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Thermodynamic Cycles
Section 3
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Thermodynamic Cycles
Air-standard analysis is used to perform elementary analysesof IC engine cycles.
Simplifications to the real cycle include:1) Fixed amount of air (ideal gas) for working fluid2) Combustion process not considered3) Intake and exhaust processes not considered4) Engine friction and heat losses not considered5) Specific heats independent of temperature
The two types of reciprocating engine cycles analyzed are:1) Spark ignition Otto cycle2) Compression ignition Diesel cycle
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SI Engine Cycle vs Thermodynamic Otto Cycle
AI
R
CombustionProducts
Ignition
IntakeStroke
FUEL
Fuel/AirMixture
Air
TC
BC
CompressionStroke
PowerStroke
ExhaustStroke
Q in Q out
CompressionProcess
Const volumeheat addition
Process
ExpansionProcess
C
onst volumeheat rejection
Process
ActualCycle
OttoCycle
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Actual SI Engine cycle
TC BC
Ignition
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Process 1 2 Isentropic compressionProcess 2 3 Constant volume heat additionProcess 3 4 Isentropic expansionProcess 4 1 Constant volume heat rejection
v 2
TC TCv
1BC BC
Qout
Q in
Air-Standard Otto cycle
3
4
2
1
vv
vv
r
Compression ratio:
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First Law Analysis of Otto Cycle
1 2 Isentropic Compression
)()( 12 mW
mQuu in
2
1
1
2
1
2
vv
T T
P P
AIR
)()( 1212 T T cuumW
vin
2 3 Constant Volume Heat Addition
mW
mQ
uu in
)()( 23
)()( 2323 T T cuumQ
vin
2
3
2
3
T
T
P
P
AIR Q in TC
11
2
1
1
2
k k r vv
T T
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3 4 Isentropic Expansion
AIR)()( 34 mW
mQ
uu out
)()( 4343 T T cuum
W v
out
4
3
3
4
3
4
v
v
T
T
P
P
4 1 Constant Volume Heat Removal
AIR Qout m
W
m
Quu out )()( 41
)()( 1414 T T cuumQ
vout
1
1
4
4
T
P
T
P
BC
1
1
4
3
3
4 1
k
k
r vv
T T
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23
1243
uuuuuu
QW
in
cycleth
23
14
23
1423 1uuuu
uuuuuu
Cycle thermal efficiency:
thin
thincycle
r r
umQ
k r r
V P Q
P imep
V V
W imep
1/
11
1 111121
Indicated mean effective pressure is:
Net cycle work:
1243 uumuumW W W inout cycle
First Law Analysis Parameters
12
1
23
14 111)()(
1 k v
v
r T T
T T cT T c
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Effect of Compression Ratio on Thermal Efficiency
Spark ignition engine compression ratio limited by T 3 (autoignition)and P 3 (material strength), both ~r k
For r = 8 the efficiency is 56% which is twice the actual indicated value
Typical SIengines9 < r < 11
k = 1.4
1
11 k
const cth r V
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Effect of Specific Heat Ratio on Thermal Efficiency
1
11 k
const cth r V
Specific heatratio (k)
Cylinder temperatures vary between 20K and 2000K so 1.2 < k < 1.4k = 1.3 most representative
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The net cycle work of an engine can be increased by either:
i) Increasing the r (1 2)ii) Increase Q in (2 3)
P
V2 V1
Q in W cycle
1
2
3
(i)
4
(ii)
Factors Affecting Work per Cycle
1
4
4
3 th
incycle
r r
V Q
V V
W imep
1121
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Effect of Compression Ratio on Thermal Efficiency and MEP
k in
r r r
V P Q
P imep 1
11111
k = 1.3
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Thermodynamic Cycles for CI engines
In early CI engines the fuel was injected when the piston reached TCand thus combustion lasted well into the expansion stroke.
In modern engines the fuel is injected before TC (about 15 o)
The combustion process in the early CI engines is best approximated bya constant pressure heat addition process Diesel Cycle
The combustion process in the modern CI engines is best approximatedby a combination of constant volume and constant pressure Dual Cycle
Fuel injection startsFuel injection starts
Early CI engine Modern CI engine
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Early CI Engine Cycle and the Thermodynamic Diesel Cycle
AI
R
CombustionProducts
Fuel injected
at TC
IntakeStroke
Air
Air
BC
CompressionStroke
PowerStroke
ExhaustStroke
Q in Q out
CompressionProcess
Const pressureheat addition
Process
ExpansionProcess
Const volumeheat rejection
Process
ActualCycle
DieselCycle
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Process 1 2 Isentropic compressionProcess 2 3 Constant pressure heat additionProcess 3 4 Isentropic expansionProcess 4 1 Constant volume heat rejection
Air-Standard Diesel cycle
Q in
Qout
2
3
vv
r c
Cut-off ratio:
v 2
TC v
1BC TC
BC
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m
V V P
m
Quu in 23223 )()(
AIR2 3 Constant Pressure Heat Addition
)()( 222333 v P uv P umQin
)()( 2323 T T chhmQ
pin
cr vv
T T
v RT
v RT
P 2
3
2
3
3
3
2
2
Q in
First Law Analysis of Diesel Cycle
Equations for processes 1 2, 4 1 are the same as those presentedfor the Otto cycle
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)()( 34m
W
m
Quu out
AIR
3 4 Isentropic Expansion
)()( 4343 T T cuumW
vout
note v 4=v 1 socr
r
v
v
v
v
v
v
v
v
v
v
3
2
2
1
3
2
2
4
3
4
r r
T T
P P
T v P
T v P c
3
4
3
4
3
33
4
44
11
4
3
3
4
k
c
k
r r
vv
T T
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23
1411hh
uu
mQ
mQ
in
out
cycle Diesel
1111
1 1c
k c
k const c Diesel
r
r
k r V
For cold air-standard the above reduces to:
Thermal Efficiency
11
1 k Ottor
recall,
Note the term in the square bracket is always larger than one so for thesame compression ratio, r , the Diesel cycle has a lower thermal efficiencythan the Otto cycle
Note: CI needs higher r compared to SI to ignite fuel
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Typical CI Engines15 < r < 20
When r c (= v 3/v2) 1 the Diesel cycle efficiency approaches theefficiency of the Otto cycle
Thermal Efficiency
Higher efficiency is obtained by adding less heat per cycle, Q in, run engine at higher speed to get the same power.
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k = 1.3
k = 1.3
The cut-off ratio is not a natural choice for the independent variablea more suitable parameter is the heat input, the two are related by:
111
111
k
in
c r V P
Q
k
k r as Q
in 0, r
c 1
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Modern CI Engine Cycle and the Thermodynamic Dual Cycle
A
IR
CombustionProducts
Fuel injectedat 15 o bTC
IntakeStroke
Air
Air
TC
BC
CompressionStroke
PowerStroke
ExhaustStroke
Q in Q out
CompressionProcess
Const pressureheat addition
Process
ExpansionProcess
Const volumeheat rejection
Process
ActualCycle
DieselCycle
Q in
Const volumeheat addition
Process
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Process 1 2 Isentropic compressionProcess 2 2.5 Constant volume heat additionProcess 2.5 3 Constant pressure heat additionProcess 3 4 Isentropic expansionProcess 4 1 Constant volume heat rejection
Dual Cycle
Q in
Q i n
Q o u t1
1
2
2
2.5
2.5
3
3
44
)()()()( 5.2325.25.2325.2 T T cT T chhuu
m
Q pv
in
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Thermal Efficiency
)()(11
5.2325.2
14
hhuu
uu
mQ
mQ
in
out
cycle Dual
1)1(11
1 1 c
k c
k cconst
Dual r k r
r v
11
1 k Otto r 1
1111 1c
k ck
const c Diesel
r r
k r V
Note, the Otto cycle (r c=1) and the Diesel cycle ( =1) are special cases:
2
3
5.2
3 andwhere P P
vvr c
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The use of the Dual cycle requires information about either:i) the fractions of constant volume and constant pressure heat addition
(common assumption is to equally split the heat addition), orii) maximum pressure P 3.
Transformation of r c and into more natural variables yields
11111 111 k r V P
Qk k r k inc
1
31 P P
r k
For the same inlet conditions P 1, V1 and the same compression ratio:
Diesel Dual Otto
For the same inlet conditions P 1, V1 and the same peak pressure P 3 (actual design limitation in engines):
otto Dual Diesel
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For the same inlet conditions P 1, V1 and the same compression ratio P 2 /P 1:
For the same inlet conditions P 1, V1 and the same peak pressure P 3:
32
141
1
Tds
Tds
in
out th
x 2.5
P max
Tmax
P o
P o
P r e s s u r e ,
P
P r e s s u r e ,
P
T e m p e r a
t u r e ,
T
T e m p e r a
t u r e ,
T
Specific Volume
Specific Volume
Entropy Entropy
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Finite Heat Release Model
In the Otto cycle it is assumed that heat is released instantaneously. A finite heat release model specifies heat release as a function of crankangle.
The cumulative heat release or burn fraction xb is given by:
n
d
sb a x
exp1)(
where = crank angle s = start of heat release d = duration of heat release
n = form factora = efficiency factor Used to fit to experimental data
0 < xb < 1
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Finite Heat Release
A typical heat release curve consists of an initial spark ignition phase,followed by a rapid burning phase and ends with burning completion phase
The curve asymptotically approaches 1 so the end of combustion is definedby an arbitrary limit, such as 90% or 99% complete combustion where xb = 0.90 or 0.99 corresponding values for efficiency factor a are 2.3 and 4.6
The rate of heat release as a function of crank angle is:
11
n
d
sb
d in
bin x
naQ
d
dxQ
d
dQ
bin dxQdQ
.99
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d dQ
V k
d dV
V P
k d dP
d dP
V d dV
P Rc
d dV
P d dQ
VdP PdV Rc
PdV Q
mR PV
d mcdT mcdU
W QdU
d
v
v
vv
1
anglecrankunit per
gasidealassuming
,change,anglecranksmallafor
cylinder in thegasthecontainingsystemclosedthetoLawFirstApplying
Finite Heat Release Model
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The cylinder volume in terms of crank angle, V( ), is
2122 )sin(cos121)( R RV r V V d d Differentiating wrt
2122 )sin(cos1sin2
RV
d
dV d
where
sl R
r
S BV d
2
rationcompressio
nt volumedisplaceme4
2
For the portion of the compression and expansion strokes with no heatrelease, where < s and > s + d dQ/d = 0
Finite Heat Release Model
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Finite Heat Release Model Results
Start of heat release:Engine 1 - 20 o bTCEngine 2 - TC
Duration 40 o
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Finite Heat Release Model Results