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JOURNALOFENGINEERINGMATHEMATICS4457–82,20022002KLUWERACADEMICPUBLISHERSPRINTEDINTHENETHERLANDSMODELLINGGASMOTIONINARAPIDCOMPRESSIONMACHINEMGMEERE1,BGLEESON1ANDJMSIMMIE2DEPARTMENTOFMATHEMATICALPHYSICS,NUI,GALWAY,IRELAND2DEPARTMENTOFCHEMISTRY,NUI,GALWAY,IRELANDRECEIVED25JULY2001;ACCEPTEDINREVISEDFORM8MAY2002ABSTRACTINTHISPAPER,AMODELWHICHDESCRIBESTHEBEHAVIOUROFTHEPRESSURE,DENSITYANDTEMPERATUREOFAGASMIXTUREINARAPIDCOMPRESSIONMACHINEISDEVELOPEDANDANALYZEDTHEMODELCONSISTSOFACOUPLEDSYSTEMOFNONLINEARPARTIALDIFFERENTIALEQUATIONS,ANDBOTHFORMALASYMPTOTICANDNUMERICALSOLUTIONSAREPRESENTEDUSINGASYMPTOTICTECHNIQUES,ASIMPLEDISCRETEALGORITHMWHICHTRACKSTHETIMEEVOLUTIONOFTHEPRESSURE,TEMPERATUREANDDENSITYOFTHEGASINTHECHAMBERCOREISDERIVEDTHERESULTSWHICHTHISALGORITHMPREDICTAREINGOODAGREEMENTWITHEXPERIMENTALDATAKEYWORDSGASDYNAMICS,RAPIDCOMPRESSIONMACHINES,SHOCKWAVES,SINGULARPERTURBATIONTHEORY1INTRODUCTION11RAPIDCOMPRESSIONMACHINESARAPIDCOMPRESSIONMACHINEISADEVICEUSEDTOSTUDYTHEAUTOIGNITIONOFGASMIXTURESATHIGHPRESSURESANDTEMPERATURES,WITHPARTICULARREFERENCETOAUTOIGNITIONININTERNALCOMBUSTIONENGINES;SEE1–3ATYPICALCOMBUSTIONENGINEISAVERYDIRTYANDCOMPLEXENVIRONMENT,ANDTHISHASPROMPTEDTHEDEVELOPMENTOFRAPIDCOMPRESSIONMACHINESWHICHENABLETHESCIENTIFICSTUDYOFCOMPRESSIONANDIGNITIONINENGINESINACLEANERANDSIMPLERSETTINGINFIGURE1WESCHEMATICALLYILLUSTRATEATWOPISTONRAPIDCOMPRESSIONMACHINE,SUCHASTHEONEINTHEDEPARTMENTOFCHEMISTRYATNUI,GALWAYHOWEVER,SINGLEPISTONMACHINES,WITHAPISTONATONEENDANDASTATIONARYSOLIDWALLATTHEOTHER,AREMORETYPICALTHEANALYSISDEVELOPEDINTHISPAPERISAPPROPRIATETOBOTHSINGLEANDTWOPISTONMACHINESTHEOPERATIONOFARAPIDCOMPRESSIONMACHINEISVERYSIMPLETHEPISTONSARESIMULTANEOUSLYDRIVENINPNEUMATICALLY,COMPRESSINGTHEENCLOSEDGASMIXTURE,THEREBYCAUSINGTHEGASPRESSURE,TEMPERATUREANDDENSITYTORISEQUICKLYINFIGURES1A,1BAND1CWESCHEMATICALLYREPRESENTARAPIDCOMPRESSIONMACHINEPRIORTO,DURING,ANDAFTERCOMPRESSION,RESPECTIVELYTHERATIOOFTHEFINALVOLUMETOTHEINITIALVOLUMEOFTHECOMPRESSIONCHAMBERFORTHEMACHINEATNUI,GALWAYISABOUT112,THISVALUEBEINGTYPICALOFOTHERMACHINESATTHEENDOFTHECOMPRESSION,THEGASMIXTUREWILLTYPICALLYHAVEBEENPUSHEDINTOAPRESSUREANDTEMPERATUREREGIMEWHEREAUTOIGNITIONCANOCCURINFIGURE2,WEDISPLAYANEXPERIMENTALPRESSUREPROFILEFORAH2/O2/N2/ARMIXTUREWHICHHASBEENTAKENFROMBRETTETAL4,WITHTHEKINDPERMISSIONOFTHEAUTHORSINTHISGRAPH,THETIMET0CORRESPONDSTOTHEENDOFTHECOMPRESSIONTIMEWENOTETHAT,FORTHEGREATERPARTOFTHECOMPRESSION,THEPRESSUREINTHECHAMBERISRISINGQUITEGENTLY,BUTTHATTOWARDSTHEENDOFTHECOMPRESSIONTHATIS,JUSTBEFORET0,THEREISASTEEPRISEINTHEPRESSUREAFTERCOMPRESSION,THEPRESSUREPROFILELEVELSOFFASEXPECTED;THEEXTREMELYSTEEPRISEATTHEENDOF58MGMEEREETALFIGURE1SCHEMATICILLUSTRATINGTHEOPERATIONOFARAPIDCOMPRESSIONMACHINE;WEHAVESHOWNTHECONFIGURATIONAPRIORTOCOMPRESSION,BDURINGCOMPRESSIONANDCAFTERCOMPRESSIONFIGURE2ANEXPERIMENTALPRESSUREPROFILEFORAGASMIXTUREH2/O2/N2/AR2/1/2/3,ASMEASUREDINTHERAPIDCOMPRESSIONMACHINEATNUI,GALWAYITISTAKENFROM4,ANDHASANINITIALPRESSUREOF005MPAANDANINITIALTEMPERATUREOF344KMODELLINGGASMOTIONINARAPIDCOMPRESSIONMACHINE59THEPROFILECORRESPONDSTOTHEIGNITIONOFTHEMIXTUREWENOTETHATTHECOMPRESSIONTIMEANDTHETIMEDELAYTOIGNITIONAFTERCOMPRESSIONAREBOTHO10MSTHEPRESSUREHISTORYISTHEONLYQUANTITYWHICHISMEASUREDINEXPERIMENTSHOWEVER,THETEMPERATUREINTHECOREAFTERCOMPRESSIONISTHEQUANTITYWHICHISOFPRIMARYINTERESTTOCHEMISTSSINCEREACTIONRATESDEPENDMAINLYONTEMPERATUREFORALMOSTALLSYSTEMS,ALTHOUGHTHEREMAYALSOBESOMEWEAKERPRESSUREDEPENDENCEMEASURINGTEMPERATUREACCURATELYINTHECORECANBEPROBLEMATICBECAUSEOFTHEPRESENCEOFATHERMALBOUNDARYLAYER;SEETHECOMMENTSBELOWONROLLUPVORTICESHOWEVER,WITHTHEEXPERIMENTALPRESSUREDATAINHAND,THECORRESPONDINGTEMPERATURESCANBEESTIMATEDUSINGTHEISENTROPICRELATIONLNP/PIINTEGRALDISPLAYTTIΓSSΓS−1DS,1WHERETI,PIARETHEINITIALVALUESFORTHECORETEMPERATUREANDPRESSURE,T,PARETHESEQUANTITIESATSOMELATERTIME,ANDΓSISTHESPECIFICHEATRATIOATTEMPERATURESINEXPERIMENTS,THEINITIALCORETEMPERATUREISTYPICALLYO300K,WHILETHECORETEMPERATUREAFTERCOMPRESSIONISUSUALLYO1000KINTHISPAPER,WESHALLCONSIDERONLYTHEBEHAVIOUROFTHEGASMIXTUREDURINGCOMPRESSION;THEPOSTCOMPRESSIONBEHAVIOURISNOTCONSIDEREDHERE,BUTTHISWILLFORMTHESUBJECTFORFUTUREWORKNEVERTHELESS,THEMODELPRESENTEDHEREDOESPROVIDEAREASONABLEDESCRIPTIONOFTHEPOSTCOMPRESSIONBEHAVIOUROFASINGLESPECIESPUREGAS,ORANINERTGASMIXTURE;SEESECTION3512THEMODELWESUPPOSETHATTHECOMPRESSIONCHAMBERISLOCATEDALONG0X2LATT0,WITHX0GIVINGTHEINITIALLOCATIONOFTHELEFTPISTONANDX2LBEINGTHEINITIALLOCATIONOFTHERIGHTPISTONITISASSUMEDTHROUGHOUTTHISPAPERTHATTHEGASMOTIONISONEDIMENSIONALSOTHATTHEFLOWVARIABLESDEPENDONLYONTHEPOSITIONXALONGTHECHAMBERANDTIMET0THISASSUMPTIONISACTUALLYQUITEASTRONGONEINTHISCONTEXTSINCEHIGHERDIMENSIONALEFFECTSAREFREQUENTLYOBSERVEDINEXPERIMENTS,ROLLUPVORTICESNEARTHECORNERREGIONSDEFINEDBYTHEPISTONHEADSANDTHECHAMBERWALLBEINGPARTICULARLYNOTEWORTHY;SEE,FOREXAMPLE,5THESEVORTICESARISEDUETOTHESCRAPINGBYTHEPISTONSOFTHETHERMALBOUNDARYLAYERATTHECHAMBERWALL,ANDTHEYCAN,ANDFREQUENTLYDO,DISTURBTHEGASMOTIONINTHECOREOFTHECOMPRESSIONCHAMBERHOWEVER,THEJUSTIFICATIONFORTHEONEDIMENSIONALMODE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