Combustion Instability PDF
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Combustion Instability M. S. Natanzon First publishedin 1986 by Mashinostroyeniye,Moscow Translated electronicallyin 1996 Editedby F. E. C.Culick California Instituteof Technology Volume222 PROGRESSIN ASTRONAUTICSANDAERONAUTICS Frank K.Lu, Editor-in-Chief UniversityofTexasat Arlington Arlington,Texas Publishedbythe AmericanInstituteofAeronauticsandAstronautics,Inc. 1801AlexanderBellDrive,Reston,Virginia20191-4344 AmericanInstituteofAeronauticsandAstronautics,Inc.,Reston,Virginia 1 2 3 4 5 Copyright#1999byM.S.NatanzonandF.E.C.Culick.PublishedbytheAmericanInstituteof AeronauticsandAstronautics,Inc.,withpermission. Dataandinformationappearinginthisbookareforinformationalpurposesonly.AIAAisnotrespon- sibleforanyinjuryordamageresultingfromuseorreliance,nordoesAIAAwarrantthatuseor reliancewillbefreefromprivatelyownedrights. ISBN10:1-56347-928-1 ISBN13:978-1-56347-928-1 PrefacetotheRussianEdition(1986) AN INSTABILITY of the combustion processes in different kinds of devices intended for burning fuel becomes apparent in the spontaneous emergence of vibrational combustion behavior, accompanied by rapid fluctuations of heat release,pressure,andvibrationsofthestructure.Vibrationalcombustionleads,as arule,todisturbancesofthenormalfunctioningofcombustionchambers,and,in anumberofcases,totheirdestruction. Special urgency accompanied the the study of vibrational combustion in the last 20 to 30 years, in connection with the creation of combustion chambers for rocketsandjetengines. Problems of vibrational combustion have been treated in the book by B.V. Rauschenbach, Vibrational Combustion, and in two books containing the results of the theoretical and applied research carried out in the USA: Theory of Combustion Instability in Liquid Propellant Rocket Engines by L. Crocco and S.-I. Cheng, and Liquid Propellant Rocket Combustion Instability edited by D.T.HarrjeandF.G.Reardon.Inspiteoftheextensiveinformationgiveninthose works,thereisaneedforfurtherilluminationoftheproblemsofvibrationalcom- bustion.Sincethetimeofthosepublications,aconsiderablenumberofarticleshas beenpublished,containingmanynewresultsofstudiesofvibrationalcombustion. Chapters 1–5 of this book are dedicated to studies of the most general laws governingvibrationalcombustion.InChapters6and7,atheorybasedonspecific models of unsteady combustion is presented. Special attention in those chapters is given to the development and application of special methods of organizing numerical experiments, allowing detailed investigation of the limits of stability andthemechanismsoffeedback.InChapter8oneofthepossiblereasonslead- ingtoapparentnonreproducibilityoftheresultsofsomeexperimentalstudiesof combustionstabilityisdescribed. Specialpositioninthisbookisgiventothedevelopmentofthephysicalnatureof processesinthesystemsinquestion,sothatthematerialshouldbeeasilyaccessible toreaderswhodonotpossessexperienceinthesubjectsofacousticsandvibration theory. The author expresses deep gratitude to corresponding member of the USSR AcademyofSciences,A.P.Vanichev,fordiscussionofthequestionsdiscussedin thisbook;toV.A.Frostforvaluableobservationsduringthereview;andalsoto B.F.Glikmana,B.N.Dubinkina,O.M.Kossova,Z.S.Lapinoy,E.V.Lebedinskiy, V.A. Mokiyenko, I.V. Merkulov, andA.E. Cham’yan, with whom collaboration overmanyyearsmadepossibleanumberofresultsgiveninthisbook. xi TableofContents PrefacetotheRussianEdition(1986)........................................ xi Editor’sPreface ................................................................ xiii PrefacetotheSecondEdition................................................. xv Bibliography.................................................................... xxv AShortBiographyContributedbyMrs.Natanzonand ProfessorV.Bazarov........................................................xxvii Chapter1. LowFrequencyOscillationsinLiquidRocket CombustionChambers..................................................... 1 Low-FrequencyOscillationsinaLiquidRocketCombustionChamber...... 2 SupplementaryAnalysisoftheMechanismofLossofStability.............. 14 InstabilityExcitedbyEntropyWaves .......................................... 23 EffectsoftheFeedSystemonStability ........................................ 33 Chapter2. PhenomenologicalModelsoftheCombustionProcess..... 43 VariableTimeLag............................................................... 44 SmoothBurnoutCurves......................................................... 49 Chapter3. The Acoustic Response of the CombustionChamber... 61 TheWaveEquationandItsSolution............................................ 62 TheAcousticResponseofaCombustionChamberwithaShort ............. SubsonicPartoftheNozzle.................................................. 66 EffectofaFiniteLengthNozzleontheAPFCoftheAcousticComponent.. 78 ExperimentalDeterminationoftheAPFCoftheAcousticComponent ...... 87 Chapter4. High-Frequency(Acoustic)OscillationsinaCombustion Chamber ..................................................................... 95 StabilityLimits .................................................................. 96 DiscussionoftheResults ....................................................... 107 Anti-PulsatingDevices.......................................................... 113 ix x Chapter5. NonlinearEffects ............................................... 125 SomeInformationfromtheTheoryofNonlinearVibrations.................. 125 NonlinearVibrationsinaCombustionChamber............................... 133 Chapter6. ApplicationoftheFrequency-ResponseMethodfor StudyingtheDynamicalPropertiesoftheCombustionZone.......... 147 BlockDiagramandtheMatrixoftheFrequencyCharacteristicsofthe...... CombustionZone;TheCharacteristicEquationofLocked ................ Construction[43]............................................................. 148 DynamicalModelfortheCombustionofFuelDropsinaFlowof........... GaseousOxidizer[44]........................................................ 154 StandardFormoftheEquationsofExcitedMotion[44]...................... 169 CalculationoftheMatrixfortheFrequencyCharacteristicsofthe........... CombustionZoneandtheFeedbackVector[44]............................ 184 Chapter7. StabilityofCombustionofFuelDropsinaFlowof GaseousOxidizer[41,45].................................................. 191 FormulasforCalculations....................................................... 191 SteadyState...................................................................... 198 StabilityLimits .................................................................. 201 AnalysisoftheMechanismsofFeedback...................................... 208 Chapter8. BifurcationsofSteadyCombustionRegimesandTheir EffectontheOnsetofHigh-FrequencyOscillations .................... 217 PhysicalPictureofthePhenomenon ........................................... 218 One-DimensionalModelofCombustionfortheGas–LiquidScheme ....... 221 TwoRegimesofCombustion(One-DimensionalModel)..................... 226 ExperimentalData............................................................... 233 ATwo-DimensionalModelDescribingBifurcationsoftheCombustion..... ZonefortheGas–GasSystem[70] .......................................... 236 References....................................................................... 245 Index............................................................................. 251 SupportingMaterials.......................................................... 000 Chapter2 Phenomenological Models of the Combustion Process THEcombustionprocessisacomplexphenomenonwhosedescriptionisbased on the physical and chemical kinetics of different kinds of “elementary” processes which ensure the conversion of fuel into combustion products. These elementaryprocessesincludeinthefirstplace,mixing,heattransfer,andchemical reactions;andinaddition,duringthecombustionofliquidpropellants,theforma- tionofdropsofliquid,theirmotion,heatingup,andevaporation.Inthecombustion process,hydrodynamicaleffectsareimportantfororganizingthecombustionpro- cessesandthestateofaggregationofcombustionproducts.Thedescriptionoffuel combustionindifferentkindsoftechnicaldevicescarries,asarule,aqualitative nature or has a sufficiently complex structure that it requires a large amount of cumbersomecalculations. Theconsiderabledifficultieswhichstandinthewayofuseofdetaileddescrip- tionsof“elementary”processeshavestimulatedtheapplicationofdifferenttypes of phenomenological models. The basis of those models is an assumed burnout curveφ(τ),thedependenceoftheproportionoffuelburnedontheperiodofcom- bustion. Methods of determining the form of the burnout curve are outside the frameworkofthemodel;itsformanddependenceonvariousfactorsarearbitrary toaconsiderableextent.Allthisnarrowsthepossibilitiesofthetheory.However this strategy significantly simplifies the analysis and, as it suggests that experi- ments be based on a similar approach, makes it possible to obtain a number of resultswhichagreewellwithexperimentaldata.Itshouldbenotedthattheselec- tionofonemodeloranotherisdeterminednotonlybythemethodoforganizing thecombustionprocess,butalsobytherangeoffrequenciesforwhichthemodel isintended. Forexample,supposethatthecombustionprocessforaliquidpropellantissuch thattheformationofthereactionproductsistheresultoftheconsecutiveactionof twowell-definedstages:fuelevaporationandturbulentcombustionofitsvapors. The first stage in this case has a long characteristic time, and the second has a shortcharacteristictime.Thenintheregionoflowfrequenciesitisnecessaryto usethephenomenologicalmodelwhichdescribesthefirststageoftheprocess;the secondstagewilltrackthelowfrequencyprocessinaquasi-stationaryfashion.In theregionofhighfrequenciesoneshoulduseaphenomenologicalmodelwhich corresponds to the second stage; the effect of the first stage will be unimportant duetoitslargeinertia. 43 44 M. S. NATANZON Themodelofconstanttimedelayusedinthepreviouschapteristhesimplest exampleofaphenomenologicalmodelwhichdescribesthedynamicsofthecom- bustion processes in the region of low frequencies. In this chapter we describe somephenomenologicalmodelsofthecombustionprocessmostfrequentlyused inthetheorystability. I. VariableTimeLag In the model of constant time delay it was assumed that τ is independent of fluctuations of pressure, initial size of the drops, and all other factors having effectsonthecombustionprocess.Thefollowingdiscussion,whichmoreprecisely formulatesthemodel,accountsforoscillationsofthetimelag. (cid:2) Withasteppedburnoutcurve,thefuelenteringatthemoment t burnsatthe timet(cid:2)+τ,whereτ inthegeneralcasedependsont: GΦ(t(cid:2))dt(cid:2) =GΓ(t(cid:2)+τ)d(t(cid:2)+τ) (2.1.1) Letuschangethezerotimereference,bysettingt =t(cid:2)+τ.Thenfrom(2.1.1)it followsdirectlythat (cid:9) (cid:10) dτ GΓ(t)=GΦ(t−τ) 1− (2.1.2) dt Linearizingthisexpression,wefind dδτ(cid:2) δGΓ(t)=δGΦ(t−τ¯)− (2.1.3) dt Therightsideof(2.1.3)containstwomembers.Thefirstdescribesfluctuationsof therateofgasformationarisingduetooscillationsintherateofconsumptionof fuel entering the combustion chamber. It coincides with the expression describ- ing fluctuations of the gas formation rate in the model with constant delay (see Eq.1.2.8).Thesecondtermisnew.Itdoesnotdependonoscillationsofreactant combustion and can be different from zero even when oscillations of consump- tion rate are absent. The loss of stability caused by this term is conventionally designatedasintrachamberinstability. Theexpressionforthetermleadingtointrachamberinstabilitycanbeobtained byanothermethod.Supposethattherearenooscillationsinthefuelconsumption, so GΦ =G¯Φ. Then the mass of liquid phase in the combustion chamber can be representedintheformQΦ =G¯Φτ.Therateofformationofcombustionproducts asaresultofachangeinthetimeτ willbeequaltotherateofchangeinthemass oftheliquidphase,withtheoppositesign:δG(cid:2)Γ =−δQ˙Φ(cid:2) =G¯Φδτ˙(cid:2).Converting thelastexpressiontodimensionlessforms,weobtainthesecondmemberofthe rightsideofexpression(2.1.3).Inordertocompleteconstructionofthemodelof combustionwiththevariabletimelag,itisnecessarytoassignthedependenceτ onthecurrenttimet. PHENOMENOLOGICAL MODELS OF THE COMBUSTION PROCESS 45 A. Dependenceofτ onPressure[1] Weassumethattherateofcombustionisdeterminedessentiallybythevalue of pressure. This dependence need not necessarily be direct: it can be realized, also, through the values of other parameters unambiguously connected with the pressure. If, as before, we approximate the smooth burnout curve by a step function, then the entire period from the moment of arrival of fuel into the combustion chamber t(cid:2) =t−τ to the combustion at the instant t should be considered as thetimeduringwhichtheprocessofpreparationtakesplace.Afterpreparationis complete,thefuelisconvertedinstantlyintocombustionproducts.Lettheextent ∗ ofpreparationbecharacterizedbyacertainparameterE,andletE bethecritical valuecorrespondingtoterminationofthepreparationtime.Assumingthattherate oftheprocessofpreparationinthefinalanalysisdependsaltogetheronlyonone parameter,thecombustionchamberpressurep,thenwefindtherelation (cid:8)t f(p)dt(cid:2) =E∗ (2.1.4) t−τ wheref(p)istherateofthepreparationprocess. Ifthepressurepisafunctionoftime,thenEq.(2.1.4)describesimplicitlythe dependenceofτ ont.DifferentiatingEq.(2.1.4)withrespecttot,weobtain (cid:8)t dτ df dp f(p) + dt(cid:2) =0 (2.1.5) dt dpdt(cid:2) t−τ Nowsetp=p¯(1+δp)andτ =τ¯ +δτ(cid:2)in(2.1.5);afterlinearizationwefind dδτ(cid:2) df¯ (cid:8)t dδp f(p¯) + p¯ dt(cid:2) =0 (2.1.6) dt dp¯ dt(cid:2) t−τ fromwhichitfollows dδτ(cid:2) =−n[δp(t)−δpt−τ¯] (2.1.7) dt The parameter n= p¯ df¯ in the last expression plays the role of an amplification f¯ dp¯ factor.Aftersubstituting(2.1.7)into(2.1.3),weobtain δGΓ(t)=δGΦ(t−τ¯)+n[δp(t)−δpt−τ¯] (2.1.8) After combining the rate of gas formation and fuel consumption δGΦ deter- mined by (2.1.8) and (1.2.6) with the equation of material balance (1.2.11), we 46 M. S. NATANZON obtain the equation describing small fluctuations of the combustion chamber pressure, (cid:5) (cid:6) τπδp˙ + h−1+n δp(t−τ¯)+(1−n)δp=0 (2.1.9) Itisnotdifficulttoseethatifthisequationisdividedby1−n,thenitwilltakethe sameformasEq.(1.1.10),inwhichtheconstantsτπ andh−1 areassignedsome effectivevalues (cid:5) (cid:6) τπ,(cid:13)Φ = (1τ−πn); h(cid:13)−Φ1 = h(−11−+n)n (2.1.10) Consequently,theD-separationisdeterminedbyEqs.(1.1.22),inwhichτπ and h should be replaced by τπ.(cid:13)Φ and h(cid:13)Φ. After solving the equations obtained in thiswayforhandτ,weobtain ω¯ 1 h=−cos ; τ¯ =− ω¯ cotω¯ (2.1.11) 1+n(cos ω¯ −1) 1−n From Eqs. (2.1.11) it is evident that even when h→∞ and, therefore, there arenooscillationsoffuelconsumption,thesystemcanlosestability.Actually,for h→∞,itisnecessaryandsufficientthatthedenominatorintheexpressionfor h be zero. From this condition and the expression for τ¯, it is easy to obtain the following equations for the stability limit in the coordinates n−τ¯ with h→∞ (intrachamberinstability): 1 ω¯(1−cosω¯) n= τ¯ = (2.1.12) (1−cosω¯) sinω¯ ItisevidentfromEqs.(2.1.12)thatfortheintrachamberinstabilitythereisacertain minimumvaluen,belowwhichthesystemisalwaysstable.Thisvalueisequalto 0.5,andattainedatω¯ =(1+2m)π,wherem=0,1,.... The region of applicability of one phenomenological model or another, as already noted, is determined not only by the method of organizing the process of combustion and by the kind of fuel, but also by the range of frequencies of thevibrations,forwhichthemodelsaretobeused.Sinceexperimenthasshown that the mechanism of liquid break-up plays the determining role for low-level frequency oscillations, one should consider that values of n in the range of low frequenciesare,asarule,lessthan0.5.Forhighfrequencies,onthecontrary,the valueofncannoticeablyexceed0.5[1,2]. FromEqs.(2.1.12)itfollowsthatwithn<0.5thestabilitylimitmustqualita- tivelyhavethesameformasinthemodelofconstantdelay.Increaseofninthis case, as analysis shows, leads to expansion of the unstable region. The limiting valueofh,higherthanwhichthesystemisstableforanyvaluesτ,isdetermined inthiscasebytheratio h =(1−2n)−1 (2.1.13) max anditapproachesinfinityforn→0.5. PHENOMENOLOGICAL MODELS OF THE COMBUSTION PROCESS 47 Ifn>0.5,thenintheplaneofparametersh−τ¯ appearregionsinwhichthe systemisunstablewithanyvaluesofh.Decreaseofthetimedelayτ¯ isthesole methodofstabilizationinthiscase. B. Dependenceofτ ontheInitialDiameterofDrops1 Theprocessesofheatingup,evaporation,andmovingthedropsaretheimpor- tantstagesinthecombustionofliquidpropellant.Withsomemethodsoforganizing the combustion processes, those processes dominate. Their role is especially importantwhenbothcomponentsentercombustionthechamberasliquid. Fromtheoreticalconsiderationsandexperimentitfollowsthattheinitialdiame- terofdropsisrelatedtoanumberoffactorswhichdominatetherateofcombustion [2, 20]. This circumstance makes it possible to formulate a model in which the timelagisafunctionoftheinitialdiameterofthedrops: τ(t)=f[a(t−τ)] (2.1.14) where a(t−τ) is the diameter of a drop at the moment t−τ of its formation. Withanincreaseintheinitialdiameterofdrops,therateofcombustiondecreases. Consequently,(df/da)>0. The initial diameter of the drops depends on the type of injection element and its mode of operation. For centrifugal and jet monopropellant injectors, the size of drops (with fixed characteristics of liquid) depends on the injection dif- ferential pressure, and on a comparatively low pressure level in the combustion chamber[21]: a=a(∆p,p) (2.1.15) where∆pistheinjectiondifferentialpressureandpisthecombustionchamber pressure. Experimental investigations show that the size of the drops decreases withanincreaseinthedifferentialandlevelofpressure. Weconsidernowacombustionchamberwithafeedsystembasedonpressurized cylindersandhavingshortsupplylines.Theinjectionpressuredifferenceinthis caseisequalto∆p=p −p.Afterusingthisexpressionfor∆pandlinearizing B Eqs.(2.1.14)and(2.1.15),weobtain (cid:9) (cid:10) 1df ∂a ∂a δτ(cid:2) =mτ¯δp(t−τ¯); m= − p¯ (2.1.16) τ¯ da ∂p ∂∆p Since ∂a and ∂a arenegative,minprinciplecanbeeitherpositiveornegative. ∂p ∂∆p However,thedependenceofthesizeofthedropsonapressuredifferenceprevails overthedependenceonthepressure(athighpressuresthedependenceofthesize ofdropsonthepressuregenerallyisabsent);thereforem>0. 1InSec.IVofChapter4aspecific(notphenomenological)dependenceisdescribedforasystemof injectionusinggaseousoxidizerandliquidfuel.
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