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International J of Engine Research1–16� IMechE 2019Article reuse guidelines:sagepub.com/journals-permissionsDOI: 10.1177/1468087419841746journals.sagepub.com/home/jer

A two-zone reaction-based combustionmodel for a spark-ignition engine

Ruixue C Li, Guoming G Zhu and Yifan Men

AbstractThis article presents a control-oriented two-zone reaction-based zero-dimensional model to accurately describe thecombustion process of a spark-ignited engine for real-time simulations, and the developed model will be used for model-based control design and validation. A two-zone modeling approach is adopted, where the combustion chamber isdivided into the burned (reaction) and unburned (pre-mixed) zones. The mixture thermodynamic properties and individ-ual chemical species in two zones are taken into account in the modeling process. Instead of using the conventional pre-determined Wiebe-based combustion model, a two-step chemical reaction model is utilized to predict the combustionprocess along with important thermodynamic parameters such as the mass-fraction-burned, in-cylinder pressure, tem-peratures, and individual species mass changes in both zones. Sensitivities of model parameters are analyzed during themodel calibration process. As a result, one set of calibration parameters is used to predict combustion characteristicsover all engine operating conditions studied in this article, which is the major advantage of the proposed method. Also,the proposed modeling approach is capable of modeling the combustion process under different air-to-fuel ratios, igni-tion timings, and exhaust-gas-recirculation rates for real-time simulations. As the by-product of the model, engine knockcan also be predicted based on the Arrhenius integral in the unburned zone, which is valuable for model-based knockcontrol. The proposed combustion model is intensively validated using the experimental data with a peak relative predic-tion error of 6.2% for the in-cylinder pressure.

KeywordsTwo-zone model, chemical reaction, real-time simulation, spark-ignition, knock prediction

Date received: 5 December 2018; accepted: 8 March 2019

Introduction

In the past decades, the model-based engine control,especially combustion control, is widely studied due tothe rapid technology development in combustion sen-sing and real-time computing power. As part of model-based engine control, major progress has been made indeveloping control-oriented engine and combustionmodels for both spark-ignition (SI) and compressionignition (CI) internal combustion engines. These devel-oped models are used for model-based design and cali-bration to efficiently reduce engine development timeand cost.1 And the control-oriented engine models arealso used for model-based control in real-time applica-tions. Furthermore, since engine dynamics, emissions,and performance can be efficiently studied throughsimulations without conducting physical experiments,it is possible to study engine operations at its opera-tional boundary when experiments are hard to conductwithout damaging the physical system.

Nowadays, the main research of SI engines is to fur-ther improve the fuel economy and reduce emissions2

that are closely related to combustion process. Model-based combustion control leads to the development ofcontrol-oriented engine combustion models to reliablypredict the in-cylinder combustion process by providingdetailed mass-fraction-burned (MFB) rate, in-cylinderpressure, and other information.1 Up to now, there areseveral widely used modeling approaches for the in-cylinder combustion process that can be classified assingle-zone, multi-zone, and multi-dimensional mod-els.3 The multi-dimensional computational fluiddynamics (CFD) models accurately describe the in-cylinder combustion process, including the flame fluiddynamics and the in-cylinder mixture characteristics,by solving a number of partial differential equations.

Department of Mechanical Engineering, Michigan State University, East

Lansing, MI, USA

Corresponding author:

Guoming G Zhu, Department of Mechanical Engineering, Michigan State

University, E148 ERC South, East Lansing, MI 48823, USA.

Email: zhug@egr.msu.edu

Some multi-dimensional CFD models predict the com-bustion process by modeling thousands of chemicalreaction species with certain reaction steps. Althoughthese models are able to predict the combustion processaccurately, they require tremendous computationalpower to even simulate the combustion process for oneengine cycle. As a result, the multi-dimensional CFDmodels are only good for offline simulations, but it isnot suitable for model-based control design and valida-tion that require real-time simulation capability. TheGT-Power engine model, widely used by automotiveindustry, uses zero-dimensional (0D)/one-dimensional(1D) modeling approach to take account of the gasflow dynamics outside of combustion chamber and 0Dsingle-zone modeling approach for the combustion pro-cess based on the empirical combustion functions suchas the Wiebe function.4 GT-power models are signifi-cantly simplified over the CFD ones, but it still cannotbe used for real-time simulations. On the other hand,the 0D single zone combustion model, such as themean-value model,5 was widely used for control designdue to its ability of conducting real-time simulations.To reduce simulation time, it uses maps to model thecomplex combustion process. As a result, the modelaccuracy is highly dependent on the calibration processand its transient responses are not accurate since thethermodynamics is not modeled. Furthermore, themean-value model is not able to predict the in-cylinderpressure, heat-release-rate (HRR), and states of chemi-cal species.

To overcome the limitations of these combustionmodels discussed above, two-zone and multi-zone mod-els are developed. Shapiro and VanGerpen6 presented atwo-zone combustion model for internal combustionengines based on the second-law of thermodynamics.Loganathan et al.7 and Saad et al.8 proposed a two-zonecombustion model for diesel engines, and Borg andAlkidas9 developed a two-zone model for SI engines tostudy engine performance. Note that in two-zone mod-els, the gas mixture are separated into burned (reaction)and unburned zones, assuming that the flame front is athin boundary layer separating the two zones. In someliterature,2,10,11 the unburned zone is further split intomultiple zones based on temperature gradients, assum-ing that the mixture in each individual zone is homoge-neous but has different thermodynamic characteristics.Rakopoulos and colleagues2,10 proposed a multi-zonecombustion model for engine transient performanceand NOx (nitric-oxide) formation of a syngas SI engine.However, the MFB rate of these discussed two- andmulti-zone models5–14 is generated by an empiricalWiebe function that needs to be calibrated as a functionof engine speed, load, and exhaust-gas-recirculation(EGR) for each engine operating condition.

To improve Wiebe-based combustion model, theflame development dynamics is introduced in the real-time combustion model recently. Hall et al.15 proposed

a control-oriented two-zone combustion model for SIengines, assuming that the burned zone is sphere-shaped with the flame front on the boundary and MFBis determined by calculating the flame speed. However,since the chemical reaction process is not modeled,thermodynamic properties of the chemical species can-not be predicted. In order to predict the combustionprocess along with key thermodynamic parameterssuch as the MFB, in-cylinder pressure, temperatures,and individual species mass changes, the reaction-basedcombustion model is developed. Jia et al.16 presented acontrol-oriented reaction-based single-zone combustionmodel for a propane-fueled homogeneous charge com-pression ignition (HCCI) engine, and Men et al.17 alsoproposed a control-oriented three-zone reaction-basedcombustion model for direct-injection (DI) dieselengines. Both discarded the empirical Wiebe-basedcombustion model and, instead, adopted single ormulti-step chemical kinetic mechanism in conjunctionwith the Arrhenius function-based reaction rates. Thereaction-based combustion model takes into accountfor chemical characteristics in the actual combustionprocess and is able to predict the MFB rate, pressurerise rate, HRR, and zone temperatures as well as theproperties of each individual chemical species with verylow computational load. Furthermore, there is littlestudy in the literature about the reaction-based two- ormulti-zone real-time model for SI engines capable ofknock prediction. Note that auto-ignition of end-gas(knock) in SI engines is a very important phenomenonand, unfortunately, there is not much results oncontrol-oriented knock modeling.

This article focuses on the development and valida-tion of a control-oriented reaction-based combustionmodel of SI engines for real-time simulations, where atwo-zone combustion model is used along with the two-step chemical reaction mechanism and the reaction rateof individual species is based on the Arrhenius function.The in-cylinder thermodynamics and combustion pro-cess are modeled between intake valve closing (IVC)and exhaust valve opening (EVO). The feasibility ofmodeling engine knock is also studied. The proposedmodel is validated against experimental data at five typ-ical operational conditions with one set of calibrationparameters and demonstrated its capability of predict-ing the combustion process accurately.

The main contribution of this article is the proposedtwo-zone two-step chemical reaction model capable ofpredicting MFB, HRR, in-cylinder pressure, along withthermodynamic properties of individual species such astheir chemical reaction rates, associated molar concen-trations, and concentration variation rates, and so on.Furthermore, the developed model is calibrated underfive different engine operating conditions using one setof calibration parameters. That is, the model does notneed to be recalibrated under different operating condi-tions, which is very important for model-based control.

2 International J of Engine Research 00(0)

Furthermore, the proposed model is also capable ofpredicting auto-ignition phenomenon (knock) in theunburned zone.

This article is organized as follows. In the next sec-tion, the outline of the proposed model is discussed byaddressing the mass and heat transfer interaction on theboundary layer of the two zones and the chemical reac-tion rates of individual species based on the Arrheniusfunction, along with thermodynamic states and proper-ties in two zones. Next, a four-cylinder SI engine experi-mental data are used to validate the developed model,and a calibration method, based on the parameter sen-sitivity analysis, is also presented. At last, the conclu-sions are drawn.

The reaction-based combustion model

The proposed model focuses on predicting the in-cylinder combustion process for a four-stroke SI enginebetween IVC and EVO. It is assumed that the fresh air,fuel, and residual-exhaust-gas (REG) from the lastcombustion cycle are homogeneously mixed in thecylinder at IVC. Between IVC and EVO, the combus-tion chamber is divided into two zones: unburned andburned (reaction). The two zones are assumed to behemisphere-shaped and both have heat losses to thecylinder wall (see Figure 1, right). Since the in-cylindercombustion chamber shape is not hemisphere (seeFigure 1, left), in the combustion model, the volumes ofburned and unburned zones indicated in the left imageof Figure 1 are matched with those in the right imageof Figure 1, respectively. An initial 1:5% of total freshair and fuel mixture is assigned to the ignition zone atIVC and after ignition it becomes the initial reactionzone mass. The REG and the rest of total fresh air andfuel mixture are assigned to both ignition and unburnedzones initially. After the spark, combustion starts in thereaction zone and the gas mixture in the unburned zoneflows into the reaction zone to burn. It is furtherassumed that the chemical reaction products stay in thereaction zone, and as a result, it expands with the flamepropagation.

The interactions between two zones include heat andmass transfer, which makes it easier to simulate theactual combustion process, flame propagation, andtemperate difference. Furthermore, different from thetraditional empirical Wiebe-based combustion model, atwo-step chemical reaction kinetic mechanism is usedin this article to calculate the chemical reaction ratesduring the combustion process. The molar concentra-tion and concentration variation rate of each individualspecies in the reaction zone during the entire combus-tion process are calculated, making it possible to studythe detailed combustion process such as MFB, HRR,chemical reaction rates, flame speed, zone tempera-tures, and pressure.

Two-zone model configuration

The combustion chamber is assumed to be divided intotwo zones as shown in Figure 1. The flame front sepa-rates the two zones, and the reaction zone is where thechemical reaction takes place and the combustion prod-ucts are assumed to stay within this zone. The pre-mixed gas mixture is in the unburned zone. From theleft image of Figure 1, the heat loss area of the reactionzone extends from the cylinder head to its wall, and thearea of unburned zone is from the piston top to cylin-der wall. This is important for modeling heat losses inthe next section. The two zones interact through theinterface with heat and mass transfer. The model inputsinclude mass of fresh air and fuel, air-to-fuel ratio(AFR), REG, chamber volume at IVC, in-cylinderpressure and temperature at IVC. As mentioned in thelast section, fresh air, fuel, and REG are pre-mixed atIVC with a known AFR, l, and the total mass in thecombustion chamber is shown in equation (1)

mtot=mair+mfuel +mREG ð1Þ

where mREG is the mass of REG trapped in the cham-ber. For the data used to calibrate this model for thetest engine, the EGR valve is closed so the EGR ratewas set to zero in this study and only REG is consid-ered. Note that if the EGR rate is not zero, only theREG mass needs to be modified based on the EGRrate. In this study, with the known IVC timing, the cal-culate REG mass is around 7% of total mass at IVC.To initiate the combustion, ignition energy is applied toa small zone (around 1.5% of the total mass for thisarticle) around the spark plug gap and the combustionis initiated when the auto-ignition condition is satisfied.The small zone is called ignition zone and the associatedmass is called initial ignition zone mass. Since auto-ignition is assumed in the ignition zone, the combustionin the ignition zone completes in one simulation stepand the ignition zone becomes the reaction zone afterthe combustion is completed and the initial ignitionzone mass becomes the initial mass of reaction zone.

The total cylinder volume Vcyl and its rate of change_Vcyl can be obtained from the typical piston motionlaw18 below

Unburned Zone

Reac�on Zone IdealizedReac�on Zone

Idealized Unburned Zone

Figure 1. Two-zone combustion model structure.

Li et al. 3

Vcyl=Vc 1+rc � 1

2R+1�cosu�

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiR2 � sin2u

ph i� �ð2Þ

_Vcyl =(rc � 1)Vc

2sinu 1+

cosuffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiR2 � sin2u

p !

du

dtð3Þ

where u is the crank angle, rc is the compression ratio,R is the ratio of connecting rod length and crankradius, and Vc is the clearance volume. Therefore,assuming the mixtures in both zones are homogeneous,the unburned zone volume Vu can be calculated firstbased on the ideal gas law, and the volume rate ofchange _Vu can be derived from Vu; see below

Vu =muRuTu

pMWu, _Vu =

dVu

dtð4Þ

where mu, Tu, and MWu are mass, temperature, andaverage molecular weight of the unburned zone mix-ture, respectively; p is the in-cylinder pressure; Ru

(8:314 J/kmol) is the universal gas constant.Therefore, the reaction zone volume Vr and its rate

_Vr can be derived as follows

Vr =Vcyl � Vu, _Vr = _Vcyl � _Vu ð5Þ

Detailed interaction between two zones is shown inFigure 2 and the associated formulas used to calculatethe mixture characteristics in each zone will be dis-cussed in the next few subsections.

Interaction between two zones

Heat and mass transfer between two zones and heatloss from two zones to the cylinder boundary are veryimportant to obtain an accurate combustion modelsince the thermodynamic properties in each zone arebased on heat loss, heat and mass transfer.

Heat transfer interface. Between IVC and SI, there is noheat and mass transfer at the interface betweenunburned and reaction (ignition) zones. As a result, themass and volume ratio of two zones remainsunchanged. With the added spark energy, the combus-tion is initiated in the reaction (ignition) zone and themodel transits to the combustion phase. During thecombustion phase, the flame stays within the reactionzone with a radius of Rr and propagates toward theunburned zone. As is shown in Figure 2, the heatrelease from the chemical reaction results in a fastincrement of the reaction zone temperature Tr, and thetemperature difference between the two zones leads tothe heat transfer _Qtr from the reaction to unburnedzone.

Physically, the mass in the interface of two zones willbe heated up much faster than the rest of mixture in theunburned zone. Therefore, it is assumed that there is avery thin virtual region (Figure 2, green area) betweentwo zones. This virtual region is homogeneous but the

temperature is much higher than that of unburnedzone. After the start of combustion (SOC), the totalheat transfer from the reaction zone is divided into twoparts, where part 1, _Qm, is used to heat the mixture inthe virtual region into Tr so that the associated mass inthis region can be moved into the reaction zone and thisprocess is called the mass transfer process; part 2, _Qt, isused to heat the mass in the rest of unburned zone.

The total heat transfer _Qtr from the reaction tounburned zone can be calculated based on the follow-ing equation

_Qtr= kcAtrTr � Tu

Rrð6Þ

where kc is the heat coefficient to be calibrated, Atr isthe effective contact area between two zones, and Rr isthe radius of the reaction zone. Since the two zone tem-peratures are assumed to be homogeneous, the tem-perature gradient distance for the heat transfer _Qtr isphysically from the center of the reaction zone to theaverage radius of the virtual region (see Figure 2,dashed line). However, the virtual region is assumed tobe a very thin layer, and as a result, the Rr is used asthe temperature gradient distance in this article.

Atr can be calculated by equation (7)

Atr=2pR2r ð7Þ

To calculate heat transfer, _Qm, assume that the tem-perature in the virtual region (Figure 2, green thinlayer) is Tg that is an average temperature of reactionand unburned zones weighted by the mass and specificheat coefficient associated with the corresponding zone

Tg =mrcp, rTr +mucp, uTu

mrcp, r +mucp, uð8Þ

where Tr, mr, and cp, r are the temperature, mass, andspecific heat of the mixtures in the reaction zone andTu, mu, and cp, u are the same parameters for theunburned zone, respectively. As a result, the heat trans-fer from the reaction zone to the virtual thin layer is asfollows

Figure 2. Interaction between reaction and unburned zones.

4 International J of Engine Research 00(0)

_Qm = kcAtrTr � Tg

Rrð9Þ

Substituting equation (8) to equation (9) yields

_Qm = kcAtrTr � Tu

Rr

mu

mr +mu=

mu

mr +mu

_Qtr ð10Þ

As a result, the remaining heat transfer _Qt used toincrease the unburned zone temperature can be derivedas follows

_Qt = _Qtr � _Qm =mr

mu +mr

_Qtr ð11Þ

As discussed in the last few subsections, it isassumed that there is no mass transfer between the twozones until combustion is initiated. Physically, the masstransfer rate is high at the beginning of combustionphase and low at the end of combustion. Equations(10) and (11) indicate that there is no temperature dif-ference between two zones before combustion is initi-ated ( _Qtr=0), resulting in _Qm =0 and _Qt =0 duringthe compression process. During the combustion phase,the fraction mu=(mu +mr) in equation (10) dominatesthe heat transfer for mass transfer. On the other hand,during the fast combustion phase, the fractionmr=(mu +mr) in equation (11) dominates the heattransfer and causes the increment of the unburned zonetemperature.

Heat loss. Heat loss affects the thermal stratificationinside the cylinder and the combustion characteristics.1

The heat loss in this model includes two parts. One isthe heat loss from the reaction zone to the cylinder headand liner, which dominates heat loss during the com-bustion phase, and is represented by _Qw1. The otherpart is the heat loss from the unburned zone to the pis-ton crown and the remaining cylinder liner and is repre-sented by _Qw2 (see Figure 2).

To reduce the computational load for this real-timecombustion model, the heat loss model is simplified andthe Woschni’s19 formula is used to calculate both heatlosses, _Qw1 and _Qw2, below

_Qw1 = hc1Aw1(Tr � Tw) ð12Þ_Qw2 = hc2Aw2(Tu � Tw) ð13Þ

where hc1 and hc2 are the heat transfer coefficients, andTr, Tu, and Tw are the temperature of the reactionzone, unburned zone, and the cylinder boundary,respectively. Note that the temperature of the cylinderboundary (cylinder head, cylinder liner, piston head) isassumed to be the same and constant. Aw1 and Aw2 arethe contacting area for the heat loss between the associ-ated zone and the cylinder boundary, respectively.

The reaction zone is a very small hemisphere initiallyand then starts expanding due to the flame propagationafter the ignition. Therefore, the contacting area Aw1

between the reaction zone and the cylinder boundaryincreases as well (see Figure 1). As the two zones are

assumed in the hemisphere shape based on the charac-teristic of the flame propagation, both physical cham-bers shown on the left of Figure 1 can be transferred tothe hemisphere shapes (shown in the right image ofFigure 1) with the matched volumes, respectively. Andit is assumed that the heat loss from the reaction zoneto the wall is only through the base surface of the reac-tion zone in hemisphere shape. As a result, Aw1 can becalculated based on the geometry below

Aw1 =pR2r ð14Þ

where Rr is the radius of the reaction zone. Note thatthe base surface of reaction zone Aw1 is not limited bythe cylinder head but will consistently increase with thevolume of the reaction zone. The contacting area fromthe unburned zone to the rest of cylinder boundary Aw2

decreases as the flame propagates and is given by

Aw2 =Ap +Ach +Ah � Aw1 ð15Þ

where Ap is the piston crown surface area; Ach is thecylinder liner area; and Ah is the cylinder head surfacearea. The heat transfer coefficients hc1 and hc2 are givenby17

hc1 =aB�0:2p0:8T�0:55r (2:28Sp)0:8 ð16Þ

hc2 =bB�0:2p0:8T�0:55u (2:28Sp)0:8 ð17Þ

where a and b are calibration constants, and Sp is thepiston velocity equal to _Vcyl=Ap.

Mass transfer. As mentioned above, the heat transferrate _Qm causes the mass transfer between two zonesduring the combustion process. If mass flow rate _mtr

were too large, the modeled combustion would beunstable, and if it were too slow, the combustion mightnot continue. Therefore, the mass transfer rate betweentwo zones is a key parameter to be modeled and it iscalculated by the following equation

_mtr =km _Qm

cp, uDT0ð18Þ

where km is the mass coefficient, and DT0 is a constantassociated with temperature increment. Both para-meters need to be calibrated based on the experimentaldata. cp, u is the specific heat of the gas mixture in theunburned zone and will be discussed in the nextsubsection.

Chemical reaction kinetic mechanism

The mixture in the unburned zone consists of fresh air,fuel, and REG trapped in the combustion chamberfrom the last cycle. The initial reaction zone size isabout 1%–2% of the total mass, and each species in themixture changes during the combustion. It is assumedthat the mixtures in both zones are homogeneous. Thecomposition and thermodynamic properties of mixtures

Li et al. 5

in both zones are determined by the in-cylinder pres-sure, zone temperature, AFR, and zone volumes.20

Based on the chemical kinetic mechanism, the proper-ties of each species in each zone can be studied. Andwith a two-step chemical reaction mechanism, thedetailed combustion process can be modeled.

Molar concentration and its concentration rate. The molarconcentration and its concentration variation rate ofeach individual species are the foundation for studyingthe chemical reaction process and thermodynamicproperties of the mixture. The mixture in the unburnedzone consists of five chemical species and they areC8H18, O2, N2, CO2, and H2O. Due to the two-stepchemical reaction mechanism, the reaction zone hasone more chemical species, CO, generated during thereaction process. Therefore, there are six species:C8H18, O2, N2, CO2, H2O, and CO in the reactionzone.

The molar concentration ½Xi� (moles per unit vol-ume) of species i is defined by

Xi½ �=Ni

Vð19Þ

where i stands for different species in each zone; Ni isthe associated molecular number of species i, whereNi =mi=MWi; and V is the associated zone volume.

The molar concentration of each species ½Xi� changesduring the combustion process and the associated zonevolume also changes due to the piston movement.Therefore, the rate of change for the molar concentra-tion ½Xi� is denoted by ½ _Xi� (kmol/m3 s) and is given by

½ _Xi�= d½Xi�dt = d(Ni=V)

dt

= 1VMWi

dmi

dt �Ni1V2

dVdt

ð20Þ

The first term on the right side of equation (20)accounts for the mass change. The mass change in azone is driven by two factors: mass transfer betweentwo zones and chemical reaction. During the chemicalreaction process, the masses of reactants and productskeep changing until the reaction is ended. The secondterm reflects the effect of volume change to the molarconcentration.

The mass change term is defined as follows

1

VMWi

dmi

dt=vi =vflow, i +vchem, i ð21Þ

where vchem, i (kmol/m3s), reaction rate, is used toreflect the effect of chemical reaction. And theArrhenius Law20 is used in this model to calculate thereaction rate vchem, i. This part will be addressed next.Note that vflow, i accounts for the effect of mass transferfor the concentration rate of species i and can be calcu-lated based on the mass transfer rate _mtr

vflow, i =

�xu, i _mtr

VuMWuin the unburned zone

xu, i _mtr

VrMWuin the reaction zone

(ð22Þ

where xu, i is the molar fraction of each species in theunburned zone, and MWu is the average molecularweight of the unburned zone mixture to be discussed inthe next subsection.

Two-step chemical reaction mechanism. A simplified butpractical chemical reaction mechanism for the combus-tion process is always desired. In the recent literature,the one-step reaction between reactants and products iswidely used to achieve this goal. However, the one-stepreaction mechanism is not able to describe the flamepropagation process from lean to rich combustion.20

The main weakness of this one-step reaction mechan-ism is the neglect of CO produced in the combustionprocess. Since in the typical hydrocarbon flames, thelarge amount of CO and H2 exists in the equilibriumwith CO2 and H2O, while the CO oxidation is also arather slow process.20,21 Therefore, a two-step chemicalreaction mechanism is proposed in the reaction zone toaccount for the influence of the CO oxidation process.The specific reaction steps are shown below

C8H18+17

2O2 �!

k18CO+9H2O ðM1Þ

CO+1

2O2�

k2

k3CO2 ðM2Þ

where the proportionality factor ki (i=1, 2, 3) is thespecific reaction rate constant dominated by thetemperature.21

The rate of the first step reaction (M1) is given by

vM1 = k1½C8H18�n1 ½O2�n2 ð23Þ

where ½C8H18� and ½O2� are molar concentrations ofspecies fuel and species O2, respectively, and n1 and n2are associated reaction order and are empiricallydetermined.

For the second step chemical reaction (M2), since itis a reversible reaction, the net reaction rate is given by

vM2 = k2½CO�n3 ½O2�n4 � k3½CO2�n5 ð24Þ

where the first and second terms on the right side arethe forward and backward reaction rates of M2, respec-tively; ½CO� and ½CO2� are molar concentrations of COand CO2, respectively; and n3;n5 are associated reac-tion order and are also empirically determined.

The specific reaction rate constant ki(i=1, 2, 3) inM1 and M2 is calculated based on the Arrhenius Law20

ki =Aie�Ea, i=RuT ð25Þ

where Ea, i is the activation energy of the reaction (J/mol), Ai is the pre-exponential factor, and Ea, i and Ai

are constant and to be calibrated. As activation energyEa and the universal gas constant Ru are constant, theactivation temperature Ta can be defined as follows

Ta, i =Ea, i

Ruð26Þ

6 International J of Engine Research 00(0)

Based on equations (23) and (24), the productionrates of each species are as follows

Ωchem = s53 2 �v23 1 ð27Þ

where Ωchem is the production rate vector of the species;s53 2 is the stoichiometric coefficient matrix; and v23 1

is the reaction rate vector for M1 and M2; see below

Ωchem =(vC8H18vO2

vCO2vH2O vCO)

T ð28Þ

and

s53 2 =

�1 0� 17

2 � 12

0 19 08 �1

266664

377775,v23 1 =

vM1

vM2

� �ð29Þ

Zone temperature. Based on the second law of thermo-dynamics, conservation of mass, conservation ofenergy, and the chemical kinetic mechanism discussedabove, the temperature rate of change for the reactionzone can be derived below

_Tr =

_Qr

Vr+RuTrS½ _Xi� � S½ _Xi��hi �

_Vr

VrS½Xi��hi + S( _Ni

�hi)Vr

S½Xi�(�cp, i � Ru)

ð30Þ

where �hi and �cp, i are molar enthalpy (kJ/kmol) andmolar specific heat (k J/kmolK) of species i, respec-tively, with respect to temperature, and _Qr(kJ/s) is thenet heat transfer rate for the reaction zone defined by

_Qr = _Qign � _Qt � _Qw1 ð31Þ

Note that _Qign(kJ/s) is the rate of provided ignitionenergy. Figure 3 provides a sample ignition energy pro-file for _Qign used in the proposed model, where the totalignition energy is 100mJ with a duration of 6 crankangle degrees (CADs).

The formula used to calculate the unburned zonetemperature Tu is similar to those for the reaction zone.However, since there is no chemical reaction in theunburned zone, the molar concentration rate (½ _Xi�) for-mula for the unburned zone needs to be modified.Note that no chemical reaction in the unburned zonemeans that the reaction rate of each species is zero(vchem, i =0). As a result, the change of molar concen-tration is caused by the mass transfer and volumechange only. Therefore, equations (20) and (21) can bemodified and combined; see below

½ _Xi�=_mi

VuMWi�Ni

_Vu

V2u

=vflow, i �Ni

_Vu

V2u

ð32Þ

Thus, the unburned zone temperature rate _Tu is

_Tu =

_Qu

Vu+RuTuS½ _Xi� � S½ _Xi��hi �

_Vu

VuS½Xi��hi + S( _Ni

�hi)Vu

S½Xi�(�cp, i � Ru)

ð33Þ

where _Qu is the unburned zone net heat transfer rateand is given by

_Qu = _Qt � _Qw2 ð34Þ

In-cylinder pressure. With the known temperature, mass,and volume of each zone, the cylinder pressure can becalculated by the ideal gas law below

p=mtotRuTavg

VcylMWmixð35Þ

where MWmix is the average molecular weight of in-cylinder mixture and can be weighted by the mass andspecific heat of the mixture in each zone; see below

MWmix =mrcp, rMWr +mucp, uMWu

mrcp, r +mucp, uð36Þ

Note that MWu and MWr are the average molecularweight of the mixture in the reaction and unburnedzones, respectively. Since the chemical reaction happensin the reaction zone, it is assumed that MWu is con-stant throughout the combustion cycle and MWr variesdue to the chemical reaction in the reaction zone. As aresult, MWu can be computed by

MWu =X

xu, iMWi ð37Þ

where xu, i and MWi are the mole fraction and molecu-lar weight of each species i, respectively, in theunburned zone and they remain unchanged throughoutthe combustion process.

However, MWr changes during the combustion pro-cess and can be derived as follows

-40 -30 -20 -10 0 10 20

0

50

100

150

200

250

300

Figure 3. Ignition energy profile used.

Li et al. 7

MWr =

PmiMWiP

mið38Þ

where i is the species in the reaction zone, and mi andMWi are the associated mass and molecular weight,respectively.

Tavg in equation (35) is the overall temperature of themixture in the cylinder weighted by the mass and spe-cific heat of mixtures in each zone; see below

Tavg=mrcp, rTr +mucp, uTu

mrcp, r +mucp, uð39Þ

where cp, r and cp, u are the specific heat of mixtures inreaction and unburned zones, respectively. cp, u and cp, rare derived as follows

cp, u =X

Xi½ � � �cp, i(Tu)Vu ð40Þ

cp, r =X

Xi½ � � �cp, i(Tr)Vr ð41Þ

where ½Xi� is the molar concentration of species i in theassociated zone; �cp, i(Tu) and �cp, i(Tr) are the molar spe-cific heat of each species i in the associated zone at cur-rent zone temperature, respectively.

Knock modeling

Due to the increased temperature and pressure in theunburned zone during the combustion process, theunburned gas (end-gas) could be auto-ignited, resultingin engine knock, which could damage the engine.Therefore, knock prediction and control are veryimportant, especially for downsized and turbochargedengines. And it is desirable to have a real-time combus-tion model capable of predicting auto-ignition (engineknock) so that it can be used for model-based knockcontrol. The commonly used criterion for the SOC inthe unburned zone is the Arrhenius integral (ARI)22

defined below

ARI=

ðuiuIVC

Aknpakn ½C8H18�bkn ½O2�ckne�

Ta, knTu (u) du ð42Þ

where auto-ignition coefficients Akn, akn, bkn, ckn andactivation temperature Ta, kn are constant and need tobe calibrated based on the experimental data to accu-rately predict engine knock.

Equation (42) can be interpreted as an integral ofchemical reaction rate of the unburned mixture (end-gas), and this equation also indicates that the ARI ispositive and the integral is monotonically increasing.The criterion for auto-ignition is at the crank anglewhen ARI reaches 1,2,22 that is

ARI=1 ð43Þ

Based on equation (43), the proposed two-zone two-step chemical reaction-based model is able to predictengine knock due to two reasons. First, the concentra-tion of fuel and oxygen used in the ARI calculation can

be obtained by the two-step chemical reaction mechan-ism proposed, and second, the known unburned mix-ture characteristics in the unburned zone makes itpossible to calculate the ARI. Furthermore, since thisproposed model is a real-time control-oriented combus-tion model, it is very useful for future model-basedknock control.

Experimental investigation

The experimental data used to validate the proposedreaction-based two-zone combustion model are col-lected from a four-cylinder, four-stroke SI enginethrough dynamometer experiments. The engine para-meters are listed in Table 1.

The test data for five typical steady-state engineoperating conditions are used for validation purposeand are summarized in Table 2, where the relativeAFR l is controlled to be close to stoichiometric. Ateach condition, 100 cycles of engine data were col-lected. In order to calibrate the ARI coefficients forauto-ignition (knock) prediction, the ignition timingunder two operational conditions (high load conditionsfor 1500 and 2000 r/min) are controlled to be near theengine knock limit.

An A&D CAS (Combustion Analysis System) isused to record in-cylinder pressure, intake manifoldpressure, ignition current, and so on. The average in-cylinder mixture temperature is calculated based on therecorded in-cylinder pressure and used to compare withthe modeled ones.

Model calibration and simulation results

This section discusses the model calibration process forkey model parameters and compares the simulation

Table 1. Test engine parameters.

Parameter Value

Bore 86 mmStroke 86 mmRod 143.6 mmCompression ratio 11:1Displacement 499.56 cm3

Intake valve closing (IVC) 190�BTDCExhaust valve opening (EVO) 156�ATDC

BTDC: before top dead center; ATDC: after top dead center.

Table 2. Engine operational conditions.

Case 1 2 3 4 5

IMEP (bar) 4.53 5.01 6.78 6.83 8.23Engine speed (r/min) 1100 1500 1500 2000 2000

IMEP: indicated mean effective pressure.

8 International J of Engine Research 00(0)

results with the experimental data. The model’s capa-bility of predicting the physical combustion process isdemonstrated by comparing the associated simulationresults of certain important combustion variables andthermodynamic states.

Model calibration

Based on the discussion at the end of ‘‘The reaction-based combustion model’’ section, there are a set ofmodel parameters to be calibrated empirically (seeTable 3). These coefficients can be grouped into twosets: low and high sensitivity parameters, where highsensitivity ones need to be calibrated carefully.

The reaction order nj (j=1, 2, 3, 4, 5) in equations(23) and (24) and the activation energy Ea, i (i=1, 2, 3)in the Arrhenius function equation (25) for calculatingreaction rates of the two-step chemical reactionmechanism have low sensitivity and these parametersdo have their recommended values verified based onstandard combustion experiments. Therefore, theseparameters are modified slightly from the reference val-ues and kept unchanged with respect to the engineoperating condition (see Table 4 for the values used).

There are seven parameters with high sensitivity andthey are as follows: a in equation (16) and b in equa-tion (17) associated with the heat loss to the cylinderboundary, kc in equation (6) with the heat transferbetween two zones, km in equation (18) for the masstransfer through the interface between two zones, andAi (i=1, 2, 3) in equation (25) for the two-step chemi-cal reaction rate.

The heat loss coefficients a and b should be cali-brated first under no combustion condition to makesure that the simulated pressure tracks the experimentalone from IVC to spark. The simulated and experimen-tal pressure curves with the calibrated heat loss coeffi-cients are compared in Figure 4, where the simulatedpressure is the dashed line and experiment one is thesolid one.

The heat transfer _Qtr between two zones is veryimportant since it affects the thermodynamic propertiesin each zone, for instance, zone temperature, in-cylinderpressure, chemical reaction rate, and so on. Also, themass transfer between the two zones is partially depen-dent on the calculated heat transfer. In this model, theheat coefficient kc in equation (6) is the key to haveaccurate heat transfer.

km in equation (18) is the mass coefficient that gov-erns the mass flow rate between two zones. It indicatesthrough the calibration process that the mass transferrate _mtr dominates the modeled combustion process.The accurate mass transfer rate from the unburned toreaction zone shall be calibrated well to ensure an accu-rate combustion process.

The pre-exponential factor Ai (i=1, 2, 3) in theArrhenius function is the main coefficient for calibrat-ing the two-step chemical reaction mechanism. If Ai

were too large, the auto-ignition could happen before

the spark, leading to pre-ignition, and if Ai were toosmall, the reaction rate would be too slow to start thecombustion process after the spark.

The high sensitivity parameters are listed in Table 5with associated calibration values. Note that these cali-brated parameters are fixed under different engine oper-ating conditions.

Thermodynamic properties

This subsection shows the model performance of pre-dicting thermodynamic states in the combustion cham-ber, such as the in-cylinder pressure, zone temperaturesand volumes, and mass transfer between two zones.Since the simulation results are pretty similar for allfive cases listed in Table 2, to simplify the presentation,only the simulation results for cases 3 and 5 are shown

Table 3. Parameters to be calibrated.

Parameter Equation Group

nj(j = 1;5) (23), (24) Low sensitivityTa, i(i = 1, 2, 3) (26)a, b (16), (17) High sensitivitykc, km (6), (18)Ai(i = 1, 2, 3) (25)

Table 4. Calibrated parameters with low sensitivity (Ta, i isdefined in equation (26)).

n1 n2 n3 n4 n5 Ta, 1 Ta, 2 Ta, 3

0.25 1.5 0.3 0.25 1 15,540 K 17,900 K 28,130 K

-150 -100 -50 0 50 100 1500

0.5

1

1.5

2

2.5

3

106

Calculated, with comb.ExperimentalCalculated, no comb.

-150 -100 -50 0 50 100 150-10

0

10

error

Figure 4. Comparison of simulated and experimental in-cylinder pressures with relative error at 1500 r/min with6.78 bar IMEP (case 3).

Li et al. 9

in detail for in-cylinder pressure and temperature. Forthe mass transfer and zone volumes, simulation resultsof case 3 are used. The results for all five cases are sum-marized in Table 6 at the end of this section.

In-cylinder pressure. The experimental data of 100 cyclein-cylinder pressure are averaged and used to comparewith the simulated one. The experimental and simulatedin-cylinder pressures of cases 3 and 5 are compared inFigures 4 and 5, respectively.

The simulated cylinder pressures in both figuresmatch the experimental data very well. The relativeerror (%) at the early stage of compression phase(within 50 CAD after IVC) is a little bit high (around7%). However, we are interested in the error betweenSOC and EVO for the developed combustion model. Inthis case, the maximum relative error for cases 3 and 5are 4:09% (occurred at around 20 CAD after top deadcenter (ATDC)) and 5:19% (occurred at around 24CAD ATDC), respectively.

Zone temperature. One of the major advantages of theproposed model is its capability of predicting tempera-tures in both zones. The simulation results of theunburned zone (Tu) and reaction zone (Tr) tempera-tures are shown in Figure 6 for case 3. The averagedtemperature over two zones is also plotted in the samefigure for comparison purpose. One can see that thetwo zone temperatures are the same before spark event,and after the spark, combustion starts and the heatrelease from chemical reaction increases the reactionzone temperature Tr, while in the unburned zone, thetemperature increment is much slower.

For the first half of combustion (between 20 CADbefore top dead center (BTDC) and 25 CAD ATDC),over 60% of the total mass in the unburned zone is

transferred to the reaction zone with very fast combus-tion in the reaction zone. As a result, the temperaturedifference between Tr and Tu increases rapidly andpeaks at around 15 CAD ATDC. The chemical energyreleased in the reaction zone gets transferred to theunburned zone and heats up the mixture in theunburned zone, leading to a mild increment of Tu atthis stage. As combustion continues, for the second-half of combustion (between 25 CAD and 40 CADATDC), over 95% of the mass in the unburned zone

Table 5. Calibrated parameters with high sensitivity.

a b km kc A1 A2 A3

0.978 2.608 5.2e-7 295 4.15e10 15.21e16 1.98e8

Table 6. Modeling error of in-cylinder pressure for all fivecases between SOC and EVO.

Case Max. relative error RMSE (bar)

1 6.20% 0.262 3.40% 0.243 4.09% 0.284 5.50% 0.235 5.19% 0.37

SOC: start of combustion; EVO: exhaust valve opening; RMSE: root

mean square error.

-150 -100 -50 0 50 100 1500

1

2

3

4106

-150 -100 -50 0 50 100 150-10

010

error

Figure 5. Comparison of simulated and experimental in-cylinder pressures at 2000 r/min with 8.23 bar IMEP (case 5).

-150 -100 -50 0 50 100 1500

500

1000

1500

2000

2500

3000Unburned Zone temperatureReaction Zone temperatureAverage temperature

Figure 6. Zone and average temperatures at 1500 r/min with6.78 bar IMEP (case 3).

10 International J of Engine Research 00(0)

flows to the reaction zone, resulting in a fast decrementof the unburned zone size. The decreased unburnedzone size with the continued heat transfer from thereaction zone increases the unburned zone temperaturequickly and reaches its peak at the end of combustion.After the end of combustion, two zone temperaturesconverge, and then decrease due to expansion.

To verify the model performance of predicting in-cylinder temperature, the simulated average tempera-ture Tavg is compared with the experimental tempera-ture calculated based on the measured in-cylinderpressure, the experimental in-cylinder temperature Texp

is derived using the ideal gas law based on the collectedexperimental in-cylinder pressure data, including therecorded in-cylinder pressure, volume of the combus-tion chamber and the total mass in the chamber. Thecomparison result and the associated error are shownin Figure 7. It shows that the simulated average in-cylinder temperature matches with the experimentaldata very well with a maximum relative error of 4:8%between SOC and EVO occurred at 45 CAD ATDC.

The simulation results for case 4 are shown in Figure 8,where the maximum relative error, in this case, is 5:7%.Therefore, the capability of predicting the mixture tem-perature and thermal stratification is demonstrated.

Mass transfer. The mass transfer between the reactionand unburned zones for case 3 is shown in Figure 9. Asdiscussed earlier, for this study, the initial unburnedzone mass is 98:5% of total mass and the reaction zoneis 1:5%. Also, the mass transfer does not occur untilcombustion is initiated.

The mass transfer rate _mtr in Figure 9 indicates thatthe mass flow rate is zero before spark. And then, aftera small ignition delay, the mass transfer starts andincreases rapidly to provide a proper amount of pre-mixed mass to the reaction zone to burn. Two masscurves of the reaction and unburned zones also indicatethat mass in both zones remains unchanged before andafter combustion. During the combustion, theunburned zone mass decreases very quickly due to massflow into the reaction zone.

Note that the mass transfer rate does not drop downto zero at the end of combustion. As shown in thezoomed plot in Figure 9, mass transfer rate goes to zerovery slowly in the combustion phase, resulting in asmall mass flow into the reaction zone. In this case,only 97% fuel is burned.

Zone volume. The volume variations in the two zonesare similar to mass case (in Figure 9) as the zone volumeis governed by the ideal gas law (see Figure 10 (case 3)).

The unburned zone volume begins decreasing afterthe SOC and the reaction zone volume keeps increasing(with an initial value of 1:5%) due to the mass transferfrom the unburned zone to the reaction zone. The reac-tion zone volume in Figure 10 also indicates that thevolume increases very quickly between SOC and middleof combustion, and after that, it increases slowly up to

-150 -100 -50 0 50 100 1500

500

1000

1500

2000

2500

ExperimentalCalculated

-150 -100 -50 0 50 100 150-10

0

10

error

Figure 7. Comparison of experimental and simulated cylindertemperatures at 1500 r/min with 6.78 bar IMEP (case 3).

-150 -100 -50 0 50 100 1500

500

1000

1500

2000

2500

ExperimentalCalculated

-150 -100 -50 0 50 100 150-10

0

10error

Figure 8. Comparison of experimental and in-cylinder averagetemperatures at 2000 r/min with 6.83 bar IMEP (case 4).

-150 -100 -50 0 50 100 1500

1

2

3

410-4

Mass in unburned zoneMass in reaction zone

-150 -100 -50 0 50 100 150

0

0.05

0.1

0.15Mass transfer rate

40 50 600

2

410-3

Figure 9. Mass transfer and its rate between two zones at1500 r/min with 6.78 bar IMEP (case 3).

Li et al. 11

96:9% of total volume at the end of combustion.During the engine expansion process, the volume has aslight increment from 96:9% to 99:3%, due to the smallmass flow into the reaction zone (see discussions in thelast subsection).

Combustion simulation results

The combustion characteristics of each chemical speciesin both zones can also be simulated in this model. Thesimulation results for case 3 are used as an example todiscuss the model’s capability of predicting the combus-tion process. To compare the combustion simulationresults, the reaction rate vchem and mass flow-in ratevflow of fuel and O2 in the reaction zone are shown inFigure 11.

As discussed earlier in ‘‘The reaction-based combus-tion model’’ section, the chemical reaction is modeled

using a two-step reaction mechanism and CO is theproducts from the first reaction step. In the secondstep, CO gets burned to produce CO2, and CO2 splitsinto CO and 0.5O2. As shown in Figure 11, the reactionrate (in (kmol/m3 s)) of fuel and O2 is negative duringthe combustion phase, indicating that the fuel and O2

are reacted. The reaction rates of these two species arezero before the SOC and then increase and reach thefirst peak, indicating that the initial mass in the reac-tion zone is burned due to the added spark energy.These two peaks are mainly caused by the extremelysmall initial reaction zone volume (1:5% of total vol-ume) at the SOC and the assumed auto-ignition in theignition zone.

After the combustion starts, the gas mixture startsflowing into the reaction zone and gets burned continu-ously; see the two smooth positive mass flow (in (kmol/m3 s)) curves of fuel and O2 in Figure 11. The reactionrates of CO, CO2, and H2O changes in a similar way(see Figure 12 for the total mass of these species).

The total mass variation trend of each individualchemical species in the entire cylinder is a good indica-tor for the actual combustion process. Since N2takes nopart in the chemical reaction, its mass keeps unchanged.The mass variations of other five chemical species (fuel,O2, CO2, H2O, and CO) during the combustion processare shown in Figure 12. It indicates that the total fuelmass in the cylinder remains constant until SOC anddrops quickly down close to zero at the end of combus-tion (around 38 CAD ATDC), indicating that the fuelis almost fully burned. The corresponding O2 mass isalso changed in a similar way and reduces very fast;however, there is a small amount of O2 left at the endof combustion since the actual l was 1:05.

The total mass of species CO increases at the begin-ning of the combustion and gets burned in the secondstep. There is a small peak before TDC, which is causedby the spark. Considering the products of the two-stepchemical reaction (CO2 and H2O), H2O is zero before

-150 -100 -50 0 50 100 1500

0.2

0.4

0.6

0.8

1

Figure 10. Individual volume fractions at 1500 r/min with6.78 bar IMEP (case 3).

-20 -10 0 10 20 30 40 50-350

-300

-250

-200

-150

-100

-50

0

50

reaction rate of Fuelreaction rate of O

2

mass flow rate of Fuelmass flow rate of O

2

0 20 40

-20

-10

0

Figure 11. Comparison of reaction rates and mass flow ratesof fuel and O2 in the reaction zone at 1500 r/min with 6.78 barIMEP (case 3).

-150 -100 -50 0 50 100 1500

2

4

6

8

10-5

-150 -100 -50 0 50 100 1500123

10-7

Figure 12. Total mass changing of species (fuel, O2, CO2, H2O,and CO) at 1500 r/min with 6.78 bar IMEP (case 3).

12 International J of Engine Research 00(0)

SOC and then increases rapidly during the combustionprocess, remains unchanged after combustion, but themass of CO2 has a slight increment after the combus-tion. By inspecting the CO curve after combustion, thetotal CO mass drops down to very low (near zero) level,and then gets burned by reacting with the remaining O2

slowly during the expansion phase, resulting in a slightincrement of CO2 during the expansion phase.

The HRR, along with heat transfer related to themass transfer ( _Qm) and heat transfer to the unburnedzone ( _Qt) is shown in Figure 13. Comparing the _Qm

and _Qt indicates that most of the total heat transfer( _Qm) is used to transfer the mass from the unburnedzone to the reaction zone during the first half of thecombustion process, where significant part of mass isprovided to the reaction zone. After that, during thelate phase of combustion, the heat transfer to theunburned zone _Qt takes a leading role, while _Qm

reduces quickly. This is reasonable since large amountof chemical energy released during combustion is trans-ferred to the unburned zone, which is one of the mainreasons for the rapid increment of unburned zonetemperature.

The simulation results for the heat loss are shown inFigure 14, where the red-line is for the heat loss fromthe reaction zone to the cylinder head and liner ( _Qw1),the blue-line is for the heat loss from the unburned zoneto the rest surface area of the cylinder boundary ( _Qw2),and the purple-line is for the total heat loss from thecombustion chamber to the cylinder liner.

_Qw1 is almost zero before the ignition because thereaction zone volume is almost zero compared with thatof the unburned zone. After the SOC, the releasedchemical energy increases the reaction zone temperaturequickly and the reaction zone expands along with theflame propagation, resulting in the rapid increment ofheat loss from the reaction zone to the cylinder bound-ary. Physically, the heat transfer area from the reaction(burned) zone to the cylinder boundary, after the

reaction (burned) zone gets large enough, is the cylinderhead surface plus cylinder liner area (Figure 1, left).However, in this article, the combustion chamber isassumed to have a hemisphere shape as shown in theright image of Figure 1, and it is also assumed that theheat transfer from the burned zone to the wall is thebase (top) area of the burned zone. With these twoassumptions, the base area is not limited by the area ofcylinder head. When the calculated area is larger thanthe cylinder head area, the extra area is considered asthe cylinder wall area.

The heat loss from the unburned zone to the cylinderboundary _Qw2, comparing with _Qw1, has an obviousdelay and increases quickly during the late combustionphase and the expansion phase due to the high tem-perature of unburned zone mixture. Since the unburnedzone becomes a very thin layer by the end of combus-tion, the heat loss from the unburned zone to the wallis quite small, comparing with the heat loss from reac-tion zone to the wall. Note that the total heat loss inFigure 14 matches with that in the literature.23–25

The calculated IMEP and crank angle, where 50%of fuel was burned (CA50) of all five cases are com-pared with experimental data in Figures 15 and 16.

A summary of model prediction error for the in-cylinder pressure is shown in Table 6, where the errorsshown cover pressure signals between SOC and EVO.Note that the error between IVC and SOC is not themain focus of this article.

Potential for knock prediction

As discussed earlier, due to the high temperature in theunburned zone, the unburned mixture (end-gas) couldbe auto-ignited, causing engine knock. The criterionused to predict knock is the auto-ignition (or SOC) atthe crank angle when the ARI reaches 1 (see equation(43)). And all the thermodynamic properties of each

-150 -100 -50 0 50 100 1500

0.5

1

1.5

2

2.5

3

Figure 14. Heat losses from reaction and unburned zones tothe cylinder boundary at 1500 r/min with 6.78 bar IMEP (case 3).

-5 0 5 10 15 20 25 30 35 400

20

40

60

80

100

Figure 13. Heat release rate and heat transfer rates betweentwo zones at 1500 r/min with 6.78 bar IMEP (case 3).

Li et al. 13

species in the unburned zone can be simulated in thismodel, so the knock phenomenon in the unburned zonecan be predicted based on equations (42) and (43).

As shown in Figure 17, the red solid-line is the simu-lated ARI for case 3 calculated from the concentrationof fuel and oxygen, zone temperature, and so on. Thered dash-line is for ARI=1. The ARI is positive andmonotonically increasing as time goes (see equation(42)) and reaches 1 at the crank angle around 33 CADATDC. The two-zone temperatures, in the black solid-line for the unburned zone and black dash-line for thereaction zone, are shown as references. It indicatesknock condition at this moment of ARI=1 withunburned zone temperature around 1900K, which isreasonable for the engine knock condition.

The reaction-based combustion model was devel-oped for predicting engine knock and associated con-trol strategy development, and it will be a useful

modeling tool for other control applications requiring areal-time combustion model. This article mainly focuseson the combustion modeling. Knock prediction is notaddressed in detail and will be the future work. We aredeveloping a real-time 1D pressure-wave model basedon this combustion model to predict the detailed knockpressure wave so that knock timing and pressure-waveintensity can be predicted.

Conclusion

A control-oriented, 0D, two-zone, reaction-based ther-modynamic model for compression, combustion, andexpansion phases of SI engines is developed, calibrated,and validated against experimental data in this article.The developed model is capable of predicting thermo-dynamic characteristics of in-cylinder chemical mix-tures, combustion process, properties of individualchemical species in both unburned and reaction zones.Furthermore, it is also able to predict the auto-ignitionin the unburned zone (engine knock).

Simulation results show that the developed two-zonecombustion model is able to predict the in-cylinderthermal states and combustion process of SI engines,such as the SOC, flame propagation process, and in-cylinder heat and mass transfer. The simulated zonetemperatures indicate how the temperature differencegoverns the heat and mass transfer rates in two zones.The proposed two-step reaction-based combustionmodel is capable of accurately predicting combustionprocess, including the mass variation and thermalproperties of each chemical species. Note that the abil-ity to simulate the zone stratification and species molarconcentrations allows predicting engine knock. Asensitivity-based calibration process divides calibrationparameters into two groups with low and high sensitiv-ity, where the default values are used for low sensitivityones and seven high sensitivity ones are carefully cali-brated using experimental data. Note that for this

-150 -100 -50 0 50 100 1500

500

1000

1500

2000

2500

0

5

10

15

20

25

Figure 17. Unburned zone ARI at 1500 r/min withIMEP = 6.78 bar (case 3).

0 2 4 6 8 100

2

4

6

8

10

Figure 15. Comparison of simulated and experimental IMEPsfor all five cases.

0 5 10 15 20 25 300

5

10

15

20

25

30

Figure 16. Comparison of simulated and experimental CA50for all five cases.

14 International J of Engine Research 00(0)

combustion model, the presented simulation resultsuses only one set of calibration parameters, whichmeans that the model does not need to be re-calibratedunder different operational conditions.

As a summary, the proposed model is able to accu-rately predict the in-cylinder temperature, pressure,MFB, HRR, and so on. The maximum relative errorfor the in-cylinder pressure is less than 6.2% under fiveoperational conditions studied with one set of calibra-tion parameters.

Declaration of conflicting interests

The author(s) declared no potential conflicts of interestwith respect to the research, authorship, and/or publi-cation of this article.

Funding

The author(s) disclosed receipt of the following finan-cial support for the research, authorship, and/or publi-cation of this article: This research was partiallysupported by Ford University Research Program underfunding number 2015-8032R.

ORCID iD

Guoming G Zhu https://orcid.org/0000-0002-2101-2698

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Appendix 1Notation

Ah cylinder head surface areaAi pre-exponential factorAp piston crown surface areaAtr effective contact area between two zonesAw1 surface area between the reaction zone

and the WallAw2 surface area between the unburned zone

and the WallB cylinder bore�cp molar specific heatcp, u specific heat of mixture in unburned zoneEa activation energy�h molar enthalpyhc1, hc2 heat coefficient for heat losskc heat coefficient for heat transferkm mass coefficient for mass transferki(T) specific reaction rate constantL piston strokema fresh air massmf fuel massmu, 0 initial mass in unburned zonemu unburned zone massmr reaction zone massmREG mass of residual gasmtot total mass in cylinder_mtr mass transfer rateMWmix molecular weight of all mixtures at IVC

MWu unburned zone molecular weightni reaction orderNi molar number of species ip cylinder pressurepIVC cylinder pressure at IVC_Qign ignition energy_Qm heat transfer rate for mass transfer_Qt heat transfer rate for heating up unburned

zone _Qtr total heat transfer rate_Qw1 heat loss from reaction zone to the wall_Qw2 heat loss from unburned zone to the wallrc compression ratioR ratio of connecting rod length to crank

radiusRr radius of reaction zoneRu universal gas constantSp piston speedTavg average temperature in cylinderTg virtual region temperatureTIVC gas temperature at IVCTu unburned zone temperatureTr reaction zone temperatureVcyl total in-cylinder volumeVIVC cylinder volume at IVCVc clearance volumeVu unburned zone volumeVr reaction zone volume½X�i molar concentration of species i½ _X�i molar concentration variation rate of

species i

u crank anglel relative air–fuel ratiovM1 reaction rate of first stepvM2 reaction rate of second step

16 International J of Engine Research 00(0)