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Numerical study of in-cylinder pressure
in an internal combustion engine
A. Shidfar *, M. Garshasbi
Department of Mathematics, Iran University of Science and Technology, Narmak, Tehran-16, Iran
Abstract
This paper presents the differential model of in-cylinder pressure in an internal com-
bustion engine. For this purpose, the compression stroke analyzed and Fourier law usedto modelling the cylinder pressure. An unknown function appears in the coefficient of
this equation. This unknown function is approximated by cubic B-spline. To estimate
unknown parameters, the modified LevenbergMarquardt algorithm is used. The
numerical solution of the direct problem is used to simulate pressure measurement.
2004 Elsevier Inc. All rights reserved.
Keywords:Cylinder pressure; Differential model; Crank angle; Internal combustion engine; Inverse
problem
1. Introduction
Cylinder pressure plays important role in engines problem. There are many
methods for measuring the cylinder pressure in an internal combustion engine.
The piezoelectric or optical pressure transducer are usually used to measure the
cylinder pressure. In this paper, we will find cylinder pressure as a function of
0096-3003/$ - see front matter 2004 Elsevier Inc. All rights reserved.
doi:10.1016/j.amc.2004.04.041
* Corresponding author.
E-mail address: [email protected](A. Shidfar).
Applied Mathematics and Computation 165 (2005) 163170www.elsevier.com/locate/amc
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crank angle, during the compression stroke. Since the thermodynamic system
which needs to be analyzed, like when analyzing the combustion process, we
study just this stroke[1,2].
2. Differential modelling of pressure
According to the Fouriers law, we have
r2T GqC
k
oT
ot ; 1
where q is the density. By using the ideal gas law, we find
TPV
mR: 2
In the cylinder, T, P, and Vare functions of crank angle. While the crank
angle is a function of time, i.e., a= a(t), then the density of in-cylinder gas is
a function of crank angle, namely q= q(a).Substitution Eq.(2)into(1)yields
r2PA GqC
k
oPA
ot : 3
During the compression stroke, both exhaust valves and intake valves are
closed. So, neglecting crevice fault, there is no mass transfer to or from com-
bustion chamber. In this stroke, no chemical reactions are taking place, then
G= 0. Crank angle is a function of time, then we have
oP
otoP
oa
da
dtdP
da
da
dt ;
oV
ot
oV
oa
da
dt
dV
da
da
dt ;
4
in the engine we have
Nomenclature
T temperature
P pressure
V volume
C specific heat
M number of pressure sensor
N number of approximation parameters
R universal gas constant
G source of energy
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da
dt We
2pN
60 ; 5
where We is the engine angular velocity, and N is the engine speed.
Now, setting Aa WeCqak
, then Eq. (3) can be rewrite in the following
form
V d2P
da2Aa
dP
da
P
d2V
da2Aa
dV
da
dP
da
dV
da 0: 6
In the compression stroke,
P cVc; 7
wherec P0Vc0, andc is polytropic exponent. Then we simplified Eq.(6)as fol-lows:
d2P
da2
1
cP
dP
da
2Aa
dP
da 0: 8
By choosing U 1c lnP, the Eq.(8)becomes
d2U
da2 c 1
dU
da
2
AadU
da
0: 9
Setting Z= exp((c 1)U), the Eq.(9)becomes
d2Z
da2Aa
dZ
da 0: 10
3. Inverse problem formulation
In this section, we consider the following problem
Z00a AaZ0a 0; 06 a6 p;
Z0 Z0; 11
Zp Zp; 12
where Za Pac1c , and Z
pand Z0 are known parameters. In this problem,
A(a) is an unknown function, but pressures measured at a number M angle
are available to be used as additional information on the pressure in the cylin-der to estimate the function A(a) [3,4]
Zan Zn; n 1; 2;. . . ;M: 13
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We use the variational formulation of the inverse problem under analysis. In
such a case the solution of the inverse problem is based on the minimization of
the residual functional defined by the following equation
JA 1
2
XMn1
Zan;A Zn2; 14
whereZ(an; A) are the pressure computed at the sensor location by solving the
direct problem(10)(12). Finally, the inverse problem the variational formula-
tion consists in minimization the residual function (14) under constrains of
problem.
4. Numerical algorithm
4.1. Method of analysis
The unknown functionA(a) is approximated in the form of cubic B-spline as
follows
Aa XNi1
qiUia; 15
whereNis the number of approximation parameters,qi(i= 1,2, . . ., N) are un-
known approximation parameters, and, Ui(i= 1,2, . . ., N) are given basis cubic
B-spline. As a result, the solving of the inverse problem reduce to the estima-
tion of a vector of parameters q= [q1, q2, . . ., qn].
Eq. (14) is minimized by differentiating it with respect to each of the un-
known parameters qi(i= 1,2, . . ., N) and then setting the resulting expression
equal to zero.
oJ
oqi
XMn1
oZnq
oqi
Zan; q Zn 0; i 1; 2;. . . ;N: 16
Here, the total number of parameters Nmust equal or exceed the number of
unknowns. In addition, the number of spatial measurement location should
also ensure uniqueness of the estimated parameters.
Eq.(16)are written in the matrix form as
oJ
oq XTT Y 0; 17
where
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T
Z1q
.
.
.
ZMq
0
BB@
1
CCA; YZ1
.
.
.
ZM
0
BB@
1
CCA 18
and
XoJ
oq
oZ1qoq1
oZ1qoq2
. . .oZ1qoqm
oZ2q
oq1
oZ2q
oq2. . .
oZ2q
oqm
.
.
.
oZMqoq1
oZMqoq2
. . . oZMqoqm
0BBBBBB@
1CCCCCCA
;
where X is sensitivity coefficient matrix with respect to q and the elements of
this matrix
Xij oZiq
oqj; i 1; 2;. . . ;M; j 1; 2;. . . ;N; 19
are the sensitivity coefficients.
4.2. Method of solution
Because the system of Eq.(16)are nonlinear, an iterative technique is nec-
essary for its solution. The modified LevenbergMarquardt algorithm is used
to solve the nonlinear least-squares Eq. (16) by iteration. This algorithm is a
combination of the Newton method which converges fast but requires a good
initial guess, and the steepest descent method which converges slowly but does
not require a good initial guess. The LevenbergMarquardt algorithm is given
byqk1 qk XTX lkI
1XY T; k 1; 2; 3;. . . ; 20
where q is estimated parameters vector, Y is measured pressure vector, lk is
damping parameters, and Xis sensitivity coefficient matrix[5,6].
Clearly, forlk! 0, Eq.(20)reduce to Newtons method and for l!1, itbecomes the steepest descent method. Calculations are started with large values
oflkand its value is gradually reduced as the solution approaches the con-
verged result.
For using this algorithm in each iteration, it is necessary to solve the directproblem (10)(12) for more times. Each time, we perturb only one of the
parameters by a small amount and compute Z. We can compute the sensitivity
coefficients for each parameter qi(i= 1,2, . . ., N). For example, with respect to
q1, we have
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oZiq
oq1Zq1 Dq1; q2;. . . ; qN; ai Zq1; q2;. . . ; qN
Dq1: 21
4.3. Numerical results
We consider one specific inverse problem. To verify the methodology con-
sidered in the present work, we consider a known function in the coefficient
of Eq.(10). The pressure reading for this inverse problem are produced numer-
ically by adding numerical noises to the exact solution of the direct problem.
The numerically produced measurement data can be expressed as follows.
Y Texact xr;
22whereTexact is the exact solution of the direct problem, ris the standard devi-
ation of the measurements, and x is a random variable generated by IMSL
subroutine DRNNOR.x has a value within2.576 and 2.576 for the 99% con-
Fig. 1. Estimation of unknown function.
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fidence bounds. For solving direct problem, we considerZ(0) = 2 andZ(p) = 9
and A(a) =a sin(a).
Knowing the simulated measured pressureY, computing the sensitivity coef-ficient oTi
oqj(i.e., X) with finite difference according to the Eq. (21), the Leven-
bergMarquardt algorithm given by Eq. (20) is used to estimate the
parameters qi(i= 1, . . ., N).
Following figures shows the ability of this method to estimate the unknown
function.Fig. 1shows the exact and estimated values ofA(a), andFig. 2shows
the exact and estimated values ofZ(a) for present example. These results cover
a broad range of property of variation. The agreement between the exact and
estimated properties is very good.
Estimated and exact pressure are compared as follows.
Fig. 2. Estimated the pressure.
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5. Conclusion
This paper presents a differential model of cylinder pressure in an internalcombustion engine that obtained from Fourier law. This model can state cyl-
inder pressure as a function of crank angle. The advantages of this model is
that it can be solved by inverse analysis with out any information about volume
of in cylinder gas or specification of gases contained in the cylinder chamber.
References
[1] J.K. Ball et al., Combustion analysis and cycle-by-cycle variation in spark ignition enginecombustion-part-1: an evaluation of combustion analysis routines by reference to model data,
in: Proceedings of the Institution of Mechanical Engineers Part D, J. Auto. Eng., 212 (D5)
(1998) 381399.
[2] John B. Heywood, Internal Combustion Engine Fundamentals, McGraw-Hill, 1988.
[3] O.M. Alifanov, E.A. Artyukhin, S.V. Rumyanstev, Extreme method for solving Ill-posed
problems with application to inverse heat transfer problems, Begell House, New York, 1995.
[4] J.P. Holman, Heat Transfer, McGraw Hill Book Company, 1996.
[5] A. Iserles, A First Course in the Numerical Analysis of Differential Equation, Cambridge
University Press, 1996.
[6] M. NecatiOzisik, Heat Conduction, John Wiley & Sons, Inc., 1993.
170 A. Shidfar, M. Garshasbi / Appl. Math. Comput. 165 (2005) 163170
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