Internal Combustion Engines - CaltechAUTHORS

1 4 InternalCombustionEnginesInternalcombust ionenginesaredevicesthatgenerateworkusin gtheproductsofcom-bustionastheworkingflu idratherthanasa ,thecombustionis carriedoutina mannerthatproduceshigh-pressurecombustio nprod-uctsthatcanbeexpandedthrougha introducesa :(1)thesparkignitionengine,whichisusedpr imarilyinautomobiles;(2)thedieselengine, whichisusedinlargevehiclesandindustrials ystemswheretheimprovementsincycleefficie ncymakeitadvantageousoverthemorecompacta ndlighter-weightsparkignitionengine;and( 3)thegasturbine, ,unburnedhydrocarbons, , , ,car-bonmonoxide, , pistonmovesupanddownina cylinder,transmittingitsmotionthrougha (topdeadcenter)B=crankanglePistonPistonr odCrankB=180 (bottomdeadcenter) ; ; ;stroke4, ,thesparkplugisfired, burningmixtureexpands, , ,highpowerisneededanda , , , 'I-,~0' ~ (f) "brakespecificfuelconsumption.

2 " ' ' , ,a ,however, , ,producingthenoisereferredtoasknock(Byet al.,1981).Onecharacteristicofthefuelcomp ositionisitstendencytoautoignite, , ,V". , ,( )Theefficiencyoftheengineisa simplefunctionofthecrankangle,(),andther atioofthelengthofthepistonrodtothatofthe crank,thatis,Vd(lV=Ve+ -1+- -cos()-2c( )wherelisthepistonrodlengthandc ()=0 ,commonlyreferredtoastopdeadcenter, ,BOC,()= (rpm) , , , ,however, ,wecanassumethatinanyonecycletheengineop eratesatconstantspeed,load, discussionofthethennodynamicsofthesparki gnitionenginecycleanddevelopa flowsthrougha carburetorandvaporizesbeforeit , , , ,thisexpan-sionrequiresthatotherelements offluid,bothburnedandunburned, result,theburningelementoffluiddoesworko ntheotherfluidinthecylinder,oW=pdV, , ,thepropagationrateforsmallpres-suredist urbancesisthespeedofsound,a,=.)

3 JyRT/M( ) 'Yistheratioofspecificheats,cilcu'andMis themolecularweightofthegas;asisoftheorde rof500to1000m cylinder10cmindiameter,thetimerequiredfo ra ,considerablyshorterthanthetimerequired ,toa firstapproximation,wemayassumethatthepre ssureis uniformthroughoutthecylinderatanyinstant oftime, "zero-dimensional"thermodynamicmodel(Lav oieetaI.,1970;BlumbergandKummer,1971).Th ismodeldescribesthethermodynamicstatesof theburnedandunburnedgasesasa functionoftime, controlvolumeenclosingallthegasesin! ,];.Similarly,massmayleavethroughtheexha ustvalveandpossiblythroughleaksata flowrate];,.Thefirstlawofthermodynamics( )forthiscontrolvolumemaybewritteninthege neralformdU----dQdWd1=];hi-].

4 H"+d1-dtwhereUisthetotalinternalenergyof thegasescontainedinthecylinderandh;andhe arethemassspecificenthalpiesoftheincomin gandexitingflows, ,W,isthatofapressureactingthrougha ~aus~valveandassumethatnoleaksoccur,noma ssentersorleavesthecylinder( ,];=Ie=0).Theenergyequationthensimplifie stod_dQdVdt(muT)=d1-PdtwhereUTisthetotal massspecificinternalenergy(includingener giesofformationofallspeciesinthecylinder ),-Qisheattransferredoutofthecharge, , ,thetimederivationsmaybeexpressedasdd- =w-dtdewherewistheenginerotationspeed(cr ankangledegreespers).ThuswehaveddQdVde(m uT)=de-pde( ) ,witha massfraction(Xofburnedgas,( )where< ) uniformintemperature( ,<uu)=u, ,a fluidelementbums, , , "((X,(X')representtheenergywhenthecombus tionhasprogressedtoburnedgasmassfraction (Xofa fluidelementthatburnedwhentheburnedgasma ssfractionwas(x'.))))

5 Averagingoverallburnedgas,wefind( )Theinternalenergyofeitherburnedorunburn edgasmaybeexpressedintermsofthespecifich eat,TUi=Llul(To)+L)c,'j(T')dT'( )Whilethespecificheatsvarywithtemperatur e,wehavealreadyseeninChapter2 thatvariationissmallovera , ,T,::::;T::::;T2,thisaveragebecomes( )Theinternalenergiesoftheburnedandunburn edportionsofthegasmaybeex-pressedinterms oftheaveragespecificheatsby( ) ( ), (Ci.,Ci.')is thetemperatureofanelementthatburnedatCi. 'ata ,( )whereSubstitutionof( ),( ),and( )intotheenergyequationyieldsd__dQdVde[m( 1-Ci.)(all+CI,"T,,)+mCi.(ab+Clb(TI, )]=de-pde( )Thetotalvolumeofburnedandunburnedgasesm ust,atalltimes,equalthevol-umeinthecylin der:( )( )( )Assumingidealgaseswithconstantcompositi on,themeanspecificvolumeoftheburnedgasis (_)~(XRbTil(Ci.))

6 ,Ci.')IRb(Th)Vh= "'--'--""-'-oPPNotingthatRh=("Ib-1)Cl'b, where"Ib=Cph/Cl'histheratioofspecifichea ts,( )maynowbesimplifiedto_pV"Ib-1 ,h(Th)=----m(l-Ci.)---cl'uT""Ih-1"Iu-1 Substitutingthisresultinto( )eliminatestheburnedgastemperaturefromth eenergyequation:~lm(l-Ci.)au+m(l-Ci.)(~) ~~U)Cl,JUpVJdQdV+ +---=de-p-e"Ih-1d( ) ,thecylinderisassumedtobefilledwitha ,cylindervolume,andgastemperatureattheti metheintakevalveclosesarePi'Vi'andTi, (atleastcomparedtothatbetweencombustionp roductsandthewall), ,thepressureinthecylindercanbedetermined fromtheformulafortherelationbetweenpress ureandvolumesinadiabaticcompression,rVT" p(O)=PilV(~)J( )Thetemperatureoftheunburnedgasthroughou tthecycleisthatdetern1inedbyadiabaticcom pressionSubstituting( )into( )anddifferentiatingyield()(-"("-1)/'1,,1 1Im(1 -a)Y"-YucT -~cpYb-1"11'PiP"IudOI()(Yu-I)

7 /1"jdY"-Yu_Pa+ma"-au-_C,'UTi--YiJ1Pi_dOP dVVdp+----+----YiJ-1dOYb-1dOdQdV=dO-PdO( )( )Thisequationmayberearrangedtoexpressthe rateofchangeofthecylinderpressureintern1 Softheconditionsattheendoftheintakestrok e,therateofvolumechange,andthecombustion andheattransferrates,thatis,QI()(-"(,,-I )/"YU'JIdYiJdVYb-YII-T ca-----P--ma"-au-C,'uidpdOYb-1dOYiJ-1 PidOdO_)--;Y"-YuYII-ITi(p)(-"(U-1l/'l"Vm (1a(,'u--+---Y"-1"IIIPPiYiJ-1( ) ,da/dO,tousethemodelof( ).Toefficientlyconverttheheatreleasedbyc ombustiontoworkonthepiston, ,sothecombustioncantakeatmosta ,tensofmilli-secondswouldberequiredforla minarflamepropagationacrossa ,therefore, ,theturbulentflamespeeddependsontheturbu lentintensity,u'.))))

8 Theturbulentintensityisgovernedbyensined esignandoperation, ratethatdependsoncombustionkineticsthrou ghthelaminarflamespeed, , ,therefore, complex, ,theflowseparates,re-sultingina highlyunsteadymotion(HoultandWong,1980). , ,theturbulencemaybecharacterizedintermso ftwoquantities:(I)theturbulentkineticene rgyperunitmass+u~)( )( )whichdescribesthelarge-scalebehavioroft heturbulence,and(2)therateofturbulentkin eticenergydissipationc=1J1~~1~11-1ax! ax!whichdecribestheeffectsofthe velocitythatisprop0l1ionaitothepistonspe edandhencetotheangularrotationspeed, , ,areusedtocreatea , ,therateofchangeofthetur-bulentkineticen ergyisa balancebetweenproductionanddissipation:p cdE,p---=pPdtwherePistherateofturbulentk ineticenergyproduction.

9 ( ) ,head, toberelatedtouIforhomogeneous,isotropict urbulence,whereAandIaretheTaylormicrosca leandintegralscale, ( )Assumingthatangularmomentumintheturbule ntfieldisconservedduringtherapidcompress ion:weseethatEisproportionaltoEL( ) ,p}3= ( ),thisyieldsor( )( )( )( ) ,theproductionofturbulentkineticenergyis muchmorerapidthanitsdissipation(Borgnakk eetal.,1980),dEkpP"",p-dtandapplying( ),theproductionofturbulentkineticenergyd uetotherapiddistortionoftheturbulentfiel dduringcompression,yields2 EkdpP"'"---3 PdtTherateequationforEkbecomesdEk2 Ekdpdt3 PdtCE~( )whereEhasbeeneliminatedusing( ).Theproductiontermgenerallydominatesdur ingthecompressionandcombustionprocessesd uetotherapidchangeindensity,so( )mayberewrittenas( ) ,(J,is,toa firstapproximation,independentofthecrank rotationspeed, ( ) ,fora givenenginegeometry,thevalueofu'atanycra nkangle,(J,isapproximatelyproportionalto theangularspeedUo~wandtheturbulentflamep ropagationvelocityincreaseswiththeengine speed.))

10 ( ) ,ifex(0)isknownforoneenginespeed, ,therefore,weshallsimplyspecifya functionalfonnforex(0)thatexhibitstheess entialfeaturesofactualcombustionprofiles ,thatis,a delayfromthetimethesparkisfireduntilthep ressureriseassociatedwithcombustionbecom esappreciable,anac-celeratingcombustionr ateuntila largefractionofthechargeisburned, ,( )where00isthecrankangleatwhichthesparkis firedandL::.0, , , ( )withex(O)givenby( ) functionoftheturbulentfield(Borgnakkeeta I.,1980).Forourpresentpurposes,itissuffi cienttoassumethattheengineisadiabatic( ,dQ/dO=0).Oncethepressureinthecylinderis knownthemeanburnedandunburnedgastemperat urescanbecalculatedusing( )and( ), isassumedthatnomixingoftheburnedgasesocc ursandthatheattransferfroma , massfractionburnedwasex'is( )Thetemperatureoftheelementimmediatelyfo llowingcombustion,T"(ex',ex'),maybeevalu atedbyapplyingthefirstlawofthennodynamic stothecombustionofaninfinitesimalmassofc harge, sufficientlysmallincrementalmass,thepres surechangeduringcombustionis ,thatis,--h"=U"+R"J:,=h"=Ub+R"T" becomes( )From( ), ( ),and( ) (1976)foranenginewitha firedat40 ,t::dl"is60.