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Tampere University of Technology
Modelling of Spray Combustion, Emission Formation and Heat Transfer in MediumSpeed Diesel Engine
CitationTaskinen, P. (2005). Modelling of Spray Combustion, Emission Formation and Heat Transfer in Medium SpeedDiesel Engine. (Tampere University of Technology. Publication; Vol. 562). Tampere University of Technology.
Year2005
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Tampereen teknillinen yliopisto. Julkaisu 562 Tampere University of Technology. Publication 562 Pertti Taskinen Modelling of Spray Combustion, Emission Formation and Heat Transfer in Medium Speed Diesel Engine Thesis for the degree of Doctor of Technology to be presented with due permission for public examination and criticism in Konetalo Building, Auditorium K1702, at Tampere University of Technology, on the 2nd of December 2005, at 12 noon. Tampereen teknillinen yliopisto - Tampere University of Technology Tampere 2005
ISBN 952-15-1476-0 (printed) ISBN 952-15-1498-1 (PDF) ISSN 1459-2045
i
ABSTRACT This thesis deals with the spray combustion, emissions (NOx and soot) formation and heat transfer
theories of phenomena and their modelling related to medium speed diesel engines. The modelling
work was done with the Marintek A/S version of the open source code KIVA-II program by
implementing new sub-models or by modifying old models of the phenomena into the code.
The aim of the work has been to develop a simulation tool for medium speed diesel engines that can
be applied later in the optimisation process of the engine economy with the allowed pollution level
by computing different cases with the different engine parameters such as compression ratio, fuel
injection timing, injection rate shaping, direction of injection, diameter of the nozzle hole etc. In
developing work of the KIVA-II code main attention was focused on the following phenomena: the
drop vaporisation under a high-pressure environment, the soot formation modelling by the Hiroyasu
TM models and the or the oxidation by the NSC model, the soot radiation modelling by the
simplified model (pure emission) or the DOM, the convective heat transfer modelling and the spray
turbulence modelling by the RNG/STD k-e turbulence models.
The high pressure drop vaporisation model was developed based on the equality of the fugacity of
the fuel in liquid and the vapour phase on the drop surface. The mass fraction of fuel vapour in the
drop surface is much larger with the high pressure model than with the original low-pressure model
yielding a more realistic ignition of the fuel vapour and air mixture and the combustion.
The original TM soot formation model of the code was a failure and this was rectified. The
Hiroyasu soot formation and the NSC soot oxidation model were added into the code and
formulated into the source term form using either the computational cell average or the EDC-
weighted values of the cell quantities in the soot transport equation. The soot emissions after
modifications were a more realistic level than in the case of the original formulation and the
models. Also the lack of an NSC soot oxidation model able to predict the soot oxidation rate
correctly was taken into account by the extra constant in the model.
The soot radiation was taken into account in the internal energy transport equation by the simplified
model (optically thin radiant media), i.e. pure emission from the radiant media or the RTE solved
by the DOM. The radiant heat flux to piston top becomes the more realistic level with the DOM
than with the simplified model compared to the experimental values of the slightly other type diesel
ii
engine than the modelled medium speed diesel engine. This shows that the absorption of soot
radiation in the radiant region must also be taken into consideration. Effect of the soot radiation on
temperature of the gas appears only in the soot region, not in the fuel vapour reaction zone where
the soot is not found. Therefore the soot radiation does not reduce maximum temperatures of the
gas in the fuel vapour reaction zone or in the nitrogen oxide (NOx) formation regions near the
reaction zone and so influence in the NOx emissions from the engine.
The original temperature wall function of the KIVA-II based on the modified Reynolds analogy
under-predicts the heat flux to wall considerably. The model was replaced by the model which was
based on the use of a one-dimensional energy equation and the correlation of dimensionless
temperature including an increasing turbulent Prandtl number near the wall. The heat flux to piston
top with the new model was a more realistic level than with the original model of the code
compared to the experimental values of the other type diesel engine.
The modified RNG k-epsilon model was developed based on the results obtained with the STD and
the basic RNG k-e models. According to the results mentioned above the STD model is too
diffusive while the basic RNG is too less diffusive in the high rate of the strain region (spray region)
and therefore the fuel vapour mixing (combustion) occurs in an un-satisfactorily way. In the
turbulence model developed the additional term of the epsilon equation was modified suitably and
therefore the spray spreading and the combustion occur more realistically compared to either the
basic RNG or the STD k-e turbulence model cases. The gas turbulence intensity was reduced in the
early phase of combustion and emphasized in the later phase of combustion compared to the
situation with the STD model. The cylinder pressure curve becomes by far the closest with the new
turbulence model than either of both the models mentioned above. In the work the failure of the
basic RNG turbulence model of the KIVA-3V was found and rectified.
iii
PREFACE This work has been carried out at the Institute of Energy and Process Engineering, Tampere
University of Technology (TUT). The work has been funded by the PROMOTOR program
(Mastering the Diesel Process (MDP)) of the National Technology Agency of Finland (Tekes) and
the CFD Graduate school program of the Aerodynamic Laboratory of Helsinki University of
Technology (HUT).
I wish to express my gratitude to Professor Reijo Karvinen, advisor of my dissertation for his
guidance during this work. I would also like to thank all the staff at the Institute of Energy and
Process Engineering.
Furthermore, I wish to extend my thanks to Dr. Eilif Pedersen at the Marintek A/S Research Centre
of the Sintef Group, Trondheim, Norway for his unique guidance with the KIVA-II code and to
Professor Martti Larmi at the Internal Combustion Engine Laboratory (ICEL) of Helsinki
University of Technology for the discussions and meetings on the MDP project. I would also like
to thank Mr. Gösta Liljenfeldt at the Wartsila Diesel Company in Vaasa for the support during the
entire co-operation time of the medium speed diesel engine process modelling and Mr. James
Rowland for the high quality reviewing the English of the manuscript.
Finally, I must thank to my roommate Licentiate of Technology Vesa Wallen, for the interesting
and inspiring discussions on the work.
Tampere, May 2005
Pertti Taskinen
iv
v
CONTENTS
ABSTRACT i
PREFACE iii
CONTENTS v
NOMENCLATURE ix
1. INTRODUCTION 1
1.1 General aspects 1
1.2 Diesel process modelling 2
1.3 Goal and outline of this thesis 5
2. THEORY OF DIESEL PROCESS MODELLING 7
2.1 Governing field equations 7
2.2 Main sub-models in diesel process modelling 8
2.2.1 Turbulence modelling 9
2.2.2 Fuel spray modelling 12
2.2.2.1 General aspects 12
2.2.2.2 Fuel jet break-up/atomisation regimes 13
2.2.2.3 Short review of the fuel spray models 15
2.2.3 Drop dynamics 21
2.2.4 Drop vaporisation 23
2.2.5 Fuel vapour combustion 27
2.2.5.1 General aspects 27
2.2.5.2 Premixed combustion 29
2.2.5.3 Diffusion combustion 30
2.2.6 Emissions modelling 39
2.2.6.1 Nitrogen oxide emissions 40
2.2.6.2 Soot emissions 41
2.2.6.2.1 Soot formation 41
2.2.6.2.2 Soot oxidation 45
2.2.6.3 Soot modelling by EDC-model formulation 47
vi
2.2.7 Heat transfer 49
2.2.7.1 Convective heat transfer 49
2.2.7.2 Heat transfer by radiation 51
3. AUTHOR’S IMPLEMENTED/DEVELOPED SUBMODELS AND
THEIR CONTRIBUTION TO THE MODELLING TOOL FOR
DIESEL PROCESS ANALYSIS 57
3.1 Sub-models in baseline Marintek KIVA-II 57
3.2 Sub-models used in current KIVA-II 57
3.3 List of author’s publications related to this work 59
4. MODELLING RESULTS AND THEIR EXPERIMENTAL
VERIFICATION 61
4.1 Turbulence results with the STD, basic RNG and modified RNG k-e models 61
4.1.1 Turbulence intensity 61
4.1.2 Turbulence kinetic energy distribution 64
4.1.3 Turbulence viscosity 66
4.1.4 Spray spreading 67
4.2 Results of drops high/low-pressure vaporisation formulation 69
4.2.1 Amount of fuel vapour in combustion chamber 70
4.2.2 Pressure of cylinder gas 71
4.2.3 Cumulative heat release 71
4.3 Effect of turbulence model on combustion results 72
4.3.1 Pressure of cylinder gas 73
4.3.2 Cumulative heat release 74
4.3.3 Temperature of gas 75
4.4 Nitrogen oxide emissions 78
4.5 Soot emissions 82
4.6 Heat transfer 88
5. CONCLUSIONS 93
6. REFERENCES 97
vii
APPENDIX A: Modelled engine specifications
Computational mesh of modelled engine
APPENDIX B: Flow chart of numerical modelling tool
viii
ix
NOMENCLATURE Latin a Parent drop radius [ ]m
a Premixed combustion model constant [ ]−a Soot formation model constant [ ]s/1
0a Soot nucleus formation model constant [ ]sgpart /
( T,fa v ) Soot absorption coefficient [ ]m/1
A Combustion model constant [ ]−
fA Hiroyasu soot formation model constant [ ]s/1
wallA Total surface area of combustion chamber [ ]2m
b Premixed combustion model constant [ ]−b Soot formation model constant [ ]spartcm /3
10 B,B Wave, drop break-up model constants [ ]−
MH B,B Spalding heat and mass transfer number [ ]−
SC
CCC
CCCC
,,,
,,,,
µηη
η
21
321
Turbulence model constant [ ]−
4321 C,C,C,C HG spray model constants [ ]−
vk
Fdb
C,C,C,C,C TAB spray model constants [ ]−
DC Drop drag coefficient [ ]−
d,vc Specific heat of drop at constant volume [ ]kgKJ /
d,pc Specific heat of drop at constant pressure [ ]kgKJ /
gas,pc Specific heat of gas at constant pressure [ ]kgKJ /
χC Time scale ratio of LFM combustion model [ ]−
MC Turbulent time scale constant of CHTC combustion model [ ]−
NSCC Extra constant in NSC soot combustion model [ ]−
x
d Diameter [ ]m
D Diffusion coefficient [ ]sm /2
RE / Activation temperature [ ]K
1E Activation energy [ ]molkJ /
f Soot formation model constant [ ]s/1
f Weighted function [ ]−
Cf Carbon factor in soot formation model [ ]−VL f,f Fugacity of fuel liquid and vapour [ ]Pa
F Aerodynamic force in TAB spray model [ ]N
TM F,F Correction factors in drop vaporisation model [ ]−g Soot formation model constant [ ]s/1
0g Soot formation model constant [ ]spartcm /3
jg Acceleration due to gravity [ ]2/ sm
Ch Heat transfer coefficient [ ]KmW 2/
h Specific enthalpy [ ]kgKJ /
i Dummy index [ ]−I Specific internal energy [ ]kgJ /
( )ω,rI Local directional intensity of radiation [ ]srmW 2/
( )TIb Intensity of black body radiation [ ]srmW 2/
( )rI i Local intensity of radiation in direction i [ ]2/ mW
j Dummy index [ ]−
wJ Convective heat flux to wall [ ]2/ mW
J Total heat flux vector [ ]2/ mW
k Dummy index [ ]−k Turbulent kinetic energy [ ]22 / sm
k Heat conductivity of gas [ ]mKW /
k TAB spray model (spring) constant [ ]mN /
fik Rate constant of forward reaction i [ ]scmmol 3/
xi
Ak , Rate constant of soot oxidation reaction Bk [ ]sPacmg 2/
Tk Rate constant of soot oxidation reaction [ ]scmg 2/
Zk Rate constant of soot oxidation reaction [ ]Pa/1
K Mass transfer coefficient [ ]smkg 2/
1K Pre-exponential factor of combustion reaction [ ]scmmol 3/
iKC
, Equilibrium constants of reactions i and ii iiKC
[ ]−
l Dummy index [ ]−l Length scale [ ]m
L Latent heat of vaporisation [ ]kgJ /
Le Lewis number [ ]−
AL , , Atomisation, turbulence and wave perturbation length scales TL WL [ ]m
IL , Intact core and break-up lengths BUL [ ]m
λL Taylor micro scale of turbulence [ ]m
m Mass [ ]kg
m Number of hydrogen atoms in fuel molecule [ ]−M Mole mass [ ]molg /
n Soot refractive index [ ]−n Number of carbon atoms in fuel molecule [ ]−N Number of drops after break-up [ ]−
0N Number of parent drops [ ]−
Oh Ohnesorge number [ ]−p Pressure [ ]Pa
Pr Prandtl number [ ]−
P~ Probability density function [ ]−
rq Radiation heat flux [ ]2/ mW
rQ Reaction enthalpy [ ]molJ /
dQ& Heat transfer rate from the gas to the drop [ ]W
*Q Heat release in fine structure [ ]kgW /
xii
r Radius of drop [ ]m
fur Stoichiometric oxygen requirement pr. unit mass of fuel [ ]−
32r Sauter mean radius [ ]m
R Universal gas constant [ ]molKJ /
ijR EDC-combustion model factor [ ]−
C,R& Soot or its nucleus oxidation rate [ ]sm3/1
f,R& Soot or its nucleus formation rate [ ]sm3/1
OH,sR& Soot oxidation rate by OH-radical [ ]sm3/1
totalR Surface mass oxidation rate of soot particle [ ]smg 2/
Re Reynolds number [ ]−
is Direction vector in direction i [ ]−
IS~ Source term of specific internal energy [ ]smJ 3/
mS~ Source term of mass [ ]smkg 3/
jUS~ Source term of momentum [ ]3/ mN
lYS~ Source term of species concentration [ ]smkg 62 /
Sc Schmidt number [ ]−Sh Sherwood number [ ]−t Time, time scale [ ]sT Temperature [ ]K
Ta Taylor number [ ]−
iu′ , Fluctuation of velocity component of gas by turbulence ju′ [ ]sm /
i,pu′ Velocity component of drop by turbulent dispersion [ ]sm /
τu Shear speed [ ]sm /
iU~ , jU~ Reynolds average velocity component of gas [ ]sm /
lkv Stoichiometric coefficient [ ]−
xiii
V Drop velocity [ ]sm /
LdV Liquid fuel molar volume [ ]molm /3
relV Drop and gas velocity difference [ ]sm /
sV Volume of the radiation layer [ ]3m
iw Weight factor in direction i [ ]−
kw Difference of weight of specie after chemical reaction k [ ]scmmol 3/
sprayW& Rate of work of spray on the turbulence [ ]3/ mskg
We Weber number [ ]−x Drop displacement from its equilibrium position [ ]m
ix Coordinate [ ]m
y Drop dimensionless displacement from its equilibrium position [ ]−y Distance from wall [ ]m+y Dimensionless distance from wall [ ]−
Y Mass fraction [ ]−Z Compressibility factor [ ]−
Greek
β Soot absorption model constant [ ]−
lβ Conversion parameter of combustion products [ ]−χ Reacting fraction of the fine structures [ ]−
heatχ Fraction of the heated fine structures [ ]−
ijδ Kronecker delta [ ]−
ε Dissipation rate of turbulent kinetic energy [ ]32 / sm
*γ Mass fraction occupied by the fine structures [ ]−
λγ Mass fraction occupied by fine structure regions [ ]−
xiv
η Ratio of turbulent to mean-strain time scale [ ]−η Parameter of EDC combustion model [ ]−η Collision efficiency [ ]−κ Von Karman constant [ ]−Λ Wave length [ ]m
µ Dynamic viscosity [ ]mskg /
ν Kinematical viscosity [ ]sm /2
ρ Density [ ]3/ mkg
σ Stefan-Boltzmann constant [ ]KmW 2/
k,σσε Turbulent Prandtl number of ε and k [ ]−τ Break-up time [ ]sτ Time scale [ ]sτ Residence time of the fine structure reactor [ ] s
*τ Residence time of the fine structure [ ]s
Cτ Characteristic time scale [ ]sΩ Solid angle [ ]sr
Subscripts
A Atomisation
C Carbon, Chemical, Critical
d Drop
e Eddy break-up
F Fuel
g Gas
l Laminar, specie l
min Minimum
n Nucleus
OH Hydroxyl radical
xv
ox Oxygen
s Soot
t Turbulent
vap Vapour
w Wall, Surface wave
∞ Ambient
Superscripts
‘ Fluctuating part of variable
* Fine structure
o Fine structure surroundings
~ Favre average
+ Drop surface
- Time average
n At time step n
Comb Combustion
Htr Heat transfer
Liq Liquid
Spray Interaction with the spray
Vap Vapour
Acronyms
AS Abramzon and Sirignano
CHTC Characteristic time combustion model
CL Cliffe-Lever
DOM Discrete ordinate method
EDC Eddy dissipation concept
FS Fine structure
xvi
HG Huh-Gosman
KH Kelvin-Helmholz
LFM Laminar flamelet model
MH Magnussen and Hjertager
NSC Nagle and Strickland-Constable
NSP Number of species component
RK Redlich-Kwong
RM Ranz-Marshall
RTE Radiative transport equation
SMR Sauter mean radius
TAB Taylor analogy break-up
TM Tesner-Magnussen
1
1. INTRODUCTION
1.1 General aspects
Medium speed diesel engines are used in ships and small power plants. High reliability, efficiency
(economy) and nowadays especially low nitrogen oxide (NOx) and particulate emissions are the
desirable features of these engines (Taskinen et al., 1997, Taskinen et al., 1998, Taskinen, 2000,
Taskinen, 2001, Weisser et al., 1997). Advantages of the diesel engine compared to the spark
ignition (SI) engine are its high fuel economy and therefore its low carbon dioxide emissions as
well as low un-burnt hydro-carbon emissions. Correspondingly a major drawback of it has been the
high particle (soot) matter emissions, but nowadays these harmful to health emissions have been
succeeded to reduce by new fuel injection and exhaust gas after treatment techniques. Nitrogen
oxide emissions depend highly on the temperature of the gas in the cylinder and its residence time
at high temperature. High speed diesel and typical SI engines produce almost the same level of
nitrogen oxide emission, while medium or low speed diesel engines may produce much larger NOx
emission due to the much longer residence (reaction) time of gas at high temperature.
The improvement of the efficiency and reduction of emission formations of diesel engines can
nowadays be done by a sophisticated numerical simulation tool and/or experimentally. The
numerical simulation of the spray combustion process of a medium speed diesel engine is quite a
new field, whereas from a small engine field a lot of references/data are available. The reason for
this is that competition in the passenger car industry is so intensively keen to develop new engines
that have both the best low emissions and economies possible. The engine process modelling saves
time and is an investment in the developing process to get the engine to the market. A numerical
simulation tool obtains solutions with the different engine construction parameters such as fuel
injection timing, duration, spray direction, nozzle hole diameter, injection pressure, compression
ratio, stroke/bore, etc. The solutions data can be utilised in the optimisation process in order to find
such a combustion system in which high efficiency is combined with the low emissions. Purely by
experiment this is not possible. However, experiments are still needed to verify the simulation
results in some cases in order to ensure that the simulation tool predicts correctly the results in other
cases.
2
1.2 Diesel process modelling
A diesel process research and development (spray combustion, emissions formation and heat
transfer) is at least partially possible to do now by a numerical simulation tool, because the
knowledge of different physical/chemical phenomena in the cylinder has currently been increased
greatly (Pedersen et al., 1995, Reitz et al, 1995, Taskinen, 2002). Especially nowadays the
computing resources have been largely increased thus enabling the simulation of more complicated
cases. In a complete diesel process modelling the following phenomena have to be modelled: fuel
spray atomisation, drops vaporisation, vapour ignition, vapour combustion by chemical
kinetics/turbulent mixing, NOx formation, soot formation and its oxidation, heat transfer by
convection and radiation. This is a very complex phenomena set and many of them are coupled
together, e.g. spray dynamics, drops vaporisation and combustion, soot formation, soot oxidation
and radiation rendering the solving process. In Fig. 1.1 is shown a general view of the KIVA-II
simulation code and its main sub-models related to the diesel process modelling (Pedersen et al.,
1995).
Fig. 1.1 KIVA-II sub-models and solver structure
3
The basic idea in the diesel process modelling which is also valid for other this kinds of processes
(SI engine, gas turbine, etc) is to solve numerically the governing equations (partial differential
equations) of the field quantities such as temperature or internal energy, species concentration, gas
velocities, gas pressure, gas turbulence kinetic energy and its dissipation rate in every
computational cells (control volumes) of the physical space considered. The governing equations
are basic physical laws such as conservation of mass, balance of linear momentum and conservation
of energy. The heart of the modelling process is often related to the source terms of the governing
equations. The source terms related to the sub-models of different physical/chemical phenomena of
the cylinder gas. The sub-models should naturally describe phenomena as near correct as possible
in order to obtain reasonable results and their behaviour correctly, when input data are varied. One
special feature in diesel process modelling is that the control volume moves, which requires special
treatment for the computational mesh.
The special features of large medium speed diesel engines compared to the high speed diesel
engines are that they are operated on a heavy fuel oil and the flow in the cylinder after an intake
stroke is nearly quiescent (non-swirl). The flow in cylinder is therefore caused merely by the spray.
This causes high demands in the fuel spray model in order to able to correctly predict fuel drops and
the vapour spreading and further the mixing with air in the combustion chamber. The dynamic
behaviour of the fuel drops is also influenced by the drag force, which depends directly on the drop
drag coefficient. The drop drag coefficient during the vaporisation process should be able to
describe as correct as possible. The standard model of Putnam used in KIVA-II (Amsden et al.,
1989) tends to over-estimate the drag and in the model does not take into account the reduction of
drag in drop boundary layer during the drop vaporisation. According to Cliffe and Lever (1986) this
effect should be taken into consideration.
The ideal gas law is widely used model to describe the equation of state in engine CFD codes. It is a
quite well valid, when the pressure of the gas is moderate and temperature of gas is high. The
conditions in medium speed diesel engine cylinder are some extended different during the
combustion process than in the high speed (small) light fuel oil used diesel engine cylinder, i.e. high
pressure of the gas all the time and a great amount of fuel vapour before early phase of the
combustion (low temperature), that the real gas effects have to take into consideration in the
equation of state. According to the literature (Leborgne et al, 1998, Reid et al., 1987) and author’s
experience the formulation of Redlich-Kwong (RK) or Peng-Robinson equation of state is the most
4
suitable and accurate enough to use and implement into the engine CFD code.
Due to low a vapour pressure of heavy fuel oil a high pressure drop vaporisation model should be
used instead of the widely used low pressure model in order to avoid too long ignition delay and a
weak early phase of combustion (Taskinen, 2000). In a high pressure formulation of drops
vaporisation the mass fraction of fuel vapour in the surface of drop is calculated based on the
equality of vapour and liquid phase fugacity in the drop and gas interface (Reid et al, 1987).
Especially with the heavy fuel oil the low mass fraction of fuel vapour causes too long an ignition
delay.
Due to a large fraction of such hydrocarbon components of the heavy fuel oil, which easily form
soot, the flame is therefore luminous and the effect of soot radiation on flame temperature in the
soot region and heat transfer to walls will be a considerable, as Abraham et al., (1999) and Kaplan
et al., (1999) have discovered. Therefore the soot radiation should be taken into consideration in
order to obtain more realistic results with the simulation code. For optically thin flames the
absorption of gas can be ignored and this leads to the pure emission model of the soot radiation. It
tends to over-predict heat fluxes to walls, if the radiation medium includes a lot of soot as in
medium speed diesel engines with the heavy fuel oil. In the diesel process modelling the emission
model of radiation is some extended used due to its simplicity and the computationally cheap
approach. In strong radiation flame cases the absorption of radiation medium has to be taken into
account and therefore a directional dependence has to be taken into account in the radiation transfer
equation (RTE). The RTE has then to solve numerically using, e.g. DOM or DTM method (Modest,
1993). A few modelling of diesel process cases have published where the DOM method has been
used in the solution of the RTE. Author has implemented and used the DOM and the emission
methods in the soot radiation modelling.
The convective heat transfer from the gas to the walls is still a dominant component of the total heat
transfer. The radiation dominates in the later phase of combustion, when the amount of soot is high
and the flow in cylinder is weak due to end of injection. Normally in the engine CFD codes
standard wall functions (velocity and temperature profiles) are used to calculate a shear stress and
convective heat transfer coefficient. They are usually based on the Reynolds analogy between the
velocity and thermal boundary layer. A great under prediction of wall heat fluxes has been found
using the traditional wall functions (Han et al., 1997). Han et al., (1997) and Kays et al. (2004) have
5
derived new equations for the heat fluxes to walls based on the one-dimensional energy equation in
the boundary layer and correlations for the turbulent Prandtl number and dimensionless temperature
of the gas. The author has used these equations in slightly modified form and also obtained better
predictions of the heat fluxes than with the standard wall functions used in KIVA-II.
The basic engine simulation codes like KIVA-II and Star-CD have typically a two-equation model
for the gas turbulence, merely the standard k-epsilon (later e) and/or the basic form RNG k-e
models. It is well known that the standard k-e model over-predicts the turbulence diffusivity of the
gas and therefore causes the over-spreading of spray (Rodi, 1996, Han et al., 1997). The basic form
RNG k-e model under-predicts the turbulence diffusivity in the high strain rate region, while in the
low strain rate region, it over-predicts and therefore causes un-realistic fuel vapour transfers. This
can be avoided by modifying suitably the additional term of the k-e equation of the basic RNG
model and the model constants (Taskinen, 2003, 2004). The spray spreading and vapour
combustion proceeds then on a more realistic way yielding almost correctly the cylinder pressure
and rate of heat release than using the basic form standard or RNG k-e model.
1.3 Goal and outline of this thesis
The goal of this work was to develop a simulation tool for a medium speed diesel process analysis
based on the MARINTEK Company version of KIVA-II code. In the developing process of the
code attention has been focused to the vaporisation of drops in high-pressure environment, gas
turbulence, soot emissions and convective/radiation heat transfer. Typically in the engine
simulation codes the drops vaporisation model based a low-pressure formulation. The soot radiation
modelling in diesel process analysis have been done a quite little and especially in medium speed
diesel analysis where this effect is more important practically nothing. In order to get more realistic
total heat fluxes into walls the effect of soot radiation has to be included into the modelling. The
convection heat transfer model of KIVA-II code was based on the standard temperature law of the
wall equations and these were improved. Turbulence models of KIVA-II yield unsatisfactory results
and the modified RNG k-e model therefore was developed based on the behaviour of the basic
RNG and STD k-e models.
The Introduction discussed the general things and background related to the medium speed diesel
process modelling. The main shortages of current engine CFD codes have been presented and how
6
they can be avoided.
In Chapter 2, the basic theory of diesel process modelling and the most important sub-models
related to the gas turbulence, fuel spray, drops vaporisation, combustion, emissions formation,
convection and radiation heat transfer are presented.
In Chapter 3, the author’s implemented/developed sub-models and their contribution to the
modelling tool for diesel process are presented.
In Chapter 4, the essential numerical simulation results with the different sub-models and their
formulations are presented and how they verified. Discussion of the simulation results and their
comparison with the available experimental values.
In Chapter 5, conclusions of the work and the estimation of their capability to predict medium
speed diesel process realistically were done. Also the improvements of the code to get better results
are presented.
7
2. THEORY OF DIESEL PROCESS MODELLING
2.1 Governing field equations
A turbulent reacting flow can be described by the continuity, Navier-Stokes or momentum, energy
conservation, species concentration and equation of state equations. These governing equations
describe the velocity, pressure, temperature and species concentration fields. They can be written in
three-dimensional case for a Newtonian fluid using Reynolds and Favre averages in the form:
Continuity
Sraym
i
i SxU
t=+
∂ρ∂
∂ρ∂ ~
(1)
Momentum
( ) ( ) Spray
Ujjij
i
i
jl
ji
jijj
i
SguuxU
xU
xxp
xUU
tU
++⎟⎟
⎠
⎞
⎜⎜
⎝
⎛−⎟
⎟⎠
⎞⎜⎜⎝
⎛++−=+ ρρ∂∂
∂∂
µ∂∂
∂∂
∂ρ∂
∂ρ∂ ''
~~~~~ (2)
Internal energy
( ) ( ) HtrI
CombI
SprayIiv
ii
i
i
i SSSTucxIk
xxU
pxIU
tI
i
~~~~~~~~'' +++⎟⎟⎠
⎞⎜⎜⎝
⎛−+
∂∂
−=+ ρ∂∂
∂∂
∂ρ∂
∂ρ∂ (3)
Species concentration
( ) ( )l
i
Ylii
lY
i
lil SYuxY
Dxx
YUtY ~~~~~
'' +⎟⎟⎠
⎞⎜⎜⎝
⎛−=+ ρ
∂∂
∂∂
∂ρ∂
∂ρ∂
NSPl ,...,1 , = (4)
Equation of state
MRTZp /ρ= (5)
8
By using the well known the Boussinesq eddy-viscosity concept the Reynolds stresses and the
turbulence scalar fluxes can be expressed as follows:
⎟⎟⎠
⎞⎜⎜⎝
⎛−+−= ij
k
k
i
j
j
itijji x
UxU
xU
kuu δ∂∂
∂∂
∂∂
µδρρ~~~
''
32
32 (6)
it
tiv x
IPr
Tuc∂∂µ
ρ~
'' −= (7)
it
tli x
YSc
Yu∂∂µ
ρ~
'' −=
SpraymS~ Spray
U jS~ Spray
IS~
CombIS~ Htr
IS~
lYS~
(8)
The source terms , and in Equations (1), (2) and (3) due to spray have been
described in KIVA-II manual (Amsden et al., 1989) while the terms , and are
described in chapters 2.2.5.2, 2.2.5.3 and 2.2.7.
The boundary conditions are needed for the velocity and temperature in this context. For velocities
the standard law of the wall equations of the KIVA-II are used (Amsden et al., 1989), where the
critical Reynolds number when the velocity profile changes from the laminar to turbulent type is
122, corresponding the dimensionless distance from the wall 11.0. The temperature boundary
conditions (temperature wall functions) are presented in Section 2.2.71 in the context of the heat
transfer. Turbulence quantities boundary conditions are presented in the next Section 2.2.1.
2.2 Main sub-models in diesel process modelling
During the diesel process cycle several the chemical/physical phenomena occur in a cylinder, such
as fuel spray atomisation in a nozzle, drops vaporisation, vapour ignition, vapour combustion
controlled by chemical kinetics/turbulent mixing, nitrogen oxide and soot emissions formation and
soot oxidation, heat transfer to walls by convection and radiation. Mathematically formulated sub-
models are needed to describe the above phenomena. The following chapters will present briefly
the most important sub-models related to the diesel process modelling.
9
2.2.1 Turbulence modelling
The modelling of the turbulent viscosity (sometimes called the eddy-viscosity) is very essential and
challenging task. The gas turbulence plays an important role in the fuel vapour mixing and
combustion and further the emission formations. From literature can be found many types of
turbulence models, but it seems that the k-epsilon (later e) model and its variant RNG (Re-
Normalization-Group) k-e model are the most popular especially in the combustion modelling cases
(Han et al., 1995, Abraham et al., 1997a). They both belong to the so-called 2-equation models
framework. They are robust and computationally much cheaper models compared to more
complicated RSM (Reynolds Stress Models) or the LES (Large Eddy Simulation) models. They
yield quite reasonable turbulence quantity results, if the situations are avoided, where they are not
able to predict correct results. The situations where their results fail can be mentioned e.g. the flow
separation/re-attachment, streamline curvature and swirl (Younis, 1997). The k-e models are not
able to predict correctly the separation and/or reattachment of the flow. The standard k-e model can
only be used for a high Reynolds number flow. Near a wall when the turbulence Reynolds number
decreases, it can be modified by additional source terms to the so-called low Reynolds number k-e
model. The source terms are activated near a wall and therefore the flow is possible to compute to
the wall without to use the law of the wall functions. Especially the heat transfer is then computed
more reliably than in the case of using the standard wall functions.
All these standard, RNG and low turbulence Reynolds number k-e models are quite similar types
and can be then presented in the same form. The turbulence kinetic energy equations are identical,
but the epsilon equations are different. In the epsilon equation of the RNG-model there is an
additional term, which changes dynamically with the rate of strain of the turbulence, providing
more accurate predictions for flows with rapid distortion and an-isotropic large-scale eddies (Han et
al., 1995). Models can be expressed with the same equations as described in below.
Turbulent kinetic energy is defined through the turbulence velocities (fluctuations from the mean
flow) as follows:
'i
'iuuk
21
= (9)
10
The transport equation of the kinetic energy
( ) ( )
spray
j
i
i
j
j
it
i
i
il
k
t
i
i
WxU
xU
xU
xU
kxk
xxkU
tk
i
&+−⎟⎟
⎠
⎞
⎜⎜
⎝
⎛+
+−⎟⎟⎠
⎞⎜⎜⎝
⎛+=+
ερ∂∂
∂∂
∂∂
µ
∂∂
ρ∂∂µ
σµ
∂∂
∂ρ∂
∂ρ∂
~~~
~
32~
(10)
The transport equation of the kinetic energy dissipation rate
( ) ( )
( ) ⎟⎟
⎠
⎞
⎜⎜
⎝
⎛+−
⎟⎟
⎠
⎞
⎜⎜
⎝
⎛+−+
⎟⎟⎠
⎞⎜⎜⎝
⎛+−−⎟⎟
⎠
⎞⎜⎜⎝
⎛+=+
sprays
j
i
i
j
j
it
i
i
i
i
il
t
i
i
WCCxU
xU
xU
CCkx
U
xUkCCCC
xxxU
ti
&ερ∂∂
∂∂
∂∂
µε∂∂
ερ
∂∂
ε∂∂εµ
σµ
∂∂
∂ερ∂
∂ερ∂
η
µε
21
131
~~~~
~
32
32~
(11)
The turbulent viscosity is calculated
ερρνµ µ
2kCtt ==
η
(12)
In the basic and modified RNG k-epsilon model the additional term C is defined as follows
(Yakhot et al., 1992; Han et al., 1995; Abraham et al., 1997a; Taskinen, 2003):
32
01
1
ηβηηη
ηηη ⋅+
⎟⎟⎠
⎞⎜⎜⎝
⎛−
=C
CC (13)
Where, ε
η kS= ( ) 21
2 ijij SSS = and and ⎟⎟⎠
⎞⎜⎜⎝
⎛
∂
∂+
∂∂
=i
j
j
iij x
UxU
S~~
21
η
(14)
The term C changes dynamically with the mean-strain rate, η . In regions of largeη , the sign of
was changed and the turbulent viscosity was decreased accordingly. Hence, they concluded that
this feature of the RNG k-epsilon model was responsible for the improvement of their modelling of
separated flows (Choudhury et al., 1993 and Han et al., 1995). According to Taskinen (2003, 2004)
ηC
11
the term control purely the largeness of the additional term and the term C prevents an
unphysical diffusion with the low
η1C η2
η values, e.g. with the basic RNG model C when . 9.0≈η 2.1≈η
The constants used in different cases of the additional term of RNG k-epsilon turbulence model are
given in Table 2.1. In the Table 2.1 symbol A=STD, B=basic RNG and C= one of the modified
RNG k-epsilon model cases. More the modified RNG model cases are discussed in the context of
the modelling results in Chapter 4.
Table 2.1. Turbulence models constants µC 1C 2C 3C η1C η2C k
σ εσ 0η β sC
A. 0.09 1.44 1.92 -1.0 - - 1.0 1.30 - - 1.5 B. 0.085 1.42 1.68 -1.0 1.0 1.0 0.72 0.72 4.38 0.012 1.5 C. 0.085 1.42 1.70 -1.0 1.0 1.5 1.0 1.0 5.0 0.014 1.5
The wall functions for and epsilon is (Amsden et al., 1989, Han et al., 1995): k
and 25.0τµ uCk ⋅= −
ykC⋅
=κ
ε µ5.175.0
( )
(15)
5.05.012
⎥⎥⎦
⎤
⎢⎢⎣
⎡ +−=
ε
µη
σκ
CCCCWhere von Karman constant
The boundary condition for k is:
0=∂∂yk
τ
(16)
The shear speed,u in Equation (15) is calculated from the velocity wall functions.
12
2.2.2 Fuel spray modelling
2.2.2.1 General aspects
Fuel spray plays an important role in a diesel combustion process. Especially in medium speed
diesel engines, where the initial flow field in a cylinder when the fuel injection starts is weak due to
low running speed and large cylinder dimensions. The gas motion and its turbulence in a cylinder
are mainly caused by fuel spray. Therefore it influences the vapour combustion, drops hitting the
piston top, nitrogen oxide and soot formation etc. greatly.
Fuel spray models describe mechanisms as to how a fuel jet and/or a drop break-up take place at the
nozzle exit or later in the combustion chamber. A part of the spray characteristics are obtained as a
result of break-up modelling, such as a drop size distribution of the product drops and the spray
angle. The very important spray quantity, spray tip penetration is calculated later, when the drops
size, gas velocities (drag) and direction are known.
In literature (Reitz et al., 1982; Corcione, Pelloni and Luppino et al., 1999; Bianchi et al., 1999)
have presented several theories for the controlling of the break-up phenomena such as an
aerodynamic, liquid turbulence or cavitation-induced mechanisms. In reality, some of them can also
appear simultaneously such as the cavitation and turbulence or aerodynamic mechanism. In spite of
the large number of studies the liquid jet break-up and atomisation are still not well understood. The
theoretical understanding of the controlling process for the break-up of low-speed jets has been
developed well, but for the high-speed atomisation jets the theories describing the jet break-up and
drops formation have been inadequate (Ramos, 1989). Several more or less complicated fuel jet
break-up/atomisation models have been introduced. Some of them able to predict quite well the fuel
spray characteristics in certain situations. When a jet velocity, nozzle diameter, liquid viscosity etc
change a little the correctness of model results deteriorates considerably. This indicates that the
break-up/atomisation model does not pose a universal character. The break-up process is very
sensitive phenomena and it depends on many factors such as nozzle diameter, fuel viscosity, fuel
flow velocity in the nozzle (Weber number), ambient gas density, etc. In certain cases where using
very high injection pressures and therefore high fuel flow velocities the nozzle flow break-up is
very fast and the break-up can be assumed to have already happened. Under these conditions the
droplet size distribution method in the nozzle exit can be applied.
13
2.2.2.2 Fuel jet break-up/atomisation regimes
When considering a liquid fuel jet so it can be identified as two different lengths of the liquid,
namely the intact core length and the break-up length as shown in Fig. 2.1. Both lengths mentioned
above depend on the jet velocity.
BUL
IL
Nozzle
Fig. 2.1 Fuel jet break-up and intact lengths
From the liquid fuel jet can be observed certain regimes as a function of jet velocity (Ramos, 1989)
as shown in Fig. 2.2.
(
Jet velocity
1) (2) (3) (4) (5)
(1) = Dripping flow (2) = Rayleigh region(laminar flow)(3) = First wind induced region
(transition) (4) = Second wind induced region
(turbulent flow) (5) = Atomisation region(fully
developed spray region)
Jet b
reak
-up
leng
th
Fig. 2.2 Break-up regions as a function of jet velocity
These regimes are labelled dripping flow, Rayleigh, first wind-induced, second wind-induced and
atomisation (Ramos (1989), Tanner et al. (1998)). The Rayleigh and first wind-induced break-up
14
regimes are well understood. In the Rayleigh regime the break-up is due to the unstable growth of
surface waves caused by surface tension and results in drops larger than the jet diameter (Ducroq,
1998). When the jet velocity increases from the value of Rayleigh regime, the first wind-induced
break-up mode becomes the main control mechanism of the break-up. In this case the force due to
the relative motion of the jet and the surrounding gas augment the surface tension force, and lead to
drop sizes of the order of the jet diameter. The break-up and intact core lengths are same in the
Rayleigh and the first wind-induced break-up regimes. Typically in diesel engines the jet velocity is
much greater than the velocities appear in Rayleigh and the first wind-induced break-up modes.
Normally the second wind-induced or using very high injection pressure the pure atomisation
regime appears in a real diesel spray. Behaviour of the break-up length is still some extended
controversy in the in second wind-induced regime as shown by dashed line in Fig. 2.2 (Ramos,
1989). The second wind-induced break-up mechanism is mainly due to aerodynamic forces that
generate instabilities into the shear layer between the jet flow and ambient gas as shown in Fig. 2.3.
Nozzle
Fig. 2.3 Instability grows in liquid/gas interface as it moves downstream
These Kelvin-Helmholz (KH) instabilities tend to grow going into the downstream forming
vortexes (Ishikawa et al., 1996). When the instabilities in the shear layer of jet flow grow into the
critical value the jet flow break-up into smaller drops whose sizes are much smaller than the jet
diameter. In the second wind-induced break-up mode the jet surface break-ups before the jet axis
and therefore the break-up length is shorter than the intact core length. both the wind induced
break-up mechanisms belong in the laminar regime, where the liquid properties and the ambient gas
conditions are factors determining the aerodynamic-induced atomisation (Bianchi et al, 1999).
In the cases where using high jet velocities and low viscosity of fuels the jet flow changes from the
laminar to turbulent mode. Thus the break-up is induced by jet internal turbulence. The break-up
15
begins in the nozzle passage from small disturbances of flow caused by turbulence and they grow
bigger going in downstream according to KH theory (Ishikawa et al. 1996). The break-up could
take place before the nozzle exit unlike in the aerodynamic model, but normally the break-up length
is greater than zero (Ramos, 1989).
When using very high injection velocities (very high injection pressure) the jet break-ups
immediately at the nozzle exit. This atomisation regime (Region 5 in Fig. 2.3) is defined as the
regime where the break-up length is zero (Ramos, 1989, Tanner et al. (1998)). The break-up
process at nozzle exit is not considered anymore, only a secondary break-up of drops is possible to
consider as Tanner et al. (1998) has done. The SMR of drops formed has to determine in some way,
e.g. by experimentally.
The break-up process is the so called cascade process where the drop break-up can take place many
times during the injection period until they reach a stable form (Tanner et al., 1998). Normally only
the primary and secondary break-up modes are considered. Actually some spray models (Wave,
TAB) do not distinguish the primary and secondary phase. The primary break-up take place in the
region close to the nozzle exit at high Weber number (>1000) while the secondary break-up take
place later on the combustion chamber at lower Weber number range (<1000). The main primary
break-up mechanism(s) can be in the laminar, turbulent, cavitation or in some unknown regimes
(Reitz et al., 1982; Bianchi et al., 1999). The secondary break-up is typically the aerodynamically
induced and therefore belongs into the laminar regime (Bianchi et al., 1999). Cavitation is a
mechanism, which augments the break-up process, but according to literature (Su, 1980; Reitz et
al., 1982) it cannot be the sole agency of break-up.
2.2.2.3 Short review of the fuel spray drop break-up models
There exist several break-up models that have been used, such as HUH&GOSMAN (HG), TAB or
WAVE. Despite the fact that several models have been developed to simulate the diesel fuel spray
drop break-up, the complexity of this process still does not allow one to provide accurate
predictions in the case of high injection pressure (Bianchi et al., 1999). Due to this reason the
hybrid models have been developed in order to able to predict more correctly all break-up
processes, the primary and secondary break-up events. In the next present sections will be discussed
16
briefly the most general and widely used the spray models.
HG MODEL
Huh and Gosman (1991) have been presented the HG model and it is a turbulence-induced spray
break-up model. The basic idea is that the turbulence fluctuations in the jet are mainly responsible
for the initial perturbations on the jet surface. These surface waves then grow according to KH
instabilities until they detach as atomised droplets (Corcione, Pelloni and Bertoni et al., 1999;
Bianchi et al. 1999). The time scale of atomisation is assumed to be a linear function of the
turbulent and the KH surface wave time scales as follows:
(17) WTA tCtCt ⋅+⋅= 41
TA LCL ⋅= 2
The length scale of the turbulence is assumed to be the dominant length scale of the atomisation
process. The atomisation length scale and the wavelength of surface perturbation waves are
expressed as a function of the turbulent length scale.
TW LCL ⋅= 3 ; (18)
Assuming that half a surface wave is detached as a drop from the jet, then
(19) WA LL ⋅= 5.0
→ 23 2 C
Substituting from Eq. (19) into Eq. (18) C ⋅=
The constant C is a correction factor that accounts for the liquid viscosity (Bianchi et al., 1999) 4
The time scale of waves derived from the KH instability theory on an infinite plane for an in-viscid
liquid is:
17
( ) ( )
5.0
3
2
2
1
⎥⎥
⎦
⎤
⎢⎢
⎣
⎡
⋅+−⎟⎟
⎠
⎞⎜⎜⎝
⎛
+
⋅=
Wgd
d
W
rel
d
gd
W
LLV
t
gρρ
σ
ρρ
ρρ (20)
The average turbulent kinetic energy and its dissipation rate in the injector can be obtained by CFD-
computing using appropriate turbulence model, by experimentally or by using a simple force
balance equation based on the pressure drop along the nozzle downstream length (Bianchi et al.,
1999).
The turbulent length and time scales are expressed with the equations as:
AVE
/AVE
TkCLεµ
23
=AVE
AVET
kCtεµ= ; (21)
In the break-up of parent droplet their diameter decreases as follows:
A
A
tL
dtdr
−= (22)
The spray angle is calculated as:
rel
AAVtL /
2tan =⎟
⎠⎞
⎜⎝⎛θ
3300 rNrN ⋅=⋅
(23)
The number of drops in the break-up during the atomisation time step is calculated based on the
mass balance before and after break-up, i.e.
(24)
18
TAB (Taylor Analogy Break-up) MODEL
In the TAB model (Taylor, 1963) droplet oscillation and distortion are modelled by using a simple
forced harmonic oscillator, based on the analogy suggested by Taylor between an oscillating and
distorting droplet and a spring-mass system. The aerodynamic force is analogous to the external
force and the surface tension is analogous to the spring restoring force while the damping force is
related to the liquid viscosity force. The governing equation of such a system is the following (O’
Rourke et al. 1987; Assanis et al. 1993; Taskinen et al., 1996; Taskinen, 1998; Bianchi et al., 1999):
(25) xdxkFxm &&& ⋅−⋅−=⋅
According to the forced harmonic oscillator analogy it can be written:
rUC
mF
lgF ⋅⋅⋅=ρ
ρ2
3rC
mk
l
lk
⋅⋅=ρ
σ2r
Cmd
l
ld
⋅⋅=ρ
µ ; ; (26)
rCxyb ⋅
=
( )
(27)
By taking into consideration Equations (26) and (27), the solution of Equation (25) is
( )
( )
⎥⎥⎥⎥⎥⎥⎥⎥
⎦
⎤
⎢⎢⎢⎢⎢⎢⎢⎢
⎣
⎡
⋅⎟⎟⎟⎟
⎠
⎞
⎜⎜⎜⎜
⎝
⎛ −++
⋅⎟⎟⎠
⎞⎜⎜⎝
⎛−
⋅+=−
tt
WeCCC
yy
tCCC
y
eWeCCC
ty
d
bk
F
bk
F
tt
bk
F d
ωω
ω
sin21
cos
2 0
0
0
&
(28)
d
relg rVWe
σ
ρ ⋅⋅=
2
221
rC
t d
dd
d ⋅⋅=
ρ
µ23
2 1
dd
dk
trC −
⋅=
ρσ
ω ; ; (29) Where
19
The break-up take place when . 1>y
The size of the product drops after break-up is randomly selected from a chi-squared distribution
function around the SMR. SMR is calculated through the energy conservation before and after
break-up as follows (Taskinen, 1998):
( )( )22232
463
21 bvb
d
dbk CCCyrCCrSMR
−⋅⋅⋅
++
=
&σ
ρ (30)
The spray angle is calculated from the geometric equation of the product drop normal and
tangential velocities (Taskinen, 1998):
g
dFbv CCC
ρρθ 2
2tan ⋅=⎟
⎠⎞
⎜⎝⎛
5=d 3/1
(31)
The original TAB model constants have obtained based on the shock wave experiments and by
matching the oscillations of the fundamental mode (O’ Rourke et al., 1987): C , =FC
0=bC 8=kC
,
, , C . 5. 0.1=v
It’s well known that the TAB model with the original constants yields too small drop size
distribution, too narrow the spray angle and too short spray tip penetration (Assanis et al., 1993;
Beatrice et al., 1995; Taskinen, 1998).
Some researchers have been modified the model constants in order to avoid shortages mentioned
above (Assanis et al., 1993, Taskinen et al., 1996, Taskinen, 1998; Bianchi et al., 1999). After
modifying the spray tip penetration and the SMR of the drop size become more realistic, but the
spray angle still remains too narrow compared to the experimental values.
20
WAVE MODEL
In the WAVE model the droplet break-up is due to aerodynamic interaction between the liquid and
gas leading to unstable KH wave growth on the surface of a cylindrical jet “blob” of liquid
(Castleman, 1932). The flow is assumed to be incompressible and a cylindrical coordinate system is
chosen which moves with the jet (Reitz et al., 1982). The linearised Navier-Stokes equations for the
surrounding gas and liquid fuel velocities and pressure perturbations can be written and solved by
introducing a velocity potential and stream functions as described in the (Reitz et al., 1982 and
Levich, 1962). Solution of the analysis leads to a dispersion equation. The dispersion equation
relates wave growth rate to its wavelength and its solution is very complicated. Only in the limiting
cases the solutions can be found (Reitz et al., 1982). The maximum growth rate and the
corresponding wavelength are related to the liquid and gas physical properties via the equations
(Beatrice et al., 1995):
( )( )( ) 6.067.1
7.05.0
87.01
4.0145.0102.9gasWe
TaOha ⋅+
⋅+⋅+=
Λ (32)
( )( )( )6.0
5.15.03
4.111
38.034.0
TaOh
Wea gasliq
⋅++
⋅+=
⎥⎥⎦
⎤
⎢⎢⎣
⎡ ⋅=Ω
σ
ρ
liqliqWeOh Re/= 5.0gasWeZ ⋅=
( ) ( )⎪⎩
⎪⎨⎧
>Λ⋅⎟⎠⎞
⎜⎝⎛ Λ⋅⋅Ω⋅⋅⋅⋅
≤Λ⋅Λ⋅=
aBaVamin
aBBr
rel 033.0233.02
00
,4/3,2/3
,
π
(33)
Where, the characteristics numbers above equations are defined as: Ohnesorge number,
and Taylor numberTa . 5.0
The model assumes that new drops of radius r are formed from blobs of radius a, with (Beatrice et
al., 1995):
(34)
The parent drop radius a changes with the following equation
21
( )τra
dtda −−
= (35)
Where the break-up time is defined as Ω⋅Λ
⋅⋅=
aB1726.3τ
0B 1B
(36)
The constants , above equations are 0.61 and 1.73, respectively.
CHI-SQUARED DROP DISTRIBUTION MODEL
Chi-squared is one basic spray model in the KIVA-II code, which assumes that the fuel-jet has
already broken-up and a so-called Chi-squared drop number distribution for the drop size exits in
the nozzle exit (Amsden et al., 1989). We do not consider a complex jet flow and its break-up
processes in different stages (primary and secondary break-ups). Also the spray angle has been
assumed known and can be taken from experiments. The directions of injected drops are distributed
uniformly in the spray angle. The amount of the number of injected drops has to be large in order to
describe the spray structure as realistic as possible. The certain sampling technique is used in
selecting randomly the radius of the injected drops from the mass distribution of drops, which the
SMR is specified. This technique is explained more detailed in reference (Amsden et al, 1989).
Using the specified spray angle taken from experimental data the effect of gas entrain into the spray
is taken into consideration more precisely than used in the break-up based models which often
under-predict the spray angle. This is very important especially in cases when the emission
formation is included in the modelling. Also the fuel vapour mixing (combustion) becomes more
realistic using the chi-squared model due to the effect mentioned. Chi-squared method is only valid
using a very high injection pressure (large Weber number) as used in medium speed diesel engines
nowadays. In this study the chi-squared model is used exclusively in the spray in order to ensure
that the other sub-models work reliable.
2.2.3 Drop dynamics
Drop trajectories in a combustion chamber were calculated based on the second law of dynamics
22
(Newton).
DROP ACCELERATION
( ) ⎟⎠⎞
⎜⎝⎛∆
++−+−+⎟⎟⎠
⎞⎜⎜⎝
⎛⋅
=∆
−+
tu
gVuUVuUr
Ct
VVi
iiiiiiid
gasDnp
np ii '
''28
31
δρρ
(37)
Where the drop drag coefficient is calculated from the CL correlation (Cliffe and Lever, 1986)
( )( )⎪⎩
⎪⎨
⎧
>
≤+⋅+=
−
1000,424.0
1000,1175.0124 131.0612.0
d
dHddDRe
ReBReReC (38)
( )Where ( )TRe
aird ˆµ=
rVuUgas '2 ρ ⋅−+⋅⋅HB ; is defined later in the Section 2.2.4
The original drop drag coefficient of KIVA-II is the model of Putnam (Amsden et al., 1989) and
according to Williams (1990) it tends to over estimate the drag, because the effect of reduction of
drag in drop and its surrounding gas interface due to vaporisation is not included.
DROP VELOCITY
⎟⎠⎞
⎜⎝⎛∆
+=∆
−+
tx
Vt
XXin
p
np
np
i
ii '1
δ (39)
⎟⎠⎞
⎝ ∆tx i'δ
⎟⎠⎞
⎜⎝⎛∆tu i'δ
⎜⎛ and Where the turbulent dispersion terms were calculated as described in reference
(Amsden et al., 1989).
23
2.2.4 Drop vaporisation
The heat and mass transfer are essential factors in drop vaporisation at high temperature and
pressure environment. Often the biggest problem is to calculate the heat and mass transfer
coefficients correctly at the drop surface. Due to well-known Hill vortex flows inside the drop, the
temperature of the drop can be assumed to be uniform. The next presented theory based mainly on
the references, (Golini et al., 1993; Leborgne et al., 1998).
The rate of drop radius change is calculated from the conservation of mass of the liquid drop:
⎟⎠⎞
⎜⎝⎛ ⋅⋅= dvap r
dtdm ρπ 3
34
& (40)
The vaporised mass can also be calculated by using the mass transfer rate from the surface of the
drop as follows:
(41) Mvap BKrm ⋅⋅⋅⋅= 24 π&
, is obtained from the Sherwood number Where, the mass transfer coefficient, K
( )M
M
vap
sBB
ShDrK
Sh+
=⋅⋅⋅
=∞
1ln20ρ
(42)
By substituting the Eq. (42) into the Eq. (41) and taking into account Eq. (40), then we get the rate
of drop radius change
( )Msd
vap BShr
Ddtdr
+⋅⋅⋅
⋅−= ∞ 1ln
2 0ρρ
(43)
The drop temperature is obtained from the energy balance equation
( ) ddcvapdvd QTThrLmTcmd
&&& =−⋅⋅⋅⋅=⋅+⋅⋅ ∞24 π (44)
24
The right hand side term represents the heat transfer rate from the gas to the drop. The heat
transfer coefficient is obtained from the Nusselt number as a same way as the mass transfer
coefficient in Equation (42).
dQ&
( )H
HcBBNu
rhNu +
=⋅⋅
=1ln2
0 (45) k∞
By substituting the Eq. (45) into the Eq. (44) resulting in the rate of drop temperature change
( ) ( ) ( )LBShcr
DBBTT
crNuk
dtdT
Mvd
vap
H
Hd
vd
d
dd
+⋅
⋅−
+−
⋅
⋅⋅= ∞
∞∞ 1ln
2
31ln23
0220
ρ
ρ
ρ (46)
In the above equations the Spalding mass and heat transfer numbers were calculated as follows
(Abramzon and Sirignano, 1989):
LeNuSh
cc
gas
d
p
p 1=Φ
PrScLe /
+
+
−
−=
s
sM
YYY
11 → ( ) 11 −+= Φ
MH BBB and (47) Where
The Lewis number in the above equation is defined as: = . In the drop boundary layer the
Lewis number at the beginning of vaporisation is high due to the lower diffusivity of the fuel
vapour in the air than the thermal diffusivity of air, but later it decreases remaining a little larger
than unity. This indicates that the heat transfer develops faster than the mass transfer in the drop
boundary layer. According to the Abramzon & Sirignano (1989) the coefficient . 2.1...05.1≈Φ
If in Equations (43) and (46) the non-vaporising Sherwood number is calculated from the Ranz-
Marshall (RM) correlation (Ranz and Marshall, 1952), the model is then the so-called Frössling
correlation for the rate of drop radius change (Faeth, 1977). For the RM correlations:
31
21
0 PrRe6.02 ⋅⋅+=Nu ; 31
21
0 Re6.02 ScSh ⋅⋅+=
( )
(48)
H
H
BB+1ln
in Equations (42 and 45) takes into consideration the Stefan flow in the drop The factor
25
boundary layer. The effect is caused for thickening the boundary layer of the drop due to
vaporisation. This is the basic drop vaporisation model in the KIVA-II.
The Abramzon&Sirignano (AS) drop vaporisation model is obtained by replacing and
with the modified Nusselt and Sherwood numbers (Abramzon and Sirignano, 1989). The AS model
based on the film-theory, that assumes that the resistance to heat or mass transfer between a surface
of the drop and the surrounding gas flow may be modelled by introducing the concept of thermal
and mass diffusion films (Bird et al, 1960, Frank-Kamenetskii, 1969). If the Lewis number is unity
the thickness of films are equal and they in a grow similar way. Due to the Stefan flow the thickness
of the films will increase and the correction factors must be introduced (Abramzon and Sirignano,
1989).
0Nu 0Sh
( ) TFNuNu /22 0* −+=
( ) MFShSh /22 0* −+=
( )
According to the Abramzon&Sirignano (1989) the modified Nusselt and Sherwood numbers can be
expressed as follows:
(49)
(50)
Where the correction factors take into consideration the increasing of film thickness due to
vaporisation. They have been derived for a case of laminar boundary layer flow past a vaporising
wedge (Abramzon and Sirignano, 1989).
( )T
TTT B
BBF ++=
1ln1 7.0
( ) ( )
(51)
M
MMM B
BBF ++=
1ln1 7.0 (52)
The drop vaporisation models based often on the so-called low-pressure formulation, i.e. for using
of Raoult’s law to calculate the mole fraction of fuel vapour at drop surface. It can be expressed as:
26
cyl
vapvap p
pX =+ (53)
The mass fraction of fuel vapour at drop surface, which is used in Equation (47)
⎟⎟⎠
⎞⎜⎜⎝
⎛−+
=+=
11
vap
cyloxF
Fl
pp
MM
MY
vapcyl pp >>
VapLiq ff = →( )
(54)
Low pressure models easily under-estimates , because . +1Y
Under the high pressure of the gas in the cylinder the low-pressure drop vaporisation model based
on Raoult’s law is no longer valid to calculate the thermodynamic equilibrium at the drop surface
(Leborgne et al., 1998). The high-pressure drop vaporisation model based on the thermodynamic
equilibrium condition, in which the fugacity of the liquid and vapour in the drop interface is equal
(Jia et al., 1993; Gradinger et al., 1998; Leborgne et al., 1998, Taskinen, 2000). In the single
component system, it can be formulated as follows:
( pTYpYdpTRpTV
p vapVvapvap
p
p
LdS
vapvapvap
,,,
exp ++ Φ⋅⋅=⎟⎟
⎠
⎞
⎜⎜
⎝
⎛∫
⋅⋅Φ⋅
+vapY
) (55)
has to be solved from Equation (55) iteratively. The liquid fuel molar volume was calculated
from the Hankinson-Brobst-Thomson method (Reid et al., 1987). The fugacity coefficients in
Equation (55) have to be calculated from the Peng-Robinson equation of state (Peng and Robinson,
1976), in which the non-sphericity of a molecule of gas is taken into consideration.
The compressibility factor in Equation (5) was calculated based on the Redlich-Kwong (RK)
equation of state (Redlich and Kwong, 1949). Accuracy of the model mentioned is enough and thus
there is no need to use a more complicated model for the compressibility factor. It can be expressed
as follows (Reid et al., 1987):
27
5.1−
+ΩΩ
−−
= rb
a TbV
bbV
VZ
lr
NSP
llr TXT ⋅∑=
=1
(56)
(57)
ll
cr T
T= b aT and , Ω , Ω , T are constants (Reid et al., 1987). Where b lc
2.2.5 Fuel vapour combustion
2.2.5.1 General aspects
In the diesel spray combustion process the premixed and the diffusion combustion phases can be
distinguished as shown in Fig. 2.4.
Fig. 2.4 Combustion phases in diesel engine
28
The fuel vapour is a mixture of hydrocarbons whose combustion can be described by the one or the
two step mechanism as follows:
One step mechanism
OHm 224 ⎠⎝nCOOmnHC mn 22 +→⎟
⎞⎜⎛ ++ (58)
Two step mechanism
22 22HmnCOOnHC mn +→+ (59) 1. Step
2221 COOCO ↔+ 222 HCOOHCO +↔+ or (60) 2. Step
OHOH 2221
↔2 +
The combustion (chemical reaction) is a molecular process and it can take place only when the
reactants are mixed on a molecular level or when, as in the case of flame in a premixed mixture,
ignition conditions are met on that level. Therefore the completion of turbulent mixing is a
prerequisite for the reaction to proceed as Chomiak (2000) has presented.
The effect of turbulence/chemical kinetics to the reaction rate can be dealt with the characteristic
reaction time as follows (Chomiak, 2000): τ r
η
Fast chemistry limit
(61) ττ <r
The chemical kinetics do not have any influences on the turbulence. The reaction region is local and
29
thin.
Moderately fast chemistry regime
lr τττη <<
η
(62)
The reaction region is still local, but it is thicker than in the previous case and wrinkled due to
turbulence. The Kolmogorov time scale, τ , is the smallest, while, lτ , is the largest turbulence time
scale.
Slow chemistry regime
lr ττ > (63)
The reaction occurs over all scales of turbulence.
2.2.5.2 Premixed combustion
The time for a reaction to occur in a turbulent region becomes the sum of the turbulent mixing time
(eddy break-up time) and chemical time as follows (Chomiak, 2000):
τ r e ct t= +
ec tt → cr t≈
(64)
The rate of fuel vapour combustion highly depends on the temperature of the gas in the premixed
phase, while the turbulence of the gas in the diffusion phase. During the ignition and the early stage
of premixed combustion when temperature of the gas is quite low the chemical kinetic control the
fuel vapour combustion rate. Then
>> τ (65)
For the chemical reaction (58) can be calculated the chemical time scale (Patterson et al., 1994)
30
( RTE
MY
MY
Ktox
ox
F
fc exp
5.175.01
1
−−
⎥⎦
⎤⎢⎣
⎡⎥⎦
⎤⎢⎣
⎡= ) (66)
The source term of the fuel vapour consumption rate in Equation (4) due to combustion as follows:
⎟⎠⎞⎜
⎝⎛−
RTEexp 1
81 1068.7 ×=K scmmol 3/ 771 =E molkJ /
⎟⎟⎠
⎞⎜⎜⎝
⎛⎟⎟⎠
⎞⎜⎜⎝
⎛⋅=
= MY
MY
KS~b
O
O
a
F
fY fl 1
2
2ρ (67)
Where , , 3. 25.0=a , 5.1=b
rYCombI QS
The heat release source term in Equation (3) related to above source term becomes as follows:
~S~
fl⋅=
= (68)
2.2.5.3 Diffusion combustion
Many chemical reactions have high rates at high temperatures and therefore it can be considered
complete as soon as the reactants are mixed. If the reaction time is negligibly short compared to the
mixing time, the turbulent mixing combustion can be approximated adequately with the fast-
chemistry assumption (Kuo, 1986, Magnussen, 1990). This feature of the complex combustion
process justifies approaching it by a simpler so called mixed burnt method.
MAGNUSSEN & HJERTAGER MODEL
The large scales turbulence eddy-break-up time in the Magnussen&Hjertager (MH) model is
calculated by an equation (Magnussen et al., 1977).
εktt er =≈ (69)
31
Basic idea in the diffusion combustion phase is to calculate the fuel vapour mixing (consumption)
rate into the air through the break-up and dissipation process of the large scale eddies of turbulence
into the molecular scale in which the mixing finally occurs.
The fuel vapour mixing rate source term with the MH model can be calculated as follows
(Magnussen et al., 1977):
⎭⎬⎫
⎩⎨ +⋅⋅⋅⋅⋅⋅⋅=
=)s/(YA,s/YA,YAmin
tS~ PrOf
eY fl
12
12
ρρρ⎧ (70)
In the model the mixing rate is limited either by the availability of the fuel or of the oxidizer (Brink,
1998). Also the combustion products influence into the mixing rate as the third term in the equation
describes. The heat release source term is similar to that in Equation (68).
MAGNUSSEN EDC MODEL
The Eddy-Dissipation Concept or EDC model of Magnussen (Magnussen, 1981a) is a further
development of the Eddy-Break-up model of the MH. This model is based on that the heat releasing
chemical reaction takes place in the intermittently distributed dissipating fine structures of
turbulence. These fine structures can typically be thin vortex sheets or vortex tubes of the flow
(Magnussen, 1990) as illustrated in Fig. 2.5. Their entire volume is only a small fraction of the
volume of the fluid. The fine structures create the reaction space for non-uniformly distributed
reactants in turbulent flow. The mixing in the highly dissipative fine structures are assumed to be
fast, and the combustion in the burning fine structures is modelled as perfectly stirred reactors
(Magnussen, 1990, Pedersen et al., 1995).
32
Fig. 2.5 Fine structure vortex
Outside of the fine structure is surroundings from the fuel vapour transfers into the fine structures as
is described in Fig. 2.6. The problem is now to know the amount of the fine structures and the mass
transfer rate between the fine structures and surrounding fluid (Magnussen, 1990).
Reactants Products
oool TY ρ,,
*m& *m&
*** ,, ρTYl
Fig. 2.6 Schematic illustration of reacting fine structure reactor
Taking into consideration that only a fraction, χ of the fine structures are burning, it can be
expressed the density weighted mass fraction of concentration of each species, i.e.:
33
( ) olYlYlY χγχγ *1**~ −+=
o
(71)
Where, * represent the burning fine structures and their surroundings.
Based on consideration of the energy transfer from bigger eddies to the fine structures, the mass
fraction of fine structures is expressed as:
3
'
**
⎟⎟⎠
⎞⎜⎜⎝
⎛=uuγ
(
(72)
The fine structure velocity scale is closely related to the Kolmogorov velocity micro scale and it
can be calculated as:
) 41
ν* 74.1 ε ⋅⋅=u
'
(73)
Where, the turbulence velocity fluctuation component, u is calculated in an isotropic turbulence
case from the turbulence kinetic energy as follows.
21
'
32
⎟⎠⎞
⎜⎝⎛= ku (74)
When treating reactions, a certain fraction of the fine structures are burning. In Equation (72), only
the fraction, χ which are sufficiently heated will react (Magnussen, 1990).
(75) ijheat R⋅= χχ
Where
34
( )( )( )( ) ( )( )oxfuprfufupr
minfuprij YrYYrY
YrYR ~1/~~1/~
1/~ 2
++++
++=
( )( )
(76)
λγ1
1
1χ
/~
~
minfupr
fupr
Yr
r
++
+⋅
( )
heat Y
Y= (77)
31*γγ λ = (78)
( )( )⎪⎩
⎪⎨⎧
>
≤=
stfufufuox
stfufufumin YYrY
YYYY
,/
, (79)
The governing equations for the reacting fine structures modelled as perfectly stirred reactors,
where the surroundings values were calculated from the average values are given as:
( ) *ˆ
~**
llll
YSYY
dtdY
=−
+τ
NSP,...,l 1= , (80)
( ) **
** 1ˆ
~Q
dtdphh
dtdh
+=−
+ρτ
( ) ( )
(81)
Where, the residence time of the fine structure is calculated from equation
( ) 21** /*141.0*1ˆ ενχγτχγτ −⋅=⋅−= (82)
If the fine structures can be assumed steady, adiabatic, the pressure change during the time step
negligible and the elemental composition identical for inlet and reactor composition, when the
average formulation is used as in Equation (71), the average reaction rate or mixing rate of specie
becomes:
35
( ) ( ) oY
llY S
YYSl
χγτ
χγ **
* 1ˆ
~~
−+−
= (83)
Because there is no a dissipative mixing process in surroundings Equation (83) is calculated as
follows:
( )τ
χγˆ
~~ *
* llY
YYSl
−= (84)
In the situations, where non-reacted fine structures are very close the burning fine structures as is in
the tail of the flame the burning fine structures may catch up reactants at a higher rate than
predicted by Equation (84) (Magnussen, 1990) and therefore, it must introduce a parameter, η ,
such that
( )τ
χγηˆ
~~ * lYS ⋅⋅=
*l
YY
l
− (85)
Where λγ
η 1= with the following limitation
( )min
minfupr
Y
YrY~
~1/~ ++≤η
Infinite fast irreversible chemistry in EDC
Often for vapour of hydrocarbon fuels the chemical reaction rate can be assumed to be infinite fast
in the burning fine structures, Equation (85) transforms into simpler form:
τχγηα
ˆ
~~ * minlY
YSl
⋅⋅⋅=
1−=
(86)
Where
= Fuel vapour, l lα
= Oxygen, l ful r−=α
36
= Product, l ( )full r+= 1βα
Infinite fast reversible chemistry in EDC
At high temperature of the gas a dissociation of the combustion products may be significant and the
irreversible of the reactions (58) are not valid. The fine structure reactor treatment can therefore be
divided into two parts, where the first reaction (59) is an irreversible and the second reactions (60)
are a reversible. Then Equation (85) becomes
( )τ
χγηˆ
~~ eq YY −* llYS l
⋅⋅=
eql
(87)
Based on the equilibrium condition after reaction (59), the equilibrium composition Y was
determined by using the equilibrium constants for the reactions (60). The non-linear equation
system obtained was solved by Newton method.
LAMINAR FLAMELET MODEL (LFM)
Peters and his group have developed and used widely this model for diesel combustion modelling
(Pitsch et al., 1995; Pitsch et al., 1996; Barths et al., 1997; Hasse et al., 1999; Hergart et al., 1999).
The basic idea of the model is to describe a turbulent flame by an ensemble of stretched laminar
layers i.e. laminar flamelets. These flamelets are thin reactive-diffusive layers embedded within an
otherwise non-reacting turbulent flow field (Peters, 1984). The governing equations of LFM based
on the mixture fraction as an independent variable, variance of the mixture fraction and the scalar
dissipation rate for the mixing process as follows:
For the reaction
21, 2,2 OFOF YYYY +→+ νν (88)
Mixture fraction is defined as:
37
2,2
2,22
1, OF
OO
YY
YY
+
+−FYZ =
ν
ν (89)
''ZThe transport equation of Favre averaged Z including its variance
( )ZDZtZ
t~~~
~∇•∇=∇•+ ρρ
∂∂ρ u (90)
Variance of the mixture fraction
( ) ( ) χρρρρ∂
∂ρ ~~2~~~
22''2''2''
−∇+∇•∇=∇•+ ZDZDZtZ
ttu (91)
Instantaneous scalar dissipation rate
2
"~~~~ Zk
C εχ χ= (92)
The conditional Favre mean scalar dissipation rate is defined by:
ZZ
Z ρρχ
χ =~
( )TYll ,=
(93)
Reactive scalars ψ have the following equation
llz
l
l
ZLetω
∂ψ∂χρ
∂∂ψ
ρ += 2
2
2
~ (94)
Surface of the flame is defined . When ( ) stZtxZ =, ( )tZ Zl ,~, χψ is known from Equation (94) the
Favre mean values of ( )txl ,ψ can be obtained at any point and time in the flow field by
38
(95) ( )txl ,ψ
kr
kikil wW ∑=
=1νω
" lklklk
( ) ( )dZtxZPtZ Zl ,;~,~,1
0χψ∫=
A great benefit in the LFM is the numerical separation of the flow dynamics and chemistry. Flame
sheet can only be stretched by a turbulent movement and the chemical structure of the flame
remains, since chemical reactions are fast enough to compensate disturbances (Pitsch et al, 1996).
The species consumption source term in Equation (94) can be expressed as:
(96)
'''
11
jkjk n
j j
jbk
n
j j
jfkk W
Yk
WY
kwνν
ρρ∏ ⎟
⎟⎠
⎞⎜⎜⎝
⎛−∏ ⎟
⎟⎠
⎞⎜⎜⎝
⎛=
==Where 'ννν −= and (97)
In Fig. 2.7 is shown schematically the RIF (Representative Interactive Flamelets) model (Pitsch et
al., 1996) that has been used in the diesel combustion modelling.
Fig. 2.7 Code structure of the RIF concept (Pitsch et al., 1996)
39
CHARACTERISTIC TIME COMBUSTION MODEL
This model is sometimes called a laminar- and –turbulent characteristic –time (LaTch) combustion
model. This model is widely used in diesel combustion modelling (Kong, 1992; Patterson et al.,
1994; Kaario et al., 2002). In this model, the rate of change of species, , due to conversion from
one chemical specie to another, is given by equation:
l
c
eqlYYdY −
c
lldt τ
−= (98)
In the model, fuel vapour combustion reaction takes place by the one step reaction (58). The
characteristic time,τ is the sum of a laminar time scale and a turbulent time scale by weighted
function, as follows: f
tlc f τττ ⋅+= (99)
Where lτ based on the Equation (66) and
ετ kCMt =
142.0=M f
(100)
Turbulent time scale constant C and the weight function, , describes the increasing
influence of turbulence on combustion after the initial phase (Patterson et al, 1994).
2.2.6 Emissions modelling
In recent years the emission of pollutants, such as NOx and soot, has become the crucial criterion in
the evaluation of combustion engines. Although large effort has been made to enhance the
efficiency and exhaust gas quality of diesel engines, the state of the art is still not satisfying. Direct
injection diesel engines have for instance very high fuel efficiency and therefore low carbon dioxide
emissions (Pitsch et al., 1996). Soot emissions from diesel engines are harmful, because they can
40
cause serious health risks (lung cancer, etc) and it also reduces engine efficiency.
2.2.6.1 Nitrogen oxide emissions
Nitrogen oxides are a significant threat to the environment, and internal combustion engines are a
major source of these pollutants. Nitrogen oxides consist of nitric oxide (NO), nitrogen dioxide
(NO2) and nitrous oxide (N2O). They are collectively referred to as NOx (Heywood, 1988; Hill et
al., 2000). The NO is the main component of NOx in diesel engines. According to Heywood (1988)
NO2 and especially N2O emissions are not significant within diesel engines. The amount of NO2
can only be significant in very lean flames (premixed) and particularly at high pressures, but in
diesel engines regardless of an elevation pressure this will be quite small. Thermal NO is the main
source of NO in the diesel process and other mechanism such as prompt and fuel NO are less
important in this context. The prompt NO is formed by the reaction of atmospheric nitrogen with
hydrocarbon radicals in fuel-rich regions of flames, which is subsequently oxidized to form NO
(Heywood, 1988; Hill et al., 2000). Although the amount of fuel-rich areas in diesel engines are
quite large, the large residence time and a low concentration of oxygen after ignition causes that
this effect will remain quite small. The thermal NO is formed from oxidation of atmospheric
nitrogen at relatively high temperatures in fuel-lean environments, and has a strong temperature-
dependence (Heywood, 1988; Hill et al., 2000). In medium speed diesel engines the long residence
time of the gas at high temperature causes a large amount of NO to be formed during the
combustion process (Taskinen, 2000). In this study widely used the Zeldo’vich-mechanism was
used for the thermal NO-modelling. It is given by:
( i ) NONN +↔2
ONOON +↔+ 2
2O
[ ]
O+
( ii ) (101)
By assuming steady state conditions for the N atoms and the O atoms are in equilibrium with .
This leads to the following expression for the NO reaction rate (Marintek report, 1995):
[ ][ ] [ ][ ][ ]⎟
⎟⎠
⎞⎜⎜⎝
⎛−⋅⋅=
22
2
2112
NONO
KKNOk
dtNOd
iiii
CCf (102)
41
When the equilibrium constants are inserted into the Equation (102), the equation for the NO
formation was obtained and is given by (Marintek report, 1995):
[ ] [ ]( ) [ ]
[ ][ ]( ) 5.0
2
2166.013
25.0
212.014
46108exp10089.2
67966exp10612.6
ONO
TT
NOT
TdtNOd
⎟⎠⎞
⎜⎝⎛−⋅⋅⋅−
⎟⎠⎞
⎜⎝⎛−⋅⋅⋅=
(103)
2.2.6.2 Soot emissions
In the soot emissions modelling both formation and oxidation (combustion) have to be modelled.
These processes occur a slight difference in time and place in the combustion chamber. Generally,
it can be said that the formation takes place naturally earlier and in the rich side of fuel vapour
region of the flame quite close the fuel spay tip, while the oxidation takes place later in the lean side
of the flame region. The formation and oxidation of the soot are both chemical kinetic controlled
processes (Kennedy, 1997) and are as phenomena very complex to study. Nowadays can be found
many good reviews of this challenging field, perhaps the Haynes and Wagner (1981) has the best
source of insight for the both phenomena mentioned above.
2.2.6.2.1 Soot formation
Despite extensive research efforts in soot formation research the formation mechanism is still
poorly understood (Haynes and Wagner, 1981; Kennedy, 1997; Kronenburg et al., 2000). The basic
theory of soot formation assumes that the formation process includes the following stages: particle
inception, surface growth of particles, particles coagulation and finally particle carbonisation
(Leung et al., 1991; Richter et al., 2000).
The first stage, particle inception or nucleation in which the first condensed material is formed. The
generally accepted theory of nucleation assumes that the precursors of soot are formed from the
heavy radical PAH and/or acetylene molecules following the decomposition reactions of fuel
vapour (Richter et al, 2000). After condense reactions the soot precursors growth to the first
42
recognisable soot particles. The size of incipient particles are still very small, the magnitude is of
the order of 1 nm. The particle nucleation depends on the fuel. Aromatic fuels (benzene, acetylene),
which have considerable amount of the aromatic PAH species easily form precursors of soot and
the tendency to form soot is high whereas with aliphatic fuels (ethylene, methane) the aromatic ring
species must first be formed from the pyrolysed products of the original fuel. These aromatic ring
species then grow in the same way as described in the aromatic fuel case (Richter et al., 2000). In
the latter case the soot tendency is much lower than in the first case (Haynes and Wagner, 1981).
The second stage, the surface growth of particles in which so called nascent soot particles grow by
the addition of PAH and/or smaller alkyl species from gas-phase to the radical sites of surface of
soot particle. In this stage the number of soot particles does not change, only the mass of the soot
particle increases (Haynes and Wagner, 1981; Richter et al., 2000).
The third stage, coagulation of particles, in which particles coagulate and coalesces via reactive
particle-particle collisions. In this process the size of the particles naturally increases and the
number of particles decreases (Haynes and Wagner, 1981; Richter et al., 2000).
The fourth stage, carbonisation of particle, in which the final processes of particle formation take
place, i.e., cyclisation, ring condensation and ring fusion attended by de-hydro-generation and
growth and alignment of poly-aromatic layers. This process converts the initially amorphous soot
material to a progressively more graphitic carbon material (Richter et al., 2000).
In order to be able to predict the soot formation exactly correct we should have to known the right
pathways of different chemical reactions in different stages. This is the so called detailed chemistry
approach and is becoming more popular, because of the knowledge of different reaction
mechanisms and their rate constants known better and computer resources to compute more
expensive cases have been increased considerably. However, despite this progress detailed
chemistry models at the moment do not necessarily yield reasonable results especially in multi-
component fuel cases (Kazakov et al., 1998). Therefore, widely used semi-empirical or
phenomenological models are still used (Kazakov et al, 1998). They have a long history of
development and use. These models describe the complex process of soot formation in terms of
several global steps. Such an approach is particularly advantageous for the practical combustions
simulations. These models are often calibrated to the certain case and their expected to reasonably
43
behave within a certain range of operating conditions. In this study the soot formation model of
Hiroyasu and Tesner-Magnussen (TM) are used.
SOOT FORMATION MODEL OF TESNER-MAGNUSSEN
The soot formation model applied in this study, is based on the work of Tesner (1971) on acetylene
laminar diffusion flames, and is generalised for other fuels by Magnussen (1981b). The soot
formation model by Tesner is a very simple model, which is based on the assumption that soot
particles grow on an active radical nucleus. These active radicals are supposed to be governed by
the following processes:
I - Spontaneous formation from the fuel molecules
II - Linear branching and linear termination
III - Termination due to the onset of the radical nucleus on the soot particles
This leads to the following simple model for the rate of source of radical nucleus:
( )
4342143421&
IIIIII, sYnYogρnYgfonR fn ⋅−−⋅+= ρ
0n
(104)
Where is the spontaneous formation of a radical nucleus from fuel molecules and can be
expressed as a simple Arrhenius equation:
⎟⎠⎞
⎜⎝⎛
⋅−⋅⋅⋅⋅=
TREexpYfa.n fuelC00 081
Cf
(105)
Factor, was introduced by Magnussen (1981b) in order to make the spontaneous formation
expression more fuel dependent. The transport equation of the Tesner&Magnussen (TM) model for
soot nucleus can be expressed as follows:
44
( ) ( ) Cnfn RRnYlScl
tSct
,,&& −+
⎟⎟⎟
⎠
⎞
⎜⎜⎜
⎝
⎛∇⎟⎟
⎠
⎞
⎜⎜
⎝
⎛+•∇µµ
nYtnY U =•∇+ ρ
∂
ρ∂ (106)
Soot formation from radical nucleus is supposed to originate from interaction between the radical
nucleus and fuel molecules. The interaction of radical nucleus and the soot particles (radical
nucleus terminate on the surface of soot particle) are assumed to form a destruction term for the
particles.
The formation rate source term of the TM model for soot can be expressed as follows:
( ) nYsbYaR fs −=,& (107)
( ) ( ) Csfs RRsYlScl
tSct
sYtsY
,,U && −+⎟⎟⎟
⎠
⎞
⎜⎜⎜
⎝
⎛∇⎟⎟
⎠
⎞
⎜⎜
⎝
⎛+•∇=•∇+µµ
ρ∂
ρ∂
CnR ,&
CsR ,&
0a 335.12 +e sgpart/
(108)
The soot model constants are given in Table 2.2. Terms and are the oxidation rate of
nucleus and soot. They are discussed later in the context of soot oxidation.
Table 2.2. TM soot formation model constants
Cf 889.0 (Heavy fuel) RE / K90000 gf − 100 s/ 1
0g 90.1 −e spartcm /3 a 50.1 +e s/ 1 b 80.8 −e spart/3 cm
SOOT FORMATION MODEL OF HIROYASU
The Hiroyasu model based on the experimental findings that the soot formation is a chemical
kinetic controlled process and it is dependent upon the amount of fuel vapour and that the higher
45
cylinder pressure promotes the formation process (Hiroyasu et al., 1983 and 1989). It is a simpler
model than the Tesner model and can be written in the Arrhenius single step form including the
appropriate model constants. The formation rate source term of the soot can be expressed as follows
(Hiroyasu et al., 1983 and 1989, Patterson et al., 1994):
(109) ( ) sfuelfffs mYRTEpAR //exp5.0, ⋅−⋅=&
The term is used as the source term in Equation (108) in the same way as the term in Equation
(107). In Table 2.3 are shown the model constants what have been used in different studies.
Table 2.3 Hiroyasu soot formation model constants
Belardini et al., 1992
Patterson et al., 1994
Hiroyasu et al., 1983
RE / 6295 6295 K 9622 K K
fA 100 s/ s/ 1 150 1 -
The soot particle mass is calculated based on the density of soot (sm =sρ 2 0. 3/ cmg
=sd 26 nm
( )
) and the size
of particle ( ).
2.2.6.2.2 Soot oxidation
The understanding of soot oxidation is also still incomplete (Haudiquert et al., 1997). Soot
oxidation occurs primarily as a result of attack by molecular oxygen and the hydroxyl radical
(Kennedy, 1997). In the flame region where the concentration of molecular oxygen is low the main
oxidation reactant is a hydroxyl radical while in the surrounding of the flame the main oxidant is
oxygen (Neoh et al., 1981; Richter et al., 2000). The soot oxidation by oxygen can be assumed to
occur partially to the carbon monoxide as follows:
COOsC + 221 (110) →
The most widely used model by oxygen oxidation is the Nagle & Strickland-Constable (NSC).
46
Originally, it describes the oxidation of pyrolytic graphite over temperature from 1100 to 2500 K
and partial pressure of oxygen from 0.1 to 0.6 bars. The NSC model is based on the concept that
there are two types of site on the carbon surface available for the oxygen attack. More reactive the
so called A sites react with oxygen to give another A sites and carbon monoxide. Type B sites are
less reactive and react with a rate of first order in the oxygen concentration producing the type A
sites plus carbon monoxide. Finally, the type A sites thermally rearranged to give type B sites. A
steady-state analysis of this mechanism yields a surface mass oxidation rate (Nagle and Strickland-
Constable, 1962; Haynes and Wagner, 1981; Patterson et al., 1994; Kennedy, 1997).
( )xpkxpkpk
R OBOz
OAtotal −⋅+⎟
⎟⎠
⎞⎜⎜⎝
⎛
⋅+
⋅= 1
1 22
2
( )
(111)
Where the proportion of A sites is given by
BTO
O
kkpp
x/
2
2
+=
The soot particle oxidation rate can be expressed as:
sss
totalWNSCC Y
dRMC
C ⋅⋅⋅=
ρ6,sR& (112)
Ak Bk Tk Zk
NSC
, , and The chemical kinetic rate constants can be found from the literature e.g. (Park
et al, 1973; Kennedy, 1997). Despite the wide acceptance of the NSC model, some reservations are
obvious. The composition of soot is not the same as the pyrolytic graphite that the NSC model
assumes. Soot oxidation in diesel engines may occur with much higher oxygen partial pressures
than 0.6 bars, which is the reliable upper limit of the model. Some investigators (Park et al., 1973;
Puri et al., 1994) have found that the NSC model under-predicts the oxidation rate at higher
temperatures while at lower temperatures it will over-predict. Therefore the extra constant,C has
been added and tested in order to see how much about the model may under-predict the soot
emissions.
47
The soot oxidation by hydroxyl radical can be assumed to occur as follows:
( ) COHOHsC +→+ 221 (113)
Some investigators (Garo et al., 1986) have argued this approach, but according to different studies
and measurements, this mechanism obviously exists especially in the flame front (Neoh et al., 1981;
Puri et al., 1994; Smooke et al., 1999). The soot oxidation by hydroxyl radical is often described
based on the kinetic theory. This approach introduces an important factor, the collision efficiency of
hydroxyl radical on the surface of the soot particle. This factor represents a reaction probability for
the oxidation reaction. The oxidation rate can be then expressed as (Puri et al., 1994):
( )50218104 .. TYYdxR OHss ⋅⋅⋅⋅⋅= −η& (114) 6,Cs
Some investigators (Neoh et al., 1981; Smooke et al., 1999) have found that the value of collision
efficiencyη is nearly constant with a certain fuel (about 0.1).
The formation of hydroxyl radical can be expressed as the following set of three bimolecular
reactions.
(115) OHOHHHOHOOH
+↔+++
22
2
HCOOHCO
HOOH
↔+↔+ 2
In reactions (110) and (113) carbon monoxide oxidises (burns) by hydroxyl radical as follows:
+→+ 2 (116)
2.2.6.3 Soot modelling by EDC-model formulation
The modelling of soot formation and oxidation can be dealt with using the average values of each
quantity in each computational cells as describing in Equations (104)-(114) or more precisely by
using the EDC-formulation for the emissions also.
48
In the EDC-model formulation the soot formation and combustion can take place in the fine
structures and also in their surroundings. Magnussen (1990) has not included the soot oxidation
(combustion) in the fine structure term into the fine structure balance equation of soot. The soot
oxidation term as the sink term of soot also has to be included into the balance equation of the fine
structure of soot. The balance equation for soot in the steady-state fine structure can then be written
as (Taskinen, 2001):
Csfsss RRYY
,*
,*
*
* ~
ˆ−=⎟⎟
⎠
⎞⎜⎜⎝
⎛−ρρτ
ηρ (117)
The soot formation rate term in the fine structure can be expressed in the case of the TM model as is
described in Equation (107) in the form:
( ) ***, nsfs YbYaR −=& (118)
When using the NSC-soot oxidation model as described in Equation (112), the soot oxidation rate
source term can be expressed in the fine structure as follows:
**,
*s
ss
totalsNSCCs Y
dRMCR
⋅⋅⋅=
ρ6 (119)
The average source term of the soot field quantity equation in the case the TM formation and NSC-
oxidation models case can be written as:
( ) (
)( )[ ]( )[ ] ⎪⎭
⎪⎬⎫
⎪⎩
⎪⎨⎧
+⋅−
−−−=
ooCsCs
oon
osns
s RR
YbYaYbYaR
ρχρχγ
ρχγρχγρ
/*/
//~
,**
,*
***
1
1**
−
+
γ (120)
The term was used in the balance equation of soot as follows:
( ) ( ) sRsYlScl
tSct
sYtsY ~~~U~~
+⎟⎟⎟
⎠
⎞
⎜⎜⎜
⎝
⎛∇⎟⎟
⎠
⎞
⎜⎜
⎝
⎛+•∇=•∇+µµ
ρ∂
ρ∂ (121)
For the nucleus of soot similar to the average source term as Equation (120) and the transport
49
equation as (121) were written. More details of the formulation mentioned above can be found from
reference (Taskinen, 2001).
2.2.7 Heat transfer
The heat transfer from the cylinder gas to the combustion chamber walls can occur by convection
and/or radiation. In small engines like passenger car engines the heat transfer by forced convection
dominates, because the flow velocities and therefore shear stresses on combustion chamber walls
are much larger than the corresponding values in large engines. Also due to fuel used the tendency
to form a large amount of soot is small in small engines and hence the soot radiation is negligible
compared to the corresponding situation in large heavy fuel engines. The discussion of heat transfer
modes in engines can then be divided into the forced convection or for short convection and the
radiation.
2.2.7.1 Convection heat transfer
The heat flux from gas to wall can be expressed as follows:
( )wgcw TThJ −= (122)
The problem now is to determine the heat transfer coefficients reliably in different situations. The
traditional models (correlations) for the heat transfer coefficients, which are based on the
dimensional-analysis, are useful from the point of view of global analysis, but they cannot provide
spatial resolution, e.g., in CFD calculations. Suitable models in CFD analysis for local heat transfer
coefficients have to been based on the solution of the one-dimensional energy equation of a
turbulent boundary layer, in which suitable correlations for the turbulent Prandtl number and
turbulent viscosity in laminar/turbulent boundary layer are used (Han et al., 1997).
In CFD codes like KIVA, the standard type temperature wall functions are used (Amsden et al.,
1989). The heat flux is given in the form:
50
( )
⎪⎪⎪
⎭
⎪⎪⎪
⎬
⎫
⎪⎪⎪
⎩
⎪⎪⎪
⎨
⎧
⎟⎟⎠
⎞⎜⎜⎝
⎛⎟⎠
⎞⎜⎝
⎛ −+
−⋅⋅⋅=
21
1
1
1
RePrPr
uuPr
uuPr
TTcuJ
l
l
wgpw gas
τ
ττρ
C
C
ReRe
ReRe
>
≤
(123)
The formula based on the modified Reynolds analogy between velocity and thermal boundary layer.
The critical value of the Reynolds number when the flow changes from the laminar to turbulent
type is 122. In this model the following assumptions are used: steady and incompressible flow, no
source terms (terms that account for pressure work, chemical heat release and sprays) and a
constant turbulent Prandtl number. Reitz (1991) and Han & Reitz (1997) have found large under-
predictions in heat fluxes to walls using the standard model of KIVA-II. Han et al., (1997) have
improved the temperature wall function model and replaced this by the standard model in KIVA3V.
The new model has been derived from the one-dimensional energy conservation equation where the
gas compressibility and the increasing of the turbulent Prandtl number in the buffer and viscous
sub-layer has been taken into consideration. The heat flux equation is given now:
( )+
⋅⋅⋅⋅=
T
TTTcuJ
wggpw
gaslnτρ
(124)
In Equation (124) the correlation of the dimensionless temperature in the study of Han et al., (1997)
has given as:
∫∫+
+
+
++−
++ +
+++=
y
ydy
yyPrdyT
40
40
01
120120025010
2
....
5.2ln1.2 +⋅= ++ yT
(125)
Han et al., (1997) has obtained for the results of Equation (125):
(126)
The Author has re-calculated Equation (125) by symbolic calculation program Maple and the result
was:
51
T (127) 24.1ln1.2 +⋅= ++ y
86.0ln1.2 +⋅= ++ yT
85.0
Equation (125) can also be roughly calculated numerically with the Simpson rule. Then the result
was:
(128)
The result of Equation (128) is quite close to the result of (127) calculated by the Maple and hence
proves that Equation (127) is the correct result.
=tPr 70.0Kays et al., (2004) have obtained using constant and =Pr
9.3ln1.2 +⋅= ++ y
the correlation:
T (129)
Author has also used and tested compromise correlation:
(130) 0.3ln1.2 +⋅= ++ yT
2.2.7.2. Heat transfer by radiation
The flames can be divided by radiation characteristics into non-luminous and luminous flames (Lee
and Tien, 1982; Modest, 1993). The non-luminous flames, where the radiant emission comes from
the radiation gases, e.g., carbon dioxide and/or water steam are usually by the radiation intensity
much weaker compared to the corresponding value in the case of luminous flame (Modest, 1993;
Leung et al., 1994; Kaplan et al., 1994; Abraham et al., 1997b). In the luminous flame the radiation
emission comes from soot particles and the radiation is spectrally continuous i.e., radiation medium
emit and absorb all wavelengths (Lee and Tien, 1982).
Especially in large medium speed diesel engines, where the engine running speed is low and heavy
fuel is used, the contribution of radiation heat transfer compared to the convective heat transfer may
52
be significant (Taskinen, 2002). The reason for the behaviour is low flow velocities in the
combustion chamber at the later phase of combustion and, therefore low convective heat transfer
combined at the same time with a high amount of soot. The effect of radiation appears from the
internal energy equation (3) or (Amsden et al., 1989) as follows:
( ) ( ) J⋅∇−
++++⋅∇−=•∇+ HtrI
CombI
SprayI SSSpI
tI ~~~U~~U~~
ρερ∂ρ∂ (131)
The total heat flux is given as:
( ) rl q+∇ (132) l
lhDTk=J ∑−∇− ρ ρρ
If soot particles are very small so that the Rayleigh limit of the interaction of the radiation and
particles is valid, i.e., the scattering of radiation is very small compared to the absorption and it can
be omitted (Lee and Tien, 1982; Modest, 1993, Kaplan et al., 1994, Abraham et al., 1997b).
Especially in the case of soot radiation a great advantage is achieved, if soot can be assumed to be
grey (Lee and Tien, 1982; Abraham, 1997). Then the integration of the radiation transport equation
(RTE) over all the wavelengths can be omitted. For the emitting, absorbing and non-scattering
radiation medium the RTE can be expressed as (Modest, 1993; Kaplan et al., 1994; Abraham et al.,
1997b):
( )Ω •∇ ( ) ( ) ( ) ( )[ ]TIrITfarI bv +Ω−=Ω ,,, (133)
The divergence of radiation heat flux in Equation (131) is obtained by integrating the RTE over the
solid angle 4π . The result is:
(134) ⎥⎦⎤
⎢⎣⎡ ΩΩ∫−⋅⋅= drITTfa gasv ),(4),(
4
0
4π
σr⋅∇ q
If the radiation medium is optically thin (an absorption path length is large) the absorption integral
can be neglected and this leads to a simplified model of the soot radiation (pure emission). In Fig.
2.8 is shown the principle of the simplified model of soot radiation.
53
Soot region
Piston top
Spray region
Fig. 2.8 Soot radiation into walls in optically thin radiation media
In this case all the radiation of soot in the flame region goes into the walls of combustion chamber.
(135) 4),(4 gasgasv TTfa ⋅⋅⋅=⋅∇ σrq
),( gasv Tfa
vf
( )Tf vv ⋅
In the absorption coefficient takes into consideration the effect of soot by the volume
fraction of soot as follows:
Tfa ⋅⋅= β66.2,
7
(136)
The model constant is (Kaplan et al, 1994). ≈β
Equation (136) is known to be the soot absorption coefficient model of Kent and Honnery, 1990
developed for ethylene-air diffusion flames.
If the flame radiation is strong (as is the case when using heavy fuel oil in medium speed diesel
engines) then the absorption into the radiation media cannot be omitted. The radiation changes the
information (heat fluxes) in the soot region by smoothing temperatures of the gas there and only the
radiation from the outer surface of the soot region goes into the walls. Fig. 2.9 illustrates the
principle of the optically thick soot region radiation.
54
Spray region
Soot region
Piston top
Fig. 2.9 Soot radiation into walls in optically thick radiation media
The RTE has to then be solved by DOM (Discrete Ordinate Method) or DTM (Discrete Transfer
Method) or other sophisticated methods (Modest, 1993; Kaplan et al., 1994; Abraham et al.,
1997b). In the case of DOM, the solid angle 4π has been divided into 24 different directions and
the control volume equations obtained are solved by iteratively taken into consideration the
radiation source term and the boundary conditions. When the radiation intensity is known in every
computational cell and in an every direction i , the divergence of radiation heat flux can then be
calculated as:
(137) rq⋅∇− ( ) ( ) ⎥⎦⎤
⎢⎣⎡∑ ⋅⋅−⋅⋅==
24
1
44,i
gasiigasv TrIwTfa σ
5236.0=iw
( ) wallsootgas AVTn /42 ⋅⋅⋅⋅ σ
n
Where the weight factor (Modest, 1993)
The radiation heat flux from the radiating regions into the surface of combustion chamber with the
simplified model is calculated from the equation (all the emitted radiation distributed smoothly into
the surface):
(138) gasvpistonr Tfaq ,4, ⋅=
In the above equation the soot refractive index, is about two (Modest, 1993). In this study value
of 1.8 was used. In the case of DOM the radiation heat flux to the piston top was calculated from
55
equation (Modest, 1993):
(139) ( )⎥⎥⎥
⎦
⎤
⎢⎢⎢
⎣
⎡
∑ ⋅⋅=
<=
24
01
,
µi
iiipistonr rIswq
56
57
3. AUTHOR’S IMPLEMENTED / DEVELOPED SUBMODELS AND THEIR CONTRIBUTION TO THE MODELLING TOOL FOR DIESEL PROCESS ANALYSIS The starting point of the work was to develop a modelling tool for a spray combustion, emission
formation and heat transfer processes in medium speed diesel engines based on the Marintek
version of the KIVA-II program (Marintek Report, 1995). The program mentioned above has been
further developed from the basic KIVA-II program (Amsden et al., 1989). Into the Marintek version
of KIVA-II have been implemented at the Marintek Research Centre the following sub-models:
Magnussen EDC-combustion model, NOx formation model and the solution method procedure for
the transport equation of an arbitrary field quantity (soot and its nucleus). The flow chart of the
updated KIVA-II program is seen in the APPENDIX B.
3.1 Sub-models in baseline Marintek KIVA-II
1. Ideal gas law for equation of state
2. Low pressure drop vaporisation model
3. Drop drag coefficient model of Putnam
4. Standard temperature wall functions for convective heat transfer
5. Standard k-epsilon turbulence model for gas
6. No soot oxidation and radiation models
3.2 Sub-models in current KIVA-II
The author has developed and/or implemented the following sub-models into the code.
1. The ideal gas law for equation of state has been replaced by the RK real gas equation of
state. This was necessary in order to obtain a more precise description of the behaviour of real
gases under diesel cylinder conditions. The equations can be found in Section 2.2.4.
58
2. The low-pressure formulated drop vaporisation model of KIVA-II has replaced by the
corresponding high-pressure model. The AS drop vaporisation model has also been
implemented into the code in order to see the difference of the drop vaporisation rates
between the original model and the AS model. In the low-pressure model due to high cylinder
pressure the mass fraction of fuel vapour in the surface of the drop is low, therefore the gas
ignition and further combustion remain poor. This shortage was avoided by changing the low-
pressure model into the high-pressure model, which was based on the equilibrium of fugacity
of the fuel vapour and liquid at the surface of the drop. The basic equations of formulation are
described in Section 2.2.4.
3. The drop drag coefficient model of the baseline KIVA-II has been replaced by the model of
CL and this was modified by the Spalding heat transfer number in order to describe the
reduction of the drop drag during the drop vaporisation as is presented in Section 2.2.3.
4. The TM soot formation model has been fixed, the Hiroyasu soot formation and the NSC
soot oxidation models have been added and formulated into the EDC form. The Hiroyasu
formation model has been implemented in order to compare the effect of the soot formation
models. The NSC soot oxidation model in a slightly modified form is necessary in order to
get more realistic soot emission levels. These have been described in Section 2.2.6.2 and
2.2.6.3.
5. The standard temperature wall functions for the convective heat transfer has replaced by the
slightly modified form of the Han & Reitz model. This was very necessary in order get more
realistic heat fluxes into the wall. This was described in Section 2.2.7.1.
6. The simplified and the DOM soot radiation models have been developed and implemented
into the code in order to be able to solve the radiation transport equation. These were
necessary in order to obtain more realistic gas temperatures in the soot region and heat fluxes
to the wall. These formulations have been described in Section 2.2.7.2.
7. The modified RNG k-epsilon turbulence model has been developed in order obtain more
realistic spray spreading, fuel vapour mixing rate and vapour combustion results. These have
been described in Section 2.2.1.
59
3.3 List of author’s publications related to this work
1. Taskinen, P. (2000): Modelling Medium Speed Diesel Engine Combustion, Soot and
NOx-emission Formations, SAE technical paper 2000-01-1886.
2. Taskinen, P. (2000): Modelling of medium speed diesel process, Topical Meeting on
Modelling of Combustion and Combustion Processes, Abo/Turku, 15-16 Nov., Finland.
3. Taskinen, P. (2001): ”Modelling of Emission Formations in a Medium Speed Diesel
Engine”, First Biennial Meeting of the NSSCI, Gothenburg, Sweden.
4. Taskinen, P. (2002): “Effect of Soot Radiation on Flame Temperature, NOx-Emission and
Wall Heat Transfer in a Medium Speed Diesel Engine”, ICE Fall Technical Conference,
ICE-Vol39, ASME2002, New Orleans, USA.
5. Taskinen, P. (2003): Modelling of Turbulence/Combustion in a Medium Speed Diesel
Engine with the RNG k-epsilon Model, 13th International Multidimensional Engine
Modelling User’s Group Meeting, Detroit, Michigan, USA.
6. Taskinen, P. (2004): Modelling of Spray Turbulence with the Modified RNG k-epsilon
Model, 14th International Multidimensional Engine Modelling User’s Group Meeting,
Detroit, Michigan, USA.
60
61
4. MODELLING RESULTS AND THEIR EXPERIMENTAL VERIFICATION
When the source terms of sub-models included governing equations of the field quantities have
been solved numerically, for the modelling results have been obtained the turbulence intensity and
viscosity of gas, pressure of cylinder gas, rate of heat release, cumulative heat release, nitrogen
oxide and soot emissions, fuel vapour concentration in cylinder, convective and radiation heat
fluxes to walls of combustion chamber as a function of crank angle.
The verification of modelling results is difficult because only the cylinder gas pressure is available
and easy to measure reliably. Other quantities such as temperature of gas, nitrogen oxide, soot or
fuel vapour concentrations are nearly impossible to measure from cylinder. The assessment of other
results such as turbulence intensity, spray spreading rate, etc., due to unavailable the experimental
results can only be done by comparing different computed results to each other.
The input data used of a modelled medium speed diesel engine and the computational grid are given
in the APPENDIX A. The grid sensitivity test has been carried out earlier in order to ensure the
results independences of the grid used.
4.1 Turbulence results with the STD, basic RNG and modified RNG k-
e models
4.1.1 Turbulence intensity
The turbulence intensity in the isotropic turbulence case is defined as:
32' ku ⋅= (140)
The turbulence models used were discussed in Section 2.2.1. They are widely used in the internal
combustion engine modelling, except the modified RNG k-e model, which is the author’s
developed model.
62
In Table 4.1 is presented the constants of calculated modified RNG k-e model cases.
Table 4.1. Modified RNG k-e model constants
η1C η2C 0η β εσσ =k µC
Case 1 1.0 1.0 4.80 0.013 1.0 0.085 Case 2 1.0 2.0 5.00 0.015 1.0 0.085 Case 3 1.0 1.5 5.00 0.014 1.0 0.085 Case 4 0.6 1.6 4.80 0.020 1.0 0.085 Case 5 1.0 2.0 4.70 0.015 1.0 0.085 Case 6 1.0 1.5 5.00 0.012 1.0 0.085
In Fig. 4.1 is presented the average turbulence intensity of the cylinder gas as a function of crank
angle with the turbulence models mentioned above.
Fig. 4.1 Average turbulence intensity of gas
The turbulence intensity and therefore the turbulence kinetic energy is the weakest with the basic
RNG k-e model (later for shortly the basic RNG model) compared to the other cases as can be seen
in Fig. 4.1 and in the colour images Figs. 4.2a-c. With the STD model the corresponding quantities
63
are the largest in the early phase of combustion (turbulence generated by the spray), while with the
modified RNG model the values are in between them. This behaviour is due to the turbulence
viscosity, which is with the basic RNG model smallest, because in high the rate of strain regions the
additional term, in the epsilon equation is negative. Therefore the epsilon tend to increase and
consequence of it, the turbulence viscosity decreases from the equilibrium value defined by
ηC
0η ,
while in low the rate of strain regions the viscosity increases from the value mentioned above.
Effect of the additional term in the case of the basic RNG model is too large and the change of sign
of the term occurs too low the rate of strain value. The well-known problem with the STD model is
that it tends to over-predict the spray spreading, while the basic RNG model under-predicts the
corresponding quantity as can be seen in Figs 4.4a-b. The Rodi’s correction (1979) of the STD
model remedies the situation to some extended by adjusting the model constant C2 as a function of
velocity gradient over the spray. The correction would reduce the effect of the sink term of C2 in
the epsilon equation, which decreases epsilon too much and therefore causes too large a turbulence
viscosity in the early phase of combustion with the standard form (without correction) STD model
as can be seen in Fig. 4.1 and as too large a spray spreading in Fig. 4.4a.
The basic idea in the developing process of the modified RNG model was to find a compromise
solution between the STD and the basic RNG models in which drawbacks of the both models are
minimised. Since the additional term in the epsilon equation is an ad hoc model (Pope, 2000) so the
term can be modified in order to find the more realistic turbulence behaviour of the gas. In low the
rate of strain regions the largeness of additional term with the basic RNG model is in the order of
magnitude of one and therefore it causes un-physical high diffusivity of the gas in these regions.
This can be prevented by parameter in Equation (13), which should be about 1.5-2.0. The
largeness of the additional term can be controlled by parameter and partly also
2ηC
1ηC β in the same
equation and the experience of earlier studies (Taskinen, 2003; Taskinen, 2004) indicated that
should be about 0.6-1.0 and
1ηC
β about 0.012-0.015. The value of 0η influences to the sign of the
additional term, i.e. how high the value of the rate of strain is needed when the additional term starts
to increase the value of the dissipation rate of turbulent kinetic energy. According to the test
computations it should be about 4.7-5.0.
In the basic RNG model a shortage of diffusivity is tried to compensate by large diffusivity
coefficients ( 39111 .== εσσ k ) while in the modified RNG model cases this type of effect has
64
been reduced and more diffusivity was obtaining to the behaviour of the model by additional term
in the epsilon equation. When choosing the parameters of the modified RNG model as in Case3 or
Case6 shown in Table 4.1, more realistic the spray behaviour was obtained as can be seen in Fig.
4.4c (Case3) compared to the baseline and other modified cases Figs. 4.4a-b. The improving in the
model behaviour due to turbulence viscosity, which is now on more realistic level than in the other
baseline cases shown in Fig. 4.3.
4.1.2 Turbulence kinetic energy distribution
The colour images of the turbulence kinetic energy at certain crank angle in Figures 4.2a-c will
show the spatial differences of quantity in the different cases.
Fig. 4.2a Turbulence kinetic energy with the STD k-e model
65
Fig. 4.2b Turbulence kinetic energy with the basic RNG k-e model
Fig. 4.2c Turbulence kinetic energy with the modified RNG k-e model (Case3)
66
Figs. 4.2a-c show how much larger the turbulent kinetic energy is in the case of STD k-e model
compared to the situation in the basic RNG k-e model. Also the distribution of it is wide spreading
in the combustion chamber as can be clearly seen. With the modified RNG the turbulent kinetic
energy is between both models mentioned above and probably more realistic level than in other
models. The behaviour of STD and the basic RNG is expected. The STD model yields too large,
while the basic RNG too small a turbulence viscosity as can be seen in Fig. 4.3 and these results
appear in all their other results, e.g. spray spreading in Figs. 4.4a-b.
4.1.3 Turbulence viscosity
The turbulence (eddy) viscosity is defined in Section 2.2.1. In Fig. 4.3 is presented the average
turbulence viscosity of the cylinder gas in different cases as a function of crank angle.
Fig. 4.3 Average turbulence viscosity of gas
In Fig. 4.3 can be seen that the eddy viscosity with the standard model is too high in the early phase
of combustion while in the later phase of combustion it is slightly too small. This can be concluded
based on the cylinder pressure curve in Fig. 4.8a-b. Better combustion results can be achieved with
67
such a turbulence model, in which the eddy diffusivity is smaller in the early phase of combustion
while in the later phase of combustion it is larger than compared to the corresponding values of the
standard k-epsilon model. In Fig. 4.3 can also be seen that increasing the value of parameter 0η in
the modified RNG model cases when the additional term changes the sign from positive to
negative, the turbulence viscosity increasing also as take places between in Case2 and Case5. This
can be concluded from Equations (10, 11, 12 and 13). In Case2 the additional term is positive with
higher
ηC
η values than in Case5 and therefore the term tends to decrease the value of the dissipation
rate of the kinetic energy. At the same time through Equation (10) the value of kinetic energy
increases and therefore the value of the turbulence viscosity increases based on Equation (12).
Reducing the largeness of the additional term as in the Case4 the turbulence model behaviour is
somewhat in the middle of the basic RNG and standard k-epsilon models. In Case3 and especially
in Case1 the additional term is slightly too large because it decreases too much for the value of
epsilon in the early phase of combustion when the fuel spray still exists (from –5 ATDC to 15
ATDC) while in the later phase of combustion (from 15 ATDC to 30 ATDC) the behaviour is
opposite. The eddy viscosity is even larger in these cases than it is in the standard k-epsilon model.
It is absolutely realistic that the eddy viscosity with the modified RNG model can be larger than the
corresponding value with the standard model in the later phase of combustion when the generation
of turbulence is small. Higher value of the eddy viscosity improves the turbulence diffusion
combustion process in the later phase of combustion to obtain it more realistically than in the
situation of the standard k-epsilon model case is. In Case2, 4 and 5 the largeness of the additional
term is correct but in Case5 the value of 0η is slightly too small.
4.1.4 Spray spreading
Colour images of the spray at certain crank angle in Figures 4.4a-c clearly shown the differences of
spreading rate in different cases. The spray penetration with the basic RNG model is much longer
than it is with the standard model, while the behaviour of the spray spreading is opposite. With the
modified model both quantities are in between of the corresponding quantities of the basic RNG
and the standard model. Difference of behaviour naturally influences to the spray combustion and
the emission formations also.
68
Fig. 4.4a Spray behaviour with the STD k-e model
Fig. 4.4b Spray behaviour with the basic RNG k-e model
69
Fig. 4.4c Spray behaviour with the modified RNG k-e model (Case3)
4.2 Results of drops high/low-pressure vaporisation formulation
In Section 2.2.4 was discussed both approaches to describe the drop vaporisation processes in a
high temperature and pressure environment. The effect is seen on the amount of fuel vapour in
cylinder and if the higher amount fuel vapour able to mix into the air and combusts, the result
appears as higher cylinder pressures. Two of the drop vaporisation models were used, the original
model of the KIVA-II (the RM correlations) and the AS model. They are both tested in the high-
pressure mode. The low vs. high-pressure mode was tested with the original model of the KIVA-II.
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4.2.1 Amount of fuel vapour in combustion chamber
In Fig. 4.5 is presented the amount of fuel vapour mass with the drop high/low-pressure
vaporisation model formulations with the original model of the KIVA-II (the RM correlations) and
the AS models as a function of crank angle.
Fig. 4.5 Amount of fuel vapour in cylinder
The effect of drop low vs. high-pressure vaporisation model on the mass of fuel vapour is clearly
seen in Fig. 4.5. Especially during the ignition of gas and early phase of combustion when the
temperature of drops and their surrounding gas are still low compared to the situation at the main
combustion phase the drops vaporisation rate with the low-pressure model remains too weak. The
shortage mentioned due to the calculation method of fuel vapour mass fraction at the surface of
drop. In high-pressure models where the fuel vapour mass fraction at the surface of drop based on
the equality of fugacity of drop in a liquid and gas phase as explained in Section 2.2.4 yield more
realistic amount of fuel vapour mass and therefore the drop vaporisation does not control the
combustion rate as takes place in the case of low-pressure model. The AS model yields a little
larger amount of fuel vapour especially at the later phase of combustion which means that the heat
and mass transfer rate are both larger than compared to the original model.
71
4.2.2 Pressure of cylinder gas
In Fig. 4.6 is presented the pressure of cylinder gas with the drop low and high-pressure
vaporisation formulated the original (the RM correlations) and the AS models. Difference of the
cylinder gas pressures with the drop low vs. high-pressure formulation is seen clearly in the figure.
Fig. 4.6 Cylinder gas pressure as a function crank angle
Naturally also in the cumulative heat release is seeing the difference of model formulation
especially at the early phase of combustion as shown in Fig. 4.7. Later when the temperature of
drops increases high enough, the low-pressure model able to produce the required amount of fuel
vapour.
4.2.3 Cumulative heat release
In Fig. 4.7 is presented the cumulative heat release with the drop high and low-pressure
vaporisation formulated methods. The “Measured” cumulative curve is too gently sloping due to its
72
calculation method. It is based on the measured cylinder pressure data and using the first law of
thermodynamics as explained later.
Fig. 4.7 Cumulative heat release
4.3 Effect of turbulence model on combustion results
Especially with the drop high-pressure vaporisation model, the combustion rate is controlled by the
turbulent mixing because there is enough fuel vapour accumulated in the cylinder, which could
combust/ignite, if it is able to mix in the air. In the case of the drop low-pressure vaporisation
model, the combustion rate may be controlled by the drops vaporisation rate especially at the early
phase of combustion. In this case the effect of turbulence on the combustion rate does not appear
reliable. In next Figures are presented the main combustion results with the STD, the basic RNG
and the modified RNG k-e models. Parameters of the cases are recorded in Table 4.1 and in all
cases the drop high-pressure vaporisation model was used.
73
4.3.1 Pressure of cylinder gas
In Figs. 4.8a-b are clearly shown the effect of turbulence model on the pressure of cylinder gas due
to different combustion rates especially between the STD and the basic RNG model cases. With the
modified RNG model cases the pressure of cylinder gas are closer with the measured curve than in
other cases mentioned above, because the fuel vapour combustion rate as a function of crank angle
is on a more realistic level compared to the baseline cases. Difference of the spray behaviour in Fig.
4.4a-c influences to the combustion results on two ways, at first the spray spreading (spray tip
penetration and spray angle) and the secondly the fuel vapour turbulent mixing (combustion) rate
are different.
The same trend sees also in cumulative heat release curves, Fig. 4.9, naturally the basic RNG is
weakest, because the pressure of cylinder gas was the lowest, while the STD model is too intense in
the early phase of combustion.
Fig. 4.8a Pressure of cylinder gas as a function crank angle
74
In Fig. 4.8b are shown the cylinder pressure curves on more precisely where the differences of the
maximum cylinder pressures in different cases appear better. From the figure can be seen that the
STD model yields too high while the basic RNG model too low the maximum cylinder pressure.
Almost all the modified RNG models yield the same maximum pressure but Case2, 3 and 4 are best
in agreement with the measured curve. In the later phase of combustion Case3 seems to be closest
to the measured curve.
Fig. 4.8b Pressure of cylinder gas as a function crank angle
4.3.2 Cumulative heat release
In Fig. 4.9 is presented the cumulative heat release with the turbulence models mentioned above as
a function of crank angle. The “Measured” curve is so called semi-empirical, because it has been
calculated based on the first law of thermodynamics and using the measured pressure of cylinder
gas shown in Fig. 4.8. Before the derivation some basic assumptions and simplifications such as a
constant heat capacity of gas and homogenous mixture of fuel and air have been done in order to
ease the calculations.
75
Fig. 4.9 Cumulative heat release
4.3.3 Temperature of gas
In Figure 4.10 is presented the effect of soot radiation model on the maximum temperature of the
gas in cylinder as a function crank angle. As can be seen the maximum temperature of the gas does
not much depend on the soot radiation, because the maximum temperature of the gas appears in the
fuel vapour combustion zone, where the amount of soot is minor. The effect of soot radiation on the
temperature of the gas appears only in the soot region. With the simplified radiation model only a
small temperature drop appears in the later phase of combustion, when the amount of soot has
become a remarkable level near the combustion zone.
The effect of the turbulence model on the temperature distributions of the gas will be seen in Figs.
4.11a-c. Because the fuel vapour mixing rate depends much on the turbulence level of the gas so it
will be expected that the turbulence models used will yield quite different temperature distribution
results. Location of the highest temperature region in the gas is quite different especially between
the STD and the basic RNG models. The penetration of spray and therefore the flame region is
76
much deeper with the STD model than the basic RNG model. This again is due to the turbulence
viscosity of the gas and therefore the behaviour of spray drops and their surrounding gas is much
more diffusive with the STD model than the corresponding behaviour with the basic RNG model.
With the modified RNG model the gas temperature distribution and spray behaviour is in the
middle of behaviours of the models above. In Figures 4.11a-c are presented colour images of the
temperature distributions of the gas in the cylinder at a certain crank angle in the cases mentioned in
Table 4.1.
Fig. 4.10 Maximum temperature of gas
77
Fig. 4.11a Temperature of gas with the STD k-epsilon model
Fig. 4.11b Temperature of gas with the basic RNG k-epsilon model
78
Fig. 4.11c Temperature of gas with the modified RNG k-epsilon model (Case3)
4.4 Nitrogen oxide emissions
Nitrogen oxide emission was modelled with the Zeldo’vich-mechanism. In Fig. 4.12 is shown the
effect of turbulence models on the average nitrogen oxide emission. Turbulence models influence
the fuel vapour mixing, velocities of the gas, spray behaviour and therefore all emissions results.
Since the Zeldo’vich mechanism tends to over-predict nitrogen oxide emission (Pitsch et al., 1996)
so with the basic RNG model the emission is in good agreement with the estimated values obtained
from Wartsila Company’s literature. This indicates that the NO-emission level in the case of the
basic RNG model is too low. With the STD model the NO-emission rate is little too intense in the
early phase of combustion due to too intense combustion rate on that phase while in the later phase
of combustion the NO-emission rate remains too low due to a too weak combustion rate. The
maximum level of NO with the STD model is probably a little too high compared to the estimated
values. Also with the modified RNG models the NO emission becomes slightly too high due to the
tendency of the Zeldo’vich NO model to over-predict the emission mentioned above.
79
The colour images 4.13a-c show clearly the effect of the turbulence model on the NO emission
distributions and locations, where the NO emission formation mainly takes place. The thermal NO
formation take places near the spray edge in the lean side of it. Largeness of the nitrogen oxide
regions in different cases can be seen clearly in the images mentioned above. With the STD model
the wideness and greatness of the NO formation is the biggest while with the basic RNG model
lowest due to spray behaviour on the combustion rate. With the modified RNG the NO behaviour is
some how in between of models behaviours of the turbulence models mentioned above.
Fig. 4.12 Effect of turbulence model on average NOx emission as a function of crank angle
80
Fig. 4.13a Nitrogen oxide distribution with the STD k-e model
Fig. 4.13b Nitrogen oxide distribution with the basic RNG k-e model
81
Fig. 4.13c Nitrogen oxide distribution with the modified RNG k-e model (Case3)
In Fig. 4.14 is shown the effect of the soot radiation model on the average nitrogen oxide emissions.
As mentioned in the context of Section 4.3.3, temperature of gas, the high temperature region of the
gas is located in a different place than the biggest soot and NO formation regions and therefore the
effect of soot radiation on the NO-emission is minor. The soot radiation reduces the temperature of
the gas only in the soot region, where the NO formation is negligible. Only a small effect between
the cases without the soot radiation and the simplified model can be noted. In the later phase of
combustion when the amount of soot is high enough the temperature reducing effect of the radiation
to the gas temperature can be a remarkable and through the gases mixing effect further the NO-
emissions. If the radiation is not included in the energy balance equation this kind of effect is
omitted. With the simplified radiation model all the radiation from the soot region goes into the wall
and therefore the cooling effect in the soot region is much greater than corresponding value in the
case of DOM radiation model, where the energy changes smoothed the temperature of the gas in the
soot region and only the radiation from the outer edge of the soot region goes into the wall.
82
Fig. 4.14 Effect of soot radiation model on average NOx emission as a function crank angle
4.5 Soot emissions
In Sections 2.2.6.2 and 2.2.6.3 were discussed the soot emissions modelling by using the EDC-
formulation in the Tesner&Magnussen (TM) and Hiroyasu soot formation and NSC soot oxidation
models. With these models in the basic and slightly modified form the following results have been
obtained shown in Figures 4.15 and 4.16a-c. The constants of soot formation models, the extra
coefficient of the NSC soot oxidation model and the formulations (average or EDC formulated cell
values) are presented in Table 4.2.
83
Table 4.2. Soot models and their constants
Case 1 Averaged Hiroyasu 30=fA s/1 , 9622=RE f / K , 01.=NSCC Case 2 Averaged TM 50.1 += ea s , /1 26=sd nm , 01.=NSCC Case 3 Averaged TM 50.1 += ea s , /1 26=sd nm , 04.=NSCC Case 4 Averaged TM 53.1 += ea s , /1 26=sd nm , 04.=NSCC Case 5 Averaged TM 50.3 += ea s , /1 26=sd nm , 04.=NSCC Case 6 EDC-formulated TM Constants as in Case 3
Case 7 EDC-formulated TM Constants as in Case3, but oxidation excluded in the FS
Case 8 EDC-formulated TM 293220 += ea . scmpart 3/ , 803 += ea . , other constants same as in the basic model, oxidation included in the FS
s/1
Case 9 EDC-formulated TM 293220 += ea . scmpart 3/ , 803 += ea . , other constants same as in the basic model, oxidation excluded in the FS
s/1
According to (Haynes and Wagner et al., 1981; Smooke et al., 1999) a typical size of the soot
particle is about 20-30 nm. In this study one size of the particle was used. According to Park et al.,
1973 and Puri et al., 1994 the NSC soot oxidation model tends to under-predict the oxidation rate
and therefore in order to improve predictivity of the model, the extra constant, , has been added
to Equations (112, 119). The values of this extra constant and particle sizes are presented in Table
4.2.
NSCC
In the case of the Hiroyasu soot formation model (Case1), Patterson et al., 1994 has used for the
pre-exponential constant value, 100 and Belardini et al., 1992, 150 , but according to my
test computations they obtain too large soot emissions. Author has used the pre-exponential
constant, 30 , which has been obtained by varying different values of it in order to get a realistic
soot emission level compared to the estimated value range obtained from literature. Author has used
the original value for the activation temperature, 9622
s/1 s/1
s/1
K , which based on the experiments
(Hiroyasu et al., 1983; Kennedy, 1997) while Belardini et al., 1992 and Patterson et al., 1994 have
used a value 6295 K . The difference of these activation temperatures causes through the kinetic
Equation (109) the difference to the kinetic rate, which is about four times larger in the case of
using activation temperature 6295 K at the gas temperature 2500 K than if the activation
temperature is 9622 K at the same gas temperature. If also the difference of the pre-exponential
84
values between the author’s case and the case of Patterson et al., 1994 is taken into account by
multiplying with the difference of the kinetic rates caused by the difference of activation
temperatures so the total difference in the soot formation rate at the gas temperature 2500 K maybe
about 15-20 times larger in the cases of Patterson et al., 1994 or Belardini et al., 1992 than in the
author case. This kind of a large difference of the formation rate raises suspicions about the model
reliability. Experimental values of the soot emissions to this engine are not available and therefore
the results assessing based merely on what other researchers have published, e.g., Patterson et al,
1994; Han et al., 1996; Montgomery et al., 1996). Estimated values in Fig. 4.15 based on these soot
modeling results. It is possible that using for the extra constant in the NSC soot oxidation model
larger value than 1.0 e.g. 4.0, a larger than 30 values for the pre-exponential constant would be
able to use. Due to lack of experiments it is impossible to know exactly how much the NSC soot
oxidation model under-predicts and then adjusts the pre-exponential constant to the correct value.
s/1
The effect of variation of the TM formation model constant, , in Equation (107) is also shown in
Fig. 4.15 (Case3, 4 and 5). Sometimes the constant mentioned has been strongly varied, e.g. in the
Fluent code for the value of has been used, but in the same time the pre-
exponential constant has reduced into the value . In cases 8 and 9 the
constants used are nearly similar than the constants used in the Fluent code. As expected higher
values of the constant yield higher soot emissions, but the effect is not a linear. The original value
based on the experiments and therefore the reasonable values that can be used should be quite close
that value without loosing the model universality. If the average formulation for the TM model was
used and the extra constant of the NSC model was four, the lowest soot emissions were naturally
obtained because the formation rate is smallest and the oxidation rate largest as is in Case3.
a
81053 ⋅= .a s/129
0 10322 ⋅= .a scmpart 3/
The effect of the value of the extra constant in the NSC soot oxidation model is also shown in Fig.
4.15 (Case2 and 3). As mentioned earlier the NSC model tends to under-predict the oxidation rate
and therefore larger values than 1.0 should be used. In this study value of 4.0 was tested and the
maximum soot emission level reduces about 25 % from the basic level ( ) as seen between
the cases, Case2 and Case3. As mentioned earlier, it is difficult to estimate how much the NSC-
model under-predicts (in some cases over-predicts) without knowing measured data and then
adjusts the extra constant precisely.
01.=NSCC
85
The effect of formulations, either the averaged or the EDC-formulated soot quantities in the
transport equations of soot and its nucleus can be also seen clearly between Case3 and Case6. In the
fine structure where the fuel vapour oxidises (combusts) and therefore temperature of the gas is
high, a lot of the soot nucleus are formed due to the high amount of pyrolysis products of fuel
vapour compared to the situation outside of the fine structure. Using the average values of the cell
in the soot formation and oxidation models, e.g. the formation rate due to chemical kinetic reaction
remains too low because the average values of the cell are quite close to the fine structure
surroundings values, e.g. the average temperature of the gas in the cell is easily 20-60 K lower than
the temperature of the gas in fine structure.
Fig. 4.15 Average soot emission as a function of crank angle
The effect of the soot oxidation term in the EDC fine structure equations can be seen between the
cases, Case6 and Case7 and also between the cases, Case8 and Case9 in Fig. 4.15. The difference of
the modelled soot emissions between the cases, where the oxidation term in the fine structure
equations is either included or excluded is surprising small. In the cases where the oxidation term is
excluded the soot emissions are only slightly higher compared to the cases where the term is
included in the fine structure balance equations. Since soot oxidation is the chemical kinetic
controlled process and therefore the oxidation rate in the fine structure is high due to high
86
temperature, if there is enough oxygen after the fuel vapour combustion. Therefore the soot
oxidation term should be included into the Equations (117,120) and the effect of it should be seen in
the soot results. Now that effect seems to be minor, although the amount of the fine structures is
known to be small. Because the TM-formation model is the global model where from the certain
initial thermodynamic state (temperature, species density) produces soot by the global chemical
kinetic reaction. Assessing of the soot result correctness is difficult, because the same soot emission
can be obtained on many constants sets of the models (formation and oxidation). If using the EDC-
formulation and the standard constant set in the TM-model and the extra constant of the NSC model
is about 4 as in Case6, the soot emissions are quite a reasonable level and in well agreement with
the estimated range. But also using the constant set used in the Fluent code some extended higher
soot emissions were obtained compared to the mentioned Case6 but still they are reasonable level.
In Fig. 4.16a-c are shown the evolution of soot distribution in the combustion chamber in soot
Case3 in the Table 4.2 with the case of the modified RNG k-e turbulence model. The highest soot
concentration appears near the tip of flame where un-burnt fuel vapour is not yet mixed into air and
is still at quite high temperature. Another place where the soot formation rate is large is the piston
top where some of spray drops reach and vaporising there slowly at the temperature of piston.
Fig. 4.16a Soot distribution at crank angle=5.0 deg. (Soot Case3, Modif. RNG k-e turb. model)
87
Fig. 4.16b Soot distribution at crank angle=10.0 deg (Case3)
Fig. 4.16c Soot distribution at crank angle=15.0 deg. (Case3)
88
4.6 Heat transfer
In Section 2.2.7 were discussed the convective and radiation heat transfer modes. In Fig. 4.17 is
shown the average heat flux to piston surface (top) with the standard and modified temperature wall
functions in convection mode and with the simplified and DOM models in the radiation mode as a
function of crank angle. The convective heat transfer models and their constants are presented in
Table 4.6.
Table 4.6. Heat transfer models and their constants
Case1 Modified temperature wall functions 0.3ln1.2 +⋅= ++ yT Case2 Modified temperature wall functions 5.2ln1.2 +⋅= ++ yT Case3 Modified temperature wall functions 24.1ln1.2 +⋅= ++ yT Case4 Standard temperature wall functions
Fig. 4.17 Average heat flux to piston top
89
The standard temperature wall functions (Case4) clearly under-predicts the convective heat flux
while in the modified cases (Case1, 2 and 3) the results are in quite well agreement with the
estimated range based on the literature (Heywood, 1988 and Han et al., 1997) as shown in Fig. 4.17.
The conditions in a diesel cylinder are totally different than in the case where the standard
temperature wall functions are purposed and predict the heat flux correctly. In the modified
temperature wall functions based on the use of the one-dimensional energy equation, where the gas
compressibility, increasing of the turbulent Prandtl number near the wall describe the temperature
gradient at the wall and therefore the heat flux more realistic than the standard model. In the
modified cases the results slightly depend on what kind of correlations for the dimensionless
temperature and turbulent Prandtl number are used. Kays et al. (2004) has used a model, which
based on the Prandtl mixing length theory and modified Reynolds analogy together with constant
turbulent Prandtl number (0.85) from which the eddy thermal diffusivity is calculated. Han et al.
(1997) has used a model, which based on using the ratio of dimensionless viscosity to turbulent
Prandtl number. The final form of this model has been constructing by a curve fitting technique and
similar way as in Kays et al. (2004) has used integration over the boundary layer thickness, which
includes the transition of the flow from the laminar to turbulent mode. In this process the model
constants have been obtained, but according to the author’s re-calculation the model constant 2.5
should be about 1.24 as shown in Equation (127). The maximum heat flux is about 10 % lower
using the original constant (Case2) than using the correct value of the constant (Case3). The Author
has taken the base of Kays et al. (2004) model, but used a same model for the turbulent Prandtl
number as Han et al. (1997) has used and then slightly modified the model constant from 3.9 to 3.0
(Case1).
Heywood (1988) has mentioned that the peak heat fluxes to combustion chamber walls are of order
10 . Han et al. (1997) has calculated and presented the heat flux to piston top value range
6-11 depending on the place of piston top. Estimation of those values correctness is
difficult without experiments, because the heat flux depends greatly on the flow and temperature
fields near the walls. Especially in medium speed diesel engines the flow field is mainly caused by
spray, while the effect of swirl is minor. Nowadays high injection pressures are used and therefore
velocities of the spray are also high, so the convection heat transfer can be a very high in the curved
region (before the bowl) of the piston surface.
2/ mMW2/ mMW
The radiant heat transfer becomes remarkable at the later phase of combustion, when the amount of
the main radiating component, soot is large enough. Heavy fuel oils have components, which easily
90
form the soot, if the conditions are suitable. Radiation from the radiating gases (water steam, carbon
dioxide) is negligible especially in a small volume of the radiating region compared to the soot
radiation (Cheung et al., 1994). In very large (slow speed) diesel engines the volume of cylinder can
be a very large (many hundreds litres) and in these cases the gas radiation should be taken into
consideration additionally to the soot radiation. Due to weak flow field in medium speed diesel
cylinder at the later phase of combustion, the fuel vapour mixing to air can be remained inadequate
in the combustion chamber and therefore a large amount of soot is formed. Also in the same time
soot particles oxidation rate can be remained too weak due to lack of oxygen. The effect still
increases the possibilities to form more radiating soot regions and further the soot emissions.
The radiant heat flux with the simplified model (pure emission) is naturally much larger than the
corresponding value with the DOM model because in this case all the radiation from the radiating
regions goes to the walls without absorption to the radiating medium while in the DOM case only
the radiation from outer surface of the radiating regions goes to the walls as can be seen in Fig.
4.17. The simplified model is therefore applicable for the cases, where the radiating medium is
optically thin. This kind of situation is typical in small high swirl light fuel oil diesel engines. If the
radiating medium is optically thick as in medium speed diesel flames, when a heavy fuel oil is used,
the absorption of the radiating medium cannot be ignored and the DOM has to be used for the
solution of the RTE. Radiation smoothes the temperatures in the radiating regions because the low
temperature regions absorb and the high temperature regions emit the radiation. Kim et al. (2002)
has also used the optically thin model and the solution of the RTE with a finite volume method for
the soot radiation. According to their results the ratio of heat loss with the solution of the RTE to
the corresponding value with the optically thin model is about 0.55. In the author’s case the ratio of
heat flux with the DOM and the simplified model is about 0.5 as seen in Fig. 4.17 and is in well
agreement with the ratio of Kim et al. (2002) mentioned above.
The peak (maximum) radiant heat flux according to Heywood (1988) is about 0.75-1.2 ,
while Cheung et al. (1994) has mentioned a similar value range of 0.75-1.44 . In Fig. 4.17
the peak radiant heat flux with the DOM is about 1.0 while with the simplified model
about 2.0 . Both predicted values are in quite well agreement with the experimental values
mentioned above. Abraham et al. (1997) has estimated that the ratio of the radiant heat flux to the
total heat flux (radiant + convection) would be about 40 % while Heywood (1988) has mentioned
the corresponding value is about 20 %. Based on the Author’s calculations in Fig. 4.17 the ratio
2/ mMW2/ mMW
2/ mMW2/ mMW
91
mentioned above would be about 11 % with the DOM and 20 % with the simplified model. In this
estimation the convective heat flux value was based on the results of the modified temperature wall
functions. If the estimation based on the results of the standard temperature wall functions the ratio
would be in some extended higher. Also the largeness of the absorption coefficient in the radiating
medium influences to the radiant heat flux values and the ratio mentioned above.
The largeness of the absorption coefficient is difficult to estimate because the values reported vary
considerably due to the engine used, air/fuel equivalence ratio etc. Cheung et al. (1994) has
calculated the peak absorption coefficient 26 at equivalence ratio 0.52 and 39 at
equivalence ratio 0.76. In the paper of Cheung et al. (1994) was also mentioned the value range 90
… 240 in a quiescent combustion chamber diesel engine. In the thesis of Sulaiman (1976) has
calculated based on the experimental data of the radiant heat flux the value of 40 at
equivalence ratio 0.46 and 25 at equivalence ratio 0.29. Heywood (1988) has calculated using
the equation of absorptivity (emissivity) vs. absorption coefficient and the measured emissivity of
the soot the absorption coefficient value 22 . Lawn et al. (1987) has reported the local
absorption coefficient value 4 in a heavy fuel oil spray combustion, which seems to be slightly
small compared to the other values mentioned above.
m/1 m/1
m/1
m/1
m/1
m/1
m/1
In Fig. 4.18 are shown the effect of soot level (oxidation rate) in the predicted average and the peak
(maximum) absorption coefficient values with the model of Kent and Honnery (Equation 136) in
the modelled medium speed diesel engine. In this engine the value of the equivalence ratio was 0.36
calculated at full load.
92
Fig. 4.18 Absorption coefficients as a function of crank angle
Comparison of the predicted peak values of the author’s calculations and the values mentioned
above, e.g. the value of 40 of the paper of Cheung et al. (1994) or the value range 90 … 240
in the same paper for a quiescent combustion chamber diesel engine as the medium speed
diesel engines especially are, it can be concluded that the results obtained are in fairly agreement
with those experimental values. The effect of the amount of soot (oxidation rate) is quite small
because the maximum soot concentration appears there, where its oxidation rate is not significant.
Estimation of the average values is difficult because the measurements are always local values. In
order to get the average value of the quantity, it should measure in many places in the combustion
chamber, which can be impossible.
m/1
m/1
93
5. CONCLUSIONS
In this thesis medium speed diesel engine spray combustion, emission (NOx and soot) formation
and heat transfer processes basic structure of modelling, the most important sub-models of
physical/chemical phenomena occurred in the cylinder and the verified modelling results obtained
have been presented. The work was done with the Marintek A/S version of the open source code
KIVA-II program by implementing new and/or modified old source files of the sub-models.
At first attention was focused on the spray combustion, improving results obtained with the
standard and the basic RNG k-e models. In medium speed diesel engines, where turbulence of the
gas is mainly generated by the spray motion, the spray model and the turbulence model paid a
decisive role in order to correctly describe the fuel vapour turbulent mixing combustion process. In
this work the modified RNG k-e model was developed based on both the basic RNG and standard
k-e models and their shortages discovered. By slightly modifying the additional term of the epsilon
equation and the model constants the diffusivity problem of both models was avoided and more
realistic results (spray spreading, fuel vapour mixing and combustion results, i.e. cylinder pressure
and cumulative heat release) were obtained compared to the results obtained with the standard or
the basic RNG k-e turbulence models.
Secondly the original and the AS drop vaporisation models in a high-pressure environment was
implemented into the code in order to obtain more realistic drops vaporisation rate results compared
to the situation when using the low-pressure model of KIVA-II. Especially in medium speed diesel
engines using heavy fuel-oils, the ignition delay becomes too long and the early phase of
combustion remains too weak using the low-pressure formulation in the calculation of the mass
fraction of the fuel vapour on the drop surface. The high-pressure model based on the equality of
the fugacity in the liquid and vapour phase and therefore it yields larger and probably more realistic
values of the fuel vapour mass fraction on the drop surface. In engine CFD codes the low-pressure
model is widely used and is accurate enough in the modelling of high speed light fuel-oil diesel
engines, but in our case the high-pressure model is necessary thus avoiding the ignition delay
problems mentioned above. The difference of the drop vaporisation rates between the original and
the AS models is small in the drops highly convective region (in the spray) but later in the slow
94
flow region near the piston top the AS model yields a larger amount of the fuel vapour than the
original model.
Thirdly in the work the soot radiation was taken into consideration in the energy balance equation
by either the simplified radiation model (pure emission) or the DOM model implemented. The
effect of soot radiation appears only in the soot region reducing temperatures there, not in the fuel
vapour reaction zone where the soot is not found and the maximum temperature of the gas appears.
Also due to same reason as above, the soot radiation does not have an influence on the nitrogen
oxide (NOx) emissions because the NOx emission forms in a slightly different place to where the
fuel vapour reaction zone is. In medium speed diesel engines where the flame is optically thick due
to the use of heavy fuel oils and therefore because a higher amount of soot is found, the flame
absorption must also be taken into consideration and the DOM method has to be used in the
solution process of the RTE. The maximum radiant heat flux with the DOM is about 50 % of the
corresponding value of the simplified model and is a reasonable level according to the experimental
values of the slightly other type diesel engine than the medium speed diesel engines. In high speed
light fuel-oil engines the simplified soot radiation model is better applicable and reliable due to the
optically thin flame than in the case of medium speed heavy fuel-oil diesel engines. The absorption
coefficients (maximum and average) with the basic form Kent and Honnery model have been
realistic levels compared to the experimental values mentioned in the literature giving more
reliability to the radiation and soot emission results.
Fourthly in the work, the TM and the Hiroyasu soot formation models and the slightly modified
(multiplied by 4.0) soot oxidation model formulated both into the EDC-form implemented into the
code. The TM formation model in the EDC-weighted form and the modified NSC oxidation model
yield a reasonable soot emission level. It seems that the NSC soot oxidation model really under-
predict the soot oxidation rate as other studies also indicate (Park et al., 1973, Puri et al., 1994). In
this study a value of 4.0 times larger than the basic rate of the NSC model was tested and according
to the soot emission results, the order of magnitude of the under-prediction is slightly below the 4.0
but in some extended larger than 1.0. It is difficult to estimate the correct value of under-prediction
without knowing the experimental value of the soot oxidation rate. Information of the real soot
emissions would help only partly in the situation, not completely because if there is a difference
between the experimental and predicted soot emissions, it is impossible to know if either the
formation rate or oxidation rate is a failure.
95
The NOx emission with the Zeldo’vich mechanism seems to be slightly over-predicted as Pitsch et
al., (1996) has also concluded. With the basic RNG k-e turbulence model the level is well in
agreement with the level of Wärtsila Diesel Co. literature but this is due to weak combustion and is
not a correct situation.
Fifthly in the work convection heat transfer model was developed/improved based on the use of the
solution of the one-dimensional energy equation and the correlation of the dimensionless
temperature near the wall as described in Han et al., (1997) and Kays et al., (2004). The standard
temperature law of the wall model under-predicts the heat flux to the wall to some extent due to
shortages of the model. According to the experimental values of the heat fluxes in the slightly other
type diesel engine than in this work considered indicated that the modification is necessary in order
to obtain more realistic the convective heat flux values to the wall.
This work clearly shows how challenging the complete diesel process modelling is and what kind of
physical/chemical phenomena must be taken into account and assumptions made in order to obtain
sensible results what can be applied in the optimisation process of the engine design parameters. In
this developing work of the KIVA-II we tried to take into consideration the special characters of
medium speed diesel engines as well as possible. The physical/chemical phenomena of the cylinder
gas and the liquid fuel are very complexed and partly un-known therefore the mathematical
equations of the phenomena described by the sub-models are always approximations which take
into consideration only the limited number of real effects of the phenomena, not completely.
Therefore in the best case the modelling results describe the situation in the cylinder only in broad
outline.
96
97
6. REFERENCES Abraham, J., and Magi, V. (1997a):”Computations of Transient Jets: RNG k-e Model Versus Standard k-e Model”, SAE Technical paper 970885. Abraham, J., and Magi, V. (1997b): Application of the Discrete Ordinates Method to Compute Radiant Heat Loss in a Diesel Engine, Numerical Heat Transfer, Part A, 31:597-610. Abramzon, B., and Sirignano, W. A. (1989): “Droplet Vaporization Model for Spray Combustion Calculations”, Int. J. Heat Mass Transfer 32(9), pp.1605-1618. Amsden, A. A., O' Rourke, P. J., and Butler, T. D. (1989): "KIVA-II: A Computer Program for Chemically Reactive Flows with Sprays", L. A. Report 111560-MS. Assanis, D., Gavaises, M. and Bergeles, G. (1993): Calibration and Validation of the Taylor Analogy Breakup Model for Diesel Spray Calculation, ASME paper 93-ICE-N. Beatrice, C., Belardini, P., Bertoli, C., Cameretti, M. C., and Cirillo, N. C. (1995): Fuel Jet Models for Multidimensional Diesel Combustion Calculation: An update, SAE technical paper 950086. Belardini, P., Bertoli, C., Ciajolo, A., D’Anna, A. and Del Giacomo, N., Three-Dimensional Calculations of DI Diesel Engine Combustion and Comparison with In-Cylinder Sampling Valve Data, SAE Technical paper 922225, 1992. Bianchi, G. M., and Pelloni, P. (1999): Modeling the Diesel Fuel Spray Breakup by Using a Hybrid Model, SAE technical paper 1999-01-0226. Bird, R. B., Steward, W. E., and Lightfoot, E. N. (1960): Transport Phenomena, Wiley, NY. Borman, G., and Nishiwaki, K. (1987): Internal-Combustion Engine Heat Transfer, Progress in Energy Combust. Science, Vol.13, pp. 1-46. Brink, A. (1998): Eddy Break-Up Based Models for Industrial Diffusion Flames with Complex Gas Phase Chemistry, Academic Dissertation, Report 98-7, Åbo Akademi, Finland Cartellieri W. P. (1987): "Status Report on a Preliminary Survey of Strategies to Meet US-1991 HD Diesel Emission Standards without Exhaust Gas after Treatment", SAE technical paper 870343. Castleman, R. A. (1932): NACA Report No. 440. Cheung, C. S., Leung, C. W., and Leung, T. P. (1994): Modelling spatial radiative heat flux distribution in a direct injection diesel engine, Proceedings of the Institute of Mechanical Engineers A, Journal of Power and Energy, vol. 208, no. A4. Chomiak, J. (2000): Turbulent Reacting Flows, Graduate course book, 3rd edition, Chalmers University of Technology, Sweden.
98
Choudhury, D., Kim, S. E., and Flannery, W. S. (1993): Calculation of Turbulence Separated Flows Using a Renormalization Group based k-epsilon Turbulence Model, FED, 149, ASME, 177-187. Corcione, F. E., Allocca, L., Pelloni, P., Bianchi, G. M., Bertoni, F. L., and Ivaldi, D. (1999): Modeling Atomization and Drop Breakup of High-Pressure Diesel Sprays, Cliffe, K. A., and Lever, D. A. (1986): The Drag on an Evaporating Fuel Droplet, IEA Combustion Research Conference, Harwell, p. 34. Ducroq, F., Borghi, R., and Delhaye, B. (1998): Development of a Spray Model for Low Weber Gasoline Jets in S.I. Engines, SAE technical paper 982608. Garo, A., Prado, G., and Lahaye, J. (1990): Chemical Aspects of Soot Partical Oxidation in a Laminar Methane-Air Diffusion Flame, Combustion and Flame, 79:226-233. Golini, S., Chiatti, G., Maggiore, M., Papetti, F., and Succi, S. (1993): Improving the Vaporization Model of Kiva-II in an Advanced Computing Environment, Computational Fluid Dynamics Journal, Vol.2, No.1. Gradinger, T. B., and Boulouchos, K. (1998): A Zero-dimensional Model for Spray Droplet Vaporization at High Pressures and Temperatures, Int. J. of Heat and Mass Transfer, 41, 2947-2959. Faeth, G. M. (1977): Progress in Energy Combust. Science, Vol. 3, 191. Frank-Kamenetskii, D. A. (1969): Diffusion and Heat Transfer in Chemical Kinetics (2nd Edition). Plenum Press, NY. Han, Z., and Reitz, R. D. (1995): “Turbulence Modeling of Internal Combustion Engines Using RNG k-e Models,” Combust. Science and Tech., Vol. 106, pp. 267-295. Han, Z., Uludogan, A., Hampson, G. J. and Reitz, R. D.: Mechanism of Soot and NOx Emission Reduction Using Multiple-Injection in a Diesel Engine, SAE technical paper 960633, 1996. Han, Z., and Reitz, R. D. (1997): “A Temperature Wall Function Formulation for Variable-Density Turbulence Flows with Application to Engine Convective Heat Transfer Modeling”, Int. J. Heat Mass Transfer, Vol. 40(3), pp.613-625. Hasse, C., Barths, H., and Peters, N. (1999): SAE technical paper 1999-01-3547. Haudiquert, M., Cessou, A., Stepowski, D., and Coppale, A. (1997): OH and Soot Concentration Measurements in a High-Temperature Laminar Diffusion Flame, Combustion and Flame, 111: 338-349. Haynes, B. S., and Wagner, H. Gg. (1981): “Soot Formation”, Progress in Energy Comb. Science, Vol. 7, pp. 229-273. Hergart, C. A., Barths, H, and Peters, N. (1999): SAE technical paper 1999-01-3550. Heywood, J. B. (1988): Internal Combustion Engine Fundamentals, McGraw-Hill Co., ISBN 0-07-100499-8.
99
Hill, S. C., and Smoot, L. D. (2000): Modeling of nitrogen oxides formation and destruction in combustion systems, Progress in Energy Combust. Science, Vol. 26, pp. 417-458. Hiroyasy, H., and Nishida, K. (1989): Simplified Three-dimensional Modeling of Mixture Formation and Combustion in a D.I. Diesel Engine, SAE technical paper 890269. Huh, K. Y., and Gosman, A. D. (1991): “A Phenomenological Model of Diesel Spray Atomization”, Proceedings of the Int. Conf. on Multiphase Flows, Sept. 24-27, Tsukuba, Japan. Ishikawa, N., Niimura, K. (1996): Analysis of Diesel Spray Structure Using Magnified Photography and PIV, SAE technical paper, 960770. Jia, H., and Gogos, G. (1993): “High Pressure Droplet Vaporization; Effects of Liquid-phase Gas Solubility”, Int. J. Heat Mass Transfer 36(18), pp. 4419-4431. Kaario, O., Larmi, M., and Tanner, F. (2002): Comparing Single-step and Multi-step Chemistry Using the Laminar and Turbulent Characteristic Time Combustion Model in Two Diesel Engines, SAE technical paper, 2002-01-1749. Kaplan, C. R., Baek, S. W., Oran, E. S., and Ellzey, J. L. (1994): Dynamics of a Strongly Radiating Unsteady Ethylene Jet Diffusion Flame, Combustion and Flame, 96:1-21. Kays, W., Crawford, M., and Weigand B. (2004): Convective Heat and Mass Transfer, Mc-Graw-Hill, 4th Edition. Kazakov, A., and Foster, D. E. (1998): Modeling of Soot Formation During DI Diesel Combustion Using a Multi-Step Phenomenological Model, SAE technical paper 982463. Kennedy, I. M. (1997): Models of soot formation and oxidation, Progress in Energy Combust. Science, Vol. 23, pp 95-132. Kent, J. H., and Honnery, D. R. (1990): Combust. Flame 79:287-298. Kim Yong-Mo, Kim Hoo-Jong, Kim Seong-Ku, Kang Sung-Mo, and Ahn Jae-Hyun (2002): Nonequilibrium and Radiative Effects on Combustion processes and Pollutant Formation in DI Diesel Engine, Twelveth Int. Multidimensional Engine Modeling User’s Group Meeting at the SAE Congress. Kong, S. C. (1992): “Modeling Ignition and Combustion process in Compression Ignited Engines”, MS Thesis, Mechanical Engineering Department, University of Wisconsin-Madison. Kronenburg, A., Bilger, R. W., and Kent, J. H. (2000): Modeling Soot Formation in Turbulent Methane-Air Jet Diffusion Flames, Combustion and Flame, 121:24-40. Kuo, K.K. (1986): Principles of Combustion, John Wiley & Sons, NY, ISBN 0-471-09852-3. Lawn, C. J., Cunningham, A. T. S., Street, P. J., Matthews, K. J., Sarjeant, M., and Godridge, A. M., (1987): The combustion of heavy fuel-oils. In: Lawn, C. J. (ed.), Principles of combustion engineering for boilers. London: Academic Press. pp. 61-196. ISBN 12-439035-8.
100
Leborgne, H., Cabot, M. S., and Berlemont, A. (1998): ”Modelling of Single Droplet Vaporization under High Pressure Conditions”, Third Int. Conf. on Multiphase Flow, ICMF98, Lyon, France, June 8-12. Lee, S. C., and Tien, C. L. (1982): Flame Radiation, Progress in Energy Combust. Science, Vol. 8, pp. 41-59. Leung, K. M., Lindstedt, R. P., and Jones, W. P. (1991): A Simplified Reaction Mechanism for Soot Formation in Nonpremixed Flames, Combustion and Flame, 87:289-305. Levich, V. G. (1962): Physiocochemical Hydrodynamics, Prentice-Hall, Englewood Cliffs, New York. Magnussen, B. F., and Hjertager, B. H. (1977): Sixteenth Symposium (International) on Combustion, The Combustion Institute, Pittsburgh, p. 719. Magnussen, B. F. (1981a): “On the structure of Turbulence and a Generalized Eddy Dissipation Concept for Chemical Reactions in Turbulent Flow”, 19th AIAA Science Meeting, St. Louis. USA. Magnussen, B. F. (1981b): “Modeling of Reaction Processes in Turbulent Flames with Special Emphasis on Soot Formation and Combustion,” Particulate Carbon Formation During Combustion (Siegla and Smith Eds.) Plenum Publishing Co. Marintek A/S Report (1995): Implementation of New Combustion and NOx formation Models into KIVA-II, MT22 F95-XXXX, 222511.00.01. Modest, M. F. (1993): Radiative heat transfer, Mc-Graw-Hill Int. Ed., ISBN 0-07-112742-9. Montgomery, D. T., Chan, M., Chang, C. T., Farrell, P. V. and Reitz, R. D.: Effect of Injector Nozzle Hole Size and Number on Spray Characteristics and the Performance of a Heavy Duty D.I. Diesel Engine, SAE Technical paper 962002, 1996. Nagle, J., and Strickland-Constable, R. F., “Oxidation of Carbon between 1000-2000 ,” Proc. of the Fifth Carbon Conf., Vol. 1, Pergamon Press, 1962.
C0
Neoh, K. G., Howard, J. B., and Sarofim, A. F. (1981): Particulate Carbon Formation During Combustion, (D. C. Siegla and G. W. Smith, Eds.), Plenum, New York, pp.261-282. O' Rourke, P. J., and Amsden, A. A. (1987): The Tab Method for Numerical Calculation of Spray Droplet Breakup, SAE technical paper 872089. Park, C., and Appleton, J. P. (1973): Shock-Tube Measurements of Soot Oxidation Rates, Combustion and Flame 20:369-379. Patterson, M. A., Kong, S-C., Hampson, G. J. and Reitz, R. D. (1994): Modeling the Effects of Fuel Injection Characteristics on Diesel Engine Soot and NOx Emissions, SAE technical paper 940523. Pedersen, E., Valland, H., and Engja, H. (1995): Modelling and Simulation of Diesel Engine Process, CIMAC paper D34.
101
Peng, D., and Robinson, D. B. (1976): A New Two-Constant Equation of State, Industrial Eng. Chem. Fund. 15:59-64. Peters, N. (1984): Progress in Energy Combust. Science, Vol. 10, pp. 319-339. Pitsch, H., Wan, Y., and Peters, N. (1995): Numerical Investigation of Soot Formation and Oxidation under Diesel Engine Conditions, SAE technical paper 952357. Pitsch, H., Barths, H., and Peters, N. (1996): Three-Dimensional Modeling of NOx and Soot Formation in DI-Diesel Engines Using Detailed Chemistry Based on the Interactive Flamelet Approach, SAE technical paper 962057. Pope, S. B., 2000, “Turbulent Flows”, Cambridge Univ. Press, ISBN 0-521-59886-9. Puri, R., Santoro, R. J., and Smyth K. C. (1994): The Oxidation of Soot and Carbon Monoxide in Hydrocarbon Diffusion Flames, Combustion and Flame 97:125-144. Ramos, J. I.: Internal Combustion Engine Modeling, Hemisphere publishing Co., 422 p, ISBN 0-89116-157-0. Redlich, O., and Kwong, J. N. S. (1949): Chem. Rev., 44:233. Reid, R. C., Prausnitz, J. M., and Poling, D. B., (1987): “The Properties of Gases and Liquids”, 4th Ed., McGraw-Hill, New York. Reitz, R. D., and Bracco, F. V. (1982): Mechanism of atomization of a liquid jet, Phys. Fluids 25(10), pp.1730-1742. Reitz, R. D. (1991): Assessment of wall heat transfer models for premixed-charge engine combustion computations, SAE technical paper 910267. Reitz, R. D., and Rutland, C. J. (1995): Development and Testing of Diesel Engine CFD models, Progress in Energy and Combustion Science 21, p.173. Richter, H., and Howard, J. B. (2000): Formation of polycyclic aromatic hydrocarbons and their growth to soot-a review of chemical reaction pathways, Progress in Energy Combust. Science 26, pp. 565-608. Rodi, W. (1979): “Turbulence Models and their Application in Hydraulics,” State of the Art Paper, Presented by the IAHR-Section on Fundamentals of Division II: Exp. and Math. Fluid Dynamics. Smooke, M. D., Mc Enally, C. S., Pfefferle, L. D., Hall, R. J., and Colket, M. B. (1999): Computational and Experimental Study of Soot Formation in a Coflow, Laminar Diffusion Flame, Combustion and Flame 117:117-139. Su, C. C. (1980): M.S. thesis, University of Princeton. Sulaiman, S. J. (1976): Convective and radiative heat transfer in a high swirl direct injection diesel engine, PhD thesis, Loughborough University of Technology.
102
Tanner, F. X., and Weisser, G. (1998): Simulation of Liquid Jet Atomization for Fuel Sprays by Means of a Cascade Drop Breakup Model, SAE Technical paper 980808. Taskinen, P., Karvinen, R., Liljenfeldt, G., and Salminen, H. J. (1996): Simulation of Heavy Fuel Spray and Combustion in a Medium Speed Diesel Engine, SAE technical paper 962053. Taskinen, P., Karvinen, R., Liljenfeldt, G., and Salminen, H. J. (1997): Combustion and NOx Emission Simulation of a Large Medium Speed Diesel Engine, SAE technical paper 972865. Taskinen, P. (1998): Effect of Fuel Spray Characteristics on Combustion and Emission Formation in a Large Medium Speed Diesel Engine, SAE technical paper 982583. Taskinen, P. (2000): Modelling Medium Speed Diesel Engine Combustion, Soot and NOx-emission Formations, SAE technical paper 2000-01-1886. Taskinen, P. (2000): Modelling of medium speed diesel process, Topical Meeting on Modelling of Combustion and Combustion Processes, Åbo/Turku, 15-16 Nov., Finland. Taskinen, P. (2001): ”Modelling of Emission Formations in a Medium Speed Diesel Engine”, First Biennial Meeting of the NSSCI, Gothenburg, Sweden. Taskinen, P. (2002): “Effect of Soot Radiation on Flame Temperature, NOx-Emission and Wall Heat Transfer in a Medium Speed Diesel Engine”, ICE Fall Technical Conference, ICE-Vol39, ASME2002, New Orleans, USA. Taskinen, P. (2003): Modeling of Turbulence/Combustion in a Medium Speed Diesel Engine with the RNG k-epsilon Model, 13th International Multidimensional Engine Modeling User’s Group Meeting, Detroit, Michigan, USA. Taskinen, P. (2004): Modeling of Spray Turbulence with the Modified RNG k-epsilon Model, 14th International Multidimensional Engine Modeling User’s Group Meeting, Detroit, Michigan, USA. Taylor, G. I. (1963): “The Shape and Acceleration of a Drop in a High Speed Air Stream, The Scientific Papers of G. I. Taylor, ed. G. K. Batchelor, Vol. 3, University Press, Cambridge. Tesner, P. A., Snegiriova, T. D., and Knorre, V. G. (1971): Kinetics of Dispersed Carbon Formation, Combustion and Flame, 17:253. Weisser, G., Tanner, F. X. and Boulouchos, K: Towards CRFD-Simulation of Large Diesel Engines: Modeling Approaches for Key Processes. 3rd International Conference, ICE97, Internal Combustion Engines: Experiments and Modeling. Capri, Italy. Williams, A. (1990): Combustion of liquid fuel sprays, Butterworth & Co, ISBN 0-408-04113-7. Yakhot, V., Orzag, S. A., Tangham, S., Gatski, T. B. and Speziale C. G. (1992): Development of Turbulence Models for Shear Flows by a Double Expansion Technique. Phys. Fluid A, 4, 1510. Yan, J. D., and Borman, G. L. (1988): Analysis and in-cylinder measurement of particulate radiant emissions and temperature in a direct injection diesel engine, SAE technical paper 881315.
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Younis, B. (1997): Lecture Notes for Course on Applied Turbulence Modelling, Helsinki University of Technology, 19-25 May.
APPENDIX A 1. Modelled engine specifications: Details of the modelled Wärtsilä W46 medium speed diesel engine are listed in Table A1.
Cylinder bore 460 mm
Stroke 580 mm
Compression ratio 14.0
Running speed 500.0 rpm
Number & size of nozzle holes 10 x 0.78 mm
Start of injection 10.deg. BTDC
Fuel injection duration 26.5 deg.
Total injected fuel mass/cycle 12.3 g
Fuel Heavy fuel (Neste Mastera)
Start of ignition 7.0deg. BTDC
Simulation begins 40.deg. BTDC
Air temperature at 20 deg. BTDC 654.0 K
Air pressure at 40 deg. BTDC 33.5 bar
Swirl ratio 0.2
Table A1. Initial conditions and operating/construction parameters of the modelled diesel engine
2. Computational mesh of modelled engine
The computational mesh consists of 45 non-equally spaced cells in the radial, 21 in the azimuthally
and 46 equally spaced cells in the axial direction. Due to piston travel the minimum number of cells
at TDC is 17. The rate change of length of cell in the radial direction is about 3 %. The angle of the
computational sector is 36 degrees. The grid is shown in Fig. A1.
Figure A1. Computational grid of modelled engine at TDC
APPENDIX B Flow chart of the updated KIVA-II modelling tool:
Figure B1. Flow chart of the KIVA-II modelling code
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