Lecture 11: More on Heat Engines• Last time we stated Carnot’s Theorem, which says that for

a given temperature difference, the most efficient heatengine possible is a reversible one

• To prove this, let’s start by asking what happens if Carnotis wrong

• That means I can make a “Varnes engine” which is notreversible but has better efficiency than a Carnot engine

• Now let’s say I use my engine to run a Carnot enginebackwards– I can do this since the Carnot engine is reversible– this really means I turn the Carnot engine into a refrigerator

• Both engines obey the 1st Law of Thermodynamics:

WC= Q

h,C! Q

c,C= e

CQ

h,C

WV= Q

h,V! Q

c,V= e

VQ

h,V

• Let’s say I set this up so that the Varnes engine providesthe power for the Carnot refrigerator. Then:

• Looking at the entire Varnes+Carnot apparatus, then, onesees– no net work input or output– a net flow of heat into the high-temperature reservoir

• In other words, this violates the 2nd Law ofThermodynamics!– that means I can’t build an engine better than Carnot’s

WC=W

V

eC

Qh,C

= eV

Qh,V

Qh,C

Qh,V

=e

V

eC

>1

Since we assumed at thestart that eV > eC

The Carnot Engine• In our idealized model of a heat engine, we do the

following:1. Asborb energy from high-temperature reservoir

(isothermal at Th)2. Do work (i.e. gas expands) without any heat transfer

(adiabatic)3. Exhaust energy to low-temperature reservoir (isothermal

at Tc)4. Compress gas to return to initial state, without heat

transfer (adiabatic)• In the Carnot engine, all of these steps are done reversibly

Carnot Efficiency• Heat input:

• Work done (by the engine):– As heat is input:

– During adiabatic expansion:

Qh= nRT

hln

VB

VA

!

"#$

%&

W1= nRT

hln

VB

VA

!

"#$

%&

W2=

1

! "1

#$%

&'(

PBV

B" P

CV

C( )

– as heat is exhausted:

– as gas is compressed

– So the total work done is:

W3= nRT ln

VD

VC

!

"#$

%&

W4=

1

! "1

#$%

&'(

PDV

D" P

AV

A( )

W =W1+W

2+W

3+W

4

= nRThln

VB

VA

!

"#$

%&+

1

' (1

!"#

$%&

PBV

B( P

CV

C( )

+nRTcln

VD

VC

!

"#$

%&+

1

' (1

!"#

$%&

PDV

D( P

AV

A( )

W = nRThln

VB

VA

!

"#$

%&+ nRT

cln

VD

VC

!

"#$

%&

+1

' (1

!"#

$%&

nRTh( nRT

c+ nRT

c( nRT

h( )

= nR Thln

VB

VA

!

"#$

%&+T

cln

VD

VC

!

"#$

%&!

"#

$

%&

• This means the efficiency is:

e =W

Qh

=

nR Thln

VB

VA

!

"#$

%&+T

cln

VD

VC

!

"#$

%&!

"#

$

%&

nRThln

VB

VA

!

"#$

%&

= 1+

Tcln

VD

VC

!

"#$

%&

Thln

VB

VA

!

"#$

%&

= 1'

Tcln

VD

VC

!

"#$

%&

Thln

VA

VB

!

"#$

%&

• Using the properties of adiabatic and isothermalexpansions, we have:

• So the efficiency is:

PAV

A= P

BV

B; P

CV

C= P

DV

D

PAV

A

!= P

DV

D

!; P

BV

B

!= P

CV

C

!

PBV

BV

A

! "1= P

DV

D

!

PBV

B

!= P

DV

DV

C

! "1

VA

! "1

VB

! "1=

VD

! "1

VC

! "1

VA

VB

=V

D

VC

e = 1!

Tcln

VD

VC

"

#$%

&'

Thln

VA

VB

"

#$%

&'

= 1!

Tcln

VA

VB

"

#$%

&'

Thln

VA

VB

"

#$%

&'

= 1!T

c

Th

The Internal Combustion Engine• The heat engine that you’re most familiar with is probably the

automotive gasoline engine– heat from burning gasoline is converted to work in the form of

motion of a car• We’ll study a somewhat idealized version of this engine• The engine goes through the following steps in each cycle:

1. Intake stroke: Piston movesdown, drawing air+gasmixture into cylinder at 1 atm• Volume of gas increases

from V2 to V1

V

P

V2 V1

A

2. Compression stroke: Intake valvecloses, and piston moves upward• air+gas compressed from V1 to V2

• temperature increases from TA to TB

• This happens quickly -- close toadiabatic since there’s little time forheat transfer

3. Combustion: Spark plug fires,causing fuel to burn• This is very quick -- can assume the

piston doesn’t move duringcombustion

• Energy |Qh| enters the system• Temperature increases from TB to TC

BANG!

V

P

V2 V1

V

P

V2 V1

A

A

B

B

C

4. Power stroke: Piston pusheddownward by expanding gas• Pretty fast -- can treat as

adiabatic• Volume increases from V2 to

V1

• Temperature drops from TCto TD

5. Exhaust valve opens• Pressure drops quickly

6. Exhaust stroke: Piston movesupward, pushing out burnedfuel and air

• This 4-stroke sequence isknown as the Otto cycle

V

P

V2 V1

V

P

V2 V1

V

P

V2 V1

A

A

A

B

B

B

C

C

C

D

D

D

Example: Efficiency of the Otto cycle• In the Otto cycle, heat enters along path BC• This is at constant volume, so we have:

• The total work done is the sum of the (positive) work doneby the gas on path CD and the (negative) work done onpath AB

• Using the equation for work in an adiabatic process fromProblem 21.53 in the text, we have:

!Eint

= Qh= nC

V!T = nC

VT

C"T

B( )

= nCV

PCV

C

nR"

PBV

B

nR

#

$%&

'(=

CVV

2

RP

C" P

B( )

W =1

! "1

#$%

&'(

PCV

C" P

DV

D+ P

AV

A" P

BV

B( )

• In terms of V1 and V2, this is:

• This means the efficiency is:

W =1

! "1

#$%

&'(

PCV

2" P

DV

1" P

BV

2+ P

AV

1( )

e =W

Qh

=

1

! "1

#$%

&'(

PCV

2" P

DV

1" P

BV

2+ P

AV

1( )

CVV

2

RP

C" P

B( )

• We can simplify this first by noting that:

so the efficiency becomes:

• From the properties of adiabatic expansion, we know that:

1

! "1=

1

CP

CV

"1

=C

V

CP"C

V

=C

V

R

e =W

Qh

=

CV

R

!

"#$

%&P

CV

2' P

DV

1' P

BV

2+ P

AV

1( )

CVV

2

RP

C' P

B( )

=P

CV

2' P

DV

1' P

BV

2+ P

AV

1

V2

PC' P

B( )

P

CV

2

!= P

DV

1

!; P

bV

2

!= P

AV

1

!

• Substituting for PB and PC we find:

e =1

V2

!

PD

V1

"

V2

" #1

$

%&'

()# P

DV

1# P

A

V1

"

V2

" #1

$

%&'

()+ P

AV

1

PD

V1

"

V2

" # PA

V1

"

V2

"

=

PD

V1

"

V2

" #V

1

V2

*

+,

-

./ # P

A

V1

"

V2

" #V

1

V2

*

+,

-

./

PD

V1

"

V2

" # PA

V1

"

V2

"

=P

D# P

A( )P

D# P

A( )

V1

"

V2

" #V

1

V2

*

+,

-

./

V1

"

V2

"

= 1#V

2

V1

$

%&'

()

" #1