Proc.Natl.Acad.Sci.USAVol.78,No.4,pp.1986-1988,April1981AppliedPhysicalSciencesFinte-imthermodynamics:Egprfoacprovedbyoptimzedpistonmotin(Otocyle/optimzedheatngies/optimalcontrl)MICHAELOZURKEWICHANDR.SBERYDepartmntofChemistryandtheJamesFranckInstiute,TheUniversityhicgo,hicgo,Ilinois60637ContributedbyR.StephenBery,ember29,1980ABSTRCThemethodsoffinite-imthermodynamicsareusdtofindtheoptimaltimepathofanOtocyclwithrictonandatlekage.Optimltyisdeinebymaximzionheorkpeycle；hesysmisconstranedooperatefxedfrequency,sotmaximumpowr-iobtained.Theeulisnimprovemntofabout10%inthefctivenes(econd-laefcncy)ofconventonalar-snusoidalengine.Finite-mhermodynamisisnxtensionithemodynamicsrlvantiprncplacosstentrpaofthesubjct,fromthemosabstrleveltohemostapplied.Thepproahibadontheconsructiofgenealizedtmmicpotentils(1)forprocescontaningtimeorrateondionsamongtconsraintsonthesystm2)andonthedetrminatonofoptmalpahshayildtheextrmorespondingohosegenerlizedpotentals.Hretofor,workonfinite-imtrmodynamicshasconcentratdonratheridealizemodels(2-7)andonexistncheorms(2),alontheabstrideofhesubjec.Thisworkisintendedstpconneingtabstrcthermmiceptsthavemrgedinfnit-imhermodynamisihprial,engineerngsiofubject,hedesgnprincplesofaelmchine.Inthirport,wetramodeloftheinternalcombustionenginecloselyrlatedotheidealOocyclebutihconstraintsnformofhewomajorlsfoundinrlengines.Woptimztheenginebycontrollingtimedependenceofthevolume-thais,thepitonmoton.Asarult,ithoutundetakingatldengineringsudy,warbletoundersandhowtloseafecdheimepahofhepistonandtoetimtheimprovementineficncyobtinablbyoptmztpistmotion.THEMODLOurmodelisbaedonthestandardfour-stroketocycle.Thisconsistofanintkestroke,acompreontroke,powetoke,andnexhausstrokeHerwbrieflydescribehebasiftursofhismodelandmhodusdtfindheoptimalpistonmoton.Adetaledprentatiwilbegivenelswhere.Wsumhathecomprionatio,fuel-to-airtio,fuelconsumption,andperiodofthecyclealrefixed.Tseonstrntservetopurposes.Frt,heyreductheoptimztionproblmtofindinghepitmotion.Alo,theyguaanttharformancecrteanotconsiderednhisanalysisarecomparablethoseforrelngine.Relxingnyofhestrantscaonlyimprovetpeformncfurr.Wekehelosestobehealekageandricton.Bothoftheseartedependentandthusafcttimerponsoftsystm.Thealekisaumdtobeproportionaltoheinsntneoussurfaceofthecylinderandtohetemperaturediferncbetwentheorkingfluidandtwl(i.e.,Newtoninheloss).Bcushistmpeaturdifrnceislargeonlyonhepowerstoke,alossincudedonlyonroke.Theictonforctkentobeproporionaltopitonvelocity,correspondingtoel-ubraedmtal-on-mealsliding；thus,hefriconaosseaditlrelad,tsquarofheveocity.Theselossarenotsameoralstroke.Thehighprsurinhepowetokemakesfrictoncoefinthigherthaniheotrstoke.Thentakestrokehascontributiondueoviscouslowthroughthevalve.Tfuncionweoptimzedistmaximumworkpercycle.Bcausebothfuelconsumptionandcycltimarfxed,hiloisequivalenttomaximizngicencyandtheavergepower.Infidgthoptimalpistonmti,wefirstepartedhpowerandnopowerstokes.Aunspecebutfxedmewalotdtohepowsokeitheminderofhecycltimegivetohenonpowrstokes.Bhporionsoftcycleroptimzeithhistimeconstraintndwertncombinedtofindtotalorkperl.Tduratonofpowrsokestnvaidandtheprocesasrepetduntilthenetworksmximum.Theoptimalpistonmotiforthenonpowrsokesakesimpleform.Becausofthequadraticveloctydependenceofriconlose,toptimummotonholdsthevelociyconsntduringmosofachstroke.Atndsofhesr,thepistacelrasnddeelratsthemaximumlowedra.Becaustfictonloserhigheontintakestroke,heoptimlsolutionaotsmortimtohisrokehantoothetwo.ThepistonvelocityasfunctioftmeihownFg.1.Thepowerstokewasmoredificultooptimzebecauseofthepresncoftheatlek.heproblmwaolvedbyusingthevaraionaltchniqueoptimalcontrolory(8).Theformalismyieldstheequationofmotionofthepisonasfourthordersetofnonlineardierntiaquaions.Tsersolvednumericly.TheresulingmotionishowniFig.1fortherecyclTasymmtrishapeofthepistonmotiontpowerstokearisefromtherade-offbetwenfriconndheatleklose.Ahebeginningofthehegaahot,cpableofyildinghigheficncy,ndtratofatlossihigh.Iistrefordvantgeoustomakethevelocityhionthispasroke.Aworkiextacd,tsecoolandherteofatekagediminiherelativetofictonaloss.ConsquentlyheoptimlpathmovesolowrltesatpowrsprocedsThesolutionswrobtinedfitwihunlimtdacelrationndtnwithlimtsonacelrationnddecelation.Thelaersuatonyieldssulfmlriothercontextundenameofturnpikesolution(9).Thesystmtritoopeateslongasposiblatisoptimalforardbackwardveloctes,byacelangnddeclrtingbetentheeveoctesthemximumtes.Inhiwy,tsystmspendsasmuchmeasposiblmovingalongisbetorturnpikepath.RESULTParmetrsforthecomputationswertakenfromref.10or,inthecaseofthefrictoncoefint,adjusedtogivefrctionalosseofthemagnitudedinrf.10.ThoseparametrsrgiveniTable1.TherultclulionsofsometypicalsargiveninTbl2,wheetyrcompardihonventionalOtocyclnginehaving