Post on 27-Feb-2018
7/25/2019 [Flip-Side] 3. Thermodynamic Cycles
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Thermodynamic Cycles
Section 3
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Thermodynamic Cycles
Air-standard analysisis used to perform elementary analysesof IC engine cycles.
Simplifications to the real cycle include:
1) Fixed amount of air (ideal gas) for woring fluid
!) Com"ustion process not considered
#) Intae and exhaust processes not considered
$) %ngine friction and heat losses not considered
&) Specific heats independent of temperature
'he two types of reciprocating engine cycles analyed are:
1) Spar ignition *tto cycle!) Compression ignition +iesel cycle
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SI Engine Cycle vs Thermodynamic Otto Cycle
,
I -
Com"ustion
roducts
Ignition
Intake
Stroke
F/%0
Fuel,ir
2ixture
,ir
'C
3C
Compression
Stroke
Power
Stroke
Exhast
Stroke
!in !ot
Compression
Process
Const volme
heat addition
Process
Expansion
Process
Co
nst volme
heat re"ection
Process
,ctual
Cycle
*ttoCycle
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Actal SI Engine cycle
TC #C
Ignition
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rocess 1! Isentropic compressionrocess !# Constant 4olume heat addition
rocess #$ Isentropic expansion
rocess $1 Constant 4olume heat re5ection
v2
TC
TCv1
#C
#C
Qout
Qin
Air-Standard Otto cycle
3
4
2
1
v
v
v
vr ==
Compression ratio:
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$irst %aw Analysis o& Otto Cycle
1! Isentropic Compression
)()( 12m
W
m
Quu in=
2
1
1
2
1
2
vv
TT
PP =
,I-
)()( 1212 TTcuum
Wv
in ==
!# Constant 6olume 7eat ,ddition
m
W
m
Q
uu in
+= )()( 23
)()( 2323 TTcuum
Qv
in ==
2
3
2
3
T
T
P
P=
,I- Qin'C
11
2
1
1
2 =
= k
k
rv
v
T
T
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#$ Isentropic %xpansion
,I-)()( 34m
W
m
Quu out+=
)()( 4343 TTcuum
Wv
out ==
4
3
3
4
3
4
v
v
T
T
P
P =
$1 Constant 6olume 7eat -emo4al
,I- Qoutm
W
m
Quu out = )()( 41
)()( 1414 TTcuum
Qv
out ==
1
1
4
4
T
P
T
P =
3C
1
1
4
3
3
4 1
=
=
k
k
rv
v
T
T
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( ) ( )( )23
1243
uu
uuuu
Q
W
in
cycle
th == ( ) ( )
23
14
23
1423 1uu
uu
uu
uuuu
=
=
Cycle thermal efficiency:
thin
thincycle
r
r
u
mQ
kr
r
VP
Q
P
imep
VV
Wimep
=
=
=
1
/
1
1
1 111121
Indicated mean effecti4e pressure is:
8et cycle wor:
( ) ( )1243 uumuumWWW inoutcycle ==
$irst %aw Analysis Parameters
1
2
1
23
14 111)(
)(1 ==
=
k
v
v
rT
T
TTc
TTc
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E&&ect o& Compression 'atio on Thermal E&&iciency
Spar ignition engine compression ratio limited "y '#(autoignition)
and #(material strength)9 "oth r
For r; < the efficiency is &=> which is twice the actual indicated 4alue
'ypical SI
engines
? @ r@ 11
; 1.$
1
11 = k
const cth
rV
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E&&ect o& Speci&ic (eat 'atio on Thermal E&&iciency
1
11 = k
const cth
rV
Speci&ic heat
ratio )k*
Cylinder temperatures 4ary "etween !AB and !AAAB so 1.! @ @ 1.$
; 1.# most representati4e
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'he net cycle wor of an engine can "e increased "y either:
i) Increasing ther(1!)ii) Increase Qin(!#D)
6! 61
QinWcycle
1
!
#
(i)
$
(ii)
$actors A&&ecting +ork per Cycle
1
$
$
#th
incycle
r
r
V
Q
VV
Wimep
=
=1
121
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E&&ect o& Compression 'atio on Thermal E&&iciency and ,EP
=
k
in
rr
r
VP
Q
P
imep 11
1111
k ./3
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Thermodynamic Cycles &or CI engines
In early CI engines the fuel was in5ected when the piston reached 'C
and thus com"ustion lasted well into the expansion stroe.
In modern engines the fuel is in5ected "efore 'C (a"out 1&o)
'he com"ustion process in the early CI engines is "est approximated "y
a constant pressure heat addition process 0iesel Cycle
'he com"ustion process in the modern CI engines is "est approximated
"y a com"ination of constant 4olume and constant pressure 0al Cycle
$el in"ection starts$el in"ection starts
Early CI engine ,odern CI engine
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Early CI Engine Cycle and the Thermodynamic 0iesel Cycle
,
I -
Com"ustion
roducts
Fuel in5ected
at 'C
Intake
Stroke
,ir
,ir
3C
Compression
Stroke
Power
Stroke
Exhast
Stroke
!in !ot
Compression
Process
Const pressre
heat addition
Process
Expansion
Process
Const volme
heat re"ection
Process
,ctual
Cycle
+ieselCycle
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rocess 1
! Isentropic compressionrocess !# Constant pressure heat addition
rocess #$ Isentropic expansion
rocess $1 Constant 4olume heat re5ection
Air-Standard 0iesel cycle
Qin
Qout
2
3
v
vrc=
CutEoff ratio:
v2
TC
v1
#CTC
#C
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( )
m
VVP
m
Quu in 23223 )()(
+=
,I-!# Constant ressure 7eat ,ddition
)()( 222333 vPuvPum
Qin ++=
)()( 2323 TTchhm
Qp
in ==
crv
v
T
T
v
RT
v
RTP ====
2
3
2
3
3
3
2
2
Qin
$irst %aw Analysis o& 0iesel Cycle
%uations for processes 1!9 $1 are the same as those presented
for the *tto cycle
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)()( 34m
W
m
Quu out+=
,I-
#$ Isentropic Expansion
)()( 4343 TTcuum
Wv
out ==
note v4=v
1so
cr
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
vP
T
vP c==3
4
3
4
3
33
4
44
11
4
3
3
4
=
=
k
c
k
rr
vv
TT
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23
1411hh
uu
mQ
mQ
in
out
cycleDiesel
==
( )
( )
=
1
1111
1
c
kc
k
const c
Diesel
r
r
krV
For cold airEstandard the a"o4e reduces to:
Thermal E&&iciency
1
11 = kOtto
r
recall9
8ote the term in the suare "racet is always larger than one so for the
same compression ratio9 r9 the +iesel cycle has a lowerthermal efficiency
than the *tto cycle
8ote: CI needs higher r compared to SI to ignite fuel
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'ypical CI %ngines
1& @ r@ !A
Ghen rc(; 4#4!)1 the +iesel cycle efficiency approaches the
efficiency of the *tto cycle
Thermal E&&iciency
7igher efficiency is o"tained "y adding less heat per cycle9 H in9
run engine at higher speed to get the same power.
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k ./3
k ./3
'he cutEoff ratio is not a natural choice for the independent 4aria"le
a more suita"le parameter is the heat input9 the two are related "y:
111
11
1
= kin
c rVP
Q
k
k
ras Q
in
0, rc
1
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rocess 1! Isentropic compression
rocess !!.& Constant 4olume heat addition
rocess !. Constant pressure heat additionrocess #$ Isentropic expansion
rocess $1 Constant 4olume heat re5ection
0al Cycle
Qin
Qin
Qout
..
1
1
1/2
1/2
3
3
)()()()( 5.2325.25.2325.2 TTcTTchhuu
m
Qpv
in +=+=
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Thermal E&&iciency
)()(
115.2325.2
14
hhuu
uu
mQ
mQ
in
out
cycle
Dual
+
==
( )
+
= 1)1(
111
1 c
k
c
kcconst
Dualrk
r
rv
1
11 = kOtto r
( )( )
= 1111
11
c
kc
kconst cDiesel
r
r
krV
8ote9 the *tto cycle (rc;1) and the +iesel cycle (;1) are special cases:
2
3
5.2
3 andwhereP
Pv
vrc ==
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'he use of the +ual cycle reuires information a"out either:
i) the fractions of constant 4olume and constant pressure heat addition
(common assumption is to equallysplit the heat addition)9 orii) maximum pressure #.
'ransformation of rcand into more natural 4aria"les yields
= 11111 1
11 krVPQ
kkr kinc
1
31PP
rk=
For the same inlet conditions 19 61and the same compression ratio:
DieselDualOtto
>>
For the same inlet conditions 19 61and the same pea pressure #
(actual design limitation in engines):
ottoDualDiesel >>
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For the same inlet conditions 19 61
and the same compression ratio P2/P1:For the same inlet conditions 19 61
and the same peak pressure P3:
=
=
32
141
1
Tds
Tds
Q
Q
in
outth
+iesel
+ual
*tto
+iesel+u
al*tto
xD J!.&D
max
'max
o
o
Pressre4
P
Pressre4
P
Temperatre4
T
Temperat
re4
T
Speci&ic 5olme
Speci&ic 5olme
Entropy Entropy
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$inite (eat 'elease ,odel
In the *tto cycle it is assumed that heat is released instantaneously.
, finite heat release model specifies heat release as a function of cran
angle.
'he cumulati4e heat release or 6rn &ractionxbis gi4en "y:
=
n
d
sb ax
exp1)(
where ; cran angles; start of heat release
d; duration of heat release
n; form factora; efficiency factor
/sed to fit to experimental data
A @ xb@ 1
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$inite (eat 'elease
, typical heat release cur4e consists of an initial spar ignition phase9
followed "y a rapid "urning phase and ends with "urning completion phase
'he cur4e asymptotically approaches 1 so the end of com"ustion is defined
"y an ar"itrary limit9 such as ?A> or ??> complete com"ustion wherexb; A.?A or A.?? corresponding 4alues for efficiency factor aare !.# and $.=
'he rate of heat release as a function of cran angle is:
( )
==
1
1
n
d
sb
d
inb
in xna
Q
d
dxQ
d
dQ
bindxQdQ=
/77
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( )
+=
+=
+=
==
=
d
dQ
V
k
d
dV
V
P
kd
dP
d
dPV
d
dVP
R
c
d
dVP
d
dQ
VdPPdVR
cPdVQ
mR
PVdmcdTmcdU
WQdU
d
v
v
vv
1
anglecrankunitper
gasidealassuming
changeanglecranksmalla!or
c"linderin thegasthecontainings"stemclosedtheto#aw$irst%ppl"ing
$inite (eat 'elease ,odel
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'he cylinder 4olume in terms of cran angle9 V(!is
( )2122 )sin(cos121)( ++= RRVrVV dd
+ifferentiating wrt
( )2122 )sin(cos1sin2
+=
RV
d
dV d
where
slR
r
"#Vd
2
rationcompressio
nt &olumedisplaceme4
2
=
=
==
For the portion of the compression and expansion stroes with no heat
release9 where sand ! s" ddQ$d'
$inite (eat 'elease ,odel
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$inite (eat 'elease ,odel 'eslts
Start o& heat release8
Engine . - 19o6TC
Engine 1 - TC
0ration 9o
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$inite (eat 'elease ,odel 'eslts