Original article

Proc IMechE Part DJ Automobile Engineering2017 Vol 231(10) 1315ndash1325 IMechE 2016Reprints and permissionssagepubcoukjournalsPermissionsnavDOI 1011770954407016670303journalssagepubcomhomepid

A control-oriented model of turbulentjet ignition combustion in a rapidcompression machine

Ruitao Song Gerald Gentz Guoming Zhu Elisa Toulson andHarald Schock

AbstractTurbulent jet ignition combustion is a promising concept for achieving high thermal efficiency and low NOx (nitrogen oxi-des) emissions A control-oriented turbulent jet ignition combustion model with satisfactory accuracy and low computa-tional effort is usually a necessity for optimizing the turbulent jet ignition combustion system and developing theassociated model-based turbulent jet ignition control strategies This article presents a control-oriented turbulent jetignition combustion model developed for a rapid compression machine configured for turbulent jet ignition combustionA one-zone gas exchange model is developed to simulate the gas exchange process in both pre- and main-combustionchambers The combustion process is modeled by a two-zone combustion model where the ratio of the burned andunburned gases flowing between the two combustion chambers is variable To simulate the influence of the turbulent jetson the rate of combustion in the main-combustion chamber a new parameter-varying Wiebe function is proposed andused for the mass fraction burned calculation in the main-combustion chamber The developed model is calibrated usingthe least-squares fitting and optimization procedures Experimental data sets with different air-to-fuel ratios in both com-bustion chambers and different pre-combustion chamber orifice areas are used to calibrate and validate the model Thesimulation results show good agreement with the experimental data for all the experimental data sets This indicates thatthe developed combustion model is accurate for developing and validating turbulent jet ignition combustion control stra-tegies Future work will extend the rapid compression machine combustion model to engine applications

KeywordsRapid compression machine turbulent jet ignition two-zone combustion model

Date received 9 October 2015 accepted 23 August 2016

Introduction

The research and development of internal combustion(IC) engines have never stopped since they wereinvented over a century ago As strict emission stan-dards appeared and the importance of vehicle fuel effi-ciency improvement were realized new combustiontechnologies were proposed and studied during the pastfew decades such as the homogeneous charge compres-sion ignition (HCCI) combustion12 Some of thesetechniques have entered production phase like theAtkinson cycle-based engines used for many hybridelectric vehicles Turbulent jet ignition (TJI) combus-tion is another promising combustion technology thathas the potential to be used in the next generation ICengines

The TJI combustion system was proposed almost acentury ago In 1918 Harry R Ricardo first developed

and patented the engine using a TJI system3 In the1970s more research efforts were devoted to the devel-opment of new TJI systems Honda developed the com-pound vortex controlled combustion (CVCC) systemthat is considered the most significant development inOttocycle engines with the TJI system4 It was able tomeet the 1975 emission standards without a catalyticconverter

A typical TJI system consists of a main-combustionchamber and a small pre-combustion chamber Its vol-ume is a few percent of that of the main-combustion

Department of Mechanical Engineering Michigan State University USA

Corresponding author

Guoming Zhu Department of Mechanical Engineering Michigan State

University E148 ERC South East Lansing MI 48824 USA

Email zhugegrmsuedu

chamber The two combustion chambers are con-nected through a few small orifices The airndashfuel mix-ture is lean in the main-combustion chamber andrelatively rich (or close to stoichiometric) in the pre-combustion chamber to make spark ignition (SI) easyConsequently the TJI system usually needs two fueldelivering systems for the two combustion chambersThe combustion process is initiated by a spark insidethe pre-combustion chamber Then the turbulent jetsof the reacting products from the pre-combustionchamber flow into the main-combustion chamber andignite the airndashfuel mixture in the main-combustionchamber

TJI combustion possesses many advantages overother combustion technologies One of the approachesto reducing the NOx (nitrogen oxides) emissions is tooperate the engine under very lean conditions with itsrelative air-to-fuel ratio (AFR) greater than 1 since theresulting relatively low temperature combustion leadsto a significant reduction of NOx formation Note thatsignificant NOx emission reduction can only beachieved at the extremely lean condition for conven-tional SI engines5 The extremely lean operation of con-ventional SI engines will lead to poor combustionstability with high occurrence of misfire due to the nar-row fuel flammability limits The lean mixture also hasvery slow laminar flame speed that often leads toincomplete combustion As a result lean operation inconventional SI engines significantly increases HC(hydrocarbon) and CO (carbon monoxide) emissionsHowever in the TJI combustion system the mixture inthe main-combustion chamber is ignited by the hot tur-bulent jet that contains much higher energy than aspark plug can provide6 As a result the lean airndashfuelmixture can be ignited and burned at a very fast ratewith high combustion stability Therefore the TJI com-bustion system is able to greatly reduce NOx emissionswhile maintaining comparatively low HC and CO emis-sions especially when the relative AFR l is greaterthan 14 According to previous research stable com-bustion can be achieved for the TJI system when l is upto 18 approaching the elimination of NOx emissions7

HCCI combustion is also able to run the engineunder very lean conditions However since HCCI com-bustion is not suitable for all engine operational condi-tions from low to high engine load mode transitionbetween SI and HCCI combustion is required Thecombustion mode transition control along with thecombustion phase control are two of the key challengesfor HCCI combustion technology89 In contrast TJIcombustion is able to cover the entire loadndashspeed rangeof a typical SI engine and the start of combustion canbe easily controlled by adjusting the spark timing in theTJI system Note that as the engine load increases theachievable lean limit decreases

The spark timing rate of combustion and othercombustion parameters in the TJI system need to beoptimized by control strategies to achieve the best fuel

efficiency with reduced emissions To develop and vali-date the TJI combustion control strategies a control-oriented TJI combustion model is also requiredToulson modeled a TJI engine using the computationalfluid dynamics (CFD) method10 Ghorbani modeled atransient turbulent jet by the probability density func-tion (PDF) method11 These investigations provideinsight into better understanding TJI combustionHowever these models are too detailed to be used formodel-based control Model-based combustion controlrequires a simple combustion model capable of captur-ing the TJI system dynamics with good accuracy andlow computational and calibration efforts12

In this article a control-oriented TJI combustionmodel is developed for a rapid compression machine(RCM) equipped with a TJI system The gas exchangeprocess in the combustion chambers before combustionis simulated by a one-zone gas exchange model It isbased on the assumption that the air and fuel are uni-formly mixed in both combustion chambers After igni-tion both combustion chambers are divided intoburned and unburned zones The ratio between theburned and unburned gases flowing through the ori-fices connecting the two combustion chambers areadjusted due to the tiny pre-combustion chamber toimprove the model accuracy To link the two combus-tion processes in both combustion chambers a newparameter-varying Wiebe function is proposed andused for the main-combustion chamber mass fractionburned (MFB) calculation The newly proposed Wiebefunction allows the combustion rate in the main-combustion chamber to vary based on the characteris-tics of turbulent jets from the pre-combustion chamberwhich is one of the key features of TJI combustion

The main contribution of this article is the develop-ment of a control-oriented TJI combustion model capa-ble of real-time simulation Especially the utilization ofthe newly proposed parameter-varying Wiebe functionmakes it possible to link the two combustion events inboth pre- and main-combustion chambers

The article is organized as follows The next sectionbriefly describes the TJI system installed on a RCMand the following section provides the governing equa-tions of the gas exchange and combustion processesThe developed model was calibrated and then validatedusing the experimental data from the RCM atMichigan State University The conclusions are drawin the last section

System description

Figure 1 shows the basic architecture of the RCMequipped with a TJI system modeled in this articleThere is an auxiliary fuel system injecting the methaneinto the pre-combustion chamber The two combustionchambers are connected by a small orifice The detailedparameters of the system are listed in Table 1

1316 Proc IMechE Part D J Automobile Engineering 231(10)

In this article methane was used as the fuel for allthe experiments Two Kistler piezoelectric pressure sen-sors were installed into the two combustion chambersfor pressure measurements During the experiment thecombustion chamber wall was heated to 80C Thecombustion chambers were firstly evacuated by avacuum pump and then filled with airndashfuel mixturewith a known AFR Then the piston rapidly com-pressed the mixture in both combustion chambers Atthe same time a charge of fuel was injected into thepre-combustion chamber see Figure 2 At the end ofcompression the piston kept still and therefore thevolume in the main-combustion chamber remained con-stant At the falling edge of the dwell control signal thespark is initiated through the spark plug inside the pre-combustion chamber and then the reacting productsfrom the pre-combustion chamber were injected intothe main-combustion chamber and ignited the airndashfuelmixture Figure 2 shows the control signals and the typ-ical pressure traces measured during the experiment

In order to have a good mixing the fuel was injectedduring the compression The fuel flow mixes withthe gas flowing through the orifice from the main-combustion chamber during the injection leading tobetter mixing Moreover to allow enough time for themixing process after fuel injection the injection timingwas set at the beginning of the compression In this waywe are able to make sure that the airndashfuel mixture in thepre-combustion chamber is close to uniformly mixed

Turbulent jet ignition combustion model

Gas exchange model

During compression methane is injected into the pre-combustion chamber The mass flow rate of the injected

methane can be calculated by the one-dimensional com-pressible flow equation13

_minj =Cd1Av1PinjffiffiffiffiffiffiffiffiffiffiffiRTinj

p cPpre

Pinj

Pinj Ppre eth1THORN

where

c xeth THORN=

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffik 2

k+1

k+1eth THORNk1eth THORN

qx 2

k+1

kk1eth THORN

x1k

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi2k

k1 1 xk1eth THORN

k

rx5 2

k+1

kk1eth THORN

8gtltgt eth2THORN

Note that the coefficient Cd1 is experimentally deter-mined Av1 is the orifice area of the fuel injector k is theratio of specific heats R is the gas constant Pinj andTinj are the upstream pressure and temperature respec-tively and Ppre is the pressure in the pre-combustionchamber

The gas exchange process between the two combus-tion chambers is modeled similarly However the pres-sure in the pre-combustion chamber can be eithergreater or less than that in the main-combustion cham-ber The mass flow rate between the two combustionchambers is calculated by the following equation

_mtur =

Cd2Av2PpreffiffiffiffiffiffiffiffiffiffiffiffiRTpre

p cPmain

Ppre

Ppre5Pmain

Cd2Av2PmainffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiRTmain

p cPpre

Pmain

Ppre Pmain

8gtgtgtltgtgtgt

eth3THORN

where Cd2 and Av2 are the discharge coefficient and thearea of the orifice connecting the two combustionchambers The subscripts pre and main denote the pre-combustion and main-combustion chamber propertiesrespectively

Before ignition the pre-combustion chamber is con-sidered as a control volume with mass and energyexchange The mass and energy conservation equationsare used to describe such a control volume

dmpre

dt= _minj _mtur

dUpre

dt= _Hinj _Htur _Qht

eth4THORN

Figure 1 Rapid compression machine

Table 1 Rapid compression machine (RCM) specifications

Parameter Value

Bore 508Stroke 202Compression ratio 851Pre-combustion chamber volume 23 cm3

Pre-combustion chamber orifice diameter 15ndash30 mm

Figure 2 Working process of the rapid compression machine(RCM)

Song et al 1317

where mpre and Upre are the mass and internal energyof the gas in the pre-combustion chamber respectively_Qht is the heat transfer rate through the chamber walland H is the enthalpy flow The subscript inj and turrepresent the properties of the gas from the fuel injectorand through the orifice connecting the two combustionchambers respectively

Assuming that the gas can be considered as an idealgas the two equations can be coupled by the ideal gaslaw below

Ppre Vpre =mpre R Tpre eth5THORN

where Vpre is the pre-combustion chamber volumeSubstituting equation (5) into equation (4) the follow-ing two equations are obtained to calculate the gaspressure and temperature

dPpre

dt=

R

Vprecvcp _minjTinj cp _mturTtur _Qht

dTpre

dt=

TpreR

PpreVprecvcp _minjTinj cp _mturTtur

cv _minj _mtur

Tpre _Qht

eth6THORN

where cp and cv are the specific heat at constant pres-sure and constant volume respectively and

Ttur =Tpre _mtur 0

Tmain _mtur40

(eth7THORN

The pre-combustion chamber volume is only around2ndash4 of the main-combustion chamber clearance vol-ume Therefore the gas flowing between the twocombustion chambers can be neglected for the main-combustion chamber model The pressure and tem-perature in the main-combustion chamber can besolved using the energy and mass conservation equa-tions These equations can be found in many other arti-cles about engine modeling and thus will not be shownhere112

Two-zone combustion model

After ignition the pre-combustion chamber is dividedinto two zones to improve the model accuracy Boththe burned and unburned zones can be regarded as con-trol volumes Besides the mass enthalpy and workexchange between the two control volumes there is alsomass and enthalpy exchange through the orifice to themain-combustion chamber see Figure 3 The burnedand unburned gases in the pre-combustion (or main-combustion) chamber are assumed to enter the burnedzone and unburned zone in the main-combustion (orpre-combustion) chamber respectively Before the igni-tion of the main-combustion chamber all the gas fromthe main-combustion chamber is considered asunburned gas The energy balance equation of theburned zone is shown in equation (8) To make theequations concise the variables in the following

equations in this subsection are for the pre-combustionchamber if not specified

cvd(mbTb)

dt+P

dVb

dt+ xb _Qht =

_Qch +mu

1 xb

dxbdt

cpTu _mturbcpTb

eth8THORN

The energy balance equation of the unburned zone isrepresented by

cvd(muTu)

dt+P

dVu

dt+ 1 xbeth THORN _Qht =

mu

1 xb

dxbdt

cpTu _mturucpTu

eth9THORN

The masses of both burned and unburned zones areobtained based on the following mass conservation law

dmb

dt= _mturb +

mu

1 xb

dxbdt

dmu

dt= _mturu

mu

1 xb

dxbdt

eth10THORN

The subscripts b and u represent the burned zoneand unburned zone Qht is the heat transfer to thechamber wall Qch is the chemical energy released bycombustion xb is the MFB mturb and mturu repre-sent the burned gas and unburned gas flowing throughthe orifice Correspondingly the area of the orifice isalso divided into two parts One for the burned gas andthe other for the unburned gas Figure 3 shows thebasic idea of the two-zone combustion model Whenthe pressure in the pre-combustion chamber is greaterthan that in the main-combustion chamber the twomass flow rates are calculated by

_mturb =abCd2Av2PpreffiffiffiffiffiffiffiffiffiRTb

p cPmain

Ppre

_mturu = 1 abeth THORNCd2Av2PpreffiffiffiffiffiffiffiffiffiRTu

p cPmain

Ppre

eth11THORN

Similar result can be obtained when the pressure in themain-combustion chamber is greater than that in thepre-combustion chamber

The coefficient ab in equation (11) is chosen as afunction of the volume fraction of the burned gas vbAssuming that the burned and unburned gases werealways well mixed ab would be always equal to vb

Figure 3 Two-zone combustion modelab area fraction for burned gas

1318 Proc IMechE Part D J Automobile Engineering 231(10)

However in reality this is not the case ab is combus-tion chamber structure dependent For our TJI systemthe spark plug is located at the top of the pre-combustion chamber see Figure 1 In this case thecombustion is initiated at the top of the pre-combustionchamber Since the orifice is at the bottom it is hardfor the burned gas to escape from the pre-combustionchamber at the early stage of the combustion As aresult the fraction of the burned gas flowing throughthe orifice to the main-combustion chamber is muchsmaller than the burned gas fraction inside the pre-combustion chamber This is why ab is smaller than vbin the pre-combustion chamber when Ppre PmainNote that ab will be determined using experimentaldata When Ppre Pmain the gas in the main-combustion chamber flows through the orifice and ab

will be determined by the burned gas fraction in themain-combustion chamber Since the combustion inthe main-combustion chamber is initiated by the turbu-lent jet (close to orifice) the orifice is surrounded bythe gas with high concentration of burned gasTherefore ab is larger than vb in the main-combustionchamber And again the actual value will be deter-mined by the experimental data This is the main reasonwhy the two-zone combustion model is used The valueof ab can be expressed by equation (12)

ab =f1 vbpre cpre

Ppre Pmain

f2 vbmain cmaineth THORN Ppre Pmain

eth12THORN

To simplify the calibration process the two func-tions f1 and f2 are approximated by second-degreeBezier curves14 Besides the control points (00) and(11) (cpre1 cpre) was added for f1 and(cmain1 cmain) for f2 as the third control points seeFigure 4 The parameters cpre and cmain are experimen-tally determined The Bezier curve guaranteesab 2 0 1frac12 as long as cpre 2 0 1frac12 and cmain 2 0 1frac12 Bychanging cpre and cmain the ratio of the burned andunburned gases flowing through the orifice can beadjusted to better match the actual physical processand thus to improve the model accuracy

Applying the principle of mass conservation theinstant fuel mass in the pre-combustion chamber canbe obtained by

dmprefueldt

=

mprefuel1 xb

dxbdt _mturu

1

l A=Feth THORNs +1

eth13THORN

where l is the relative AFR and A=Feth THORNs is the stoichio-metric AFR Note that only the fuel from the unburnedzone is considered

From equations (11) and (13) it can be observedthat the total amount of fuel burned inside the pre-combustion chamber is highly influenced by ab

The rate of chemical energy release (CER) isobtained by the following relationship

_Qch =hpreQLHVmprefuel1 xb

dxbdt

eth14THORN

where the combustion efficiency hpre is experimentallydetermined and QLHV is the lower heating value of thefuel

The rate of heat transfer to the combustion chamberwall can be modeled by the following equation15

_Qht =Aprehc Tpre Tw

eth15THORN

where Apre is the pre-combustion chamber surface areaTw is the mean wall temperature and hc is the heat-transfer coefficient calibrated by the experiment

After the ignition in the pre-combustion chamberthe combustion in the main-combustion chamber willnot be initiated until the generation of the turbulentjet from the pre-combustion chamber Before the igni-tion of the main-combustion chamber the mass flowfrom the burned zone of the pre-combustion chamberto the main-combustion chamber is neglected Theamount of the fuel in the main-combustion chamberis calculated by

dmmainfueldt

= _mturu1

lpre A=Feth THORNs + 1

eth16THORN

where mmainfuel is the fuel mass in the main-combustion chamber

After ignition in the main-combustion chamber theburned zone is created Different from the two-zonecombustion model in a conventional SI engine thecombustion model of the main-combustion chamberneeds to consider the gas flowing through the orificeinto the pre-combustion chamber The mass and energyconservation equations for burned and unburned zonesare very similar to those of the pre-combustion cham-ber model presented in this subsection and are omittedhere The major difference is that the total volume ofthe main-combustion chamber is varying

Mass fraction burned model

The MFB in the pre-combustion chamber is obtainedfrom the Wiebe function16Figure 4 The value of ab in the two cases

Song et al 1319

xb =1 exp a t tignDtd

m+1

eth17THORN

The coefficients a and m are chosen to be 6908and 2 respectively tign is the start of ignition and Dtdis the burn duration that is calibrated by AFR beforeignition

At the early stage of the combustion in the main-combustion chamber the rate of combustion is deter-mined by not only the gas properties in themain-combustion chamber but also the turbulent jetfrom the pre-combustion chamber This is due to thefact that the turbulent jets create multiple and distribu-ted ignition sites which increases the overall flamefront area in the main-combustion chamber Moreoverthese turbulent jets increase the turbulence intensity inthe main-combustion chamber and thus the flame frontpropagation speed After the turbulent jet disappearsthe rate of combustion reduces gradually to a relativelylow level and its characteristics are mainly determinedby the gas properties only in the main-combustionchamber Here the term lsquointensityrsquo of the turbulent jetis used to describe the resulting increment of the com-bustion rate in the main-combustion chamber due tothe turbulent jet Since the intensity of the turbulent jetis determined by the combustion processes in bothcombustion chambers estimating the rate of combus-tion before ignition is difficult and requires significantcalibration effort Therefore adjusting the rate of com-bustion according to the turbulent jet intensity duringthe combustion process is preferred for the TJI com-bustion model The conventional single-Wiebe functionis not suitable for our combustion model The multi-Wiebe function is a possible approach for modeling theMFB However this requires determining all associatedparameters before ignition occurs Therefore a newparameter-varying Wiebe function is proposed andused in this article see equation (18)

x0b teth THORN=1 exp a ttign b teth THORNDtd

h im+1

tign b teth THORN= t0 R tt0b teth THORN 1frac12 dt

8lt eth18THORN

where t0 and Dtd are determined by the spark timingand the AFR in the main-combustion chamber Thecoefficients a and m are chosen to be 6908 and 2respectively

If a m and Dtd are the same in equations (17) and(18) it can be proved that for any given tign b = tign

dx0bdt

=dxbdt b teth THORN eth19THORN

In other word the combustion rate calculated by thenew Wiebe function is b(t) times larger than that calcu-lated by the conventional Wiebe function Thereforethe intensity of the turbulent jet can be mathematicallyexpressed by b teth THORN The combustion model is able toadjust the rate of combustion by making b(t) a functionof some characteristics of the turbulent jet Moreover

b(t) can be changed at any time during the combustionprocess As long as b teth THORN is greater than 0 the combus-tion rate is greater than 0 and x0b teth THORN tends to 1 as t goesto infinity From the available experimental results it isfound that the rate of combustion in the main-combustion chamber is highly related to the mass flowrate of the turbulent jets from the pre-combustionchamber As a result it is assumed that the intensity ofthe turbulent jet can be linked to its mass flow rateAlthough this assumption provides a good matchbetween the modeled and available experimentalresults it is important to find an accurate method tocalculate the intensity of the turbulent jet in the futurewhen more experimental data are available Since theinfluence of the turbulent jet on the combustion in themain-combustion chamber is also delayed b teth THORN is mod-eled to be proportional to the mass flow rate of the tur-bulent jet with first-order dynamics see equation (20)An offset is used such that b(t)=1 when the mass flowrate of the turbulent jet is zero

b teth THORN=b frac12( _m+tur fl)(t)+1

( _m+tur fl)(t)=

R t0

_m+tur(t)fl(t t)dt eth20THORN

Here is the convolution operator and _m+tur and fl are

defined as follows

_m+tur =

_mtur _mtur500 _mtur 0

fl(t)=vce

vctu(t)

eth21THORN

Note that u(t) denotes unit step The convolution of_m+tur with the exponential decay function fl represents

the first-order dynamics and is used to emulate the timedelay To be more specific fl is the time response of alow-pass filter with cutoff frequency vc The parametersb and wc are experimentally determined When _mtur40we have b teth THORN=1 which means that the combustion ratewill not be altered if there is no turbulent jet from thepre-combustion chamber

Model calibration

The combustion model was calibrated using the experi-mental data collected from the RCM at Michigan StateUniversity described in the lsquoSystem descriptionrsquo sectionThe model is firstly calibrated using two experimentaldata sets and validated using another data set Then tofurther validate the model more data sets with differ-ent pre-combustion chamber orifice sizes are used Theexperimental set-ups for the first three cases can befound in Table 2 where cases 1 and 2 are used formodel calibration and case 3 for model validation

The first step is to calibrate the heat transfer modelTo do this the net heat release (NHR) rate in themain-combustion chamber needs to be calculated froman inverse thermodynamic calculation based on theexperimental pressure data1217 However this calcula-tion cannot be completed without knowing the mass

1320 Proc IMechE Part D J Automobile Engineering 231(10)

flow rate between the two combustion chambersFortunately the mass flow between the two combus-tion chambers mainly influences the combustion pro-cess in the pre-combustion chamber Its effect on themain-combustion chamber is limited For calibrationpurpose the mass flow can be neglected when calculat-ing the NHR rate in the main-combustion chamberThe result of the inverse calculation is shown in theupper plot of Figure 5 The NHR rate is the sum of theCER rate and the heat transfer rate from the cylinderwall The heat transfer model can be calibrated assum-ing that the heat transfer rate is dominant where theNHR rate is negative The calculated heat transfer rateis shown in the bottom plot of Figure 5

The next step is to calibrate Dtd for the main-combustion chamber After the heat transfer model iscalibrated the CER rate can be obtained by subtract-ing the heat transfer rate from the NHR rate shown asthe dotted line in Figure 5 This allows one to calculatethe MFB of the main-combustion chamber which isthe solid line in Figure 6 According to the later stageof the MFB curve (shown in Figure 6) the parameterDtd in the Wiebe function can be determined by a lin-ear least-squares fitting procedure18 The dotted line inFigure 6 is the curve fitting result The parameter b teth THORNis set to be 1 during this calibration procedure

Because the heat transfer coefficient hc of the pre-combustion chamber is assumed to be equal to that of

the main-combustion chamber The heat transfer rateto the pre-combustion chamber wall is determined Thisallows one to calibrate the other unknown parametersin Table 3 using a nonlinear least-squares optimizationprocedure Among these parameters cpre cmain vc andb in equations (12) and (20) remain constant for all thefirst three experimental cases The results of the linearleast-squares fitting and nonlinear least-squares optimi-zation procedures are shown in Table 3

The nonlinear least-squares optimization problem issolved by the nonlinear least-squares solver in Matlabusing the Trust-Region-Reflective algorithm This algo-rithm minimizes the following function

Xni=1

(Pprei Pprei)2 +

Xni=1

(Pmaini Pmaini)2

eth22THORN

where n is the total number of the data points Ppreiand Pmaini are the experimental pressure points Pprei

Table 2 Experimental set-up

Parameter Case

1 2 3 4 5 6

Orifice diameter (mm) 15 15 15 20 20 25Main-chamber AFR 183 210 183 15 125 15Pre-chamber fuel addition (mg) 066 089 089 0 0 0Pre-chamber AFR (calculated) 098 090 085 15 125 15

AFR air-to-fuel ratio

Figure 5 Inverse thermodynamic calculation results

Figure 6 MFB obtained from inverse thermodynamiccalculation and least-squares fitting result

Table 3 Calibration results for cases 1 and 2

Parameter Case 1 Case 2

cpre 0655 0655cmain 001 001vc 3263 3263b 172 172

Pre-chamber Dtd (ms) 417 394hpre 0941 0884

Main-chamber Dtd (ms) 193 286hmain 0912 0975

Song et al 1321

and Pmaini represent the modeled pressure points i isthe data index

Model validation and simulation results

After the calibration procedure the model is then vali-dated by the third experimental data set listed inTable 2 The simulation parameters for the thirdexperimental case are determined based on the follow-ing simple assumptions Since the AFR in the main-combustion chamber of the third case is the same asthe first one Dtd and hmain of the main-combustionchamber are assumed to be the same as in the first caseIn the pre-combustion chamber the other combustionparameters are assumed to vary linearly with the AFRfor the three experimental cases because their AFRs inthe pre-combustion chamber are within a relativelysmall range Based on the calculated AFRs in Table 2the coefficients Dtd and hpre of the pre-combustionchamber are calculated and listed in Table 4 The para-meters cpre cmain vc and b remain the sameFigures 7ndash9 show the comparison between the modeledand experimental pressure traces in two combustionchambers for the three experimental cases The calcu-lated pressure traces for the first two cases match theexperimental pressure traces very well since their para-meters are obtained by the calibration procedureAlthough the combustion parameters for the third case

are calculated based on the very simple assumptionsdiscussed above the agreement between the modeledand measured pressure traces is satisfactory The rela-tive errors on the pressure traces are always below10 The errors mainly occur after 31 ms in Figure 9These errors are mainly caused by the simple assump-tions used for determining the simulation parametersfor case 3 In reality the combustion process in themain-combustion chamber does not only depend onthe parameters in the main-combustion chamber Therelationship between the combustion parameters andthe AFR in the pre-combustion chamber is also notexactly linear Once more experimental data are avail-able further calibration can be done and model accu-racy can be improved

One of the main differences between TJI combustionand conventional SI combustion is that the turbulentjet increases the rate of combustion in TJI combustionThe experimental and calculated NHR rates are com-pared in Figure 10 We can find a peak on the NHRrate curve at the early stage of the combustion whichis caused by the turbulent jet As an example Figure 11shows how the model simulates the rate of combustion

Table 4 Simulation parameters

Parameter Case 3

cpre 0655cmain 001vc 3263b 172

Pre-chamber Dtd (ms) 380hpre 0848

Main-chamber Dtd (ms) 193hmain 0912

Figure 7 Experimental and calculated pressure traces forexperimental case 1

Figure 8 Experimental and calculated pressure traces forexperimental case 2

Figure 9 Experimental and calculated pressure traces forexperimental case 3

1322 Proc IMechE Part D J Automobile Engineering 231(10)

according to the mass flow rate of the turbulent jetThe top plot of Figure 11 shows the mass flow ratethrough the orifice connecting the two combustionchambers The curve in the middle plot of Figure 11 isb teth THORN in equation (20) The bottom plot of Figure 11 isthe calculated CER rate According to the simulationresults the MFB model with the parameter-varyingWiebe function successfully links the combustion pro-cesses in two combustion chambers

The value of the coefficient ab during the simulationis shown in Figure 12 where the dashed line is thevalue of ab calculated by assuming Ppre is greater thanPmain and the dash-dotted line is the value of ab calcu-lated by assuming Ppre is less than Pmain The actualvalue of ab used for the combustion calculation that isshown by the solid line is on the dashed line at begin-ning because the pre-combustion chamber is ignitedfirst After the switch line (see Figure 12) the main-combustion chamber pressure becomes larger than thatof the pre-combustion chamber ab jumps from thedashed line to the dash-dotted line

To further validate the model 12 more experimentaldata sets were used The orifice diameter varied from20mm to 30mm The relative AFRs in the two com-bustion chambers varied from 09 to 15 To simplifythe presentation the pressure traces were plotted onlyfor the first three cases see Figures 13ndash15 The experi-mental set-ups for the three cases are shown in Table 2as cases 4-6 The associated calibration results can befound in Table 5 For the other cases the calculated10ndash50 burn duration (Burn1050) and 50ndash90 burnduration (Burn5090) of the main-combustion chamber

Figure 10 Experimental and calculated net heat release rates

Figure 11 Chemical energy release rate calculation

Figure 12 The calculated value of ab during combustion

Figure 13 Experimental and calculated pressure traces forexperimental case 4

Figure 14 Experimental and calculated pressure traces forexperimental case 5

Song et al 1323

were compared with the experimental values seeFigures 16ndash17 Due to large variations of orifice areasand AFRs of these data sets the model parametersneed to be re-calibrated However b and wc were keptunchanged for the cases with the same orifice sizes likecases 4 and 5 It was also found that cpre and cmain werevery similar for all the experimental cases This indi-cates that these two parameters are mainly associated

with the combustion chamber structure In Table 5 thepre-combustion chamber burn durations are quite dif-ferent from the previous experimental cases 1ndash3 This isdue to the difference in the orifice area

To conclude the proposed model is able to fit theexperimental data sets with large ranges of AFRs inboth combustion chambers and different pre-combustion chamber orifice areas This indicates thatthe developed combustion model has the potential tobe used for the development of a TJI engine model

Conclusion

This article presents a control-oriented TJI combustionmodel for the RCM at Michigan State University Anewly proposed parameter-varying Wiebe combustionmodel is used to link the combustion processes in bothpre- and main-combustion chambers The developedmodel can be calibrated using a simple and systematiccalibration procedure based on the experimental dataThe model validation process shows good agreementbetween the modeled and experimental pressure traceswhich indicates that the developed model is capable ofaccurately capturing TJI combustion dynamics Thevalidation results also indicate that the model is able topredict the combustion process that is not used to cali-brate the model parameters This shows that the devel-oped model has the potential to be used for studyingTJI combustion engines and developing associated con-trol strategies Although only methane is used as fuel inthis article this model can be extended to other gaseousfuels with new calibrations However for liquid fuelthe model structure may need to be changed particu-larly a gasndashfuel mixing model will be required Futurework will extend the modeling work for TJI enginesusing gaseous fuel Note that in this case the pistondynamics and gas exchange (intake and exhaust) mod-els need to be added

Table 5 Calibration results for cases 4ndash6

Parameter Case 4 Case 5 Case 6

cpre 0651 0651 0651cmain 0 0 0vc 3829 3829 5064b 0824 0824 0764

Pre-chamber Dtd (ms) 197 228 179hpre 0980 0939 0901

Main-chamber Dtd (ms) 165 112 140hmain 0927 0905 0932

Figure 15 Experimental and calculated pressure traces forexperimental case 6

Figure 16 Experimental and calculated Burn1050 in the main-combustion chamberBurn1050 10ndash50 burn duration

Figure 17 Experimental and calculated Burn5090 in themain-combustion chamberBurn5090 50ndash90 burn duration

1324 Proc IMechE Part D J Automobile Engineering 231(10)

Declaration of conflicting interests

The author(s) declared no potential conflicts of interestwith respect to the research authorship andor publi-cation of this article

Funding

The author(s) disclosed receipt of the following finan-cial support for the research authorship andor publi-cation of this article This research is co-sponsored bythe US National Science Foundation and Departmentof Energy under contract number CBET-1258581

References

[1] Zhang S Zhu G and Sun Z A control-oriented chargemixing and two-zone HCCI combustion model IEEE T

Veh Technol 2014 63(3) 1079ndash1090[2] Yang X and Zhu G SI and HCCI combustion mode

transition control of an HCCI capable SI engine IEEET Contr Syst T 2013 21(5) 1558ndash1569

[3] Turkish MC 3-valve stratified charge engines Evolve-ment analysis and progression SAE technical paper741163 1974

[4] Date T Yagi S Ishizuya A et al Research and develop-ment of the Honda CVCC engine SAE Technical Paper1974

[5] Toulson E Schock HJ and Attard WP A review of pre-chamber initiated jet ignition combustion systems SAEtechnical paper 2010-01-2263 2010

[6] Song R Gentz G Zhu G et al A control-oriented jetignition combustion model for an SI engine In ASME

2015 dynamic systems and control conference ColumbusOhio 28ndash30 October 2015 ppV001T11A001ndashV001T11A001 American Society of Mechanical Engineers

[7] Toulson E Huisjen A Chen X et al Visualization ofpropane and natural gas spark ignition and turbulent jetignition combustion SAE Int J Engines 2012 5(4)

1821ndash1835[8] Yang X and Zhu GG A control-oriented hybrid com-

bustion model of a homogeneous charge compressionignition capable spark ignition engine P I Mech Eng

DmdashJ Aut 2012 226(10) 1380ndash1395[9] Stanglmaier RH and Roberts CE Homogeneous charge

compression ignition (HCCI) Benefits compromisesand future engine applications SAE technical paper1999-01-3682 1999

[10] Toulson E Watson HC and Attard WP Modeling alter-native prechamber fuels in jet assisted ignition of gasolineand LPG SAE technical paper 2009-01-0721 2009

[11] Ghorbani A Steinhilber G Markus D et al Numericalinvestigation of ignition in a transient turbulent jet bymeans of a PDF method Combust Sci Technol 2014186(10-11) 1582ndash1596

[12] Canova M Garcin R Midlam-Mohler S et al Acontrol-oriented model of combustion process in aHCCI diesel engine In Proceedings of the 2005

American control conference Portland Oregon 8ndash10

July 2005 vol 7 pp4446ndash4451 American AutomaticControl Council

[13] Guzzella L and Onder C Introduction to modeling and

control of internal combustion engine systems Berlin Hei-delberg Springer Science amp Business Media 2009

[14] Mortenson ME Mathematics for computer graphics

applications South Norwalk CT Industrial Press 1999[15] Heywood JB Internal combustion engine fundamentals

Vol 930 New York McGraw-Hill 1988[16] Ghojel J Review of the development and applications of

the Wiebe function A tribute to the contribution of Ivan

Wiebe to engine research Int J Engine Res 2010 11(4)297ndash312

[17] Zhao H and Ladommatos N Engine combustion instru-

mentation and diagnostics Warrendale PA Society of

Automotive Engineers 2001[18] Hellstrom E Stefanopoulou A and Jiang L A linear

least-squares algorithm for double-Wiebe functions

applied to spark-assisted compression ignition J Eng

Gas Turb Power 2014 136(9) 091514

Appendix

Abbreviations

TJI Turbulent jet ignitionRCM Rapid compression machineAFR Air-to-fuel ratioIC Internal combustionHCCI Homogeneous charge compression

ignitionCVCC Compound vortex controlled combustionSI Spark ignitionCFD Computational fluid dynamicsPDF Probability density functionMFB Mass fraction burnedNHR Net heat releaseCER Chemical energy release

List of simulation parameters

Parameter Symbol

Fuel injector discharge coefficient Cd1

Pre-chamber orifice discharge coefficient Cd2

Mass flow rate correcting parameter cpre

Mass flow rate correcting parameter cmain

Heat transfer coefficient hc

Lower heating value of the fuel QLHV

Wall temperature Tw

Coefficient of parameter-varying Wiebe function vc

Coefficient of parameter-varying Wiebe function bStart of ignition tign

Wiebe function coefficient aWiebe exponent mBurn duration in Wiebe function DtdPre-chamber combustion efficiency hpre

Main-chamber combustion efficiency hmain

Song et al 1325

In this article methane was used as the fuel for allthe experiments Two Kistler piezoelectric pressure sen-sors were installed into the two combustion chambersfor pressure measurements During the experiment thecombustion chamber wall was heated to 80C Thecombustion chambers were firstly evacuated by avacuum pump and then filled with airndashfuel mixturewith a known AFR Then the piston rapidly com-pressed the mixture in both combustion chambers Atthe same time a charge of fuel was injected into thepre-combustion chamber see Figure 2 At the end ofcompression the piston kept still and therefore thevolume in the main-combustion chamber remained con-stant At the falling edge of the dwell control signal thespark is initiated through the spark plug inside the pre-combustion chamber and then the reacting productsfrom the pre-combustion chamber were injected intothe main-combustion chamber and ignited the airndashfuelmixture Figure 2 shows the control signals and the typ-ical pressure traces measured during the experiment

In order to have a good mixing the fuel was injectedduring the compression The fuel flow mixes withthe gas flowing through the orifice from the main-combustion chamber during the injection leading tobetter mixing Moreover to allow enough time for themixing process after fuel injection the injection timingwas set at the beginning of the compression In this waywe are able to make sure that the airndashfuel mixture in thepre-combustion chamber is close to uniformly mixed

Turbulent jet ignition combustion model

Gas exchange model

During compression methane is injected into the pre-combustion chamber The mass flow rate of the injected

methane can be calculated by the one-dimensional com-pressible flow equation13

_minj =Cd1Av1PinjffiffiffiffiffiffiffiffiffiffiffiRTinj

p cPpre

Pinj

Pinj Ppre eth1THORN

where

c xeth THORN=

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffik 2

k+1

k+1eth THORNk1eth THORN

qx 2

k+1

kk1eth THORN

x1k

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi2k

k1 1 xk1eth THORN

k

rx5 2

k+1

kk1eth THORN

8gtltgt eth2THORN

Note that the coefficient Cd1 is experimentally deter-mined Av1 is the orifice area of the fuel injector k is theratio of specific heats R is the gas constant Pinj andTinj are the upstream pressure and temperature respec-tively and Ppre is the pressure in the pre-combustionchamber

The gas exchange process between the two combus-tion chambers is modeled similarly However the pres-sure in the pre-combustion chamber can be eithergreater or less than that in the main-combustion cham-ber The mass flow rate between the two combustionchambers is calculated by the following equation

_mtur =

Cd2Av2PpreffiffiffiffiffiffiffiffiffiffiffiffiRTpre

p cPmain

Ppre

Ppre5Pmain

Cd2Av2PmainffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiRTmain

p cPpre

Pmain

Ppre Pmain

8gtgtgtltgtgtgt

eth3THORN

where Cd2 and Av2 are the discharge coefficient and thearea of the orifice connecting the two combustionchambers The subscripts pre and main denote the pre-combustion and main-combustion chamber propertiesrespectively

Before ignition the pre-combustion chamber is con-sidered as a control volume with mass and energyexchange The mass and energy conservation equationsare used to describe such a control volume

dmpre

dt= _minj _mtur

dUpre

dt= _Hinj _Htur _Qht

eth4THORN

Figure 1 Rapid compression machine

Table 1 Rapid compression machine (RCM) specifications

Parameter Value

Bore 508Stroke 202Compression ratio 851Pre-combustion chamber volume 23 cm3

Pre-combustion chamber orifice diameter 15ndash30 mm

Figure 2 Working process of the rapid compression machine(RCM)

Song et al 1317

where mpre and Upre are the mass and internal energyof the gas in the pre-combustion chamber respectively_Qht is the heat transfer rate through the chamber walland H is the enthalpy flow The subscript inj and turrepresent the properties of the gas from the fuel injectorand through the orifice connecting the two combustionchambers respectively

Assuming that the gas can be considered as an idealgas the two equations can be coupled by the ideal gaslaw below

Ppre Vpre =mpre R Tpre eth5THORN

where Vpre is the pre-combustion chamber volumeSubstituting equation (5) into equation (4) the follow-ing two equations are obtained to calculate the gaspressure and temperature

dPpre

dt=

R

Vprecvcp _minjTinj cp _mturTtur _Qht

dTpre

dt=

TpreR

PpreVprecvcp _minjTinj cp _mturTtur

cv _minj _mtur

Tpre _Qht

eth6THORN

where cp and cv are the specific heat at constant pres-sure and constant volume respectively and

Ttur =Tpre _mtur 0

Tmain _mtur40

(eth7THORN

The pre-combustion chamber volume is only around2ndash4 of the main-combustion chamber clearance vol-ume Therefore the gas flowing between the twocombustion chambers can be neglected for the main-combustion chamber model The pressure and tem-perature in the main-combustion chamber can besolved using the energy and mass conservation equa-tions These equations can be found in many other arti-cles about engine modeling and thus will not be shownhere112

Two-zone combustion model

After ignition the pre-combustion chamber is dividedinto two zones to improve the model accuracy Boththe burned and unburned zones can be regarded as con-trol volumes Besides the mass enthalpy and workexchange between the two control volumes there is alsomass and enthalpy exchange through the orifice to themain-combustion chamber see Figure 3 The burnedand unburned gases in the pre-combustion (or main-combustion) chamber are assumed to enter the burnedzone and unburned zone in the main-combustion (orpre-combustion) chamber respectively Before the igni-tion of the main-combustion chamber all the gas fromthe main-combustion chamber is considered asunburned gas The energy balance equation of theburned zone is shown in equation (8) To make theequations concise the variables in the following

equations in this subsection are for the pre-combustionchamber if not specified

cvd(mbTb)

dt+P

dVb

dt+ xb _Qht =

_Qch +mu

1 xb

dxbdt

cpTu _mturbcpTb

eth8THORN

The energy balance equation of the unburned zone isrepresented by

cvd(muTu)

dt+P

dVu

dt+ 1 xbeth THORN _Qht =

mu

1 xb

dxbdt

cpTu _mturucpTu

eth9THORN

The masses of both burned and unburned zones areobtained based on the following mass conservation law

dmb

dt= _mturb +

mu

1 xb

dxbdt

dmu

dt= _mturu

mu

1 xb

dxbdt

eth10THORN

The subscripts b and u represent the burned zoneand unburned zone Qht is the heat transfer to thechamber wall Qch is the chemical energy released bycombustion xb is the MFB mturb and mturu repre-sent the burned gas and unburned gas flowing throughthe orifice Correspondingly the area of the orifice isalso divided into two parts One for the burned gas andthe other for the unburned gas Figure 3 shows thebasic idea of the two-zone combustion model Whenthe pressure in the pre-combustion chamber is greaterthan that in the main-combustion chamber the twomass flow rates are calculated by

_mturb =abCd2Av2PpreffiffiffiffiffiffiffiffiffiRTb

p cPmain

Ppre

_mturu = 1 abeth THORNCd2Av2PpreffiffiffiffiffiffiffiffiffiRTu

p cPmain

Ppre

eth11THORN

Similar result can be obtained when the pressure in themain-combustion chamber is greater than that in thepre-combustion chamber

The coefficient ab in equation (11) is chosen as afunction of the volume fraction of the burned gas vbAssuming that the burned and unburned gases werealways well mixed ab would be always equal to vb

Figure 3 Two-zone combustion modelab area fraction for burned gas

1318 Proc IMechE Part D J Automobile Engineering 231(10)

However in reality this is not the case ab is combus-tion chamber structure dependent For our TJI systemthe spark plug is located at the top of the pre-combustion chamber see Figure 1 In this case thecombustion is initiated at the top of the pre-combustionchamber Since the orifice is at the bottom it is hardfor the burned gas to escape from the pre-combustionchamber at the early stage of the combustion As aresult the fraction of the burned gas flowing throughthe orifice to the main-combustion chamber is muchsmaller than the burned gas fraction inside the pre-combustion chamber This is why ab is smaller than vbin the pre-combustion chamber when Ppre PmainNote that ab will be determined using experimentaldata When Ppre Pmain the gas in the main-combustion chamber flows through the orifice and ab

will be determined by the burned gas fraction in themain-combustion chamber Since the combustion inthe main-combustion chamber is initiated by the turbu-lent jet (close to orifice) the orifice is surrounded bythe gas with high concentration of burned gasTherefore ab is larger than vb in the main-combustionchamber And again the actual value will be deter-mined by the experimental data This is the main reasonwhy the two-zone combustion model is used The valueof ab can be expressed by equation (12)

ab =f1 vbpre cpre

Ppre Pmain

f2 vbmain cmaineth THORN Ppre Pmain

eth12THORN

To simplify the calibration process the two func-tions f1 and f2 are approximated by second-degreeBezier curves14 Besides the control points (00) and(11) (cpre1 cpre) was added for f1 and(cmain1 cmain) for f2 as the third control points seeFigure 4 The parameters cpre and cmain are experimen-tally determined The Bezier curve guaranteesab 2 0 1frac12 as long as cpre 2 0 1frac12 and cmain 2 0 1frac12 Bychanging cpre and cmain the ratio of the burned andunburned gases flowing through the orifice can beadjusted to better match the actual physical processand thus to improve the model accuracy

Applying the principle of mass conservation theinstant fuel mass in the pre-combustion chamber canbe obtained by

dmprefueldt

=

mprefuel1 xb

dxbdt _mturu

1

l A=Feth THORNs +1

eth13THORN

where l is the relative AFR and A=Feth THORNs is the stoichio-metric AFR Note that only the fuel from the unburnedzone is considered

From equations (11) and (13) it can be observedthat the total amount of fuel burned inside the pre-combustion chamber is highly influenced by ab

The rate of chemical energy release (CER) isobtained by the following relationship

_Qch =hpreQLHVmprefuel1 xb

dxbdt

eth14THORN

where the combustion efficiency hpre is experimentallydetermined and QLHV is the lower heating value of thefuel

The rate of heat transfer to the combustion chamberwall can be modeled by the following equation15

_Qht =Aprehc Tpre Tw

eth15THORN

where Apre is the pre-combustion chamber surface areaTw is the mean wall temperature and hc is the heat-transfer coefficient calibrated by the experiment

After the ignition in the pre-combustion chamberthe combustion in the main-combustion chamber willnot be initiated until the generation of the turbulentjet from the pre-combustion chamber Before the igni-tion of the main-combustion chamber the mass flowfrom the burned zone of the pre-combustion chamberto the main-combustion chamber is neglected Theamount of the fuel in the main-combustion chamberis calculated by

dmmainfueldt

= _mturu1

lpre A=Feth THORNs + 1

eth16THORN

where mmainfuel is the fuel mass in the main-combustion chamber

After ignition in the main-combustion chamber theburned zone is created Different from the two-zonecombustion model in a conventional SI engine thecombustion model of the main-combustion chamberneeds to consider the gas flowing through the orificeinto the pre-combustion chamber The mass and energyconservation equations for burned and unburned zonesare very similar to those of the pre-combustion cham-ber model presented in this subsection and are omittedhere The major difference is that the total volume ofthe main-combustion chamber is varying

Mass fraction burned model

The MFB in the pre-combustion chamber is obtainedfrom the Wiebe function16Figure 4 The value of ab in the two cases

Song et al 1319

xb =1 exp a t tignDtd

m+1

eth17THORN

The coefficients a and m are chosen to be 6908and 2 respectively tign is the start of ignition and Dtdis the burn duration that is calibrated by AFR beforeignition

At the early stage of the combustion in the main-combustion chamber the rate of combustion is deter-mined by not only the gas properties in themain-combustion chamber but also the turbulent jetfrom the pre-combustion chamber This is due to thefact that the turbulent jets create multiple and distribu-ted ignition sites which increases the overall flamefront area in the main-combustion chamber Moreoverthese turbulent jets increase the turbulence intensity inthe main-combustion chamber and thus the flame frontpropagation speed After the turbulent jet disappearsthe rate of combustion reduces gradually to a relativelylow level and its characteristics are mainly determinedby the gas properties only in the main-combustionchamber Here the term lsquointensityrsquo of the turbulent jetis used to describe the resulting increment of the com-bustion rate in the main-combustion chamber due tothe turbulent jet Since the intensity of the turbulent jetis determined by the combustion processes in bothcombustion chambers estimating the rate of combus-tion before ignition is difficult and requires significantcalibration effort Therefore adjusting the rate of com-bustion according to the turbulent jet intensity duringthe combustion process is preferred for the TJI com-bustion model The conventional single-Wiebe functionis not suitable for our combustion model The multi-Wiebe function is a possible approach for modeling theMFB However this requires determining all associatedparameters before ignition occurs Therefore a newparameter-varying Wiebe function is proposed andused in this article see equation (18)

x0b teth THORN=1 exp a ttign b teth THORNDtd

h im+1

tign b teth THORN= t0 R tt0b teth THORN 1frac12 dt

8lt eth18THORN

where t0 and Dtd are determined by the spark timingand the AFR in the main-combustion chamber Thecoefficients a and m are chosen to be 6908 and 2respectively

If a m and Dtd are the same in equations (17) and(18) it can be proved that for any given tign b = tign

dx0bdt

=dxbdt b teth THORN eth19THORN

In other word the combustion rate calculated by thenew Wiebe function is b(t) times larger than that calcu-lated by the conventional Wiebe function Thereforethe intensity of the turbulent jet can be mathematicallyexpressed by b teth THORN The combustion model is able toadjust the rate of combustion by making b(t) a functionof some characteristics of the turbulent jet Moreover

b(t) can be changed at any time during the combustionprocess As long as b teth THORN is greater than 0 the combus-tion rate is greater than 0 and x0b teth THORN tends to 1 as t goesto infinity From the available experimental results it isfound that the rate of combustion in the main-combustion chamber is highly related to the mass flowrate of the turbulent jets from the pre-combustionchamber As a result it is assumed that the intensity ofthe turbulent jet can be linked to its mass flow rateAlthough this assumption provides a good matchbetween the modeled and available experimentalresults it is important to find an accurate method tocalculate the intensity of the turbulent jet in the futurewhen more experimental data are available Since theinfluence of the turbulent jet on the combustion in themain-combustion chamber is also delayed b teth THORN is mod-eled to be proportional to the mass flow rate of the tur-bulent jet with first-order dynamics see equation (20)An offset is used such that b(t)=1 when the mass flowrate of the turbulent jet is zero

b teth THORN=b frac12( _m+tur fl)(t)+1

( _m+tur fl)(t)=

R t0

_m+tur(t)fl(t t)dt eth20THORN

Here is the convolution operator and _m+tur and fl are

defined as follows

_m+tur =

_mtur _mtur500 _mtur 0

fl(t)=vce

vctu(t)

eth21THORN

Note that u(t) denotes unit step The convolution of_m+tur with the exponential decay function fl represents

the first-order dynamics and is used to emulate the timedelay To be more specific fl is the time response of alow-pass filter with cutoff frequency vc The parametersb and wc are experimentally determined When _mtur40we have b teth THORN=1 which means that the combustion ratewill not be altered if there is no turbulent jet from thepre-combustion chamber

Model calibration

The combustion model was calibrated using the experi-mental data collected from the RCM at Michigan StateUniversity described in the lsquoSystem descriptionrsquo sectionThe model is firstly calibrated using two experimentaldata sets and validated using another data set Then tofurther validate the model more data sets with differ-ent pre-combustion chamber orifice sizes are used Theexperimental set-ups for the first three cases can befound in Table 2 where cases 1 and 2 are used formodel calibration and case 3 for model validation

The first step is to calibrate the heat transfer modelTo do this the net heat release (NHR) rate in themain-combustion chamber needs to be calculated froman inverse thermodynamic calculation based on theexperimental pressure data1217 However this calcula-tion cannot be completed without knowing the mass

1320 Proc IMechE Part D J Automobile Engineering 231(10)

flow rate between the two combustion chambersFortunately the mass flow between the two combus-tion chambers mainly influences the combustion pro-cess in the pre-combustion chamber Its effect on themain-combustion chamber is limited For calibrationpurpose the mass flow can be neglected when calculat-ing the NHR rate in the main-combustion chamberThe result of the inverse calculation is shown in theupper plot of Figure 5 The NHR rate is the sum of theCER rate and the heat transfer rate from the cylinderwall The heat transfer model can be calibrated assum-ing that the heat transfer rate is dominant where theNHR rate is negative The calculated heat transfer rateis shown in the bottom plot of Figure 5

The next step is to calibrate Dtd for the main-combustion chamber After the heat transfer model iscalibrated the CER rate can be obtained by subtract-ing the heat transfer rate from the NHR rate shown asthe dotted line in Figure 5 This allows one to calculatethe MFB of the main-combustion chamber which isthe solid line in Figure 6 According to the later stageof the MFB curve (shown in Figure 6) the parameterDtd in the Wiebe function can be determined by a lin-ear least-squares fitting procedure18 The dotted line inFigure 6 is the curve fitting result The parameter b teth THORNis set to be 1 during this calibration procedure

Because the heat transfer coefficient hc of the pre-combustion chamber is assumed to be equal to that of

the main-combustion chamber The heat transfer rateto the pre-combustion chamber wall is determined Thisallows one to calibrate the other unknown parametersin Table 3 using a nonlinear least-squares optimizationprocedure Among these parameters cpre cmain vc andb in equations (12) and (20) remain constant for all thefirst three experimental cases The results of the linearleast-squares fitting and nonlinear least-squares optimi-zation procedures are shown in Table 3

The nonlinear least-squares optimization problem issolved by the nonlinear least-squares solver in Matlabusing the Trust-Region-Reflective algorithm This algo-rithm minimizes the following function

Xni=1

(Pprei Pprei)2 +

Xni=1

(Pmaini Pmaini)2

eth22THORN

where n is the total number of the data points Ppreiand Pmaini are the experimental pressure points Pprei

Table 2 Experimental set-up

Parameter Case

1 2 3 4 5 6

Orifice diameter (mm) 15 15 15 20 20 25Main-chamber AFR 183 210 183 15 125 15Pre-chamber fuel addition (mg) 066 089 089 0 0 0Pre-chamber AFR (calculated) 098 090 085 15 125 15

AFR air-to-fuel ratio

Figure 5 Inverse thermodynamic calculation results

Figure 6 MFB obtained from inverse thermodynamiccalculation and least-squares fitting result

Table 3 Calibration results for cases 1 and 2

Parameter Case 1 Case 2

cpre 0655 0655cmain 001 001vc 3263 3263b 172 172

Pre-chamber Dtd (ms) 417 394hpre 0941 0884

Main-chamber Dtd (ms) 193 286hmain 0912 0975

Song et al 1321

and Pmaini represent the modeled pressure points i isthe data index

Model validation and simulation results

After the calibration procedure the model is then vali-dated by the third experimental data set listed inTable 2 The simulation parameters for the thirdexperimental case are determined based on the follow-ing simple assumptions Since the AFR in the main-combustion chamber of the third case is the same asthe first one Dtd and hmain of the main-combustionchamber are assumed to be the same as in the first caseIn the pre-combustion chamber the other combustionparameters are assumed to vary linearly with the AFRfor the three experimental cases because their AFRs inthe pre-combustion chamber are within a relativelysmall range Based on the calculated AFRs in Table 2the coefficients Dtd and hpre of the pre-combustionchamber are calculated and listed in Table 4 The para-meters cpre cmain vc and b remain the sameFigures 7ndash9 show the comparison between the modeledand experimental pressure traces in two combustionchambers for the three experimental cases The calcu-lated pressure traces for the first two cases match theexperimental pressure traces very well since their para-meters are obtained by the calibration procedureAlthough the combustion parameters for the third case

are calculated based on the very simple assumptionsdiscussed above the agreement between the modeledand measured pressure traces is satisfactory The rela-tive errors on the pressure traces are always below10 The errors mainly occur after 31 ms in Figure 9These errors are mainly caused by the simple assump-tions used for determining the simulation parametersfor case 3 In reality the combustion process in themain-combustion chamber does not only depend onthe parameters in the main-combustion chamber Therelationship between the combustion parameters andthe AFR in the pre-combustion chamber is also notexactly linear Once more experimental data are avail-able further calibration can be done and model accu-racy can be improved

One of the main differences between TJI combustionand conventional SI combustion is that the turbulentjet increases the rate of combustion in TJI combustionThe experimental and calculated NHR rates are com-pared in Figure 10 We can find a peak on the NHRrate curve at the early stage of the combustion whichis caused by the turbulent jet As an example Figure 11shows how the model simulates the rate of combustion

Table 4 Simulation parameters

Parameter Case 3

cpre 0655cmain 001vc 3263b 172

Pre-chamber Dtd (ms) 380hpre 0848

Main-chamber Dtd (ms) 193hmain 0912

Figure 7 Experimental and calculated pressure traces forexperimental case 1

Figure 8 Experimental and calculated pressure traces forexperimental case 2

Figure 9 Experimental and calculated pressure traces forexperimental case 3

1322 Proc IMechE Part D J Automobile Engineering 231(10)

according to the mass flow rate of the turbulent jetThe top plot of Figure 11 shows the mass flow ratethrough the orifice connecting the two combustionchambers The curve in the middle plot of Figure 11 isb teth THORN in equation (20) The bottom plot of Figure 11 isthe calculated CER rate According to the simulationresults the MFB model with the parameter-varyingWiebe function successfully links the combustion pro-cesses in two combustion chambers

The value of the coefficient ab during the simulationis shown in Figure 12 where the dashed line is thevalue of ab calculated by assuming Ppre is greater thanPmain and the dash-dotted line is the value of ab calcu-lated by assuming Ppre is less than Pmain The actualvalue of ab used for the combustion calculation that isshown by the solid line is on the dashed line at begin-ning because the pre-combustion chamber is ignitedfirst After the switch line (see Figure 12) the main-combustion chamber pressure becomes larger than thatof the pre-combustion chamber ab jumps from thedashed line to the dash-dotted line

To further validate the model 12 more experimentaldata sets were used The orifice diameter varied from20mm to 30mm The relative AFRs in the two com-bustion chambers varied from 09 to 15 To simplifythe presentation the pressure traces were plotted onlyfor the first three cases see Figures 13ndash15 The experi-mental set-ups for the three cases are shown in Table 2as cases 4-6 The associated calibration results can befound in Table 5 For the other cases the calculated10ndash50 burn duration (Burn1050) and 50ndash90 burnduration (Burn5090) of the main-combustion chamber

Figure 10 Experimental and calculated net heat release rates

Figure 11 Chemical energy release rate calculation

Figure 12 The calculated value of ab during combustion

Figure 13 Experimental and calculated pressure traces forexperimental case 4

Figure 14 Experimental and calculated pressure traces forexperimental case 5

Song et al 1323

were compared with the experimental values seeFigures 16ndash17 Due to large variations of orifice areasand AFRs of these data sets the model parametersneed to be re-calibrated However b and wc were keptunchanged for the cases with the same orifice sizes likecases 4 and 5 It was also found that cpre and cmain werevery similar for all the experimental cases This indi-cates that these two parameters are mainly associated

with the combustion chamber structure In Table 5 thepre-combustion chamber burn durations are quite dif-ferent from the previous experimental cases 1ndash3 This isdue to the difference in the orifice area

To conclude the proposed model is able to fit theexperimental data sets with large ranges of AFRs inboth combustion chambers and different pre-combustion chamber orifice areas This indicates thatthe developed combustion model has the potential tobe used for the development of a TJI engine model

Conclusion

This article presents a control-oriented TJI combustionmodel for the RCM at Michigan State University Anewly proposed parameter-varying Wiebe combustionmodel is used to link the combustion processes in bothpre- and main-combustion chambers The developedmodel can be calibrated using a simple and systematiccalibration procedure based on the experimental dataThe model validation process shows good agreementbetween the modeled and experimental pressure traceswhich indicates that the developed model is capable ofaccurately capturing TJI combustion dynamics Thevalidation results also indicate that the model is able topredict the combustion process that is not used to cali-brate the model parameters This shows that the devel-oped model has the potential to be used for studyingTJI combustion engines and developing associated con-trol strategies Although only methane is used as fuel inthis article this model can be extended to other gaseousfuels with new calibrations However for liquid fuelthe model structure may need to be changed particu-larly a gasndashfuel mixing model will be required Futurework will extend the modeling work for TJI enginesusing gaseous fuel Note that in this case the pistondynamics and gas exchange (intake and exhaust) mod-els need to be added

Table 5 Calibration results for cases 4ndash6

Parameter Case 4 Case 5 Case 6

cpre 0651 0651 0651cmain 0 0 0vc 3829 3829 5064b 0824 0824 0764

Pre-chamber Dtd (ms) 197 228 179hpre 0980 0939 0901

Main-chamber Dtd (ms) 165 112 140hmain 0927 0905 0932

Figure 15 Experimental and calculated pressure traces forexperimental case 6

Figure 16 Experimental and calculated Burn1050 in the main-combustion chamberBurn1050 10ndash50 burn duration

Figure 17 Experimental and calculated Burn5090 in themain-combustion chamberBurn5090 50ndash90 burn duration

1324 Proc IMechE Part D J Automobile Engineering 231(10)

Declaration of conflicting interests

The author(s) declared no potential conflicts of interestwith respect to the research authorship andor publi-cation of this article

Funding

The author(s) disclosed receipt of the following finan-cial support for the research authorship andor publi-cation of this article This research is co-sponsored bythe US National Science Foundation and Departmentof Energy under contract number CBET-1258581

References

[1] Zhang S Zhu G and Sun Z A control-oriented chargemixing and two-zone HCCI combustion model IEEE T

Veh Technol 2014 63(3) 1079ndash1090[2] Yang X and Zhu G SI and HCCI combustion mode

transition control of an HCCI capable SI engine IEEET Contr Syst T 2013 21(5) 1558ndash1569

[3] Turkish MC 3-valve stratified charge engines Evolve-ment analysis and progression SAE technical paper741163 1974

[4] Date T Yagi S Ishizuya A et al Research and develop-ment of the Honda CVCC engine SAE Technical Paper1974

[5] Toulson E Schock HJ and Attard WP A review of pre-chamber initiated jet ignition combustion systems SAEtechnical paper 2010-01-2263 2010

[6] Song R Gentz G Zhu G et al A control-oriented jetignition combustion model for an SI engine In ASME

2015 dynamic systems and control conference ColumbusOhio 28ndash30 October 2015 ppV001T11A001ndashV001T11A001 American Society of Mechanical Engineers

[7] Toulson E Huisjen A Chen X et al Visualization ofpropane and natural gas spark ignition and turbulent jetignition combustion SAE Int J Engines 2012 5(4)

1821ndash1835[8] Yang X and Zhu GG A control-oriented hybrid com-

bustion model of a homogeneous charge compressionignition capable spark ignition engine P I Mech Eng

DmdashJ Aut 2012 226(10) 1380ndash1395[9] Stanglmaier RH and Roberts CE Homogeneous charge

compression ignition (HCCI) Benefits compromisesand future engine applications SAE technical paper1999-01-3682 1999

[10] Toulson E Watson HC and Attard WP Modeling alter-native prechamber fuels in jet assisted ignition of gasolineand LPG SAE technical paper 2009-01-0721 2009

[11] Ghorbani A Steinhilber G Markus D et al Numericalinvestigation of ignition in a transient turbulent jet bymeans of a PDF method Combust Sci Technol 2014186(10-11) 1582ndash1596

[12] Canova M Garcin R Midlam-Mohler S et al Acontrol-oriented model of combustion process in aHCCI diesel engine In Proceedings of the 2005

American control conference Portland Oregon 8ndash10

July 2005 vol 7 pp4446ndash4451 American AutomaticControl Council

[13] Guzzella L and Onder C Introduction to modeling and

control of internal combustion engine systems Berlin Hei-delberg Springer Science amp Business Media 2009

[14] Mortenson ME Mathematics for computer graphics

applications South Norwalk CT Industrial Press 1999[15] Heywood JB Internal combustion engine fundamentals

Vol 930 New York McGraw-Hill 1988[16] Ghojel J Review of the development and applications of

the Wiebe function A tribute to the contribution of Ivan

Wiebe to engine research Int J Engine Res 2010 11(4)297ndash312

[17] Zhao H and Ladommatos N Engine combustion instru-

mentation and diagnostics Warrendale PA Society of

Automotive Engineers 2001[18] Hellstrom E Stefanopoulou A and Jiang L A linear

least-squares algorithm for double-Wiebe functions

applied to spark-assisted compression ignition J Eng

Gas Turb Power 2014 136(9) 091514

Appendix

Abbreviations

TJI Turbulent jet ignitionRCM Rapid compression machineAFR Air-to-fuel ratioIC Internal combustionHCCI Homogeneous charge compression

ignitionCVCC Compound vortex controlled combustionSI Spark ignitionCFD Computational fluid dynamicsPDF Probability density functionMFB Mass fraction burnedNHR Net heat releaseCER Chemical energy release

List of simulation parameters

Parameter Symbol

Fuel injector discharge coefficient Cd1

Pre-chamber orifice discharge coefficient Cd2

Mass flow rate correcting parameter cpre

Mass flow rate correcting parameter cmain

Heat transfer coefficient hc

Lower heating value of the fuel QLHV

Wall temperature Tw

Coefficient of parameter-varying Wiebe function vc

Coefficient of parameter-varying Wiebe function bStart of ignition tign

Wiebe function coefficient aWiebe exponent mBurn duration in Wiebe function DtdPre-chamber combustion efficiency hpre

Main-chamber combustion efficiency hmain

Song et al 1325

where mpre and Upre are the mass and internal energyof the gas in the pre-combustion chamber respectively_Qht is the heat transfer rate through the chamber walland H is the enthalpy flow The subscript inj and turrepresent the properties of the gas from the fuel injectorand through the orifice connecting the two combustionchambers respectively

Assuming that the gas can be considered as an idealgas the two equations can be coupled by the ideal gaslaw below

Ppre Vpre =mpre R Tpre eth5THORN

where Vpre is the pre-combustion chamber volumeSubstituting equation (5) into equation (4) the follow-ing two equations are obtained to calculate the gaspressure and temperature

dPpre

dt=

R

Vprecvcp _minjTinj cp _mturTtur _Qht

dTpre

dt=

TpreR

PpreVprecvcp _minjTinj cp _mturTtur

cv _minj _mtur

Tpre _Qht

eth6THORN

where cp and cv are the specific heat at constant pres-sure and constant volume respectively and

Ttur =Tpre _mtur 0

Tmain _mtur40

(eth7THORN

The pre-combustion chamber volume is only around2ndash4 of the main-combustion chamber clearance vol-ume Therefore the gas flowing between the twocombustion chambers can be neglected for the main-combustion chamber model The pressure and tem-perature in the main-combustion chamber can besolved using the energy and mass conservation equa-tions These equations can be found in many other arti-cles about engine modeling and thus will not be shownhere112

Two-zone combustion model

After ignition the pre-combustion chamber is dividedinto two zones to improve the model accuracy Boththe burned and unburned zones can be regarded as con-trol volumes Besides the mass enthalpy and workexchange between the two control volumes there is alsomass and enthalpy exchange through the orifice to themain-combustion chamber see Figure 3 The burnedand unburned gases in the pre-combustion (or main-combustion) chamber are assumed to enter the burnedzone and unburned zone in the main-combustion (orpre-combustion) chamber respectively Before the igni-tion of the main-combustion chamber all the gas fromthe main-combustion chamber is considered asunburned gas The energy balance equation of theburned zone is shown in equation (8) To make theequations concise the variables in the following

equations in this subsection are for the pre-combustionchamber if not specified

cvd(mbTb)

dt+P

dVb

dt+ xb _Qht =

_Qch +mu

1 xb

dxbdt

cpTu _mturbcpTb

eth8THORN

The energy balance equation of the unburned zone isrepresented by

cvd(muTu)

dt+P

dVu

dt+ 1 xbeth THORN _Qht =

mu

1 xb

dxbdt

cpTu _mturucpTu

eth9THORN

The masses of both burned and unburned zones areobtained based on the following mass conservation law

dmb

dt= _mturb +

mu

1 xb

dxbdt

dmu

dt= _mturu

mu

1 xb

dxbdt

eth10THORN

The subscripts b and u represent the burned zoneand unburned zone Qht is the heat transfer to thechamber wall Qch is the chemical energy released bycombustion xb is the MFB mturb and mturu repre-sent the burned gas and unburned gas flowing throughthe orifice Correspondingly the area of the orifice isalso divided into two parts One for the burned gas andthe other for the unburned gas Figure 3 shows thebasic idea of the two-zone combustion model Whenthe pressure in the pre-combustion chamber is greaterthan that in the main-combustion chamber the twomass flow rates are calculated by

_mturb =abCd2Av2PpreffiffiffiffiffiffiffiffiffiRTb

p cPmain

Ppre

_mturu = 1 abeth THORNCd2Av2PpreffiffiffiffiffiffiffiffiffiRTu

p cPmain

Ppre

eth11THORN

Similar result can be obtained when the pressure in themain-combustion chamber is greater than that in thepre-combustion chamber

The coefficient ab in equation (11) is chosen as afunction of the volume fraction of the burned gas vbAssuming that the burned and unburned gases werealways well mixed ab would be always equal to vb

Figure 3 Two-zone combustion modelab area fraction for burned gas

1318 Proc IMechE Part D J Automobile Engineering 231(10)

However in reality this is not the case ab is combus-tion chamber structure dependent For our TJI systemthe spark plug is located at the top of the pre-combustion chamber see Figure 1 In this case thecombustion is initiated at the top of the pre-combustionchamber Since the orifice is at the bottom it is hardfor the burned gas to escape from the pre-combustionchamber at the early stage of the combustion As aresult the fraction of the burned gas flowing throughthe orifice to the main-combustion chamber is muchsmaller than the burned gas fraction inside the pre-combustion chamber This is why ab is smaller than vbin the pre-combustion chamber when Ppre PmainNote that ab will be determined using experimentaldata When Ppre Pmain the gas in the main-combustion chamber flows through the orifice and ab

will be determined by the burned gas fraction in themain-combustion chamber Since the combustion inthe main-combustion chamber is initiated by the turbu-lent jet (close to orifice) the orifice is surrounded bythe gas with high concentration of burned gasTherefore ab is larger than vb in the main-combustionchamber And again the actual value will be deter-mined by the experimental data This is the main reasonwhy the two-zone combustion model is used The valueof ab can be expressed by equation (12)

ab =f1 vbpre cpre

Ppre Pmain

f2 vbmain cmaineth THORN Ppre Pmain

eth12THORN

To simplify the calibration process the two func-tions f1 and f2 are approximated by second-degreeBezier curves14 Besides the control points (00) and(11) (cpre1 cpre) was added for f1 and(cmain1 cmain) for f2 as the third control points seeFigure 4 The parameters cpre and cmain are experimen-tally determined The Bezier curve guaranteesab 2 0 1frac12 as long as cpre 2 0 1frac12 and cmain 2 0 1frac12 Bychanging cpre and cmain the ratio of the burned andunburned gases flowing through the orifice can beadjusted to better match the actual physical processand thus to improve the model accuracy

Applying the principle of mass conservation theinstant fuel mass in the pre-combustion chamber canbe obtained by

dmprefueldt

=

mprefuel1 xb

dxbdt _mturu

1

l A=Feth THORNs +1

eth13THORN

where l is the relative AFR and A=Feth THORNs is the stoichio-metric AFR Note that only the fuel from the unburnedzone is considered

From equations (11) and (13) it can be observedthat the total amount of fuel burned inside the pre-combustion chamber is highly influenced by ab

The rate of chemical energy release (CER) isobtained by the following relationship

_Qch =hpreQLHVmprefuel1 xb

dxbdt

eth14THORN

where the combustion efficiency hpre is experimentallydetermined and QLHV is the lower heating value of thefuel

The rate of heat transfer to the combustion chamberwall can be modeled by the following equation15

_Qht =Aprehc Tpre Tw

eth15THORN

where Apre is the pre-combustion chamber surface areaTw is the mean wall temperature and hc is the heat-transfer coefficient calibrated by the experiment

After the ignition in the pre-combustion chamberthe combustion in the main-combustion chamber willnot be initiated until the generation of the turbulentjet from the pre-combustion chamber Before the igni-tion of the main-combustion chamber the mass flowfrom the burned zone of the pre-combustion chamberto the main-combustion chamber is neglected Theamount of the fuel in the main-combustion chamberis calculated by

dmmainfueldt

= _mturu1

lpre A=Feth THORNs + 1

eth16THORN

where mmainfuel is the fuel mass in the main-combustion chamber

After ignition in the main-combustion chamber theburned zone is created Different from the two-zonecombustion model in a conventional SI engine thecombustion model of the main-combustion chamberneeds to consider the gas flowing through the orificeinto the pre-combustion chamber The mass and energyconservation equations for burned and unburned zonesare very similar to those of the pre-combustion cham-ber model presented in this subsection and are omittedhere The major difference is that the total volume ofthe main-combustion chamber is varying

Mass fraction burned model

The MFB in the pre-combustion chamber is obtainedfrom the Wiebe function16Figure 4 The value of ab in the two cases

Song et al 1319

xb =1 exp a t tignDtd

m+1

eth17THORN

The coefficients a and m are chosen to be 6908and 2 respectively tign is the start of ignition and Dtdis the burn duration that is calibrated by AFR beforeignition

At the early stage of the combustion in the main-combustion chamber the rate of combustion is deter-mined by not only the gas properties in themain-combustion chamber but also the turbulent jetfrom the pre-combustion chamber This is due to thefact that the turbulent jets create multiple and distribu-ted ignition sites which increases the overall flamefront area in the main-combustion chamber Moreoverthese turbulent jets increase the turbulence intensity inthe main-combustion chamber and thus the flame frontpropagation speed After the turbulent jet disappearsthe rate of combustion reduces gradually to a relativelylow level and its characteristics are mainly determinedby the gas properties only in the main-combustionchamber Here the term lsquointensityrsquo of the turbulent jetis used to describe the resulting increment of the com-bustion rate in the main-combustion chamber due tothe turbulent jet Since the intensity of the turbulent jetis determined by the combustion processes in bothcombustion chambers estimating the rate of combus-tion before ignition is difficult and requires significantcalibration effort Therefore adjusting the rate of com-bustion according to the turbulent jet intensity duringthe combustion process is preferred for the TJI com-bustion model The conventional single-Wiebe functionis not suitable for our combustion model The multi-Wiebe function is a possible approach for modeling theMFB However this requires determining all associatedparameters before ignition occurs Therefore a newparameter-varying Wiebe function is proposed andused in this article see equation (18)

x0b teth THORN=1 exp a ttign b teth THORNDtd

h im+1

tign b teth THORN= t0 R tt0b teth THORN 1frac12 dt

8lt eth18THORN

where t0 and Dtd are determined by the spark timingand the AFR in the main-combustion chamber Thecoefficients a and m are chosen to be 6908 and 2respectively

If a m and Dtd are the same in equations (17) and(18) it can be proved that for any given tign b = tign

dx0bdt

=dxbdt b teth THORN eth19THORN

In other word the combustion rate calculated by thenew Wiebe function is b(t) times larger than that calcu-lated by the conventional Wiebe function Thereforethe intensity of the turbulent jet can be mathematicallyexpressed by b teth THORN The combustion model is able toadjust the rate of combustion by making b(t) a functionof some characteristics of the turbulent jet Moreover

b(t) can be changed at any time during the combustionprocess As long as b teth THORN is greater than 0 the combus-tion rate is greater than 0 and x0b teth THORN tends to 1 as t goesto infinity From the available experimental results it isfound that the rate of combustion in the main-combustion chamber is highly related to the mass flowrate of the turbulent jets from the pre-combustionchamber As a result it is assumed that the intensity ofthe turbulent jet can be linked to its mass flow rateAlthough this assumption provides a good matchbetween the modeled and available experimentalresults it is important to find an accurate method tocalculate the intensity of the turbulent jet in the futurewhen more experimental data are available Since theinfluence of the turbulent jet on the combustion in themain-combustion chamber is also delayed b teth THORN is mod-eled to be proportional to the mass flow rate of the tur-bulent jet with first-order dynamics see equation (20)An offset is used such that b(t)=1 when the mass flowrate of the turbulent jet is zero

b teth THORN=b frac12( _m+tur fl)(t)+1

( _m+tur fl)(t)=

R t0

_m+tur(t)fl(t t)dt eth20THORN

Here is the convolution operator and _m+tur and fl are

defined as follows

_m+tur =

_mtur _mtur500 _mtur 0

fl(t)=vce

vctu(t)

eth21THORN

Note that u(t) denotes unit step The convolution of_m+tur with the exponential decay function fl represents

the first-order dynamics and is used to emulate the timedelay To be more specific fl is the time response of alow-pass filter with cutoff frequency vc The parametersb and wc are experimentally determined When _mtur40we have b teth THORN=1 which means that the combustion ratewill not be altered if there is no turbulent jet from thepre-combustion chamber

Model calibration

The combustion model was calibrated using the experi-mental data collected from the RCM at Michigan StateUniversity described in the lsquoSystem descriptionrsquo sectionThe model is firstly calibrated using two experimentaldata sets and validated using another data set Then tofurther validate the model more data sets with differ-ent pre-combustion chamber orifice sizes are used Theexperimental set-ups for the first three cases can befound in Table 2 where cases 1 and 2 are used formodel calibration and case 3 for model validation

The first step is to calibrate the heat transfer modelTo do this the net heat release (NHR) rate in themain-combustion chamber needs to be calculated froman inverse thermodynamic calculation based on theexperimental pressure data1217 However this calcula-tion cannot be completed without knowing the mass

1320 Proc IMechE Part D J Automobile Engineering 231(10)

flow rate between the two combustion chambersFortunately the mass flow between the two combus-tion chambers mainly influences the combustion pro-cess in the pre-combustion chamber Its effect on themain-combustion chamber is limited For calibrationpurpose the mass flow can be neglected when calculat-ing the NHR rate in the main-combustion chamberThe result of the inverse calculation is shown in theupper plot of Figure 5 The NHR rate is the sum of theCER rate and the heat transfer rate from the cylinderwall The heat transfer model can be calibrated assum-ing that the heat transfer rate is dominant where theNHR rate is negative The calculated heat transfer rateis shown in the bottom plot of Figure 5

The next step is to calibrate Dtd for the main-combustion chamber After the heat transfer model iscalibrated the CER rate can be obtained by subtract-ing the heat transfer rate from the NHR rate shown asthe dotted line in Figure 5 This allows one to calculatethe MFB of the main-combustion chamber which isthe solid line in Figure 6 According to the later stageof the MFB curve (shown in Figure 6) the parameterDtd in the Wiebe function can be determined by a lin-ear least-squares fitting procedure18 The dotted line inFigure 6 is the curve fitting result The parameter b teth THORNis set to be 1 during this calibration procedure

Because the heat transfer coefficient hc of the pre-combustion chamber is assumed to be equal to that of

the main-combustion chamber The heat transfer rateto the pre-combustion chamber wall is determined Thisallows one to calibrate the other unknown parametersin Table 3 using a nonlinear least-squares optimizationprocedure Among these parameters cpre cmain vc andb in equations (12) and (20) remain constant for all thefirst three experimental cases The results of the linearleast-squares fitting and nonlinear least-squares optimi-zation procedures are shown in Table 3

The nonlinear least-squares optimization problem issolved by the nonlinear least-squares solver in Matlabusing the Trust-Region-Reflective algorithm This algo-rithm minimizes the following function

Xni=1

(Pprei Pprei)2 +

Xni=1

(Pmaini Pmaini)2

eth22THORN

where n is the total number of the data points Ppreiand Pmaini are the experimental pressure points Pprei

Table 2 Experimental set-up

Parameter Case

1 2 3 4 5 6

Orifice diameter (mm) 15 15 15 20 20 25Main-chamber AFR 183 210 183 15 125 15Pre-chamber fuel addition (mg) 066 089 089 0 0 0Pre-chamber AFR (calculated) 098 090 085 15 125 15

AFR air-to-fuel ratio

Figure 5 Inverse thermodynamic calculation results

Figure 6 MFB obtained from inverse thermodynamiccalculation and least-squares fitting result

Table 3 Calibration results for cases 1 and 2

Parameter Case 1 Case 2

cpre 0655 0655cmain 001 001vc 3263 3263b 172 172

Pre-chamber Dtd (ms) 417 394hpre 0941 0884

Main-chamber Dtd (ms) 193 286hmain 0912 0975

Song et al 1321

and Pmaini represent the modeled pressure points i isthe data index

Model validation and simulation results

After the calibration procedure the model is then vali-dated by the third experimental data set listed inTable 2 The simulation parameters for the thirdexperimental case are determined based on the follow-ing simple assumptions Since the AFR in the main-combustion chamber of the third case is the same asthe first one Dtd and hmain of the main-combustionchamber are assumed to be the same as in the first caseIn the pre-combustion chamber the other combustionparameters are assumed to vary linearly with the AFRfor the three experimental cases because their AFRs inthe pre-combustion chamber are within a relativelysmall range Based on the calculated AFRs in Table 2the coefficients Dtd and hpre of the pre-combustionchamber are calculated and listed in Table 4 The para-meters cpre cmain vc and b remain the sameFigures 7ndash9 show the comparison between the modeledand experimental pressure traces in two combustionchambers for the three experimental cases The calcu-lated pressure traces for the first two cases match theexperimental pressure traces very well since their para-meters are obtained by the calibration procedureAlthough the combustion parameters for the third case

are calculated based on the very simple assumptionsdiscussed above the agreement between the modeledand measured pressure traces is satisfactory The rela-tive errors on the pressure traces are always below10 The errors mainly occur after 31 ms in Figure 9These errors are mainly caused by the simple assump-tions used for determining the simulation parametersfor case 3 In reality the combustion process in themain-combustion chamber does not only depend onthe parameters in the main-combustion chamber Therelationship between the combustion parameters andthe AFR in the pre-combustion chamber is also notexactly linear Once more experimental data are avail-able further calibration can be done and model accu-racy can be improved

One of the main differences between TJI combustionand conventional SI combustion is that the turbulentjet increases the rate of combustion in TJI combustionThe experimental and calculated NHR rates are com-pared in Figure 10 We can find a peak on the NHRrate curve at the early stage of the combustion whichis caused by the turbulent jet As an example Figure 11shows how the model simulates the rate of combustion

Table 4 Simulation parameters

Parameter Case 3

cpre 0655cmain 001vc 3263b 172

Pre-chamber Dtd (ms) 380hpre 0848

Main-chamber Dtd (ms) 193hmain 0912

Figure 7 Experimental and calculated pressure traces forexperimental case 1

Figure 8 Experimental and calculated pressure traces forexperimental case 2

Figure 9 Experimental and calculated pressure traces forexperimental case 3

1322 Proc IMechE Part D J Automobile Engineering 231(10)

according to the mass flow rate of the turbulent jetThe top plot of Figure 11 shows the mass flow ratethrough the orifice connecting the two combustionchambers The curve in the middle plot of Figure 11 isb teth THORN in equation (20) The bottom plot of Figure 11 isthe calculated CER rate According to the simulationresults the MFB model with the parameter-varyingWiebe function successfully links the combustion pro-cesses in two combustion chambers

The value of the coefficient ab during the simulationis shown in Figure 12 where the dashed line is thevalue of ab calculated by assuming Ppre is greater thanPmain and the dash-dotted line is the value of ab calcu-lated by assuming Ppre is less than Pmain The actualvalue of ab used for the combustion calculation that isshown by the solid line is on the dashed line at begin-ning because the pre-combustion chamber is ignitedfirst After the switch line (see Figure 12) the main-combustion chamber pressure becomes larger than thatof the pre-combustion chamber ab jumps from thedashed line to the dash-dotted line

To further validate the model 12 more experimentaldata sets were used The orifice diameter varied from20mm to 30mm The relative AFRs in the two com-bustion chambers varied from 09 to 15 To simplifythe presentation the pressure traces were plotted onlyfor the first three cases see Figures 13ndash15 The experi-mental set-ups for the three cases are shown in Table 2as cases 4-6 The associated calibration results can befound in Table 5 For the other cases the calculated10ndash50 burn duration (Burn1050) and 50ndash90 burnduration (Burn5090) of the main-combustion chamber

Figure 10 Experimental and calculated net heat release rates

Figure 11 Chemical energy release rate calculation

Figure 12 The calculated value of ab during combustion

Figure 13 Experimental and calculated pressure traces forexperimental case 4

Figure 14 Experimental and calculated pressure traces forexperimental case 5

Song et al 1323

were compared with the experimental values seeFigures 16ndash17 Due to large variations of orifice areasand AFRs of these data sets the model parametersneed to be re-calibrated However b and wc were keptunchanged for the cases with the same orifice sizes likecases 4 and 5 It was also found that cpre and cmain werevery similar for all the experimental cases This indi-cates that these two parameters are mainly associated

with the combustion chamber structure In Table 5 thepre-combustion chamber burn durations are quite dif-ferent from the previous experimental cases 1ndash3 This isdue to the difference in the orifice area

To conclude the proposed model is able to fit theexperimental data sets with large ranges of AFRs inboth combustion chambers and different pre-combustion chamber orifice areas This indicates thatthe developed combustion model has the potential tobe used for the development of a TJI engine model

Conclusion

This article presents a control-oriented TJI combustionmodel for the RCM at Michigan State University Anewly proposed parameter-varying Wiebe combustionmodel is used to link the combustion processes in bothpre- and main-combustion chambers The developedmodel can be calibrated using a simple and systematiccalibration procedure based on the experimental dataThe model validation process shows good agreementbetween the modeled and experimental pressure traceswhich indicates that the developed model is capable ofaccurately capturing TJI combustion dynamics Thevalidation results also indicate that the model is able topredict the combustion process that is not used to cali-brate the model parameters This shows that the devel-oped model has the potential to be used for studyingTJI combustion engines and developing associated con-trol strategies Although only methane is used as fuel inthis article this model can be extended to other gaseousfuels with new calibrations However for liquid fuelthe model structure may need to be changed particu-larly a gasndashfuel mixing model will be required Futurework will extend the modeling work for TJI enginesusing gaseous fuel Note that in this case the pistondynamics and gas exchange (intake and exhaust) mod-els need to be added

Table 5 Calibration results for cases 4ndash6

Parameter Case 4 Case 5 Case 6

cpre 0651 0651 0651cmain 0 0 0vc 3829 3829 5064b 0824 0824 0764

Pre-chamber Dtd (ms) 197 228 179hpre 0980 0939 0901

Main-chamber Dtd (ms) 165 112 140hmain 0927 0905 0932

Figure 15 Experimental and calculated pressure traces forexperimental case 6

Figure 16 Experimental and calculated Burn1050 in the main-combustion chamberBurn1050 10ndash50 burn duration

Figure 17 Experimental and calculated Burn5090 in themain-combustion chamberBurn5090 50ndash90 burn duration

1324 Proc IMechE Part D J Automobile Engineering 231(10)

Declaration of conflicting interests

The author(s) declared no potential conflicts of interestwith respect to the research authorship andor publi-cation of this article

Funding

The author(s) disclosed receipt of the following finan-cial support for the research authorship andor publi-cation of this article This research is co-sponsored bythe US National Science Foundation and Departmentof Energy under contract number CBET-1258581

References

[1] Zhang S Zhu G and Sun Z A control-oriented chargemixing and two-zone HCCI combustion model IEEE T

Veh Technol 2014 63(3) 1079ndash1090[2] Yang X and Zhu G SI and HCCI combustion mode

transition control of an HCCI capable SI engine IEEET Contr Syst T 2013 21(5) 1558ndash1569

[3] Turkish MC 3-valve stratified charge engines Evolve-ment analysis and progression SAE technical paper741163 1974

[4] Date T Yagi S Ishizuya A et al Research and develop-ment of the Honda CVCC engine SAE Technical Paper1974

[5] Toulson E Schock HJ and Attard WP A review of pre-chamber initiated jet ignition combustion systems SAEtechnical paper 2010-01-2263 2010

[6] Song R Gentz G Zhu G et al A control-oriented jetignition combustion model for an SI engine In ASME

2015 dynamic systems and control conference ColumbusOhio 28ndash30 October 2015 ppV001T11A001ndashV001T11A001 American Society of Mechanical Engineers

[7] Toulson E Huisjen A Chen X et al Visualization ofpropane and natural gas spark ignition and turbulent jetignition combustion SAE Int J Engines 2012 5(4)

1821ndash1835[8] Yang X and Zhu GG A control-oriented hybrid com-

bustion model of a homogeneous charge compressionignition capable spark ignition engine P I Mech Eng

DmdashJ Aut 2012 226(10) 1380ndash1395[9] Stanglmaier RH and Roberts CE Homogeneous charge

compression ignition (HCCI) Benefits compromisesand future engine applications SAE technical paper1999-01-3682 1999

[10] Toulson E Watson HC and Attard WP Modeling alter-native prechamber fuels in jet assisted ignition of gasolineand LPG SAE technical paper 2009-01-0721 2009

[11] Ghorbani A Steinhilber G Markus D et al Numericalinvestigation of ignition in a transient turbulent jet bymeans of a PDF method Combust Sci Technol 2014186(10-11) 1582ndash1596

[12] Canova M Garcin R Midlam-Mohler S et al Acontrol-oriented model of combustion process in aHCCI diesel engine In Proceedings of the 2005

American control conference Portland Oregon 8ndash10

July 2005 vol 7 pp4446ndash4451 American AutomaticControl Council

[13] Guzzella L and Onder C Introduction to modeling and

control of internal combustion engine systems Berlin Hei-delberg Springer Science amp Business Media 2009

[14] Mortenson ME Mathematics for computer graphics

applications South Norwalk CT Industrial Press 1999[15] Heywood JB Internal combustion engine fundamentals

Vol 930 New York McGraw-Hill 1988[16] Ghojel J Review of the development and applications of

the Wiebe function A tribute to the contribution of Ivan

Wiebe to engine research Int J Engine Res 2010 11(4)297ndash312

[17] Zhao H and Ladommatos N Engine combustion instru-

mentation and diagnostics Warrendale PA Society of

Automotive Engineers 2001[18] Hellstrom E Stefanopoulou A and Jiang L A linear

least-squares algorithm for double-Wiebe functions

applied to spark-assisted compression ignition J Eng

Gas Turb Power 2014 136(9) 091514

Appendix

Abbreviations

TJI Turbulent jet ignitionRCM Rapid compression machineAFR Air-to-fuel ratioIC Internal combustionHCCI Homogeneous charge compression

ignitionCVCC Compound vortex controlled combustionSI Spark ignitionCFD Computational fluid dynamicsPDF Probability density functionMFB Mass fraction burnedNHR Net heat releaseCER Chemical energy release

List of simulation parameters

Parameter Symbol

Fuel injector discharge coefficient Cd1

Pre-chamber orifice discharge coefficient Cd2

Mass flow rate correcting parameter cpre

Mass flow rate correcting parameter cmain

Heat transfer coefficient hc

Lower heating value of the fuel QLHV

Wall temperature Tw

Coefficient of parameter-varying Wiebe function vc

Coefficient of parameter-varying Wiebe function bStart of ignition tign

Wiebe function coefficient aWiebe exponent mBurn duration in Wiebe function DtdPre-chamber combustion efficiency hpre

Main-chamber combustion efficiency hmain

Song et al 1325

However in reality this is not the case ab is combus-tion chamber structure dependent For our TJI systemthe spark plug is located at the top of the pre-combustion chamber see Figure 1 In this case thecombustion is initiated at the top of the pre-combustionchamber Since the orifice is at the bottom it is hardfor the burned gas to escape from the pre-combustionchamber at the early stage of the combustion As aresult the fraction of the burned gas flowing throughthe orifice to the main-combustion chamber is muchsmaller than the burned gas fraction inside the pre-combustion chamber This is why ab is smaller than vbin the pre-combustion chamber when Ppre PmainNote that ab will be determined using experimentaldata When Ppre Pmain the gas in the main-combustion chamber flows through the orifice and ab

will be determined by the burned gas fraction in themain-combustion chamber Since the combustion inthe main-combustion chamber is initiated by the turbu-lent jet (close to orifice) the orifice is surrounded bythe gas with high concentration of burned gasTherefore ab is larger than vb in the main-combustionchamber And again the actual value will be deter-mined by the experimental data This is the main reasonwhy the two-zone combustion model is used The valueof ab can be expressed by equation (12)

ab =f1 vbpre cpre

Ppre Pmain

f2 vbmain cmaineth THORN Ppre Pmain

eth12THORN

To simplify the calibration process the two func-tions f1 and f2 are approximated by second-degreeBezier curves14 Besides the control points (00) and(11) (cpre1 cpre) was added for f1 and(cmain1 cmain) for f2 as the third control points seeFigure 4 The parameters cpre and cmain are experimen-tally determined The Bezier curve guaranteesab 2 0 1frac12 as long as cpre 2 0 1frac12 and cmain 2 0 1frac12 Bychanging cpre and cmain the ratio of the burned andunburned gases flowing through the orifice can beadjusted to better match the actual physical processand thus to improve the model accuracy

Applying the principle of mass conservation theinstant fuel mass in the pre-combustion chamber canbe obtained by

dmprefueldt

=

mprefuel1 xb

dxbdt _mturu

1

l A=Feth THORNs +1

eth13THORN

where l is the relative AFR and A=Feth THORNs is the stoichio-metric AFR Note that only the fuel from the unburnedzone is considered

From equations (11) and (13) it can be observedthat the total amount of fuel burned inside the pre-combustion chamber is highly influenced by ab

The rate of chemical energy release (CER) isobtained by the following relationship

_Qch =hpreQLHVmprefuel1 xb

dxbdt

eth14THORN

where the combustion efficiency hpre is experimentallydetermined and QLHV is the lower heating value of thefuel

The rate of heat transfer to the combustion chamberwall can be modeled by the following equation15

_Qht =Aprehc Tpre Tw

eth15THORN

where Apre is the pre-combustion chamber surface areaTw is the mean wall temperature and hc is the heat-transfer coefficient calibrated by the experiment

After the ignition in the pre-combustion chamberthe combustion in the main-combustion chamber willnot be initiated until the generation of the turbulentjet from the pre-combustion chamber Before the igni-tion of the main-combustion chamber the mass flowfrom the burned zone of the pre-combustion chamberto the main-combustion chamber is neglected Theamount of the fuel in the main-combustion chamberis calculated by

dmmainfueldt

= _mturu1

lpre A=Feth THORNs + 1

eth16THORN

where mmainfuel is the fuel mass in the main-combustion chamber

After ignition in the main-combustion chamber theburned zone is created Different from the two-zonecombustion model in a conventional SI engine thecombustion model of the main-combustion chamberneeds to consider the gas flowing through the orificeinto the pre-combustion chamber The mass and energyconservation equations for burned and unburned zonesare very similar to those of the pre-combustion cham-ber model presented in this subsection and are omittedhere The major difference is that the total volume ofthe main-combustion chamber is varying

Mass fraction burned model

The MFB in the pre-combustion chamber is obtainedfrom the Wiebe function16Figure 4 The value of ab in the two cases

Song et al 1319

xb =1 exp a t tignDtd

m+1

eth17THORN

The coefficients a and m are chosen to be 6908and 2 respectively tign is the start of ignition and Dtdis the burn duration that is calibrated by AFR beforeignition

At the early stage of the combustion in the main-combustion chamber the rate of combustion is deter-mined by not only the gas properties in themain-combustion chamber but also the turbulent jetfrom the pre-combustion chamber This is due to thefact that the turbulent jets create multiple and distribu-ted ignition sites which increases the overall flamefront area in the main-combustion chamber Moreoverthese turbulent jets increase the turbulence intensity inthe main-combustion chamber and thus the flame frontpropagation speed After the turbulent jet disappearsthe rate of combustion reduces gradually to a relativelylow level and its characteristics are mainly determinedby the gas properties only in the main-combustionchamber Here the term lsquointensityrsquo of the turbulent jetis used to describe the resulting increment of the com-bustion rate in the main-combustion chamber due tothe turbulent jet Since the intensity of the turbulent jetis determined by the combustion processes in bothcombustion chambers estimating the rate of combus-tion before ignition is difficult and requires significantcalibration effort Therefore adjusting the rate of com-bustion according to the turbulent jet intensity duringthe combustion process is preferred for the TJI com-bustion model The conventional single-Wiebe functionis not suitable for our combustion model The multi-Wiebe function is a possible approach for modeling theMFB However this requires determining all associatedparameters before ignition occurs Therefore a newparameter-varying Wiebe function is proposed andused in this article see equation (18)

x0b teth THORN=1 exp a ttign b teth THORNDtd

h im+1

tign b teth THORN= t0 R tt0b teth THORN 1frac12 dt

8lt eth18THORN

where t0 and Dtd are determined by the spark timingand the AFR in the main-combustion chamber Thecoefficients a and m are chosen to be 6908 and 2respectively

If a m and Dtd are the same in equations (17) and(18) it can be proved that for any given tign b = tign

dx0bdt

=dxbdt b teth THORN eth19THORN

In other word the combustion rate calculated by thenew Wiebe function is b(t) times larger than that calcu-lated by the conventional Wiebe function Thereforethe intensity of the turbulent jet can be mathematicallyexpressed by b teth THORN The combustion model is able toadjust the rate of combustion by making b(t) a functionof some characteristics of the turbulent jet Moreover

b(t) can be changed at any time during the combustionprocess As long as b teth THORN is greater than 0 the combus-tion rate is greater than 0 and x0b teth THORN tends to 1 as t goesto infinity From the available experimental results it isfound that the rate of combustion in the main-combustion chamber is highly related to the mass flowrate of the turbulent jets from the pre-combustionchamber As a result it is assumed that the intensity ofthe turbulent jet can be linked to its mass flow rateAlthough this assumption provides a good matchbetween the modeled and available experimentalresults it is important to find an accurate method tocalculate the intensity of the turbulent jet in the futurewhen more experimental data are available Since theinfluence of the turbulent jet on the combustion in themain-combustion chamber is also delayed b teth THORN is mod-eled to be proportional to the mass flow rate of the tur-bulent jet with first-order dynamics see equation (20)An offset is used such that b(t)=1 when the mass flowrate of the turbulent jet is zero

b teth THORN=b frac12( _m+tur fl)(t)+1

( _m+tur fl)(t)=

R t0

_m+tur(t)fl(t t)dt eth20THORN

Here is the convolution operator and _m+tur and fl are

defined as follows

_m+tur =

_mtur _mtur500 _mtur 0

fl(t)=vce

vctu(t)

eth21THORN

Note that u(t) denotes unit step The convolution of_m+tur with the exponential decay function fl represents

the first-order dynamics and is used to emulate the timedelay To be more specific fl is the time response of alow-pass filter with cutoff frequency vc The parametersb and wc are experimentally determined When _mtur40we have b teth THORN=1 which means that the combustion ratewill not be altered if there is no turbulent jet from thepre-combustion chamber

Model calibration

The combustion model was calibrated using the experi-mental data collected from the RCM at Michigan StateUniversity described in the lsquoSystem descriptionrsquo sectionThe model is firstly calibrated using two experimentaldata sets and validated using another data set Then tofurther validate the model more data sets with differ-ent pre-combustion chamber orifice sizes are used Theexperimental set-ups for the first three cases can befound in Table 2 where cases 1 and 2 are used formodel calibration and case 3 for model validation

The first step is to calibrate the heat transfer modelTo do this the net heat release (NHR) rate in themain-combustion chamber needs to be calculated froman inverse thermodynamic calculation based on theexperimental pressure data1217 However this calcula-tion cannot be completed without knowing the mass

1320 Proc IMechE Part D J Automobile Engineering 231(10)

flow rate between the two combustion chambersFortunately the mass flow between the two combus-tion chambers mainly influences the combustion pro-cess in the pre-combustion chamber Its effect on themain-combustion chamber is limited For calibrationpurpose the mass flow can be neglected when calculat-ing the NHR rate in the main-combustion chamberThe result of the inverse calculation is shown in theupper plot of Figure 5 The NHR rate is the sum of theCER rate and the heat transfer rate from the cylinderwall The heat transfer model can be calibrated assum-ing that the heat transfer rate is dominant where theNHR rate is negative The calculated heat transfer rateis shown in the bottom plot of Figure 5

The next step is to calibrate Dtd for the main-combustion chamber After the heat transfer model iscalibrated the CER rate can be obtained by subtract-ing the heat transfer rate from the NHR rate shown asthe dotted line in Figure 5 This allows one to calculatethe MFB of the main-combustion chamber which isthe solid line in Figure 6 According to the later stageof the MFB curve (shown in Figure 6) the parameterDtd in the Wiebe function can be determined by a lin-ear least-squares fitting procedure18 The dotted line inFigure 6 is the curve fitting result The parameter b teth THORNis set to be 1 during this calibration procedure

Because the heat transfer coefficient hc of the pre-combustion chamber is assumed to be equal to that of

the main-combustion chamber The heat transfer rateto the pre-combustion chamber wall is determined Thisallows one to calibrate the other unknown parametersin Table 3 using a nonlinear least-squares optimizationprocedure Among these parameters cpre cmain vc andb in equations (12) and (20) remain constant for all thefirst three experimental cases The results of the linearleast-squares fitting and nonlinear least-squares optimi-zation procedures are shown in Table 3

The nonlinear least-squares optimization problem issolved by the nonlinear least-squares solver in Matlabusing the Trust-Region-Reflective algorithm This algo-rithm minimizes the following function

Xni=1

(Pprei Pprei)2 +

Xni=1

(Pmaini Pmaini)2

eth22THORN

where n is the total number of the data points Ppreiand Pmaini are the experimental pressure points Pprei

Table 2 Experimental set-up

Parameter Case

1 2 3 4 5 6

Orifice diameter (mm) 15 15 15 20 20 25Main-chamber AFR 183 210 183 15 125 15Pre-chamber fuel addition (mg) 066 089 089 0 0 0Pre-chamber AFR (calculated) 098 090 085 15 125 15

AFR air-to-fuel ratio

Figure 5 Inverse thermodynamic calculation results

Figure 6 MFB obtained from inverse thermodynamiccalculation and least-squares fitting result

Table 3 Calibration results for cases 1 and 2

Parameter Case 1 Case 2

cpre 0655 0655cmain 001 001vc 3263 3263b 172 172

Pre-chamber Dtd (ms) 417 394hpre 0941 0884

Main-chamber Dtd (ms) 193 286hmain 0912 0975

Song et al 1321

and Pmaini represent the modeled pressure points i isthe data index

Model validation and simulation results

After the calibration procedure the model is then vali-dated by the third experimental data set listed inTable 2 The simulation parameters for the thirdexperimental case are determined based on the follow-ing simple assumptions Since the AFR in the main-combustion chamber of the third case is the same asthe first one Dtd and hmain of the main-combustionchamber are assumed to be the same as in the first caseIn the pre-combustion chamber the other combustionparameters are assumed to vary linearly with the AFRfor the three experimental cases because their AFRs inthe pre-combustion chamber are within a relativelysmall range Based on the calculated AFRs in Table 2the coefficients Dtd and hpre of the pre-combustionchamber are calculated and listed in Table 4 The para-meters cpre cmain vc and b remain the sameFigures 7ndash9 show the comparison between the modeledand experimental pressure traces in two combustionchambers for the three experimental cases The calcu-lated pressure traces for the first two cases match theexperimental pressure traces very well since their para-meters are obtained by the calibration procedureAlthough the combustion parameters for the third case

are calculated based on the very simple assumptionsdiscussed above the agreement between the modeledand measured pressure traces is satisfactory The rela-tive errors on the pressure traces are always below10 The errors mainly occur after 31 ms in Figure 9These errors are mainly caused by the simple assump-tions used for determining the simulation parametersfor case 3 In reality the combustion process in themain-combustion chamber does not only depend onthe parameters in the main-combustion chamber Therelationship between the combustion parameters andthe AFR in the pre-combustion chamber is also notexactly linear Once more experimental data are avail-able further calibration can be done and model accu-racy can be improved

One of the main differences between TJI combustionand conventional SI combustion is that the turbulentjet increases the rate of combustion in TJI combustionThe experimental and calculated NHR rates are com-pared in Figure 10 We can find a peak on the NHRrate curve at the early stage of the combustion whichis caused by the turbulent jet As an example Figure 11shows how the model simulates the rate of combustion

Table 4 Simulation parameters

Parameter Case 3

cpre 0655cmain 001vc 3263b 172

Pre-chamber Dtd (ms) 380hpre 0848

Main-chamber Dtd (ms) 193hmain 0912

Figure 7 Experimental and calculated pressure traces forexperimental case 1

Figure 8 Experimental and calculated pressure traces forexperimental case 2

Figure 9 Experimental and calculated pressure traces forexperimental case 3

1322 Proc IMechE Part D J Automobile Engineering 231(10)

according to the mass flow rate of the turbulent jetThe top plot of Figure 11 shows the mass flow ratethrough the orifice connecting the two combustionchambers The curve in the middle plot of Figure 11 isb teth THORN in equation (20) The bottom plot of Figure 11 isthe calculated CER rate According to the simulationresults the MFB model with the parameter-varyingWiebe function successfully links the combustion pro-cesses in two combustion chambers

The value of the coefficient ab during the simulationis shown in Figure 12 where the dashed line is thevalue of ab calculated by assuming Ppre is greater thanPmain and the dash-dotted line is the value of ab calcu-lated by assuming Ppre is less than Pmain The actualvalue of ab used for the combustion calculation that isshown by the solid line is on the dashed line at begin-ning because the pre-combustion chamber is ignitedfirst After the switch line (see Figure 12) the main-combustion chamber pressure becomes larger than thatof the pre-combustion chamber ab jumps from thedashed line to the dash-dotted line

To further validate the model 12 more experimentaldata sets were used The orifice diameter varied from20mm to 30mm The relative AFRs in the two com-bustion chambers varied from 09 to 15 To simplifythe presentation the pressure traces were plotted onlyfor the first three cases see Figures 13ndash15 The experi-mental set-ups for the three cases are shown in Table 2as cases 4-6 The associated calibration results can befound in Table 5 For the other cases the calculated10ndash50 burn duration (Burn1050) and 50ndash90 burnduration (Burn5090) of the main-combustion chamber

Figure 10 Experimental and calculated net heat release rates

Figure 11 Chemical energy release rate calculation

Figure 12 The calculated value of ab during combustion

Figure 13 Experimental and calculated pressure traces forexperimental case 4

Figure 14 Experimental and calculated pressure traces forexperimental case 5

Song et al 1323

were compared with the experimental values seeFigures 16ndash17 Due to large variations of orifice areasand AFRs of these data sets the model parametersneed to be re-calibrated However b and wc were keptunchanged for the cases with the same orifice sizes likecases 4 and 5 It was also found that cpre and cmain werevery similar for all the experimental cases This indi-cates that these two parameters are mainly associated

with the combustion chamber structure In Table 5 thepre-combustion chamber burn durations are quite dif-ferent from the previous experimental cases 1ndash3 This isdue to the difference in the orifice area

To conclude the proposed model is able to fit theexperimental data sets with large ranges of AFRs inboth combustion chambers and different pre-combustion chamber orifice areas This indicates thatthe developed combustion model has the potential tobe used for the development of a TJI engine model

Conclusion

This article presents a control-oriented TJI combustionmodel for the RCM at Michigan State University Anewly proposed parameter-varying Wiebe combustionmodel is used to link the combustion processes in bothpre- and main-combustion chambers The developedmodel can be calibrated using a simple and systematiccalibration procedure based on the experimental dataThe model validation process shows good agreementbetween the modeled and experimental pressure traceswhich indicates that the developed model is capable ofaccurately capturing TJI combustion dynamics Thevalidation results also indicate that the model is able topredict the combustion process that is not used to cali-brate the model parameters This shows that the devel-oped model has the potential to be used for studyingTJI combustion engines and developing associated con-trol strategies Although only methane is used as fuel inthis article this model can be extended to other gaseousfuels with new calibrations However for liquid fuelthe model structure may need to be changed particu-larly a gasndashfuel mixing model will be required Futurework will extend the modeling work for TJI enginesusing gaseous fuel Note that in this case the pistondynamics and gas exchange (intake and exhaust) mod-els need to be added

Table 5 Calibration results for cases 4ndash6

Parameter Case 4 Case 5 Case 6

cpre 0651 0651 0651cmain 0 0 0vc 3829 3829 5064b 0824 0824 0764

Pre-chamber Dtd (ms) 197 228 179hpre 0980 0939 0901

Main-chamber Dtd (ms) 165 112 140hmain 0927 0905 0932

Figure 15 Experimental and calculated pressure traces forexperimental case 6

Figure 16 Experimental and calculated Burn1050 in the main-combustion chamberBurn1050 10ndash50 burn duration

Figure 17 Experimental and calculated Burn5090 in themain-combustion chamberBurn5090 50ndash90 burn duration

1324 Proc IMechE Part D J Automobile Engineering 231(10)

Declaration of conflicting interests

The author(s) declared no potential conflicts of interestwith respect to the research authorship andor publi-cation of this article

Funding

The author(s) disclosed receipt of the following finan-cial support for the research authorship andor publi-cation of this article This research is co-sponsored bythe US National Science Foundation and Departmentof Energy under contract number CBET-1258581

References

[1] Zhang S Zhu G and Sun Z A control-oriented chargemixing and two-zone HCCI combustion model IEEE T

Veh Technol 2014 63(3) 1079ndash1090[2] Yang X and Zhu G SI and HCCI combustion mode

transition control of an HCCI capable SI engine IEEET Contr Syst T 2013 21(5) 1558ndash1569

[3] Turkish MC 3-valve stratified charge engines Evolve-ment analysis and progression SAE technical paper741163 1974

[4] Date T Yagi S Ishizuya A et al Research and develop-ment of the Honda CVCC engine SAE Technical Paper1974

[5] Toulson E Schock HJ and Attard WP A review of pre-chamber initiated jet ignition combustion systems SAEtechnical paper 2010-01-2263 2010

[6] Song R Gentz G Zhu G et al A control-oriented jetignition combustion model for an SI engine In ASME

2015 dynamic systems and control conference ColumbusOhio 28ndash30 October 2015 ppV001T11A001ndashV001T11A001 American Society of Mechanical Engineers

[7] Toulson E Huisjen A Chen X et al Visualization ofpropane and natural gas spark ignition and turbulent jetignition combustion SAE Int J Engines 2012 5(4)

1821ndash1835[8] Yang X and Zhu GG A control-oriented hybrid com-

bustion model of a homogeneous charge compressionignition capable spark ignition engine P I Mech Eng

DmdashJ Aut 2012 226(10) 1380ndash1395[9] Stanglmaier RH and Roberts CE Homogeneous charge

compression ignition (HCCI) Benefits compromisesand future engine applications SAE technical paper1999-01-3682 1999

[10] Toulson E Watson HC and Attard WP Modeling alter-native prechamber fuels in jet assisted ignition of gasolineand LPG SAE technical paper 2009-01-0721 2009

[11] Ghorbani A Steinhilber G Markus D et al Numericalinvestigation of ignition in a transient turbulent jet bymeans of a PDF method Combust Sci Technol 2014186(10-11) 1582ndash1596

[12] Canova M Garcin R Midlam-Mohler S et al Acontrol-oriented model of combustion process in aHCCI diesel engine In Proceedings of the 2005

American control conference Portland Oregon 8ndash10

July 2005 vol 7 pp4446ndash4451 American AutomaticControl Council

[13] Guzzella L and Onder C Introduction to modeling and

control of internal combustion engine systems Berlin Hei-delberg Springer Science amp Business Media 2009

[14] Mortenson ME Mathematics for computer graphics

applications South Norwalk CT Industrial Press 1999[15] Heywood JB Internal combustion engine fundamentals

Vol 930 New York McGraw-Hill 1988[16] Ghojel J Review of the development and applications of

the Wiebe function A tribute to the contribution of Ivan

Wiebe to engine research Int J Engine Res 2010 11(4)297ndash312

[17] Zhao H and Ladommatos N Engine combustion instru-

mentation and diagnostics Warrendale PA Society of

Automotive Engineers 2001[18] Hellstrom E Stefanopoulou A and Jiang L A linear

least-squares algorithm for double-Wiebe functions

applied to spark-assisted compression ignition J Eng

Gas Turb Power 2014 136(9) 091514

Appendix

Abbreviations

TJI Turbulent jet ignitionRCM Rapid compression machineAFR Air-to-fuel ratioIC Internal combustionHCCI Homogeneous charge compression

ignitionCVCC Compound vortex controlled combustionSI Spark ignitionCFD Computational fluid dynamicsPDF Probability density functionMFB Mass fraction burnedNHR Net heat releaseCER Chemical energy release

List of simulation parameters

Parameter Symbol

Fuel injector discharge coefficient Cd1

Pre-chamber orifice discharge coefficient Cd2

Mass flow rate correcting parameter cpre

Mass flow rate correcting parameter cmain

Heat transfer coefficient hc

Lower heating value of the fuel QLHV

Wall temperature Tw

Coefficient of parameter-varying Wiebe function vc

Coefficient of parameter-varying Wiebe function bStart of ignition tign

Wiebe function coefficient aWiebe exponent mBurn duration in Wiebe function DtdPre-chamber combustion efficiency hpre

Main-chamber combustion efficiency hmain

Song et al 1325

xb =1 exp a t tignDtd

m+1

eth17THORN

The coefficients a and m are chosen to be 6908and 2 respectively tign is the start of ignition and Dtdis the burn duration that is calibrated by AFR beforeignition

At the early stage of the combustion in the main-combustion chamber the rate of combustion is deter-mined by not only the gas properties in themain-combustion chamber but also the turbulent jetfrom the pre-combustion chamber This is due to thefact that the turbulent jets create multiple and distribu-ted ignition sites which increases the overall flamefront area in the main-combustion chamber Moreoverthese turbulent jets increase the turbulence intensity inthe main-combustion chamber and thus the flame frontpropagation speed After the turbulent jet disappearsthe rate of combustion reduces gradually to a relativelylow level and its characteristics are mainly determinedby the gas properties only in the main-combustionchamber Here the term lsquointensityrsquo of the turbulent jetis used to describe the resulting increment of the com-bustion rate in the main-combustion chamber due tothe turbulent jet Since the intensity of the turbulent jetis determined by the combustion processes in bothcombustion chambers estimating the rate of combus-tion before ignition is difficult and requires significantcalibration effort Therefore adjusting the rate of com-bustion according to the turbulent jet intensity duringthe combustion process is preferred for the TJI com-bustion model The conventional single-Wiebe functionis not suitable for our combustion model The multi-Wiebe function is a possible approach for modeling theMFB However this requires determining all associatedparameters before ignition occurs Therefore a newparameter-varying Wiebe function is proposed andused in this article see equation (18)

x0b teth THORN=1 exp a ttign b teth THORNDtd

h im+1

tign b teth THORN= t0 R tt0b teth THORN 1frac12 dt

8lt eth18THORN

where t0 and Dtd are determined by the spark timingand the AFR in the main-combustion chamber Thecoefficients a and m are chosen to be 6908 and 2respectively

If a m and Dtd are the same in equations (17) and(18) it can be proved that for any given tign b = tign

dx0bdt

=dxbdt b teth THORN eth19THORN

In other word the combustion rate calculated by thenew Wiebe function is b(t) times larger than that calcu-lated by the conventional Wiebe function Thereforethe intensity of the turbulent jet can be mathematicallyexpressed by b teth THORN The combustion model is able toadjust the rate of combustion by making b(t) a functionof some characteristics of the turbulent jet Moreover

b(t) can be changed at any time during the combustionprocess As long as b teth THORN is greater than 0 the combus-tion rate is greater than 0 and x0b teth THORN tends to 1 as t goesto infinity From the available experimental results it isfound that the rate of combustion in the main-combustion chamber is highly related to the mass flowrate of the turbulent jets from the pre-combustionchamber As a result it is assumed that the intensity ofthe turbulent jet can be linked to its mass flow rateAlthough this assumption provides a good matchbetween the modeled and available experimentalresults it is important to find an accurate method tocalculate the intensity of the turbulent jet in the futurewhen more experimental data are available Since theinfluence of the turbulent jet on the combustion in themain-combustion chamber is also delayed b teth THORN is mod-eled to be proportional to the mass flow rate of the tur-bulent jet with first-order dynamics see equation (20)An offset is used such that b(t)=1 when the mass flowrate of the turbulent jet is zero

b teth THORN=b frac12( _m+tur fl)(t)+1

( _m+tur fl)(t)=

R t0

_m+tur(t)fl(t t)dt eth20THORN

Here is the convolution operator and _m+tur and fl are

defined as follows

_m+tur =

_mtur _mtur500 _mtur 0

fl(t)=vce

vctu(t)

eth21THORN

Note that u(t) denotes unit step The convolution of_m+tur with the exponential decay function fl represents

the first-order dynamics and is used to emulate the timedelay To be more specific fl is the time response of alow-pass filter with cutoff frequency vc The parametersb and wc are experimentally determined When _mtur40we have b teth THORN=1 which means that the combustion ratewill not be altered if there is no turbulent jet from thepre-combustion chamber

Model calibration

The combustion model was calibrated using the experi-mental data collected from the RCM at Michigan StateUniversity described in the lsquoSystem descriptionrsquo sectionThe model is firstly calibrated using two experimentaldata sets and validated using another data set Then tofurther validate the model more data sets with differ-ent pre-combustion chamber orifice sizes are used Theexperimental set-ups for the first three cases can befound in Table 2 where cases 1 and 2 are used formodel calibration and case 3 for model validation

The first step is to calibrate the heat transfer modelTo do this the net heat release (NHR) rate in themain-combustion chamber needs to be calculated froman inverse thermodynamic calculation based on theexperimental pressure data1217 However this calcula-tion cannot be completed without knowing the mass

1320 Proc IMechE Part D J Automobile Engineering 231(10)

flow rate between the two combustion chambersFortunately the mass flow between the two combus-tion chambers mainly influences the combustion pro-cess in the pre-combustion chamber Its effect on themain-combustion chamber is limited For calibrationpurpose the mass flow can be neglected when calculat-ing the NHR rate in the main-combustion chamberThe result of the inverse calculation is shown in theupper plot of Figure 5 The NHR rate is the sum of theCER rate and the heat transfer rate from the cylinderwall The heat transfer model can be calibrated assum-ing that the heat transfer rate is dominant where theNHR rate is negative The calculated heat transfer rateis shown in the bottom plot of Figure 5

The next step is to calibrate Dtd for the main-combustion chamber After the heat transfer model iscalibrated the CER rate can be obtained by subtract-ing the heat transfer rate from the NHR rate shown asthe dotted line in Figure 5 This allows one to calculatethe MFB of the main-combustion chamber which isthe solid line in Figure 6 According to the later stageof the MFB curve (shown in Figure 6) the parameterDtd in the Wiebe function can be determined by a lin-ear least-squares fitting procedure18 The dotted line inFigure 6 is the curve fitting result The parameter b teth THORNis set to be 1 during this calibration procedure

Because the heat transfer coefficient hc of the pre-combustion chamber is assumed to be equal to that of

the main-combustion chamber The heat transfer rateto the pre-combustion chamber wall is determined Thisallows one to calibrate the other unknown parametersin Table 3 using a nonlinear least-squares optimizationprocedure Among these parameters cpre cmain vc andb in equations (12) and (20) remain constant for all thefirst three experimental cases The results of the linearleast-squares fitting and nonlinear least-squares optimi-zation procedures are shown in Table 3

The nonlinear least-squares optimization problem issolved by the nonlinear least-squares solver in Matlabusing the Trust-Region-Reflective algorithm This algo-rithm minimizes the following function

Xni=1

(Pprei Pprei)2 +

Xni=1

(Pmaini Pmaini)2

eth22THORN

where n is the total number of the data points Ppreiand Pmaini are the experimental pressure points Pprei

Table 2 Experimental set-up

Parameter Case

1 2 3 4 5 6

Orifice diameter (mm) 15 15 15 20 20 25Main-chamber AFR 183 210 183 15 125 15Pre-chamber fuel addition (mg) 066 089 089 0 0 0Pre-chamber AFR (calculated) 098 090 085 15 125 15

AFR air-to-fuel ratio

Figure 5 Inverse thermodynamic calculation results

Figure 6 MFB obtained from inverse thermodynamiccalculation and least-squares fitting result

Table 3 Calibration results for cases 1 and 2

Parameter Case 1 Case 2

cpre 0655 0655cmain 001 001vc 3263 3263b 172 172

Pre-chamber Dtd (ms) 417 394hpre 0941 0884

Main-chamber Dtd (ms) 193 286hmain 0912 0975

Song et al 1321

and Pmaini represent the modeled pressure points i isthe data index

Model validation and simulation results

After the calibration procedure the model is then vali-dated by the third experimental data set listed inTable 2 The simulation parameters for the thirdexperimental case are determined based on the follow-ing simple assumptions Since the AFR in the main-combustion chamber of the third case is the same asthe first one Dtd and hmain of the main-combustionchamber are assumed to be the same as in the first caseIn the pre-combustion chamber the other combustionparameters are assumed to vary linearly with the AFRfor the three experimental cases because their AFRs inthe pre-combustion chamber are within a relativelysmall range Based on the calculated AFRs in Table 2the coefficients Dtd and hpre of the pre-combustionchamber are calculated and listed in Table 4 The para-meters cpre cmain vc and b remain the sameFigures 7ndash9 show the comparison between the modeledand experimental pressure traces in two combustionchambers for the three experimental cases The calcu-lated pressure traces for the first two cases match theexperimental pressure traces very well since their para-meters are obtained by the calibration procedureAlthough the combustion parameters for the third case

are calculated based on the very simple assumptionsdiscussed above the agreement between the modeledand measured pressure traces is satisfactory The rela-tive errors on the pressure traces are always below10 The errors mainly occur after 31 ms in Figure 9These errors are mainly caused by the simple assump-tions used for determining the simulation parametersfor case 3 In reality the combustion process in themain-combustion chamber does not only depend onthe parameters in the main-combustion chamber Therelationship between the combustion parameters andthe AFR in the pre-combustion chamber is also notexactly linear Once more experimental data are avail-able further calibration can be done and model accu-racy can be improved

One of the main differences between TJI combustionand conventional SI combustion is that the turbulentjet increases the rate of combustion in TJI combustionThe experimental and calculated NHR rates are com-pared in Figure 10 We can find a peak on the NHRrate curve at the early stage of the combustion whichis caused by the turbulent jet As an example Figure 11shows how the model simulates the rate of combustion

Table 4 Simulation parameters

Parameter Case 3

cpre 0655cmain 001vc 3263b 172

Pre-chamber Dtd (ms) 380hpre 0848

Main-chamber Dtd (ms) 193hmain 0912

Figure 7 Experimental and calculated pressure traces forexperimental case 1

Figure 8 Experimental and calculated pressure traces forexperimental case 2

Figure 9 Experimental and calculated pressure traces forexperimental case 3

1322 Proc IMechE Part D J Automobile Engineering 231(10)

according to the mass flow rate of the turbulent jetThe top plot of Figure 11 shows the mass flow ratethrough the orifice connecting the two combustionchambers The curve in the middle plot of Figure 11 isb teth THORN in equation (20) The bottom plot of Figure 11 isthe calculated CER rate According to the simulationresults the MFB model with the parameter-varyingWiebe function successfully links the combustion pro-cesses in two combustion chambers

The value of the coefficient ab during the simulationis shown in Figure 12 where the dashed line is thevalue of ab calculated by assuming Ppre is greater thanPmain and the dash-dotted line is the value of ab calcu-lated by assuming Ppre is less than Pmain The actualvalue of ab used for the combustion calculation that isshown by the solid line is on the dashed line at begin-ning because the pre-combustion chamber is ignitedfirst After the switch line (see Figure 12) the main-combustion chamber pressure becomes larger than thatof the pre-combustion chamber ab jumps from thedashed line to the dash-dotted line

To further validate the model 12 more experimentaldata sets were used The orifice diameter varied from20mm to 30mm The relative AFRs in the two com-bustion chambers varied from 09 to 15 To simplifythe presentation the pressure traces were plotted onlyfor the first three cases see Figures 13ndash15 The experi-mental set-ups for the three cases are shown in Table 2as cases 4-6 The associated calibration results can befound in Table 5 For the other cases the calculated10ndash50 burn duration (Burn1050) and 50ndash90 burnduration (Burn5090) of the main-combustion chamber

Figure 10 Experimental and calculated net heat release rates

Figure 11 Chemical energy release rate calculation

Figure 12 The calculated value of ab during combustion

Figure 13 Experimental and calculated pressure traces forexperimental case 4

Figure 14 Experimental and calculated pressure traces forexperimental case 5

Song et al 1323

were compared with the experimental values seeFigures 16ndash17 Due to large variations of orifice areasand AFRs of these data sets the model parametersneed to be re-calibrated However b and wc were keptunchanged for the cases with the same orifice sizes likecases 4 and 5 It was also found that cpre and cmain werevery similar for all the experimental cases This indi-cates that these two parameters are mainly associated

with the combustion chamber structure In Table 5 thepre-combustion chamber burn durations are quite dif-ferent from the previous experimental cases 1ndash3 This isdue to the difference in the orifice area

To conclude the proposed model is able to fit theexperimental data sets with large ranges of AFRs inboth combustion chambers and different pre-combustion chamber orifice areas This indicates thatthe developed combustion model has the potential tobe used for the development of a TJI engine model

Conclusion

This article presents a control-oriented TJI combustionmodel for the RCM at Michigan State University Anewly proposed parameter-varying Wiebe combustionmodel is used to link the combustion processes in bothpre- and main-combustion chambers The developedmodel can be calibrated using a simple and systematiccalibration procedure based on the experimental dataThe model validation process shows good agreementbetween the modeled and experimental pressure traceswhich indicates that the developed model is capable ofaccurately capturing TJI combustion dynamics Thevalidation results also indicate that the model is able topredict the combustion process that is not used to cali-brate the model parameters This shows that the devel-oped model has the potential to be used for studyingTJI combustion engines and developing associated con-trol strategies Although only methane is used as fuel inthis article this model can be extended to other gaseousfuels with new calibrations However for liquid fuelthe model structure may need to be changed particu-larly a gasndashfuel mixing model will be required Futurework will extend the modeling work for TJI enginesusing gaseous fuel Note that in this case the pistondynamics and gas exchange (intake and exhaust) mod-els need to be added

Table 5 Calibration results for cases 4ndash6

Parameter Case 4 Case 5 Case 6

cpre 0651 0651 0651cmain 0 0 0vc 3829 3829 5064b 0824 0824 0764

Pre-chamber Dtd (ms) 197 228 179hpre 0980 0939 0901

Main-chamber Dtd (ms) 165 112 140hmain 0927 0905 0932

Figure 15 Experimental and calculated pressure traces forexperimental case 6

Figure 16 Experimental and calculated Burn1050 in the main-combustion chamberBurn1050 10ndash50 burn duration

Figure 17 Experimental and calculated Burn5090 in themain-combustion chamberBurn5090 50ndash90 burn duration

1324 Proc IMechE Part D J Automobile Engineering 231(10)

Declaration of conflicting interests

The author(s) declared no potential conflicts of interestwith respect to the research authorship andor publi-cation of this article

Funding

The author(s) disclosed receipt of the following finan-cial support for the research authorship andor publi-cation of this article This research is co-sponsored bythe US National Science Foundation and Departmentof Energy under contract number CBET-1258581

References

[1] Zhang S Zhu G and Sun Z A control-oriented chargemixing and two-zone HCCI combustion model IEEE T

Veh Technol 2014 63(3) 1079ndash1090[2] Yang X and Zhu G SI and HCCI combustion mode

transition control of an HCCI capable SI engine IEEET Contr Syst T 2013 21(5) 1558ndash1569

[3] Turkish MC 3-valve stratified charge engines Evolve-ment analysis and progression SAE technical paper741163 1974

[4] Date T Yagi S Ishizuya A et al Research and develop-ment of the Honda CVCC engine SAE Technical Paper1974

[5] Toulson E Schock HJ and Attard WP A review of pre-chamber initiated jet ignition combustion systems SAEtechnical paper 2010-01-2263 2010

[6] Song R Gentz G Zhu G et al A control-oriented jetignition combustion model for an SI engine In ASME

2015 dynamic systems and control conference ColumbusOhio 28ndash30 October 2015 ppV001T11A001ndashV001T11A001 American Society of Mechanical Engineers

[7] Toulson E Huisjen A Chen X et al Visualization ofpropane and natural gas spark ignition and turbulent jetignition combustion SAE Int J Engines 2012 5(4)

1821ndash1835[8] Yang X and Zhu GG A control-oriented hybrid com-

bustion model of a homogeneous charge compressionignition capable spark ignition engine P I Mech Eng

DmdashJ Aut 2012 226(10) 1380ndash1395[9] Stanglmaier RH and Roberts CE Homogeneous charge

compression ignition (HCCI) Benefits compromisesand future engine applications SAE technical paper1999-01-3682 1999

[10] Toulson E Watson HC and Attard WP Modeling alter-native prechamber fuels in jet assisted ignition of gasolineand LPG SAE technical paper 2009-01-0721 2009

[11] Ghorbani A Steinhilber G Markus D et al Numericalinvestigation of ignition in a transient turbulent jet bymeans of a PDF method Combust Sci Technol 2014186(10-11) 1582ndash1596

[12] Canova M Garcin R Midlam-Mohler S et al Acontrol-oriented model of combustion process in aHCCI diesel engine In Proceedings of the 2005

American control conference Portland Oregon 8ndash10

July 2005 vol 7 pp4446ndash4451 American AutomaticControl Council

[13] Guzzella L and Onder C Introduction to modeling and

control of internal combustion engine systems Berlin Hei-delberg Springer Science amp Business Media 2009

[14] Mortenson ME Mathematics for computer graphics

applications South Norwalk CT Industrial Press 1999[15] Heywood JB Internal combustion engine fundamentals

Vol 930 New York McGraw-Hill 1988[16] Ghojel J Review of the development and applications of

the Wiebe function A tribute to the contribution of Ivan

Wiebe to engine research Int J Engine Res 2010 11(4)297ndash312

[17] Zhao H and Ladommatos N Engine combustion instru-

mentation and diagnostics Warrendale PA Society of

Automotive Engineers 2001[18] Hellstrom E Stefanopoulou A and Jiang L A linear

least-squares algorithm for double-Wiebe functions

applied to spark-assisted compression ignition J Eng

Gas Turb Power 2014 136(9) 091514

Appendix

Abbreviations

TJI Turbulent jet ignitionRCM Rapid compression machineAFR Air-to-fuel ratioIC Internal combustionHCCI Homogeneous charge compression

ignitionCVCC Compound vortex controlled combustionSI Spark ignitionCFD Computational fluid dynamicsPDF Probability density functionMFB Mass fraction burnedNHR Net heat releaseCER Chemical energy release

List of simulation parameters

Parameter Symbol

Fuel injector discharge coefficient Cd1

Pre-chamber orifice discharge coefficient Cd2

Mass flow rate correcting parameter cpre

Mass flow rate correcting parameter cmain

Heat transfer coefficient hc

Lower heating value of the fuel QLHV

Wall temperature Tw

Coefficient of parameter-varying Wiebe function vc

Coefficient of parameter-varying Wiebe function bStart of ignition tign

Wiebe function coefficient aWiebe exponent mBurn duration in Wiebe function DtdPre-chamber combustion efficiency hpre

Main-chamber combustion efficiency hmain

Song et al 1325

flow rate between the two combustion chambersFortunately the mass flow between the two combus-tion chambers mainly influences the combustion pro-cess in the pre-combustion chamber Its effect on themain-combustion chamber is limited For calibrationpurpose the mass flow can be neglected when calculat-ing the NHR rate in the main-combustion chamberThe result of the inverse calculation is shown in theupper plot of Figure 5 The NHR rate is the sum of theCER rate and the heat transfer rate from the cylinderwall The heat transfer model can be calibrated assum-ing that the heat transfer rate is dominant where theNHR rate is negative The calculated heat transfer rateis shown in the bottom plot of Figure 5

The next step is to calibrate Dtd for the main-combustion chamber After the heat transfer model iscalibrated the CER rate can be obtained by subtract-ing the heat transfer rate from the NHR rate shown asthe dotted line in Figure 5 This allows one to calculatethe MFB of the main-combustion chamber which isthe solid line in Figure 6 According to the later stageof the MFB curve (shown in Figure 6) the parameterDtd in the Wiebe function can be determined by a lin-ear least-squares fitting procedure18 The dotted line inFigure 6 is the curve fitting result The parameter b teth THORNis set to be 1 during this calibration procedure

Because the heat transfer coefficient hc of the pre-combustion chamber is assumed to be equal to that of

the main-combustion chamber The heat transfer rateto the pre-combustion chamber wall is determined Thisallows one to calibrate the other unknown parametersin Table 3 using a nonlinear least-squares optimizationprocedure Among these parameters cpre cmain vc andb in equations (12) and (20) remain constant for all thefirst three experimental cases The results of the linearleast-squares fitting and nonlinear least-squares optimi-zation procedures are shown in Table 3

The nonlinear least-squares optimization problem issolved by the nonlinear least-squares solver in Matlabusing the Trust-Region-Reflective algorithm This algo-rithm minimizes the following function

Xni=1

(Pprei Pprei)2 +

Xni=1

(Pmaini Pmaini)2

eth22THORN

where n is the total number of the data points Ppreiand Pmaini are the experimental pressure points Pprei

Table 2 Experimental set-up

Parameter Case

1 2 3 4 5 6

Orifice diameter (mm) 15 15 15 20 20 25Main-chamber AFR 183 210 183 15 125 15Pre-chamber fuel addition (mg) 066 089 089 0 0 0Pre-chamber AFR (calculated) 098 090 085 15 125 15

AFR air-to-fuel ratio

Figure 5 Inverse thermodynamic calculation results

Figure 6 MFB obtained from inverse thermodynamiccalculation and least-squares fitting result

Table 3 Calibration results for cases 1 and 2

Parameter Case 1 Case 2

cpre 0655 0655cmain 001 001vc 3263 3263b 172 172

Pre-chamber Dtd (ms) 417 394hpre 0941 0884

Main-chamber Dtd (ms) 193 286hmain 0912 0975

Song et al 1321

and Pmaini represent the modeled pressure points i isthe data index

Model validation and simulation results

After the calibration procedure the model is then vali-dated by the third experimental data set listed inTable 2 The simulation parameters for the thirdexperimental case are determined based on the follow-ing simple assumptions Since the AFR in the main-combustion chamber of the third case is the same asthe first one Dtd and hmain of the main-combustionchamber are assumed to be the same as in the first caseIn the pre-combustion chamber the other combustionparameters are assumed to vary linearly with the AFRfor the three experimental cases because their AFRs inthe pre-combustion chamber are within a relativelysmall range Based on the calculated AFRs in Table 2the coefficients Dtd and hpre of the pre-combustionchamber are calculated and listed in Table 4 The para-meters cpre cmain vc and b remain the sameFigures 7ndash9 show the comparison between the modeledand experimental pressure traces in two combustionchambers for the three experimental cases The calcu-lated pressure traces for the first two cases match theexperimental pressure traces very well since their para-meters are obtained by the calibration procedureAlthough the combustion parameters for the third case

are calculated based on the very simple assumptionsdiscussed above the agreement between the modeledand measured pressure traces is satisfactory The rela-tive errors on the pressure traces are always below10 The errors mainly occur after 31 ms in Figure 9These errors are mainly caused by the simple assump-tions used for determining the simulation parametersfor case 3 In reality the combustion process in themain-combustion chamber does not only depend onthe parameters in the main-combustion chamber Therelationship between the combustion parameters andthe AFR in the pre-combustion chamber is also notexactly linear Once more experimental data are avail-able further calibration can be done and model accu-racy can be improved

One of the main differences between TJI combustionand conventional SI combustion is that the turbulentjet increases the rate of combustion in TJI combustionThe experimental and calculated NHR rates are com-pared in Figure 10 We can find a peak on the NHRrate curve at the early stage of the combustion whichis caused by the turbulent jet As an example Figure 11shows how the model simulates the rate of combustion

Table 4 Simulation parameters

Parameter Case 3

cpre 0655cmain 001vc 3263b 172

Pre-chamber Dtd (ms) 380hpre 0848

Main-chamber Dtd (ms) 193hmain 0912

Figure 7 Experimental and calculated pressure traces forexperimental case 1

Figure 8 Experimental and calculated pressure traces forexperimental case 2

Figure 9 Experimental and calculated pressure traces forexperimental case 3

1322 Proc IMechE Part D J Automobile Engineering 231(10)

according to the mass flow rate of the turbulent jetThe top plot of Figure 11 shows the mass flow ratethrough the orifice connecting the two combustionchambers The curve in the middle plot of Figure 11 isb teth THORN in equation (20) The bottom plot of Figure 11 isthe calculated CER rate According to the simulationresults the MFB model with the parameter-varyingWiebe function successfully links the combustion pro-cesses in two combustion chambers

The value of the coefficient ab during the simulationis shown in Figure 12 where the dashed line is thevalue of ab calculated by assuming Ppre is greater thanPmain and the dash-dotted line is the value of ab calcu-lated by assuming Ppre is less than Pmain The actualvalue of ab used for the combustion calculation that isshown by the solid line is on the dashed line at begin-ning because the pre-combustion chamber is ignitedfirst After the switch line (see Figure 12) the main-combustion chamber pressure becomes larger than thatof the pre-combustion chamber ab jumps from thedashed line to the dash-dotted line

To further validate the model 12 more experimentaldata sets were used The orifice diameter varied from20mm to 30mm The relative AFRs in the two com-bustion chambers varied from 09 to 15 To simplifythe presentation the pressure traces were plotted onlyfor the first three cases see Figures 13ndash15 The experi-mental set-ups for the three cases are shown in Table 2as cases 4-6 The associated calibration results can befound in Table 5 For the other cases the calculated10ndash50 burn duration (Burn1050) and 50ndash90 burnduration (Burn5090) of the main-combustion chamber

Figure 10 Experimental and calculated net heat release rates

Figure 11 Chemical energy release rate calculation

Figure 12 The calculated value of ab during combustion

Figure 13 Experimental and calculated pressure traces forexperimental case 4

Figure 14 Experimental and calculated pressure traces forexperimental case 5

Song et al 1323

were compared with the experimental values seeFigures 16ndash17 Due to large variations of orifice areasand AFRs of these data sets the model parametersneed to be re-calibrated However b and wc were keptunchanged for the cases with the same orifice sizes likecases 4 and 5 It was also found that cpre and cmain werevery similar for all the experimental cases This indi-cates that these two parameters are mainly associated

with the combustion chamber structure In Table 5 thepre-combustion chamber burn durations are quite dif-ferent from the previous experimental cases 1ndash3 This isdue to the difference in the orifice area

To conclude the proposed model is able to fit theexperimental data sets with large ranges of AFRs inboth combustion chambers and different pre-combustion chamber orifice areas This indicates thatthe developed combustion model has the potential tobe used for the development of a TJI engine model

Conclusion

This article presents a control-oriented TJI combustionmodel for the RCM at Michigan State University Anewly proposed parameter-varying Wiebe combustionmodel is used to link the combustion processes in bothpre- and main-combustion chambers The developedmodel can be calibrated using a simple and systematiccalibration procedure based on the experimental dataThe model validation process shows good agreementbetween the modeled and experimental pressure traceswhich indicates that the developed model is capable ofaccurately capturing TJI combustion dynamics Thevalidation results also indicate that the model is able topredict the combustion process that is not used to cali-brate the model parameters This shows that the devel-oped model has the potential to be used for studyingTJI combustion engines and developing associated con-trol strategies Although only methane is used as fuel inthis article this model can be extended to other gaseousfuels with new calibrations However for liquid fuelthe model structure may need to be changed particu-larly a gasndashfuel mixing model will be required Futurework will extend the modeling work for TJI enginesusing gaseous fuel Note that in this case the pistondynamics and gas exchange (intake and exhaust) mod-els need to be added

Table 5 Calibration results for cases 4ndash6

Parameter Case 4 Case 5 Case 6

cpre 0651 0651 0651cmain 0 0 0vc 3829 3829 5064b 0824 0824 0764

Pre-chamber Dtd (ms) 197 228 179hpre 0980 0939 0901

Main-chamber Dtd (ms) 165 112 140hmain 0927 0905 0932

Figure 15 Experimental and calculated pressure traces forexperimental case 6

Figure 16 Experimental and calculated Burn1050 in the main-combustion chamberBurn1050 10ndash50 burn duration

Figure 17 Experimental and calculated Burn5090 in themain-combustion chamberBurn5090 50ndash90 burn duration

1324 Proc IMechE Part D J Automobile Engineering 231(10)

Declaration of conflicting interests

The author(s) declared no potential conflicts of interestwith respect to the research authorship andor publi-cation of this article

Funding

The author(s) disclosed receipt of the following finan-cial support for the research authorship andor publi-cation of this article This research is co-sponsored bythe US National Science Foundation and Departmentof Energy under contract number CBET-1258581

References

[1] Zhang S Zhu G and Sun Z A control-oriented chargemixing and two-zone HCCI combustion model IEEE T

Veh Technol 2014 63(3) 1079ndash1090[2] Yang X and Zhu G SI and HCCI combustion mode

transition control of an HCCI capable SI engine IEEET Contr Syst T 2013 21(5) 1558ndash1569

[3] Turkish MC 3-valve stratified charge engines Evolve-ment analysis and progression SAE technical paper741163 1974

[4] Date T Yagi S Ishizuya A et al Research and develop-ment of the Honda CVCC engine SAE Technical Paper1974

[5] Toulson E Schock HJ and Attard WP A review of pre-chamber initiated jet ignition combustion systems SAEtechnical paper 2010-01-2263 2010

[6] Song R Gentz G Zhu G et al A control-oriented jetignition combustion model for an SI engine In ASME

2015 dynamic systems and control conference ColumbusOhio 28ndash30 October 2015 ppV001T11A001ndashV001T11A001 American Society of Mechanical Engineers

[7] Toulson E Huisjen A Chen X et al Visualization ofpropane and natural gas spark ignition and turbulent jetignition combustion SAE Int J Engines 2012 5(4)

1821ndash1835[8] Yang X and Zhu GG A control-oriented hybrid com-

bustion model of a homogeneous charge compressionignition capable spark ignition engine P I Mech Eng

DmdashJ Aut 2012 226(10) 1380ndash1395[9] Stanglmaier RH and Roberts CE Homogeneous charge

compression ignition (HCCI) Benefits compromisesand future engine applications SAE technical paper1999-01-3682 1999

[10] Toulson E Watson HC and Attard WP Modeling alter-native prechamber fuels in jet assisted ignition of gasolineand LPG SAE technical paper 2009-01-0721 2009

[11] Ghorbani A Steinhilber G Markus D et al Numericalinvestigation of ignition in a transient turbulent jet bymeans of a PDF method Combust Sci Technol 2014186(10-11) 1582ndash1596

[12] Canova M Garcin R Midlam-Mohler S et al Acontrol-oriented model of combustion process in aHCCI diesel engine In Proceedings of the 2005

American control conference Portland Oregon 8ndash10

July 2005 vol 7 pp4446ndash4451 American AutomaticControl Council

[13] Guzzella L and Onder C Introduction to modeling and

control of internal combustion engine systems Berlin Hei-delberg Springer Science amp Business Media 2009

[14] Mortenson ME Mathematics for computer graphics

applications South Norwalk CT Industrial Press 1999[15] Heywood JB Internal combustion engine fundamentals

Vol 930 New York McGraw-Hill 1988[16] Ghojel J Review of the development and applications of

the Wiebe function A tribute to the contribution of Ivan

Wiebe to engine research Int J Engine Res 2010 11(4)297ndash312

[17] Zhao H and Ladommatos N Engine combustion instru-

mentation and diagnostics Warrendale PA Society of

Automotive Engineers 2001[18] Hellstrom E Stefanopoulou A and Jiang L A linear

least-squares algorithm for double-Wiebe functions

applied to spark-assisted compression ignition J Eng

Gas Turb Power 2014 136(9) 091514

Appendix

Abbreviations

TJI Turbulent jet ignitionRCM Rapid compression machineAFR Air-to-fuel ratioIC Internal combustionHCCI Homogeneous charge compression

ignitionCVCC Compound vortex controlled combustionSI Spark ignitionCFD Computational fluid dynamicsPDF Probability density functionMFB Mass fraction burnedNHR Net heat releaseCER Chemical energy release

List of simulation parameters

Parameter Symbol

Fuel injector discharge coefficient Cd1

Pre-chamber orifice discharge coefficient Cd2

Mass flow rate correcting parameter cpre

Mass flow rate correcting parameter cmain

Heat transfer coefficient hc

Lower heating value of the fuel QLHV

Wall temperature Tw

Coefficient of parameter-varying Wiebe function vc

Coefficient of parameter-varying Wiebe function bStart of ignition tign

Wiebe function coefficient aWiebe exponent mBurn duration in Wiebe function DtdPre-chamber combustion efficiency hpre

Main-chamber combustion efficiency hmain

Song et al 1325

and Pmaini represent the modeled pressure points i isthe data index

Model validation and simulation results

After the calibration procedure the model is then vali-dated by the third experimental data set listed inTable 2 The simulation parameters for the thirdexperimental case are determined based on the follow-ing simple assumptions Since the AFR in the main-combustion chamber of the third case is the same asthe first one Dtd and hmain of the main-combustionchamber are assumed to be the same as in the first caseIn the pre-combustion chamber the other combustionparameters are assumed to vary linearly with the AFRfor the three experimental cases because their AFRs inthe pre-combustion chamber are within a relativelysmall range Based on the calculated AFRs in Table 2the coefficients Dtd and hpre of the pre-combustionchamber are calculated and listed in Table 4 The para-meters cpre cmain vc and b remain the sameFigures 7ndash9 show the comparison between the modeledand experimental pressure traces in two combustionchambers for the three experimental cases The calcu-lated pressure traces for the first two cases match theexperimental pressure traces very well since their para-meters are obtained by the calibration procedureAlthough the combustion parameters for the third case

are calculated based on the very simple assumptionsdiscussed above the agreement between the modeledand measured pressure traces is satisfactory The rela-tive errors on the pressure traces are always below10 The errors mainly occur after 31 ms in Figure 9These errors are mainly caused by the simple assump-tions used for determining the simulation parametersfor case 3 In reality the combustion process in themain-combustion chamber does not only depend onthe parameters in the main-combustion chamber Therelationship between the combustion parameters andthe AFR in the pre-combustion chamber is also notexactly linear Once more experimental data are avail-able further calibration can be done and model accu-racy can be improved

One of the main differences between TJI combustionand conventional SI combustion is that the turbulentjet increases the rate of combustion in TJI combustionThe experimental and calculated NHR rates are com-pared in Figure 10 We can find a peak on the NHRrate curve at the early stage of the combustion whichis caused by the turbulent jet As an example Figure 11shows how the model simulates the rate of combustion

Table 4 Simulation parameters

Parameter Case 3

cpre 0655cmain 001vc 3263b 172

Pre-chamber Dtd (ms) 380hpre 0848

Main-chamber Dtd (ms) 193hmain 0912

Figure 7 Experimental and calculated pressure traces forexperimental case 1

Figure 8 Experimental and calculated pressure traces forexperimental case 2

Figure 9 Experimental and calculated pressure traces forexperimental case 3

1322 Proc IMechE Part D J Automobile Engineering 231(10)

according to the mass flow rate of the turbulent jetThe top plot of Figure 11 shows the mass flow ratethrough the orifice connecting the two combustionchambers The curve in the middle plot of Figure 11 isb teth THORN in equation (20) The bottom plot of Figure 11 isthe calculated CER rate According to the simulationresults the MFB model with the parameter-varyingWiebe function successfully links the combustion pro-cesses in two combustion chambers

The value of the coefficient ab during the simulationis shown in Figure 12 where the dashed line is thevalue of ab calculated by assuming Ppre is greater thanPmain and the dash-dotted line is the value of ab calcu-lated by assuming Ppre is less than Pmain The actualvalue of ab used for the combustion calculation that isshown by the solid line is on the dashed line at begin-ning because the pre-combustion chamber is ignitedfirst After the switch line (see Figure 12) the main-combustion chamber pressure becomes larger than thatof the pre-combustion chamber ab jumps from thedashed line to the dash-dotted line

To further validate the model 12 more experimentaldata sets were used The orifice diameter varied from20mm to 30mm The relative AFRs in the two com-bustion chambers varied from 09 to 15 To simplifythe presentation the pressure traces were plotted onlyfor the first three cases see Figures 13ndash15 The experi-mental set-ups for the three cases are shown in Table 2as cases 4-6 The associated calibration results can befound in Table 5 For the other cases the calculated10ndash50 burn duration (Burn1050) and 50ndash90 burnduration (Burn5090) of the main-combustion chamber

Figure 10 Experimental and calculated net heat release rates

Figure 11 Chemical energy release rate calculation

Figure 12 The calculated value of ab during combustion

Figure 13 Experimental and calculated pressure traces forexperimental case 4

Figure 14 Experimental and calculated pressure traces forexperimental case 5

Song et al 1323

were compared with the experimental values seeFigures 16ndash17 Due to large variations of orifice areasand AFRs of these data sets the model parametersneed to be re-calibrated However b and wc were keptunchanged for the cases with the same orifice sizes likecases 4 and 5 It was also found that cpre and cmain werevery similar for all the experimental cases This indi-cates that these two parameters are mainly associated

with the combustion chamber structure In Table 5 thepre-combustion chamber burn durations are quite dif-ferent from the previous experimental cases 1ndash3 This isdue to the difference in the orifice area

To conclude the proposed model is able to fit theexperimental data sets with large ranges of AFRs inboth combustion chambers and different pre-combustion chamber orifice areas This indicates thatthe developed combustion model has the potential tobe used for the development of a TJI engine model

Conclusion

This article presents a control-oriented TJI combustionmodel for the RCM at Michigan State University Anewly proposed parameter-varying Wiebe combustionmodel is used to link the combustion processes in bothpre- and main-combustion chambers The developedmodel can be calibrated using a simple and systematiccalibration procedure based on the experimental dataThe model validation process shows good agreementbetween the modeled and experimental pressure traceswhich indicates that the developed model is capable ofaccurately capturing TJI combustion dynamics Thevalidation results also indicate that the model is able topredict the combustion process that is not used to cali-brate the model parameters This shows that the devel-oped model has the potential to be used for studyingTJI combustion engines and developing associated con-trol strategies Although only methane is used as fuel inthis article this model can be extended to other gaseousfuels with new calibrations However for liquid fuelthe model structure may need to be changed particu-larly a gasndashfuel mixing model will be required Futurework will extend the modeling work for TJI enginesusing gaseous fuel Note that in this case the pistondynamics and gas exchange (intake and exhaust) mod-els need to be added

Table 5 Calibration results for cases 4ndash6

Parameter Case 4 Case 5 Case 6

cpre 0651 0651 0651cmain 0 0 0vc 3829 3829 5064b 0824 0824 0764

Pre-chamber Dtd (ms) 197 228 179hpre 0980 0939 0901

Main-chamber Dtd (ms) 165 112 140hmain 0927 0905 0932

Figure 15 Experimental and calculated pressure traces forexperimental case 6

Figure 16 Experimental and calculated Burn1050 in the main-combustion chamberBurn1050 10ndash50 burn duration

Figure 17 Experimental and calculated Burn5090 in themain-combustion chamberBurn5090 50ndash90 burn duration

1324 Proc IMechE Part D J Automobile Engineering 231(10)

Declaration of conflicting interests

The author(s) declared no potential conflicts of interestwith respect to the research authorship andor publi-cation of this article

Funding

The author(s) disclosed receipt of the following finan-cial support for the research authorship andor publi-cation of this article This research is co-sponsored bythe US National Science Foundation and Departmentof Energy under contract number CBET-1258581

References

[1] Zhang S Zhu G and Sun Z A control-oriented chargemixing and two-zone HCCI combustion model IEEE T

Veh Technol 2014 63(3) 1079ndash1090[2] Yang X and Zhu G SI and HCCI combustion mode

transition control of an HCCI capable SI engine IEEET Contr Syst T 2013 21(5) 1558ndash1569

[3] Turkish MC 3-valve stratified charge engines Evolve-ment analysis and progression SAE technical paper741163 1974

[4] Date T Yagi S Ishizuya A et al Research and develop-ment of the Honda CVCC engine SAE Technical Paper1974

[5] Toulson E Schock HJ and Attard WP A review of pre-chamber initiated jet ignition combustion systems SAEtechnical paper 2010-01-2263 2010

[6] Song R Gentz G Zhu G et al A control-oriented jetignition combustion model for an SI engine In ASME

2015 dynamic systems and control conference ColumbusOhio 28ndash30 October 2015 ppV001T11A001ndashV001T11A001 American Society of Mechanical Engineers

[7] Toulson E Huisjen A Chen X et al Visualization ofpropane and natural gas spark ignition and turbulent jetignition combustion SAE Int J Engines 2012 5(4)

1821ndash1835[8] Yang X and Zhu GG A control-oriented hybrid com-

bustion model of a homogeneous charge compressionignition capable spark ignition engine P I Mech Eng

DmdashJ Aut 2012 226(10) 1380ndash1395[9] Stanglmaier RH and Roberts CE Homogeneous charge

compression ignition (HCCI) Benefits compromisesand future engine applications SAE technical paper1999-01-3682 1999

[10] Toulson E Watson HC and Attard WP Modeling alter-native prechamber fuels in jet assisted ignition of gasolineand LPG SAE technical paper 2009-01-0721 2009

[11] Ghorbani A Steinhilber G Markus D et al Numericalinvestigation of ignition in a transient turbulent jet bymeans of a PDF method Combust Sci Technol 2014186(10-11) 1582ndash1596

[12] Canova M Garcin R Midlam-Mohler S et al Acontrol-oriented model of combustion process in aHCCI diesel engine In Proceedings of the 2005

American control conference Portland Oregon 8ndash10

July 2005 vol 7 pp4446ndash4451 American AutomaticControl Council

[13] Guzzella L and Onder C Introduction to modeling and

control of internal combustion engine systems Berlin Hei-delberg Springer Science amp Business Media 2009

[14] Mortenson ME Mathematics for computer graphics

applications South Norwalk CT Industrial Press 1999[15] Heywood JB Internal combustion engine fundamentals

Vol 930 New York McGraw-Hill 1988[16] Ghojel J Review of the development and applications of

the Wiebe function A tribute to the contribution of Ivan

Wiebe to engine research Int J Engine Res 2010 11(4)297ndash312

[17] Zhao H and Ladommatos N Engine combustion instru-

mentation and diagnostics Warrendale PA Society of

Automotive Engineers 2001[18] Hellstrom E Stefanopoulou A and Jiang L A linear

least-squares algorithm for double-Wiebe functions

applied to spark-assisted compression ignition J Eng

Gas Turb Power 2014 136(9) 091514

Appendix

Abbreviations

TJI Turbulent jet ignitionRCM Rapid compression machineAFR Air-to-fuel ratioIC Internal combustionHCCI Homogeneous charge compression

ignitionCVCC Compound vortex controlled combustionSI Spark ignitionCFD Computational fluid dynamicsPDF Probability density functionMFB Mass fraction burnedNHR Net heat releaseCER Chemical energy release

List of simulation parameters

Parameter Symbol

Fuel injector discharge coefficient Cd1

Pre-chamber orifice discharge coefficient Cd2

Mass flow rate correcting parameter cpre

Mass flow rate correcting parameter cmain

Heat transfer coefficient hc

Lower heating value of the fuel QLHV

Wall temperature Tw

Coefficient of parameter-varying Wiebe function vc

Coefficient of parameter-varying Wiebe function bStart of ignition tign

Wiebe function coefficient aWiebe exponent mBurn duration in Wiebe function DtdPre-chamber combustion efficiency hpre

Main-chamber combustion efficiency hmain

Song et al 1325

according to the mass flow rate of the turbulent jetThe top plot of Figure 11 shows the mass flow ratethrough the orifice connecting the two combustionchambers The curve in the middle plot of Figure 11 isb teth THORN in equation (20) The bottom plot of Figure 11 isthe calculated CER rate According to the simulationresults the MFB model with the parameter-varyingWiebe function successfully links the combustion pro-cesses in two combustion chambers

The value of the coefficient ab during the simulationis shown in Figure 12 where the dashed line is thevalue of ab calculated by assuming Ppre is greater thanPmain and the dash-dotted line is the value of ab calcu-lated by assuming Ppre is less than Pmain The actualvalue of ab used for the combustion calculation that isshown by the solid line is on the dashed line at begin-ning because the pre-combustion chamber is ignitedfirst After the switch line (see Figure 12) the main-combustion chamber pressure becomes larger than thatof the pre-combustion chamber ab jumps from thedashed line to the dash-dotted line

To further validate the model 12 more experimentaldata sets were used The orifice diameter varied from20mm to 30mm The relative AFRs in the two com-bustion chambers varied from 09 to 15 To simplifythe presentation the pressure traces were plotted onlyfor the first three cases see Figures 13ndash15 The experi-mental set-ups for the three cases are shown in Table 2as cases 4-6 The associated calibration results can befound in Table 5 For the other cases the calculated10ndash50 burn duration (Burn1050) and 50ndash90 burnduration (Burn5090) of the main-combustion chamber

Figure 10 Experimental and calculated net heat release rates

Figure 11 Chemical energy release rate calculation

Figure 12 The calculated value of ab during combustion

Figure 13 Experimental and calculated pressure traces forexperimental case 4

Figure 14 Experimental and calculated pressure traces forexperimental case 5

Song et al 1323

were compared with the experimental values seeFigures 16ndash17 Due to large variations of orifice areasand AFRs of these data sets the model parametersneed to be re-calibrated However b and wc were keptunchanged for the cases with the same orifice sizes likecases 4 and 5 It was also found that cpre and cmain werevery similar for all the experimental cases This indi-cates that these two parameters are mainly associated

with the combustion chamber structure In Table 5 thepre-combustion chamber burn durations are quite dif-ferent from the previous experimental cases 1ndash3 This isdue to the difference in the orifice area

To conclude the proposed model is able to fit theexperimental data sets with large ranges of AFRs inboth combustion chambers and different pre-combustion chamber orifice areas This indicates thatthe developed combustion model has the potential tobe used for the development of a TJI engine model

Conclusion

This article presents a control-oriented TJI combustionmodel for the RCM at Michigan State University Anewly proposed parameter-varying Wiebe combustionmodel is used to link the combustion processes in bothpre- and main-combustion chambers The developedmodel can be calibrated using a simple and systematiccalibration procedure based on the experimental dataThe model validation process shows good agreementbetween the modeled and experimental pressure traceswhich indicates that the developed model is capable ofaccurately capturing TJI combustion dynamics Thevalidation results also indicate that the model is able topredict the combustion process that is not used to cali-brate the model parameters This shows that the devel-oped model has the potential to be used for studyingTJI combustion engines and developing associated con-trol strategies Although only methane is used as fuel inthis article this model can be extended to other gaseousfuels with new calibrations However for liquid fuelthe model structure may need to be changed particu-larly a gasndashfuel mixing model will be required Futurework will extend the modeling work for TJI enginesusing gaseous fuel Note that in this case the pistondynamics and gas exchange (intake and exhaust) mod-els need to be added

Table 5 Calibration results for cases 4ndash6

Parameter Case 4 Case 5 Case 6

cpre 0651 0651 0651cmain 0 0 0vc 3829 3829 5064b 0824 0824 0764

Pre-chamber Dtd (ms) 197 228 179hpre 0980 0939 0901

Main-chamber Dtd (ms) 165 112 140hmain 0927 0905 0932

Figure 15 Experimental and calculated pressure traces forexperimental case 6

Figure 16 Experimental and calculated Burn1050 in the main-combustion chamberBurn1050 10ndash50 burn duration

Figure 17 Experimental and calculated Burn5090 in themain-combustion chamberBurn5090 50ndash90 burn duration

1324 Proc IMechE Part D J Automobile Engineering 231(10)

Declaration of conflicting interests

The author(s) declared no potential conflicts of interestwith respect to the research authorship andor publi-cation of this article

Funding

The author(s) disclosed receipt of the following finan-cial support for the research authorship andor publi-cation of this article This research is co-sponsored bythe US National Science Foundation and Departmentof Energy under contract number CBET-1258581

References

[1] Zhang S Zhu G and Sun Z A control-oriented chargemixing and two-zone HCCI combustion model IEEE T

Veh Technol 2014 63(3) 1079ndash1090[2] Yang X and Zhu G SI and HCCI combustion mode

transition control of an HCCI capable SI engine IEEET Contr Syst T 2013 21(5) 1558ndash1569

[3] Turkish MC 3-valve stratified charge engines Evolve-ment analysis and progression SAE technical paper741163 1974

[4] Date T Yagi S Ishizuya A et al Research and develop-ment of the Honda CVCC engine SAE Technical Paper1974

[5] Toulson E Schock HJ and Attard WP A review of pre-chamber initiated jet ignition combustion systems SAEtechnical paper 2010-01-2263 2010

[6] Song R Gentz G Zhu G et al A control-oriented jetignition combustion model for an SI engine In ASME

2015 dynamic systems and control conference ColumbusOhio 28ndash30 October 2015 ppV001T11A001ndashV001T11A001 American Society of Mechanical Engineers

[7] Toulson E Huisjen A Chen X et al Visualization ofpropane and natural gas spark ignition and turbulent jetignition combustion SAE Int J Engines 2012 5(4)

1821ndash1835[8] Yang X and Zhu GG A control-oriented hybrid com-

bustion model of a homogeneous charge compressionignition capable spark ignition engine P I Mech Eng

DmdashJ Aut 2012 226(10) 1380ndash1395[9] Stanglmaier RH and Roberts CE Homogeneous charge

compression ignition (HCCI) Benefits compromisesand future engine applications SAE technical paper1999-01-3682 1999

[10] Toulson E Watson HC and Attard WP Modeling alter-native prechamber fuels in jet assisted ignition of gasolineand LPG SAE technical paper 2009-01-0721 2009

[11] Ghorbani A Steinhilber G Markus D et al Numericalinvestigation of ignition in a transient turbulent jet bymeans of a PDF method Combust Sci Technol 2014186(10-11) 1582ndash1596

[12] Canova M Garcin R Midlam-Mohler S et al Acontrol-oriented model of combustion process in aHCCI diesel engine In Proceedings of the 2005

American control conference Portland Oregon 8ndash10

July 2005 vol 7 pp4446ndash4451 American AutomaticControl Council

[13] Guzzella L and Onder C Introduction to modeling and

control of internal combustion engine systems Berlin Hei-delberg Springer Science amp Business Media 2009

[14] Mortenson ME Mathematics for computer graphics

applications South Norwalk CT Industrial Press 1999[15] Heywood JB Internal combustion engine fundamentals

Vol 930 New York McGraw-Hill 1988[16] Ghojel J Review of the development and applications of

the Wiebe function A tribute to the contribution of Ivan

Wiebe to engine research Int J Engine Res 2010 11(4)297ndash312

[17] Zhao H and Ladommatos N Engine combustion instru-

mentation and diagnostics Warrendale PA Society of

Automotive Engineers 2001[18] Hellstrom E Stefanopoulou A and Jiang L A linear

least-squares algorithm for double-Wiebe functions

applied to spark-assisted compression ignition J Eng

Gas Turb Power 2014 136(9) 091514

Appendix

Abbreviations

TJI Turbulent jet ignitionRCM Rapid compression machineAFR Air-to-fuel ratioIC Internal combustionHCCI Homogeneous charge compression

ignitionCVCC Compound vortex controlled combustionSI Spark ignitionCFD Computational fluid dynamicsPDF Probability density functionMFB Mass fraction burnedNHR Net heat releaseCER Chemical energy release

List of simulation parameters

Parameter Symbol

Fuel injector discharge coefficient Cd1

Pre-chamber orifice discharge coefficient Cd2

Mass flow rate correcting parameter cpre

Mass flow rate correcting parameter cmain

Heat transfer coefficient hc

Lower heating value of the fuel QLHV

Wall temperature Tw

Coefficient of parameter-varying Wiebe function vc

Coefficient of parameter-varying Wiebe function bStart of ignition tign

Wiebe function coefficient aWiebe exponent mBurn duration in Wiebe function DtdPre-chamber combustion efficiency hpre

Main-chamber combustion efficiency hmain

Song et al 1325

were compared with the experimental values seeFigures 16ndash17 Due to large variations of orifice areasand AFRs of these data sets the model parametersneed to be re-calibrated However b and wc were keptunchanged for the cases with the same orifice sizes likecases 4 and 5 It was also found that cpre and cmain werevery similar for all the experimental cases This indi-cates that these two parameters are mainly associated

with the combustion chamber structure In Table 5 thepre-combustion chamber burn durations are quite dif-ferent from the previous experimental cases 1ndash3 This isdue to the difference in the orifice area

To conclude the proposed model is able to fit theexperimental data sets with large ranges of AFRs inboth combustion chambers and different pre-combustion chamber orifice areas This indicates thatthe developed combustion model has the potential tobe used for the development of a TJI engine model

Conclusion

This article presents a control-oriented TJI combustionmodel for the RCM at Michigan State University Anewly proposed parameter-varying Wiebe combustionmodel is used to link the combustion processes in bothpre- and main-combustion chambers The developedmodel can be calibrated using a simple and systematiccalibration procedure based on the experimental dataThe model validation process shows good agreementbetween the modeled and experimental pressure traceswhich indicates that the developed model is capable ofaccurately capturing TJI combustion dynamics Thevalidation results also indicate that the model is able topredict the combustion process that is not used to cali-brate the model parameters This shows that the devel-oped model has the potential to be used for studyingTJI combustion engines and developing associated con-trol strategies Although only methane is used as fuel inthis article this model can be extended to other gaseousfuels with new calibrations However for liquid fuelthe model structure may need to be changed particu-larly a gasndashfuel mixing model will be required Futurework will extend the modeling work for TJI enginesusing gaseous fuel Note that in this case the pistondynamics and gas exchange (intake and exhaust) mod-els need to be added

Table 5 Calibration results for cases 4ndash6

Parameter Case 4 Case 5 Case 6

cpre 0651 0651 0651cmain 0 0 0vc 3829 3829 5064b 0824 0824 0764

Pre-chamber Dtd (ms) 197 228 179hpre 0980 0939 0901

Main-chamber Dtd (ms) 165 112 140hmain 0927 0905 0932

Figure 15 Experimental and calculated pressure traces forexperimental case 6

Figure 16 Experimental and calculated Burn1050 in the main-combustion chamberBurn1050 10ndash50 burn duration

Figure 17 Experimental and calculated Burn5090 in themain-combustion chamberBurn5090 50ndash90 burn duration

1324 Proc IMechE Part D J Automobile Engineering 231(10)

Declaration of conflicting interests

The author(s) declared no potential conflicts of interestwith respect to the research authorship andor publi-cation of this article

Funding

The author(s) disclosed receipt of the following finan-cial support for the research authorship andor publi-cation of this article This research is co-sponsored bythe US National Science Foundation and Departmentof Energy under contract number CBET-1258581

References

[1] Zhang S Zhu G and Sun Z A control-oriented chargemixing and two-zone HCCI combustion model IEEE T

Veh Technol 2014 63(3) 1079ndash1090[2] Yang X and Zhu G SI and HCCI combustion mode

transition control of an HCCI capable SI engine IEEET Contr Syst T 2013 21(5) 1558ndash1569

[3] Turkish MC 3-valve stratified charge engines Evolve-ment analysis and progression SAE technical paper741163 1974

[4] Date T Yagi S Ishizuya A et al Research and develop-ment of the Honda CVCC engine SAE Technical Paper1974

[5] Toulson E Schock HJ and Attard WP A review of pre-chamber initiated jet ignition combustion systems SAEtechnical paper 2010-01-2263 2010

[6] Song R Gentz G Zhu G et al A control-oriented jetignition combustion model for an SI engine In ASME

2015 dynamic systems and control conference ColumbusOhio 28ndash30 October 2015 ppV001T11A001ndashV001T11A001 American Society of Mechanical Engineers

[7] Toulson E Huisjen A Chen X et al Visualization ofpropane and natural gas spark ignition and turbulent jetignition combustion SAE Int J Engines 2012 5(4)

1821ndash1835[8] Yang X and Zhu GG A control-oriented hybrid com-

bustion model of a homogeneous charge compressionignition capable spark ignition engine P I Mech Eng

DmdashJ Aut 2012 226(10) 1380ndash1395[9] Stanglmaier RH and Roberts CE Homogeneous charge

compression ignition (HCCI) Benefits compromisesand future engine applications SAE technical paper1999-01-3682 1999

[10] Toulson E Watson HC and Attard WP Modeling alter-native prechamber fuels in jet assisted ignition of gasolineand LPG SAE technical paper 2009-01-0721 2009

[11] Ghorbani A Steinhilber G Markus D et al Numericalinvestigation of ignition in a transient turbulent jet bymeans of a PDF method Combust Sci Technol 2014186(10-11) 1582ndash1596

[12] Canova M Garcin R Midlam-Mohler S et al Acontrol-oriented model of combustion process in aHCCI diesel engine In Proceedings of the 2005

American control conference Portland Oregon 8ndash10

July 2005 vol 7 pp4446ndash4451 American AutomaticControl Council

[13] Guzzella L and Onder C Introduction to modeling and

control of internal combustion engine systems Berlin Hei-delberg Springer Science amp Business Media 2009

[14] Mortenson ME Mathematics for computer graphics

applications South Norwalk CT Industrial Press 1999[15] Heywood JB Internal combustion engine fundamentals

Vol 930 New York McGraw-Hill 1988[16] Ghojel J Review of the development and applications of

the Wiebe function A tribute to the contribution of Ivan

Wiebe to engine research Int J Engine Res 2010 11(4)297ndash312

[17] Zhao H and Ladommatos N Engine combustion instru-

mentation and diagnostics Warrendale PA Society of

Automotive Engineers 2001[18] Hellstrom E Stefanopoulou A and Jiang L A linear

least-squares algorithm for double-Wiebe functions

applied to spark-assisted compression ignition J Eng

Gas Turb Power 2014 136(9) 091514

Appendix

Abbreviations

TJI Turbulent jet ignitionRCM Rapid compression machineAFR Air-to-fuel ratioIC Internal combustionHCCI Homogeneous charge compression

ignitionCVCC Compound vortex controlled combustionSI Spark ignitionCFD Computational fluid dynamicsPDF Probability density functionMFB Mass fraction burnedNHR Net heat releaseCER Chemical energy release

List of simulation parameters

Parameter Symbol

Fuel injector discharge coefficient Cd1

Pre-chamber orifice discharge coefficient Cd2

Mass flow rate correcting parameter cpre

Mass flow rate correcting parameter cmain

Heat transfer coefficient hc

Lower heating value of the fuel QLHV

Wall temperature Tw

Coefficient of parameter-varying Wiebe function vc

Coefficient of parameter-varying Wiebe function bStart of ignition tign

Wiebe function coefficient aWiebe exponent mBurn duration in Wiebe function DtdPre-chamber combustion efficiency hpre

Main-chamber combustion efficiency hmain

Song et al 1325

Declaration of conflicting interests

The author(s) declared no potential conflicts of interestwith respect to the research authorship andor publi-cation of this article

Funding

The author(s) disclosed receipt of the following finan-cial support for the research authorship andor publi-cation of this article This research is co-sponsored bythe US National Science Foundation and Departmentof Energy under contract number CBET-1258581

References

[1] Zhang S Zhu G and Sun Z A control-oriented chargemixing and two-zone HCCI combustion model IEEE T

Veh Technol 2014 63(3) 1079ndash1090[2] Yang X and Zhu G SI and HCCI combustion mode

transition control of an HCCI capable SI engine IEEET Contr Syst T 2013 21(5) 1558ndash1569

[3] Turkish MC 3-valve stratified charge engines Evolve-ment analysis and progression SAE technical paper741163 1974

[4] Date T Yagi S Ishizuya A et al Research and develop-ment of the Honda CVCC engine SAE Technical Paper1974

[5] Toulson E Schock HJ and Attard WP A review of pre-chamber initiated jet ignition combustion systems SAEtechnical paper 2010-01-2263 2010

[6] Song R Gentz G Zhu G et al A control-oriented jetignition combustion model for an SI engine In ASME

2015 dynamic systems and control conference ColumbusOhio 28ndash30 October 2015 ppV001T11A001ndashV001T11A001 American Society of Mechanical Engineers

[7] Toulson E Huisjen A Chen X et al Visualization ofpropane and natural gas spark ignition and turbulent jetignition combustion SAE Int J Engines 2012 5(4)

1821ndash1835[8] Yang X and Zhu GG A control-oriented hybrid com-

bustion model of a homogeneous charge compressionignition capable spark ignition engine P I Mech Eng

DmdashJ Aut 2012 226(10) 1380ndash1395[9] Stanglmaier RH and Roberts CE Homogeneous charge

compression ignition (HCCI) Benefits compromisesand future engine applications SAE technical paper1999-01-3682 1999

[10] Toulson E Watson HC and Attard WP Modeling alter-native prechamber fuels in jet assisted ignition of gasolineand LPG SAE technical paper 2009-01-0721 2009

[11] Ghorbani A Steinhilber G Markus D et al Numericalinvestigation of ignition in a transient turbulent jet bymeans of a PDF method Combust Sci Technol 2014186(10-11) 1582ndash1596

[12] Canova M Garcin R Midlam-Mohler S et al Acontrol-oriented model of combustion process in aHCCI diesel engine In Proceedings of the 2005

American control conference Portland Oregon 8ndash10

July 2005 vol 7 pp4446ndash4451 American AutomaticControl Council

[13] Guzzella L and Onder C Introduction to modeling and

control of internal combustion engine systems Berlin Hei-delberg Springer Science amp Business Media 2009

[14] Mortenson ME Mathematics for computer graphics

applications South Norwalk CT Industrial Press 1999[15] Heywood JB Internal combustion engine fundamentals

Vol 930 New York McGraw-Hill 1988[16] Ghojel J Review of the development and applications of

the Wiebe function A tribute to the contribution of Ivan

Wiebe to engine research Int J Engine Res 2010 11(4)297ndash312

[17] Zhao H and Ladommatos N Engine combustion instru-

mentation and diagnostics Warrendale PA Society of

Automotive Engineers 2001[18] Hellstrom E Stefanopoulou A and Jiang L A linear

least-squares algorithm for double-Wiebe functions

applied to spark-assisted compression ignition J Eng

Gas Turb Power 2014 136(9) 091514

Appendix

Abbreviations

TJI Turbulent jet ignitionRCM Rapid compression machineAFR Air-to-fuel ratioIC Internal combustionHCCI Homogeneous charge compression

ignitionCVCC Compound vortex controlled combustionSI Spark ignitionCFD Computational fluid dynamicsPDF Probability density functionMFB Mass fraction burnedNHR Net heat releaseCER Chemical energy release

List of simulation parameters

Parameter Symbol

Fuel injector discharge coefficient Cd1

Pre-chamber orifice discharge coefficient Cd2

Mass flow rate correcting parameter cpre

Mass flow rate correcting parameter cmain

Heat transfer coefficient hc

Lower heating value of the fuel QLHV

Wall temperature Tw

Coefficient of parameter-varying Wiebe function vc

Coefficient of parameter-varying Wiebe function bStart of ignition tign

Wiebe function coefficient aWiebe exponent mBurn duration in Wiebe function DtdPre-chamber combustion efficiency hpre

Main-chamber combustion efficiency hmain

Song et al 1325