# A Novel Laminar Flame Speed Correlation for the Refinement of the Flame Front Description in a Phenomenological Combustion Model for Spark-Ignition Engines.

IntroductionIn order to meet the upcoming stringent regulations on pollutant and C[O.sub.2] emissions [1,2], the car manufacturers are increasingly dealing with innovative and sometimes very complex technical solutions aimed at reducing the environmental impact of internal combustion engines. In this regard, new technologies have been investigated over the years, among which are turbocharging/turbocompounding [3,4], variable valve actuation systems [5], variable compression ratio [6], water injection [7], and exhaust gas recirculation [8,9]. However, these engine developments must also comply with a number of customers' requirements, such as vehicle drivability and high engine performance.

Innovative technical solutions can be investigated using both numerical and experimental approaches. The increasing number of engine control parameters results in more expensive and time-consuming experimental campaigns. Therefore, numerical simulation is becoming an increasingly attractive investigation tool, as it leads to a considerable reduction in engine development cost and time.

As known, pure 0D numerical approaches cannot describe the wave propagation in the engine pipe systems. Instead, 3D models, because of their relevant computational burden, are usually applied to the flow simulation in limited portions of the engine (intake airbox, after-treatment device, and cylinder) and for a reduced set of operating conditions. As a practical compromise, 1D simulations combine a sufficiently accurate prediction of the unsteady flow in the intake and exhaust pipes with a reliable description of in-cylinder processes (turbulence, combustion, heat transfer) when supplemented with appropriate phenomenological submodels. The development of such sub-models represents a key aspect of the 1D engine modeling. As far as spark ignition (SI) engines are concerned, a reliable model for the propagation of the turbulent premixed flame is required.

The premixed combustion in a turbulent flow field is a very complex and not yet fully understood process, involving the interaction of different phenomena that affect the turbulent flame dynamics. It is widely recognized that combustion in an SI engine mainly occurs in the so-called corrugated flamelet regime, where the flame front thickness is smaller than the finest turbulence length scale. In this regime, turbulence does not affect the inner flame structure, and therefore the chemical reactions and molecular diffusion processes inside the flame can be described by the laminar flame speed (LFS). The effect of turbulence, whose intensity is usually greater than the LFS, is to wrinkle and distort an essentially laminar flame front, which strongly increases the burning rate.

However, even in a laminar flame, various instabilities may appear, which are related to the molecular diffusion of radical species and heat ahead of the flame front. These instabilities are responsible for flame deformation and displacement, thus modifying the LFS. Moreover, since in an SI engine the flame propagates in a quasi-spherical shape, a flame stretch also occurs, causing local variations of the flame speed.

Various combustion models for SI engines have been proposed in the past [10,11], some of which are implemented in proprietary and commercial 1D simulation codes. These models mainly differ in the description of the laminar-to-turbulent flame transition and of the process that causes the turbulence-induced burning rate increase [12,13]. Whichever the combustion model is, the LFS plays a critical role in the burning rate prediction. Given its importance, the LFS has been largely investigated, as highlighted by the literature review [14, 15, 16, 17, 18, 19, 20, 21].

As an example, in [14, 15, 16, 17] experimentally derived LFS correlations were proposed for different blends of air and reference hydrocarbons. They result from the analytical fitting of experimental data obtained for wide ranges of pressure, p, temperature, T, and composition (equivalence ratio, [phi], and mixture dilution). The mathematical formulation of the pioneering correlations of Metghalchi and Keck [14] and Rhodes and Keck [15] is based on the so-called power law formula. This formulation does not consider any cross-influences between p and T, and the exponents of the individual terms depend monotonically on the equivalence ratio. These oversimplifications were removed in novel formulations proposed by different researchers [16, 17].

Regardless of the mathematical expression for the LFS correlation, the application of an experimentally derived formulation may result in relevant uncertainties under typical engine-like operations. This is because the experimental tests could only be performed in a limited range of equivalence ratios and at low pressure and temperature (usually below 15 bar and 600 K, respectively), due to technical [18, 19] and ignitability [14] limitations. These experimental investigations [14, 18, 19] were performed in preheated closed vessels with optical access, where the laminar burning speed of spherically expanding flames in premixed reactants were measured.

To overcome these limitations, alternative LFS models based on reaction kinetics calculations were presented, such as the ones proposed by Bougrine et al. [19], Hann et al. [21], and Bozza et al. [22]. These models allow extending the ranges of the independent variables, leading to more accurate LFS correlations for engine-like operating conditions. However, the accuracy and the computational burden of such models strongly depend on the number of species and reactions of the adopted kinetic scheme [23, 24, 25]. If the correlation aims to mimic the behavior of a commercial gasoline, the computations are usually carried out for a surrogated fuel (a blend of reference primary fuels, such as n-heptane, iso-octane, toluene, and methanol). The composition of the surrogated fuel can influence significantly the LFS predictions, and its most appropriate selection remains an open issue in the current literature [26].

Another critical issue affecting the combustion evolution, especially in its early stage of quasi-laminar flame propagation, is the flame stretch. It is a measure of the flame surface distortion resulting from its motion and the underlying flow field. For weakly stretched flames, theoretical studies have shown that the flame speed-stretch relationship is linear, the proportionality coefficient being the Markstein length. In these studies, the whole flame, including both preheat and reaction zones, is considered as a surface dividing the fresh unburned mixture from the hot combustion products. Markstein [27] introduced for the flame speed a dependence on the local curvature through a phenomenological parameter nowadays known as the Markstein length. More recently, the use of asymptotic approaches has provided a more rigorous foundation for the flame stretch problem [28]. The asymptotic models consider the flame as a thin region dominated by transport processes and containing a much thinner reaction layer. Starting from this flame representation, the flame speed is expressed as a linear function of the local stretch rate, K. Besides a curvature term, as postulated by Markstein, the stretch rate includes a dependence on the hydrodynamic strain experienced by the flame. The asymptotic theories have been further extended and distinct expressions have been provided for the burned and unburned Markstein lengths [29]. The flame stretch has also been studied from an experimental point of view. As an example, Wu and Law [30] performed measurements of flame stretch for various Lewis numbers, equivalence ratios, and fuels. All the theoretical and experimental studies have highlighted the LFS variations induced by the flame stretch, which can affect the progress of combustion in typical SI engine operations.

Based on the above literature overview, both the LFS correlation and the flame stretch description represent critical issues for a phenomenological combustion model.

The main purpose of the present work is the introduction of a novel LFS correlation for gasoline/air mixtures derived from a 1D chemical kinetics model. The new correlation is compared with the well-known Metghalchi and Keck formulation [14], which represents the state-of-the-art in most commercial 1D simulation codes. In addition, a refined model of the flame stretch based on the asymptotic approach of Matalon et al. [29] is introduced, and its influence on the prediction of the combustion process is discussed. The present analysis concerns a small-size naturally aspirated engine, equipped with an external exhaust gas recirculation (EGR) circuit and an intake-exhaust phased variable valve timing (VVT) device.

The article is organized as follows. First, the new LFS correlation is introduced and compared with the one by Metghalchi and Keck in wide variation ranges of T, p, [phi], and residual gas content. Then, the flame stretch formulation is described in detail. Mention is made of the measurements performed on the considered engine and used for the validation of the proposed LFS correlation. A 1D engine model, developed in the GT-Power[TM] framework, is presented with special reference to the phenomenological sub-models of turbulence, combustion, and heat transfer. The engine model is tuned at full load using both LFS correlations, and the identified tuning constants are provided for each of them. The model accuracy is then assessed at part loads, which leads to comments on the potentials and limitations of the proposed LFS correlation. The influence of the flame stretch on the model predictions is discussed in a separate section, and the conclusions of the work are finally drawn.

Laminar Flame Speed Correlations

Two different LFS correlations for gasoline have been considered in order to assess their effectiveness in reproducing the combustion process and performance of an SI engine. The first one, labeled as "Cor A", is experimentally derived and it is based on the well-known correlation of Metghalchi and Keck [14]. The second one, labeled as "Cor B", is a correlation developed in the present work that fits the values of 1D LFSs resulting from chemical kinetics computations.

Both correlations express the LFS ([S.sub.L]) according to the power law form of Equation 1, where [S.sub.L0] is the flame speed at reference conditions T=[T.sub.0] and p=[p.sub.0] for given fuel type and equivalence ratio, while [EGR.sub.factor] is an LFS reduction term that accounts for the inert gases contained in the unburned mixture:

[S.sub.L]=[S.sub.L0][([T/[T.sub.0]]).sup.[alpha]][([p/[p.sub.0]]).sup.[beta]] [EGR.sub.factor] Eq. (1)

For "Cor A", exponents [alpha] and [beta] depend only on [phi], and [EGR.sub.factor] is a function of the residual gas molar fraction, [x.sub.i], raised to a constant exponent, [delta]. The equations for "Cor A" are:

[S.sub.L0] = [B.sub.m]+ [B.sub.[phi]][([phi]-[[phi].sub.m]).sup.2] Eq. (2)

[mathematical expression not reproducible] Eq. (3)

The values of the constants, listed in Table 1, are the default ones used in software GT-Power[TM]. The original values proposed by Rhodes and Keck [15] for fuel RMFD-303 have been slightly modified to reflect the results reported in [16].

This correlation was based on experimental tests, where the operating parameters could span the following ranges [14]:

* Pressure, p: 0.4 - 50.7 atm

* Temperature, T: 300 - 700 K

* Equivalence ratio, [phi]: 0.8 - 1.4

* EGR molar fraction, [x.sub.r]: 0 - 0.2

Correlation "Cor B", introduced in the present work, has been obtained by fitting the results of a chemical kinetics solver (CANTERA) applied to 1D planar laminar flames. The actual gasoline is modeled by a toluene reference fuel (TRF), composed of iso-octane, n-heptane, and toluene. The volume fractions of the three components of the surrogate fuel are selected to match the RON and MON of a typical commercial gasoline, according to the methodology reported in [26]. Various oxidation mechanisms for gasoline surrogates exist in the literature, and they are classified as simplified [23], semi-detailed [24], and detailed kinetics [25]. Due to the large number of flame speed data (more than 6000 evaluations) needed to achieve a quite general correlation, a compromise between accuracy and computational effort is required. For this reason, it was decided to use the semi-detailed mechanism developed by Andrae et al. [31], which well describes both low and high temperature kinetics. This mechanism includes 5 elements, 137 species, and 633 reactions. The LFS calculations are performed for several values of pressure, p, temperature, T, equivalence ratio, [phi], RON, MON, and exhaust gas mass fraction, [x.sub.r], in the following ranges:

* Pressure, p: 1 - 20 bar

* Temperature, T: 323 - 923 K

* Equivalence ratio, [phi]: 0.5 - 1.5

* RON: 90 - 98

* MON: 80 - 94

* EGR mass fraction, [x.sub.r]: 0 - 0.2

The composition of the residual gas is established in each condition (TRF blend, [phi], T, p) as the equilibrium composition of the combustion products. The LFS results are fitted by a function of [phi], T, p, [x.sub.r], and TRF composition. An important preliminary finding is that, in the considered ranges of RON and MON, the LFS mainly depends on the fuel sensitivity, S = RON - MON, rather than on the specific TRF blend. This entails a strong simplification of the data fitting, leading to a correlation that can still be expressed in power law form, Equation 1. In this case, the expressions for the coefficients are more complex compared to "Cor A", and [EGR.sub.factor] depends also on T, p, and [phi], according to the equation:

[mathematical expression not reproducible] Eq. (4)

The coefficients of "Cor B" are expressed as polynomials in [phi] according to Equations 5-9. The polynomial coefficients are generally functions of T, p, and S, which causes a cross-dependency of the correlation terms on all the considered independent variables. The values of the correlation constants that provide the best fitting of the LFS data are listed in Table 2. The average fitting error is 1.75% and the maximum error reaches 6.0%, only for some extreme values of the variables.

Figure 1 shows the comparison between the LFS values resulting from the 1D flame simulations and the predictions of "Cor B", together with the regression line and an almost unitary value of the coefficient of determination ([R.sup.2]). This result highlights the fitting capability of the proposed correlation, which is not so obvious considering the large number of independent variables, their wide variation ranges, and the different compositions of the fuel blend:

[S.sub.L0] ([phi],S) = [6.summation over (j=1)][A.sub.j][[phi].sup.(j-1)] [A.sub.j] = [A.sub.j1] + [A.sub.j2] S Eq. (5)

[alpha]([phi],T,p) = [4.summation over (j=1)][[alpha].sub.j][[phi].sup.(j-1)] [[alpha].sub.j] = [[alpha].sub.j1] + [[alpha].sub.j2][T/[T.sub.0]]+[[alpha].sub.j3]] ([[p.sub.0]/p) Eq. (6)

[beta]([phi],S) =[5.summation over (j=1)][[beta].sub.j][[phi].sup.(j-1)]) [[beta].sub.1] = [[beta].sub.11] +[[beta].sub.12] (1/S) Eq. (7)

[[gamma].sub.EGR] ([phi],T) = [3.summation over (j=1)][[gamma].sub.EGR,j][[phi].sup.(j-1)]) [[gamma].sub.EGR,j] = [[gamma].sub.EGR,j1] + [[gamma].sub.EGR,j2](T/[T.sub.0]) Eq. (8)

[[delta].sub.EGR] ([phi],p) = [4.summation over (j=1)][[delta].sub.EGR,j][[phi].sup.(j-1)]) [[delta].sub.EGR,j] = [[delta].sub.EGR,j1] + [[delta].sub.EGR,j2]([p.sub.0]/p) Eq. (9)

Regarding the thermodynamic conditions of the unburned mixture, multiple temperature ranges have been considered, depending on the pressure level. In all cases the minimum temperature is 323 K, whereas the maximum value is 923 K for pressures lower than 3 bar, 873 K for pressures between 3 and 5 bar, 723 K for pressures between 5 and 10 bar, and 673 K for pressures between 10 bar and the maximum value of 20 bar. This variability of the temperature range arises from the impossibility of simulating stable flames at simultaneously high pressures and temperatures in the whole variation ranges of fuel composition, equivalence ratio, and residual gas content. However, LFS simulations are possible up to pressures and temperatures as high as 100 bar and 1000 K for near-stoichiometric air/fuel mixtures or very low EGR fractions. In these conditions, the results reported in the next section show that the present correlation is capable of predicting quite accurately the LFSs resulting from the reaction mechanism even for pressures and temperatures far beyond the maximum values for which "Cor B" was derived.

As a final remark, an interesting characteristic of the proposed correlation is that it does not require the specification of the surrogate fuel composition, but only the RON and MON of the actual gasoline. This simplifies the setup of the 1D engine simulation while properly accounting for the effects of the fuel composition.

Assessment of the LFS Correlations

Before discussing the impact of the considered LFS correlations on the combustion process under engine operations, which is also affected by turbulence, the LFSs predicted by "Cor A" and "Cor B" are compared at varying pressure, temperature, equivalence ratio, and residual gas content. The considered fuel is a commercial gasoline, surrogated in "Cor B" by a TRF having RON=95 and MON=87. Both ambient (p=1 bar, T=300 K) and engine-like conditions (p=50-100 bar, T=800-1000 K) are examined in the comparison. Figure 2, Figure 3, and Figure 4 show the effects of temperature, pressure, and equivalence ratio, respectively. It can be observed that the two correlations provide similar results only under ambient conditions, whereas the differences in their predictions increase with rising pressure and temperature. As shown in Figure 4, the LFS peak predicted by both correlations occurs for rich mixtures. Under ambient conditions, the maximum is reached at [phi] [congruent to] in both cases, but for higher pressure and temperature values, including those found in typical engine operations, the LFS peak predicted by "Cor A" shifts to higher values of [phi], whereas this does not occur for "Cor B".

Figures 2-4 report also the values of LFS resulting from Andrae's reaction mechanism for a comparison with the predictions of "Cor B". Note that the considered pressure and temperature ranges well extended beyond the nominal validity ranges of the present correlation. In most cases, an excellent agreement is found between the predictions of "Cor B" and the results of the reaction kinetics simulations, both inside and outside the nominal validity range of the correlation. Inaccurate predictions are observed in Figure 4 only for rich mixtures and high values of pressure and temperature.

Other differences between "Cor A" and "Cor B" emerge when considering their response to the variation of residual gas content, as shown in Figure 5. Both correlations predict a decrease in the LFS at increasing residual gas mass fraction. However, [EGR.sub.factor] in "Cor A" does not depend on p, T, and [phi], and it well agrees with the predictions of "Cor B" only for ambient conditions and stoichiometric mixture. For typical engine operations (high p and T, [phi] = 1), Figure 5 shows that "Cor B" predicts a smaller decrease in LFS at increasing [x.sub.r], while rich mixtures ([phi] = 1.3) cause a stronger reduction in [EGR.sub.factor] As well as the previous figures, Figure 5 reports also the results of reaction kinetics simulations. The comparison between "Cor B" predictions and reaction kinetics computations shows that the correlation applies quite well also for EGR mass fractions above 0.2. The agreement is acceptable also under engine-like operations, although a slight systematic overestimation of [EGR.sub.factor] is observed.

Flame Stretch

The considered LFS correlations provide the velocity of one-dimensional planar flames relative to the unburned mixture. In an SI engine, the flame geometry and dynamics are much more complex, mainly due to the flame stretch. This phenomenon consists in local modifications of the flame speed caused by two effects: the preferential diffusion, which results in local variations of flame temperature and burning rate (depending on the effective Lewis number), and the flow divergence, which causes an increase in the flame speed at the upstream boundary of the preheat zone with respect to the speed of the reaction zone [30, 32].

The speed of a stretched flame can be related to that of a planar flame introducing a flame-thickness-dependent coefficient that accounts for the effects of diffusion and chemical reactions in the flame zone. This coefficient, known as the Markstein length, L, depends on fuel type and reactivity, and mixture composition [27].

In an SI engine, the flame front exhibits a shape that is very different from a planar one, and hence it propagates relative to the unburned gas with a displacement speed, [S.sub.d], that differs from the planar flame speed, [S.sub.L], according to the equation:

[S.sub.d] = [S.sub.L] - LK Eq. (10)

where K is the flame stretch rate, originally defined by Karlovitz [33] as the Lagrangian time derivative of the logarithm of the flame surface area, A:

K = [d/dt] (ln A) = [1/A] [dA/dt] Eq. (11)

The flame stretch rate accounts for the effects of the curved flame motion and the velocity gradients inside the flame [29], as shown by the equation:

K = k[S.sub.L] + [K.sub.s] Eq. (12)

where [kappa] is the flame curvature and [K.sub.S] is the hydrodynamic strain. In the case of an outwardly propagating quasi-spherical flame, such as the one occurring in an SI engine, the flame curvature and strain are:

K= 2/[r.sub.f] [K.sub.s] = 2 ([sigma]- 1)[S.sub.L] /[r.sub.f] Eq. (13)

where [r.sub.f] is the flame radius and [sigma] = [[rho].sub.u]/[[rho].sub.b] = [T.sub.b]/[T.sub.u] is the thermal expansion parameter. Substituting expressions (13) into Equation 12 leads to the following form for the total stretch rate:

K = [2[sigma][S.sub.L]/[r.sub.f]] Eq. (14)

According to the description provided in [334], a real flame of finite thickness [[delta].sub.th] can be divided into a number of layers characterized by different temperatures. Each of them propagates with a different speed, which can be computed using Equation 10 once the appropriate Markstein length, changing through the flame front, is selected. Since this detailed flame description has to be synthesized here in a 0D phenomenological framework, the flame layer where the fuel oxidation is completed is chosen as representative of the modeled zero-thickness flame front. This layer, whose Markstein length is labelled as [L.sub.b], is located on the burned side of the real flame front and propagates with speed:

[S.sub.L,stretched] = [S.sub.L] - [L.sub.b]K Eq. (15)

The linear relation expressed by Equation 15 applies properly for low turbulence intensities, namely, in the corrugated flamelet regime. As the turbulence intensity increases, the relation between flame speed and stretch becomes increasingly nonlinear [35]. This behavior is not taken into account in the present paper, but it will be considered in a future development of the model. According to the asymptotic theory of Matalon et al. [29], the expression for [L.sub.b] is:

[mathematical expression not reproducible] Eq. (16)

where [[delta].sub.th] = [[lambda].sub.u]/([c.sub.pu][[rho].sub.u] [S.sub.L]) is the flame thermal thickness and [lambda] is the thermal conductivity of the mixture scaled with respect to its value in the unburned gas. Term [alpha] in Equation 16 has the expression [34]:

[mathematical expression not reproducible] Eq. (17)

where Ze = [E.sub.a] ([T.sub.b] - [T.sub.u])/[RT.sup.2.sub.b] is the Zel'dovich number, which depends on the apparent activation energy [E.sub.a], and [Le.sub.eff] is the effective Lewis number. The apparent activation energy is defined by the relationship:

[[E.sub.a]/2R] [[partial derivative] ln ([[rho].sub.u][S.sub.L])/[partial derivative](1/[T.sub.b])] Eq. (18)

where R is the universal gas constant. [E.sub.a] is expressed as a function of p, [phi], [x.sub.r], and S by means of a specially developed correlation based on 1D flame computations by CANTERA. For any given set of pressure, equivalence ratio, EGR mass fraction, and fuel sensitivity, the temperature of the unburned mixture is perturbed (it is assumed that [E.sub.a] does not depend on [T.sub.u]) and the derivative on the right-hand side of Equation 18 is computed. The values of [E.sub.a] so obtained are finally fitted by the following four variables polynomial:

[mathematical expression not reproducible] Eq. (19)

The values of the constants that minimize the interpolation error are reported in Table 3.

As far as the computation of the other parameters in Equation 17 is concerned, [Le.sub.eff] is evaluated as a function of equivalence ratio [29] and temperatures [T.sub.u] and [T.sub.b], while the thermal conductivity is assumed to vary linearly with the temperature.

In order to check the consistency of the [L.sub.b] estimation, the values provided by Equation 16 were compared with the predictions of the experimental correlation proposed by Galmiche et al. [36]. The comparison, not reported here for the sake of brevity, showed a reasonable agreement between the two approaches.

Experimental Tests

The considered engine is a naturally aspirated port fuel injection (PFI) SI engine, with a typical pent-roof architecture of the combustion chamber. Each cylinder is equipped with a centered spark-plug, a standard ignition system, and two valves. The valves are controlled by a single VVT system acting on both intake and exhaust sides. An external cooled EGR (eEGR) circuit allows recirculating the exhaust gas to the intake manifold. The recirculated gas temperature is lowered by a water-cooled heat exchanger and the amount of eEGR is adjusted by an automatically controlled throttle valve.

An extensive experimental campaign was carried out, investigating the whole operating plane of the engine. Operations with and without EGR actuation were considered. For all the analyzed operating conditions, the fuel is metered to realize a stoichiometric air-to-fuel (A/F) ratio. The spark advance is generally set to ensure engine operation at maximum brake torque. Only at low speeds and high loads it is delayed to avoid knocking combustion. No particular constraint is applied to limit the maximum in-cylinder pressure and exhaust gas temperature. The VVT device is controlled so as to maximize the torque at full load, whereas the valve timing is delayed with decreasing load to promote the internal EGR, thus reducing the need for intake throttling. This latter strategy is adopted when the external EGR device cannot ensure an adequate recirculation level, namely, for very low brake mean effective pressures (BMEPs). The considered engine speeds range from 1000 to 6500 rpm, with a maximum BMEP of about 12 bar at 3500 rpm.

The instrumentation of the test bench enabled measuring both the overall engine performance (fuel flow rate, A/F ratio, torque, fuel consumption) and the in-cylinder pressure cycles. The latter were post-processed using an inverse analysis to obtain the burning rate profiles and the characteristic combustion angles, namely, [MFB.sub.10], [MFB.sub.50], and [MFB.sub.90].

Engine Model

The tested engine has been schematized and simulated in the GT-power environment. The simulation is based on a 1D model of the unsteady flow inside the intake and exhaust pipes and a 0D model of the in-cylinder processes. The engine schematic includes cylinders, intake and exhaust pipe systems, airbox, throttle valve, and EGR circuit (composed of a valve and an EGR cooler).

The experimental A/F ratio and valve timing are used as input data to the simulations. According to the experiments, the throttle valve is fully opened for the full load analysis, while it is adjusted by a proportional integral derivative (PID) controller to match the measured BMEP for the part load calculations. The EGR valve is controlled so as to reproduce the measured external EGR rate. The spark advance is automatically modified by another controller targeting the experimental angle of 50% mass fraction burned ([MFB.sub.50]).

A refined model of the heat transfer inside the cylinder and exhaust pipes is introduced, applying a wall temperature solver based on a finite element approach. Concerning the in-cylinder heat transfer, a Woschni-like correlation is used, while convective, radiative, and conductive heat transfer modes are considered for the exhaust pipes. The combustion process is modeled using a two-zone "fractal approach" [37], where the burning rate is computed as:

[[d.sub.mb]/dt]= [[rho].sub.u][A.sub.T][S.sub.L] Eq. (20)

[[rho].sub.u] being the unburned gas density, [A.sub.T] the area of the turbulent flame front, and [S.sub.L] the stretched LFS. The LFS is calculated using both correlations "Cor A" and "Cor B", including or not the effects of the flame stretch. The properties of the commercial gasoline used in the experimental campaign are reproduced by assigning its RON and MON values in "Cor B" (95 and 87, respectively).

Based on the fractal geometry theory [39,40], the ratio of the turbulent flame area, [A.sub.T], to the laminar one, [A.sub.L], is assumed to vary according to the relation:

[mathematical expression not reproducible] Eq. (21)

where [[GAMMA].sub.max] and [[GAMMA].sub.min] are the length scales of the maximum and minimum flame wrinkling, respectively, and [D.sub.3] is the fractal dimension. Flame front area [A.sub.L] is computed using a CAD software package to model the intersection of a smooth spherically shaped surface with the actual 3D geometry of the combustion chamber. The estimation of [[GAMMA].sub.max], [[GAMMA].sub.min], and [D.sub.3] is based on the turbulence sub-model detailed in [41,42].

Tuning and Validation of the Engine Model

The first stage of the model tuning is the identification of the constants of the turbulence sub-model, according to a hierarchical 1D/3D methodology as deeply discussed in previous works by the present authors [41, 42]. The constants are adjusted by fitting the results of 3D simulations for various engine operating conditions. In a previous research, a number of 3D simulations were carried out using the commercial software STAR-CD v4.28, as described in detail in [43]. The computational domain includes the combustion chamber and the in-head intake and exhaust ports. Base and minimum cell sizes are 0.6 mm and 0.1 mm, respectively. The total number of fluid cells at the top dead center (TDC) is nearly 350k, and the addition of mesh layers at increasing cylinder volume leads to a maximum of about 700k cells at the bottom dead center. The simulations are based on a RANS approach coupled with a k-[epsilon] re-normalization group (RNG) turbulence model [44] adapted for compressible flows. The time-varying boundary conditions at the intake and exhaust ports are derived from 1D simulations under motored conditions. The tuning of the 0D turbulence model and the investigated operating conditions for the considered engine are detailed in [42], where a satisfying agreement between the 0D and mass-averaged 3D turbulence parameters is found.

Concerning the combustion model, it was tuned at full load for both correlations "Cor A" and "Cor B" to reproduce the experimental burning rates and in-cylinder pressure cycles. In the full-load calculations, the effect of the flame stretch on the LFS is not considered, since preliminary simulations showed that it had a negligible influence on the results. The combustion model includes three tuning constants [37], which independently affect the different stages of the combustion. The first constant, named [c.sub.trans], controls the transition between an initially laminar flame and a fully turbulent combustion by progressively increasing fractal dimension [D.sub.3] in Equation 21. [D.sub.3] changes over time as a function of a characteristic time scale weighted by [c.sub.trans]. The duration of the laminar-to-turbulent transition lengthens as [c.sub.trans] increases. During the transition, [D.sub.3] changes from a value representative of laminar combustion (2.0) to a value typical of fully developed flames (up to 2.35). This latter depends on the LFS and turbulence intensity, and it is estimated using the empirical correlation proposed in [38]. The second constant, [c.sub.wrk], is a multiplier of maximum wrinkling scale [[GAMMA].sub.max], and its increase causes an overall speed-up of the combustion process. The third constant, [x.sub.wc], controls the impact of the wall-combustion stage. The burning rate slowdown, typical of this stage, is mimicked by a progressive transition to a laminar flame. The weight of the wall-combustion is increased according to the ratio between the surface area of the combustion chamber wetted by the flame and the total surface area of the chamber. A higher value of [x.sub.wc] results in a faster turbulent-to-laminar transition and hence in a longer combustion tail. The combustion constants have been identified considering only full load operations over the whole variation range of the engine speed. The tuning procedure has been performed separately for each of the three combustion parameters, because they affect only locally different stages of combustion. The optimal values of these constants have been selected as the ones that produce the best fitting of experimental data of combustion core duration, [MFB.sub.10-50] ([c.sub.wrk]), initial combustion stage duration, [MFB.sub.0-10] ([c.sub.trans]), and combustion tail duration, [MFB.sub.50-90] ([x.sub.wc]). Single values of [c.sub.wrk], [c.sub.trans], and [x.sub.wc] were identified at full load as the best compromise over the entire range of engine speeds. More details about the role of these constants in the combustion model and the tuning procedure are reported in [37].

The tuning of the combustion parameters was performed separately for each of the considered LFS correlations, and the optimal values of the constants are reported in Table 4 for both "Cor A" and "Cor B". It is observed that [x.sub.wc] takes the same value for both correlations, whereas [c.sub.trans] and [c.sub.wrk] exhibit slightly different values, just to compensate the difference in the LFS predictions.

Figure 6 shows the comparison between experimental and predicted values of the characteristic combustion angles at full load and various engine speeds. The numerical values, computed using either "Cor A" or "Cor B", are observed to be in good agreement with the experimental data for both correlations. This means that they respond in a similar way to the variations in pressure, temperature, residual gas content, and equivalence ratio that typically occur at full load operations. However, some differences are detected at high engine speeds. In fact, the predictions of spark, [MFB.sub.10] and [MFB.sub.90] over 4500 rpm, resulting from the use of "Cor B", turn out to be more accurate than those obtained via "Cor A". Since the two correlations behave in a very similar way at full load, the validation of the engine model reported below refers only to the use of "Cor B". The model ability to simulate the engine breathing results from the experimental/numerical comparison of the volumetric efficiency shown in Figure 7. Moreover, the good agreement between predicted and measured BMEPs (Figure 8) and in-cylinder pressure peaks (Figure 9) shows that the model is capable of reproducing properly the combustion process and the wall heat transfer.

A further insight into the model reliability is provided by the numerical/experimental comparisons of the in-cylinder pressure cycles and burning rates at three different engine speeds reported in Figures 10 to 12. A satisfactory agreement is found for all the considered speeds, despite the significant variations in BMEP, in-cylinder pressure, temperature, EGR rate, and turbulence intensity.

Only at 1500 rpm the model predictions are somewhat inaccurate, probably due to a slight overestimation of heat transfer or turbulence intensity. Summarizing, all these results prove that the proposed approach is capable of reproducing the engine operations at full load, without the need of any case-dependent tuning and regardless of the adopted LFS correlation.

Model Assessment at Part Load

After ascertaining the prediction capabilities of the model at full load, it is used to compute the engine performance and combustion parameters over its whole operating range. The two LFS correlations are compared once again to assess their relative impact on the combustion simulation under part load conditions. Coherently with the full load calculations, the effect of the flame stretch on the LFS is not taken into account at this stage. The present assessment includes operations with and without the application of an external EGR. Note that all the following results are obtained without modifying the values of the model tuning constants identified at full load operations. Figure 13a shows the contour plot in the engine operating plane of the percent difference between experimental and numerical brake-specific fuel consumptions (BSFCs), computed according to the definition below:

[DELTA]BSFC%=[[BSFC.sub.exp] - [BSFC.sub.model]/[BSFC.sub.exp]]*100 Eq. (22)

The contour plot of the measured BSFC is shown in Figure 13b for completeness. Both pictures refer to EGR-free operating conditions and to simulations performed using only "Cor B", as the Metghalchi correlation provides very similar BSFC levels. The error map in Figure 13a shows a satisfactory global accuracy of the BSFC predictions, which mainly depends on the good reproduction of engine breathing, combustion evolution, and wall heat transfer also under part load operations. The largest errors are observed at medium loads and low speeds, where the BSFC is overestimated by up to 5%, and at high speeds, where the BSFC is underestimated by up to 6%. These disagreements are probably due to an over-estimation of the heat transfer at low engine speeds and an underestimation of the pumping work at high speeds.

A similar comparison is reported in Figure 14 for the case of activated eEGR circuit. The experimental mass percentages of recirculated gas are shown in Figure 14c. Looking at the latter, it can be noted that the highest eEGR rates are applied at low speeds and medium-high loads, resulting in a maximum BSFC reduction of 10 g/kWh, that is, about 4% (compare Figure 13b with Figure 14b). At increasing engine speed, the EGR rate is reduced because of the unfavorable pressure gradient between the intake and exhaust pipes. After testing the overall accuracy of the model also at part load, the relative impact of two LFS correlations is assessed with reference to the prediction of the characteristic combustion durations [MFB.sub.10-50] and [MFB.sub.0-10]. The maps in Figure 15a and Figure 15b show the angular differences between experimental and predicted [MFB.sub.10-50] for "Cor A" and "Cor B", respectively, and for EGR-free operations. Similar data are plotted in Figure 16a and Figure 16b for the case with eEGR. For completeness, Figure 15c and Figure 16c display the values of [MFB.sub.10-50] resulting from the measurements. As a first consideration, the comparison of the experimental combustion durations with and without eEGR (Figure 15c and Figure 16c) shows that the exhaust gas recirculation causes, as expected, some increase in the combustion duration. This effect is more pronounced for the highest eEGR percentages, namely, at low speeds and low-to-medium loads (Figure 14c).

Looking at the results for "Cor A" in Figure 15a and Figure 16a, the combustion duration is correctly predicted in most of the engine operating plane. However, at decreasing BMEP, the model overestimates the [MFB.sub.10-50] duration, especially for eEGR operations, where a maximum error of 4 CADs is obtained at medium/high speeds and low load. When using "Cor B" (Figure 15b and Figure 16b), the prediction errors decrease, being mostly in the range [+ or -] 1 CAD. In summary, both LFS correlations prove to be sufficiently accurate over the whole engine operating plane, although "Cor B" leads to better predictions in the most critical conditions (high eEGR rates and very low engine loads). Moving to the analysis of the combustion process first stage, the simulated durations from the spark event to the [MFB.sub.10] angle are compared with the experimental data in Figure 17 and Figure 18, for operations without and with eEGR, respectively. As expected, the measurements show that the combustion duration significantly lengthens at decreasing BMEP (Figure 17c), especially in the presence of exhaust gas recirculation (Figure 18c). The differences between experimental and numerical data in Figure 17a and Figure 17b, which refer to eEGR-free operations, show that the model provides accurate predictions only for load levels over 6 bar BMEP, whereas it underestimates the combustion duration for lower engine loads. For very low BMEPs, errors as high as 22 CADs are obtained. In the presence of eEGR (Figure 18a and b), the model exhibits a similar behavior, but a significant underestimation of [MFB.sub.0-10] is also observed in the map region with the highest EGR rates, namely, at medium-high loads and low speeds.

The analysis of the characteristic combustion durations leads to two main outcomes:

* The model predicts the combustion core ([MFB.sub.10-50]) with good accuracy for both LFS correlations. Better results are provided by "Cor B".

* The model underpredicts the duration of the initial combustion stage ([MFB.sub.0-50]), with higher errors at decreasing engine load and increasing eEGR rate. The LFS correlations behave similarly to each other.

Based on the above observations, a model refinement is required to improve the [MFB.sub.0-50] prediction at low loads. This issue is addressed by introducing the stretch effect in the LFS calculation, as discussed in the next section. In the following analysis, only "Cor B" is considered, since this correlation showed to provide a more accurate description of the combustion core.

Impact of Flame Stretch on the Model Predictions

Accounting for the flame stretch effect in the LFS calculation is expected to cause a reduction in the combustion speed when the flame radius is small and consequently the stretch rate is highest. To clarify the influence of the flame stretch, Figure 19 shows the comparison between the stretched and unstretched LFSs at varying crank angle for different BMEPs. It can be noted that the influence of the stretch is strongest at the combustion start and it becomes vanishingly small as the combustion proceeds. In addition, Figure 19 highlights that the stretch effect is stronger at very low loads, where it is expected to improve substantially the numerical estimation of the combustion duration.

To assess a possible improvement of the model formulation, the calculations over the whole engine operating plane are repeated considering the flame stretch effect. The resulting maps of [MFB.sub.0-10] are shown in Figure 20a for EGR-free operations and in Figure 20b for the case with eEGR. Comparing these results with those obtained ignoring the flame stretch (Figure 17b and Figure 18b, respectively), it is observed that significant improvements arise for loads lower than 4 bar BMEP. Except for very low engine speeds, the model continues to underestimate the [MFB.sub.0-10] duration, with errors smaller than 6-8 CADs in most cases (see Figure 20a and b). However, these results are considerably better than the ones obtained ignoring the stretch effect, which exhibit errors of 12-20 CADs at low engine loads (see Figure 17b and Figure 18b). The predicted maps of [MFB.sub.10-50] (Figure 15b and Figure 16b) and [MFB.sub.0-10] (Figure 20a and b) show that the present model is able to reproduce quite accurately the variations in in-cylinder pressure and temperature, equivalence ratio, turbulence intensity, and residual gas content over most of the engine operating plane. However, some inaccuracies emerge in the prediction of the eEGR influence on the early stage of the combustion process, as highlighted by the comparison of Figure 20a and b. In fact, the latter shows errors of 6-8 CADs in the map region of higher eEGR (medium/high loads), whereas in the same operating conditions without eEGR (Figure 20a), the error drops to 0-2 CADs. This is probably due to a poor description of the ignition process, which is not directly modeled. This problem could be overcome by introducing a dedicated model, such as the ones proposed in [45,46]. However, these models require additional information concerning, for instance, the spark discharge duration and power, which are not directly available in most practical applications. Despite these difficulties, an ignition model will be investigated in future developments of the present work.

As a final check on the applicability of the present flame speed and stretch models, the combustion regimes in all the operating conditions experienced by the engine (both with and without eEGR) are reported in the Borghi diagrams shown in Figure 21. The plots depict the ratios of characteristic velocities (turbulence intensity, u', over LFS, [S.sub.L]) and lengths (turbulence integral length scale, [L.sub.t], over flame thickness, [[delta].sub.f]) halfway through the combustion process ([MFB.sub.50] events). It can be observed that most points lie within the corrugated flamelet regime, where the assumptions of the present models are valid. A few points fall close or just within the thin reaction regime, and they correspond to high engine speeds (large values of u') and/or large residual gas fractions (small values of [S.sub.L]). Under such borderline operating conditions, where the combustion regime is expected to change, the predicted combustion durations, especially [MFB.sub.10-50], continue to agree well with the experimental data.

To test the robustness of the proposed model, it was applied to the simulation of the operating map of a different engine, widely investigated by the authors in previous works [47, 48]. This is a downsized twin-cylinder turbo-charged SI engine, equipped with a variable valve actuation (VVA) system on the intake side. The combustion model was tuned at full load and the identified set of tuning constants was applied to the simulation of all the engine operating conditions. The results obtained are plotted in Figure 22 and Figure 23, which show the maps of experimental [MFB.sub.10-50] and [MFB.sub.0-10] (Figure 22a and Figure 23a), and the corresponding prediction errors (Figure 22b and Figure 23b). As highlighted in Figure 22a, the combustion core duration, [MFB.sub.10-50], does not change significantly with engine BMEP and speed. The model is able to capture this behavior (see Figure 22b) mainly thanks to a correct estimation of the impact of the in-cylinder thermodynamic state and turbulence on the combustion process. The duration of the early combustion stage, [MFB.sub.0-10], lengthens at increasing engine speed and decreasing load (Figure 23a). The simulation proves to reproduce quite well this behavior, although an ever-larger underestimation of [MFB.sub.0-10] with increasing combustion duration is observed in Figure 23b.

Additional insight into the model reliability is provided by the results in Figure 24, where experimental/numerical comparisons of burn rates are shown for this second engine at 3000 rpm and different loads (4.5, 7.0, and 12.5 bar BMEP), with and without eEGR activation. The model shows to be sensitive to the burning rate slowdown due to the exhaust gas recirculation, although to a slightly lesser extent compared to experimental data.

As a final remark, the good predictive capability of the present model is not an obvious achievement, considering the relevant variations in the in-cylinder pressure and temperature, and internal EGR over the whole operating plane of the engine, and the use of a single set of tuning constants.

Conclusions

This article presents a novel LFS sub-model aimed at improving the predictions of a phenomenological combustion model for spark ignition engines.

In order to obtain experimental data to be compared with the model results, a small naturally aspirated SI engine has been fully characterized at the test bench, measuring its global performance parameters and the in-cylinder pressure cycles. Tests have been carried out over the whole operating plane of the engine, with and without the activation of an external EGR circuit.

A 1D model of the considered engine has been developed and supplemented with refined in-cylinder sub-models of turbulence, combustion, and heat transfer. In particular, two LFS correlations have been considered. The first one derives from the historical correlation by Metghalchi and Keck, which is based on experimental data, while the second one is presented here for the first time and is based on the computation of 1D LFSs via a chemical kinetics solver. The effect of the flame stretch is also taken into account by using an appropriate formulation from an asymptotic theory and by introducing a novel correlation for the apparent activation energy used to compute the Markstein length.

The model has been tuned and validated at full load operation, and predictions of comparable accuracy have been obtained by using either of the LFS correlations. Then, the model has been applied to the simulation of the engine operating conditions over the whole speed/BMEP plane. The comparison between numerical and experimental data shows that the proposed LFS correlation provides a better estimation of the combustion core duration ([MFB.sub.10-50]), whereas a poor simulation of the early combustion stage ([MFB.sub.0-10]) at low engine loads is obtained for both LFS correlations when ignoring the flame stretch effect. The introduction of an appropriate flame stretch model leads to much better predictions at part loads, especially regarding the duration of the initial combustion stage. The effectiveness and robustness of the proposed approach have also been verified in the simulation of a different engine.

In summary, the main contribution of this work is the introduction of a novel correlation for the LFS, which has proven to improve the predictions of a phenomenological combustion model for SI engines, especially during the core stage of the burning process. In addition, a refined formulation for the flame stretch has been presented, which improves the simulation of the initial combustion stage at part load. As a next step of this research activity, an ignition model will be implemented with the aim of describing the early combustion stage on a more physical basis.

Contact information

V. De Bellis, Researcher

vincenzo.debellis@unina.it

E. Malfi, PhD Candidate

enrica.malfi@unina.it

L. Teodosio, Research Fellow luigi.teodosio@unina.it

University of Naples "Federico II" Naples, Italy +39-081-7683264

P. Giannattasio, Full Professor pietro.giannattasio@uniud.it

F. Di Lenarda, MScEng fdl.coder@gmail.com

University of Udine Udine, Italy

+39-0432-558012

Acronyms

0D/1D/3D - Zero/One/Three-dimensional

A/F - Air-to-fuel ratio

AFTDC - After firing top dead center

BMEP - Brake mean effective pressure

BSFC - Brake-specific fuel consumption

CAD - Crank angle degree/Computer aided design

EGR - Exhaust gas recirculation

LFS - Laminar flame speed

MFB - Mass fraction burned

MON - Motor octane number

PFI - Port fuel iniection

PID - Proportional integral derivative controller

RANS - Reynolds-averaged Navier-Stokes

RNG - Re-normalization group

RON - Research octane number

SI - Spark ignition

TDC - Top dead center

TRF - Toluene reference fuel

VVA/VVT - Variable valve actuation/timing

Symbols

A - Flame front area

[B.sub.m], [B.sub.[phi]] - Metghalchi correlation coefficients

[c.sub.p] - Specific heat at constant pressure

[c.sub.trans] - Laminar-turbulent transition multiplier

[c.sub.wrk] - Flame wrinkling multiplier

[D.sub.3] - Flame front fractal dimension

[E.sub.a] - Apparent activation energy

K- Flame stretch rate

[K.sub.s] - Hydrodynamic strain

L - Markstein length

[L.sub.t] - Turbulence integral length scale

m - Mass

p - Pressure

[r.sub.f] - Flame radius

R - Universal gas constant

[S.sub.L] - Planar laminar flame speed

[S.sub.d] - Flame displacement speed

t - time

T - Temperature

u' - Turbulence intensity

[x.sub.r], [x.sub.r] - Residual mass/molar fraction

[x.sub.wc] - Wall-combustion tuning constant

Ze - Zel'dovich number

Greeks

[alpha] - Temperature exponent in [S.sub.L] correlation

[beta] - Pressure exponent in [S.sub.L] correlation

[gamma] - Residuals exponent in [S.sub.L] correlation

[[GAMMA].sub.max] - Maximum scale of flame front wrinkling

[[GAMMA].sub.min] - Minimum scale of flame front wrinkling

[delta] - Residuals multiplier in [S.sub.L] correlation

[[delta].sub.f] - Flame front thickness

[[delta].sub.th] - Flame front thermal thickness

[kappa] - Flame curvature

[lambda] - Thermal conductivity

[nu] - Kinematic viscosity

[phi], [[phi].sub.m] - Air-to-fuel equivalence ratio

[rho] - Density

[sigma] - Thermal expansion parameter

Subscripts

0 - Reference condition

0/10/50/90 - Referred to 0/10/50/90% of MFB

b - Burned

exp - Experimental

L - Laminar

T - Turbulent

u - Unburned

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Vincenzo De Bellis, Enrica Malfi, and Luigi Teodosio, University of Naples Federico II, Italy

Pietro Giannattasio and Fabio Di Lenarda, University of Udine, Italy

History

Received: 30 Dec 2018

Revised: 28 Feb 2019

Accepted: 27 Mar 2019

e-Available: 25 Apr 2019

Keywords

0D combustion model, Fractal combustion model, Laminar flame speed correlation, Flame stretch, EGR

Citation

De Bellis, V., Malfi, E., Teodosio, L., Giannattasio, P. et al., "A Novel Laminar Flame Speed Correlation for the Refinement of the Flame Front Description in a Phenomenological Combustion Model for Spark-Ignition Engines," SAE Int. J. Engines 12(3):251-270, 2019,

doi:10.4271/03-12-03-0018.

TABLE 1 Coefficients and reference conditions for correlation "Cor A". Coefficients [B.sub.m] [m/s] 0.35 [[beta].sub.0] -0.357 [B.sub.[phi]] [m/s] -0.549 [[beta].sub.1] 0.14 [[phi].sub.m] 1.1 [[beta].sub.2] 2.77 [[alpha].sub.0] 2.4 [gamma] 2.06 [[alpha].sub.1] -0.27 [delta] 0.77 [[alpha].sub.2] 3.51 Reference conditions [T.sub.0] [K] 298 [p.sub.0] [bar] 1.013 TABLE 2 Coefficients and reference conditions for correlation "Cor B". [mathematical expression not reproducible] Coeffiicients [A.sub.11] -2.3832 [A.sub.12] 0.1877 [A.sub.21] 15.636 [A.sub.22] -1.1768 [A.sub.31] -40.547 [A.sub.32] 2.8624 [alpha] Coefficients [[alpha].sub.11] 6.2174 [[alpha].sub.11] 0.1307 [[alpha].sub.13] -0.8983 [[alpha].sub.21] -13.125 [[alpha].sub.22] 1.0024 [[alpha].sub.23] 3.8909 [beta] Coefficients [[beta].sub.11] 0.8055 [[beta].sub.12] 0.1295 [[beta].sub.2] -5.2798 EGR Coefficients [[gamma].sub.EGR,11] 6.0973 [[gamma].sub.EGR,12] -2.1178 [[gamma].sub.EGR,21] -10.696 [[gamma].sub.EGR,22] 4.3429 [[gamma].sub.EGR,31] 6.8101 [[gamma].sub.EGR,32] -2.6495 [[delta].sub.EGR,11] -2.4850 Reference conditions [T.sub.0] [K] 323 [mathematical expression not reproducible] Coeffiicients [A.sub.11] [A.sub.41] [A.sub.12] [A.sub.42] [A.sub.21] [A.sub.51] [A.sub.22] [A.sub.52] [A.sub.31] [A.sub.61] [A.sub.32] [A.sub.62] [alpha] Coefficients [[alpha].sub.11] [[alpha].sub.31] [[alpha].sub.11] [[alpha].sub.32] [[alpha].sub.13] [[alpha].sub.33] [[alpha].sub.21] [[alpha].sub.41] [[alpha].sub.22] [[alpha].sub.42] [[alpha].sub.23] [[alpha].sub.43] [beta] Coefficients [[beta].sub.11] [[beta].sub.3] [[beta].sub.12] [[beta].sub.4] [[beta].sub.2] [[beta].sub.5] EGR Coefficients [[gamma].sub.EGR,11] [[delta].sub.EGR,12] [[gamma].sub.EGR,12] [[delta].sub.EGR,21] [[gamma].sub.EGR,21] [[delta].sub.EGR,22] [[gamma].sub.EGR,22] [[delta].sub.EGR,31] [[gamma].sub.EGR,31] [[delta].sub.EGR,32] [[gamma].sub.EGR,32] [[delta].sub.EGR,41] [[delta].sub.EGR,11] [[delta].sub.EGR,42] Reference conditions [T.sub.0] [K] [p.sub.0] [bar] [mathematical expression not reproducible] Coeffiicients [A.sub.11] [A.sub.41] [A.sub.12] [A.sub.42] [A.sub.21] [A.sub.51] [A.sub.22] [A.sub.52] [A.sub.31] [A.sub.61] [A.sub.32] [A.sub.62] [alpha] Coefficients [[alpha].sub.11] [[alpha].sub.31] [[alpha].sub.11] [[alpha].sub.32] [[alpha].sub.13] [[alpha].sub.33] [[alpha].sub.21] [[alpha].sub.41] [[alpha].sub.22] [[alpha].sub.42] [[alpha].sub.23] [[alpha].sub.43] [beta] Coefficients [[beta].sub.11] [[beta].sub.3] [[beta].sub.12] [[beta].sub.4] [[beta].sub.2] [[beta].sub.5] EGR Coefficients [[gamma].sub.EGR,11] [[delta].sub.EGR,12] [[gamma].sub.EGR,12] [[delta].sub.EGR,21] [[gamma].sub.EGR,21] [[delta].sub.EGR,22] [[gamma].sub.EGR,22] [[delta].sub.EGR,31] [[gamma].sub.EGR,31] [[delta].sub.EGR,32] [[gamma].sub.EGR,32] [[delta].sub.EGR,41] [[delta].sub.EGR,11] [[delta].sub.EGR,42] Reference conditions [T.sub.0] [K] [p.sub.0] [bar] TABLE 3 Coefficients of the correlation for the apparent activation energy. [A.sub.E,j1] [MJ/kmol] [A.sub.E,j2] [MJ/kmol] [A.sub.E,11] 1919.9 [A.sub.E,12] -20.666 [A.sub.E,21] -8705.1 [A.sub.E,22] 128.99 [A.sub.E,31] 15554 [A.sub.E,32] -269.09 [A.sub.E,51] -11343 [A.sub.E,42] 222.81 [A.sub.E,51] 2911.2 [A.sub.E,52] -63.394 EGR coefficient Sensitivity coefficient [[gamma].sub.E,EGR] 0.4615 [[gamma].sub.E,S] 0.0087 TABLE 4 Values of the tuning constants of the combustion model. Cor A Cor B [c.sub.trans] 1.2 1.1 [c.sub.wrk] 0.58 0.7 [X.sub.wc] 0.5 0.5