Sixteenth International Multidimensional Engine Modeling User’s Group Meeting
at the SAE Congress, April 2, 2006, Detroit, Michigan
A G-equation Combustion Model Incorporating Detailed Chemical
Kinetics for PFI/DI SI Engine Simulations
Long Liang∗, Rolf D. Reitz
Engine Research Center, University of Wisconsin-Madison, Madison, WI 53706, USA
Jianwen Yi, Claudia O. Iyer
Ford Research and Advanced Engineering, Dearborn, MI 48124, USA
A G-equation-based combustion model incorporating detailed chemical kinetics has been developed and
implemented in KIVA-3V for Spark-Ignition (SI) engine simulations for better predictions of flame propagation and pollutant formation. A progress variable concept is introduced into the turbulent flame speed
correlation to account for the laminar to turbulent evolution of the spark kernel flame. The flame front
in the spark kernel stage is tracked using the Discrete Particle Ignition Kernel (DPIK) model. In the
G-equation model, it is assumed that after the flame front has passed, the mixture within the mean
flame brush tends to local equilibrium. The subgrid-scale burnt/unburnt volumes of the flame-containing
cells are tracked for the primary heat release calculation. An iso-octane kinetic mechanism coupled with
a reduced NOX mechanism is used to describe the chemical processes in the post-flame region and the
potential heat release from the end gas. The integrated model was used to simulate the combustion process in a Ford four-valve single-cylinder SI engine, which is equipped with both Port-Fuel-Injection (PFI)
and Direct-Injection (DI) fuel systems. For both PFI and DI operational modes, good agreement with
experimental in-cylinder pressure, heat release rates and engine-out NOX was obtained for different spark
timings and internal residual levels.
1
Introduction
mental engine combustion process and to further improve the versatility of multidimensional models, attention is being given to models incorporating comprehensive elementary chemical kinetic mechanisms.
The objective of the current work is to incorporate
detailed chemical kinetics into the G-equation-based
turbulent combustion model which was implemented
into the KIVA-3V code by Tan et al. [5] [7]. Specifically, detailed fuel oxidation mechanisms coupled
with a reduced NOX mechanism are applied behind
the mean flame front for modeling post flame combustion and NOX formation. The chemical kinetic
mechanisms are also applied in front of the flame front
for potential capability of predicting the compression
autoignition of the end-gas. In the course of coupling
detailed chemistry with the G-equation combustion
model for the primary heat release calculation within
the flame front, it was required to revisit and improve laminar and turbulent flame speed correlations
for better description of the turbulent flame propagation process.
The in-cylinder turbulent combustion in SI engines
is a complicated aero-thermo-chemical process especially due to the turbulence and chemistry interactions on tremendously different time-scale and
length-scale levels. In this paper, we present a
G-equation-based flamelet combustion model incorporating detailed chemical kinetics for both PortFuel-Injection (PFI) and Direct-Injection (DI) SparkIgnition (SI) engine combustion simulations. The
level set method is a powerful tool for describing interface evolution. With its application to combustion, Williams [1] first suggested a transport equation
of a non-reactive scalar, G, for laminar flame propagation. Peters [2] [3] subsequently extended this approach to the turbulent flame regime. The turbulent
G-equation concept has been successfully applied to
SI engine combustion simulations by Dekena et al. [4],
Tan [5] and Ewald et al. [6].
In recent years, to better understand the funda∗
Corresponding author. Email address: [email protected]
1
2
Model Formulation
2.1
the present model to account for the laminar to turbulent evolution of the spark kernel flame. This term
can be obtained by solving Eq. (2) assuming a uniform turbulence profile [3]. Physically, the additional
exponential term can be interpreted as a progress
variable which accounts for the increasing disturbing
effect of the surrounding eddies on the flame front
surface as the ignition kernel grows from the laminar
flame stage into the fully developed turbulent stage.
In the present study, Cm2 in Eq. (3) is selected as a
tunable model constant for different engines considering uncertainties due to other sub-models and/or
mesh resolution. However, for the same Ford engine
studied in this paper, Cm2 is fixed over all operating
conditions.
Laminar flame speed SL0 is one of the key scaling
factors in Eq. (3). Metghalchi et al. [9] suggested a
correlation for SL0 as a function of equivalence ratio
φ, temperature and pressure of the unburnt mixture
given by:
G-equation description of turbulent
flame propagation
In the flamelet modeling theory of premixed turbulent combustion by Peters [3], two regimes of practical
interest, i.e., the corrugated flamelet regime and the
thin reaction zone regime, were described by the same
group of level set equations, including the transport
, and its variance,
equations for the Favre averaged, G
G2 , and a model equation for the turbulent/laminar
flame surface area ratio, which in turn gives an algebraic solution for steady-state planar turbulent flame
speed, ST0 . The set of equations used in the current
implementation is [5] [7]:
∂G
= ρu ST0 |G|−D
(1)
+(
vf −vvertex )·G
κ|G|
T
∂t
ρ
2
∂G
Tu α p β
2
0
+ (
vf − vvertex ) · G
(
) (
) (1 − 2.1 · Ydil )
(4)
SL0 = SL,ref
∂t
Tu,ref
pref
ρu
2 ) + 2D (G)
2 − cs G2 (2) where the subscript ref means the reference condi= · ( DT G
T
ρ
k
tion of 298 K and 1 atm. Ydil is the mass fraction of
0
and fuel-type independent expodiluent. The SL,ref
0
nents α and β are correlated as functions of φ as:
ST
= 1 + 1 − exp(−Cm2 · t/τ )
SL0
0
= BM + B2 (φ − φM )2
(5)
SL,ref
2
2
a4 b l
ul
a4 b3 l
α = 2.18 − 0.8(φ − 1)
(6)
+ ( 3 )2 + a4 b23 0
(3)
· −
2b1 lF
2b1 lF
SL lF
β = −0.16 + 0.22(φ − 1)
(7)
f is
where is the tangential gradient operator, v
the fluid velocity vector, vvertex is the vertex moving
velocity. DT is the turbulent diffusivity, and a4 ,b1 ,b3
and cs are constants from the turbulence model or
experiment (cf. [3]). and k are the Favre mean turbulence kinetic energy and its dissipation rate from
the RNG k- model [8]. u is the turbulence intensity.
SL0 is the unstretched laminar flame speed. l and lF
are the turbulence integral length scale and laminar
flame thickness, respectively. κ
is the mean flame
front curvature. These equations together with the
Reynolds averaged Navier-Stokes equations and the
turbulence modeling equations form a complete set to
describe premixed turbulent flame front propagation.
One significant advantage of the G-equation formulation of turbulent premixed flames is the absence
of chemistry source terms in the transport equations.
As a consequence, the turbulent flame speed ST0 plays
a crucial role as a predetermined input. Compared
with the correlation for ST0 derived by Peters [2], an
exponentially increasing term is added in Eq. (3) in
Since Eq. (5) is invalid for very lean and very rich
mixtures (by predicting negative flame speed values), which is unacceptable in modeling DI operating
conditions, a formula proposed by Gülder [10] was
adopted in this study:
0
= ωφη · exp(−ξ(φ − σ)2 )
SL,ref
(8)
Due to the relatively coarse mesh resolution in
engine simulations, the growth of the ignition kernel
is tracked by using the DPIK model [5], where the
flame front position is marked by Lagrangian particles. The kernel growth rate is:
ρu
drk
= (Splasma + ST )
dt
ρk
(9)
where rk is the kernel radius, ρu and ρk are the local
unburnt gas density and the gas density inside the
kernel, respectively. The plasma velocity Splasma is
given as [5]:
Splasma =
2
Q̇spk · ηef f
− hu ) + p · ρu /ρk ]
4πrk2 [ρu (uk
(10)
where Q̇spk is the electrical energy discharge rate,
ηef f is the electrical energy transfer efficiency due
to heat loss to the spark plug. hu is the enthalpy
of unburnt mixture, uk is the internal energy of the
mixture inside the kernel. To account for turbulent
strain and curvature effects on the kernel flame, the
unstretched laminar flame speed SL0 in Eq. (4) was
multiplied by a stretch factor I0 , which takes the folFigure 1: Numerical descriptions of the turbulent flame
lowing form according to Herweg et al. [11]:
structure and the flame containing cells.
I0 = 1 − (lF /15l)1/2 (u /SL0 )3/2 − 2 · (lF /rk )(ρu /ρk )
(11)
where the second and third terms on the right hand
side represent the contributions due to turbulent
strain and due to the geometrical curvature of the
kernel, respectively. Note that the mean curvature
effects are also considered in the G-equation combustion model by the last term of Eq. (1). The transition
from the kernel model to the turbulent G-equation
combustion model follows the same criterion as the
one used in the previous work by Tan and Reitz [5],
i.e., the transition is controlled by a comparison of
the kernel radius with a critical size which is proportional to the locally averaged turbulence integral
length scale,viz.,
rk ≥ Cm1 · l = Cm1 · 0.16k3/2 /
As shown in Fig. 1, in this method, in order to
predict Yi,u for all the species in the flame-containing
cells, the sub-grid scale unburnt/burnt volumes partitioned by the mean flame front are tracked based
on geometrical information. As the mean flame front
sweeps forward, the mixture behind the sweeping volume tends to local equilibrium following a constant
pressure, constant enthalpy process. Yi,u can be calculated as follows:
(1) Determine the equilibrium species mass fractions Yi,b and the equilibrium flame temperature. In
this study, an element potential method-based code
by Pope [13] was used for the chemical equilibrium
calculation.
(2) Calculate the burnt gas density and the burnt
species densities based on the equation of state and
(12) the Yi,b from step (1):
(14)
ρi,b = Yi,b · (pM Wmix,b )/(Tb )
where Cm1 is a model constant.
Although elements of the above G-equation de- where M W
mix,b is the average molecular mass of the
scription were originally developed for premixed burnt mixture, and is the universal gas constant.
flames, it is also successfully applied to partially pre(3) Calculate the unburnt species densities ρi,u
mixed flames in the DI operating mode in this study. based on species mass conservation:
2.2
ρi,u = (ρi Vi4 − ρi,b Vb )/Vu
Primary heat release within the turbulent flame brush
(15)
(4) Finally, determine the unburnt species mass
fractions:
In the present implementation of the G-equation
=
ρ
/
ρi,u
(16)
Y
i,u
i,u
model, it is assumed that after the flame front has
i
passed, the mixture within the turbulent flame brush
In KIVA, the heat release rate due to the chemtends to the local and instantaneous thermodynamic
istry
source term is directly related to the species conequilibrium. Based on this assumption, an updated
version
rate [12]. It needs to be noted that the four
method is suggested to calculate the species conversion rate and the associated primary heat release at species associated with the NOX formation mechanism, i.e., NO, NO2 , N, and N2 O, are excluded from
the flame front, viz.,
the equilibrium calculation due to their relatively
(13) short residence time within the flame front, and the
dρi /dt = ρu (Yi,u − Yi,b )ST0 Af,i4 /Vi4
relative slow rate of the NOX chemical reactions.
where i4 is the computational cell index in KIVA [12].
Yi,u and Yi,b are the mass fractions of species i with
2.3 Post-flame heat release and pollutant
respect to the unburnt and burnt mixtures, respecformation
tively. Af is the mean flame front area and V is the
In this study, the computational cells ahead and becell volume.
hind the propagating flame front are modeled as Well
3
constants Cm1 = 2.0 and Cm2 = 1.0 were held fixed
(see Eqs. (12) and (3)).
The predicted in-cylinder pressure traces match
the measured data well in terms of peak pressure
and combustion phasing in the PFI operating mode
with spark timing sweeps, as shown in Fig. 2. Figure 3 shows the evolution of the mean turbulent
= 0 iso-surface) in the PFI mode with
flame front (G
spark timing = -40 ◦ ATDC and engine speed 1500
rev/min). As seen, in the PFI mode, the flame propagates throughout the whole cylinder reaching the
wall at about 20 ◦ ATDC.
Stirred Reactors (WSR). A detailed PRF chemical kinetic mechanism was applied to account for the further oxidation of CO and other intermediate species,
such as small hydrocarbon molecules and the species
in the H2 -O2 system. To consider the effects of turbulent mixing, the reaction rates could be adjusted by
considering the eddy turnover time as an additional
timescale, and by combining this timescale with the
kinetic timescale. However, this was not done in the
present study, i.e., only kinetic rates were used. A
nine-reaction reduced NOX mechanism was coupled
with the hydrocarbon oxidation mechanism for predicting the formation of NO and NO2 [14].
3
Results and Discussion
The engine studied is a Ford four-valve single-cylinder
SI gasoline engine which features a pentroof combustion chamber and a converging-shaped piston bowl.
The engine is equipped with both PFI and DI fuel
systems. The DI system employs the wide-spacing
arrangement, with a centrally mounted spark plug
and an intake-side-mounted swirl-type injector. The
test data used in this work was reported by Muñoz
et al. [15]. The specifications of the engine and the
modeled operating conditions are listed in Table 1.
Figure 2: Measured (EXPT) and predicted (SIMU) incylinder pressure in PFI mode.
Table 1: Ford engine specifications and operating conditions [15].
Bore / Stroke
89 mm / 79.5 mm
Compression Ratio
12
Engine Speed
1500 rev/min
PFI Mode
Spark Timings (◦ ATDC)
-44, -40, -36, -32
Internal Residual
28%
MAP
65kPa
DI Mode
◦
Spark Timings ( ATDC)
-32, -28, -24, -20
Internal Residual
6%
MAP
75kPa
End-of-Injection (◦ ATDC) -72
(a)
(b)
(c)
(d)
Figure 3: Evolution of the mean turbulent flame front
= 0 iso-surface) in PFI mode (Spark timing =
The computational mesh contains around 170,000 (shaded G
◦
cells, including the intake and exhaust manifolds -40 ATDC, Engine speed=1500 rev/min).
Compared to PFI cases, it is more challenging to
accurately predict the pressure evolution and heat release in the DI mode. The equivalence ratio of the
stratified charged mixture varies from very rich to
very lean. Therefore the laminar flame speed correlation needs to be reliable over a wide range of equiv-
and the cylinder. A reduced 25-species, 51-reaction
iso-octane mechanism [16] including the NOX reactions was used to model the post-flame chemistry
and the low temperature chemistry in the end gas.
CHEMKIN II was used to solve the detailed chemical
kinetic equations. In all simulated cases, the model
4
the dark lines denote stoichiometric conditions, the
bright lines represent the flame front surfaces. According to the simulation, most of the NO is formed
around the stoichiometric lines, while CO is mainly
generated within the fuel rich region, as expected.
alence ratios, and also as a function of temperature
and pressure of the unburnt mixture. As seen in Fig.
4, the pressure and heat release rate predicted by the
present model agree with the measured data reasonably well for all spark timings.
Figure 6 shows the comparison of normalized
engine-out NOX in both the PFI and DI modes. Although there are discrepancies in absolute values, the
general trends as functions of spark timing are well
captured.
(a)
(b)
(a)
(b)
Figure 6: Measured and predicted engine-out NOX . (a)
PFI mode; (b) DI mode.
(c)
(d)
Figure 4: Comparisons of in-cylinder pressure and heat
release rate in the DI mode.
(a)
(b)
Figure 7: Calculated evolution of kernel radius. (a) PFI
mode; (b) DI mode.
During the calibration of the present ignition and
combustion models, it was found that the stretch effects on the kernel flame, as described by Eq. (11),
played an important role in the prediction of combustion phasing. Referring to Eq. (11), the stretch effect
due to flame curvature (the last term on the right
hand side) decays very quickly with the growth of rk ,
while the strain effect due to strong turbulence and a
Figure 5: In-cylinder NO and CO mass fraction contours
at 20 ◦ ATDC (DI, spark timing = -32 ◦ ATDC).
Figures 5 (a) and (b) show the spatial distribution
of NO and CO species mass fractions at 20 ◦ ATDC for
the DI case with spark timing = -32 ◦ ATDC, where
5
[6] J. Ewald and N. Peters. A level set based
flamelet model for the prediction of combustion in spark ignition engines. 15th International Multidimensional Engine Modeling User’s
Group Meeting, Detroit, MI, 2005.
thick laminar flame structure (the second term on the
right side) might significantly delay the kernel flame’s
fast propagation, as in the PFI case with spark timing
= -44 ◦ ATDC shown in Fig. 7 (a). (Here for the first
10 ◦ CA from spark timing, the kernel flame was sustained mainly by the plasma velocity). Comparing
Figs. 7 (a) and (b), the stretch effects in the DI cases
are generally less significant compared to those in the
PFI cases, mainly because of the lower internal residual fractions and correspondingly higher unstretched
laminar flame speeds in the DI cases.
4
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6