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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.
May8,2013 Markovchick ASEN6367 Schoomaker Snow
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
May8,2013 Markovchick ASEN6367 Schoomaker Snow
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
May8,2013 Markovchick ASEN6367 Schoomaker Snow
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
May8,2013 Markovchick ASEN6367 Schoomaker Snow
AsitcanbeseeninthefigureabovethedefinitionforelementisbasedonfirsttakingthecornernodesintheCCWdirectionandfollowedbythemid‐sidenodesstartingwiththefirstnodesCCWmid‐sidenode.
AfirstcutatthisproblemcamewithonlymodelingtheouterG11andthefuelgraininordertoestablishabaselineproblem.ThisformulationwasdoneusingthefollowingMathematicascripttodevelopthemesh.Afterthisisdonesuccessfullytheboundaryconditionsareappliedandthentheoverallsystemissolvedforasawhole.
Theformulationofthemeshisbrokeninto3divisionsshowninFigure9.Eachsectionofmeshisgeneratedandthenstitchedtogethermanuallybyalteringelementnodedefinitionsforthecontactingelements.Thisisbeforethenozzleisthenaddedusingthesameprocedureasdescribed.
Figure9Meshdivisionsandinterfacezones
Forthesoftwareportionofthisprojectthemainmoduleisbrokenupbelowforpresentationpurposesbasedontheapplicationofeachportion.
Theproblemdefinitionisgeneratedinthispreliminaryportion,wherevariousmaterialpropertiesaredefined.Foreachsectiontworadiiaredefinedaswellastheycoordinateofthebeginningandtheycoordinateoftheendheight.Morezonesaredefinedintheactualcode,butonlyafewareshownhereforbrevity.
May8,2013 Markovchick ASEN6367 Schoomaker Snow
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
May8,2013 Markovchick ASEN6367 Schoomaker Snow
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.
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