis of molecular

motions.

A connection between entropy and

the number of microstates (W) for a

given macrostate is

S = kB ln W

The more microstates that

correspond to a given

macrostate, the greater the

entropy of that macrostate.

This shows that entropy is a measure

of disorder.

Section 22.8

Heat Death of the Universe

Ultimately, the entropy of the Universe should reach a maximum value. At this value, the Universe will be in a state of uniform temperature and density. All physical, chemical, and biological processes will cease.

The state of perfect disorder implies that no energy is available for doing work. This state is called the heat death of the Universe.

Big Bang Cosmogenesis

The Stellar Age

Stars Rule for a Trillion Years

In about 5 billion years,

Our Sun will swell into a cool Red Giant,

engulfing Mercury, Venus and possibly Earth!

The Degenerate Age After about 100 trillion, years, the stars are dead.

By 1037 years, the material in the "Local Galaxy"

consists of isolated stellar remnants and black holes.

Everything is cold and dark.

The Black Hole Age

All the stars turn into black holes.

After about 10100 years, all the black holes are gone.

The Dark Era

The remaining black holes evaporate: first the small ones, and

then the supermassive black holes. All matter that used to

make up the stars and galaxies has now degenerated into

photons and leptons.

Boltzmann was subject to rapid alternation of depressed moods with elevated, expansive or irritable moods, likely the symptoms of undiagnosed bipolar disorder. On September 5,

1906, while on a summer vacation in Duino, near Trieste, Boltzmann hanged himself during an attack of depression. He is buried in the Viennese Zentralfriedhof; his tombstone

bears the inscription.