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Springer-VerlagBerlinHeidelberg2012 2SteadyOne-DimensionalDetonations35 D media at restmedia at rest(downstreamsonic planeshock frontexothermicreaction 30 – 50 p115 – 25 p1p1 translational/rotational equilibrium (10 20 – 60GPa10 – 40GPa vibrational redistribution/bond breaking (10 0GPa sonic plane sonic plane unsteady expansion shock front shock front pressurepressure Fig.2.1.SchematicoftheZNDstructureofadetonationwave,forbothgaseousdetonationatatmosphericconditionsandcondensedphasedetonation,etc.)occursaswellasslowercondensationanddiusion-controlledpro-cessessuchascarbonparticulateformation.AcurrentpictureoftheNon-EquilibriumZND(NEZND)structureofcondensedphasedetonationwaveshasbeenpresentedbyTarver[]andinChap.ofthisvolume.Thispictureisfurthercomplicatedbythefactthatallknowngaseousdetonationsarehydrodynamicallyunstable(Chap.ofthisvolume)andthisinstabilityismanifestedasatransient,three-dimensionalcellularstructure(seeChaps.inthisvolume).Condensed-phaseexplosivesaresimilarlycomplex.Mostliquidexplosivesdisplayastructureoftransversewavesthatarebelievedtobesimilartothoseingaseousdetonation,onlyonamuch“nerscale.Solidexplosivesasusedinmostapplicationsarepolycrystalline, 40A.Higginsbetweenand),thelocationofthemaximumreactionrate,orthein”ectionpointofthetemperaturepro“le.Asactivationenergyisincreasedfrom =10to =25,thereactionzoneincreasesbyanorderofmagnitudeinlength.Increasingfrom =25to =50resultsinanadditionaltwoordersofmagnitudeincreaseinreactionzonelength.Thelengthscaleofthesereactionzonesissomewhatarbitrary,duetotakingthevalueofasunity.Whatismoresigni“cantistheincreas-sharpnessofthereactionpro“leasactivationenergyisincreased.Aloweractivationenergyresultsinarelativelygradualincreaseintemperaturebegin-ningimmediatelyaftertheshock,whilethehighactivationenergyresultsinalongplateauofnearlyconstantconditionsfollowedbyacomparativelyrapidenergyrelease.Thisfeatureofhighactivationenergyre”ectstheunderly-ingtemperaturesensitivityofchemicalreactionsthatmustbeactivatedandisoneofthemostsigni“cantandreoccurringthemesinthedevelopmentofdetonationtheory.Thetemperaturepro“lesinFig.2.2featuremaximathatoccurbeforetheendofthereactionzone;thisismostclearlyvisibleinthetemperaturepro“leforthe =50caseduetothesteepnessofthepro“lebutitispresentinallcasesplotted.Beyondthismaximum,exothermicheatreleaseinthe”owhastheeectofloweringthetemperature.Thiscounterintuitiveresultisaconsequenceoftheheatreleaseresultingingreater”owaccelerationandpres-sure/densityreductionthanitscontributiontoraisingthestatictemperatureofthe”ow.ThistemperaturedecreasewithheatadditionoccurswhentheMachnumberisintherange  1;thisresultiswellestablishedincompressibleone-dimensional”owwithheataddition(Rayleigh”ow).Itisinterestingtoexplorewhathappensifacalculationofthereactionzonestructureisinitializedwithanon-CJdetonationvelocity.ThisisdoneinFig.2.3for =25,wherethesolutionwasinitializedwiththepost-shockstateforshockMachnumbers20%greaterand20%lessthantheCJMachnumber.Forthestrongershockcase,thereactionzonestructurecanbesolvedfor,andasthereactionprogressvariablereaches=1,the”owisseentoremainsubsonic.Thissolutioncorrespondstoastrongdetonation,i.e.,thebranchoftheproductHugoniotcurveabovetheCJdetonationpointonthe(p,v)plane.Thiscaseisalsoreferredtoasanoverdrivendetonation,sincethedetonationisbeingforcedtopropagateataspeedgreaterthantheCJspeed,andtheheatreleaseisinsucienttochokethe”ow.Ifthesolu-tionisinitializedwithasub-CJdetonationvelocity,thenumericalintegrationencountersasingularityasthe”owbecomessonicwhilethereactionrateisstill“nite(1)in(2.16).Inthiscase,theheatreleaseissucienttochokethe”owbeforeaddingthefullheatrelease,andfurtherheatadditiontoasonic”owisnotpermittedinasteadysolution.Thissingularitywillappearforanyinitialconditionthatisevenin“nitesimallylessthantheCJspeed.Asimilarresultisobtainedifthewavespeedis“xedandtheheatreleaseincreasedincrementally.Theappearanceofamathematicalsingularitymay 2SteadyOne-DimensionalDetonations41 X Mach number M0.20.30.40.50.70.80.91.01.1 singularityX singularity 80% 120% X -0.50.00.51.0 singularity X Distance xp/p1102103104105Distance x102103104105Distance x102103104105Distance x1021031041051020304050607080 singularity Fig.2.3.StructureofZNDdetonationforreactionzonestructureinitializedwithaCJdetonation,adetonationoverdrivenat120%,andashockat80%showingpressure,temperature,reactionprogressvariable,and”owMachnumber =25,=10,2.Thesub-CJshockdoesnotcorrespondtoasolutionofconservationlaws,resultinginasingularityappearinginthenumericalintegrationappearalarming,butrecallthattheCJdetonationistheminimumvelocitywaveconsistentwiththegoverningconservationlaws.Thefactthatasin-gularityisencounteredinnumericallyintegratingthroughthereactionzoneinitializedwithaspeedlessthantheminimumspeedsimplyre”ectsthatanattemptisbeingmadetosolvea”owusinginitialconditionsforwhichnosteadysolutionexists.2.3PathologicalDetonationsInsteadoftheonestepreactionAB,consideratwo-stepreactionwithAreactingtoformB,whichinturnreactstoformCReaction1:AB(2.19)Reaction2:BC(2.20) 2SteadyOne-DimensionalDetonations43numberbasedontheequilibriumheatreleaseisequil781.Initial-izingtheintegrationof(2.23)withthevonNeumannstatebasedonthisshockMachnumberresultsinasingularitybeingencountered,asshowninFig.2.4.ItisnecessarytoincreasetheshockMachnumbertoeigen(orapproximately6%greaterthantheCJspeedbasedonequilibrium)inorderto“ndasolutionthatdoesnotencounterasingularitywithinthereac-tionzonestructure.Thisvaluecanonlybefoundbytrialanderrorintegrationofthegoverningordinarydierentialequations.ThetermChapman…Jouguetisstillusedtorefertothissolution,sinceitfeaturesasonicsurfaceconsistentwithJouguetsoriginalcriterion.Todierentiatethesetwocases,theywillbereferredtoastheCJequilibriumsolutionandtheCJeigenvaluesolution.Finally,toeliminateanyconfusionastowhythepresenceofanendother-micreactionresultsinagreaterthanequilibriumdetonationvelocity,notethatthedetonationvelocityassociatedwiththe“rststageexothermicreac-tionalonewouldbe)=9486.Theendothermicreactionreducestheeectiveheatreleaseanddetonationvelocityincomparisontothisvalue,butthepathologicalbehaviorresultsinadetonationvelocitythatisgreaterthantheCJequilibriumsolutionbasedonthetotalheatrelease.Numericalintegrationofordinarydierentialequationswithapotentialsingularityembeddedwithinthesolutionisanotoriouslydicultproblem.Itmaybenecessarytointerveneinthenumericalalgorithmusedinordertoselectasolution,sinceuponencounteringthesonicpoint,thedierentialequationsbecomeindeterminate.Alternatively,itispossibletostartatthesonicpointandintegrateforward,towardtheshock.Eveninthiscase,iter-ationisstillrequiredto“ndasolutionconsistentwiththeinitialconditionsupstreamofthewave,sincethethermodynamicstateofthesonicpointisnotknownapriori.Forthesolutionofthegoverningordinarydierentialequationsinitializedeigen207,thesolutionforthereactionzonestructureisshowninFig.2.4.Coincidentwiththesonicpoint,thelocalheatreleaserateisseentogotozeroduetoanexactbalancebetweentherateofexothermicheatadditionandendothermicheatremoval.ThisconditionisreferredtoasthegeneralizedChapman…Jouguetcondition,aterm“rstusedbyEyringetal..27].Theconditionofthelocalheatreleaseratebeingzerobehaveslikeaquasi-equilibriumconditionin(2.23),enablingthesolutiontopasssmoothlythroughthesonicpointandproceeddownstreamasasupersonic”ow(recallthatheatremovalacceleratesasupersonic”owawayfromsonic).Thiscaseisshownastheheavy,solidlineinFig.2.4ThesonicpointofthegeneralizedCJconditionisasaddlepoint,soinprincipleitisalsopossibleforthesolutionatthesonicpointtoreturntothesubsonicbranchunderthein”uenceofsubsequentendothermicreaction,asshownbythethin,dashedlinesinFig.2.4.Thissolution,however,featuresanonphysicalkinkŽinthe”owproperties.Thus,itcouldbesuggestedthatthesolutionthatpassessmoothlyfromthesubsonictothesupersonicbranchisthecorrectone,butthiscannotbeprovenrigorously.Asmentionedpreviously, 44A.Higgins X –0.50.00.51.01.5 singularityCJ EigenvalueCJ Equilibriumsupersonicbranchsubsonicbranch X Distance xMach number M102103104105Distance x102103104105Distance x102103104105Distance x1021031041050.00.51.01.5 singularityCJ EigenvalueCJ Equilibriumsubsonicbranchsupersonicbranch X p/p1102030405060 singularityCJ EigenvalueCJ Equilibriumsupersonicbranchsubsonicbranch X T/T105101520 singularityCJ EigenvalueCJ Equilibriumsubsonicbranchsupersonicbranchq./cpT1sonic pt. in eigenvalue soln.sonic pt. in eigenvalue soln.sonic pt. in eigenvalue soln.sonic pt. in eigenvalue soln. Fig.2.4.ReactionzonestructureofZNDdetonationwithatwo-step,exother-mic/endothermicsystem,showingpressure,temperature,heatreleaserate,and”owMachnumberfor RT1=Ea2 =25,=20,2.ThereactionzoneinitializedwithashockbasedontheCJequilibriumsolutionencountersasingular-ityinthenumericalintegration.Theeigenvaluesolutioncanpasssmoothlythroughthesonicpoint,resultinginsupersonic”owattheendofthereactionzone(weakdetonation)similarindeterminacyoccursincompressibleisentropic”owwhenaninitiallysubsonic”owbecomessonicatanareaminimum(i.e.,atathroat).Iftherequirementthatthe”owpropertiesatthethroatvarysmoothlyisimposed,the”owshouldtransitiontoasupersonicsolutionlocallyatthethroat,andfromthere,the”owcaneithercontinuesupersonicallyorreturntothesub-sonicbranchviaashockwave,dependingonthedownstreamboundarycon-dition(e.g.,backpressure).Forboththedetonationwithcompetingreactionsandisentropic”owwithareachange,theissueofresolvingwhichbranchofthesolutionfollowingthesonicpointisthecorrectsolutionisultimatelydeterminedbyconsideringthedownstreamboundarycondition.Establishinghowthe”owexitingthesonic 2SteadyOne-DimensionalDetonations45 CJ EquilibriumQ = 0(inert shock) weakdetonation 00.20.40.60.81 InitialconditionQ = 10(equilibrium)Q = 5(partially reacted)Q = 11.35(partially reacted)Q = 20(exothermic only)v2/v1Generalized CJ(q. = 0)von Neumannstates strong detonation Fig.2.5.Equilibriumandeigenvaluesolutionsforatwo-step,exother-mic/endothermicsystemvisualizedinthe(p,v)plane( RT1=Ea2 =25,=20,surfaceofsuchadetonationmatcheswithadownstreamconditionimposedbyapistonbecomesquiteinvolvedasanumberofpossiblescenariosneedtobeconsidered.ThereaderisreferredtothebookbyFickettandDavis[foracompleteexpositiononthisproblem.FickettandDavisalsoconsiderthecaseoftwoexothermicreactions,whichturnsouttobequalitativelysimilartoasingleexothermicreactionanddoesnotexhibitpathologicalbehavior.Forthemostfrequentlyencounteredcaseofadownstreampistonatrest(correspondingtoadetonationinaclose-endedtube,forexample),the”owpassessmoothlytothesupersonicsolutionbeyondthesonicpoint.Inthiscase,theleadingedgeoftherarefactionwavelagsbehindthesonicsurfaceandtheendofthereactionzone,resultinginaregionofuniformsupersonic”owbetweentheendofthereactionzoneandtheleadingrarefactionthatwillincreaseinsizeasthedetonationpropagates.ThesescenariosarediscussedfurtherinAppendixTheeigenvaluesolutioncanalsobeinterpretedinthe(p,v)planewiththeuseofpartiallyreactedHugoniots,as“rstproposedbyvonNeumann[Fig.2.5,theHugoniotandRayleighlinefortheequilibriumsolutionareshownasdashedlines.TheHugoniotsforthevalueofheatreleaseatanintermediate Formally,aHugoniotisalocusofpossibleequilibriumendstatesforasteadywave,soapartiallyreactedHugoniotŽisamisnomer. 46A.Higginsvalue(=5)andatthesonicpointoftheeigenvaluesolution(=1135)areplottedasdashed-dottedandsolidlines,respectively.TheRayleighlinefortheeigenvaluesolutionisseentobetangenttothesolidHugoniotcurveatthepointwherethenetrateofheatreleaseiszero(thegeneralizedCJcondition).Inactuality,thereisacontinuousseriesofpartiallyreactedHugoniots,andthecorrectonetouseforthedetonationsolutioncannotbedetermineduntilthekineticrateequationhasbeenintegrated,aswasdoneabove.StartingfromthevonNeumannpoint,asthereactionprogresses,thesolutionproceedsdowntheRayleighlineasindicatedbythedirectionalarrows.Atthesonicpointoftheeigenvaluesolution,thesolutionisindeterminate.Asendothermicreactionsdominateovertheexothermicbeyondthesonicpoint,thesolutionmaycontinuedowntheRayleighline(supersonicbranch),ormayproceedbackuptheRayleighline(subsonicbranch).EitherbranchofthesolutioneventuallyreachestheequilibriumHugoniot.Theupperintersectionpointcanberecognizedasastrongdetonationandthelowerintersectionasaweakdetonation.Asdiscussedabove,theweaksolutionismorelikelytobereal-ized,andthuspathologicaldetonationsareanexampleofaweakdetonationsolution.Ifthereactionzonecalculationisinitializedwiththepost-shockstatecorrespondingtotheCJequilibriumsolution(dashedstraightline),thenasthetangencycondition(sonic”ow)isencounteredforthe“rsttime,thenetheatreleaserateisstillpositive.Asnetexothermicreactioncontinues,thereisnolongeranintersectionbetweentheRayleighlineandthepartiallyreactedHugoniot,resultinginthesingularityencounteredinFig.2.4.Thus,itisnotpossibletoconstructatrajectoryofpartiallyreactedstatestoreachtheCJequilibriumsolution.DetonationsforwhichtheequilibriumCJdet-onationsolutionmaynotbethecorrectsolution(orpermissiblesolution)duetocompetingeectsinthereactionzonemayoccurinsystemswithrealchemistryaswell(Sect.2.52.4DetonationswithSourceTermsInordertodiscusslimitstodetonationpropagation,lossesmustbeintroducedintothegoverningconservationequations.InasystemwithoutlossesandgovernedbyanArrhenius-typereactionrate,detonationpropagationisalwayspossibleduetothefollowingmechanism.AnArrhenius-governedsystemwillinevitablyreacttoequilibrium,andthetemperatureincreasefromevenaweakshockwavewillacceleratethereaction,resultinginawaveofexothermicenergyreleasetravelingwiththeshocksomedistancebehindit.Inthecaseofaveryslowreaction,thisfrontofexothermicitymaybespatiallyseparatedfromtheshockbyvastdistances,butintheabsenceoflossessuchasheattransferorfriction,theenergyreleasedwilleventuallyfeedintosupportingtheshock,ultimatelyresultinginadetonation.(RecallthataCJdetonationistheminimumvelocityallowedforasteady,compressivecombustionwave; 2SteadyOne-DimensionalDetonations47lowervelocitycombustionwavesarenotpermittedbytheconservationlaws.)ThisconceptissometimesreferredtointheRussiandetonationliteratureasKharitonsprinciple,namelythatanymediacapableofexothermicreactioniscapableofsupportingdetonationwavepropagationintheabsenceoflossesosses72].Thus,anydiscussionofthelimitstodetonationpropagationmustincludetheeectsoflosses.Thesteady,one-dimensionalmass,momentum,andenergyequationsincludingfrictionandheattransfersourcetermsinthewave-“xedreferenceframeare)=0(2.26)(2.27) (2.28)whereisasourcetermofmomentumandisasourcetermofenergy.Thevelocityisthevelocityofthewallinthewave-“xedframe,whichisequalinmagnitudetothevelocityofthewaveinthelaboratory-“xedframe,.Aswrittenhere,andarethevolumetricsourceterms,withunitsof[Nm]and[Wm],respectively.Amasssourcetermcouldalsobeincludedtoaccountfor,forexample,masslossintoaporouswall;however,thiseectisusuallytreatedbyintroducingareadivergenceintothegoverningequations,asdiscussedinSect.2.6.Thesemomentumandenergysourceswillberelatedtowallfrictionandheattransfercoecientsbelow.Notetheappearanceofthefrictiontermintheenergyequation,representingtheworkdonebyfriction.Thesigni“canceofthistermwillbeelaborateduponinSect.2.4.2below.FollowingthedevelopmentofSect.2.2,adierentialequationforthe”owvelocityinthereactionzonecanbefound dx=q cpT+q cpT+f(u1(Š1)Šu) c2 (2.29)Notethatif=0and=0,(2.29)revertsbackto(2.16).Heattransfertothewallisaheatloss(0),soinspecting(2.29)revealsthattheeectofheatlossissimilartothatofendothermicreactionsstudiedintheprevioussection.HeatlosseswillresultinthedetonationpropagatingatspeedslessthantheidealCJdetonationvelocity(i.e.,adetonationwithoutlosses),anddeterminingthesolutionwillnecessitateiteratingonthepropagationvelocityuntilaregularsolutionofthereactionzonestructurecanbefound.Theeectoffrictionisnotasintuitive,duetothetermsofmixedsigninvolving2.29);however,itwillbeshownthatfrictionalsoresultsinavelocityde“citincomparisontotheidealCJvelocity. 2SteadyOne-DimensionalDetonations57verylarge(i.e.,beyondvaluesthatcorrespondtorealreactivesystems)inordertoseethisconvergence.InSect.2.6,thesamerelationrelatingvelocityde“cittoactivationenergywillbefoundtoapplytodetonationswithfrontcurvatureduetolateral”owdivergencefromyieldingcon“nement.Detonationswithlargefrictionalandheattransferlosses,aswouldbeencounteredinporousmediaforexample,havebeenextensivelystudiedthe-oreticallybySivashinskyandcolleaguesinrecentyears[].Theirworkhasfurtherelucidatedthestructureofthereactionfrontandclassi-“edthepossiblemodesofpropagation,particularlylowvelocity(includingsubsonic)regimes.Theirmodeling,similartothatpresentedhere,usedasingle-stepArrheniuskineticsreactionmechanism.Asdiscussedinthenextsubsection,thisreactionmodelislikelynotrelevanttotheactualmechanismofburninginreactivewaveswithlargelosses.Modelsthatattempttoincor-poratethecontributionofturbulentcombustiontothereactionmechanismbyusinggreatlyexaggeratedvaluesoftransportproperties(e.g.,eectivetur-bulentdiusivity,etc.)mayprovideafuturedirectiontolinkthesemodelswithexperimentalresults[2.4.5ExperimentswithLossesGaseousdetonationsinchannelsthatexhibitlargevelocityde“citsduetomomentumandheattransferlossestowallroughnessorobstacleshavebeenextensivelystudiedandarereferredtoasquasi-detonations.Quasi-steadypropagationatvelocitiesaslowas50%oftheidealCJvelocityhasbeenobserved.Reviewsofresultsofthesestudiescanbefoundin[]andChap.of[].Itisunlikelythatplanarshock-initiatedhomogeneousreactionscon-tributesigni“cantlytotheenergyreleaseinsuchdetonations.Calculationsperformedwithdetailedchemistryshowthatthereactionratesforthelowshockvelocitiesinvolvedaremuchtooslowtoresultinexothermicreactionsonthetimescalesoftheobservedquasi-detonations.AseminalstudybyShchelkinhelkin78]suggestedthatshockre”ectionoobstaclesinthetubemaygeneratelocalhotspotsthatinitiatereactionandthusenablethedetonationtocontinuepropagatingatspeedsforwhichaplanarshockwouldnotbeofsucientstrengthtosustainpropagation.Muchoftheframeworkforexplainingdeto-nationphenomenaingaseousandcondensedphaseexplosiveshassincebeenbuiltaroundthishotspotŽidea.Inaddition,itislikelythatinteractionsofthepost-shock”owwiththeobstacles,theboundarylayerthatformsonthetubewall,andtheturbulentnatureofthequasi-detonationfrontitselfallcontributetoburningthemixtureandsustainingthefront.Themechanismofpropagationofveryhighspeed(300ms)turbulent”ameshasnotbeenconvincinglyelucidatedandcurrentlycomprisesanomanslandŽbetweenpremixedturbulentcombustionanddetonation,whichincludesthetransitionbetweenthetwo(de”agrationtodetonationtransition)[].Assuch,mod-elstoaddressthereactionmechanismareadhocandsemi-empiricalinnature 2SteadyOne-DimensionalDetonations59aregeometrydependent,asseenintheresultswiththesquaretubewithstaggeredpillars,whichwhileexhibitingacriticalvelocitydidnotexhibitatransitiontothelowvelocitychokingregime.Similarstudiesofdetonationsingaseousmixtures“llingtheinterstitialspacesinpackedbedsofinertsphericalbeadsbyMakrisetal.[]andPinaevandLyamin[]failedtoidentifyacriticalvelocityforalargenumberofdierentexplosivemixtures,withacontinuousspectrumofpropagationvelocityfromtheidealCJvelocitydownto30%oftheCJvelocity.Itislikelythattheroleofturbulentmixing-drivencombustionwavesaccountsforthegeometry-dependingvelocityde“citandcriticalbehaviorthatcannotbepredictedbythesimplehomogeneousreactionmechanismswithexponentialtemperaturedependenceusedinthischapter.2.5SystemswithRealChemistryInordertohaveamoreaccuratemodelforrealsystems,itisnecessarytotakeintoaccountthedetailsofthechemicalreactions,ratherthansimplytreatingtheenergyreleaseofreactionasexternalheatadditionaswasdoneinpriorsections.Inthissection,amodelofasteady,one-dimensionaldetonationinareactingsystemofidealgaseswillbedeveloped,followingthedevelopmentsfoundin[].Thisanalysisbeginswiththesamedierentialformoftheconservationequations(2.1)…(2.3)andadditionalconservationequationsforeachindividualchemicalspecies(2.40)sincespecieswillbeproducedandconsumedwhilechemicalreactionsproceed.subscriptin(2.40)denotesaparticularchemicalelementorcompound.Thetermistherateofspeciesproductionviachemicalreaction(units:kmol/m-s),andisthemolecularweightofspecies.Itisconvenienttointroducethemassfractionofaspecies ,whereisthedensityofthemixture.UndertheassumptionoftheDaltonmodelofpartialpressure,theidealgaslawappliestoeachindividualspecies .Theaveragemolecularweightofthemixture(whichwill,ingeneral,notremainconstantasthemixturereacts)isgivenby W\biYi (2.41)Theenthalpytermcontainsboththesensibleenthalpyandthelatententhalpyofformation.Thus,itisnotnecessarytointroduceaheatreleasetermintotheenergyequationtoaccountforexothermicchemicalreaction.Speci“cally,isgivenby(2.42) 64A.Higginswhererefisthereferencepressureusedinde“ningtheequilibriumconstantbasedonpartialpressures.Equilibriumconstantsarederivedfromthermodynamicdataasfollows:=exp (2.67)where¯istheGibbsfunctiononapermolebasisofspeciesevaluatedatthetemperatureandatthereferencepressureref=1atm.Thus,theequi-libriumconstants,inbeingderivedfromfundamentalthermodynamicdata,areknownwithmuchgreatercon“dencethankineticrateconstants.Thisapproachthenensuresthattheoverallkineticmechanismisconsistentwiththehigher-con“denceequilibriumconstants.Aconsequenceofthisassumptionisthatitispossibletostartwithacompletelyarbitraryorincorrectreactionmechanismandstillreachthecorrectequilibriumcompositionofareactinggasmixture.Thus,correctlyreproducingtheCJdetonationvelocityviaaZNDcalculationwithdetailedchemistryshouldnotbetakenasvalidationofthekineticmechanism.Theoverallconversionoffuelandoxidizerintoproductsisdescribedbyakineticmechanism,consistingofelementaryreactions.Formostcombustiblemixtures,itisnecessarytoconsidernumerouspossiblereactions.Evenasim-pleŽsystemlikehydrogen/oxygenistypicallymodeledwith20orsoelemen-taryreactions,whilemechanismsforhydrocarbonfuels(e.g.,methane)canconsiderreactionsnumberinginthehundredsorthousands.Foramechanismconsistingof,...,Melementaryreactionsthatdescribesthereactionof,...,Nspecies,theoverallproductionrateofspeciesisgivenby(2.68)(2.69)whereisthestoichiometriccoecientofspeciesintheforwardreactionandisthecorrespondingstoichiometriccoecientforthebackwardreac-tion.Ifaspeciesisabsentfromareaction,itsstoichiometriccoecientisThetimerateofchangeofconcentrationisoftenexpressedinthecom-bustionliteratureas (2.70)Thisexpressionisstrictlyonlyvalidforaconstantvolume(density)reac-tion.Ingeneral,reacting”owsarevariabledensity,sospeciesconcentrationscanchangeduetochangesindensityaswellasreaction,andthus(2.70)is 66A.HigginsTable2.1.Carbonmonoxide/oxygenreactionmechanism ReactionAnE(kJ/kmol) CO+O+M+M1.80.009,970CO+O+O2.50.00200,000O+O+M+M1.21.000 kmolkmolroleofhydrogeninconvertingtheCOintoCO.Thus,inanyrealsystemwitheventracecontaminationofhydrogen,water,orhydrocarbons,themechanism2.72)…(2.74)isnotthedominantreactionpath.Forthepurposesofamodelsystemtoexplorenumerically,however,thedryoxidationofcarbonmonoxideisconvenienttouse.InordertoinitializethecalculationofthereactionzonestructureatthevonNeumannpoint,itisnecessarytospecifythepropagationvelocityofthedetonationwave.Thiswasdoneintwodierentways:anequilibriumsolu-tionbasedonacontrolvolumeenclosingthewaveandaZNDsolutionwhichthestructureofthereactionzoneissolved.Theequilibriumsolutionbasedonacontrolvolumewasdeterminedthroughiterationupontheinte-gralconservationlaws(2.1)…(2.3)assumingchemicalequilibriumattheexitplaneuntiltheminimumwavevelocitysolutionwasfound.Thereactionzonestructurewasnotconsideredinthiscase.Thespeedofsoundwasnotexplic-itlyintroducedinthesecalculations,butthesolutionfoundcorrespondstosonicout”ow(i.e.,”owvelocityequalstheequilibriumspeedofsound).Thissolutionmethodologycanbeshowntoagreewiththatusedinwell-knownequilibriumprogramssuchastheNASACEAprogram[],andgeneratesthesameresultswithinnumericalprecision.TheZND-basedsolutionwasfoundbynumericallyintegratingthereactionzonestructureusingthe(2.58)…(2.71).Theinitialvelocityofthewavewasiter-ateduponuntilasolutionthathadsonicout”owandthatdidnotencounterasingularitywasfound.Notethatthesonicout”owconditionfoundisthatde“nedbyusingtheequilibriumspeedofsound.Theout”owwithrespecttothefrozenspeedofsoundwasstillsubsonic(Mach0.963).Therefore,thissolutiondoesnotsatisfythegeneralizedCJcondition(2.59TheresultsofthetwosolutionmethodologiesarecomparedinTable2.2Thepropagationvelocityofthedetonationwavefoundbythesetwomethodsagreestosixsigni“cantdigits,andthe”owpropertiesattheexitstateagreetowithin“veorsixsigni“cantdigits.Thus,itcanbeconcludedthatthesetwomethods“ndthesamesolutionforthedetonationwave.ThestructureofthewaveisshowninFig.2.10.Thereactionzonelength,asde“nedbyidentifyingthelocationofmaximumthermicityorthein”ectionpointintemperature,isontheorderof10…15cm.However,thesoniccondition(usingtheequilibriumsoundspeed)isonlyapproachedasymptotically.Thus, 2SteadyOne-DimensionalDetonations69existsinthiscasethatpermitsthesolutiontosatisfythegeneralizedCJcriterion.Thisresultisduetothefactthatthethermicityparameterisalwayspositiveandonlyapproacheszeroasymptotically.Inorderforthesolutiontopassthroughsonicviaasaddlepoint,thethermicitymustpassthroughzero.Thus,forthecarbonmonoxide/oxygensystemdiscussedhere,thereisnoambiguitysincetheequilibriumandtheZND-basedmethodsgeneratethesameuniquesolution.2.5.3Hydrogen/ChlorineSystemIdenti“cationofarealcombustiblemixturethatexhibitsthetypeofpatho-logicalbehaviordiscussedinSect.2.3isaninterestingproblemthathasbeenthesubjectofperiodicinvestigationsincetheintroductionoftheZNDmodelinthe1940s.ZeldovichandRatner[]pointedoutthatthereactionofHandCltoformHClcanoccurviatheNernstchainreactionmuchmoreread-ilythanthedissociationofCl,whichhasarelativelyhighactivationenergy.Thus,thehighlyexothermicreactionformingHClmaybefollowedbyanendothermicdissociationreactioninCl,resultinginthetypeofexother-mic/endothermicreactionnecessaryforpathologicalbehavior.Subsequentstudiesofexperimentalsystemsexhibitingpathologicalbehaviorhavetendedtofocusonthissystem.The“rstdetailedchemicalkineticcalculationsofaZNDdetonationin/ClwereperformedbyGu´enocheetal.[]usingthemechanismgiveninTable2.3.Thesecalculationsarereproducedhere.Afuel-lean(chlorine-rich)mixturewithfuelequivalenceratio66atrelativelylowpres-sure(33kPa)wasselectedhereforasamplecalculationinordertoaccentuatethedierencebetweentherapid,exothermicformationofHClandtheslowerdissociationoftheexcessCl.Thegoverningdierentialequa-tions(2.58)…(2.71)wereintegratedcoupledwiththekineticmechanisminTable2.3.TheinitialshockvelocitywasiterateduponuntilasolutionthatTable2.3.Hydrogen/chlorinereactionmechanism ReactionAnEkmol)(kJ/kmol) H+H+M+M1.01.000HCl,Cl,Cl1.000H0.600H2Cl+M62.07238815H,Cl,HCl,H2.07238857ClHCl+MH+Cl+M62.00427765H,Cl,HCl,H,ClCl+HHCl+H40.0022H+ClHCl+Cl60.684 kmol 70A.HigginsTable2.4.Hydrogen/chlorinedetonation Initialcomposition(Moles)H Initialstate(kPa)3(K)300(ms)278 DetonationsolutionEquilibriumEigenvalue(ZND) (ms)1,320.71,527.34.7355.476vonNeumannstate(kPa)86.469116.0618.06634.852(K)1,374.11,724.4(ms)232.96251.88(ms)586.81655.230.39700.3844Chapman…Jouguetstate(kPa)47.07360.30014.13618.108(K)2,075.942,989.09(ms)748.4883.3(ms)789.0883.3(ms)748.4N/A0.94851.00001.0000N/A passesthroughthefrozensonicpointwasidenti“ed.ThissolutionispresentedinTable2.4andFig.2.12,showingthethermodynamicproperties,speciesconcentrations,thermicityparameter,andthe”owMachnumber(usingthefrozenspeedofsound).AshypothesizedbyZeldovichandRatner,theH/Clsystemdoesexhibitanovershootintheheatreleasefollowedbyanendother-micphase.Thisismostclearlyseenbyexaminingthetemperatureandthethermicityparameter,whichpassesthroughzeroandbecomesnegativeatthesamepointwherethe”owbecomesfrozensonic,asrequiredbythegeneralizedCJcondition.(Notethefactthatthethermicitybecomesnegativepreventstheuseofalog-scaleonthe-axis,incomparisontoFig.2.10.)Thehistoryofspeciesconcentrationsthroughthereactionzoneveri“esthattherapidforma-tionofHClfollowedbyaslowerdissociationofClintoClisthesourceoftheexothermic/endothermicnatureofthereaction.Thissolution,resultingfromiterationuponthereactionzonestructureuntilatrajectorythatsatis“esthegeneralizedCJconditionwasfound,istheeigenvaluesolution.Theeigenvaluesolutionfoundaboveiscomparedtotheequilibriumsolu-tion,suchaswouldbefoundviatheNASACEA[]orotherchemicalequilibriumsoftware,inTable2.4.Asigni“cantdierencebetweenthetwosolutionsisfound,witha15%discrepancyinthedetonationpropagationvelocitieswiththeequilibriumvelocitybeinglower.Thisresultisunlikethat 2SteadyOne-DimensionalDetonations71 Mole Fraction10-310-210-1100Cl2H2ClH X 100HClsonic surface Flow Mach Number (frozen sound speed)0.20.40.60.81.01.21.41.61.8sonic (frozen) Thermicity (1/s) Distance (m)Pressure (atm)Temperature (K)10-410-310-210-1100Distance (m)10-410-310-210-1100Distance (m)10-410-310-210-110000.511.521600180020002200240026002800300032003400PressureTemperature Fig.2.12.StructureofHdetonation,showingpressure,temperature,molefractions,thermicity,and”owMachnumber.ThelocationofthesonicpointisdenotedasaverticaldashedlineobtainedwithCO/Odetonationsconsideredinthepriorsection,forwhichtheequilibriumandZNDapproacheswereseentogeneratethesamesolution(withinnumericalprecision).IfaZNDcalculationoftheH/Clreactionzoneisinitializedwiththepost-shockvonNeumannconditionsfortheCJequilib-riumsolution,asisdoneinFig.2.13,thecalculationencountersasingularity.Thereactionzoneisnowlongerduetothelowerpost-shocktemperature,butwhentheexothermicreactionbeginstoreleasesigni“cantheatintothe”ow,itresultsinsonic”owbeingencounteredbeforethethermicitygoingtozero,hencecausingthesingularity.Qualitatively,thisisidenticaltothebehaviorseeninSect.2.3withanarti“ciallyconstructedtwo-stependother-mic/exothermicsystem.Indeed,anyattempttoinitializeasolutionwithashockvelocityevenincrementallybelowtheeigenvaluesolutionfortheH/Clsystemresultsinasingularity.Solutionsinitializedwithafastershockremainsubsonicwithrespecttothefrozenspeedofsound(i.e.,thestrongdetonationsolution).Finally,notethatitisnotpossibletode“neanequilibriumsoundspeedforthe(nonequilibrium)sonicpointoftheeigenvaluesolution. 76A.Higginswavefrombehindanddisruptingthereactionzone.However,asadetona-tionpropagates,thegradientofexpansionbehindthewavebecomesmoregradual,andthusthefrequencyofrarefactionwavesmightbeexpectedtoapproachthelow(equilibrium)limit.Further,whiletheinitialleadingedgeofacenteredrarefactionwouldpropagateatthefrozenspeedofsound,thisacousticwaveisquicklyattenuated(decaysexponentially)andthemajorityoftherarefactionisgovernedbytheequilibriumspeedofsound.Thisaspectofrarefactionspropagatinginreactive”owwasexploredindetailbyWoodandParker[].Ifthisisthecase,thentheequilibriumsolutionwouldagreewiththeconceptualpictureoftheCJdetonationbeingisolatedfromdownstreamdisturbancesbyasonicplane.Forasystemwithpathologicalheatrelease,suchasthehydrogen/chlorinesystemconsideredinSect.2.5.3,itisnotpossibletoobtainasolutionforthereactionzonestructurethatisinitializedwiththeCJequilibriumvalueofdet-onationspeed(seeFig.2.13).Thiscasecanbevisualizedonthe(p,v)planebytheinabilityofasequenceofintermediateHugoniotstoprovideapathalongtheRayleighlinetoastateoftangencytotheequilibriumHugoniot(seeFig.2.5).Theonlypermittedsteadysolutionwithasonicpointistheeigenvaluesolutionwithafrozensonicpointembeddedinthesolutionandanexit”owthat,withrespecttotheequilibriumHugoniot,isaweakdeto-nation.Similarly,inanysystemwithlossessuchasheattransferorfriction,asdiscussedinSect.2.4orwithlaterallydivergent”ow,aswillbediscussedinSect.2.6,itisthefrozensoundspeedthatde“nesthesaddlepointoftheeigenvaluesolution.Thissituationgivesrisetoanapparentparadox,whereinthelimitingsolu-tionasthelossmechanismdecreasestozeromaynotconvergetotheideal,planarsolution,sincetheformerisde“nedbythefrozenspeedofsoundandthelatterusestheequilibriumspeedofsound.Forexample,forthecaseofadetonationinatubewithlossesatthewalls,astheradiusofthetubeincreasestoin“nity,thesolutionforthedetonationvelocitywillnotagreewiththeplanarsolution(i.e.,lim).Inordertoresolvethisapparentinconsistency,apossiblestrategywouldbetosolvethefull,unsteadyEulerequationswithoutimposingaparticularcriterion,andseewhichsolu-tionevolves(thisapproachwasdonebySharpeandFalle[]forpatho-logicaldetonationsandbyDionneetal.[]fordetonationswithfrictionandheattransfer,asdiscussedinSect.2.4.3).SuchacalculationwasperformedbySharpe[]foramodelsystemwithasingle-stepreversiblechemistryoriginallyproposedbyFickettandDavis[](FickettandDavisconsideredasystemwithzeroactivationenergy,whileSharpeconsideredasystemwithlowactivationenergythatgaveastableZNDsolution).Sharpe[]showedthat,foraplanardetonationinitiatedbyanoverdrivenblastwaveintheabovesystem,afteralongpropagationtime,thewavevelocityandreactionzonestructureapproachedthatofanequilibriumsonicCJdetonation.However,ifevenanin“nitesimaldegreeoflosswasintroduced(curvature,inthecaseofSharpescalculations),thelong-termevolutionofthesolutionconverged 78A.Higginswhichthecriticaldimensionis“rstencountered.Ingaseousexplosivescon-tainedinrigidtubes,near-limitbehaviorismorecomplex,sincethewavecaninteractwiththetubewallinordertosustainpropagationintransientmodesknownasspinningandgallopingdetonations(seediscussioninChap.of[Thevelocityde“citin“nite-sizedchargesandtheexistenceofacriti-caldimension(usually,thediameterofacylindricalcharge)areheavilyuti-lizedinresearchondetonationwavesincondensed-phaseexplosivesasameanstoprobethereactionzonestructureandthermodynamicpropertiesofthedetonationproducts.Duetotheextremelyshortreactiontimescalesincondensed-phasedetonations(typically,sub-microsecondtonanosecond),theopaquenatureofdetonatingmedia,andtheextremepressuresgenerated,insitumeasurementsofanydetonationpropertiesareextraordinarilydicult.Asaresult,almostallmodelsforcondensed-phaseddetonationsutilizedataderivedfromthevelocityde“citand/orthecurvatureoftheshockfrontasthechargediameterisdecreased,ormeasurementofthecriticaldiameteratwhichdetonationfailureoccurs.Theradialexpansionofdetonationproductsinacylindricalchargecon“nedwithaductilematerial(e.g.,copper)isalsothebasisofmostsemi-empiricalmodelsfortheequationofstateofthedetonationproducts.Fordetonationsingas-phasemediaontheorderofatmosphericpressure,perfectlyrigidcon“nementispossible,forexample,usingasteel-walledtube.However,theperipheryofthegaseouschargestillin”uencesthedetonationwaveviafrictionandheattransfertothewall.InSect.2.4,theseeectsweretreatedasvolumetriclossesthatwerespreaduniformlyacrossthecrosssectionofthetube.Inreality,thein”uenceofthewallisconveyedtothedetonation”owviaaboundarylayerregionnearthewall.Thegrowthoftheboundarylayerresultsinadiverging”owinthereactionzone,qualitativelysimilartothediverging”owthatoccursinacondensed-phasedetonationduetoyieldingcon“nement.Itmayappearcounterintuitivethatgrowthofaboundarylayeralongthewallresultsinadiverging”owinthedetonationreactionzone;notethatinthewave-“xedreferenceframe,thewallshavetheeectofacceleratingthe”owwhilealsocoolingandincreasingthedensityofgasintheboundarylayer.Bycontinuity,thiseectrequiresthatthecoreofthe”owdivergesoutwardtoaccountforthemasseectivelyremovedbytheboundarylayer.BoundarylayersofthistypearesometimescallednegativeboundarylayersWithinthecontextofaquasi-one-dimensionalapproximation,detonationswithdivergent”owcanbetreatedwiththereactionzoneequationsderivedpreviouslyinSect.2.2.Inparticular,(2.10),reproducedhereslightlymodi“ed dx=q cpT\nŠu1 AdA dx (2.75)modelstheeectofdiverging”owviathe AdA term.Inthissection,thesteadyreactionzonestructureofdetonationwaveswithdiverging”owisfurther 80A.Higgins(0)��Ld))whereisthelengthofthereactionzone,thentheareadivergencetermsdierbyanumericalfactoroftwoforthetwogeometries2-DSlab: AdA dxt(x) (0)(2.80)AxisymmetricTube: AdA dx2d(x) (0)(2.81)Thismeansthatthesolutionforthereactionzonestructureinthetwogeome-triesshouldbethesame,providedthediameteroftheaxisymmetriccylinderisscaledbytwicethethicknessofthe2Dslab.Earlymodelsfordetonationin“nite-diameterchargesbyJones[]andEyringetal.[]usedphenomenologicaldescriptionsforthestreamtubeareadivergence(i.e.,nozzle”owmodels).Whilenotrigorous,thesemodelsareofinterestintheirprovidingaphysicalpictureofthe”ow“eldsthatresultfromtheboundaryofthechargeexpandingoutwardastheexplosivereacts.ThemodelsdevelopedbyJonesandelaborateduponbyEryingetal.assumedaPrandtl…Meyer(P…M)expansionfanoriginatingwhereanormaldetonationencounteredtheedgeofthecharge.TheinitialstatewastakenastheCJstatefortheideal,constantareadetonation(notethatusingthesubsonicvonNeumannstateisnotanoptionsincetheP…Mfunctionisonlyde“nedforsupersonic”ow).Foranuncon“nedcharge,theP…Mfanattheedgewasmatchedtoastreamtubecontainingthecoreofthedetonation”ow.Foraheavilycon“nedcharge,theP…Mfanwasmatchedtoanobliqueshocktransmittedintothecon“ningmaterialtodeterminethedivergenceangle,whichwastakenasconstant.Eyringetal.wentontodevelopheuristicmodelsincorporatingthefactthat,ifthe”owattheboundaryofthechargeisdivergingoutwardfollowingtheleadingshock,theshockfrontitselfcannotbe”atandmustbeobliqueattheboundary.Inotherwords,theleadingshockfrontiscurved(normalonthecentralaxisofthechargeandincreasinglyobliquetowardtheedges),similartothemeniscusofaliquidsurfaceinacapillarytube.Eyringetal.werecarefultopointoutthatfrontcurvatureisanecessaryconsequenceofdiverging”ow;curvatureand”owdivergencearedierentmanifestationsofthesamephenomenon,notseparateeectsthatshouldbesuperimposedinmodels.Theoreticalinvestigationsintothe1960scontinuedtopresupposefunctionalformsforthestreamtubeareapro“le,oftenoutofanalyticconvenienceratherthanfromphysicalconsiderationss26,100].2.6.2RadialFlowDerivativeBeginningwiththeworkofWoodandKirkwood[],amorerigorousapproachtosolveforthereactionzoneofadetonationwavewithdivergent”owwasdevelopedthatexaminedthetwo-dimensional”ow(eitherrectangu-laroraxisymmetric)alongthecentralaxialstreamlineofthereactionzone. 2SteadyOne-DimensionalDetonations81Utilizingthefactthatthe”owissymmetricaboutthisline,thecontinuityequationfortwo-dimensional,steadycompressible”owŠ=0(2.82)canbesimpli“edforrectangularcoordinates x+(v) =0(2.83)sincethetransverse”owvelocityiszeroalongthisstreamline,asfollows: x+v (2.84)Notethat,whilethetransversevelocityiszeroalongthecentralstreamline=0),thederivativeofthevelocityintheradialdirectionhasanonzerovalue .Likewisethecontinuityequationforaxisymmetric”owincylindricalcoordinates x+1 rv =0(2.85)specializedtothecentralstreamlinevialHopitalsruleyields x+2v (2.86)-momentumequationinrectangular uv y=Šp (2.87)andcylindricalcoordinatesru ruv r=Šrp (2.88)bothrevert(viasymmetry)tothefamiliarformofthemomentumequationwhenappliedalongtheaxialstreamline x+uu (2.89)Returningtocontinuity,(2.84)and(2.86)canbewrittenas x+v (2.90)wheredenotesthetransverse-directionintherectangulargeometryandradial-directioninthecylindricalgeometry,andhasthevalueof1and 82A.Higgins2fortherectangularandcylindricalgeometries,respectively.Bycomparing2.90)with(2.4),thederivativeofthetransverse/radialvelocityterm( canberelatedtotheareadivergenceofastreamtube AdA dx= uv (2.91)Thisequivalencecanalsobedemonstratedbyconsideringanarbitrarilysmallstreamtubethatenclosesthe”owalongthechargeaxis,andusingthederiva-tiveoftheradial”owvelocitytoapproximatethedivergenceofthestreamtubeboundary.Asthestreamtubeshrinkstotheaxis,thecorrespondence2.91)becomesexact.2.6.3ShockFrontCurvatureThederivativeoftheradial”owvelocitycanalsoberelatedtotheradiusofcurvatureoftheleadingshockfrontbyusingthegeometricconstructionintheshock-attachedreferenceframeshowninFig.2.16.The”owvelocityapproachingtheshockisthedetonationpropagationvelocity().Notethatthisconstructionappliestothe2Dslabandaxisymmetricgeometriesequally.Usingthefactthatthecomponentof”owvelocityparalleltoashockfrontdoesnotchangeasthe”owcrossestheshock,asrequiredbyconservationofmomentum,itispossibletoexpresstheradialcomponentofvelocityas(2.92) DDu\\ =D sin\bu\t = D cos\bu\tv\b u\\centerlineRshock frontvon Neumann point x y, r two-dimensional slabaxisymmetric cylinder Fig.2.16.Geometricconstructionrelatingshockfrontradiusofcurvaturetothederivativeoftheradial”owvelocity rorv atthecenterlineimmediatelybehindtheshock 2SteadyOne-DimensionalDetonations83Performingapartialdierentiationofthevelocitywithrespecttotheangle shock(cos2 shock(2.93)wheretheshockŽsubscriptdenotesthatthedierentiationwasperformedalongtheshockfront.Thedierentiationcanbeconvertedtoadierentialwithrespectto(denotedas)asfollows: shock shock shock shock (2.94)Takingthelimitapproachingthecentralaxis (0) (2.95)where(0)istheaxialvelocityatthevonNeumannpointalongthecentralaxis.Thus,theradiusofthecurvatureoftheshockfrontisdirectlyrelatedtothederivativeoftheradial”owimmediatelybehindtheshockfront.ThisrelationappliesstrictlyonlyatthevonNeumannpoint(i.e.,immediatelyaftertheleadingshock);however,lackingfurtherinformation,thisderivativeisoftentakenasconstantthroughthereactionzone.Theshockradiuscanberelatedtothecurvatureoftheshockfrontasfollows: (2.96)whereagainis1forthetwo-dimensionalslabgeometry(cylindricalcurvatureoftheshockfront)and2fortheaxisymmetriccylindergeome-try(sphericalcurvatureoftheshockfront).Thefactthat,foralocallysteadydetonationfront,theshockcurvatureuniquelydeterminestheeigenvaluevelocityofpropagationprovidesthemeanstoconstructthedetonationfrontshapeandtrajectoryasthedetonationpropagates,forexample,throughacomplexchargegeometry.Thistechnique,calledDetonationShockDynamicsisthesubjectofChap.ofthisvolume.2.6.4Con“nementInteractionviaNewtonianTheoryInordertoassociatethesedivergent”owmodelswithexperimentalresultsormakequantitativepredictionsofvelocityde“citsorcriticaldiameter,itisnecessarytolinkthe”owdivergencetotheoveralldimension(diameterorthickness)ofthechargeandthepropertiesofthecon“nement.Forcondensed-phasedetonations,solvingforthisinteractioncanbechallenging;thistopiciselaborateduponinChap.ofthepresentvolume.Asanillustrativenumericalexamplerelevanttogaseousdetonations,asimpleNewtonianmodelfortheinteractionofthediverging”owinthereactionzonewiththecon“nementwillbedevelopedfurtherinthissection.Beinganozzle”ow-typemodel,itdoesnotexplicitlyincludethecurvatureoftheshockfront.However,this 84A.Higginsmodel(originallyproposedbyTsugeetal.[])isinstructiveinthatittreatstheevolvinginteractionofthereacting”owwiththecon“nementviaananalyticallytractablemethodwithoutresortingtoempiricalinputoranassumedformulafortheareadivergence.TheNewtonianmodelforhypersonic”owassumesthata”owencounteringaninclinedsurfacelosesthecomponentofvelocitynormaltothesurfacebutretainsthetangentialvelocity.Inotherwords,a”owencounteringaninclinedsurfaceslidesalongthesurface,andthechangeinmomentum”uxofthe”owdeterminesthatthepressureonthesurfacemustvaryas(2.97)whereisthesurfaceinclinationangletothe”owwithfreestreamdensity,andvelocity.OriginallyproposedbyNewton,this”owmodelhasbeenshowntoberemarkablyaccurateinpredictingsurfacepressuresforslenderbodiesinhypersonic”ow[].Themodelisinvokedheretotreatthe”owofthecon“nementmaterialasitencounterstheexpanding”owofreactinggas,therebylinkingthepressureofthereacting”owtothedivergenceangleofthestreamtubeenclosingthat”ow.Sincethestreamtubecannotsupportapressuredierence,thereacting”owmustlocallymatchthepressureofthecon“nementandtherebytheslopeofthestreamtubeboundary.Using2.97)forthe”owofcon“nementmaterial dx2 1+\ndz (2.98)whereŽdenotesthepropertiesofthecon“nementbeforeinteractionwiththedetonationwaveandŽiseithertheradiusorthehalf-thicknessthestreamtube.Solvingfortheslopeofthestreamtube dx=\f\r\r p(x)Špc cu2c 1Šp(x) (2.99)Thisexpressionisusedto“ndtheareadivergencetermasafunctionofthelocalpressureinthestreamtube2-DSlab: AdA dx=t t=y y=1 y\f\r\r p(x)Špc cu2c 1Šp(x)Špc (2.100)AxisymmetricTube: AdA dx=2d d=2r r=2 r\f\r\r p(x)Špc cu2c 1Šp(x)Špc (2.101)Theseexpressionsfor AdA canbeuseddirectlyintheODEgoverningthereactionzonestructure(2.75).Sincepressurenowappearsexplicitly,thisODEmustbeintegratedcoupledtothemomentumequation(2.5)reproducedhere 2SteadyOne-DimensionalDetonations85 dx=Šudu dx=Šm Adu (2.102)whichcanberelatedtothelocalstreamtubedimensionasfollows:2-DSlab: dx=Š1u1y1 ydu (2.103)AxisymmetricTube: dx=Š1u1\br1 r\t2du (2.104)whereandarethedensityandvelocityupstreamofthedetonation)andandaretheinitialhalf-thicknessandradiusoftheexplo-sive,respectively.ThissetofODEswasintegratedtoobtainthestructureofadetonationwithyieldingcon“nement.Thepropertiesoftheexplosivewerethesamethathavebeenusedpreviouslyinthischapter(=10,andasingle-stepArrheniusreactionwith =25).Theinertcon“nementwasassumedtobeatthesameinitialpressureastheexplosive,butwithadensity2.5timesgreaterthantheexplosive().Thisdensityratioapproximatelycor-respondstothedensityratioofairtostoichiometrichydrogen/oxygen,whichwillbeusedasanillustrativeexperimentlaterinthissection.AsexpectedfromtheformofthegoverningODE(2.75),wheretheeectofareadivergencefromtheyieldingcon“nementisseentocompetewithexothermicheatrelease,itisnecessarytoiterateuponthedetonationvelocityto“ndaneigenvaluesolutionthatcanpasssmoothlythroughthesonicpointwithoutencounter-ingasingularity.Aswasseenpreviouslywiththeeectofheattransferandfriction,thereisacriticalamountoflossthatthedetonationcansustain,inthiscase,resultingfromtheexplosivechargebeingtoothinandlosingtoomuchmomentumtothedivergenceofthe”ow.Forchargesthinnerthanthiscriticalthickness,nosteadysolutionwithasonicpointcanbefound.Thestructureofthereactionzoneforthecriticalslabthicknessatwhichfailureoccurs(i.e.,atthecriticalturningpoint)isshowninFig.2.17asthicklinesincomparisontotheidealCJsolutionforanin“nitethicknesscharge(thinlines).Asmallschematicdonetoscaleisincludedtoshowtheactualamountofareadivergenceobservedatthecriticalturningpoint.Alldimen-sionsarenormalizedbythehalfreactionthicknessoftheidealCJdetona-tion().Thelocationofreactioncompletionorthesonicsurfacecannotbede“nedfortheidealCJdetonation.Thesonicsurfaceforthe“nite-sizechargewithareadivergence,however,canbedeterminedfromtheeigenvaluesolution.Notethattheareadivergencefromtheshocktothesonicplaneismoderate(30%increaseinarea)andthat,evenforthesmallestchargethick-nessthatcansupportdetonation,thethicknessoftheexplosiveslabisstillmorethanthreetimesthelengthofthedetonationfront(i.e.,thedistancefromtheshocktothesonicsurface).IncomparisontotheidealCJsolution,thelocationofthepeakinexothermicityandhalfreactionlengthhavemorethanquadrupledduetothelowerpost-shocktemperature,andtheexother-micityisstill“niteasthesolutionpassesthroughsonic.Atthesonicsurface 88A.HigginsStewart[],andKleinetal.[].Theanalysisby[]and[]yieldsaresultforthecriticalvelocityidenticaltothatpreviouslyfoundbyZeldovichandKompaneetsfordetonationswithheatloss(2.38),reproducedherecrt DCJ1Š1 21 \bEa (2.105)Theanalysisof[]producesaslightlymorecomplexexpressiondependentupon,whichnonethelessisequivalentto(2.105).ThisrelationisplottedinFig.2.18asadashedline.WhileexhibitingqualitativelysimilartrendstotheresultsobtainedbyintegratingthegoverningODEsto“ndtheeigenvaluedetonationvelocity,quantitativeagreementisonlyfoundforveryhighvaluesofactivationenergy(wheretheasymptoticanalysisisintendedtoapply).2.6.5ComparisonstoExperimentExperimentswithcolumnsofdetonablegas(hydrogen/oxygen)con“nedbyinertgas(nitrogen)havebeenperformedby[].Somecharacter-isticimagesofsuccessfulpropagationandfailureofpropagationareshowninFig.2.19;furtherexperimentsarediscussedinChap.ofthisvolume.Theexpansionofthedetonationproductsandtheobliqueshockbeingdrivenintothecon“ninggasareclearlyvisible,asisthecurvatureoftheshockfront.Quantitativecomparisonofthemeasureddetonationvelocityinthese“nite-diametercolumnsofdetonablegaswithZND-typecalculationusingtheNew-tonianmodelforinteractionwiththecon“nement(aswasdoneinSect.2.6.4butwithamoredetailedchemistrymodel)wasdonebyTsugeetal.[andshowedgeneralagreementwiththeexperiments.Theexperimentaldata,however,hadrelativelylargeerrorbars,andmoreaccuratevelocitymeasure-mentswouldnecessitatemeasuringpropagationinlongercolumnsofdeton-ablegas,whichisadicultexperimenttoprepare. PropagationFailure bdbdacac Fig.2.19.Experimentalphotographsofdetonationsin“nite-diameter,uncon“nedcolumnsofhydrogen/oxygen[ 2SteadyOne-DimensionalDetonations89PerhapsthemostimpressivedemonstrationlinkingtheZNDmodeltoanexperimentalresultisthestudyofgaseousdetonationsinporous-walledtubesandchannelsofRadulescu[].Duetothe”owintotheporouswalls,thereactionzoneofthedetonationinsuchatubeorchannelexperiencesadiver-gent”ow,qualitativelysimilartothedivergenceresultingfromyieldingcon-“nementorboundarylayersdiscussedearlierinthissection.Thisdivergenceresultsinvelocityde“citsand,forasucientlysmalltubediameter,failureofthedetonation.Theexperimentalmeasurementsofthevelocityde“citandfailurediameterinmixturesofhydrogen/oxygenandacetylene/oxygenwithlargeamountsofargondilutionexhibitedgoodcorrelationwithaZNDcalcu-lationwithdetailedchemistry(similartothatinSect.2.5)whichincludestheeectof”owdivergence(asisdoneinthissection).Theradial”owderiva-tive(or,equivalently,theareadivergence)wasestimatedbyassumingthatthe”owintotheporouswallwaschoked(sonic)attheporeopenings,withthenetout”owbeingscaledbytheporosityofthewall.Comparisonsofthismodeldemonstratedgoodagreementwiththeexperimentalresults,includingpredictionofthecriticaldiameteratwhichdetonationfailurewasobserved(seeFig.2.20).Whatismore,studiesinporous-walledtubesandporous-lined L*/dD/DCJ10-410-310-20.600.700.800.901.10 2H2 + O2ModelExp.C2H2 + 2.5O2 + 75%ArModelExp. Fig.2.20.Velocityde“citsobservedforgaseousdetonationspropagatinginporous-walledtubes,comparedtoaZNDmodelwithareadivergencedueto”owintothetubewalls.ThetubediameterisnormalizedbythecomputedreactionzonelengthoftheZNDdetonationwithdivergence,.Intheexperiments,initialpressurewaschangedinordertovaryreactionzonelengthwhilethetubediameterwasconstant[ 92A.Higginsnozzlethroat).SimilartransonicŽcombustionregimesarebelievedtooccurinramaccelerators[].Adynamicalsystemsanalysisofthecriticalpointsinthesetypeof”owswasrecentlyperformedbyDeSterck[Researcherswhohavelaboredtodeveloptheframeworkforunderstand-ingdetonationwavesshouldderivesatisfactionthattheireortsmay“ndawiderrangeofapplicationthanthenarrow“eldforwhichthatframeworkwasoriginallydeveloped.AcknowledgmentsThischapterwasdevelopedoutofdiscussionswithJimmyVerreault,OrenPetel,Fran¸cois-XavierJett´e,PatrickBatchelor,andDavidMack.VincentTanguaycontributedtotheanalysisoftheinclusionoftheworkdonebyfrictionindetonationsandtheTaylorwaveanalysisintheappendix.Jean-PhilippeDionnesdoctoraldissertationprovidedatemplateformuchofthischapter.JennyChaoandMateiRadulescuarethankedforsharingtheirexper-imentaldata.FanZhangandCraigTarverprovidedhelpfulandinsightfulcommentary.A.1AppendixA:GasdynamicsofDetonationProductsTheexistenceofsonic”owattheexitplaneofadetonation,whichcomprisestheclassicalChapman…Jouguetconditionanditsgeneralizationtononidealdetonations,meansthatthedetonationwaveisdecoupledfromtheexpansionofthedetonationproductsinthewakeofthewave.Thus,thedetailsoftheexpansiondonotneedtobeexplicitlyconsideredinsolvingforthedetonationwavepropertiesorstructure.Theexpansionofthegasthathasbeenprocessedbythedetonation(i.e.,theburnedgas)canbeofinterestinitsownrightforsomeapplications.Considerationofhowthedetonationexitstateismatchedwiththedownstreamexpansioncanalsoprovidesomeguidanceindeterminingthepossiblenon-CJexitstatesthatmightberealizable.ThegasdynamicsoftheunsteadyexpansionofdetonationproductswastreatedbyG.I.Taylorinaseminalwork[],andthesolutionhefoundisoftenreferredtoastheTaylorwave.Inthisappendix,theTaylorwave”ow“eldsolutionisbrie”ydevelopedusingthemethodofcharacteristicsfortheplanarcase(Sect.A.1.1matchingoftheTaylorwavewithdierentbranchesofdetonationsolutionsisthenexamined(Sect.A.1.2),and“nallyamoreformalsimilaritysolutionforthecylindricalandsphericalcase(Sect.A.1.3)isdeveloped.A.1.1PlanarDetonationTaylorstreatmentofthedynamicsoftheproductsemergingfromapla-nardetonationbeginsbyassumingthatthedetonationwavepropagatesata 2SteadyOne-DimensionalDetonations93steadyspeed,withtheproductsleavingthewaveatconstantconditions.Itisfurtherassumedthattherearenoshockwavesinthe”owdownstreamofthedetonation.Thus,sinceallthe”oworiginatesatthesamestate(denotedstate2Žtobeconsistentwiththenotationinthischapter)andremainsisentropic,theentire”owhasthesamevalueofentropy(i.e.,the”owishomentropic).Thismeansthat,ateverypointinthe”ow,characteristicwavesorig-inatingfromthedetonationbutpropagatingintheoppositedirectionasthewavehavethesamevalueoftheRiemanninvariant: Š1c=u2Š2 (2.106)wheretheprime()onisusedtodenotethatitisthevelocityintheunsteady,laboratory-“xedreferenceframe,inordertodierentiateitfromthesteady”owvelocityusedthroughoutthischapter.From(2.106),itcanbeshownthatateverypointintheproducts,andarelinearlyrelated.Thisresultmeansthatcharacteristics(alongwhich constant)thatpropagateinthedetonationproductstowardthewavemusthaveconstantvaluesofandandarethereforestraightlinesinthe(x,tplane.Onepossiblesolutionthatcanbeconstructedwithstraightcharac-teristicsisacenteredrarefactionfanoriginatingattheorigin.Thedetonationisalsoassumedtooriginateattheoriginandhasnegligiblethicknesscom-paredtothedomainofinterest.Thus,the”owpatternbecomesself-similar,asseeninFig.2.21.Atanypointinsidethecenteredrarefaction,thevalue x detonation wave’ + characteristics characteristics C+ = constantC = constant Fig.2.21.Centeredrarefactionsolutionfortheexpansionofproductsbehindadetonationwave,showingtheandcharacteristics 94A.Higginsmustequalthevalueof atthatpoint.Usingthisproperty,alongwiththeconstantcharacteristicsemanatingfromthedetonation,the”owvelocityandsoundspeedcanbesolvedfor:x,t +1\bx tŠc2\t+Š1 (2.107)x,t +1\bx tŠu\t+2 (2.108)Notethatpositionandtimealwaysappearasthecombination ,verifyingthatthisisasimilaritysolution.Theotherthermodynamicproperties(pres-sure,density,temperature)canbedeterminedfromthesoundspeedbyusingtheisentropicrelations.Thevelocitypro“lebehindthewaveisplottedinFig.2.22.Forthisparticularplot,theconditionsatthedetonationexitplaneweretakenastheCJconditionsinthelimitoflargeheatrelease D=1 +1,limQc2 D= +1,limQ2 1=+1 ,limQp2 1D2=1 (2.109)whereavalueof2wasusedforFig.2.22.Notethatthevelocityinthecenteredrarefactionfangoestozeroandreversesdirection,reachinganescapespeedof -0.500.51 Fig.2.22.Flow“eldbehindaplanarCJdetonationwave(Taylorwave)showingpressureand”owvelocityforbothclose-endedandopen-endedtubes( 96A.Higgins xtCJ detonation wave weak detonation wave u’+c = Du’+c DD strong detonation wave u’+c� DD strong detonation wave u’+c = DCJ DetonationWeak DetonationStrong Detonation (supported)Strong Detonation decayingto CJ Detonationwallwallpiston at restpistonpistonpistonstops uniform region rarefactiondeceleratesdetonation x x x t t t dcba Fig.2.23.Expansionofcombustionproductsbehindadetonationwaveforthecasesof()CJdetonation,()weakdetonation,()strong(piston-supported)detonation,and()apiston-supporteddetonationinwhichthepistonstopsandtherarefactionsgeneratedovertakedetonation,deceleratingituntilthedetonationisparallelwithcharacteristics(CJdetonation)detonation,establishingaCJdetonation.Thissimplepicture,illustratedinFig.2.23d,perhapsprovidesamoresatisfyingexplanationforthelegitimacyoftheCJconditionofsonic”owattheexitofanunsupporteddetonationwave.A.1.3CylindricalandSphericalDetonationsToconstructasolutionforthedynamicsoftheproductsbehindasphericaldetonationoriginatingfromapoint(oracylindricaldetonationinitiatedalongaline),theconservationofmassandmomentumforsphericalandcylindricalsymmetryareusedasfollows: 2SteadyOne-DimensionalDetonations99outthatinitiationofasphericaldetonationwillalwayslikelyrequiresomedegreeofinitialoverdrivethatwillresultinthedetonationbeinginitiallyonthestrongbranchofthesolution()andwillonlyasymptoticallyapproachanidealCJdetonationasthewaveexpands,sothattheissueofthein“nitegradientsinthesolutionisavoided[A.2AppendixB:CriticalSonicPointwithFrictionForadetonationwithfriction(andnoheattransfer),thecriticalconditionatwhichthewavepropagationvelocityequalsthevelocityofthecombustionproductscanbesolvedforanalyticallyasfollows[].Themomentumandenergyequations(2.27)and(2.28)canbeequated(excludingtheheatlossterm)viathefrictionfactor u1h+1 (2.117)andintegratedtoyield u1h+1 (2.118)whereisaconstantofintegrationthatcanbeevaluatedusingconditionsupstreamofthedetonationwave(,and).Atthecriticalsoniccondition,the”owexitingthewaveissonicandequaltothevelocityofthe”owapproachingthewave(inthelaboratory-“xedframe,thisisequivalenttothe”owthathasbeenprocessedbythewavebeingatrest),crs(2.119)wherethe2Žsubscriptdesignatestheexitstate.Theconditionthatexitvelocityisthesameasthein”owvelocitymeansthatthedensityisconstantacrossthewave().TheserelationsallowthecriticalMachnumberofpropagationtobesolvedforcrs c1= (2.120)Sincethedensityisconstantacrossthewave,thiscriticalsonicconditioncorrespondstoafrontofconstantvolumeexplosionpropagatingthroughthemediumwithapressureincreasegivenby p1=pCV (2.121)Atthiscriticalsoniccondition,the”owvelocitybeingzerorelativetothewallattheexitofthewavemeansthatfrictionnolongerin”uencesthe”ow 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