REACTION-BASEDKNOCKPREDICTIVEMODELINGANDMODEL-BASED STOCHASTICKNOCKLIMITCONTROLOFSPARK-IGNITIONENGINES By RuixueLi ADISSERTATION Submittedto MichiganStateUniversity inpartialful˝llmentoftherequirements forthedegreeof MechanicalEngineeringDoctorofPhilosophy 2020 ABSTRACT REACTION-BASEDKNOCKPREDICTIVEMODELINGANDMODEL-BASED STOCHASTICKNOCKLIMITCONTROLOFSPARK-IGNITIONENGINES By RuixueLi Thisdissertationstudiesthespark-ignition(SI)engineknockphenomenon,abnormalcombustion duetotheauto-ignitionofend-gasaheadofthepropagated˛amefront,resultingintherapid chemicalenergyreleasewithaggressivecombustion,limitingthefurtherimprovementofthermal e˚ciencyandevendamagingtheenginemechanically.Acontrol-orientedcombustionandpressure wavemodelwithsatisfactoryaccuracyandlowcomputationale˙ortisanecessityfortheknock controlstrategydesign.Thisdissertationdevelopsacontrol-orientedknockpredictivemodelthat includesatwo-zonereaction-basedcombustionmodelandapressurewavemodel.Thisknock predictivemodeliscapableofaccuratelydescribingthecombustionprocessofaspark-ignited engineandpredictthein-cylinderpressureoscillationsunderknockingcombustioninreal-time. Basedonthismodel,afeedforwardandfeedbackstochasticknocklimitcontrolstrategyisdeveloped toreducetheknockcyclicvariabilityandcontroltheknockmean-intensitybelowadesiredupbound whilekeepingsparktimingasclosetoenginemaximumbraketorque(MBT)timingaspossible. Acontrol-orientedtwo-zonereaction-basedmodeltoaccuratelydescribethecombustionpro- cessofaSIengineis˝rstdeveloped.Insteadofusingtheconventionalpre-determinedWiebe-based combustionmodel,atwo-stepchemicalreactionmodelisutilizedtopredictthecombustionpro- cessalongwithimportantthermodynamicparameterssuchasthemass-fraction-burned,in-cylinder pressure,temperaturesandindividualspeciesmasschangesinbothzones.Sensitivitiesofmodel parametersareanalyzedduringthemodelcalibrationprocess.Asaresult,onesetofcalibration parametersareusedtopredictcombustioncharacteristicsoverallengineoperatingconditions studiedinthispaper,whichisthemajoradvantageoftheproposedmethod.Also,theproposed modelingapproachiscapableofmodelingthecombustionprocessforreal-timesimulations.As theby-productofthemodel,engineknockcanalsobepredictedbasedontheArrheniusintegral intheunburnedzone,whichisvaluableformodel-basedknockcontrol.Theproposedcombustion modelisintensivelyvalidatedusingtheexperimentaldatawithapeakrelativepredictionerrorof 6.2%forthein-cylinderpressure. Basedonthisvalidatedcombustionmodel,acontrol-orientedpressurewavemodelforSIen- ginesisfurtherdeveloped.Thismodeliscapableofpredictingthein-cylinderpressureoscillations underknockingcombustioninreal-timeandcanbeusedforthemodel-basedknockprediction andcontrol.Apressurewaveequationincludingtheknockdeadeningbehaviorisproposed,sim- pli˝ed,andusedtocalculatethepressureperturbationsgeneratedbytheknockingcombustion. Theboundaryandinitialconditionsatknockonsetareanalyzedandtheanalyticsolutionofthe pressurewaveequationisobtained.Themodeliscalibratedandvalidatedovertwodi˙erentengine operatingconditionsatknocklimit.Thechemicalkinetic-basedArrheniusintegral(ARI)andthe KI20areusedastheevaluationmethodsforknockonsetandintensityprediction,andtheknock frequencyisstudiedwithafastFouriertransformofthe˝lteredin-cylinderpressureoscillations. Especially,theknockcharacteristicsassociatedwithgasmixturepropertiesatintakevalveclosing isanalyzedbasedontheexperimentaldataandtheire˙ecttoknockcycle-to-cyclevariationisalso studiedfortheproposedmodel. Inaddition,thisdissertationstudiesthecorrelationbetweenin-cylindermixturetemperatureat intakevalveclosingandtheengineknock,alongwithknockcyclicvariabilitybasedontheknock predictivemodel.Astrongcorrelationbetweentheintaketemperatureandknockintensityhasbeen obtainedandvalidatedbasedonthesimulationinvestigationandexperimentdataobtainedatknock limit.Therefore,amodel-basedfeedforwardandfeedbackstochasticknocklimitcontrolstrategy isdevelopedtoreducetheknockcycle-to-cyclevariabilityandmaintaintheknockmean-intensity withinadesiredupboundbycontrollingthesparktimingasclosetoMBTtimingaspossible. Thecontrolperformanceisvalidatedwiththesimulationresultstoshowthecapabilityofthe model-basedfeedforwardandfeedbackstochasticknocklimitcontrolinsigni˝cantlyreducingthe knockcyclicvariabilityandimprovingtheknockintensitydistributionforthebestfueleconomy. Copyrightby RUIXUELI 2020 ACKNOWLEDGMENTS Thisdissertationwouldnothavebeenpossiblewithoutthehelpofsomanypeople.Iwouldlike toexpressmyimmeasurableappreciationanddeepestgratitudetoallthepeoplewhoseassistance wereamilestoneinthecompletionofmyPh.D.study. Foremost,Iwishtoexpressmydeepestgratitudetomyadvisor,Prof.Guoming(George) Zhu,forhispersistentsupport,professionalguidanceandprofoundencouragementofmyPh.D. studyandresearch.Itismyfortunetojoinhisgroup,studyanddomyresearchworkwithhis patience,enthusiasmandimmenseknowledge.Hiscomprehensiveknowledgeinadvancedcontrol theoryandpracticalexperienceinautomotiveindustryinspiredmypassionforcontroltheories andapplicationsinautomotive.Besides,IwouldnothavebeenpossibletoworkinIsuzuNorth AmericaandPilotSystemsInternationalLLC.asacontrolengineerinternfor12monthswithout hisrecommendationandguidance.Itiswhole-heartedlyappreciatedthathisgreatadviceformy studyprovedmonumentaltowardsthesuccessofmyPh.D.study.WhatIlearnedfromhimis invaluableformyfuturecareerandlife! Iwishtoexpressmysincereappreciationtomycommitteemembers:Dr.HaroldSchock,Dr. HassanKhalil,andDr.RanjanMukherjee,fortheirencouragementandinsightfulcommentsof mycourses,researchanddissertation.Dr.HaroldSchockgavemethegreatestguidanceinthe internalcombustionengineandsimulationusingGT-SUITE.Dr.HassanKhalilandDr.Ranjan Mukherjeeprovidedmesolidknowledgeoflinear/nonlinearcontrolandadaptivecontrol,whichis thefoundationofmyresearchworkandindustryinternshipexperience. Iwishtoexpressmywarmestregardstothelabengineersandmyfellowlabmates,withwhom IamhonoredtohaveworkedduringmyPh.DstudyinEnergyandAutomotiveResearchLab.My specialgratitudegoestoDr.TianyiHeforhiskindnessandhelpinthecontrolsystemdesign ofmyPh.DprojectandmyinternshipprojectinPilotSystemsInternationalLLC.Ialsowishto thankDr.YifanMenandDr.RuitaoSongfortheirinvaluableassistanceinthereaction-based combustionmodeling,calibrationandbenchtesting.I'dliketoexpressmyappreciationtoour v electricalengineer,KevinMoran,andmechanicalengineer,TomStuecken,andDr.Sedigheh Tolou,fortheirhelpforthebenchsetupandtesting.ManythankstoDr.YingxuWang,Dr.Huan Li,Dr.ChengshengMiao,AnujPal,WenpengWei,JianTang,DaweiHuandShenQu,fortheir kindlyhelpandwonderfultimewespenttogether. IwishtosendmyspecialappreciationtotheengineersandinternswhomIworkedwith inPilotSystemsInternationalLLC.andIsuzuNorthAmerica,PowertrainandVehicleResearch Development.IwishtothankDr.JimWinkelmanandStuartSteelfortheirprofessionalknowledge, practicalindustryexperienceandconsistentsupportintheprojectofroadnoisecancellationcontrol systemdesignofsuspensionsystem.IalsowishtothankmysupervisorandmentorinIsuzuNorth America:TimVictorandThirumalHarikrishnanfortheirguidanceandsupportintheprojectof oxygenfractionestimationofdieselengineair-pathsystem. Lastly,tomylovingandsupportiveparents,brother,andsweetestcatSally.Thisjourneywould nothavebeenpossiblewithouttheirlove,understandingandencouragement. vi TABLEOFCONTENTS LISTOFTABLES ....................................... ix LISTOFFIGURES ....................................... x CHAPTER1INTRODUCTION ............................... 1 1.1Motivation.......................................1 1.2ResearchOverview..................................2 1.2.1Control-OrientedCombustionModelforSIEngines.............2 1.2.2KnockPredictionandModelingforSIEngines...............6 1.2.3Model-BasedKnockLimitControlforSIEngines..............11 1.3DissertationContributions..............................14 1.4DissertationOrganization...............................14 CHAPTER2TWO-ZONEREACTION-BASEDCOMBUSTIONMODELING ...... 16 2.1Two-ZoneModelCon˝guration............................17 2.2InteractionbetweenTwoZones............................19 2.2.1HeatTransferInterface............................19 2.2.2HeatLoss...................................21 2.2.3MassTransfer.................................23 2.3ChemicalReactionKineticMechanism.......................24 2.3.1MolarConcentrationanditsConcentrationRate:..............24 2.3.2Two-StepChemicalReactionMechanism:..................25 2.3.3ZoneTemperature...............................27 2.3.4In-CylinderPressure.............................29 2.4ModelCalibration...................................30 2.4.1ExperimentalInvestigation..........................30 2.4.2ModelCalibration...............................31 2.5ModelValidationandSimulationResults.......................33 2.5.1ThermodynamicProperties..........................33 2.5.2CombustionSimulationResults.......................40 2.6Summary.......................................46 CHAPTER3REAL-TIMEPRESSUREWAVEMODELINGFORKNOCKPREDIC- TIONANDCONTROL ............................ 49 3.1DerivationofPressureWaveModel.........................50 3.2BoundaryandInitialConditions...........................53 3.2.1Bessel'sEquation...............................54 3.2.2TimeFunction.................................55 3.2.3Coe˚cient < ................................56 3.2.4PressureRateatKnockOnset........................57 3.3SolutionofPressureWaveEquationforKnockPrediction..............58 vii 3.4EvaluationMethodsofKnockPhenomenon.....................59 3.4.1KnockOnset.................................59 3.4.2KnockIntensity................................60 3.5ModelCalibrationandValidation...........................60 3.5.1ExperimentSetupandModelCalibration..................60 3.5.2KnockOnsetandIntensityPrediction....................61 3.5.3FFTAnalysisofIn-CylinderPressureWaves.................66 3.5.4KnockCycle-to-CycleVariability......................68 3.6Summary.......................................76 CHAPTER4MODEL-BASEDSTOCHASTICKNOCKLIMITCONTROL ....... 78 4.1Reaction-BasedKnockPredictiveModel.......................78 4.2Model-BasedPredictionofKnockCyclicVariability................78 4.2.1KnockIntensity-KI20(MAPO).......................78 4.2.2KnockPredictiveModelCalibrationandValidation.............80 4.2.3IntakeTemperaturewithKnockCycle-to-CycleVariability.........81 4.2.4SparkTimingwithKnockCycle-to-CycleVariability............82 4.3StochasticKnockLimitControlandResultsDiscussion...............86 4.3.1ControlObjectives..............................86 4.3.2Model-BasedFeedforwardKnockLimitControl...............87 4.3.2.1ControlAlgorithm.........................87 4.3.2.2ResultsandDiscussion.......................89 4.3.3Closed-LoopStochasticKnockLimitControl................91 4.3.3.1ControlAlgorithm.........................91 4.3.3.2ResultsandDiscussion.......................95 4.4Summary.......................................100 CHAPTER5CONCLUSIONSANDFUTUREWORK ................... 102 5.1Conclusions......................................102 5.2RecommendationsforFutureWork..........................104 BIBLIOGRAPHY ........................................ 105 viii LISTOFTABLES Table2.1:Testengineparameters................................31 Table2.2:Engineoperationalconditions............................31 Table2.3:Parameterstobecalibrated.............................32 Table2.4:Calibratedparameterswithlowsensitivity.....................32 Table2.5:Calibratedparameterswithhighsensitivity....................33 Table2.6:Modelingerrorofin-cylinderpressureforall˝vecasesbetweenSOCandEVO.46 Table3.1:RootsofBesselfunctionsandthederivativefororder a =0 and 1 [1]......54 Table3.2:Engineoperatingconditions(atknocklimit)....................61 Table3.3:Calibratedparameters................................61 Table3.4:Initialconditionsatknockonsetforthe˝rstknockingcycleofcases1and2...64 Table3.5:Simulationresultsfor6enginecycleswithmonotonicallydecreasing ) ˚+˘ ....72 Table3.6:Simulationresults(20consecutivecycles)usingpredicted ) ˚+˘ (case2).....77 Table4.1:Engineoperatingconditions.............................81 Table4.2:Statisticanalysisfortheimpactofsparktimingtotheknockcycle-to-cycle variability.......................................85 Table4.3:Stochasticanalysisofcycle-to-cycleknockintensity................97 ix LISTOFFIGURES Figure1.1:Atwo-zonecombustionmodelforSIengines..................5 Figure1.2:Experimentalin-cylinderpressureand˝lteredpressureoscillationsunder knockcondition..................................7 Figure1.3:Time-basedexperimentalin-cylinderpressurewhentheengineisoperatedat theknocklimit(steady-state)...........................8 Figure1.4:Cycle-basedexperimentalin-cylinderpressurewhentheengineisoperated atknocklimit(steady-state)............................9 Figure1.5:Experimentalin-cylinderpressureandMFBrateforaheavyknockcycle....10 Figure2.1:Two-zonecombustionmodelstructure......................17 Figure2.2:Interactionbetweenreactionandunburnedzones.................20 Figure2.3:Ignitionenergypro˝leused............................28 Figure2.4:Comparisonofsimulatedandexperimentalin-cylinderpressureswithrelative errorat1500rpmwith6.78barIMEP(case3)..................34 Figure2.5:Comparisonofsimulatedandexperimentalin-cylinderpressuresat2000rpm with6.83barIMEP(case4)............................35 Figure2.6:Comparisonofsimulatedandexperimentalin-cylinderpressuresat2000rpm with8.23barIMEP(case5)............................36 Figure2.7:Zoneandaveragetemperaturesat1500rpmwith6.78barIMEP(case3)....37 Figure2.8:Comparisonofexperimentalandsimulatedcylindertemperaturesat1500rpm with6.78barIMEP(case3)............................38 Figure2.9:Comparisonofexperimentalandin-cylinderaveragetemperaturesat2000rpm with6.83barIMEP(case4)............................39 Figure2.10:Masstransferanditsratebetweentwozonesat1500rpmwith6.78barIMEP (case3)......................................40 Figure2.11:Individualvolumefractionsat1500rpmwith6.78barIMEP(case3)......41 x Figure2.12:Comparisonofreactionratesandmass˛owratesoffueland O 2 inthereaction zoneat1500rpmwith6.78barIMEP(case3)..................42 Figure2.13:Totalmasschangingofspecies(fuel, O 2 , CO 2 , H 2 O and CO )at1500rpm with6.78barIMEP(case3)............................43 Figure2.14:PredictedMFBrateat2000rpmwith6.83barIMEP(case4)..........44 Figure2.15:Heatreleaserateandheattransferratesbetweentwozonesat1500rpmwith 6.78barIMEP(case3)..............................45 Figure2.16:Heatlossesfromreactionandunburnedzonestothecylinderboundaryat 1500rpmwith6.78barIMEP(case3)......................46 Figure2.17:ComparisonofsimulatedandexperimentalIMEPsforall˝vecases.......47 Figure2.18:ComparisonofsimulatedandexperimentalCA50forall˝vecases.......48 Figure3.1:0-Dreaction-basedtwo-zonecombustionmodelandthepressurewavemodel.50 Figure3.2:Connectionbetweenthetwo-zonereaction-basedcombustionmodelandpres- surewavemodel..................................51 Figure3.3:Experimentalin-cylinderpressureand˝lteredpressurewaveofthe˝rstknock cycleat1500rpmwithIMEP=7.5bar(case1).................62 Figure3.4:SimulatedARIanditsrateintheunburnedzoneofthe˝rstknockcycleat 1500rpmwithIMEP=7.5bar(case1).....................64 Figure3.5:Calculatedin-cylinderpressurewaveat1500rpmwithIMEP=7.5bar(case1)65 Figure3.6:Experimentalin-cylinderpressureand˝lteredpressurewaveofatypical knockcycleat2000rpmwithIMEP=8.23bar(case2).............66 Figure3.7:SimulatedARIanditsrateintheunburnedzoneforthe˝rstknockingcycle at2000rpmwithIMEP=8.23bar(case2)...................67 Figure3.8:Simulatedin-cylinderpressureandpressurewavewitha3 ˘ 10kHzband-pass ˝lterat2000rpmwithIMEP=8.23bar(case2).................68 Figure3.9:FFTanalysisoftheexperimentalin-cylinderpressurewaveat1500rpm, IMEP=7.5bar(case1)..............................69 Figure3.10:FFTanalysisoftheexperimentalin-cylinderpressurewaveat2000rpm, IMEP=8.23bar(case2)..............................70 xi Figure3.11:Experimentalin-cylinderpressureof6consecutivecyclesunderknocking combustion,engineoperatedat1500rpm,IMEP=7.5bar(case1)........71 Figure3.12:MAPOof20consecutiveenginecyclescalculatedbasedontheexperimental in-cylinderpressureat2000rpm,IMEP=8.23bar(case2).SD:Standard deviation.....................................72 Figure3.13:Reactionzonetemperature(6cycles)withmonotonicallydecreasing ) ˚+˘ at 2000rpm,IMEP=8.23bar(case2).......................73 Figure3.14:Curve˝ttingforpredictingthemixturetemperatureatIVCofnextcyclebased ontheexhausttemperatureatcurrentcycle...................75 Figure3.15:PredictedMAPOsof20consecutiveenginecycles...............76 Figure4.1:Correlationdiagramofreaction-basedtwo-zonecombustionmodel,thepres- surewavemodel,andmodel-basedstochasticknocklimitcontrol........79 Figure4.2:Experimentalin-cylinderpressureandband-pass˝lteredpressurewaveof oneenginecycle..................................80 Figure4.3:Fittedcorrelationcurvebetweencurrentcycle ) 4E> ( : ) andnextcycle ) 8E2 ( : +1) 82 Figure4.4:Intaketemperature ) 8E2 withknockintensity...................83 Figure4.5:Interpretedmapforthecorrelationofintaketemperature ) 8E2 andknock intensityaslongasthesparktiming.......................84 Figure4.6:Thein˛uenceofsparktimingtoknockintensitycycle-to-cyclevariability...85 Figure4.7:Model-basedstochasticfeedforwardknocklimitcontroldiagram........88 Figure4.8:Model-basedstochasticfeedforwardknocklimitcontrolperformance......90 Figure4.9:Cycle-basedsparktimingforknocklimitandMBTtimingcontrol.......91 Figure4.10:Detailsofknockintensityundermodel-basedstochasticfeedforwardknock limitcontrol....................................92 Figure4.11:Comparisonofknockintensitybeforeandafterthecompensationofspark timingforeachcycle................................93 Figure4.12:Gaussiandistributioncomparisonforknockintensitywithandwithoutthe proposedfeedforwardknocklimitcontrolalgorithm...............94 xii Figure4.13:Model-basedclosed-loopstochasticknocklimitcontroldiagram........95 Figure4.14:Bu˙erformationforthestatisticalanalysisofknockintensitydistribution....96 Figure4.15:Time-basedsparktimingwiththeclosed-loopstochasticknocklimitcontrol algorithm.....................................97 Figure4.16:Closed-loopstochasticknocklimitcontrolperformance.............98 Figure4.17:ComparisonofknockintensitydistributionPDFwithdi˙erentcontrolmethods99 Figure4.18:KnockintensitydistributionPDFwithclosed-loopstochasticknocklimitcontrol100 xiii CHAPTER1 INTRODUCTION 1.1Motivation Nowadays,energycrisisandenvironmentalissuesaretwoofthebiggestchallengesworldwide thatmotivatetheautomotiveindustrytoimprovetheenginee˚ciencyandreduceemissions,that aretwomaingoalsforoptimizinginternalcombustion(IC)engineperformances.Over90 % of passengercarsaroundtheworldarepoweredbytheICengines,wherespark-ignited(SI)engines consistofabout85 % ofthem.Especially,thefuelconsumptionofSIenginesare20 ˘ 30%higher thanthedieselengines[2].Therefore,themainresearchofSIenginesistofurtherimprovethe fueleconomyandreduceemissions[3]thatarecloselyrelatedtocombustionprocess.However, operatingengineatitsknocklimitturnsouttobeoneofthemainchallengesinimprovingthe thermale˚ciencyofSIengines.Especially,SIengineswithhighcompressionratio,turbocharger, andevenreduceddisplacementcombustionchamberbecomethetendencyinrecentyearstoimprove theenginefuele˚ciency.Theincreasedboostlevelorhighcompressionratioraisestheprobability ofengineknockevent[2,4]. Knockisanabnormalcombustionphenomenoninthespark-ignited(SI)engineresultedby theauto-ignitionofend-gas(unburned)inthepropagating˛amefront[5,6].Thechemical reactionenergyaggressivelyreleasedbytheauto-ignitedend-gasandthemaincombustionbythe sparkignitionwillcausetheunevenheatdistributionsinthecombustionchamber,leadingtohigh frequencyshockwaves,sharppressurerise,andhightemperaturethatmaydamagetheengine. Knockathighenginespeedcancauserapidenginedamageandlessdamagingbutismorelikelyto causedriverannoyanceatlowenginespeed[7].Especially,undercertainoperationalconditions, theMBTtimingismoreadvancedthantheknockboundarysparktiming,itisnotabletooperate theignitionatMBTtimingwithoutengineknock.Intheseconditions,itisdesirabletooperatethe engineattheknocklimittoproducethebestfuele˚ciency.Therefore,theengineknockprediction 1 andsparktimingcontroltooperatetheengineatknocklimitorMBTtimingtakeansigni˝cant roleinimprovingSIenginefueleconomy. Sparktimingcontrolisextensivelystudiedintheliterature(see[8,9])tooperatetheengine closetoitsMBTtimingwithoutknockcombustion.Atraditionalmethodforknocklimitcontrolis toadvanceorretardthesparktimingbyintegralcontrolbasedontheerrorbetweenthemeasured knockintensityandthepre-de˝nedknockintensitythreshold[9],resultinginoperatingtheengine inandoutofknockcombustionunsmoothly.The˛uctuationsofenginesparktimingoverengine cyclescannotguaranteesatisfactoryknocklimitcontrolperformanceofoperatingtheengineatits MBTtimingascloseaspossible.Theanalysisofexperimentalknockdataindicatesthatengine knockisastochasticphenomenonandknockcontrolshouldnotonlylimittheengineknockintensity belowitsdesiredlevelbutalsoreducetheknockcycle-to-cyclevariability. Therefore,themotivationofthisdissertationistodevelopmodel-basedstochasticknocklimit controlalgorithmstooperatetheSIengineatborder-lineknocklimitundertheseconditions thattheMBTtimingislimitedbyknock,andtocontroltheknockintensitywithinthedesired boundwithminimumcycle-to-cycleknockvariability.Thefoundationforthecontroldesignis aknockpredictivemodelthatisabletoaccuratelypredictthecombustionprocess,knockmajor characteristics(knockonsettiming,frequency,intensity,cyclicvariability)inreal-time.However, thereislimitedstudyinliteratureaboutthereaction-basedreal-timemodelforspark-ignition(SI) enginescapableofknockprediction.Notethatauto-ignitionofend-gas(knock)inSIenginesisa veryimportantphenomenon,andunfortunately,thereisnotmanyresultsoncontrol-orientedknock modeling.Sothedevelopmentofknockcontrolorientedcombustionandknockpredictionmodel, andmodel-basedstochasticknocklimitcontrolaretwomaingoalsofthisdissertation. 1.2ResearchOverview 1.2.1Control-OrientedCombustionModelforSIEngines Inthepastdecades,themodel-basedenginecontrol,especiallycombustioncontrol,iswidelystudied duetotherapidtechnologydevelopmentincombustionsensingandreal-timecomputingpower.As 2 partofmodel-basedenginecontrol,majorprogresshasbeenmadeindevelopingcontrol-oriented engineandcombustionmodelsforbothspark-ignition(SI)andcompressionignition(CI)internal combustionengines[10].Thesedevelopedmodelsareusedformodel-baseddesignandcalibration toe˚cientlyreduceenginedevelopmenttimeandcost[11].Thecontrol-orientedenginemodels areusedformodel-basedcontrolinreal-timeapplications.Furthermore,sinceenginedynamics, emissions,andperformancecanbee˚cientlystudiedthroughsimulationswithoutconducting physicalexperiments,itispossibletostudyengineoperationsatitsoperationalboundarywhen experimentsarehardtoconductwithoutdamagingthephysicalsystem. Model-basedcombustioncontrolleadstothedevelopmentofcontrol-orientedenginecom- bustionmodelstoreliablypredictthein-cylindercombustionprocessbyprovidingdetailedmass- fraction-burned(MFB)rate,in-cylinderpressure,andotherinformation[11].Uptonow,there areseveralwidely-usedmodelingapproachesforthein-cylindercombustionprocessthatcanbe classi˝edassingle-zone,multi-zone,andmulti-dimensionalmodels[12].Themulti-dimensional computational˛uiddynamics(CFD)modelsaccuratelydescribethein-cylindercombustionpro- cess,includingthe˛ame˛uiddynamicsandthein-cylindermixturecharacteristics,bysolvinga numberofpartialdi˙erentialequations.Somemulti-dimensionalCFDmodelspredictthecombus- tionprocessbymodelingthousandsofchemicalreactionspecieswithcertainreactionsteps.One ofthemostpopularapproachesistheCHEMKIN-CFDmoduleforANSYSFLUENTthatusesthe multi-dimensionalmodeltosimulatedetailedchemicalmechanismsandchemicallyreacting˛ows inamultidimensionalcylinderandtotraceindividualspeciesinformation. Althoughthesemodelsareabletopredictthecombustionprocessaccurately,theyrequire tremendouscomputationalpowertoevensimulatethecombustionprocessforoneenginecycle. Asaresult,themulti-dimensionalCFDmodelsareonlygoodforo˙-linesimulations,butitisnot suitableformodel-basedcontroldesignandvalidationthatrequirereal-timesimulationcapability. TheGT-PowerandRicardoWaveenginemodels,widelyusedinautomotiveindustry,use0-D/1-D modelingapproachtotakeaccountofthegas˛owdynamicsoutsideofcombustionchamberand 0-Dsingle-zonemodelingapproachforthecombustionprocessbasedontheempiricalcombustion 3 functionssuchastheWiebefunction[13,14,15,16].GT-powermodelsaresigni˝cantlysimpli˝ed overtheCFDones[17]butitstillcannotbeusedforreal-timesimulations.Ontheotherhand,the 0-Dsinglezonecombustionmodel,suchasthemean-valuemodel[18],waswidelyusedforcontrol designduetoitsabilityofconductingreal-timesimulations.Toreducesimulationtime,itusesmaps tomodelthecomplexcombustionprocess.Asaresult,themodelaccuracyishighlydependent onthecalibrationprocessanditstransientresponsesarenotaccuratesincethethermodynamicsis notmodeled.Furthermore,themean-valuemodelisnotabletopredictthein-cylinderpressure, heat-release-rate(HRR),andstatesofchemicalspecies. Toovercomethelimitationsofthesecombustionmodelsdiscussedabove,two-zoneandmulti- zonemodelsaredeveloped.ShapiroandGerpen[19]presentedatwo-zonecombustionmodelfor internalcombustionenginesbasedonthesecond-lawofthermodynamics.Loganathan[20]and Saad[21]proposedatwo-zonecombustionmodelfordieselengines,andBorg[22]developeda two-zonemodelforSIenginestostudyengineperformance.Notethatintwo-zonemodels,thegas mixtureareseparatedintoburned(reaction)andunburnedzones,assumingthatthe˛amefrontis athinboundarylayerseparatingthetwozones,asshowninFigure1.1foratwo-zonecombustion modelofSIengines.Insomeliterature[3,23,24],theunburnedzoneisfurthersplitinto multiplezonesbasedontemperaturegradients,assumingthatthemixtureineachindividualzone ishomogeneousbuthasdi˙erentthermodynamiccharacteristics.RakopoulosandMichos[3,23] proposedamulti-zonecombustionmodelforenginetransientperformanceandNOx(Nitric-oxide) formationofasyngasSIengine.However,theMFBrateofthesediscussedtwo-andmulti-zone models[18,19,20,21,22,23,24,25,26,27]isgeneratedbyanempiricalWiebefunctionthat needstobecalibratedasafunctionofenginespeed,load,exhaust-gas-recirculation(EGR),etc. foreachengineoperatingcondition. ToimproveWiebe-basedcombustionmodel,the˛amedevelopmentdynamicsisintroducedin thereal-timecombustionmodelrecently.Hall,etal.[28]proposedacontrol-orientedtwo-zone combustionmodelforSIengines,assumingthattheburnedzoneissphere-shapedwiththe˛ame frontontheboundaryandMFBrateisdeterminedbycalculatingthe˛amespeed.However,since 4 Figure1.1:Atwo-zonecombustionmodelforSIengines thechemicalreactionprocessisnotmodeled,thermodynamicpropertiesofthechemicalspecies cannotbepredicted.Inordertopredictthecombustionprocessalongwithkeythermodynamic parameterssuchastheMFBrate,in-cylinderpressure,temperaturesandindividualspeciesmass changes,thereaction-basedcombustionmodelisdeveloped.JiaandWang[29]presentedacontrol- orientedreaction-basedsingle-zonecombustionmodelforapropane-fueledHomogeneousCharge CompressionIgnition(HCCI)engine,andMenandZhu[30]alsoproposedacontrol-orientedthree- zonereaction-basedcombustionmodelfordirect-injection(DI)dieselengines.Bothdiscardedthe empiricalWiebe-basedcombustionmodel,andinstead,adoptedsingleormulti-stepchemical kineticmechanisminconjunctionwiththeArrheniusfunction-basedreactionrates.Thereaction- basedcombustionmodeltakesintoaccountforchemicalcharacteristicsintheactualcombustion processandisabletopredicttheMFBrate,pressureriserate,HRR,andzonetemperaturesaswell asthepropertiesofeachindividualchemicalspecieswithverylowcomputationalload.Therefore, thereaction-basedcombustionmodelcanbefurtherdevelopedforthereal-timeknockprediction 5 andmodel-basedknockcontrol. 1.2.2KnockPredictionandModelingforSIEngines Theauto-ignitionoftheend-gasintheunburnedmixtureswillresultintheshockwavesinthe combustionchamber,ledtothepressureoscillations,rapidpressureandtemperatureraise.As showninFigure1.2fortheexperimentalin-cylinderpressure(bluesolid-line)andtheband-pass ˝lteredpressureoscillations(red-solidline)foronecyclewhentheengineisoperatedatknock condition.Aftertheknockonset,thein-cylinderpressurerapidlyraisesupto35barduetothe aggressiveknockingcombustion.Andthemaximumpressureoscillationamplitudeis3bar.The temperatureoftheunburnedmixtureisbelow1000Kbeforeknockonset,andthenitcanbeup to3000K.Therefore,thepredictionandcontrolofengineknockisveryimportanttopreventthe enginedamage,andmeetthefueleconomyandemissionregulations. Engineknockphenomenonisextensivelystudiedinpastdecadesfromtheknockdetection methodsto3-Dmodelingoftheknockingcombustion.Knockdetectionisthe˝rststeptostudy knockcharacteristicsandmanymethodsareproposedforknockdetection[31,32,33,34].The pressuresensorisoneofthemostpopularmethodsusedtodetectengineknockinthelab, accelerometer(alsocalledknocksensor)iswidelyusedinproductionvehiclesfordetectingengine knockbutthemechanicalvibrationcausedbyenginevalveeventsandinteractionamongdi˙erent combustionchambersaddssigni˝cantnoiseintotheknocksensor,sometimesmakingknock detectionimpossible.Somenewmethodsareproposedinthepastdecade.Zhu,Haskara,and Winkelman[35]developedanignitioncoilbasedionizationdetectioncircuitfordetectingthe engineknock.Theknockonsetpredictionisanotherimportanttopicforimprovingengineknock control[36,37,38].Twopredictionmethodsarewidelyusedinpastdecades.Oneisauto-ignition delaymodel[39]basedoncalculatedauto-ignitiondelaytime g de˝nedasdurationbetweenthe endofcompressionandknockonsettime.Thepressureandtemperatureoftheunburnedmixtures arerequiredbutnochemicalreactionand˛amedynamicsareconsidered.Theothermethod isthechemicalkineticmethodbasedontheArrheniusintegralutilizingthechemicalreaction 6 Figure1.2:Experimentalin-cylinderpressureand˝lteredpressureoscillationsunderknock condition rateandconcentrationofthespeciesinvolvedintheauto-ignition.Bothmethodshavetheir advantagesanddisadvantages.Theauto-ignitiondelaymethodhasbeenimproved[40,38,41]and iswidelyusedduetotheitssimplicityandlowcomputingcost.However,withoutconsideringthe importantchemicalreactionprocessinthecombustionchamber,theauto-ignitionprocesscannot bepredictedaccurately.TheArrheniusintegralmethodiswidelyusedin3-Dcomputational˛uid dynamic(CFD)modelsforpredictingtheknockonsetbutthechemicalreactionstepsandspecies involvedcanbefairlylarge(uptothousands),resultinginthehighcomputationload.Itisnot suitableforthemodel-basedknockcontroldesign.Soareal-timemodelthatisabletopredictthe 7 Figure1.3:Time-basedexperimentalin-cylinderpressurewhentheengineisoperatedatthe knocklimit(steady-state) enginecombustionprocessandalsoknockphenomenonishighlydesired. Engineknockphenomenoniscomplexandnotconsistent.In-cylinderpressureoscillation informationhasgreatpotentialinidentifyingmajorknockcharacteristics[42]suchasknock frequencyandintensityanditiswidelystudied.AsshowninFigure1.2,thepressureoscillations aregeneratedaftertheknockonset,andthe˝lteredpressureoscillationwave(redsolid-line)canbe furtheranalyzedtoobtainthefrequency,intensityandknockonsettiming.Moreover,theengine knockhasadistinguishingcharacteristics:thecycle-to-cyclevariability[43,44,45].Asshownin Figure1.3foratime-basedin-cylinderpressureof25continuouscycles.Itisobviousthatthepeak pressureisnotconsistentatknockconditionwhentheengineisoperatedatsteady-state.Tobetter demonstratetheknockintensityvariability,thecycle-basedexperimentalin-cylinderpressureof 11continuouscyclesareshowninFigure1.4.Itindicatesthattheknockintensityisnotconsistent butshowsstrongvariability,andthepeakpressurelocation(PPL)movestobeclosedtoMBT aslongastheknockintensityincreases.Thedetailsofcycle # 11,thathastheheaviestknock intensity,areshowninFigure1.5.Therefore,thestudyofpressureoscillationsaswellastheknock 8 Figure1.4:Cycle-basedexperimentalin-cylinderpressurewhentheengineisoperatedatknock limit(steady-state) cycle-to-cyclevariabilityareveryimportantfortheknockpredictionmodeling. Katsumata,MorikawaandTanabe[46]investigatedtheend-gasshockwavesunderknocking combustionusingahigh-speeddirectandschlierenphotographymethod.KawaharaandTomita [47]visualizedtheauto-ignitionofend-gasandpressurewavesinahydrogenspark-ignitionengine usingahigh-speedcamera,andfurthermore,theyevenobservedthecycle-to-cycleimagesofthe auto-ignitedkernelintheend-gasregion.Thevisualizationresultsindicatethatlargeamount ofunburnedmixturesgeneratesstrongshockwavescausedbytheauto-ignitedkernelexplosion. Theseopticaldiagnosticsmethodsareveryhelpfultounderstandthephysicalprocessofknocking combustion.Modelingin-cylinderpressurewavesunderknockingcombustionandassociated numericalsimulationsarewidelyusedtostudytheknockphenomenon.YaoandXu[48]developed 9 Figure1.5:Experimentalin-cylinderpressureandMFBrateforaheavyknockcycle a3-DCFDmodeltostudythepropagationandre˛ectionofpressurewavesgeneratedbytheend- gasauto-ignition.Theyusedthe3-Dsimulationresultstooptimizethecombustionchamber shape.TerashimaandKoshi[49]proposeda1-Dpressurewavemodelincludinglargedetailed chemicalkineticmechanismstostudythepressurewavegeneratedduringend-gasauto-ignition andtheresultsdemonstratethein˛uenceofwalltemperaturetotheknockintensity.Richard [50]developeda1-DCFDmodeltopredictthecycle-to-cyclevariabilityinSIenginesbasedon theanalysisofLarge-EddySimulation(LES).These1-Dand3-Dmodelsareabletopredictthe in-cylinderpressureoscillationaccuratelybutwithtremendouscomputationalload.Therefore, theyaregoodforo˙-linestudyingknockcharacteristicsbutnotsuitableformodel-basedknock predictionandcontroldesign. 10 Toovercomethelimitationsofthe1-Dand3-Dmodelsdiscussedaboveandreducethe computationload,simpli˝ed0-Dpressurewavemodelisrequired.However,therearefewliterature intheareaofreal-timepressureoscillationmodeling.Draper[51]isapioneerofapplyingthe acousticwaveequationtointernalcombustionenginestostudythepressurewavecharacteristics inthecombustionchamberbutthepressureoscillationmagnitudedecaybehavior,causedbythe energylossandpistonmovement,wasnotconsidered.Notethatthedeadeningbehaviorisanatural processandmustbeconsideredforaccuratelypredictingthereal-timepressureoscillations.Based onDraper'swork,BrecqandCorre[52]proposedapressureenvelopcurvetopredictthepressure oscillationpeaks.Althoughthisenvelopcurveshowsthemagnitudedecaybehavior,itisacurve ˝ttingmodelthatisnotabletopredicttheactualpressureoscillations.Basedontheenvelopcurve model,animproved3-DpressurewavemodelisproposedbyDiGaeta[53]todescribetheactual pressureoscillationsandmagnitudedecaybehavior.Ageneralsolutiontothiswaveequationis providedbutthe3-Dmodelisstilltoocomplextobeusedformodel-basedknockpredictionand controldesign.Therefore,aphysical-basedreal-timepressurewavemodelthatiscapabletopredict thepressureoscillationsresultedbyknockingcombustionisimportantforknockcontroldesign. 1.2.3Model-BasedKnockLimitControlforSIEngines Theknockintensityiscommonlyusedtopresenttheengineknockseverity.Ingeneral,theknock intensityiscalculatedbyprocessingtheknocksensorsignalwithaband-pass˝lterandthen integratingwithinapre-de˝nedknockwindow.Asdiscussedinthelastsection,theengineknock phenomenonhasasigni˝cantcharacteristics:theknockintensityofeachcycleisrandomlyvarying cycle-to-cyclewithminimalcycliccorrelationevenundersteady-stateoperationalcondition,as showninFigure1.3.Duetothiscyclicknockvariability,theknockcontrolobjectivegenerally focusesonthestochasticknockintensitycontrolwithstatisticalanalysisapproach.Inliterature [35,9,49,7,54,55,56,57,58,59,60],manydi˙erentstochasticknockcontrolstrategies,suchas thelikelihood-basedcontrol,thefeedforwardorclosed-loopknocklimitcontrol,thelearning-map basedmethod,Bayesianapproach,areproposedtocontroltheknockintensitydistributionpresented 11 withthemeanvalueandstandarddeviation.Zhuetal.[35,9]unitizedtheionizationsignalto detecttheknockintensity,andproposedastochasticclosed-loopsparktimingmanagementsystem tomaintaintheknockintensitywithinthecon˝dencelevelwithMBTtimingcontrolandknock limitcontrol.Stotsky[56]proposedaclosed-loopknockcontrolalgorithmbasedonthestatistic approachtocontrolthesparktimingmovingupanddownateachcyclebasedonthefeedback errorofaregulationvariableandit'stargetedvalue.Inrecentlyyears,theon-boardlearning-based stochasticknocklimitcontrol[55,54]hasbeenstudiedaswellintheliteraturetocontroltheknock intensitydistributionbasedonthereal-timeexperimentaldataadaptationandestimation. However,mostofthestochasticknockcontrolalgorithmsproposedintheliteraturearebasedon thereal-timeknocksensormeasurementandknockintensityestimation.Therefore,asigni˝cant numberofcyclesdataaregenerallyrequiredtoobtaintheaccuratemeanvalueandstandard deviationforstatisticalanalysisbeforethesparktimingcompensationbytheknockcontroller. Thisprocessistime-costandthecontrolupdatingistooslowtooperatethesparktimingatthe border-lineknocklimitwithminimumknockcycle-to-cyclevariabilityandbestfueleconomy. Toovercomethelimitationsoftheexperimentaldata-basedstochasticknocklimitcontrol,the model-basedstochasticknocklimitcontroldemonstratesitsadvantages.Jonesetal.[7]developed amodel-basedBayesianknockeventcontrollerandthecontrol-orientedmodelwasanempirical modelthatcancapturethecharacteristicsofknockintensitydistribution(meanvalueanddeviation) aftertheextensivelycalibrationwithexperimentaldata.However,themodelcannotpredictthe physicalprocessfortheknockcombustionanditrequirestremendouscalibrationstoimprovethe modelaccuracy.Thenaphysical-basedknockpredictivemodelthatcanaccuratelydescribethe physicalcombustionprocessunderknockconditionandtheimportantknockcharacteristicsin real-timewillreducethecalibrationcostandisthefoundationforthemodel-basedstochasticknock limitcontrolstrategydesign. Asfortheknockcycle-to-cyclevariability,itisin˛uencedbymanyfactors,butthein-cylinder mixturesintaketemperatureatintakevalveclosing(IVC)isthemajorone.Zhouetal.[61]studied thepercentagesofknockandcycle-to-cyclevariationwithdi˙erentintaketemperaturesbasedon 12 extensiveexperimentdata.Theresultsindicatedthatthehigherintaketemperaturewillincrease theknockoccurrencepercentageandsigni˝cantlyin˛uencetheknockcycle-to-cyclevariability. Therefore,it'simportantthatthecontrol-orientedknockpredictivemodelhasthecapabilityto predictthecycle-to-cycleknockvariability,besidesthepredictionofknockonsettimingand intensityofindividualcycles. Thisdissertationfocusesonthedevelopmentofthereal-timecombustionmodelandpressure wavemodelandmodel-basedknockpredictionandcontrol.Acontrol-orientedreaction-based combustionmodelofSIenginesforreal-timesimulationsisdevelopedandvalidated˝rst,where atwo-zonecombustionmodelisusedalongwiththetwo-stepchemicalreactionmechanism andthereactionrateofindividualspeciesisbasedontheArrheniusfunction.Thein-cylinder thermodynamicsandcombustionprocessaremodeledbetweenIVCandexhaustvalveopening (EVO).Theproposedcombustionmodelisvalidatedagainstexperimentaldataat˝vetypical operationalconditionswithonesetofcalibrationparametersanddemonstrateditscapabilityof predictingthecombustionprocessaccurately.Basedonthisreaction-basedcombustionmodel,a control-orientedknockpressurewavemodelcapableofpredictingthemajorcharacteristicsofknock phenomenoninreal-timeisdevelopedandvalidated.Theproposed0-Dreaction-basedreal-time knockpressurewavemodelcapableofpredictingtheknockonsettimingandin-cylinderpressure wave(usedtopredictknockfrequencyandintensity)underknockcombustion.Furthermore,the cycle-by-cycleknockvariabilityisalsodemonstratedbytheproposedmodel. Consequently,thisdissertationstudiesthestochasticknocklimitcontrolbasedonthecontrol- orientedphysical-basedknockpredictivemodelthatconsistsofthecombustionandthepressure wavemodels.Comparingwiththeexperimentaldata-basedknockcontrolmethods,thephysical model-basedknockpredictionandcontrolhasthesigni˝cantimprovementofcontrolcompensation e˚ciencywithminimumtimecost,andmakesitpossibletocompensatethesparktimingofeach cycleandoperatetheengineattheborder-lineknocklimitasclosetoMBTtimingaspossible. 13 1.3DissertationContributions Themaincontributionsofthisdissertationare: (1)Theproposed0-Dtwo-zonetwo-stepchemicalreactioncombustionmodelcapableof predictingMFB,HRR,in-cylinderpressure,alongwiththermodynamicpropertiesofindividual speciessuchastheirchemicalreactionrates,associatedmolarconcentrations,andconcentration variationrates,andsoon.Furthermore,thedevelopedmodeliscalibratedunder˝vedi˙erentengine operatingconditionsusingonesetofcalibrationparameters.Thatis,themodeldoesnotneedtobe recalibratedunderdi˙erentoperatingconditions,whichisveryimportantformodel-basedcontrol. (2)Theproposed0-Dreaction-basedreal-timeknockpressurewavemodelcapableofpredicting theknockonsettimingandin-cylinderpressureoscillationwaveunderknockingcombustion thatcanbeusedtopredictknockfrequencyandintensity.Especially,thecycle-by-cycleknock variabilityisalsodemonstratedbytheproposedpressurewavemodel. (3)Thedevelopmentofthemodel-basedstochasticknocklimitcontrolstrategytomaintainthe knockintensitywithinadesiredboundde˝nedbythetargetedmeanvalueandstandarddeviation ofcyclicknockintensityasclosetoMBTtimingaspossible.Astrongcorrelationbetweencycle- to-cycleknockvariabilityandin-cylindermixturesintaketemperatureisobtainedbasedonthe knockpredictivemodel.Andtwodi˙erentmodel-basedstochasticknocklimitcontrolstrategies areproposedandvalidatedtomaintaintheknockintensitybelowthedesignedboundascloseto MBTtimingaspossible. 1.4DissertationOrganization Thisdissertationisorganizedasfollows.InChapter2,acontrol-orientedtwo-zonereaction- basedcombustionmodelisdeveloped,calibratedandvalidatedagainsttheexperimentaldata collectedfrom5typicalengineoperatingconditions.Indetail,theoutlineoftheproposedmodelis discussedbyaddressingthemassandheattransferinteractionontheboundarylayerofthetwozones andthechemicalreactionratesofindividualspeciesbasedontheArrheniusfunction,alongwith thermodynamicstatesandpropertiesintwozones.Next,afourcylinderSIengineexperimental 14 dataisusedtovalidatethedevelopedmodel,andacalibrationmethod,basedontheparameter sensitivityanalysis,isalsopresented.TheninChapter3,azero-dimensionalreal-timepressure wavemodelisdevelopedandvalidatedbasedonthecombustionmodeldevelopedinChapter2for knockprediction.Indetail,a0-Dpressurewaveequationformodelingthein-cylinderpressure oscillationsisderived˝rst.Thentheboundaryandinitialconditionsusedforderivingthepressure waveequationisfurtherdiscussed.Next,theArrheniusintegralforpredictingknockonsetand knockintensityarediscussedandtheproposedmodeliscalibratedusingtheexperimentaldata fromafourcylinderSIengineunderknockingcombustion.Thesimulationresultsarepresented toshowthemodel'scapabilityofpredictingtheknockonset,knockfrequencyandintensityunder di˙erentengineoperatingconditions.Theknockcycle-to-cyclevariabilityisalsoanalyzedusing bothexperimentaldataandmodelsimulationresults,andsimulationresultscon˝rmitscapability ofpredictingthecycle-by-cyclevariability.InChapter4,thecorrelationsofknockcycle-to-cycle variabilitywithin-cylindermixturesintaketemperatureaslongasthesparktimingarestudied separatelybasedontheknockpredictivemodel.Thenaknockpredictivemodel-basedfeedforward stochasticknocklimitcontrolstrategyisproposedandthecontrolperformanceinoperatingthe engineattheborder-lineknocklimitwithMBTtimingconstraint,andreducingtheknockcycle-to- cyclevariabilityisvalidatedwithsimulationresults.Furthermore,aclosed-loopcontrolstrategy isfurtherproposed,includingthefeedforwardcontrolstrategyandaPIcontrollerinthefeedback looptocompensatethesparktimingwiththefeedbackerrorofthreestandarddeviationcon˝dence limitandapre-de˝nedtargetedthreshold.Thecontrolperformanceofthefeedforwardandclosed- loopstochasticknocklimitcontrolstrategiesarecompared.Theconclusionsandfurtherworkare discussedinChapter5. 15 CHAPTER2 TWO-ZONEREACTION-BASEDCOMBUSTIONMODELING Theproposedmodelfocusesonpredictingthein-cylindercombustionprocessforafour-strokeSI enginebetweenIVCandEVO.Itisassumedthatthefreshair,fuel,andresidual-exhaust-gas(REG) fromthelastcombustioncyclearehomogeneouslymixedinthecylinderatIVC.BetweenIVCand EVO,thecombustionchamberisdividedintotwozones:unburnedandburned(reaction).The twozonesareassumedtobehemisphere-shapedandbothhaveheatlossestothecylinderwall;see therightdrawinginFigure2.1.Sincethein-cylindercombustionchambershapeisnothemisphere (seetheleftdrawinginFigure2.1),inthecombustionmodel,thevolumesofburnedandunburned zonesindicatedintheleftdrawinginFigure2.1arematchedwiththeseintherightdrawingin Figure2.1,respectively.Aninitial 1 Ł 5% oftotalfreshairandfuelmixtureisassignedtotheignition zoneatIVCandafterignitionitbecomestheinitialreactionzonemass.TheREGandtherestof totalfreshairandfuelmixtureareassignedtobothignitionandunburnedzonesinitially.After thespark,combustionstartsinthereactionzoneandthegasmixtureintheunburnedzone˛ows intothereactionzonetoburn.Itisfurtherassumedthatthechemicalreactionproductsstayinthe reactionzone,andasaresult,itexpandswiththe˛amepropagation. Theinteractionsbetweentwozonesincludeheatandmasstransfer,whichmakesiteasierto simulatetheactualcombustionprocess,˛amepropagation,andtemperatedi˙erence.Furthermore, di˙erentfromthetraditionalempiricalWiebe-basedcombustionmodel,atwo-stepchemicalre- actionkineticmechanismisusedinthispapertocalculatethechemicalreactionratesduringthe combustionprocess.Themolarconcentrationandconcentrationvariationrateofeachindividual speciesinthereactionzoneduringtheentirecombustionprocessarecalculated,makingitpossible tostudythedetailedcombustionprocesssuchasMFB,HRR,chemicalreactionrates,˛amespeed, zonetemperatures,andpressure. 16 Figure2.1:Two-zonecombustionmodelstructure 2.1Two-ZoneModelCon˝guration ThecombustionchamberisassumedtobedividedintotwozonesasshowninFigure2.1.The ˛amefrontseparatesthetwozones,andthereactionzoneiswherethechemicalreactiontakes placeandthecombustionproductsareassumedtostaywithinthiszone.Thepre-mixedgasmixture isintheunburnedzone.FromtheleftdrawinginFigure2.1,theheatlossareaofthereactionzone extendsfromthecylinderheadtoitswall;andtheareaofunburnedzoneisfromthepistontopto cylinderwall.Thisisimportantformodelingheatlossesinthenextsection.Thetwozonesinteract throughtheinterfacewithheatandmasstransfer.Themodelinputsincludemassoffreshairand fuel,air-to-fuelratio(AFR),REG,chambervolumeatIVC,in-cylinderpressureandtemperatureat IVC.Asmentionedinthelastsection,freshair,fuel,andREGarepremixedatIVCwithaknown AFR, _ ,andthetotalmassinthecombustionchamberisshownbelowin(2.1). < C>C = < 08A + < 5D4; + < 'ˆ˝ (2.1) where < 'ˆ˝ isthemassofresidualexhaustgastrappedinthechamber.Forthedatausedto calibratethismodelforthetestengine,theEGR(exhaustgasrecirculation)valveisclosedsothe 17 EGRratewassettozerointhisstudyandonlyREGisconsidered.NotethatiftheEGRrateisnot zero,onlytheREGmassneedstobemodi˝esbasedontheEGRrate.Inthisstudy,withtheknown IVCtiming,thecalculateREGmassisaround 7% oftotalmassatIVC.Toinitiatethecombustion, ignitionenergyisappliedtoasmallzone(around1.5%ofthetotalmassforthispaper)aroundthe sparkpluggapandthecombustionisinitiatedwhentheauto-ignitionconditionissatis˝ed.The smallzoneiscalledignitionzoneandtheassociatedmassiscalledinitialignitionzonemass.Since auto-ignitionisassumedintheignitionzone,thecombustionintheignitionzonecompletesinone simulationstepandtheignitionzonebecomesthereactionzoneafterthecombustioniscompleted andtheinitialignitionzonemassbecomestheinitialmassofreactionzone. Thetotalcylindervolume + 2H; anditsrateofchange ¤ + 2H; canbeobtainedfromthetypical pistonmotionlaw[62]below. + 2H; = + 2 1+ A 2 1 2 h ' +1 cos \ p ' 2 sin 2 \ i (2.2) ¤ + 2H; = ( A 2 1) + 2 2 sin \ 1+ cos \ p ' 2 sin 2 \ ! 3\ 3C (2.3) where \ isthecrankangle; A 2 isthecompressionratio; ' istheratioofconnectingrodlengthand crankradius;and + 2 istheclearancevolume.Therefore,assumingthemixturesinbothzonesare homogeneous,theunburnedzonevolume + D canbecalculated˝rstbasedontheidealgaslaw,and thevolumerateofchange ¤ + D canbederivedfrom + D ;seebelow. + D = < D ' D ) D ?", D Œ ¤ + D = 3+ D 3C (2.4) where < D , ) D ,and ", D aremass,temperatureandaveragemolecularweightoftheunburned zonemixture,respectively; ? isthein-cylinderpressure; ' D ( 8 Ł 314 J/K-mol)istheuniversalgas constant. Therefore,thereactionzonevolume + A anditsrate ¤ + A canbederivedas + A = + 2H; + D Œ ¤ + A = ¤ + 2H; ¤ + D (2.5) 18 DetailedinteractionbetweentwozonesisshowninFigure2.2andtheassociatedformulasused tocalculatethemixturecharacteristicsineachzonewillbediscussedinthenextfewsubsections. 2.2InteractionbetweenTwoZones Heatandmasstransferbetweentwozonesandheatlossfromtwozonestothecylinderboundary areveryimportanttoobtainanaccuratecombustionmodelsincethethermodynamicpropertiesin eachzonearebasedonheatloss,heatandmasstransfer. 2.2.1HeatTransferInterface BetweenIVCandsparkignition,thereisnoheatandmasstransferattheinterfacebetween unburnedandreaction(ignition)zones.Asaresult,themassandvolumeratiooftwozonesremain unchanged.Withtheaddedsparkenergy,thecombustionisinitiatedinthereaction(ignition)zone andthemodeltransitstothecombustionphase.Duringthecombustionphase,the˛amestays withinthereactionzonewitharadiusof ' A andpropagatestowardstheunburnedzone.Asis showninFigure2.2,theheatreleasefromthechemicalreactionresultsinafastincrementofthe reactionzonetemperature ) A ,andthetemperaturedi˙erencebetweenthetwozonesleadstothe heattransfer ¤ & CA fromthereactiontounburnedzone. Physically,themassintheinterfaceoftwozoneswillbeheatedupmuchfasterthantherest ofmixtureintheunburnedzone.Therefore,itisassumedthatthereisaverythinvirtualregion (thegreenareainFigure2.2)betweentwozones.Thisvirtualregionishomogeneousbutthe temperatureismuchhigherthanthatofunburnedzone.Afterthestartofcombustion,thetotal heattransferfromthereactionzoneisdividedintotwoparts,wherepartone, ¤ & < ,isusedtoheat themixtureinthevirtualregioninto ) A sothattheassociatedmassinthisregioncanbemovedinto thereactionzoneandthisprocessiscalledthemasstransferprocess;parttwo, ¤ & C ,isusedtoheat themassintherestofunburnedzone. Thetotalheattransfer ¤ & CA fromthereactiontounburnedzonecanbecalculatedbasedonthe followingequation. 19 Figure2.2:Interactionbetweenreactionandunburnedzones ¤ & CA = : 2 CA ) A ) D ' A (2.6) where : 2 istheheatcoe˚cienttobecalibrated; CA isthee˙ectivecontactareabetweentwo zones; ' A istheradiusofthereactionzone.Sincethetwozonetemperaturesareassumedtobe homogeneous,thetemperaturegradientdistancefortheheattransfer ¤ & CA isphysicallyfromthe centerofthereactionzonetotheaverageradiusofthevirtualregion;seethedash-lineinFigure 2.2.However,thevirtualregionisassumedtobeaverythinlayer,andasaresult,the ' A isused asthetemperaturegradientdistanceinthispaper. CA canbecalculatedby(2.7)below. CA =2 c' A 2 (2.7) Tocalculateheattransfer, ¤ & < ,assumethatthetemperatureinthevirtualregion(greenthin layerinFigure2.2)is ) 6 thatisanaveragetemperatureofreactionandunburnedzonesweighted 20 bythemassandspeci˝cheatcoe˚cientassociatedwiththecorrespondingzone. ) 6 = < A 2 ?ŒA ) A + < D 2 ?ŒD ) D < A 2 ?ŒA + < D 2 ?ŒD (2.8) where ) A , < A and 2 ?ŒA arethetemperature,massandspeci˝cheatofthemixturesinthereaction zone; ) D , < D and 2 ?ŒD arethesameparametersfortheunburnedzone,respectively.Asaresult,the heattransferfromthereactionzonetothevirtualthinlayeris ¤ & < = : 2 CA ) A ) 6 ' A (2.9) Substituting(2.8)to(2.9)yields ¤ & < = : 2 CA ) A ) D ' A < D < A + < D = < D < A + < D ¤ & CA (2.10) Asaresult,theremainingheattransfer ¤ & C usedtoincreasetheunburnedzonetemperaturecan bederivedas ¤ & C = ¤ & CA ¤ & < = < A < D + < A ¤ & CA (2.11) Asdiscussedinthelastfewsubsections,itisassumedthatthereisnomasstransferbetweenthe twozonesuntilcombustionisinitiated.Physically,themasstransferrateishighatthebeginningof combustionphaseandlowattheendofcombustion.Equations(2.10)and(2.11)indicatethatthere isnotemperaturedi˙erencebetweentwozonesbeforecombustionisinitiated( ¤ & CA =0 ),resulting in ¤ & < =0 and ¤ & C =0 duringthecompressionprocess.Duringthecombustionphase,thefraction < D € ( < D + < A ) in(2.10)dominatestheheattransferformasstransfer.Ontheotherhand,duringthe fastcombustionphase,thefraction < A € ( < D + < A ) in(2.11)dominatestheheattransferandcauses theincrementoftheunburnedzonetemperature. 2.2.2HeatLoss Heatlossa˙ectsthethermalstrati˝cationinsidethecylinderandthecombustioncharacteristics [11].Theheatlossinthismodelincludestwoparts.Oneistheheatlossfromthereactionzone 21 tothecylinderheadandliner,whichdominatesheatlossduringthecombustionphase,andis representedby ¤ & F 1 .Theotherpartistheheatlossfromtheunburnedzonetothepistoncrown andtheremainingcylinderliner,andisrepresentedby ¤ & F 2 ;seeFigure2.2. Toreducethecomputationalloadforthisreal-timecombustionmodel,theheatlossmodelis simpli˝edandtheWoschni'sformula[63]isusedtocalculatebothheatlosses, ¤ & F 1 and ¤ & F 2 below. ¤ & F 1 = 2 1 F 1 ( ) A ) F ) (2.12) ¤ & F 2 = 2 2 F 2 ( ) D ) F ) (2.13) where 2 1 and 2 2 aretheheattransfercoe˚cients; ) A , ) D and ) F arethetemperatureofthe reactionzone,unburnedzoneandthecylinderboundary,respectively.Notethatthetemperature ofthecylinderboundary(cylinderhead,cylinderliner,pistonhead)isassumedtobethesameand constant. F 1 and F 2 arethecontactingareafortheheatlossbetweentheassociatedzoneand thecylinderboundary,respectively. Thereactionzoneisaverysmallhemisphereinitially,andthenstartsexpandingduetothe ˛amepropagationaftertheignition.Therefore,thecontactingarea F 1 betweenthereactionzone andthecylinderboundaryincreasesaswell;seeFigure2.1.Asthetwozonesareassumedinthe hemisphereshapebasedonthecharacteristicofthe˛amepropagation,bothphysicalchambers shownontheleftofFigure2.1canbetransferredtothehemisphereshapes(shownintheright drawingofFigure2.1)withthematchedvolumes,respectively.Anditisassumedthattheheat lossfromthereactionzonetothewallisonlythroughthebasesurfaceofthereactionzonein hemisphereshape.Asaresult, F 1 canbecalculatedbasedonthegeometrybelow. F 1 = c' A 2 (2.14) where ' A istheradiusofthereactionzone.Notethatthebasesurfaceofreactionzone F 1 isnot limitedbythecylinderheadbutwillconsistentlyincreasewiththevolumeofthereactionzone. 22 Thecontactingareafromtheunburnedzonetotherestofcylinderboundary F 2 decreasesasthe ˛amepropagatesandisgivenby F 2 = ? + 2 + F 1 (2.15) where ? isthepistoncrownsurfacearea; 2 isthecylinderlinerarea;and isthecylinder headsurfacearea.Theheattransfercoe˚cients 2 1 and 2 2 aregivenby[30] 2 1 = U 0 Ł 2 ? 0 Ł 8 ) 0 Ł 55 A (2 Ł 28 ( ? ) 0 Ł 8 (2.16) 2 2 = V 0 Ł 2 ? 0 Ł 8 ) 0 Ł 55 D (2 Ł 28 ( ? ) 0 Ł 8 (2.17) where U and V arecalibrationconstants;and ( ? isthepistonvelocityequalto ¤ + 2H; € ? . 2.2.3MassTransfer Asmentionedabove,theheattransferrate ¤ & < causesthemasstransferbetweentwozonesduring thecombustionprocess.Ifmass˛owrate ¤ < CA weretoolarge,themodeledcombustionwouldbe unstable;andifitweretooslow,thecombustionmightnotcontinue.Therefore,themasstransfer ratebetweentwozonesisakeyparametertobemodeledanditiscalculatedbythefollowing equation. ¤ < CA = : < ¤ & < 2 ?ŒD ) 0 (2.18) where : < isthemasscoe˚cient;and ) 0 isaconstantassociatedwithtemperatureincrement. Bothparametersneedtobecalibratedbasedontheexperimentaldata. 2 ?ŒD isthespeci˝cheatof thegasmixtureintheunburnedzoneandwillbediscussedinthenextsubsection. 23 2.3ChemicalReactionKineticMechanism Themixtureintheunburnedzoneconsistsoffreshair,fuel,andREGtrappedinthecombustion chamberfromthelastcycle.Theinitialreactionzonesizeisabout1 ˘ 2%ofthetotalmass,andeach speciesinthemixturechangesduringthecombustion.Itisassumedthatthemixturesinbothzones arehomogeneous.Thecompositionandthermodynamicpropertiesofmixturesinbothzonesare determinedbythein-cylinderpressure,zonetemperature,AFR,andzonevolumes[64].Basedon thechemicalkineticmechanism,thepropertiesofeachspeciesineachzonecanbestudied.And withatwo-stepchemicalreactionmechanism,thedetailedcombustionprocesscanbemodeled. 2.3.1MolarConcentrationanditsConcentrationRate: Themolarconcentrationanditsconcentrationvariationrateofeachindividualspeciesarethe foundationforstudyingthechemicalreactionprocessandthermodynamicpropertiesofthemixture. Themixtureintheunburnedzoneconsistsof5chemicalspeciesandtheyare C 8 H 18 , O 2 , N 2 , CO 2 ,and H 2 O .Duetothetwo-stepchemicalreactionmechanism,thereactionzonehasone morechemicalspecies, CO ,generatedduringthereactionprocess.Therefore,thereare6species: C 8 H 18 , O 2 , N 2 , CO 2 , H 2 O ,and CO inthereactionzone. Themolarconcentration [ - 8 ] (molesperunitvolume)ofspecies 8 isde˝nedby » - 8 ¼ = # 8 + (2.19) where 8 standsfordi˙erentspeciesineachzone; # 8 istheassociatedmolecularnumberofspecies 8 ,where # 8 = < 8 €", 8 and + istheassociatedzonevolume. Themolarconcentrationofeachspecies [ - 8 ] changesduringthecombustionprocessandthe associatedzonevolumealsochangesduetothepistonmovement.Therefore,therateofchangefor themolarconcentration [ - 8 ] isdenotedby [ ¤ - 8 ] (kmol/m 3 s)andgivenby 24 [ ¤ - 8 ]= 3 [ - 8 ] 3C = 3 ( # 8 €+ ) 3C = 1 +", 8 3< 8 3C # 8 1 + 2 3+ 3C (2.20) The˝rsttermontherightsideof(2.20)accountsforthemasschange.Themasschangeina zoneisdrivenbytwofactors:masstransferbetweentwozonesandchemicalreaction.Duringthe chemicalreactionprocess,themassesofreactantsandproductskeepchanginguntilthereactionis ended.Thesecondtermre˛ectsthee˙ectofvolumechangetothemolarconcentration. Themasschangetermisde˝nedbelow. 1 +", 8 3< 8 3C = l 8 = l 5;>FŒ8 + l 24<Œ8 (2.21) where l 24<Œ8 (kmol/m 3 s),reactionrate,isusedtore˛ectthee˙ectofchemicalreaction.And theArrheniusLaw[64]isusedinthismodeltocalculatethereactionrate l 24<Œ8 .Thispartwill beaddressednext.Notethat l 5;>FŒ8 accountsforthee˙ectofmasstransferfortheconcentration rateofspecies 8 andcanbecalculatedbasedonthemasstransferrate ¤ < CA . l 5;>FŒ8 = 8 > > > >< > > > > : G DŒ8 ¤ < CA + D ", D intheunburnedzone G DŒ8 ¤ < CA + A ", D inthereactionzone (2.22) where G DŒ8 isthemolarfractionofeachspeciesintheunburnedzone; ", D istheaveragemolecular weightoftheunburnedzonemixturetobediscussedinthenextsubsection. 2.3.2Two-StepChemicalReactionMechanism: Asimpli˝edbutpracticalchemicalreactionmechanismforthecombustionprocessisalways desired.Inrecentliterature,theone-stepreactionbetweenreactantsandproductsiswidelyused toachievethisgoal.However,theone-stepreactionmechanismisnotabletodescribethe˛ame propagationprocessfromleantorichcombustion[64].Themainweaknessofthisone-step reactionmechanismistheneglectof CO producedinthecombustionprocess.Sinceinthetypical 25 hydrocarbon˛ames,largeamountof CO and H 2 existsintheequilibriumwith CO 2 and H 2 O ,while the CO oxidationisalsoaratherslowprocess[64,65].Therefore,atwo-stepchemicalreaction mechanismisproposedinthereactionzonetoaccountforthein˛uenceofthe CO oxidationprocess. Thespeci˝creactionstepsareshownbelow. C 8 H 18 + 17 2 O 2 k 1 ! 8CO+9H 2 O (M1) CO+ 1 2 O 2 k 2 Ž Ÿ k 3 CO 2 (M2) wheretheproportionalityfactor : 8 ( 8 =1 Œ 2 Œ 3 )isthespeci˝creactionrateconstantdominatedby thetemperature[65]. Therateofthe˝rststepreaction(M1)isgivenby l " 1 = : 1 [C 8 H 18 ] = 1 [O 2 ] = 2 (2.23) where [C 8 H 18 ] and [O 2 ] aremolarconcentrationsofspecies fuel andspecies O 2 ,respectively; and = 1 and = 2 areassociatedreactionorderandareempiricallydetermined. Forthesecondstepchemicalreaction(M2),sinceitisareversiblereaction,thenetreaction rateisgivenby l " 2 = : 2 [CO] = 3 [O 2 ] = 4 : 3 [CO 2 ] = 5 (2.24) wherethe˝rstandsecondtermsontherightsidearetheforwardandbackwardreactionratesof M2,respectively; [CO] and [CO 2 ] aremolarconcentrationsof CO and CO 2 ,respectively;and = 3 ˘ = 5 areassociatedreactionorderandarealsoempiricallydetermined. Thespeci˝creactionrateconstant : 8 ( 8 =1 Œ 2 Œ 3 )inM1andM2arecalculatedbasedonthe ArrheniusLaw[64]. : 8 = 8 4 ˆ 0Œ8 €' D ) (2.25) 26 where ˆ 0Œ8 istheactivationenergyofthereaction(J/mol); 8 isthepre-exponentialfactor; ˆ 0Œ8 and 8 areconstantandtobecalibrated.Asactivationenergy ˆ 0 andtheuniversalgasconstant ' D areconstant,theactivationtemperature ) 0 canbede˝nedas ) 0Œ8 = ˆ 0Œ8 ' D (2.26) Basedon(2.23)and(2.24),theproductionratesofeachspeciesare 24< = s 5 2 ! 2 1 (2.27) where 24< istheproductionratevectorofthespecies; s 5 2 isthestoichiometriccoe˚cient matrix;and ! 2 1 isthereactionratevectorforM1andM2;seebelow. 24< =( l C 8 H 18 l O 2 l CO 2 l H 2 O l CO ) | (2.28) and s 5 2 = 2 6 6 6 6 6 6 6 6 6 6 6 6 6 6 4 10 17 2 1 2 01 90 8 1 3 7 7 7 7 7 7 7 7 7 7 7 7 7 7 5 Œ ! 2 1 = 2 6 6 6 6 6 4 l " 1 l " 2 3 7 7 7 7 7 5 (2.29) 2.3.3ZoneTemperature Basedonthesecond-lawofthermodynamics,conservationofmass,conservationofenergy,and thechemicalkineticmechanismdiscussedabove,thetemperaturerateofchangeforthereaction zonecanbederivedbelow. ¤ ) A = ¤ & A + A + ' D ) A [ ¤ - 8 ] [ ¤ - 8 ] 8 ¤ + A + A [ - 8 ] 8 + ( ¤ # 8 8 ) + A [ - 8 ]( 2 ?Œ8 ' D ) (2.30) 27 Figure2.3:Ignitionenergypro˝leused where 8 and 2 ?Œ8 aremolarenthalpy(kJ/kmol)andmolarspeci˝cheat(kJ/kmol K)ofspecies 8 , respectively,withrespecttotemperature;and ¤ & A (kJ/s)isthenetheattransferrateforthereaction zonede˝nedby ¤ & A = ¤ & 86= ¤ & C ¤ & F 1 (2.31) Notethat ¤ & 86= (kJ/s)istherateofprovidedignitionenergy.Figure2.3providesasampleignition energypro˝lefor ¤ & 86= usedintheproposedmodel,wherethetotalignitionenergyis100mJwith adurationof6crankangledegrees(CADs). Theformulausedtocalculatetheunburnedzonetemperature ) D issimilartothoseforthe reactionzone.However,sincethereisnochemicalreactionintheunburnedzone,themolar concentrationrate( [ ¤ - 8 ] )formulafortheunburnedzoneneedstobemodi˝ed.Notethatnochemical reactionintheunburnedzonemeansthatthereactionrateofeachspeciesiszero( l 24<Œ8 =0 ). Asaresult,thechangeofmolarconcentrationiscausedbythemasstransferandvolumechange only.Therefore,(2.20)and(2.21)canbemodi˝edandcombined;seebelow. 28 [ ¤ - 8 ]= ¤ < 8 + D ", 8 # 8 ¤ + D + 2 D = l 5;>FŒ8 # 8 ¤ + D + 2 D (2.32) Thus,theunburnedzonetemperaturerate ¤ ) D is ¤ ) D = ¤ & D + D + ' D ) D [ ¤ - 8 ] [ ¤ - 8 ] 8 ¤ + D + D [ - 8 ] 8 + ( ¤ # 8 8 ) + D [ - 8 ]( 2 ?Œ8 ' D ) (2.33) where ¤ & D istheunburnedzonenetheattransferrateandisgivenby ¤ & D = ¤ & C ¤ & F 2 (2.34) 2.3.4In-CylinderPressure Withtheknowntemperature,mass,andvolumeofeachzone,thecylinderpressurecanbecalculated bytheidealgaslawbelow. ? = < C>C ' D ) 0E6 + 2H; ", <8G (2.35) where ", <8G istheaveragemolecularweightofin-cylindermixtureandcanbeweightedbythe massandspeci˝cheatofthemixtureineachzone;seebelow. ", <8G = < A 2 ?ŒA ", A + < D 2 ?ŒD ", D < A 2 ?ŒA + < D 2 ?ŒD (2.36) Notethat ", D and ", A aretheaveragemolecularweightofthemixtureinthereactionand unburnedzones,respectively.Sincethechemicalreactionhappensinthereactionzone,itis assumedthat ", D isconstantthroughoutthecombustioncycleand ", A variesduetothe chemicalreactioninthereactionzone.Asaresult, ", D canbecomputedby ", D = X G DŒ8 ", 8 (2.37) where G DŒ8 and ", 8 arethemolefractionandmolecularweightofeachspecies 8 ,respectively,in theunburnedzoneandtheyremainunchangedthroughoutthecombustionprocess. 29 However, ", A changesduringthecombustionprocessandcanbederivedas ", A = P < 8 ", 8 P < 8 (2.38) where 8 isthespeciesinthereactionzone; < 8 and ", 8 aretheassociatedmassandmolecular weight,respectively. ) 0E6 in(2.35)istheoveralltemperatureofthemixtureinthecylinderweightedbythemass andspeci˝cheatofmixturesineachzone;seebelow ) 0E6 = < A 2 ?ŒA ) A + < D 2 ?ŒD ) D < A 2 ?ŒA + < D 2 ?ŒD (2.39) where 2 ?ŒA and 2 ?ŒD arethespeci˝cheatofmixturesinreactionandunburnedzones,respectively. 2 ?ŒD and 2 ?ŒA arederivedas 2 ?ŒD = X » - 8 ¼ 2 ?Œ8 ( ) D ) + D (2.40) 2 ?ŒA = X » - 8 ¼ 2 ?Œ8 ( ) A ) + A (2.41) where [ - 8 ] isthemolarconcentrationofspecies 8 intheassociatedzone; 2 ?Œ8 ( ) D ) and 2 ?Œ8 ( ) A ) arethemolarspeci˝cheatofeachspecies 8 intheassociatedzoneatcurrentzonetemperature, respectively. 2.4ModelCalibration 2.4.1ExperimentalInvestigation Theexperimentaldatausedtovalidatetheproposedreaction-basedtwo-zonecombustionmodelis collectedfroma4-cylinder,four-strokeSIenginethroughdynamometerexperiments.Theengine parametersarelistedinTable2.1. Thetestdatafor˝vetypicalsteady-stateengineoperatingconditionsareusedforvalidation purposeandissummarizedinTable2.2,wheretherelativeair-fuelratio _ iscontrolledtobeclose tostoichiometric.Ateachcondition,100cyclesofenginedatawerecollected.Inordertocalibrate 30 Table2.1:Testengineparameters. ParameterValue Bore86 mm Stroke86 mm Rod143.6 mm Compressionratio11:1 Displacement499.56 cm 3 Intakevalveclosing(IVC)190 ° BTDC Exhaustvalveopening(EVO)156 ° ATDC theARIcoe˚cientsforauto-ignition(knock)prediction,theignitiontimingundertwooperational conditions(highloadconditionsfor1500and2000rpm)arecontrolledtobeneartheengineknock limit. Table2.2:Engineoperationalconditions. Case12345 IMEP[bar]4.535.016.786.838.23 EngineSpeed[rpm]11001500150020002000 AnA & DCAS(CombustionAnalysisSystem)isusedtorecordin-cylinderpressure,intake manifoldpressure,ignitioncurrent,etc.Theaveragein-cylindermixturetemperatureiscalculated basedontherecordedin-cylinderpressureandusedtocomparewiththemodeledones. 2.4.2ModelCalibration Thissectiondiscussesthemodelcalibrationprocessforkeymodelparametersandcomparesthe simulationresultswiththeexperimentaldata.Themodel'scapabilityofpredictingthephysical combustionprocessisdemonstratedbycomparingtheassociatedsimulationresultsofcertain importantcombustionvariablesandthermodynamicstates. BasedonthediscussionattheendofReaction-BasedCombustionModelsection,thereare asetofmodelparameterstobecalibratedempirically;seeTable2.3.Thesecoe˚cientscanbe 31 groupedintotwosets:lowandhighsensitivityparameters,wherehighsensitivityonesneedtobe calibratedcarefully. Thereactionorder = 9 ¹ 9 =1 Œ 2 Œ 3 Œ 4 Œ 5 º inequations(2.23)and(2.24)andtheactivationen- ergy ˆ 0Œ8 ¹ 8 =1 Œ 2 Œ 3 º intheArrheniusfunction(2.25)forcalculatingreactionratesofthetwo-step chemicalreactionmechanismhavelowsensitivityandtheseparametersdohavetheirrecommended valuesveri˝edbasedonstandardcombustionexperiments.Therefore,theseparametersaremod- i˝edslightlyfromthereferencevaluesandkeptunchangedwithrespecttotheengineoperating condition;seeTable2.4forthevaluesused. Table2.3:Parameterstobecalibrated. ParameterEquationGroup = 9 ( 9 =1 ˘ 5) (2.23),(2.24) lowsensitivity ) 0Œ8 ( 8 =1 Œ 2 Œ 3) (2.26) U , V (2.16),(2.17) highsensitivity : 2 , : < (2.6),(2.18) 8 ( 8 =1 Œ 2 Œ 3) (2.25) Table2.4:Calibratedparameterswithlowsensitivity = 1 = 2 = 3 = 4 = 5 ) 0Œ 1 ) 0Œ 2 ) 0Œ 3 0.251.50.30.25115540K17900K28130K Therearesevenparameterswithhighsensitivityandtheyare: U in(2.16)and V in(2.17) associatedwiththeheatlosstothecylinderboundary, : 2 in(2.6)withtheheattransferbetween twozones, : < in(2.18)forthemasstransferthroughtheinterfacebetweentwozones,and 8 ( 8 =1 Œ 2 Œ 3 )in(2.25)forthetwo-stepchemicalreactionrate. Theheatlosscoe˚cients U and V shouldbecalibrated˝rstundernocombustionconditionto makesurethatthesimulatedpressuretrackstheexperimentalonefromIVCtospark.Thesimulated andexperimentalpressurecurveswiththecalibratedheatlosscoe˚cientsarecomparedinFigure 2.4,wherethesimulatedpressureisthedashed-lineandexperimentoneisthesolidone. 32 Theheattransfer ¤ & CA betweentwozonesisveryimportantsinceita˙ectsthethermodynamic propertiesineachzone,forinstance,zonetemperature,in-cylinderpressure,chemicalreactionrate, andsoon.Also,themasstransferbetweenthetwozonesispartiallydependentonthecalculated heattransfer.Inthismodel,theheatcoe˚cient : 2 in(2.6)isthekeytohaveaccurateheattransfer. : < in(2.18)isthemasscoe˚cientthatgovernsthemass˛owratebetweentwozones.It indicatesthroughthecalibrationprocessthatthemasstransferrate ¤ < CA dominatesthemodeled combustionprocess.Theaccuratemasstransferratefromtheunburnedtoreactionzoneshallbe calibratedwelltoensureanaccuratecombustionprocess. Thepre-exponentialfactor 8 ¹ 8 =1 Œ 2 Œ 3 º intheArrheniusfunctionisthemaincoe˚cientfor calibratingthetwo-stepchemicalreactionmechanism.If 8 weretoolarge,theauto-ignitioncould happenbeforethespark,leadingtopre-ignition;andif 8 weretoosmall,thereactionratewould betooslowtostartthecombustionprocessafterthespark. ThehighsensitivityparametersarelistedinTable2.5withassociatedcalibrationvalues.Note thatthesecalibratedparametersare˝xedunderdi˙erentengineoperatingconditions. Table2.5:Calibratedparameterswithhighsensitivity UV: < : 2 1 2 3 0.9782.6085.2e-72954.15e1015.21e161.98e8 2.5ModelValidationandSimulationResults 2.5.1ThermodynamicProperties Thissubsectionshowsthemodelperformanceofpredictingthermodynamicstatesinthecombustion chamber,suchasthein-cylinderpressure,zonetemperaturesandvolumes,andmasstransfer betweentwozones.Sincethesimulationresultsareprettysimilarforall5caseslistedinTable2.2, tosimplifythepresentation,onlythesimulationresultsforcases3and5areshownindetailfor in-cylinderpressureandtemperature.Forthemasstransferandzonevolumes,simulationresults 33 Figure2.4:Comparisonofsimulatedandexperimentalin-cylinderpressureswithrelativeerrorat 1500rpmwith6.78barIMEP(case3) ofcase3areused.Theresultsforall˝vecasesaresummarizedinTable2.6attheendofthis section. In-CylinderPressure: Theexperimentaldataof 100 cyclein-cylinderpressureareaveragedandusedtocomparewith thesimulatedone.Theexperimentalandsimulatedin-cylinderpressuresofcases3,4and5are comparedinFigures2.4,2.5and2.6,respectively. Thesimulatedcylinderpressuresforthethreecasesmatchtheexperimentaldataverywell. Therelativeerror( % )attheearlystageofcompressionphase(within50CADafterIVC)isa littlebithigh(around 7% ).However,weareinterestedintheerrorbetweenSOCandEVOforthe 34 Figure2.5:Comparisonofsimulatedandexperimentalin-cylinderpressuresat2000rpmwith 6.83barIMEP(case4) developedcombustionmodel.Inthesecases,themaximumrelativeerrorforcases3,4and5are 4 Ł 09% (occurredataround20CADaftertopdeadcenter(ATDC)), 5 Ł 3% (occurredataround0 CADATDC)and 5 Ł 19% (occurredataround24CADATDC),respectively. ZoneTemperature: Oneofthemajoradvantagesoftheproposedmodelisitscapabilityofpredictingtemperaturesin bothzones.Thesimulationresultsoftheunburnedzone( ) D )andreactionzone( ) A )temperatures areshowninFigure2.7forcase3.Theaveragedtemperatureovertwozonesisalsoplottedin thesame˝gureforcomparisonpurpose.Onecanseethatthetwozonetemperaturesarethe samebeforesparkevent,andafterthespark,combustionstartsandtheheatreleasefromchemical 35 Figure2.6:Comparisonofsimulatedandexperimentalin-cylinderpressuresat2000rpmwith 8.23barIMEP(case5) reactionincreasesthereactionzonetemperature ) A ,whileintheunburnedzone,thetemperature incrementismuchslower. Forthe˝rsthalfofcombustion(between20CADbeforetopdeadcenter(BTDC)and25CAD ATDC),over 60% ofthetotalmassintheunburnedzoneistransferredtothereactionzonewith veryfastcombustioninthereactionzone.Asaresult,thetemperaturedi˙erencebetween ) A and ) D increasesrapidlyandpeaksataround 15 CADATDC.Thechemicalenergyreleasedinthereaction zonegetstransferredtotheunburnedzoneandheatsupthemixtureintheunburnedzone,leading toamildincrementof ) D atthisstage.Ascombustioncontinues,forthesecond-halfofcombustion (between25CADand40CADATDC),over 95% ofthemassintheunburnedzone˛owstothe reactionzone,resultinginafastdecrementoftheunburnedzonesize.Thedecreasedunburned zonesizewiththecontinuedheattransferfromthereactionzoneincreasestheunburnedzone 36 Figure2.7:Zoneandaveragetemperaturesat1500rpmwith6.78barIMEP(case3) temperaturequicklyandreachesitspeakattheendofcombustion.Aftertheendofcombustion, twozonetemperaturesconverge,andthen,decreaseduetoexpansion. Toverifythemodelperformanceofpredictingin-cylindertemperature,thesimulatedaverage temperature ) 0E6 iscomparedwiththeexperimentaltemperaturecalculatedbasedonthemeasured in-cylinderpressure,theexperimentalin-cylindertemperature ) 4G? isderivedusingtheidealgas lawbasedonthecollectedexperimentalin-cylinderpressuredata,includingtherecordedin-cylinder pressure,volumeofthecombustionchamberandthetotalmassinthechamber.Thecomparison resultandtheassociatederrorareshowninFigure2.8.Itshowsthatthesimulatedaveragein- cylindertemperaturematcheswiththeexperimentaldataverywellwithamaximumrelativeerror of 4 Ł 8% betweenSOCandEVOoccurredat 45 CADATDC.Thesimulationresultsforcase4 areshowninFigure2.9,wherethemaximumrelativeerror,inthiscase,is 5 Ł 7% .Therefore,the capabilityofpredictingthemixturetemperatureandthermalstrati˝cationisdemonstrated. 37 Figure2.8:Comparisonofexperimentalandsimulatedcylindertemperaturesat1500rpmwith 6.78barIMEP(case3) MassTransfer: Themasstransferbetweenthereactionandunburnedzonesforcase3areshowninFigure2.10. Asdiscussedearlier,forthisstudy,theinitialunburnedzonemassis 98 Ł 5% oftotalmassandthe reactionzoneis 1 Ł 5% .Also,themasstransferdoesnotoccuruntilcombustionisinitiated. Themasstransferrate ¤ < CA inFigure2.10indicatesthatthemass˛owrateiszerobeforespark. Andthen,afterasmallignitiondelay,themasstransferstartsandincreasesrapidlytoprovidea properamountofpre-mixedmasstothereactionzonetoburn.Twomasscurvesofthereaction andunburnedzonesalsoindicatethatmassinbothzonesremainsunchangedbeforeandafter combustion.Duringthecombustion,theunburnedzonemassdecreasesveryquicklyduetomass ˛owintothereactionzone. Notethatthemasstransferratedoesnotdropdowntozeroattheendofcombustion.Asshown inthezoomedplotinFigure2.10,masstransferrategoestozeroveryslowlyinthecombustion 38 Figure2.9:Comparisonofexperimentalandin-cylinderaveragetemperaturesat2000rpmwith 6.83barIMEP(case4) phase,resultinginasmallmass˛owintothereactionzone.Inthiscase,only 97% fuelisburned. ZoneVolume: Thevolumevariationsinthetwozonesaresimilartomasscase(inFigure2.10)asthezonevolume isgovernedbytheidealgaslaw;seeFigure2.11(case3). Theunburnedzonevolumebeginsdecreasingafterthestartofcombustionandthereactionzone volumekeepsincreasing(withaninitialvalueof 1 Ł 5% )duetothemasstransferfromtheunburned zonetothereactionzone.ThereactionzonevolumeinFigure2.11alsoindicatesthatthevolume increasesveryquicklybetweenSOCandmiddleofcombustion,andafterthat,itincreasesslowly upto 96 Ł 9% oftotalvolumeattheendofcombustion.Duringtheengineexpansionprocess,the volumehasaslightincrementfrom 96 Ł 9% to 99 Ł 3% ,duetothesmallmass˛owintothereaction zone(seediscussionsinthelastsubsection). 39 Figure2.10:Masstransferanditsratebetweentwozonesat1500rpmwith6.78barIMEP(case3) 2.5.2CombustionSimulationResults Thecombustioncharacteristicsofeachchemicalspeciesinbothzonescanalsobesimulatedinthis model.Thesimulationresultsforcase3areusedasanexampletodiscussthemodel'scapability ofpredictingthecombustionprocess.Tocomparethecombustionsimulationresults,thereaction rate l 24< andmass˛ow-inrate l 5;>F offueland O 2 inthereactionzoneareshowninFigure 2.12. AsdiscussedearlierintheReaction-BasedCombustionModelsection,thechemicalreactionis modeledusingatwo-stepreactionmechanismand CO istheproductsfromthe˝rstreactionstep. Inthesecondstep, CO getsburnedtoproduce CO 2 ,and CO 2 splitsinto CO and 0 Ł 5O 2 .Asshown inFigure2.12,thereactionrate(in[kmol/m 3 s])offueland O 2 isnegativeduringthecombustion phase,indicatingthatthefueland O 2 arereacted.Thereactionratesofthesetwospeciesarezero beforetheSOC,andthen,increaseandreachthe˝rstpeak,indicatingthattheinitialmassinthe 40 Figure2.11:Individualvolumefractionsat1500rpmwith6.78barIMEP(case3) reactionzoneisburnedduetotheaddedsparkenergy.Thesetwopeaksaremainlycausedbythe extremelysmallinitialreactionzonevolume( 1 Ł 5% oftotalvolume)atthestartofcombustionand theassumedauto-ignitionintheignitionzone. Afterthecombustionstarts,thegasmixturestarts˛owingintothereactionzoneandgetsburned continuously;seethetwosmoothpositivemass˛ow(in[kmol/m 3 s])curvesoffueland O 2 in Figure2.12.Thereactionratesof CO , CO 2 ,and H 2 O changesinasimilarway;seeFigure2.13 forthetotalmassofthesespecies. Thetotalmassvariationtrendofeachindividualchemicalspeciesintheentirecylinderisa goodindicatorfortheactualcombustionprocess.Since N 2 takesnopartinthechemicalreaction, itsmasskeepsunchanged.Themassvariationsofother˝vechemicalspecies(fuel, O 2 , CO 2 , H 2 O ,and CO )duringthecombustionprocessareshowninFigure2.13.Itindicatesthatthetotal fuelmassinthecylinderremainsconstantuntilSOCanddropsquicklydownclosetozeroatthe endofcombustion(around38CADATDC),indicatingthatthefuelisalmostfullyburned.The 41 Figure2.12:Comparisonofreactionratesandmass˛owratesoffueland O 2 inthereactionzone at1500rpmwith6.78barIMEP(case3) corresponding O 2 massisalsochangedinasimilarwayandreducesveryfast,however,thereisa smallamountof O 2 leftattheendofcombustionsincetheactual _ was 1 Ł 05 . Thetotalmassofspecies CO increasesatthebeginningofthecombustionandgetsburnedin thesecondstep.ThereisasmallpeakbeforeTDC,whichiscausedbythespark.Considering theproductsofthetwo-stepchemicalreaction( CO 2 and H 2 O ), H 2 O iszerobeforeSOCand thenincreasesrapidlyduringthecombustionprocess,remainsunchangedaftercombustion,but themassof CO 2 hasaslightincrementafterthecombustion.Byinspectingthe CO curveafter combustion,thetotal CO massdropsdowntoverylow(nearzero)level,andthen,getsburnedby reactingwiththeremaining O 2 slowlyduringtheexpansionphase,resultinginaslightincrement of CO 2 duringtheexpansionphase. ThepredictedMFBofcase4isshowninFigure2.14.Theslopofthiscurveshowsthatthe burnrateisveryfastfromSOCtoaround20CADATDC,where50 % oftotalfuelisburned.After 42 Figure2.13:Totalmasschangingofspecies(fuel, O 2 , CO 2 , H 2 O and CO )at1500rpmwith 6.78barIMEP(case3) that,theburnrateslowsdownuntiltheendofcombustion. Theheatreleaserate,alongwithheattransferrelatedtothemasstransfer( ¤ & < )andheattransfer totheunburnedzone( ¤ & C )isshowninFigure2.15.Comparingthe ¤ & < and ¤ & C indicatesthatmost ofthetotalheattransfer( ¤ & < )isusedtotransferthemassfromtheunburnedzonetothereaction zoneduringthe˝rsthalfofthecombustionprocess,wheresigni˝cantpartofmassisprovidedto thereactionzone.Afterthat,duringthelatephaseofcombustion,theheattransfertotheunburned zone ¤ & C takesaleadingrolewhile ¤ & < reducesquickly.Thisisreasonablesincelargeamountof chemicalenergyreleasedduringcombustionistransferredtotheunburnedzone,whichisoneof themainreasonsfortherapidincrementofunburnedzonetemperature. ThesimulationresultsfortheheatlossareshowninFigure2.16,wherethered-lineisfor theheatlossfromthereactionzonetothecylinderheadandliner( ¤ & F 1 ),theblue-lineisforthe heatlossfromtheunburnedzonetotherestsurfaceareaofthecylinderboundary( ¤ & F 2 ),andthe 43 Figure2.14:PredictedMFBrateat2000rpmwith6.83barIMEP(case4) purple-lineisforthetotalheatlossfromthecombustionchambertothecylinderliner. ¤ & F 1 isalmostzerobeforetheignitionbecausethereactionzonevolumeisalmostzerocompared withthatoftheunburnedzone.Afterthestartofcombustion,thereleasedchemicalenergy increasesthereactionzonetemperaturequicklyandthereactionzoneexpandsalongwiththe ˛amepropagation,resultingintherapidincrementofheatlossfromthereactionzonetothe cylinderboundary.Physically,theheattransferareafromthereaction(burned)zonetothecylinder boundary,afterthereaction(burned)zonegetslargeenough,isthecylinderheadsurfaceplus cylinderlinerarea;seetheleftdrawingofFigure2.1.However,inthispaper,thecombustion chamberisassumedtohaveahemisphereshapeasshownintherightdrawingofFigure2.1,and itisalsoassumedthattheheattransferfromtheburnedzonetothewallisthebase(top)areaof theburnedzone.Withthesetwoassumptions,thebaseareaisnotlimitedbytheareaofcylinder head.Whenthecalculatedareaislargerthanthecylinderheadarea,theextraareaisconsideredas thecylinderwallarea. 44 Figure2.15:Heatreleaserateandheattransferratesbetweentwozonesat1500rpmwith6.78bar IMEP(case3) Theheatlossfromtheunburnedzonetothecylinderboundary ¤ & F 2 ,comparingwith ¤ & F 1 , hasanobviousdelayandincreasesquicklyduringthelatecombustionphaseandtheexpansion phaseduetothehightemperatureofunburnedzonemixture.Sincetheunburnedzonebecomesa verythinlayerbytheendofcombustion,theheatlossfromtheunburnedzonetothewallisquite small,comparingwiththeheatlossfromreactionzonetothewall.Notethatthetotalheatlossin Figure2.16matchestheseinliterature[66,67,68]. Thecalculatedindicatedmeane˙ectivepressure(IMEP)andcrankangle,where 50% offuel wasburned(CA50)ofall˝vecasesarecomparedwithexperimentaldatainFigures2.17and2.18. Asummaryofmodelpredictionerrorforthein-cylinderpressureisshowninTable2.6,where theerrorsshowncoverpressuresignalsbetweenSOCandEVO.NotethattheerrorbetweenIVC andSOCisnotthemainfocusofthispaper. 45 Figure2.16:Heatlossesfromreactionandunburnedzonestothecylinderboundaryat1500rpm with6.78barIMEP(case3) Table2.6:Modelingerrorofin-cylinderpressureforall˝vecasesbetweenSOCandEVO CaseMax.relativeerrorRMSE[bar] 16.20 % 0.26 23.40 % 0.24 34.09 % 0.28 45.50 % 0.23 55.19 % 0.37 2.6Summary Acontrol-oriented0-Dreaction-basedtwo-zonethermodynamicmodelforcompression,com- bustionandexpansionphasesofspark-ignitionengineshasbeenanalyticallydeveloped,imple- mentedinMatlab/Simulink,calibratedandvalidatedagainstexperimentaldatainthischapter.The proposedmodelwillbemainlyusedforpredictingthermodynamiccharacteristicsofin-cylinder 46 Figure2.17:ComparisonofsimulatedandexperimentalIMEPsforall˝vecases. mixtures,combustionprocess,propertiesofeachindividualchemicalspeciesinbothreactionand unburnedzones.Theexperimentalvalidationshowsthattheproposedtwo-zonemodelstructure isabletopredictthein-cylinderSIcombustionprocessaccuratelywithasimplecalibration.Es- pecially,itisabletosimulatethein-cylinderthermalstrati˝cationcharacteristics.Thiscon˝rms thattheproposedtwo-zonemodelstructurewithatwo-stepchemicalreactionmechanismcanbe usedformodel-basedcombustioncontrol;andfurthermore,theapplicationoftwo-stepchemical reactionmechanismtotheproposedmodelmakesitpossibletotraceindividualspeciesstateand topredictthemass-fraction-burned,heatreleaserate,andin-cylinderpressureetc.inreal-time. Themaximumrelativeerrorforthein-cylinderpressureislessthan6.2%under˝veoperational conditionsstudiedwithonesetofcalibrationparameters.Therefore,thee˙ectofSIenginecon- trolinputstoin-cylindercombustionprocesscanbepredictedandcanbeusedformodel-based combustionandknockcontrol. 47 Figure2.18:ComparisonofsimulatedandexperimentalCA50forall˝vecases. 48 CHAPTER3 REAL-TIMEPRESSUREWAVEMODELINGFORKNOCKPREDICTIONAND CONTROL Thein-cylinderpressureunderknockingcombustioncanbede˝nedas ? 2H; = ? + ? (3.1) where ? 2H; isthein-cylinderpressure; ? istheaveragein-cylinderpressurewithoutknockandit canbepredictedbythereaction-based0-Dtwo-zonecombustionmodeldevelopedinChapter2; ? isthepressurewavegeneratedbytheshockwavesduetoknockingcombustionanditiszero beforetheknockonset.Aknockpressurewavemodelwillbedevelopedinthischaptertopredict ? .Combinedwiththecombustionmodel,thein-cylinderpressureunderknockingcombustion canbepredictedandanalyzedforthestudyofknockcharacteristics;seeFigure3.1. Forthefurtherdevelopmentofthepressurewavemodel,theinitialconditionsatknockonset, suchasthein-cylinderpressure,volumesandtheirratesofthetwozones,arerequiredandthey canbeobtainedbasedonthetwo-zonereaction-basedcombustionmodel.AsdiscussedinChapter 2,theaveragein-cylinderpressurewithoutknockingcombustion( ? )isobtainedbasedontheideal gaslawshownin(2.35),andvolumesandtheirratesofthetwozonescanbecalculatedbasedon thefollowingtwoequations. + D = < D ' D ) D ?", D Œ ¤ + D = 3+ D 3C (3.2) + A = + 2H; + D Œ ¤ + A = ¤ + 2H; ¤ + D (3.3) where < D , ) D ,and ", D aremass,temperatureandaveragemolecularweightoftheunburnedzone mixture,respectively; + A and + 2H; arethevolumeofthereactionzoneandthecylinder,respectively. Therelationbetweenthetwo-zonereaction-basedcombustionmodelandtheproposedpressure wavemodelisshowninFigure3.2. 49 Figure3.1:0-Dreaction-basedtwo-zonecombustionmodelandthepressurewavemodel 3.1DerivationofPressureWaveModel Thegeneral3-Dwaveequationwas˝rstappliedtotheinternalcombustionenginebyDraper [51]tostudythein-cylinderpressureoscillations.Andthepressurewaveisaphysical˛uidmotion thatcanbedescribedbythepressurewaveequationinrectangularcoordinates;see(3.4)below. m 2 ? mG 2 + m 2 ? mH 2 + m 2 ? mI 2 = 1 2 2 m 2 ? mC 2 (3.4) where ? isthein-cylinderpressureperturbations;and 2 isthesoundofspeed.Assumingthat mixtureintheunburnedzoneisidealandadiabaticgas,thesmall-amplitudesoundspeedtheory [69]canbeappliedand 2 isaconstant,leadingtotheadiabaticsound 2 = s W ? 0 d 0 (3.5) where W istheheatcapacityratio,assumedtobeaconstantfortheidealgas.Notethatthezero subscriptindicatesthatthepressureanddensityin(3.5)aretakenattheequilibriumconditions. 50 Figure3.2:Connectionbetweenthetwo-zonereaction-basedcombustionmodelandpressure wavemodel This3-Dwaveequation(3.4)canbewritteninthecylindricalcoordinatestodescribethe pressureoscillationsconveniently;see(3.6)below. m 2 ? mA 2 + 1 A m ? mA + 1 A 2 m 2 ? m\ 2 + m 2 ? mI 2 = 1 2 2 m 2 ? mC 2 (3.6) Draper[51]gaveageneralsolutionofthispressurewaveequationtostudythepressure-wave frequenciesandvibrationmodes.Butthemagnitudedecayofthein-cylinderpressurewave,a veryimportantknockproperty,wasnotconsidered.Tostudytheactualpressurewavesunder knockingcombustion,Gaetaect.[53]proposedaDampedWaveEquation(DWE)byintroducing 51 atime-dependentdissipationterminDraper'swaveequation;see(3.7)below. m 2 ? mA 2 + 1 A m ? mA + 1 A 2 m 2 ? m\ 2 + m 2 ? mI 2 = 1 2 2 m 2 ? mC 2 + f m ? mC ! (3.7) where f m ? mC isthedampingtermusedtodescribethepressureoscillationsdeadeningbehaviordue totheprogressiveenergylossofthein-cylinderpressurewavecausedbytheheattransfer,friction andpistonmovement.And f isthedampingcoe˚cienttobecalibrated. Thischapterfocusesondevelopingacontrol-orientedreal-timeknockpressurewavemodel capableofpredictingthemaincharacteristicsofengineknock,suchastheknockonsettiming, intensity,andfrequency.Asimpli˝edpressurewaveequationisproposed,assumingthatthe pressurewaveisuniforminthecircumferentialandaxialdirections.Theassumptionofuniformity incircumferentialdirectionisduetothepre-assumedknockonsetlocationisat60%ofthe unburnedzoneradius.Therefore,thein-cylinderpressureperturbation ? becomesafunctionof radiuslocation A andtime C .Thatis ? = ? ( AŒC ) ;see(3.8)below. m 2 ? mA 2 + 1 A m ? mA = 1 2 2 m 2 ? mC 2 + f m ? mC ! (3.8) Assumingthat C and A in(3.8)areindependent,thewaveequationcanbesolvedusingseparating variablemethodandFourierseries.Forthemethodofseparatingvariables,thesolutionofwave equation(3.8)canbewrittenintheformbelow ? ( AŒC )= ' ( A ) ) ( C ) (3.9) Thatisaproductoftwofunctions, ' ( A ) and ) ( C ) ,andeachdependingononlyoneofthe twovariables A and C .Di˙erentiatingequation(3.8)yieldsthefollowingtwoordinarydi˙erential equations(ODEs) ) 00 + f) 0 + _ 2 ) =0 (3.10) and ' 00 + 1 A ' 0 + _ 2 2 2 ' =0 (3.11) 52 where _ 2 istheseparationconstant.Solving(3.10)and(3.11)leadstothegeneralsolutionsof ) ( C ) and ' ( A ) below. ) ( C )= 4 f 2 C » cos lC + sin lC ¼ (3.12) where and areconstantsdeterminedbytheinitialandboundaryconditions; l isthevibration frequency: l = _ r 1 ( f 2 _ ) 2 (3.13) Equation(3.11)canbereducedtoBessel'sequationbysetting : = _€2 and B = :A .Then,the derivativeof ' canbeobtainedbythechainrule ' 0 = : 3' 3B 0=3' 00 = : 2 3 2 ' 3B 2 (3.14) Substituting(3.14)into(3.11)yieldsthefollowingBessel'sequationwithorder a =0 below. 3 2 ' 3B 2 + 1 B 3' 3B + ' =0 (3.15) Solutionsofequation(3.15)aretheBesselfunction ˜ 0 and . 0 ofthe˝rstandsecondkind.But . 0 becomesin˝niteat0andcannotbekept.Therefore,thesolutionof(3.15)is ' ( A )= ˜ 0 ( B )= ˜ 0 ( :A ) , where ˜ 0 istheBesselfunctionofthe˝rstkindwithorderof0,whichwillbediscussedindetail inthenextsection. 3.2BoundaryandInitialConditions Engineknockistheauto-ignitionofunburnedmixtureinthecombustionchamber,sotheinitial conditionsfortheknockeventcanbegroupedintotwoparts:beforeandatknockonsetthepressure wavemagnitudeiszeroandatthemomentrightafterknockonsetthepressurewaverateisnon-zero intheunburnedmixtureandzerointheburnedmixture;see(3.16)and(3.17). ? ( AŒC =0)=0 ŒA 2 [0 Œ' 2 ] (3.16) and m ? mC AŒC =0 = 8 > > > >< > > > > : 0 inthereactionzone 5 ( A ) intheunburnedzone (3.17) 53 where ' 2 istheradiusoftheenginecylinder;and 5 ( A ) isthepressurewaverateatknockonset. Fortheboundaryconditions,itisassumedthatthecylinderwallisarigidbody,sothepressure oscillationrateatthewalliszero;see(3.18). m ? mA A = ' 2 ŒC =0 forall C 0 (3.18) 3.2.1Bessel'sEquation Thesolution ˜ 0 toBessel'sequation(3.15)hasin˝nitenumberofrootsdenotedby U 0 Œ< ( < = 1 Œ 2 ŒŁŁŁ ) ,anditsderivative ¤ ˜ 0 alsohasin˝nitenumberofrootsdenotedby V 0 Œ< ( < =1 Œ 2 ŒŁŁŁ ) .The rootsofBesselfunctionsandtheirderivativesfororders0and1areshowninTable3.1.Itindicates thattherootsarenotregularlyspacedontheaxis.Notethat < isthecorrespondingvibration normalmodeandtherootsinthethirdandfourthcolumnswillbeusedinthispaper. Table3.1:RootsofBesselfunctionsandthederivativefororder a =0 and 1 [1] <˜ 0 ( U 0 Œ< ) ˜ 1 ( U 1 Œ< ) ¤ ˜ 0 ( V 0 Œ< ) ¤ ˜ 1 ( V 1 Œ< ) 1 2.40483.83173.83171.8412 2 5.52017.01567.01565.3314 3 8.653710.173510.17358.5363 4 11.791513.323713.323711.7060 5 14.930916.470616.470614.8636 Tosatisfytheboundarycondition(3.18),thepressurewaveequation(3.9)canbederivedas m ? mA A = ' 2 ŒC = ) ( C ) ¤ ˜ 0 ( :A ) A = ' 2 =0 Ł (3.19) Thetimefunction ) ( C ) isnon-zero,andtherefore,thederivativeofBesselfunctionis ¤ ˜ 0 ( :' 2 )=0 (3.20) Sincethederivative ¤ ˜ 0 ofBesselfunctionwithorder a =0 hasin˝nitenumberofpositiveroots (Table3.1),theboundaryconditionfor(3.17)canbesatis˝edand(3.20)leadsto :' 2 = V 0 Œ< thus : = : < = V 0 Œ< ' 2 Œ< =1 Œ 2 ŒŁŁŁ (3.21) 54 whereroots( V 0 Œ< )aregiveninTable3.1. Therefore,thesolutionofODE(3.11)satisfyingtheboundarycondition(3.18)is ' ( A )= ˜ 0 ( : < A )= ˜ 0 V 0 Œ< ' 2 A Œ< =1 Œ 2 ŒŁŁŁ (3.22) Notethatalthoughtheknockonsetlocationisassumedat60%ofunburnedzoneradius,the solution ' ( A ) variesasafunctionbasedonradius A duetotheexpansionofthereactionzoneduring combustion. 3.2.2TimeFunction Bysatisfyingthe˝rstinitialcondition(3.16),thetimefunction ) ( C ) provideinitialcondition ) ( C =0)= =0 .Thus,thegeneralsolutionofthetimefunction ) ( C ) is ) ( C )= 4 f 2 C sin lC (3.23) where isaconstantand 6 =0 ; f isthedampingcoe˚cienttobecalibrated;andthewave frequencyis l€ (2 c ) Hz,where l iscomputedby l = l < = _ < s 1 f 2 _ < 2 Œ_ < = : < 2 (3.24) Notethattheadiabaticsoundspeed 2 ,whichisaconstant,canbefurtherderivedbasedonthe small-amplitudeofsoundspeedtheoryandidealgaslaw 2 = s W ? 0 d 0 = r W ' D ", D ) (3.25) where ' D istheuniversalgasconstant; ", D istheaveragemolarweightofunburnedzone mixtures;and ) istheaveragetemperatureofend-gasattheequilibriumcondition.Thesound ofspeedisusedforcalculatingthepressurewavefrequency.Itisfoundexperimentallythatthe in˛uenceoftemperature ) totheknockfrequencyissmallenoughandcanbeneglected,comparing withthee˙ectofenginegeometry.Furthermore,theknockingcombustionoccursrapidlyandthe temperaturechangecanalsobeneglected.Therefore, ) isassumedtobeconstant(constantsound ofspeed)andbecomesacalibrationparameterbasedontheexperimentalknockfrequency. 55 Thedampingcoe˚cient f isaboundedconstant.Sinceparameter l in(3.24)isde˝nedasa positiverealnumber,thefollowingrootsolutionshouldbepositive 1 f 2 _ < 2 ¡ 0 (3.26) andtherefore, 0 f 2 _ < =2 : < 2 = 2 2 ' 2 V 0 Œ< 2 2 ' 2 min V 0 Œ< (3.27) BasedonthecharacteristicsofBesselfunction(min V 0 Œ< = V 0 Œ 1 =3 Ł 8317 ),dampingcoe˚cient f isup-boundedby f 2 2V 0 Œ 1 €' 2 .Then,thecorrespondinggeneralsolutionofODE(3.10)is ) < ( C )= < 4 f 2 C sin l < C (3.28) Hence,thegeneralsolutionofthepressurewaveequation(3.8)satisfyingtheboundarycondition (3.18)andthe˝rstinitialcondition(3.16)is ? ( AŒC )= 1 X < =1 ' < ( A ) ) < ( C ) = 1 X < =1 < 4 f 2 C sin ¹ l < C º ˜ 0 V 0 Œ< ' 2 A (3.29) 3.2.3Coe˚cient < Knockisanauto-ignitionoftheunburnedend-gasinthecombustionchamber,sothepressure waveequationforknockpredictionisfortheunburnedzone.Therefore,the˝rstinitialcondition in(3.17)regardingpressurewavesinthereactionzoneisignored.Asaresult,coe˚cient < can bedeterminedbyusingthesecondinitialconditionin(3.17).Di˙erentiating(3.29)withrespectto C andusinginitialcondition(3.17)leadto m ? mC AŒC =0 = 1 X < =1 < l < ˜ 0 V 0 Œ< ' 2 A = 5 ( A ) (3.30) ThisistheFourier-Besselseriesrepresenting 5 ( A ) intermsof ˜ 0 ( V < A€' 2 ) .Andthecorresponding coe˚cients < canbedeterminedas < = 2 l < ' 2 2 ˜ 1 2 ( V 0 Œ< ) Z ' 2 0 A5 ( A ) ˜ 0 V 0 Œ< ' 2 A 3A (3.31) 56 where ˜ 1 ( V < ) istheBesselfunctionofthe˝rstkindwithorder a =1 .Sincethesquareofnormof theBesselfunction ˜ 0 is ˜ 0 V 0 Œ< ' 2 A 2 = Z ' 2 0 A˜ 2 0 V 0 Œ< ' 2 A 3A = ' 2 2 2 ˜ 1 2 V 0 Œ< (3.32) Therefore,term ˜ 2 1 ( V 0 Œ< ) canbeobtainedby ˜ 2 1 ( V 0 Œ< )= 2 ' 2 2 Z ' 2 0 A˜ 2 0 V 0 Œ< ' 2 A 3A (3.33) Then,substitutingequation(3.33)into(3.31),thecoe˚cient < canbewrittenas < = Z ' 2 0 A5 ( A ) ˜ 0 V 0 Œ< ' 2 A 3A l < Z ' 2 0 A˜ 2 0 V 0 Œ< ' 2 A 3A (3.34) where A isintheintervalof [0 Œ' 2 ] . 3.2.4PressureRateatKnockOnset Themassconservationequationforthein-cylinder˛ow˝eldcanbewrittenas m ? mC + 2 2 r ( d u )=0 (3.35) where d isthemassdensity; u isthevelocityvector;and ? isthein-cylinderpressureperturbations underknockcombustion.Usingthisequationfortheunburnedzonewhereknockoccursand calculatingtheintegraloverthetotalunburnedzonevolumeyield, Z + D m ? mC 3+ D + Z + D 2 2 r ( d u ) 3+ D =0 (3.36) anditcanbefurthersimpli˝edto m ? mC = 2 2 d ¤ + D + D (3.37) Combiningequations(3.25)and(3.37),thein-cylinderpressurerateatknockonset,denoted by 5 ( A ) ,canbefound;seebelow. 57 m ? mC AŒC =0 = 5 ( A )= W ? 0 1 + DŒ 0 d + AŒ 0 d C (3.38) where W istheheatcapacityratio; ? 0 istheaveragein-cylinderpressureatknockonset; + DŒ 0 and + AŒ 0 arethevolumeoftheunburnedandreactionzoneatknockonset,respectively.Allthese variablescanbeobtainedbythetwo-zonecombustionmodeldevelopedinChapter2. 3.3SolutionofPressureWaveEquationforKnockPrediction Basedontheseresultsfromtheprevioussection,thegeneralsolutionofthepressurewave equationsatisfyingtheboundaryandinitialconditionsisinaseriesform;see(3.29),andeachterm intheequationcanbecalculated,where < isthevibrationmodeofthein-cylinderpressurewave. Therefore,equation(3.29)isageneralsolutioncoveringallvibrationmodesfrom < =1 to 1 .The detailsfordi˙erentvibrationmodescanbefoundinreference[70]. Inthisdissertation,the˝rstvibrationmode( < =1 )isusedformodelingtheknockpressure wave,andtheassociatedsolutionis ? ( AŒC )= 1 4 f 2 C sin ¹ l 1 C º ˜ 0 V 0 Œ 1 ' 2 A (3.39) wherecoe˚cient 1 is 1 = W ? 0 ¤ + AŒ 0 Z ' 2 0 A˜ 0 V 0 Œ 1 ' 2 A 3A l 1 + DŒ 0 Z ' 2 0 A˜ 2 0 V 0 Œ 1 ' 2 A 3A (3.40) Notethat ¤ + AŒ 0 = 3+ AŒ 0 €3C ; V 1 isthe˝rst(minimum)rootofthederivativeofBesselfunction ˜ 0 ; and l 1 isthedampedfrequency.Combiningequations(3.21)and(3.24), l 1 canbefound;see below. l 1 = s V 0 Œ 1 2 2 2 ' 2 2 f 4 (3.41) Therefore,thein-cylinderpressure ? 2H; canbeobtained,basedonequation(3.1).Itcanbefurther analyzedtopredicttheknockcharacteristics,suchastheknockonsettiming,frequency,intensity andeventhecycle-to-cyclevariability. 58 3.4EvaluationMethodsofKnockPhenomenon 3.4.1KnockOnset TheLivengood-Wucorrelation[36]waswidelyusedasanempiricalmethodtopredicttheknock timingofspark-ignitionengines.Andbasedontheproposedreaction-basedcombustionmodel,a chemicalkinetic-basedmethodbasedontheArrheniusintegral(ARI)isdevelopedfromLivengood- Wucorrelationandusedinthispapertopredictthestartofcombustion(SOC)intheunburned zone,whereARIisde˝nedbelow. ARI= Z \ 8 \ ˚+˘ :=: ? 2H; 0 :=: [C 8 H 18 ] 1 :=: [O 2 ] 2 :=: 4 ) 0Œ:=: ) D ( \ ) d \ (3.42) where :=: , 0 :=: , 1 :=: , 2 :=: areauto-ignitioncoe˚cientstobecalibratedbasedonexperimental dataunderknockingcombustion; [C 8 H 18 ] and [O 2 ] arethemolarconcentrationoffuelandoxygen, respectively;and ) D istheunburnedzonetemperature.Notethatthemolarconcentrationandzone temperaturesareprovidedbythereaction-basedtwo-zonecombustionmodeldevelopedearlier; ) 0Œ:=: istheactivationtemperaturede˝nedby ) 0Œ:=: = ˆ 0Œ:=: ' D (3.43) ˆ 0Œ:=: (J/mol)in(3.43)istheactivationenergyofthechemicalreactionintheunburnedzone;and ' D istheuniversalgasconstant.Notethat ˆ 0Œ:=: isaconstanttobecalibrated. Equation(3.42)canbeinterpretedasanintegralofchemicalreactionrateoftheunburned mixture(end-gas);andthisequationalsoindicatesthattheARIispositiveandtheintegralis increasingmonotonically.Thecriterionforauto-ignitionisatthecrankanglewhenARIreaches one,thatis '˚ =1 Ł (3.44) 59 3.4.2KnockIntensity Engineknockpropertiesaregenerallycharacterizedintermsofknockonsettiming,knockintensity andfrequencybasedonin-cylinderpressurewaves.Inreality,theknockonsettimingisthemost importantfactorandisinvestigatedbymanyliterature.Theknockintensityandfrequencyare theothertwoimportantfactorstodescribetheknockseverityandphysicalcharacteristics.There arethreewidelyusedmeasuresforknockintensityandtheyareMaximumAmplitudeofPressure Oscillations(MAPO),IntegralofModulusofPressureGradient(IMPG),andIntegralofModulus ofPressureOscillation(IMPO),wheretheMAPOmethodisusedintherestofpapertorepresent theknockintensityduetoitssimpleness;seebelow. MAPO = 1 # # X 1 max \ 0 ; \ 0 + Z j ? j (3.45) where ? isthepressureperturbations; # isthenumberofpressurewavepeaks; \ 0 isthecrank angleofknockonset; Z istheknockwindowlength,anditisde˝nedtobe20CADfromknock onset.SoMAPOcanbepresentedasKI20aswell. 3.5ModelCalibrationandValidation 3.5.1ExperimentSetupandModelCalibration Theexperimentaldatausedtocalibrateandvalidatetheproposedreaction-based0-Dpressurewave modeliscollectedfroma4-cylinder,four-strokeSIenginethroughdynamo-meterexperiments. TheengineparametersarelistedinTable2.1. Thetestdatafortwotypicalsteady-stateengineoperatingconditions(highloadat1500rpmand 2000rpm)areusedforvalidationandissummarizedinTable3.2,wheretherelativeair-fuelratiois controlledtobeclosetostoichiometric.Notethattheignitiontimingunderthesetwooperational conditionsarecontrolledtobeneartheengineknocklimit,makingitpossibletovalidatethe pressurewavemodel. AnA & DCAS(combustionanalysissystem)isusedtorecordthein-cylinderpressure,intake manifoldpressure,ignitioncoildwellcurrent,etc.ThecalibratedparametersarelistedinTable 60 Table3.2:Engineoperatingconditions(atknocklimit). Case12 IMEP[bar]7.58.23 EngineSpeed[rpm]15002000 3.3.ParameterslistedinTable3.3canbeclassi˝edintotwogroups.The˝rstgroupconsists of :=: Œ0 :=: Œ1 :=: Œ2 :=: and ˆ :=: ,whicharerelatedtotheknockonsetprediction;andthe secondgroupparameters, f and ) ,arerelatedtoknockfrequency.The˝rstgroupparametersare calibratedbasedontheknockonsettimingobtainedfromexperimentaldata,where 0 :=: ˘ ˆ 0Œ:=: arecoe˚cientscorrespondingtothechemicalreactionsofknockingcombustionandhavelow sensitivitytoenginegeometryandoperatingconditions,andtheyarecalibrated˝rst. :=: has ahighsensitivityandiscalibratedbykeepingtheotherfourparametersconstant. f isusedto describethepressurewavedeadeningbehavioranditisbounded;see(3.26)and(3.27),and ) istheaveragetemperatureofunburnedzonemixture,andwasassumedtobeconstant.Basedon (3.25), ) isusedtocalculatethesoundofspeedandthen,combiningwith(3.41),toobtainthe knockfrequency.Asasummary, f and ) arecalibratedbasedontheknockfrequencyobtained fromtheexperimentdata.Notethattheseparameterskeepunchangedunderdi˙erentoperating conditions,whichisoneofthemajoradvantagesoftheproposedpressurewavemodelandcanbe convenientlyusedforthemodel-basedknockcontrol. Table3.3:Calibratedparameters. :=: 0 :=: 1 :=: 2 :=: ˆ 0Œ:=: f ) 3e70.5e-100.251.51.8691(J/mol)0.0152080K 3.5.2KnockOnsetandIntensityPrediction Generally,theknockpressurewaveisdominatedbyits˝rstandsecondvibrationmodes,andthe ˝rstmodefrequencyisaround6kHzandthesecondmodefrequencyisaround12.5kHz.Inthe 61 Figure3.3:Experimentalin-cylinderpressureand˝lteredpressurewaveofthe˝rstknockcycleat 1500rpmwithIMEP=7.5bar(case1) simulationstudy,the˝rstmodefrequencyisinvestigatedinthispaper.Itiswell-knownthatthe in-cylinderpressuresignalcontainsrichinformationofkeyknockcharacteristics.Figure3.3shows theexperimentalin-cylinderpressureofthe˝rstknockcycleanditsbandpass-˝lteredpressure signalsat1500rpmwithIMEPof7.5bar.Theblacksolid-linepresentstheun˝lteredin-cylinder pressuresignalunderknockingcombustionandthereddash-dottedlinepresentsthepressurewave obtainedusingaband-pass˝lterof3 ˘ 10kHz(˝rstknockmode).Theknockwindowisbetween knockonsetto20CADafterknockonset,whichisusedtocalculateMAPO(KI20).Asaresult, knockonsettimingandintensitycanbeobtainedusingthein-cylinderpressuresignal,andthe knockfrequencycanbefoundbyprocessingthe˝lteredpressurewaveusingfastFouriertransform (FFT),whichwillbediscussedlater.Themodel'sabilityofpredictingtheknockonsettiming, knockintensityandfrequencywillbediscussedinnexttwosubsectionsbyanalyzingthesimulated in-cylinderpressures. 62 Thesecondpartformodelvalidationisregardingtheknockcycle-to-cyclevariability,which iskindofrandombutcouldberelatedtothemixturepropertiesatintakevalveclose(IVC).Fora ˝xedengineoperatingcondition,theknockintensitycouldchangecycle-by-cycleandvaryfrom lowtohigh.Althoughitisdi˚culttopredicttherandomknockintensityvariationaccurately,the correlationbetweenknockintensityandthein-cylindermixturepropertiesatIVCwillbestudied usingthesimulatedpressuresignalfromtheproposedmodelinthethirdsubsection. Inthissubsection,thecapabilityofestimatingknockonsetandintensityisdemonstratedunder twoengineoperatingconditions:case1(1500rpm,7.5bar,Figure3.3)andcase2(2000rpm,8.2 bar,Figure3.6),wherethe˝rstknockcyclepressuresignalsforbothcasesareused.Themain reasontousethe˝rstknockcycledataisthatthein-cylindermixturepropertiesatIVCarerelatively consistentforthe˝rstknockcycleandafterknockoccursthepropertiesatIVCarein˛uencedby thepreviousknockcycle. Tovalidatetheproposedmodel,˝rst,theexperimentalin-cylinderpressureofcase1isprocessed usingaband-pass˝lterof3 ˘ 10kHz,wherethe˝rstknockcycledataisshowninFigure3.3.The blacksolid-lineistherecordedin-cylinderpressureofthe˝rstknockcycle,andthereddash-dotted lineisthepressurewaveobtainedusingaband-pass˝lter.Figure3.3showsthattheknockonset timingis21.83CADATDCandtheknockintensityMAPOcalculatedbasedonthe˝lteredpressure wavewithintheknockwindowis0.7519bar. Forpredictingknockonsettimingbasedonthedevelopedmodel,Equations(3.42)and(3.44) areusedtosimulatetheunburnedzoneArrheniusintegral(ARI)andtheassociatedrateforcase1; seeFigure3.4.NotethattheupperplotinFigure3.4istheArrheniusintegralrateintheunburned zoneandtheloweroneisthecorrespondingARI.TherateofARIisameasurerepresentingthe knockseverity.Forinstance,asteepincrementofARIrateindicatesaheavyknockcycle.Based onearlydiscussion,ARIispositiveandmonotonicallyincreasing(seeEquation(3.42)).The calculatedARIcon˝rmsitandincreasesfrom0andreaches1(markedwithareddot)at21.3CAD ATDC,indicatingknockonsettimingof21.3CADATDC.Notethattheexperimentaldatashows aknockonsettimingof21.83CADATDC,andtherefore,thepredictingerrorisabout2.43%. 63 Figure3.4:SimulatedARIanditsrateintheunburnedzoneofthe˝rstknockcycleat1500rpm withIMEP=7.5bar(case1) Theinitialconditionsatknockonset,suchasthein-cylinderpressureanditsrate,theunburned zonevolumeandthevolumerateofthereactionzone,arerequiredtomodeltheknockpressure waves.Withthepredictedknockonsettiming,theinitialconditionscanbeobtainedviathe two-zonereaction-basedcombustionmodeldevelopedearlier;seeTable3.4. Table3.4:Initialconditionsatknockonsetforthe˝rstknockingcycleofcases1and2. ? 0 [ bar ] + D [ m 3 ] ¤ + A [ m 3 € s ] m? mC AŒC =0 [ bar € s ] case134.451.186e-50.037961.1696e5 case232.161.382e-50.036141.1791e5 Thesimulatedin-cylinderpressureandpressurewaveareshowninFigure3.5,wherethe 64 Figure3.5:Calculatedin-cylinderpressurewaveat1500rpmwithIMEP=7.5bar(case1) predictedin-cylinderpressureis˝lteredwitha3 ˘ 10kHzband-pass˝lter(sameoneusedfor processingtheexperimentaldata).Next,theknockintensityiscalculatedanditis0.7999barwith apredictingerrorof6.38%overtheexperimentaldatashowninFigure3.3. Similarly,theproposedmodelissimulatedat2000rpmusingtheboundaryconditionsshown inTable3.4.Case2experimentalin-cylinderpressureofthe˝rstknockcycleis˝lteredusingthe sameband-pass˝lterandtheresultsareshowninFigure3.6,indicatingaknockonsettimingof24 CADATDCandtheMAPOof1.173bar. Forpredictingtheknockonsettimingunderthisoperatingcondition,thesimulatedARIand itsrateintheunburnedzoneareshowninFigure3.7.TheARIincreasesfrom0andreaches1 at23.6CADATDC,indicatingtheknockonsettimingof23.6CADATDC.Comparingwiththe experimentalonsettimingof24CADATDC(seeFigure3.6),thepredictingerrorisonly1.67%. Thesimulatedin-cylinderpressureandpressurewaveobtainedusingthesameband-pass˝lter areshowninFigure3.8.ThecalculatedMAPOis1.1969barwithapredictingerrorof2.037%, 65 Figure3.6:Experimentalin-cylinderpressureand˝lteredpressurewaveofatypicalknockcycle at2000rpmwithIMEP=8.23bar(case2) comparingwiththeexperimentaldatainFigure3.6. Asasummary,themodel'scapabilityofpredictingtheknockonsettimingandintensityunder twodi˙erentengineoperatingconditionsaredemonstratedbycomparingwiththeexperiment results.Themaximumpredictingerrorforknockintensityis6.38%(case1)andmaximum predictionerrorforknockonsettimingis2.43%(case1). 3.5.3FFTAnalysisofIn-CylinderPressureWaves Theknockpressurewavefrequencyismainlydeterminedbyenginegeometryandremainsun- changedthroughoutthecombustionprocessunderdi˙erentoperatingconditions.Ingeneral,the fastFouriertransform(FFT)isawell-knownmethodforanalyzingtheknocksignalinfrequency domain.To˝ndtheknockfrequency,FFTisusedtoanalyzebothexperimentalandsimulated in-cylinderpressurewavesforcases1and2.TheFFTresultsofexperimentalsimulateddataof 66 Figure3.7:SimulatedARIanditsrateintheunburnedzoneforthe˝rstknockingcycleat2000 rpmwithIMEP=8.23bar(case2) cases1and2areshowninFigures3.9and3.10.Thesetwo˝guresindicatethattheexperimental knockfrequencyis6.303kHzandremainsunchangedunderdi˙erentengineoperatingconditions whilethepredictedknockfrequencyforbothcases(seeFigures3.9and3.10)are6.310kHz.Note thatthemagnitudeofthepressurewavesintwo˝guresaredi˙erentduetothedi˙erenceinknock intensity. Theseresultsmatchwiththeexperimentalones,andthepredictingerrorofknockfrequencyis lessthan0.2%,indicatingthattheproposedmodelisabletopredictknockfrequencyaccurately. 67 Figure3.8:Simulatedin-cylinderpressureandpressurewavewitha3 ˘ 10kHzband-pass˝lterat 2000rpmwithIMEP=8.23bar(case2) 3.5.4KnockCycle-to-CycleVariability Theengineknockphenomenonshowscycle-to-cyclevariabilityevenunderasteady-stateengine operatingcondition.Figure3.11showstheexperimentalin-cylinderpressuresfor6consecutive enginecyclesunderknockingcombustionwhentheengineisoperatedat1500rpmwithIMEPof 7.5bar(case1inTable2.2).Itisclearthattheknockintensitychangescycle-by-cycle,wherecycle # 1haslightknock,cycle #2 heavyknock,cycles #3 to #5 mediumknock,andcycle # 6normal combustion(withoutknock-baselinecycle),wherethebaselinecyclepressurehasasmallrateof riseaftertheignitionandthepeakpressureisaround27bar.Comparingwiththebaselinecycle, theknockcyclesshowsfasterpressurerateofriseandtheheavyknockcyclehasthemoststeep rateofrise.Thedi˙erenceofpeakin-cylinderpressurebetweenheavyknockandbased-linecycles canbeupto17bar. Theexperimentalin-cylinderpressuresunder2000rpmwith8.23barIMEP(case2)are 68 Figure3.9:FFTanalysisoftheexperimentalin-cylinderpressurewaveat1500rpm,IMEP=7.5 bar(case1) analyzedaswell.AsshowninFigure3.12,20consecutiveenginecyclesunderknockingcombustion conditionarebandpass-˝lteredandthecorrespondingMAPOsarecalculated.Theknockintensity plotshowninFigure3.12indicatescycle-to-cyclevariability.Althoughitiskindofrandombut witharepeatablepattern,whereahighknockintensitycycleisalwaysfollowedbyalowknock intensityoneformostofthecases.NotethatMAPOpatternrepeatsbutitsvalueisdi˙erent, indicatingthattheknockservilityvaries,andtheMAPOcandropdowntozero(noknock),after repeatingthepatternafewtimes;seecycles #14 and #19 .Furthermore,themeanvalueand standarddeviationofknockintensityforthese20consecutiveenginecyclesare1.0009barand 0.5713bar,respectively. Thecycle-to-cyclevariabilityofknockphenomenonmayberelatedtothein-cylindermixtures propertiesatIVC,whichisin˛uencedbythelastcombustioncycle.Therefore,thecorrelation betweencycle-to-cycleknockvariabilityandmixturepropertiesatIVCwillbestudiedinthis 69 Figure3.10:FFTanalysisoftheexperimentalin-cylinderpressurewaveat2000rpm,IMEP=8.23 bar(case2) sectionusingthecalibratedpressurewavemodel. Inthisstudy,theexternalEGRvalvewasclosedforenginetestsandhencethereisnoexternal EGR.Asaresult,themixtureatIVCconsistsofthepre-mixedfreshair,fuel,andthetrapped residualgasfromthelastcycle.Basedonourstudy,themixturesproperties,especiallythemixture temperatureatIVC( ) ˚+˘ ),hassigni˝cantin˛uencetothecombustioncharacteristics,heattransfer andgastemperatureatexhaustvalveopen(EVO).Foraknockcycle,theknockingcombustionin theunburnedzoneleadstohighpressurerateofriseandrapidheatlosstothewall.Asaresult, thenetheattransferrateintheunburnedzone ¤ & D willbesmallerthanthenormalcombustion cycle.BasedonEquation(2.33),theunburnedzonetemperatureatEVO( ) ˆ+$ )willdecrease, too.Therefore,themixturetemperatureatIVCforthefollowingcyclewilldecreaseduetothe reducedtemperatureofthetrappedresidualgas,whichfurthera˙ectsthenextcombustioncycle. Thiscouldbeoneofthereasonsforthecycle-to-cycleknockvariability. 70 Figure3.11:Experimentalin-cylinderpressureof6consecutivecyclesunderknocking combustion,engineoperatedat1500rpm,IMEP=7.5bar(case1) 71 Figure3.12:MAPOof20consecutiveenginecyclescalculatedbasedontheexperimental in-cylinderpressureat2000rpm,IMEP=8.23bar(case2).SD:Standarddeviation Toverifythisassumptionandstudythein˛uenceof ) ˚+˘ totheknockintensityand ) ˆ+$ ,6 combustioncyclesaresimulatedat2000rpmwith8.23barIMEPforagivensetofmonotonically decreasing ) ˚+˘ ,alongwiththeproposedpressurewavemodel,togeneratepressureandits wavesignals.TheassociatedresultsareshowninTable3.5withthehighest ) ˚+˘ of389K andlowest349K.TheexhausttemperateatEVO( ) ˆ+$ )ispredictedbythetwo-zonereaction- basedcombustionmodel(seeEquation(2.33))developedearlier.Thepredictedunburnedzone temperatureareshowninFigure3.13. Table3.5:Simulationresultsfor6enginecycleswithmonotonicallydecreasing ) ˚+˘ . Cycles ) ˚+˘ [K] ) ˆ+$ [K]MAPO[bar] #1 38915372.3759 #2 37615491.7679 #3 37015581.5773 #4 36615650.9912 #5 35815860.8211 #6 34916340 AsshowninTable3.5andFigure3.13,the˝rstcyclestartswiththehighest ) ˚+˘ of389K,the 72 Figure3.13:Reactionzonetemperature(6cycles)withmonotonicallydecreasing ) ˚+˘ at2000 rpm,IMEP=8.23bar(case2) combustionisveryfast,leadingtohighreactionrateandunburnedzonetemperature,resultingin fastheatlosstothecylinderwallandlowrestnetheattransferintheunburnedzone(small ¤ & D ). Basedonequation(2.33),theexhausttemperaturewilldecrease.Inaddition,knockingcombustion couldalsodestroythecylinder-walloil˝lmandleadlargerheattransfer(loss),comparingtothe normalcombustioncycle.Basedontheenergybalance,increasedenergy(heat)lostduringknock combustionresultsinreducedexhausttemperature.Therefore,the˝rstcyclehasthelowestexhaust temperature( ) ˆ+$ ),comparingwith5othercycles. Alittlebitlower ) ˚+˘ isgiventothenext(second)cycle,andthesimulationresultsindicate that ) ˆ+$ increasesby12Kandthecorrespondingknockintensity(MAPO)decreasesby25.5 % . ThistrendcontinuesasthetemperatureatIVCreduces.Therefore,thesimulationresultscon˝rm thehypothesisthat ) ˚+˘ ishighlycorrelatedtoknockintensity.Thatis,thehighthe ) ˚+˘ is,the 73 hightheknockintensity. Inaddition,fortheenginecyclesunderknockcombustionata˝xedoperatingcondition,the mixturetemperatureatIVCofcycle : +1 ( ) ˚+˘ ( : +1) )couldbein˛uencedbytheresidualgas (exhaust)temperatureofcycle : ( ) ˆ+$ ( : ) ).Generally,theknockvariabilitycouldbecausedby manyfactors,suchasthemixturepropertiesatIVC,hotspots,theconcentrationoffuel,airand residualgasthroughoutthechamber,locationsofauto-ignition.Amongthem, ) ˚+˘ isoneof themainfactors.Inordertostudythecycle-to-cycleknockvariabilitydueto ) ˚+˘ variation,the experimentalin-cylinderpressuresof50consecutiveenginecyclesandtheproposedpressurewave modelareusedtopredictthenextcycle ) ˚+˘ ( : +1) basedontheexhausttemperature ) ˆ+$ ( : ) ofcurrentcycle.Assumingthat ) ˚+˘ isonlya˙ectedbytheexhausttemperatureoftheprevious cycle,50enginecyclesaresimulatedtogeneratein-cylinderpressureanditsknockwave.Foreach cycle, ) ˚+˘ isoptimizedbyminimizingthepredictionerrorofin-cylinderpressure.Theresultsare showninFigure3.14,whereeachblue-squaremarkerrepresentsonecycleexperimentaldatapoint. theerrorbarpresentsthein˛uenceofotherfactors.Notethattherearemanyrepeatedtemperate points ) ˚+˘ ( : +1) andthenumberofmarkersinFigure3.14islessthan50. AsshowninFigure3.14,theexperimental ) ˚+˘ ( : +1) and ) ˆ+$ ( : ) showanobviouscorrelation, andthe˝ttedcurve(redsolid-line)canbeusedtopredictthe ) ˚+˘ ofnextcyclebasedontheexhaust temperatureofcurrentcycle.However,the˝ttedcurveonlyre˛ectsthein˛uenceof ) ˚+˘ dueto thelastcycleexhausttemperature.Inordertohaveanaccuratepredictionof ) ˚+˘ ,thein˛uence ofotherfactorsareincludedbyaddingarandomvariabletothis˝ttedcurve;seebelow. ) ˚+˘ ( : +1) = 5 25 ( ) ˆ+$ ( : ))+ k (3.46) where 5 25 isthe˝ttedcurve(redsolid-line)inFigure3.14,and k isazero-meanrandomvariable withnormaldistribution,thatis, k ˘N (0 Œ 30) . Therefore,thecorrelationbetweenknockintensityandmixturetemperatureatIVCisfurther studiedbasedon(3.46),wherethesimulatedcurrentcycleexhausttemperatureisusedto˝ndthe mixturetemperatureatIVCforthenextcycle.Forthesimulationstudy,themixturetemperature atIVCforthe˝rstcycleisassignedto348K(relativelow)andthepredictedexhausttemperature 74 Figure3.14:Curve˝ttingforpredictingthemixturetemperatureatIVCofnextcyclebasedonthe exhausttemperatureatcurrentcycle (usingtheproposedpressurewavemodel)is1634Kandnotethatthiscycleisnotknockingwith MAPOof0.Basedon(3.46),thepredicted ) ˚+˘ ofthesecondcycleis390.25K.Therefore, thesecondcyclesimulationisbasedon ) ˚+˘ ( : =2) of390.25K.Thisprocesscontinuesfor20 enginecycles;seeTable3.6forcalculated ) ˚+˘ andsimulatedMAPOresults.AsshowninTable 3.6andFigure3.15,itisobviousthattheheavyknockcycle(highknockintensity)isfollowed byalightknockcycle(lowknockintensity)formostcases.Thissimulationresultsmatchthe experimentalresults(showninFigure3.12)inthetermsofknockintensitymeanandstandard deviation(SD),wherethemeanandSDfromsimulationsare0.9741and0.7605,respectivelyand thesefromexperiments(showninFigure3.12)are1.0009and0.5713,respectively.Thecapability oftheproposedpressurewavemodelinpredictingcycle-to-cyclevariabilitybasedonin-cylinder mixturetemperatureatIVCisdemonstrated.Webelievethatthedi˙erentcouldbecausedbythe un-modeledmixturecharacteristicsatIVC. 75 Figure3.15:PredictedMAPOsof20consecutiveenginecycles Insummary,thesimulationresultswiththegivenmonotonicallydecreasing ) ˚+˘ con˝rms thatthehigh ) ˚+˘ is,thehighknockintensityis.Also,heavyknockingcombustionleadstolow exhausttemperature,andhence,lowmixturetemperatureatIVCforthenextcycle.Thesimulation resultsof20consecutiveenginecyclesshowthatcurrentcycle ) ˚+˘ isa˙ectedbythelastcycle knockintensityandvariationof ) ˚+˘ isoneofthemaincauseforcycle-to-cycleknockvariability ata˝xedengineoperatingcondition. 3.6Summary Acontrol-orientedknockpressurewavemodel,basedonoutputsofatwo-zonereaction-based combustionmodel,forspark-ignited(SI)enginesisdevelopedtopredictknockingcombustionin thisChapter.Thedevelopedmodeliscalibratedandvalidatedbyexperimentaldata.Thesimulation resultscon˝rmmodel'scapableofpredictingthekeyknockcharacteristicssuchasknockonset timing,knockfrequencyandintensity.Themaximumpredictionerrorundertwoengineoperating conditionsislessthan2.5%andthemaximumpredictionerrorforknockintensity(MAPO)is lessthan6.38%;theknockfrequencypredictionisevensmall(lessthan0.2%).Inaddition,the capabilityofpredictingthecycle-to-cycleknockvariabilityisstudiedbycorrelatingtheknock 76 Table3.6:Simulationresults(20consecutivecycles)usingpredicted ) ˚+˘ (case2) Cycles ) ˚+˘ [K] ) ˆ+$ [K]MAPO[bar] #1 34816340 #2 390.2515362.4512 #3 357.4915880.7986 #4 370.0915571.5429 #5 351.8615930.4073 #6 372.1915541.7591 #7 350.6416040.3985 #8 371.4915551.6881 #9 357.8715870.8043 #10 363.2015700.9509 #11 347.8716350 #12 387.6715402.3088 #13 355.8715910.7139 #14 360.1815760.8915 #15 356.9115900.7453 #16 351.5215940.4014 #17 368.4915621.3589 #18 347.5216350 #19 378.5115461.9567 #20 350.4716080.3052 intensitytothein-cylindermixturetemperatureatintakevalveclosing.Simulationresultscon˝rms thehypothesisthatthein-cylindermixturestemperatureatIVCisoneofthekeyfactorleading tocycle-to-cycleknockvariability,andtheproposedmodelisabletoreplicatethisphenomenon, whereahighknock-intensitycycleisoftenfollowedbyalowknock-intensitycycle. 77 CHAPTER4 MODEL-BASEDSTOCHASTICKNOCKLIMITCONTROL 4.1Reaction-BasedKnockPredictiveModel Theknockpredictivemodelusedforthemodel-basedstochasticknocklimitcontrolisconsisted withtworeal-timemodels:the0-Dreaction-basedtwo-zonecombustionmodelandthepressure wavemodel,thataredevelopedinChapters2and3,respectively.Thecorrelationsofthetwomodels andthemodel-basedknocklimitcontroldesignareshowninFigure4.1.Asdiscussedearlier,the reaction-basedcombustionmodelisdevelopedtopredictthephysicalcombustionprocessofSI engineinreal-time.Theinputsaretheinitialconditionsatintakevalveclose(IVC);seeFigure4.1. Importantly,thecombustionmodeliscapableofpredictingthepropertiesofspeciesinvolvedinthe chemicalreactionsandthethermodynamicconditionsinthecombustionchamber,whicharethe inputsofthepressurewavemodel.Thepressurewavemodelisdevelopedbasedonthesimpli˝ed pressurewaveequations,andthecorrespondingboundaryandinitialconditionstopredictthe in-cylinderpressureoscillationsduetotheshockwavesgeneratedinthechamberbytheknocking combustion.Basedonthesetwomodels,themajorcharacteristicsofknockingcombustion,such astheknockonsettiming,knockintensity,andcycle-to-cyclevariability,canbepredictedandused forthemodel-basedstochasticknocklimitcontroldesign,whichwillbediscussedinthisChapter. 4.2Model-BasedPredictionofKnockCyclicVariability 4.2.1KnockIntensity-KI20(MAPO) Knockintensityisthemostimportantcriteriontodescribetheengineknockphenomenon.The experimentalin-cylinderpressureofanenginecycleunderknockingcombustionisshowninFigure 4.2(theblacksolid-line).Inordertocalculatetheknockintensity,thisin-cylinderpressureis˝ltered witha6thorderbutterworthband-pass˝lter;seetheredsolid-lineinFigure4.2.Theamplitudesof 78 Figure4.1:Correlationdiagramofreaction-basedtwo-zonecombustionmodel,thepressurewave model,andmodel-basedstochasticknocklimitcontrol ˝lteredin-cylinderpressurewavekeepdecreasingduetotheknockintensitydecay.Topresentthe knockintensityappropriately,MAPO(KI20)thatistocalculatetheaverageamplitudeof˝lteredin- cylinderpressurewaveinapre-de˝nedknockwindowisusedinthisdissertation.Thecalculation formulaofMAPOisshownin(3.45).Notethattheknockwindowisde˝nedfromknockonsetto 20CADafteritinthisdissertation.Therefore,tobetterindicatetheknockintensitywiththe20 79 Figure4.2:Experimentalin-cylinderpressureandband-pass˝lteredpressurewaveofoneengine cycle CADknockwindow,theknockintensityinthisChapterispresentedasKI20insteadofMAPO. 4.2.2KnockPredictiveModelCalibrationandValidation Thereaction-basedcombustionmodelandpressurewavemodeldiscussedinChapters2and3 werecalibratedandvalidatedwiththeexperimentdataobtainedfromtheenginebenchshownin Table2.1.Theenginewasrunat6di˙erentconditions,asshowninTable4.1.Notethatcases 4and6wererunatknocklimitwhiletheothercasesaregeneralconditions.Thereaction-based combustionmodelwascalibratedandvalidated˝rstwiththeexperimentdatafromcases1,2,3 and5,andtheresultsdemonstratedgoodmodelaccuracy.Basedonthecombustionmodel,the pressurewavemodelwasfurthercalibratedandvalidatedwiththeexperimentdatafromcases4 and6.Themodelcalibrationcoe˚cientsandvalidationresultscanbefoundinChapters2and3. Theresultsindicatedthecapabilityofthepressurewavemodeltopredictthein-cylinderpressure 80 Table4.1:Engineoperatingconditions. Case 1234(knock)56(knock) IMEP[bar] 4.535.016.787.56.838.23 EngineSpeed[rpm] 110015001500150020002000 underknockingconditionwithhighaccuracyofknockonsettiming,frequencyandintensity,which arethreemajorcharacteristicsofengineknockduringonecycle.Andthisisthefoundationfor model-basedknockcontrol. 4.2.3IntakeTemperaturewithKnockCycle-to-CycleVariability Engineknocknotonlyhasthecharacteristicspresentingwithknockonsettiming,frequencyand intensitybutalsoshowsastrongcycle-to-cyclevariabilitythatmakesthecontrolofknockdi˚cult. Byanalyzingtheexperimentdataofin-cylinderpressuremeasuredatcases4and6showninTable 4.1,itturnsoutthattheknockintensitypresentedwithKI20hasastrongcycle-to-cyclevariability evenatasteady-stateengineoperatingcondition.Therearemanyfactorsresultintheknock cycle-to-cyclevariability,but ) 8E2 takesanimportantrole.Thehighintakemanifoldtemperature willleadtoaheavyknockingcombustion,withthehightemperatureinthechamberandheatloss tothewall,andledtothelowexhausttemperature.Lowexhausttemperaturewillconsequently reducetheintaketemperatureofnextcycleduetotheREGtrappedinthechamberandthenfurther in˛uencetheknockintensityofnextcycle.Theknockpredictionmodeldescribedinlastsection wasusedtostudythecorrelationof ) 8E2 andtheexhausttemperatureandknockintensityofeach cycle.Itwasveri˝edthattheexhausttemperatureofeachcyclehasastrongcorrelationwith theintakemanifoldtemperature ) 8E2 ofnextcycle.Andanexperimentdatabased˝ttingcurve hasbeenfoundtopredictthe ) 8E2 ofnextcyclewiththeestimatedexhausttemperatureofcurrent cycle,whichcanbeutilizedtostudythecorrelationofknockcycle-to-cyclevariabilityandintake temperature ) 8E2 ;seeFigure4.3. Thebluesquaresrepresenttheexperimentdatabasedexhausttemperature( ) 4E> )withthe 81 Figure4.3:Fittedcorrelationcurvebetweencurrentcycle ) 4E> ( : ) andnextcycle ) 8E2 ( : +1) corresponding ) 8E2 ofnextcycle.Theredsolid-lineisthe˝ttedcurve.Basedonthis˝ttedcurve, theintakemanifoldtemperatureofeachcyclecanbepredictedbasedontheexhausttemperature ofthelastcycle. 4.2.4SparkTimingwithKnockCycle-to-CycleVariability Afterthecorrelationof ) 8E2 andknockintensityKI20andcycle-to-cyclevariabilityhavebeen obtained,thein˛uenceofsparktimingand ) 8E2 totheknockcycle-to-cyclevariabilityhasbeen studiedaswell.Notethatthestatisticmethodhasbeenutilizedtoanalyzetheknockcycle-to-cycle variability.Especially,theexperimentaldatashowsthattheknockintensitydistribution˝tsthe Gaussiancurve,sothemeanvalueanddeviationofknockintensityhavebeenusedtopresentthe knockcycle-to-cyclevariabilityinthisdissertation. First,thecorrelationofknockintensity(KI20)andin-cylindermixturetemperatureatIVC (Tivc)isstudiedaswellusingtheknockpredictivemodel.Sixtycontinuousenginecyclesare 82 Figure4.4:Intaketemperature ) 8E2 withknockintensity simulated,andthesparktimingofthe˝rst30cyclesis20CADBTDCwhiletheother30cycles haveasparktimingof15CADBTDC.Foreachcycle,thein-cylindermixturetemperatureatIVC ( ) 8E2 )ispredictedbasedontheexhausttemperaturefromthecombustionmodelandthe˝tted curveshowninFigure4.3.Theknockintensity(KI20)ofeachcycleiscalculatedbyprocessing thein-cylinderpressuresignalfromtheknockpredictivemodel.Andtheresultsareshownin Figure4.4.Itindicatesthattheknockintensityhasastrongpolynomialcorrelationwith ) 8E2 ,and advancingsparktimingincreasesknockintensity. Theextensivesimulationsareexecutedoverdi˙erentsparktiming(from10CADBTDCto 20CADBTDC,with1CADincrement).Basedonthesimulationresults,Figure4.4hasbeen extendedunderdi˙erentsparktiming,showninFigure4.5.Therefore, ) 8E2 canbeusedtopredict theknockintensity,whichisthefoundationforthemodel-basedstochasticknocklimitcontrol designinnextsection. Second,theknockpredictivemodelisusedtostudythein˛uenceofsparktimingtotheknock 83 Figure4.5:Interpretedmapforthecorrelationofintaketemperature ) 8E2 andknockintensityas longasthesparktiming cycle-to-cyclevariability.Thesimulationresultsofthesixtycontinuouscycleswiththespark timingof20CADBTDCand15CADBTDCareanalyzedtostudythisimpactandtheresultis showninFigure4.6. AsshowninFigure4.6,theredbarsaretheKI20of30continuousenginecycleswithspark timingof20CADBTDCwhilethebluebarsareforanother30continuouscycleswithretarded sparktimingof15CADBTDC.Itisobviousthattheknockintensitieswithretardedsparktiming aremoregentlethantheresultswithadvancedsparktiming.Sincetheexperimentaldatastudies showthattheknockintensitydistribution˝tstheGaussiancurve,sothemeanvalue,standard deviationandthethreestandarddeviationcon˝denceintervaluplimit(CIL 3 f )areusedtoevaluate theimpactofsparktimingtotheknockcycle-to-cyclevariability. Themeanvalueandstandarddeviationforthesparktimingof20CADBTDCare2.3169bar and1.0291barwhiletheyare1.4527barand0.6481barfor15CADBTDC.With5degreesretarded 84 Figure4.6:Thein˛uenceofsparktimingtoknockintensitycycle-to-cyclevariability Table4.2:Statisticanalysisfortheimpactofsparktimingtotheknockcycle-to-cyclevariability. casemean[bar]std[bar]CIL 3 f [bar] SPK=20[CADBTDC]2.31691.02915.4042 SPK=15[CADBTDC]1.45270.64813.3970 CIL 3 f :threestandarddeviationcon˝denceintervaluplimit sparktiming,themeanvalueofKI20reducedby37.3%andthestandarddeviationreducedby 37.02%.Thethreestandarddeviationcon˝denceintervaluplimit(CIL 3 f )forsparktimingof20 CADBTDCis5.4042barwhilethecasewith15CADBTDCis3.3970bar,moved2.0072barto left;seeTable4.2.Insummary,theknockintensityofeachcyclecanbecontrolledbytuningthe sparkingtimingofthecorrespondingcycle,andtheknockcycle-to-cyclevariabilitycanbefurther reducedaswell. 85 4.3StochasticKnockLimitControlandResultsDiscussion 4.3.1ControlObjectives Inthisdissertation,thestochasticknocklimitcontrolstrategyisdesignedbasedontheknock predictivemodel,wherethecontrolinputoftheknockpredictivemodelisthesparktimingofeach cycleandtheoutputisthepredictedknockintensityoftheassociatedcycle.Itisassumedthatthe knockintensity˝tstheGaussiandistribution,andthestochasticknocklimitcontrolperformance isevaluatedbyanalyzingitsmeanvalue,standarddeviationandCIL 3 f ofknockintensityover certaincycles.Moreover,theknocklimitandMBTtimingconstraintofeachcycleareconsidered aswellforthebestfueleconomy. Therearethreeobjectivesforthestochasticknocklimitcontroldesign. 1. ThemeanvalueoftheknockintensityshouldbebelowthedesiredlimitKI20 34B8A4 ,where KI20 34B8A4 isde˝nedtobe1(KI20 34B8A4 =1 bar)inthisstudy. 2. Sinceanalysisindicatesthattheknockintensity˝tstheGaussiandistribution,thesecond controlobjectiveistoreducethecycle-to-cycleknockintensity.Inthispaper,thisintervalis de˝nedasthethreestandarddeviationintervalwiththeup-boundpresentedasCIL 3 f .Both theknocklimitcontrolandMBTtimingcontrolareconsideredateachcyclebycompensating sparktiming.Forthecyclethattheknockintensityiswithinthedesiredlimit,thespark timingshouldbegraduallyadvancedtotheMBTtiming. 3. Thethreestandarddeviationcon˝denceintervalup-boundCIL 3 f shouldbecloseto1,which guaranteestheknockintensitydistributionstayingwithinadesiredboundwithminimum cycle-to-cyclevariability. Therefore,afeedforwardknocklimitcontrolalgorithmisproposed˝rsttocontroltheknock intensitycycle-by-cycletoreducethecyclicvariation.Basedonthisfeedforwardcontrolalgorithm, aclosed-loopstochasticknocklimitcontrolalgorithmisfurtherproposed.Thetwocontrol 86 algorithmsarebasedontheknockpredictivemodeldevelopedearlierandthetwo˝ttedcurves showninFigures.4.3and4.4areusedtocompensatethesparktimingcycle-by-cycle. 4.3.2Model-BasedFeedforwardKnockLimitControl 4.3.2.1ControlAlgorithm Basedonthediscussioninthelastsection,theengineknockcycle-to-cyclevariabilityhasastrong correlationwithintaketemperature ) 8E2 .Underknockcondition,the ) 8E2 ofeachcycleisin˛uenced bytheknockingcombustionandexhausttemperateofthelastcycle.The˝ttedcurveshownin Figure4.3canbeusedtopredictthe ) 8E2 ofnextcycle.Withthepredicted ) 8E2 andthe˝ttedcurve showninFigure4.4,theknockintensityofnextcyclecanbepredicted.Notethatthe˝ttedcurve showninFigure4.4hasbeenextendedunderdi˙erentsparktimingbysweepingthesparktiming from10to20CADBTDCwith1CADincrement;seeFigure4.5.Therefore,thesparktimingcan beretardedforknocklimitcontroloradvancedforMBTtimingcontrol.Afeedforwardknocklimit controlalgorithmhasbeenproposedinthissectiontocontroltheknockintensitycycle-by-cycle; seeFigure4.7. AsshowninFigure4.7,theinitialsparktiming( \ B?:Œ) )isde˝nedbasedonacalibratedtable asafunctionofcurrentengineoperatingconditiontoachieveMBTtiming.With \ B?:Œ) andother initialinputs,suchas ) 8E2 Œ? 8E2 Œ˙'Œ< 5 ,etc.,theexhausttemperatureofcurrentcycle ) 4E> ( : ) canbeobtainedfromtheknockpredictivemodel.Withthe ) 4E> ( : ) ,theintaketemperatureofnext cycle ) 8E2Œ? ( : +1) canbepredictedbasedonthe˝ttedcurveshowninFigure4.3.Notethata zero-meanrandomvariable q withnormaldistributiondisturbance(seeequation(4.1))isaddedto thepredicted ) 8E2Œ? ( : +1) torepresentthein˛uenceofotherfactorstotheengineknock. ) 8E2 ( : +1)= ) 8E2Œ? ( : +1)+ qŒq 2N (0 Œ 30) (4.1) Asaresult,thenextcycleknockintensityKI20 ? ( : +1) canbepredictedbasedonthe˝ttedcurve showninFigure4.5basedon \ B?:Œ) andestimated ) 8E2 ( : +1) .Asaresult,therearetwocasesfor 87 Figure4.7:Model-basedstochasticfeedforwardknocklimitcontroldiagram sparktimingcontrol:knocklimitcontrolandMBTtimingcontrol. KnockLimitControl: Ifthepredictedknockintensityisgreaterthanthedesiredknock intensitylimit(KI20 ? ( : +1) ¡ KI20 34B8A4 ),thesparktimingofthenextcyclewillberetarded.The compensationofsparktiming,representedby \ B?:ŒA4C0A3 ,canbeobtainedbasedon ) 8E2 ( : +1) , KI20 34B8A4 =1barandthemapinFigure4.5. MBTTimingControl: Ifthepredictedknockintensityiswithinthedesiredlimit(KI20 ? ( : + 1) KI20 34B8A4 ),therewillbenoknocksparktimingcorrectionandthecalibratedengineMBT 88 timingshouldbeused.TheMBTtimingisobtainedbythepeakcylinderpressurelocation(PPL).In general,thePPLshouldbemaintainedat15CADATDC(desiredMBTtiming).Thecompensation ofMBTsparktiminginthiscaseisrepresentedby \ B?:Œ03E0=24 .Notethatthecompensated MBTsparktimingshouldnotbegreaterthantheknocklimitedsparktiming. \ B?:Œ03E0=24 can beobtainedbasedon ) 8E2 ( : +1) ,KI20 34B8A4 =1barandthemapinFigure4.5. 4.3.2.2ResultsandDiscussion TovalidatethecontrolperformanceoftheproposeddiagramshowninFigure4.7,abaseline simulationof30continuousenginecycleswithoutcontrolwasconducted˝rst,andthenthe feedforwardcontrolwasturnedonforanother20continuouscycles.Theresultsareshownin Figure4.8.Notethatthe˝rst30cycleshasaconstantsparktimingof20CADBTDC,andthe sparktimingforthecyclesfrom31to50iscontrolledcycle-by-cyclebasedonthecontrolscheme inFigure4.7. Thesparktimingfromcycles31to50(totalof20cycles)withfeedforwardcontrolisshown inFigure4.9,wheretheredbarspresentthesparktimingofeachcyclewithoutthecompensation \ B?:Œ˙˙ ,whilethebluebarsarewiththefeedforwardcompensation.Thedi˙erencebetweenred andbluebarsistheadvancedorretardedsparktiminggeneratedbythefeedforwardcontrolscheme showninFigure4.7.Forthesecyclesthatpredictedknockintensityaregreaterthanthedesired level,thesparktimingisretardedwithadownwardarrowpresentedinFigure4.9(forexample, cycles31,32,35,etc.).Forthesecyclesthatthepredictedknockintensityisbelowthedesired level,thesparktimingisadvancedbutnotovertheknocklimittobeasclosetotheMBTtiming aspossible;seeupwardarrowinFigure4.9(forexample,cycles36,37,38,etc.).Forthesecycles, withthepredictedknockintensityveryclosetothedesiredlevelbutnotoverit,thesparktiming willnotbecompensated( \ B?:Œ˙˙ =0 )(forexample,cycles33and34). Thdetailsofknockintensityfromcycles31to50areshowninFigure4.10.Figures4.8 and4.10demonstratethattheknockcycle-to-cyclevariationandmeanvaluehavebesigni˝cantly improved.AsshowninFigure4.8,themeanvalueandstandarddeviationoftheknockintensity 89 Figure4.8:Model-basedstochasticfeedforwardknocklimitcontrolperformance ofthe˝rst30cyclesare2.26barand1.2715bar,respectively.Therefore,thecorresponding con˝dencelimitCIL 3 f is6.0745bar.Withthestochasticfeedforwardcontrol,themeanvalueand standarddeviationforcycles31to50arereducedto0.9828barand0.1218bar,respectively.The correspondingCIL 3 f isreduceddownto1.3482bar,witha77.81 % improvement.Figure4.10 showsthattheknockintensityofeachcycleismaintainedclosedtothedesiredlevelwiththespark timingcompensatedforeachcyclebytheproposedfeedforwardcontrol. Theknockintensityofeachcyclebeforeandafterthecompensationofsparktiming( \ B?:Œ˙˙ ) isalsocompared,andtheresultisshowninFigure4.11.Thebluesolid-linewithmarkerpresentsthe knockintensityofeachcyclewithoutthecompensationofsparktimingwhiletheredsolid-linewith markeriswithcompensatedsparktiming.Basedonthediscussionoffeedforwardcontrolscheme showninFigure4.7,theknockintensitywithoutsparktimingcompensationisKI20 ? ( : +1) that ispredictedbasedonthe \ B?: ( : ) and ) 8E2 ( : +1) .Ifthesparktimingisnotcompensated(without \ B?:Œ˙˙ ), \ B?: ( : +1)= \ B?: ( : ) andtheknockintensityofnextcycleKI20 ( : +1)= KI20 ? ( : +1) , presentedwithabluemarkerinFigure4.11.Withthefeedforwardcontrol,thesparktimingwill becompensatedand \ B?: ( : +1)= \ B?: ( : )+ \ B?:Œ˙˙ ,withwhichthepredictedknockintensity 90 Figure4.9:Cycle-basedsparktimingforknocklimitandMBTtimingcontrol KI20 ? ( : +1) willbereducedtoKI20 ( : +1) ,presentedwitharedmarkerinFigure4.11.Figure 4.11demonstratesthesigni˝cantimprovementofknockintensityofeachcyclewiththeproposed feedforwardcontrolalgorithmshowninFigure4.7. Inaddition,withtheGaussiandistributionassumption,thecomparisonofbothprobability densityfunction(PDF)forthe˝rst30cycleswithoutcontrolandanother20cycleswithfeedforward controlisshowninFigure4.12.Itindicatesthat99.7 % ofknockintensityiswithintheintervalof [0.6174,1.3482]barafterthefeedforwardcontrolisturnon,thatisasigni˝cantimprovementof knockcycle-to-cyclevariability. 4.3.3Closed-LoopStochasticKnockLimitControl 4.3.3.1ControlAlgorithm Thedetailsofknockintensityofthecycles31to50inFigure4.8withtheproposedfeedforward controlalgorithmareshowninFigure4.10.Itindicatestheknockintensityvariationhasbeen 91 Figure4.10:Detailsofknockintensityundermodel-basedstochasticfeedforwardknocklimit control improvedsigni˝cantlyandthemeanvaluehasbeencontrolledwithinthedesiredbound.However, therearecertaincyclesthatknockintensityislightlyoverthedesiredbound,evenwiththeopen- loopsparktimingcompensation.Thisisresultedbythefactorsotherthan ) 8E2 ,thatisnotincluded inthisknockpredictivemodel;seetheerrorbarshowninFigure3.14.Thein˛uenceofthesefactors areconsideredusingazero-meandisturbancetobeaddedtopredicttheintaketemperature ) 8E2 . Therefore,aclosed-loopstochasticknocklimitcontrolalgorithmisdevelopedbasedonFigure4.7 toimprovethecontrolperformanceofknockintensityofeachcyclebyfurtherreducingthemean valueofknockintensityandalsoregulatingthecon˝dencelimitCIL 3 f tobeclosetoKI20 34B8A4 . Theproposedclosed-loopstochasticknocklimitcontrolalgorithmisshowninFigure4.13. Fortheproposedclosed-loopstochasticknocklimitcontrolalgorithm,thesparktimingofeach cycleiscompensatednotonlybythefeedforwardcontroldevelopedinthelastsectionbutalsoby 92 Figure4.11:Comparisonofknockintensitybeforeandafterthecompensationofsparktimingfor eachcycle aPIcontrollerinthefeedbackloop;see(4.2)andFigure4.13. \ B?: = \ B?:Œ) + \ B?:Œ˙˙ + \ B?:Œ˙ (4.2) where \ B?:Œ) isthesparktimingbasedonacalibratedtableasafunctionofcurrentengine operatingconditiontoachieveMBTtiming.Itremainsconstantoverenginecyclesunderasteady- stateenginecondition; \ B?:Œ˙˙ isthesparktimingcompensationbythefeedforwardcontrol algorithmdevelopedearliershowninFigure4.7; \ B?:Œ˙ isthesparktimingcompensationfrom thePIcontrollerinthefeedbackcontrolloopandwillbediscussedinthissection;and \ B?: isthe actualsparktimingusedasancontrolinputtothedevelopedknockpredictivemodeltopredictthe in-cylinderpressureandexhausttemperatureofindividualcycle. AsshowninFigure4.13,thecompensatedsparktiming \ B?: andintaketemperature ) 8E2 will bethetwoimportantinputstotheknockpredictivemodel,acombinationofthereaction-based combustionmodelandpressurewavemodel.Themodelhastwooutputs:exhausttemperature 93 Figure4.12:Gaussiandistributioncomparisonforknockintensitywithandwithouttheproposed feedforwardknocklimitcontrolalgorithm ( ) 4E> )andin-cylinderpressure( ? 2H; ),where ) 4E> isusedforthefeedforwardcontroldiscussedin theprevioussectionandin-cylinderpressure ? 2H; willbeprocessedbyabutterworthband-pass ˝ltertogeneratetheknockintensityKI20ofthecurrentcycle.ThenKI20ofeachcyclewillbe storedinabu˙erforthestatisticalanalysis.Notethatthebu˙erstoredtheknockintensityoftotal Ncontinuouscyclesasa˝rstin,˝rstout(FIFO)queue:thenewpredictedknockintensitywillbe storedasthelatestvalueintheFIFOqueueandthepreviousvaluestoredasKI20(0)willbepop outsothatthetotallengthinthebu˙erismaintainedasNcycles;seeFigure4.14. BasedonthestoredlastNcyclesknockintensities,theknockintensitydistributioncanbe analyzedtoachievethemeanvalue,standarddeviationandCIL 3 f .Therefore,theerrorbetween thedesiredknockintensity(KI20 34B8A4 )andthecon˝dencelimit(CIL 3 f )willberegulatedwith aPIcontrollerinthefeedbackloop;seetheyellowsolidlineinFigure4.13.Thesparktiming 94 Figure4.13:Model-basedclosed-loopstochasticknocklimitcontroldiagram compensationbythisloopiscalculatedby \ B?:Œ˙ = ? 4 ( g )+ 8 ) B g X 9 =1 4 ( 9 ) (4.3) where ? and 8 aretwocontrolcoe˚cientsforthePIcontroller; ) B isthetimestep.Similar withthefeedforwardcontrolloopwhichcompensatesthesparktimingforeveryenginecycle,the feedbackcontrolloopalsoexecutesateverycycle.AndthelengthofcyclesNinthebu˙eris de˝nedas20cyclesinthepaper.So \ B?:Œ˙ compensatesthesparktimingwithtimestepof ) B . ) B = 720 6 # 4 (4.4) where # 4 istheenginespeed. 4.3.3.2ResultsandDiscussion Theclosed-loopstochasticknocklimitcontrolperformanceisvalidatedbyconducting130contin- uousenginecycles.Forthe130cycles,the˝rst30cyclesareconductedwithoutthetwoproposed 95 Figure4.14:Bu˙erformationforthestatisticalanalysisofknockintensitydistribution controlalgorithms,andthenthefeedforwardcontrolalgorithmshowninFigure4.7isturnedon fromcycles31to50.Afterthat,theclosed-loopcontrolalgorithmshowninFigure4.13isturned onfromcycles51to130tocompensatethesparktimingateverycycle.Similartothefeedfor- wardcontrolalgorithm,thecontrolperformanceoftheclosed-loopcontrolalgorithmisevaluated basedonthestatisticanalysisoftheknockintensityovercertaincycles:themeanvalue,standard deviationandcon˝dencelimit. Thetime-basedsparktimingofthe130continuousenginecyclesisshowninFigure4.15and thesimulationresultsofknockintensityareshowninFigure4.16.Sincethesparktimingisnot compensatedby \ B?:Œ˙˙ and \ B?:Œ˙ forthe˝rst30cycles,itremainsunchanged(20CAD BTDC)asshowninFigure4.15.Asdiscussedinlastsection,theknockintensityoverthisduration showsahighdeviationandthemeanvalueisoverthedesiredknockintensitylimit.Withthe feedforwardcontrol,theknockintensityofeachcycleisimprovedsigni˝cantlyandthemeanvalue iscontrolledwithinthebound,andCIL 3 f isreducedtobeclosetothetargetat1bar.However, therearesomecycleswithknockintensityslightlyoverthebound.Therefore,boththefeedback 96 Figure4.15:Time-basedsparktimingwiththeclosed-loopstochasticknocklimitcontrol algorithm Table4.3:Stochasticanalysisofcycle-to-cycleknockintensity. Conditions Cycles ` KI20 [bar] f KI20 [bar] CIL 3 f [bar] WOTcontrol 1-30 2.26 1.2715 6.0745 Feedforwardcontrol 31-50 0.9828 0.1218 1.3482 Closed-loopcontrol 31-130 0.9432 0.0187 0.9993 andfeedforwardcontrolloopsstarttocompensatethesparktimingfromcycle51.Figure4.16a showsthattheknockcycle-to-cyclevariabilityhasbeenfurtherimproved.Theperformanceoftwo controlalgorithmsiscomparedinFigure4.16b.Itindicatesthatthemeanvalueofknockintensity isfurtherreducedwiththeclosed-loopstochasticknocklimitcontrolandCIL 3 f iscontrolledto beclosedtothedesiredknockintensitybound,whichdemonstratestheimprovementofknock cycle-to-cyclevariability. Thestatisticanalysisresultsofthe130cyclesaresummarizedinTable4.3.Itindicatesthatthe meanvalueofknockintensityhasbeenreducedfrom2.26barto0.9828barwiththefeedforward control.Thenwiththeclosed-loopcontrol,themeanvaluehasbeenfurtherreduceddownto 0.9432barafteranother80continuouscycles.Notethatthemeanvalueisslightlyreducedwith theclosed-loopcontrol.ThereasonisthattheMBTtimingcontrolregulatestheknockintensity 97 (a) (b) Figure4.16:Closed-loopstochasticknocklimitcontrolperformance tobeclosetotheboundforthe50cycleswithpredictedknockintensitybelowthedesiredknock intensitybound.Especially,comparingwiththefeedforwardcontrol,theknockintensitystandard deviationwiththeclosed-loopcontrolissigni˝cantlyreducedfrom0.1218barto0.0187bar,with 86.4%improvement.Thecon˝dencelimitCIL 3 f isdrivenfrom1.3482barto0.9993bar,very closetoKI20 34B8A43 andthethreestandarddeviationinterval( [ ` KI20 3 f KI20 Œ` KI20 +3 f KI20 ] ) 98 Figure4.17:ComparisonofknockintensitydistributionPDFwithdi˙erentcontrolmethods getsnarrow.Thisresultsindicatethatthedevelopedclosed-loopcontrolshowninFigure4.13 isabletocontroltheknockintensityofindividualcyclewithinthebound,andlimittheknock intensitydistributioninasmallintervalclosetotheboundandtheknockcycle-to-cyclevariability isminimized. Theprobabilitydensityfunction(PDF)ofthe130cyclesisstudiedaswelltoevaluatetheclosed- loopstochasticknocklimitcontrolperformanceinreducingtheknockcycle-to-cyclevariability;see Figure4.17,whichshowstheknockintensitydistributionof130cycleswithoutandwithdi˙erent controlmethod.Figure4.18showsthedetailsforthecyclesthatthesparktimingiscompensated bytheproposedtwocontrolalgorithms.Theblacksolid-lineinFigure4.17isthePDFcurve forthe˝rst30cycleswithoutcontrol,andtheredsolidlineisforthefeedforwardcontrol,the bluesolidlineisfortheclosed-loopcontrol.Thetwoblackdash-dotlinespresentthecon˝dence limitofcycles1to30and31to130,respectively.ItisobviousthatCIL 3 f hasbeenreduced from6.0745barto0.9993barafter130cycles,andisveryclosetothedesiredknockintensity 99 Figure4.18:KnockintensitydistributionPDFwithclosed-loopstochasticknocklimitcontrol bound.Theoverallimprovementis83.5 % andonly0.15 % ofcycleshavetheknockintensity greaterthan0.9993bar.Figure4.18showstheknockintensitydistributioncomparisonwithtwo controlmethods.Itindicatesthatthemeanvalueisgentlymovedtoleftandthecon˝denceinterval getsnarrowsigni˝cantly.Thecon˝dencelimitCIL 3 f isreducedfrom1.3482barto0.9993bar with25.87 % improvement.Thisresultshowsthatthecapabilityofclosed-loopcontrolhasabetter performancethanthefeedforwardcontrolforreducingtheknockcycle-to-cyclevariability. 4.4Summary Amodel-basedstochasticfeedforwardandclosed-loopknocklimitcontrolstrategyisdemon- stratedinthischaptertoregulatethecycle-by-cycleknockintensitywithinthedesiredlimitandalso maintaintheknockintensitydistributionwithinthethreetimesofstandarddeviationcon˝dence interval.Theknockpredictivemodelisbasedona0-Dtwo-zonereaction-basedcombustionmodel andthepressurewavemodeldevelopedandvalidatedearlier.Theindividualcyclesparktimingis 100 retardedoradvancedbasedontheknocklimitcontrolorMBTtimingcontrolinthefeedforward controlstrategy,respectively.AndaPIcontrollerisdesignedfortheclosed-loopknocklimitcon- troltofurtherimprovethecontrolperformance.Thecontrolperformanceisdemonstratedthrough simulationstudiesover130continuousenginecycles.Withtheproposedstochasticfeedforward knocklimitcontrolstrategy,themeanvalueofknockintensityisreducedfrom2.26barto0.9828 bar.Thethreetimesofstandarddeviationcon˝dencelimitCIL 3 f isreducedfrom6.0745barto 1.3482bar,witha77.81 % improvement.Withtheproposedclosed-loopstochasticknocklimit control,themeanvaluehasbeenlightlyreducedfrom0.9828barto0.9432bar,andCIL 3 f is signi˝cantlyreducedfrom1.3482barto0.9993bar,witha25.87 % improvement,comparingwith thefeedforwardcontrol.Theseresultsindicatethecapabilityoftheproposedtwocontrolalgo- rithmsforimprovingtheknockcycle-to-cyclevariabilityandfuele˚ciencyduetocontrollingthe sparktimingclosetoMBT.Especially,theclosed-loopstochasticknocklimitcontrolhasabetter performancecomparingwiththefeedforwardone. 101 CHAPTER5 CONCLUSIONSANDFUTUREWORK 5.1Conclusions Theconclusionsofthisdissertationcanbesummarizedasfollows: 1. Acontrol-oriented,zero-dimensional,two-zone,reaction-basedcombustionmodelforcom- pression,combustionandexpansionphasesofsparkignitionenginesisdeveloped,calibrated, andvalidatedagainstexperimentaldatainthisdissertation.Thedevelopedmodeliscapable ofpredictingthermodynamiccharacteristicsofin-cylinderchemicalmixtures,combustion process,propertiesofindividualchemicalspeciesinbothunburnedandreactionzones.Fur- thermore,itisalsoabletopredicttheauto-ignitionintheunburnedzone(engineknock). Simulationresultsshowthatthedevelopedtwo-zonecombustionmodelisabletopredict thein-cylinderthermalstatesandcombustionprocessofsparkignitionengines,suchasthe startofcombustion,˛amepropagationprocess,andin-cylinderheatandmasstransfer.The proposedcombustionmodeliscapableofaccuratelypredictingcombustionprocess,includ- ingthemassvariationandthermalpropertiesofeachchemicalspecies.Notethattheability tosimulatethezonestrati˝cationandspeciesmolarconcentrationsallowspredictingengine knock.Asensitivitybasedcalibrationprocessdividescalibrationparametersintotwogroups withlowandhighsensitivity,wherethedefaultvaluesareusedforlowsensitivityonesand sevenhighsensitivityonesarecarefullycalibratedusingexperimentaldata.Especially,the presentedsimulationresultsusesonlyonesetofcalibrationparameters,whichmeansthat themodeldoesnotneedtobere-calibratedunderdi˙erentoperationalconditions. 2. Areal-timepressurewavemodelisdevelopedutilizingtheresultsfromthetwo-zonereaction- basedcombustionmodeltopredicttheknockcharacteristicsofSIengines.Usingthe chemicalkineticcharacteristicsoftheunburnedmixtures,theknockonsettimingcanbe 102 predictedusingthechemical-basedArrheniusintegral(ARI).Apressurewaveequation, includingthepressureoscillationmagnitudedecaybehavior,isobtainedbysimplifyingthe 3-Dwaveequationforreal-timesimulations,wheretheinitialandboundaryconditionsforthe knockcombustionareobtainedfromthereaction-basedcombustionmodel.Theproposed in-cylinderpressureperturbationsignaliscombinedwiththepressurefromthereaction- basedcombustionmodelforthecompositepressuresignalunderknockcombustion.The capabilityofthisproposedmodelisvalidatedusingtheexperimentaldataforknockonset timing,knockintensityandfrequency.First,theARIiscalculatedandindicatesthattheknock onsettimingisat21.04CADafterTDCfromexperimentaldataand21.06CADafterTDC fromsimulateddatawithapredictionerroroflessthan1.00 % .Second,theexperimentaland simulatedin-cylinderpressuresignalsareusedtocalculatetheknockintensity,wherethe knockintensityis1.0088and1.0143barfromexperimentalandsimulateddata,respectively. Thepredictionerrorislessthan7.0 % .Last,themodel'sabilityofpredictingtheknock frequencyisvalidatedbyFFTanalysisanditshowsthattheknockfrequenciesare6.303 kHzand6.330kHzforexperimentalandsimulateddata,respectively,wheretheprediction errorislessthan0.5 % .sasummary,theproposedin-cylinderpressurewavemodelisable toaccuratelypredictthekeyknockcharacteristicssuchastheknockonsettiming,knock intensityandfrequency. 3. Amodel-basedfeedforwardandclosed-loopstochasticknocklimitcontrolalgorithmis proposedtocontroltheengineknockintensitycycle-by-cycleandreducetheknockcyclic variability.ThefeedforwardcontrolalgorithmincludestheknocklimitcontrolandMBT timingcontroltocompensatethesparktimingofindividualcycle.Theclosed-loopcontrol algorithmisdevelopedbasedonthefeedforwardandfeedbackcontrollers.Withthefeedfor- wardcontrol,themeanvalueofknockintensitydistributionisreducedfrom2.26to0.9828 barandthethreestandarddeviationup-boundlimitCIL 3 f isreducedfrom6.0745to1.3482 bar,witha77.81 % improvement,indicatingthesigni˝cantimprovementinreducingthe knockcycle-to-cyclevariabilitywithMBTtimingconstraint.Withtheclosed-loopcontrol, 103 themeanvalueofknockintensityhasbeenfurtherreducedandCIL 3 f hasbeenfurther reducedby7.42 % aswell,indicatingtheclosed-loopstochasticknocklimitcontrolalgo- rithmhasimprovedperformanceforreducingtheknockcyclicvariabilitycomparingwith thefeedforwardcontrolonlycase. 5.2RecommendationsforFutureWork Thefollowingfutureworkisrecommendedfortheknockpredictivemodel-basedstochastic knocklimitcontrol: 1. Forthetwo-zonereaction-basedcombustionmodel,itcanbeextendedinthefutureby modelingtheair-pathsystemofSIengineswithEGRandturbocharger.Thiswillbea signi˝cantimprovementforthecontrol-orientedSIenginemodelanditwillhaveanextensive applicationforthemodel-basedenginecontrol. 2. Fortheknockpredictivemodel: a) The˝ttedcurveforintaketemperaturepredictionofeachcyclebasedontheexhaust temperaturecanbefurtherimprovedandvalidatedbasedontheexperimentdataof extensiveengineconditions. b) Theinterpreted3-DmapshowninFigure4.5canbefurthercalibratedwiththeon- boardmachinelearningmethodinthefuturebenchtestingtoimprovethemodel-based stochasticknocklimitcontrolperformance. 3. 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