Numerical Study of in-cylinder Pressure

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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
mailto:[email protected]:[email protected]


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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

164 A. Shidfar, M. Garshasbi / Appl. Math. Comput. 165 (2005) 163170
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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

A. Shidfar, M. Garshasbi / Appl. Math. Comput. 165 (2005) 163170 165
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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.


Numerical Study of in-cylinder Pressure

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