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Proc.Natl.Acad.Sci.USAVol.78,No.4,pp.1986-1988,April1981AppliedPhysicalSciencesFinite-timethermodynamics:Engineperformanceimprovedbyoptimizedpistonmotion(Ottocycle/optimizedheatengines/optimalcontrol)MICHAELMOZURKEWICHANDR.S.BERRYDepartmentofChemistryandtheJamesFranckInstitute,TheUniversityofChicago,Chicago,Illinois60637ContributedbyR.StephenBerry,December29,1980ABSTRACTThemethodsoffinite-timethermodynamicsareusedtofindtheoptimaltimepathofanOttocyclewithfrictionandheatleakage.Optimalityisdefinedbymaximizationoftheworkpercycle；thesystemisconstrainedtooperateatafixedfrequency,sothemaximumpower-isobtained.Theresultisanimprovementofabout10%intheeffectiveness(second-lawefficiency)ofaconventionalnear-sinusoidalengine.Finite-timethermodynamicsisanextensionofconventionalthermodynamicsrelevantinprincipleacrosstheentirespanofthesubject,fromthemostabstractleveltothemostapplied.Theapproachisbasedontheconstructionofgeneralizedthermodynamicpotentials(1)forprocessescontainingtimeorrateconditionsamongtheconstraintsonthesystem(2)andonthedeterminationofoptimalpathsthatyieldtheextremacorrespondingtothosegeneralizedpotentials.Heretofore,workonfinite-timethermodynamicshasconcentratedonratheridealizedmodels(2-7)andonexistencetheorems(2),allontheabstractsideofthesubject.Thisworkisintendedasastepconnectingtheabstractthermodynamicconceptsthathaveemergedinfinite-timethermodynamicswiththepractical,engineeringsideofthesubject,thedesignprinciplesofarealmachine.Inthisreport,wetreatamodeloftheinternalcombustionenginecloselyrelatedtotheidealOttocyclebutwithrateconstraintsintheformofthetwomajorlossesfoundinrealengines.Weoptimizetheengineby"controlling"thetimedependenceofthevolume-thatis,thepistonmotion.Asaresult,withoutundertakingadetailedengineeringstudy,weareabletounderstandhowthelossesareaffectedbythetimepathofthepistonandtoestimatetheimprovementinefficiencyobtainablebyoptimizingthepistonmotion.THEMODELOurmodelisbasedonthestandardfour-strokeOttocycle.Thisconsistsofanintakestroke,acompressionstroke,apowerstroke,andanexhauststroke.Herewebrieflydescribethebasicfeaturesofthismodelandthemethodusedtofindtheoptimalpistonmotion.Adetailedpresentationwillbegivenelsewhere.Weassumethatthecompressionratio,fuel-to-airratio,fuelconsumption,andperiodofthecycleallarefixed.Theseconstraintsservetwopurposes.First,theyreducetheoptimizationproblemtofindingthepistonmotion.Also,theyguaranteethattheperformancecriterianotconsideredinthisanalysisarecomparabletothoseforarealengine.Relaxinganyoftheseconstraintscanonlyimprovetheperformancefurther.Wetakethelossestobeheatleakageandfriction.Bothoftheseareratedependentandthusaffectthetimeresponseofthesystem.Theheatleakisassumedtobeproportionaltotheinstantaneoussurfaceofthecylinderandtothetemperaturedifferencebetweentheworkingfluidandthewalls(i.e.,Newtonianheatloss).Becausethistemperaturedifferenceislargeonlyonthepowerstroke,heatlossisincludedonlyonthisstroke.Thefrictionforceistakentobeproportionaltothepistonvelocity,correspondingtowell-lubricatedmetal-on-metalsliding；thus,thefrictionallossesaredirectlyrelated,tothesquareofthevelocity.Theselossesarenotthesameforallstrokes.Thehighpressuresinthepowerstrokemakeitsfrictioncoefficienthigherthanintheotherstrokes.Theintakestrokehasacontributionduetoviscousflowthroughthevalve.Thefunctionwehaveoptimizedisthemaximumworkpercycle.Becausebothfuelconsumptionandcycletimearefixed,thisalsoisequivalenttomaximizingbothefficiencyandtheaveragepower.Infindingtheoptimalpistonmotion,wefirstseparatedthepowerandnonpowerstrokes.Anunspecifiedbutfixedtimetwasallottedtothepowerstrokewiththeremainderofthecycletimegiventothenonpowerstrokes.Bothportionsofthecyclewereoptimizedwiththistimeconstraintandwerethencombinedtofindthetotalworkpercycle.Thedurationtofthepowerstrokewasthenvariedandtheprocesswasrepeateduntilthenetworkwasamaximum.Theoptimalpistonmotionforthenonpowerstrokestakesasimpleform.Becauseofthequadraticvelocitydependenceofthefrictionlosses,theoptimummotionholdsthevelocityconstantduringmostofeachstroke.Attheendsofthestroke,thepistonacceleratesanddeceleratesatthemaximumallowedrate.Becausethefrictionlossesarehigherontheintakestroke,theoptimalsolutionallotsmoretimetothisstrokethantotheothertwo.ThepistonvelocityasafunctionoftimeisshowninFig.1.Thepowerstrokewasmoredifficulttooptimizebecauseofthepresenceoftheheatleak.Theproblemwassolvedbyusingthevariationaltechniqueofoptimalcontroltheory(8).Theformalismyieldstheequationofmotionofthepistonasafourthordersetofnonlineardifferentialequations.Theseweresolvednumerically.TheresultingmotionisshowninFig.1fortheentirecycle.Theasymmetricshapeofthepistonmotiononthepowerstrokearisesfromthetrade-offbetweenfrictionandheatleaklosses.Atthebeginningofthestrokethegasesarehot,capableofyieldinghighefficiency,andtherateofheatlossishigh.Itisthereforeadvantageoustomakethevelocityhighonthispartofthestroke.Asworkisextracted,thegasescoolandtherateofheatleakagediminishesrelativetofrictionallosses.Consequentlytheoptimalpathmovestolowervelocitiesasthepowerstrokeproceeds.Thesolutionswereobtainedfirstwithunlimitedaccelerationandthenwithlimitsonaccelerationanddeceleration.Thelattersituationyieldsaresultfamiliarinothercontextsunderthenameof"turnpike"solution(9).Thesystemtriestooperateaslongaspossibleatitsoptimalforwardandbackwardvelocities,byacceleratinganddeceleratingbetweenthesevelocitiesatthemaximumrates.Inthisway,thesystemspendsasmuchtimeaspossiblemovingalongitsbestorturnpikepath.RESULTSParametersforthecomputationsweretakenfromref.10or,inthecaseofthefrictioncoefficient,adjustedtogivefrictionallossesofthemagnitudecitedinref.10.ThoseparametersaregiveninTable1.TheresultsofthecalculationsofsometypicalcasesaregiveninTable2,wheretheyarecomparedwiththeconventionalOttocycleenginehavingthesamecompressionratiobutastandardnear-sinusoidalmotion.Theeffectiveness(theratiooftheworkdonetothereversiblework,alsocalledthesecond-lawefficiency)isslightlyhigherfortheoptimizedenginewhosepiston-accelerationislimitedto5x103m/sec2，themaximumoftheconventionalengineofthefirstrow.Ifthepistonisallowedtohave4timesthe