lecture 11. real heat engines and refrigerators (ch. 4) stirling heat engine internal combustion...
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Lecture 11. Real Heat Engines and refrigerators (Ch. 4)
• Stirling heat engine
• Internal combustion engine (Otto cycle)
• Diesel engine
• Steam engine (Rankine cycle)
• Kitchen Refrigerator
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
Te 1max
T
S
TH TC
1
23
4
On the S -T diagram, the work done is the area of the loop:
entr
opy
cont
aine
d in
ga
s
PdVTdSdU 0
The heat consumed at TH (1 – 2) is the area surrounded by the broken line:
CHHH SSTQ 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
34
absorbsheat
rejects heat
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
41
2 3
V2 V1
T1
T2
Page 133, Pr. 4.21
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
41
2 3
V2 V1
T1
T2
Efficiency of Stirling Engine
T
41
2 3
V2 V1
T1
T2
1-2 002
312121212 WTTnRTTCQ V
2-3 23231
22223 0ln
2
1
2
1
WQV
VnRT
V
dVnRTPdVQ
V
V
V
V
3-4 002
334212134 WTTnRTTCQ V
4-1 41412
11141 0ln
1
2
1
2
WQV
VnRT
V
dVnRTPdVQ
V
V
V
V
max1212
2
1212
12212
4123
2312 1
/ln2
3
/ln
/ln23
1
eVVTT
T
VVTT
VVTTT
W
Q
eH
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.
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
igni
tion
exha
ust
expansion
work done by gascompressionwork done on gas
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.
Otto cycle (cont.)
S
V
12
3 4
V1 V2
S2
S1
CQHQ
0Q
0Q
1
2 1
1 2
1 1V T
eV T
1
2
V
V- the compression ratioThe 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)
2V- minimum cylinder volume
- maximum cylinder volume
1V
P
V
1
2
3
4
igni
tion
exha
ust
expansion
work done by gascompressionwork done on gas
V1 V2
Patm intake/exhaust
0
Diesel engine
c
h
1Q
eQ
3
2
3
2
1
2
1
111
1
VV
VV
Ve
V
(Pr. 4.20)
4 1 c 4 14 1: isochoric, ,2
fV V Q nR T T
1 11 1 2 21 2 : adiabatic, TV T V
h 3 2
22 3: isobaric,
2
fQ nR T T
1 13 3 4 43 4 : adiabatic, TV T V
Steam engine (Rankine cycle)
heat
work
hea
t
hot reservoir, TH
cold reservoir, TC
Sadi Carnot
Boiler
Condenser
HQ
CQ
Tur
bin
ePum
pP
V
water
steamWater+steam
1
2
3
4
Turbine
Boiler
condense
processes at P = const,
Q=dHPump
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
Steam engine (Rankine cycle)
heat
work
hea
t
hot reservoir, TH
cold reservoir, TC
Sadi Carnot
Boiler
Condenser
HQ
CQ
Tur
bin
ePum
pP
V
water
steamWater+steam
1
2
3
4
Turbine
Boiler
condense
processes at P = const,
Q=dHPump
gas gas liquid3 4 3 4 4
gas liquid4 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 1H H H H
eH H H H
4.12
2 1Here , water is almost incompressible.H H
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
liq
uid
gas liquid+gas
1
23
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 HCOP
Q Q H H H H H H
liquid liquid gas3 4 3 4 4
2 1 2 2 2 2
, 1
,
H H H x H x H
S S T H T P