Thermodynamics and Statistical
Mechanics
Heat Engines and Refrigerators
Thermo & Stat Mech Spring 2006 Class 5
1
Carnot Cycle
A Carnot cycle is an idealized reversible cycle
that operates between two heat reservoirs at
temperatures T1 and T2, where T2 > T1. It can
operate as a heat engine, or a refrigerator.
Thermo & Stat Mech - Spring 2006
Class 5
2
Carnot Cycle
Thermo & Stat Mech - Spring 2006
Class 5
3
Carnot Cycle (Heat Engine)
Thermo & Stat Mech - Spring 2006
Class 5
4
Carnot Cycle
Thermo & Stat Mech - Spring 2006
Class 5
5
Carnot Cycle
For the entire cycle, Q = W + DU, and DU = 0.
The total heat equals the total work, Q = W.
|W| = |Q2| – |Q1|
Conservation of energy.
|Q2| = |Q1| + |W|
Thermo & Stat Mech - Spring 2006
Class 5
6
Carnot Cycle
For an isothermal process:
1
PV  constant, so P 
V
For an adiabatic process:
1
PV  constant, so P  
V

Thermo & Stat Mech - Spring 2006
Class 5
(  1)
7
Path ab
Isothermal at temperature T2 .
dQ  dU  PdV  CV dT  PdV  PdV
nRT2
dV
dQ  dW 
V
 Vb 
dV
 nRT2 ln 
W2  Q2  nRT2
V
 Va 
a
b
Thermo & Stat Mech - Spring 2006
Class 5
8
Paths bc, cd, and da
b  c) Adiabatic : TV
 1
 1
 constant. T2Vb
 1
 T1Vc
 Vc 
c  d ) Isothermal : W1  Q1  nRT1 ln 
 Vd 
d  a ) Adiabatic : TV  1  constant. T1Vd 1  T2Va 1
Thermo & Stat Mech - Spring 2006
Class 5
9
Carnot Cycle
Divide ab by cd and then bc by da, and
 1
 1
T2Vb
T1Vc
Vb Vc

 1 
 1 , so
T2Va
T1Vd
Va Vd
 Vb 
nRT2 ln 
Q2
Va  T2



Q1
 Vc  T1
nRT1 ln 
 Vd 
Thermo & Stat Mech - Spring 2006
Class 5
Q T
10
Thermal Efficiency (h)
W Q2  Q1
Q1
h

 1
Q2
Q2
Q2
T1 T2  T1
h  1 
T2
T2
If T1 = 0, h = 1 (100%)
Thermo & Stat Mech - Spring 2006
Class 5
11
Refrigerator (Heat Pump)
Run the cycle in reverse. Do work |W|
on the system, remove |Q1| from the
low temperature reservoir, and put |Q2|
into the high temperature reservoir. As
in a heat engine,
|Q2| = |Q1| + |W|.
Thermo & Stat Mech - Spring 2006
Class 5
12
Coefficient of Performance (for
Refrigerator)
Q1
Q1
T1
c


W Q2  Q1 T2  T1
T2  T1
h
T2
Thermo & Stat Mech - Spring 2006
Class 5
13
Coefficient of Performance (for
Heat Pump)
Q2
Q2
T2
g


W Q2  Q1 T2  T1
T2  T1
h
T2
Thermo & Stat Mech - Spring 2006
Class 5
14
Otto Cycle (Gasoline Engine)
Thermo & Stat Mech - Spring 2006
Class 5
15
Otto Cycle
W = |Q2| – |Q1|
W Q2  Q1
Q1
h

 1
Q2
Q2
Q2
|Q2| = CV(Tc – Tb)
|Q1| = CV(Td – Ta)
Thermo & Stat Mech - Spring 2006
Class 5
16
Otto Cycle
Q1 Td  Ta

Q2 Tc  Tb
Td  Ta
h 1
Tc  Tb
Thermo & Stat Mech - Spring 2006
Class 5
17
Otto Cycle
 1
 1
 1
 1
and TdV1  TcV2
Subtract, and get,
 1
 1
(Td  Ta )V1  (Tc  Tb )V2
Then,
 1
 1
 V2 
(Td  Ta ) V2
  1   
(Tc  Tb ) V1
 V1 
TaV1
 TbV2
Thermo & Stat Mech - Spring 2006
Class 5
18
Otto Cycle
 V2 
Td  Ta
h 1
1  
Tc  Tb
 V1 
 1
V1 is called the compression ratio.
r
V2
1
h  1   1
r
Thermo & Stat Mech - Spring 2006
Class 5
19
Diesel Cycle
Thermo & Stat Mech - Spring 2006
Class 5
20