Master Thesis CEMIL Final

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Numerical Modeling of Diesel SprayFormation and Combustion

Cemil Bekdemir

WVT 2008.15

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Numerical Modeling of Diesel Spray

Formation and Combustion

master thesis

Cemil Bekdemir

Supervisor: dr.ir. L.M.T. Somers

Exam committee: prof.dr. L.P.H. de Goeydr.ir. L.M.T. Somersdr.ir. C.C.M. Luijtendr. C.W.M. van der Geld (Process Technology)dr.ir. N.A. Beishuizen (University of Twente)

Eindhoven University of TechnologyFaculty of Mechanical EngineeringSection Combustion Technology

Copyright c© 2008 by Cemil Bekdemir

All rights reserved. No part of this publication may be reproduced, stored in a retrievalsystem, or transmitted, in any form, or by any means, electronic, mechanical, photocopying,recording, or otherwise, without the prior permission of the author.

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vi

Abstract

Nowadays diesel engine related research, driven by environmental issues and global energydemand, is becoming more and more important. For that reason, rapid development of highefficiency and low emission engines in a numerical fashion is needed instead of conventionalprototyping. In modern diesel engines spray formation and combustion are the main processesthat challenge engineers. In this study in the first place diesel spray formation in a constantvolume chamber is modeled accurately and efficiently with a combination of a 1D spray modeland Fluent. Then, ignition and combustion by means of a tabulated chemistry method (FGM)is applied.

Liquid spray formation concern a lot of physics, starting from breakup of the liquid core intodroplets short after the nozzle exit, called primary breakup. In a second stage the formeddroplets breakup into smaller droplets, called secondary breakup. In automotive applications,with high ambient pressures and temperatures, the fuel droplets evaporate during their pathuntil the liquid length is reached. From then on the evaporated fuel penetrates further intothe surrounding gas, and at some point the spray auto-ignites.

Spray modeling can be done with phenomenological and CFD models, from which the lastone is more complex and expensive. But, in order to do a full engine simulation includingcombustion 3D CFD is required. Therefore, Fluent and its Euler-Lagrange spray modelis used. However, comparisons with liquid length and spray length measurements on theEHPC and data from Sandia show unsatisfactory correspondence. Also major numericaldisadvantages exist, like restricted mesh refinement possibilities and lack of parallelization ofthe computation.

These problems are circumvented by implementing a 1D Euler-Euler spray model of Versaevelet al in Matlab. From that, source terms are extracted for 3D calculations in Fluent. The re-sults are validated with experimental data of IFP and Sandia. The newly created combinationof the 1D spray model with 3D Fluent gives a better overal performance in spray length andshape compared to the Euler-Lagrange model of Fluent. At the same time the mentioned nu-merical drawbacks are taken care of, since there is no complex interaction between gas phaseand liquid droplets any more. Improvements for the future should be updated source termsduring the CFD calculation to make simulations with variable volume combustion chamberspossible and sound spray angle prediction methods/relations should be developed.

Finally, an attempt to include ignition and combustion is made. The detailed, though time-efficient, tabulated chemistry approach called FGM is used for this purpose. This approachis based on the flamelet concept and the manifold is preprocessed with igniting non-premixedflamelets. Now, Fluent solves four (mixture fraction, progress variable and their variances

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because of turbulence interaction) predefined scalar transport equations that are lookup in-dices for the 4D table. The spray length and shape are still well predicted, and the sprayauto-ignites. But combustion temperatures stay low, while the flame does not extinguish.This observation seems to be caused by the PDF integration procedure during the generationof the FGM, and it should be investigated first by simulating laminar spray combustion. Yet,the implementation (spray model-CFD interaction, FGM data handling) on its own seems towork well.

Contents

1 Introduction 1

2 Fundamentals of Liquid Sprays 3

2.1 Spray Regimes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3

2.2 Breakup Regimes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4

2.2.1 Primary Breakup . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5

2.2.2 Secondary Breakup . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8

2.2.3 Atomization in Diesel Sprays . . . . . . . . . . . . . . . . . . . . . . . 8

2.3 Direct Injection Diesel Combustion . . . . . . . . . . . . . . . . . . . . . . . . 10

3 Classification of Models 13

3.1 Thermodynamic Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13

3.2 Phenomenological Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13

3.3 CFD Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14

4 CFD Spray Model 15

4.1 Fluent . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15

4.1.1 Transport Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15

4.1.2 Discrete Phase Model . . . . . . . . . . . . . . . . . . . . . . . . . . . 17

4.1.3 Model Settings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22

4.2 Results and Parametric Dependencies . . . . . . . . . . . . . . . . . . . . . . 24

4.2.1 Spray Length . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25

4.2.2 Liquid Length . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28

4.3 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29

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x CONTENTS

5 CFD and Phenomenological Spray Model 31

5.1 Phenomenological Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32

5.2 Results; 1D Spray Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36

5.2.1 Spray Length . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37

5.2.2 Liquid Length . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38

5.2.3 Sensitivity Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40

5.3 Phenomenological Model into CFD . . . . . . . . . . . . . . . . . . . . . . . . 42

5.3.1 Source Terms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42

5.3.2 Interaction between Models . . . . . . . . . . . . . . . . . . . . . . . . 44

5.4 Results; 3D Spray Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45

5.5 Achievements and Future Work . . . . . . . . . . . . . . . . . . . . . . . . . . 47

6 Modeling Spray Combustion with Tabulated Chemistry 49

6.1 FGM Approach . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49

6.1.1 The Flamelet Concept . . . . . . . . . . . . . . . . . . . . . . . . . . . 50

6.1.2 FGM Generation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50

6.1.3 Implementation into CFD . . . . . . . . . . . . . . . . . . . . . . . . . 55

6.2 Results; 3D Spray Model with FGM Combustion Modeling . . . . . . . . . . 56

6.2.1 Spray Length without Combustion . . . . . . . . . . . . . . . . . . . . 57

6.2.2 Ignition Behavior . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57

6.2.3 Combustion Temperature . . . . . . . . . . . . . . . . . . . . . . . . . 59

6.3 Improvement Points . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60

7 Conclusions and Recommendations 61

7.1 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61

7.2 Recommendations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62

Nomenclature 63

A Transport Equations for the Mean and Variance of Z and PV 65

Acknowledgements 67

Bibliography 69

Chapter 1

Introduction

This study contributes to the development of diesel spray formation and combustion models.The motivation is based on the public need for maintaining, or even improving, currentprosperity, while preserving the environment and the health of mankind. In daily practicethis means, amongst others, that one has to comply with stringent regulations concerninginternal combustion engine emissions. These emissions include pollutants like nitrogen oxides(NOx) and soot. More and more also the emission of carbon dioxide (CO2) is restricted dueto its involvement with the reinforced greenhouse effect. Another implication of this publicneed, together with an increase of the global energy demand, is the approaching depletion offossil fuels, which makes the efficient use of organic fuels necessary.

To efficiently deal with fossil sources and to reduce harmful emissions, diesel engine relatedresearch is of upmost importance, since this type of engine is mainly used for heavy dutytransport purposes and also increasingly for passenger cars. Conventional engine design ap-proaches that rely on prototype development are too time-consuming and expensive, whiledevelopment of predictive and efficient computational tools would represent a significant stepforward in the ability to rapidly design high efficiency, low emission engines [FCD+07]. Mod-ern diesel technology consists of direct liquid fuel injection under high pressure, that formsa non-homogenous mixture leading to relatively high levels of soot. This spray formationprocess may seem straightforward, but in reality it is dauntingly complex [flu06b]. And alsothe combustion presents especially difficult and complex challenges [RR95].

The aim of this study is in the first place to accurately and efficiently model non-reacting dieselspray formation. A commercial CFD package (Fluent) with a dedicated spray model (DPM),and a 1D spray model from literature are used for this purpose. The 1D model of Versaevel etal [VMW00] is relatively simple compared to DPM, but has some major advantages when usedin 3D simulations. Validation of these spray simulations is done with constant volume, highpressure cell measurements from several engine research groups (TU/e, Sandia, IFP). Whenspray formation results are satisfactory, the second aim is to include ignition and combustionby means of a detailed, though efficient, tabulated chemistry method called FGM.

First of all, a literary study on spray fundamentals is reviewed in Chapter 2, with specialattention to fuel sprays in engine like conditions. Then, numerical models are classified intothree groups of varying complexity levels in Chapter 3. Furthermore, in Chapter 4 inert

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2 CHAPTER 1. INTRODUCTION

sprays are modeled in a CFD environment with the standard available DPM model, and theresults are discussed. Subsequently, a 1D spray model is implemented and validated andthen coupled to 3D CFD in Chapter 5, and the results are discussed and compared with theformer spray model results. In Chapter 6 a first attempt to add ignition and combustion tothe transient and turbulent spray model is considered, and improvement points are discussed.Finally, the conclusions and recommendations are given in Chapter 7.

Chapter 2

Fundamentals of Liquid Sprays

The concept of injecting liquid through a small hole may seem a trivial process, but thephysics of spray formation proves to be extremely complex. Although the analysis of liquidspray formation is a science discipline on its own, understanding some of its physical aspectsis already valuable for numerical modeling. In this chapter the fundamentals of liquid spraysin general, like spray regimes, droplet formation and breakup regimes, are presented. Theseconsiderations imply no assumptions about the liquid, so they hold for a liquid spray ingeneral. But in this study particularly fuel sprays in modern diesel engines are relevant,therefore in the last section some remarks concerning direct injection diesel spray formationand combustion are included.

The presented information in this chapter is mostly based on the books of Baumgarten [Bau06]and Stiesch [Sti03].

2.1 Spray Regimes

Diesel engine sprays are usually of the full-cone type. This means that in the idle modethe fuel is blocked from the upstream side of the nozzle and during injection the core of thespray is more dense than the outer regions. In spark-ignition engines however, gasoline iscommonly injected with a hollow-cone type of injector, which will not be discussed in thisstudy. See Figure 2.1 for a schematic drawing of a full-cone spray. The liquid spray can becharacterized by distinguishing five spatial regimes. Starting from the nozzle exit first thereis an intact liquid core. A few nozzle diameters further downstream in the so-called churningflow the liquid consists of ligaments (blobs). These liquid parts are like large droplets withsizes comparable to the nozzle diameter. Then the ligaments breakup into many smallerdroplets in the thick zone where the volume and mass fraction of the liquid phase is high.

Further downstream the breakup process of droplets goes on and in the same time more andmore of the surrounding gas is entrained into the spray area. This results in a divergingbehavior with a characteristic spray angle. The regimes after the thick zone are the thin zone(low volume but still high mass fraction of liquid) and the dilute zone (negligible volume andlow mass fraction of liquid), respectively.

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4 CHAPTER 2. FUNDAMENTALS OF LIQUID SPRAYS

Figure 2.1: Full-cone spray example; definition of spray angle, spray length and appearance of differentregimes [Bau06]

In a hot environment the position at which all liquid droplets are evaporated is called theliquid length. From (automotive) experiments this liquid length is found to be approximatelyconstant after a short development time. From that point on the evaporated fuel continuesto penetrate the surrounding gas and is denoted as vapor length. In a typical diesel injectiontimeframe (few milliseconds) the vapor length does not reach a steady state.

2.2 Breakup Regimes

The disintegration of liquid jets is described by two main mechanisms. The first mechanismis the breakup of the intact liquid core into droplets and is called primary breakup. Thismechanism is characterized by the droplet size and the breakup length, which is defined asthe length of the intact liquid core. The second mechanism is the breakup of droplets intosmaller ones, which is called secondary breakup. Here the size of the droplets is a characteristicparameter. Both breakup length and droplet size are dependent on the properties of the liquidand the surrounding gas. At least as important is the relative velocity between the liquid andthe surrounding gas.

The primary breakup is the most important mechanism in fuel injection systems, because itdetermines the size of the droplets that separate from the liquid core, hence therefore alsodetermines evaporation behavior and it marks the starting point for further breakup intosmaller droplets (secondary breakup). It is also far more difficult to analyse primary breakupboth experimentally and numerically. In the following the breakup regimes are treated inmore detail, but just for clearness the scheme in Figure 2.2 can be kept in mind to have thebig picture right.

2.2. BREAKUP REGIMES 5

Primary

Breakup

Secondary

Breakup

Spray

Breakup

Second wind-

induced regime

Rayleigh

regime

First wind-

induced regime

Atomization

regime

Turbulence Cavitation

Aerodynamic

droplet breakup

regimes

Figure 2.2: Breakup scheme; treated regimes

2.2.1 Primary Breakup

The primary breakup mechanism concerns the breakup of the intact liquid core and can bedivided into four regimes. Namely, the Rayleigh regime, the first and second wind-inducedregimes and last but not least the atomization regime. In order to make a quantitativeclassification of the regimes the Ohnesorge number Oh is introduced:

Oh =√

Wel

Rel. (2.1)

Herein the Weber number Wel and the Reynolds number Rel are defined as:

Wel =u2 D ρl

σ, (2.2)

Rel =u D ρl

µl, (2.3)

ρ is the density, σ is the surface tension, µ is the dynamic viscosity, u is the jet velocity andD is the diameter of the nozzle. The subscript l denotes the properties of the liquid. TheWeber number is the ratio between inertial (or aerodynamic) and surface tension forces. TheReynolds number is the ratio between inertial and viscous forces. Substitution of (2.2) and(2.3) into equation (2.1) gives:

Oh =µl√

σ ρl D. (2.4)


Thus, the Ohnesorge number is a ratio between viscous forces and surface tension forces.Now all relevant liquid properties are incorporated, so the various regimes can be classifiedin the space Oh as function of the jet velocity, or alternatively Rel. See Figure 2.3 for theso-called Ohnesorge diagram. In this figure the four regimes and also the relevant zone fordiesel injection applications are indicated.

Figure 2.3: Ohnesorge diagram; four primary breakup regimes, and region for diesel injection applica-tion indicated [Sch03]

It appears that also the density of the surrounding gas influences the breakup process. Byincreasing gas density enhanced atomization is achieved. Therefore the ratio of the gas andliquid densities is used to span a three dimensional space. A schematic representation of sucha three dimensional Ohnesorge diagram is seen in Figure 2.4.

Figure 2.4: 3D Ohnesorge diagram; primary breakup regimes as function of the Ohnesorge number(Z = Oh), Reynolds number and gas to liquid density ratio [Bau06]


Now the four regimes are described in more detail by increasing jet velocity. See also thecorresponding illustrations in Figure 2.5.

Rayleigh regime Breakup at low jet velocity due to axisymmetric oscillations initiated byliquid inertia and surface tension forces. Ddroplet > Dnozzle, the breakup length Ljet is longand by increasing jet velocity u also Ljet increases.

First wind-induced regime Liquid inertia and surface tension forces are amplified byaerodynamic forces. The relevant Weber number for this regime is:

Weg =u2

rel D ρg

σ. (2.5)

Here urel is the relative velocity between liquid and surrounding gas and the subscript gdenotes the gas properties. Ddroplet ≈ Dnozzle, Ljet > Dnozzle and by increasing jet velocityu the breakup length Ljet decreases.

Second wind-induced regime The flow in the nozzle is turbulent. Instable growth ofshort wavelength surface waves initiated by the turbulence and amplified by aerodynamicforces. Ddroplet < Dnozzle and by increasing jet velocity u the breakup length Ljet decreases.

Atomization regime Breakup at surface directly at the nozzle hole, so the intact corelength Ljet goes to zero. Conical spray develops immediately after leaving the nozzle. Ddroplet ¿Dnozzle, relevant regime for diesel sprays.

Figure 2.5: Schematic representations of the primary breakup regimes: (a) Rayleigh regime, (b) wind-induced regime, (c) atomization regime [Sti03]


2.2.2 Secondary Breakup

The secondary breakup mechanism concerns the breakup of droplets due to aerodynamicforces that are induced by the relative velocity between the droplets and the surrounding gas.On the gas-liquid interface instable growth of waves occur, while in the same time surfacetension counteracts the disintegration process. Similar to the first wind-induced regime forthe liquid core the gas Weber number is the relevant dimensionless quantity to identify theprocess, with the only difference that the nozzle diameter D in equation (2.5) is replaced withthe droplet diameter before breakup d :

Weg =u2

rel d ρg

σ. (2.6)

Decreasing the droplet diameter d raises the surface tension force σ. This means that thecritical relative velocity, the relative velocity at which breakup takes place, must be higher.Weg in equation (2.6) is used to separate the droplet breakup regimes. The values at whichtransitions from one regime to another occur, are determined experimentally. A schematicrepresentation of different droplet breakup processes are depicted in Figure 2.6. In the samefigure the corresponding Weber numbers are indicated according to Wierzba [Wie90], but alsoother classifications exist in literature. In engine sprays all droplet breakup regimes occur at

vibrational breakup ( 12)We »

catastrophic breakup ( 100)We >

stripping breakup ( 100)We <

bag/streamer breakup ( 50)We <

bag breakup ( 20)We <

Figure 2.6: Droplet breakup regimes and corresponding transition Weber numbers according to Wierzba[Wie90][Bau06]

the same time. Near the nozzle the Weber number is high, so most of the breakup takes placeat the nozzle exit. Further downstream the Weber number is lower due to smaller dropletdiameters and lower relative velocities. Therefore the breakup far from the nozzle is muchless.

2.2.3 Atomization in Diesel Sprays

Modern injectors for diesel engines have nozzle diameters of 200 µm or less, and the lengthof the nozzle hole is approximately 1 mm. Injection pressures up to 200 MPa are used and


therefore the jet velocity u reaches values of 500 m/s and more. These conditions resultin an atomization regime for the primary breakup mechanism. Some possible sources foratomization are shortly treated in the following.

Aerodynamic shear Aerodynamic shear forces amplify the surface waves created by theturbulence in the nozzle hole. The waves separate from the jet and form droplets. Thereare two reasons why this aerodynamic source is less important. First, this process is timedependent, but it is known from experiments that jets break immediately at the exit of thenozzle. Second, aerodynamic breakup is a surface effect, so it cannot explain disintegrationof the inner structure.

Relaxation of velocity profile At the wall inside the nozzle a no-slip boundary conditionsexists, forcing the flow towards a Poiseuille velocity profile. When the liquid exits the nozzle,the velocity profile will transform into a uniform one. In order to realize that the outer regionof the liquid accelerates, which may cause instabilities and ultimately result in breakup intodroplets. However, in modern diesel engines the length to diameter ratio of the nozzle hole istypically small ([ L

D ]nozzle = 5), so probably the flow in the nozzle has no time to develop.

Turbulence The presence of radial turbulent velocity fluctuations in the jet results, if strongenough to overcome the surface tension, in formation of droplets. Turbulence-induced primarybreakup is considered one of the most important mechanisms in high pressure applications.

Cavitation Cavitation is the transition from liquid to gas due to the decrease of staticpressure below the vapor pressure. The curved streamlines at the upstream edge of the noz-zle result in a radial pressure gradient. So, at places where the pressure is lower than thevapor pressure, cavitation bubbles are formed. These bubbles in the liquid flow contributeto primary breakup since they implode when they enter the high pressure environment. Pa-rameters that influence cavitation are the upstream nozzle edge and the angle between theinjector needle axis and the nozzle hole axis. A sharper edge results in stronger cavitation,which in turn results in smaller ligaments and a larger cone angle. If the angle between theneedle and the hole is too large, the flow in the nozzle and also the spray is asymmetricdue to the asymmetry of the streamlines. Although cavitation is strongly dependent on theinjector/nozzle geometry, the cavitation number K is an important dimensionless parameterto predict the inception of cavitation. The cavitation number is defined as follows:

K =p1 − pvap

p1 − p2≈ p1

p1 − p2. (2.7)

The indices 1 and 2 refer to the upstream and downstream pressures respectively and pvap isthe vapor pressure of the liquid. Since in automotive applications pvap ¿ p1 the vapor pressuremay be eliminated from equation (2.7). K is defined such that it decreases with increasingcavitation intensity. To include the influence of the geometry, an empirical criterium is usedto decide when cavitation occurs. This and more about computational considerations aretreated in the following chapters.


2.3 Direct Injection Diesel Combustion

Until now a qualitative description of spray formation is given without taking combustioninto account. In DI diesel engines however, liquid fuel is injected into an environment at highpressure and temperature that causes the fuel to evaporate and finally to burn. Here somegeneral remarks are reproduced from the conceptual model of Dec [Dec97] that is derivedfrom laser-sheet imaging.

As mentioned before, from the start of injection first the spray penetrates into the combustionchamber, whereby due to entrainment of ambient gas a spreading angle is observed. Shortafter the start of injection the liquid fuel breaks up into ligaments (primary breakup) andfurther into smaller droplets (secondary breakup), that in the mean time are heated andevaporated by the high temperature entrained gas. The point at which all fuel dropletsare evaporated is called the liquid length. This short recapitulation is shown in Figure 2.7as function of time. The liquid length is depicted with the dark color and has a value ofapproximately 20 mm, that can occur for diesel like fuels at typical engine conditions. In theregion direct after the liquid length an excess of fuel exists.

Figure 2.7: Temporal sequence of schematics showing how a DI diesel spray evolves from the start ofinjection [Dec97]

In time, somewhere in the spray the equivalence ratio and temperature reach values appro-priate for ignition. Then the premixed part of the spray auto-ignites and a flame establishesat the so-called flame lift-off length. This flame is still rich in fuel and produces lots of sootprecursors that move downstream and form soot particles. Finally, most of the soot and itsprecursors combust in a diffusion flame surrounding the spray. This hot reaction layer is themain location of NOx formation [Lui08]. See Figure 2.8 for the combustion evolution. Againthe liquid length and mixed gas region in front of it are shown, but now also the flame lift-offand soot concentration are indicated.

2.3. DIRECT INJECTION DIESEL COMBUSTION 11

Figure 2.8: Temporal sequence of schematics showing how DI diesel combustion evolves in the earlypart of the mixing-controlled burn [Dec97]


Chapter 3

Classification of Models

In-cylinder processes in engines can be modeled on different complexity levels. Some mainmodel categories are shown here in order to understand the applicability and shortcomingsfor modeling sprays.

3.1 Thermodynamic Models

Thermodynamic models originate from the laws of thermodynamics. In these models thereis no flow field present in the computational domain. Instead, the whole domain is ideallymixed, so the temperature, pressure, species etc. are distributed homogeneously. Due to therelative simplicity of the models small computational effort is sufficient. Apart from empiricalrelations, like for heat release rate, subprocesses cannot be incorporated. This means thatwith these simple models important engine parameters like pollutant formation due to localmixing behavior cannot be predicted. Therefore this class of models is not considered in therest of this study.

3.2 Phenomenological Models

In phenomenological models the computational domain is divided into multiple zones whichcan have different temperatures, compositions etc. . But also in this model class there is noflow field present. Spray modeling with phenomenological models is very common, because ofthe small computational effort needed, although more than for thermodynamic models. Theyare mostly applied to compare spray and liquid lengths for sprays in constant volume vesselswithout an initial flow field. As soon as mixing behavior in internal combustion engines isconsidered, the (turbulence) interaction of sprays with the surrounding flow is important, somore advanced models are needed.

Existing phenomenological spray models are roughly distinguished into two modeling schools,namely the droplet limited evaporation and the mixing limited evaporation [Lui08]. In thedroplet limited case, as the name says, the droplet evaporation process is the limiting factorfor spray formation. Hereby the local conditions around the droplet are important, whereas

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14 CHAPTER 3. CLASSIFICATION OF MODELS

in the mixing limited case the vapor-liquid (thermodynamic) equilibrium is assumed to settleimmediately, which makes the amount of air entrained a direct limiting factor. The immediatethermodynamic equilibrium assumption makes the mixing limited models only applicable forhigh ambient temperatures and densities, such as in modern diesel engines. The dropletlimited models are valid for any conditions, but are also more complex. In Chapter 4 anexample of the droplet limited approach is used and in Chapter 5 a mixing limited model isstudied in detail.

3.3 CFD Models

In CFD (Computational Fluid Dynamics) models the computational domain is divided intocells of specific sizes for which standard mass, momentum and energy equations are solved.Submodels are added to incorporate for example turbulence and spray formation. Interactionbetween the flow field and the spray is reached by solving the equations in an iterative fashion.A possible disadvantage could be that relative small errors in submodels may result in largeerrors in the overall result. Also the complexity of the model that makes the necessity of highCPU power inevitable can be a serious disadvantage. Figure 3.1 gives an indication of thecomputational expenses needed for each model class, note the logarithmic scale on the y-axis.

Figure 3.1: Computational effort needed for model classes ranked by complexity [Sti03]

Commercial codes like Fluent and StarCD and open source codes like KIVA supply submodelsthat are specially developed for sprays under engine like conditions. In the CombustionTechnology group at the Eindhoven University of Technology, mainly Fluent is used for heatand flow problems in engine applications, therefore in this study Fluent is the first choice tomodel diesel fuel sprays. More about Fluent and its spray submodel is presented in Chapter4.

Chapter 4

CFD Spray Model

Considering multidisciplinary heat and flow problems, with complex interaction between fluidphases and a variety of (moving) geometries, the best way to reach numerical solutions isprobably to use a software package that can solve transport equations for all kind of flowsunder user specified constraints. Specific geometries/constraints should of course be definedusing a proper user interface and results should be easy to postprocess. Fluent is an exampleof a commercial code that more or less meets the mentioned demands, hence the choice forFluent to model engine sprays in this study.

This (and also the next) chapter deals with evaporating non-reacting fuel injections only, sincespray formation modeling is best achieved when other modeling features that are necessaryfor practical engine simulations with burning sprays and moving pistons are absent. In theengine community it is very common to investigate diesel injection in a constant volume, highpressure cell. Through the years experimental techniques are developed to measure importantspray properties like liquid and vapor penetrations and spray angles. Spray penetration anddispersion are needed to promote fuel-air mixing, but impingement and collection of liquidfuel on in-cylinder walls (wall wetting) can lead to greater harmful emissions [Sie99]. Thefact that many research groups worldwide (like TU/e, IFP, Sandia) do such experiments withinert sprays gives the opportunity to use those measured data for validation purposes.

In the following, first the solved equations in Fluent are presented, with special attentionfor the spray submodel. Next, the model settings and the resulting solutions are discussed.Finally, some model dependencies such as mesh size and timestep are investigated.

4.1 Fluent

4.1.1 Transport Equations

Fluent solves a number of transport equations depending on the user’s specific problem setup.In this section an overview is given of the (general) continuity, momentum, energy, speciesand turbulence equations [flu06b]. Additional models and settings that are required to dealwith sprays are treated in the next sections.

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16 CHAPTER 4. CFD SPRAY MODEL

Continuity equation The general continuity equation is written as follows:

∂ρ

∂t+ ~∇ · (ρ~v) = Sm, (4.1)

where Sm is a mass source from the discrete phase due to evaporation of droplets. It can alsobe a user-defined mass source.

Momentum equations The momentum equation that is solved in this study is:

∂

∂t(ρ~v) + ~∇ · (ρ~v~v) = −~∇p + ~∇ · τ + ~F . (4.2)

Here p is the static pressure, τ is the stress tensor and ~F is a body force due to interaction ofthe discrete phase with the continuous phase and/or a user-defined momentum source. Thegravity term in the momentum equation is neglected because of the minimal contributioncompared to the high momentum injection event.

Energy equation The energy equation in Fluent is written as follows:

∂

∂t(ρE) + ~∇ · [~v(ρE + p)] = ~∇ ·

(k + kt)~∇T −

∑

j

hj~Jj + (τ · ~v)

+ Se, (4.3)

where the term between the brackets on the right hand side consists of energy transfer due toconduction, species diffusion and viscous dissipation, respectively. Se is a user-defined energysource. Energy E is defined as follows:

E = h− p

ρ+

~v · ~v2

. (4.4)

Herein h is the enthalpy for ideal gases, and is written as a summation of mass fractions timesspecies’ enthalpy:

h =∑

j

Yjhj . (4.5)

It is important, especially for Chapter 5, to state that the enthalpy is calculated by integrat-ing the specific heat from Tref to the instantaneous temperature T , whereby the referencetemperature in Fluent is 298.15 K:

hj =∫ T

Tref

cp,j dT, Tref = 298.15K. (4.6)

Species transport equations In spray simulations there are at least two different species,one species is in the gas phase (oxidizer) and an other one is injected (fuel), which afterevaporation goes into the gas phase where it can mix with the oxidizer. N − 1 transportequations for N species are solved because the sum of fractions must equal one. The transportequation for the ith species is as follows:

∂

∂t(ρYi) + ~∇ · (ρ~vYi) = −~∇ · ~Ji + Si. (4.7)

4.1. FLUENT 17

Si is again a source from the liquid droplet phase that is activated when evaporation occurs.Also user-defined sources are included in this term. Species transport due to diffusion iscalculated via the diffusion flux ~Ji. For turbulent flows this flux is:

~Ji = −(

ρDi,m +µt

Sct

)~∇Yi, (4.8)

where Di,m is the diffusion coefficient of the ith species in the mixture. µt is the turbulentdynamic viscosity and Sct is the turbulent Schmidt number:

Sct =µt

ρDt, (4.9)

which is equal to 0.7 by default. Dt is the turbulent diffusivity.

Turbulence equations Turbulence is dealt with the transport equations for the turbulentkinetic energy k and its dissipation rate ε. Here the realizable k-ε model is preferred becauseit is more suitable for axisymmetric jets than the standard one [flu06b].

∂

∂t(ρk) + ~∇ · (ρk~v) = ~∇ ·

[(µ +

µt

σk

)~∇k

]+ µtS

2 − ρε, (4.10)

∂

∂t(ρε) + ~∇ · (ρε~v) = ~∇ ·

[(µ +

µt

σε

)~∇ε

]+ ρC1Sε− ρC2

ε2

k +√

νε. (4.11)

4.1.2 Discrete Phase Model

Fluent provides a model that is specially developed for spray simulations, or more generalsuspended particle trajectory simulations. This is the Discrete Phase Model (DPM) and itis based on the so-called Euler-Lagrange method. In the computational domain there aretwo separate phases present, namely the continuous and the discrete phase (particles). Thetransport equations from the previous section are solved for the continuous phase only andthe motion of particles is dealt with particle trajectory calculations. Through an iterativesolution procedure the mass, momentum and energy interaction between both phases can berealized. Some important aspects of the DPM model are presented in this section. For moreinformation the reader is referred to the Fluent user’s guide [flu06b, Chapter 22].

Atomizer In order to simulate spray formation, (discrete) liquid particles have to be in-troduced to interact with the present (continuous) gas phase. As described in Chapter 2, indiesel sprays the primary breakup takes place in the atomization regime. So, it is assumedthat there is no liquid core; all the liquid is formed into droplets immediately after the exit ofthe nozzle hole. That is where the so-called atomizer model comes into play. The atomizercreates initial conditions, that depend on the internal nozzle flow, for further particle trajec-tory calculations by defining initial droplet diameter, velocity and the cone angle of the spray.Here the procedure to determine the internal nozzle flow state and its consequence for thecalculation of initial quantities is presented without going into the details.

In Fluent’s Plain-Orifice Atomizer Model three kinds of internal nozzle flows are defined,namely single-phase, cavitating and flipped flows. Figure 4.1 shows schematic cross-section


drawings of those possible nozzle flows. The upstream radius r, hole diameter d and lengthL of the nozzle are geometrical details that are used as parameters in empirical relations.On the righthand side also the corresponding criteria based on the cavitation number K (seeequation (2.7)) are given. Kincep and Kcrit are the cavitation number at which inceptionoccurs and the critical cavitation number, respectively. These cavitation numbers can beobtained with empirical relations based on experimental data. Cavitating nozzle flow is themain regime that occurs in today’s high pressure diesel injectors.

Figure 4.1: Plain-Orifice Atomizer, possible nozzle flows with cavitation number criteria [flu06b]

Once the internal flow state of the nozzle is known, the calculation of the initial dropletdiameter and velocity for the cavitating case proceeds according to the scheme in Figure 4.2.Given the predetermined cavitation number, the case data (r, d, p1 and p2; see Figure 4.1for definitions), and material properties in the upper left frame, the discharge coefficient Cd

and the initial velocity u0 are calculated. Then, via the effective mass flow rate meff and theeffective nozzle diameter deff an initial droplet diameter is obtained.

To complete the initialization of the droplets, apart of the size and velocity, the initial direction(cone angle) should be defined. This is again done by an empirical relation, but now for thecone angle. The cone angle, from now on also-called spray angle, is twice the angle betweenthe outer boundary of the spray and the main spray axis. In literature there are several sprayangle relations proposed. In Chapter 5 three of those empirical relations are compared, butfor now only the relation of Reitz & Bracco, that is used in the DPM model, is presented:

tan

(θ

2

)=

2π

3CA

√3ρa

ρf, (4.12)

4.1. FLUENT 19

Cd meff deff

d0

u0K, r, d, p1, p2

, , , pvaprlm s

.

Figure 4.2: Atomizer scheme; the way initial velocity and diameter are calculated

withCA = 3 +

L

3.6d. (4.13)

This angle relation seems complete because it takes into account fluid property as well asnozzle geometry information. However, later on it will be clear that also other relations withsimilar appearances give rather different results.

Particle motion The trajectory calculation of a discrete phase particle is done by inte-grating the force balance on the droplet. The force balance in vector notation is written asfollows:

d~vp

dt= FD(~v − ~vp) +

ρp − ρ

ρp~g +

ρ

ρp~vp · (~∇~v), (4.14)

where the left hand term is the acceleration of the particle in question, the term with FD isthe drag force on the particle. FD is defined as:

FD =18µρpd2

p

CDRe

24. (4.15)

The drag coefficient CD is determined from the dynamic drag model that accounts for theeffects of droplet distortion, linearly varying the drag between that of a sphere and a disk[flu06b]. The term with ~g in equation (4.14) is the contribution of the gravitational accel-eration. In this study the gravitational effect is neglected because of the very low mass ofthe droplets and the short injection times. The last term is an additional force that arisesdue to pressure gradients in the fluid. However this contribution is accounted for, in caseof a relative large constant volume pressure cell, gradients of pressure are not that large, inparticular when non-reacting sprays are concerned.

Phase coupling While the discrete particle phase is always influenced by the continuousphase solution (one-way coupling), the other way around (two-way coupling) is just providedas an option. In the one-way coupling case the continuous phase is solved first thereafter theparticle trajectory calculation is performed. When two-way coupling is applied an iterativeprocedure is followed. Then, after the particle trajectory calculation the continuous flow fieldis solved again with updated source terms until convergence is reached. See Figure 4.3 fora graphical representation of the procedure. Because the discrete phase during an injectionevent possesses high momentum, thus affects the continuous phase considerably, the two-waycoupling is turned on.


continuous phase flow field calculation

particle trajectory calculation

update continuous phase source terms

one-way coupling

two-way coupling

Figure 4.3: Solving procedure for one-way and two-way coupling

Source terms The exchange of mass, momentum and energy between the continuous anddiscrete phases is computed by the change of the concerning quantity as a particle passesthrough a computational cell, see Figure 4.4. These changes act as sources in the continuousflow calculation. So in case of a non-reacting spray in a hot environment, the mass andmomentum sources are positive. But the energy source is usually negative (energy sink)because the fuel with a relative low pre-injection temperature has to be heated and possiblyevaporated. For non-reacting droplets Fluent makes a distinction between three modes ofheating/vaporization. The first one is heating without vaporization until the user-definedvaporization temperature is defined. From then, the droplets can heat up and vaporize atthe same time. The vaporization temperature is of course an artificial boundary betweenheating only and vaporization, because liquid can vaporize at any temperature, hence theconcept of vapor pressure. But in this way the mass exchange calculation due to vaporizationat low temperatures can be neglected to save time. Finally, when the user-defined boilingtemperature is reached all added heat to the particles is used for vaporization, so the droplettemperature does not change any more.

Collision and breakup During the motion of droplets throughout the domain they canbreakup into smaller droplets in several different ways as treated in the chapter on sprayfundamentals. But the droplets can also collide with each other. Both phenomena are coveredin Fluent.

Fluent provides two droplet breakup models, the Taylor Analogy Breakup (TAB) model forlow Weber number injections and the Wave breakup model for high Weber number injections.In typical fuel injection systems (We À 100) the Wave breakup model of Reitz [Rei87] isthe best option. The Wave breakup model considers the breakup due to the relative velocitybetween the gaseous and liquid phases. The shear-off of child droplets from the parent dropletis induced by the growth of Kelvin-Helmholtz instabilities on the liquid surface. The rate ofchange of the droplet radius and the resulting child droplet size are related to the frequencyΩ and wavelength Λ of the fastest growing surface wave [KSR99]. Expressions for Ω and Λ asfunction of Weber numbers We and Ohnesorge number Oh are fitted to numerical stabilityanalysis solutions. Further details about the stability analysis do not add significant value to

4.1. FLUENT 21

Figure 4.4: Discrete phase particle traveling through a continuous phase cell, exchanging mass, mo-mentum and energy [flu06b]

the rest of this study, for more details the reader is referred to Reitz [Rei87] or the Fluent user’sguide [flu06b]. A schematic representation of the breakup model is depicted in Figure 4.5.Herein, r is the radius of the newly formed droplet as functions of the wavelength Λ and themodel constant B0 (= 0.61 [Rei87]). η is an infinitesimal axisymmetric surface displacementimposed on the steady motion and acts as a basis for the stability analysis.

Figure 4.5: Sketch of the Wave breakup principle [KSR99]

Collisions between droplets may be important in regions with a high particle density. Lookingat the whole computational domain there are N droplets that result in 1

2N2 possible collisionpairs. This implies the need for huge computing capacity since millions of droplets exist,or more practical, there is need for a simplified method. The last approach is achievedthrough the use of O’Rourke’s algorithm that reduces the computational expense for practicalsprays through the introduction of parcels, considering only in-cell collisions and stochasticestimation. The so-called parcel is a group of particles that have identical properties likedroplet diameter, temperature and velocity. Now, for example just several thousands of


parcels exist that in collision calculations are treated as they where single droplets, reducingthe amount of possible collision pairs tremendously.

In contrast to breakup, collisions decrease the amount of particles because of coalescence. Butalso inelastic collisions (bouncing) are possible that do not change the number of particles.Once it is determined that two parcels collide, the following step is to decide whether theresult should be coalescence or bouncing. This is done by comparing

√y(r1 +r2), where y is a

random number between 0 and 1 and r is the droplet radius, with a critical value expressionas function of the droplet radii and the relative Weber number. If the value is below critical,then the collision results in coalescence of the droplets and the new properties are calculatedfrom basic conservation laws. Otherwise the collision is inelastic and the new properties arecalculated from momentum and kinetic energy conservation, but part of the kinetic energy isalso dissipated.

Limitations So far the DPM model of Fluent seems to contain all necessary modelingfeatures to capture most of the spray physics present in diesel injection systems, but thereare also some major shortcomings from a computational point of view. Apart from issuesthat are described in the next sections, probably the most important drawback of the DPMmodel arises from the Euler-Lagrange approach assumption that at most 10 to 12 volumepercent of a cell should contain discrete phase particles. Otherwise the discrete phase wouldoccupy a significant amount of the continuous phase volume, whereas in the continuous phasecalculation the volume is constant and equal to that of the user-specified size. This would giveerroneous interaction sources between the two phases. In practice this restriction means thatcomputational cells, especially near the nozzle exit, must be big enough. This is the pointwhere a tradeoff have to be made between relative large cells in favor of the DPM model onone hand, and small cells to solve the high velocity flow field as accurate as possible on theother hand. A direct consequence of large cells is the cell shape and orientation dependencyof the results [Sti03].

Additionally, when cell sizes are decreased to improve the flow field resolution, the statistics(related to the amount of parcels) would run into convergence problems [SBK+04]. This hasto do with the low number of parcels per cell, therefore the total amount of parcels should beincreased, leading to a huge number of parcels and therefore also very high computing times.

Despite the known limitations of the DPM model, it is worthwhile investigating to whatextent these limitations restrict the reach of the ultimate goal; modeling direct injection ofa reactive spray in the variable volume of an auto-ignition engine. Therefore the followingsections show the applied Fluent settings and the resulting solutions.

4.1.3 Model Settings

In the section with the results of the DPM spray model, comparisons with experimental datagained from high pressure cell setups with a constant volume are done. This approach tovalidate spray models is very common, because in a constant volume cell the mean pressurestays approximately constant even when combusting sprays are considered. Important fea-tures like spray angle and penetration are then relatively easy to measure due to the controlledand reproducible conditions. In this study experimental data from several research groups is

4.1. FLUENT 23

used. For more information on their specific experimental layout and measuring techniquessee [Pet07][KDFS+06, Eindhoven High Pressure Cell], [VVB98, Institut Francais du Petrole]and [ECN, Sandia National Laboratories].

First of all a mesh is constructed to define the constant volume environment wherein the sprayis simulated. This is done with the default drawing and mesh creating tool of Fluent, namedGambit. Two different base meshes are created taking the model related cell size restrictioninto account. In typical high pressure fuel injection cases a cell size near the nozzle of 1 mm3

is common and gives the most realistic results [Sti03]. The first base mesh consists of squarecells. In order to reduce computational expense and because of symmetry, only a quarter ofthe spray is captured, see Figure 4.6 for a three dimensional view. The base square mesh

Figure 4.6: Square mesh, 14 th piece of the spray

has 90,000 cells which all are 1 mm3 big. The other base mesh is a 18th slice of a cilinder. For

the same magnitude of mean cell dimensions near the nozzle this mesh contains much lesscells than the square one, namely 6,000. See Figure 4.7 for a three dimensional picture.

Additional variations to the base mesh configurations are only in cell size. Especially thesquare mesh is used for cell size variation in different directions because this is much easierto accomplish than for the slice mesh. Boundary conditions on the mesh surfaces are set asfollows. The cutoff cross-sections are symmetry boundaries. All other surfaces have adiabaticconstraints.

The meshes do not include detailed interior geometry of the experimental constant volumeapparatus. This is allowed because the high pressure cells have cube or cilinder like volumesthat are much bigger than the space occupied by the spray.

In the next section only simulation results for heptane sprays are considered. By choosing asingle-component fuel like heptane, all temperature dependent material properties are defined


Figure 4.7: Slice mesh, 18 th slice of a cilinder

relatively simple. The specific heat, thermal conductivity, viscosity, vapor pressure and surfacetension of the liquid heptane are defined as function of temperature. Also the specific heatof all gaseous species is set temperature dependent. All these data are gained from thethermophysical database from DIPPR [DIfPP].

In the DPM model two-way coupling of the phases and droplet collision and breakup areenabled. The spray origin, spray direction, initial temperature, nozzle diameter and with themodeled section corresponding mass flow are prescribed.

4.2 Results and Parametric Dependencies

For now only evaporating, non-reacting heptane sprays are considered. Validation of the DPMmodel is done with an in-house measurement in the Eindhoven High Pressure Cell (EHPC)and with a measurement of Sandia National Laboratories. These measurements contain spraylength, liquid length and spray angle information. Well developed experimental techniquesexist to measure these quantities, and they represent easy to compare physical properties ofthe spray. Details of the experiments that are necessary for numerical modeling are listed inTable 4.1. Herein the corner radius is defined as in Figure 4.1. Ideally, things like temperaturefield and mass fractions of species should be measured to make more accurate comparisonswith (CFD) spray models. But unfortunately these field quantity measurements are notcommon practice at the moment.

In the following successively the spray length and liquid length are investigated, and in thesame time some encountered model dependencies are indicated. The spray angle remainsuntreated since it is an input to the DPM model, so validation on the basis of angles hasno meaning. However, in Chapter 5 a closer look is taken at the influence of certain angle

4.2. RESULTS AND PARAMETRIC DEPENDENCIES 25

relations.

Table 4.1: Experimental data of EHPC and Sandia measurements

EHPC Sandiafuel heptane heptanenozzle diameter [µm] 177 100nozzle length [mm] 1 0.4corner radius [µm] 0.07× d = 12.4 sharp edgemass flow per nozzle [kg/s] 0.0088 0.0028ambient temperature [K] 951 1000initial fuel temperature [K] 323 373ambient air composition[mass fraction CO2/H2O/N2] 0.130/0.060/0.810 0.065/0.038/0.897ambient pressure [MPa] 2.45 4.33

4.2.1 Spray Length

Spray lengths (vapor lengths) are experimentally determined with a Schlieren imaging tech-nique [Pet07][NS96][VVB98]. This is a line of site technique that makes use of the deflectionof light that travels through a medium with density gradients. While this is an appropri-ate method to measure spray lengths, it makes direct comparison with numerical results nottrivial. In order to make validation possible, a numerical technique is developed by Huijnen[Hui07]. Using this technique an image is constructed with virtual rays of light that travelthrough the 3D density domain that is extracted from the model results. The position onthe resulting image where the rays are most far from the nozzle exit is considered to markthe end of the spray in the length direction. Due to the imitation of the real Schlieren recordwith virtual rays of light, this numerical method is called the virtual Schlieren technique.

Mesh dependency Mainly the EHPC heptane measurement, indicated as a dotted line inFigure 4.8, is used to investigate the sensitivity of the DPM model for various settings. First,as expected, the DPM model appears to be highly dependent on cell sizes in the (square)mesh. The red solid lines are spray lengths as function of time for 1 mm3 cubic cells and theblack dotted lines are for the 1×1×0.2 mm3 cells. The last ones are only 0.2 mm in the z-direction at the nozzle exit, but gradually increase to 1.5 mm to the end of the computationaldomain. Also the result for a 2 mm3 mesh is shown, see the star markers. The difference withthe two other configurations is very large, and maybe more important, they do not followthe same increasing trend. Now imagine the spray in a combustion chamber model with amoving piston. Then, either a computationally very expensive and sophisticated adaptivemesh can be generated to align cells near the spray such that the initiated flow enters thecells perpendicular, or the mesh stays aligned with the liner. In the second case, the sprayaxis is chosen to have an angle of 15 degrees compared to the z-axis. The spray length for theunaligned case is indicated in Figure 4.8 with the diamond markers. Not seen in this figure


is the observed asymmetry of the spray around its own main axis due to the imposed anglein combination with the relative large cells.

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 20

10

20

30

40

50

60

70

80

90

100Penetration lengths (EHPC)

Time [ms]

SL

[mm

]

Cells ∆t [s] Alignment

1 mm3 10−6 yes

1x1x0.2 mm3 10−6 yes

1 mm3 10−5 yes

1x1x0.2 mm3 10−5 yes

2 mm3 10−6 yes

1 mm3 10−6 noEHPC measurement

Figure 4.8: Case: EHPC heptane. Spray length as function of time for various mesh configurationsand solver timesteps

Solver timestep Second, the timestep of the time-dependent solver gives rise to very largedifferences in spray length. In Figure 4.8 the circular markers are solutions with a timestepof 10−5 s and the square markers indicate the 10−6 s results. Decreasing the timestep to5 × 10−7 s leads to little improvement while computational expenses increase tremendously.Even a smaller timestep like 10−7 s is tried, but it gives alternately flipped and cavitatingnozzle flows, and the solution does not converge at all. This is remarkable because the internalnozzle flow is determined with empirical relations that does not depend on the solver timestep,but depend on nozzle geometry and fluid properties. Anyway, even the best result (1 mm3

cells and timestep of 10−6 s) is still far off from the experimental curve.

Physical properties Spray formation includes thermodynamic interaction between twophases with large temperature differences in a high pressure environment. Therefore materialproperties play an important role in spray modeling, even in the case of inert sprays. Thematerial properties in Fluent are set as function of temperature with data from the thermo-physical database of DIPPR [DIfPP]. Especially the specific heat, vapor pressure and boilingpoint (see the definition under the caption ’Source terms’ in Section 4.1.2) are key propertiesthat have a big influence on the results. The improved results for the spray length are shown

4.2. RESULTS AND PARAMETRIC DEPENDENCIES 27

in Figure 4.9 with red solid lines, once for the square mesh (square markers) and once for theslice mesh (triangular markers) configuration. In approximately the first 0.5 ms the modelestimates too large lengths. This is due to the assumed constant massflow in the numericalcase, whereas in (EHPC) practice the massflow takes some time to develop after the injectionstarts [SSB05]. To account for this phenomenon the massflow is gradually increased. This isdone manually, because unfortunately there is no other/easier way to do so in Fluent. Thedotted blue lines in Figure 4.9 are the results for the unsteady massflow cases. One can seethat the start of the injection is predicted much better, but thereafter, as expected, the pen-etration lags behind the measured curve. And once again the square and slice meshes showsubstantial differences.

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 20

10

20

30

40

50

60

70

80

90Penetration lengths (EHPC)

Time [ms]

SL

[mm

]

Square mesh, massflow constantSlice mesh, massflow constantSquare mesh, massflow(t)Slice mesh, massflow(t)EHPC measurement

Figure 4.9: Case: EHPC heptane. Spray length as function of time for the square and slice meshes,and with timedependent massflow

From many simulations of the Sandia case (see Table 4.1) similar trends as for the EHPCare found, therefore only the best practice result is shown in Figure 4.10. From the formerconsiderations best practice means the 1 mm3 square mesh with a solver timestep of 10−6 s,and of course also this time temperature dependent material properties are used. Now, themassflow is kept constant because the development time of the Sandia injection is just lessthan 0.1 ms [ECN]. Overall one can conclude that also for the Sandia spray the results arenot satisfactory.


0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 20

10

20

30

40

50

60

70

Penetration lengths (Sandia)

Time [ms]

SL

[mm

]

Square meshSandia measurement

Figure 4.10: Case: Sandia heptane. Spray length as function of time for the square mesh

4.2.2 Liquid Length

Liquid length (LL) is defined as the position at which all fuel is just evaporated. In anexperimental setup the liquid length is measured using the scattering of light caused by smalldroplets. In Fluent it is determined simply by the position of the droplets-parcel that is mostfar from the nozzle exit. The amount of parcels is a key issue here, recall the definition ofa parcel from Section 4.1.2 under the caption ’Collision and breakup’. Suppose that too fewparcels exist in the domain at an arbitrary moment in time. The parcel furthest in the domainprescribes the liquid length at that time. Now suppose that one timestep later that parcel doesnot exist anymore because it is evaporated. Because of the little amount of parcels present,one can imagine that the next furthest parcel is still far upstream compared to the evaporatedone. This causes the liquid length to vary in time as depicted with square markers in Figure4.11, where 10 new particle streams (or parcels) are introduced every timestep. Increasingthis amount finally makes the liquid length to stay at a steady value, see the 100 and 1000particle stream solutions marked with circles and stars, respectively.

Figure 4.11 is only shown to state the influence of the chosen amount of particle streams. TheEHPC heptane measurement does not include liquid length information, so the comparisoncan only be done for the Sandia experiment. But also the IFP heptane case, which is consid-ered in the next chapter (see Table 5.1), is already compared concerning the liquid length inorder to give a more complete picture of the reliability of the DPM model. Sandia reports asteady liquid length of 9.2 mm, whereas the DPM model calculates a length of approximately

4.3. DISCUSSION 29

2.5 mm. This large difference comes from the predicted droplet diameter in Fluent whichis too small, in the order of 0.01 µm whereas in practice the droplets have diameters in theorder of 10 µm. In the first place the diameter after injection is too small, and secondly afterbreakup the droplets become even smaller. The IFP heptane measurement is considerablydifferent from Sandia. This results in a cavitating nozzle flow instead of a flipped nozzle flowas in the Sandia simulation and therefore one order of magnitude larger droplets. One wouldalso expect a smaller liquid length this time, because the droplets are still relatively small,but remarkably the liquid length reaches values larger than 16 mm whereas the experimentalvalue is 10.8 mm.

0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 50

1

Liquid Lengths (EHPC)

Time [ms]

LL

particle stream: 10particle stream: 100particle stream: 1000

Figure 4.11: Case: EHPC heptane. Liquid length as function of time for various particle stream(amount of parcels per timestep) settings

4.3 Discussion

Fluent’s DPM model is extensively used to model evaporating, but inert heptane sprays. Thisis done with special attention for temperature dependent material properties and for manydifferent setups, including various meshes, solver timesteps and amount of parcels. The resultsare compared with a measurement on the EHPC and experimental data from Sandia. Fromthese comparisons it is found that the DPM model gives unsatisfactory results concerningspray and liquid lengths. Nevertheless, some best practice setups resulted from this study,which are at least valid for heptane sprays in engine like conditions. That are 1 mm3 cellsaligned with the spray axis and solved with a solver timestep of 10−6 s, and injection of 1000


parcels per timestep. Besides these numerical features, setting material properties as functionof temperature is probably the most important.

From a numerical point of view there are also major disadvantages. One of them is theimposed limitation to mesh refinement which is far from desirable when detailed in-cylindermixture formation and combustion are to be modeled. The second disadvantage is that thediscrete phase part of the calculations cannot be parallelized, while those detailed investiga-tions require fine resolution, thus expensive simulations.

Chapter 5

CFD and Phenomenological SprayModel

From the previous chapter, about the 3D spray model of Fluent, called DPM, clearly followsthat practical engine sprays are difficult to solve accurately with an Euler-Lagrange method.The results are highly mesh and timestep dependent and often convergence problems mayoccur. Also the statistical approach with parcels is a source of problems due to large com-puting times when parcels accumulate in the domain. Many authors tried to overcome theseproblems by fine tuning the submodels for specific cases, but this is obviously not the wayto go due to the fundamental discrepancy between, limitation to cell sizes and lack of paral-lelization possibilities, and solving in-cylinder velocity fields and turbulence with increasinglyfiner meshes.

However, others also developed (semi) Euler-Euler methods to model spray formation. One ofthem is the so-called ICAS (Interactive Cross section Averaged Spray) model [WP97][SBK+04]that combines a 1D (Eulerian) spray model with the existing Langrangian CFD model. Theregion near the nozzle exit is then covered with the 1D spray model and somewhere furtherdownstream droplet source terms are introduced to incorporate 3D spray formation in theCFD environment. By introducing parcels anyway, the influence of in-cylinder flow to sprayformation would still be captured [Sti03], but to do so the Euler-Euler region should be notmore than a few millimeters from the nozzle exit, because further downstream (behind LL)there are no droplets anymore. This makes complex, adaptive meshes necessary, and theDPM model of Fluent would still be needed. However, with the results of the precedingchapter in mind, the CFD code should ideally be used for gas phase calculations only in orderto get rid of the complex discrete phase interaction. Therefore a 1D model that covers thecomplete spray region is presented in this chapter. This model is coupled to Fluent withappropriate source terms for mass (fuel vapor), momentum and energy. Initially, just likethe DPM model, the continuous phase exists of solely ambient gas. When fuel vaporizes it isintroduced into the computational domain through a fuel vapor source term. Whereas in theDPM model real droplets and their trajectories are calculated to determine source terms, inthe 1D model this is done relatively simple.

In the following, first the phenomenological spray model proposed by Versaevel et al [VMW00]is treated extensively. The model is then implemented in Matlab and validated with mea-

31

32 CHAPTER 5. CFD AND PHENOMENOLOGICAL SPRAY MODEL

surements of IFP [VVB98] and Sandia [ECN][Sie99]. Last but not least, source terms areextracted from the 1D model and put into Fluent via UDFs (User-Defined Function), and theresulting 3D solutions are also compared with measurements.

5.1 Phenomenological Model

The 1D quasi steady spray model of Versaevel et al [VMW00] is an extension of the earlierefforts of Naber et al [NS96] and Siebers [Sie99]. Naber and Siebers developed a 1D modelfor non-vaporizing spray penetration first, and later Siebers added some thermodynamics todistinct liquid penetration from vapor penetration. But Siebers’ contribution is based onthe assumption that only at the steady liquid length position thermodynamic equilibriumexists. This approach implies that no temperature information is available, except at theliquid length position. Also the composition of the spray volume between the nozzle exit andliquid length is unknown. Versaevel et al overcame this shortcoming by introducing a voidfraction m that couples the mass, momentum and energy equations.

The model is based on five basic assumptions from which the first four are the same as in thework of Naber et al [NS96] and Siebers [Sie99].

• no velocity slip between gas and liquid phases

• whole system at constant pressure

• uniform velocity, density and temperature profiles

• constant spray angle

• whole system at thermodynamic equilibrium

The last assumption is necessary to gain information about the temperature and compositionof the spray in the entire 1D domain. The mass, momentum and energy equations arederived by considering a control volume as defined in Figure 5.1. The spray is described inone direction due to the constant angle and the axisymmetry. The figure shows that fromfuel injection (mfl,0) into the x-direction the spray diverges due to air entrainment (ma) intothe spray volume. Air entrainment is controlled by the prescribed spray angle (θ

2). For thispurpose an angle relation is chosen, like is done with the DPM model in the previous chapter.At the liquid length just enough hot air is entrained into the spray to evaporate all liquid fuel,so from that point on the fuel penetrates the surrounding gas as a vapor. deff in Figure 5.1 isthe effective orifice diameter defined as: deff =

√Cad, with d the geometric orifice diameter

and Ca the area contraction coefficient.

Now the mass, momentum and enthalpy balances as defined by Versaevel et al [VMW00] areshown, preceded by the description in words.

Mass balance Liquid fuel mass flow rate at the injector hole = remaining liquid fuel massflow rate at x + gaseous evaporated fuel mass flow rate at x

ρfl,0A0vfl,0 = ρfl[1−m(x)]A(x)v(x) + Yfg(x)ρg(x)m(x)A(x)v(x). (5.1)

5.1. PHENOMENOLOGICAL MODEL 33

control volume

gas and liquid gas onlyma

ma

mfl,0

/2

df

x

.

.

.

q

deff

Figure 5.1: Definition of the control volume used in the derivation of the 1D model [VMW00]

Here, ρ is the density, A represents the cross-section area of the spray, v is the spray velocity,m is the void fraction and Y stands for mass fraction. The subscripts fl, 0, fg and g refer tothe liquid fuel, the nozzle exit, the gaseous fuel and the whole gaseous phase, respectively.

Momentum balance Liquid fuel x-momentum flow rate at the injector hole = remainingliquid fuel x-momentum flow rate at x + gaseous mixture x-momentum flow rate at x

ρfl,0A0v2fl,0 = ρfl[1−m(x)]A(x)v(x)2 + ρg(x)m(x)A(x)v(x)2 (5.2)

Enthalpy balance Liquid fuel enthalpy flow rate at the injector hole + entrained air en-thalpy flow rate = liquid fuel enthalpy flow rate at x + fuel vapor enthalpy flow rate at x +air enthalpy flow rate at x

mfl,0hfl(Tfl,0) + ma(x)ha(Ta) = ρfl[1−m(x)]A(x)v(x)hfl(T (x))+ (5.3)

Yfg(x)ρg(x)m(x)A(x)v(x)hfg(T (x)) + ma(x)ha(T (x))

Here, m is the mass flow rate, h represents the enthalpy and T is the temperature. Thesubscript a refers to the ambient gas.

The quantities that depend on the spatial coordinate x are indicated explicitly. Only thespray cross-section area A(x) is known on beforehand through the applied correlation for thespray angle θ:

A(x) = π

[deff

2+ x tan

(θ

2

)]2

. (5.4)


Now the empirical angle relation of Siebers is used, like Versaevel et al, to compare the results.Siebers’ experimental fit is as follows:

tan(

θ

2

)= ac

(ρa

ρfl

)0.19

− 0.0043(

ρfl

ρa

)0.5 , (5.5)

where the factor ac is a fitting parameter. It must be said that the parameter ac is the largestpoint of concern of this phenomenological model for conventional diesel sprays. The choice ofa proper coefficient is explained later on. In the next section some common angle relations arecompared to investigate the predictive capacities and give recommendations for improvement.

The coupled non-linear balance equations (5.1), (5.2) and (5.3) are written in an other formto make it suitable for solving with a non-linear solver. With the nozzle exit velocity of thefuel:

vfl,0 =mfl,0

ρfl,0A0, (5.6)

and with the velocity at any axial position:

v(x) =ma(x)

ρg(x)[1− Yfg(x)]A(x)m(x), (5.7)

the mass balance equation is written as:

Γ(x) =ma(x)mfl,0

=m(x)[1− Yfg(x)]ρg(x)

ρfl[1−m(x)] + m(x)Yfg(x)ρg(x). (5.8)

In the same manner with equations (5.6) and (5.7) the momentum balance is written as:

Γ(x)2 =m(x)2[1− Yfg(x)]2ρg(x)2A(x)

A0ρfl,0([1−m(x)]ρfl + m(x)ρg(x)). (5.9)

Also the enthalpy equation is rewritten, using equation (5.7) and (5.8). Additionally anenthalpy relation that relates liquid and gaseous enthalpies through the heat of vaporizationLv is applied:

hfl(T (x)) = hfg(T (x))− Lv(T (x)), (5.10)

then the enthalpy equation is written as:

Γ(x)[ha(Ta)− ha(T (x))] =

(Yfg(x)Γ(x)1− Yfg(x)

− 1

)Lv(T (x)) + Lv(Tfl,0) + [hfg(T (x))− hfg(Tfl,0)].

(5.11)Together with a constant pressure ideal gas relation for the density of the total gaseous phase:

ρg(x) = ρa(T (x))Ta

T (x)1

Yfg(x)MaMf

+ [1− Yfg(x)], (5.12)

wherein M is the molecular mass, the coupled non-linear equations (5.8), (5.9) and (5.11)form the ingredients to describe the 1D spray.

The solution procedure is split into two spatial parts, the liquid length part and the vaporlength part. First the part from the nozzle exit to the liquid length is considered. Here

5.1. PHENOMENOLOGICAL MODEL 35

are three unknowns, namely the temperature T (x), the massflow ratio Γ(x) and the voidfraction m(x). Yfg(x) is not considered as an unknown because of the imposed thermodynamicequilibrium, that states that the fuel partial pressure is equal to the fuel vapor saturationpressure:

psat(T (x)) = paXfg(x). (5.13)

Then using the relation between molar and mass fractions Xfg(x) = Mg

MfYfg(x), Yfg becomes:

Yfg(x) =1(

pa

psat(T (x)) − 1)

MaMf

+ 1, (5.14)

thus Yfg(x) is calculated when a solution for T (x) is found. And then also ρg(x) is definedvia equation (5.12).

Once the void fraction reaches unity in x-space, the rest of the spray in increasing x-directionis solved for the fuel vapor with the same three equations (5.8), (5.9) and (5.11) as for theliquid length part. Now the unknowns are also the temperature T (x) and massflow ratioΓ(x), but instead of the void fraction m(x) this time the gaseous fuel mass fraction Yfg(x) isunknown. Downstream of the liquid length position the void fraction is by definition constantand equal to unity, provided that the fuel does not condensate. ρg(x) is again determinedafterwards via equation (5.12).

So far the 1D spray model is steady, thus the coupled equations can be solved for any chosenx-position. Once all quantities are solved for the whole x-domain the next step is to introducetimedependence, in order to mimic the unsteady behavior of a practical spray. This is accom-plished by following the assumption of Naber et al [NS96] that the velocity v(x), determinedwith equation (5.7), is equal to the velocity of the spray tip. In this way one can constructan unsteady penetration length SL(t) of the spray from the steady velocity profile v(x):

dSL

dt= v(x). (5.15)

The phenomenological spray model is completely implemented in Matlab. The standard non-linear solver of Matlab is used for this purpose. Material properties, except the liquid fueldensity, are temperature dependent, and are obtained from the thermophysical database ofDIPPR [DIfPP]. The current Matlab model works with a run-file with input fields that arerequired to be filled in, in order to start a well defined simulation. These inputs are thecase setup data as is shown in Table 5.1 and model related inputs like the preferred spatialresolution and initial guesses of the three unknowns. Furthermore one can choose to solvefor the steady liquid length or transient spray length only, or both at the same time. Andfinally, the user can turn on the ’plot during calculation’ option. If plotting is turned on, sixquantities (m, T , Γ, v, Yfg and ρg) are shown as function of x during the whole simulation,otherwise the progress of the simulation is indicated with a progress bar and the simulationtime is much shorter. An example output to the screen during calculation is shown in Figure5.2. For each defined x-position the quantities are computed and added to the figure untilthe loop over all meshpoints is finished.

The curves in Figure 5.2 give some understanding about what happens from x = 0 at thenozzle exit up to the point that the void fraction reaches a value of 1. For example the


0 5 10 15 20 25 300

0.2

0.4

0.6

0.8

1

1.2

1.4

x [mm]

m [-]

ac = 0.2537 dx = 0.1 mm exitflag = 1

0 5 10 15 20 25 30350

400

450

500

550

600

650

700

x [mm]

T [K

]

status: FINISHED

0 5 10 15 20 25 300

2

4

6

8

10

x [mm]

G[-

]

0 5 10 15 20 25 300

100

200

300

400

500

x [mm]

v [m

/s]

0 5 10 15 20 25 300

0.1

0.2

0.3

0.4

0.5

x [mm]

Yfg

[-]

0 5 10 15 20 25 300

10

20

30

40

50

60

70

x [mm]

ρg

[kg/m

3]

d = 200 mm

Cd

= 0.8

T0

= 373 K

Ta

= 800 K

Pinj

= 80 MPa

Pa

= 6.2783 MPa

ρa

= 25 kg/m3

Figure 5.2: Example output to screen of the 1D spray model during calculation

mass fraction of gaseous fuel increases due to evaporation and has its maximum at the liquidlength. From then on, only gaseous fuel is present while air entrainment keeps on going, soYfg decreases. A discontinuity at the liquid length does not appear in the velocity curve, itdeclines asymptotically from a maximum velocity at the nozzle exit to zero. At the end of thesimulation also curves of the transient vapor length (SL) and liquid length (LL) are plottedin a separate window, see the next section for the results.

5.2 Results; 1D Spray Model

In this section IFP [VVB98] and Sandia [Sie99][ECN] measurements are used for validationpurposes. These are all for single component fuels that are well documented, so thermo-physical data needed for the numerical model is found in literature. The most importantexperimental setup information, needed as input for the spray model, is included in Table5.1. Also now, penetration lengths form the basis for validation for the same reasons as men-tioned in Chapter 4. But this time more attention is paid to the effect of different spray anglerelations.

5.2. RESULTS; 1D SPRAY MODEL 37

Next, the spray length is shortly viewed. The rest of the comparison is mostly done forthe steady liquid length because in literature only very few measurements of transient spraylengths for single component fuels is available. Finally, the sensitivity of the model to mea-surement errors and physical property prediction errors is analyzed.

Table 5.1: Experimental data of IFP and Sandia measurements

IFP Sandiafuel heptane, dodecane cetanenozzle diameter d [µm] 200 246contraction coefficientsCv/CA/Cd 0.8/1.0/- 0.81/-/0.78initial fuel temperature Tfl,0 [K] 373 438ambient temperature Ta [K] 800 850/1000/1150ambient density ρa [kg/m3] 25.0 7.3/14.8/30.2injection pressure pinj [MPa] 80 136ambient air compositionmass fraction CO2/H2O/N2 0.017/0.114/0.869 0.065/0.038/0.897

5.2.1 Spray Length

Regarding the fitting coefficient ac in equation (5.5), validation of the model by Versaevel etal [VMW00] with experimental data from IFP [VVB98], results in the choice of two differentvalues. One of them gives good agreement with the experiments concerning the unsteadyvapor length, and has a value of 0.2537. Versaevel et al argue that this value is in goodcorrespondence with the findings of Naber and Siebers [NS96]. But for predicting the steadyliquid length this value is too large. To obtain the best fit for the experimental data, Versaevelet al propose an ac of 0.104 for steady liquid length calculations. Therefore the model is runtwo times, once for each ac, and from the ac = 0.2537 solution the timedependent spraylength is constructed as shown in equation (5.15). Subsequently, from the philosophy thatthe unsteady paths of the vapor and liquid phases should match until the steady liquid lengthis reached, the ac = 0.104 solution is only used to determine the steady value of the liquidlength.

In Figure 5.3 the results for an IFP case with dodecane is shown. The vapor length as ismeasured by IFP is indicated with the dotted line, which is a fit of Verhoeven et al [VVB98]through measured points. The 1D model results are indicated with the black and red line forac = 0.25 and ac = 0.29 respectively. Versaevel et al choose ac = 0.25 as a best fit coefficient,but from the figure it is seen that ac = 0.29 fits better. Therefore in the rest of this studythe angle relation coefficient ac for the vapor length calculations is set to 0.29. The unsteadyliquid length, indicated with the dotted blue line, follows the same curve as the unsteadyvapor length as mentioned before. In correspondence with practice, when the steady value of20.7 mm is reached the length remains constant. More vapor length comparisons, for heptaneand different ambient densities, between the 1D model and measurements are not shown butcan be found in the paper of Versaevel et al [VMW00]. There it is shown that overall good


agreement is achieved except for low ambient temperatures because of the mixing limitedassumption of this model.

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

5

10

15

20

25

30

35

40

45

50Penetration lengths (dodecane)

SL

and

LL [m

m]

Time [ms]

SL with ac = 0.29LLSL with ac = 0.25fit through IFP measurements

Figure 5.3: Case: IFP dodecane. Spray length and liquid length as function of time

5.2.2 Liquid Length

Now the focus is on the liquid length prediction, which is for this study more important thanthe vapor length because the liquid length part is used to couple spray formation with CFDcalculations. Figure 5.4 contains steady liquid lengths for various ambient densities. Theexperiment points (black stars) are again from IFPs dodecane measurements. The results ofthe 1D model of Versaevel et al (red squares) are within 10 % of the experiments, while the1D model of Siebers (blue circles) [Sie99] predicts liquid lengths that are up to 50 % too large.This would make no sense since the findings of Siebers in his paper show that his model isalso within 10 % accurate, but in his case compared with Sandia data. Sandia’s experimentaldata for cetane are found in Figure 5.5 (stars). These liquid length measurements are plottedas function of the ambient temperature and for three different ambient densities. Also thenumerical results of the Versaevel and Siebers 1D models are shown, with squares and circlesrespectively. Now Siebers’ model deviates just up to 10 %, while Versaevel’s model is up to 20% off. Both models are based on basic conservation laws and most assumptions are common,so the striking differences could only have to do with the capability to predict an appropriatespray angle for a certain case.

In literature there are three commonly used angle relations for vaporizing sprays. One is


10 12 14 16 18 20 22 24 26 28 30 3215

20

25

30

35

40Liquid lengths (dodecane)

ρa [kg/m3]

LL [m

m]

Versaevel (Matlab model)IFP measurementSiebers (Matlab model)

Figure 5.4: Case: IFP dodecane. Liquid length comparison between Versaevel and Siebers models forvarious ambient densities

from Siebers as is used in this chapter until now. And the others are from Reitz & Braccoand Hiroyasu & Arai. These relations are conveniently arranged in Table 5.2. The thingthat attracts attention from this table is the dissimilarity of quantity dependencies. Siebers’empirical relation is a function of fuel and ambient densities only, whereas Reitz & Braccoalso include nozzle geometry information. Hiroyasu & Arai’s relation is a function of thenozzle diameter, ambient properties and the pressure drop over the nozzle. The impact ofthese differences on the spray angle is made visible in Figure 5.6. Herein the full spray angles

Table 5.2: Three commonly used angle relations

Siebers tan(θ2) = 0.26

[(ρa

ρfl

)0.19 − 0.0043(

ρfl

ρa

)0.5]

Reitz & Bracco tan(θ2) =

(2

3πCA

) (3ρa

ρfl

)0.5, CA = 3 + L

3.6d

Hiroyasu & Arai θ = 0.05[

ρa(Pinj−Pa)d2

µ2a

]0.25


800 850 900 950 1000 1050 1100 1150 120010

20

30

40

50

60

70

Ta [K]

LL [m

m]

Liquid lengths (cetane)

ρa = 7.3 kg/m3

ρa = 14.8 kg/m3

ρa = 30.2 kg/m3

Siebers (Matlab model)Sandia measurementVersaevel (Matlab model)

Figure 5.5: Case: Sandia cetane. Liquid length comparison between Versaevel and Siebers models forvarious ambient temperatures and densities

θ are plotted as function of several ambient temperatures and densities. By creating thisfigure special attention is paid to the definition of the spray angle, in the sense that someauthors use the half spray angle whereas others prefer the full spray angle. So the largevariations in the angles is solely due to the lack of predictive ability of the relations in casesfor which their are not fitted on. Also Jung et al [JA01] observed this discrepancy (they onlylooked at Reitz & Bracco and Hiroyasu & Arai). They choose to work with Reitz & Bracco’srelation because their compared experimental data was a bit closer to that model than thatof Hiroyasu & Arai and the measured angles did not show any dependency on the injectionpressure. Obviously, a good functioning angle relation is important, therefore Siebers’ formulafitted to IFP measurements is continued to use in the rest of this study. But for future spraystudies better and generic angle predictions may be important, therefore it is recommendedto do an extensive (experimental) study to improve existing empirical relations or come withother solutions. In the short term it may be more important to modify an existing formulato fit experiments on the EHPC. Subsequently any proposed angle prediction model can beput into the spray model in a straightforward fashion.

5.2.3 Sensitivity Analysis

The last 1D analysis before the phenomenological model is used to model 3D sprays in aCFD environment is a sensitivity analysis. This is very relevant considering the high temper-


800 850 900 950 1000 1050 1100 1150 120010

15

20

25

30

35Comparison between angle relations

Ta [K]

Ful

l spr

ay a

ngle

θ [d

egre

es]

Sandia cetane measurement conditions

SiebersReitz & BraccoHiroyasu & Arai

ρa = 14.8 kg/m3

ρa = 30.2 kg/m3

Figure 5.6: Three commonly used angle relations applied to the Sandia cetane case

ature and pressure levels at which spray measurements are done and material properties aremodeled. The question is to what extent the computed temperature in the spray, the liquidlength and the vapor length are influenced by varying measured quantities and thermody-namic properties. The measured quantities Ta and ρa (pa actually) typically have an error of10 %, and the ambient gas composition can vary more than that because it is not measureddirectly. And also the spray angle θ is varied 10 % to investigate the influence of possiblymalfunctioning angle relations. The results in Table 5.3 are for the indicated percentages ofthe reference values for the IFP dodecane case. The deviations from the reference tempera-ture, liquid length (LL) and vapor length (SL) are positive or negative referring to larger orsmaller values, respectively. The same holds for the material properties, but these are varied5 %. Sensitivity to material properties at elevated temperatures and pressures is importantbecause often there is simply not sufficient temperature dependent fuel information presentin literature, especially for practical (multi-component) fuels. Most remarkable values in thetable are found at the sensitivity of LL for variations in the ambient temperature. This is dueto the importance of the energy equation for the liquid length part of the spray where evapo-ration of liquid fuel is a crucial process. Concerning the spray angle, or indirectly the amountof air entrainment, one can see that the liquid length varies approximately with the samepercentage. The sensitivity to thermodynamic properties corresponds well to the findings ofvan Erp [vE07], who did similar work with Naber and Siebers spray model.


Table 5.3: Sensitivity analysis

[%] T [%] LL [%] SL [%]Measurement errors

Ta 90 −9.54 +28.50 +1.45110 +9.51 −18.84 −1.28

ρa 90 −1.87 +4.83 +3.90110 +1.80 −4.35 −3.42

YCO2 , YH2O 50 −1.09 +2.90 −0.17200 +2.01 −5.31 +0.20

θ 90 −2.47 +11.11 +5.02110 +2.34 −9.18 −4.35

Property errorsρfl 95 +1.03 −3.86 −0.38

105 −1.01 +3.38 +0.36cp,fg 95 +0.73 −1.93 +0.14

105 −0.65 +1.45 −0.14psat 95 +0.21 +1.45 +0.01

105 −0.21 −1.45 −0.01Lv 95 +0.57 −1.45 +0.08

105 −0.58 +0.97 −0.08

5.3 Phenomenological Model into CFD

The 1D phenomenological spray model discussed in the previous section is, in contrary tothe earlier model of Naber and Siebers, suitable to apply in combination with a 3D CFDcode. To accomplish such an interaction, source terms are extracted from the 1D model andare assigned to the corresponding transport equations in Fluent. Subsequently the combinedmodel is validated through spray length comparison with experimental data.

5.3.1 Source Terms

Three source terms are needed to describe spray formation, namely mass, momentum andenergy sources. The coupling via source terms is realized only in the domain between thenozzle exit and the liquid length (x < LL). Further downstream (x > LL) all liquid fuel isevaporated, so there is only gaseous phase involved in the flow that already is covered withthe transport equations in Fluent. In the following, each source term is treated one after theother.

Mass source The mass source is related to the evaporation of fuel in the x < LL part.Because of evaporation, gaseous heptane mass is introduced in the computational domain.

5.3. PHENOMENOLOGICAL MODEL INTO CFD 43

Following Versaevel et al, the liquid fuel mass flow rate Mfl [kg/s] is:

Mfl(x) = ρfl[1−m(x)]A(x)v(x), (5.16)

differentiation to x and division by −A(x) gives the mass source needed in [kg/m3s]:

Msource(x) = − 1A(x)

dMfl

dx. (5.17)

This equation is based on the change of liquid mass, which is the opposite of gaseous masschange, hence the minus sign.

Momentum sources Momentum consists of three components, one in each spatial direc-tion. Only one component is provided by the 1D model, and as long as the spray in Fluent isoriented along a major axis (x, y or z) no extra measures have to be taken. The x-momentumsource from the 1D model is then set as the source for one of the other directions in 3D, as isdone in this study. But unfortunately when real in-cylinder sprays are simulated, the sprayaxis generally does not match with the main axes. Some mathematics is included to makethe necessary rigid body rotation possible, see the next section for details. Now for one di-rection the same approach as for the mass source gives the liquid fuel momentum flow rateTfl [kgm/s2]:

Tfl(x) = Mfl(x)v(x), (5.18)

again, differentiation and division gives the momentum source [N/m3] in x-direction:

Tsource(x) = − 1A(x)

dTfl

dx. (5.19)

Energy source The derivation of a source term for the energy equation is not that straight-forward as for the mass equation. In the first place there is a difference in the definition ofenergy/enthalpy between the 1D model and Fluent. The 1D model works on an enthalpybasis, but in Fluent energy is used and for liquid fuel it is defined as follows:

Efl(x) = hfl(x)− pa

ρfl+

v(x)2

2. (5.20)

The same definition also holds for the gaseous fuel:

Efg(x) = hfg(x)− pa

ρfg(x)+

v(x)2

2. (5.21)

With equations (5.16) and (5.20) the liquid fuel energy flow rate [J/s] is determined:

Efl(x) = Mfl(x)Efl(x) (5.22)

The second difficulty is due to the added gaseous fuel mass into the computational domain.Whenever this happens, the mass enters the domain without thermal energy, so the energysource must have an additional term to account for the temperature at which the liquid fuelhas evaporated. Then, the full energy source [J/m3s] reads as follows:

Esource(x) = − 1A(x)

dEfl

dx+ Msource(x)Efg(x) (5.23)


5.3.2 Interaction between Models

In this study the approach is limited to a constant pressure assumption, just like with thederivation of the 1D model. Such an assumption is allowed as long as the (non-reacting)spray is analyzed in a relatively large constant volume environment. From a numerical pointof view this is beneficial because the 1D model in Matlab is ran, source terms for the 3Dsimulation are calculated and saved in a data-file as plain text. This procedure is followedjust one time for each different case setup. From then on, everything is done with Fluent.

Fluent is a commercial CFD package from which the source code is not accessible. Customsettings that are not adjustable via the graphical or text user interface are only accessedby means of user-defined functions (UDF) [flu06a]. UDFs are C language scripts that arecompiled in Fluent before the initialization of the case. Even with the use of UDFs thepossibilities are restricted to special formats, called macro’s, that are predefined by Fluent.Hence, UDFs are the only way to provide sources of mass, momentum and energy to Fluent.In the header part of the c-file the work directory, spray origin, spray direction and theinitial ambient conditions are defined. The rest of the c-file is universal, thus does not needadaptations for each case. The interaction between files and models is schematically depictedin Figure 5.7.

m-file

Run

data

file***

c-file

UDF

1D spray model

Matlab

3D spray model

Fluent

manual input

case setup*

model settings**

* d, Cd, T0, Ta, Pinj, Pa or ρa, Yi

** dx, N, initial guess of Γ, m and T at nozzle exit

ac for spray angle relation

*** mass, momentum and enthalpy sources

spray geometry information

(spray angle and unsteady LL)

Figure 5.7: Followed procedure from manual input to 3D model in Fluent

As is mentioned in the previous section about momentum sources, the spray axis is generallynot the same as one of the main axes. Therefore rotations are performed to virtually coincidede 1D spray axis with the real spray axis in the 3D domain. The z-axis in Fluent is chosen tobe the default spray direction. Other directions are defined by elementary rotations aroundthe x-axis first and then around the new, after the first rotation obtained, y-axis, such thatthe z-axis after two rotations points in the desired spray direction.


5.4 Results; 3D Spray Model

For simulations with the combined 1D spray model and Fluent, the same slice mesh config-uration and boundary conditions as shown in Figure 4.7 in Section 4.1.3 is used, but nowwith smaller cells. In radial direction the cell length is approximately 0.09 mm, because fromAbraham’s work [Abr97] follows that at least 2 cells should be present in the nozzle holegeometry in order to have mesh independent results near the nozzle. In axial direction thecells also have lengths of 0.09 mm at the nozzle exit. To save simulation time (domain islargest in axial direction) these cells gradually increase to at most 0.5 mm at the end of thedomain. But according to Versaevel et al further refinement to half of the axial and radialcell size barely (approximately 1 %) influences the results.

Another numerical consideration is the solver timestep. The results shown in this sectionare for a timestep of 10−5 s. Smaller steps like 10−6 and 5 × 10−7 s give only little (1-2 %)improvement, except for the first part (around 0.1 ms) where the spray length is predicted upto 10 % better. Nevertheless a timestep of 10−5 s is preferred at this stage because of, again,computational expenses.

The results in Figure 5.8 represent the spray lengths for the IFP dodecane case. The IFPmeasurement is indicated with the dotted black line and the 1D model result is indicatedwith the red line. The circular markers are spray lengths from the combined 3D model, thatproves to be consistent with the experimental values. Leaving the timedependent massflowat the start of injection out of consideration, the maximal deviation is 6.3 %.

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

5

10

15

20

25

30

35

40

45

50case: IFP dodecane

Time [ms]

SL

[mm

]

3D model1D modelfit through IFP measurements

Figure 5.8: Spray length as function of time with the 3D model, compared to 1D and measurement


A similar picture is observed for the IFP heptane case in Figure 5.9. But now, also FluentDPM results are shown, indicated with the stars. The DPM model is run with the squaremesh configuration (1 mm3 cells) and 10−6 s as the solver timestep. The liquid length is inthis case far off from the experimental value. The spray length however corresponds better tothe IFP measurements. Still, the DPM model gives up to 18.3 % deviation whereas the 3Dmodel is 8.6 % off. And, as mentioned earlier, the error of the 3D model prediction decreasesto about 6.7 % for smaller timesteps and decreases maybe even slightly more when a higherresolution mesh is applied.

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

5

10

15

20

25

30

35

40

45

50case: IFP heptane

Time [ms]

SL

[mm

]

3D modelFluent DPM1D modelfit through IFP measurements

Figure 5.9: Spray length as function of time with the 3D model, compared to DPM, 1D and measure-ment

The correctness of the 3D model prediction is much better visualized with the contours offuel mass fraction at the spray cross-section shown in Figure 5.10. The upper spray is a DPMsimulation result and the other one is gained with the 3D model, both at 1 ms. Apart from theobvious spray length difference, the shape/width of the sprays are also dissimilar. It is clearlyseen that the DPM spray is much wider than the more realistic 3D model spray, especiallynear the nozzle, because there are 1 mm3 large cells present in the DPM case. The result isthat evaporated heptane is introduced in a larger cell, and therefore spreads the results out,hence the lower absolute mass fraction values.

5.5. ACHIEVEMENTS AND FUTURE WORK 47

Figure 5.10: Case: IFP heptane. Contours of fuel mass fraction gained with Fluent’s DPM model andthe implemented 3D spray model. Note the minimum/maximum values between the brackets at theright hand side, the colorbar is scaled for each separately

5.5 Achievements and Future Work

Summarizing the achievements in this chapter, the 1D mixing limited spray model proposedby Versaevel et al is successfully implemented in Matlab. The results for spray and liquidlengths correspond well with the measurements of IFP. Next, source terms are extracted fromthe 1D results and are via UDFs used in 3D simulations with Fluent. This leads to similarspray lengths as for the 1D model and the overall performance (spray length, spray shape)is much better than the solution of the DPM model from Chapter 4. Furthermore, togetherwith the proper mesh resolution behavior (higher resolution gives better solutions), the abilityto parallelize is a major advantage compared with Fluent’s DPM model, which uses only oneCPU to do all discrete phase calculations no matter how many CPUs are available. Theconsequence is large computation times despite the relatively small amount of cells.

Possible improvements to the 1D/3D spray model for the future may be the following, inorder of decreasing importance:

• Make 3D spray model pressure dependent in order to simulate spray formation in avariabele volume combustion chamber. The best way to do so, is probably by convertingthe 1D model script (m-file) into C language. Then the 1D model, which generatessources for the 3D calculation (matter of seconds), can update the sources online at


each timestep by, for example, looking at the average pressure in the volume at thatmoment.

• Sound spray angle prediction. Not really an improvement to the model itself, butspray angle prediction certainly has a large effect on the final results as extensivelydiscussed in Section 5.2. Therefore the quest for proper and generic angle predictionmethods/relations should be promoted.

• Include radial distribution of velocity, density and temperature profiles. For example theuse of a Gaussian radial distribution under the assumption of a fully-developed turbulentflow as is done by Pastor et al [PLGP08]. This issue becomes particularly importantwhen other than conventional, circular nozzle hole, injectors would be modeled with thedeveloped 1D to 3D infrastructure.

Chapter 6

Modeling Spray Combustion withTabulated Chemistry

All non-premixed combustion phenomena go hand in hand with mixing processes. A chal-lenging example from a numerical point of view is diesel spray formation and combustion.In the preceding chapters a 3D Euler-Euler spray model is implemented and validated forevaporating inert fuel sprays. This model is mesh and solver timestep independent, and issuitable for parallel simulations. So, regarding the status of modeling the mixing process,additional modeling features, which may require fine spatial and time resolutions, can beincluded. In this chapter an attempt is made to add combustion, more specific, the emphasisis on the application of FGMs (Flamelet Generated Manifolds) in modeling of the turbulentcombustion of a transient spray.

In the following, first the principle of flamelets and its use for modeling combustion withtabulated chemistry is mentioned. Then, the procedure of 4D FGM generation is described.Subsequently, its implementation into Fluent and from that arising necessary adaption to theway of implementation of the spray model is shown. Finally, some results and emerged issuesthat still have to be solved are discussed.

6.1 FGM Approach

Many combustion models exist, varying from heat release rate functions to detailed chemicalkinetics with finite-rate Arrhenius reaction sources. These examples are two extremes of sim-ple and detailed combustion modeling, respectively. Simple models are known for their shortcomputing times but also for their lack of accuracy. Detailed models however, can be veryaccurate, but unfortunately also computationally very expensive. To overcome the imprac-tical computing times, while solving the combustion process still with high detail (dependson used reaction scheme), an approach with tabulated chemistry is applied. This so-calledFGM approach is developed by van Oijen [vO02] for laminar premixed flames, and makes useof 1D laminar flamelet data to tabulate composition, density, temperature etc. as functionof local control variables. Later, Ramaekers [Ram05] extended the application to turbu-lent partially-premixed combustion by choosing one control variable describing non-premixed

49

50CHAPTER 6. MODELING SPRAY COMBUSTION WITH TABULATED CHEMISTRY

(mixture fraction Z) and one describing premixed (reaction progress variable PV ) combus-tion, and by PDF (Probability Density Function) integration to account for turbulence.

6.1.1 The Flamelet Concept

The flamelet concept views the turbulent flame as an ensemble of thin, laminar, locally 1Dflames, called flamelets, embedded within the turbulent flow field [flu06b]. Furthermore,the concept is based on the assumption that the smallest turbulent time and length scalesare much larger than the chemical ones, and there exists a locally undisturbed sheet wherechemical reactions occur [SRM06].

A counterflow diffusion flame is a common laminar flame that is used to represent a flamelet ina turbulent flow. The counterflow geometry consists of two opposed flows, with in this studypure fuel at one side and oxidizer (mixture of ambient air) at the other side. Increasing flowvelocities and/or decreasing the distance between the flows results in straining of the flame andincreasingly departs from chemical equilibrium until the flame extinguishes at some point. Innon-premixed combustion it is common practice to introduce the mixture fraction Z definedas:

Z =sYFu − YO2 + YO2,2

sYFu,1 + YO2,2, (6.1)

where s is the stoichiometric mass fraction and the subscripts 1 and 2 refer to the constantmass fraction in the original fuel and oxidizer streams, respectively. In the fuel stream themixture fraction is equal to unity and monotonically decreases to zero at the oxidizer stream.In the 1D laminar flamelet concept this property of the mixture fraction is used to study theconcerned scalars as function of Z instead of the spatial coordinate x. See reference [dGSB+07]for a descriptive derivation or van Oijen [vO02] for a complete derivation. An additionalcontrol variable, called the reaction progress variable PV , is introduced to parameterize theprogress of the irreversible combustion process. In this study a combination of CO2, CO andCH2O mass fractions is chosen as a reaction progress variable:

PV =YCO2

MCO2

+YCO

MCO+

YCH2O

MCH2O. (6.2)

This decision is based on successful auto-ignition modeling efforts at the University of Karl-sruhe [SHS+08], and on common hydrocarbon chemistry knowledge that formaldehyde (CH2O)is an intermediate that has a driving force on reactions in the early stages of combustion atrelative low temperatures. Later in the combustion process CO becomes more important,ultimately (ideally) all carbon atoms end in CO2 molecules.

The succes of this concept is related to the fact that all occurring compositions tend tohave a common, low-dimensional, attractor in composition space, a so-called intrinsic low-dimensional manifold (ILDM) [MP92]. Hence, the complex chemistry is reduced and com-pletely described by the mixture fraction Z and the reaction progress variable PV .

6.1.2 FGM Generation

The low-dimensional manifold is now tabulated as function of the two spanning variables Zand PV . This reduction allows the flamelet calculation to be preprocessed and all species

6.1. FGM APPROACH 51

mass fractions Yi, temperature T , density ρ, source of the progress variable sPV etc. to betabulated as function of the two control variables in a 2D look-up table. By preprocessingthe chemistry, computational costs are reduced considerably [flu06b]. The flamelet equationswith counterflow boundary conditions are solved with CHEM1D [che], which is a specializedone-dimensional laminar flame code developed at the Eindhoven University of Technology.

The turbulence-chemistry interaction is accounted for by integrating the quantities ϕ in the2D table with a β-PDF function as follows:

ϕ =∫ 1

0

∫ 1

0ϕ(Z, PV ) P (Z)P (PV ) dZdPV. (6.3)

Note that this explicit formulation assumes that Z and PV are statistically independent.The overtilde stands for Favre (mass) averaged. Both control variables are from then ondescribed with a mean value (Z, P V ) and a variance (Z”2, PV ”2), so a quantity is definedby the probability of occurrence for several states instead of one fixed state. Details on theapplication of the β-PDF can be found in [Ram05]. The chemistry is in this way extended to a4D look-up table with the means and variances of the two control variables as the parameters(look-up indices).

The way(s) a FGM is constructed in this study is depicted schematically in Figure 6.1. First aheptane flamelet database at constant pressure is calculated with CHEM1D, making use of areduced TRF (Toluene Reference Fuels) mechanism from which the original C1-C3 chemistryis replaced by that of the GRI 3.0 mechanism [SGF+] to reduce the total number of species,ultimately leading to 248 elementary reactions between 48 species. For more informationabout reduced TRF mechanisms the reader is referred to for instance [ABCK07].

FGMs can be generated in many ways. For stationary flames, there is a classical way withsteady flamelets only, where a loop over strainrates is performed until the flame extinguishes.An illustrative example is shown in Figure 6.2, see the gray area between the solution forthe lowest strainrate and the solution at wich the strainrate reached its maximum beforeextinction. For the stationary TRF mechanism solutions the quenching part of the wellknown S-curve is plotted in Figure 6.3; maximum flame temperature as function of one overthe strainrate. Temperatures up to 2400 K are observed for small strainrates and at a value ofapproximately 14600 s−1 the maximum temperature drops beneath 1700 K prior to extinction.

But now the situation is unsteady and initially non-reacting, so to cover the ignition pro-cess the table should also contain information in the area beneath the quenching strainratesolution. Several ways exist to fill this gap in the Z-PV plane. One way is to solve a timede-pendent flamelet with a higher strainrate than the highest possible non-quenching strainrate,in this way forcing the flame to extinguish and in the mean time sampling data to fill the gap.Another approach, that is more appropriate for this study, is solving timedependent flameletsfrom a mixed, but non-reacting initial state in a counterflow setup. The ignition behavior isfollowed in time until a steady flame is reached. This process for heptane ignition is shownin Figure 6.4 for PV as function of Z at several moments in time. It is seen that the steadysolution (the one with the widest base) has a lower peak than few solutions earlier in time.So, the mixture ignites, reaches high values for the progress variable in Z-space and before asteady solution is reached, the progress variable slightly decreases. The third possibility is toreproduce ignition of mixtures covering the entire Z-space with PSR (Perfectly Stirred Re-


actor) auto-ignition calculations as Michel et al [MCV07] applied to laminar diffusion flamesrecently. All three methods to fill the Z-PV gap are depicted schematically in Figure 6.2.In this study however, only the quenching and igniting flamelet approaches, as is shown inFigure 6.1, are used.

igniting flame

Z-PV domain filled with , from

initial pure mixing solution ,

igniting flame , using the

timedependent solver

TRF mechanism

48 species

248 reactions

quenching flame

Z-PV domain filled with

stationary loop over strainrates ,

and with timedependent

quenching flame

CHEM1D

solves:

flamelet equations

(constant pressure )

4D FGM– table

(turbulence included )

2D data integrated with PDF

functions. ρ, sPVm, sPVv, Yi and T

as function of mmmmmmmm

2D FGMZ, PV – table

(laminar)

interpolated laminar flamelet data ρ,

sPV, Yi and T as function of

Z and PV

±%, ,"2 "2

PV ,PVZ Z

± "2

sPV ,sPV

±%, ,"2 "2

PV ,PVZ Z

Figure 6.1: FGM construction scheme


0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

Flamelet database generation

Z [−]

PV

[−]

igniting PSRs

Lowest strainrateSteady solutions regionHighest non−quenching strainrateTimedependently extinguishing or igniting flameletPerfectly Stirred Reactors (PSRs) before ignition

igniting flamelet

extinguishing flamelet

Figure 6.2: Ways to generate a ’full’ flamelet database


10−5

10−4

10−3

10−2

10−1

100

101

1600

1700

1800

1900

2000

2100

2200

2300

2400

2500Maximum temperature as function of strainrate

Strainrate−1, 1/a [s]

Tem

pera

ture

[K]

Tmax

Figure 6.3: S-curve, quenching part

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

1

2

3

4

5

6

Z [−]

PV

[−]

Progress variable as function of mixture fraction

Z = (s*YFu

− YO

2

+ YO

2,2

) / (s*YFu,1

+ YO

2,2

)

PV = YCO

2

/MCO

2

+ YCO

/MCO

+ YCH

2O

/MCH

2O

ignition

Z = (s*YFu

− YO

2

+ YO

2,2

) / (s*YFu,1

+ YO

2,2

)

PV = YCO

2

/MCO

2

+ YCO

/MCO

+ YCH

2O

/MCH

2O

ignition

Z = (s*YFu

− YO

2

+ YO

2,2

) / (s*YFu,1

+ YO

2,2

)

PV = YCO

2

/MCO

2

+ YCO

/MCO

+ YCH

2O

/MCH

2O

ignition

Z = (s*YFu

− YO

2

+ YO

2,2

) / (s*YFu,1

+ YO

2,2

)

PV = YCO

2

/MCO

2

+ YCO

/MCO

+ YCH

2O

/MCH

2O

ignition

Z = (s*YFu

− YO

2

+ YO

2,2

) / (s*YFu,1

+ YO

2,2

)

PV = YCO

2

/MCO

2

+ YCO

/MCO

+ YCH

2O

/MCH

2O

ignition

Z = (s*YFu

− YO

2

+ YO

2,2

) / (s*YFu,1

+ YO

2,2

)

PV = YCO

2

/MCO

2

+ YCO

/MCO

+ YCH

2O

/MCH

2O

ignition

Z = (s*YFu

− YO

2

+ YO

2,2

) / (s*YFu,1

+ YO

2,2

)

PV = YCO

2

/MCO

2

+ YCO

/MCO

+ YCH

2O

/MCH

2O

ignition

Z = (s*YFu

− YO

2

+ YO

2,2

) / (s*YFu,1

+ YO

2,2

)

PV = YCO

2

/MCO

2

+ YCO

/MCO

+ YCH

2O

/MCH

2O

ignition

Z = (s*YFu

− YO

2

+ YO

2,2

) / (s*YFu,1

+ YO

2,2

)

PV = YCO

2

/MCO

2

+ YCO

/MCO

+ YCH

2O

/MCH

2O

ignition

Z = (s*YFu

− YO

2

+ YO

2,2

) / (s*YFu,1

+ YO

2,2

)

PV = YCO

2

/MCO

2

+ YCO

/MCO

+ YCH

2O

/MCH

2O

ignition

Z = (s*YFu

− YO

2

+ YO

2,2

) / (s*YFu,1

+ YO

2,2

)

PV = YCO

2

/MCO

2

+ YCO

/MCO

+ YCH

2O

/MCH

2O

ignition

Z = (s*YFu

− YO

2

+ YO

2,2

) / (s*YFu,1

+ YO

2,2

)

PV = YCO

2

/MCO

2

+ YCO

/MCO

+ YCH

2O

/MCH

2O

ignition

Z = (s*YFu

− YO

2

+ YO

2,2

) / (s*YFu,1

+ YO

2,2

)

PV = YCO

2

/MCO

2

+ YCO

/MCO

+ YCH

2O

/MCH

2O

ignition

Z = (s*YFu

− YO

2

+ YO

2,2

) / (s*YFu,1

+ YO

2,2

)

PV = YCO

2

/MCO

2

+ YCO

/MCO

+ YCH

2O

/MCH

2O

ignition

Z = (s*YFu

− YO

2

+ YO

2,2

) / (s*YFu,1

+ YO

2,2

)

PV = YCO

2

/MCO

2

+ YCO

/MCO

+ YCH

2O

/MCH

2O

ignition

Z = (s*YFu

− YO

2

+ YO

2,2

) / (s*YFu,1

+ YO

2,2

)

PV = YCO

2

/MCO

2

+ YCO

/MCO

+ YCH

2O

/MCH

2O

ignition

Z = (s*YFu

− YO

2

+ YO

2,2

) / (s*YFu,1

+ YO

2,2

)

PV = YCO

2

/MCO

2

+ YCO

/MCO

+ YCH

2O

/MCH

2O

ignition

Z = (s*YFu

− YO

2

+ YO

2,2

) / (s*YFu,1

+ YO

2,2

)

PV = YCO

2

/MCO

2

+ YCO

/MCO

+ YCH

2O

/MCH

2O

ignition

Z = (s*YFu

− YO

2

+ YO

2,2

) / (s*YFu,1

+ YO

2,2

)

PV = YCO

2

/MCO

2

+ YCO

/MCO

+ YCH

2O

/MCH

2O

ignition

Figure 6.4: Ignition from an initially mixing-only solution in a counterflow setup


6.1.3 Implementation into CFD

The implementation of the 4D FGM in Fluent, in order to do turbulent spray combustion sim-ulations, is once again done with UDFs. Remember from Section 5.3.2 that UDFs are the onlyway to do serious user-defined adjustment in Fluent. This time there are much more UDFsinvolved than for the implementation of the 1D spray model, because the FGM approachrequires four variables to be solved, namely: Z, P V , Z”2 and PV ”2. These four scalars aresolved with user-defined scalar transport equations, in addition to the standard continuity,momentum and turbulence equations. Compared to inert spray formation calculations, thereare no species transport equations and energy equation solved any more. All species concen-trations and corresponding temperatures are in principle known from the flamelet databasefor any mixture fraction and progress variable combination. But only important species, likethose defining the reaction progress variable, are tabulated in the 2D and 4D FGM togetherwith essential quantities like mean densities and temperatures, to save FGM generation timeand memory.

The transport equations for Z, P V , Z”2 and PV ”2 are derived by Ramaekers [Ram05], butthese are for stationary flames without spray formation in contrary to the diesel injection case.Therefore some modifications are needed. In the first place all four equations are extendedwith an unsteady term. And secondly a source term for the mean mixture fraction is addedsince the mixture fraction is not a conserved scalar anymore. Here, a short derivation is given.The transport equations for fuel and oxidizer are:

∂

∂t(ρYFu) + ~∇ · (ρ~vYFu)− ~∇ · (ρD~∇YFu) = ρFu + SFu, (6.4)

∂

∂t(ρYO2) + ~∇ · (ρ~vYO2)− ~∇ · (ρD~∇YO2) = ρO2 + 0 = sρFu, (6.5)

respectively. s is the stoichiometric mass fraction, ρ is a source due to reactions and SFu isan injection mass source. Substraction of equation (6.5) from equation (6.4) multiplied withs, gives:

∂

∂t(ρ[sYFu − YO2 ]) + ~∇ · (ρ~v[sYFu − YO2 ])− ~∇ · (ρD~∇[sYFu − YO2 ]) = sSFu. (6.6)

Together with the definition of the mixture fraction:

Z =sYFu − YO2 + YO2,2

sYFu,1 + YO2,2=

sYFu − YO2

sYFu,1 + YO2,2+ Zst, (6.7)

with Zst the constant mixture fraction at the location where stoichiometric conditions exist,the transport equation for Z becomes:

∂

∂t(ρZ) + ~∇ · (ρ~vZ)− ~∇ · (ρD~∇Z) =

s

sYFu,1 + YO2,2SFu. (6.8)

From this equation with a source term on the right hand side, transport equations for Z andZ”2 can be derived like Ramaekers did. However, to evaluate the terms SFu and Z”SFu thatappear in those equations, PDF integration is required. Remember from equation (6.3) thatthe PDFs are functions of either Z or PV , but the mass source from the 1D spray model is not.


The mass source is preprocessed and tabulated as function of position and time only, so thereis no coupling with, for example, variations in ambient temperature and density. Therefore,the term on the right hand side is considered to be the source for the mean mixture fraction,and the variance of the mixture fraction is assumed to be unaffected by the fuel injection.See Appendix A for an overview of the solved transport equations.

Furthermore, the same momentum sources are set as before. The 4D FGM and the spraymodel sources are read into Fluent prior to initialization of the problem. From then on thesimulation is ready to start. During computations, at each timestep, the four user-definedscalars are solved and used to look-up the corresponding density ρ, source of the mean progressvariable (sPV ) and the source of its variance (sPV ”2). These are passed to the Fluent solverin order to update the corresponding equations for the new timestep. In Figure 6.5 a schemeof this CFD-FGM interaction is shown. In the mean time all preferred quantities like thetemperature and mass fractions of certain species are saved to allocated memory in Fluent,so they can be postprocessed to view contour plots for instance. However, these quantitiesare not used for any purpose during the simulation itself.

CFD

solves :

u, p, Zm, Zv, PVm, PVv

4D FGM

look up tabulated quantities at :

Zm, Zv, PVm, PVv

Zm, Zv, PVm, PVv

ρ, sPV, sPVv± "2

sPV ,sPV

±%, ,"2 "2

PV ,PVZ Z

±%, ,"2 "2

PV ,PVZ Z

±%, ,"2 "2

PV ,PVZ Z

Figure 6.5: CFD-FGM interaction

6.2 Results; 3D Spray Model with FGM Combustion Model-ing

A reacting spray, modeled with the FGM approach as discussed in the preceding section, isapplied to simulate the IFP heptane experiment. The details of this experiment are shownin Table 5.1, however, the initial ambient gas composition is different now. There is also 0.21(mass fraction) O2 present, while the fractions of CO2 and H2O are the same. In the paper ofVerhoeven et al [VVB98] heptane ignition and combustion results are not presented, mainlythe dodecane case is treated elaborately. But for now, the lack of detailed reaction mechanismsfor relative heavy hydrocarbons like dodecane prevents the combustion simulation of suchfuels with FGMs. Although the simulation results with heptane, for which detailed reactionmechanisms are available, are not compared directly with measurements (often ignition delayand flame lift-off length) at the moment, they are still very important to monitor the FGMspray combustion modeling status in a quantitative fashion. This monitoring is done first bychecking the spray length without combustion and then by looking at the ignition behaviorand the combustion temperatures.

6.2. RESULTS; 3D SPRAY MODEL WITH FGM COMBUSTION MODELING 57

6.2.1 Spray Length without Combustion

Now the FGM approach is implemented, and with that also the way the spray model originallywas implemented, is changed. An important first result to check is therefore the, by nowfamiliar, spray length as function of time for a not yet ignited fuel spray. Figure 6.6 is similarto Figure 5.9, actually it should be the same if the implementation is passed successfully. Theblack circular marks indicate the spray length gained from the 3D spray model with FGM.It is seen that now, the length corresponds even better with the original 1D model and themaximum deviation from the experimental values is just 4.8 %, whereas with the original 3Dmodel this was 6.7 %. So one can say that concerning the spray formation the model is stillfunctioning satisfactory.

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

5

10

15

20

25

30

35

40

45

50case: IFP heptane

Time [ms]

SL

[mm

]

3D model with FGM (no combustion)Fluent DPM1D modelfit through IFP measurements

Figure 6.6: Spray length as function of time with the 3D model with FGM, compared to DPM, 1D andmeasurement

6.2.2 Ignition Behavior

Due to the unsteady nature of a diesel injection event, ignition modeling is at least as im-portant as combustion modeling. Following the FGM approach, besides combustion, ignitionshould be covered inherently. But not surprisingly this depends on the way the FGM is gen-erated. The extinguishing flamelet approach, as presented in Section 6.1.2, is applied anddoes not lead to ignition of the spray. Instead, only local temperatures slightly above theinitial ambient temperature are found, and the source of the reaction progress variable is not


big enough to end in total ignition within a few milliseconds. A FGM constructed with anigniting flamelet database is also applied, and this time the result is auto-ignition of the wholespray in very short time.

The evolution from the early stage of ignition to further combustion of the spray is shownat four moments in time in Figure 6.7. The upper half of the plots represent the values ofthe progress variable and the lower parts are contours of temperature. Several interestingobservations are done from this figure. First, at 0.025 ms, the outer edge of the spray showsactivity by means of an increased (and still increasing) progress variable and a modest increasein temperature. This activity is particularly present close to the spray axis and at the tip ofthe spray. Further in time, at 0.05 ms, the outer contour of the igniting spray is clearly seendue to high values of PV and T , but reactions are still located at the diffusive edge of thespray. In the next plot, at 0.1 ms, the combustion region is expanded to the inner volumeand the maximum temperature continues to rise. From 0.2 ms on the general picture doesnot change much.

Figure 6.7: Case: IFP heptane. Temporal sequence of progress variable and temperature contoursshowing the early ignition process resulting in total combustion

The diffusion flame at the spray edge stays the hottest region, as can be expected on thebasis of the used non-premixed flamelet database. But in Section 2.3 on the conceptualdiesel combustion model it is stated that in DI diesel injection, regions with premixed andnon-premixed combustion can be distinguished. The next step may be the creation of apartially-premixed database in order to capture the presence of both regimes at the sametime. Again, there are no measurements for this case to compare, but until now there are no

6.2. RESULTS; 3D SPRAY MODEL WITH FGM COMBUSTION MODELING 59

strange things happening.

6.2.3 Combustion Temperature

Despite the good looking ignition and flame geometry, temperatures do not reach realisticvalues above 2000 K. Therefore also a much simpler combustion model is used that is availablein Fluent, called eddy-dissipation model [flu06b]. This model is, like the flamelet approach,a mixing-limited combustion model, with the differences that only the global reaction fromfuel and O2 to CO2 and H2O is considered and immediate reaction is assumed. So, it isnot expected that this simple model can be as accurate as the much more detailed FGMmethod, but it gives nice material to compare regarding for example temperature profiles.Such a comparative picture is given in Figure 6.8, taken at 1 ms. In contrast to the eddy-dissipation model, a flame lift-off settles automatically using tabulated chemistry. It is seenthat the eddy-dissipation (upper spray) solution gives temperatures up to 2400 K whereasthe FGM solution stay around 1300-1400 K. This outcome is remarkable when rememberingthe laminar quenching temperature of about 1650 K shown in Figure 6.3 and knowing thatthe FGM flame does not extinguish at all but burns at those low temperatures.

Figure 6.8: Case: IFP heptane. Contours of temperature gained with the eddy-dissipation model andthe implemented FGM model. Note the minimum/maximum values between the brackets at the righthand side, the colorbar is scaled for each separately

It is thought that this occurrence is caused by the PDF integration procedure. In order toconfirm this view a similar heptane spray at 1 atmosphere (atm) instead of the original highpressure of 60 atm is simulated, because the reaction layer thickness will increase at lowerpressures and therefore high peak temperatures will spread out over a larger region resultingin lower peak temperatures. In this way the averaging PDF integration should have lessinfluence. From new CHEM1D calculations indeed follows that the reaction layer thickness


increases from tens of micrometers to the order of millimeters. Since flamelets at 373 K fueland 800 K oxidizer temperature does not auto-ignite, the quenching method is performed.With the obtained quenching flamelet database a FGM is generated and used in manuallyignited spray simulations. From simulations with the original case it is found that manualignition does not influence the ’steady’ combustion behavior and corresponding temperaturesafter few milliseconds. The temperatures at 1 atm ambient pressure stay at values of 1400 K,where the maximum possible temperature according to the FGM is 1810 K. At the original 60atm case the maximum in the table is 2230 K. The temperatures from the 1 atm simulationare closer to the tabulated maximum temperatures, so relatively the FGM combustion modelperforms better now. This is also supported by the fact that only 0.052 % of the Z, P V ,Z”2 and PV ”2 combinations in the FGM give temperatures above 2000 K. Concluding, thesimulations at 1 atmosphere give an affirmative answer to the notion that PDF integration isa source for the low temperatures.

6.3 Improvement Points

So far, the attempted application of a FGM to model the combustion of a transient andturbulent fuel spray is partially succeeded. Spray formation is still predicted accurately andignition follows automatically from the igniting flamelet database. However, the spray auto-ignites too early and (therefore) also too close to the nozzle. Combustion temperatures aretoo low, (partly) due to PDF integration of the laminar 2D table. Yet, the implementation(spray model-CFD interaction, FGM data handling) on its own seems to work well.

In the future, first the PDF integration process should be avoided by simulating 3D sprayswith a laminar flamelet database. When this approach gives promising results, one can focusto the averaging procedure in more detail. If not, then it should be considered to go back toa 1D laminar case to get ignition and combustion right, without the difficulties introducedby working in a not entirely transparant 3D CFD environment with complex data handling.

Ultimately, when serious improvements are achieved, direct validation with measurementslike ignition delay time and flame lift-off length can be performed.

Chapter 7

Conclusions and Recommendations

7.1 Conclusions

In this study first Fluent’s Euler-Lagrange DPM model is used to model evaporating, inertheptane sprays, since heptane is considered as a surrogate for diesel from which all materialproperties are well known. Special attention is paid to model the fuel properties as func-tion of temperature, making use of the thermophysical database of DIPPR. The results arecompared with spray length and liquid length measurements on the EHPC and from Sandia,because these are common spray characterization quantities. Numerical parameters like meshconfigurations, solver timestep and amount of parcels are varied to investigate the influenceon spray length and liquid length. From the comparisons it is found that the DPM modelgives erroneous results, but more significant are the numerical inadequacies. One of themis the imposed limitation to mesh refinement which is far from desirable when detailed in-cylinder mixture formation and combustion are to be modeled. The second disadvantage isthat the discrete phase part of the calculations cannot be parallelized, while those detailedinvestigations require fine resolution, thus expensive simulations. Nevertheless, some bestpractice setups resulted from this study, which are at least valid for heptane sprays in enginelike conditions. That are 1 mm3 cells aligned with the spray axis and solved with a solvertimestep of 10−6 s, and injection of 1000 parcels per timestep.

In the second stage, recognizing the limitations of the DPM model, a 1D mixing limited spraymodel proposed by Versaevel et al is successfully implemented in Matlab. The results for sprayand liquid lengths correspond well with the measurements of IFP. Source terms are extractedfrom this 1D results in order to use in 3D Euler-Euler simulations with Fluent. This leads tosimilar spray lengths as for the 1D model and the overall performance (spray length, sprayshape) is much better than the solution of Fluent’s Euler-Lagrange approach. Furthermore,together with the proper mesh resolution behavior (higher resolution gives better solutions),the ability to parallelize is a major advantage compared with the DPM model, which usesonly one CPU to do all discrete phase calculations no matter how many CPUs are available.The consequence is large computation times despite the relatively small amount of cells.

Finally, having achieved sound mixture formation modeling, ignition and combustion of thefuel spray is considered. The flamelet concept is applied for this purpose. A laminar flamelet

61

62 CHAPTER 7. CONCLUSIONS AND RECOMMENDATIONS

database for heptane is created and PDF integrated, resulting in a 4D FGM. This so-calledtabulated chemistry approach is known for its time-efficiency and at the same time highaccuracy. With this method the implementation of spray sources and solved equations aremodified, but spray formation is still predicted accurately. So far, this first attempt to applyFGMs to model the combustion of a transient and turbulent fuel spray is partially succeeded.Ignition follows automatically from the igniting flamelet database and the whole spray contin-ues to burn. However, the spray auto-ignites too early and combustion temperatures are toolow, (partly) due to PDF integration of the laminar 2D table. Yet, the created infrastructure(spray model-CFD interaction, FGM data handling) on its own seems to work well.

7.2 Recommendations

Directions for future research, taking this study as a starting point, are recapitulated, bothfor spray formation as for spray combustion modeling, respectively.

Possible improvements to the 1D/3D spray model may be the following, in order of decreasingimportance:

• Make 3D spray model pressure dependent in order to simulate spray formation in avariabele volume combustion chamber. The best way to do so, is probably by convertingthe 1D model script (m-file) into C language. Then the 1D model, which generatessources for the 3D calculation (matter of seconds), can update the sources online ateach timestep by, for example, looking at the average pressure in the volume at thatmoment.

• Sound spray angle prediction. Not really an improvement to the model itself, butspray angle prediction certainly has a large effect on the final results as extensivelydiscussed in Section 5.2. Therefore the quest for proper and generic angle predictionmethods/relations should be promoted.

• Include radial distribution of velocity, density and temperature profiles. For example theuse of a Gaussian radial distribution under the assumption of a fully-developed turbulentflow as is done by Pastor et al [PLGP08]. This issue becomes particularly importantwhen other than conventional, circular nozzle hole, injectors would be modeled with thedeveloped 1D to 3D infrastructure.

Considering spray combustion, first the PDF integration process should be avoided by sim-ulating 3D sprays with a laminar flamelet database. When this approach gives promisingresults, one can focus to the averaging procedure in more detail. If not, then it should beconsidered to go back to a 1D laminar case to get ignition and combustion right, without thedifficulties introduced by working in a not entirely transparant 3D CFD environment withcomplex data handling.

Ultimately, when serious improvements are achieved, direct validation with measurementslike ignition delay time and flame lift-off length can be performed.

Nomenclature

Greek symbolsΓ ratio of mass flow rates [-]ε turbulent kinetic energy dissipation rate []θ full spray angle [degrees]Λ wavelength [m]µ dynamic viscosity [kg/ms]ρ density [kg/m3]ρ mass source [kg/m3s]σ surface tension [kg/s2]Ω frequency [Hz]

Sub-/superscripts0 initial and/or at nozzle exita ambient gaseff effectivef fuelfg gaseous fuelfl liquid fuelg whole gaseous phasel liquidt turbulent˜ Favre (mass) average

63

Latin symbolsA cross-section area [m2]Cd discharge coefficient [-]cp specific heat [J/kgK]d,D diameter [m]D diffusion coefficient [m2/s]E energy [J/kg]E energy flow rate [J/s]F force [N]~g gravitational acceleration [m/s2]h enthalpy [J/kg]~J diffusion flux [kg/ms]k turbulent kinetic energy []K cavitation number [-]L length [m]LL liquid length [m]Lv latent heat of vaporization [J/kg]m void fraction [-]m, M mass flow rate [kg/s]M molecular mass [kg/mol]N amount of droplets [-]Oh Ohnesorge number [-]p pressure [Pa]P Probability Density Function [-]PV reaction progress variable [-]r radius [m]Re Reynolds number [-]s stoichiometric mass fraction [-]Sc Schmidt number [-]SL spray/vapor length [m]sPV source of progress variable [kg/m3s]t time [s]T temperature [K]T momentum flow rate [kgm/s2]u, v velocity [m/s]~v velocity vector [m/s]x coordinate on spray axis [-]X mole fraction [-]Y mass fraction [-]We Weber number [-]Z mixture fraction [-]

64

Appendix A

Transport Equations for the Meanand Variance of Z and PV

The steady terms of the following equations are adopted from Ramaekers [Ram05], exceptfor the mean mixture fraction equation. The mean mixture fraction equation is derived inSection 6.1.3.

Mean mixture fraction Z:

∂

∂t(ρZ) + ~∇ · (ρ~vZ)− ~∇ · (ρ[D + DT ]~∇Z) =

s

sYFu,1 + YO2,2SFu. (A.1)

Variance of mixture fraction Z”2:

∂

∂t(ρZ”2) + ~∇ · (ρ~vZ”2)− ~∇ · (ρ[D + DT ]~∇Z”2) = 2ρ

(DT [~∇Z”2]2 − εZ”2

k

). (A.2)

Mean reaction progress variable P V :

∂

∂t(ρP V ) + ~∇ · (ρ~vPV )− ~∇ · (ρ[D + DT ]~∇P V ) = ωPV = sPV . (A.3)

Variance of reaction progress variable PV ”2:

∂

∂t(ρPV ”2)+~∇·(ρ~vPV ”2)−~∇·(ρ[D+DT ]~∇PV ”2) = 2ρ

(DT [~∇PV ”2]2 − εPV ”2

k

)+2PV ”ωPV ,

(A.4)where the last term represents the tabulated source of the progress variable variancesPV ”2 = PV ”ωPV .

65

66APPENDIX A. TRANSPORT EQUATIONS FOR THE MEAN AND VARIANCE OF Z AND PV

Acknowledgements

Many people contributed to the realization of this Master’s thesis. I would like to thankeveryone who helped me in one way or another, and a few people in particular.

First of all, my supervisor Bart Somers for his coaching and for the freedom he gave me indoing my research. Especially his enthusiasm during discussions about combustion modelingwere motivating. Concerning spray formation modeling, the enthusiasm and help of CarloLuijten was motivating in the same way. And Philip de Goey for his sincere interest in theproject and encouragement. The informal discussions at the ”Spray Meetings” I had withPhilip, Bart and Carlo resulted in better understanding from both sides. Thank you all forthat.

Thanks to Giel Ramaekers for his support by implementing the FGM approach into Flu-ent. For several weeks/months I worked together with Justin Gerritzen (spray modeling inFluent) and Jos Reijnders (phenomenological spray model), thank you both for the smoothcooperation.

Last but not least, I would like to thank the other master students in the hallway for thepleasant times and excellent atmosphere. Once again special thanks to Jos Reijnders andDennis Verwaaijen, who were my roommates for the whole and half year respectively, for thedaily help and above all the many amusing moments.

67

68

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