1999-01-3548

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1999-01-3548

A Simple Approach to Studying the Relation between Fuel Rate

Heat Release Rate and NO Formation in Diesel Engines

Rolf EgnelLund Institute of Technology

Copyright © 1999 Society of Automotive Engineers, Inc.

ABSTRACT

Modern diesel engine injection systems are largely com-puter controlled. This opens the door for tailoring the fuelrate. Rate shaping in combination with increased injec-tion pressure and nozzle design will play an important

role in the efforts to maintain the superiority of the dieselengine in terms of fuel economy while meeting futuredemands on emissions.

This approach to studying the potential of rate shaping inorder to reduce NO formation is based on three sub-models. The first model calculates the fuel rate by usingstandard expressions for calculating the areas of thedimensioning flow paths in the nozzle and the corre-sponding discharge coefficients. In the second sub-model the heat release rate is described as a function ofavailable fuel energy, i.e. fuel mass, in the cylinder. Thethird submodel is the multizone combustion model thatcalculates NO for a given heat release rate underassumed air /fuel ratios.

When studying the potential of fuel rate shaping, differentheat release rates are generated by using Vibe functions.The NO formation, convective heat losses, IMEP, indi-cated thermal efficiency and fuel rate are calculated forevery given heat release rate.

The calculations show that by increasing the length of theinjection period and maintaining a smooth and even fuelrate, the NO formation could be reduced with very limitedreduction of the indicated thermal efficiency. In the exam-ple given in the paper, NO is reduced by about 20% with

an increase in the fuel consumption of only 1%.

INTRODUCTION

The increasing demands on the environmental propertiesof diesel engines call for a deeper understanding of thelimits of the typical diffusive type of combustion that ischaracteristic of the diesel engines of today.

Recent reports, [1] and [2], on the combustion and emis-sion formation in diesel engines claim that the NO forma-

tion takes place on the core of the spray at close tostoichiometric air/fuel ratio. This conclusion is supportedby previous work of the author [3] where the engine-ouNO concentration is calculated for different assumptionson the average equivalence ratio. Comparing the calcu-lated results with measured data for several load cases

revealed that model prediction was close to measurements if the average equivalence ratio in the zones wheremost of the NO is formed was slightly higher than 1.

In this paper results are presented from calculationswhere the average equivalence ratio is kept constant forvarious inputs of the heat release rates (HRR). The ratesare given as the sum of two Vibe functions, one repre-senting the premixed, and the other the diffusive, com-bustion.

In order to find a connection between the HRR and thefuel rate, measurements were performed at 6 differentconditions where the fuel nozzles and load conditions

were altered. The fuel rates were calculated with themodel shown below. This model is based on the flowequation according to Bernoulli, a geometrical model ofthe flow path in the nozzle and a mathematical expres-sion [4] for the discharge coefficient where the influenceof Reynolds number, hydraulic diameter and length of thepath are given.

The calculated fuel rate is compared with the HRR calcu-lated from the pressure trace, and it is found that HRRcan reasonably well be described by a very simple differential equation where the HRR is a given as a function othe available fuel energy, i.e. the fuel mass, in the cylin

der. This approach is actually a simplified version of themodel presented in [5]. The difference is that the influence of turbulence on the HRR is not explicitly accountedfor.

The derived expression for the relation between the fuelrate and the HRR is used to connect the given HRR, pro-vided by the Vibe functions, to the corresponding fuelrate. Thus it is possible to connect the fuel rate to the NOformation.

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MODELS

FUEL RATE MODEL – The flow dimensioning areas inthe flow path in the nozzle are the truncated cone shapedarea between the needle and the needle seat and thehole area. The geometry of the former varies with theneedle lift.

Figure 1. Flow paths in a nozzle [6]

The mantel surface area of the truncated cone can becalculated by using the following parameters shown inFigure 1 where:

H is the needle liftR is the base circle radius of the truncated cone

α is half the tip angle of the needle

By using standard geometrical relationships it can beshown that the truncated cone area, i.e. the flow areabetween the needle and the seat, can be expressed inthe following way:

(Eq. 1)

The total area of the holes is not varied and is given by

the following expression:

(Eq. 2)

Where:

is the number of holes. [-]

is the diameter of the holes. [m]

is a correction factor for the hole diameter. [-]

The effective flow area is calculated by multiplying thegeometric area by the discharge coefficient, C d, which iscalculated with the following expression [4]:

for

(Eq. 3

(Eq. 4

Where:

is the length of the hole or the valve seat [m].

is the hydraulic diameter of the hole or the valve sea[m]. For the holes, D is the nominal hole diameter multiplied by the correction factor: .

is the Reynolds number, i.e. , where:

is the speed of the flow [m/s]

is the kinematic viscosity [m2 /s]

The pressure drop across the valve seat is given by:

(Eq. 5

The pressure drop across the holes is given by:

(Eq. 6

Where:

is the injection pressure [Pa]

is the pressure in the sac [Pa]

is the pressure in the cylinder [Pa]

is the speed through the valve seat [m/s]

is the speed through the holes [m/s]

is the density of the fuel [kg/m3]

Due to the incompressibility of the fuel the mass

flow through the valve se at,, and the holes, is assumed to be equal:

(Eq. 7

(Eq. 8

(Eq. 9

)sin()]cos()sin(2[ α α α π H H R Acone −=

25.0)(2 ××××= holes Dholesholes DCorr N A π

holes N

holes D

DCorr

L

D

DCorr

Reν

CD=Re

C

ν

2

2

conesacinj

C PP

×=−

ρ

2

2

holescylsac

C PP

×=−

ρ

injP

sacP

cylP

coneC

holesC

ρ

conem&

holesm&

coneconeconecone ACd C m ×××= ρ &

holesholesholesholes ACd C m ×××= ρ &

holescone mm && =

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The calculated injected fuel mass per injection,, is given by the following expression:

(Eq. 10)

Where:

is the time at start of injection [s]

is the time at end of injection [s]

The injection pressure and the pressure in the cylinderare given at every time step of calculation. The pressurein the sac is calculated by iteration where the speeds,Reynolds numbers and discharge coefficients at the seatand the holes are calculated. The needle lift is also givenat every time step and thus the mantel surface area of thetruncated cone can be calculated.

When the areas of the flow paths, the corresponding dis-charge coefficients, the flow speeds and the densities aregiven, the mass flow, i.e. the flow rate, can be calculated

using equations (7) or (8).

Model Verification – The fuel rate model was verified bycomparing calculated and measured accumulated fuelflow data. The measurements were performed on a sin-gle cylinder supercharged heavy-duty diesel engine witha displacement of two liters. The compression ratio was18:1 and a unit injector system was used. During the testseries, consisting of 6 runs, the engine speed was keptconstant at 1200 rpm. The fuel pressure, the cylinderpressure and the needle lift were recorded during thetest. The following input to the calculations was kept con-stant at all runs:

R = 0.6e-3 [m]

α = 47.5 [o]

N holes = 6 [-]

D cone = half the gap between the needle and the seat

Lholes = 0.6e-3 [m]

Lcone = 0.01e-3 [m]

ν = 3.0e-6 [m2 /s]

ρ = 840 [kg/m3]

The test variables are shown in Table 1 below. IP standsfor Input and refers to the run number.

Table 1. Test conditions and input for calculations

D holes is the nominal hole diameter

H max is the maximum needle lift

C 1000 is the plunge speed at 1000 rpm

Delta is the measured mass of fuel injected at eachcycle

CorrD1 is the correction factor for Dholes to get the mea-sured fuel mass when using the model. See equation (2).

CorrD2 is the correction factor for Dholes to get the measured fuel mass when using the model while keepingCd cone=1.

When using the nominal hole diameter the accumulatedfuel mass, i.e. Delta, is underestimated with the flowmodel at all runs. There are many possible reasons fothis, one being that the holes are somewhat bigger in

reality than the nominal diameter. The variable Corr D1represents the increase of the hole diameter necessaryto get the measured fuel mass when using the flowmodel. The figures correspond quite well with the uppermanufacturing tolerance of the holes.

Another explanation is that the discharge coefficient ofthe valve seat, i.e. Cd cone, is under-estimated. With thegiven input, the value is close to 0.8 at maximum flow. Asthe flow path in the gap is constantly narrowing it is possible that the contraction of the flow is much less. For thatreason the fuel flows were recalculated with the assumption that Cd cone = 1. It was then found that the model

overpredicted the flow and, in order to get agreemenwith measured data, a correction factor that reduced thehole diameters was introduced. This factor CorrD2 for thedifferent nozzles is also shown in Table 1.

Since the fuel flow model does not include any pressureloss due to turbulence in the sac, it is very likely that thedischarge coefficient of the valve seat is closer to 1 thanthe model predicts. Some typical calculated results fromthe flow model are shown in the Figures below. Cd cone =1 and CorrD2 is used when generating the curves.

DeltaCalc

∫ = EOI

a

t

t

holes dt t m DeltaCalc )(&

α t

EOI t

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Figure 2. Cylinder, Injection and Sac Pressures vs CAD.IP 1

Figure 2 shows the measured injection and cylinder pres-sures and the calculated sac pressure. As can be seen,

the sac pressure is close to the cylinder pressure at thefirst crank angles when the needle is opening, e.g. theflow is mainly controlled by the valve seat area. It canalso be seen that injection pressure falls rapidly and evenbelow cylinder pressure at the end of injection. As will beshown in the next figure, this results in a negative fuelflow which is not likely to happen. The reason is a bumpin the needle lift curve.

Figure 3. Calculated injected fuel flow. IP 1

The last two figures, 3 and 4, show the calculated fuelflow and accumulated fuel mass during the injectionperiod. The star in Figure 4 shows the measured fuelmass, i.e. Delta. The correction factor CorrD2 was varieduntil the Delta and the calculated fuel mass, DeltaCalc,agreed. In this particular case, e.g. IP=1, the value of thecorrection factor was found to be 0.956. This means thatthe nominal hole diameter had to be reduced by 4.4%.

Figure 4. Accumulated injected fuel mass versus CAD.IP 1

The conclusions from using the fuel flow model are that

the discharge coefficient at the valve seat is probablyunderestimated and that the pressure loss due to turbulence in the sac should be accounted for in order to getbetter agreement with measured data.

HEAT RELEASE RATE (HRR)- FUEL RATE (FR)MODEL – When examining different experimental caseswhere the HRR and FR were calculated it, was observedthat the following equation gave a reasonable relationshipbetween the two rates:

(Eq. 11)

Where:

is the heat release rate, HRR, [W]

is the accumulated released heat, HR, [J]

is the energy of the accumulated injected fuel [J]

is an empirical constant [1/s]

The fuel energy can be expressed in the following way:

(Eq. 12)

Where:

is the lower heating value of the fuel [W/kg]

is the fuel rate [kg/s]

is the time at start of injection [s]

is the time of the calculation step in question [s]

)( QQk dQ fuel −=

dQ

Q

fuelQ

k

∫ =t

t

u fuel dt t m H Q

α

)(&

u H

m&

α t

t

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Figure 5. Measured and calculated HRR. IP 1






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The interpretation of equation (11) is that the HRR is pro-portional to the mass of injected fuel not yet combustedin the cylinder. The figures above show the results ofHRR calculations based on the fuel rate calculations dis-cussed above, i.e. based on input data given in Table 1and in the text below the table.

The net portion of the “measured” HRR, ,is calculatedwith an expression derived from the first law of thermody-namics. Any effects of blowby, crevices and the enthalpy

of the injected fuel are neglected.

(Eq. 13)

Where:

is the net heat release [J]

is the ratio of specific heats [-]

is the pressure [Pa]

is the volume [m3]

The ratio of specific heats is calculated by using theactual fluid composition according to the multi-zonemodel used. The method is thoroughly described in [3].

The heat transfer is calculated with the standard expres-sion for convective heat transfer, i.e.:

(Eq. 14)

Where:

is the heat transferred to the wall by convection

[J]is the heat transfer coefficient [W/K m2]

is the area of the wall [m2]

is the temperature of the gas [K]

is the temperature of the wall [K]

The heat transfer coefficient is calculated by theexpression proposed by Woschni as presented in [7].

The sum of and is the gross heat release which isthe quantity labeled “measured” in the previous figures.

Different values of the empirical constant in equation(11) were tested, and it was found that k =500 [1/s] gavereasonable agreement with the HRR derived from thecylinder pressure measurements. Thus, this value wasused in the calculation of the results shown in the figuresabove.

The constant can be interpreted as a spray or atomiza-tion characteristic number. The higher the value of k thesmaller the droplets and the better the penetration, i.e.factors that control the rate of combustion. For a givenspray can probably be related to the swirl ratio or the

turbulence. Heat release rates for different values of k areshown in Figure 11 below.

Figure 11. The influence on calculated HRR of theconstant k. The curves correspond to k values:

100, 300, 500, 1000, 1500, 2000 and 3000.

The Fuel Rate to Heat Release Rate model does notinclude any ignition delay calculations at the moment.This means that the heat release is assumed to start assoon the fuel enters the cylinder. Or, if the heat release isgiven, the fuel injection starts at the time of combustioninitiation. This weakness of the model does not play anyimportant role when the ignition delay is short, as it wasin the cases studied. If, however, the model should beused in combustion cases with long ignition delays, itmust include a submodel for the delay.

NO CALCULATION – The model used for NO calculationis described in depth in [3]. Basically it is a multizonecombustion model where the HRR is used as input. Atevery time step of calculation the amount of air and fuethat corresponds to the released heat during the step inquestion is combusted in a simple combustion modelThe local temperatures are determined by minimizingGibb’s free energy. NO formation in the combustion andpost combustion zones is calculated with the Zeldovichmechanism.

The combustion and post combustion zones could bedivided into different subzones with different assumed

equivalence ratios. In this work, however, it is assumedthat the combustion influencing the NO formation takesplace at an average equivalence ratio close to one (1.05)This ratio was found when comparing calculated andmeasured NO emissions. Another assumption in themodel is that the post combustion zones do not mix withother zones or the fresh charge as long as the NO formation continues.

nQ

dQn

dt p

dV

dt V

dp

dt =

−+

−γ

γ γ 1

1

1

nQ

γ

p

V

dQ

dt h Area T T

ht

c wall gas wall= × −( )

Qht

hc

Areawall

T gas

T wall

hc

nQ Qht

k

k

k

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VIBE FUNCTION REPRESENTATION OF HRR – Vibefunctions are frequently used to describe the heatrelease in IC engines. In this model the sum of two Vibefunctions is used to represent the heat release in a dieselengine. The first describes the premixed and the secondthe diffusive part of the heat release.

A Vibe function has the general form:

(Eq. 15)

Where:

is the fraction of the total released heat atCAD [-].

is the crank angle at which the heat releasestarts [CAD].

is the duration of the heat release [CAD].

and are constants [-]

By using Vibe functions for the premixed and the diffusivestages of combustion, the total released heat can be rep-resented by the following expression:

(Eq. 16)

Where:

is the released heat at crank angle [J]

is the released heat at the end of combustion [J]

is the value of the Vibe function for thediffusive released heat at crank angle [-]

is the value of the Vibe function for thepremixed released heat at crank angle [-]

is the premixed heat release fraction of thereleased heat by diffusive combus-tion at the end of com-bustion. [-]

A typical HRR for a diesel engine generated with Vibefunctions is shown in Figures 12 and 13 below. The fol-lowing input was used:

Table 2. Input to the HRR shown in Figures 12 and 13below

The figure below shows the contribution of the premixedHRR and the diffusive HRR on the total heat release ratedQ. The input is chosen to get a reasonable correspondence to a measured HRR curve. This is shown in Figure13 below. The proportions between the energy releasedat premixed and diffusive phases of combustion are notcorrelated to the fuel flow during the actual ignition delay

time.

Figure 12. Typical HRR curve generated with Vibefunctions

Separating the two combustion modes could be impor-tant when looking at NO formation. According to [1] thepremixed combustion takes place at such a high equiva-lence ratio that NO formation is not likely to occur. As can

be seen in Table 2 and Figure 12, the premixed and diffu-sive combustion phases are slightly overlapping (0.7CAD). Experiments with optical access have to be per-formed in order to clarify to what degree this is true.

)exp(1)(

1

0

+

∆ΘΘ−Θ

−−=Θ

m

a X

X ( )ΘΘ

Θ0

∆Θ

a m

( ) )1 /()]()([ +Θ+Θ=Θfrac pm fracdiff tot k X k X QQ

)(ΘQ Θ

tot Q

)(Θdiff X

Θ)(Θ pm X

frack

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Figure 13. Measured and Vibe function generated HRR.

When generating the pressure trace on which the “mea-sured” HRR curve was based, the test engine was, in thiscase, equipped with a Common Rail injection system.The injection period was 15 CAD. The calculated NOemission for the measured HRR case and from the Vibefunction HRR agreed quite well. The Vibe function basedNO was 97% of the calculated value based on measuredHRR data. The corresponding figures for the calculatedIMEP and ITE (Indicated Thermal Efficiency) were 95%in both cases. This shows that it is possible to representdiesel engine combustion HRR with the sum of two Vibefunctions.

PARAMETRIC STUDY

In order to study the effect on NO, IMEP, ITE of different

heat release rates and to calculate the correspondingfuel rates, a set of calculations was performed where dif-ferent HRR were generated by changing the constant inequation (15) for the diffusive case.

Figure 14. Vibe function based heat release rates.varied from 10 (the top curve) to 3.

By reducing the value from 10 (which was used in Figure13) to 3 in steps of 1, 8 different HRR curves were cre-ated. All data in Table 2, except for the diffusive com-bustion, were kept constant. The different HRR curvesare shown in Figure 14. As can be seen, the premixedportion of the heat release is not affected by the parametric change. This could of course be questioned since thedifferent HRR curves could be accomplished with fuelrate shaping as will be discussed below. Thus the fuelrate during the ignition delay could also be shaped and

the premixed portion would be more related to the fuerate during the diffusive phase. However, since this para-metric study is chosen to demonstrate the generaapproach it is of pedagogic value to simplify the parametric change.

Figure 15. Pressure curves calculated with the HRRcurves shown in the Figure 14 above.

The pressure curves corresponding to the HRR shown inFigure 14 can be seen in the figure above. The maximumpressure is reduced from about 180 bar to 130 bar. Thelocation of the peak pressure is also changed.

Figure 16. Normalized NO emissions versus

a

diff a

a

diff a

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For this reason it can be expected that the chosen heatrelease rates will have a considerable influence on NOformation, IMEP and ITE.

Figure 16 shows the relative NO versus the constant inthe Vibe expression for the diffusive combustions. By rel-ative is meant the normalized value expressed as percent. The calculated NO emission is almost halved overthe parameter range. This is mainly due to the reductionof the maximum temperature.

Figure 17. Relative IMEP versus

Figure 17 shows the impact on IMEP. The change of thepressure curve results in a reduction of IMEP by 12%.However, as can be seen in Figure 18 below, this doesnot give the same reduction of the Indicated Thermal Effi-

ciency (ITE). Over the parameter range only about 4.5%of the baseline efficiency is lost. This somewhat surpris-ing result is explained by a decrease in thermal losses.

Figure 18. Relative ITE versus

Figure 19. Qnet and Qgross versus

The heat release rates created with the Vibe functionsare the net HRR, i.e. the apparent heat release rate. In

the model this HRR is used to calculate the pressure andthe temperature in the cylinder. These quantities are thenused to calculate the convective heat losses using theexpression proposed by Woshni. The method isdescribed in [7]. The sum of the net heat release and theconvective heat losses is the gross heat release which isused when calculating the ITE.

When the constant in the Vibe expression for the diffu-sive combustion is decreased, the temperaturedecreases, which reduces the heat losses and, conse

quently, the gross heat release. This is shown in Figure19. Thus the decrease of IMEP is followed by a reductionof the supplied energy, which explains why ITE is lessaffected than IMEP.

Figure 20. Relative ITE versus relative NO

a

diff a

diff a

diff a

a

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By combining the results shown in Figures 16 and 18, thetrade-off between NO and ITE can be illustrated. This isdone in Figure 20 above. As can be seen, the loss of effi-ciency when NO is reduced by 45% is about 4.5%. Thecorresponding value for a reduction of 20% is about 1%.

The presented parametric study is chosen to illustrate theapproach. No attempts have been made to find the opti-mum HRR in order to minimize NO with the least loss ofITE.

Figure 21. Calculated fuel rates for the HRRs given inFigure 15

The final figure in this paper, i.e. Figure 21, shows thefuel rates calculated for the Vibe function created HRRs.Different values of in equation (11) were tested to get areasonable fuel rate in all cases. At lower - values, neg-

ative fuel flows were needed to obtain the given HRR.The fuel rates in Figure 21 were generated with= 2200, which is considerably higher than the value

that matched the measured data when the fuel ratemodel was verified. See “Model Verification” above. Thismeans that, in order to use the fuel rate to create adesired HRR, a change in the spray characteristics and/ or the flow pattern may be necessary.

As the fuel flow model does not take ignition delay intoaccount all fuel, that in the real case would have beeninjected during the delay period, is now supplied to thecylinder during the premixed phase of combustion. Thiscan be seen in Figure 21. The premixed combustion wasnot changed during the parametric study, see Table 2.Thus the fuel rate is the same for all cases at the begin-ning of the injection period.

When the diffusive HRR is flattened out by the reductionof the constant in the Vibe expression for the diffusivecombustion, the fuel rate also becomes more even, ascan be seen in Figure 21.

The conclusion from the parametric study is that the NOformation could be reduced, with limited effects on fuel

consumption, by improving the conditions for fast com-bustion and injecting the fuel slowly and evenly.

SUMMARY

This paper is focused on a simple approach to studyingthe relation between fuel rate, heat release rate and NOformation in diesel engines. Models are presented for:

1. Fuel flow calculations

2. The connection between fuel flow and heat releaserate

3. Creation of heat release rates with Vibe functions

4. NO calculations based on heat release rates [3]

In order to illustrate the approach, the different sub- models are connected and used in a parametric study wherethe shape of the diffusive heat release rate is changed.The effect of these changes on NO, indicated thermalefficiency, indicated mean effective pressure and fuel rateare presented.

CONCLUSION

The general conclusion from the work presented in thispaper is that fuel rate shaping could be effective in reducing NO formation with limited effects on fuel consump-tion. In the parametric study, where the diffusivecombustion rate was changed, it was found that IMEPdecreased when NO was reduced, but the effect on thethermal efficiency was less due to lower thermal losses

The verification of the fuel rate model revealed that thedischarge coefficient in the valve seat path is probably

higher than the model predicts. The model value is about0.8 at maximum flow, but the value is most likely closer to1.

The very simple differential equation used to describe theconnection between fuel rate and heat release rate givesreasonable results in 6 different cases. The modelassumes that the heat release rate is proportional to theavailable fuel energy in the cylinder. The proportionaconstant could be interpreted as a measure of the size othe combustion zone for a given amount of fuel mass inthe cylinder. This size depends on the spray propertiesand the gas motions in the cylinder.

The faster the combustion takes place, the easier it willbe to tailor the heat release with the shape of the fuelrate.

ACKNOWLEDGMENTS

The author would like to thank Scania CV AB for supply-ing the measured data for the diesel engines. The projechas been financed by the National Council for TechnicaResearch and Vehicle Engineering. The author is verygrateful for this support

k k

k

a

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REFERENCES

1. Dec, J.D. “A Conceptual Model of Diesel CombustionBased on Laser-Sheet Imaging” SAE Paper 970873

2. Flynn, P.F. “Diesel Combustion: An Integrated View Com-bining Laser Diagnostics, Chemical Kinetics, and EmpiricalValidation. SAE-Paper 1999-01-0509.

3. Egnell, R. “Combustion Diagnostics by Means of MultizoneHeat Release Analysis and NO Calculation”. SAE-Paper981424

4. Merritt, H.E. “Hydraulic Control Systems” John Wiley &

Sons, Inc. 19675. Chmela, F.G. and Orthaber, G.C. “Rate of Heat Release

Prediction for Direct Injection Diesel Engines Based onPurely Mixing Controlled Combustion. SAE-paper 1999-01-0186.

6. Gåsste, J. “Measurements and Calculations of the Flow outof Diesel Nozzles” Internal report No. 98. Internal Combus-tion Engines, Department of Machine Design, Royal Insti-tute of Technology. 1997

7. Heywood, J. B. ”Internal Combustion Fundamentals”McGraw-Hill series in mechanical engineering. 1988.

CONTACT

The author, Rolf Egnell, has been working in an industry-sponsored project concerning NOx formation in dieselengines since 1996. He has also been doing research ondirect injected, natural gas fueled Otto engines. RolfEgnell can be contacted through the following emailaddress: [email protected].

1999-01-3548

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