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Cylinder air/fuel ratio estimation using net heat release data

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Control Engineering Practice 11 (2003) 311–318
Cylinder air/fuel ratio estimation using net heat release data
Per Tunest
(
al
a
, J. Karl Hedrick
b,
*
a
Lund Institute of Technology, Div. Combustion Engines, P.O. Box 118, SE-221 00 Lund, Sweden
b
Department of Mechanical Engineering, University of California, Berkeley, 6189 Etcheverry Hall, Berkeley, CA 94720-1740, USA
Received 26 October 2001; accepted 14 December 2001
Abstract
An estimation model which uses the net heat release proﬁle for estimating the cylinder air/fuel ratio of a spark ignition engine isdeveloped. The net heat release proﬁle is computed from the cylinder pressure trace and quantiﬁes the conversion of chemical energyof the reactants in the charge into thermal energy. The net heat release proﬁle does not take heat- or mass transfer into account.Cycle-averaged air/fuel ratio estimates over a range of engine speeds and loads show an RMS error of 4.1% compared tomeasurements in the exhaust.
r
2002 Elsevier Science Ltd. All rights reserved.
Keywords:
Engine control; Engine modelling; Pressure; Least-squares estimation; Least-squares identiﬁcation
1. Introduction
Increasingly high demands on pollutant control onautomotive engines have been pushing the level of sophistication of engine control modules (ECM) higherand higher for more than twenty years from now. Thethree-way catalytic converter, which completely dom-inates the after treatment of exhaust gases fromautomotive spark ignition (SI) Engines, mandates thatthe air/fuel ratio (AFR) is kept in a very narrow bandaround stoichiometric. A modern ECM handles thistask very well under steady-state conditions by estimat-ing the air/fuel ratio using an oxygen sensor in theexhaust gas stream. The control problem becomes moredifﬁcult during transient operation, though. The in-herent time delay in the feedback system due to theplacement of the oxygen sensor in the exhaust imposes abandwidth limitation on the closed-loop system. Thisbandwidth limitation will result in AFR excursionsduring fast transients, which in turn will result inincreased pollutant emissions.A sensor inside the combustion chamber providinginformation that replaces the oxygen sensor signal,would drastically reduce the time delay in the feedbacksystem, and could consequently reduce the AFRexcursions during fast transients. In this paper, cylinderpressure is used as a source of feedback. Manyapproaches in how best to utilize cylinder pressurefeedback have been proposed (Powell, 1993; Wibberley& Clark, 1989; Pestana, 1989). Most approaches thathave previously been proposed require statistics over alarge number of cycles, which defeats the purpose of trying to reduce the time delay in the feedback system.For example, it has been found that, using the indicatedmean effective pressure (imep), up to 300 engine cyclesare required to achieve acceptable repeatability andaccuracy (Brunt & Emtage, 1996). Tunest
(
al, Lee,Wilcutts, and Hedrick (2000) presents an ad hoc attemptat higher bandwidth feedback using cylinder pressure.This paper develops a method to estimate the AFR inan SI engine from cylinder pressure measurements. Themethod is developed from a well-established empiricalmodel for the dependence of laminar ﬂame speed ontemperature, pressure, and AFR, and relates this modelto the heat-release rate during the rapid burn phase,which is obtained from the cylinder-pressure-based netheat release proﬁle.Since the actual ﬂame speed in an SI engine dependson the turbulence intensity, a turbulence model also hasto be included. This model includes a simple turbulencemodel implicitly, by assuming that the turbulenceintensity is a function of engine speed (Heywood, 1988).An AFR estimator which is able to estimate cylinderAFR from cylinder pressure measurements over a widerange of operating points is developed. The variance of
*Corresponding author. Tel.: +1-510-642-0870; fax: +1-510-642-6163.
E-mail addresses:
per.tunestal@vok.lth.se (P. Tunest
(
al),khedrick@me.berkeley.edu (J.K. Hedrick).0967-0661/02/$-see front matter
r
2002 Elsevier Science Ltd. All rights reserved.PII: S 0967 -0661(02 )0 0045-X
an individual cycle estimate is very high due to therandom nature of the amount of residual gas in thecylinder, as well as the turbulent ﬂow ﬁeld which willcause the ﬂame development to be different from cycleto cycle. Cycle-averaged AFR estimates show an RMSerror of only 4.1% though. The identiﬁcation andvalidation are based on experiments performed at theUniversity of California, Berkeley. The experimentalsetup is essentially the same as in Tunest
(
al, Wilcutts,Lee, and Hedrick (1999), and the work is based onTunest
(
al (2000).
2. Review of the concepts of ﬂame and ﬂame speed
The following section is a review of the concepts of ﬂame and ﬂame speed, and is included for completeness.The presentation is largely based on Heywood (1988).
2.1. Deﬁnition of ﬂame
A ﬂame is a combustion reaction which propagatessubsonically through space. For motion of the reactionzone to be well-deﬁned, it is assumed that the thicknessof the reaction zone is small compared to the dimensionsof the space it is conﬁned to. Propagation of the reactionzone refers to its motion relative to the unburned gasahead of it, and thus a propagating ﬂame can very wellbe stationary with respect to the observer.Two different classes of ﬂames can be distinguishedbased on where the mixing of fuel and oxidizer (air)takes place. If fuel and oxidizer are uniformly mixedwhen entering the reaction zone, a
premixed
ﬂameresults. A
diffusion
ﬂame results if fuel and oxidizer haveto mix as the reaction is taking place. Similarly, ﬂamescan be characterized based on the gas ﬂow character-istics in the reaction zone. Flames can be either
laminar
(stream lined ﬂow), or
turbulent
(vortex motion). Flamescan be classiﬁed as
unsteady
or
steady
depending onwhether their overall motion or structure change withtime or not. Finally, the initial phase of the fuel, when itenters the reaction zone can be used for classiﬁcation of the ﬂame. It can be either
solid
,
liquid
or
gaseous
.The ﬂame in an SI engine is premixed, turbulent, andunsteady, and the fuel is gaseous when it enters thereaction zone.
2.2. Laminar and turbulent ﬂame speed
The laminar ﬂame speed is deﬁned as the velocity,relative and normal to the ﬂame front, with which theunburned gas moves into the front and is converted intoproducts under laminar ﬂow conditions (see Fig. 1).This deﬁnition srcinates from experiments with bur-ners, where the ﬂame is stationary, the unburned gasmoves into the ﬂame, and the burned gas moves out of and away from the ﬂame. The density of the burned gasis, in general, lower than the density of the unburnedgas, and consequently the burned gas will have a highervelocity than the unburned gas.A situation more relevant to engines, is that theunburned gas is stationary, and the reaction zonepropagates through the gas. In this case the laminarﬂame speed is actually the speed at which the ﬂame frontpropagates (see Fig. 2). The laminar ﬂame speed underengine pressure and temperature is of the order 0
:
5 m
=
s
:
In reality, the unburned gas in an engine actually movesaway from the ﬂame front, due to the expansion of theburned gas and compression of the unburned gas.One might be tempted to claim that the laminar ﬂamespeed is of limited interest for internal combustionengines, since the ﬂow conditions in any practical engineare highly turbulent. It turns out however, that oneimportant way of modeling the turbulent ﬂame devel-opment (see Section 2.2) which actually takes place in aninternal combustion engine, includes the laminar ﬂameconcept as a submodel.The presence of turbulence aids in propagating theﬂame, and since the local gas velocities due to turbulencecan be signiﬁcantly higher than the laminar ﬂame speed,turbulence can drastically increase the actual speed with
Fig. 1. Illustration of a stationary laminar ﬂame with unburned gasentering the reaction zone at the laminar ﬂame speed,
S
L
¼
v
u
;
andleaving the reaction zone at a velocity,
v
b
>
v
u
:
Fig. 2. Illustration of a propagating laminar ﬂame. The ﬂame frontmoves at the laminar ﬂame speed,
S
L
;
into the stationary unburnedgas. Due to the expansion, the burned gas is pushed backwards out of the reaction zone at a velocity,
v
b
:
P. Tunest
(
al, J.K. Hedrick / Control Engineering Practice 11 (2003) 311–318
312
which the reaction zone propagates. This speed is calledthe turbulent ﬂame speed. Groff and Matekunas (1980)indicates that the turbulent ﬂame speed is proportionalto the laminar ﬂame speed, and a factor which increasesmonotonically with the turbulence intensity.
3. Flame speed models
3.1. A laminar ﬂame speed model
The laminar ﬂame speed of a premixed gasoline/airﬂame increases monotonically with temperature, de-creases monotonically with pressure, and peaks forequivalence ratios slightly rich of stoichiometric. Thisbehavior can be modeled empirically by
S
L
¼
S
L
;
0
T
u
T
0
b
p p
0
m
;
ð
1
Þ
where
S
L
;
0
is the laminar ﬂame speed at
T
0
¼
298 K and
p
0
¼
1 atm
:
T
u
represents the temperature of theunburned gas ahead of the ﬂame, and
p
is the pressure.The parameters
b
and
m
are empirical model parametersand depend slightly on the equivalence ratio.The laminar ﬂame speed at normal pressure andtemperature,
S
L
;
0
can be modeled by
S
L
;
0
¼
B
m
þ
B
f
ð
f
f
m
Þ
2
:
ð
2
Þ
Here,
B
m
represents the maximum ﬂame speed attainedat equivalence ratio
f
m
;
and
B
f
quantiﬁes the depen-dence of ﬂame speed on equivalence ratio. The followingvalues were experimentally identiﬁed by Metgalchi andKeck (1982) for the laminar ﬂame speed of a gasoline/airﬂame:
f
m
¼
1
:
21
;
B
m
¼
0
:
305 m
=
s
;
B
f
¼
0
:
549 m
=
s
:
The range of equivalence ratios relevant for an SI engineintended for stoichiometric operation is roughly
½
0
:
9
;
1
:
1
;
which means that the laminar ﬂame speedvaries approximately
7
8
%
relative to the stoichiometricvalue.
3.2. Modeling the turbulent ﬂame speed
It is discussed in Heywood (1988) that the rapid burnangle
D
a
b
;
i.e. the crank angle interval between 10
%
and90
%
heat release, increases only slightly with enginespeed. This implies that the turbulence intensityincreases with engine speed, which causes an increasein the ﬂame speed. Measurements in Hires, Tabaczyns-ki, and Novak (1978) indicate that the rapid burn angle,with inlet pressure and equivalence ratio constant,follows approximately a power law with respect toengine speed,
N
;
according to
D
a
b
B
N
0
:
37
:
ð
3
Þ
Since the way the turbulent ﬂow characteristics changewith engine speed depends on the engine geometry, theexponent in the previous equation is considered as anunknown,
Z
;
which has to be determined from experi-ments. Thus,
D
a
b
¼
kN
Z
ð
4
Þ
for some constant
k
:
The time it takes for the ﬂame to propagate from thespark gap to the farthest end of the combustion chambercan be expressed as
D
t
b
¼
sB u
f
;
ð
5
Þ
where
B
is the cylinder bore,
s
is a factor which dependson the location of the spark gap, and
u
f
is the ﬂamespeed. A simple relationship based on the engine speedrelates
D
t
b
and
D
a
b
;
D
a
b
¼
d
a
d
t
D
t
b
¼
2
p
N
60
D
t
b
:
ð
6
Þ
Thus, the ﬂame speed can be expressed in terms of therapid burn angle,
u
f
¼
2
p
sBN
60
D
a
b
:
ð
7
Þ
Combining (4) and (7) yields
u
f
¼
u
0
N N
0
1
Z
;
ð
8
Þ
where
u
0
¼
2
p
sBN
1
Z
0
60
k
ð
9
Þ
and
N
0
is some arbitrary engine speed. It should benoted here that
u
0
will depend on equivalence ratio,pressure, and temperature in the same way as thelaminar ﬂame speed, according to (2). Now, assumingthat
f
is close enough to
f
m
;
u
0
¼
u
00
T
u
T
0
b
p p
0
m
:
ð
10
Þ
4. Incorporating the cylinder-pressure-based heat releaseinto the ﬂame speed models
4.1. Relating burn rate to ﬂame speed
A common simpliﬁcation when modeling ﬂamepropagation in a spark-ignition engine is to assume thatthe ﬂame front propagates spherically outward from thespark plug. Due to the ﬁnite nature of the combustionchamber, the ﬂame-front area quickly assumes a nearlyconstant value (see Fig. 3),
A
f
:
Thus, the enﬂamed
P. Tunest
(
al, J.K. Hedrick / Control Engineering Practice 11 (2003) 311–318
313
volume will grow according tod
V
f
d
t
¼
A
f
u
f
:
ð
11
Þ
The rate at which fuel is consumed can now be expressedusing (11) and the ideal gas law.d
m
f
d
t
¼
m
f
V
d
V
f
d
t
¼
m
f
m
a
m
a
V
d
V
f
d
t
E
1AFR
pRT A
f
u
f
:
ð
12
Þ
An approximation has been made in the application of the ideal gas law. The fuel mass has been neglected inassuming
pV
¼
m
a
RT
:
ð
13
Þ
Since the AFR in a gasoline engine is nominally 14
:
7
;
this is a reasonable approximation.The angular velocity,
o
e
;
of the engine relates crank-angle derivatives and time derivatives according tod
m
f
d
a
¼
d
m
f
d
t
o
e
:
ð
14
Þ
Introducing
N
as the engine speed in revolutions perminute yieldsd
m
f
d
a
¼
60
A
f
u
f
p
2
p
N
AFR
RT
:
ð
15
Þ
Using (8) for the ﬂame speed, (15) can be rewritten asd
m
f
d
a
¼
60
A
f
u
0
pN
Z
2
p
N
1
Z
0
AFR
RT
:
ð
16
Þ
Using (10) and consolidating the multiplicative con-stant, an expression for the burn rate as a function of pressure, temperature, and engine speed is obtainedd
m
f
d
a
¼
bp
1
þ
m
T
b
1
N
Z
AFR
1
;
ð
17
Þ
for some constant
b
:
4.2. Relating burn rate to average heat release rate
During the rapid burn phase of the cycle, the heat-release rate is essentially constant in the crank-angledomain. Thus, during the bulk of the combustion event,the heat-release rate can approximately be expressed asd
Q
ch
d
a
E
Q
tot
D
a
b
:
ð
18
Þ
The chemical energy released when combusting a unitmass of fuel with air is the lower heating value,
Q
LHV
:
Thus, the rate of fuel conversion can be expressed as
D
m
f
D
a
¼
1
Q
LHV
Q
tot
D
a
b
:
ð
19
Þ
Combining the two expressions for the burn rate, (17),provides an equation from which the AFR can bedetermined. The pressure and temperature at the start of combustion are determined by a compression polytropefrom the inlet pressure and temperature. Thus, theinlet pressure and temperature,
p
0
;
T
0
;
can be usedaccording to
bp
1
þ
m
0
T
b
10
N
Z
AFR
1
¼
1
Q
LHV
Q
tot
D
a
b
:
ð
20
Þ
Isolating AFR yieldsAFR
¼
bQ
LHV
D
a
b
Q
tot
p
1
þ
m
0
T
b
10
N
Z
:
ð
21
Þ
So, with
c
¼
bQ
LHV
;
AFR
¼
c
D
a
b
Q
tot
p
1
þ
m
0
T
b
10
N
Z
;
ð
22
Þ
where
c
;
m
;
b
;
and
Z
are unknown constants, which haveto be determined from experiments.
5. Identiﬁcation of model parameters
5.1. Identiﬁcation method
The experimental setup used for this work does notallow control of the intake air temperature, and thusonly the dependence of AFR on burn rate, pressure, andengine speed are investigated. In the identiﬁcationexperiments, the coolant temperature is held constant,in order to prevent any inﬂuence from this temperature.In order to identify the unknown parameters of (22),it is rewritten by taking logarithmsln AFR
D
a
b
=
Q
tot
¼
ln
ð
cT
b
10
Þ þ ð
1
þ
m
Þ
ln
p
0
þ ð
Z
Þ
ln
N
:
ð
23
Þ
Fig. 3. Two-dimensional illustration of ﬂame propagation from thespark plug. The circle segments represent the ﬂame front as itpropagates radially outward from the spark gap. All ﬂame frontsmarked with solid lines are essentially of equal length.
P. Tunest
(
al, J.K. Hedrick / Control Engineering Practice 11 (2003) 311–318
314

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