2. Carnot Cycle
1 – 2 isothermal expansion (in contact with TH)
2 – 3 isentropic expansion to TC
3 – 4 isothermal compression (in contact with TC)
4 – 1 isentropic compression to TH
(isentropic adiabatic+quasistatic)
Efficiency of Carnot
cycle for an ideal gas:
(Pr. 4.5) H
C
T
T
e
1
max
T
S
TH
TC
1
2
3
4
On the S -T diagram, the work done is
the area of the loop:
entropy
contained
in
gas
PdV
TdS
dU 0
The heat consumed at TH (1 – 2) is the area
surrounded by the broken line:
C
H
H
H S
S
T
Q
S - entropy
contained in gas
- is not very practical (too slow), but operates at the maximum efficiency allowed by
the Second Law.
TC
P
V
TH
1
2
3
4
absorbs
heat
rejects heat
3. Stirling heat engine
Stirling engine – a simple, practical heat
engine using a gas as working substance. It’s
more practical than Carnot, though its
efficiency is pretty close to the Carnot
maximum efficiency. The Stirling engine
contains a fixed amount of gas which is
transferred back and forth between a "cold"
and and a "hot" end of a long cylinder. The
"displacer piston" moves the gas between the
two ends and the "power piston" changes the
internal volume as the gas expands and
contracts.
T
4
1
2 3
V2
V1
T1
T2
Page 133, Pr. 4.21
4. Stirling heat engine
The gases used inside a Stirling engine never leave the engine. There
are no exhaust valves that vent high-pressure gasses, as in a gasoline or
diesel engine, and there are no explosions taking place. Because of this,
Stirling engines are very quiet. The Stirling cycle uses an external heat
source, which could be anything from gasoline to solar energy to the heat
produced by decaying plants. Today, Stirling engines are used in some
very specialized applications, like in submarines or auxiliary power
generators, where quiet operation is important.
T
4
1
2 3
V2
V1
T1
T2
5. Efficiency of Stirling Engine
T
4
1
2 3
V2
V1
T1
T2
1-2 0
0
2
3
12
1
2
1
2
12
W
T
T
nR
T
T
C
Q V
2-3 23
23
1
2
2
2
23 0
ln
2
1
2
1
W
Q
V
V
nRT
V
dV
nRT
PdV
Q
V
V
V
V
3-4 0
0
2
3
34
2
1
2
1
34
W
T
T
nR
T
T
C
Q V
4-1 41
41
2
1
1
1
41 0
ln
1
2
1
2
W
Q
V
V
nRT
V
dV
nRT
PdV
Q
V
V
V
V
max
1
2
1
2
2
1
2
1
2
1
2
2
1
2
41
23
23
12 1
/
ln
2
3
/
ln
/
ln
2
3
1
e
V
V
T
T
T
V
V
T
T
V
V
T
T
T
Q
Q
Q
Q
W
Q
e
H
In the Stirling heat engine, a working substance, which may be assumed to be an
ideal monatomic gas, at initial volume V1 and temperature T1 takes in “heat” at
constant volume until its temperature is T2, and then expands isothermally until its
volume is V2. It gives out “heat” at constant volume until its temperature is again T1
and then returns to its initial state by isothermal contraction. Calculate the efficiency
and compare with a Carnot engine operating between the same two temperatures.
6. Internal Combustion Engines (Otto cycle)
Otto cycle. Working substance – a mixture of air
and vaporized gasoline. No hot reservoir – thermal
energy is produced by burning fuel.
0 1 intake (fuel+air is pulled into the cylinder
by the retreating piston)
1 2 isentropic compression
2 3 isochoric heating
3 4 isentropic expansion
4 1 0 exhaust
P
V
1
2
3
4
ignition
exhaust
V1
V2
Patm
intake/exhaust
0
- engines where the fuel is burned inside the engine
cylinder as opposed to that where the fuel is burned
outside the cylinder (e.g., the Stirling engine). More
economical than ideal-gas engines because a small
engine can generate a considerable power.
7. Otto cycle (cont.)
S
V
1
2
3 4
V1
V2
S2
S1
C
Q
H
Q
0
Q
0
Q
1
2 1
1 2
1 1
V T
e
V T
1
2
V
V
- the compression ratio
The efficiency:
(Pr. 4.18)
= 1+2/f - the adiabatic exponent
For typical numbers V1/V2 ~8 , ~ 7/5 e = 0.56, (in reality, e = 0.2 – 0.3)
(even an “ideal” efficiency is smaller than the second law limit 1-T1/T3)
2
V
- minimum cylinder volume
- maximum cylinder volume
1
V
P
V
1
2
3
4
ignition
exhaust
V1
V2
Patm
intake/exhaust
0
8. Diesel engine
c
h
1
Q
e
Q
3
2
3
2
1
2
1
1
1
1
1
V
V
V
V
V
e
V
(Pr. 4.20)
4 1 c 4 1
4 1: isochoric, ,
2
f
V V Q nR T T
1 1
1 1 2 2
1 2: adiabatic, TV T V
h 3 2
2
2 3: isobaric,
2
f
Q nR T T
1 1
3 3 4 4
3 4: adiabatic, T V T V
9. Steam engine (Rankine cycle)
heat
work
heat
hot reservoir, TH
cold reservoir, TC
Sadi Carnot
Boiler
Condenser
H
Q
C
Q
Turbine
Pump
P
V
water
steam
Water+steam
1
2
3
4
Turbine
Boiler
condense
processes
at P = const,
Q=dH
Pump
NOT an ideal gas!
3 2
4 1
,
1 2: isothermal adiabatic
2 3: isobaric,
3 4: adiabatic
4 1: isobaric,
P P
h
h
H U PV Q H
Q H H
Q H H
10. Steam engine (Rankine cycle)
heat
work
heat
hot reservoir, TH
cold reservoir, TC
Sadi Carnot
Boiler
Condenser
H
Q
C
Q
Turbine
Pump
P
V
water
steam
Water+steam
1
2
3
4
Turbine
Boiler
condense
processes
at P = const,
Q=dH
Pump
gas gas liquid
3 4 3 4 4
gas liquid
4 4 4
1
1
S S S x S x S
H x H x H
4 1 4 1
3 2 3 1
1 1
H H H H
e
H H H H
4.12
2 1
Here , water is almost incompressible.
H H
11. Kitchen
Refrigerator
A liquid with suitable characteristics (e.g., Freon) circulates
through the system. The compressor pushes the liquid
through the condenser coil at a high pressure (~ 10 atm).
The liquid sprays through a throttling valve into the
evaporation coil which is maintained by the compressor at
a low pressure (~ 2 atm).
cold reservoir
(fridge interior)
T=50C
hot reservoir
(fridge exterior)
T=250C
P
V
liquid
gas
liquid+gas
1
2
3
4
compressor
condenser
throttling
valve
evaporator
processes
at P = const,
Q=dH
The enthalpies Hi can be found in tables.
1 4 1 4
2 3 1 4 2 1
C
H C
Q H H H H
COP
Q Q H H H H H H
liquid liquid gas
3 4 3 4 4
2 1 2 2 2 2
, 1
,
H H H x H x H
S S T H T P
