Multi-Zone Quasi-Dimensional Combustion Models for Spark-Ignition Engines

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2013-24-0025

Multi-zone quasi-dimensional combustion models for Spark-Ignition engines

Leslie Kaprielian1, Marc Demoulin1, Paola Cinnella2, Virginie Daru2 1Renault SAS, 2Arts et Métiers ParisTech

Copyright © 2012 SAE International

ABSTRACT

The present work aims at improving the predictive capabilities of quasi-dimensional combustion models for fast and accurate automated design of spark engines. The models are based on mass and energy conservation principles supplemented by sub models based on experimental correlations. Here, we improve the accuracy of the classical two-zone model by means of two successive modifications. First, we generate a three-zone model by introducing a reacting zone near the walls. In the third zone, the gases burn at a lower temperature than in the main reacting zone, due to heat losses to the walls. Secondly, a multi-zone model is built by dynamically adding new reacting zones at given crank-angle intervals. The use of multiple zones allows to take into account temperature and concentrations gradients in the flame. To validate our models, the energy release rates and pressures time histories predicted by the three-zone and by the multi-zone models are compared to experimental data and to the standard two-zone approach for several operating conditions.

INTRODUCTION

The current challenge for the automotive industry is to build always more efficient but less pollutant internal combustion engines. This requires an optimization of the combustion process. Several technological solutions have been proposed to reach this goal: downsizing, variable compression ratio and variable valve actuation. In order to validate these technological solutions, efficient simulation tools are needed in early design stages to evaluate engine cycle performance. First developed in the 1970, the quasi-dimensional models are based on the use of mass and energy conservation equations, integrated in time and applied to pre-defined control volumes (or zones) of the combustion chamber. To describe mass and energy transfer in each zone and between one zone and another, sub models often based on experimental correlations are adopted. Although not as accurate as more complete three-dimensional models based on the solution of the Navier-Stokes equations in the combustion chamber, these quasi-dimensional models have the advantage of an extremely low computational cost, which enables fast simulations of a full

engine cycle with few design data and coupling with automated optimization tools. This is why these cheaper models are recently coming back to interest the engine community, that sees an opportunity of replacing time-consuming tridimensional models at least at the earliest stages of the design process. On the other hand, the accuracy of tridimensional models is very much dependent on the combustion model used, which is often of poor quality, especially concerning the combustion near the walls [1]. However, in order to apply with quite enough confidence quasi-dimensional models to engine design, some of their main drawbacks need to be improved. First of all, the hypothesis that gas mixture concentrations and thermodynamic variables like temperature are uniform in each zone causes a major loss of predictivity. Furthermore, the predictivity of these models depends strongly on the particular way of choosing chamber subzones, on the sub-models in use, and on their calibration from available experimental data.

The two-zone combustion model is the most used in the industry and the most often described in the literature [2] [3] [4] [5] [6]. The combustion chamber is divided in two zones, a fresh gas zone and a burned gas zone, separated by a thin flame front. The temperature and the chemical composition are homogeneous in each zone. In order to take into account the gradients of temperature and chemical species concentration in the chamber, multi-zone models were introduced. First, a three-zone model has been proposed by Heywood [1] [7]: the chamber is divided in three zones: a fresh gas zone, an adiabatic core of burned gas and a thermal boundary layer of burned gas with an imposed temperature profile. This thermal law takes into account the temperature gradient between the burned gas zone and the walls. The burned mass rate is approximated by a Wiebe function [8]. The introduction of the third zone improves predictions of NO emissions but the attenuation effects of the reaction rate of combustion by the thermal boundary layer are not modeled in a physical way. To solve this problem, Heywood has also proposed a two-zone combustion model, in which the burned mass rate is modeled by two different laws depending on the flame position in the chamber [1]. A turbulent entrainment model by Keck and al. [9] is used to quantify the created burned gases. When the flame is near the engine’s walls, an

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attenuation exponential term is added to take into account walls effects on the flame. More recently, Boiarduc and al. [10] have pointed out significant errors on the heat release rate starting when half of gases is burned. The correct modeling of the final phase of combustion is a major difficulty. In the aim to improve the pollutant calculation, Tinaut and al. [11] have developed a combustion model in which the chamber is divided in seven zones: a fresh gas zone, a burned gas zone and five zones in which hydrocarbons can be found (crevices, crankcase zone...). The inconvenient is that the attenuation of combustion by walls is not modeled. Furthermore, in all models named above, the gradients of temperature and species concentrations in the combustion flame are not taken into account. Rakopoulos and al. [12] [13] have developed a multi-zone model in which burned gas zones independent of each other are added at given crank-angle intervals. However, they don’t take into account the flame quenching and the mix between chemical components at the interface of burned gas zones. Hvezda [14] has developed a three-zone combustion model, in which mass flows between the two burned gas zones are modeled. They are approximated with a statistic turbulent flame velocity model supplemented by a corrective coefficient.

In this paper, we propose a multi-zone model that aims to improve the calculation of cylinder temperature and, consequently, improve predictions of heat release rate, pollutants emissions and knock models. This multi-zone model takes into account the temperature and concentration gradients in flame. To this purpose, the classical two-zone model is initially transformed into a three-zone model: this includes a fresh gas zone, a core of reacting zone and a reacting zone near the walls. In the last zone, gases burn more slowly than gases in the reacting core. We don’t impose an arbitrary law for the temperature. Starting with this three-zone model, we successively add dynamically new reacting zones, which interact one with each other. The proposed models are validated by comparing heat release rates and pressure distributions predicted by the two-zone, the three-zone and the multi-zone models against experimental data.

THE THREE-ZONE MODEL

Classical quasi-dimensional models consider a combustion chamber divided in two zones by a flame front. The first zone is an admitted zone containing fresh gases (air), fuel and burned gases (exhaust gas recirculation EGR and residual burned gases RBG) with mass , , respectively. The second zone is a reacting zone containing burning fresh gases and fuel as well as resulting burned gases with masses

, , respectively. A physical model is used to calculate the burned mass rate, which depends on the quantity of reacting species and on the temperature.

In the present three-zone model, a new reacting zone is added near the walls. The aim is to take into account the temperature gradient between the middle of the chamber and the area near

the walls. Similarly to the main reacting zone, this secondary zone contains burning fresh gases and fuel, as well as burned gases, with mass , , respectively. The temperature in this zone may be different from that of the main reacting zone: namely, if there are heat losses at the walls, the temperature of the reacting zone near the walls is lower than that of the main one and the gases burn more slowly. We have defined the right boundary conditions to ensure that the model is relevant even when heat losses to the walls are not simulated.

Figure 1. Schematic of the three zones in the combustion chamber.

Each zone is uniform in temperature and in composition. The pressure is homogeneous in the chamber. Each zone is defined by its mean thermodynamic properties. The enthalpy and the specific heat are approximated through polynomials depending on the temperature of the considered zone. They interpolate data from the JANAF tables [1].

Governing equations

The equations are presented in their general form during the combustion process.

In the equations below, the index k can take three values: GF for the fresh gases, GB for the burned gases and Fu for fuel. The index i denotes the number of the zone (i=1, i=2 or i=3).

First, we write the mass conservation equations:

In the admitted zone:

(1)

where is the total entrained mass by the flame front.

In the two reacting zones:

(2)

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(3)

(4)

In the above equations, is the entrained mass of the component k in the reacting zone i. The terms , define the burned mass of fresh gases and fuel. The mass of burned products is the quantity .

Second, we apply the first principle of Thermodynamics to each zone:

∑ (5)

∑ (6)

∑ (7)

where is the internal energy, is the pressure and is the

volume on the i-th zone, so that represents the work of

pressure forces, represents heat losses at solid walls. The

term ∑ in equation (5) defines the energy lost in the

admitted zone due to entrainment. The terms ∑ and

∑ in equations (6) and (7) define the energies

added in the two reacting zones, due to the coming of entrained gases. The energetic transfers between the two

reacting zones are quantified by the term . The term

ensures that the model is relevant even when heat losses to the walls are not simulated: the equality of the reacting zones temperatures is imposed when walls are adiabatic. Energy terms are evaluated using suitable sub-models, described later.

Finally, we write the perfect gases law in each zone and the volume conservation law:

(8)

(9)

In the above equations, , , are respectively the temperature, the volume and the total mass of the zone i. The cylinder volume is known function of time, depending on the engine geometry and the rotation speed.

We give in Appendix A the final formulation of the differential equations system. The system is numerically integrated to give the pressure in the cylinder, the temperature of each zone, the concentration of components in each zone and the volume occupied by each zone.

Sub models

Burned Mass Rate

We use a turbulent entrainment model by Keck and al. [9] [15] to calculate the burned mass rate. The combustion is described in tow steps: in the first step, the presence of fresh gases and of fuel initiates the combustion. In the second step, the combustion reaction takes place during a certain time.

The mass rate of the gas component k entrained by the flame front is:

(10)

In the preceding equation, is the flame speed, is the mean flame surface, and is the gas density of the gas component k.

Equation (10) gives the mass coming in the total reacting zone. In our model, there are two reacting zones: a reacting core and a zone near the walls. The entrained mass is distributed between these two reacting zones through a distribution ratio, which evolves in time because of flame propagation. Combustion attenuation due to heat losses at the walls increases when the flame propagates, because of the greater amount of reacting gases in contact with the walls.

We define the repartition ratio as:

(11)

where is the initially admitted mass of component k. The ratio is equal to 1 when there is no more admitted zone.

The entrained mass in the zone near the walls is defined by:

(12)

The mass rate of the gas component k entrained in the core of reacting zone is:

(13)

This rate decreases with the flame propagation.

The burned mass of fresh gases and fuel in the two reacting zones is respectively:

(14)

(15)

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The reacting mass of fresh gases (fuel) is and is the combustion characteristic time in the reacting zone i.

The production of burned gases is also quantified by:

(16)

Flame surface

The mean flame surface is calculated by a geometrical flame model. The flame is defined like a sphere centered on the sparkplug. The flame radius is first calculated as a function of piston height and burned gases volume. Then, the surface of a half-sphere truncated by walls is given by a function depending on piston height and flame radius.

Combustion characteristic time

The combustion characteristic time is the ratio between an integral scale length and the laminar flame speed (i=2 or 3).

(17)

Scale is often computed as a function of the cylinder volume [10]:

. / (18)

The model used to estimate is given later.

Finally, is an empirical closure coefficient that should be calibrated from experiments.

Laminar speed

The laminar speed is derived from experimental measurements carried out by Metghalchi and Keck in a pressure vessel [16]:

. (19)

where is the temperature of the zone i, and is the cylinder pressure.

For the considered study:

0.305 0.549 1.21

2.18 0.8 1

0.16 0.22 1

298

1.013

where is the fuel air equivalence ratio, is the initially admitted mass of burned gases, is the initially admitted mass, is the dilution rate.

Flame speed

For computing the flame speed, we use a classical statistical model [17] [18]. The flame speed is the sum of an average velocity and a fluctuation term ′:

′ (20)

where is a calibration coefficient. The laminar speed is calculated as seen previously, with the temperature of the admitted zone .

The turbulent intensity ′ is derived from a zero dimensional turbulence model [19]. This model is based on energy balances during the combustion phase.

(21)

(22)

In the preceding equations, is the total mass in the cylinder, is the kinetic energy of the mean flow field, is the kinetic

energy of the turbulent flow field and /

is the

dissipation rate of the turbulent kinetic energy.

The production of turbulent energy is quantified as:

0.3307 .

In this work, the model doesn’t take into account turbulence production by tumble and swirl mechanisms.

Heat losses

Heat losses at the walls are for the most due to convection and are estimated through the model by Woschni [20] [21]. This model gives an empirical expression of the convection coefficient :

. . . . (23)

In the above equation, is the engine bore, is a calibration constant taken equal to 110, is the temperature of the i-th zone, is the cylinder pressure, is the velocity of gases.

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The aerodynamic of gases depends on the piston dynamics and on a velocity scale characteristic of combustion:

. . (24)

where is the mean piston velocity, and is a velocity term calculated from pressure changes due to the combustion.

(25)

where is the displaced volume, , , are respectively the pressure, the temperature and the gases volume at the beginning of combustion, and is the motored cylinder pressure.

Finally, heat losses due to convection in zone i are given by:

(26)

where is the contact surface between the gases and the wall, and is a mean wall temperature. In our model, we distinguish three different walls: the piston head, the cylinder head and the liner. We calculate the contact surface for each wall using information about the mean flame surface. The temperature of each wall is fixed as a model input and is derived from experimental measurements. Databases provide wall temperatures for any operating conditions of a given engine.

Energy transfers between the two reacting zones

We suppose that energy transfers between the two reacting zones are for the most due to convection. They are approximated through a model by Woschni described earlier, supplemented by a corrective coefficient .

(27)

where (respectively ) is the temperature of the core of the reacting zone (respectively the reacting zone near the walls),

is a convection coefficient calculated as seen previously, with the temperature . We calculate the surface

using information about the mean flame surface.

Boundary condition between the two reacting zones

We impose as boundary condition that the two reacting zones have the same temperature when heat losses at the walls are not simulated.

The term is calculated by setting:

We have also:

∑

∑ ) ∑

∑ ) (28)

where and are respectively the total number of moles and the mean specific heat of the core of the reacting zone (the reacting zone near walls). The terms , are respectively the internal energy of the core (zone near walls) and the enthalpy of the admitted zone. The molecular weight of the component k is defined by .

THE N-ZONE MODEL

To take into account temperature and concentration gradients inside the chamber, additional zones are introduced dynamically at given crank-angle (CA) intervals. ‘N’ reacting zones exist. Each reacting zone is in turn composed by two sub-zones: a warmer core and a cooler zone near the walls.

Figure 2. Schematic of the N zones in the combustion chamber.

The admitted zone contains fresh gases, fuel and burned gases, with masses , ,, respectively, and is characterized by its temperature and its volume .The core of the j-th reacting zone contains masses

, , , , , of the same species, and is

characterized by a temperature 2, and volume 2, . The j-th near-wall reacting zone contains masses

, , , , , , and has temperature , and the volume , . The pressure is uniform throughout the chamber. Thermodynamic properties are calculated from JANAF tables.

Governing equations

The basic principle is the same used in the three-zone model. In the equations below, the index k can take three values: GF for the fresh gases, GB for the burned gases and Fu for fuel.

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The index j denotes the number of the reacting zone (j=1,…, N). The index i can take two values: 2 for the core of the reacting zone and 3 for the reacting zone near the walls.

First, we write the mass conservation principle in the admitted zone, in the core of reacting zone j and in the reacting zone near the walls j (j=1,…, N-1). In these reacting zones, there is no more entrainment, but only combustion.

(29)

, , (30)

, , (31)

, , (32)

In the preceding equations, is the total entrained mass, , , , , , , ) are respectively the burned mass of fresh gases and fuel in the core of the reacting zone (the reacting zone near the walls) j. The resulting burned mass in this zone is , , .

In the latest reacting zones, there is entrainment of admitted gases and combustion.

, , (33)

, , (34)

, , (35)

In the above equations, (respectively, ) is the entrained mass of the component k in the core of the reacting zone (respectively in the near-wall reacting zone).

Second, we apply the first principle of Thermodynamics to each zone:

∑ (36)

For the N-1 already existing reacting zones:

, , , , (37)

, , , , , (38)

For the latest reacting zones:

, , ∑ , , (39)

, , , ∑ , ,

(40)

where 2, respectively, 3, is the internal energy of the j-

th core reacting zone (respectively, the near-wall reacting

zone), , (respectively, , is the pressure forces

work, , is heat losses in the reacting zone near the walls

j, is heat losses in the admitted zone. The term

∑ in the equation (36) is the energy lost in the

admitted zone due to entrainment. The terms ∑ and

∑ in the equations (39) and (40) define the energies

added in the latest reacting zones. The energetic transfers between the reacting zone near the walls j and the core of the

reacting zone j are quantified by the term , . The term , ensures that the model is relevant even when heat losses

to the walls are not simulated by imposing the temperatures equality of the core of the reacting zone and near-walls reacting zone j when walls are adiabatic.

Finally, we write the perfect gases law in each zone and the conservation volumes law:

(41)

, , , , (42)

, , , , (43)

∑ , ∑ , (44)

The mixing between reacting zones is not currently simulated. In the equations above, the mass transfers between two consecutive reacting zones is not taken into account. Further work should be undertaken to correctly simulate this ‘mixing’ due to gases aerodynamic. Results from 3D purely aerodynamic simulations (no combustion) can be used for the calculation of mass flows between these reactive zones, 3D calculations providing reliable estimates of the flow aerodynamic behavior. This work is not presented in this article.

We present in Appendix B the final formulation of the differential equations system.

The sub models described in the three-zone model are adapted in the multi-zone model. There are the same sub models, with the right temperatures used.

MODEL CALIBRATION

In order to correlate to the experimental results, we have to calibrate the combustion model:

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is a parameter of the flame speed, seen in the

description of the flame speed sub model. It plays an important role on the initiation of the combustion.

is a parameter of the combustion characteristic time. It allows to correct the duration of the combustion reaction.

The coefficient allows to correct the energy transfers between a core of a reacting zone and the corresponding near-walls zone. It avoids having a core of reacting zone too hot and a near-walls zone to cool.

We compare the energy release rates from models to some reference data for the model calibration. These reference data are given by a quasi-dimensional tool named SIMCYDE (‘Simulation de cycles diesel et essence’).

Model calibration: presentation of the reference data

Model calibration is based on data provided by SIMCYDE, a quasi-dimensional tool used internally in Renault SA. This tool can be used in three ways briefly described hereafter.

The first way is the combustion analysis mode. The aim is to calculate the instantaneous heat release rate starting from an experimental pressure distribution. Required inputs are: engine geometry, the experimental pressure distribution, experimental walls temperatures, the injected fuel mass, the admitted air mass, the concentrations of hydrocarbon, the exhaust gas recirculation rate, the upstream and downstream conditions of valves. Knowing the introduced energy, global heat losses on an engine cycle are calculated from the first principle of Thermodynamics. A Woschni model is then introduced to calculate instantaneous heat losses. The heat flux coefficient is adjusted in order to get the same heat losses on the whole cycle. The instantaneous heat release rate is finally calculated from the first principle of Thermodynamics. The outputs are therefore the instantaneous heat release rate and the corrected instantaneous heat losses.

The second way is the simulation mode. This option allows to simulate an engine cycle. Required inputs are: engine geometry, the injected fuel mass, the admitted air mass, the concentrations of hydrocarbon, the exhaust gas recirculation rate, the upstream and downstream conditions of valves. The heat release rate is approximated by a model [12] [5] [13]. Given the heat release rate and the corrected heat losses, the pressure, temperature and chemical species time histories are calculated.

The third way is a nested analysis/simulation mode. Combustion analysis gives the instantaneous heat release rate and the instantaneous downstream pressure of exhaust valves. These data will be inputs of the simulation. The simulation

option recalculates the RBG rate and the temperatures of the walls, through a thermal conductance model. These data are inputs of the analysis. Convergence is achieved when the P-V diagram of the calculated mean cycle matches the P-V diagram of the experimental mean cycle. This option allows to correct inputs of analysis and simulation.

We use as reference data the results from the third way. A set of calibration parameters is found under a given engine operating condition. The combustion coefficients keep constant values for the two-zone and three-zone models. The combustion duration and the flame speed have to be recalibrated for the multi-zone model. The parameter keeps a constant value, equal to 1.1, whatever the operating condition.

NUMERICAL RESULTS

The models are correlated and validated by comparing simulated pressures and simulated instantaneous heat release rate with reference data. We use experimental pressures and heat release rates output from SIMCYDE as reference data. The numerical results presented below are promising. But we have still to improve the comparative studies. Comparisons with other engines and other operating conditions will be carried out.

Engine test characteristics

The engine tested is a single-cylinder, and four-stroke SI engine. It is pressure-charged and fuelled with gasoline. The main specifications of this engine are shown in Table 1.

Table 1. Main specifications of test engine

Bore 72.2mm

Stroke 73.2mm

Rod length 123.55mm

Number of cylinders 1

Calorific value 43080kJ/kg

For all presented comparisons, the calculations are made with an angular step ∆ 0.001° .

Full load conditions

The first test is carried out under full load conditions. We give the details of the engine operating condition in Table 2.

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Table 2. Full load conditions

Mass initially injected of air/fuel

398.4mg/27.625mg

Initial mass of exhaust gas recirculated

0mg

Initial mass of residual burned gas

12.125mg

Ignition time 5.12°CA

Speed 4000 rpm

Compression ratio 10.132:1

BMEP 14 bar

Temperature of the cylinder head/the piston head/the liner

255.56K/370.29K/173.57K

The cylinder pressure and the cylinder temperature calculated with SIMCYDE at the beginning of combustion are used as initial conditions for the models under investigations. When combustion starts, 29.4 and all zone-temperatures are equal to 553 .

We compare the results from the two-zone model to reference data. We plot the heat release rate and the log(P)-log(V) diagram during the combustion phase.

Figure 3. Heat release rate comparison from two-zone model to the SIMCYDE data.

Figure 4. logP-logV comparison from two-zone model to the SIMCYDE data.

Figures 3 and 4 show that the two-zone model gives reasonably good results for this full load condition. However, the calculated heat release rate has a higher maximum than the heat release rate from SIMCYDE. More gases burn earlier in the two-zone model. The maximal instantaneous value of the heat release rate from SIMCYDE is 87% of the maximal instantaneous value of heat release rate from the two-zone model.

We compare the results from the three-zone model to reference data.

Figure 5. Heat release rate comparison from three-zone model to the SIMCYDE

Figure 5 displays the results provided by the three-zone model. Thanks to a better modeling of combustion temperature close to the walls, the three-zone model improves results of the two-zone one, namely the peak heat release rate (see Fig. 6). Moreover, the slope of the heat release curve after the peak is slower, and the combustion is predicted to take a longer time than by the two-zone model.

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Figure 6. Comparison between heat release rates from the two-zone and three-zone models

The mean error on the peak heat release rate calculation is about 0.3% with the three-zone model, instead of 0.4% with the two-zone model.

Figure 7. logP-logV comparison from three-zone model to the SIMCYDE data.

The comparisons presented in figures 5 and 7 confirm that the results from the three-zone model are better than the results from the two-zone model: the end of combustion is better correlated.

Finally, we compare the results from the multi-zone model to reference data.

Figure 8. Heat release rate comparison from multi-zone model to the SIMCYDE data, N=3.

Figure 9. logP-logV comparison from multi-zone model to the SIMCYDE data, N=3.

Finally, in Figs 8 and 9 we provide the heat release rate obtained with the n-zone model, as well as the log(P)-log(V) diagram during the combustion phase. For this full-load case, no real improvement is noticed.

Actually, at full load regime, combustion occurs at optimal conditions (good flame homogeneity). For these conditions, the three-zone model is often sufficient to correctly simulate the heat release rate. However, at low load regimes, combustion conditions are not optimal: gases are diluted by EGR, the BRG rate is high and the cylinder temperature is lower. To simulate accurately the combustion under low load, the multi-zone model can be useful.

Low load conditions

We give the details of the engine operating conditions in Table 3.

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Table 3. Low load conditions

Mass initially injected of air/fuel

106.66mg/7.34mg

Initial mass of exhaust gas recirculated

0mg

Initial mass of residual burned gas

16.77mg

Ignition time 22.4°CA

Speed 2000 rpm

BMEP 3.4 bar

Compression ratio 9.308:1

Temperature of the cylinder head/the piston head/the liner

138.22K/181.6K/106.71K

When the combustion starts, 5.08 and all zone-temperatures are equal to 448.32 .

We compare the heat release rate and the pressure from the two-zone model to SIMCYDE data.

Figure 10. Heat release rate comparison from two-zone model to the SIMCYDE data.

Figure 10 shows that the two-zone model overestimates the heat release rate. The ratio between the maximal values of the simulated heat release rate and of the heat release rate from SIMCYDE is 95%.

Figure 11. logP-logV comparison from two-zone model to the SIMCYDE data.

Figure 11 shows that the pressure is overestimated during the second phase of combustion (expansion). Heat losses are underestimated in our model compared to SIMCYDE, because of a higher cylinder temperature at the top dead center (TDC).

Then, we compare the heat release rate and the pressure from the three-zone model to SIMCYDE data.

Figure 12. Heat release rate comparison from three-zone model to the SIMCYDE data.

Figure 12 shows that the heat release rate is now better correlated to the SIMCYDE data. The curves have the same maximal value. With this set of calibration parameter, the starting of combustion is faster in our model than in SIMCYDE but the end of combustion is in good agreement.

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Figure 13. logP-logV comparison from three-zone model to the SIMCYDE data.

Similarly to the two-zone model, the pressure (see figure 13) is still overestimated during the expansion.

Finally, we compare the heat release rate and the pressure from the multi-zone model to SIMCYDE data.

Figure 14. Heat release rate comparison from multi-zone model to the SIMCYDE data, N=4.

Figure 15. logP-logV comparison from multi-zone model to the SIMCYDE data, N=4.

Figures 14 and 15 show the results provided by the multi-zone model: these are found to be in good agreement with SIMCYDE. The heat release rate and the pressure are well predicted. Comparison of figure 12 and figure 14 shows that no major improvement is obtained for the heat release rate calculation when using the multi-zone model. This low load

operating condition is characterized by low RBG (RBG rate is about 12%) and no EGR: as a consequent no degradation of the combustion conditions is observed and the three-zone model is still sufficient to correctly model the combustion.

The most significant improvement is observed on the pressure distribution: the pressure is no longer overestimated during the expansion, since the cylinder temperature is better calculated.

SUMMARY/CONCLUSIONS

This work has presented an improved quasi-dimensional model using multiple reacting subzones to take into account temperature and concentration gradients in the combustion chamber as well as near-wall reacting zones taking into account attenuation of combustion due to wall heat losses. Comparisons with experimental measurements and with a classical two-zone model show that introducing a near-wall zone is essential to improve predictions of the heat release rate, already at full load conditions. In this favorable case, even a three-zone model can predict with reasonable accuracy the heat release rate and pressure distribution in the cylinder. At partial load conditions that are not too severe, however, the use of multiple zones becomes crucial to improve the prediction of the pressure distribution, even if the heat release rate is still relatively well predicted by a three-zone model. In order to better demonstrate the interest of the proposed multi-zone model, comparisons under different EGR rate and under other operating conditions will be carried out and inserted in the final paper. As a future work, we plan to include in the n-zone model the mixing between neighbor reacting zones, due to gas Aerodynamics. To this purpose, we plan to use results from 3D aerodynamic simulations.

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[14] J. Hvezda, «Multi zone Models of Combustion and Heat Transfer Processes in SI Engines,» SAE Technical Paper No 2011-37-0024, 2011.

[15] R. Tabaczynski, C. Ferguson and K. Radhakrishnan, «A turbulent entrainment model for spark ignition engine combustion,» SAE Technical Paper No 770647, 1977.

[16] M. Metghalchi and J. Keck, «Burning velocities of mixtures of air with methanol, isooctane and indolene at high pressure and temperature,» Combustion and Flame, vol. 48, pp. 191-210, 1982.

[17] R. Borghi and M. Destriau, La combustion et les flammes, Technip, 1995.

[18] L. Landry, «Etude expérimentale des modes de combustion essence sous forte pression et forte dilution,» PhD Thesis, Université d'Orléans, 2009.

[19] J. Heywood and S. Poulos, «The effect of chamber geometry on spark ignition engine combustion,» SAE Technical Paper No 830334, 1983.

[20] K. Sihling and G. Woschni, «Experimental investigation of the instantaneous heat transfer in the cylinder of a high diesel engine,» SAE Technical Paper No 790833, 1979.

[21] K. Huber, G. Woschni and K. Zeilinger, «Investigations on Heat Transfer in Internal Combustion Engines under Low Load and Motoring Conditions,» SAE Technical Paper No 905018, 1990.

[22] J. Guibet, Carburants et moteurs. Technologie-Energie-Environnement, Paris: Edition Technip, 1997.

[23] B. Boust, «Etude expérimentale et modélisation des pertes thermiques pariétales lors de l'intéraction flamme-paroi instationnaire,» PhD Thesis, Université de Poitiers, 2006.

[24] P. Guibert, «Modélisation du cycle moteur, approche zéro dimensionnelle,» Technique de l'Ingénieur, BM 2510, 2005.

[25] R. R. Raine, C. R. Stone and J. Gould, «Modeling of Nitric Oxide Formation in Spark Ignition Engines with a Multizone Burned Gas,» Combustion and Flame , vol. 102, pp. 241-255, 1995.

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ACKNOWLEDGMENTS

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DEFINITIONS/ABBREVIATIONS

, ,, Admitted mass of fresh gases, fuel, burned gases (kg)

, , , , , Mass of fresh gases, fuel, burned gases in the j-th core of reacting zone (kg)

, , , , , Mass of fresh gases, fuel, burned gases in the j-th reacting zone near walls (kg)

, , , , , Temperature and volume of the j-th core of reacting zone (reacting zone near walls) (K)

, , , , Specific gas constant (J/kg/K)

R Specific gas constant (J/mol/K)

Specific enthalpy (J/kg)

Mean specific heat (J/mol/K) of the zone i

, Number of mole of the component k, total number of mole in the zone i (mol)

Internal energy (J/mol)

Molecular weight of the component k (kg/mol)

Mean specific heats ratio

Total entrained mass (kg)

, , , Mean specific heat (J/mol/K) of the j-th reacting zone (reacting zone near walls)

, , , , , Number of mole of the component k and total number of mole in the j-th reacting zone (reacting zone near walls) (mol)

, , Internal energy in the j-th reacting zone (reacting zone near walls) (J/mol)

, , Mean specific heats ratio

, , , Heat losses (J)

, Energetic transfers between the j-th core of reacting zone and near-walls reacting zone (J)

EGR Exhaust Gas Recirculation

BRG Burned Residual Gas

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APPENDIX

In the equations below, the index k can take three values: GF for the fresh gases, GB for the burned gases and Fu for fuel.

A. Final formulation of the governing equations of the three-zone model

The index i denotes the number of the zone (i=1, i=2 or i=3).

We note that equations are written in moles. Starting from the first principle of Thermodynamics and the perfect gases law applied in each zone, we isolate the derivative of the temperature of a zone.

.

∑ ∑ )

.∑ ∑ )

.∑ ∑ )

By adding the equations (5), (6) and (7), the variation of the cylinder pressure is expressed.

In the equation below:

11 1

1 1 1 1 1 1 1

11

11 1

11

11 1

11

1 1 1 1 1 1

1 1 1 1 1

The volumes are calculated with the equation (8) applied in each zone and with the equation (9):

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B. Final formulation of the governing equations of the multi-zone model

The index j denotes the number of the reacting zone (j=1,…, N). The index i can take two values: 2 for the core of the reacting zone and 3 for the reacting zone near the walls.

Starting from the first principle of Thermodynamics and the perfect gases law applied in each zone, we isolate the derivative of the temperature of a zone.

.∑ ∑ )

For the N-1 already existing reacting zones:

,

, . ,

,, , ∑ ,

,, , )

,

, . ,

,, , ∑ ,

,, , )

For the latest reacting zones:

,

, . ,

,, , ∑ ,

,, , ∑ )

,

, . ,

,, , ∑ ,

,, , ∑ , )

By adding all first thermodynamic principles, we have the variation of the cylinder pressure. In the equation below:

,, ,

, , , ,

,, ,

, , , ,

1

∑ ,

,∑ ,

,∑ 1

,

1,

, , , ,

1 1

,

1

,

1

,,

,,

1

,

1

,

1

,,

,, 1

11

1

,

, 11

1

,1

1

,

1

,,

,,

1

,1

1

,

1

,,

,,

1 1

,

1

,

1

,,

1

,

1

,

1

,,

1

,

1

,

,

The volumes are calculated with the equations (42), (43) and (44):

,, ,

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,, ,

, ,

Multi-Zone Quasi-Dimensional Combustion Models for Spark-Ignition Engines

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