Xin Q. Diesel Engine System Design
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318 Diesel engine system design
© Woodhead Publishing Limited, 2011
d
d
=
1d
d
(
)
+
d
d
+
d
T
m
hu
– hu –
m
RT
Q
out
ex
RT
ex
RT
3
T
3
T
3
3
3
RT
3
RT
ff
d
ff
d
ff
=
ff
=
mc
ff
mc
v
ff
v
33
ff
33
mc
33
mc
ff
mc
33
mc
v33v
ff
v33v
f
loss
llossl
d
f
È
1d
È
1d
Î
ff
Î
ff
Í
È
Í
È
1d
È
1d
Í
1d
È
1d
Î
Í
Î
ff
Î
ff
Í
ff
Î
ff
˘
˚
˙
˘
˙
˘
˚
˙
˚
4.35
where
d
d
=
–
m
Ap
NR
NR
T
p
p
out
Ap
out
Ap
Ee
Ee
NR
Ee
NR
NR
Ee
NR
x
EexEe
ex
ex
out
3
33
Ap
33
Ap
Ap
out
Ap
33
Ap
out
Ap
3
T
3
T
3
3
6
2
1
f
k
2
k
2
k
-
◊
Ê
ËËË
Ê
Ë
Ê
Ë
Ê
Ë
Ê
Ê
Á
Ê
ËËË
Á
ËËË
Ê
Ë
Ê
Ë
Ê
Ë
Ê
Á
Ê
Ë
Ê
Ë
Ê
Ë
Ê
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
3
3
k
k
k
ex
ex
ex
p
p
out
–
+1
4.36
Note that the above equation is valid for subsonic ow conditions, for
example. If the ow is supersonic, the supersonic ow equation should be
used accordingly. Applying the ideal gas law to the gas in the exhaust control
volume (the port and the manifold) gives the following:
p
3
V
3
= m
3
R
ex
T
3
4.37
The exhaust port heat loss can be modeled by forced convective water
cooling. The heat transfer of the exhaust manifold is often modeled by natural
convection heat transfer. T
3
, m
3
, and p
3
are solved by using the governing
equations 4.33, 4.35, and 4.37 as a function of crank angle or time. The
equations describing the gas dynamics in the intake port and manifold can
be formulated in a similar way. The pressure and ow boundary conditions
at the outlet of the exhaust manifold are modeled by the turbine ow
characteristics (to be detailed in the next section). The pressure and ow
boundary conditions at the inlet of the intake manifold are modeled by the
compressor ow characteristics (to be detailed in the next section). In the
mean-value model, the ow boundary conditions at the outlet of the intake
manifold and at the inlet of the exhaust manifold are modeled by using the
engine volumetric ef ciency model which links the intake manifold pressure
to the engine gas ow rate (to be detailed in the next section). In the crank-
angle-based high delity model, the boundary conditions at the cylinder
side for the intake/exhaust manifold control volumes are calculated by (or
coupled with) the detailed in-cylinder cycle process model introduced in
the previous section.
When the manifold pipe is short, the ‘ lling-and-emptying’ method may
give satisfactory simulation results. If the gas pressure wave propagation and
re ection along the intake or exhaust pipe need to be calculated for manifold
tuning design or valve timing analysis, the partial differential equations
of one-dimensional gas pressure wave dynamics must be solved with the
method of characteristics lines or nite volume or nite difference numerical
methods (Benson, 1982; Horlock and Winterbone, 1986). Comprehensive
theories of engine unsteady one-dimensional gas wave dynamics and the
applications in manifold design were provided by Annand and Roe (1974)
and Winterbone and Pearson (1999, 2000). Engine manifold unsteady gas
Diesel-Xin-04.indd 318 5/5/11 11:47:29 AM
319Fundamentals of dynamic and static engine system designs
© Woodhead Publishing Limited, 2011
wave dynamics was also researched by Blair (1991), Peters and Gosman
(1993), Arias et al. (2000), Chalet et al. (2006) and Royo et al. (1994). The
more complicated three-dimensional CFD modeling of manifold ows is
usually a specialized component-level analysis, and is generally outside the
scope of engine system design analysis.
4.4 Mathematical formulation of static engine
system design
4.4.1 Predicting hardware performance
In order to understand engine air system design and the factors affecting
pumping loss, the mathematical formulation of the entire engine gas ow
network (Fig. 4.3) needs to be derived in a simple format so that a closed-
form solution can be analyzed to facilitate an intuitive understanding.
After the instantaneous cylinder pressure p is solved by the in-cylinder
cycle process, if the engine mechanical friction work W
f,E
is known (de ned
as negative value), the engine brake work W
E
of total n
E
cylinders can be
calculated by using the following formula:
Wp
VW
Wp
E
Wp
j
n
jj
VW
jj
VW
fE
VW
fE
VW
E
Wp =Wp
d
Wp dWp
jj
d
jj
d
Wp dWp
VW +VW
=1
S
dS d
Wp dWpSWp dWp
Ú
Wp
Ú
Wp
d
Ú
d
Wp dWp
Ú
Wp dWp
d◊ d
jj
d
jj
◊
jj
d
jj
fE,fE
4.38
Equation 4.38 is the link between the in-cylinder cycle process and the
performance of other subsystems in the engine gas ow network.
In the p–V diagram, the area encompassed by the curve of ‘in-cylinder
pressure vs. instantaneous volume’ during intake and exhaust strokes is
the work from pumping loss (Fig. 4.4). The PMEP in this book, unless
especially mentioned, corresponds to the gross p – V integral from 180°
to 540° (i.e., including both areas of B and C in Fig. 4.4). However, the
actual net pumping loss usually should be only the area B (Pierik and
Burkhard, 2000). This distinction is important when evaluating variable valve
actuation systems. The characteristic in-cylinder pressure differential for the
pumping loss is Dp
cyl
= p
exhaust
– p
intake
, where p
intake
= p
IntakeManifold
– Dp
in
and p
exhaust
= p
ExhaustManifold
+ Dp
ex
. The Dp
in
is the pressure drop across the
ow restrictions of the intake manifold, ports, and valves. The Dp
ex
is the
pressure drop across the ow restrictions of the exhaust valves, ports, and
manifold. The pumping loss indicator Dp
cyl
can be further derived as Dp
cyl
=
(p
ExhaustManifold
– p
IntakeManifold
) + (Dp
in
+ Dp
ex
) = (engine delta P) + (Dp
in
+
Dp
ex
). The engine delta P is de ned as the exhaust manifold pressure minus
the intake manifold pressure. It is noted that the pumping loss consists of
two parts: the engine delta P, and the ow restrictions, which are related
to volumetric ef ciency. Both parts are equally important in diesel engine
system design. Engine delta P is related to turbocharger and EGR circuit,
Diesel-Xin-04.indd 319 5/5/11 11:47:30 AM
© Woodhead Publishing Limited, 2011
Ambient
Ambient
Ambient
CAC, IT
CAC
EBP
valve
Compressor
Compressor
EGR valve
EGR valve
EGR cooler
EGR cooler
Ambient
Turbine
Turbine
Ambient
Intake
manifold
Intake
restriction
Intake
restriction
Intake
manifold
Intake
manifold
Intake
valve
Intake
valve
Intake
valve
Engine
cylinder
Engine
cylinder
Engine
cylinder
Exhaust
valve
Exhaust
valve
Exhaust
valve
Exhaust
manifold
Exhaust
manifold
Exhaust
manifold
Exhaust
restriction
Exhaust
restriction
Exhaust
restriction
p
1
p
3
p
3
p
1
p
1
p
2
p
4
p
4
p
2
p
2a
p
2a
p
3
p
1a
T
1
T
3
T
3
T
1
T
1
T
2
T
4
T
4
T
2
T
2a
T
2a
T
3
Ambient
(a) Naturally aspirated engine
(b) Turbocharged engine with high-pressure-loop EGR system
(c) Turbocharged engine with low-pressure-loop EGR system
h
T
h
T
h
C
h
C
T
CACout
T
CACout
N
C
, N
T
; W
˙
C
, W
˙
T
N
C
, N
T
; W
˙
C
, W
˙
T
4.3 Schematic diagrams of engine system layout.
Diesel-Xin-04.indd 320 5/5/11 11:47:30 AM
321Fundamentals of dynamic and static engine system designs
© Woodhead Publishing Limited, 2011
Logp–LogV diagram
EngCylinder
This entire area B represents
net pumping loss work
These two areas reflect volumetric efficiency
Volume/V
max
Average intake
manifold pressure
Average exhaust
manifold pressure
6 7 8 9
10
–1
2 3 4 5 6 7 8 9
10
0
Pressure (bar)
2
10
2
9
8
7
6
5
4
3
2
10
1
9
8
7
6
5
4
3
2
10
0
4.4 Pumping loss, engine delta P and manifold gas wave dynamics
(GT-POWER simulation).
In-cylinder
Intake port
Intake manifold
Exhaust port
Exhaust manifold
Internal residual
fraction is caused by
reverse flow from
cylinder to intake port
Internal residual fraction
is caused by: (1) flow from
exhaust port to intake
port; (2) reverse flow from
exhaust port to in-cylinder
–180 Compression 0 Power 180 Exhaust 360 Intake 540
BDC TDCF BDC TDC BDC
Crank angle (degree)
Pressure (bar)
5
4
3
2
1
0
Area B+C
represents 360°
gross pumping
loss work.
Area B represents
net pumping
loss work (also
called intersection
pumping integral)
Area A:
1
-
2
-
3
Area B:
3
-
4
-
5
Area C:
2
-
3
-
5
2
3
5
4
1
Diesel-Xin-04.indd 321 5/5/11 11:47:30 AM
322 Diesel engine system design
© Woodhead Publishing Limited, 2011
while volumetric ef ciency is related primarily to the designs of cylinder
head, valvetrain, and manifolds.
The system of equations governing the steady-state engine performance
is derived below. Starting with a simpler example of the naturally aspirated
non-EGR engine (Fig. 4.3(a)), at a given engine speed, fuel rate, and brake
power, the four unknowns to be solved are the engine air ow rate
m
ai
r
,
the exhaust manifold gas temperature T
3
, the inlet pressure p
1
, and the
exhaust manifold pressure p
3
. T
3
can be solved based on the engine energy
balance of thermodynamic rst law. The air ow rate
m
ai
r
is solved based
on the de nition of engine volumetric ef ciency by assuming the volumetric
ef ciency is a known value. The pressure p
1
is solved by using the intake
ow restriction characteristics (i.e., a curve of pressure drop vs. volume
ow rate). The pressure p
3
is solved by using the exhaust ow restriction
characteristics.
For turbocharged EGR engines (Fig. 4.3(b) and (c)), the situation is
much more complex. At a given engine speed, fuel rate and brake power,
assuming the turbo characteristics (turbo maps) are known, there are 18
unknowns to be solved:
m
ai
r
,
m
EG
R
EGREG
T
3
, p
4
, T
1,ROA
, p
1
, p
2
, T
2
, p
3
, T
4
, N
C
,
N
T
, h
C
, h
T
, p
2a
, T
2a
(for the high-pressure-loop EGR system) or T
1
(for the
low-pressure-loop EGR system), T
CACout
,
and T
EGRcoolerGasOut
. The detailed
explanation of the high-pressure loop and low-pressure loop EGR systems
is provided in Chapter 13 and Fig. 13.4. The engine delta P is p
3
– p
2a
. In
order to understand the system performance, a set of 18 equations is formed
(equations 4.39–4.56) based on the thermal, ow or ef ciency characteristics
of each element or device. It should be noted that the formulated equations
also serve as a foundation for developing future real-time model-based
algorithms in advanced electronic controls.
Unlike using the detailed differential equations of the in-cylinder cycle
process (equations 4.7, 4.10 and 4.12) for high- delity modeling, the engine
in the air system circuit is treated as a single lumped element here in terms
of energy balance so that the exhaust enthalpy or the exhaust manifold gas
temperature can be calculated by the following:
QW
HQ
Q
fuel
QW
fuel
QW
Ei
QW
Ei
QW
HQ
Ei
HQ
ne
HQ
ne
HQ
x
HQ
x
HQ
nexne
HQ
ne
HQ
x
HQ
ne
HQ
bas
ec
oolant
mi
QW =QW
QW QW
Ei
Ei
QW
Ei
QW QW
Ei
QW
+
Ei
+
Ei
HQ +HQ
ec
ec
oolant
oolant
HQ HQ
bas
bas
ec
ec
+
D
HQDHQ
Ei
D
Ei
HQ
Ei
HQDHQ
Ei
HQ
ec-ec
ec
ec-ec
ec
HQ
ne
HQ
–
HQ
ne
HQ
sss
cellaneous
4.39
where
D
H
DHD
in-ex
is the rise in gas enthalpy rate from the intake manifold to
the exhaust manifold,
Q
bas
ec
oolant
ec-ec
is the base engine heat rejection to the
coolant (if water cooled) or the cooling air (if air cooled), and
Q
miscell
aneous
miscellaneousmiscell
,
the miscellaneous heat losses, is treated as a known quantity here and is
detailed in Chapter 12. Equation 4.39 is used to solve for the exhaust
manifold gas temperature T
3
when the engine brake power
W
E
W
E
W
is assumed
to be known or calculated from equation 4.38.
It should be noted that the exhaust temperature measured by a thermocouple
is different from the time-averaged and the mass-averaged temperatures. This
Diesel-Xin-04.indd 322 5/5/11 11:47:32 AM
323Fundamentals of dynamic and static engine system designs
© Woodhead Publishing Limited, 2011
issue may cause large errors in engine cycle simulations. Proper care should
be taken to ensure correct temperature input. More detailed discussions on
this issue can be found in Caton (1982), Heywood (1988), Kar et al. (2004,
2006), and Son and Kolasa (2007).
The four-stroke engine may also be treated as a single lumped element
in the air system circuit network for overall breathing performance by using
the de nition of intake manifold air–EGR mixture non-trapped volumetric
ef ciency shown as follows:
h
r
vo
l
mixt
ur
e
aE
E
ai
rE
GR
a
m
NV
aE
NV
aE
E
NV
E
mm
rE
mm
rE
TR
a
TR
a
=
NV
NV
=
(
ai
(
ai
rE
(
rE
mm( mm
ai
mm
ai
(
ai
mm
ai
rE
mm
rE
(
rE
mm
rE
+
mm+ mm
rE
mm
rE
+
rE
mm
rE
)
2
2
2
2
TR
2
TR
(
(
gas
ggasg
aE
E
pN
aE
pN
aE
V
E
V
E
pN
2
pN
4.40
where h
vol
is the engine volumetric ef ciency with a reference gas density
de ned at the intake manifold, N
E
is the engine crankshaft speed (revolution
per second), V
E
is the engine displacement, T
2a
and p
2a
are at the intake
manifold, p
2a
= p
2
– Dp
IntakeThrottle
– Dp
CAC
,
m
ai
r
is the engine fresh air
mass ow rate, and
m
EG
R
EGREG
is the EGR mass ow rate. The h
vol
is related to
valve size, valve timing, valve lift pro le, port ow coef cient, and manifold
design. The h
vol
is also affected by intake manifold temperature, engine delta
P and the consequent internal residual gas fraction. The trapped residue
gas is affected by engine delta P and gas scavenging. In fact, it is very
dif cult to use hand calculations to compute h
vol
. Note that equations 4.39
and 4.40 describe the engine macro behavior, and they are essentially the
lumped simpli ed forms of the detailed in-cylinder cycle process described
in equations 4.7, 4.10, and 4.12.
Volumetric ef ciency is a very important system parameter. More
information on engine volumetric ef ciency can be found from Livengood et
al. (1952), Fukutani and Watanabe (1979), Roussopoulos (1990), and Smith
et al. (1999). Thorough elaborations on volumetric ef ciency are provided
in Chapters 9 and 13.
The lumped intake air ow restriction at the compressor inlet is given
by:
mf
mf
Cp
pT
mf
ai
mf
rd
mf
rd
mf
Cp
rd
Cp
,i
Cp
,i
Cp
Cp
rd
Cp
,i
Cp
rd
Cp
Cp
nt
Cp
am
bien
ta
pT
ta
pT
mb
ient
mf =mf
mf
rd
mf =mf
rd
mf
(
mf (mf
rd
(
rd
mf
rd
mf (mf
rd
mf
)
pT )pT
ta
)
ta
pT
ta
pT )pT
ta
pT
mb
)
mb
ient
)
ient
11
pT
11
pT
11
Cp
11
Cp
rd11rd
mf
rd
mf
11
mf
rd
mf
Cp
rd
Cp
11
Cp
rd
Cp
Cp
,i
Cp
11
Cp
,i
Cp
Cp
rd
Cp
,i
Cp
rd
Cp
11
Cp
rd
Cp
,i
Cp
rd
Cp
11
Cp
nt
Cp
11
Cp
nt
Cp
am11am
bien11bien
ta11ta
pT
ta
pT
11
pT
ta
pT
ta11ta
pT
ta
pT
11
pT
ta
pT
rd
(
rd11rd
(
rd
mf
rd
mf (mf
rd
mf
11
mf
rd
mf (mf
rd
mf
–
11
–
ta
–
ta11ta
–
ta
pT
ta
pT,,pT
ta
pT
11
,,
11
Cp
11
Cp,,Cp
11
Cp
am11am
,,
am11am
bien11bien
,,
bien11bien
ta11ta
,,
ta11ta
pT
ta
pT
11
pT
ta
pT,,pT
ta
pT
11
pT
ta
pT
ta
–
ta11ta
–
ta
,,
ta
–
ta11ta
–
ta
4.41
where f
1
is a known function (e.g., second-order polynomial) and C
d,int
is a
lumped ow restriction coef cient of the intake system at the compressor
inlet, including the restrictions of the air lter and any regulating valves, for
example, an intake throttle valve used in a low-pressure loop EGR system.
Equation 4.41 is used to solve p
1
. It should be noted that the intent here is
not to provide a detailed precise form of the function f. The goal is rather
to illustrate how to formulate the air system design problem mathematically
to match the number of unknowns with the number of equations in order to
facilitate a clear and intuitive understanding. The mathematical formulation of
Diesel-Xin-04.indd 323 5/5/11 11:47:33 AM
324 Diesel engine system design
© Woodhead Publishing Limited, 2011
the design topic here cannot become over-constrained or under-constrained.
Such a mathematical formulation is important for identifying various air
system control ‘knobs’ and understanding their performance behavior.
The intake fresh air temperature at the compressor inlet is given by:
T
1,ROA
= T
ambient
+ DT
ROA
4.42
where DT
ROA
is a ‘rise-over-ambient’ (ROA) value of the air temperature
increase from the ambient to the compressor inlet. DT
ROA
is related to
intake pipe insulation and vehicle underhood thermal management. In a
high-pressure loop EGR system, T
1
= T
1,ROA
, where T
1
is the compressor
inlet gas temperature. In a low-pressure EGR system, T
1
> T
1,ROA
due to
EGR mixing.
The following equation of lumped exhaust ow restriction at the turbine
outlet is used to solve the turbine outlet pressure p
4
:
mm
mm
fC
exh
mm
exh
mm
Cf
mm
Cf
mm
ue
mm
ue
mm
CfueCf
mm
Cf
mm
ue
mm
Cf
mm
lL
mm
lL
mm
ubeO
ilCons
de
x
dexde
mm =mm
mm mm
Cf
Cf
+
Cf
+
Cf
+
mm +mm
mm
lL
mm +mm
lL
mm
lL
lL
mm
lL
mm mm
lL
mm
ubeO
ubeO
ilCons
ilCons
=
fC(fC
,
de,de
fC
2
fC
ha
hhah
mb
ient
pp
ha
pp
ha
T
ha
,
ha
44
ha44ha
mb44mb
ient44ient
ha
pp
ha44ha
pp
ha
T
44
T
ha
pp
ha
–
ha
pp
ha44ha
pp
ha
–
ha
pp
ha
,
44
,
)
4.43
where
m
exh
is the exhaust mass ow rate,
m
C
is the compressor mass ow
rate (note that in a high-pressure loop EGR system,
mm
Ca
mm
Ca
mm
ir
mm =mm
mm
Ca
mm =mm
Ca
mm
),
m
fuel
is the
fuel ow rate, and C
d,exh
is a lumped exhaust restriction ow coef cient.
The lumped EGR circuit ow restriction is given by
mf
mf
Cp
pT
mf
EG
mf
Rd
mf
Rd
mf
Cp
Rd
Cp
mf
EG
mf
Rd
mf
EG
mf
Cp
EG
Cp
RE
Cp
RE
Cp
Cp
EG
Cp
RE
Cp
EG
Cp
GRin
lE
pT
lE
pT
pT
GR
pT
pT
out
pT
EGRcool
pT
EGRcool
pT
e
mf =mf
mf
Rd
mf =mf
Rd
mf
(
mf (mf
Rd
(
Rd
mf
Rd
mf (mf
Rd
mf
Cp
RE
Cp, Cp
RE
Cp
lE
–
lE
pT ,pT
lE
,
lE
pT
lE
pT ,pT
lE
pT
pT
GR
pT ,pT
GR
pT
pT
out
pT ,pT
out
pT
,
Cp
,
Cp
Rd3Rd
mf
Rd
mf
3
mf
Rd
mf
Rd
(
Rd3Rd
(
Rd
mf
Rd
mf (mf
Rd
mf
3
mf
Rd
mf (mf
Rd
mf
rO
rrOr
ut
)
4.44
The pressure p
EGRinl
refers to the gas pressure at the EGR circuit inlet, and
p
EGRout
refers to the gas pressure at the EGR circuit outlet. For example,
in a high-pressure-loop (HPL) EGR system, p
EGRinl
= p
ExhaustManifold
. The
p
ExhaustManifold
is equal to the turbine inlet pressure p
3
plus any pressure
drop from the exhaust manifold or the EGR pickup location to the turbine
inlet. In a HPL EGR system, p
EGRout
= p
2a
, and the engine delta P is the
EGR driving force. In a low-pressure loop (LPL) EGR system, p
EGRinl
is a
pressure somewhere between p
4
and p
ambient
, depending on where the EGR
ow is picked up between different aftertreatment devices, and p
EGRinl
is a
function of C
d,exh
and p
4
or a function of C
d,int
and p
1
. In a LPL EGR system,
p
EGRout
= p
1
. C
d,EGR
is a lumped ow restriction coef cient including the ow
restrictions of the EGR cooler and the tubing (i.e., xed ow restrictions)
as well as the ow restriction due to EGR valve opening (i.e., adjustable
ow restriction). The discussion below is focused mainly on the HPL EGR
system as an example.
The compressor outlet air temperature T
2
can be calculated based on the
de nition of compressor isentropic ef ciency (h
C
, de ned on a total-to-total
basis for temperatures and pressures) as below, assuming a single-stage
compressor or two-stage compressor without inter-stage cooling:
h
kk
C
pp
cc
kk
cc
kk
=
()
pp()pp
– 1
()
TT()TT
– 1
(
–1)
kk
–1)
kk
cc
–1)
cc
kk
cc
kk
–1)
kk
cc
kk
/
kk
/
kk
cc
/
cc
kk
cc
kk
/
kk
cc
kk
21
pp
21
pp
()
21
()
pp()pp
21
pp()pp
21
()
21
()
TT()TT
21
TT()TT
()/()
pp()pp/pp()pp
pp()pp
21
pp()pp/pp()pp
21
pp()pp
()/()
TT()TT/TT()TT
()
21
()/()
21
()
TT()TT
21
TT()TT/TT()TT
21
TT()TT
4.45
Diesel-Xin-04.indd 324 5/5/11 11:47:34 AM
325Fundamentals of dynamic and static engine system designs
© Woodhead Publishing Limited, 2011
The turbine outlet gas temperature T
4
can be calculated based on the
de nition of turbine isentropic ef ciency (h
T
, de ned on a total-to-static
basis) as below:
h
kk
T
TT
pp
tt
kk
tt
kk
=
1 – (
)
1 – (
(
–1)
kk
–1)
kk
tt
–1)
tt
kk
tt
kk
–1)
kk
tt
kk
/
kk
/
kk
tt
/
tt
kk
tt
kk
/
kk
tt
kk
43
TT
43
TT
43
pp
43
pp
/
TT/TT
43
/
43
TT
43
TT/TT
43
TT
/)
pp/)pp
43
/)
43
pp
43
pp/)pp
43
pp
4.46
The compressor power on a total-to-total basis is given by
W
mc
T
p
p
C
W
C
W
Cp
mc
Cp
mc
C
C
cc
=
– 1
,
(
–1)
cc
–1)
cc
/
cc
/
cc
1
T
1
T
2
1
h
kk
cc
kk
cc
kk
cc
kk
cc
kk
cc
kk
cc
–1)
kk
–1)
cc
–1)
cc
kk
cc
–1)
cc
/
kk
/
cc
/
cc
kk
cc
/
cc
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
˘
˚
˙
˘
˙
˘
˙˙˙
˚
˙
˚
˙
˚
˙
˚
˙
˙
˙
˙
˙
˙
˙
The turbine power is on a total-to-static basis given by
Wm
Wm
cT
p
p
TT
Wm
TT
Wm
Tp
cT
Tp
cT
T
cT
T
cT
tt
Wm = Wm
Wm
TT
Wm = Wm
TT
Wm
1 –
,
(
–1)
tt
–1)
tt
/
tt
/
tt
h
Wm
h
Wm
TT
h
TT
Wm
TT
Wm
h
Wm
TT
Wm
kk
tt
kk
tt
kk
tt
kk
tt
kk
tt
kk
tt
–1)
kk
–1)
tt
–1)
tt
kk
tt
–1)
tt
/
kk
/
tt
/
tt
kk
tt
/
tt
3
cT
3
cT
4
3
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
˘
˚
˙
˘
˙
˘
˙˙˙
˚
˙
˚
˙
˚
˙
˚
˙
˙
˙
˙
˙
˙
˙
The turbocharger power balance is given by
WW
CT
WW
CT
WW
TC
m
ech
WW =WW
WW
CT
WW =WW
CT
WW
,
h
CT
h
CT
, where h
TC,mech
is the turbocharger mechanical ef ciency if it has not been included in the
turbine ef ciency. The turbocharger power balance can be expanded as
follows:
1–
+
(
–1)
/
,
p
p
m
m
c
cc
–1)
cc
–1)
/
cc
/
CT
TC
m
ech
T
C
2
1
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
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
kk
–1)
kk
–1)
/
kk
/
cc
kk
cc
–1)
cc
–1)
kk
–1)
cc
–1)
/
cc
/
kk
/
cc
/
hh
CT
hh
CT
h
CT
h
CT
pT
ppTp
pC
c
T
T
p
p
tt
pT,pT
pC,pC
Ê
Ë
Á
Ê
Á
Ê
Ë
Á
Ë
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
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
3
T
3
T
1
T
1
T
4
3
1–
(
–1)
tt
–1)
tt
/
tt
/
tt
kk
tt
kk
tt
kk
tt
kk
tt
kk
tt
kk
tt
–1)
kk
–1)
tt
–1)
tt
kk
tt
–1)
tt
/
kk
/
tt
/
tt
kk
tt
/
tt
ÍÍÍ
È
Í
È
Í
È
Í
È
Í
Î
Í
Î
ÍÍÍ
Í
ÍÍÍ
˘
˚
˙
˘
˙
˘
˙
˚
˙
˚
˙
˙
˙
= 0
= 0 4.47
Note that the measured gas stand turbine ef ciency is usually the product
of the isentropic ef ciency h
T
multiplied by the mechanical ef ciency of
the turbocharger h
TC,mech
. The overall turbocharger ef ciency is de ned as
h
TC
= h
C
h
T
h
TC,mech
. Assuming the piston blow-by is negligible, the turbine
ow rate and the compressor ow rate are related by
mm
mm
TC
mm
TC
mm
fuel
mm
fuel
mm
WG
mm =mm
mm
TC
mm =mm
TC
mm
TC
TC
mm
TC
mm mm
TC
mm
+
mm – mm
,
where
m
WG
is the turbine wastegate or bypass ow rate. It should be noted
that there is a humidity effect on the c
p
in equation 4.47 due to the water
vapor in the atmosphere. The reader is referred to Israel and Hu (1993) for
their discussions on the humidity effect on a composite gas constant and
air density. The composite gas constant is a function of barometric pressure
and relative humidity.
The compressor speed needs to be equal to the turbine speed, and this
relationship is given as
N
C
= N
T
4.48
The compressor ef ciency map can be described as a function (e.g., a
sixth-order polynomial) of corrected compressor ow rate and pressure
Diesel-Xin-04.indd 325 5/5/11 11:47:36 AM
326 Diesel engine system design
© Woodhead Publishing Limited, 2011
ratio, as shown in any compressor map. This can be expressed as
h
CC
corr
fm
CC
fm
CC
pp
=
CC
=
CC
(
CC
(
CC
fm (fm
CC
fm
CC
(
CC
fm
CC
)
,
42
CC42CC
corr42corr
fm
42
fm
CC
fm
CC42CC
fm
CC
pp
42
pp
fm (fm
42
fm (fm
CC
fm
CC
(
CC
fm
CC42CC
fm
CC
(
CC
fm
CC
,
42
,
,42,
1
fm
fm
/
pp/pp
4.49
The turbine ef ciency map with a given effective area can be described as
a function (e.g., a sixth-order polynomial) of corrected turbine ow rate and
pressure ratio, as shown in any turbine map. This can be expressed as
h
TT
corr
fm
TT
fm
TT
fm
TT
fm
TT
pp
=
TT
=
TT
TT
,
TT
TT
fm
TT
,
TT
fm
TT
)
pp )pp
,
53
TT53TT
fm
53
fm
TT
fm
TT53TT
fm
TT
fm
53
fm
TT
fm
TT53TT
fm
TT
TT
fm
TT
,
TT
fm
TT53TT
fm
TT
,
TT
fm
TT
53
corr53corr
pp
53
pp
,53,
4
)
4
)
(/
fm(/fm
)(/ )
pp )pp(/pp )pp
53
(/
53
TT53TT
(/
TT53TT
corr53corr
(/
corr53corr
fm
53
fm(/fm
53
fm
TT
fm
TT53TT
fm
TT
(/
TT
fm
TT53TT
fm
TT
TT
fm
TT53TT
fm
TT
(/
TT
fm
TT53TT
fm
TT
,
53
,(/ ,
53
,
TT
,
TT53TT
,
TT
(/
TT
,
TT53TT
,
TT
corr
,
corr53corr
,
corr
(/
corr
,
corr53corr
,
corr
TT
fm
TT
,
TT
fm
TT53TT
fm
TT
,
TT
fm
TT
(/
TT
fm
TT
,
TT
fm
TT53TT
fm
TT
,
TT
fm
TT
)
53
)(/ )
53
)
pp )pp
53
pp )pp(/pp )pp
53
pp )pp
,53,
(/
,53,
,
,
,
53
,
,
,(/ ,
,
,
53
,
,
,
fm(/fm
fm(/fm
4.50
The compressor speed map can also be described as a function of corrected
compressor ow rate and pressure ratio as follows:
Nf
mp
p
CC
Nf
CC
Nf
mp
CC
mp
mp
corr
mp
Nf =Nf
Nf
CC
Nf =Nf
CC
Nf
(
Nf (Nf
Nf
CC
Nf (Nf
CC
Nf
)
,
mp
,
mp
62
mp
62
mp
CC62CC
Nf
CC
Nf
62
Nf
CC
Nf
mp
CC
mp
62
mp
CC
mp
mp
corr
mp
62
mp
corr
mp
(
62
(
Nf (Nf
62
Nf (Nf
CC
(
CC62CC
(
CC
Nf
CC
Nf (Nf
CC
Nf
62
Nf
CC
Nf (Nf
CC
Nf
mp, mp
62
mp, mp
mp
,
mp
62
mp
,
mp
1
/
4.51
The turbine speed map with a given effective area is also a function of
corrected turbine ow rate and pressure ratio as follows:
Nf
mp
p
Nf
TT
Nf
mp
corr
mp
Nf =Nf
Nf
TT
Nf =Nf
TT
Nf
(
Nf (Nf
Nf
TT
Nf (Nf
TT
Nf
)
,
mp
,
mp
73
Nf
73
Nf
mp
73
mp
TT73TT
Nf
TT
Nf
73
Nf
TT
Nf
mp
TT
mp
73
mp
TT
mp
mp
corr
mp
73
mp
corr
mp
(
73
(
Nf (Nf
73
Nf (Nf
TT
(
TT73TT
(
TT
Nf
TT
Nf (Nf
TT
Nf
73
Nf
TT
Nf (Nf
TT
Nf
mp, mp
73
mp, mp
mp
,
mp
73
mp
,
mp
4
/
4.52
In fact, the turbine ef ciency can also be described as a parabolic function of
speed ratio based on its aerodynamic performance. Different turbine pressure
ratios can form a family of parabolic curves. Denoting the subscript ‘o’ for
the total state and ‘s’ for the static state, the speed ratio is de ned by
v
C
dN
hh
T
T
TT
dN
TT
dN
os
hh
os
hh
0
34
hh
34
hh
os34os
hh
os
hh
34
hh
os
hh
2
=
(
hh
34
hh – hh
34
hh
hh
os
hh
34
hh
os
hh – hh
os
hh
34
hh
os
hh
)
p
dN
p
dN
The EGR and air mixing in the intake manifold for a HPL EGR system can
be described by
mc
Tm
Tm
cT
ai
mc
ai
mc
rp
mc
rp
mc
ai
r
T
EG
Rp
cT
Rp
cT
EGRpEG
EG
cT
EG
cT
R
cT
R
cT
EGREG
cT
EG
cT
R
cT
EG
cT
T
CACout
T
CACout
T
EGRcool
T
EGRcool
T
,
d
d
Tmd Tm
Tm
d
Tm
Tm+ Tm
d
cTdcT
00
Ú
Ú
mc
Ú
mc
rp
Ú
rp
mc
rp
mc
Ú
mc
rp
mc
00
Ú
00
,
erGasOut
EGRcoolerGasOutEGRcool
eerGasOute
EGRcooleEGRcoolerGasOutEGRcooleEGRcool
a
mm
cT
ai
rE
mm
rE
mm
GR
pm
ix
T
Ú
Rp
Ú
Rp
00
Ú
00
Ú
=
(
mm( mm
ai
(
ai
mm
ai
mm( mm
ai
mm
rE
(
rE
mm
rE
mm( mm
rE
mm
+
mm+ mm
mm
rE
mm+ mm
rE
mm
)d
cT)dcT
pm
)d
pm
cT
pm
cT)dcT
pm
cT
ix
)d
ix
cT
ix
cT)dcT
ix
cT
)d
Ú
)d
Ú
pm,pm
pm
)d
pm,pm
)d
pm
(
(
0
Ú
0
Ú
2
T
2
T
4.53a
The EGR and air mixing at the compressor inlet for a LPL EGR system
can be described by
mc
Tm
Tm
cT
ai
mc
ai
mc
rp
mc
rp
mc
ai
r
T
EG
Rp
cT
Rp
cT
EGRpEG
EG
cT
EG
cT
R
cT
R
cT
EGREG
cT
EG
cT
R
cT
EG
cT
T
RO
AE
T
AE
T
ROAERO
GRc
ool
e
,,
Rp,,Rp
,,
ai,,ai
r,,r
EG,,EG
Rp,,Rp
EGRpEG,,EGRpEG
d
d
Tmd Tm
Tm
d
Tm
Tm+ Tm
d
cTdcT
,
00
1
T
1
T
Ú
Ú
mc
Ú
mc
rp
Ú
rp
mc
rp
mc
Ú
mc
rp
mc
00
Ú
00
rG
rrGr
as
rGasrG
Ou
t
mm
cT
ai
rE
mm
rE
mm
GR
pm
ixture
T
Ú
Rp
Ú
Rp
AE
Ú
AE
Ú
Rp
Ú
Rp
Rp,,Rp
Ú
Rp,,Rp
00
Ú
00
Ú
=
(
mm( mm
ai
(
ai
mm
ai
mm( mm
ai
mm
rE
(
rE
mm
rE
mm( mm
rE
mm
+
mm+ mm
mm
rE
mm+ mm
rE
mm
)d
cT)dcT
pm
)d
pm
cT
pm
cT)dcT
pm
cT
ixture
)d
ixture
cT
ixture
cT)dcT
ixture
cT
)d
Ú
)d
Ú
pm,pm
pm
)d
pm,pm
)d
pm
(
(
0
Ú
0
Ú
1
T
1
T
4.53b
The effectiveness of the charge air cooler (CAC) is de ned as
Diesel-Xin-04.indd 326 5/5/11 11:47:38 AM
327Fundamentals of dynamic and static engine system designs
© Woodhead Publishing Limited, 2011
e
CA
C
CACout
CACcoolin
g
TT
CACout
TT
CACout
TT
CACcoolin
TT
CACcoolin
=
TT – TT
TT – TT
2
TT
2
TT
2
TT
2
TT
4.54
The effectiveness of the EGR cooler is de ned as
e
EGRcooler
EGRcoolerGasOut
EGRcool
TT
EGRcoolerGasOut
TT
EGRcoolerGasOut
TT
EGRcool
TT
EGRcool
=
TT – TT
TT – TT
3
TT
3
TT
3
TT
3
TT
ant
EGRcoolantEGRcool
aanta
EGRcoolaEGRcoolantEGRcoolaEGRcool
In
antInant
le
t
4.55
A detailed discussion on cooler effectiveness and design parameters is
provided in Chapter 12.
The lumped ow restriction of the charge air cooler and its downstream
intake throttle valve (if any) is characterized by a lumped ow coef cient
C
d,CAC-IT
as
mf
mf
Cp
pT
mf
ai
mf
rd
mf
rd
mf
Cp
rd
Cp
Cp
CA
Cp
Cp
CI
Cp
Ta
Cp
Ta
Cp
pT
Ta
pT
Cp
CI
Cp
Ta
Cp
CI
Cp
CACout
pT
CACout
pT
mf =mf
mf
rd
mf =mf
rd
mf
(
mf (mf
rd
(
rd
mf
rd
mf (mf
rd
mf
Ta
–
Ta
pT, pT
,
Cp
,
Cp
82
Cp
82
Cp
rd82rd
mf
rd
mf
82
mf
rd
mf
Cp
rd
Cp
82
Cp
rd
Cp
Cp
CA
Cp
82
Cp
CA
Cp
Cp
CI
Cp
82
Cp
CI
Cp
Ta82Ta
Cp
Ta
Cp
82
Cp
Ta
Cp
Cp
CI
Cp
Ta
Cp
CI
Cp
82
Cp
CI
Cp
Ta
Cp
CI
Cp
rd
(
rd82rd
(
rd
mf
rd
mf (mf
rd
mf
82
mf
rd
mf (mf
rd
mf
Cp
Ta
Cp, Cp
Ta
Cp
82
Cp
Ta
Cp, Cp
Ta
Cp
Cp
,
Cp
82
Cp
,
Cp
pT
2
pT
pT
Ta
pT
2
pT
Ta
pT
Cp
CI
Cp
82
Cp
CI
Cp
-
Cp
CI
Cp
82
Cp
CI
Cp
)
4.56
After the actual turbine mass ow rate
m
T
is solved by the above 18
equations, the turbine effective cross-sectional area A
T
can be calculated
using equation 4.57 for the compressible ow. This is provided as below
for an axial ow turbine as a simpli ed illustration:
mA
mA
p
RT
p
p
p
p
TT
mA
TT
mA
ex
RT
ex
RT
t
t
t
mA = mA
mA
TT
mA = mA
TT
mA
–
2/
◊◊
◊◊
-
◊
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
Ê
3
3
RT
3
RT
4
3
4
3
2
1
k
2
k
2
k
k
ËËË
Ê
Ë
Ê
Ë
Ê
Ë
Ê
Ê
Á
Ê
ËËË
Á
ËËË
Ê
Ë
Ê
Ë
Ê
Ë
Ê
Á
Ê
Ë
Ê
Ë
Ê
Ë
Ê
ˆ
¯
ˆ
¯
ˆ
ˆ
˜
ˆ
¯
˜
¯
ˆ
¯
ˆ
˜
ˆ
¯
ˆ
(+
1)
kk
(+
kk
(+
1)
kk
1)
tt
(+
tt
(+
1)
tt
1)
kk
tt
kk
(+
kk
(+
tt
(+
kk
(+
1)
kk
1)
tt
1)
kk
1)
/
kk
/
kk
tt
/
tt
kk
tt
kk
/
kk
tt
kk
4.57
where p
3
and T
3
are de ned in the total state. Note that equation 4.57 is only
valid for subsonic ow conditions when
p
p
t
tt
4
3
2
>
+ 1
/(
tt
/(
tt
–1)
k
kk
tt
kk
tt
/(
kk
/(
tt
/(
tt
kk
tt
/(
tt
Ê
Ë
Á
Ê
Á
Ê
Ë
Á
Ë
ˆ
¯
˜
ˆ
˜
ˆ
¯
˜
¯
4.58
However, when the following condition applies as
p
p
t
tt
4
3
2
≤
+ 1
/(
tt
/(
tt
–1)
k
kk
tt
kk
tt
/(
kk
/(
tt
/(
tt
kk
tt
/(
tt
Ê
Ë
Á
Ê
Á
Ê
Ë
Á
Ë
ˆ
¯
˜
ˆ
˜
ˆ
¯
˜
¯
4.59
the gas ow becomes supersonic, and the turbine ow equation is described
as
mA
mA
p
RT
TT
mA
TT
mA
ex
RT
ex
RT
t
t
t
t
mA =mA
mA
TT
mA =mA
TT
mA
+ 1
+ 1
(
–1)
◊
Ê
Ë
Á
Ê
Á
Ê
Ë
Á
Ë
ˆ
¯
˜
ˆ
˜
ˆ
¯
˜
¯
◊
3
3
RT
3
RT
1
2
2
k
k
2
k
2
k
k
/
4.60
The ow equation for radial ow turbines, which are used in most automotive
diesel engines, is much more complex and requires the modi cation of an
enthalpy factor with respect to the term p
4
/p
3
in equation 4.57. The turbine
effective area A
T
is the product of the following two contributing factors:
Diesel-Xin-04.indd 327 5/5/11 11:47:39 AM