Xin Q. Diesel Engine System Design
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308 Diesel engine system design
© Woodhead Publishing Limited, 2011
S
j
j
j
in
in
ex
ex
h
m
h
m
h
m
·
d
d
=
·
d
d
+
·
d
d
ff
in
ff
in
h
ff
h
=
ff
=
·
ff
·
d
ff
d
f
4.5
where m
in
is the intake gas mass owing into the cylinder, m
ex
is the exhaust
gas mass owing out of the cylinder, and h
in
and h
ex
are the speci c enthalpies
of the gases at the intake and exhaust valves. Since the speci c internal
energy for ideal gases can be expressed as u = u(T, ~), where ~ is the excess
air–fuel ratio (de ned as actual air–fuel ratio divided by stoichiometric
air–fuel ratio, or the reciprocal of the ‘equivalence ratio’), the following
relationship is obtained:
d
d
=
·
d
d
+
·
d
d
=
·
d
d
+
uu
Tu
+
Tu
+
c
T
v
ff
=
ff
=
·
ff
·
d
ff
d
T
ff
T
V
V
ff
=
ff
=
·
ff
·
d
ff
d
c
ff
c
v
ff
v
∂
uu∂uu
∂
ff
∂
ff
∂
Tu∂Tu
∂
∂∂∂
∂
u
V
V
f
·
d
d
4.6
Substituting these relationships into equation 4.1, the energy conservation
equation is converted to the following form for solving the in-cylinder gas
temperature T:
d
d
=
d
d
+
d
d
–
d
d
+
d
T
mc
Q
Q
p
V
h
m
v
fuel
wall
in
f
ff
+
ff
+
d
ff
d
f
1
mc◊mc
¥
in
iini
ex
ex
h
m
u
m
m
u
d
+
d
d
–
d
d
–
·
d
d
ff
ex
ff
ex
h
ff
h
+
ff
+
d
ff
d
fV
m
fV
m
–
fV
–
V
f
∂
∂
fV
∂
fV
Ê
Ë
Á
Ê
Á
Ê
Ë
Á
Ë
ˆ
¯
˜
ˆ
˜
ˆ
¯
˜
¯
4.7
The mass conservation equation can be expressed as
d
d
=
d
d
+
d
d
+
d
d
m
mm
dmmd
m
in
mm
in
mm
ex
fuel
B
ff
=
ff
=
d
ff
d
ff
+
ff
+
d
ff
d
4.8
where m
fuelB
is the fuel mass injected into the cylinder. If the total injected
fuel mass per engine cycle is m
fuelC
and the fraction of the fuel burnt is
de ned as X
fuel
= m
fuelB
/m
fuelC
, then the following is obtained:
d
d
=
d
d
m
m
X
dXd
fuel
B
fuelBfuel
fuel
C
fuel
X
fuel
X
ff
=
ff
=
d
ff
d
m
ff
m
fuel
ff
fuel
C
ff
C
◊
4.9
The mass conservation equation is then converted to the following form:
d
d
=
d
d
+
d
d
+
d
d
m
mm
dmmd
m
X
dXd
in
mm
in
mm
ex
fuel
C
fuel
X
fuel
X
ff
=
ff
=
d
ff
d
ff
+
ff
+
d
ff
d
m
ff
m
fuel
ff
fuel
C
ff
C
4.10
The heat release rate from the burnt fuel in equation 4.7 is
d
d
=
d
d
d
d
Qm
dQmd
qm
=qm =
X
dXd
fuel
Qm
fuel
Qm
fuel
B
fuelBfuel
qm
LHV
qm
qm
co
qm
mf
qm
mf
qm
=qm =
mf
=qm =
uelC
mfuelCmf
fuel
X
fuel
X
ff
=
ff
=
d
ff
d
h
qm
h
qm
◊◊
qm◊◊qm
qm
LHV
qm◊◊qm
LHV
qm
◊
fff
h
◊◊
q
◊◊q◊◊
LHV
◊◊
LHV
◊◊
co
m
4.11
Diesel-Xin-04.indd 308 5/5/11 11:47:23 AM
309Fundamentals of dynamic and static engine system designs
© Woodhead Publishing Limited, 2011
where q
LHV
is the lower heating value of the fuel, and h
com
is the combustion
ef ciency (h
com
= 1 means complete combustion).
The ideal gas law applied to the in-cylinder gas gives the equation of
state:
pV = mR
gas
T 4.12
where R
gas
is the gas constant. In the system of equations 4.7, 4.10 and 4.12,
dQ
wall
/df, h
in
(dm
in
/df), and h
ex
(dm
ex
/df) are functions of gas temperature
and pressure; ∂u/∂~ and c
v
are functions of gas temperature and constituents;
dX
fuel
/df, dV/df, m
fuelC
, q
LHV
and h
com
are treated as known inputs. Therefore,
the three unknowns, in-cylinder gas pressure p, temperature T, and mass m
can be solved by a time-marching numerical integration. Equations 4.7 and
4.10 can be further simpli ed for different stages during an engine cycle,
as explained below.
4.2.4 Simplifi cation of governing equations for each
stage of the cycle process
Compression stage (from intake valve closing to start of combustion)
In this stage, dm
in
/df = 0, dm
ex
/df = 0, dm
fuel
/df = 0, and d~/df = 0.
Therefore, dm/df = 0 and dQ
fuel
/df = 0. The energy conservation equation
4.7 becomes
d
d
=
d
d
–
d
d
T
Q
p
V
wall
ff
d
ff
d
ff
=
ff
=
mc
ff
mc
v
ff
v
f
1
mc
ff
mc◊mc
ff
mc
Ê
Ë
ff
Ë
ff
Ê
Ë
Ê
Ê
Á
Ê
Ë
Á
Ë
ff
Ë
ff
Á
ff
Ë
ff
Ê
Ë
Ê
Á
Ê
Ë
Ê
ˆ
¯
ˆ
¯
ˆ
ˆ
˜
ˆ
¯
˜
¯
ˆ
¯
ˆ
˜
ˆ
¯
ˆ
4.13
Combustion stage (from start of combustion to end of combustion)
In this stage, dm
in
/df = 0 and dm
ex
/df = 0. Equation 4.3 becomes dm/df
= dm
fuelB
/df = m
fuelC
(dX
fuel
/df). Ignoring the impact of ~ on u, the energy
conservation equation 4.7 becomes
d
d
=
d
d
+
d
T
mc
mq
X
dXd
Q
v
fuel
mq
fuel
mq
mq
Cc
mq
mq
om
mq
LHV
fuel
X
fuel
X
wa
l
f
mq
h
mq
mq
Cc
mq
h
mq
Cc
mq
f
1
mc◊mc
()
mq()mq
u()u
mq
Cc
mq()mq
Cc
mq
mq
om
mq()mq
om
mq
LHV
()
LHV
h
()
h
mq
h
mq()mq
h
mq
mq
Cc
mq
h
mq
Cc
mq()mq
Cc
mq
h
mq
Cc
mq
◊-()◊-
mq◊-mq()mq◊-mq
LHV
◊-
LHV
()
LHV
◊-
LHV
lll
p
V
d
–
d
d
ff
p
ff
p
–
ff
–
d
ff
d
È
Î
Í
È
Í
È
Î
Í
Î
˘
˚
˙
˘
˙
˘
˚
˙
˚
4.14
Expansion stage (from end of combustion to exhaust valve opening)
In this stage, dm
in
/df = 0, dm
ex
/df = 0, dm
fuelB
/df = 0, and d~/df = 0.
Therefore, dm/df = 0 and dQ
fuel
/df = 0. The energy conservation equation
4.7 becomes
d
d
=
d
d
–
d
d
T
Q
p
V
wall
ff
d
ff
d
ff
=
ff
=
mc
ff
mc
v
ff
v
f
1
mc
ff
mc◊mc
ff
mc
Ê
Ë
ff
Ë
ff
Ê
Ë
Ê
Ê
Á
Ê
Ë
Á
Ë
ff
Ë
ff
Á
ff
Ë
ff
Ê
Ë
Ê
Á
Ê
Ë
Ê
ˆ
¯
ˆ
¯
ˆ
ˆ
˜
ˆ
¯
˜
¯
ˆ
¯
ˆ
˜
ˆ
¯
ˆ
4.15
Diesel-Xin-04.indd 309 5/5/11 11:47:24 AM
310 Diesel engine system design
© Woodhead Publishing Limited, 2011
Exhaust stage (from exhaust valve opening to intake valve opening)
In this stage, dm
in
/df = 0, dm
fuelB
/df = 0 and d~/df = 0. Equation 4.10 becomes
dm/df = dm
ex
/df. The energy conservation equation 4.7 becomes
d
d
=
d
d
–
d
d
+
d
d
T
Q
p
V
hu
m
wall
ex
ff
=
ff
=
d
ff
d
mc
ff
mc
v
ff
v
ff
+
ff
+
d
ff
d
ex
ff
ex
1
mc
ff
mc◊mc
ff
mc
È
()
hu()hu
– hu – () – hu –
hu
ex
hu()hu
ex
hu
ff
()
ff
hu
ff
hu()hu
ff
hu
– hu –
ff
– hu – () – hu –
ff
– hu –
ex
ff
ex
()
ex
ff
ex
hu
ex
hu
ff
hu
ex
hu()hu
ex
hu
ff
hu
ex
hu
ÎÎÎ
ff
Î
ff
Î
ff
Î
ff
Í
È
Í
È
ÎÎÎ
Í
ÎÎÎ
ff
Î
ff
Î
ff
Î
ff
Í
ff
Î
ff
Î
ff
Î
ff
˘
˚
˙
˘
˙
˘
˚
˙
˚
4.16
Intake stage (from exhaust valve closing to intake valve closing)
In this stage, dm
ex
/df = 0 and dm
fuelB
/df = 0. Equation 4.10 becomes dm/df
= dm
in
/df. Ignoring the impact of ~ on u, the energy conservation equation
4.7 becomes
d
d
=
d
d
–
d
d
+
d
d
T
Q
p
V
hu
m
wall
in
ff
=
ff
=
d
ff
d
mc
ff
mc
v
ff
v
ff
+
ff
+
d
ff
d
in
ff
in
1
mc
ff
mc◊mc
ff
mc
È
()
hu()hu
– hu – () – hu –
hu
in
hu()hu
in
hu
ff
()
ff
hu
ff
hu()hu
ff
hu
– hu –
ff
– hu – () – hu –
ff
– hu –
in
ff
in
()
in
ff
in
hu
in
hu
ff
hu
in
hu()hu
in
hu
ff
hu
in
hu
ÎÎÎ
ff
Î
ff
Î
ff
Î
ff
Í
È
Í
È
ÎÎÎ
Í
ÎÎÎ
ff
Î
ff
Î
ff
Î
ff
Í
ff
Î
ff
Î
ff
Î
ff
˘
˚
˙
˘
˙
˘
˚
˙
˚
4.17
Valve overlap stage (from intake valve opening to exhaust valve closing)
In this stage, dm
fuelB
/df = 0 and dQ
fuel
/df = 0. Equation 4.10 becomes dm/df
= dm
in
/df + dm
ex
/df. Ignoring the impact of ~ on u, the energy conservation
equation 4.7 becomes
d
d
=
d
d
–
d
d
+ (
)
d
d
T
Q
p
V
hu
– hu –
m
wall
hu
in
hu
in
ff
d
ff
d
ff
=
ff
=
mc
ff
mc
v
ff
v
ff
+ (
ff
+ (
)
ff
)
d
ff
d
hu
ff
hu
– hu –
ff
– hu –
in
ff
in
hu
in
hu
ff
hu
in
hu
1
mc
ff
mc◊mc
ff
mc
+ (
++ (+
)
d
d
hu
– hu –
m
ex
hu
ex
hu
ex
f
È
Î
ff
Î
ff
Í
È
Í
È
Î
Í
Î
ff
Î
ff
Í
ff
Î
ff
˘
˚
˙
˘
˙
˘
˚
˙
˚
4.18
4.2.5 Key sub-models
Instantaneous in-cylinder volume
In equation 4.7, the instantaneous volume dV/df can be derived from engine
geometry as
V
BS
S
f
EE
BS
EE
BS
E
S
E
S
CC
f
CC
f
=
+
+
–
–
CC-CC
p
f
2
BS
2
BS
41
2
2
1
1
W-
41W-41
Ê
Ë
Ë
Ê
Ë
Ê
Ê
Á
Ê
Ë
Á
Ë
Ë
Á
Ë
Ê
Ë
Ê
Á
Ê
Ë
Ê
ˆ
¯
ˆ
¯
ˆ
ˆ
˜
ˆ
¯
˜
¯
ˆ
¯
ˆ
˜
ˆ
¯
ˆ
co
s
f
s
f
111
1
2
f
f
CC
f
CC
f
CC
f
CC
f
CC-CC
CC-CC
2
–
◊
È
Î
Î
Í
È
Í
È
Î
Í
Î
Î
Í
Î
˘
˚
˙
˘
˙
˘
˚
˙
˚
Ï
BS
Ï
BS
BS
EE
BS
Ï
BS
EE
BS
Ì
EE
Ì
EE
BS
EE
BS
Ì
BS
EE
BS
41
Ì
41
BS
EE
BS
Ï
BS
EE
BS
Ì
BS
EE
BS
Ï
BS
EE
BS
Ó
41
Ó
41
41
Ì
41
Ó
41
Ì
41
¸
˝
¸
˝
¸
˛
˝
˛
˝
si
n
f
4.19
dV
d
BS
f
f
EE
BS
EE
BS
CC
f
CC
f
CC
f
CC
f
f
d
f
d
p
f
f
f
=
+
()
f
()
f
CC-CC
CC-CC
2
2
BS
2
BS
2
82
f
82
f
82
f
f
82
f
f
+
82
+
()2()
1
si
82
si
82
82
si
82
n
82
n
82
f
82
f
n
f
82
f
si
n
si
n
-◊
f-◊f
CC
-◊
CC
f
CC
f-◊f
CC
f
È
ÎÎÎ
82
Î
82
Î
82
Î
82
Í
82
Í
82
È
Í
È
82
Í
82
ÎÎÎ
Í
ÎÎÎ
82
Î
82
Î
82
Î
82
Í
82
Î
82
Î
82
Î
82
82
Í
82
Í
82
Í
82
82
Í
82
Í
82
Í
82
˘
˚
˙
˘
˙
˘
˙
˚
˙
˚
˙
˙
˙
4.20
where W is the engine geometric compression ratio, B
E
is the cylinder bore
diameter, S
E
is the engine stroke, and f
C-C
is the ratio between the connecting
rod length (the ‘center-to-center’ length) and the crank radius.
Diesel-Xin-04.indd 310 5/5/11 11:47:25 AM
311Fundamentals of dynamic and static engine system designs
© Woodhead Publishing Limited, 2011
Heat transfer through cylinder walls
The heat transfer terms in the energy conservation equation 4.7 can be
derived in the following form:
d
d
=
d
=3
,
Q
Q
N
A
wall
wall
i
E
i
gw
A
gw
A
al
l
ff
=
ff
=
ff
d
ff
d
=3
ff
=3
i
ff
i
a
SS
SS
d
SS
d
d
SS
d
=SS =
,
SS
,
Q
SS
Q
N
SS
N
wall
SS
wall
i
SS
i
ff
SS
ff
d
ff
d
SS
d
ff
d
SS
-
SS
◊
1
SS
1
SS
6
SS
6
SS
iw
iiwi
al
li
TT
iw
TT
iw
(
iw
(
iw
TT( TT
iw
TT
iw
(
iw
TT
iw
)
iw
)
iw
al
)
al
li
)
li
TT )TT
iw
TT
iw
)
iw
TT
iw
,
li,li
li
)
li,li
)
li
TT-TT
iw
TT
iw
-
iw
TT
iw
4.21
where a
g
is the instantaneous spatial-average heat transfer coef cient from the
in-cylinder gas to the inner cylinder wall, N
E
is the engine speed (rpm), A
wall
is the heat transfer area, T
wall
is the spatial-average temperature of the cylinder
wall surface, and i = 1, 2, 3 refers to the cylinder head, the piston, and the
liner, respectively. The cylinder liner heat transfer area is A
wall,3
= pB
E
[L
clear
+ y(f)], where L
clear
is the clearance height, and y(f) is given by:
y
S
ff
f
E
S
E
S
ff
CC
ff
CC
ff
CC
ff
C
f
C
f
()
–
ff
–
ff
1 –
ff
--
ff
ff
CC
ff
--
ff
CC
ff
CC--CC
ff
CC
ff
--
ff
CC
ff
-
ff
ff
ff
ff
ff
1 +
ff
1 +
–
ff
–
ff
–
ff
ff
ff
–
ff
ff
ff
S
ff
S
E
ff
E
S
E
S
ff
S
E
S
()
ff
()
=
ff
=
2
ff
2
ff
11
ff
11
ff
Ê
ff
Ê
ff
Ë
ff
Ë
ff
ff
Ê
ff
Ë
ff
Ê
ff
Ê
Á
Ê
ff
Ê
ff
Á
ff
Ê
ff
Ë
Á
Ë
ff
Ë
ff
Á
ff
Ë
ff
ff
Ê
ff
Ë
ff
Ê
ff
Á
ff
Ê
ff
Ë
ff
Ê
ff
ˆ
ff
ˆ
ff
11
ˆ
11
ff
11
ff
ˆ
ff
11
ff
¯
ff
¯
ff
ff
--
ff
¯
ff
--
ff
ff
¯
ff
ff
ff
ff
¯
ff
ff
ff
ff
ˆ
ff
¯
ff
ˆ
ff
ff
ˆ
ff
˜
ff
ˆ
ff
11
ˆ
11
˜
11
ˆ
11
ff
11
ff
ˆ
ff
11
ff
˜
ff
11
ff
ˆ
ff
11
ff
ff
¯
ff
˜
ff
¯
ff
ff
¯
ff
˜
ff
¯
ff
ff
ff
ff
¯
ff
ff
ff
˜
ff
ff
ff
¯
ff
ff
ff
ff
ˆ
ff
¯
ff
ˆ
ff
˜
ff
ˆ
ff
¯
ff
ˆ
ff
ff
co
ff
ff
ff
ff
co
ff
ff
ff
ff
s
ff
ff
ff
ff
s
ff
ff
ff
CCC
22
◊
È
ff
È
ff
Î
ff
Î
ff
ff
Í
ff
ff
È
ff
Í
ff
È
ff
Î
Í
Î
ff
Î
ff
Í
ff
Î
ff
˘
˚
˙
˘
˙
˘
˚
˙
˚
si
22
si
22
n
22
n
22
f
4.22
The a
g
is critical for heat transfer calculation. The development history of
the engine heat transfer theory has gone through three stages: (1) empirical
modeling (pioneered by the Nusselt correlation in 1923); (2) semi-empirical
similarity theory (represented by the Woschni correlation); and (3) the
multi-zone CFD simulation since the 1980s. To date, the widely accepted
and successful approach is still the Woschni correlation. It was developed in
1965 and was based on a correlation in the form of Nu
= 0.035Re
0.8
Pr
0.333
,
where Nu is the Nusselt number, Re is the Reynolds number, and Pr is the
Prandtl number. It assumes that forced convection has a dominant effect on
cylinder heat transfer as follows:
a
g
E
mp
ec
pv
TK
T
T
ec
T
ec
=
)
E
)
E
mp
)
mp
KB )KB
E
KB
E
)
E
KB
E
pv )pv
TK +TK
)
–
)
TK
–
TK
1
)
1
)
KB )KB
1
KB )KB
0
)
0
)
214
)
214
)
0
786
0
TK
0
TK
525
TK
525
TK
2
)
.
)
..
TK
..
TK
TK
0
TK
..
TK
0
TK
..
786..786
)( )
Ê
..
Ê
..
Ë
Ê
Ë
Ê
Ê
Á
Ê
..
Ê
..
Á
..
Ê
..
Ë
Á
Ë
Ê
Ë
Ê
Á
Ê
Ë
Ê
ˆ
¯
ˆ
¯
ˆ
ˆ
˜
ˆ
˜
ˆ
˜
ˆ
¯
˜
¯
˜
¯
˜
¯
ˆ
¯
ˆ
˜
ˆ
¯
ˆ
˜
ˆ
¯
ˆ
˜
ˆ
¯
ˆ
4.23
In 1970 an improved formula was given by Woschni as:
a
g
E
mp
bc
cyl
bc
Bp
E
Bp
E
TC
vC
TV
bc
TV
bc
cyl
TV
cyl
pV
bc
pV
bc
bc
pV
bc
=
Bp Bp
E
Bp
E
E
Bp
E
TC TC
vC+ vC
–
Bp
–
Bp
0.
80
TC
80
TC
.5
TC
.5
TC
3
TC
3
TC
820
820
02
Bp
02
Bp
12
mp12mp
vC
12
vC
mp
vC
mp12mp
vC
mp
12
vC vC
12
vC vC
mp
vC
mp
mp
vC
mp12mp
vC
mp
mp
vC
mp
vC+ vC
12
vC+ vC
Bp
.
Bp
Bp
02
Bp
.
Bp
02
Bp
TC
80
TC
–
TC
80
TC
(((
)
pp
– pp –
mo
t
È
TC
È
TC
Î
TC
Î
TC
TC
Í
TC
TC TC
Í
TC TC
TC
È
TC
Í
TC
È
TC
Î
Í
Î
TC
Î
TC
Í
TC
Î
TC
˘
˚
˙
˘
˙
˘
˚
˙
˚
08
.
08.08
4.24
in the unit of W/(m
2
.K).
In the above formula, p is the in-cylinder gas pressure (MPa), T is the
instantaneous in-cylinder bulk gas temperature (K), T
ec
is the in-cylinder gas
temperature at the end of compression, B
E
is the cylinder diameter (m), and v
mp
is the mean piston speed (m/s). Moreover, p
bc
, T
bc
, and V
bc
are the in-cylinder
pressure, temperature, and volume at the beginning of the compression stroke.
V
cyl
is the cylinder displacement (m
3
), and p
mot
is the in-cylinder pressure
at the motoring condition without combustion. K
1
is a scavenging constant,
Diesel-Xin-04.indd 311 5/5/11 11:47:26 AM
312 Diesel engine system design
© Woodhead Publishing Limited, 2011
K
2
is a combustion constant, C
1
is a gas velocity coef cient, given by C
1
= 6.18 + 0.2085(B
E
w
p
/v
mp
) for intake and exhaust strokes, and C
1
= 2.28
+ 0.154(B
E
w
pw
/v
mp
) for compression and expansion strokes, where w
pw
is
the paddle wheel angular velocity (in radian per second) in the steady- ow
swirl test. C
2
is a combustion chamber shape coef cient in the expansion
stroke. C
2
= 0.00324 m/(K.s) for direct injection combustion chambers;
C
2
= 0.00622 m/(K.s) for indirect injection combustion chambers; C
2
= 0
for intake, exhaust, and compression strokes. The last term in the Woschni
correlation attempted to re ect the effects of radiation and combustion.
The Woschni correlation has several limitations, for example: (1) zero-
dimensional homogeneous heat transfer was assumed; (2) the heat transfer
equivalent diameter was treated as a constant B
E
rather than an instantaneous
parameter within an engine cycle; and (3) the modeling of radiation heat
transfer was primitive. In fact, accurately determining a
g
is very dif cult,
from either theoretical computation or experimental measurement. Fortunately,
the impact of the accuracy of heat transfer modeling on the accuracy of
computing engine power and gas ow rate is not extremely critical. For
engine system-level cycle simulations, the main objective of accurately
computing cylinder heat transfer is to predict the base engine heat rejection
for cooling system design and exhaust manifold gas temperature. It is usually
suf cient to apply a multiplier to the Woschni heat transfer coef cient a
g
to calculate cylinder heat transfer more accurately. The multiplier is tuned
based on the energy balance analysis of engine test data in order to capture
the engine fundamental characteristics of the percentage of fuel energy lost
to cylinder heat rejection. This topic is further explained in Chapter 12.
More information on engine cylinder heat transfer can be found in Woschni
(1967), Hohenberg (1979), Jennings and Morel (1991), Hansen (1992),
Imabeppu et al. (1993), Alkidas (1993), Shayler et al. (1996, 1997), Bohac
et al. (1996), Wolff et al. (1997), Franco and Martorano (1999), Luján et
al. (2003), Zeng and Assanis (2004), and Schubert et al. (2005).
Gas masses fl owing into and out of the cylinder through engine valves
With the simpli ed assumptions of subsonic one-dimensional isentropic
ow for intake valves, the intake mass ow rate entering the cylinder is
given by:
d
d
=
– 1
m
CA
p
NR
NR
T
p
p
in
fi
CA
fi
CA
ni
CA
ni
CA
ni
p
ni
p
n
Ei
Ei
NR
Ei
NR
NR
Ei
NR
ni
T
ni
T
n
in
in
in
f
k
k
fi,fi
6
2
k
2
k
◊
Ê
Ë
Ê
Ë
Ê
Ê
Á
Ê
Ë
Á
Ë
Ê
Ë
Ê
Á
Ê
Ë
Ê
ˆˆˆ
¯
ˆˆˆ
¯
ˆˆˆ
ˆˆˆ
˜
ˆˆˆ
¯
˜
¯
ˆˆˆ
¯
ˆˆˆ
˜
ˆˆˆ
¯
ˆˆˆ
Ê
Ë
Ê
Ë
Ê
Ê
Á
Ê
Ë
Á
Ë
Ê
Ë
Ê
Á
Ê
Ë
Ê
ˆ
¯
ˆ
¯
ˆ
ˆ
˜
ˆ
¯
˜
¯
ˆ
¯
ˆ
˜
ˆ
¯
ˆ
È
Î
Í
È
Í
È
Í
Î
Í
Î
Í
Í
Í
˘
˚
˙
˘
˙
˘
˙
˚
˙
˚
˙
˙
˙
2
k
k
k
in
in
in
p
p
in
–
+1
4.25
where N
E
is the engine speed (rpm), C
f,in
is the intake valve ow coef cient,
A
in
is the intake valve instantaneous ow area, p
in
and T
in
are the pressure
Diesel-Xin-04.indd 312 5/5/11 11:47:27 AM
313Fundamentals of dynamic and static engine system designs
© Woodhead Publishing Limited, 2011
and temperature in the intake port just before the intake valve, respectively,
R
in
is the gas constant, k
in
is the ratio of speci c heat capacities of the intake
gas ow, and p is the in-cylinder pressure.
The exhaust mass ow rate out of the cylinder is given below. When
p
p
ex
ex
ex
ex
>
+
Ê
Ë
Á
Ê
Á
Ê
Ë
Á
Ë
ˆ
¯
˜
ˆ
˜
ˆ
¯
˜
¯
-
2
k
k
k
1
1
4.26
the valve gas ow is subsonic and can be described as
d
d
=
m
CA
p
NR
NR
T
p
p
ex
fe
,fe,
CA
fe
CA
xe
CA
xe
CA
fexefe
CA
fe
CA
xe
CA
fe
CA
x
xexxe
Ee
Ee
NR
Ee
NR
NR
Ee
NR
x
EexEe
ex
ex
ex
f
k
k
-
◊
-
Ê
Ë
Ê
Ë
Ê
Ê
Á
Ê
Ë
Á
Ë
Ê
Ë
Ê
Á
Ê
Ë
Ê
ˆ
6
2
k
2
k
1
¯¯¯
ˆ
¯
ˆ
¯
ˆ
¯
ˆ
ˆ
˜
ˆ
¯¯¯
˜
¯¯¯
ˆ
¯
ˆ
¯
ˆ
¯
ˆ
˜
ˆ
¯
ˆ
¯
ˆ
¯
ˆ
-
Ê
Ë
Ê
Ë
Ê
Ê
Á
Ê
Ë
Á
Ë
Ê
Ë
Ê
Á
Ê
Ë
Ê
ˆ
¯
ˆ
¯
ˆ
ˆ
˜
ˆ
¯
˜
¯
ˆ
¯
ˆ
˜
ˆ
¯
ˆ
È
Î
Í
È
Í
È
Í
Î
Í
Î
Í
Í
Í
˘
˚
˙
˘
˙
˘
˙
˚
˙
˚
˙
˙
˙
2
k
k
k
ex
ex
ex
p
p
ex
+1
4.27
When
p
p
ex
ex
ex
ex
≤
2
+ 1
–1
k
k
k
Ê
Ë
Á
Ê
Á
Ê
Ë
Á
Ë
ˆ
¯
˜
ˆ
˜
ˆ
¯
˜
¯
4.28
the valve gas ow is supersonic and can be described as
d
d
=
–
+ 1
–1
m
CA
p
NR
ex
fe
,fe,
CA
fe
CA
xe
CA
xe
CA
fexefe
CA
fe
CA
xe
CA
fe
CA
x
xexxe
Ee
Ee
NR
Ee
NR
NR
Ee
NR
x
EexEe
ex
ex
fk
fk
=
fk
=
NR
fk
NR
NR
fk
NR
T
fk
T
Ee
fk
Ee
NR
Ee
NR
fk
NR
Ee
NR
NR
Ee
NR
fk
NR
Ee
NR
x
fk
x
EexEe
fk
EexEe
k
6
fk
6
fk
2
1
Ê
Ë
fk
Ë
fk
Á
Ê
Á
Ê
Ë
Á
Ë
fk
Ë
fk
Á
fk
Ë
fk
ˆ
¯
˜
ˆ
˜
ˆ
¯
˜
¯
◊◊
2
k
2
k
2
k
ex
ex
+ 1
4.29
where C
f,ex
is the exhaust valve ow coef cient, A
ex
is the exhaust valve
instantaneous ow area, and p
ex
is the pressure in the exhaust port just
behind the exhaust valve. Figure 4.2 shows typical engine valve ow rates
in ring operation.
The valve and port flow coefficients C
f,in
and C
f,ex
can usually be
obtained from the experimental ow bench test of the cylinder head. They
are monotonic functions of normalized valve lift (Stone, 1999). The ow
coef cients are determined by port design, valve diameter, valve seat angle,
and ow recirculation effects between the valve and the cylinder bore. More
detailed information about the valve/port ow coef cients can be found in
Old eld and Watson (1983), Agnew (1994), Danov (1997), Mattarelli and
Valentini (2000), and Bohac and Landfahrer (1999).
Thermodynamic properties of in-cylinder gases
The in-cylinder speci c internal energy is a function of temperature and
species including the combustion products. For in-depth discussion of its
calculation, the reader is referred to Stone (1999). Ferguson and Kirkpatrick
Diesel-Xin-04.indd 313 5/5/11 11:47:28 AM
314 Diesel engine system design
© Woodhead Publishing Limited, 2011
(2000) presented a method to calculate the thermodynamic properties of the
combustion gas under chemical equilibrium.
Analysis of cylinder pressure and heat release rate for system design
The combustion heat release rate can be calculated by using a known measured
cylinder pressure trace and the energy conservation equation:
d
d
=
d
d
–
d
d
–
d
d
Q
UW
dUWd
Q
fuel
wall
ff
=
ff
=
d
ff
d
ff
–
ff
–
d
ff
d
4.30
where Q
fuel
= m
fuelC
X
fuel
q
LHV
h
com
. Or, alternatively the heat release rate may be
speci ed as an input by using an empirical formula. The widely used Wiebe
semi-empirical formula (also known as the Wiebe function) was developed
based on the homogeneous chain reaction theory and gasoline engine test
data. In the Wiebe function, the percentage of burned fuel X
fuel
represents
the combustion rate as follows:
X
fuel
X
fuel
X
C
co
m
SO
C
co
m
=
– e
–
+1
1
ff
SO
ff
SO
–
ff
–
f
d
D
Ê
Ë
Ê
Ë
Ê
Ê
Á
Ê
Ë
Á
Ë
Ê
Ë
Ê
Á
Ê
Ë
Ê
ˆ
¯
ˆ
¯
ˆ
ˆ
˜
ˆ
¯
˜
¯
ˆ
¯
ˆ
˜
ˆ
¯
ˆ
4.31
d
d
=
+ 1
e
X
dXd
C
fuel
X
fuel
X
co
m
co
m
SO
C
co
m
f
d
f
ff
–
ff
–
f
d
DD
co
DD
co
m
DD
m
f
DD
f
Ê
Ë
DD
Ë
DD
Ê
Ë
Ê
Ê
Á
Ê
Ë
Á
Ë
DD
Ë
DD
Á
DD
Ë
DD
Ê
Ë
Ê
Á
Ê
Ë
Ê
ˆ
¯
ˆ
¯
ˆ
ˆ
˜
ˆ
¯
˜
¯
ˆ
¯
ˆ
˜
ˆ
¯
ˆ
◊
–––
+1
C
co
m
SO
C
co
m
ff
–
ff
–
SO
ff
SO
f
d
D
Ê
Ë
Ê
Ë
Ê
Ê
Á
Ê
Ë
Á
Ë
Ê
Ë
Ê
Á
Ê
Ë
Ê
ˆ
¯
ˆ
¯
ˆ
ˆ
˜
ˆ
¯
˜
¯
ˆ
¯
ˆ
˜
ˆ
¯
ˆ
4.32
0.3
0.25
0.2
0.15
0.1
0.05
0
–0.05
Engine valve mass fl ow rate (kg/s)
–90 0 90 180 270 360 450 540 630
Crank angle (degree)
Intake valve fl ow
rate at 2000 rpm
rated power
Intake valve fl ow
rate at 1200 rpm
peak torque
Intake valve fl ow
rate at 2300 rpm
no load
Exhaust valve
fl ow rate at 2000
rpm rated power
Exhaust valve
fl ow rate at 1200
rpm peak torque
Exhaust valve
fl ow rate at 2300
rpm no load
4.2 Illustration of engine valve fl ows in fi ring operation.
Diesel-Xin-04.indd 314 5/5/11 11:47:28 AM
315Fundamentals of dynamic and static engine system designs
© Woodhead Publishing Limited, 2011
where d is a non-dimensional shape factor characterizing the instantaneous
change of fuel concentration of the effective burning portions during the
combustion process. The d value is a function of engine type and speed,
and affects the shape of the heat release rate. A smaller d produces a faster
burning rate, hence the peak of the normalized heat release rate occurs earlier.
In the formula, Df
com
is the combustion duration in the unit of degree crank
angle, f
SOC
is the crank angle at the start of combustion (SOC), and C
com
is a constant. If X
fuel
= 0.999 (i.e., 99.9% of the fuel is burnt at the end of
combustion), C
com
= 6.908. The start of combustion is determined by the
start of fuel injection and the ignition delay. The start of injection is usually
dened as the moment when the injection needle has lifted a specied distance
from its seat. Practically, the start of combustion is regarded as the moment
when the heat release rate becomes zero or when the accumulated heat
release rate reaches a minimum, or approximately as the moment when the
rst-order time derivative of the cylinder pressure trace reaches a minimum
after the start of injection.
Ignition delay consists of both physical and chemical processes. In the
physical processes, the fuel sprays break up, vaporize and mix with the air. In
the chemical processes, pre-ame oxidation of the premixed fuel occurs, and
localized ignition in multiple areas within the combustion chamber happens.
Ignition delay is dependent upon in-cylinder pressure and temperature, mean
piston speed, and fuel cetane number. A high cetane number reduces ignition
delay and may prevent diesel knock. In the empirical modeling of diesel
engine combustion, many efforts were made to predict the reaction rate of
the fuel burning process (which is split into the premixed and the diffusion
phases) based on ignition delay models or the more fundamental Arrhenius
equation of reaction rate. Moreover, there are many models regarding fuel
propagation rate and the diffusion of the oxygen into the fuel jet (Stone,
1999). More information on diesel engine ignition delay can be found in
Ryan (1987), Rosseel and Sierens (1996), and Kamimoto et al. (1998).
Bypassing the complex details of the combustion processes, the Wiebe
function provides a simple, convenient but still effective way to calculate the
in-cylinder thermodynamic bulk pressure and temperature. The Df
com
, f
SOC
and d in the Wiebe function can be easily determined by using experimental
diagrams of the cylinder pressure trace and heat release rate analysis. These
parameters can then be used as input data in cycle simulations. The formulae
in equations 4.31 and 4.32 are based on a single Wiebe function that can
simulate the heat release rate of low- and medium-speed diesel engines
reasonably well. For more complex heat release rates which characterize
the diesel premixed and diffusion combustion phases, two or three Wiebe
functions can be linearly super-positioned together.
Predicting Df
com
, f
SOC
,
and d for other operating conditions based on a set
of known Df
com
, f
SOC
,
and d values for a given condition was attempted by
Diesel-Xin-04.indd 315 5/5/11 11:47:28 AM
316 Diesel engine system design
© Woodhead Publishing Limited, 2011
previous researchers, but still remains a signicant challenge for modern EGR
diesel engines equipped with advanced fuel systems. A practical approach is
to rely on engine testing to obtain the experimental data for Df
com
, f
SOC
,
and
d in the entire engine speed–load domain or at various ambient conditions,
and then to empirically interpolate or surface-t them to obtain the required
values. Another approach is to rely on advanced combustion simulation (e.g.,
KIVA) to deduce certain general trends or phenomenological correlations
of heat release rates.
Diesel engine combustion heat release rate has been extensively researched
and modeled. Abundant information can be found in Meguerdichian and
Watson (1978), Watson et al. (1980), Miyamoto et al. (1985), Grimm and
Johnson (1990), Sierens et al. (1992), Tuccillo et al. (1993), Oppenheim et
al. (1997), Homsy and Atreya (1997), Kamimoto et al. (1997), Ladommatos
et al. (1998), Egnell (1999), Brunt and Platts (1999), Assanis et al. (2000),
Nieuwstadt et al. (2000), Lakshminarayanan et al. (2002), Schihl et al. (2002),
Hountalas et al. (2004), Cesario et al. (2004), Friedrich et al. (2006), Ponti
et al. (2007), Manente et al. (2008), Nuszkowski and Thompson (2009), and
Thor et al. (2009). Diesel engine combustion analysis is summarized in Hsu
(2002) and Borman and Ragland (1998).
4.3 Engine manifold filling dynamics and dynamic
engine system design
The thermodynamic processes of the transient performance of the intake and
exhaust systems are introduced in this section. The engine manifold lling
dynamics predicts the dynamic instantaneous values of the gas pressure,
temperature, and ow rate in the intake manifold and the exhaust manifold
as a function of time or crank angle. It is the foundation of dynamic engine
system design in the time domain or crank angle domain, which is related to
transient engine performance and engine controls. More information about
the mathematical model formulation of dynamic engine system design can
be found from the references provided in Chapter 14 (transient performance
and electronic controls) in the sections regarding the mean-value models
used by the engine controls community.
If the engine cylinder itself is the only subject of study for understanding
the in-cylinder cycle process, a simplied approach to solve the above
governing differential equations of the in-cylinder cycle process in the
previous section is to assume the pressures in the intake port and the exhaust
port are known constants as inputs. However, a more realistic and complex
approach, often required by dynamic engine system design or high-delity
static (steady state) engine system design, is to model them as instantaneous
(i.e., dynamic) unknowns by using the equations that govern the gas ows
in the intake and exhaust systems. Such an instantaneous calculation can be
Diesel-Xin-04.indd 316 5/5/11 11:47:28 AM
317Fundamentals of dynamic and static engine system designs
© Woodhead Publishing Limited, 2011
coupled with the equations of the in-cylinder cycle process if the purpose is
for high- delity static or dynamic engine system design. The instantaneous
manifold dynamics calculation can also be carried out independently, as
normally practiced by the engine controls community, to decouple it from
or bypass the equations of the in-cylinder cycle process, if the purpose is for
approximate dynamic engine system design or electronic controls. When the
crank-angle-based details of the in-cylinder cycle process are bypassed, the
cylinder is treated as a cycle-average ‘mean value’ object, for example for
the parameters of engine volumetric ef ciency (representing the breathing or
mass conservation feature of the engine), exhaust manifold gas temperature
(representing the energy conservation feature of the engine) and engine
brake power (derived from the p–V diagram and the mechanical friction
of the engine). The mean-value model is a low- delity model in terms of
the in-cylinder cycle process. The high- delity static engine system design
refers to the simulation feature of including the manifold gas dynamics
model to couple with the in-cylinder cycle process model. The high- delity
dynamic engine system design refers to the simulation feature of including
the in-cylinder cycle process to couple with the manifold gas dynamics.
They share commonality in the logics and the methodology in diesel engine
system design. Their difference is that the simulations in static system design
can be conducted on a non-real-time basis, but the dynamic system design
preferably needs to be conducted on a real-time basis. The details of the
crank-angle-resolution real-time model used for high- delity dynamic engine
system design are elaborated in Chapter 14.
If the gas properties in the intake and exhaust ducts are assumed to change
only as a function of time or crank angle, and their variations along the duct
dimension are ignored, then the zero-dimensional ‘ lling-and-emptying’
method can be used as follows. Denoting p
3
, T
3
, m
3
,V
3
, and A
3inl
as the gas
pressure, the temperature, the mass, the volume, and the inbound effective
ow area in the exhaust port and manifold, respectively; p, T, and m as the
gas pressure, the temperature, and the mass, respectively, in the cylinder
from which the gas is owing into the exhaust control volume; and p
3out
,
T
3out
, m
3out
, and A
3out
as the gas pressure, the temperature, the mass, and
the outbound effective ow area at the downstream of the exhaust control
volume (e.g., owing to the turbine), respectively, the following equations
are obtained:
Mass conservation:
d
d
=
d
d
+
d
d
m
m
m
out
33
=
33
=
d
33
d
+
33
+
d
33
d
m
33
m
m
33
m
ff
=
ff
=
d
ff
d
f
4.33
Energy conservation:
d(
)
d
=
d
d
+
d
d
+
d
d
mu
m
h
m
h
Q
out
loss
33
mu
33
mu
3
3
ff
=
ff
=
d
ff
d
ff
+
ff
+
d
ff
d
h
ff
h
3
ff
3
4.34
Substituting equation 4.33 into 4.34 gives
Diesel-Xin-04.indd 317 5/5/11 11:47:29 AM