May8,2013 Markovchick ASEN6367 Schoomaker Snow

FiniteElementAnalysisofanAxisymmetricHybridRocketCombustionChamber

Markovchick

Schoomaker

Snow

May8th,2013

May8,2013 Markovchick ASEN6367 Schoomaker Snow

1.ObjectiveThepurposeofthisFEMprojectwastoassistwithfurtheranalysisofCUGraduateProjectHySoR(HybridSoundingRocket).Theprojectcurrentlyisinthetestingiterationasitmovestowardsalaunchreadyvehicle.Manyoftheissuesassociatedwiththeprogressionoftheprojectdealwithmanythermaluncertaintiesthatarepresentinthecombustionchamber.Thisprojectwasdesignedinordertoprovideadditionalinsightintothethermalelementsthatarepresentintheoverallrocketsystem.

2.BackgroundandProblemDescriptionHySoRisahybridsoundingrocketdesignedtotakea2kgpayloadtoanaltitudeof10km.AsystemdiagramisshowninFigure1below.Theprojecthasbeengoingonforabout3yearswithmanydesigniterations.Itemsthataretoodifficultorcostlytoanalyzearetested.Unfortunately,testingwillshowwhetherornotthecomponentfailed,buttheactualcausecanbesomewhatconvoluted.

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Figure 1: Rocket Structural Layout Severalissueshavebeenfoundduringtestingofthehybridrocket.Atstatictestfire2,aburnthroughofthecombustionchamberoccurred.Thiscausedtherockettosplitinhalf.Duetoapoorthermalprotectiondesign,theheatcausedmeltinginthecombustionchamberwall,ultimatelyresultinginachamberbreach.Atstatictestfire5,thenozzleandnozzleretainerpoppedoffabout8secondsintotheburn.Thisisatimeafterthepeakpressure,soonewouldthinkthatithadalreadysuccessfullysurvivedthroughatimeperiodofgreateststructuralstress.Thisistrueunlessanotherfactorisatworksuchasthermalexpansion.Thermalexpansion’seffecton

Oxidizer Plumbing

Fuel and Combustion Chamber

Payload Ring Top Tank Ring

Longeron

Bottom Tank Ring

U-Channel

Injector Ring

Load Cell Bracket

Nozzle Ring

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thenozzleretainerwasnotanalyzedinprevioussemestersandappearstobetheleadingculpritinthefailureastheworstcasechamberpressurewouldhavehadtobemultipliedbyafactoroffivetocauseanozzlefailureifitactedalone.However,withthehightemperaturesofthecombustionchamberandrelativelylowtemperatureoftheexteriorcomponents,athermalexpansionmismatchisamplifiedandlargestressescanoccur.At8seconds,thenozzleandnozzleretainerarestillheatingupbeforereachingsteadystate,sothelargestthermalstressisnoseenyet.Thisgivesmorecredibilitytothetheorythatthermalexpansioncausedthefailure.

Inordertocorrectlydesignforthethermalexpansion,thetemperaturevaluesofthepartsinquestionneedtobeknown.Thisisthemotivationfortheproblem.Thermocoupleswereplacedonthecombustionchamberandnozzleattachmentduringthetesttolearnabouttheirtemperaturevaluesandpossiblydetectasourceoffailure.Theirlocationsareshowninthefigurebelow.

Figure 2: Thermocouple Positions Thesetemperaturevalueswillbeusedasboundaryconditionsforthefiniteelementmodel.Likewise,knowingtheambienttemperature,theheatfluxtoambientairviaconvectionisestimated.Withthesetwovalues,thecombustionchambertemperaturecanbeestimatedalongwiththematerialin‐between.Combustionchambertemperatureisalsoimportantforpredictingrocketthrust.Mostfirstorderpropulsionestimatesaremadeusingisentropicrelations.Theserelations

May8,2013 Markovchick ASEN6367 Schoomaker Snow

coupletemperatureandpressuretohelpdeterminenominaloperationvaluesatdifferentareasoftherocketandalsothrust.Nowthatsufficientmotivationhasbeenestablishedtheproblemcanbefullydevelopedandmodeled.

3.PreliminaryModelFormulations

SingleDimensionModelFormulationA1‐Dmodelwascreatedusingthebasicdimensionsofthecombustionchamber,simplifyingeverypartintoatube.Thecombustionchamberwasthenseparatedintozoneswheretherewasonlyasinglematerialatagivenradius,sothateachzonecouldbeconstructedfromsimpletubesofdifferentmaterialsthatwereallthesamelength.Thethermalresistancewasthencalculatedforeachmaterialineachzoneaccordingtotheequationforanannulartubegivenbelow.

Thethermocoupledatafromtestfirenumberfivewasthenenteredasaconstantexternaltemperatureforeachcorrespondingapplicablezone.Theheatfluxcalculatedfromeachthermocouplewasalsomultipliedbytheexternalchamberareaandappliedasthetotalheatflowinthezone.Mid‐layerandinternaltemperatureswerethencalculatedaccordingtotheequationbelow.

Figure3

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Theresultsofthe1‐DmodelareshowninFigure3.Knowninputsaregiveninblackandoutputsinred.AlltemperaturesareindegreesC.Ambienttemperatureforthethermocoupledatawas5°C.

Thismodelislimitedbytheassumptionthateachzoneisdiscontinuousandtemperatureeffectsalongthelengthofthecombustionchamberarelargelyignored.Thereforeeachzonetemperatureshouldbeconsideredamaximumasthethermocoupletemperaturesusedwerethemaximumavailableinaparticularzone,especiallysincecontactresistancewasignored.Also,thecomplexgeometryofthenozzledefiesanyconvenientanalyticalthermalresistanceformulation.Thelastmajorlimitationofthismodelistheexclusionoftimeeffects.Thethermocoupledatawherethepeaktemperaturecamefromwasbothtransientandduringafailedrun.

The1‐Dpredictivemodelfoundthatthehighesttemperaturesattheinteriorwallofthefuelpelletcouldbeashighas1600°C.Thisismuchlessthanthe3000°Cpredictedbyanearlierchemicalmodel.Thiscouldbeduetomanyeffects,buttheparalysisofthefuelpellet,thesimultaneousvaporizationandchemicaldecompositionpriortocombustion,istheprimarysuspecttransportingthermalenergyoutofthesystem.Theconvectiveeffectsofthecombustionchamberflowweretoocomplextomodelbutalsoplayapartinthelowerthanexpectedtemperature.Thelargedifferencebetweenthetwointernaltemperaturesleadustobelievethatsettingtheinternalboundaryconditiontothe3000°Cofthechemicalmodelwouldbeinaccurate.Instead,theinternaltemperaturesduringthetestwererecreatedusingthethermocoupledataavailableastheexternalboundaryconditions.

Otherinsightsgleanedfromthe1‐Dmodelwerethatthestructureofthecombustionchamberdidnotgethotenoughtowarranttemperaturedependantmaterialsproperties.Thiswouldhavemeantperformingmultiplerunsuntilthetemperaturesfoundonaveragematchedthematerialspropertiesused.Thenozzletemperaturespredicted,64°Catthethroat,aregrosslyincorrect.Thisonlyhighlightsthelargevariationsinflowandthermalpropertiesacrossthenozzleaswellasthefactthatthenozzleisloosingmaterialatanunknownrate,sincethenozzlefromthelasttestfiringwasnotabletoberecovered.

May8,2013 Markovchick ASEN6367 Schoomaker Snow

ANSYSModelAnANSYSmodelwasalsocreatedthatmodeledthecentralsectionofthefuelpellet.Thisanalysiswasconservative,ignoringtheeffectsofnaturalconvection,contactresistance,andtheheatofablationofthefuelpellet.Twotimestheinternalconvectioncoefficientexpectedwasused.Thetemperaturesfoundusingthismodel,includinganinternalmaximumtemperatureofaround1000°CasshowninFigure4,wereinlinewiththeresultsoftheaxisymmetricmodel.

Figure 4: Thermocouple Positions

4.FEMModeling

ThermalSet‐UpInordertosolvetheproblem,Dr.Felippa’selastostaticaxisymmetricprogramsareusedasabaseline.However,therearesubtledifferencesbetweentheelastostaticandthermalproblems.ThisbecomesevidentwhenTontidiagramsofthegoverningequationsarelookedatinthefollowFigures5and6below.

May8,2013 Markovchick ASEN6367 Schoomaker Snow

Figures5and6­TontiDiagrams

BylookingattheTontiDiagrams,thedatafieldsareshowntobeinthesamelocationsinbothdiagrams.UsingtheelastostaticaxisymmetricprobleminputsandtheTontiDiagrams,thethermalinputsarefound.Heatgenerationhisusedwherebodyforcesbareusedintheelastostaticproblem.Heatfluxqisusedwheresurfacetractionstareusedintheelastostaticproblem.FinaltemperaturesTareusedwheredisplacementsuareusedintheelastostaticproblem.Insteadofanelasticity

May8,2013 Markovchick ASEN6367 Schoomaker Snow

matrixE,aconductivitymatrixkisused.Theformationoftheconductivitymatrixisessentiallythesameastheelasticitymatrix.Theelement’sthermalconductivityisusedinsteadofYoung’sModulus.ThermalproblemsdonothaveaPoisonRatioanalogso0isusedforthatvalue.IfthermalproblemsdidhaveaPoisonRatio,thenatemperaturedifferentialinonedirectionwouldinduceafluxinanorthogonaldirectionwhichdoesnotoccurinheattransfer.

Thefinalpartofthemodelthatneedstoberemediedbetweenthethermalandtheelastostaticproblemisthatelastostaticproblemscanhavedisplacementsin3directionswhereasatemperatureisascalarandneedstobethesameatthenoderegardlessofwhichdirection(s)heatfluxisgoing.Inordertoconstraintheprobleminthismanner,multifreedomconstraintsareinstated.Basedonhowtheproblemiscurrentlyconstructed,therewillbearadialtemperatureTrandaverticaltemperatureTzjustlikearadialdisplacementurandaverticaldisplacementuz.Theradialtemperaturemustequaltheverticaltemperature.Toaddthisconstraint,rowsandcolumnsareaddedtothestiffnessmatrixsuchthatk*(Tr‐Tz)=0.Apositiveandnegative1areaddedasextrarowsinthestiffnessmatrixforeachnodeinthemesh.Tokeepthestiffnessmatrixsymmetric,apositiveandnegative1areaddedasextracolumns.Inordertomakethecalculationsstillworkoutandnotdisturbtheentiremodulezerosareaddedtotheendofthetemperaturevectorforeachcolumnaddedtothestiffnessmatrix(thetotalnumberofzerosisequaltothetotalnumberofnodes).Likewise,zerosareaddedtotheendoftheheatfluxvectorforeachrowaddedtothestiffnessmatrix(thetotalnumberofzerosisequaltothetotalnumberofnodes).Thismethodislikeahybridbetweenthemaster‐slaveandLagrangeMultipliermethodsofaddingmultifreedomconstraints.Moreworkcanlikelybeputinheretohelpoptimizeaddingtheseconstraintstothethermalproblem.AsamplematrixofimplementationisshowninFigure7.

Figure7­SampleMatrixImplementation

Nowtheproblemcanbesolvedjustlikeanyelastostaticproblembycreatingthemesh,definingthematerialproperties(thermalconductivity)andaddingtheboundaryconditions.

May8,2013 Markovchick ASEN6367 Schoomaker Snow

SoftwareFormulationForthisprojectmanyoftheelementsusedinthesoftwareformulationweretakenfromthecoursematerialprovidedbyDr.Felippa.ManyofthemoduleswereadaptedtofittheneedsoftheHySoRmodelandproducethedesiredresults.

ForthesoftwarecomponentsMathematicawasused.Similarproblemsofthisnaturehadbeenperformedwithmanyofthemodelsforstructuralproblemspreviously.Asdiscussedpreviously,theformulationofthemodelchangedtheapplicationofmaterialtypesandthewayboundaryconditionsweretreatedcomparedwithastructuralproblem.

Nowinsteadofapplyingforcesandfixingdisplacements,nodeswereheldbytemperatureandheatfluxesasboundaryconditions.Thisrequiredanunderstandingoftheoverallrocketsysteminordertoapplytheappropriateboundaryconditions.Thelargestunknownthatwasposedanissuetothisprojectwasdefiningtheexactvaluefortheheatfluxtobeappliedtotheinternalnodesofthecombustionchamber.Theoreticallythefluxisafunctionofconvectionaswellastheheatproducedbythechemicalreaction.Afterthefluxwasdeterminedforsolvingtheproblemtheset‐upoftheproblemcouldtakeplace.

Forthisprojectthecombustionchamberwasassumedtobeaxi‐symmetric.Thisassumptionneglectedthecross‐portdesignforthefuel‐grain,butshouldnothaveanysignificanteffectsontheoverallanalysis.Thechallengefordevelopingthemeshforthisproblemcamefromthevariousmaterialtypespresentfortheelements.Additionallytherewereacoupledifferentlayerstothemeshofvariousthicknessesthatneededtobemanuallymeshedtogethermeaningnodalnumberswerenotalwaysinorder.

Tostartthismesh8nodeelementswereusedinordertoestablishthemesh.AsimpleschematicoftheelementandthemethodforcallingandestablishingthenodecoordinatesandelementsisshownbelowinFigure8.

Figure8­Nodesandelementdefinition

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AsitcanbeseeninthefigureabovethedefinitionforelementisbasedonfirsttakingthecornernodesintheCCWdirectionandfollowedbythemid‐sidenodesstartingwiththefirstnodesCCWmid‐sidenode.

AfirstcutatthisproblemcamewithonlymodelingtheouterG11andthefuelgraininordertoestablishabaselineproblem.ThisformulationwasdoneusingthefollowingMathematicascripttodevelopthemesh.Afterthisisdonesuccessfullytheboundaryconditionsareappliedandthentheoverallsystemissolvedforasawhole.

Theformulationofthemeshisbrokeninto3divisionsshowninFigure9.Eachsectionofmeshisgeneratedandthenstitchedtogethermanuallybyalteringelementnodedefinitionsforthecontactingelements.Thisisbeforethenozzleisthenaddedusingthesameprocedureasdescribed.

Figure9­Meshdivisionsandinterfacezones

Forthesoftwareportionofthisprojectthemainmoduleisbrokenupbelowforpresentationpurposesbasedontheapplicationofeachportion.

Theproblemdefinitionisgeneratedinthispreliminaryportion,wherevariousmaterialpropertiesaredefined.Foreachsectiontworadiiaredefinedaswellastheycoordinateofthebeginningandtheycoordinateoftheendheight.Morezonesaredefinedintheactualcode,butonlyafewareshownhereforbrevity.

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Figure10­Problemdefinition

Thenodecoordinatesaregeneratedandthentheelementnodesforthesubsequentelementsarecreated.ThemoduleswerewrittenbytheprojectteamtoworkspecificallyfortheHySoRmodel.Thesetakeinspecificnodeinputstogeneratetheelementnodes.

Figure11­Nodesandelementdefinition

May8,2013 Markovchick ASEN6367 Schoomaker Snow

Afterthenodesandelementsaredefinedthematerialtypesneedtobeassigned.Sincemultiplematerialtypesareused,theorderoftheelementtypescorrespondstothematerialtype.Thismakesforassigningmaterialtypesstraightforward.ThisisshowninFigure12.

Figure12­Nodesandelementdefinition

Afterthesestepsaremade,manualnodedefinitionsareassignedtotheelementsthatarealignedbetweenzones.Thentheringanalysisdriverisusedinordertoperformthenecessarycalculationsandanalysisfortheproject.WhenthesestepsaretakenrunningthecodeHySoRthermalresultsaregenerated.Theywillbediscussedinthefollowingsection.

May8,2013 Markovchick ASEN6367 Schoomaker Snow

5.ResultsandDiscussion

Asmentionedpreviouslyinthisreport,heatpathsandtheoverallthermalperformanceofthesystemisgreatimportanceforthisproject.Manyofthedesignedelementsinthisprojectarederivedfromthermalrequirements.Withabetterunderstandingoftheoverallthermalperformanceoftherocketwilltranslatetoabetterrocketdesignforfuturesemesters.TogetabigpictureofthethermalresultstheycanbeseeninFigure13.

Thiswasgeneratedtogiveanoverallideaofthetotalheatpresentintherocketstructure.Oneareathatwasofspecificinterestwasthenozzlelocatedatthebottomofthefigure.Itwasnotedduringthetestthatwhilethenozzleareadidgethotterduringthetestitwasn'tanythingoutoftheordinaryoranythingthatwouldmakeonesuspectthatthealuminumwouldyieldthewaythatitdid.Soitwasinterestingtoseewhenrunningthesimulationsthatchangingtheparametersontheinsideofthenozzle(mainlyalteringtheflux)eventoveryhighvaluesthatmorethanlikelywouldnotbeseenatatestdidnotcausesignificantheatspreadingtotheoutersurfaceofthenozzle.Thisbacksuptheexperimentalresultsthatweregathered.Theinitialfailurereasoningofthenozzlewasthermalweakening,butwiththisanalysisitledtheteaminanotherdirection.

Takingthesupposedtemperatureofthenozzlethethermalexpansionofthegraphitenozzlewasanalyzed.Withthisthermalexpansionthegraphitewasseentogrowbyasmallamount,butdefinitelynoticeablechangeinsize.Thiswouldthenneedtobeimplementedintoananalysisforfuturework.DuetolackoftimethisnewanalysiswasdoneinANSYSwithdisplacementsusedforBC'sinthemodel.TheanalysisissummarizedinFigure14shown.Itcanbeseenwithayield

strengthof40,000psiforaluminumthatthecurrentnozzleretainerwillfail.

Figure13

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Figure14­Nozzleretainerfailure

Thetemperaturevariationfromthefuelgraintotheouterwallisofimportance.BasedontheresultsfromtheFEMsoftwarethetemperaturevariesfromtheinnerwalltotheouterwallasshowninFigure15below.

Figure15­Nodesandelementdefinition

Asitcanbeseenthetemperaturedistributionisnotasextremeasoncepredictedinpreviousmodels.Theuseofthismodelisimportantinbackingupthedesignworkthathadbeendoneinprevioussemesters.

Forfutureworkthethermalsystemisdeemedadequatebesidethefindingsforthenozzleretainer.Theuseofthecurrentcomponentsissufficientforasuccessfulrockettest.Thismodelwasextremelyusefulinhelpingtodeterminingproblemareasandanunderstandingofthethermalbehavioroftherocket.Futureworkcouldbepossibleonthismodeltomakeitevenmoresophisticated.

May8,2013 Markovchick ASEN6367 Schoomaker Snow

6.RecommendationsandFutureWork

Allthreeanalysiswereinconcurrencethattheinternalcombustionchambertemperatureswerelowerthantheexpected3000°C.Thiswarrantslookingbackonthecombustionchemistryandexpectedflowpropertiesinsidethecombustionchamberanddevelopingnewfiguresmoreinlinewiththeresultsseen.Thesenewflowpropertieswillbecriticaltotheredesignofthenozzlethatisrequiredafterthefailureexperienceduringstatictestfire5.ThethermalresultsofthisanalysiscouldalsobeusedtomoreaccuratelydefinethestressescausedbytheCTEmismatchbetweenthegraphitenozzleandaluminumretainer.

Thisanalysiscouldbeextendedandmademorecertaininseveraldifferentways.First,oncemoreaccurateflowconditionsinsidethechamberareestimated,theboundaryconditionscanbechangedtoforcedconvectioninsidethecombustionchamberandnaturalconvectionoutsideofit.Amodelsuchasthiswouldprovidestrongevidenceforthevalidityoftheflowconditionsused.Thisproblemisextremelydifficulthowever,sinceboththeflowpropertiesandcombustionconditionsdonothavesimpleanalyticallyderivedcharacteristics.Thefiniteelementmodelcouldbefurtherrefinedbyreducingelementsize,includingcontactresistance,andincludingsmallerfeaturessuchasepoxylayersthatwerepreviouslysimplifiedoutoftheproblem.

Second,theprogramcouldcombinethermalandmechanicalanalysis;iterativelysolvingfortemperatures,displacements,andstresses.Thiscouldprovidemoreinsightintothenozzleretainerfailure,sincethermaleffectsaretheprinciplesuspectofthefailure.Mathematicallybothproblemsarewelldefined,yettheMathematicaprogramusedfortheaxisymmetricanalysiswouldrequiresomemodificationtointegratetwodifferentprimaryvariables.

Lastly,aftertheprevioustwocomponentshavebeencompleted,theanalysiscouldbemadetimedependant.Thiswillgiveaclearerpictureofhowthefuelpelletburnsduringthecourseofalaunchandhowthemaximumtemperatureinthecombustionchamberchangesandmigrates.Asimplewaytomodelthefuelpelletburningcouldbeanelementdeletionmethod,afteranelementoffuelhasabsorbedasetamountofenergy,itisdeletedfromthemeshrepresentingtheburningawayofthatpieceoffuel.Thisanalysiswillalsoprovideinformationonwhattoexpectthermallyandmechanicallyneartheendofthefuelburn,whentheburningfuelandcompositecasinghavenearlynoseparation.Previousteststhathavereachedthispointhaveseencrackinginthecompositecasingwherethealuminumcollarsextendoverthecasing.