A Specific Heat Ratio Model for Single-Zone Heat Release Models

16 Pages·0.583 MB·English

by  Klein M., Eriksson L.

2004-01-1464
A Specific Heat Ratio Model for Single-Zone Heat
Release Models
MarcusKleinandLarsEriksson
VehicularSystems,DeptofEE
Linko¨pingsUniversitet
SWEDEN
Email:{klein,larer}@isy.liu.se
Copyright(cid:176)c 2004SocietyofAutomotiveEngineers,Inc.
ABSTRACT be well tuned. These parameters, such as heat transfer coeffi-
cients, γ and b in the linear γ-model (1) and so on, can be
300
Theobjectiveistoinvestigatemodelsofthespecificheatratio
tunedusingwellknownmethods[4]. Thiscanbedonee.g. by
forthesingle-zone heatreleasemodel, andfindamodelaccu-
minimizingthepredictionerrorofthemeasuredcylinderpres-
rateenoughtointroduceamodelingerrorlessthanorintheor-
sure, i.e. by minimizing the difference between the modeled
derofthecylinderpressuremeasurementnoise,whilekeeping
andmeasuredcylinderpressure. Thisusuallyendsupinabsurd
thecomputationalcomplexityataminimum.Basedonassump-
andnonphysicalvaluesofγ ,asitbecomeslargerthan1.40,
300
tionsoffrozenmixturefortheunburnedmixtureandchemical
whichisthevalueofγ forpureair. Thereforeabettermodel
300
equilibriumfortheburnedmixture,thespecificheatratioiscal-
of γ(T,p,λ) is sought. A correct model of γ(T,p,λ) is also
culatedusingafullequilibriumprogramforanunburnedanda
desirableinordertoavoidbadlytuned(biased)parameters.
burned air-fuel mixture, and compared to already existing and
The objective is to investigate models of the specific heat
newlyproposedapproximativemodelsofγ.
ratio for the single-zone heat release model, and find a model
A two-zone mean temperature model, Matekunas pressure
accurateenoughtointroduceamodelingerrorlessthanorinthe
ratiomanagementandtheVibefunctionareusedtoparameter-
order of the cylinder pressure measurement noise, while keep-
izethemassfractionburned. Themassfractionburnedisused
ingthecomputationalcomplexityataminimum.
to interpolate the specific heats for the unburned and burned
mixture, andthenformthespecificheatratio, whichrendersa
small enough modeling error in γ. The specific heats for the
Outline
unburnedmixtureiscapturedwithin0.2%byalinearfunction,
andthespecificheatsfortheburnedmixtureiscapturedwithin Inthefollowingsectionthreeexistingapproximativeγ-models
1%byahigher-orderpolynomialforthemajoroperatingrange aredescribed. Thenbasedonchemicalequilibriumareference
ofasparkignited(SI)engine. modelforthespecificheatratioisdescribed.Thereafter,theref-
erencemodeliscalculatedforanunburnedandaburnedair-fuel
mixture respectively, and compared to these existing approxi-
1 INTRODUCTION mative models in the two following sections. With the knowl-
edgeofhowtodescribeγ fortheunburnedandburnedmixture
Theaccuracywithwhichtheenergybalancecanbecalculated
respectively,thefocusisturnedtofindingaγ-modelduringthe
for a combustion chamber depends in part on how accurately
combustion process, i.e. for a partially burned mixture. This
changes in the internal energy of the cylinder charge are rep-
isdoneinsection6, whereanumberofapproximativemodels
resented. Themostimportantthermodynamicpropertyusedin
areproposedandtheseareevaluatedintermsoftherootmean
calculating the heat release rates in engines is the ratio of spe-
cificheats,γ(T,p,λ)= cp [1,2,3]. squareerrorrelatedtothereferenceγ-modelfoundfromchem-
cv icalequilibriumandthecorrespondinginfluenceonthecylinder
Gatowskiet.al.[1]developedasingle-zoneheatreleasemodel
pressure.
basedontheFirstLawofthermodynamicsthathasbeenwidely
used, where the specific heat ratio is represented by a linear
functioninmeanchargetemperatureT
2 EXISTINGMODELSOFγ
γ (T)=γ +b(T −300) (1)
lin 300
Thecomputationaltimeinvolvedinrepeateduseofafullequi-
This allows a critical examination of the burning process by librium program, such as CHEPP [5], can be substantial, and
analysis of the heat release. In order to compute the heat re- thereforeapproximatefitsofthethermodynamicpropertieshave
leasecorrectly,theparametersinthesingle-zonemodelneedto beendeveloped. Threesuchfitswillnowbedescribed.
1
LinearmodelinT given in [kJ/(kg of original air) K]. Krieger and Borman sug-
gestedthatthecorrectiontermsu andR shouldaccount
corr corr
The specific heat ratio during the closed part of the cycle, i.e.
fordissociation, sotheyarenon-zeroforT > 1450Kandare
when both intake and exhaust valves are closed, is most fre-
givenby:
quentlymodeledaseitheraconstant,orasalinearfunctionof
temperature. The later model is used in [1], where it is stated u (T,p,λ)=c exp(D(λ)+E(T,λ)+F(T,p,λ)) (6a)
corr u
that the approximation is in parity with the other approxima-
tions for this family of single-zone heat-release models. The
linearfunctioninT canbewrittenas D(λ)=d0+d1λ−1+d3λ−3 (6b)
γ (T)=γ +b(T −300) (2)
lin 300 e +e λ−1+e λ−3
E(T,λ)= 0 1 3 (6c)
Depending on which temperature region and what air-fuel ra- T
tioλweareinterestedin, theslopebandconstantγ in(2)
300
have to be adjusted. Concerning the temperature region, this
f +f λ−1
shortcoming can be avoided by increasing the complexity of F(T,p,λ)=(f +f λ−1+f λ−3+ 4 5 )ln(f p)
0 1 3 T 6
the model and use a second (or higher) order polynomial for
(6d)
γ (T). Thishasbeendoneinforexample[6]. Suchanexten-
lin
sion reduces the need for having different values of γ300 and (cid:181) (cid:182)
b for different temperature regions. Later on, γlin(T) is cal- R (T,p,λ)=c exp r lnλ+ r1+r2/T +r3ln(f6p)
culated in a least squares sense for both burned and unburned corr r 0 λ
mixtures. (7)
whereT isgiveninKelvin(K)andpinbar. Thevaluesofthe
coefficients are given in Table 11. For a fuel of composition
SegmentedlinearmodelinT CnH2n, the stoichometric fuel-air ratio is 0.0676. Therefore,
equations(3)-(5)shouldbedividedby(1+0.0676λ−1),toget
Accordingto[2],thecommonlymadeassumptionthatγlin(T) the internal energy per unit mass of products. In general, the
isconstantoralinearfunctionofmeantemperatureisnotsuf- errorinuwasfoundtobelessthan2.5percentinthepressure
ficiently accurate. Instead, they propose a segmentation of the andtemperaturerangeofinterest,andlessthan1percentover
closed part of the engine cycle into three segments; compres- mostoftherange. Amodelofγ isthenfoundas
sion, combustion and post-combustion (expansion). Both the
compression and post-combustion are modeled by linear func- γ = cp =1+ R (8)
tions of T, while the combustion event is modeled by a con- KB cv cv
stantγ. Theyfurtherstatethatwiththeseassumptions,theone-
whereRisgivenby(5)andc = ∂u isfoundbydifferentiat-
zone analysis framework will provide accurate enough predic- v ∂T
ing(3)withrespecttoT.
tions.Unfortunately,theγ-modelproposedby[2]hasdisconti-
nuitiesandthereforethismodelwillnotbeconsideredlateron
when searching for a γ-model describing the partially burned
Summaryofexistingγ-models
mixture.
Apparently there are ambiquities in which model structure to
useforγ,thereforeγ(T,p,λ)iscalculatedforadequatetemper-
PolynomialmodelinpandT ature and pressure regions for both unburned and burned mix-
ture,assumingthatthecylinderchargeisfrozenforT ≤ 1700
Thethirdmodelisapolynomialmodeloftheinternalenergyu
Kandatequilibriumotherwise. Thisinordertofindoutwhat
developed in [7] for combustion products of C H , e.g. iso-
n 2n
model structure of γ that is accurate enough for our purposes.
octane. Forweakandstoichometricmixtures(λ ≥ 1),asingle
One such purpose is to estimate parameters in the single-zone
set of equations could be stated, whereas different sets where
model such as heat transfer coefficients, burn rate parameters
foundforeachλ<1. Themodelofuforλ≥1isgivenby:
andsoon,usingthemeasuredcylinderpressure. Thisrequires
B(T) amodelofthecylinderpressure,andthereforetheimpacteach
u(T,p,λ)=A(T)− +u (T,p,λ) (3)
λ corr approximative γ-model has on the cylinder pressure is moni-
givenin[kJ/(kgoforiginalair)],where tored. Allthermodynamicpropertiesdependontheair-fuelra-
tio λ, but for notational convenience this dependence is here
A(T)=a T +a T2+...+a T5 (4a)
1 2 5 afterleftoutwhenthereisnoexplicitdependence.
B(T)=b +b T +...+b T4 (4b)
0 1 4
3 CHEMICALEQUILIBRIUM
Thegasconstantwasfoundtobe:
Assuming that the air-fuel mixture is frozen (T < 1700 K) or
0.020
R(T,p,λ)=0.287+ λ +Rcorr(T,p,λ) (5) atequilibrium(T ≥ 1700K)ateveryinstant,thespecificheat
2
ratioandotherthermodynamicpropertiesofvariousspeciescan Property Constant Slope NRMSE RMSE
be calculated using the Matlab package CHEPP [5]. The fuel γu [-] 1.3488 −13.0·10−5 0.19% 0.0024
lin
iso-octane,C H reactswithairaccordingto: clin [J/(kgK)] 1051.9 0.387 0.15% 1.78
8 18 p,u
clin [J/(kgK)] 777.0 0.387 0.20% 1.78
v,u
1
C H +(O +3.773N )−→
λ(8+18/4) 8 18 2 2 Table1 Coefficients, normalized RMSE and RMSE in linear
x1O+x2O2+x3H +x4H2+x5OH approximationsofγ,mass-specificcv andcp,fortem-
peratureregionT ∈[300,1000]Kandλ=1
+x H O+x CO+x CO +x NO+x N (9)
6 2 7 8 2 9 10 2
where the products given on the right hand side are chosen by
theuserandλistheair-fuelratio(AFR).Thecoefficientsx are
i
foundbyCHEPPandwhenscaledproperlywithλtheyreveal Iso−octane; Unburned mixture @l =1
1.36
the mole fraction of specie i that the mixture consists of at a
CHEPP
given temperature, pressure and air-fuel ratio. The mixture is Linear
assumedtoobeytheGibbs-Daltonlaw,whichstatesthatunder 1.34
theideal-gasapproximation,thepropertiesofagasinamixture
arenotinfluencedbythepresenceofothergases,andeachgas [−]1.32
component in the mixture behaves as if it exists alone at the go −
mixturetemperatureandmixturevolume[8,Ch12].Therefore, at rati 1.3
thepropertiescanbeaddedtogetherase.g.in: e
h
c
u(T,p)=(cid:88)xi(T,p)ui(T) (10) Specifi1.28
i
1.26
whereu istheinternalenergyfromspeciex anduisthetotal
i i
internalenergy.
1.24
300 400 500 600 700 800 900 1000
Temperature [K]
4 UNBURNEDMIXTURE
Figure1:Specificheatratioforunburnedstoichometricmixture
Firstofall, thespecificheatratioforanunburnedfrozenmix- usingCHEPPandthecorrespondinglinearfunctionoftemper-
tureofiso-octaneiscomputedusingCHEPPinthetemperature ature.
region T ∈ [300,1000] K, which is valid for the entire closed
partofamotoredcycle. Thespecificheatratioforfuel-airratio
λ = 1 is shown in figure 1 as a function of temperature, to-
Thecoefficientsin(2)varywithλasshowninthetwoup-
getherwithitslinearapproximation(2)inaleastsquaressense.
per plots of figure 3. Both the constant γ and the slope b
The linear approximation γu is fairly good for λ = 1. Ac- 300
lin becomesmaller as the fuel-airratio becomes stronger (richer).
tually,thespecificheatsc andc fromwhichγ isformed,are
p v From the bottom plot of figure 3, which shows the RMSE for
fairlywelldescribedbylinearfunctionsoftemperature.Table1
different AFR:s, it can be concluded that the linear approxi-
summarizes the root mean square error (RMSE), normalized
mation γu (T) is better the leaner the mixture is, at least for
RMSE (NRMSE) and the coefficients of the respective linear lin
λ∈[0.8,1.2].
function for γ, mass-specific c and c for temperature region
v p
Sinceitisalwaysdesirabletohaveassimplemodelsaspos-
T ∈ [300,1000]Kandλ = 1. TheRMSEofγu isdefined
lin sible, an important question is: –Would it inflict a major de-
as
(cid:118)
RMSE =(cid:117)(cid:117)(cid:116) 1 (cid:88)M (γ(T )−γu (T ))2 (11) swcirtehptahnecyairt-ofufiexlrtahteios?loTpheiscoisefifincvieesntitgbataenddbyletseotntilnygγt3h0e0svloaprye
M j lin j btothevalueforλ = 1,andfindthecoefficientγ inaleast
j=1 300
squares sense. The slope is fixedat λ = 1, since for spark ig-
whereM arethenumberofsamples[9,p.429]. TheNRMSEis nitedenginesthisistheregionwheretheengineshouldbeop-
thenfoundbynormalizingRMSEwiththemeanvalueofγ(T). eratingmostofthetime,ifcontrolledcorrectly. Thisapproach
leads to figure 4, where the coefficient γ becomes approx-
300
RMSE
NRMSE = (cid:80) (12) imately the same as when letting the slope vary. The relative
M1 Mj=1γ(Tj) differenceislessthan0.1%forλ ∈ [0.8,1.2]. FortheRMSE
an increase for λ (cid:54)= 1 is expected, but the the increase is not
Besides temperature, the specific heat ratio also vary with verysignificantatall,forλ ∈ [0.8,1.12]therelativedifference
AFR,asshowninfigure2whereλisvariedbetween0.8(rich) inRMSEislessthan5%andforλ∈[0.94,1.06]itislessthan
and1.2(lean). Forcomparison,γ(T)isalsoshownforλ=∞, 1%. Thissuggeststhatatleastforλ ∈ [0.94,1.06], thelinear
i.e.pureairwhichcorrespondstofuelcut-off. approximation with fixed slope set at λ = 1, can be used as a
3
Iso−octane; Unburned mixture Iso−octane; Linear model of g for unburned mixture
1.4
1.36
Fix slope
Free slope
l =¥ −]1.35
[0
0
g− [−]1.35 g31.34
o
ati
at r l =1.2 0.8 0.85 0.9 0.95 1 1.05 1.1 1.15 1.2
ecific he 1.3 3x 10−3 Fix slope
Sp l =0.975 Free slope
l =1 −]
l =1.025 l =0.8 E [2.5
S
M
R
2
1.25
300 400 500 600 700 800 900 1000
Temperature [K] 0.8 0.85 0.9 0.95 1 1.05 1.1 1.15 1.2
Air−fuel ratio − l [−]
Figure2:Specificheatratioforunburnedstoichometricmixture
Figure4:Upper:Theconstantvalueγ in(2)asafunctionof
using CHEPP for various AFR:s as functions of temperature. 300
λforunburnedmixtureatequilibriumwithfixedandfreeslope
AFR=∞correspondstopureair.
b respectively. Bottom: RMSE for γu (T) for fixed and free
lin
slopecoefficient.
Iso−octane; Linear model of g for unburned mixture
1.36
[−]01.35 temperatureT,butγ alsodependsupontheair-fuelratioλand
30 pressure p as shown in figure 5 and figure 6 respectively. For
g1.34
the same deviation from λ = 1, rich mixtures tend to deviate
x0 .180−4 0.85 0.9 0.95 1 1.05 1.1 1.15 1.2 more from the stoichometric mixture, than lean mixtures do.
K] Thepressuredependenceofγ isonlyvisibleforT > 1500K,
1/−1.26
[1−1.28 andahigherpressuretendtoretardthedissociationandyielda
b
e −1.3 higherγ.
op−1.32
Sl−1.34
x0 .180−3 0.85 0.9 0.95 1 1.05 1.1 1.15 1.2
3 Iso−octane; Burned mixture @p=7.5 bar
E [−]2.5 1.4
MS 2 1.35
R l =1.2
1.50.8 0.85 0.9 0.95 1 1.05 1.1 1.15 1.2 1.3 l =0.8
Air−fuel ratio − l [1] [−]
go − 1.25 l =0.8
Fofigλurfeor3u:nUbuprpneerd: mThixetucroensattaenqtuvialilbureiuγm30.0Miindd(l2e):aTshaefvuanlucteioonf at rati 1.2
he l =1.2
theslopecoeffientbin(2)asafunctionofAFR.Bottom: Root c
meansquareerror(RMSE)forγluin(T). pecifi1.15 l =0.975
S 1.1 l =1
l =1.025
1.05
modelofγ(T)withgoodaccuracyfortheunburnedmixture.
1
500 1000 1500 2000 2500 3000 3500
Temperature [K]
5 BURNEDMIXTURE
Figure 5: Specific heat ratio for burned mixture at various
Thespecificheatratioγ foraburnedmixtureofiso-octaneis AFR:sat7.5barusingCHEPP.
computedusingCHEPPintemperatureregionT ∈[500,3500]K
and pressure region p ∈ [0.25,100] bar, which covers most of
theclosedpartofafiringcycle.Fortemperaturesabove1700K, Tomodelthespecificheatratiowithalinearfunctionγb (T)
lin
the mixture is assumed to be at equilibrium and frozen other- oftemperatureandtherebyneglectthedependenceofpressure,
wise. The specific heat ratio is strongly dependent on mixture willofcourseintroduceamodelingerror. Thismodelingerror
4
Iso−octane; Burned mixture @l =1 Region T ∈ γ300 b
1.4 A [500,3500] 1.3695 −9.6·10−5
2 bar
7.5 bar B [500,3000] 1.3726 −9.9·10−5
1.35 35 bar C [500,2700] 1.3678 −9.4·10−5
1.3 D [500,2500] 1.3623 −8.8·10−5
[−] E [1200,3000] 1.4045 −11.4·10−5
g− 1.25
o
ati Table2 Coefficients in linear approximation γb (T) found
at r 1.2 in(2)forλ=1andp=7.5bar. lin
e
h
c
cifi1.15
e
p
S
1.1
the model captures the modes of γ(T) quite well. By increas-
1.05
ingthecomplexityofthemodelevenmore,anevenbetterfitis
1 found. ThishasbeendoneintheKrieger-Bormanpolynomial,
500 1000 1500 2000 2500 3000 3500
Temperature [K] andforthisexampleitcapturesthethebehaviorofγ(T)wellfor
temperaturesbelow2700Kasseeninfigure7andintable3. In
Figure 6: Specific heat ratio for burned stoichometric mixture table3,theNRMSEandmaximumrelativeerror(MRE)forthe
usingCHEPPatvariouspressures. linear approximation γb and the Krieger-Bormanpolynomial
lin
γ at various temperature regions are given. As expected,
KB
Region T ∈ γb γ
dependsonwhichtemperature(andpressure)regionthelinear lin KB
MRE NRMSE MRE NRMSE
functionisestimatedfor, sincedifferentregionswillyielddif-
A [500,3500] 2.0% 0.97% 2.0% 0.56%
ferent coefficient values in (2). In figure 7 γ is computed at
B [500,3000] 1.6% 0.95% 0.7% 0.20%
{λ = 1,p = 7.5bar}forT ∈ [500,3500]K,andcomparedto
C [500,2700] 1.9% 0.90% 0.3% 0.17%
the corresponding linear function γb (2) and the polynomial
lin D [500,2500] 2.4% 0.74% 0.3% 0.17%
γ developedby[7]givenin(8).
KB
E [1200,3000] 1.6% 0.74% 0.7% 0.21%
Iso−octane; Burned mixture @l =1, p=7.5 bar Table3 Maximum relative error (MRE) and normalized root
1.4
CHEPP meansquareerror(NRMSE)fordifferenttemperature
Linear
1.35 KB poly regionsatλ=1andp=7.5bar.
1.3
−]
[
g− 1.25
o theKrieger-Bormanpolynomialoutranksthelinearapproxima-
at rati 1.2 tion in every chosen temperature region, since the NRMSE is
e smaller. Comparing just the MRE:s could result in false con-
h
cific 1.15 clusions. TakethetemperatureregionAforinstance,wherethe
pe respective MRE are approximately the same. One could then
S
1.1 conclude that the models describe γ equally well, but in fig-
ure7itwasclearlyvisiblethatγ isthebetterone,whichis
1.05 KB
alsotheconclusionwhencomparingtherespectiveNRMSE.
1 In table 4, the NRMSE and MRE for the Krieger-Borman
500 1000 1500 2000 2500 3000 3500
Temperature [K] polynomialγKB(T,p,λ)whenvaryingtheair-fuelratioλclosely
aroundstoichometricisdisplayed. Forλ ≥ 1(lean),γ fits
KB
Figure 7: Specific heat ratio for burned stoichometric mixture the equilibrium γ better than for λ < 1, a tendency which
usingCHEPP,thecorrespondinglinearfunctionγb andγ is most evident when comparing the NRMSE for temperature
lin KB
foundusingKrieger-Bormanspolynomial. region B. For temperature region A the difference for differ-
ent λ is less striking, since the γ does not fit γ as well
KB
forT > 3000K.ThereforetheKrieger-Bormanpolynomialis
The linear approximation γb (T) does not capture the be- preferably only to be used on the lean side. On the rich side
lin
havior of γ(T) for λ = 1 very well. The coefficients for the andclosetostoichometric(within2.5%),theKrieger-Borman
linear model γb (T) vary for the specific temperature region. polynomialdoesnotintroduceanerrorlargerthanthelinearap-
lin
They are given for temperature regions A to E in Table 2. A proximationgivenintable3,andγ shouldthereforebeused
KB
secondorderpolynomialshowsthesamebehaviorasthelinear inthisoperatingrange.
case,butwhentheorderofthepolynomialisincreasedtothree, If a linear model of γ is requested, the performance of the
5
Reg- γ @λ=0.975 γ @λ=1 γ @λ=1.025
KB KB KB
c (T,p,x )
ion MRE NRMSE MRE NRMSE MRE NRMSE γ (T,p,x )= p b (13c)
A 1.9% 0.86% 2.0% 0.56% 2.1% 0.59% CE b cv(T,p,xb)
B 1.8% 0.73% 0.7% 0.20% 0.7% 0.28% wherex isprovidedbytheVibefunction(38)usingthepres-
b
sureratiomanagementdevelopedbyMatekunas[10],morethor-
Table4 Maximum relative error (MRE) and normalized root oughly described in B. The single-zone (T), burned zone (T )
b
mean square error (NRMSE) for different tempera- andunburnedzone(T )temperaturesaregivenbythetemper-
u
ture regions for γKB(T,p,λ) at p = 7.5 bar and aturesmodels(28)and(34)respectively. Thetwotemperature
λ={0.975, 1, 1.025} modelsaredescribedinA. Thefirstistheordinarysingle-zone
temperaturemodelandthesecondisatwo-zonemeantemper-
aturemodeldevelopedby[11]. Themassspecificheatsin(13)
arecomputedusingCHEPP[5]andγ thenformstherefer-
CE
encemodel.
linearmodelcouldbeenhancedbyproperselectionoftemper-
Tocomputeγ iscomputationallyheavy. Evenwhenthe
ature region. However, the MRE does not decrease for every CE
specificheatsarecomputedbefore-handatanumberofoperat-
reduction in interval, as seen when comparing MRE:s for re-
ingpoints,thecomputationalburdenisstillheavyduetothenu-
gionsDandBintable3. Thus,thetemperatureregionshould
meroustablelook-upsandinterpolationsrequired. Therefore,a
bechosenwithcare:
computationallymoreefficientmodelwhichretainsaccuracyis
• When using the single-zone temperature T to describe sought for. A number of γ-models will therefore be described
the specific heat ratio of the burned mixture, tempera- and compared to γCE found from (13), in terms of normal-
ture region B is preferable, since during the closed part ized root mean square error (NRMSE) and maximum relative
T ≤3000K. error (MRE) for γ, and in terms of RMSE for the correspond-
ingcylinderpressures. Thelastonewillgiveameasureofthe
• When using the burned-zone temperature Tb in a two- impact that a certain model error has on the cylinder pressure
zonemodel,thentemperatureregionEisrecommended, andhelptofindaγ-modelaccurateenoughforthesingle-zone
since for most cases Tb ∈ [1200,3000]. The limits are model. The computional efficiency is evaluated by comparing
found by evaluating a number of experimental cylinder the required simulation time of the cylinder pressure given a
pressuretracesusing(28)and(34). BychosingregionE burnratetraceandaspecificγ-model.
insteadofregionB,theNRMSEisreducedby25%. The eight γ-models are divided into three subgroups; The
firstgroupisbasedupontheburnedmixtureonly. Thesecond
isbasedoninterpolationofthespecificheatratiosdirectly,and
6 PARTIALLYBURNEDMIXTURE thethirdgroup,towhich(13)belongs,isbasedoninterpolation
ofthespecificheats,fromwhichtheratioisdetermined.
ThespecificheatratioγasafunctionofmixturetemperatureT
andair-fuelratioλforunburnedandburnedmixtureofairand
iso-octane has been investigated in the two previous sections.
Burnedmixture
During the closed part of a motored engine cycle, these inves-
tigations would be enough since the models of the unburned Thefirstsubgrouprepresentsthein-cylindermixtureasasingle
mixture will be valid for the entire region. When considering zone of burned mixture with single-zone temperature T, com-
firing cycles on the other hand, neither the unburned nor the putedin(28). Thefirstmodel,denotedB ,isthelinearapprox-
1
burnedmixtureapproachwillbevalidfortheentirecombustion imationin(2),wherethecoefficientsareoptimizedfortemper-
chamberduringthehighpressurepart. atureregionT ∈[500,3000](regionBinTable2)
Therefore,tofindthespecificheatratioforthesingle-zone
B : γ (T)=γb (T)=γ +b(T −300) (14)
modelforapartiallyburnedmixture, themassfractionburned 1 B1 lin 300
tracexbisusedtointerpolatethe(mass-)specificheatsoftheun- Thismodelstructureisusedin[1],althoughthecoefficientval-
burnedandburnedzonestofindthesingle-zonespecificheats. uesdiffersomewhatcomparedtotheonesfoundinTable2.
The specific heat ratio is then found as the ratio between the Thesecondmodel,denotedB ,istheKrieger-Bormanpoly-
2
interpolatedspecificheats. nomialdescribedin(3)
The single-zone specific heats are found from energy bal-
B(T)
ance between the single-zone and the two-zone model, from B : u=A(T)− −→ γ (T)=γ (T) (15)
2 λ B2 KB
which the single-zone specific heat ratio γ can be stated:
CE
withoutthecorrectiontermfordissociation.TheKrieger-Borman
polynomialisusedinmodelB aswell,
3
cp(T,p,xb)=xbcp,b(Tb,p)+(1−xb)cp,u(Tu) (13a) B(T)
B : u=A(T)− +u (T,p,λ)
3 λ corr
−→ γ (T,p)=γ (T,p) (16)
B3 KB
c (T,p,x )=x c (T ,p)+(1−x )c (T ) (13b) withthecorrectiontermfordissociationincluded.
v b b v,b b b v,u u
6
Interpolationofspecificheatratios Simulated cylinder pressure
2
Thesecondsubgroupusesatwo-zonemodel,i.e.aburnedand
1.8
an unburned zone, and finds the specific heat ratio γ (T ) and
b b
γ (T ) for each zone respectively, where the temperatures are 1.6
u u
givenbythetwo-zonemeantemperaturemodel(34). Themass
a]1.4
fraction burned trace xb is used to find the single-zone γ by MP
interpolatingγ andγ . e [1.2
b u ur
Thefirstmodel,denotedC1,interpolateslinearapproxima- ess 1
tionsfortheunburnedandburnedmixture. Thelinearfunctions pr
are optimized for temperature region T ∈ [300,1000] for the der 0.8
n
unburnedmixture,andtemperatureregionT ∈[1200,3000]for Cyli0.6
theburnedmixture. Theresultingγcanthereforebewrittenas
0.4
C : γ (T,x )=x γb (T )+(1−x )γu (T ) (17)
1 C1 b b lin b b lin u 0.2
where the coefficients for the linear functions are given in Ta- 0
−100 −50 0 50 100
ble2andTable1respectively. Crank angle [deg ATDC]
The second model was proposed in [12, p.423], here de-
notedC ,andisbasedonainterpolationoftheinternalenergy
2
ucomputedfromtheKrieger-Bormanpolynomial. Figure 8: Simulatedcylinderpressure using nominal valuesin
table10.
B(T)
C : u=A(T)−x −→ γ (T,x ) (18)
2 b λ C2 b
Evaluationofproposedγ-models
Thismodelincludesneitherdissociationnortheinternalenergy
oftheunburnedmixture. The cylinder pressure given in figure 8 is used as an example
An improvement of model C1 is expected when substitut- toseetheimpactthateachmodelhaveonspecificheatratioγ
ing the linear model for the burned mixture with the Krieger- and cylinder pressure respectively. The γ-models in the three
Bormanpolynomial. Thisnewmodel isdenoted C3 andis de- subgroupspreviouslydescribedarecomputedandcomparedto
scribedby γ (13) in figure 17 as functions of crank angle, where ref-
CE
erence model γ is the dashed line and the solid line corre-
C : γ (T,p,x )=x γ (T ,p)+(1−x )γu (T ) (19) CE
3 C3 b b KB b b lin u sponds to each specific model. By visual inspection it seems
Thefourthmodelinterpolatesγu(Tu)andγb(Tb,p)foundfrom that models C1, C3, C4 and D1 are able to retain the accuracy
CHEPP, ofmodelD2,whichisconfirmedbythenormalizedrootmean
squareerrorsinγ giveninTable5. Thespecificheatratiosfor
C4 : γC4(T,p,xb)=xbγb(Tb,p)+(1−xb)γu(Tu) (20) eachmodelisalsogiveninfigure18asfunctionsofsingle-zone
temperatureT.
andthismodelisdenotedC .Thismodelwillreflectthemodel-
4
ingerrorintroducedbyinterpolatingthespecificheatratiosdi-
Model NRMSE: RMSE: MRE: Time
rectlyinsteadofusingthedefinitionthroughthespecificheats.
γ [%] p[kPa] γ [%] [s]
B (14) 2.3 34.0 6.0 3.8
1
B (15) 3.5 24.9 9.4 4.1
Interpolationofspecificheats 2
B (16) 1.9 24.7 7.3 4.2
3
Thelastsubgroupusesatwo-zonemodel, i.e.aburnedandan C1 (17) 0.57 2.6 2.7 4.7
unburnedzone,justasthesecondsubgroup,anddeterminesthe C2 (18) 3.7 69.7 10.0 4.9
specific heats instead of the specific heat ratios for each zone. C3 (19) 0.52 3.6 2.8 5.1
Theunburnedandburnedzonespecificheatsaretheninterpo- C4 (20) 0.58 2.6 3.0 381.9
latedtogetthesingle-zonespecificheats. Thefirstmodel,de- D1 (21) 0.34 0.2 1.6 5.2
notedD1, usestheKrieger-Bormanpolynomialfortheburned D2 (13) 0.0 0.0 0.0 384.2
zonetofindc (T ,p)andc (T ,p), andthelinearapproxi-
p,b b v,b b
Table5 Evaluationofγ-models.
mations of c (T ) and c (T ) given in Table 1 for the un-
p,u u v,u u
burnedzone:
x cKB(T ,p)+(1−x )clin(T )
D : γ (T,p,x )= b p,b b b p,u u
1 D1 b x cKB(T ,p)+(1−x )clin(T ) Theinfluenceofdifferentγ-modelsonthecylinderpressure
b v,b b b v,u u
(21) areshowninfigure19,wherethedifferencebetweenthesimu-
Thereferencemodelgivenin(13)belongstothisgroupandis latedcylinderpressureformodelD andeachmodelisplotted.
2
denotedD . Alargemodelingerrorinγ willintroducealargedifferencein
2
7
the cylinder pressures, as seen for models B , B , B and C . pressureduringthecombustionisapproximately7.5bar,which
1 2 3 2
But even a small modeling error in γ(T) apparently induces a is the pressure that γb is optimized for. But for operating
lin
smallbutcrucialfaultduringthehighcompressionphaseofthe pointswhenthisapproximationisnotvalid,C ≺C .
3 1
closed part of the engine cycle. The RMSE of the measure- ComparingRMSEforthecylinderpressurep,theordering
mentnoiseisapproximately6kPainthenon-filteredcaseand oftheγ-modelsbecomes:
approximately2kPainthefilteredcase. Ifthemeasuredcylin-
D ≺D ≺{C ,C ,C }≺B ≺B ≺B ≺C (23)
derpressureisusedtocalculatee.g.theheatreleased,thiswill 2 1 1 3 4 3 1 2 2
be done using a filtered cylinder pressure and then it is only i.e.thesameasin(22).
modelD thatintroducesasmallermodelingerrorintermsof
1
RMSE.Thus,theotherγ-modelswillintroduceamodelinger- Summary Theresultscanbesummarizedas:
rorwhichissignificantlylargerthanthemeasurementnoiseas
• The choice between C and C is not clear, since one is
seeninTable5,andtherebyaffecttheaccuracyoftheparameter 1 3
better than the other for a certain operating range. Still,
estimates.
the operating range where model C is the better one, is
Thesameapproachhasbeenmadeforthesimulatedcylin- 3
thelargestandthereforemodelC isrecommendedwhen
derpressurefromninedifferentoperatingpoints,wherep ∈ 3
IVC a interpolation of the specific heat ratios directly is re-
[0.25,2]barandT ∈[325,372]K.Theparametersforeach
IVC quired.
cycleisgivenintable12aswellasthecorrespondingcylinder
pressuresinfigure16.TheoperatingrangeinpandT thatthese • ComparingmodelsC andD ,itisobviousthatinterpo-
4 2
cyclescoverisgiveninfigure9,wheretheupperplotshowsthe latingthespecificheatratiosinsteadofthespecificheats
range covered for the unburned mixture, and the lower shows causesalargemodelingerror.
the range covered for single-zone (solid) and burned (dashed)
• If only single-zone temperatures are allowed, model B
mixture. According to [13, p.109], the temperature region of 3
isthebetterone.
interestforanSIengineis400to900Kfortheunburnedmix-
ture;fortheburnedmixture,theextremeendstatedareapprox- • ModelC haspoorperformance.
2
imately{2800 K, 3.5 MPa}and{1200 K, 0.2 MPa}. Of
• Model D is required to get a cylinder pressure RMSE
course,notallpointsintherangearecoveredbutthecyclesat 1
thatisinthesameorderofsizeasthemeasurementnoise.
handcovertheextremesoftherangeofinterest.
• The computational effort is approximately the same for
10 allmodelsexceptD andC .
2 4
a] Unburned zone
P Orderingofthemodelsbytheirperformanceandwithcompu-
M
ure [ 5 tationalefficiencyinascendingorder:
s
Pres D2 ≺D1 ≺C3 ≺B3 ≺B1 (24)
3000 400 500 600 700 800 900 1000 Anoteontimecomplexity Introducingthismodelimprove-
Temperature [K] mentofthespecificheatratiototheGatowskietal.single-zone
10 heatreleasemodelissimple,anditdoesnotincreasethecom-
Burned zone
Pa] Single zone putational burden immensely. The computational effort is in-
M creasedbyafactor2comparedtothelinearγ-modelwhensim-
ure [ 5 ulating the Gatowski et al. single-zone heat release model, ac-
s
s cordingtotable5wherethecomputionaltimesforD andB
e 1 1
Pr aredisplayed.
0
0 500 1000 1500 2000 2500 3000 3500
Temperature [K] A note on crevice volume modeling Note that introducing
aγ-modeldifferentfromthelinearmodelusedin[1],willalso
Figure9: OperatingrangeinpandT. Upper: Unburnedzone.
affecttheamountofenergyleftoraddedtothesystemwhena
Lower: Singlezone(solid)andburnedzone(dashed).
masselemententersorleavesthecrevicevolume. Thisenergy
termhastoberestatedforeveryγ-modelathandexceptmodel
B .
ThesesimulationsaresummarizedintermsofNRMSEfor 1
γ (table 13, RMSE for p in table 14 and MRE for γ in ta-
A note on fuel composition A small, and by no means ex-
ble 16. When comparing the NRMSE for γ, the ordering of
haustivesensitivity analysis is made for fuels such as methane
theγ-modelsare:
andtwocommercialfuelsinappendixC. Thisinordertoseeif
theresultsarevalidforotherfuelsthaniso-octane. Itisfound
D ≺D ≺{C ,C ,C }≺B ≺B ≺B ≺C (22)
2 1 1 3 4 3 1 2 2
that the hydrocarbon ratio needs to be close the one for iso-
where B ≺ C means that model B is better than C . For octane(2.25),althoughanexactlimitcannotbegivenwithout
2 2 2 2
{C ,C ,C } it is found that C ≺ C when the mean cylinder furtherstudies.
1 3 4 1 3
8
Influence of residual gas The influence of the residual gas thecombustion,themassspecificheatsforthesinglezonewill
onthespecificheatratiohassofarbeenneglected. Introducing coincidewith theones for theburnedzone inaccordance with
theresidualgasmassfractionx , thesingle-zonespecificheat themodelassumptions,andthusnodifferenceinγ.
r
ratioγ in(13)isreformulatedas:
CE
Iso−octane; In−cylinder mixture for x ˛ [0, 0.15]
c (T,p,x ,x )=x c (T ,p)+(1−x ) (c (T ) +x c (T ,p)) x 10−3 r
p b r b p,b b b p,u u r p,b u 8
(25a)
7
−]
cv(T,p,xb,xr)=xbcv,b(Tb,p)+(1−xb) (cv,u(Tu)+xrcv,b(Tu,p))go [6
(25b) ati
at r5
e
c (T,p,x ,x ) h
γCE(T,p,xb,xr)= cvp(T,p,xbb,xrr) (25c) ecific 4 xr=0.15
p
wherethemodelassumptionsare: e in s3 xr=0.10
c
• the residual gas is homogenously distributed throughout en
er2
thecombustionchamber Diff
1
• the residual gas is described by a burned mixture at the
x=0.05
appropriatetemperatureandpressure r
0
−100 −50 0 50 100
• aresidualgasmasselementintheunburnedzoneassumes Crank angle [deg ATDC]
theunburnedzonetemperatureT
u
Figure11:Differenceinspecificheatratioγ forresidualgas
CE
• whenaresidualgasmasselementcrossestheflamefront, fractionx =[0.05 0.1 0.15]comparedtox =0.
r r
it enters the burned zone and assumes the burned zone
temperature T . Note that the pressure is assumed to be
b
homogenousthroughoutallzones. Asimplemodeloftheinfluenceofx onγ istomodelthe
r
influenceasalinearfunctionofx duringtheclosedpart,i.e.
r
Infigure10,thespecificheatratioγ iscomputedaccording
CE
to (25) for residual gas fractions xr = [00.050.10.15] given γ(T,p,xb,xr)=γCE(T,p,xb)+bxr =γCE(T,p,xb)+γbias(xr)
the cylinder pressure in figure 8. It shows that the larger the (26)
residual gas fraction, the larger the γ. The difference in γ for whereγ (T,p,x )isgivenby(13). Sincex isconstantdur-
CE b r
ing a cycle, the term bx can be considered as a constant bias
r
Iso−octane; In−cylinder mixture for x ˛ [0, 0.15] γ (x ) that changes from cycle to cycle. At the operating
r bias r
1.35 point in figure 11, the RMSE in γ is reduced from 2·10−3 to
x=0.15
r 0.8·10−3byusing(26).
x=0
1.3 r
−] 7 CONCLUSIONS
[
g−
o 1.25 Basedonassumptionsoffrozenmixturefortheunburnedmix-
ati
at r ture and chemical equilibrium for the burned mixture [7], the
he specific heat ratio is calculated using a full equilibrium pro-
c 1.2
cifi gram [5] for an unburned and a burned air-fuel mixture, and
e
p comparedtoseveralapproximativemodelsofγ.Itisshownthat
S
specific heat ratio for the unburned mixture is captured within
1.15
0.2 % by a linear function in mean charge temperature T, and
the burned mixture is captured within 1 % by a higher-order
polynomial in cylinder pressure p and T developed in [7] for
1.1
−100 −50 0 50 100
the major operating range of a spark ignited (SI) engine. If a
Crank angle [deg ATDC]
linearmodelisrequiredfortheburnedmixture,thetemperature
Figure 10: Specific heat ratio γ for residual gas fraction regionshouldbechosenwithcare.
CE
x =[0 0.05 0.1 0.15]. Withtheknowledgeofhowtodescribeγ fortheunburned
r
and burned mixture respectively, the focus is turned to finding
a γ-model during the combustion process, i.e. for a partially
x = 0 and x = [0.05 0.1 0.15] is shown in figure 11. The burnedmixture. Thisisdonebyfirstfindingthemassfraction
r r
differenceislargestduringcompressionandcombustion. After burnedx usingtheMatekunaspressureratiomanagement,then
b
9
representingx bytheVibefunctionandfinallybyinterpolating [9] F.Gustafsson. AdaptiveFilteringandChangeDetection.
b
the specific heats for the unburned and burned mixture using Wiley,2000.
x . It is found that interpolating the linear specific heats for
b
[10] F.A.Matekunas. Modesandmeasuresofcycliccombus-
theunburnedmixtureandthehigher-orderpolynomialspecific
tionvariability. 1983. SAETechnicalPaper830337.
heatsfortheburnedmixture,andthenformingthespecificheat
ratio
[11] I. Andersson. Cylinder pressure and ionization current
γ(T,p,x )= cp(T,p,xb) = xbcKp,bB +(1−xb)clpi,nu (27) mhiocudlealrinSgyfsotremspsa,rk20ig0n2i.tedLeiUng-TinEeKs.-LTIeCch-2n0ic0a2l:3re5p,oTrth,eVseis-
b c (T,p,x ) x cKB +(1−x )clin
v b b v,b b v,u No.962.
rendersasmallenoughmodelingerrorinγ.Thismodelingerror [12] R. Stone. Introduction to Internal Combustion Engines.
issmallenoughsinceitcausesarootmeansquareerrorof7.4 SAEInternational,3rdedition,1999.
kPainthecylinderpressurepfortheworstcase,anerrorwhich
is in the same order as 6 kPa rendering from the measurement [13] J. B. Heywood. Internal Combustion Engine Funda-
noise. mentals. McGraw-Hill series in mechanical engineering.
Introducingthismodelimprovementofthespecificheatra- McGraw-Hill,1988.
tiototheGatowskietal.single-zoneheatreleasemodelissim-
[14] Y.NilssonandL.Eriksson. Anewformulationofmulti-
ple,anditdoesnotincreasethecomputationalburdenimmensely.
zone combustion engine models. IFAC Workshop: Ad-
Infact,itisincreasedonlybyafactor2comparedtothelinear
vancesinAutomotiveControl,pages629–634,Karlsruhe,
γ-model when simulating the Gatowski et al. single-zone heat
Germany,2001.
releasemodel.
[15] L.Eriksson. SparkAdvanceModelingandControl. PhD
thesis, Linko¨ping University, May 1999. ISBN 91-7219-
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