.Universit` adiGenovaDipartimento di Ingegneria delle Costruzioni, dellAmbiente e del TerritorioViaMontallegro116145GenovaItalye-mail: matteo.bargiacchi@alice.itUNIVERSIT`ADEGLISTUDIDIGENOVAFACOLT`ADIINGEGNERIATesidiLaureaSpecialisticainINGEGNERIAMECCANICAAnalysisandcontrolofthermoacousticinstabilitiesingasturbinesystemsRelatore CandidatoChiar.moProf. AlessandroBottaro MatteoBargiacchiCorrelatoreDott. GiulioMoriAnnoAccademico2009/2010RiassuntoLenormativeambientalidegliultimiannihannospostatoilimitisulleemis-sioni inquinanti verso valori sempre pi` u stringenti. Basse emissioni di ossidi diazotopossonoessereottenutemedianteunacombustionepremiscelatapovera.Sfortunatamentequestesonopropriolecondizioniidealipercuisipossonoin-staurareinstabilit`atermoacustichecheportanoadunacombustioneinecien-tegenerandovibrazioni cos` intensedamettereinseriopericololaresistenzastrutturaledellamacchina. Comprenderelanaturadiquestifenomeni`emoltoimportante ed `e attualmente uno dei maggiori argomenti di studio riguardo allosviluppoditurbineheavy-dutyeaeronautiche. Inquestatesi`estatosviluppatoeimplementatounmodellolineareaparametri concentrati chiamatoLOMTIingradodifarelucesuiparametricheguidanoquestiprocessisici. Inmododarenderei risultati sucientementeadabili`estataportataatermineunacalibrazionedellageometriabasatasusimulazioni3DconmodelliFEM.Introduzione. Lhumming`eunfenomenocaratterizzatodauttuazioni dirilasciotermicochegeneranoondedi pressioneequindi vibrazioni. Questeonde, viaggiandoindirezionecirconferenzialeinunacameradi combustioneanulare o riettendosi sulle sue pareti, ritornano ad inuenzare la zona di am-ma innescando un processo di retroazione. Diversi criteri sono gi` a stati propostiin letteratura gi`a nel XIX secolo, primo fra tutti il criterio di Rayleigh; purtrop-po, anche se formalmente corretti nellambito delle loro ipotesi, essi non sono difacile implementazione a causa della loro natura integrale. Un modello ispiratoallamodernateoriadel controllodei sistemi`estatosviluppatonellambitodiunprogettodicollaborazionefrailDICATeAnsaldoEnergia. QuestocodiceimplementatoinMATLABprendeil nomedi LOMTI(LowOrderModel forThermoacoustic Instabilities). Il LOMTI `e un modello a parametri concentratilinearecherappresentaunacongurazionerealedellacameradi combustionediunturbogascomeunasuccessionedicondottiinterconnessi.iIlmodellomatematico. I gradi di libert` a del sistema sono il minor numerodi variabili di stato interne xi in grado di rappresentarlo univocamente a tempossato. Lacameradi combustionedel turbogas`eanalizzatamedianteunsetdi equazioni rappresentativedel sistemacontenenti sololevariabili xiched`aluogoallespressione:[M]x = 0 (1)dove[M] `elamatricequadratadeicoecientidix.In accordo con lapproccio a parametri concentrati si ipotizza un usso mediodallecaratteristichecostanti inogni condotto(plenum, 24premixersecame-radi combustione). Quindi ogni grandezzaG(x, r, , t) viene scompostainunapartecostanteG(diversaper ogni condotto) einunaparteuttuanteG

(x, r, , t) 0or q

0 (2.1)whereisthedomaininanalysis,TPistheperiodofexcitableharmonics,p

istheperturbationofthepressureeld,q

istheperturbationofheatreleaserate.Ifviscousdissipationistakenintoaccount,equation(2.1)becomes:__TP0p

(x, t)q

(x, t)dtdV>__TP0(x, t)dtdV (2.2)whereisthelocalviscousheatingterm.Just to understand the phenomenon let us forget the dissipation term (x, t)in equation (2.2) applying the rst ideal inequality to Rijke tube conguration.Considering p

and q

as sinusoidal waves of period TP= 1s diering by a phasepqonly,wecanarguethatRayleighscriterionissatisedwhen [ pq [< /2.Letusanalyzeafewinterestingcases:pq= 0: amplication is maximum because p

is completely superimposedtoq

thusq

p

> 0 x.pq= /4: amplicationiszerobecauseforhalfacycleheatisgivenforq

p

< 0andfortheotherhalfq

p

> 0. Seegure2.3.pq=/2: amplicationisnegativebecauseq

p

_p

v

dA. (2.6)Figure 2.7 - Generic domain for the mod-iedRayleighcriterion.Equation (2.6) means that the usual Rayleighs integral must be greater thanthenetuxof acousticenergyacrossthesurfaceforinstabilitiestoarise.3Such a criterion becomes useful only with advanced LES analysis able to predicttheselosses.Nevertheless thereis noevidencethat esinequation(2.4) is therelevantenergythat must betakenintoaccount. It is possibletoshowthat suchadenitionlooses consistencyinanon-isentropicoweld. This means thatanotherquantityshouldbeusedinthecaseof reactingowstocharacterizethe global amount of uctuations properly and that this energy should includeentropyuctuations. Moreover, theentropyeldisnotconstantoverspacewhencombustionoccursrenderingequation(2.4)inconsistent.Thisassumptionleadstoanewdenitionoftherelevantenergy,asocalleductuationenergy:etott+ (p

v

) =T

T[q

+ (TT)] pRcps

v

s +v

(

v) (2.7)whereetotisdenedasetot= v

v

2+p22 c2+S2 p2Rcp(2.8)3Theuxterminequation(2.6)iscompletelydierentfromther.h.s. ofequation(2.2)whichincludesonlyviscouslosses.11CHAPTER2. ABRIEFLITERATUREREVIEWNote that the fact that entropy is included does not mean that entropy uc-tuationsareaddedtotheanalysissincethezeroMachnumberapproximationremovesentropywavesthatcannotpropagatewithoutanonzeromeanow.Nevertheless anon-isentropicsteadyoweld(=0) couldbetakenintoaccount.Integrating over the domain equation (2.7) we get the extended Rayleigh likecriterion:_(T

q

T pRcpS

v

S)dV>_p

v

dA. (2.9)What reported so far might appear as a useless complication without an orderof magnitudeestimateof thedierentterms. Itispossibletoshowthattheadditional termrelatedtothenon-uniformentropyeldispotentiallylargerthantheclassical Rayleighsterm. Thisargumentissupportedbynumericalresults that yieldvalues of entropyuctuations oftenlarger thanacousticalones. Several studiesalreadypresentintheliteraturehavedevelopedpurelyacousticmodels. Duetorecentevidencewecannottrusttheseresultswhencombustionoccurs. Sincethepresenceofentropyuctuationsrequiresanon-zeromeanow, theMachnumberassumesadoubleimportanceconsideringthatitsgeneral inuenceonstabilityhasalreadybeendemonstrated[1] (seesection3.1).Figure2.8- Discretizationof aAE94.3Agasturbinecombustor[24].Figure 2.9 - Discretization of asimpledomain[14].Inthepresentworknointegral criterionisappliedsinceitwouldrequiredaspatial andtemporal discretizationof thewholedomain, likeingure2.8122.3. FLAMEDYNAMICSor2.9. Theaimof thepresentresearchistodevelopandvalidateasimpletoolabletopredictrapidlytheinuenceofgeometricalparametersontheon-setof instabilities. Inthismodel thegeometryisdenedbyfewparametersstraightforwardlymodiable(only10eectivevaluesintheAE94.3Agastur-bine model). It is trivial that a three dimensional approach would lead to largetimeforthesettingof everysimulationwithdierentgeometrical congura-tion since every geometry would require hours for generating the CAD and theadequatemesh. Oursimulations,describelater,requireonlyminutesofCPUtimeonadesktopcomputer.2.3 FlamedynamicsAn important step for a correct modelling and to be able to envisage passiveor activecontrol techniquestofull-scalecombustorsisthedevelopment andanalysisoflow-orderdynamical modelsthatcontainthefundamental physicsofthesystem. Itwill beclearinchapter4thatthecoreofthesystemistheame model, in other words the equation that describe the interaction betweentheheatreleaseandtheperturbations. Theoreticalandnumericaltechniquesinnon-lineardynamical systemstheoryhavebeendevelopedintheresearchcommunityover the past tenyears andcanbe eectivelyappliedtothesereducedordermodels.The interactionbetweenturbulence andcombustionis tobe investigatedinorder toachieveawareness onthedominatingprocesses. Without goingtoodeeplyintothetheoreticalanalysis,wedistinguishthreedierentkindsofinteractions:a reactionsheetsb ameletsineddiesc distributedreactionsDistributedreactionsoccurwhentheturbulenceintegral scalel0issmallerthen the scale relative to combustion forcing turbulence time to be larger thenthechemicalone. Thisfactimpliesthatchemicalkineticsisinuencedbythefeaturesoftheow. Flameletsineddiesoccurswhenlk< L< l0;inthiscasecombustionmaytakeplaceinsidesmall enoughvortical structures. Reactionsheetsoccurwhentheturbulencescalelkisgreaterthenthescalerelativeto13CHAPTER2. ABRIEFLITERATUREREVIEWFigure 2.10 - Important parame-ters characterizing turbulent pre-mixedcombustion. Conditionssat-isfying the Williams-Klimov crite-rion for the existence of wrinkledames lieabovetheredline(lk=L), and conditions satisfying theDamkohlercriterionfordistributedreactions fall below the blue line(l0=L). Hypothesis: Le =1,Sc=1 =1. Integral scaleisdependent ontheKolmogorovone:l0= lk Re3/4l0.combustion. Reactionsdevelopinarangeinternal totheKolmogorovscale,thusturbulenceisableonlytowrinklethesurfaceoftheame.Combustion itself is not inuenced but a new ame speed that takes into ac-count turbulence is to be dened (ST= SL+u

for example, with STand SL theturbulentandlaminaramespeed,andu

auctuatingvelocity). Distributedreactions are useless for engineering applications since they would require highow velocities in small diameters (in order to get large u

and small l0 ) generat-ing unacceptable losses. Flamelets in eddies assume interest in few applicationssuch as some four-stroke internal combustion engines. Gas turbine combustiontsinthereactionsheetsduetothehighDamk ohlernumbertogetherwithreducedturbulenceReynoldsnumbers.Thesignicantinuenceofunsteadyphenomenaconnectedwithturbulenceinteractionwithamefronts makes CFDsimulations far frombeingtrivial.Itisveryimportanttoknowtheameshapeanditspositionsincefromthisinformation we can infer parameters that can be tuned in order to get a model142.3. FLAMEDYNAMICSabletodescribeproperlytheinteractionsbetweenaameandacousticwavesoccurring in the combustion system. Simulations have already been performedwith classical turbulence models based on the Reynolds-Averaged Navier StokesEquationsandLESsimulations. Resultsweresignicantlydierentaccordingtopreviousstatements: itisclearthatamodelthatcutsotheintegralscalethatwouldhaveainuenceontheamefrontishardlysuitabletopredictinareliablewaytheamefrontitself. FurthermoreLESsimulationsonsuchacomplicated geometry with coupled combustion models require large computa-tionaltimeandresources.4Theheatreleaserate[W] ofaturbulentpremixedamemaybeexpressedaccordingto[23]as:uASTHi,where uistheunburntgasdensity, Aistheamesurfacearea, STistheturbulentburningvelocityorturbulentamespeed, Hiistheheatofreactionperunitmass.The turbulent ame speed can be dened using dimensionless groups (Reynolds,Damk ohler and Lewis numbers) and relating it to the laminar ame speed SL:STSL= f(Re, Da, Le, ...).2.3.1 FlametransferfunctionsAn operational transfer function is simply a mathematical model that relatesathermodynamicvariabletoanotherone. Consideringaninputsignal Aeitand an output one Beit+we dene the transfer function as the ratio betweenoutput andinput. Therefore B/Ais the gainof the transfer functionand isthephaseangle. Inthismannerthetimedependenceisnotneededinafeedback-loop analysis because all the equations have the same time dependenceeit.4Combustion process can be preliminarily modelled as a double stage combustion process,asCH4 +32O2 CO + 2H2O,followedbyCO +12O2 CO215CHAPTER2. ABRIEFLITERATUREREVIEWAametransferfunctionisneededinordertodescribetheame-acousticinteraction. Threephysicalmechanismaretobeidentied:1. amefrontkinematics,2. amespeedmodications,3. equivalenceratioperturbations.For turbulent premixed ames, the direct inuence of pressure, temperature,anddensity variations onthe heat release rate is usually consideredto be neg-ligiblysmall. InrealitytemperatureandpressurecouldinuencethelaminaramespeedSL.5Inthiscasetheamefrequencyresponseorametransferfunction,denedasthenormalizedratioofheatreleaserateandvelocityuc-tuations,completelycharacterizesthedynamicresponseofaame toacousticperturbations.F() =Q

/ Qu

/ u(2.10)Fromequation(2.10)severalsimplecriteriahavebeendeveloped. Thevali-dationofsuchamodelisinpracticeoftendicult,becausemanyparameterscannotbedeterminedfromrstprinciples. Instead, parametersmustbead-justedtomatchexperimental or numerical data. Inrecent years, detailedlaser diagnosticstudies ingas turbinerelevant combustors havecontributedsignicantlytoabetter understandingof processes like ame stabilization,combustioninstabilities, pollutantformationandnite-ratechemistryeects.Frequentlyusedmeasuringtechniqueswereparticleimagevelocimetry(PIV)or laser Doppler velocimetry(LDV) for theoweld, laser induceduores-cence(LIF)forameradicals(OHchemiluminescenceingure2.11), trac-ers, pollutants or temperature, coherent anti-Stokes Raman scattering (CARS)fortemperature, andlaserRamanscatteringforspeciesconcentrations. Nu-merical datawhichareoftenlimitedinscopeaswell asaccuracy, suchthatsignicantuncertaintiesstill remain. Despitetheseproblemsitispossibletoget information above certain parameters just from global conservation laws inaquasi-steadylimit[23], i.e. thestatethatanyphysicallyvalidmodel reachwhen 0.5ForhydrocarbonsSL T0.875i T0.5u T0.875b eEa/(2RTb)p0.125wheresubscriptsarerespectivelyignition,unburnedandburned,see[37].162.3. FLAMEDYNAMICS(a) (b)Figure2.11-ImagesofOHPLIFfromoscillatingame[41].(a): meanOHdistribution. (b): singleshotmeasurement.Let us consider a general ame transfer function that correlates heat releaserelativeuctuationsQ

/ QwiththeNquantitiesGithatcouldhaveinuenceonit.Q

Q= 1G

1G1+ 2G

2G2+ + iG

iGi+ + NG

NGN(2.11)Thenweareabletosaythat:lim0N

i=1i= K (2.12)where Ksatises the quasi-steady limit inferable from global conservation laws.This is useful toverifythe validityof anytransfer function. For example,assumingastifuel injection m

F=0, theheatreleaseQ

V[W] mustbezerowhen 0becauseQV= mFHi.Diculties in implementing such models are obvious since too many param-eters are to be taken into account simultaneously, even if they are correlated byanequation(onlyoneequationsuchas(2.12)). Tooverrunthisproblemsev-eral simplied models have been developed considering only the most pertinentquantitiesanddisregardingallothers.Research on ame dynamics is made considering the inuence of equivalenceratiooname dynamics as ingure 2.12. Direct numerical simulationareperformedtounderstandtheinteractionof acousticwaveswithame(gure2.13). As stated previously the experimental analysis is extremely complicated17CHAPTER2. ABRIEFLITERATUREREVIEWsinceitisverydiculttogetagoodinstrumentationof acritical oweld(temperatures are very high) and to get measurement of certain quantities likeheatreleaseuctuations.Figure 2.12 - PerturbedequivalenceratiodistributioninaVame[22].Figure2.13- Acoustically-inducedpres-sureuctuation(dashedlines). Thein-stantaneous position of the ame front isshown with isolevels of heat release (blacksolidlines)[27].Thereforefewempirical modelsremain. Letuslistsomeof themfromthesimplesttothemoreelaborated:Q

Q= 0 (2.13)Q

Q=m

i mi(2.14)Q

Q= m

i miei(2.15)Q

Q=

ii= u

i uisin()ei(2.16)Q

Q= pp

i pieiu+ uu

i aieiu+ vv

i aieiv+ ww

i aieiw(2.17)182.3. FLAMEDYNAMICSModel (2.13) makes thehypothesis of amekinematics independent fromoweld. Asrstapproximationthisisveryusefulbecauseitcutsoalltheeigenmodes connected with the ame. In eect such a model is a sub case of alltheothermodelswheni=0. Inmodel(2.14)theheatreleaseisconsidereddirectly dependent from the mass ow rate perturbation at the reference pointi,that could be fuel injection point,for example.6Model (2.15) due to Croccois a sort of milestone in transfer function evolution since it represents the delaytimeof perturbationsfromareferencepointtotheamesheet. Atpresentthisisthemostfrequentlyusedmodel sinceitseemstorepresentfairlywellthe physics of the system. Unfortunately, parameters and have to be tunedaccuratelyotherwiseresults arefar frombereliable. All of theseequationshaveacommonhypothesis: aat thiname. Thethinassumptionat rstglanceseems tobereasonablyacceptablesincetheamethickness Lhaveanorder of magnitude of 103102m, small enoughif comparedwithcombustorgeometry. Analyzingbetterthishypothesis,ifweconsiderthattheame is interacting with waves, we have to perform a dimensional analysis thatcompares wavelength with L. The -model in (2.15) is suitable to representthephysicsofthesystemonlyifL . Thus, respectivelyforacousticandconvectivewaves,letusevaluatethislimitconsideringveryprudentvaluesforthespeedofsound,themeanowvelocityandtheamethickness:ac=csf=f csL800m/s2cm= 40 KHzcv= uf=f uL20m/s2cm= 1 KHzSincehummingisaphenomenonoccurringatlowfrequency 1 KHz,thethinameassumptionis agoodapproximationbeyondanydoubt evenfor convective waves. Onthe contrarythe at ame assumptionseems tobe veryweakbecause reactions are distributedover averycomplexthree-dimensional surfacedistributedoveranotnegligiblelengthasingure2.14.Model(2.16)proposedin[21]triestomodelathree-dimensionalamesurfacethrough a varying value of . Finally model (2.17) considers several dependencyassociating to every quantity proper values for and . This is a more completemodel but it is very hard to calibrate since there are several degree of freedom.6Suchamodelwasusedinpreviouswork. Seesection3.119CHAPTER2. ABRIEFLITERATUREREVIEWFigure 2.14 - Three dimensional amesurface,LESsimulation. Figure2.15-SwirlerTakingaglanceatalltheissuespresented,itisclearthatastrongphysicalmodelwhereempiricalparametersarenotcriticalisstillfarfromhavingbeendeveloped. Until suchatheoryisavailableresultsfromalumpedapproacharehardlyreliableandappropriatetuningof themodel parametersmustbeconstantlycarriedout against full simulations or/andexperimental results.Thus, almostall eortsatthistimetendtowardsthedevelopmentofagoodtransferfunction.The shape of the ame surface is connected with the indispensable anchoringoftheameitself. Togainthisconditiontheamespeedmustbetakenintoaccount: the turbulent ame velocityfor methane inair is about ST10m/s. Since meanowvelocityineachcombustor is about 7times faster,ame stabilizer are required. Inorder toachieve stability, the presence ofrecirculationisasimplyandsafewaytogetintheoweldasurfacewhere u = ST. RecirculationisobtainedduetotheconcurrentpresenceofasuddencrosssectionalareaincreaseandaswirlerthatprovidesaSwirlnumberSw =(Moment of momentum)/(MomentumRadius) greater thentheminimumrequired(about 0.5). These devices provide alobedshape of the ame asshowningure2.14.Increasingmeanowvelocitywill shift theamedownstream. Asecondconditionforstabilityisthatthetimeowisincontactwithhotrecirculating202.3. FLAMEDYNAMICSuid(c)mustbegreaterthanthetimerequiredforignition(i); thesetwocharacteristictimesare:c LREC udUBlowOffi TS2Lwhere SLis the laminar ame velocity(about 1m/s) andTthe thermaldiusivity.Inordertopreventashbacksgridsareusuallyplacedatthebeginningofthe premix. Thank to quenching phenomena the upstream transmission of theameisinhibited. Gridsthatgeneratepressuredroparealreadymodelledinthepresentcode.Ingure2.15onlythediagonalswirlerisvisible. Indeedneartheaxisthereisalsoanaxialswirlerthatprocessesabout10%ofthetotalmassowrate.Finally, neartheaxisthereisapilotdiusionamethatgrantsignitionforeveryoperatingcondition. Thedownstreamshiftingof theamewhenthevelocityintheburnersincreasescannotbeneglectedwhenchoosingthege-ometry. Infactthecouplingbetweenazimuthalmodesattheamemightbeshifteddownstream, wheretheradiusissmaller. Atpresentnodataallowtoappreciatetheappropriatenessofthisstatement.21Chapter3StartingpointThisthesisispartof abroaderprojectconductedbyAnsaldoEnergiaforthe design of new gas turbines, known as AE94.3A. Currently, research evolvesalongseveraldirections,sincemanyaspectsmustbeinvestigated,suchasthephysicsof theame, itsinteractionswithmeanowperturbations, dampingandcontrollingdevicesabletoavoidinstabilities, 3Dcomputationsuseful tohave a better idea of phenomena, etc. The collaboration with DICAT begun in2008 and is focused on providing a code able to evaluate the inuence of severalparameters on the onset of instabilities. This code should be useful in the designphase in order to understand where and what to modify when instabilities arise.Furthermore,aamemodelistobeinvestigatedandcalibrated.3.1 IncreasingcomplexitySince the complete problemhas never beentreatedcompletelyintheliteraturesmall stepstowardamorecomplexsetofequationsarere-quired. Analytical treatment is developed as faraspossibleinordertoachieveamorecompletecomprehension of the problem and to gain a bet-tercomputationaleciencywhenpossible. Ev-erystephas beenvalidatedwithexperimentaldataandCFDnumericalresultsfromtheliter-ature.Figure 3.1 - Steps currentlycarriedout.23CHAPTER3. STARTINGPOINTThe Matlab environment has been chosen for the development of the codesbecause of its easy and straightforward programming. In gure 3.1 the increas-ingcomplexityofthesubsequentmodelsisshown.Thebasicideaistoanalyzethecombustorof agasturbinethroughasetof equationssuitedtorepresentthesystem. Theapproachchosenisatime-domainstatespacerepresentationderivedfrommoderncontrol theory. Theinternal state variables are the smallest possible subset of system variables thatcan represent the entire state of the system at any given time. The variables areexpressed as vectors and the dierential and algebraic equations that representthesystemarewritteninmatrixform(thelatter onlybeingpossiblewhenthedynamical systemislinearoritislinearized). Thenumberof unknownvariables (i.e. the length of the relative vector) must be equal to the number ofequations (i.e. to the rank of the relative matrix) in order to have a well posedproblem. Sometimes,andthisisthecasehere,theequationsavailablearelessthanrequiredandtransferfunctionsarerequired. Atransferfunctionisanequationthat couplestwoor morevariables. It isimportant tounderstandthat using a transfer function form instead of physical principles may cause thelossof informationof thesystem, andmayprovideadescriptionof asystemwhichisstable,whenthestate-spacerealizationisunstableatcertainpoints.Internalvariablesforagasturbinesystemaretheperturbationsofthermo-dynamicalvariablessuchaspressure,velocityorheatrelease. Thecombustoris composed by several ducts such as an annular diuser at the compressor exit,24premixersandanannularcombustionchamberjustbeforetheturbineen-try. Since we use a lumped approach the system is uniquely represented by theamplitudeoftheperturbationsofthethermodynamicalvariablesineachducttogetherwithheatreleaseperturbationattheame. Equationsarededucedfromconservationprinciplesatthejunctionsof eachduct(jumpconditions),together with well posed boundary conditions. Furthermore a transfer functionthatcouplesheatreleasewithothervariablesisrequiredtoclosetheproblem.Thepresentmatrixhasmorethanonehundredrowsandifvorticitytermsaregoingtobeincludedthenumberofunknownsshouldstillgrowinfurthermodels. Since the complexity of an immediate approach to the complete prob-lemislarge, easiermodelshavebeentreatedbeforethemorecomplexones.Letustakeanoverviewofthesemodelssynthesizedingure3.1.243.1. INCREASINGCOMPLEXITYRijketube. Therstmodelproducedwasasimpleone-dimensionalsystemconsistingof twoductsof thesamecrosssectional areaseparatedbyathinameataxedpoint. Twobasicamemodelswerechosenandfuelinjectionis locatedexactlyat theamecoordinate. Inthis casetheinuenceof theMachnumber(gure3.3A)andofheatreleasemagnitude(gure3.3B)hasbeeninvestigated. Moreoveritallowedtodetectimmediatelyhowcriticaltheamemodelis.Figure3.2-Firstone-dimensional modelParametricstudyonMachnumber(Ma1)andonheatreleasemagnitude(T02/T01)wereperformedinordertoachieveawarenessof theirinuenceoneigenfrequencies. Theseresultswherevalidatedin[3].Singleburner. Amorecomplexgeometrywaseventuallyelaboratedtoap-proachareal burner geometry(lower right part of gure3.1). Threeductswereconsidered: alargeplenumthatcollectsairfromacompressor; asmallpremixer where fuel and oxidizer are mixed separated by the combustion cham-berbyastillthinamelocatedexactlyattheareaincrease. Amorecomplexamemodel wasrequiredinordertotakeintoaccountthatfuel isinjectedfarupstreamfromthecombustionprocess. Afurtherparameterwasaddedtohavethepossibilitytovarycontinuouslythesensitivityoftheheatreleaseperturbationtomeanowperturbations. Thesocalled-modelwas:1Q

= Qm

i miei(3.1)wherem

iand miaremassowratescalculatedat areadecreasewherethepremixerbegins. Theparameter isthesocalledtimelag, i.e. thetimeaperturbations generated were fuel is injected takes to reach the ame. The -modelinuencewashighlightedbyresultsingures3.4and3.5:1Seesection4.6foramorecompletetreatmentoftheametransferfunctionmodel.25CHAPTER3. STARTINGPOINTSo, evenif itcouldbeasuitablemodel, itneedsacareful calibrationthatrequires experimental data or CFD numerical solutions obtained with the same,oratleastcompatible,boundaryconditionsandassumptions.2Dmodel: annular duct. Sinceit is knownfromexperiencethat ther-moacoustic instabilities occurs innowadays combustionchambers primarilyalongthecircumferential direction, themodel aboveomitsthemostinterest-ingeigenfrequencies. Thereforethefollowingstepsconsistindevelopingatwodimensional model that takes into account also the azimuthal coordinate. Theexpressionsfortheperturbationsincludealsoaneintermwherenrepresenttheindexofeachazimuthalwave. Obviouslythismodelincorporatesthepre-viousonesinceann = 0modehasnocircumferentialoscillationandthusitispurelyaxial.Parallel burners. Again, fromexperience, itisknownthataddingCBO

sattheendof someburnersispositiveforstability.2Thus, themodellingofall 24burnersisrequired. Jumpconditionsbecomemorecomplexsinceeachburnercommunicatesonlywithasectionofotherducts. Moreover,theequa-tions suddenlybecomemuchmorethanbeforeandthecomputationof thedeterminantofthematrixbeginstobemuchslowerthaninpreviousmodels.Modal coupling. Adding CBO

s in the combustion chamber geometry breaksthe axisymmetry. Modes of oscillation are no longer suitable to be representedbypuretones. AFourierseriesexpansionisthusneeded, withall themodescoupled to one another. This is the actual model used in further test campaigns.Seesection4.3fortheanalyticaltreatment.3Dmodel: radialdependency. Finally,aradialdependencyistakenintoaccount to completely represent the system with all its features. See chapter 7fortheanalyticaltreatment.2Seesection4.3.263.1. INCREASINGCOMPLEXITYFigure3.3- Minimumfrequencyfor(a):T02/T01=6; (b): Ma1=0. o: noheatreleaseperturbation. o: heat releaseperturbationproportional tomassowrateperturbation< q

= cp(T02T01)( 1u

1 +

1 u1) >.27CHAPTER3. STARTINGPOINTFigure3.4-Comparisonbetweenresultsobtainedfor=0(o)and=1(+)andresonancemodefrom[5]for = 0(o)and = 1(x).Figure3.5-Sensitivityofamodeofoscillationto choice28Chapter4ThegoverningequationsThe spread of disturbances, in the absence of heat release, is exactly describedbytheacousticdAlembertequation. Accordingtoreality, thepresenceof aameandameanowconvectivetermsaretakenintoaccountmodifyingtheenergyequationandincludingamaterialderivative. Moreover,manysystemsof heatreleasegeneratesignicantpressuredropduetofrictionthatcertaincomponents,likegrids,exertontheuid.HypothesisVolumeforcesgnegligible.Ideal gas1: =0; cp, cvconstantindependentfromtemperature; p=RT.Lumpedparameters: ineveryductthevaluesofthemeanowvariablesareconsideredspatiallyconstant.Linearapproach: itisassumedthatallquantitiesarethesumofacon-stant part andavariableonewherethelatter has alower scale, soagenericGcanbewrittenasG + G

(t)whereGistheconstantpartas-sociatedwithmeanowandG

thevariablepart,i.e. theperturbation.Thisassumptionallowstoneglectnonlinearterms(productof pertur-bations),providedthatG

> cmodalcouplingisinhibited.39CHAPTER4. THEGOVERNINGEQUATIONS4.4 Equationsforthemeanowcomputation4.4.1 PerfectgasequationsAperfectgasequationiswrittenineachduct:p = rT, i.e. 4perfectgasequations.4.4.2 MassuxconservationequationsThemassuxconservation,writtenlocallyateachpremixerinlet,imposes m1= Nb m2,AttheexitofpremixerswithCBO,wehave m2= m2b,andinthecombustionchamber m1= m3, i.e. 3massuxconservationequations.4.4.3 EnergyconservationequationsTheenergyconservationbetweentheplenumandthepremixersgives m1H1= Nb m2H2,with Hthe enthalpy dened as H= cpT +U2/2. Between the CBOs and theircorrespondingpremixers,wehave m2H2= m2bH2b,andatthecombustionchamberinlet: m3H3= m1H1 + A3QT, i.e. 3energyconservationequations.404.5. PERTURBATIONSEQUATIONS4.4.4 Isentropic/gridconditionsThe turbine conguration includes a grid at each premixer inlet. Its inuenceis neglected in the present computation, so that the isentropic condition usualforanareadecreaseapplies:p1/p2= (1/2);if thegridhadnotbeenneglected, thefollowinggrid-typeequationcouldbeused:p1p2=12 k122u22,wherek12isalossparameter, i.e. 1isentropic/gridequation.4.4.5 BordaequationsWewriteaBordaequationateachCBOinlet: m2u2 m2bu2b= A2b (p2bp2) ,and a Borda-like equation for the set of premixers/CBOs reaching the combus-tionchamber: m3u3Ncbo

k=1 mk2buk2b

l ml2ul2=A3Nb_Ncbo

k=1pk2b +

lpl2_A3p3,wherel NbrepresentstheindexoftheburnerswithoutCBO, i.e. 2Bordaequations.Thetotal numberofequationsisthenNmeaneq=13. Onceapropernumberof boundaryconditionsisknownitispossibletosolvethenon-linearsystemof equationsthanktoanoptimizationtool. Theunknownsmostsuitabletobetakenasboundaryconditionsaretheinletpressure,temperatureandmassowrateandtheametemperature. SinceLOMTI is alumpedmodel theinletvaluesareexactlythevaluesinsidetheentireplenump1, T1and m1aswell astheametemperaturerepresentsthemeantemperatureof theentirecombustionchamberT3.4.5 PerturbationsequationsThe set of equations for the perturbations are the linearized equations of themeanowcomputationsshownbefore. Letuswritethemexplicitly.41CHAPTER4. THEGOVERNINGEQUATIONS4.5.1 MassuxconservationequationsAstheperturbationintheplenumdependson(seegure6.1), thelocalmassuxconservationequationateachpremixerinletbecomesd1R1_i+ /Nbi/Nb(1u1)

d =_ mi2_

, 1 i Nb.AttheexitofpremixerswithCBO,themassconservationsimplyis_ mi2_

=_ mi2b_

, 1 i Ncbo,andinthecombustionchamberd3R3_i+ /Nbi/Nb_i3ui3_

d =_ mik_

, 1 i Nb,wherek=2fortheburnerswithoutCBOandk=2bfortheburnerswithCBO

s. Sowehave (2Nb + Ncbo)massuxconservationequations.4.5.2 EnergyconservationequationsAsforthemassux, wewritelocallyanenergyconservationequationateachpremixerinlet:d1R1_i+ /Nbi/Nb(1u1H1)

d =_ mi2Hi2_

, 1 i Nb.AttheexitofpremixerswithCBO,theenergyequationreads_ mi2Hi2_

=_ mi2bHi2b_

, 1 i Ncbo,andinthecombustionchamberd3R3_i+ /Nbi/Nb(3u3H3)

d =_ mikHik_

+A3Nb_Qi_

, 1 i Nb, .i.e.(2Nb + Ncbo)energyconservationequations.424.5. PERTURBATIONSEQUATIONS4.5.3 IsentropicconditionsAsforthemeanow, wewritefortheperturbationalinearizedisentropicconditionateachpremixerinlet:_p1/pi2_

=__1/i2__

, 1 i Nb, i.e. Nbisentropicequations.4.5.4 BordaequationsSimilarly,theBordaequationsarethelinearizedformofthoseusedforthemeanow. AteachCBOinlet:_ mi2ui2_

_ mi2bui2b_

= Ai2b__pi2b_

_pi2_

_, 1 i Ncbo,andforeachductreachingthechamber:_ mikuik_

+A3Nbp

k= d3R3_i+ /Nbi/Nb__3u23_

+ p

3_, 1 i Nb, i.e. (Nb + Ncbo)Borda-likeequations.Inordertocompletetheproblemtheacousticboundaryconditionsattheinletoftheplenumandattheoutletofthecombustionchamberarerequiredtogetherwithaamemodelforeachburner.4.5.5 InletandoutletconditionsThreekindsofinletconditionsareimplemented:aclosed-endinlet/outletwhereu

= 0(inniteimpedance);anopen-endinlet/outletwherep

= 0(zeroimpedance);achokedinlet/outletwhere m

= 0.Ineachcase, wesupposethattheinletconditionisveriedindependentlybyeachoftheNbmodesintheplenumandwehave: Nbinletconditions; Nboutletconditions.43CHAPTER4. THEGOVERNINGEQUATIONS4.5.6 FlametransferfunctionUnder the hypothesis that ames issuing fromneighbouring injectors donotinteract,weusetheso-calledISAAC(IndependenceSectorAssumptioninAnnular Combustor) assumption to dene the ame transfer function. It statesthattheheatreleaseuctuationsinagivensector[ofthecombustionchamber]areonlydrivenbytheuctuatingmassowratesduetothevelocityperturbationsthroughitsownswirler.Thismeansthatalocal transferfunctioniswrittenforeachoftheNbameswithaspecicperturbationinheat release(Qj)

, 1 j Nb, andwithadierentreferencepointeachtime,chosenattheinletofitsownpremixer:(Qj)

Q=Ng

gFg()(Gj)

Gj1 j Nb,wheregrepresentstheindexrelatedtotheNgquantityGthatappearsintheametransferfunction. i.e. NbequationstodenetheNbpartsofQ

.4.6 FlamemodelTheacousticofthecombustionchamberisstrictlycorrelatedwiththeun-steadyheat release. Therefore acorrect estimate of the ame dynamics isneededtoprovideaccuratepredictions. Theactual amemodel isbasedonatransferfunctionwheretheperturbationsofmassowrate(orvelocity)arelinked to the heat release uctuations at a reference point close to the combus-torinlet.Modellingaamedynamicsisdicultsinceseveral mechanismsenterthepicture, such as chemical kinetics, convective and diusion terms, fuel injection,turbulentbehaviourandameanchoring. Thetheorymaybecomesuddenlyinextricable [11], and the huge quantity of unknown parameters yields actuallytoauselessmodel.Someassumptions arerequiredinorder toobtainasimpleandavailablemodel. Neglectingthephysiological ones (suchas homogeneous mixingbe-tweenspecies andinstantaneous reactions) themainassumptionis that we444.6. FLAMEMODELareconsideringtheameas athinat interfaceseparatingpremixers fromcombustionchamber. Thuswelinkunsteadyheatreleasetomassowrateuctuationsaccordingtothesocalled-model:Q

=Q mi

mieiThis methodneeds togather apriori informationcomingfromnumericalsimulations or experimental dataonlyfor thetwoparameter and. Theimportanceoftheseparametershasbeenhighlightedinchapter3.Validityof the model. The time lag is, bydenition, the convectiontimebetweenapointi, locatedupstreamof theame, andtheameitself.Generallythepositionof this point couldvaryinpractical applications. Inactualmodelsthefuelinjectionpointisoftenusedasareference. LetLifbethiscritical distance; ithasbeenshown[17] thattheametransferfunctionproblemisill-denedifLifisnotsmallenough. Fortunately,itispossibletoquantifythislimitusingtheHelmholtznumberdenedby:Hn=LifcsThisnumbercouldbearrangedasfollows:Hn=Lifsfs=2LifsTheoretically Hnhas to be the smallest possible. In practice, there are prob-lemstoperformmeasurementsneartheame, alsobecauseof itsoscillatingposition. However values of theHelmholtznumber less Hn=102arere-quiredtoensurethattheestimatedparametersandareindependentfromtheoutletconditions, andarethusintrinsiccharacteristicsofthecombustionchamber.Obviouslynomeasurementisperformedsinceourmodel iscompletelynu-merical. But just for this fact when mass ow perturbation (or velocity one) isevaluatedatthebeginningof thepremixersanumerical erroroccursmakingthe Helmholtz number limit a real restriction. An explanation of this statementis shown in gure 4.2 where frequency is evaluated with a 5% of approximation45CHAPTER4. THEGOVERNINGEQUATIONSbetweenblackandredline. It is clear that signicant errors ariseonlyforhigherfrequencyFigure4.2-Errorforlowanhigherfrequencies.Animportant thingtonote here is that lowvalues of are requiredtosatisfythiscriterion. IfLifisnotsmallenoughtherewillbealimitationonthe allowable values of the resonant frequencies, thus restricting the validity ofthemodeltolow-frequencymodes. IneectconsideringAE94.3AgasturbineLifis equal to the length of premixers thus Lif= 142 mm and cs= 528 m/s.Thereforeshouldbesmallerthan37rad/s,i.e. f< 6 Hz!According to these statements and seems not to be intrinsic parameters ofthe ame, vanishing all the eorts made to obtain reliable results. A parametricstudyonLifisperformedinsection6.4.46Chapter5CalibrationofLOMTIparametersagainsta3DCOMSOLsimulationFigure5.1-Geometryfor3DsimulationFigure 5.2 presents a comparisonof the computational domains usedinCOMSOL(left) andinLOMTI (right). Fromgure5.2(a), it appears thatthemostcomplexparttomodel inLOMTIslumpedparametersapproachisthe plenum. Only the volume coloured in blue, along which air travels from thecompressorexittotheburnersinlet, issuitabletoberepresentedinLOMTI.This means that the outer part of the plenum (colored in pink in gure 5.2(a))47CHAPTER5. CALIBRATIONONAFEMCODEwill not bemodelled. Besides, thereal plenumcontains somenarrowgapswhichcannotobviouslybeincludedinLOMTI(seealsosection5.5). Thege-ometricalparameterschosenforthepremixersarethosealreadyusedinSesta[16]. Finally,asarstapproximation,thecombustionchamberisrepresentedbyasingleannularduct(volumecolouredingreeningure5.2).(a) (b)Figure5.2-(a)GeometryforCOMSOL.(b)GeometryforLOMTI.The input parameters used for the computation in COMSOL are reported intable5.1andtheinputparametersusedinLOMTIshouldbethesame. Itisnoproblemtousethesameinlet/outletboundaryconditionsandtoconsiderornotthepresenceofameperturbations. However,forconvergencereasons,it is not possible to run LOMTI with a zero Mach number (zero mass ow rate)and alternative mean ow parameters must be computed. Using the proceduredescribedin[32], themeanowparameters arecomputedbyimposingthepressureandthetemperatureintheplenumequaltothoseusedinCOMSOLandchoosingasmall inlet mass owrate equal to19.85kg/s. The meanparametersobtainedarereproducedintable5.2forthecalibratedgeometry(seesection5.5,tables5.4and5.6). ThemaximumMachnumberobtainedis0.005,andwethusexpectitnottoinuencesignicantlythesolution1.1TheMachnumberinthepremixersinrealturbineconditionsisabout0.13soitisherereducedto3%ofitsoriginalvalue.48Plenumandburnerstemperature 683.15KPlenumandburnerspressure 16.43barFlametemperature 1736KMassowrate 0Boundaryconditions(inlet/outlet) u

= 0Noameperturbation k = 0Table5.1-InputparametersfortheCOMSOLcomputations.duct p[bar] [kg/m3] T[K] u[m/s] cs[m/s] Maplenum16.4300 8.379 683.15 0.51 523.95 0.97 103burners16.4297 8.379 683.15 2.90 523.95 5.54 103chamber16.4297 3.297 1736.20 1.741 835.28 2.08 103Table5.2- MeanowparametersusedinLOMTIforthecalibratedgeometry(seesection5.5,tables5.4and5.6). ThedatainredarethoseavailablefromCOMSOL.Theamemeanheatrelease,Q = 6 074kW/m2,hasbeenchoseninordertogetthecorrecttemperatureinthechamber.Several geometrical parameters have to be chosen in order to calibrate LOMTIsgeometry: thelength,radiusandthicknessoftheplenumandthecombustionchamber, and the length and cross sectional area of the premixers, i.e. 8 param-eters. Thismeansthatthecalibrationcampaignrequiresplentyofcomputa-tions and of computation time to obtain simply-to-understand informationandtrends. So, startingfromthearbitraryinitial geometryusedin[32], wehavetotesttheinuenceof thesequantitiesonebyone, assumingthatthetrends obtained would be similar if any of the other quantities were modied.2Letusrstanalyzetheresultswewanttoobtain. The3DCOMSOLsimu-lationsarecarriedoutinitiallywithoutameperturbationandwithoutmeanow; thereforeall modesobtainedareneutral (i.e. withzerogrowthrates).DuetothenonzeroMachnumberapproximationusedinLOMTI,themodesobtainedwillnotbeexactlyneutralbutstable.2Onlythreetestsperparameterrunforall thecombinationswouldgive38simulations,whichwouldrequiremorethan103CPUhours.49CHAPTER5. CALIBRATIONONAFEMCODEWiththeoriginalgeometryfrom[32],wegetthesamenumberofmodesintheselectedrangeoffrequencyaswithCOMSOL.Thesemodesareshowningure5.3and,asexpected,areslightlystable.Figure5.3-FrequenciescalculatedwithLOMTIintheinitial non-calibratedge-ometry,intheabsenceofamedisturbances.5.1 InuenceoftheplenumgeometryStarting from 0.8 m, the inuence on the frequencies of increasing Lplenumisshown in gure 5.5. The frequencies obtained increase with the plenum length,butnomonotoniceectisnotedonthegrowthrates.Figure 5.4 - Plenum(yellowarea)inAE.95.AgasturbinecombustorFigure 5.5 - Eect of the variation ofplenumacoustical lengthonmodeof oscillation. o : Lplenum= 1.24 m;o: Lplenum=1.2m; o: Lplenum=1m;o: Lplenum= 0.8m;505.1. INFLUENCEOFTHEPLENUMGEOMETRYTheplenumradiushasbeenoriginallyarbitrarilysetto3.249m. However,this value does not correspond to the real geometry, since the maximum radialextensionintheplenumissomewhatlessthan2m. Thechoiceoftheplenumradius is critical. Our current model considers the plenum (and the combustionchamber)asathinannularduct. Inordertomaintainthevolumeoftheductclosetothereal one, reducingitsradiusmeansincreasingitsthickness. Thereal volumeis about Vreal=10.692m3. Sincewehavetogainasocalledacoustical lengththat shouldbesmaller thanthereal one, wecouldexpectthatapropervolumeshouldbesmallerthanVreal.CurrentGeometry ModiedGeometryPlenumlengthLplenum1.24m 0.8mPlenumradiusRplenum3.249m 2mPlenumthicknessdplenum0.614m 1.3mPlenumvolume 14.073m38.821m3rplenum/dplenum18% 65%Table5.3-ComparisonbetweenpossiblegeometriesInordertoobtaincorrectvaluesfortheazimuthal modes, itseemstobeimportant tohaveaproper valuefor theradius (that means thecircumfer-ential extensionthroughwhichanazimuthal modecanbeestablished). Thisobservation seems to lead to a sort of deadlock impossible to overcome withouttakingintoaccounttheradial dependency. Letusstill analyzetheeectofradial extensionontheeigenfrequencies. Ingure5.6itisclearhowcriticalthe choice of the radius is. Some frequencies such as those at about 60 Hzandat135 Hzexperiencealargealteration, andmoveoutof thewindowchosenfortheparametricstudy. Theeectof varyingtheplenumthicknessislesscritical,asshowningure5.7.51CHAPTER5. CALIBRATIONONAFEMCODEFigure 5.6 - Eect of the variation ofplenumradiusonmodeofoscillation. o: Rplenum= 3.249m; o: Rplenum= 3.2m; o : Rplenum= 2.5 m; o : Rplenum= 2m;Figure 5.7 - Eect of the variation ofplenumthicknessonmodeofoscillation.o: dplenum= 0.614m;o: dplenum= 0.8m;o: dplenum= 1m;o: dplenum= 1.3m;5.2 InuenceofthepremixersgeometryInthenal calibratedgeometry, thelengthandcross-sectional areaof thepremixers (yellow area in gure 5.8) will remain the original ones already used inthe computations for Sesta,i.e. Lpremixer= 0.142 m and Apremixer= 0.034 m2.However, inordertoprovideanextensivestudyofthegeometricparameters,theeectof increasingLpremixerandApremixerispresentedingures5.9and5.10, respectively. Theresultsaresimilar tothoseobtainedintheprevioussectionwhenvaryingtheplenumgeometry, i.e. themodes growthrates donot varymonotonicallywiththe length/areaof the premixers, whereas thefrequenciesdo: theydecrease(increase)whenthepremixerslength(section)increases.525.2. INFLUENCEOFTHEPREMIXERSGEOMETRYEven if we have decided to keep theoriginal premixers geometry, these g-uresprovideaveryimportantresult.The length scales associated to thepremixers are very small compared tothoseofplenumandchamber,soonecould expect a minor eect on the fre-quencieswhentheyvary. However,itappears that doublingthepremixerslength or cross-sectional area can shiftsome frequencies by 40% of their origi-nal value. This means that it is crucialto choose the premixers geometry cor-rectlytoobtainaccurateandreliableresults.Figure5.8-Apremixer(yellowarea)intheAE.95.AgasturbinecombustorFigure 5.9 - Eect on the frequenciesof varying the premixers length. o:Lpremixer=0.142 m; o: Lpremixer=0.15 m; o: Lpremixer= 0.2 m; o:Lpremixer= 0.3m.Figure 5.10 - Eectonthefrequenciesofvaryingthepremixerscrosssectionarea.o: Apremixer= 0.034m2;o: Apremixer=0.04 m2; o: Apremixer=0.07 m2; o:Apremixer= 0.1m2.53CHAPTER5. CALIBRATIONONAFEMCODE5.3 Inuenceof thecombustionchamber ge-ometryThecombustionchamber(coloredinyellowingure5.11)ismodelledasasinglethinannularduct; thisappearstobeamorereasonableapproximationthan in the case of the plenum. Varying the annulus length has little inuenceonthefrequencies(seegure5.12);decreasingthelengthby30%modiesthefrequenciesonlyby15%intheworstcase. Theconclusionisthesamefortheannuluswidth, whosevariationsonlyslightlymodifytheresults, asshowningure5.14.Figure5.11-Combustionchamber(yel-lowarea) inAE.95.Agas turbinecom-bustorFigure 5.12- Eect of the variationofcombustionchamber lengthonmode ofoscillation. o: Lcc= 1.24m;o: Lcc=1.2m;o: Lcc= 1m;o: Lcc= 0.8m;As in the plenum case, the most inuential geometric parameter is the cham-ber radius Rcc. It appears in gure 5.13 that some frequencies strongly decreasewhenRccincreases. Thefrequencyshiftcanbelargerthan40%whenRccde-creasesby35%.545.4. CALIBRATIONOFTHEGEOMETRYFigure 5.13- Eect of the variationofcombustionchamber radius onmodeofoscillation. o : Rcc= 3.249 m; o : Rcc=3.2m;o: Rcc= 2.5m;o: Rcc= 2m;Figure 5.14- Eect of the variationofcombustionchamber thickness onmodeof oscillation. o: dcc=0.614m; o:dcc= 0.8m;o: dcc= 0.5m;5.4 CalibrationofthegeometryTheparametricstudyof theprevious sections nowallows us tocalibratethe geometric parameters in order to t as much as possible COMSOL results.Asalreadystated, wedonotmodifythepremixersoriginal geometrysothatonlytheplenumandchamberparametershavetobechosen. Besides,consid-eringtheslightinuenceoftheplenum/chamberthicknessonthefrequencies,wechoosenot tousethemfor calibration: theywill becalculatedas func-tions of thelength/radius of theannuli inorder toremainclosetotherealplenum/chamber volumes. Theonlyrestrictionisthat thechosenthicknessremainssmall comparedtotheductradiussothatthethinannulusapproxi-mationholdsandradialdependencecanbeneglected.Choosingtheplenumlengthis tricky, sinceinternal reections cancreateadditional paths for the waves. Consequently, additional characteristic lengthsarecreated, asforinstancetheonedisplayedingure5.15. Goingonwiththis analysis, the plenum radius should be smaller than the real one, or better,than the maximum radius. This is because no obstruction seems to modify thecircumferential path available. This deduction leads us to take a plenum radiuscloseto2m.55CHAPTER5. CALIBRATIONONAFEMCODEFigure5.15- Apossiblepathforwavesintheplenum, shownwithablackseg-mentedline.As the model of the combustionchamberisrelativelyclosetotherealone, its length Lcccan be xed tobeequal tothereal one. As intheplenumcase, the combustioncham-ber radius chosen is smaller thanthe real one, i.e. smaller than 1.7m, as arst approximation. Start-ingfromthesepreliminaryconsidera-tions, andknowingfromthelastsec-tionstheinuenceofeachparameter,a long and extensive test campaignhas been run in order to x the geom-etryforLOMTI.Thenalresultsarepresentedinthenextsection, forthecase=0(i.e. noameperturba-tion).5.5 ModeshapeanalysisThissectionpresentsacomparisonof theshapeof thepressureperturba-tionforthefrequenciesobtainedwithCOMSOLandwithLOMTIinthenalcalibratedgeometry. Thegeometricparameters chosenfor theplenumandthechamberarepresentedintables5.4and5.6respectively. Themodesob-tained with COMSOL are displayed in gures 5.16-5.28, and compared to thoseobtainedwithLOMTIwhenrelevant.Tobetterunderstandtheresults,werecalltheexpressionofageneralpres-sureperturbationinLOMTI[32]:p

(x, , t) =Nn/2

n=Nn/2+1Anei(kx+t+n).Since the conguration is axisymmetric, no coupling between modes arises [15],which means that pure modes are easily recognizable in the mode shape visual-ization. It is thus possible to clearly identify for instance a n=2modein thecombustion chamber or a n=4modein the plenum where nis the numberofmaxima(orminima)alongthecircumference.565.5. MODESHAPEANALYSISIn COMSOL results, modes at 105, 150.1, 150.4, 150.5, 196, and 202 Hz havebeenidentiedascharacteristicof theplenumconguration. Thegeometricparametersintable5.4havebeenchoseninordertotLOMTIresultswiththese modes; the plenum thickness has been chosen in order to have the correctvolumeintheplenum.Asimilaranalysishasbeendevelopedforthecombustionchambergeome-try,consideringthechambermodesat116and193 Hz. Sincebothmodesareazimuthal,theyarestronglyinuencedbytheradiusvalue. Unfortunately,noradius value allowing to match both frequencies has been detected. The expla-nationprobablyresidesinLOMTIcurrentlumpedparameterapproximation:onlyonecharacteristicradiusisdenedforthecombustionchamber,whereasthereal chambergeometryisof coursemorecomplex. Thismismatchcouldpossiblybeovercomebysplittingthecombustionchamberintoseveralducts,withdierentadequatecharacteristicgeometries. Thenalgeometryintable5.6hasbeenchosentotthefrequencyat193 Hz.OptimalGeometryPlenumlengthLplenum2.3mPlenumradiusRplenum1.7mPlenumthicknessdplenum0.435mPlenumvolumeVplenum10.68m3Table5.4-Final calibratedgeometryfortheplenum.OptimalGeometryPremixerslengthLpremixer0.142mPremixersareaApremixer0.034m2Table5.5-Final calibratedgeometryforthepremixers.57CHAPTER5. CALIBRATIONONAFEMCODE.OptimalGeometryCombustionchamberlengthLcc1.3mCombustionchamberradiusRcc1.55mCombustionchamberthicknessdcc0.355mCombustionchambervolumeVcc4.495m3Table5.6-Final calibratedgeometryforthecombustionchamber.Figure 5.16 - Mode shape for the mode n = 0,obtained at 72 Hz with COMSOL (top)and70HzwithLOMTI(bottom).585.5. MODESHAPEANALYSISFigure 5.17 - Mode shape for the mode n = 1,obtained at 88 Hz with COMSOL (top)and56HzwithLOMTI(bottom).Figure5.18- Modeshapeforthemoden=2, obtainedat 105HzwithCOMSOL(top)and105HzwithLOMTI(bottom).59CHAPTER5. CALIBRATIONONAFEMCODEFigure5.19- Modeshapeforthemoden=1, obtainedat 116HzwithCOMSOL(top)and102HzwithLOMTI(bottom).Figure5.20- Modeshapeforthemoden=0, obtainedat 125HzwithCOMSOL(top)and139HzwithLOMTI(bottom).605.5. MODESHAPEANALYSISFigure5.21-Modeshapeforthemoden=2, obtainedat150.1HzwithCOMSOL(top)and156HzwithLOMTI(bottom).Figure5.22-Modeshapeforthemoden=1, obtainedat150.4HzwithCOMSOL(top)and150HzwithLOMTI(bottom).61CHAPTER5. CALIBRATIONONAFEMCODEFigure5.23-Modeshapeforthemoden=3, obtainedat150.5HzwithCOMSOL(top)and152HzwithLOMTI(bottom).Figure5.24- Modeshapeforthemoden=2, obtainedat 193HzwithCOMSOL(top)and191HzwithLOMTI(bottom).625.5. MODESHAPEANALYSISFigure5.25- Modeshapeforthemoden=4, obtainedat 196HzwithCOMSOL(top)and200HzwithLOMTI(bottom).Figure5.26- Modeshapeforthemoden=3, obtainedat 202HzwithCOMSOL(top)and196HzwithLOMTI(bottom).63CHAPTER5. CALIBRATIONONAFEMCODEAll in all, 11 of the 16 modes obtained with COMSOL have been found withLOMTI, forfrequenciesclosetothoseobtainedwithCOMSOL. Thismeansthat 5 of the COMSOL modes have not been detected. Their mode shapes aredisplayedingures5.27and5.28. Examiningthe3modesingure5.27,itisclearwhyLOMTIhasnotdetectedthem.Thesemodesareindeedconnedinaverysmall cornerof theplenum, ataverysmall length-scale, sotheycouldnot andshouldnothaveacor-respondentmodeinthelumpedparameterapproximation. Consideringtheirlocationintheplenum, theyareanywayveryunlikelytoyieldhumming. Itislessclearwhythe2modesingure5.28havenotbeenfound. Thesetwomodes are clearlyplenummodes andthere is noobvious reasonwhythesemodesshouldnotbedetectedwhentheotherplenummodesare. AsolutioncouldbetodividetheplenuminLOMTIinseveralducts,asexplainedaboveforthechamber, inthehopethatthesemodeswouldthusemerge. Itisverysignicant,nonetheless,that allthe combustionchamber modes are present inLOMTIscomputations.Figure 5.29 summarizes the results, highlighting the frequencies obtained/notobtainedwithLOMTIintheoptimalconguration.645.5. MODESHAPEANALYSISFigure 5.27 - ModeshapeofthemodesobtainedwithCOMSOLat165,188and206,andnotdetectedwithLOMTI.65CHAPTER5. CALIBRATIONONAFEMCODEFigure5.28- Modeshapeof themodesobtainedwithCOMSOLat 53and60, andnotdetectedwithLOMTI.Figure5.29- o: LOMTI modes intheoptimal conguration; COMSOLmodeswithverygoodagreement; COMSOLmodeswithworseagreement; COMSOLmodesundetected665.6. COMPARISONWITHFLAMEPERTURBATIONS5.6 Comparison between COMSOL and LOMTIwithaswitchedonameOnce the geometry is xed we are able to perform a further comparison witha switched on ame with reasonable condence. Again we have to set the sameboundaryconditionsandoperatingparametersbetweenthetwoapproaches.Moreoverthesameametransferfunctionistobeusedinordertocompareresults. Diering from the usual approach the reference point for this equationis now chosen at the beginning of the combustion chamber. This does not implythat the reference is placed just at the ame because in COMSOL the ame isnotmodelledasaatsheetbutitisathree-dimensionalsurfaceevolvingintothecombustionchamber. The-modelreads:Q

(t) = c2 1u

3(t ), (5.1)wherethe relationis: = c2 Q u( 1).Inthepresent settings, avalueof =0.2is equivalent to=1. Theeective value of the coecient in the transfer function is signicantly aectedbyQ that has to be tuned with the Mach number in order to obtain the properametemperature. ThereforeitisnotstrictlyrequiredthatthevalueofinLOMTI equalize the value of [ c2 Q]/[ u( 1)] present in COMSOL. On thecontrarythevalueof hasadierentmeaningbecauseitrepresentsthetimethe perturbations take to travel from the beginning of the combustion chambertotheamelocatedinapositioninternal tothechamberitself. Inpreviousworkthe time lagrepresentedonlythe time necessarytotravel throughapremixer.WehavejustarguedthattheapproachusedinCOMSOLisverydierentfromLOMTIs. Furthermore the temperature eldis not uniformineachduct andthus it is possible tomodel athree dimensional ame front withCOMSOLas showningure5.30. EventhoughtheameobtainedfromaRANS simulation is not completely reliable since turbulence aects signicantlythereactionrate, thedierencefromthethinatameusedinLOMTIandthespatiallydistributedtemperatureeldusedinCOMSOLisevident.67CHAPTER5. CALIBRATIONONAFEMCODEFigure5.30-TemperatureeldfromaRANSsimulation.In table 5.7 results obtained matching all the parameters present in LOMTIandCOMSOLareshown.Number COMSOL LOMTI Agreementwithoutame1 52+i - undetected2 58+i - undetected3 84+i 54-130i good4 90 48+45i bad5 102+5i 103-27i good6 112-48i 142+42i bad7 121-i 118+45i bad8 147+3i 149+2i good9 144+9i 150+4i good10 155-16i 157+16i good11 164+3i - undetectable12 187+2i - undetectable13 192 188-85i good14 198+20i 201+18i good15 201-i 186+44i good16 206+i - undetectableTable5.7-ComparisonbetweenLOMTIandCOMSOL. Therst columncontainsCOMSOLresultsfrom[25]. ThesecondoneshowsLOMTIresultswith= 1. Thelast column describes the agreement obtained that particular modes in the case withoutame of the previous section. Blue values are those likelytohave anagreementbetweenthetwodierenttoolsevenwithaswitchedoname.685.6. COMPARISONWITHFLAMEPERTURBATIONSComputational timerequiredis veryhighat lowMachnumber sincethevalues of the determinant become very large or very small, far from zero growthrate. A campaign looking for the best value of would be very time consumingsince very small frequency domains should be explored in many successive steps.From preliminary studies it is known that this problem is due to the low valueof theMachnumber. Asalreadydoneforsimplergeometriesletusanalyzetheinuenceof theMachnumberoneigenfrequencies. Resultsingure5.31showthatthemeanowcouldbeneglectedasarstapproximation. Ontheotherhandthepresenceof meanowgeneratesasignicantspeedupinthecomputations, and, giventhefactthattheresultsarebutmildlyaectedbythe presence of a mean ow, it becomes advantageous to include it, in any case.Figure5.31- ComparisonbetweenlowMachnumberandnonzeromeanow( =7ms,= 1,noCBO

s).Itisnowpossibletosearchforthebestvalueofwhichprovidesthebestmatchbetweenthetwoapproaches. Sincetheameislocatedintwodierentpointsandhavedierentthicknessinthetwomodels, thevalueof shouldnotbethesame. Infactthebestagreementshowningure5.33isobtainedforavalueof= 0.2(LOMTI)insteadof= 1(COMSOL).Moreover, small variationsofand signicantlymodifyseveral eigenfre-quencies. Weareforcedtoinferthatsuchatransferfunctionistooweaktorepresent a complex conguration like a gas turbine combustor because the stillarbitrary choice of ame parameters strongly aects results making them fairlyunreliable. Things could be possibly improved if we could consider and (or)asafunctionof,possiblyusingLESresultsfromCERFACS.Seesection6.5formoreonthispoint.69CHAPTER5. CALIBRATIONONAFEMCODEFigure5.32-ComparisonbetweenLOMTI( =7ms, =1, noCBO

s, nonzeromeanow) andCOMSOL[25]. Someof theazimuthal modes havealsoanaxialcomponent.Figure5.33-ComparisonbetweenCOMSOLresultsobtainedin[25] with =7msandLOMTI(= 7ms,= 0.2,noCBO

s,nonzeromeanow).70Chapter6ParametricstudiesOncethegeometryisxedweareable to take advantage of LOMTIsfeatures. Letustrytoassesstheef-fectof importantparameterspresentinthemodel. Theseparametersare:combustion chamber size(lengthandradius),amemodelsparameters-,amemodelsreferencepoint,amemodelitself.Simulationsareperformedwiththerealgasturbinegeometryandswitchedon ame perturbations. The conguration used is shown in gure 6.1. This setup has been experimentally proven to be the best conguration concerning theappearanceofthermoacousticinstabilities.6.1 CombustionchambersizeEvenif there is noreal possibilitytostraightforwardlychange the entiregeometry of the annular combustor, the inuence of general geometrical values,suchasplenumlengthandradius, hasjustbeendetectedinchapter5while71CHAPTER6. PARAMETRICSTUDIESFigure6.1-CBO

scongurationinAE94.3Agasturbine(nottoscale).Burners1,2,23,24arewithoutCBO

s.calibrating the model on COMSOLs results. However for real conguration (20CBO

s and ame perturbations) the inuence of combustion chamber geometryis analyzed in gure 6.3 and 6.4. This study has been performed in order to gaininformationaboutfuturedevelopmentsofthecombustionchambergeometry.Analyzinggure6.2, thenewmodel of combustionchamber (A5) is longer thantheprevious one(A4) andits meanradius is smaller.Concerning its thickness nosensible dierences are de-tected. Furthermore, thethickness d aects signi-cantlyonlythe meanowparameters; in the eigen-modes evaluation d is al-ways multiplied by the per-tinentradiussothatitsef-fectcanbeincorporatedinthevariationsofthecham-bermeanradius.Figure6.2-ComparisonbetweenA4andA5com-bustionchambers.In order to obtain sensible variations of the eigenmodes, the length and radiusofthechamberhavebeenmodiedbeyondtherealpossibilitiestochangethereal geometry. Theresults for varyinglengths andradii of thecombustionchamberareshowningures6.3and6.4respectively.726.1. COMBUSTIONCHAMBERSIZEFigure6.3-o: Lcc= 1.3m;o: Lcc= 1.5m;o: Lcc= 1.7m;Zoomsfora,b,c,dareasareshowningures6.5and6.6.Modesshownwithablackcircleareexactlyneutral andhardlydetectablebyourgraphical approach; thusthepossibilitythattheyareonlynumericalsingularitiesshouldbetakenintoaccount. Ingure6.7contoursof real andimaginarypartof thedeterminantareshown: probablysomenumerical ap-proximationleadtoaperturbationinsomeregionsofthecomplexplanethatarerecognizedbythegraphicalsolveraseigenmodes.Nevertheless, eveniftheywerereal solutions, theirzerogrowthrateallowsto disregard them due to the presence of viscous forces not modelled in LOMTIthat would possibly force them into the stable half-plane. Two couples of eigen-modesmustbecarefullytracked: oneisfoundcloseto70 Hzandtheotherisinthevicinityof 187 Hz. Therstoneseemstohaveonlybenetswiththemodiedgeometry. Thelatterhasdierentresponsesforindependentvaria-tions of the length and radius, thus no trend is easy to detect. A further studyhas been performed considering only this mode, and by varying simultaneouslylengthandradius. Ingure6.8itisshownthatareductioninradiustogetherwithanincreaseoflengthyieldanenhancementoftheamplicationfactorofthisunstablemode.73CHAPTER6. PARAMETRICSTUDIESFigure6.4-o: Rcc= 1.55m;o: Rcc= 1.5m;o: Rcc= 1.3m;Thisparametricanalysishasbeencarriedouttoinfertrends. Sincethereareonlysmall changesfromtheA4totheA5gasturbinemodels, weexpectthattheeectsonthermoacousticstabilitymodesareminor. Unstablemodesaretoofarfromzerogrowthratetobedampedonlythankstoanincreaseof4%ofthecombustionchamberlength.6.2 FlameparameterIt has alreadybeenstatedhowcritical couldbetheamemodel chosen.Theclosureequationthatdescribestheinteractionsbetweenheatreleaseandperturbations has been discussed in chapter 4. In this section we want to pointoutitsimportanceandconsistencyfocusingonthe-modelusedsofar.Simulationsforthecompletecomplexplanewouldbehighlytimeexpensiveandprobablywouldnotyieldremarkableresults. Thusonlythen=2modeat187 Hzwill betrackedduring variation. Thechoiceofthismodeisduetoitsrelevanceinlimitedexperimental dataatdisposal. Infactitisknownthat instabilities ariseinthecombustionchamber inazimuthal directionatfrequencynotfarfrom170 Hz, thereforethismodeisthemostsuitabletotthevibrationsnoticedinrealgasturbine.746.2. FLAMEPARAMETERFigure6.5-ZoomforareaaandbFigure6.6-ZoomforareacanddThewaytochoosethetimelagis averyhardtask. Simulations runatCERFACSonpremixersusedinAE94.3Agasturbinehaveprovidedavalueforanacoustical closeto1ms. Thevalueusedsofaris7ms. Thiswasevaluatedfrom:=Lpremixer + LCBOUpremixer=0.142m + 0.054m68m/s 3 ms;however, sinceswirlisveryhighinthepremixers, theowtakemoretimetoreachtheamessurface,thatalreadyisnotlocatedattheendofCBO

sbutalittledownstream. Thustimelagwaschosenhigherthen3msinordertorepresentthishighlyswirledow.75CHAPTER6. PARAMETRICSTUDIESFigure6.7-: realdet(X). : imagdet(X). Positiveandnegativehalf-planeFigure6.8-Modeat187Hzwithlengthandradiussimultaneouslyvarying.Ingure6.10theinuenceof isshown. Asingularresultoccurs: indeedtherearetwocouplesofeigenmodesforeachvalueof. Modeshapesaresosimilar that theycanbeeasilyconfusedwithoneanother (ineect fromapreviousanalysismodesfor=7 msand=1 mswererecognizedasbeingthesame). Thisfactgeneratesfurtherdoubtsontheconsistencyof thethinameapproximation.Wecouldbeinducedtothinkthat thevalueof aspeciceigenfrequency=r+ iishouldbeunchangedif wechoosetwovaluesof shiftedonlybytheperiodof thateigenfrequency, i.e. T=2/r. Performingsomeeasyalgebraicpassagesitbecomesclearthatthisstatementhasnomathematical766.2. FLAMEPARAMETERconrmation when modes are not neutral. Imposing 1= and 2= 2/rweseekfortheconditionthatallowstowrite:ei(t1)= ei(t2)$$$$ei(t)=$$$$ei(t) ei 2r.This expressionbecomes anidentity only if = r +ii= krwhere we havedenedk 0, 1, 1, 2, 2, . . . . Thusimustbezeroandmodesmustbeneutral in order to assert that could be shifted by the specic period in ordertorecoverthesameeigenmode.Figure6.9-Modesnear90Hz.77CHAPTER6. PARAMETRICSTUDIESFigure6.10-Modesnear170Hz.Furthersimulationshavebeendonefordierentvaluesofdemonstratingthatfor=0and =1msor =7msthesecoupleof modesmeetinasingleplace: thisresultsdoesnotallowtostatethatthissplittingisdueonlytotheametrasferfunctionchosenbecausefor= 4.5msthisfactdoesnotoccur. Thestrongsensitivityofeigenmodestotheameparametersdoesnotallow to rely on any results obtained in terms of frequency and growth rate butonlytoinfertrends.6.3 FlameparameterThe inuence of is evaluated in the following. No general trend arises fromthese results. Modes near 150 Hz (n = 1, 2, 3 located in the plenum) are weaklydependentontheametransferfunctionparameters.786.4. TRANSFERFUNCTIONSREFERENCEPOINTFigure6.11-Modesnear170Hzfordierentvaluesof.Figure6.13-.6.4 TransferfunctionsreferencepointToachieveinformationabouttheproblemdiscussedinsection4.6,wehavealso performed a parametric study on the distance Lifbetween the ame and79CHAPTER6. PARAMETRICSTUDIESFigure 6.12 - Modes near 170Hz for dierent values of tracked from dierent valuesof.thereferencepoint whereperturbations present inthetransfer functionareevaluated. Onlythreedierenteigenmodesaretrackedfromareferencepointlocated exactly at the ame (end of premixers) and another one located at thepremixerentry. Thesemodesarechosenaccordingtotheirrepresentativeness.Bothaxial andazimuthal modesarepresentandthelatterhavebeenchosenamongtwodierentclasses: therstisstrictlydependentonametransferfunctionparametersandthesecondseemstobequiteindependentfromtheseones. Surprisingly, nodierenceresultsfromverydierentreferencepoints.This is a good news since it demonstrates that the choice of the reference pointdoesnotinuencesomuchtheresults. Thisisprobablyduetothefactthatpressuremodeshapesinthepremixersdonotundergogreatchangesintheiramplitudealongtheduct.806.5. INFLUENCEOFTHETRANSFERFUNCTIONSFigure 6.14 - EigenfrequenciesforincreasingLiffromzerototheentirepremixerslength.6.5 InuenceofthetransferfunctionsFinally, a comparison between dierent transfer functions is carried out. Letusrstanalyzethedierencesbetweenequation(6.1)and(6.2). Again, onlythree representative eigenmodes are chosen. The frequency at 150 Hzis clearlynotmuchinuencedbyametransferfunctionchoiceasshowningure6.15.Since in every simulation it has been noticed to not vary signicantly its positionweareinducedtoconsideritasageometricalmode. Thesameconsiderationisvalidalsoforalltheothersub-stablemodes.Q

Q= m

mei(6.1)Q

Q= u

uei(6.2)The evaluation of the proper and is a challenging task. LES simulationsperformedatCERFACSprovidevaluesoftheseparametersdependingonthefrequency.81CHAPTER6. PARAMETRICSTUDIESFigure6.15-TF1: equation(6.1);TF2: equation(6.2).This dierent approachinevitablyleads toanewaspect of spectra. Letusrstconsidergure6.10. When variestheeigenfrequencyshiftsinthefrequency-growthrateplane. Letusimposethatwhen =1or =2theeigenfrequencies obtained are respectively f1and f2(let us not account for theamplicationfactorforthetimebeing). Ifthevaluesof= (f)obtainedatCERFACSforf1andf2aresimilarto1and2thentwosimultaneouseigen-frequencieswillbedetectedatthesametime. Sinceweknowfromexperiencethatsmallvariationsimplysmallfrequencyvariationsthissituationisnotauselessbutarealcasethatleadstosomethinglikeacontinuousspectrum.This must not induce us to think that varying the value of could generate asort of closed loop following one eigenfrequency. If varies and the eigenmodeshiftssomewherethesolutioncouldgenerateanopenpathasshowningure6.18.Sesta. Apreliminarytestwasdoneonadierentgeometricalconguration.AcodebasedonanexperimentalplantlocatedinSestahasalreadybeenim-plemented. Thisplanthasthepurposetotestdierenttypeof burnersandgivesalsoinformationsontheirthermoacousticbehaviour. Aschemeof thisplantisshowningure6.17.826.5. INFLUENCEOFTHETRANSFERFUNCTIONSFigure6.16- valuescalculatedatCERFACSforburnerswithCBO

s. Greensquares are numerically computed values and the blue line is a parabola tted on thesevalues.Considerationsoncontinuousspectraareconrmedbyresultsingure6.18whereasingleeigenmodegoesthroughthedomainoutliningthreepeaksat65 Hz, 98 Hzand141 Hz. Duringitsmovement, themodeshapevariescon-tinuouslyasconrmedbygures6.20to6.22.It could be interesting the fact that a mode at 141 Hzhas just been detectedinaprevioustestwithaxedvalueof =1.59ms(gure6.19). Inthistestonlythiseigenmodewasfoundandthustherelativecontinuousspectrumiseasyrecognizable.83CHAPTER6. PARAMETRICSTUDIESFigure6.17-ModelledgeometryoftheburnerinSestausedinLOMTI.Figure 6.18 - Eigenmodes for evalu-ated from tting values obtained at CER-FACS.Figure 6.19- AE94.3Aeigenmodes fordierent valuesof and =1.59from[31].Turbine. LetusnowapplytheresultsfromCERFACSshowningure6.16tothepresentcongurationoftheAE95.3Aturbine. Theresultingspectrumisshowningure6.23.Sinceelevenmodesarepresentinthiscongurationitisclearhowdicultitistorecognizecontinuousspectralikethoseobservedingure6.18. Eveniftheywereeectivelypresent, thediscretecomputationresultsinacloudofmodes. Tryingtoplotmodeshapesof someof theseeigenfrequenciesweareabletoreproducethoseobservedinpreviouscalibrations. Forexample,letusevaluatemodeshapesfor peaksat 56 Hz or at 200 Hz. Theyareshowningure6.24and6.25. Thesemodeshapesareverysimilartogures5.17and5.25. Butactuallyweareabletoplotmodeshapesforeveryeigenfrequency,846.5. INFLUENCEOFTHETRANSFERFUNCTIONSeven if it is not an eigenfrequency that renders the determinant zero;thereforeinthegreatquantityofmodesfound, probablyweareabletondouteverymodeshapewewant. Sowehavetorecognizeapathforeachmodeinthedomaininordertodevelopsomecondenceintheresults.In gure 6.26 is plotted a comparison between results obtained with two dif-ferent transfer functions. TF2 is the usual transfer function used at CERFACS;TF4isthetransferfunctionusedatPoliBaforCOMSOLsimulations.Figure6.26-AE94.3Aeigenmodesfor evaluatedfromttingvaluesobtainedatCERFACS.TF= 2: Q

/Q = u

u ei. TF= 4: Q

/Q = c21u

3(t )Alsointhiscaseacloudof modesappearsrenderingtheidenticationofinterestingdisturbanceshapesextremelydicult. Theresultof thisstudyisthatemployingand functionof inthecodecomplicatesthingsratherthanclarifythem.85CHAPTER6. PARAMETRICSTUDIESFigure6.20-Modeshapesforthepeakat65Hz.866.5. INFLUENCEOFTHETRANSFERFUNCTIONSFigure6.21-Modeshapesforthepeakat98Hz.87CHAPTER6. PARAMETRICSTUDIESFigure6.22-Modeshapesforthepeakat141Hz.886.5. INFLUENCEOFTHETRANSFERFUNCTIONSFigure6.23-AE94.3Aeigenmodesfor evaluatedfromttingvaluesobtainedatCERFACS.Figure6.24-Modeshapeformodeat56Hz ( evaluatedfromttingvaluesobtainedatCERFACS).Figure 6.25 - Mode shape for mode at 200Hz ( evaluatedfromttingvaluesobtainedatCERFACS).89Chapter7Annularductswithnon-negligiblethicknessLetusconsideracylindrical ductof radiusRandthicknessd0, wherem,nisobtainedfromdJndr(m,nR)=0. These94cut-ofrequencieshavebeencalculatedin[33],andaregivenintable7.1.n plenum chamber0 602.57Hz 1126.97Hz1 604.60Hz 1130.02Hz2 610.72Hz 1139.22HzTable7.1- Cut-ofrequencies infunctionof azimuthal wavenumbernforradialmodem = 1.Results for m > 0 conrm the damping phenomenon of the duct. For examplein gures from 7.4 to 7.6 the lowest frequencies of oscillation in the plenum forn = 0, 1, 2andm = 1areshown.(a) (b)Figure7.4-Modeshapeform = 1andn = 0at603.8Hz,GR = 1.7s1.Itisremarkablethefactthatall thesemodesareslightlyhigherthantherelativecutofrequency. Thisisverycomfortablesinceitimpliesthattheycouldbesafelydisregardedduetotheirveryhighfrequencyandtheirconse-quentabsenceofrelationtothehummingphenomena. Ontheotherhand, itisimportanttobeabletoaccountforthem,sincetheyareverylikelyrelatedtoaphenomenonknownasscreech[34].95CHAPTER7. ANNULARDUCTSWITHNON-NEGLIGIBLETHICKNESS(a) (b)Figure7.5-Modeshapeform = 1andn = 1at605.8Hz,GR = 1.75s1.(a) (b)Figure7.6-Modeshapeform = 1andn = 2at612.1Hz,GR = 1.8s1.96Chapter8ProposalforanewtransferfunctionSince the only weakness of this lumped approach seems to be the ame modelperhapsanewtransferfunctionbasedonempiricaldatashouldbedevelopedbasedonknownphysical principles. Previousmodelshavealreadybeenana-lyzedinsubsection2.3.1andtheinuenceof dierenttransferfunctionshasbeenevaluatedinsection6.5.8.1 DeducingtheexpressionAs just stated we try to derive a new transfer function based on consolidatedprinciples. Theheat releaseper unit of areaQ[W/m2] is well knownfromequation(8.1):Q = mFHiA. (8.1)where mFisthefuel masslowrate, AistheameareaandHiisthelowerheatofcombustion.Since in the lumped approximation the ame is actually located at the com-bustionchamberinterfaceswitheachburnerandthethermodynamicvariableof thecombustionchamberaretunedinordertomatchtheempirical ametemperatureweassumethat Ais equal toAflame=Accat thecombustionchamberinletsection. Thisisastronghypothesiswhichdoesnotmatchwiththeeectiveameshape; despitethis, itcertainlyrepresentstheexactamemodelthatthelumpedapproximationhasimposed.97CHAPTER8. PROPOSALFORANEWTRANSFERFUNCTIONImposing mflame= mair+ mF=flameuflameAcc, anddeningtheair/fuelratio = mair/ mFwegettheexpression:Q =Hiflameuccflame + 1. (8.2)Weremarkthefactthatallthequantitiesarecalculatedatthecombustionchamberinletwheretheameisactuallylocatedinthepresentmodel. Oper-ating the usual decomposition, simplifying the proper terms and linearizing thelower order ones we get the expression where no direct inuence of Hiappears(andthusthefeaturesofthefuelarenotsignicant):Q

Q=

flame flame+u

flame uflame

flame flame + 1(8.3)DensityandvelocityarealreadypresentinLOMTI. Thiscannotbesaidfortheair/fuelratioatthecombustionchamberinlet. Analyzingthephysicsofthephenomenonitisclearthattheuctuationofthisquantityisdrivenbythepressureuctuationsatthefuelinjectorsthatarelocatedinthemiddleofeachdiagonalswirler.1Theideaistorelatetheair/fuelratioperturbationattheametothepressureperturbationattheoriceofeachinjector(withorwithoutCBO

s). Itisclearthatthisvaluehastobeshiftedbythetimeitspends to travel from the injectors to the ame (ei(f)).2Indeed the pressureof the fuel supplyducts couldbe assumedconstant andindependent fromouter pressure perturbations of the air ux. Therefore, the air/fuel ratio at theinjectorsislinkedwiththefuelandairpressuresaccordingtoDaltonslawofpartialpressuresthatstates: thetotalpressureexertedbyagaseousmixtureisequaltothesumofthepartialpressuresofeachindividualcomponentinagasmixture. In mathematical terms, for fuel and air only, it reads: ptotal= pF+pair,or yF= pF/ptotalwhere ptotalis the mean pressure at the injector itself, i.e. themeanpressureineachpremixerduct(p2).Since the thermal power produced by the ame is, more or less, all convertedinenthalpyof themixture, wecanwrite, neglectingthekineticenergy, that mFHi= mH0 mF( + 1)cP(Tcc Tpremixer).3Sincethemethaneheatof1Thepossibleinuenceofthepilotdiusionamesustainedbytheaxialswirlerisdisre-garded.2Notethatthisrepresentsonlyaphasedisplacementsinceeit ei= ei(t).3Tcc= Tflame.988.1. DEDUCINGTHEEXPRESSIONcombustionisapproximately50MJ/Kgandthetemperaturesareknown, avalueof 35isunequivocallyobtained. Thisvalueconrmsthechemicalapproachthat imposes aleancombustionwheretheequivalenceratiois setapproximatelyto 0.53. InthiscaseisdenedasST/where, forareactionlikeCxHy + (x +y4)(O2 +7921N2) xCO2 +y2H2O + (x +y4N2),astoichiometricair/fuelratiolikeST=_x +y4_ 10021PMairPMCxHy,results. ForthecombustionofmethaneinairST 17.167,thus = mair mF=ST=17.1670.53 32.From the value of we are able to unequivocally dene the value of pFthatis considered unaected by any uctuation; then in its expression all the valuescouldbereplacedwiththeirmeanquantity: =mairmF=nairnFPMairPMF=pairpFPMairPMF=ptotalpFpFPMairPMF=_p2pF1_ PMairPMF.ImposingPMair/PMF==28.84/16 =1.8025for methane we get theexpression:pF= p2 + .Then we have to relate the pressure uctuations at the injector with the air/fuelratiouctuations. Againfromtheexpressionofweget: =_p2pF1_.Aftertheusualdecompositionitbecomes:

+

=_ p2pF+p

inj,pF1_.wherethevalueofpFhastobereplacedwithitsexpression. Theexpressionfor

= f(p

inj)becomes:99CHAPTER8. PROPOSALFORANEWTRANSFERFUNCTION

= ( + )p

inj p2. (8.4)We can briey discuss on its physical meaning arguing that a positive value forp

, i.e. apressureincreaseofthemixtureinthepremixer, yieldsaminorfuelmassowrateandthenanaugmentedair/fuelratioorratherapositivevaluefor

. The value of

already obtained reaches the ame,i.e. the combustionchamberinlet,afteraconvectivedelaytimerepresentedbytheusualfunctionei. Thenalexpressionfortheproposedtransferfunctionisthen:Q

Q=

flame 3+u

flame u3 + + 1 p

inj p2ei, (8.5)heresubscripts2and3standforpremixersandcombustionchamberrespec-tivelywhileflameandinjmeanthecombustionchamberinletandtheinjec-tion point. It is remarkable the fact that ( +)/( +1) is approximately equalto unit in the case of lean combustion of methane and air, thus its contributionmustbetakenintoaccount. Letusfurtheranalyzethisterm: itpracticallytakes theplaceof theunknownparameter that is sonolonger unknown.ReplacingwithST/letusplotthistermasafunctionof.Figure 8.1- Inuence of onthe transfer functiontermcontaining multiplyingp

. Theblackcrossdenotesthevalueof =1.024for=0.53usedfortheleancombustion.From gure 8.1 it is clear that the evaluation of is not a critical issue sincethevalueof( + )/( + 1)remainsveryclosetooneevenifthevalueofvariesfromzero(nofuel)toone(stoichiometriccombustion). Itseemstobealineartrendbutactuallylim+=(notethatthevalidityofthemodel1008.1. DEDUCINGTHEEXPRESSIONbreaks o for > 1) thus it is a bounded set. Nevertheless we have just arguedthat its value is always near to one,forcing us to take it into account for everypossibleload.Thisnewametransferfunctionhastheaimtoovercomesomeinconsisten-cieswithpreviousmodels. Althoughthereisnounknownmultiplyingfactorsuchas, thepresenceofthedelaytime isnotavoided. Stressingthefactthatwearetryingtorepresentthelumpedatamethisvalueof shouldbe exactly equal to the ratio between the distance of the fuel injectors from theame and the mean ow velocity in each premixer duct. In this case assumesclearlyonlyaconvectivemeaning, thereforethebestwaytodetermineitisbyevaluatingthetimeaparticleemploystotravel fromtheinjectiontotheamesurfacewheretherateof reactionismaximumaswecanseeingure8.2. Sincetherearemorepathlinesthatcanbeevaluatedameanvalueofthistimesshouldbeconsidered.Figure 8.2 - Resultsfrompathlinescomputationforparticlesreleasedinthediagonal(left) andintheaxial (right) swirler. Thepeaks denotethetimewhenaparticleexperiencedthemaximumrateofreaction,i.e. whenitpassthroughtheamefront.In the calibration phase not much importance was ascribed to the premixersdimensions. In such a model, where the delay time is calculated from geometri-cal values, the length of the premixers assumes a new importance. Furthermore,itiswell knownthatthemixtureuxintheseductsisconsiderablyswirled,thus a correction that takes into account this feature could be required in ordertoobtainreliableresults.44Theswirl numberSw=(Moment of momentum)/(MomentumRadius) 0.5thusthevalueoftheconvectivedelaytimeshouldbeapproximatelydoubled.101CHAPTER8. PROPOSALFORANEWTRANSFERFUNCTIONTheeventual weaknessesof thisamemodel arethesameencounteredinthe lumpedapproachthat inevitablyconsiders aat regionof heat releaseconnedinathinsheetatthepremixers/combustionchamberinterface. Theuctuatingquantitiesarealwayscalculatedattheameitself whenpossibleorrelatedtootherphysicalvariablesaccordingtowellknownprinciples. Theonlyfurtherhypothesisistoconsiderthefuel pressureindependentfromtheunsteadyoweldat the injectors. Nevertheless this seems tobe agoodapproximationconsideringthe mechanismof fuel supplythat consists inarotatingvalvemodulatingthemassowrateof fuel injectedinthediagonalswirlers. Thereforenofurtherhypothesisisdonetoobtainthisametransferfunction.If itwouldbepossibletocontrol thevalueof pFrenderingittimedepen-dent,this approachcouldgive informationonthe designof active controlthatmodulatefuelinjectioninordertogetanegativevalueofthep

(x, t)q

(x, t)termintheRayleighscriterionforallregimes.8.2 ResultsfortheturbinecongurationIn gure 8.3 results for transfer function (8.5) are shown. Only two unstableeigenmodesarefoundat106and195 Hz(showningure8.4). Furthermoretheyseemnottobegreatlyinuencedbyvariations.Figure 8.3 - Resultsfordref=12Lpremixer. ThetypicalcongurationofAE94.3Agasturbinewith20CBO

sischosen. Theacousticboundaryconditionsareu

= 0.1028.2. RESULTSFORTHETURBINECONFIGURATIONFigure8.4-Modeshapefor106 Hz + i70 s1and195 Hz + i23 s1eigenmodesre-spectively.Further analysis varying the reference point conrm its weak inuence as justdeducedfromthepreviousstudyshowningure6.14. Anexplanationcouldbe found analyzing the pressure mode shape in each premixer: actually p

doesnotvarysignicantlyalongtheducts, henceareferencepointlocatedinthemiddle of the duct experiences very similar pressure oscillations if compared toanotheroneatanyotherlocation(inparticularifweareprimarilyinterestedintheirphase).It is remarkable the fact that we obtainonlytwounstable modes as weexpect fromexperimental results. Neverthelessthesemodesseemnot tobegreatly inuenced by the parameter that should be related to the appearanceof the humming phenomenon. At rst, we could assume that this fact nds anexplanationinthepressuremodeshapesinthepremixers. Performingatime-spaceequivalencewegobacktothestatement donefor thereferencepointinuence. Butthisargumentshouldbevalidalsoforthecommonlyemployedtransfer function, andthisisnot true. Thereforeweareforcedtoconcludethat p

2has a lower order of magnitude than m

3. According to this, the leadingtermseemstobethemassowrateperturbationinthecombustionchamberitself. Evenif pressureperturbationsattheinjectorsmodifytheequivalenceratio of the mixture, the fuel mass ow rate at the ame is primarily inuenced103CHAPTER8. PROPOSALFORANEWTRANSFERFUNCTIONbytheoverall massowrateattheameitself. Finallytheheatreleaserateis dueonlytothefuel reachingtheamefront. Weareforcedtoconcludethat therelationbetweenandp

2isonlyarenement of themodel. Theresultingsimpliedtransferfunctionwherepressureinuenceattheinjectorsisneglectedis:Q

Q=

flame 3+u

flame u3, (8.6)and results are shown in gure 8.5. The unstable modes are only slightly inu-encedbythissimplication. Thisresultsconrmthehypothesisjustoutlined.Neverthelesstransferfunction(8.6)doesnotensureanyspeed-upinthecom-putationoftheeigenvalues,thereforeequation(8.5)shouldbepreferred.Figure8.5-Comparisonbetweenresultsfor= 4msingure8.3andthesimpliedtransferfunction(8.6).104Chapter9Conclusionsandfuturedevelopments9.1 ConclusionsThegeometricparametersusedinLOMTItorepresentanANSALDOtur-binecongurationhavebeencalibratedinordertoreproduceresultsobtainedwithCOMSOLina3Dsectoroftheturbine,withoutameanow.Considering the drastic geometric ap-proximationsusedinLOMTI, thematch-ingof LOMTIresultswithCOMSOLs isverygood, bothwithandwithout ameand the capacity of LOMTIs simple modeltoobtainanalogous results withamuchsmaller computational time is doubtlesssatisfying. Further development couldbe done by increasing the number ofducts representing plenum and combustionchamber inorder to improve the matchwiththereal geometryasshowningure9.1.Figure9.1- Twopieces congura-tionforthecombustionchamber.Severalparametricstudieshavebeencarriedout. Theinuenceofcombus-tionchambersizehasbeencheckedinordertoprovideinformationsonnewdevelopments of the AE94.3A gas turbine. The inuence of and parameters105CHAPTER9. CONCLUSIONSANDFUTUREDEVELOPMENTShas been evaluated,also with frequency dependence. Dierent FTF have beentestedwithdierentreferencepoints.Constantspecicheatsapproximationhasbeenremovedinordertoobtaina more accurate model. cpand cvare now functions of temperature and chem-ical compositionineachduct. Finallythethinductapproximationhasbeenremoved to demonstrate that it provides an accurate representation because ofthe presence of cuto frequencies that allow to disregard high radial frequencies.9.2 Futuredevelopments9.2.1 FurtherdevelopmentoftheFTFInorder toovercometheat ameapproximationit will benecessarytodeveloparenedametransferfunctionfromequation(8.5). Inthatmodelthe surface of the ame is approximatedbythe cross sectional areaof thecombustion chamber applying the well known relationship m = uA. Howevertheamesurfaceitselfisinuencedbyacoustics,sothattherelationbelowismoreappropriatesinceittakesintoaccountthesevariations: m = uSTAflame. (9.1)SinceuisthedensityofunburnedgasinthepremixerswehavetodealonlywiththeturbulentamespeedST. Unfortunately, therelationshipsavailablearefullofempiricalparameters[39]orcontainunknowntermssuchas m

Finequation (9.2) that is the negative production of fuel inside the ame volume,denedas mF/(Aflame):STSL= 1 +u

rmsSL,SL=2T2ucP( + 1) m

F.(9.2)Amodel [40] thatrelatestheamesurfacevariationstothesupplyvelocityucanbeusedtodeneanoscillatingame. Howeverthismodelrequirestheevaluation of the characteristic time that represents the average time for gasexitingtheburnertobeconsumedbytheame. Forclarityinequation(9.3),thenotationisusedtomakedistinctionwiththeusualtimelag.1069.2. FUTUREDEVELOPMENTSA

flameAflame=11 + iu

u. (9.3)Imposingei=(1 + i)1wecanwriteequation(9.3)accordingtotheusualexpressionsforthetransferfunctions:A

flameAflame=11 + 22u

ueiatan(). (9.4)Directlylinearizingtheequation(8.1)weobtaintheexpression:Q

Q= m

2 m2A

flameAflame

flame flame + 1. (9.5)In this expression m2 is no longer evaluated at the ame front but it represents amass ow rate upstream, hence a multiplying term eiis required. Combiningtheequations(8.4),(9.3)and(9.5)wegetthecompleteexpression:Q

Q=_ m

ref m2 + + 1p

inj p2_ei11 + 22u

ueiatan().It is immediately evident that the number of parameters has grown since bothand aretobeevaluated, andthereferencepoint must bechosen. Ul-timately, acareful studyof equation(9.3)shouldbecarriedoutinordertodetermineitsrangeofapplicability.9.2.2 ComputationaltimeForsimplecongurationstherearenoinconvenienceswiththeMatlaben-vironment: thedeterminantofthematrixgeneratedfromthesetofequationscanbecalculatedforeach=r+ iinaveryshorttime. Thecoderunsinafewsecondsandnoproblemsconcerningspeeduparise. Thecomputa-tional timerequiredcouldbecomenolonger negligibleinmorecomplicatedcongurations. Sincemorethan60%of elapsedtimeisspentcomputingthedeterminant,thecodecouldbeupgradedtoaMEXFilethatusesacompiledprogram(C++,FORTRAN)forcalculatingthedeterminantforeachvalueofinordertoobtainasignicantspeedup.9.2.3 PassivecontrolsSeveral solutionsfordampinginstabilitieswithoutdesigningactivecontroltechniqueshavebeentestedinrecentyears. Disruptingaxisymmetryadding107CHAPTER9. CONCLUSIONSANDFUTUREDEVELOPMENTSCBO

shasbeenmentionedinsection4.3. Thisistheeasiestandlessintru-sivewaytodamptheinstabilities. OtherdevicesfocusonasingleresonantfrequencysuchasHelmholtzresonators.Helmholtz resonators are studied todampstandingwavesoccurringinsidethecombustionchamber. The characteristicfrequency, suchas amass-springsystem,is:R= 2fR= cs_AnVLnwhere An and Ln are respectively the crosssectional areaandthelengthof theneckof theHelmholtzresonator, andV isthevolumeoftheentirecavity.Figure9.2- Academical Helmholtzresonators.Wenowrapidlycommentontheeectof addingHelmholtzresonatorstothegeometry. Helmholtzresonatorsaredampingdevicesconsistingofalargevolumeconnectedtothecombustorviaashortneck.1AttachingaHelmholtzresonator destroys theaxisymmetryof thegeometry, leadingtomodal cou-pling.2FurthermoretheresonatoritselfisabletomodifythepressureeldatitsneckconnectiondampingthecharacteristicfrequencyfR.The locationof the Helmholtz resonators has amajor inuence ontheireectiveness. AsingleHelmholtzresonatordoesnotabsorbenergyfromacir-cumferential wavebutonlyreectanequal andoppositewave, generatingapressurenodeattheneckconnection. Thus, inordertodampdangerousaz-imuthal waves multiple resonators distributed in optimal locations are required.Theoptimalplacementis:j=n_j 1N+ P_wherej istheindexof theN>1resonators, nisthecircumferential wavenumber and Pan arbitrary positive integer. The main problem with Helmholtzresonators is the dimension of the cavity required to damp typical low frequency1Problems in placing Helmholtz resonators in the combustion chamber could arise becauseoflmcooling,assemblingandmaintenance.2The interruption of axisymmetry has already been noticed adding CBO

s to some burner.1089.2. FUTUREDEVELOPMENTSinstabilities. Consideringa100 Hzinstabilitythevolumerequiredwouldbeapproximately3dm3thatcouldbeunsuitedforthecombustorsdesign.A specic resonator is able to damp only a single frequency fR. Thus it couldbe very dicult to capture the exact frequency since the ideal resonator workswithnotolerance. ImplementingthepresenceofsuchanHelmholtzresonatorinamodel(COMSOLorLOMTI)meanstoimposeaspecicimpedanceIasboundarycondition. ThisimpedancewouldbecomposedonlybyareactivetermiXwherei2= 1andXproceedsfromalgebraicrelationshipwithfRand further parameters. But the eective physics of the phenomenon must alsotake into account viscosity. A viscous term is always present and it extends theeld of frequencies where the resonator works. Whereas an ideal resonator hasthe capability to completely damp its characteristic frequency, a real resonatorhas aminor eect as viscosityis stronger but its range is expanded. If amajor viscous termis desired, agridcouldbeplacedat theresonator neckthatcanamplifytheeectiveviscousterm. Theequivalentimpedanceisnowcomposedbyanadditional dissipative termRdependingon andonthegeometrical valuesof theperforatedplate(suchasthenumberof holesandtheir diameter). It is not trivial to choose the optimal arrangement concerningtheeectivevaluesoftheresonatorsdimensions(onlyratiosbetweenAn, LnandV aregiven)andtheoptimalperforatedplate.109AppendixACombustionprocessesandrelatedemissionsCombustionprocessescanbeclassiedaccordingtodierentaspectsthatcharacterizethem. Themostinterestingclassicationconcernstheinitialcon-ditionsofmixingbetweenairandfuel,i.e. premixedordiusionames.Aameispremixedwhen,priortocombustion,fuelandoxidizeraremixedtogether. Combustionoccurs inahomogeneous environment, withaamefrontwhichseparatesthereactantsfromproductsandignitioninoneormorepoints. Aameisdenedadiusionamewhenthemixingbetweenfuelandoxidizeroccursinthesameareainwhichitdevelopscombustion.Theconditionsofmixingofreagentshavemanyrepercussionsbothonthecombustionprocesses andthe formationof pollutants. We are particularlyinterestedintheeectontheformationofNOX. LetusbrieyanalyzewhyNOXaregenerated. Theformationof NO(approximately90%of NOX)isstronglyaectedbytemperature. Infact, oxygenandnitrogenarepresent,withoutgivingrisetoreactionsofanykind,intheatmosphere,butformationofNOoccurswhenT> 1600K.111APPENDIX A. COMBUSTION PROCESSES AND RELATED EMISSIONSTemperature[C] Equilibriumconcentration[ppm] Timetoget500ppm[s]27 1.11010527 0.77 1316 550 13701538 1380 1621760 2600 1.101980 4150 0.117TableA.1-TimeofformationforNOfromN2andO2.NOXformation processes are quite complex and are developed through manychemicalequilibriumandnon-equilibriumreactionsthatoccurinprocessesofpre, post-combustion and combustion. Considering only balance reactions, themain mechanism of formation of NOis proposed by Zeldovich [38]. It is basedontwobasicreactionsofbalance:N2 + O NO + NN+ O2NO + O.The rates of these two steps are known accurately,and the second is fast com-pared with the rst. Hence the rate at which NO is produced depends stronglyon the concentration of O atoms which in turn is highly sensitive to the kineticsofthecombustionprocess. Theygiverisetoachainreactionwhichcanbesosynthesized:N2 + O22NO.Atlowtemperatures, equilibriumisstronglyshiftedtotheleftaccordingtotableA.1. ButthekineticsisstronglyaectedbytemperatureasshowninArrheniusequation:t[NO] = 61016 T0.5eq eEa

uTeq [O2eq]0.5 [N2eq]whereEaistheactivationenergyequalto19.8MJKginthiscase.InZeldovichsreaction,temperatureandresidencetimeofgasatthattem-perature play a fundamental role. Indeed, if the gas temperature drops quicklyafterfuel combustion(eg. fortheirexpansionintherststageof TG), theequilibrium achieved at high temperature is frozen and NOXdo not have time112toreturntomolecularnitrogenandoxygen. Thereforetheattain