Proceedings of the ASME Internal Combustion Engine Division 2009 Fall Technical Conference
ICEF2009
September 27-30, 2009 , Lucerne, Switzerland
ICEF2009-14098
ICES2008
Modifications to Quasi-dimensional Burning Model of Spark Ignition Engines
M.R. Modarres Razavi and A. Hosseini and M. Dehnavi
Department of Mechanical Engineering, Ferdowsi University of Mashhad
Mashhad, Iran, 91775-1111
E-mail: m-razavi@um.ac.ir, hosseini.amir65@gmail.com, dehnavi.ma@gmail.com
ABSTRACT
The way in which position of spark plug affects
combustion in a spark ignition engine can be analyzed by
using two-zone burning model. The purpose of this paper is
to extract correlations to simulate the geometric interaction
between the propagating flame and the general cylindrical
combustion chamber. Eight different cases were recognized.
Appropriate equations to calculate the flame area (Af), the
burned and the unburned volume (Vb & Vu) and the heat
transfer areas related to the burned and unburned regions
were derived and presented for each case using Taylor
expansion in order to replace numerical solution with
trigonometric algebraic functions.
INTRODUCTION
Over the past decade, strict environmental legislation
has driven a significant research effort to reduce the
environmental impacts of fuel combustion in internal
combustion engines. To accomplish this goal, an
understanding of the complex chemical and physical
processes that occur during combustion is required [1- 3].
However, detailed measurements in real combustion systems
are costly and difficult due to short time scales in
temperature and pressure [1]. Therefore, to provide insight
into the non-linear interactions that occur during combustion
and obtain parameters that cannot be measured directly,
numerical models for combustion have been developed [2,
4].
The codes employed to simulate internal combustion engines
include quasi-dimensional models and two or three
dimensional codes. Although two or three dimensional
codes, which are classified as CFD codes, permit to simulate
very well the physical phenomena involved in engines, the
long time needed for calculation is one of their shortages [2,
5]. Quasi-dimensional models have reasonable accuracy and
fast computation time [6]. During the combustion period in
an IC engine, different in-cylinder conditions can lead to
different heat fluxes. The flux varies substantially with
location: regions of the chamber that are contacted by
rapidly moving high-temperature burned gases generally
experience the highest fluxes. In regions of high heat flux,
thermal stress must be kept below levels that would cause
fatigue tracking. Spark plug and valves must be kept cool to
avoid pre-ignition and knock problems which result from
overheated spark plug electrodes or exhaust valve. Therefore,
solving these engine heat transfer problems is obviously a
major design task.
Heat transfer affects engine performance, efficiency, and
emissions. For a given mass of fuel within the cylinder, higher
heat transfer to the combustion chamber walls will lower the
average combustion gas temperature and pressure, and reduce
the work per cycle transferred to the piston. Thus specific
power and efficiency are affected by the magnitude of engine
heat transfer. Heat transfer between the unburned charge and
the chamber walls in spark-ignition engines affects the onset
of knock which, by limiting the compression ratio, also
influences power and efficiency [7].
Thus, predicting engine behavior over a wide range of design
operating variables contributes to screen concepts prior to
major hardware problems, to determine trends and trade-offs ,
and, if the model is sufficiently accurate, to optimize design
and control. It can also provide a rational basis for design
innovation. For the processes that govern engine performance
and emissions, two basic types of models have been
developed. These can be categorized as thermodynamic and
fluid dynamic in nature, depending on whether the equations
which give the model its predominant structure are based on
energy conservation or on a full analysis of the fluid motion.
Other labels given to thermodynamic energy-conservationbased models are: zero-dimensional (since in the absence of
any fluid modeling, geometric features of the fluid motion
cannot be predicted), phenomenological (since additional
detail beyond the energy conservation equations is added for
each phenomenon in turn), and quasi-dimensional (where
specific geometric features, e.g. the spark-ignition engine
flame or the diesel fuel spray shapes, are added to the basic
Thermodynamic approach) [6,7]. The structure of
thermodynamic-based simulations of IC engines is depicted
in Fig. 1 [7].
Figure 1. Logic structure of thermodynamic-based simulations
of internal combustion engine operating cycle
SPARK-IGNITION ENGINE MODELS
During combustion, which starts with the spark discharge in
spark-ignition engines, the actual processes to be modeled
become much more complex. Many approaches to predicting
the burning or chemical energy release rate have been used
successfully to meet different simulation objectives. The
simplest approach has been to use a one-zone model where a
single thermodynamic system represents the entire
combustion chamber contents and the energy release rate is
defined by empirically based functions specified as part of
the simulation input. At the other extreme, quasi-geometric
models of turbulent premixed flames are used with a twozone analysis of the combustion chamber contents-an
unburned and a burned gas region-in more sophisticated
simulations of spark-ignition engine.
Features of spark-ignition engine combustion process that
permit major simplifying assumptions for thermodynamic
modeling are:
(1) the fuel, air, residual gas charge is essentially uniformly
premixed;
(2) the volume occupied by the reaction zone where the fuelair oxidation process actually occurs is normally small
compare with the clearance volume-the flame is a thin
reaction sheet even though it becomes highly wrinkled and
convoluted by the turbulent flow as it develops; thus
(3) for thermodynamic analysis, the contents of the
combustion chamber during combustion can be analyzed as
two zones-an unburned and a burned zone. In the absence of
strong swirl, the surface which defines the leading edge of the
flame can be approximated by a portion of the surface of a
sphere. Thus the mean burned gas front can also be
approximated by a sphere. The flame is initiated at the spark
plug; however, it may move away from the plug during the
early stages of its development [7, 8].
Then, for a given combustion chamber shape and assumed
flame center location (e.g. spark plug), the spherical burning
area Ab, the burned gas volume Vb,, and the combustion
chamber surface “wetted” by the unburned gases can be
calculated for a given flame radius rb and piston position
(defined by crank angle) from purely geometric consideration.
The practical importance of such “model” calculations is that:
(a) the mass burning rate for a given burning speed S b (which
depends on local turbulence and mixture composition) is
proportional to the spherical burning area (Ab);
(b) the heat transfer occurs largely between the burned gases
and the walls and is proportional to the chamber surface area
wetted by the burned gases [7].
The purpose of this study is to extract correlations to simulate
combustion as a functions of engine parameters, mainly spark
eccentricity (the distance between the location of spark plug
and the center of disc combustion chamber), which reduce the
time of calculation to a great extent while maintaining
accuracy of the aforementioned codes.
EFFECTS OF SPARK ECCENTRICITY ON FLAME
PROPAGATION
Spark position has an important role in combustion of SI
engines. The case in which spark is located at the center of the
disk-shaped combustion chamber has been studied before [8].
The four modes and their corresponding correlations are
shown in Fig.2.
Figure 2. Four different modes of flame front interactions with
combustion chamber walls and piston area, with no eccentricity
B
rf   ;   rf  hgap
2
(case1)
(1)
B
rf   ;   rf  hgap
2
(case2)
(2)
B
rf   ;   rf  hgap
2
B
rf   ;   rf  hgap
2
(case3)
(3)
(case4)
(4)
where rf is the flame radius, B is the cylindrical bore and
hgap is the height of combustion chamber.
However, due to some technical considerations, it is
necessary that spark plug have eccentricity. Given different
values of eccentricity, cylinder bore, and combustion
chamber height, eight geometry interaction modes occur [2].
These modes are depicted in Fig.3.
Figure 3. Eight different modes of flame front interactions
with combustion chamber walls and piston area, with spark
eccentricity
B
rf  hgap  ;   rf   e
2
B
rf  hgap  ;   rf   e
2
(case1)
(5)
(case2)
(6)
B
B
rf  hgap ; rf   e; rf   2
2
2
(case3)
(7)
B
rf  hgap ; rf   e
2
(case4)
B
B
rf  hgap ; rf   e ; rf   2
2
2
B
rf  (h2gap  (  e)2 )
2
(8)
(9)
(case6)
B
rf  hgap  ;   rf   2
2
B
rf  (h2gap  (  e)2 )
2
(case7)
(10)
(11)
(case8)
(12)
According to the cases above, the correlations for Ab and Vb
using MATHEMATICA software to obtain the volume created
by intersection of cylinder and sphere, are presented in
Appendix.
MODIFICATION OF QUASI-DIMENSIONAL MODEL OF
AN SI ENGINE WITH SPARK PLUG ECCENTRICITY
As it can be seen above, most of the equations have the term
in common. The correlations obtained for
calculation of burned and unburned regions, and flame area by
using this method, include integrals which are hyperbolic due
to presence of the term
in g(θ) . Since these
integrals lack primitive functions, they have no analytical
solution and need to be solved numerically for each crank
angle and require a great amount of calculations. To simplify
these equations and reduce the time needed for calculations,
hyperbolic integrals were replaced by trigonometric algebraic
functions. These functions were derived by Taylor expansion
of hyperbolic statements with
. The best
replacement for this statement can be obtained by its Newton
binomial expansion. Newton binomial expansion is defined as:
m (m  1) x 2 m (m  1)(m  2) x 3

 ...
2!
3!
k
(13)
m (m  1)(m  2)...(m  k  1)x
... 
 ...
k!
(1  x) m  1  mx 
Using this expansion
is simplified to
. In order to minimize the errors,
g2(θ)
was computed first (Eq.14) and then the Newton
binomial expansion was applied (Eq. 15).
B2
g2 (x)  e2 cos2 (x)   e2 sin2 (x)
4
B2 2 2
 e sin (x)
4
B2
g2 (x)  e2 cos2 (x)   e2 sin2 (x)
4
 B e2

 2e cos(x)   sin2 (x)
2
B


 2e cos(x)
(case5)
B
B
rf   2  ;     rf   2
2
2
B
rf  (h2gap  (  e)2 )
2
B
B
rf   e, rf  (h2gap  (  e)2 )
2
2
(14)
(15)
To compare the exact and approximate results for g and g2,
they have been plotted as functions of crank angle in Fig. 4 and
5.
Now,
can be calculated as:
B2
2e3
 e2 )  (Be 
)cos(x)
4
B
2e3
 (2e2 )cos2 (x)  (
)cos3 (x)
B
where A, B and C are:
rf 2  g2  (rf 2 
3
(16)
After simplification, the above equation is:
rf 2  g2  a0  a1 cos(x)  a2 cos2 (x)  a3 cos3 (x)
(17)
where
B2
2e3
2e3
a0  rf 2   e2 ; a1  Be 
; a2  2e2 ; a3  
4
B
B
Euler equation (Eq.18) can be employed to reduce the
exponent of the above equations. Using this equation, we
have (Eq.19).
1
(18)
cos(x)  (eix  e ix )
2
a0 cos(x)  a1 cos2 (x)  a2 cos3 (x)
 c0 cos(x)  c1 cos(2 x)  c2 cos(3x)
(19)
Applying these modifications, the term which made
relations analytically unsolvable is converted to an algebraic
equation:
3
2
2
 (rf  g (x))2 dx 
K0 x  K1 sin(x)  K2 sin(2x)  K3 sin(3x)
e
e
2   4  
2e  4B e
B
B
A
   2  
2
3
4Brf  B
r 
4 f  1
B
3
2
(26)
3
e
4  
4e3
B
B

2
2
3
4Brf  B
r 
4 f  1
B
(27)
e
2  
2e3
B
C

2
4Brf 2  B3
 rf 
4  1
B
(28)
Similarly, other equations with rational exponents were
computed using (Eq. 15).
As it can be seen above, most of the equations have the term
in common. The correlations obtained
for calculation of burned and unburned regions, and flame area
by using this method, include integrals which are hyperbolic
due to presence of the term
in g(θ).
(20)
Using Newton binomial expansion and considering the
desirable accuracy, a number of first nonzero terms of
Taylor series were determined. The final form is:
3
2
2
 (rf  g (x))2 dx 
K0 x  K1 sin(x)  K2 sin(2x)  K3 sin(3x)
(21)
In the above equations, K0, K1, K2, K3 are defined as:
3 A2 B 2 C 2
1 3
3
K 0  (    1)  ( A2B  ABC )
8 2 2 2
16 4
2
3
3
K 1  A  B(C  A)
2
8
3
1
1
1
 (AC 2  CB2  A3  AB2  CA2 )
32
2
2
2
3
3 A
3 B2
K 2  B  A(  C )  B(  AC  A2  C 2 )
4
16 2
64 2
(22)
Figure 4. Comparison of the results obtained using Newton
binomial expansion with exact values of g(θ).
(23)
It can be noted that the results obtained by the method
employed here are in good agreement with the exact values.
(24)
Tables 1 and 2 represent the values of approximate and exact
functions in terms of Princeton engine specifications, used in
reference [9], whose bore is 105 mm.
Considering case 3 (Eq. 7) and three different values for
eccentricity, 0.1 B, 0.25B, and 0.3 B [9], values of flame
1
1
K 3  C  AB
2
8
1 3
1
3
 ( AB2  A3  BCA2  3CB2  C 3 )
96 2
2
2
radius are 1.05, 2.625 and 3.15 cm respectively.
(25)
0.21
0.64
0.78
Replacing the correlation of Af with different values of
eccentricity, it can be noticed that the error is only a
function of the dimensionless term e/B (Tables 3, 4 and 5).
Determining the correction function in terms of e/B, error
of Af which was represented in Table 1, can be corrected to
the values of Table 6.
Af )Exact  Af )App.
e
 4.134( )  3.831
100
B
(29)
Percentage of error
Eccentricity value
-3.4252
-3.4252
-3.4252
-3.4252
-3.4252
Percentage of error
2.625
3
3.75
4.5
5
Flame radius
Eccentricity value
Eccentricity
(fraction of bore)
0.25
0.25
0.25
0.25
0.25
7.875
9
11.25
13.5
15
2.76822.76822.76822.76822.7682-
Table 5. Effect of dimensionless (e/B) on accuracy
of Af computation for e = 0.30B
10.5
12
15
18
20
0.3
0.3
0.3
0.3
0.3
3.15
3.6
4.5
5.4
6
Percentage of error
Error
(%)
6.3
7.2
9
10.8
12
Flame radius
3.08
6.2
6.6
10.5
12
15
18
20
Eccentricity value
Table2. Burned volume (Vb) for Case (3)
Eccentricity
Exact
Approximate
(fraction of B)
value
value (cm3)
(cm3)
0.1
420.30
421.22
0.25
542.38
545.84
0.30
576.78
581.29
Error
(%)
Eccentricity
(fraction of bore)
Table1. Flame area (Af) for Case (3)
Eccentricity
Exact
Approximate
(fraction of B)
value
value (cm2)
2
(cm )
0.1
122.67
118.88
0.25
120.06
112.60
0.3
119.46
111.54
Piston bore
To determine the precision of this method, the above values
were replaced in Eq. 14 and 15. Comparing the results with
those of numerical ones indicates that these equations are
acceptably accurate.
1.05
1.2
1.5
1.8
2
Table 4. Effect of dimensionless (e/B) on accuracy
of Af computation for e = 0.25B
Piston bore
Figure 5. Comparison of the results obtained using Newton
binomial expansion with exact values of g2(θ).
0.1
0.1
0.1
0.1
0.1
Flame radius
10.5
12
15
18
20
Eccentricity
(fraction of bore)
Piston bore
Table 3. Effect of dimensionless (e/B) on accuracy
of Af computation for e = 0.1B
8.4
9.6
12
14.4
16
2.61302.61302.61302.61302.6130-
Table 6. Corrected flame area (Af) using error
function (Eq. 38) for Case (3)
CONCLUSION
In this study, algebraic equations have been proposed to
modify the quasi-dimensional burning model of SI engines
with spark plug eccentricity. These equations, obtained by
Taylor series expansion, provide a fast response compared
to the integral forms which requires too demanding
computational numerical solutions.
It can be noted that using this method gives accurate results
which fit desirably on the exact values.
REFERENCES
1.
2.
3.
4.
5.
6.
7.
8.
9.
Tallio K.V., Colella P., “A Multi-Fluid CFD
Turbulent Entrainment Combustion Model:
Formulation and One-Dimensional Results” SAE
972880 (1997).
Modarres-Razavi M.R, “The Effects of Spark Plug
Position on Combustion of Spark Ignition Engines”,
13th International Conference on thermal
Engineering and Thermogrammetry (THERMO) ,
Budapest, 2003.
Bazari, Z. A DI Diesel Combustion and Emission
Predictive Capability for use in Cycle Simulation,
SAE 920462, (1992).
WEN Hua, LIU Yong-chang, WEI Ming-rui,
ZHANG Yu-sheng, “Multidimensional modeling of
Dimethyl Ether (DME) spray combustion in DI
diesel engine. Journal of Zhejiang University
SCIENCE, 2005 6A(4):276-282
Fiveland, S.B. & Assanis, D.N., “Development of a
Two-zone HCCI Combustion Model. Accounting
for Boundary Layer Effects”, SAE 2001-01-1028.
Verhelst S., Sierens R., “A quasi-dimensional model
for the power cycle of a hydrogen-fuelled ICE”,
International Journal of Hydrogen Energy 32 (2007)
3545 – 3554.
Heywood, J. B., “Internal Combustion Engine
Fundamentals” , McGraw Hill Inc. (1988).
Chin, Y.W, Mattews, R.D., Nicholas, S.P. &
Kiehne, T.M., “Use of Fractal Geometry to Model
Turbulent Combustion in a SI Engines”,
Combustion Science and Technology, 86 pp 1-30,
(1992).
Amooshahi, .H.R., ”Study of Combustion Fractal
Flame Model and Plug Location in SI Engine”. MSc
thesis, Mechanical Engineering Dept., Ferdowsi
University of Mashhad, Iran. March, 2000.
Eccentricity
(fraction of
B)
Exact
value
(cm2)
Approximate
value (cm2)
Error
(%)
0.10
0.25
0.30
122.67
120.06
120.74
123.56
120.35
120.92
0.73
0.24
0.15
NOMENCLATURE
B
e
rf
Vb
Af
Bore diameter (cm)
Spark plug eccentricity(cm)
Flame radius(cm)
Burned volume(cm3)
Flame area(cm2)
APPENDIX:
Correlations for flame area and burned volume, using Mathematica software, are presented here:
Af  2  πrf2 
Af  2  π rf hgap 
Af
Af
Af
Af
1
 2 π

 2  πrf   rf rf2  g2 (θ ) 2 dθ 


θ1
π
1


2
2
2


 2  π rf hgap   rf rf  g (θ ) dθ 


θ1
1
 2 π

2
2
 2  πrf   rf rf  g (θ ) 2 dθ 
0


θ2
1


2
2
 2 rf hgapθ2   rf rf  g (θ ) 2 dθ 
θ1


π
1


2
2
Af  2  π rf hgap   rf rf  g (θ ) 2 dθ 
0


θ2
1


Af  2 rf hgapθ2   rf rf2  g2 (θ ) 2 dθ 
0


 π rf3 
Vb  2 

 3 
3
 π rf2 π hgap

Vb  2 


2
6


(Case 1)
(A-1)
(case 2)
(A-2)
(case 3)
(A-3)
(case 4)
(A-4)
(case 5)
(A-5)
(case 6)
(A-6)
(case 7)
(A-7)
(case 8)
(A-8)
(Case 1)
(A-9)
(case 2)
(A-10)
3
 3 π

2
2
2
r

g
(
θ
)


f
 π rf

Vb  2 
 
dθ 
3 θ1
3


3


2
2
2
2
π
2
rf  g (θ ) 
 rf hgap 



Vb  2  π hgap  
dθ 
 
6  θ1
3
2


3
 3 π

2
2
2
r

g
(
θ
)
π
r


f


Vb  2  f   
dθ 
3
3
0


2

 rf2 hgap
 π hgap 2
Vb  2 θ2 hgap  
g (θ )dθ

2
2
2


 0
3
θ2
 
 rf2  g2 (θ )2
3
θ1


dθ 

3


2
2
2
2
π
2
rf  g (θ )
 rf hgap 



Vb  2  π hgap  
dθ 
 
6  0
3
2


2

 rf2 hgap
 θ2 hgap 2
Vb  2 θ2 hgap  
g (θ )dθ
 
2
6
2

 0

π

θ2
r

2
f
 g (θ ) 
2
3
3
2


dθ 

(case 3)
(A-11)
(case 4)
(A-12)
(case 5)
(A-13)
(case 6)
(A-14)
(case 7)
(A-15)
(case 8)
(A-16)