Sig

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Modeling and Estimation

for Heat Release Analysis

of SI Engines

Markus Klein

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Single-ZoneCylinder Pressure Modeling and Estimation

for Heat ReleaseAnalysis ofSI Engines

Copyright c

2007MarkusKlein

markus.s.klein@gmail.com

http://www.fs.isy.liu.se/

DepartmentofElectricalEngineering,

LinköpingUniversity,

SE58183Linköping,

Sweden.

ISBN 978-91-85831-12-8 ISSN0345-7524

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Cylinderpressuremodelingandheatreleaseanalysisaretoday

impor-tantandstandardtoolsforengineersandresearchers,whendeveloping

andtuningnewengines. Beingabletoaccuratelymodelandextract

in-formationfromthecylinderpressureisimportantfortheinterpretation

andvalidityoftheresult.

Therstpartofthethesistreatssingle-zonecylinderpressure

mod-eling, wherethespecic heatratiomodelconstitutesakeypart. This

model component is therefore investigated more thoroughly. For the

purpose of reference, the specic heat ratio is calculated for burned

andunburnedgases,assumingthattheunburnedmixtureisfrozenand

thattheburnedmixtureisatchemicalequilibrium. Useofthereference

modelinheatreleaseanalysisistootimeconsumingandthereforeaset

ofsimplermodels,bothexistingandnewlydeveloped,arecomparedto

thereferencemodel.

Atwo-zonemeantemperaturemodelandtheVibefunctionareused

toparameterizethemassfractionburned. Themassfractionburnedis

usedtointerpolatethespecicheatsfortheunburnedandburned

mix-ture,andtoformthespecicheatratio,whichrendersacylinder

pres-sure modelingerror in thesameorder asthe measurementnoise,and

fteen times smallerthan themodel originally suggestedin Gatowski

etal.(1984). Thecomputationaltimeisincreasedwith40%compared

totheoriginalsetting,butreducedbyafactor70comparedto

precom-puted tablesfrom thefullequilibriumprogram. Thespecicheats for

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andthespecicheatsfortheburnedmixturearecapturedwithin 1%

by higher-orderpolynomialsforthe major operatingrange ofaspark

ignited(SI)engine.

Inthesecond part,four methods forcompression ratioestimation

basedoncylinderpressuretracesaredevelopedandevaluatedforboth

simulatedandexperimental cycles. Threemethods relyuponamodel

of polytropic compression for the cylinder pressure. It is shown that

they give a good estimate of the compression ratio at low

compres-sion ratios, although the estimatesare biased. A method basedon a

variable projectionalgorithmwith alogarithmicnorm ofthe cylinder

pressure yields the smallest condence intervalsand shortest

compu-tational time for these three methods. This method is recommended

when computationaltime isanimportant issue. Thepolytropic

pres-suremodellacksinformationaboutheat transferandthereforethe

es-timationbiasincreaseswiththecompressionratio. Thefourthmethod

includes heat transfer, crevice eects, and acommonly used heat

re-lease model for ring cycles. This method estimates thecompression

ratio more accurately in terms of bias and variance. The method is

more computationally demanding and thus recommended when

esti-mationaccuracyisthemostimportantproperty. Inorderto estimate

thecompressionratioasaccuratelyaspossible,motoredcycleswithas

high initialpressureaspossibleshouldbeused.

The objective in part 3 is to develop an estimation tool for heat

releaseanalysis that isaccurate, systematic and ecient. Two

meth-ods that incorporate prior knowledge ofthe parameternominal value

and uncertaintyin asystematic mannerare presentedand evaluated.

Method 1is basedonusing asingularvalue decomposition of the

es-timated hessian, to reduce the number of estimated parameters

one-by-one. Thenthe suggestednumberof parametersto use isfound as

theoneminimizingtheAkaikenalpredictionerror. Method 2usesa

regularizationtechniquetoincludethepriorknowledgeinthecriterion

function.

Method2givesmoreaccurateestimatesthanmethod1. Formethod

2,priorknowledgewithindividuallysetparameteruncertaintiesyields

moreaccurateandrobustestimates. Onceachoiceofparameter

uncer-taintyhasbeendone,nouserinteractionisneeded. Method 2isthen

formulatedforthreedierentversions,whichdierinhowthey

deter-minehowstrong theregularizationshould be. Thequickest versionis

basedonad-hoctuning andshould beusedwhencomputationaltime

isimportant. Anotherversionismoreaccurateandexibletochanging

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Den svenskatiteln påavhandlingen ärEn-zons-modelleringoch

esti-meringföranalysavfrigjordvärmeibensinmotorer 1

.

Förbränningsmotorerharvaritdenprimäramaskinenföratt

gene-reraarbeteifordonimeränhundraår,ochkommerattvara

högintres-santäveni fortsättningen främstp.g.a.bränslets högaenergidensitet.

Emissionskravfrån främstlagstiftare,prestandakravsåsom eektoch

bränsleförbrukning från potentiella kunder, samtden konkurrenssom

ges av nyateknologier såsombränsleceller fortsätter att driva

teknik-utvecklingenavförbränningsmotorerframåt.

Teknikutvecklingen möjliggörs av att ingenjörer och forskare har

möttdessakravgenomt.ex.grundforskningpåförbränningsprocessen,

nyaellerförbättradekomponenterimotorsystemetochnyateknologier

såsom variabla ventiltider och variabelt kompressionsförhållande. De

tvåsistnämndaärexempelpåteknologiersomdirektpåverkar

tryckut-vecklingeni cylindern,därdet ärviktigt att fånoggrannkunskapom

hur förbränningsprocessen fortlöper och hur bränslets kemiska energi

frigörs som värme och sedan omvandlas till mekaniskt arbete. Detta

kallasanalysavfrigjordvärmeochkopplardirekttillmotorns

emissio-ner,eektochbränsleförbrukning.

En analys av frigjord värme möjliggörs av att man: 1) kan mäta

cylindertrycket under förbränningsprocessen;2) harmatematiska

mo-deller för hur cylindertrycket utvecklassom funktion av kolvrörelsen

och förbränningsprocessen; samt 3) har metodik för att beräkna den

1

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frigjordavärmen genom att anpassaden valdamodellen till

cylinder-trycksmätningen.

Sensorerna för att mäta trycket i cylindern har den senastetiden

blivit bådenoggrannareoch robustaremotden extrema miljöoch de

snabbatryck-ochtemperaturförändringarsomsensornutsättsför

un-dervarjecykel.Idaganvändscylindertryckssensorerenbartilabb-och

testmiljö, mest beroende på att sensorn är relativt dyr, men det

på-går enteknikutveckling avsensorernasomsiktar påatt dei framtida

fordonävenskallsittai produktionsmotorer.

Matematiska modeller av cylindertrycksutvecklingen och metoder

förattberäknadenfrigjordavärmenbehandlasidennaavhandling.

Avhandlingens innehåll och kunskapsbidrag

Avhandlingenbeståravtredelarochdetsammanhållandetematär

cy-lindertryck.Den första delenbehandlar en-zons-modellering av

cylin-dertrycksutvecklingen,därblandningenavluftochbränsleicylindern

behandlas som homogen. Detta antagande möjliggör en kort

beräk-ningstid förden matematiskamodellen. Ett bidragi avhandlingen är

identieringavden viktigastemodellkomponentenienallmänt

veder-tagenen-zons-modellsamtenmodellförbättringfördennakomponent.

Iden andradelen studeraskompressionsförhållandet,vilketär

för-hållandetmellanstörstaochminstacylindervolymunderkolvens

rörel-se.Dennastorhetärdirektkoppladmotorns verkningsgradoch

bräns-leförbrukning. Avsnittet beskriveroch utvärderarmetoderför att

be-stämma kompressionsförhållandeti en motor utgående från

cylinder-trycksmätningar. Dessa metoder appliceras och utvärderas på en av

SaabAutomobileutveckladprototypmotormedvariabelt

kompressions-förhållande.

Närmanberäknarparametrarimatematiskamodellerdär

paramet-rarna har fysikalisk tolkning,såsom temperatur, har användarenofta

förkunskapoch erfarenhetom vilka värdendessa böranta. Den

tred-je delen utvecklar ett verktyg för hur användaren kan väga in sådan

förkunskap när parametrarnas värden beräknas utgående från

cylin-dertrycksmätningar och den valda matematiskamodellen. Särskilt en

avmetodernager en brakompromissmellan användarensförkunskap

ochmätdata.Dettaverktygkananvändasföranalysavfrigjordvärme

ochmotorkalibrering,somettdiagnosverktygochsomett

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This workhas been carriedoutat the department ofElectrical

Engi-neering,divisionofVehicularSystemsatLinköpingsUniversitet,

Swe-den. It was nanciallyfunded by the Swedish Agency for Innovation

Systems (VINNOVA) throughtheexcellence center ISIS (Information

Systems for Industrial control and Supervision), and by the Swedish

Foundation for Strategic Research SSF through the research center

MOVII.Theirsupportisgratefullyacknowledged.

My timeasaPhD student hasbroadenmy mindand beenenjoyable

and Iwould thankallmycolleagues,formerandpresent,atVehicular

Systemssinceyourarealljointlythecauseofthat.

I wouldalso liketo thankmysupervisorsDr. LarsEriksson and

Pro-fessor Lars Nielsen for their guidance and for letting me join the

re-searchgroupofVehicularSystems. Dr.Erikssonhasinspiredmegreatly

throughinterestingdiscussionsandthroughhistrueenthusiasmfor

en-gine research. I haveenjoyed working with you. My co-authorsProf.

LarsNielsen, YlvaNilssonand Dr. JanÅslund arealsoacknowledged

for fruitfulcollaborations. Theyhave,aswellasDr. Ingemar

Anders-son,Dr.PerAndersson,Dr.GunnarCedersund,Dr.ErikFrisk,Martin

Gunnarsson, PerÖbergand manyothers, contributed tomy research

byinsightfuldiscussionsandcreativeeorts. MartinGunnarssonisalso

acknowledgedfor keepingthe enginelab running,as well asCarolina

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deservesanextramentionsincehehasconstantlybeenwithinrangeof

myquestions andhehasalwayshelpedoutwithouthesitation.

Dr.LarsErikssonhasproof-readtheentirethesis,whileDr.ErikFrisk,

Dr. JanÅslund andPerÖberghaveproof-readpartsof it. I amvery

grateful forthetime andeortyouhaveputinto mywork. Dr.

Inge-marAnderssonhasinspiredmetonishthethesis,andProfessorLars

Nielsen has helped me to stay focused during the nal writingof the

thesis.

ToallthepeoplewhohavehostedmeduringthemiddlepartofmyPhD

studies, among those Andrej Perkovic and Maria Asplund, Dan and

TottisLawesson,FredrikGustavssonandLouiseGustavsson-Ristenvik,

PontusSvensson,MathiasandSara Tyskeng,andespeciallyJan

Bru-gård,Iwishtodirectmysinceregratitudeforroomingmebutalsofor

making sure I had an enjoyable stay. The year I shared house with

IngemarandJanneisamemoryforlife.

Thesupportfrom myfamily, myin-lawfamilyandall myfriendshas

beeninvaluable.

This work would neverhave been accomplishedwithout the

indefati-gablelove,supportandencouragementofSoa. FinallyIwouldliketo

thankNora,Samuel,EbbaandSoa,whomIlovemorethanlifeitself,

forallthejoyandadventuresyoubringalong.

Linköping,July2007

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1 Introduction 1

1.1 Outline . . . 3

1.2 Contributions . . . 6

1.3 Publications. . . 6

I Modeling 9 2 Anoverview ofsingle-zone heat-releasemodels 11 2.1 Modelbasisandassumptions . . . 12

2.2 Rassweiler-Withrowmodel. . . 15

2.3 Apparentheat releasemodel . . . 18

2.4 Matekunaspressureratio . . . 20

2.5 Gatowskietal. model . . . 20

2.6 Comparisonofheatreleasetraces. . . 24

2.7 Summary . . . 26

3 Heat-releasemodelcomponents 27 3.1 Pressuresensormodel . . . 27

3.1.1 Parameterinitializationpressureoset

∆p

. . . 28

3.1.2 Parameterinitializationpressuregain

K

p

. . . 28

3.1.3 Crankanglephasing . . . 29

3.1.4 Parameterinitializationcrank angleoset

∆θ

. 30 3.2 Cylindervolumeandareamodels . . . 31

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3.2.1 Parameterinitializationclearancevolume

V

c

. 32

3.3 Temperaturemodels . . . 32

3.3.1 Single-zonetemperaturemodel . . . 33

3.3.2 Parameter initialization cylinder pressure at IVC

p

IV C

. . . 33

3.3.3 Parameterinitializationmeancharge tempera-tureat IVC

T

IV C

. . . 33

3.4 Crevicemodel. . . 35

3.4.1 Parameterinitializationcrevicevolume

V

cr

. . 36

3.4.2 Parameterinitializationcylindermeanwall tem-perature

T

w

. . . 36

3.5 Combustionmodel . . . 37

3.5.1 Vibefunction . . . 37

3.5.2 Parameterinitializationenergyreleased

Q

in

. . 38

3.5.3 Parameterinitialization angle-related parame-ters{

θ

ig

,

∆θ

d

,

∆θ

b

} . . . 39

3.6 Engineheattransfer . . . 39

3.6.1 Parameterinitialization{

C

1 , C

2

} . . . 42

3.7 Thermodynamicproperties . . . 42

3.7.1 Parameterinitialization

γ

300

and

b

. . . 42

3.8 Summaryofsingle-zoneheat-release models . . . 43

3.9 Sensitivityinpressureto parameterinitialization . . . . 45

4 A specic heat ratio model for single-zoneheat release models 49 4.1 Outline . . . 50

4.2 Chemicalequilibrium. . . 50

4.3 Existingmodelsof

γ

. . . 52

4.3.1 Linearmodelin

T

. . . 52

4.3.2 Segmentedlinearmodelin

T

. . . 52

4.3.3 Polynomialmodel in

p

and

T

. . . 53

4.4 Unburnedmixture . . . 54

4.4.1 Modeling

λ

-dependencewithxed slope,

b

. . . . 57

4.5 Burnedmixture . . . 58

4.6 Partiallyburnedmixture. . . 63

4.6.1 Referencemodel . . . 63

4.6.2 Groupingof

γ

-models . . . 64

4.6.3 Evaluationcriteria . . . 67

4.6.4 Evaluationcoveringoneoperatingpoint . . . 68

4.6.5 Evaluationcoveringalloperatingpoints . . . 72

4.6.6 Inuenceof

γ

-modelsonheatreleaseparameters 76 4.6.7 Inuenceofair-fuel ratio

λ

. . . 77

4.6.8 Inuenceofresidualgas . . . 79

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4.7 Summaryandconclusions . . . 83

II Compression Ratio Estimation 85 5 Compressionratio estimation with focus on motored cycles 87 5.1 Outline . . . 88

5.2 Cylinderpressuremodeling . . . 89

5.2.1 Polytropicmodel . . . 89

5.2.2 Standardmodel. . . 90

5.2.3 Cylinderpressurereferencing . . . 90

5.3 Estimationmethods . . . 91

5.3.1 Method 1Sublinearapproach . . . 91

5.3.2 Method 2Variableprojection . . . 92

5.3.3 Method3Levenberg-Marquardtandpolytropic model . . . 93

5.3.4 Method 4Levenberg-Marquardtandstandard model . . . 94

5.3.5 Summaryofmethods . . . 94

5.4 Simulationresults . . . 95

5.4.1 Simulatedenginedata . . . 95

5.4.2 ResultsandevaluationformotoredcyclesatOP2 96 5.4.3 Resultsand evaluationfor motoredcycles at all OP. . . 98

5.4.4 SensitivityanalysisatOP2 . . . 101

5.4.5 Resultsandevaluationforredcyclesat OP2. . 103

5.5 Experimental results . . . 103

5.5.1 Experimental enginedata . . . 103

5.5.2 ResultsandevaluationforOP2 . . . 105

5.5.3 ResultsandevaluationforallOP . . . 108

5.6 Conclusions . . . 112

III Prior Knowledge based Heat Release Analy-sis 115 6 Usingpriorknowledgeforsingle-zoneheatrelease anal-ysis 117 6.1 Outline . . . 119

6.2 Cylinderpressuremodeling . . . 120

6.2.1 Standardmodel. . . 120

6.2.2 Cylinderpressureparameters . . . 120

6.3 Problemillustration . . . 120

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6.4.1 Method1SVD-basedparameterreduction . . 124

6.4.2 Method2Regularizationusingpriorknowledge 130 6.4.3 Howtodeterminethepriorknowledge? . . . 139

6.4.4 Summarizingcomparisonofmethods1and2 . . 141

7 Resultsand evaluationfor motoredcycles 143 7.1 Simulationresultsmotoredcycles . . . 143

7.1.2 Parameterpriorknowledge . . . 144

7.1.3 Method1Resultsandevaluation . . . 146

7.1.5 Summaryforsimulationresults . . . 169

7.2 Experimentalresultsmotoredcycles . . . 174

7.3 Summaryofresultsformotoredcycles . . . 184

8 Resultsand evaluationfor red cycles 187 8.1 Simulationresultsredcycles . . . 187

8.1.5 Summaryforsimulationresults . . . 208

8.2 Experimental resultsredcycles . . . 209

8.3 Summaryofresultsforredcycles . . . 220

8.4 FutureWork . . . 221

9 Summaryand conclusions 223 9.1 A specic heat ratio model forsingle-zone heat release models . . . 223

9.2 Compressionratioestimation . . . 224

9.3 Priorknowledgebasedheat releaseanalysis . . . 226

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A Aspecic heat ratio modelfurther details 237

A.1 Temperaturemodels . . . 237

A.1.1 Single-zonetemperaturemodel . . . 237

A.1.2 Two-zonemeantemperaturemodel. . . 238

A.2 SAAB2.3LNAGeometricdata. . . 239

A.3 Parametersinsingle-zonemodel . . . 240

A.4 Creviceenergyterm . . . 242

A.5 Simpleresidualgasmodel . . . 245

A.6 Fuelcompositionsensitivityof

γ

. . . 246

A.6.1 BurnedmixtureHydrocarbons . . . 246

A.6.2 BurnedmixtureAlcohols . . . 247

A.6.3 Unburnedmixtures. . . 250

A.6.4 Partiallyburnedmixture inuence oncylinder pressure . . . 250

A.7 Thermodynamicpropertiesforburnedmixture . . . 253

A.8 Thermodynamicpropertiesforpartially burnedmixture 254 B Compressionratio estimationfurther details 263 B.1 Taylorexpansionsforsublinearapproach. . . 263

B.2 VariableProjectionAlgorithm. . . 264

B.3 SVCGeometricdata . . . 265

B.4 Parametersinsingle-zonemodel . . . 265

C Prior knowledge approachfurther details 267 C.1 Levenberg-Marquardtmethod . . . 267

C.1.1 Minimizingpredictionerrorsusingalocaloptimizer268 C.2 Linearexample . . . 272

C.2.1 Linearexampleformethods1and2 . . . 274

C.3 Motivation forM2:3+ . . . 275

C.4 L850Geometricdata. . . 279

C.5 Parametersinsingle-zonemodelmotoredcycles. . . . 279

C.6 Parametersinsingle-zonemodelredcycles . . . 279

C.7 Complementaryresultsforpriorknowledgeapproach . . 282

D Notation 289 D.1 Parameters . . . 289

D.1.1 Heattransfer . . . 289

D.1.2 Enginegeometry . . . 290

D.1.3 Enginecycle . . . 290

D.1.4 Thermodynamicsandcombustion . . . 291

D.1.5 Parameterestimation . . . 292

D.2 Abbreviations . . . 293

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1

Introduction

Internal combustion engines havebeen the primary machine for

gen-erating work inmobile applications formorethanacentury, theyare

also continuing to be of high interest due to the high energy density

of the fuelsand their possibility togivegoodtotal fuelconsumption.

Continuous improvements and renements are made to meet the

in-creasing performance demands from customers and legislators, where

bothemissionsandtotalsystemeconomyareimportant.

Emission regulations from the legislators provide a hard limit on

the designthey must be met. Today the state-of-the-art technology

forachievinglowemissionsfromcombustionengines,isthegasoline

en-gineequippedwithathree-waycatalyst(TWC).Regulationsfordiesel

enginesarealsocontinuouslybeingmadestricterto reachthoseofthe

gasolineenginewithaTWC.

Developmentandcompetitionbetweenmanufacturersstrivestomeet

the needs ofcustomers and deliverproducts with better performance

bothwithrespecttopowerandfuelconsumption. Emerging

technolo-gieslikethegasturbineandnowthefuelcellposepossibilitiesandgive

a healthy competition, which also drivesthe technology development

ofcombustionenginesforward.

Engineershavemetthechallengesposedbystricteremission

regula-tionsthroughforexamplefundamentalresearchoncombustion,adding

new componentsto morecomplex systems,aswell asoptimizationof

total systemperformance. Engine systemsare becoming increasingly

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also requiredto handleand integratethese technologies. Some

exam-plesofpromisingtechniquesforsparkignited(SI)enginesarevariable

valveactuationandvariablecompressionratio. Bothoftheseexemplify

technologiesthatcontrolthedevelopmentofthein-cylinderprocess

di-rectly and whereit isof importance to get accurateknowledge about

thecombustionprocess. Thecombustionprocessandotherin-cylinder

processesaredirectlyreectedin themeasuredcylinderpressure,and

usedasastandardtoolfortuningandoptimizingengineperformance.

Thisisofcoursealsoimportantforconventionalengines.

In-cylinder pressuremodeling and estimation

The in-cylinder pressure is important since it contains information

abouttheworkproductionin thecombustionchamberandthus gives

important insight into the control and tuning of the engine. Being

able to accurately model and extract information from the cylinder

pressureis importantforthe interpretationand validity ofthe result.

Researchersandengineersstrivetoextractasmuchinformationas

pos-siblefromthecombustionchamberthroughthein-cylinderpressureand

models of dierent complexity exist for interpretation of the cylinder

pressures. Here thefocusis on single-zonemodels that treats the

in-cylinder contentsas asingle zoneand single uid. These models can

describethecylinderpressurewellandhavealowcomputational

com-plexity, which is also animportant parameterwhen analyzing engine

data.

Dueto theshorttime scalesoftheprocessasequenceof

measure-ments on an engine gives hugeamounts of data. These large sets of

datahavetobeanalyzedeciently,systematically,and withgood

ac-curacy. in-cylinderpressureanalysis,themostecientmodelsarethe

singlezonemodels,andtheaccuracyoftheseisthetopicofthethesis.

The foundation for the analysis of themodel is the rst law of

ther-modynamics where the relation between work, volume, pressureand

temperatureisdescribed throughtheratio ofspecic heats. Analyses

thathavebeenperformedshowthatthespecicheatratioisofhigh

im-portanceforthemodelandthereforethismodelcomponentisstudied

in great detail. Therefore,therstpartofthe thesisisonsingle-zone

heat releasemodeling, where thespecic heat ratiomodel constitutes

akeypart.

In-cylinder pressuremodels in general havea number of

parame-ters, thathavetobedetermined. Foranaccuratein-cylinderpressure

analysis itisnecessary,but notsucientaswillbeshownlaterin the

thesis, that thedierence betweenthemeasuredand modeled data is

small. To minimize this dierence, in agiven measure, requires that

theparametersareestimated, andthisistheestimationproblem.

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mean-ing, the userusually hasan expectation orprior knowledge of either

whatvaluestheparametersshouldhave,oratleasttherangeinwhich

they should be in. The usermightevenknow which parametersthat

are most certain. These are examplesof information that come from

our prior knowledge. Inthe third part of the thesis it is shown how

suchinformationcanbeincorporatedin theestimationproblem.

Compressionratio estimation

Thethemeinthethesisiscylinderpressureandthesecondparttreats

compressionratioestimationbasedonmeasuredcylinderpressuredata.

Thisparticularproblemisdirectlymotivatedbythevariable

compres-sionengine,where thecompressionratiocanbechangedcontinuously

to eliminate an important design trade-o made in conventional

en-gines. Highcompression ratiosgivegoodengineeciencybut athigh

loadsahighcompressionratiocanresultinenginedestructionthrough

engineknock. Insuchanenginethecompressionratioischanged

con-tinuouslytogetthebestperformancefromtheengine. Whentheengine

is drivenatlowloads ahighcompressionratiois selectedfor good

ef-ciency and at high loads a low compression ratio is used to reduce

engineknock. Compressionratioestimationis studiedforseveral

rea-sonswhere themostimportantis fordiagnosticpurposes. A toohigh

compression ratiocan leadtoenginedestructionwhileatoolow

com-pressionratiogivesatoohighfuelconsumption.

Four dierent methods for compression ratio estimation are

pro-posedandevaluated. Theresearchwasmotivatedbythevariable

com-pressionengine,butthemethodsaregenerallyapplicableandcanalso

beusedonconventionalenginestogetabettervalueofthecompression

ratiofromexperimental data.

1.1 Outline

An outlineofthethesisin termsofshortsummariesofeachchapteris

givenbelowandindicatesthescopeofeachchapter. Thenotationused

is summarized in appendix D, where theparameters are givenin

ap-pendixD.1,andtheabbreviationsaresummarizedinappendixD.2. In

thethesisvariousevaluationcriteriaareused,andtheyaresummarized

in appendixD.3.

Chapter 2: Anoverview of single-zoneheat-release models

Thischapterservesasanintroductiontosingle-zoneheatrelease

mod-eling. Firstthebasisandassumptionsmadeforsingle-zoneheatrelease

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mod-elsarepresented. Thesearecomparedwithrespecttotheircomputed

heat releasetracegivenacylinderpressuretrace.

Chapter 3: Heat-releasemodelcomponents

The model components used in the most descriptive single-zoneheat

releasemodel in chapter 2, theGatowski et al.(1984)model, are

de-scribed. Themodel componentsof the other three heat release

mod-els form a subsetof these. Foreach model component, a method to

initializethemodelcomponentparametersisgiven. Thesensitivityin

cylinderpressureforeachoftheseparametersistheninvestigated. The

chapterendswithasummaryoftheequations,parameters,inputsand

outputsoftheGatowskietal. model.

Chapter 4: A specic heat ratio model for single-zone heat

release models

Anaccuratespecicheatratiomodelisimportantforanaccurateheat

releaseanalysis. Thissincethespecic heat ratiocouplesthesystems

energyto other thermodynamicquantities. This chapter therefore

in-vestigates models of the specic heat ratio for the single-zone heat

release model developed by Gatowskiet al. (1984). The objective is

to ndamodelaccurateenoughto onlyintroduceacylinderpressure

modelingerrorintheorderofthecylinderpressuremeasurementnoise,

while keepingthecomputationalcomplexityat aminimum. Basedon

assumptionsoffrozenmixturefortheunburnedmixtureandchemical

equilibriumfortheburnedmixture,thespecicheatratioiscalculated

usingafullequilibriumprogramforanunburnedandaburnedair-fuel

mixture,andcomparedtoalreadyexistingandnewlyproposedmodels

of

γ

. It is assumed that ageneralsingle-zone heat releasemodel can beusedasareferencemodel.

Theevaluationisperformedintermsofmodelingerrorin

γ

andin cylinder pressure. Theimpact each

γ

-model hason the heat release, in termsof estimated heat releaseparametersin the Vibefunction is

illustrated. Theinuenceoffuelcomposition,air-fuelratioandresidual

gascontentisalsoinvestigated.

LargepartsofthematerialinthischapterandinappendixAhave

previouslybeenpublishedinKleinandEriksson(2004c)andKleinand

Eriksson(2004a). AppendixAcontainsfurtherdetailsand

argumenta-tionthatsupportthedevelopmentofthespecicheatratiomodels,and

givesabackgroundandathoroughexplanation ofsomeofthedetails

(21)

Chapter 5: Compressionratioestimationwith focuson

mo-tored cycles

Thepurposeofthischapteristoestimatethecompressionratiogiven

a cylinder pressure trace, in order to diagnose the compression ratio

if it e.g. gets stuck at a too high or too low ratio. Four methods

for compressionratioestimationbasedoncylinderpressuretracesare

developed and evaluated for both simulatedand experimental cycles.

A sensitivity analysis ofhowthemethods performwhen subjectedto

parameterdeviationsincrankanglephasing,cylinderpressurebiasand

heattransferisalsomade.

In appendix B further details and argumentation on compression

ratio estimation formotored cyclesare given, andit servesasa

com-plement to this chapter. Chapter 5 together with appendix B is an

editedversionofKleinetal.(2006),whichitselfisbasedonKleinetal.

(2004)andKleinandEriksson(2005b).

Chapter 68: Using prior knowledge for single-zone heat

re-lease analysis

Twomethods thattakeparameterpriorknowledgeintoaccount,when

performing parameter estimation on the Gatowski et al. model, are

presented. The application in mind is atool forcylinder pressure

es-timation that is accurate, systematic and ecient. The methods are

describedindetail,anditisshownhowtoincorporatetheprior

knowl-edgeinasystematicmanner. Guidelinesofhowtodeterminetheprior

knowledgefor aspecic applicationare then given. Theperformance

of the two methods is evaluated for both simulated and

experimen-tally measured cylinder pressure traces. These evaluations are made

in chapter 7: Results andevaluation for motoredcycles and in

chap-ter 8: Results and evaluation for red cycles. Appendix C contains

further details and argumentation that support the developmentand

evaluationofthetwoparameterestimationmethods.

Experimentaland simulatedengine data

DuringtheprojectdierentengineshavebeenavailableintheVehicular

Systems enginelaboratory. Thereforethesimulatedandexperimental

dataused inthechaptersarefromdierentengines.

Chapters 24 use a naturally aspirated

2.3L

engine from SAAB, and its geometry is given in appendix A.2. In chapter 5 the SAAB

Variable Compression(SVC)engineis used, with the geometry given

in appendixB.3. Theresultsforchapters68arebasedondatafrom

(22)

1.2 Contributions

Thefollowinglistsummarizesthemaincontributionsofthisthesis:

•

Theinterrelationbetweenmodelsin thesingle-zoneheatrelease modelfamilyis shown. A methodforndingnominal valuesfor

allparametersthereinissuggested.

•

Itisshown thatthespecic heatratiomodelisthemost impor-tantcomponentincylinderpressuremodeling.

•

Theimportanceofusingthecylinderpressureerrorasameasure ofhowwellaspecic heatratiomodelperformsispinpointed.

•

A new specic heat ratio model to be used primarily in single-zoneheatreleasemodels. This modelcaneasilybeincorporated

withthe Gatowskiet al.-model, and reduces themodeling error

to be of the same order as the cylinder pressure measurement

noise.

•

Four methods forestimating the compression ratioindex, given a cylinder pressure trace are proposed. One method is

recom-mendedfor its accuracy, while another is preferablewhen

com-putationaleciencyisimportant.

•

Twomethodsofusingpriorknowledgeapplied tothein-cylinder pressureestimationproblemarepresentedandevaluated. Forthe

second method, it isshownthat priorknowledgewith

individu-allyset parameteruncertainties yieldsmoreaccurateandrobust

estimates.

1.3 Publications

Intheresearchworkleadingtothisthesis,theauthorhaspublisheda

licentiatethesisandthefollowingpapers:

Journal papers:

•

M. Klein, L.Eriksson, and Y. Nilsson (2003). Compression es-timation from simulated and measured cylinder pressure. SAE

2002TransactionsJournalofEngines,2002-01-0843,111(3),2003.

•

M.KleinandL.Eriksson(2005a). Aspecicheatratiomodelfor single-zoneheatreleasemodels. SAE2004 Transactions Journal

ofEngines, 2004-01-1464,2005.

•

M.Klein, L.Eriksson andJ. Åslund (2006). Compressionratio estimationbasedoncylinderpressuredata. Control Engineering

(23)

Conference papers:

•

M. Klein, L. Eriksson and Y. Nilsson (2002). Compression es-timationfrom simulated and measuredcylinder pressure.

Elec-tronic engine controls, SP-1703, SAE Technical Paper

2002-01-0843. SAEWorldCongress,Detroit, USA,2002.

•

M. Klein and L. Eriksson (2002). Models, methods and per-formance when estimating the compression ratio based on the

cylinder pressure. Fourth conference on Computer Science and

SystemsEngineeringinLinköping(CCSSE),2002.

•

M. Klein and L. Eriksson (2004c). A specic heat ratio model forsingle-zoneheatreleasemodels. Modeling of SIengines,

SP-1830,SAETechnicalPaper2004-01-1464.SAEWorldCongress,

Detroit,USA,2004.

•

M.Klein, L.Eriksson andJ. Åslund (2004). Compressionratio estimation based on cylinder pressure data. In proceedings of

IFAC symposium onAdvancesin Automotive Control,Salerno,

Italy,2004.

•

M. Klein (2004). A specic heat ratio model and compression ratioestimation. Licentiatethesis,VehicularSystems,Linköping

University,2004. LiU-TEK-LIC-2004:33,ThesisNo. 1104.

•

M.KleinandL.Eriksson(2004b). Acomparisonofspecicheat ratiomodelsforcylinderpressuremodeling. Fifth conferenceon

Computer Science and Systems Engineering in Linköping

(CC-SSE),2004.

•

M. Klein and L. Eriksson (2005b). Utilizing cylinder pressure data for compression ratio estimation. IFAC World Congress,

Prague,CzechRepublic,2005.

Thefollowingconferencepapershavealsobeenproducedbytheauthor

during theproject,buttheyarenotexplicitlyincludedinthethesis:

•

M.KleinandL.Nielsen(2000). Evaluating someGain Schedul-ingStrategies in Diagnosisof aTankSystem. Inproceedings of

IFACsymposiumonFaultDetection,SupervisionandSafetyfor

TechnicalProcesses,Budapest, Hungary,2000.

•

M. Klein and L. Eriksson (2006). Methods for cylinder pres-surebasedcompressionratioestimation. ElectronicEngine

Con-trol, SP-2003, SAE Technical paper 2006-01-0185. SAE World

(24)

(25)

(26)

(27)

2

An overview of single-zone

heat-release models

Whenanalyzingtheinternalcombustionenginethein-cylinderpressure

hasalwaysbeenanimportantexperimentaldiagnostic,duetoitsdirect

relation to the combustion and work producing processes(Chunand

Heywood,1987;CheungandHeywood,1993). Thein-cylinderpressure

reectsthecombustionprocess, thepistonworkproduced onthegas,

heattransfertothechamberwalls,aswellasmassowsinandoutof

creviceregionsbetweenthepiston,ringsandcylinderliner.

Thus, whenanaccurate knowledgeofhowthecombustionprocess

propagates throughthe combustion chamberis desired, each of these

processesmustberelatedto thecylinder pressure,so thecombustion

process can be distinguished. The reductionof the eects of volume

change,heat transfer,andmass losson thecylinderpressureiscalled

heat-releaseanalysis andisdonewithin theframeworkoftherstlaw

of thermodynamics. In particular during the closed part of the

en-gine cycle when the intakeand exhaust valves are closed. The most

common approach is to regardthecylinder contents asasinglezone,

whose thermodynamicstateand propertiesare modeledasbeing

uni-form throughout the cylinder and representedby average values. No

spatialvariationsareconsidered,andthemodelisthereforereferredto

as zero-dimensional. Models for heat transferand crevice eects can

easilybeincludedinthisframework. Anotherapproachistodoamore

detailed thermodynamic analysis byusing amulti-zone model, where

thecylinderisdividedintoanumberofzones,dieringincomposition

(28)

temper-ature, andthe pressureisthe samefor allzones, see for e.g.(Nilsson

andEriksson,2001).

Thegoalofthischapteristoshowthestructureofdierent

single-zone heat-release model families and how they are derived. The

dis-cussionofdetailsinthemodelcomponentsarepostponedtochapter3,

sincetheymightdistractthereadersattentionfromthegeneral

struc-tureofthemodelfamily. Chapter3givesamorethoroughdescription

ofthemodel components.

Single-zonemodelsforanalyzingtheheat-releaserateand

simulat-ing the cylinder pressure are closely connected; they share the same

basic balanceequation and canbeinterpreted aseach others inverse.

Theyarebothdescribedbyarstorderordinarydierentialequation

that hastobesolved. Inheatreleasemodelsapressuretraceis given

as input and the heat releaseis the output, while in pressuremodels

a heat release trace is the input and pressure is the output. For a

givenheat-releasemodel anequivalentpressuremodel isobtained by

reordering thetermsin the ordinarydierentialequation. Since they

aresocloselyconnecteditisbenecialtodiscussthem together.

2.1 Model basis and assumptions

The basis for the majority of the heat-release models is the rstlaw

of thermodynamics; the energy conservation equation. For an open

systemitcanbestatedas

dU =

Q −

W +

X

i

h

i

dm

i

,

(2.1)

where

dU

isthechangeininternalenergyofthemassinthesystem,

Q

istheheattransportedtothesystem,

W

istheworkproducedbythe systemand

P

i

h

i

dm

i

istheenthalpyuxacrossthesystemboundary. Possiblemassows

dm

i

are: 1)owsinandoutofthevalves;2)direct injectionoffuelintothecylinder;3)owsinandoutofcreviceregions;

4) pistonring blow-by. Themassow

dm

i

ispositiveforamassow into the system and

h

i

is the mass specic enthalpy of ow

i

. Note that

h

i

isevaluated atconditionsgivenby thezonethemasselement leaves.

Asmentionedearlier,single-zonemodelsisourfocusatthemoment,

sowewill now look into thosein moredetail. Somecommonly made

assumptionsforthesingle-zonemodelsare:

•

the cylinder contents and the state is uniform throughout the entirechamber.

(29)

Wp

Qhr

Qht

Mcr

Piston

Unburned

Burned gas

Spark plug

gas

Figure 2.1: Schematic of thecombustion processin thecylinder, that

denesthesignconventionusedin thepressureandheat-release

mod-els.

•

the heat released from the combustion occurs uniformly in the chamber.

•

thegasmixtureisanidealgas.

Considerthecombustionchambertobeanopensystem(single zone),

with the cylinder head, cylinder wall and piston crown asboundary.

Figure 2.1 showsa schematic of the combustion chamber, where the

signconventionsusedin pressureandheat-releasemodelsaredened.

The change in heat

Q

consists of thereleased chemical energyfrom thefuel

Q

ch

,whichisaheataddingprocess,andtheheattransferto the chamberwalls

Q

ht

, which isaheat removingprocess. Theheat transport is therefore represented by

Q =

Q

ch

−

Q

ht

. Note that theheat transfercoolsthegasesat mosttimes,but in someinstances

it heats the air-fuel mixture. The only work considered is the work

donebytheuidonthepiston

W

p

anditisconsideredpositive, there-fore

W =

W

p

. The rst lawof thermodynamics (2.1) canthen be rewrittenas

Q

ch

= dU

s

+

W

p

−

X

i

h

i

dm

i

+

Q

ht

.

(2.2)

The pistonwork

W

p

isassumed to bereversibleand canbewritten as

W

p

= pdV

. For anidealgas,thechange insensible energy

dU

s

is afunctionofmeanchargetemperature

T

only,thus:

(30)

whichin itsdierentiatedformbecomes:

dU

s

= m

tot

c

v

(T )dT + u(T )dm

tot

,

(2.4)

where

m

tot

is the charge mass, and

c

v

=

∂u

∂T

V

is the mass specic

heat at constant volume. The mean temperature is found from the

idealgaslawas

T =

pV

m

tot

R

,anditsdierentiatedformis

dT =

1 m

tot

R

(V dp + pdV − RT dm

tot

),

(2.5)

assuming

R

to be constant. Forreading convenience, thedependence of

T

in

c

p

,

c

v

and

γ

isoftenleftoutin thefollowingequations. Equa-tion(2.2)cannowberewrittenas

Q

ch

=

c

v

R

V dp +

c

v

+ R

R

p dV + (u − c

v

T )dm

tot

−

X

i

h

i

dm

i

+

Q

ht

,

(2.6)

using equations (2.4) and (2.5). The specic heat ratio is dened as

γ =

c

p

c

v

and with theassumptionof anidealgas themassspecic gas

constant

R

canbewrittenas

R = c

p

−c

v

,yieldingthatthemassspecic heat atconstantvolumeisgivenby

c

v

=

R

γ − 1

.

(2.7)

The mass specic heat is the amount of energy that must be added

or removedfrom the mixture to change its temperature by 1 K at a

given temperature and pressure. It relates internal energy with the

thermodynamicstatevariables

p

and

T

,andisthereforeanimportant partoftheheat releasemodeling. Inserting(2.7)into(2.6)resultsin

Q

ch

=

1 γ − 1

V dp+

γ

γ − 1

p dV +(u−

RT

γ − 1

)dm

tot

−

X

i

h

i

dm

i

+

Q

ht

.

(2.8)

Fromthisequation,fourdierentsingle-zonemodelswithvariouslevels

of complexitywill bederived. Tobeginwiththeisentropicrelation is

derived,thenthepolytropicmodelisformulatedandthismodelforms

thebasisforcalculatingthemassfractionburnedwiththe

Rassweiler-Withrowmethod (Rassweilerand Withrow,1938). Secondly,amodel

forcomputing theapparentheatreleaserstproposedin Kriegerand

Borman (1967)will bederived. Thirdly,the pressureratiodeveloped

byMatekunas(1983)isshortlysummarized. Finally,amodelincluding

(31)

The isentropic processand isentropic relation

Inmanysituationsrealprocessesarecomparedtoidealprocessesandas

comparisontheisentropicprocessisnormallyused. Fromtheisentropic

processanisentropicrelationcanbefoundbyintegratingtherstlaw

ofthermodynamics(2.8). Theassumptionsare:

•

Nomasstransfer:Creviceeectsandleakagestothecrankcase (of-tencalled blow-by)arenon-existent,i.e.

dm

tot

= dm

i

= 0

.

•

Neitherheat transfernorheat release:

- Heattransferis notexplicitlyaccountedfor,i.e.

Q

ht

= 0

, andthus

Q =

Q

ch

−

Q

ht

=

Q

ch

.

- Using the fact that there is no release of chemical energy

during the compression phase prior to the combustion or

duringtheexpansionphaseafter thecombustion, therefore

Q = 0

fortheseregions.

•

Thespecicheat ratio

γ

isconstant.

Thersttwoassumptionsyieldthat (2.8)canbeexpressedas:

dp = −

γ p

_V

dV.

(2.9)

From (2.9) and the last assumption above the isentropic relation is

foundbyintegratingas

pV

γ

= C =

constant (2.10)

bynotingthat

γ

isconsideredtobeconstant.

2.2 Rassweiler-Withrow model

TheRassweiler-Withrowmethodwasoriginallypresentedin 1938and

many still use the method for determining the mass fraction burned,

due to itssimplicity andit beingcomputationallyecient. Themass

fractionburned

x

b

(θ) =

m

b

(θ)

m

tot

istheburnedmass

m

b

(θ)

normalizedby thetotalchargemass

m

tot

,anditcanbeseenasanormalizedversionof theaccumulatedheat-releasetrace

Q

ch

(θ)

suchthatit assumesvalues in theinterval [0, 1]. Therelation betweenthemass fraction burned

and the amount of heat released can be justied by noting that the

energyreleasedfrom asystemis proportionaltothemassoffuelthat

isburned. Theinputtothemethodisapressuretrace

p(θ

j

)

wherethe crank angle

θ

ateach sample

j

isknown(or equivalently; thevolume is known at each sample)and theoutput isthe massfraction burned

(32)

Acornerstoneforthemethodisthefact that pressureandvolume

data during compression and expansion canbe approximated by the

polytropicrelation

pV

n

=

constant

.

(2.11)

This expression comesfrom the isentropic relation(2.10) but

γ

is ex-changedforaconstantexponent

n ∈ [1.25, 1.35]

. Thishasbeenshown to giveagood t to experimental data forbothcompression and

ex-pansionprocessesin anengine(Lancaster etal.,1975). Theexponent

n

is termed the polytropic index. It diers from

γ

sincesome of the eects of heat transferare includedimplicitly in

n

. It is comparable to the averagevalue of

γ

u

forthe unburned mixtureduring the com-pression phase, priorto combustion. But due to heat transfer to the

cylinderwalls,index

n

isgreaterthan

γ

b

fortheburnedmixtureduring expansion(Heywood,1988,p.385).

Whenconsideringcombustionwhere

Q =

Q

ch

6= 0

,equation(2.8) can berewrittenas

dp =

n − 1

V

Q −

n p

V

dV = dp

c

+ dp

v

,

(2.12) where

dp

c

is the pressure change due to combustion, and

dp

v

is the pressure change due to volume change, compare

dp

in (2.9). In the Rassweiler-Withrow method (Rassweilerand Withrow,1938), the

ac-tualpressurechange

∆p = p

j+1

−p

j

duringtheinterval

∆θ = θ

j+1

−θ

j

, is assumed to be made upof a pressurerisedue to combustion

∆p

c

, andapressureriseduetovolumechange

∆p

v

,

∆p = ∆p

c

+ ∆p

v

,

(2.13)

whichisjustiedby(2.12). Thepressurechangeduetovolumechange

duringtheinterval

∆θ

isapproximatedbythepolytropicrelation(2.11), whichgives

∆p

v

(j) = p

j+1,v

− p

j

= p

j

V

j

V

j+1

n

− 1

.

(2.14)

Applying

∆θ = θ

j+1

− θ

j

,(2.13)and(2.14)yieldsthepressurechange due tocombustionas

∆p

c

(j) = p

j+1

− p

j

V

j

V

j+1

n

.

(2.15)

By assumingthat thepressurerisedue to combustion in theinterval

∆θ

isproportionaltothemassofmixturethatburns,themassfraction burnedattheendofthe

j

'thintervalthusbecomes

x

b,RW

(j) =

m

b

(j)

m

b

(total)

=

P

j

k=0

∆p

c

(k)

P

M

k=0

∆p

c

(k)

,

(2.16)

(33)

−100

−50

0

50

100

0

0.5

1

1.5

2 Crank angle [deg ATDC]

Cylinder pressure [MPa]

−100

−50

0

50

100

0

0.5

1 Crank angle [deg ATDC]

Mass fraction burned [−]

Figure 2.2: Top: Fired pressure trace (solid) and motored pressure

trace(dash-dotted). Bottom: Calculatedmassfraction burnedprole

usingtheRassweiler-Withrowmethod.

where

M

is the total number of crank angle intervals and

∆p

c

(k)

is found from(2.15). Theresultfrom amassfractionburnedanalysisis

shownin gure 2.2, where themass fraction burnedproleis plotted

togetherwith thecorrespondingpressuretrace. Intheupperplottwo

cylinder pressuretraces,one from aredcycle(solid)and onefrom a

motoredcycle(dash-dotted)aredisplayed. Whenthepressurerisefrom

thecombustionbecomesvisible,i.e. itrisesabovethemotoredpressure,

the mass fraction burned prole starts to increase above zero. The

massfractionburnedproleincreasesmonotonouslyasthecombustion

propagates throughthecombustion chamber. Equations(2.15)-(2.16)

form theclassicalRassweiler-Withrowmassfractionburnedmethod.

If instead a heat-release trace is sought, the pressurechange due

to combustion in(2.12),

dp

c

=

n−1

V

Q

, canberewrittenand approxi-matedby

∆Q

RW

(j) =

V

j+1/2

n − 1

∆p

c

(j),

(2.17) wherethevolume

V

duringinterval

j

isapproximatedwith

V

j+1/2

(the volumeatthecenteroftheinterval),and

∆p

c

(j)

isfoundfrom(2.15). Theheat-releasetraceisthenfoundby summation. This methodwill

be called the Rassweiler-Withrow heat release method. The

calcu-lated heatreleaseapproximatesthereleasedchemicalenergyfrom the

fuelminusenergy-consumingprocessessuchastheheattransfertothe

(34)

−100

−50

0

50

100

0

200

400 Crank angle [deg ATDC]

Heat released Q [J]

−100

−50

0

50

100

0

0.5

1 Crank angle [deg ATDC]

Mass fraction burned [−]

Figure 2.3: Calculated heat-release trace (upper) and mass fraction

burned trace (lower), using the apparent heat release (solid) and

Rassweiler-Withrow(dash-dotted)methods.

where non-existent,the heatreleasewould correspond directly to the

amountofenergyaddedfrom thechemicalreactions. Theheat-release

tracefor thesamedata asin gure2.2 isdisplayedin theupperplot

ofgure2.3asthedash-dottedline.

2.3 Apparent heat release model

The work by Krieger and Borman (1967) was derived from the rst

law of thermodynamics and called the computation of apparentheat

release. It is also called the computation of net heat release. The

method takes neither heat transfer nor crevice eects into account,

thus

Q

ht

is lumped into

Q =

Q

ch

−

Q

ht

and

dm

tot

= dm

i

= 0

in (2.8). Hence,theapparentheatrelease

Q

canbeexpressedas:

Q

AHR

=

1 γ(T ) − 1

V dp +

γ(T )

γ(T ) − 1

p dV,

(2.18)

which is the sameexpression as theRassweiler-Withrowmethod was

based upon (2.12), but assuming that

γ(T ) = n

. The mass fraction burned

x

b,AHR

iscomputedbyintegrating(2.18)andthennormalizing withthemaximumvalueoftheaccumulatedheatrelease

Q

AHR

,i.e.

x

b,AHR

(θ) =

Q

AHR

(θ)

max Q

AHR

=

R

Q

AHR

dθ

max Q

AHR

.

(2.19)

(35)

Method

θ

10 θ

50 θ

85 ∆θ

b

Rassweiler-Withrow -6.4 9.8 26.9 33.3

Apparentheatrelease -4.5 11.0 25.2 29.8

Table2.1: Crankanglepositionsfor10%,50% and85% mfbaswell

astherapidburnangle

∆θ

b

= θ

85 − θ

10

,allgivenindegreesATDCfor themassfractionburnedtracein gure2.3.

TheRassweiler-Withrowmethodin(2.16)isadierenceequation,and

thiscausesanquantizationeectcomparedtotheordinarydierential

equation givenin(2.18). Thenetheat-releasetraceandmassfraction

burnedprolefromtheKriegerandBormanmodelaresimilartothose

from theRassweiler-Withrowmethod,theformerbeingphysicallythe

moreaccurateone. Oneexampleisgiveningure2.3,wheretheupper

plotshowsthenetheat-releasetracesandthelowerplotshowsthemass

fractionburnedtraces,fromthecylinderpressureingure2.2. Forthis

particular case, the Rassweiler-Withrow method yields a slowerburn

ratecomparedtotheapparentheatreleasemethodforthesamedata.

This is reected in the crank angle for 50 % mfb

θ

50

, which is 11.0 [degATDC]fortheapparentheatreleasemethodand9.8[degATDC]

forthe Rassweiler-Withrowmethod. Table2.1 summarizesthe

crank-anglepositionsfor10%,50%and85%mfbaswellastherapidburn

angle duration

∆θ

b

, and shows that the Rassweiler-Withrowmethod yieldsashorterburndurationforthisparticularcase. Therapidburn

angledurationisdened as

∆θ

b

= θ

85 − θ

10

.

Theshorterburndurationisalsoreectedintheheatreleasetrace,

and the dierence is due to the assumptions on

n

and

V

j+1/2

in the Rassweiler-Withrowmethod. The massfraction burnedprole is

cal-culated assumingthat themass of burnedmixture is proportionalto

theamountofreleasedchemicalenergy.

Pressure simulation

An ordinarydierentialequationforthepressurecanbesimulatedby

solving(2.18)forthepressuredierential

dp

:

dp =

(γ(T ) − 1)

Q − γ(T ) p dV

V

.

(2.20)

Whenperformingaheat-releaseanalysis thepressureis usedasinput

and the heat release is given asoutput, and when the pressuretrace

isbeingsimulatedtheheat-releasetraceisgivenasinput. Thereforea

cylinderpressuresimulationbasedon(2.20),canbeseenastheinverse

of the heat release analysis (2.18). The only additional information

(36)

2.4 Matekunas pressure ratio

Thepressureratioconceptwasdevelopedby Matekunas(1983)andit

isacomputationallyecientmethodtodetermineanapproximationof

themassfractionburnedtrace. Thepressureratioisdenedastheratio

of thecylinderpressurefrom aredcycle

p(θ)

andthe corresponding motoredcylinderpressure

p

0 (θ)

:

P R(θ) =

p(θ)

p

0 (θ)

− 1.

(2.21)

Thepressureratio(2.21)isthennormalizedbyitsmaximum

P R

N

(θ) =

P R(θ)

max P R(θ)

,

(2.22)

whichproducestracesthataresimilartothemassfractionburned

pro-les. The dierence betweenthem has been investigated in Eriksson

(1999),andfortheoperatingpointsused,thedierenceinpositionfor

P R

N

(θ) = 0.5

wasin theorderof 1-2degrees. Thissuggests

P R

N

(θ)

can be used as the mass fraction burned trace

x

b,MP R

. The cylin-der pressure in the upper plot of gure 2.4 yields the pressure ratio

PR (2.21)giveninthemiddleplot,andanapproximationofthemass

fraction burnedin thelowerplot.

2.5 Gatowski et al. model

Amorecomplexmodelistoincorporatemodelsofheattransfer,crevice

eects and thermodynamic properties of thecylinder chargeinto the

energy conservationequation (2.8). Thiswasdonein Gatowskiet al.

(1984),whereaheat-releasemodelwasdevelopedandappliedtothree

dierentenginetypes,among thoseaspark-ignitedengine.

Crevice eect model

Crevicesaresmall,narrowvolumesconnectedtothecombustion

cham-ber. Duringcompressionsomeofthechargeowsintothecrevices,and

remain there until the expansionphase, when most of the charge

re-turnstothecombustionchamberandsomechargestaysinthecrevices.

Also,asmallpartofthechargeinthecylinderblowsbythetoppiston

ring, before it either returns to thecylinder orends up in the

crank-case,aphenomenatermedblow-by. Sincetheamecannotpropagate

into thecrevices,thechargeresidingin thecrevicesisnotcombusted.

The temperaturein thecrevicesareassumed to beclose tothe

(37)

−100

−50

0

50

100

0

1

2 Pressure [MPa]

−100

−50

0

50

100

0

1

2

3 PR [−]

−100

−50

0

50

100

0

0.5

1 Crank angle [deg ATDC]

x

b,MPR

[−]

Figure 2.4: Top: Fired pressure trace (solid) and motored pressure

trace (dash-dotted), same asin the upper plot of gure2.2. Middle:

Matekunaspressureratio

P R(θ)

(2.21). Bottom:Computedmass frac-tionburnedproleusing (2.22).

1988, p.387). This hastheresultthat during theclosed phasea

sub-stantialamountofchargecouldbetrappedin thecrevices. According

to Gatowski et al. (1984), the crevicevolumes constitute asmuch as

1-2percentoftheclearancevolumeinsize. Itisalsoshownthatdueto

thetemperaturedierenceinthecylinderandinthecrevices,asmuch

as10(mass)percentofthechargecouldthenbetrappedincrevicesat

peakpressure.

Themodel in Gatowskiet al.(1984)assumesthat allcrevicescan

be modeled as a single aggregateconstant volume

V

cr

, and that the charge in the crevice assumes the wall temperature

T

w

and has the same pressure as the combustion chamber. The ideal gas law thus

givesthefollowingexpressionforthemassinthecrevice

m

cr

=

p V

cr

R T

w

=⇒ dm

cr

=

V

cr

R T

w

dp,

(2.23)

where itisassumedthat

T

w

and

R

areconstant.

Here,wewillonlyconsider spark-ignitionengineswith apremixed

air-fuel chargeduring the closed partof the enginecycle. Blow-byis

alsoneglected,hencetheonlymassowoccurringistheoneinandout

ofthecreviceregion. Massbalancethusyields

dm

tot

= dm

i

= −dm

cr

.

(2.24)

(38)

thenequation(2.8)canthenberewrittento:

Q

ch

=

1 γ−1

V dp +

γ

γ−1

p dV + (

RT

γ−1

+ h

0 _{− u)}

V

cr

RT

w

dp +

Q

ht

=

_γ−1

1 V dp +

_γ−1

γ

p dV + (

_γ−1

T

+ T

0 ₊

u

0 −u

R

)

V

cr

T

w

dp +

Q

ht

.

(2.25)

Togetacylinderpressuremodel,equation(2.25)canbesolvedforthe

pressuredierentialyieldingthefollowingexpression:

dp =

Q

ch

−

γ

γ−1

p dV −

Q

ht

1 γ−1

V +

V

cr

T

w

T

γ−1

+ T

0 +

u

0 _−u

R

.

(2.26) The enthalpy

h

0

is evaluated at cylinder conditions when the crevice

mass owisoutof thecylinder(

dm

cr

> 0

),and atcreviceconditions otherwise.

Heat transfer model

TheheattransfermodelreliesuponNewton'slawofcooling

˙

Q

ht

= h

c

A ∆T = h

c

A (T − T

w

),

(2.27)

and Woschni (1967) found acorrelation between the convectionheat

transfercoecient

h

c

andsomegeometricandthermodynamic proper-ties

1

,

h

c

=

0.013B

−0.2

_p

0.8 _C

1 u

p

+

C

2 (p−p

p

ref

0 V

)T

ref

V

s

0.8 T

0.55 .

(2.28)

Woschni'sheat transfercorrelation model will be furtherdiscussed in

section3.6. Notethatwhensimulatingheattransferinthecrankangle

domain,

Q

ht

dθ

=

Q

ht

dt

dθ

= ˙

Q

ht

60 2πN

(2.29)

should be used, where theenginespeed

N

[rpm] is assumedconstant in thelast equality.

Model ofthermodynamic properties

Theratioofspecicheats

γ(T )

ismodeledasalinearfunctionof tem-perature

γ

lin

(T ) = γ

300 + b (T − 300).

(2.30) In Gatowski et al. (1984) it is stated that this component is

impor-tant, since it captures how the internal energy varies with

temper-ature. This is an approximation of the thermodynamic properties

1

Thevalueofthe rstcoecientdiersfromtheonein(Woschni,1967),since

(39)

but it isfurther stated that thisapproximationisconsistentwith the

other approximations made in the model. Using

γ(T ) =

c

p

(T )

c

v

(T )

and

R(T ) = c

p

(T ) −c

v

(T )

,togetherwiththelinearmodelof

γ(T )

in(2.30), givesthefollowingexpressionfor

c

v

(T )

:

c

v

(T ) =

R

γ(T ) − 1

=

R

γ

300 + b(T − 300) − 1

.

(2.31)

The onlything remaining in (2.25) to obtainafull descriptionof the

model, is an expression for

u

0 − u

. Remembering that

c

v

= (

∂u

∂T

)

V

,

u

0 _−u

canbefoundbyintegrating

c

v

. Thisdescribesthesensibleenergy changeforamassthatleavesthecreviceand entersthecylinder. The

integrationisperformedas:

u

0 _{− u =}

R

T

0 T

c

v

dT

=

R

b

{ln(γ

300 + b (T

0 _{− 300) − 1) − ln(γ}

300 + b (T − 300) − 1)}

=

R

_b

ln

γ

0 lin

−1

γ

lin

−1

,

(2.32)

where equation(2.31)isused.

Gross heat-releasesimulation

Inserting equations(2.23) to(2.32) into (2.8),yieldsthe following

ex-pressionforthereleasedchemicalenergy:

Q

ch

=

1 γ − 1

V dp +

γ

γ − 1

p dV +

Q

ht

+ (c

v

T + R T

0 ₊

R

b

ln

γ

0 − 1

γ − 1

)

V

cr

R T

w

dp,

(2.33) whichisreformulatedas

Q

ch

=

1 γ − 1

V dp +

γ

γ − 1

p dV

|

{z

}

dQ

net

+

Q

ht

+

(

1 γ − 1

T + T

0 ₊

1 b

ln

γ

0 _{− 1}

γ − 1

)

V

cr

T

w

dp

|

{z

}

dQ

crevice

.

(2.34)

Thisordinarydierentialequationcaneasilybesolvednumericallyfor

theheat-releasetrace,ifacylinderpressuretraceisprovided,together

withaninitialvaluefortheheatrelease.Giventhecylinderpressurein

gure2.2, theheat-releasetracegivenin gure2.5 iscalculated. The

(40)

−100

−50

0

50

100

0

50

100

150

200

250

300

350

400

450

500 Crank angle [deg ATDC]

Heat released Q [J]

Q

_net

Q

ht

Q

_gross

Q

_crevice

Figure2.5: Heat-releasetracefromtheGatowskimodelgiventhe

cylin-derpressurein gure2.2.

during theenginecycle. Thedash-dottedline showstheheatreleased

if not considering the crevice eect, and the dashed line shows the

net heatrelease,i.e.when neitherheattransfernorcreviceeects are

considered. Forthisparticularcase,theheattransferisabout70Jand

thecreviceeect isabout30J,i.e. approximately14and6percentof

thetotalreleasedenergyrespectively.

Cylinder pressuresimulation

Reordering(2.34),givesanexpressionforthepressuredierentialas

dp =

Q

ch

−

γ

γ−1

p dV −

Q

ht

1 γ−1

V +

V

cr

T

w

T

γ−1

+

1 b

ln

γ

0 −1

γ−1

+ T

0 .

(2.35)

Thisordinarydierentialequationcaneasilybesolvednumericallyfor

thecylinderpressure,ifaheat-releasetrace

Q

ch

isprovided,together withaninitialvalueforthecylinderpressure.

2.6 Comparison of heat release traces

The single-zone heat release models presented in the previous

sec-tionsallyielddierentheatreleasetracesforagivencylinderpressure

trace. Thisisshowningure2.6,wheretheheatreleasetracesforthe

(41)

−30

−20

−10

0

10

20

30

40

50

60

0

200

400 Crank angle [deg ATDC]

Heat released Q [J]

−30

−20

−10

0

10

20

30

40

50

60

0

0.5

1 Crank angle [deg ATDC]

Mass fraction burned [−]

Figure 2.6: Upper: Heat-release traces from the three methods,

Gatowski (solid), apparent heat release (dashed) and

Rassweiler-Withrow (dash-dotted), given the cylinder pressure in gure 2.2.

Lower: Mass fraction burned tracescorrespondingto the upper plot

withtheadditionoftheMatekunaspressureratio(dotted).

displayedintheupperplot. Asexpected,theaccumulatedheatrelease

ishigherfortheGatowskimodelsinceitaccountsforheattransferand

creviceeects. Themassfractionburnedtracesdonotdierasmuch,

as displayed in the lowerplot of gure2.6. Forthis operatingpoint,

theapparentheatreleasemodelproducesamassfractionburnedtrace

moreliketheonefoundbytheGatowskimodel,asshownbycomparing

the burn angles given in table 2.2. Note that the heat releasetraces

from the Rassweiler-Withrow, apparent heat release and Matekunas

modelsaresetconstantwhentheyhavereachedtheirmaximumvalues

in gure 2.6. If not, their behaviorwould be similar to thenet heat

releasetrace

Q

net

givenin gure2.5.

Method

θ

10 θ

50 θ

85 ∆θ

b

Rassweiler-Withrow -6.4 9.8 26.9 33.3

Apparentheatrelease -4.5 11.0 25.2 29.8

Matekunas -6.9 9.2 24.0 30.9

Gatowskiet.al. -5.1 10.4 24.4 29.5

Table2.2: Crankanglepositionsfor10%,50% and85% mfbaswell

astherapidburnangle

∆θ

b

= θ

85 − θ

10

,allgivenindegreesATDCfor themassfractionburnedtracesinthelowerplotofgure2.6.

(42)

2.7 Summary

A numberof single-zoneheat releasemodels havebeenderived

start-ing from therstlawofthermodynamics. Thefour modelsdescribed

are then compared and their specic model assumptions are pointed

out. Themost elaborate oneis the Gatowski et al. model, which

in-cludes heat transfer described by Woschni's heat transfer correlation

and crevice eects. This model also assumes that the specic heat

ratio for the cylinderchargecan be described by alinear function in

temperature. Theother threemodels, theRassweiler-Withrowmodel,

the Matekunas pressure ratio and the apparent heat release model,

areall morecomputationallyecientthan theGatowskiet al.model,

merelysincetheylackthemodelingofheattransferandcreviceeect,

aswellashavingaconstantspecicheat ratiofor thersttwocases.

Thiscomputationaleciencyofcoursecomestoacostofless

descrip-tivemodels. Themodel componentsintheGatowskiet al.modelwill

(43)

3

Heat-release model

components

Single-zone zero-dimensional heat-release models were introduced in

the previous chapter, where their structure and interrelations were

discussed. Now the attention is turned to the details of the various

componentsgiven in chapter 2,and especiallythose for theGatowski

et al.-model are treatedmore fully here. Some of the model

compo-nentshavealreadybeenintroducedinsection2.5, butallcomponents

will bemorethoroughlyexplainedandcomparedto othermodel

com-ponentsinsections3.13.7. Itisalsodescribedhowtondinitialvalues

for allparameters. Thesevaluesare usedasinitial valueswhen using

thesingle-zonemodelsforparameterestimation. Inchapters 58they

willalsobeusedasnominalvaluesforxedparameters,i.e.parameters

thatarenotestimated. TheequationsthatformthecompleteGatowski

etal. single-zoneheatreleasemodelareemphasizedbyboxes,andthe

modelissummarizedinsection3.8. Insection3.9thecylinderpressure

sensitivitytotheinitialvaluesoftheparametersisbrieyinvestigated.

3.1 Pressure sensor model

Thein-cylinderpressureismeasuredusingawater-cooledquartz

pres-suretransducer,apiezoelectricsensorthatbecomeselectricallycharged

whenthereisachangeintheforcesactinguponit. Piezoelectric

trans-ducers react to pressurechanges by producing a charge proportional

to thepressurechange. This chargeis then integrated by the charge