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JournalofEngineeringMathematics4457–82,2002.©2002KluwerAcademicPublishers.PrintedintheNetherlands.ModellinggasmotioninarapidcompressionmachineM.G.MEERE1,B.GLEESON1andJ.M.SIMMIE2DepartmentofMathematicalPhysics,NUI,Galway,Ireland2DepartmentofChemistry,NUI,Galway,IrelandReceived25July2001acceptedinrevisedform8May2002Abstract.Inthispaper,amodelwhichdescribesthebehaviourofthepressure,densityandtemperatureofagasmixtureinarapidcompressionmachineisdevelopedandanalyzed.Themodelconsistsofacoupledsystemofnonlinearpartialdifferentialequations,andbothformalasymptoticandnumericalsolutionsarepresented.Usingasymptotictechniques,asimplediscretealgorithmwhichtracksthetimeevolutionofthepressure,temperatureanddensityofthegasinthechambercoreisderived.Theresultswhichthisalgorithmpredictareingoodagreementwithexperimentaldata.Keywordsgasdynamics,rapidcompressionmachines,shockwaves,singularperturbationtheory1.Introduction1.1.RAPIDCOMPRESSIONMACHINESArapidcompressionmachineisadeviceusedtostudytheautoignitionofgasmixturesathighpressuresandtemperatures,withparticularreferencetoautoignitionininternalcombustionenginessee1–3.Atypicalcombustionengineisaverydirtyandcomplexenvironment,andthishaspromptedthedevelopmentofrapidcompressionmachineswhichenablethescientificstudyofcompressionandignitioninenginesinacleanerandsimplersetting.InFigure1weschematicallyillustrateatwopistonrapidcompressionmachine,suchastheoneinthedepartmentofChemistryatNUI,Galway.However,singlepistonmachines,withapistonatoneendandastationarysolidwallattheother,aremoretypical.Theanalysisdevelopedinthispaperisappropriatetobothsingleandtwopistonmachines.Theoperationofarapidcompressionmachineisverysimplethepistonsaresimultaneouslydriveninpneumatically,compressingtheenclosedgasmixture,therebycausingthegaspressure,temperatureanddensitytorisequickly.InFigures1a,1band1cweschematicallyrepresentarapidcompressionmachinepriorto,during,andaftercompression,respectively.TheratioofthefinalvolumetotheinitialvolumeofthecompressionchamberforthemachineatNUI,Galwayisabout112,thisvaluebeingtypicalofothermachines.Attheendofthecompression,thegasmixturewilltypicallyhavebeenpushedintoapressureandtemperatureregimewhereautoignitioncanoccur.InFigure2,wedisplayanexperimentalpressureprofileforaH2/O2/N2/ArmixturewhichhasbeentakenfromBrettetal.4,withthekindpermissionoftheauthors.Inthisgraph,thetimet0correspondstotheendofthecompressiontime.Wenotethat,forthegreaterpartofthecompression,thepressureinthechamberisrisingquitegently,butthattowardstheendofthecompressionthatis,justbeforet0,thereisasteepriseinthepressure.Aftercompression,thepressureprofilelevelsoffasexpectedtheextremelysteepriseattheendof58M.G.Meereetal.Figure1.Schematicillustratingtheoperationofarapidcompressionmachinewehaveshowntheconfigurationapriortocompression,bduringcompressionandcaftercompression.Figure2.AnexperimentalpressureprofileforagasmixtureH2/O2/N2/Ar2/1/2/3,asmeasuredintherapidcompressionmachineatNUI,Galway.Itistakenfrom4,andhasaninitialpressureof005MPaandaninitialtemperatureof344K.Modellinggasmotioninarapidcompressionmachine59theprofilecorrespondstotheignitionofthemixture.WenotethatthecompressiontimeandthetimedelaytoignitionaftercompressionarebothO10ms.Thepressurehistoryistheonlyquantitywhichismeasuredinexperiments.However,thetemperatureinthecoreaftercompressionisthequantitywhichisofprimaryinteresttochemistssincereactionratesdependmainlyontemperatureforalmostallsystems,althoughtheremayalsobesomeweakerpressuredependence.Measuringtemperatureaccuratelyinthecorecanbeproblematicbecauseofthepresenceofathermalboundarylayerseethecommentsbelowonrollupvortices.However,withtheexperimentalpressuredatainhand,thecorrespondingtemperaturescanbeestimatedusingtheisentropicrelationlnp/piintegraldisplayTTiγssγs−1ds,1whereTi,piaretheinitialvaluesforthecoretemperatureandpressure,T,parethesequantitiesatsomelatertime,andγsisthespecificheatratioattemperatures.Inexperiments,theinitialcoretemperatureistypicallyO300K,whilethecoretemperatureaftercompressionisusuallyO1000K.Inthispaper,weshallconsideronlythebehaviourofthegasmixtureduringcompressionthepostcompressionbehaviourisnotconsideredhere,butthiswillformthesubjectforfuturework.Nevertheless,themodelpresentedheredoesprovideareasonabledescriptionofthepostcompressionbehaviourofasinglespeciespuregas,oraninertgasmixtureseeSection3.5.1.2.THEMODELWesupposethatthecompressionchamberislocatedalong00.Thisassumptionisactuallyquiteastrongoneinthiscontextsincehigherdimensionaleffectsarefrequentlyobservedinexperiments,rollupvorticesnearthecornerregionsdefinedbythepistonheadsandthechamberwallbeingparticularlynoteworthysee,forexample,5.Thesevorticesariseduetothescrapingbythepistonsofthethermalboundarylayeratthechamberwall,andtheycan,andfrequentlydo,disturbthegasmotioninthecoreofthecompressionchamber.However,thejustificationfortheonedimensionalmodelstudiedhereistwofoldithecornervorticescanbesuccessfullysuppressedbyintroducingcrevicesatthepistonheadswhichswallowthethermalboundarylayerasthepistonsmoveinseeLee6,renderingthegasmotionawayfromthechamberwallsonedimensionaltoagoodapproximation,and,iithestudyoftheonedimensionalmodelprovidesausefulpreliminarytothestudyofhigherdimensionalmodels.Wenowgivethegoverningequationsforouronedimensionalmodel.Areasonablycompletederivationofthegoverningequationsforamulticomponentreactinggascanbefoundintheappendicesof7thesestandardderivationsarenotreproducedhere.Themodelwhichweshallstudyincludesanumberofsimplifyingassumptionsandthesewillbeclearlystatedastheyarise.Theequationexpressingconservationofmassisgivenby∂ρ∂t∂∂xρv0,60M.G.Meereetal.whereρρx,tandvvx,tarethedensityandthevelocityofthegas,respectively,atlocationxandtimet.Itshouldbeemphasizedthatthesequantitiesrefertoagasmixture,sothatifthereareNdifferentspeciesinthemixturethenρNsummationdisplayi1ρi,whereρiρix,tisthedensityofspeciesi.Also,thevelocityvabovereferstothemassaveragedvelocityofthemixture,thatis,vNsummationdisplayi1Yivi,whereYiρi/ρandvivix,tarethemassfractionandvelocity,respectively,ofspeciesisee7.Neglectingbodyforcesandviscouseffects,theequationexpressingconservationofmomentumisgivenby∂v∂tv∂v∂x−1ρ∂p∂x,whereppx,tisthepressure.Weassumethatthegasmixtureisideal,sothattheequationofstateisgivenbypRMρT,2whereTTx,tisthetemperature,Ristheuniversalgasconstant8314JK−1mol−1,andMistheaveragemolecularmassofthemixture.ThislastquantityisgivenbyMNsummationdisplayi1niWimA,whereniandWigivethenumberfractionandmolecularweight,respectively,ofspeciesi,mistheatomicmassunit166110−27kgandAisAvogadrosnumber60221023moleculesmol−1.Theequationexpressingconservationofenergyisgivenbysee7or8ρparenleftbigg∂u∂tv∂u∂xparenrightbigg−Mparenleftbigg∂q∂xp∂v∂xparenrightbigg,3whereuux,tistheinternalenergyofthegasmixtureandqqx,tistheheatflux.WealsohavethethermodynamicidentityuNsummationdisplayi1hiYi−Mp/ρ,4wheretheenthalpieshihiTaregivenbyhiThiT0integraldisplayTT0cp,isds,i1,2,...,N,5
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