piston bowl combustion simulation from fuel spray …1083220/fulltext01.pdf · 2017-03-20 ·...
TRANSCRIPT
Degree Project in Mechanical engineeringSecond Cycle, 30 Credits
September 2016
PISTON BOWL COMBUSTION SIMULATION
From Fuel Spray Calibration to Emissions Minimization
Diego Garcia Pardo
FRIENDSHIP SYSTEMS
Contents
1 PRINCIPLES OF CFD MODELLING 1
1.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
1.2 Eulerian and Lagrangian Fluid Description . . . . . . . . . . . . . . . . . . . . . . . . . . . 2
1.3 The Navier-Stokes Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2
1.4 Principles of Turbulence Modelling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4
1.4.1 Basics of RANS Modelling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4
1.4.2 RNG k− ε RANS Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6
2 SPRAY MODELLING 7
2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7
2.2 Parcel Grouping . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8
2.3 Spray Break Up . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9
2.4 Droplet Drag Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11
2.5 Spray Vaporizing Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12
2.6 Sandia Spray A . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13
2.6.1 Initial and Boundary Conditions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13
2.6.2 Grid Configuration and Road Map . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14
2.6.3 Parcel Number: Lagrangian Phase Numerical Dependency Study . . . . . . . . . . 15
2.6.4 Grid Study and Collision Mesh . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18
2.6.5 Calibration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20
2.6.6 Break Up Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28
2.6.7 Correlations and Analytical Estimates . . . . . . . . . . . . . . . . . . . . . . . . . . 32
2
3 Case Study: Diesel ULPC Piston Bowl Optimization 36
3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36
3.2 ULPC Piston Bowl . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37
3.2.1 Pilot and Post Injection . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37
3.3 Exhaust Gas Recirculation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37
3.4 CFD Methodology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38
3.4.1 Computational Model: Sector Mesh and Flow Initialization . . . . . . . . . . . . . 39
3.4.2 CFD Set Up . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41
3.4.3 Combustion Modelling and Emissions . . . . . . . . . . . . . . . . . . . . . . . . . . 41
3.4.4 Baseline Case and Validation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42
3.4.5 About Grid Dependency . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43
3.4.6 Turbulence Model Constants . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47
3.4.7 Spray Model Constants . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48
3.5 NOx and Soot Constrained Optimization . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49
3.5.1 Parametrization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49
3.5.2 Optimization Algorithm . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51
3.5.3 Analysis of the Optimization Results . . . . . . . . . . . . . . . . . . . . . . . . . . 52
3.5.4 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57
3.6 Unexplored Optimization Posibilities: Fuel Injection Curve . . . . . . . . . . . . . . . . . . 58
3
Diego García PardoFRIENDSHIP SYSTEMS
Potsdam
4
Aknowledgements
The current pollution policies in all European and American countries are forcing the industry to movetowards a more efficient and environmentally friendly engines. On the other hand, customers requiremaintaining the power and fuel consumption. Lowering mainly nitrous oxides (NOx) and carbon par-ticles (Soot) is therefore a challenging task with a very strong impact on mainly the automotive andaeronautical market.
The purpose of the current work is to research the pollution production of automotive diesel enginesand optimize the fuel injection and piston geometry to lower the emissions. The interaction betweenfuel and air as well as the combustion are the two main physical and chemical processes governing thepollutants formation. Converged-CFD will be the CFD tool employed during the analysis of the previ-ous problems.
The fuel-air interaction is related to jet break up, vaporization and turbulence. The strong dependenceon the surrounding flow field of the previous processes require the equations to be solved numericallywithin a CFD code. The fuel is to be placed in a combustion chamber (piston) where the spray will affectthe surrounding flow field and ultimately the combustion process.
In order to accurately represent the nature of the processes, the current work is divided into two mainchapters. Spray modelling and Combustion Modelling. The first will help to accurately model the dis-crete phase (fuel spray) and the vapour entrainment. The second chapter, combustion modelling willretrieve the knowledge gain in the first part to accurately represent the fuel injection in the chamber aswell as the combustion process to ultimately model the pollutants emissions.
Finally, a piston bowl optimization will be performed using the previous analysed models and give theindustry a measure of the potential improvement by just adjusting the fuel injection or by modifyingthe piston bowl geometry.
PREFACE
The current pollution policies in all European and American countries are forcing the industry to movetowards a more efficient and environmentally friendly engines. On the other hand, customers requiremaintaining the power and fuel consumption. Lowering mainly nitrous oxides (NOx) and carbon par-ticles (Soot) is therefore a challenging task with a very strong impact on mainly the automotive andaeronautical market.
The purpose of the current work is to research the pollution production of automotive diesel enginesand optimize the fuel injection and piston geometry to lower the emissions. The interaction betweenfuel and air as well as the combustion are the two main physical and chemical processes governing thepollutants formation. Converged-CFD will be the CFD tool employed during the analysis of the previ-ous problems.
The fuel-air interaction is related to jet break up, vaporization and turbulence. The strong dependenceon the surrounding flow field of the previous processes require the equations to be solved numericallywithin a CFD code. The fuel is to be placed in a combustion chamber (piston) where the spray will affectthe surrounding flow field and ultimately the combustion process.
In order to accurately represent the nature of the processes, the current work is divided into two mainchapters. Spray modelling and Combustion Modelling. The first will help to accurately model the dis-crete phase (fuel spray) and the vapour entrainment. The second chapter, combustion modelling willretrieve the knowledge gain in the first part to accurately represent the fuel injection in the chamber aswell as the combustion process to ultimately model the pollutants emissions.
Finally, a piston bowl optimization will be performed using the previous analysed models and give theindustry a measure of the potential improvement by just adjusting the fuel injection or by modifyingthe piston bowl geometry.
PREFACIO
Las actuales políticas medioambientales tanto en países Europeos como Americanos está forzando a laindustria a producir motores de combustión más eficientes y limpios. Los clientes requieren mantenerla potencia y el consumo de combustible. Es por esto que la minimización de la emisión de óxidos denitrógeno NOx y de part’iculas de carbono (Soot) es una tarea cuanto menos desafiante con un granimpacto en el mercado automovilístico y aeronáutico.
El propósito de este trabajo es realizar una investigació sobre las emisiones de motores diésel y optimizarla inyección de combustible y la geometria del bowl del piston. La interacción entre el combustible y elaire así como el proceso de combustión son los dos principales procesos químicos y físicos que gobier-nan la formación de humos tóxicos. ConvergeCFD será utilizado como herramienta para llevar acabolas simulaciones CFD durante el análisis de los procesos previamente descritos.
La interacción entre el aire y el combustible se debe a los modelos de ruptura del jet líquido, vapor-ización y valores de turbulencia. La fuerte dependencia en el fluido que rodea al spray require que lasecuaciones sean resueltas numéricamente en un software CFD. El combustible liquido será inyectado enuna cámara de combustion (piston) donde el liquido se atomizará, evaporará y finalmente igniciará.
Con el objetivo de estudiar de forma exhaustiva los procesos que tendrán lugar, el trabajo se divide enun primer capitulo en el que se analiza el comportamiento de los sprays de combustible y un segundocapítulo en el que el proceso de combustión se valida para finalmente proceder con la el proceso deoptimización. El primero ayudará a entender la relación entre la fase discreta del fluido (spray liquido)así como la penetración del vapor en la cámara de combustión. En el segundo capítulo, se describirátanto el proceso de combustión como la formación de compuestos tóxicos.
Finalmente la geometría del bowl del piston será parametrizada y optimizada con el objetivo de medirla capacidad de mejora simplemente modificando la geometría del pistón y la estrategia de inyección.
Chapter 1
PRINCIPLES OF CFD MODELLING
1.1 Introduction
Fluid mechanics is a wide and complex branch of engineering. The applicability of the laws of physicsin this field has been shown to be especially difficult. The behaviour of gases and liquids is still limiteddue to the complexity of the equations describing the flow. The main sets of equations that can be foundare either Lattice-Boltzmann equations and the Navier-Stokes equations.
The applicability of the previous equations is widely discussed in [3]. The main parameter affecting tothe applicability of both sets is the so called mean free path, λ. The continuum hypothesis is based onthis parameter. It requires the mean free path to be much smaller than the actual length scale of theflow being described. Therefore one necessary requirement (but not sufficient [3]) is that the Knudsennumber:
Kn =λ
L 1
The mean free path of air at standard conditions can be estimated (as shown in [2]) by:
λ ≈(
ρNAm
)−1/3= 3× 10−9m
Where NA and m represent the Avogadro number and the molecular weight.
Computational Fluid Dynamics (from now on simply CFD), attempts to solve these equations numerically.The increase in the available computational power in the recent years has opened many possibilitieswithin the world of research and engineering. This project will make use of the Navier-Stokes equationsto model fuel sprays. This project will seek to compare and asses the predictive capabilities of CFD bycomparing the results with experimental data.
1
1.2 Eulerian and Lagrangian Fluid Description
The mathematical description of the behavior of a fluid can be done by either tracking particles withinthe fluid (Lagrangian approach) or by looking at a specified location over time (Eulerian approach).
It is usually found in bibliography the following comparison:
UL(x0, t) = UE(~x(t), t) (1.1)
Where x0 represents a given particle. The time derivative of both should be the same, therefore leadingto the following relationship
∂
∂tUL =
∂u∂t
(1.2)
∂
∂tUE =
∂u∂t
+∂u∂x
∂x∂t
=∂u∂t
+ u∂u∂x
(1.3)
Last equation is the definition of the so called material derivative, which in its most general form isgiven by;
DDt≡ ∂
∂t+ uj
∂
∂j(1.4)
1.3 The Navier-Stokes Equations
The description of the flow field requires a set of equations describing the mass, momentum and energyconservation. That is, the flow physics are derived from first principles.
The momentum conservation equations are the so called Navier-Stokes Equations. The derivation canbe found in almost any introductory books to fluid mechanics.
Below, the Navier-Stokes equations are presented in conservative form. They include the terms thataccount for a mass, momentum and energy transfers from only the Lagrangian to Eulerian phase [13].
Mass conservation
∂ρ
∂t+
∂
∂xi(ρui) = SI (1.5)
Where SI accounts for the mass transfer from the Lagrangian phase (fuel injection as it will be shown).
Momentum conservation
∂ (ρui)
∂t+
∂(ρuiuj
)∂xj
= − ∂P∂xi
+∂σi,j
∂xj+ SI I
i (1.6)
2
Where SI I accounts for the momentum transfer from the Lagrangian phase. The stress tensor is definedas:
σi,j = µ
(∂ui∂xj
+∂1uj
∂xi
)+
(µb −
23
µ
)δi,j
∂uk∂xk
(1.7)
In the CFD Software used in the present work, the bulk viscosity, µb is set to zero [13].
Energy conservation
∂ρe∂t
+∂ρeuj
∂xj= −P
∂uj
∂xj+ σij
∂ui∂xj
+∂
∂xj
(K
∂T∂xj
)+
∂
∂xj
(ρD ∑
mhm
∂Ym
∂xj
)+ SI I I (1.8)
Where SI I I accounts for the energy transfer from the Lagrangian phase. Besides, ∂∂xj
(ρD ∑
mhm
∂Ym∂xj
)represents for the energy transport due to species diffusion as stated in [13]. The species mass fractionis Ym = Mm
Mtot= ρm
ρtotand the D = ν
Sc represents the molecular mass diffusion coefficient1.
The previous equations have to be supplemented with additional state equations:
P = ρRT and e = e0 + cvT (1.9)
The Source term for mass, momentum and energy will be presented later in the chapter describing spraymodelling, Ch.2.
1Sc ≡ Schmidt Number
3
1.4 Principles of Turbulence Modelling
Solving the Navier-Stokes equations is a challenging task usually performed numerically. There is only afew known analytical solutions. The numerical solutions are however very expensive computationallyspeaking. The cost of solving the Navier-Stokes equations is directly related the size of the smallesteddies in the flow field. The scale is given by the so called Kolmogorov scale. From dimensional analysis[4]:
η =(
ν3/ε)1/4
(1.10)
It is observed that smallest scales are independent from the geometrical boundaries and depend solelyon the viscosity, ν and dissipation of turbulent kinetic energy, ε. Also as reported in [4] it is possible toestimate the dissipation order of magnitude as a function of velocity (large eddie velocity), u′ and thegeometrical length scale, Λ:
ε ∼ (u′)3
Λ(1.11)
The cost of evaluating numerically the Navier-Stokes Equations comes directly from the combination ofthe last two equations as:
Λη
=Λ
(ν3/ε)1/4 = Λ
(u′3/Λ
)1/4
ν3/4 = Re3/4Λ (1.12)
If we consider a 3 dimensional flow with a grid fine enough to resolve the kolmogorov scales, then:
Ngrid points ∝ Re9/4Λ = Re2.25
Λ (1.13)
Real application usually involve dealing with Reynolds numbers large enough to support alternativesto the Direct Numerical Solution (usually abbreviated as DNS) of the Navier-Stokes equations.
The golden rule in engineering, if you can’t solve the problem, solve a simpler one fits perfectly in this situa-tion. Nowadays there exists many turbulence models that lower the cost of performing a DNS analysis.Some of these models are listed in [4] (LES, DES, Reynolds Stress Model, Hybrid RANS-LES, RANS,Spalart-Allmaras, etc..).
1.4.1 Basics of RANS Modelling
The present work is mainly focus in RANS Modelling where the Navier Stokes instantaneous velocityis decomposed as u = U + u′. After the decomposition, an average is taken over the equations. Forexample the RANS Momentum equation is then:
∂
∂t(ρU) +
∂
∂xj
(ρUiUj
)= − ∂P
∂xi+
∂σij
∂xj− ∂
∂xj
(ρu′iu
′j
)+ SI I (1.14)
4
Where the average of the stress tensor is then:
σij = µ
(∂Ui∂xj
+∂Uj
∂xi
)+ (−2
3µ)
∂Uk∂xk
δij (1.15)
The so called closure problem in turbulence modelling is shown here. The turbulent stress tensor,(
ρu′iu′j
)needs to be modelled in order to be able to solve the so called RANS equations. The eddy viscositymodels assume a general invariant form of the turbulent stress tensor as [13]:
− ρu′iu′j = 2µTSij −
23
δij
(ρK + µT +
∂Ui∂xi
)≡ τij (1.16)
Again, at the cost of removing the turbulent stress tensor a new unknown has been introduced: theturbulent viscosity
νT ∼ Λu′
The turbulent viscosity is now another unknown in the model. The challenge is now to model a constantrather than a second order tensor. Analogously, the turbulent heat conductivity κT or the turbulentdiffusion DT need to be modelled. At this point it is easy to understand the need of validation for theRANS-Based CFD simulations given the number of assumptions taken.
Following the derivation in [4], the eddies length scale can be estimated as Λ ∼ K3/2
ε and the veloc-ity scale u′ ∼ K1/2 finally leading to the need of an equation for the turbulent kinetic energy, K anddissipation ε.
In any introductory book for turbulence modelling it is possible to find a derivation for such equa-tions. Here the final shape of the compressible standard K− ε equations employed in ConvergeCFD areretrieved[13]:
DDt
(ρK) = τij∂ui∂xj
+∂
∂xj
(µ
Prk
∂K∂xj
)− ρε +
Cs
aSSTK (1.17)
DDt
(ρε) =∂
∂xj
(µ
Prε
∂ε
∂xj
)+ Cε,3ρε
∂ui∂xi
+
(Cε,1τij
∂ui∂xj− Cε,2ρε + CsSTK
)ε
K+ STε (1.18)
The terms STK and STε represent the interaction with the Lagrangian phase. The constant as is linked tothe turbulence generated by the Lagrangian phase. Empirically, aS has been set to 1.5, but it allows somecalibration. Finally the turbulent stress tensor is remodelled and integrated in the RANS Equations aspart of the viscosity, diffusivity and conductivity:
µT = CµρK2
ε(1.19)
DT =
(1
ScT
)µT (1.20)
κT =
(1
PrT
)µTcP (1.21)
5
The calibration of the constants Cε,i allows to modify the rates of production and dissipation of turbulentkinetic energy. Particularly, Cε,1 is intrinsically related to the production of kinetic energy and Cε,2 to thedissipation. Cε,3 is constant related to the compressibility of the fluid. It models the turbulent dissipationdue to the "dilatation" or "elasticity" of a fluid element.
1.4.2 RNG k− ε RANS Model
In principle, for all the simulations present in this report, the RNG k− ε RANS model will be employed.This model is derived using a statistical technique called Re normalization Group. The main differencebetween the standard and the RNG version of the k− ε model is that the values of the constants becomefunctions of flow parameters (length scales). However, It is not possible to say that the RNG turbulencemodel always correlates better than the standard k− ε model.
In the standard k− ε model, the eddy viscosity is found from a single turbulent length scale. In realitythis is not true. All scales of motion affect the turbulent diffusion and hence, the turbulent viscosity.The RNG procedure allows to take this into account. The RNG k − ε equations implemented in Con-vergeCFD are shown below:
D(ρk)Dt
=∂
∂xj
((µ +
µt
σk
)∂k∂xj
)+ Pk − ρε (1.22)
D(ρε)
Dt=
∂
∂xj
((µ +
µt
σε
)∂ε
∂xj
)+(cε1Pk − c∗ε2
ρε) ε
k(1.23)
c∗ε2= cε2 +
cµη3 (1− η/η0)
1 + βη3 (1.24)
η =(2SijSij
)1/2 kε→ Sij= mean flow strain rate tensor (1.25)
In the formulation of the dissipation rate equation, it is possible to observe the differences with respectto the standard model. Mainly the the parameter Cε2becomes now C∗ε2
and therefore itself depends onthe turbulent kinetic energy and dissipation present in the flow field through the function η
6
Chapter 2
SPRAY MODELLING
2.1 Introduction
The fuel injection into the combustion chamber is a classical jet break up problem within fluid mechan-ics. The complexity lies on the strong interaction (shear and normal forces) between the fuel and thesurrounding fluid (air in this case). The difficulty increases even more given the interaction betweendroplets and their evaporation.
Figure 2.1: Fuel Injection evolution.Source: Oregon State University
A correct prediction of the spray penetration, opening and evap-oration rate is of extremely importance for the follow up work ofcombustion modelling.
Given the large number of droplets (For a typical diesel engine([12]), N ∼ 107) created after the jet break up, it has been cho-sen to model the flow using a discrete phase approach. Thismeans that it is possible to avoid resolving the free surface ofeach droplet in the flow field and therefore reducing the com-putational cost per simulation significantly. The droplets are allcontained inside parcels where the characteristics of each dropletare equal. This further reduces the cost when dealing with thepossible collisions between droplets.
The reduction in the CPU cost however is then balanced by theusage of Lagrangian models with the corresponding difficultywhen interacting between the Eulerian phase. In [9] it is pre-sented the dependency of the Eulerian phase mesh on the Lagrangian phase giving rise to the so calledgrid-dependent models (as it will be shown). The dependency on the grid lies on the representation ofthe collision between parcels. The droplets are only allowed to collide if they are contained in the samecell.
The first attempt to avoid the grid dependency is given by the O’Rourke Method. However as seen in
7
[10] the CPU Cost increases with the square of the number of parcels reducing its applicability to realcases. Schmidt in [10] proposes a new method with a cost that increases linearly with the number ofparcels in the flow.
2.2 Parcel Grouping
As it was seen in the first chapter 1, a given fluid flow can be described by looking at a specific particle(Lagrangian approach) or by looking at a specific location in space (Eulerian approach).
In spray simulations, the liquid jet is represented by a continuous injection of spherical droplets with adiameter equal to the nozzle of the injector. The correct physical behavior of these droplets is then a keypart in this work.
This droplets are then subjected to drag, collision, coalescence, wall interaction and evaporation. Thisphysical behavior is entirely modeled in the Lagrangian phase.
The computational cost of tracking each particle is extremely expensive and only viable in researchprojects where the CPU cost or time is not an issue. In the present work, the droplets are groupedtogether within parcels containing identical droplets (radius, temperature, velocity, etc...). This is alsoremarked in the software manual [13] . Below the concept of parcel is pictured for clarification. To theleft it is represented the actual injected droplets and to the right their representation in the CFD code.
Figure 2.2: Parcel Grouping
8
2.3 Spray Break Up
A typical Spray is formed after the break up of a liquid jet due to the aerodynamic forces, cavitation,shearing and other instabilities growing in the jet and in the surrounding fluid. Within this work, it hasbeen chosen to model the break up process using the Kelvin-Helmholtz wave model and the Rayleigh-Taylor Model to account for the liquid core and droplet instabilities.
Kelvin-Helmholtz Break Up Model
The Kelvin-Helmholtz assumes that a liquid axisymmetric jet surface can be described by:
η = η0eikz+ωt (2.1)
The derivations presented by Reitz [15] find the dispersion relation ωKH = ωKH(kKH). Most impor-tantly, It is found the dispersion relation for the most unstable wave (ΩKH = ΩKH(KKH)), where ΩKHis the growth rate of the most unstable wave. At the same time, KKH = 2π
ΛKHprovide us with the wave-
length of the fastest growing wave [13]. The results presented below are implemented in the CFD Code.
ΛKHrp
= 9.02
(1 + 0.45Z0.5
l) (
1 + 0.4T0.7)1 + 0.87We0.87
g(2.2)
ΩKH
(ρLr3
p
σ
)0.5
=0.34 + 0.38We1
g.5
(1 + Zl) (1 + 1.4T0.6)(2.3)
Where σ is the surface tension of the fluid, rp is the drop radius of a parent parcel, Zl =√
Wel/Rel isthe Ohnesorge number and T = Zl
√Weg the Taylor number. The subindices l or g indicate whether
the properties are referred to the gaseous or liquid phase of the flow domain. The Webber Number,
We represents the ratio between the inertial forces and surface tension: We =ρlU2rp
σ . Usually in sprayregimes, the Webber number is very large1 We & 100
The Kelvin-Helmholtz model assumes that a parent parcel breaks into a new parcel of radius rc accord-ing to the following derivation:
rc = B0ΛKH (2.4)
Where B0 is model constant, candidate for calibration. The standard model assumes B0 = 0.68.
The break up process forms new parcels at a rate given by :
drdt
= − r− rc
τKH(2.5)
τKH =3.726B1rΩKHΛKH
=3.726B1r
UKH(2.6)
1The Webber number seems not be upper bounded. In a typical CFD Spray Simulation for a fuel injector Wemax ≈ 106
9
Again, B1 is model constant subjected to calibration. The standard model sets B1 = 40. However inthe CFD code [13], the default value is set to B1 = 7, which is a value in good agreement with typicalDiesel Sprays. Again we see the magnitude of the changes occurring in this constants, spanning almostchanges of one order of magnitude.
Finally, the newly created droplets are assigned a velocity given by:
v = C1ΛKHΩKH (2.7)
Where C1 is a model constant with a value of C1 = 0.188 as expressed in [13].
Rayleigh Taylor Breakup Model
The Kelvin-Helmholtz (KH) breakup model is employed inside the intact core of the fluid region. Thebreakup of the droplets beyond the intact core is linked to the Rayleigh-Taylor (RT) mechanism [9]. TheRayleigh-Taylor Break up model accounts in particular for the break up due to the rapid deceleration ofthe particle in the fluid domain due to the drag forces [13].
The wavelength and frequency of the most unstable RT waves are given by:
ΛRT = 2π
√3σ
a(ρl − ρg)(2.8)
ΩRT =
√√√√ 23√
3σ
[a(ρl − ρg
)]3/2
ρl + ρg(2.9)
The child droplet size and the characteristic break up time is then given by:
rc =πCRTKRT
(2.10)
τRT =Cτ
ΩRTv (2.11)
For each of the models, the breakup distance is now defined according to: L = Uτ.
LKH = B1r0
√ρlρg
(2.12)
LRT = CBLd0
√ρlρg
(2.13)
For the RT Mechanism to be coherent with the KH model, CBL = B1/2. This however is not typically thecase and the calibration of the liquid penetration usually requires tuning both constants independently.This is also stated in [13].
10
2.4 Droplet Drag Models
The Lagrangian phase (droplets/parcels) exchanges momentum with the surrounding fluid due to thedrag forces acting on the particles. The drag force of a droplet is modelled by taking into account it shape(Webber number dependency) and the relative velocity and state with the surrounding fluid (ReynoldsNumber).
Neglecting body forces (specifically, gravity), the force on a droplet reduced to the aerodynamic drag ofthe form:
Md∂vi∂t
= ρlV∂vi∂t
=12
ρg A f CD|Ui|Ui (2.14)
Where Ui = ui + u′i − vi (relative velocity between fluids), V is the droplet volume and finally A f = πr2d
is the droplet cross section area2. This leads to:
∂vi∂t
=38
ρg
ρlCD|Ui|Ui
r(2.15)
This is the motion equation employed in ConvergeCFD [13] at all times. The drag coefficient, CD ismodelled using the TAB/Dynamic drag model [13]. This model performs an analogy with a spring-mass system. Therefore, it allows to compute the drag force taking into account the deformation of thedroplet.
In this spring-mass system, the droplet drag is represented by the external force, the damping of thesystem by the viscosity and the restoring force by the surface tension.
F− kx− cx = mx (2.16)
The force, F, the spring constant, K and the viscosity coefficient, c are as follows:
Fm
= CFρg|Ui||Ui|
ρlr0(2.17)
km
= Ckσ
ρlr30
(2.18)
cm
= Cdµl
ρlr20
(2.19)
The drag coefficient of a sphere can be approximated by:
CD,sphere =
0.424 if Re > 1000
24Re
(1 + 1
6 Re2/3)
if Re ≤ 1000(2.20)
In order to account for the droplet distortion, the sphere drag is corrected [13] using:
2Spherical Droplet Assumption at this step, therefore V = 43 πr3
d
11
CD = CD,sphere (1 + 2.632y) (2.21)
Where droplet state is represented by the normalized value y = xCbr0
The constants CF = 13 , Ck = 8, Cd = 5 and Cb = 1
2 have been tuned empirically.
2.5 Spray Vaporizing Models
For the simulations contained in this report, the so called Frossling correlation is employed to model thechanges in droplet radius due to evaporation. This correlation dictates [13]:
dr0
dt= −
ρgD2ρlr0
BdShd (2.22)
Where r0 is the droplet radius, Bd is parameter that relates the amount of fuel vapour at the surface tothe total amount of vapour and Shd is the Sherwood Number.
The Sherwood number is itself dependent of the Reynolds number based on the droplet relative velocityto the mean flow velocity and on the Schmidt number. Here the Schmidt number is defined as:
Sc =µair
ρgasD(2.23)
Where D is the vapour diffusivity in the flow. One of the possible tuning parameters in the Vapourmodels is precisely the diffusion term which is obtained by evaluating [13]:
ρgasD = 1.293D0
(Tgas + 2Td
31
273
)n0−1(2.24)
The values of D0 and n0 are difficult to define universally for a combinations of different materials andthermodynamic state of the gas. Typically D0 = 4.16× 10−5 and n0 = 1.6 for Diesel fuels being injectedin a hot chamber.
The spray behaviour is in general less sensitive to the vaporizing model than other tuning candidatesas the break up model or the turbulence model. For this reason, we will conserve the the vaporizationmodel standard values for the present work.
12
2.6 Sandia Spray A
Spray A is the name given to the publicly available diesel spray data at Sandia’s National laboratoriesweb page in the US. Spray A is a Diesel Spray Injected into a stagnant pressurized and heated chamberto resemble as much as possible the conditions found within an internal combustion engine.
Our aim is to perform a CFD simulation of the spray and tune the break up model in order to prop-erly capture the vapour and liquid penetration. The experimental data set can be identified using thecodename: bkldaAL1
2.6.1 Initial and Boundary Conditions
The control volume is entirely a set of walls with constant fixed temperature of 900 K. The chamber isfilled with an inert gas with the following composition3:
Species ConcentrationO2 0.0%N2 89.71%
H2O 6.52%CO2 3.77%
The temperature of the gas is different from that of the walls and is initialized to : Tchamber = 829.5K.This is all in accordance to the boundary conditions reported by Sandia data set bkldaAL1.
The initial mean velocity fluctuation is urms = 0.2[m/s]. We will assume here isotropy (urms = vrms =wrms) in order to estimate the value of the turbulent kinetic energy:
k =32
u2rms = 0.06 [m2/s2] (2.25)
At the same time, the turbulent dissipation is calculated as:
ε = Cµk2
νt(2.26)
However the turbulent viscosity is still unknown. The results from Reitz and Abani in [6] are retrievedwhere the turbulent viscosity in the mixing layer between the vaporizing jet and the stagnant air isreported as:
νt = Ctπ0.5Uinj
deq
2(2.27)
Where Ct is a constant with a value of Ct = 0.0161 and deq = dnozz
√ρfuelρgas
as reported in [18]. Finally, the
3To prevent the fuel from autoignition, oxygen has been removed
13
initial turbulent dissipation in the chamber is given by:
ε = 0.0085 [m2/s3] (2.28)
Spray Geometry Representation
The spray does not have any geometrical input to the CFD Simulation. The fuel is injected in the cham-ber through a circular surface with a diameter equal to that of the nozzle: dnozz = 0.084[mm]. Thedischarge coefficient of the injector is equal to Cd = 0.89
2.6.2 Grid Configuration and Road Map
The mesh configuration chosen for the simulation is represented in the figure 2.3. There exists twoconical regions of constant refinement to properly capture the jet break up and the vapour penetration.At the same time, there is an Adaptive Mesh Refinement (AMR) updated every 10 time steps. The AMRrefines cells within the grid according to velocity levels and vapour concentration.
Figure 2.3: Mesh Set-Up
In ConvergeCFD, one "level of refinement" or "Scale" is a division of a cell into 8 smaller cells (4 Cellsin 2D, 8 in 3D) occupying the same volume as the original one. A higher level of refinement implies anexponential increase of N = 4L cells. The refinement level has a dramatic impact on the simulation timesince each refinement level will half the time step in order to fulfil the CFL criteria.
Figure 2.4: Grid Refinement Levels
14
In order to set up the simulation, one must identify which are the number of parcels required, therelationship with the Eulerian grid, and the grid dependency. In order to meet our goal and obtain gridindependent and calibrated results, the following roadmap is followed. The RNG K − ε model will beemployed.
Figure 2.5: Spray CFD Road Map
The report will start by analysing the dependency among phases, that is, the parcel number. The parcelnumber is the analogous parameter to the grid size in the Lagrangian phase. The higher this number is,the more accurate the prediction will be.
Achieving parcel number independency will allow to study the effects of the grid size in a safer manner.The uncertainty from the Lagrangian-Eulerian coupling is removed. As it will be shown, the liquidjet calibration is the last calibration step. The turbulence model affects both the liquid jet and vapourpenetration whereas the break-up model will have a negligible effect on the vapour phase (at least inthe range considered).
2.6.3 Parcel Number: Lagrangian Phase Numerical Dependency Study
The number of injected parcels is itself dependent on the cell size and therefore must increase withfiner grids. Typically the number of parcels at a given location (cell) at a given time step should remainconstant under further refinement.
In the case of halving the volume of the cells4, then the total number of parcels should be twice the initialnumber (assuming that the actual collision rate gives a good representation of the particle behaviour).Given a cell of side length a, the volume is given by: Vcell = a3. If the volume is halved, then the cellside length becomes:
a′ =(
12
)1/3a ≈ 0.79 a (2.29)
4Increasing the number of cells by a factor of 2
15
Increasing the number of cells by two5, the number of parcels should also be multiplied by a factor of×2. The higher the number of parcels the more accurate the Lagrangian phase is represented.
The previous relationship provides a scaling relationship between the grid and the parcel number. How-ever, in order to asses how many parcels there has to be initially, a study of the effect of the parcel numbermust be carried out. Below, keeping the cell size constant, a sweep over the number of parcels injectedis plotted.
Time [ms]0 0.5 1 1.5 2
Liqu
id P
enet
ratio
n [m
m]
-20
0
20
40
60
NP = 1.2× 103
NP = 9.4× 103
NP = 2.5× 104
NP = 7.5× 104
NP = 6× 105
NP = 1× 106
Time [ms]0 0.5 1 1.5 2
Vap
our
Pen
etra
tion
[mm
]-5
0
5
10
15
Parcel Number []104 105 106
Avg
. Liq
uid
Pen
etra
tion
[mm
]
5
6
7
8
9
Std
. dev
iatio
n fr
om m
ean
[mm
]
0
0.5
1
1.5
2
Average Liquid PenetrationAccepted IndependenceStd. Deviation from Mean
Figure 2.6: Parcel Number Study (Base Grid a = 4mm)
The number of parcels affects both the vapour penetration and liquid penetration6. In the Eulerian phase(vapour), the results seem to overpredict the penetration for a low number of parcels. This seems to berelated to the inertia of bigger parcels sizes and less number of collisions. A lower number of parcelsimplies fewer collisions and therefore a major part of the momentum is spent only to counteract thedrag.
The liquid penetration is a very noisy signal for low parcel number. This is because the low numberof parcels leads to stronger but less frequent collisions. A higher number of parcels predicts a highernumber of collisions. A higher number of collisions leads to a much more uniform description of thespray.
It is particularly interesting how for a large number of parcels, the vapour penetration decreases, how-ever the liquid penetration remains almost constant in average. Extremely low parcel numbers (NP =1.2× 103) are believed to have a lower liquid penetration due to faster evaporation (proportional to sur-face area of droplets). A faster evaporation also leads to fewer collision which also points in the directionof larger vapour penetration.
5The base grid side length is 79% of the original one6Vapour penetration is defined by integrating the mass in the domain from the nozzle to the tip of the jet up to account 99%.
The liquid penetration is however defined based on the 95%
16
In order to better understand this phenomena, one can plot the turbulent kinetic energy levels for dif-ferent parcel numbers. The following plot reveals a much stronger interaction for high parcel numbersbetween the spray and the gas. Therefore the turbulent kinetic energy levels are higher. Specially,around the injector nozzle region (top of the figure). In Fig.2.7 the turbulent kinetic energy at 0.3 and 1.5ms is shown. Note that the contours are shown in logarithmic color scale.
Figure 2.7: Colors in Log Scale. Turbulent Kinetic Energy Levels for NP = 1.2× 103 (left) and NP =1× 106 (right) at t = 0.3 [ms] (top row) and t = 1.5 [ms] (bottom row). Base Grid a = 4mm
The larger number of parcels and therefore collisions will also open the spray which decreases thevapour penetration.
17
Grid Label Cell Size (a) Parcel Number Total Cells (End Of Simulation) Wall Clock Time1 5.04 3.75× 104 8.5× 104 0.18 h2 4.00 7.50× 104 1.66× 105 0.48 h3 3.17 1.50× 105 4.4× 105 1.37 h4 2.52 3.00× 105 7.1× 105 4.49 h5 2.00 6.00× 105 1.1× 105 10.36 h6 1.59 1.20× 106 3.7× 106 27.54 h
Table 2.1: Grid and Parcel Number Link. Cell number and Simulation Time
2.6.4 Grid Study and Collision Mesh
The dependency in the grid will be analysed monitoring our main calibration values which are thevapour and liquid penetration. The number of parcels is related to the grid size as given by Eq.2.29. Anincrease by ×2 in the number of cells requires doubling the number of parcels7.
The previous table is used to design the grid dependency study. The vapour and liquid penetrationfigures are shown below.
Time [ms]0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6
Vap
our
Pen
etra
tion
[mm
]
-20
0
20
40
60
a = 5.04 mma = 4.00 mma = 3.17 mma = 2.52 mma = 2.00 mma = 1.59 mm
Time [ms]0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6
Liqu
id P
enet
ratio
n [m
m]
-5
0
5
10
15
Figure 2.8: Grid Convergence
The vapour advances faster at the very beginning of the simulation for fine grids. Later its speed lowersand coarser grids keep on penetrating the chamber further distances. For the case of the liquid penetra-tion, there is a clear peak, overshoot in the very early stage until the liquid phase achieves equilibrium.The peak is more pronounced for finer grids whereas coarser grids tend to smooth this initial overshoot.
It is interesting to note, that the overshoot in the liquid phase (at around 0.1 [ms]) also converges with
7Wall Clock Time based on 16-core Workstation, CPU:Intel Xeon E5-2698V3
18
finer meshes. There is almost no differences in the grids 4 to 6.
The computational cost of these simulations scales quickly due to the transient nature of the previoussimulations. The main cost rocketing parameters are the time step based on the so called collision mesh.During an early stage of the analysis, it would be interesting to observe how the results vary using acollision mesh.
For this purpose, an analysis of the behaviour of the spray with and without this approach is performed.For the collision mesh the distance computed is always 8 times smaller (3rd level of refinement, 23 =8) than the cell in which the parcel is found. In the following picture it is plotted the average liquidpenetration versus the number of cells in the grid (taken into account the increase in parcel number atthe same time) and the Vapour penetration. Grids 2 to 5 are considered in this analysis.
Time [ms]0 0.5 1 1.5
Vap
our
Pen
etra
tion
[mm
]
0
10
20
30
40
50
60
NTC Collision MeshNo Collision Mesh
Time [ms]0 0.5 1 1.5
Liqu
id P
enet
ratio
n [m
m]
0
5
10
15
20
25
30Collision Mesh on Grids 1-4
NTC Collision MeshNo Collision Mesh
Figure 2.9: Convergence based on NTC Collision Mesh
Let us now define LP as the mean normalized liquid penetration (normalized with respect to the pen-etration of grid #2). In the horizontal axis, N = Cellsi/Cells1 which is the increment in the total cellswith respect to cells in the coarsest grid. Regarding the vapour penetration, it is known from the analyt-ical results of Reitz [6] and the correlations given by Hiroyasu and Arai [8] that the vapour penetration:s ∝ t1/2. In this fashion, a least squares fit is performed over each of the CFD simulation Vapour pene-tration. The convergence of the parameter mi is monitored.
s = mit1/2 (2.30)
In Fig.2.9 It is possible to observe that the collision mesh affects mostly to the liquid phase. This isnot surprising since the collision mesh only acts on the Lagrangian phase which is the liquid jet. Thecollision mesh allows to have a better convergence and smoother results.
The convergence of our monitoring parameters can be assessed by looking at Fig.2.10. Not using acollision mesh clearly affects to the convergence of the simulations on finer grids. Not using a collision
19
mesh, parcels can only collide with those within the same cells. In finer grids, the number of parcels percell will be reduced and therefore lowering the number of collisions. On the other hand, on coarser gridsthe Eulerian phase cannot be properly simulated. At the same time, parcels can collide with parcels inanother cell even if they are actually colliding.
N
2 4 6 8 10 12 14
LP
0.5
0.6
0.7
0.8
0.9
1
1.1Liquid Penetration
NTC Collision MeshNo Collision Mesh
Grid Number (coarse to fine)1 2 3 4
LP
0.4
0.5
0.6
0.7
0.8
0.9
1
NTC Collision MeshNo Collision Mesh
N
2 4 6 8 10 12 14
mi
0.5
0.6
0.7
0.8
0.9
1
1.1Vapour Penetration
NTC Collision MeshNo Collision Mesh
Grid Number (coarse to fine)1 2 3 4
mi
0.8
0.85
0.9
0.95
1
1.05
1.1
NTC Collision MeshNo Collision Mesh
Figure 2.10: NTC Collision Mesh (black) effect on grid convergence
2.6.5 Calibration
In the previous figures, the grid has been shown to achieve convergence for our criteria. However,given the nature of our modelling approach, the RANS turbulence model as well as the Lagrangianspray model contain several constants which cannot be determined analytically. It is expected that ourmodel requires of certain level of calibration given that the current models cannot reproduce completelythe physics involved in this simulation. This is clearly shown when comparing the CFD results withthe experimental values of the Sandia Spray A, data set: bkldaAL1. This is further evidenced in the nextfigure:
By looking at the results of Fig.2.10 and Fig.2.9 together with the wall clock times shown in table2.1, itwas decided that grid #3 provides sufficient accuracy at a reasonable CPU cost.
Turbulence Modelling
Following our road map specified in Fig.2.5, the RNG K − ε model will be studied for the previousconditions using grid #3. S. B. Pope in [16] achieved jet calibration modifying only the turbulent modelconstant Cε1 which so far has taken the standard value for the RNGK− ε Model. For the present work,both the constants Cε1 and Cε2 will be analysed together with their effects on the liquid and vapourpenetration.
20
0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6-20
0
20
40
60
Experimental DataCFD Grid #3
-0.2 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6-5
0
5
10
15
Experimental DataCFD Grid #3
Figure 2.11: Converged CFD and Experimental Data
The domain of study cannot be well justified before running the simulation. It may happen that thedomain of study becomes also part of this iterative process. The goals to achieve are in based on themean liquid penetration and maximum vapour penetration.
The RNG K− ε equations implemented in ConvergeCFD are shown below:
D(ρk)Dt
=∂
∂xj
((µ +
µt
σk
)∂k∂xj
)+ Pk − ρε (2.31)
D(ρε)
Dt=
∂
∂xj
((µ +
µt
σε
)∂ε
∂xj
)+(cε1Pk − c∗ε2
ρε) ε
k(2.32)
c∗ε2= cε2 +
cµη3 (1− η/η0)
1 + βη3 (2.33)
η =(2SijSij
)1/2 kε
(2.34)
Let us now roughly estimate the magnitude of the production and dissipation terms:
Pk ≡ u′iu′j∂Ui∂xj≈ 2νtSijSji ∼ Cµ
k2
ε
( Uinj
dnozz
)2
(2.35)
ε ∼ k3/2
dnozz(2.36)
(2.37)
From where it is possible to estimate the relative importance of the terms as:
Pkε∼ Cµ
U2inj
k(2.38)
21
Following the last equation, one could in principle say that k ∼ U2inj and therefore, Pk
ε ∼ Cµ. Thishowever is a rather simple analysis which can only provide some rough guidance for the calibration.
The coefficient Cε1affecting to the production term in the dissipation rate equation will lead to smallerand slower changes in the dissipation in the flow field. If Pε < 1 then: (Cε1P− Cε2 ρε) becomes lessnegative for increasing values of Cε1 . This in principle would point out in the direction of increasing thevapour penetration distance as the dissipation does not feel fast enough the perturbation caused by thespray. The same argument for lower values than the standard ones of Cε2are also valid. This approachwas also suggested by S.B Pope [16]. In this publication Pope modifies8 Cε1such that Cε1=1.6 in orderto solve the turbulent jet penetration under prediction It is possible to think of this coefficients as "flowdamping" coefficients.
Based on the previous analysis, the following experiments will be carried out:
Design Point 1 2 3 4 5 6Cε1 1.2 1.2 1.2 1.5 1.5 1.5Cε2 1.2 1.5 1.8 1.2 1.5 1.8
Vapour [mm] 50.93 40.84 35.89 94.36 55.37 43.36Liquid [mm] 10.0 7.22 5.63 24.6 11.51 9.09
Table 2.2: RNG K− ε Calibration Domain
Time [ms]0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6
Vap
our
Pen
etra
tion
[mm
]
-20
0
20
40
60
80
100
C1 = 1.2 C2 = 1.2
C1 = 1.2 C2 = 1.5
C1 = 1.2 C2 = 1.8
C1 = 1.5 C2 = 1.2
C1 = 1.5 C2 = 1.5
C1 = 1.5 C2 = 1.8
Std Values
Time [ms]-0.2 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6
Liqu
id P
enet
ratio
n [m
m]
-5
0
5
10
15
20
25
30
C1 = 1.2 C2 = 1.2
C1 = 1.2 C2 = 1.5
C1 = 1.2 C2 = 1.8
C1 = 1.5 C2 = 1.2
C1 = 1.5 C2 = 1.5
C1 = 1.5 C2 = 1.8
Std Values
Figure 2.12: Turbulence Model Analysis
The time evolution of the spray given the previous constants values can be visualized in Fig.2.12. Mainlyat this step, the goal is to match as good as possible the vapour penetration curve. The values cor-responding to (Cε1 , Cε2) = (1.5, 1.5) seem to match the best the vapour penetration without stronglycompromising the liquid penetration curve.
The liquid penetration peak at approximately 0.1 ms seems to be independent of the turbulence model8This paper is related to the standard k− ε model
22
(at least on the range considered). It will always overshoot for all the turbulence model combinationsstudied. The magnitude of the peak can be correlated with increasing turbulence production (which canbe identified by lowering the cε2 or increasing Cε1 ).
One must note that the combination of (Cε1 , Cε2) = (1.5, 1.2) that is, increasing the production ratewhile decreasing the dissipation rate constants critically increases the vapour penetration and liquidpenetration values.
Below, the turbulence and dissipation levels at the end of the simulation is shown:
Figure 2.13: Turbulent Kinetic Energy Levels at t=1.5 [ms]
Figure 2.14: Turbulent Dissipation Levels at t=1.5 [ms]
It is possible to appreciate how lowering both Cε1and Cε2 (left most column) has only a scaling effecton the turbulence penetration. The structure of the spray remains similar to the one computed usingstandard values. It is possible to conclude that by increasing the Cε2constant, the destruction of dissi-pation is increased and therefore the turbulent kinetic energy is larger mixing more the flow with itssurroundings. As a consequence, a wider spray appears.
At the right most column, the value of Cε1 is then increased while keeping Cε2at its lowest value consid-
23
ered. This modification leads to a flow field more turbulent where the flow is relatively slow to adapt tothe changes produced by the fuel spray. Besides, the higher levels of turbulence and therefore smallerscales present lead to believe that the structure of the flow field is different enough to believe this mightbe a result of a numerical artefact. Mainly the grid is not fine enough for this level of turbulence.
Figure 2.15: Std RNG Vs. Calibrated For Vapour Penetration, t = 1.5[ms]
The last figure shows the comparison between the standard spray and the calibrated for vapour pene-tration one. A more elongated and narrow cone is shown in the figures.
24
Vapour Jet Asymptotic Slope
One of the main difficulties in the simulation is to match the actual asymptotic slope of the vapour jet.Following the results of [8], one could plot the vapour penetration curve against the square root of thetime. This helps identifying the asymptotic slope of the penetration as shown below:
√
t [ms0.5]
0 0.2 0.4 0.6 0.8 1 1.2 1.4
Vap
our
Pen
etra
tion
[mm
]
0
20
40
60
80
100
120
EXPC1 = 1.2 C2 = 1.2
C1 = 1.2 C2 = 1.5
C1 = 1.2 C2 = 1.8
C1 = 1.5 C2 = 1.2
C1 = 1.5 C2 = 1.5
C1 = 1.5 C2 = 1.8
Std Values
Figure 2.16: Asymptotic Tendency of Vapour Penetration Curves. (- -) Asymptotic Tendency
This can be understood as a multiobjective search algorithm. On one hand the penetration curve of thespray should be matched as good as possible but on the other hand, the tendency of the curve at largetimes should be understood as measure of the quality of the calibration too.
However, given the unlikelihood to meet both criteria, a higher weight is given to the goal of matchingthe penetration as good as possible during the simulated time. In practise, the fuel sprays deal withhigh temperature combustion and the distance that they traverse before the fuel ignites is relativelylower than those spanned in this figures.
25
Liquid and Vapour Simultaneous Matching
Using the previous data, it is possible to extract even more information. From a mathematical point ofview, there exists 7 design points which could in principle be used to reproduce a response cubic surface.This is an interpolating surface using MatLab cubic method.
Vapour Penetration [mm]
Cǫ1
[]1.2 1.25 1.3 1.35 1.4 1.45 1.5
Cǫ2 []
1.2
1.3
1.4
1.5
1.6
1.7
1.8
40
50
60
70
80
90Liquid Penetration [mm]
Cǫ1
[]1.2 1.25 1.3 1.35 1.4 1.45 1.5
Cǫ2 []
1.2
1.3
1.4
1.5
1.6
1.7
1.8
10
15
20
Cǫ1
[]1.2 1.25 1.3 1.35 1.4 1.45 1.5
Cǫ2 []
1.2
1.3
1.4
1.5
1.6
1.7
1.8
sv = 56.7 [mm]
sl = 10.1 [mm]
Std. RNG
Figure 2.17: Set of Possible Solutions for the calibration. Extracted contour lines from response surface.
Ideally, the calibration lines for the liquid and vapour penetration would cross each other and wouldideally give us a set of values for the turbulence model that could be used for both liquid and vapourphases calibration. It is advised to the reader that the previous response surfaces originates from rela-tively small number of points and therefore interpolated values might not be correct.
By measuring the vapour penetration at 0.2, 0.4 and 1.5 ms one can asses the accuracy of calibrationconstants. Let us define an error metric as:
error ≡∑ |sv − svexp | → at t = [0.2, 0.4, 1.5] ms (2.39)
The next figure will show the value of the error metric for all design points and standard calibration.
26
Design Point1 2 3 4 5 6 STD
erro
r
5
10
15
20
25
30
35
40
45
Figure 2.18: Error Metric in order to choose Calibration Constants
From now on, the set of constants (Cε1 , Cε2) = (1.5, 1.5) will be chosen in our simulations as they arethe ones to predict the best the vapour penetration curve in general. The comparison with experimentaldata might be visualized in a better way below:
Time [ms]0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6
Vap
our
Pen
etra
tion
[mm
]
-20
0
20
40
60
C1 = 1.5 C2 = 1.5
Std. RNGExperimental Data
Time [ms]-0.2 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6
Liqu
id P
enet
ratio
n [m
m]
0
5
10
15
20
C1 = 1.5 C2 = 1.5
Std. RNGExperimental Data
The same misprediction of the spray penetration using the standard RNG K − ε model is shown in[17]. This seems to be a problem related to the RANS models. In [19] LES Simulations resolving smallerturbulent scales seem to be the only possibility to accurately predict the vapour penetration of a fuelspray [19]. The computational cost is extremely high (between 25 to 30 million cells are needed justfor the spray). The alternative is to use a calibrated RANS Model. It predicts more accurately thevapour penetration (even though the matching is not perfect), but the set-up of the turbulence modelwill be valid only for the current operating conditions (chamber pressure, temperature and backgroundturbulence levels mainly).
Argonne National Laboratories (USA) and ConvergentScience (USA) show in [11] the deficits of RANSmodelling for spray simulations. The liquid penetration is usually not an issue after break up calibration(except for the overshooting peak in the beginning) whereas the vapour penetration is usually not well
27
predicted. The more CPU expensive solution but much more accurate without requiring any calibrationis to perform an LES simulation.
By using a more accurate turbulence model (LES) a better prediction of the turbulence levels is obtainedand therefore a much better spray penetration prediction. The cost of an LES is extremely high andtypically not in the range of industrial application.
2.6.6 Break Up Model
The break up model employed in the simulations has been previous described in Sec.2.1. In order tocalibrate the mean liquid penetration, the size and time constants affecting the primary break up arehere studied using grid #4 as described Table2.1.
Let us recall here the equations describing the characteristic break time and droplet radius:
rc = B0ΛKH (2.4)
τKH =3.726B1rΩKHΛKH
=3.726B1r
UKH(2.6)
drdt
= − r− rc
τKH(2.5)
The standard break up model according to [9] dictates that B0 = 0.68, B1 = 40. However the range ofvalues that B1 can take depends on the fuel as well as operating conditions. According to [13], the timeconstant B1 can take values between 5 and 100.
The constant B0 will affect to the size of the children parcels formed after break. This constant thereforewill affect to the size more than the speed at which the break up occurs. However one can rearrange theprevious equations to lead to:
1ΛKHΩKH
drdt
= − r− B0ΛKH3.726B1r
= r = r/ΛKH = −r− B0
3.726B1r=
1ΩKH
drdt
(2.40)
In principle, one would be interested in noting the most sensitive parameter. In this sense one can definea coordinate system based on the values of B0 and B1. The biggest changes, that is, the steepest slope ofthe response surface to 1
ΩKHdrdt can be found by computing the gradient over this coordinates as:
(∂
∂B0~u0 +
∂
∂B1~u1
)1
ΩKH
drdt
=1
3.726B1r~u0 +
(1
3.726B21− B0
3.726B21 r
)~u1 (2.41)
Using the previous equation, one must first identify the order of magnitude of r = r/ΛKH . For thatpurpose, the results from Eq.2.2 are here retrieved:
r =ΛKH
r= 9.02
(1 + 0.45Z0.5
l) (
1 + 0.4T0.7)1 + 0.87We0.87
g(2.2)
In principle, according to the previous equation r ≈ 10−4. Based on this number it would be possible todesign a set of experiments following the gradient vector equation in Eq.2.41 as:
28
B0
0 1 2 3 4 5 6 7 8
B1
0
1
2
3
4
5
6
7Max Slope From Std. Values
Steepest PathStd. Values (Start Point)Considered Combinations
time [ms]0 0.05 0.1 0.15 0.2 0.25 0.3
Liqu
id P
enet
ratio
n [m
m]
0
2
4
6
8
10
12
14
16
18
[B0,B1] = [0.68,7.00]
[B0,B1] = [2.62,6.53]
[B0,B1] = [4.56,5.36]
[B0,B1] = [6.50,2.70]
Figure 2.19: Break Up Model Sensitivity on Liquid Phase
It is possible to observe a strong variation in the overshoot of the liquid jet and therefore on the maxi-mum penetration achieved by the liquid spray. Increasing the constant B0 while decreasing the constantB1 increases the size of the particles along the jet. This is mainly due to Eq. 2.4 which determines thesize of children parcels after break up. The jet breaks but the new particles are still quite big. Recallthat bigger spheres suffer less drag9 and therefore a higher overshoot is achieved. This is however justan academical exercise since in our aim to match the experimental data, the interest is to decrease suchovershoot.
Figure 2.20: Calibrated Model Turbulence Model. Time t = 0.1[ms]. Upper Figure [B0, B1] = [0.68, 7],Lower Figure [B0, B1] = [6.50, 2.70]. (Note non linear color scale to properly identify the changes)
The red color in the particles identify a larger parcel radius which is approximately 10 times larger thanthe one with standard values shown in the upper figure. This seems to be in agreement with B0 valuewhich is approximately 10 times larger in the lower row.
Following the results at the left of Fig.2.19, in principle one should traverse the curve in the oppositedirection (that is decreasing B0 and increasing B1). The size constant B0 cannot be reduced much more,
9Drag model is linked to sphere drag model. Higher Reynolds Number is linked to lower drag values
29
in exchange to that, the increase in the constant B1 is mainly the only possibility that help reducing theovershooting. Below an analysis of this two constants is presented:
Time [ms]0 0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.16 0.18 0.2
s l [mm
]
0
5
10
15Varying B
0
B0 = 0.10
B0 = 0.68
B0 = 2.05
B0 = 4.00
Time [ms]0 0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.16 0.18 0.2
s l [mm
]
0
5
10
15Varying B
1
B1 = 4.00
B1 = 7.00
B1 = 12.00
B1 = 20.00
B0
0 0.5 1 1.5 2 2.5 3 3.5 4
Max
Liq
uid
Pen
etra
tion
[mm
]
10.5
11
11.5
12
12.5
B1
4 6 8 10 12 14 16 18 20
10.5
11
11.5
12
12.5
Figure 2.21: Break Up Model Effects on Liquid Phase
The main effect of the variation of the constants is in the size of the droplets. A bigger value of B1and B0 leads to a smaller overshoots (around t = 0.07 [ms]) due to a slower decrease in the dropletsize. The larger the surface area of the droplets the faster the evaporation and therefore the mean liquidpenetration is achieved in a smoother way and earlier time. At the same time, this is reflected in thevapour penetration curve as:
Time [ms]0 0.5 1 1.5
s v [mm
]
0
10
20
30
40
50Varying B
0
B0 = 0.10
B0 = 0.68
B0 = 2.05
B0 = 4.00
Time [ms]0 0.5 1 1.5
s v [mm
]
0
10
20
30
40
50Varying B
1
B1 = 4.00
B1 = 7.00
B1 = 12.00
B1 = 20.00
B0
0 0.5 1 1.5 2 2.5 3 3.5 4
Max
Vap
our
Pen
etra
tion
[mm
]
42
42.5
43
43.5
44
44.5
45
B1
4 6 8 10 12 14 16 18 20
42
42.5
43
43.5
44
44.5
45
Figure 2.22: Break Up Model Effects on Vapour Phase
It is possible to identify different key features from the modification of both constants. Reducing theconstant B0 achieves the creation of much smaller particles with respect to the standard model. Moreinterestingly is the physical process involving high B1 values. This constant does not generate so manychild parcels and empties the core of the spray thus leading to also less overshooting. The increase inthe characteristic time maintains big parcels with high number of droplets within them without actuallycolliding with each other.
30
Figure 2.23: Particle Size. Time t = 0.1[ms]. Upper Row [B0, B1] = [0.68, 7], Middle row [B0, B1] =[0.1, 7], Bottom Row: [B0, B1] = [0.68, 20]
The best calibration found so far employs a lower value than the standard one for the Break Up sizeconstant B0 = 0.1 and keeps constant the standard time constant. Correcting the overshoot was notpossible.
Time [ms]0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6
Vap
our
Pen
etra
tion
[mm
]
-20
0
20
40
60
ExperimentalCalibrated RNG and KHRT
Time [ms]-0.2 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6
Liqu
id P
enet
ratio
n [m
m]
-5
0
5
10
15
ExperimentalCalibrated RNG and KHRT
Figure 2.24: Calibrated Turbulence and Break Up Model. [cε1 , cε2 , B0, B1] = [1.5, 1.5, 0.1, 7]
31
2.6.7 Correlations and Analytical Estimates
The Navier-Stokes equations can be simplified in order to estimate the penetration of a jet. A self-similarjet penetration can be obtained following the derivation of [6]. This self-similar approach is dependenton the liquid properties during the injection.
Uv(x, r) = min
Uinj,3U2
injd2eq
32νtx(
1 +3U2
injd2eqr2
256ν2t x2
)2
(2.42)
Eq. 2.42 depends on the equivalent diameter which is a function of the liquid density which is ill definedas the temperature changes with the penetration distance. For the next plot a value of fuel liquid densityof ρl = 425Kg/m3 will be used in accordance to the material properties at 600 Kelvin. This tempera-ture was chosen as the average between the injection temperature and the chamber temperature. Theequivalent diameter and the turbulent viscosity are defined as:
deq = dnozz
√ρlρg
(2.43)
νt =12
Ct√
πUinjdeq (2.44)
Where Ct was reported by Abraham [7] to be Ct = 0.0161. For the present case (Sandia Spray A) it hasbeen found that a value of Ct = 0.0191 gives the best possible match with the experimental data for theself similar jet.
The density of the fuel changes dramatically due to the fuel thermal expansion. The density changes arerelated to the coefficient of thermal expansion as:
β = −1ρ
∂ρ
∂T(2.45)
Temperature [K]0 100 200 300 400 500 600 700
Liqu
id D
ensi
ty [K
g/m
3]
300
400
500
600
700
800
900
1000C
12H
26 Material Properties
Figure 2.25: Fuel Material Properties
32
Finally, using Eq.2.42, the vapour penetration of the spray tip can be computed as:
t(x) =∫ x
x=0
dxU(x, r = 0)
(2.46)
Besides, Hiroyasu and Arai [8] worked on the experimental correlation of fuel sprays discharging in ahot, inert and stagnant chamber. They found the following correlations which serve to appreciate theimportance of some typical spray parameters as the discharging pressure and the chamber temperature(gas density).
s =
0.39√
2∆Pρl
t 0 ≤ t ≤ tb
2.95(
∆Pρg
)0.25 √dnozzt tb ≤ t
(2.47)
The vapour penetration prediction through the different models are plotted in Fig. 2.26 together withthe penetration prediction given by the calibrated CFD Model.
Time [ms]0 0.5 1 1.5
Vap
our
Pen
etra
tion
[mm
]
0
10
20
30
40
50
60
Experimental PenetrationSelf-Similar JetHiroyasu and AraiCFD Calibrated
Figure 2.26: Self-Similar Jet, Statistical Correlation and CFD Prediction
33
Real Injection Profiles
Real Injection Profiles are non constant. In order to handle them it is possible to use the Duhamelsuperposition principle as shown in the following derivation.
Following the derivation given in [6], it is possible to define a so called effective injection speed to correctEq. 2.42. In this fashion, it is possible to express the equation as:
Uv =3Ue f f deq
Kif x > xb (2.48)
At this point, it is interesting to insert a momentum balance to account for the drag of the droplets as:
mdvdt
=12
ρgπD2
4CD(u− v)|u− v| (2.49)
Where v represents the velocity of the gas and u the velocity of the droplet. Under the assumption oflow Reynolds Number, the drag of a sphere can be approximated as:
CD =24Re
(2.50)
Being possible now to rewrite Eq. 2.49 as:
dvdt
=
(18µ
ρl D2CDRe
24
)(u− v) =
1τv
(u− v) (2.51)
Since the ratio CD Re24 is basically a constant near unity as shown in Eq. 2.50, the previous equation can be
integrated to lead to:
v = u(1− e−t/τv) (2.52)
Looking back now at non-constant injection profiles, the effective injection velocity will take the shapeof Eq. 2.52. According to [6], the variations in the injection profile can be accounted by Duhamel super-position as:
Ue f f = Uinj(t = 0) +n
∑k=1
A(x, t− tk)(∆Uinj)k (2.53)
A(x, t− tk) = 1− exp(− t− tk
τv,k
)(2.54)
At the same time, in [6] it is proposed to generalize the response time as τvk = Stx
Uk. St stands for the
Stokes number.
Now it is possible to rewrite the previous equations in integral for by:
(∆Uinj)k ≡(∆Uinj)k
∆tk∆tk (2.55)
34
Meaning that:
Ue f f (x, t) = Uinj(x, t = Break) +∫ τ=t
τ=Break
[1− exp
(− t− τ
τav
)](dUdτ
)dτ (2.56)
In [6] it is proposed to use an average response time based on:
τav =Stdeq
Uav(2.57)
Where deq is given by Eq. 2.43 and the average velocity Uav = Uinj. The Stokes Number is a constant thatrequires some fine tuning. Here a value of St = 50 as employed in [6] has been used. Error minimizationby varying the Stokes number is left for future work.
The previous method allows to have a non-uniform injection profile which can handle a realistic injec-tion profile as it really happens in fuel injectors.
35
Chapter 3
Case Study: Diesel ULPC Piston BowlOptimization
3.1 Introduction
Diesel engines are commonplace across the industry due to their economic and reliability advantages.In this case-study, a multi-objective genetic algorithm is employed to optimize the injection strategy, fuelspray orientation, and piston bowl geometry using Converge CFD and CAESES, for the flow simulationand automated optimization respectively. The goal of the optimization is to reduce NOx and SOOTemissions.
The Ultra Low Particulate Combustion (ULPC) Piston Bowl is analyzed in this study. These types of ge-ometries increase the local air to fuel ratio, improving the mixing and finally leading to lower pollutantemissions. For the analysis, a variable parametric geometry of the piston bowl with volume control isgenerated in CAESES. CAESES drives the automated execution of Converge CFD computations includ-ing the consideration of physics parameters such as injection timing and spray orientation.
The current optimization strategies focus on the geometry or in the injection strategy alone. The effectsof spray orientation are then not well considered for different types of geometries. In [21], it is analysedthe effects of the bowl radius (in plan view) and depth on the emissions obtaining a reduction of 60% inPM. In [22] the effects of the lobe diameter with respect to the depth are analysed. The results concludethat a higher temperature in the cylinder reduces the final emissions of Soot, however it increases theamount of NOx. The injection strategy and fuel spray orientation have a strong impact on the pollutantsconsidering that this are ultimately determined by the thermodynamic state of the piston and thereforethe combustion process. In [23] a 4D DOE analysis is carried out. The spray angle and the start of injection(SOI) analysed together with changes in the diameter of the bowl and lobe radius. It is concluded mostrelevant parameters are the diameter of the bowl, the spray angle and start of injection. The number ofnozzles in the injector has almost no effect on the emissions.
Summarizing, there is evidence enough in the literature to substantiate an optimization that considersthe injection timing, cone angle and geometrical parameters together.
36
3.2 ULPC Piston Bowl
3.2.1 Pilot and Post Injection
The pilot injection is meant to produce and initial flame ahead of the injector in order to heat the sur-rounding air. The goal with the pilot injection is to increase the chamber temperature at TDC beyond theauto-ignition temperature of Diesel1. At the same time, a uniform increase of the chamber temperaturewill help to avoid engine "knock"2 as well as reduce the evaporation time reducing wall films. This inthe end leads to lower emissions.
Figure 3.1: Pilot and Post Injection Examples. Picture taken from www.dieselnet.com
In [29], the effects of the dwell time between the pilot, main and post injections is deeply analysed. Thistriggers another design variable in the optimization process of ICE. It is reported that late injectionstogether with high dwell times reduce the formation of NOx. Soot and unburned hydrocarbons seem tobe larger in this case therefore making one of the optimization objectives be agains the other one. Thiswill be further discussed within the optimization subsection.
3.3 Exhaust Gas Recirculation
The Exhaust Gas Recirculation, EGR is way typically found in diesel cars to recirculate part of the ex-haust gases to displace part of the air entering the chamber during the intake stroke. Diesel runningalways lean achieves higher temperatures than petrol engines. The main consequence is a higher NOxproduction. In order to reduce the emissions, the exhaust gases reduce the amount of fresh air thus slow-ing down the combustion process and lowering the maximum temperatures achieved in the chamber.Lowering the temperature decreases the NOx at the expense of increasing inefficiencies and therefore,increasing the emissions of carbon particles (SOOT).
In order to asses the composition of the gases in the chamber at IVC, one can use a 1-step combustionchemical reaction which provides sufficiently accuracy for this calculation.
C7H16 + ξ
(O2 +
7921
N2
)−→ α1CO2 + α2H2O + α3N2 (3.1)
The solution leads to ξ = 11, α1 = 7, α2 = 8 and α3 = 82.761. This is the so called stoichiometric ratiobetween fuel and fresh air (here consider to be a unique composition of oxygen and nitrogen). In orderto calculate the species mass fractions χi, the next step is to calculate the weight of each of the productsmi of the previous equations.
1Diesel Auto Ignition temperature, depending on the exact composition and thermodynamic state is approximately 2202localized uncontrolled fuel ignition
37
CO2 H20 N244 18 28
Table 3.1: Molecular Weights in [g/mol]
The mass fraction of each species under stochiometric conditions is given by:
χi =αimi
∑ mi(3.2)
For a given EGR mass fraction ψ, one defines the mass fraction of exhaust gases within the chamber bycomputing:
χi = ψχi (3.3)
And the mass fraction of pure air will be given by:
χO2 = (1− ψ)× 0.21 (3.4)
χN2 = (1− ψ)× 0.79 (3.5)
The current project will deal with an EGR fraction of ψ = 0.25
3.4 CFD Methodology
Internal combustion engines for utility or heavy-duty vehicles typically repeat a 4-stroke cycle. TheIntake stroke is the first one occurring in the cycle. Fresh Air enters the chamber when the piston is atTDC (Top-Dead-Center, closest to the valves). The cylinder then expands dragging the air along with it.After reaching the bottom, it starts going up again. At some point when going up again, the valves fullyclose. This instant of time is called IVC (Inlet Valve Close).
From IVC to EVO (Exhaust Valve Open) is considered to be the Power Stroke. When the piston is closethe valves again at TDC (top dead center), the injector releases the spray of fuel and the diesel will autoignite due to the chamber pressure and temperature. The release of energy will push the piston downproducing mechanical work.
Slightly before getting to BDC (bottom dead center), the exhaust valves will open again allowing torelease the burnt gases and finishing the cycle.
Figure 3.2: Internal Combustion Engine Cycles
38
3.4.1 Computational Model: Sector Mesh and Flow Initialization
The power stroke spans from the start of the compression until almost the end of the expansion processof the internal combustion engine. The analysis of the power stroke can be strongly simplified with-out decreasing strongly the accuracy of the simulation by employing a periodic control volume. Thesimplification can be better understood by looking at Fig.3.3 and Fig.3.4
Figure 3.3: SOI (Start Of Injection) Full Geometry (This is not CFD)
For the particular scenario of Diesel engines, the inlet vales usually tend to produce a swirling motionwithin the cylinder.3. At this point it is not the scope of the project to asses the differences or the errordue to simplifying the simulation in this way. Petersen and Miles in [24] performed a set of experimentalmeasurements at Sandia National Laboratories confirming the swirling nature of the flow in the chamberprior to combustion.
For the simulation, the flow will be initialize using the so called wheel-paddle method. The flow initializa-tion might be part of the validation process since a perfect description of the flow at the beginning of thesimulation might not be available. Reuss et Al. [25] investigated the importance of the flow initializationin combustion simulations. It has a strong effect on the combustion efficiency and fuel mixing with thesurrounding air and therefore on CFD simulations.
The magnitude of the swirl flow within the piston is usually identified with the so called swirl ratio
swirl ratio ≡ Rs =Ω f low
Ωcrank(3.6)
However, a simple wheel flow4 (i.e vtheta(r) = Ω r ∀z) usually over predicts the speed of the flownear the walls of the cylinder. Petersen et Al [24] solve5 the angular momentum of the Navier-Stokesequations to give:
vθ(r, t) =ΩrbRsα
4J2(α)· J1(α
rrb) · exp
[−α2
r2b
νt(t− tIVC)
](3.7)
3In petrol engines the typical motion goal is "tumble" motion4Assume the z-coordinate is always aligned with the axis of the cylinder5This solution is subject to assumptions such as axisymmetric flow or axial uniformity (∂/∂z = 0) among others
39
Figure 3.4: Sector Volume during Power Stroke
Where J1 and J2 are the first and second order Bessel functions, Ω the crankshaft angular velocity, rb thebore radius, Rs is the swirl ratio, νt is a viscosity value that whose calibration allows to partially considerturbulent effects ([24]). The constant α determines the value of the flow next to the walls.
In order to fulfill the non-slip condition:α = 3.8317 (3.8)
However this value is typically not found in flow initialization since it leads to non realistic flow be-haviours. Typically the swirl profile value used in flow initialization is approximately α = 3.11 (at leastaccording to [13])
Radial Location [mm]0 10 20 30 40 50 60 70
v θ [m
/s]
0
5
10
15
20
25Angular Speed Profiles
α = 2.8α = 3.11α = αn.s
Cylinder Wall
Figure 3.5: Initial Velocity Profiles at t = tIVC
40
3.4.2 CFD Set Up
For our computational model, following the recommendations of [13], the swirl profile at initializationis set to α = 3.11. The turbulence will be modelled using RNG k− ε model. The fuel spray is modelledfollowing the study performed in Part 2, that using a KH-RT break up model with a NTC collision mesh.The injection is performed injecting droplets with the diameter of the nozzle. The Frossling correlationwill be employed to model the evaporation rate of the spray droplets.
The Number of Parcels injected follows from the study of Convergent Inc. with Caterpillar Inc. andArgonne National Laboratories in [26]. The number of parcels at a given minimum grid size is given bythe publication and the following scaling law is used when correcting the mesh size:
Np = 50× 104 × 2( dxb
dx −1)
(3.9)
Where dxb = 2× 10−3 [m] is the baseline grid size for Np = 50× 104 following [26]
3.4.3 Combustion Modelling and Emissions
The following simulations will make use of a set of 4 additional models to allow the flame burn (Shellauto-ignition model), the CTC model however predicts the speed at which the chemical reaction occurs.At the same time, Hiroyasu Soot Model will be employed to analyse the value of Soot at the end of thecombustion process and Zeldovich model will be employed to account for NOx production.
The previous models are tuned for their usage in Diesel Engines. One must realize that the numberof chemical paths to produce a given chemical specie is enormous. The Zeldovich Model for exampleaccounts for the so called thermal formation of NOx, that is:
N2 + O NO + N (3.10)
N + O2 NO + O (3.11)
N + OH NO + H (3.12)
However, the Zeldovich model will not take into account the formation of pollutants that followed, forexample these reactions:
N2O + O N2 + O2 (3.13)
N2O + O 2NO (3.14)
N2O + H N2 + OHNNH N2 + H (3.15)
NNH + O NH + NO (3.16)
41
3.4.4 Baseline Case and Validation
The experimental data published in [21] together with the detailed description of the engine set up allowto reproduce the results with very few unknowns which can be derived from the values present in thepublication. The engine specifications are:
Stroke 100 mmBore Diameter 125 mmEngine Speed 1900 RPMSwirl Ratio 1.4Compression Ratio 17.4Intake Manifold Pressure 2.8 barTemperature at Manifold 356 KAir Fuel Ratio 21.3
Table 3.2: Engine Characteristics
The current validation case has an EGR value of 25% thus providing the following mass fraction com-position of each specie at initial time:
CO2 H20 O2 N22.47% 1.15% 18.57% 77.81%
Table 3.3: Initial Species Composition
The fuel injection curve is also one of the parameters published in [21]. The injection is done in threedifferent steps: Pilot, Main and Post injection.
CAD [deg]-20 -10 0 10 20 30 40
Nor
mal
ized
Inj.
Rat
e
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
Figure 3.6: Fuel Injection History
The validation will focus in matching the pressure and the heat release rate traces. The experimentalvalue of such is also published in [21]. This two ultimately determine the emission values as all chemicalreactions depend mainly on the rate of combustion given by the heat release rate.
42
3.4.5 About Grid Dependency
Combustion CFD simulations have been traditionally grid dependent due to the Lagrangian-Eulerianspray models. The grid dependency problems have been previously discussed in the Part.2 of the cur-rent document.
The grid convergence was previously achieved on grids strongly refined around the nozzle. The sizeof the cells around the injector had to be approximately the same as the diameter of the nozzle. Thisimposes a strong CPU cost on the simulations as the CFL condition increases dramatically the numberof time steps to perform on a given time span. It must be noted that not only the CFL condition limitsthe time step but also the analogous CFL condition for the collision mesh.
Typically Internal Combustion Engine (ICE) model makers rely on experimental data to adequately tunethe spray models and grid sizes in order to achieve a realistic flow description without achieving gridconvergence. In [26], grid convergence is achieved using the spray model set up described in Part.2(mainly NTC Collision Mesh algorithm and Adaptive Mesh Refinement) together with fully detailedchemistry (200 species and about 2000 chemical reactions). In order to achieve grid convergence, boththe chemical reactions and the Lagrangian phase must be able to converge at some point and producethe same set of chemical species after the combustion. This increases the predictive capabilities of themodels and decrease the number of tuning factors.
In [26] the cell count in a single sector cylinder simulation with fully detailed chemistry is around 10million cells. The piston bowl of this case study shares some of the engine specifications in [26], howeverthe simulation time reported is around 90h to complete on a 64 core workstation.The time requirementsstrongly limit our capacity to perform an engine optimization, which in the end is our future goal. Thegoal will be to match the heat release rate as well as the pressure trace curves varying the mesh on arange which is known to be grid dependent.
For this project, the CTC Model will be used to calculate the characteristic time required to achievechemical equilibrium. The chemical equilibrium is dependent on both chemical kinetics and local tur-bulence levels (i.e RANS modelling). The CTC model is used together with the Shell model. This lastmodel is employed to predict the fuel ignition.
The heat release rate ultimately determines the chemical reactions rate and therefore the production ofpollutants. The following sections will study the combustion process from −147 < CAD < 60 whichspans the time window for which experimental data is available.
In order to evaluate the effects of the grid size, the time evolution of the validation data, pressure andheat release rate are plotted. At the same time it is of interest to monitor the changes in SOOT and NOxwhich are the goal of the present optimization study.
In in Fig.3.7 the grid effects are shown. It is obvious that the grid has a strong effect on the flame frontand therefore on the chemical reaction rates which ultimately determine the emissions amount. Theinitial flame front (first row) in the finest case is more elongated and does not interact with the pistonlower wall. A finer grid seems to increase the rate at which the flame propagates in the domain.
One of the main differences is in the secondary combustion region, that is after the fuel is directed abovethe lip. It is possible to appreaciate a much smaller flame zone compared to coarser meshes. A fastercombustion in the fine grid reduces the amount of oxygen therefore lowering the local equivalence ratioand therefore reduces the flame volume in this region.
43
Figure 3.7: Temperature Contours at CAD = [6 15 30]
Let us name the grid from left to right as A,B and C. Below more representative information about thegrid size can be found:
A B CBase Grid 2.8 mm 2.23 mm 2.23Min Cell Size 0.7 mm 0.55 mm 0.28 mmPeak Cell Count (millions) 0.11 0.13 0.45
Table 3.4: Engine Characteristics
The grid affects to the flame structure, evolution and temperature. This is clearly evidenced when look-ing at the pressure trace and heat release rates:
44
CAD [deg]-20 -10 0 10 20 30 40 50 60
Mea
n C
ylin
der
Pre
ssur
e [M
Pa]
2
4
6
8
10
12
14
16
Grid AGrid BGrid C
CAD [deg]-20 -10 0 10 20 30 40 50 60
Hea
t Rea
leas
e R
ate
[J/d
eg]
0
50
100
150
Cum
ulat
ive
Rea
leas
ed [J
]
0
1000
2000
3000
4000
Figure 3.8: Pressure trace,Heat Release Rates (-) and Cumulative Heat Released (- -)
The previous figure shows how a fine grid using the CTC combustion model clearly leads to faster com-bustion that releases more energy than on medium or coarser grid. In the sector simulations performedin [21] and [9], constant grid sizes were approximately 1mm. This has served as a basis for the currentsimulation.
It is interesting to identify a higher pressure trace with a fine grid which is of course correlated with amuch faster heat release rate (HRR). The total heat released (HR) is however approximately the same inall cases. The differences are mainly due to combustion efficiency differences. Besides, the HRR peakduring the pilot injection is lower in the fine grid case and higher for coarser grids. This evidences againthe dependency of the combustion model on the grid sizes.
The grid dependency can also be observed in the pollutant formation. Below the accumulated value ofNOx and SOOT is plotted:
CAD [deg]-20 -10 0 10 20 30 40 50 60
Mas
s [K
g]
×10-7
0
1
2
3
4
5
6
7
8
SOOT Grid ANOx Grid ASOOT Grid BNOx Grid BSOOT Grid CNOx Grid C
Figure 3.9: Pollutant Formation (Only until 60 CAD. EVO is at 135 CAD)
45
The higher temperature predicted by the fine grid also leads to a higher NOx formation. The SOOThowever is approximately equal for the medium (B) and fine grids (C). It is possible to observe thatthere is no time offset in the formation peaks as it happened in the heat release rate curves.
Let us now plot below the comparison between the pressure and heat release rates between the experi-mental data and our CFD calculations.
Time CAD-30 -20 -10 0 10 20 30 40 50 60
Pre
ssur
e [b
ar]
40
60
80
100
120
140
160 ExperimentalGrid AGrid BGrid C
Time CAD-30 -20 -10 0 10 20 30 40 50 60
HR
R [J
/deg
]
0
50
100
150
ExperimentalGrid AGrid BGrid C
Figure 3.10: Grid Comparison with Experimental Pressure and HRR traces
The previous results are at least to this author unexpected. The standard RNG model and standardbreak up models seems to provide an almost perfect match with the pressure curves. The heat releaserate is however more difficult to match as it strongly depends on the chemical reactions. Using thecurrent CTC + Shell model it is believed that it is unlikely that the results can be improved much more.
Most of the discrepancies occur near TDC (top dead center) where the combustion begins. there is adecrease in the pressure trace of the experimental data likely due to the fact that the combustion wasnot able to maintain the pressure level. Later the combustion becomes more intense and the pressurerises again. The behaviour of the pressure trace in the very end of the time spanned by the simulation isalmost fixed by thermodynamics, therefore it is basically grid independent.
Regarding the heat release rate, the pilot injection is over predicted. However the post injection barelyincreases the pressure. This might be due difficulties of modelling the re-ignition process. Using fullydetailed chemistry it is expected that this problem could be easily solved. Also it must be noted thatGrid C (the finest) predicts an earlier ignition and coarser grids (Grid B and more extremely grid, A)predict later and less violent combustion process.
However, as it was shown in Fig.3.8, the total heat releases is basically the same, it is only the chemicalpath what is modified.
It is concluded that the set up involved using Grid B is accurate enough for the calculations that are ofour interest.
46
Figure 3.11: Grid B. Flame Front at 15 CAD. Temeperature given in Kelvin
3.4.6 Turbulence Model Constants
A study trying to improve the results from the grid calibration by varying the turbulence model wasperformed. Variation of the turbulence constants Cε1and Cε2between 1.3 and 1.8 (both of them) is per-formed. The effect on the pressure trace and heat release rate is shown below
CAD [deg]-30 -20 -10 0 10 20 30 40 50 60
Pre
ssur
e [B
ar]
0
50
100
150
Varying Cǫ2
Experimental DataCǫ1 = 1.3
Cǫ1 = 1.55
Standard
CAD [deg]-30 -20 -10 0 10 20 30 40 50 60
Pre
ssur
e [B
ar]
0
50
100
150
Varying Cǫ1
Experimental DataCǫ2 = 1.3
Cǫ2 = 1.55
Cǫ2 = 1.8
Standard
CAD [deg]-30 -20 -10 0 10 20 30 40 50 60
HR
R [J
/deg
]
0
50
100
150
Experimental DataCǫ2 = 1.3
Cǫ2 = 1.55
Cǫ2 = 1.8
Standard
CAD [deg]-30 -20 -10 0 10 20 30 40 50 60
HR
R [J
/deg
]
0
50
100
150
Experimental DataCǫ1 = 1.3
Cǫ1 = 1.55
Standard
At the same time, it would be interesting to monitor what is the effect on the production of pollutants,mainly NOx and Soot.
47
CAD [deg]-30 -20 -10 0 10 20 30 40 50 60
SO
OT
[Kg]
×10-6
0
1
2
3
4
5
6Varying C
ǫ1
Cǫ2 = 1.3
Cǫ2 = 1.55
Cǫ2 = 1.8
Standard
CAD [deg]-30 -20 -10 0 10 20 30 40 50 60
NO
x [Kg]
×10-7
0
1
2
3
4
5
6
Cǫ2 = 1.3
Cǫ2 = 1.55
Cǫ2 = 1.8
Standard
CAD [deg]-30 -20 -10 0 10 20 30 40 50 60
SO
OT
[Kg]
×10-6
0
1
2
3
4
5
6Varying C
ǫ2
Cǫ1 = 1.3
Cǫ1 = 1.55
Standard
CAD [deg]-30 -20 -10 0 10 20 30 40 50 60
NO
x [Kg]
×10-7
0
1
2
3
4
5
6
Cǫ1 = 1.3
Cǫ1 = 1.55
Standard
It is not possible to find a much better combination of Cε1and Cε2 than the standard one for the heatrelease rate evolution. The pressure trace seems to be however almost independent of the turbulencemodel. This is probably due to the fact that no extremely big changes in the combustion and therefore inthe Heat Release Rate occur. However one can note that higher values of Cε2seem to have a stabilizingeffect over the combustion, therefore contributing to raise the maximum temperature and it is thoughtthat because of that, there is a higher NOx prediction.
There exists many chemical "paths" through which NOx can be produced. According to [30], the NOxproduction will always be increased by an increase in the temperature of the flame, independently of thedominant chemical path. This seems to justify the changes in the amount of pollutants being formed.
It is interesting to note in the previous figure how an increase (in this case due to turbulence modifica-tion) in the NOx leads to a decrease in Soot and vice-versa. This is a first indicative of the difficulty tominimize simultaneously both of them.
It has been decided that the standard turbulence model is the one performing better for our interest.
3.4.7 Spray Model Constants
Following the same procedure as with the spray modelling, it is of our interest, at least academicallyspeaking to analyse the effect of the break up model in the combustion process. Let us perform a studyof the size and time constants analogously to the previous one performed in the Sandia Spray A Section.
48
Time CAD-30 -20 -10 0 10 20 30 40 50 60
Pre
ssur
e [B
ar]
0
50
100
150 Experimental(B0,B1)=(0.4,7)
(B0,B1)=(2,7)
(B0,B1)=(0.68,20)
Standard
Time CAD-30 -20 -10 0 10 20 30 40 50 60
HR
R [J
/deg
]
0
50
100
150
Experimental(B0,B1)=(0.4,7)
(B0,B1)=(2,7)
(B0,B1)=(0.68,20)
Standard
Figure 3.12: Effects of Spray Break Up Model on Combustion
The previous results show clear increase in the heat release rate due to a more violent combustion hap-pening due to larger drops in the case of increase the time constant B1. The particles reduce their size ina slower way therefore given up higher amount of concentrated fuel vapour thus increasing the violenceof the combustion process. This is also reflected in the pressure peak.
An increase in the break up constants leads to a longer penetration values since the evaporation andignition process take longer to occur. Based only on the previous results one could in principle expectthat an intermediate value of the size and time constant could help to obtain a better correspondencebetween CFD and experimental results. Besides, an increase in the constants leads to much longerpenetration which leads to unphysical spray behaviour.
The standard spray model will me employed together with the standard turbulence model.
3.5 NOx and Soot Constrained Optimization
The main objective is to optimize the piston bowl to lower the emissions by improving the combustionefficiency. The improvement is given by modifying the geometrical parameters and the spray orienta-tion.
However the designs variations can vary the volume of the bowl. This affects to the compression ratiowhich is a number fixed by structural constraints as well as fuel efficiency. In the optimization process,the compression ratio is kept constant to 17.4 as described in Table 3.2.
3.5.1 Parametrization
The Piston Bowl geometry is parametrized considering variations in 4 different parameters.
49
Figure 3.13: Axisymmetrical Bowl Parametric Cross Section. Baseline (Black), Parametric Design Can-didate (red)
A variation of the "Bowl Radius" is represented in the left most column of the previous figure. Thesecond picture from the left represents a variation in the depth of the bowl. The third one from the leftincreases the entrainment of the "secondary lip" typical from ULPC piston bowls whereas the last one isa variation of the so called "inner diameter".
Additionally, a variation of the spray direction is also taken into account. This parameter is shown inFig.3.14
Figure 3.14: Spray Angle Parametrization
The spray direction it is weakly dependent on other parameters. The direction is measured from theinjector position at SOI (start of Injection) towards the secondary lip. This direction is thus altered byparametric variation of the geometry. The spray direction parameter is only an offset of with respect tothis direction. Before, the domain of study here proposed is shown:
Parameter Lower Bound Baseline Value Upper BoundBowl Radius 4 6 8
Depth 4 6 8Secondary Lip 0 3 6
Inner Diameter Scale 0.9 1 1.1Spray Offset Angle 0 0 30
However this changes almost always have to be corrected so that the compression ratio is fulfilled. Theparameters chosen to produce such change are:
50
The left most picture represents a variation in the arc composing the bowl. This is accomplished byvarying the tension of the tangency between the arc of the bowl and the guide line from the centre ofthe piston. The centre figure represents a vertical scaling. This parameter is only allowed to makesmall variations so that it does not affect to the depth design parameter. Finally the right most columnrepresents a displacement of the bowl arc centre horizontally.
Parameter Lower Bound Baseline Value Upper BoundTangency Tension -3 0 2
Vertical Scaling 0.95 1 1.05Horizontal Offset 3 3 10
3.5.2 Optimization Algorithm
The optimization algorithm is a surrogate multi-objective genetic algorithm. This algorithm is specially wellsuited for this kind of simulations where the time of computations specially expensive.
The algorithm operates in the following way:
1. The algorithm generates an initial pool of designs. Generally, the number of initial designs is thesquare of the number of parameters (In this case N = 52 = 25). However this is know to be alower bound for the number of initial designs. Therefore it was chosen to compute 31.
For each design, the algorithm first triggers the adjustment to fulfill the compression ratio.Such algorithm is a Nelder Mead Simplex which contrary to gradient methods, tries to find asolution far from the boundaries of the parameters chosen to fulfill the compression ratio.
2. The algorithm, once it has the initial pool of designs, it generates an initial response surface basedon the Kriging method.
3. Based on this response surface, new global minimum candidates are computed, 4 in this case. Thealgorithm is also tuned to select design points which are at least 10% far from each other in thedesign space.
4. The new computed designs allow to improve the quality of the response surface and next 4 designcandidates are chosen. The process repeats steps 2-4 until pareto front convergence is achieved.
51
3.5.3 Analysis of the Optimization Results
The Pareto Front
The pareto front is a curve describing the set of designs who best accomplished a given goal. This is acurve that typically appears in optimization algorithms when two objectives compete with each other.In order to minimize SOOT one would probably have to expect large values of NOx and viceversa. Theevolution of the optimization algorithm is shown below:
NOx []0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
SO
OT
[]
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1Pareto Front After 4 Generations
BaselineInitial SamplingGen 1Gen 2Gen 3Gen 4Pareto Front
Figure 3.15: Pareto Front after 4 Generations
The squares in blue represent the initial pool of design found while sampling the domain whereas thedashed (- -) lines represents the actual evolution of the algorithm. The response surface of the algorithmestimates the location of the minimum. However the actual value of NOx and Soot is corrected byrunning the CFD calculation and improving at the same time the response surface.
It is possible to appreciate that the algorithm tries to improve on both NOx and Soot direction. This isevidenced in generations 3 and 4 of the genetic algorithm.
The previous results show a genetic algorithm which has not been able to improve much more the paretofront algorithm. This indicates that more generations should be run in order to obtain a smoother andbetter posed pareto front.
Designs in the Pareto Front
In order to understand why this improvement has taken place, an analysis starting from the geometryvariations should be done.
The left most design is the best performing design in order to reduce NOx. The main characteristic ofthis design is its more pronounced secondary lip with a smaller radius. The so called corner design is theclosest to the (0,0) in the NOx-Soot map shown in Fig.3.15. This design features a smaller bowl radius
52
Figure 3.16: Pareto Front Designs. Baseline in orange. From left to right column: Minimum NOx, CornerValue, Minimum Soot
with a slightly more pronounced angle from the bowl axis to the tangency point. Finally the best Sootperforming design features a deeper bowl with a reduced inner diameter which pushes the secondarylip to the interior of the bowl.
However the Best for Soot and Best for NOX designs share one thing in common, the spray angle offset.In both cases the spray is pointing towards the secondary lip, whereas for the corner design, the sprayis pointing towards the bowl.
53
Figure 3.17: Left: Best for NOx. Middle : Corner Design. Right : Best for Soot. Time 30 CAD
The previous designs seem to be mostly affected by the spray direction. The best design for NOxspraystowards the lower side of the bowl. The best design for Soot reduction however sprays towards the linerupper part of the bowl. The corner design, that is the one that reduces both the Soot and the NOxchoosesto spray towards the secondary lip splitting the flame front between an upper and lower side.
54
Parameter Best NOx Corner Best SootBowl Radius 5.99 4.41 5.66
Depth 6.22 6.12 4.05Secondary Lip 5.45 2.48 0.7
Inner Diameter Scale 1.04 1.01 1.06Spray Offset Angle 16.77 25.64 4.3
Table 3.5: Parameter Specification for Pareto Designs
The flame front chooses to generate a vortical motion in the best for NOxdesign (left most column). Thisflame front keeps a larger volume of air at around 1600 K. This is not the extreme value at the flamefront but hot enough to stop soon enough the generation of nitrous oxides. The nitrous oxides seemto be generated at the hottest portions of air. Relatively fine flame fronts seem to help reducing thegeneration of NOx.
The right most case has a more chaotic behaviour. The flame is more turbulent and generates a thickerflame that increases the average temperature in the burning zone, this hotter region enhances the speedat which chemical reactions occurred leading to higher NOxamounts.
However, the best for NOxdesign seals the burning zone from the surrounding air due to its morelaminar alike structure (not so much chaotic). The sealing reduces the amount of oxygen (equivalenceratio) and therefore a higher amount of soot is created.
It is possible to observe a strong correlation between the local equivalence ratio and the generation ofSoot. This is strongly evidenced for the best NOxdesign which accumulates almost 5 times the stoichio-metric fuel to air ration near the axis of the cylinder.
Below the temporal evolution of NOx and Soot is shown.
Metric Baseline Best NOx Corner Best SootNOx 1 0.16 0.34 1.49Soot 1 1.95 0.22 0.06
Table 3.6: Improvement With Respect To Baseline
One must not truly believe the optimization results as this may have a strong impact on other quantitiessuch as carbon monoxide (CO), carbon dioxide (CO2) or hydrocarbons (HC). For this reason one would
55
CAD-150 -100 -50 0 50 100 150
Soo
t [m
g]
0
0.5
1
1.5
BaselineBest NOX
Corner ValueBest Soot
CAD-150 -100 -50 0 50 100 150
NO
x [mg]
0
0.1
0.2
0.3
0.4
0.5
0.6
BaselineBest NOX
Corner ValueBest Soot
Figure 3.18: Pareto Front Emissions
like to analyse the time evolution of not only the goal value (NOxand Soot) but also other pollutants.
At the same time the temporal evolution can provide information about the combustion and oxidationprocesses. For example in Fig.3.18, the combustion process for the best soot design finish earlier burningall the fuel and therefore the soot curve becomes flat earlier than the rest. The best for NOxhoweverkeeps on decreasing at a steady slow rate which leads to think that incomplete combustion might behappening.
CAD-150 -100 -50 0 50 100 150
CO
[mg]
0
50
100
150
200
BaselineBest NOX
Corner ValueBest Soot
CAD-150 -100 -50 0 50 100 150
CO
2 [m
g]
0
100
200
300
400
CAD-150 -100 -50 0 50 100 150
HC
[mg]
0
10
20
30
40
BaselineBest NOX
Corner ValueBest Soot
Figure 3.19: Effect on other pollutants
Similarly, the best for NOxor soot designs have some relations that are worth remarking. The best forsoot design leads to almost zero carbon monoxide emissions. This is a measure of the completeness ofthe combustion process. This is also why a higher amount of CO2 is produced. The corner design alsoachieves complete combustion and exactly same amount of CO2 emissions, whereas the baseline and
56
the best NOxdesign seem to reach the same amount of CO and CO2.
Finally the amount of hydro carbons being released from the best NOxdesign leads to believe that thisdesign is not worth the reduction of NOx.
3.5.4 Conclusions
The validation of diesel combustion simulation can be done without requiring an excessive amountcomputational power but at the cost of requiring experimental data from a test engine. Therefore, thereare advantages and disadvantages to this method.
In order to have simulation that cover all the chemical reaction and the turbulence modelling one canalways keep increasing the accuracy of the models up to LES and fully detailed chemistry. Mainly, thecost of LES restricts the optimization possibilities and number of iterations required. It is the opinionof this author that RANS modelling can be sufficient for piston bowl design. Besides detailed detailedchemistry is being foreseen by the industry for the very near future.
Detailed chemistry differs from fully resolved chemistry in the number of species being considered.The number of species contained within a fuel molecule can be around 7000. Detailed chemistry al-lows to choose the most relevant chemical paths decreasing the cost while maintaining the accuracy ofthis type of simulations. Modelling emissions strongly depends on models which require of some finetuning. Detailed chemistry removes such tuning tasks increasing the predictive capabilities of the CFDsimulations.
The optimization algorithm relies on purely automatic analysis. It is believed that the best for NOxdesignis not a reliable design. The strong deviation of the equivalence ratio from its stoichiometric value nearthe axis zone reveals a strong risk of knocking. This design needs to be tested under slightly differentoperating conditions in order to evaluate the stability of the design.
The dramatic increase in the soot generation while minimizing the NOx(almost 2 times) is a strongvalue which is most likely related to the fact that not so much fuel has been burned. Leading to higheramounts of unburned fuel (Soot) and lower NOxamounts. Optimization in order to minimize NOxhasshown to be much more difficult than expected at the beginning.
The corner design shares many geometrical similarities together with the baseline design, specially thespray being directed towards the bowl instead of the secondary lip. This seems to be one of the mostrelevant parameters in the optimization.
57
3.6 Unexplored Optimization Posibilities: Fuel Injection Curve
One of the critical parameters of a combustion process is the rate of injection of fuel, which ultimatelydetermines the injection pressure (and therefore vaporization rate) as well as timing of the fuel curve.The fuel injection at the same time, determines the heat release rate whose shape has a strong effect onthe power and pollutant emission.
The heat release rate can be estimated on hand by the combustion efficiency (η) which in turn is afunction of the local equivalence ratio6 and the adiabatic index γ = cp/cv
∂Qch∂t≈ η(φ(~X, t), γ(t)) m ρE (3.17)
On the other hand, the first Law of Thermodynamics allow us to express [1]:
δQch = dU + δW + δQht + ∑ hidmi (3.18)
The main difficulty of the previous equation is usually related to the accuracy to which we can deter-mine the parameters entering into the previous equation. U represents the internal energy, W the workperformed by the piston, Qht the heat transfer across the walls and ∑ hidmi the energy leakage across thecrevice and valves. The unusual terms in the first law of thermodynamics are due to the "open" natureof this system. This is represented in Fig.3.20:
Figure 3.20: Heat Generation and Losses. Combustion Stroke
du = mcv(T)dT + udm (3.19)
∑ hidmi = hcrdmcr + h f dm f (3.20)
∆W = pdV (3.21)
∆Qht ≈ Ahc(T − Tw) = Ahc(PVRg− Tw) (3.22)
6The local equivalence ratio is defined as φ ≡ χ f uel /χox
(χ f uel /χox)st, that is the ratio of fuel to oxidizer agent with respect to the stoichio-
metric ratio of fuel to oxidizer agent.
58
Figure 3.21: Heat Generation and Losses. Combustion Stroke
Typically under combustion, the chamber would experiment a pressure rise due to an increase in thetemperature. This would exacerbate the heat losses due to flow leakage in the crevice region. Howeverthe increase in pressure would improve the sealing in the valve reducing the leakages and therefore onecan usually neglect leakages in the valve region:
∑ hidmi = hcrmcr + h f dm f ≈ hcrdmcr (3.23)
Under this assumption, in the changes of internal energy:udm = −udmcr. Introducing at the same timethe ideal gas law, one obtains:
∆Qch =cv
RVdp +
( cv
R+ 1)
pdV + (hcr − u)dmcr + ∆Qht (3.24)
A better measure of the evolution in time is usually given in terms of the so called crank angle degree,θ which is given by the engine rotational speed (RPM’s). The relationship between the crank angle andtime is given by the simple relationship expressed below.
∂Qch∂t
= t = ωθ = 1ω
∂Qch∂θ
(3.25)
The crank angle degree is measured (θ = 0) from Top Dead Center (that is, when the cylinder is com-pressed the most)
In [1], the author uses a empirical correlation in order to evaluate the term: (hcr − u)dmcr. Also Eq.3.24will be differentiated with respect to time in order to obtain the heat release rate. In this fashion the timederivative is given by:
Qch∂θ
=γ
γ− 1p
∂V∂θ
+1
γ− 1V
∂P∂θ
+Qleak
∂θ+
∂Qht∂θ
(3.26)
Where Qleak is given by the empirical relation in [1]
∂Qleak∂θ
≈ Vcr
[T′
Tw+
TTw(γ− 1)
+1
bTwln(
γ− 1γ′ − 1
)]∂p∂θ
(3.27)
59
In principle one could solve for the rate of fuel mass being injected however the approximation turnsout to be really crude given the difficulty of estimating the previous equations. That is:
η(φ(~X, t), γ(t)) m ρE ≈γ
γ− 1p
∂V∂θ
+1
γ− 1V
∂P∂θ
+Qleak
∂θ+
∂Qht∂θ
(3.28)
The previous equation is difficult to model due to the variation in time of the pressure. The heat transferthrough the walls is also another difficult term to estimate. Mainly the convection and radiation willdominate the heat transfer through the walls. Possible analytical solutions can exist neglecting the heattransfer and assuming either a constant pressure (which is somehow true in some cases for an interval oftime spanning a few crank degrees from TDC). Another possibility is to assume an evolution described
in a way analogous to an isentropic expansion, that is ∂P∂θ = ∂
∂θ
[PTDC
(VTDC
Vθ
)γ]
.
In all of the previous cases, the heat transfer plays a very important role which can overcome the impor-tance of the pressure and volume evolution. This sits out of the scope of the current report and can beused as a basis for a future study.
60
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