Thetheoryof compression ignition engines93mRsc=(1.6b)P(m,),2Inpractice1.1<Rsc<1.6Forfour-stroke engines,particularly those with small valveoverlap,e.g.in road traction,it is safe to assume that all the airdelivered to the engine is trapped in the cylinder, i.e, n 1.However,due tocharge heating during the gas exchange processand adverse pressure conditions in the cylinder, it is likely thatthe volumetric efficiency nvol defined assweptAOTvolume of air trapped under inletVmanifold conditions(b)↑vol =swept cylinder volumeRT/Vawep=ma(1.7)102p(a)(where T and p are respectively the inlet manifold temperature(°K)and pressure (bar))is considerably less than unity.Clearlyfor the highest specific output, both the relative air fuel ratio eand the volumetric efficiency should be as close to unity aspossible.1.3Air standard cyclesIt will beclearfrom theforegoing sections thatthereal processesin the diesel engine cylinder,particularlythoseoffuel preparation,Vcombustion and gas exchangeare extremely complexandrequire(c)sophisticated computational techniques which are discussed inFigure1.7Air standard cycles:(a)Constant pressure cycle:1,2,3a number of specialisttexts.(b) Constant volume cycle: (c) Dual combustion or composite cycleAir standard cycles which are discussed in most elementarytextbooks,provide a useful basis for comparing actual engine2-3,and theblowdown-gas exchange sequence onceagain byperformance expressed intermsof indicated mean effectiveconstant volume heat rejection 4-1.Again compression 1-2pressure (pind, eqn (1.1) and indicated thermal efficiency (n..and expansion 3-4areisentropic.eqn(1.4a) with corresponding values for highly idealized cycles,Traditionally this is the reference cycle for spark ignitionbased on certain drastic simplifying assumptions as follows:(SI) engines, but it has distinct validity as a reference cycle fordiesel engines,particularlyunder lightload conditions when(a) the mass of working fluid remains constant throughout thethe heat release period is short so that the assumptions of zerocycle, i.e. gas exchange and fuel addition are ignored:heatrelease durationimpliedbythe constantvolumeprocess(b) the working fluid throughout the cycle is pure air treated as2-3 does not introduce excessive errors.a perfect gas;(c)thecombustionandgasexchangeprocessesarereplacedby(c)The‘dual combustion'orcompositecycle(Figure1.7c)external heattransferto or from the working fluid underidealized,This represents a combination of the constantpressure ande.g. constant volume or constant pressure conditions;constant volume cycles and is intended to provide a closer(d)compressionandexpansionprocessesaretreatedasadiabaticapproximation to actual diesel cycles than either of the aboveand reversible,i.e.heattransfer and friction effects are completelyideal cycles. It is particularly appropriate where comparisonsneglected;are to be made with actual diesel cycles on the basis of the(e)atanypoint of theworking cycle,cylinderchargepressuremaximum cylinder pressure Pmax obtained during the heat releaseand temperature are completely uniform,i.e.spatial variationsperiod, i.e.for engines operating in the mid-to full load range.intheir values as for instance during combustion or scavenging,are completely neglected1.3.1Theoretical expressions for air standard cyclesThe most commonly used air-standard cycles are as follows(Figures 1.7a, b and c):In thefollowing derivations it will be assumed that thecompressionratioCRcorrespondstotheeffectivecompression(a)Theconstantpressureordieselcycle(FigureI.7a)ratio (CR)efr of the engine, eqn (1.3a), and that the isentropicHere combustion is simulatedby constant pressure heat additionindex y i.e.the specifc heat ratio for air as a perfect gas, has the(2-3),andblowdown.followedbyscavenge,byconstantvolumeconstant value y= 1.4.heat rejection 4-1.Compression 1-2 and expansion 3-4followthe isentropic state relationships fora perfect gas.This particular1.3.7.1Theconstantpressureordiesel cycle(Figure1.7a)cycle has, in the past, been used as a reference cycle for theFrom basic engineering thermodynamics'classical'Diesel engine with airblast injection giving a ratherlong injection and hence heat release period, corresponding to+p,V,-P2V22-3.It has,however, little relevanceto the modern diesel cycle.compression workwp2 =(i)y-1(b)TheconstantvolumeorOnocycle(FigureI.7b)(note this is negative)Here combustion is simulated by constant volume heat release
#sc - 7?V (L6b) Oa )t In practice 1.1 < Rsc < 1.6. For four-stroke engines, particularly those with small valve overlap, e.g. in road traction, it is safe to assume that all the air delivered to the engine is trapped in the cylinder, i.e. T] tr = 1. However, due to charge heating during the gas exchange process and adverse pressure conditions in the cylinder, it is likely that the volumetric efficiency T]vol defined as volume of air trapped under inlet manifold conditions vo1 swept cylinder volume PT I = Wa T^T-/Vswept (1-7) 10-P/ (where Tandp are respectively the inlet manifold temperature ( 0K) and pressure (bar)) is considerably less than unity. Clearly for the highest specific output, both the relative air fuel ratio £ and the volumetric efficiency should be as close to unity as possible. 1.3 Air standard cycles It will be clear from the foregoing sections that the real processes in the diesel engine cylinder, particularly those of fuel preparation, combustion and gas exchange are extremely complex and require sophisticated computational techniques which are discussed in a number of specialist texts.1 ' 2 ' 3 Air standard cycles which are discussed in most elementary textbooks, provide a useful basis for comparing actual engine performance expressed in terms of indicated mean effective pressure (pind, eqn (1.1) and indicated thermal efficiency (T]1, eqn (1.4a) with corresponding values for highly idealized cycles, based on certain drastic simplifying assumptions as follows: (a) the mass of working fluid remains constant throughout the cycle, i.e. gas exchange and fuel addition are ignored; (b) the working fluid throughout the cycle is pure air treated as a perfect gas; (c) the combustion and gas exchange processes are replaced by external heat transfer to or from the working fluid under idealized, e.g. constant volume or constant pressure conditions; (d) compression and expansion processes are treated as adiabatic and reversible, i.e. heat transfer and friction effects are completely neglected; (e) at any point of the working cycle, cylinder charge pressure and temperature are completely uniform, i.e. spatial variations in their values as for instance during combustion or scavenging, are completely neglected. The most commonly used air-standard cycles are as follows (Figures 1.7'a, b and c): (a) The constant pressure or diesel cycle (Figure 1.7a) Here combustion is simulated by constant pressure heat addition (2-3), and blowdown, followed by scavenge, by constant volume heat rejection 4-1. Compression 1-2 and expansion 3-4 follow the isentropic state relationships for a perfect gas. This particular cycle has, in the past, been used as a reference cycle for the 'classical' Diesel engine with air blast injection giving a rather long injection and hence heat release period, corresponding to 2-3. It has, however, little relevance to the modern diesel cycle. (b) The constant volume or Otto cycle (Figure 1.7b) Here combustion is simulated by constant volume heat release (C) Figure 1.7 Air standard cycles: (a) Constant pressure cycle; (b) Constant volume cycle; (c) Dual combustion or composite cycle 2-3, and the blowdown-gas exchange sequence once again by constant volume heat rejection 4-1. Again compression 1-2 and expansion 3-4 are isentropic. Traditionally this is the reference cycle for spark ignition (SI) engines, but it has distinct validity as a reference cycle for diesel engines, particularly under light load conditions when the heat release period is short so that the assumptions of zero heat release duration implied by the constant volume process 2-3 does not introduce excessive errors. (c) The 'dual combustion' or composite cycle (Figure 1.7c) This represents a combination of the constant pressure and constant volume cycles and is intended to provide a closer approximation to actual diesel cycles than either of the above ideal cycles. It is particularly appropriate where comparisons are to be made with actual diesel cycles on the basis of the maximum cylinder pressure /?max obtained during the heat release period, i.e. for engines operating in the mid-to full load range. 1.3.1 Theoretical expressions for air standard cycles In the following derivations it will be assumed that the compression ratio CR corresponds to the effective compression ratio (CR)eff of the engine, eqn (1.3a), and that the isentropic index y, i.e. the specifc heat ratio for air as a perfect gas, has the constant value J= 1.4. 1.3.1.1 The constant pressure or diesel cycle (Figure Ua) From basic engineering thermodynamics: + P1Vi - p2 V2 compression work W\2 = ; (i) (note this is negative) swept volume
10DieselEngineReferenceBook0.7Welt(vii)(n;)cp=Q23CR=200.6(vii)eventually reduces to18-BY-11(n:)cp = I (1.8)16Limiting air-fuel ratio14.(CR)y-JY(β-)0.5(equation 1.9)12The volume ratio βis an indication of the air-fuel ratio A/F at10NeNDwhich the engine is operating,since to a firstapproximation0.48CR=6Q23= mCp(T3- T2)= P2V2(β-1)0.3= m(CV)(vili)where m is the mass of fuel burnt0.2P,V,But mair=m, =(ix)RT,0.1Y(β- 1)PaV2P,Vy-1whence A/F=RT,(CV)10213A567Cut-off ratio βCVFigure 1.81Constant pressure cycle. Indicated efficiency vs cut-off(1.9)ratio (eqn 1.8)RTCRY(β-1)y-1constant pressure workAssuming that the limiting air-fuel ratio is the stoichiometricW23=P2 (V-V2) =P2V2(β- 1)(ii)ratio (A/F)stoich eqn (1.5c),it is possible to find a limiting valueofthevolumeratioβforanygivencompressionratioCRfromVeqn (1.9).This is shown inFigure1.8 indicating the behaviourwhere β= volume ratioV2ofeqn(1.8)withdifferentvaluesof compressionratioCRand'cut off'ratioβ,including theposition of the"limiting line'forconstantpressureheattransferstoichiometric combustion.Indicated efficiency (n,)cp is seen to increase rapidly withQ23 = mCp(T3 - T2)volumetric compression ratio CR andtodecrease with increasingvalues of the cut off ratio β, i.e.with decreasing air-fuel ratio,(beinga minimum,for any valueof CR on the limit line,and amaximumfora cut offratioβ=1.Efficiency is not the only consideration appertaining to cycles.Specific output also hastobe taken into account so thatthe=P2V2β-11(iii)relationship between indicated efficiency,specific output andcompression ratio is equallyimportant.The specific outputisbest measured in terms of the mean effectivepressure definedexpansion work Wy=DV-_Vβ-pM(iv)by eqn (1.1) relative to the trapped pressure pi.The calculationy-1Y-Iis as follows:nettworkWne=W12+W23+W34For any assumed value of the cut-off ratio β the equivalent air-fuel ratio A/F may be calculated from eqn (1.9).giving theheat=PiV-p2V2+PaVeB-p4viinput to the cycle as-1y-1PIVCVCV(i)+ p2V2(β- 1)(v)Qin=mA/FRTA/FbutWith indicated efficiency (n)cp from eqn (1.8),the indicatedworkoutputofthecycle isgivenby= PI(CR)Y. V2 =P2 = p3 =CRJdW=Qm(n)cp(ii)() =p()and the mean effective pressure,from eqn (1.1)becomesP4 = p3[=P2(vi)Tp,VCV(ni)cpJawJdwRT,A/FSubstituting from (vi) for p2,V and p4 in (v) and (ii) andPind =writing for the ideal efficiency of the constantpressure (CP)VsweptV(1-v.(1-cycle:CRCR
Cut-off ratio ft Figure 1.8 Constant pressure cycle. Indicated efficiency vs cut-off ratio (eqn 1.8) constant pressure work W23 = P2 (V3 - V2 ) = P2V2 (P - 1) (ii) V3 where /3 = volume ratio —— constant pressure heat transfer Q23 = mCp (T3 - T2 ) P2V2 ( 7 .YV3 = \ -KfT [T^T *}{v; - 1 J^ = p2v2 ^-I1J(JS-O №) P3V3 -P4V4 P2V2 I-P4Vi expansion work Vr34 = —: = — (iv) nett work Wnett = W12+ W23 + W34 = PiVi -P2V2 P2V2 I-P4Vi 7-1 7-1 + P2V2OS-I) (v) but A » 2 =P3=P,(£)r ^1(CK)', V 2 = (^) ( v3 y f/?v2 y fjs y P4=P3 ^_J =P2^_ J = p2^_J (v,) Substituting from (vi) for p2, V2 and p4 in (v) and (iii) and writing for the ideal efficiency of the constant pressure (CP) cycle: W (Hi ) CP = ^ (ViO (vii) eventually reduces to (l" )CT = I - bM W^) (L8) The volume ratio /3 is an indication of the air-fuel ratio A/F at which the engine is operating, since to a first approximation Q23 = mCp(T3 - T2) =P2V2~ (ft - O = mf(CV) (viii) where mf is the mass of fuel burnt But mair = m, = ^L (ix) ^ 1V1 /IPM^P-V whence A/F= ^i-/ L_ My-'oi i (19) ICK J «*£<,_„ Assuming that the limiting air-fuel ratio is the stoichiometric ratio (A/F)stoich eqn (1.5c), it is possible to find a limiting value of the volume ratio /3 for any given compression ratio CR from eqn (1.9). This is shown in Figure 1.8 indicating the behaviour of eqn (1.8) with different values of compression ratio CR and 'cut off ratio j3, including the position of the 'limiting line' for stoichiometric combustion. Indicated efficiency (r/^cp is seen to increase rapidly with volumetric compression ratio CR and to decrease with increasing values of the cut off ratio /3, i.e. with decreasing air-fuel ratio, being a minimum, for any value of CR on the limit line, and a maximum for a cut off ratio j8 = 1. Efficiency is not the only consideration appertaining to cycles. Specific output also has to be taken into account so that the relationship between indicated efficiency, specific output and compression ratio is equally important. The specific output is best measured in terms of the mean effective pressure defined by eqn (1.1) relative to the trapped pressure ^1. The calculation is as follows: For any assumed value of the cut-off ratio ft the equivalent airfuel ratio A/F may be calculated from eqn (1.9), giving the heat input to the cycle as O -m CV _ PiV, CV m ^ in " m ~A/F ~ ~RT]~ ~A/F (l) With indicated efficiency (T]J)CP from eqn (1.8), the indicated work output of the cycle is given by JdW=Gi n (Tj 1 ) C P (H) and the mean effective pressure, from eqn (1.1) becomes PiVi CV (n . _ IdW _ IdW RT, A/F(7?i)cp ^ ind ~ V f \ \~ ( \ \ Vswept y f j _ J_ y I I _ _L 1 I CR) 1 I CR j HIND Limiting air-fuel ratio (equation 1.9)
Thetheoryof compressionignition engines11Substituting from (ii) in (i) and (ii) and writing for the ideal20efficiencyof the constantvolume(CV)cycle18CR = 20Wert16Limiting air-fuel(iv)(ni)cv=ratio (equation 1.9)14Q231512(iv)eventuallyreducesto10(ni)cy=/-(1.11)CRRPnd10PEquation (I.11)demonstrates that the efficiency of the constantvolumecycleisa functionofcompressionratioCRonly,andCR=6unlike the constant pressure cycle,independent of the level ofheat addition, as expressed by the pressure ratio p/p2=T,/T2=α (seeFigure 1.10).5It is generally quoted in support of arguments to raisecompression ratio in spark ignition (SD) engines.-D1.3.1.3Thedualcombustionorcompositecycle234567(Figure 1.7c)Cut-off ratio βAs already stated, this cycle tends to approximate more closelyFigure1.9Constant pressure cycle.Indicatedmean effective pressureto actual diesel cycles than either the pure constant pressure orvs cut-off ratio (eqn 1.10)constant volume cycles as described above. It lends itselfparticularly well to the representation of limited maximumCVcylinderpressure,as expressedbythepressureratioα=ps/p1AVF(n)cPoften specified in real diesel cycles, and to assessment of theRT.Piador(1.10)effectof increased orretarded heat release,asexpressed mainlyp1CP1-bythevolumeratioβ=V/V,CRThe evaluation of cycle efficiencyfollows a similar patternto that adopted above:Equation(1.10)isrepresentedbyFigure1.9andshowsthat,forany given compression ratio CR, efficiency decreases withnett cycle work =W2+W34 +W45increasing specific output,with a minimum valueagain onthelimit line. PiVi-P2Va2+p3Vs(β-1)+P3V4-psVsFigure 1.9 may be used both for naturally aspirated engines(i)y-1Y-1forwhichthetrappedpressureP,is approximatelyequaltoatmosphericpressurep.as well asfor supercharged enginesConstantvolume heattransferwitha supercharge (orboost)pressureratio given approximatelybyR= (p//pa) (>1).0.71.3.1.2TheconstantvolumeorOttocycle(Figure1.7b)As already stated this cycle has only limited applicability to0.6diesel engines,mainly under part loadconditions.Heat transfernowoccurs under constant volume conditions,both for the'combustion process2-3andthegas exchangeprocess 4-1.0.5Nett cycle work( aae eWner = Wi2 + W34P,V-paV40.4PV-P2V2(i)Y-1-1constant volumeheattransfer0.3Q23 =mC,(T, -T2)= PaV_RRT-T0.2P2V(α-1)(ii)Y-10.1whereα=PP2Vi1110But p2 =pI(CR), V2 == V3048121620CRCompression ratio (CR)Figure 1.10Constantvolume cycle.Indicated efficiency vs(ili)P4=p=piα=p2(CRChcompression ratio
Cut-off ratio /3 Figure 1.9 Constant pressure cycle. Indicated mean effective pressure vs cut-off ratio (eqn 1.10) CV 1 „ (a^ _W^f^' ( ">° ('-£) Equation (1.10) is represented by Figure 1.9 and shows that, for any given compression ratio CR, efficiency decreases with increasing specific output, with a minimum value again on the limit line. Figure 1.9 may be used both for naturally aspirated engines for which the trapped pressure p\ is approximately equal to atmospheric pressure pa as well as for supercharged engines with a supercharge (or boost) pressure ratio given approximately by RE = (Pi/Pa) (> O. 1.3.1.2 The constant volume or Otto cycle (Figure 1.7b) As already stated this cycle has only limited applicability to diesel engines, mainly under part load conditions. Heat transfer now occurs under constant volume conditions, both for the 'combustion' process 2-3 and the 'gas exchange' process 4-1. Nett cycle work HU = Wl 2 + W34 = P1V1 -P2 ^ 2 P3V3 -P4V4 (n y- 1 7-1 constant volume heat transfer Q23=mC^-T2}-^^-l)T2 = frj(a-l> («) ft-, where a = -^- Pi But p2 = p} (CR)Y, V2 = ^ = V3 p< = P*(ZR) = ^a (^) =*<* (ill) Substituting from (iii) in (i) and (ii) and writing for the ideal efficiency of the constant volume (CV) cycle w (Hi)CV = ^ Ql3 W (iv) eventually reduces to (TJi)CV = I -(c^)7 O- 1 1 ) Equation (1.11) demonstrates that the efficiency of the constant volume cycle is a function of compression ratio CR only, and unlike the constant pressure cycle, independent of the level of heat addition, as expressed by the pressure ratio p-Jpi - T3/T2 - a (see Figure 1.10). It is generally quoted in support of arguments to raise compression ratio in spark ignition (SI) engines. 13.13 The 'dual combustion or composite cycle (Figure Uc) As already stated, this cycle tends to approximate more closely to actual diesel cycles than either the pure constant pressure or constant volume cycles as described above. It lends itself particularly well to the representation of limited maximum cylinder pressure, as expressed by the pressure ratio a = p$lp2 often specified in real diesel cycles, and to assessment of the effect of increased or retarded heat release, as expressed mainly by the volume ratio ft = V4IV3,. The evaluation of cycle efficiency follows a similar pattern to that adopted above: nett cycle work = W12 + W34 + W45 = ^^+P3v3 (/M)+^^ (1) Constant volume heat transfer Compression ratio (CR) Figure 1.10 Constant volume cycle. Indicated efficiency vs compression ratio Limiting air-fuel ratio (equation 1.9) Thermal efficiency (77,) Pjnd Pl