Degree Project in Mechanical engineering
Second Cycle, 30 Credits
September 2016
PISTON BOWL COMBUSTION SIMULATION
From Fuel Spray Calibration to Emissions Minimization
Diego Garcia Pardo
FRIENDSHIP SYSTEMS
Contents
1
2
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 − e RANS Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
6
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
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
Unexplored Optimization Posibilities: Fuel Injection Curve . . . . . . . . . . . . . . . . . .
58
3.5
3.6
3
Diego García Pardo
FRIENDSHIP SYSTEMS
Potsdam
4
Aknowledgements
The current pollution policies in all European and American countries are forcing the industry to move
towards a more efficient and environmentally friendly engines. On the other hand, customers require
maintaining the power and fuel consumption. Lowering mainly nitrous oxides ( NOx ) and carbon particles (Soot) is therefore a challenging task with a very strong impact on mainly the automotive and
aeronautical market.
The purpose of the current work is to research the pollution production of automotive diesel engines
and optimize the fuel injection and piston geometry to lower the emissions. The interaction between
fuel and air as well as the combustion are the two main physical and chemical processes governing the
pollutants formation. Converged-CFD will be the CFD tool employed during the analysis of the previous problems.
The fuel-air interaction is related to jet break up, vaporization and turbulence. The strong dependence
on the surrounding flow field of the previous processes require the equations to be solved numerically
within a CFD code. The fuel is to be placed in a combustion chamber (piston) where the spray will affect
the 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 main
chapters. Spray modelling and Combustion Modelling. The first will help to accurately model the discrete phase (fuel spray) and the vapour entrainment. The second chapter, combustion modelling will
retrieve the knowledge gain in the first part to accurately represent the fuel injection in the chamber as
well 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 the
industry a measure of the potential improvement by just adjusting the fuel injection or by modifying
the piston bowl geometry.
PREFACE
The current pollution policies in all European and American countries are forcing the industry to move
towards a more efficient and environmentally friendly engines. On the other hand, customers require
maintaining the power and fuel consumption. Lowering mainly nitrous oxides ( NOx ) and carbon particles (Soot) is therefore a challenging task with a very strong impact on mainly the automotive and
aeronautical market.
The purpose of the current work is to research the pollution production of automotive diesel engines
and optimize the fuel injection and piston geometry to lower the emissions. The interaction between
fuel and air as well as the combustion are the two main physical and chemical processes governing the
pollutants formation. Converged-CFD will be the CFD tool employed during the analysis of the previous problems.
The fuel-air interaction is related to jet break up, vaporization and turbulence. The strong dependence
on the surrounding flow field of the previous processes require the equations to be solved numerically
within a CFD code. The fuel is to be placed in a combustion chamber (piston) where the spray will affect
the 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 main
chapters. Spray modelling and Combustion Modelling. The first will help to accurately model the discrete phase (fuel spray) and the vapour entrainment. The second chapter, combustion modelling will
retrieve the knowledge gain in the first part to accurately represent the fuel injection in the chamber as
well 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 the
industry a measure of the potential improvement by just adjusting the fuel injection or by modifying
the piston bowl geometry.
PREFACIO
Las actuales políticas medioambientales tanto en países Europeos como Americanos está forzando a la
industria a producir motores de combustión más eficientes y limpios. Los clientes requieren mantener
la potencia y el consumo de combustible. Es por esto que la minimización de la emisión de óxidos de
nitrógeno NOx y de part’iculas de carbono (Soot) es una tarea cuanto menos desafiante con un gran
impacto en el mercado automovilístico y aeronáutico.
El propósito de este trabajo es realizar una investigació sobre las emisiones de motores diésel y optimizar
la inyección de combustible y la geometria del bowl del piston. La interacción entre el combustible y el
aire así como el proceso de combustión son los dos principales procesos químicos y físicos que gobiernan la formación de humos tóxicos. ConvergeCFD será utilizado como herramienta para llevar acabo
las 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, vaporización y valores de turbulencia. La fuerte dependencia en el fluido que rodea al spray require que las
ecuaciones sean resueltas numéricamente en un software CFD. El combustible liquido será inyectado en
una 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 en
un primer capitulo en el que se analiza el comportamiento de los sprays de combustible y un segundo
capítulo en el que el proceso de combustión se valida para finalmente proceder con la el proceso de
optimización. 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 describirá
tanto 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 medir
la 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 physics
in this field has been shown to be especially difficult. The behaviour of gases and liquids is still limited
due to the complexity of the equations describing the flow. The main sets of equations that can be found
are either Lattice-Boltzmann equations and the Navier-Stokes equations.
The applicability of the previous equations is widely discussed in [3]. The main parameter affecting to
the applicability of both sets is the so called mean free path, λ. The continuum hypothesis is based on
this parameter. It requires the mean free path to be much smaller than the actual length scale of the
flow being described. Therefore one necessary requirement (but not sufficient [3]) is that the Knudsen
number:
λ
Kn = 1
L
The mean free path of air at standard conditions can be estimated (as shown in [2]) by:
λ≈
N
ρ A
m
−1/3
= 3 × 10−9 m
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 possibilities
within the world of research and engineering. This project will make use of the Navier-Stokes equations
to model fuel sprays. This project will seek to compare and asses the predictive capabilities of CFD by
comparing 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 within
the 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 leading
to the following relationship
∂
∂u
UL =
∂t
∂t
∂
∂u ∂u ∂x
∂u
∂u
UE =
+
=
+u
∂t
∂t
∂x ∂t
∂t
∂x
(1.2)
(1.3)
Last equation is the definition of the so called material derivative, which in its most general form is
given by;
∂
∂
D
≡
+ uj
(1.4)
Dt
∂t
∂j
1.3
The Navier-Stokes Equations
The description of the flow field requires a set of equations describing the mass, momentum and energy
conservation. That is, the flow physics are derived from first principles.
The momentum conservation equations are the so called Navier-Stokes Equations. The derivation can
be found in almost any introductory books to fluid mechanics.
Below, the Navier-Stokes equations are presented in conservative form. They include the terms that
account for a mass, momentum and energy transfers from only the Lagrangian to Eulerian phase [13].
Mass conservation
∂ρ
∂
+
(ρui ) = S I
∂t
∂xi
(1.5)
Where S I accounts for the mass transfer from the Lagrangian phase (fuel injection as it will be shown).
Momentum conservation
∂σi,j
∂P
∂ (ρui ) ∂ ρui u j
+
=−
+
+ SiI I
∂t
∂x j
∂xi
∂x j
2
(1.6)
Where S I I accounts for the momentum transfer from the Lagrangian phase. The stress tensor is defined
as:
! ∂1u j
∂ui
2
∂u
σi,j = µ
+
+ µb − µ δi,j k
(1.7)
∂x j
∂xi
3
∂xk
In the CFD Software used in the present work, the bulk viscosity, µb is set to zero [13].
Energy conservation
∂u j
∂ρe ∂ρeu j
∂u
∂
+
= −P
+ σij i +
∂t
∂x j
∂x j
∂x j
∂x j
∂T
K
∂x j
!
∂
+
∂x j
∂Ym
ρD ∑ hm
∂x j
m
!
Where S I I I accounts for the energy transfer from the Lagrangian phase. Besides,
+ SI I I
∂
∂x j
(1.8)
m
ρD ∑ hm ∂Y
∂x
m
j
represents for the energy transport due to species diffusion as stated in [13]. The species mass fraction
Mm
m
ν
represents the molecular mass diffusion coefficient1 .
is Ym = M
= ρρtot
and the D = Sc
tot
The previous equations have to be supplemented with additional state equations:
P = ρRT
and
e = e0 + c v T
(1.9)
The Source term for mass, momentum and energy will be presented later in the chapter describing spray
modelling, Ch.2.
1 Sc
≡ Schmidt Number
3
1.4
Principles of Turbulence Modelling
Solving the Navier-Stokes equations is a challenging task usually performed numerically. There is only a
few known analytical solutions. The numerical solutions are however very expensive computationally
speaking. The cost of solving the Navier-Stokes equations is directly related the size of the smallest
eddies in the flow field. The scale is given by the so called Kolmogorov scale. From dimensional analysis
[4]:
1/4
η = ν3 /e
(1.10)
It is observed that smallest scales are independent from the geometrical boundaries and depend solely
on the viscosity, ν and dissipation of turbulent kinetic energy, e. Also as reported in [4] it is possible to
estimate the dissipation order of magnitude as a function of velocity (large eddie velocity), u0 and the
geometrical length scale, Λ:
( u 0 )3
e∼
(1.11)
Λ
The cost of evaluating numerically the Navier-Stokes Equations comes directly from the combination of
the last two equations as:
1/4
u03 /Λ
Λ
Λ
=Λ
= Re3/4
=
Λ
1/4
η
ν3/4
(ν3 /e)
(1.12)
If we consider a 3 dimensional flow with a grid fine enough to resolve the kolmogorov scales, then:
2.25
Ngrid points ∝ Re9/4
Λ = ReΛ
(1.13)
Real application usually involve dealing with Reynolds numbers large enough to support alternatives
to 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 situation. 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 velocity
is decomposed as u = U + u0 . After the decomposition, an average is taken over the equations. For
example the RANS Momentum equation is then:
∂σij
∂
∂P
∂ 0 0
∂
ρUi Uj = −
+
−
ρui u j + S I I
(ρU ) +
∂t
∂x j
∂xi
∂x j
∂x j
4
(1.14)
Where the average of the stress tensor is then:
σij = µ
∂Uj
∂Ui
+
∂x j
∂xi
!
2 ∂U
+ (− µ) k δij
3 ∂xk
(1.15)
The so called closure problem in turbulence modelling is shown here. The turbulent stress tensor, ρui0 u0j
needs to be modelled in order to be able to solve the so called RANS equations. The eddy viscosity
models assume a general invariant form of the turbulent stress tensor as [13]:
∂Ui
2
− ρui0 u0j = 2µ T Sij − δij ρK + µ T +
≡ τij
(1.16)
3
∂xi
Again, at the cost of removing the turbulent stress tensor a new unknown has been introduced: the
turbulent viscosity
νT ∼ Λu0
The turbulent viscosity is now another unknown in the model. The challenge is now to model a constant
rather than a second order tensor. Analogously, the turbulent heat conductivity κ T or the turbulent
diffusion DT need to be modelled. At this point it is easy to understand the need of validation for the
RANS-Based CFD simulations given the number of assumptions taken.
3/2
Following the derivation in [4], the eddies length scale can be estimated as Λ ∼ K e and the velocity scale u0 ∼ K1/2 finally leading to the need of an equation for the turbulent kinetic energy, K and
dissipation e.
In any introductory book for turbulence modelling it is possible to find a derivation for such equations. Here the final shape of the compressible standard K − e equations employed in ConvergeCFD are
retrieved[13]:
∂u
∂
D
(ρK ) = τij i +
Dt
∂x j
∂x j
D
∂
(ρe) =
Dt
∂x j
µ ∂e
Pre ∂x j
µ ∂K
Prk ∂x j
!
!
− ρe +
∂u
+ Ce,3 ρe i +
∂xi
Cs
STK
aS
∂u
Ce,1 τij i − Ce,2 ρe + Cs STK
∂x j
(1.17)
!
e
+ STe
K
(1.18)
The terms STK and STe represent the interaction with the Lagrangian phase. The constant as is linked to
the turbulence generated by the Lagrangian phase. Empirically, aS has been set to 1.5, but it allows some
calibration. Finally the turbulent stress tensor is remodelled and integrated in the RANS Equations as
part of the viscosity, diffusivity and conductivity:
K2
µ T = Cµ ρ
e
1
DT =
µT
Sc T
1
κT =
µT cP
Pr T
5
(1.19)
(1.20)
(1.21)
The calibration of the constants Ce,i allows to modify the rates of production and dissipation of turbulent
kinetic energy. Particularly, Ce,1 is intrinsically related to the production of kinetic energy and Ce,2 to the
dissipation. Ce,3 is constant related to the compressibility of the fluid. It models the turbulent dissipation
due to the "dilatation" or "elasticity" of a fluid element.
1.4.2
RNG k − e RANS Model
In principle, for all the simulations present in this report, the RNG k − e RANS model will be employed.
This model is derived using a statistical technique called Re normalization Group. The main difference
between the standard and the RNG version of the k − e model is that the values of the constants become
functions of flow parameters (length scales). However, It is not possible to say that the RNG turbulence
model always correlates better than the standard k − e model.
In the standard k − e model, the eddy viscosity is found from a single turbulent length scale. In reality
this 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 − e equations implemented in ConvergeCFD are shown below:
!
D (ρk)
∂
µt ∂k
=
µ+
+ Pk − ρe
(1.22)
Dt
∂x j
σk ∂x j
!
e
D (ρe)
∂
µt ∂e
=
µ+
+ ce1 Pk − c∗e2 ρe
(1.23)
Dt
∂x j
σe ∂x j
k
cµ η 3 (1 − η/η0 )
1 + βη 3
1/2 k
η = 2Sij Sij
→ Sij = mean flow strain rate tensor
e
c∗e2 = ce2 +
(1.24)
(1.25)
In the formulation of the dissipation rate equation, it is possible to observe the differences with respect
to the standard model. Mainly the the parameter Ce2 becomes now Ce∗2 and therefore itself depends on
the 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 mechanics. The complexity lies on the strong interaction (shear and normal forces) between the fuel and the
surrounding fluid (air in this case). The difficulty increases even more given the interaction between
droplets and their evaporation.
A correct prediction of the spray penetration, opening and evaporation rate is of extremely importance for the follow up work of
combustion modelling.
Given the large number of droplets (For a typical diesel engine
([12]), N ∼ 107 ) created after the jet break up, it has been chosen to model the flow using a discrete phase approach. This
means that it is possible to avoid resolving the free surface of
each droplet in the flow field and therefore reducing the computational cost per simulation significantly. The droplets are all
contained inside parcels where the characteristics of each droplet
are equal. This further reduces the cost when dealing with the
Figure 2.1: Fuel Injection evolution.
possible collisions between droplets.
Source: Oregon State University
The reduction in the CPU cost however is then balanced by the
usage of Lagrangian models with the corresponding difficulty
when interacting between the Eulerian phase. In [9] it is presented the dependency of the Eulerian phase mesh on the Lagrangian phase giving rise to the so called
grid-dependent models (as it will be shown). The dependency on the grid lies on the representation of
the collision between parcels. The droplets are only allowed to collide if they are contained in the same
cell.
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 real
cases. Schmidt in [10] proposes a new method with a cost that increases linearly with the number of
parcels 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 a
diameter equal to the nozzle of the injector. The correct physical behavior of these droplets is then a key
part in this work.
This droplets are then subjected to drag, collision, coalescence, wall interaction and evaporation. This
physical behavior is entirely modeled in the Lagrangian phase.
The computational cost of tracking each particle is extremely expensive and only viable in research
projects where the CPU cost or time is not an issue. In the present work, the droplets are grouped
together within parcels containing identical droplets (radius, temperature, velocity, etc...). This is also
remarked in the software manual [13] . Below the concept of parcel is pictured for clarification. To the
left 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 has
been chosen to model the break up process using the Kelvin-Helmholtz wave model and the RayleighTaylor 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 = η0 eikz+ωt
(2.1)
The derivations presented by Reitz [15] find the dispersion relation ωKH = ωKH (k KH ). Most importantly, It is found the dispersion relation for the most unstable wave (ΩKH = ΩKH (KKH )), where ΩKH
is the growth rate of the most unstable wave. At the same time, KKH = Λ2π
provide us with the waveKH
length of the fastest growing wave [13]. The results presented below are implemented in the CFD Code.
1 + 0.45Zl0.5 1 + 0.4T 0.7
ΛKH
= 9.02
rp
1 + 0.87We0.87
g
!0.5
ρ L r3p
0.34 + 0.38We1g .5
ΩKH
=
σ
(1 + Zl ) (1 + 1.4T 0.6 )
(2.2)
(2.3)
√
Where σ is the surface tension of thepfluid, r p is the drop radius of a parent parcel, Zl = Wel /Rel is
the Ohnesorge number and T = Zl We g 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 =
regimes, the Webber number is very large1 We & 100
ρl U 2 r p
σ .
Usually in spray
The Kelvin-Helmholtz model assumes that a parent parcel breaks into a new parcel of radius rc according to the following derivation:
rc = B0 ΛKH
(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 :
dr
r − rc
=−
dt
τKH
3.726B1 r
3.726B1 r
τKH =
=
ΩKH ΛKH
UKH
1 The
Webber number seems not be upper bounded. In a typical CFD Spray Simulation for a fuel injector Wemax ≈ 106
9
(2.5)
(2.6)
Again, B1 is model constant subjected to calibration. The standard model sets B1 = 40. However in
the CFD code [13], the default value is set to B1 = 7, which is a value in good agreement with typical
Diesel Sprays. Again we see the magnitude of the changes occurring in this constants, spanning almost
changes of one order of magnitude.
Finally, the newly created droplets are assigned a velocity given by:
v = C1 ΛKH ΩKH
(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. The
breakup of the droplets beyond the intact core is linked to the Rayleigh-Taylor (RT) mechanism [9]. The
Rayleigh-Taylor Break up model accounts in particular for the break up due to the rapid deceleration of
the particle in the fluid domain due to the drag forces [13].
The wavelength and frequency of the most unstable RT waves are given by:
s
3σ
a(ρl − ρ g )
(2.8)
v
u
u 2 a ρ − ρ 3/2
g
l
=t √
ρl + ρ g
3 3σ
(2.9)
Λ RT = 2π
Ω RT
The child droplet size and the characteristic break up time is then given by:
πCRT
K RT
Cτ
=
v
Ω RT
rc =
τRT
For each of the models, the breakup distance is now defined according to: L = Uτ.
r
ρl
LKH = B1 r0
ρg
r
ρl
L RT = CBL d0
ρg
(2.10)
(2.11)
(2.12)
(2.13)
For the RT Mechanism to be coherent with the KH model, CBL = B1 /2. This however is not typically the
case 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 the
drag 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 (Reynolds
Number).
Neglecting body forces (specifically, gravity), the force on a droplet reduced to the aerodynamic drag of
the form:
Md
∂vi
∂v
1
= ρl V i = ρ g A f CD |Ui |Ui
∂t
∂t
2
(2.14)
Where Ui = ui + ui0 − vi (relative velocity between fluids), V is the droplet volume and finally A f = πrd2
is the droplet cross section area2 . This leads to:
∂vi
3 ρg
|U |U
=
CD i i
∂t
8 ρl
r
(2.15)
This is the motion equation employed in ConvergeCFD [13] at all times. The drag coefficient, CD is
modelled using the TAB/Dynamic drag model [13]. This model performs an analogy with a springmass system. Therefore, it allows to compute the drag force taking into account the deformation of the
droplet.
In this spring-mass system, the droplet drag is represented by the external force, the damping of the
system by the viscosity and the restoring force by the surface tension.
F − kx − c ẋ = m ẍ
(2.16)
The force, F, the spring constant, K and the viscosity coefficient, c are as follows:
ρ g |Ui ||Ui |
F
= CF
m
ρ l r0
(2.17)
k
σ
= Ck 3
m
ρ l r0
(2.18)
µ
c
= Cd l 2
m
ρ l r0
(2.19)
The drag coefficient of a sphere can be approximated by:
(
CD,sphere =
0.424
1 + 61 Re2/3
24
Re
if Re > 1000
if Re ≤ 1000
In order to account for the droplet distortion, the sphere drag is corrected [13] using:
2 Spherical
Droplet Assumption at this step, therefore V = 43 πrd3
11
(2.20)
CD = CD,sphere (1 + 2.632y)
Where droplet state is represented by the normalized value y =
The constants CF = 13 , Ck = 8, Cd = 5 and Cb =
2.5
1
2
(2.21)
x
Cb r0
have been tuned empirically.
Spray Vaporizing Models
For the simulations contained in this report, the so called Frossling correlation is employed to model the
changes in droplet radius due to evaporation. This correlation dictates [13]:
ρg D
dr0
=−
B Sh
dt
2ρl r0 d d
(2.22)
Where r0 is the droplet radius, Bd is parameter that relates the amount of fuel vapour at the surface to
the 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 velocity
to the mean flow velocity and on the Schmidt number. Here the Schmidt number is defined as:
Sc =
µ air
ρ gas D
(2.23)
Where D is the vapour diffusivity in the flow. One of the possible tuning parameters in the Vapour
models is precisely the diffusion term which is obtained by evaluating [13]:
ρ gas D = 1.293D0
Tgas + 2Td 1
3
273
n0−1
(2.24)
The values of D0 and n0 are difficult to define universally for a combinations of different materials and
thermodynamic state of the gas. Typically D0 = 4.16 × 10−5 and n0 = 1.6 for Diesel fuels being injected
in a hot chamber.
The spray behaviour is in general less sensitive to the vaporizing model than other tuning candidates
as the break up model or the turbulence model. For this reason, we will conserve the the vaporization
model 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 laboratories
web page in the US. Spray A is a Diesel Spray Injected into a stagnant pressurized and heated chamber
to 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 properly capture the vapour and liquid penetration. The experimental data set can be identified using the
codename: 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 is
filled with an inert gas with the following composition3 :
Species
O2
N2
H2 O
CO2
Concentration
0.0%
89.71%
6.52%
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=
3 2
u
= 0.06 [m2 /s2 ]
2 rms
(2.25)
At the same time, the turbulent dissipation is calculated as:
e = Cµ
k2
νt
(2.26)
However the turbulent viscosity is still unknown. The results from Reitz and Abani in [6] are retrieved
where the turbulent viscosity in the mixing layer between the vaporizing jet and the stagnant air is
reported as:
deq
νt = Ct π 0.5 Uinj
(2.27)
2
q
ρ
Where Ct is a constant with a value of Ct = 0.0161 and deq = dnozz ρfuel
as reported in [18]. Finally, the
gas
3 To
prevent the fuel from autoignition, oxygen has been removed
13
initial turbulent dissipation in the chamber is given by:
e = 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 chamber through a circular surface with a diameter equal to that of the nozzle: dnozz = 0.084[mm]. The
discharge 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 two
conical 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 AMR
refines 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 Cells
in 2D, 8 in 3D) occupying the same volume as the original one. A higher level of refinement implies an
exponential increase of N = 4 L cells. The refinement level has a dramatic impact on the simulation time
since 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, the
relationship with the Eulerian grid, and the grid dependency. In order to meet our goal and obtain grid
independent and calibrated results, the following roadmap is followed. The RNG K − e model will be
employed.
Figure 2.5: Spray CFD Road Map
The report will start by analysing the dependency among phases, that is, the parcel number. The parcel
number 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 liquid
jet calibration is the last calibration step. The turbulence model affects both the liquid jet and vapour
penetration whereas the break-up model will have a negligible effect on the vapour phase (at least in
the 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 with
finer grids. Typically the number of parcels at a given location (cell) at a given time step should remain
constant under further refinement.
In the case of halving the volume of the cells4 , then the total number of parcels should be twice the initial
number (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 cell
side length becomes:
1/3
1
a0 =
a ≈ 0.79 a
(2.29)
2
4 Increasing
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. However, in order to asses how many parcels there has to be initially, a study of the effect of the parcel number
must be carried out. Below, keeping the cell size constant, a sweep over the number of parcels injected
is plotted.
15
Vapour Penetration [mm]
Liquid Penetration [mm]
60
40
N P = 1.2× 10 3
N P = 9.4× 10 3
20
N P = 2.5× 10 4
N P = 7.5× 10 4
0
N P = 6× 10 5
10
5
0
N P = 1× 10 6
-5
Avg. Liquid Penetration [mm]
0
0.5
1
Time [ms]
1.5
2
0
0.5
1
Time [ms]
1.5
2
9
2
8
1.5
Average Liquid Penetration
Accepted Independence
Std. Deviation from Mean
7
6
1
0.5
5
104
105
Std. deviation from mean [mm]
-20
0
106
Parcel Number []
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 be
related to the inertia of bigger parcels sizes and less number of collisions. A lower number of parcels
implies fewer collisions and therefore a major part of the momentum is spent only to counteract the
drag.
The liquid penetration is a very noisy signal for low parcel number. This is because the low number
of parcels leads to stronger but less frequent collisions. A higher number of parcels predicts a higher
number of collisions. A higher number of collisions leads to a much more uniform description of the
spray.
It is particularly interesting how for a large number of parcels, the vapour penetration decreases, however 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 surface area of droplets). A faster evaporation also leads to fewer collision which also points in the direction
of larger vapour penetration.
5 The
base grid side length is 79% of the original one
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%
6 Vapour
16
In order to better understand this phenomena, one can plot the turbulent kinetic energy levels for different parcel numbers. The following plot reveals a much stronger interaction for high parcel numbers
between 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 1.5
ms 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 the
vapour penetration.
17
Grid Label
1
2
3
4
5
6
Cell Size (a)
5.04
4.00
3.17
2.52
2.00
1.59
Parcel Number
3.75 × 104
7.50 × 104
1.50 × 105
3.00 × 105
6.00 × 105
1.20 × 106
Total Cells (End Of Simulation)
8.5 × 104
1.66 × 105
4.4 × 105
7.1 × 105
1.1 × 105
3.7 × 106
Wall Clock Time
0.18 h
0.48 h
1.37 h
4.49 h
10.36 h
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 the
vapour and liquid penetration. The number of parcels is related to the grid size as given by Eq.2.29. An
increase 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 penetration
figures are shown below.
Vapour Penetration [mm]
60
40
20
a = 5.04 mm
a = 4.00 mm
a = 3.17 mm
a = 2.52 mm
a = 2.00 mm
a = 1.59 mm
0
-20
0
0.2
0.4
0.6
0.8
Time [ms]
1
1.2
1.4
1.6
0
0.2
0.4
0.6
0.8
Time [ms]
1
1.2
1.4
1.6
Liquid Penetration [mm]
15
10
5
0
-5
Figure 2.8: Grid Convergence
The vapour advances faster at the very beginning of the simulation for fine grids. Later its speed lowers
and coarser grids keep on penetrating the chamber further distances. For the case of the liquid penetration, 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
7 Wall
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 previous
simulations. 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 a
collision 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 liquid
penetration versus the number of cells in the grid (taken into account the increase in parcel number at
the same time) and the Vapour penetration. Grids 2 to 5 are considered in this analysis.
Vapour Penetration [mm]
60
NTC Collision Mesh
No Collision Mesh
50
40
30
20
10
0
0
0.5
1
1.5
1
1.5
Time [ms]
Collision Mesh on Grids 1-4
Liquid Penetration [mm]
30
NTC Collision Mesh
No Collision Mesh
25
20
15
10
5
0
0
0.5
Time [ms]
Figure 2.9: Convergence based on NTC Collision Mesh
Let us now define L̃ P as the mean normalized liquid penetration (normalized with respect to the penetration of grid #2). In the horizontal axis, Ñ = Cellsi /Cells1 which is the increment in the total cells
with respect to cells in the coarsest grid. Regarding the vapour penetration, it is known from the analytical 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 penetration. The convergence of the parameter mi is monitored.
s = mi t1/2
(2.30)
In Fig.2.9 It is possible to observe that the collision mesh affects mostly to the liquid phase. This is
not surprising since the collision mesh only acts on the Lagrangian phase which is the liquid jet. The
collision 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 a
collision 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 per
cell will be reduced and therefore lowering the number of collisions. On the other hand, on coarser grids
the Eulerian phase cannot be properly simulated. At the same time, parcels can collide with parcels in
another cell even if they are actually colliding.
Liquid Penetration
1.1
1
NTC Collision Mesh
No Collision Mesh
1
0.9
mi
0.9
L̃P
Vapour Penetration
1.1
NTC Collision Mesh
No Collision Mesh
0.8
0.8
0.7
0.7
0.6
0.6
0.5
0.5
2
4
6
8
10
12
14
2
Ñ
6
8
10
12
14
Ñ
1
1.1
NTC Collision Mesh
No Collision Mesh
0.9
NTC Collision Mesh
No Collision Mesh
1.05
1
mi
0.8
L̃P
4
0.7
0.95
0.6
0.9
0.5
0.85
0.4
0.8
1
2
3
Grid Number (coarse to fine)
4
1
2
3
Grid Number (coarse to fine)
4
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 Lagrangian
spray model contain several constants which cannot be determined analytically. It is expected that our
model requires of certain level of calibration given that the current models cannot reproduce completely
the physics involved in this simulation. This is clearly shown when comparing the CFD results with
the experimental values of the Sandia Spray A, data set: bkldaAL1. This is further evidenced in the next
figure:
By looking at the results of Fig.2.10 and Fig.2.9 together with the wall clock times shown in table2.1, it
was 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 − e model will be studied for the previous
conditions using grid #3. S. B. Pope in [16] achieved jet calibration modifying only the turbulent model
constant Ce1 which so far has taken the standard value for the RNGK − e Model. For the present work,
both the constants Ce1 and Ce2 will be analysed together with their effects on the liquid and vapour
penetration.
20
60
40
20
0
Experimental Data
CFD Grid #3
-20
0
0.2
0.4
0.6
0.8
1
1.2
1.4
1.6
15
10
5
0
Experimental Data
CFD Grid #3
-5
-0.2
0
0.2
0.4
0.6
0.8
1
1.2
1.4
1.6
Figure 2.11: Converged CFD and Experimental Data
The domain of study cannot be well justified before running the simulation. It may happen that the
domain of study becomes also part of this iterative process. The goals to achieve are in based on the
mean liquid penetration and maximum vapour penetration.
The RNG K − e equations implemented in ConvergeCFD are shown below:
!
D (ρk)
∂
µt ∂k
µ+
+ Pk − ρe
=
Dt
∂x j
σk ∂x j
!
e
D (ρe)
∂
µt ∂e
=
µ+
+ ce1 Pk − c∗e2 ρe
Dt
∂x j
σe ∂x j
k
cµ η 3 (1 − η/η0 )
1 + βη 3
1/2 k
η = 2Sij Sij
e
c∗e2 = ce2 +
(2.31)
(2.32)
(2.33)
(2.34)
Let us now roughly estimate the magnitude of the production and dissipation terms:
Pk ≡
e∼
∂U
ui0 u0j i
∂x j
k2
≈ 2νt Sij S ji ∼ Cµ
e
Uinj
dnozz
2
k3/2
dnozz
(2.35)
(2.36)
(2.37)
From where it is possible to estimate the relative importance of the terms as:
2
Uinj
Pk
∼ Cµ
e
k
21
(2.38)
2 and therefore, Pk ∼ C . This
Following the last equation, one could in principle say that k ∼ Uinj
µ
e
however is a rather simple analysis which can only provide some rough guidance for the calibration.
The coefficient Ce1 affecting to the production term in the dissipation rate equation will lead to smaller
and slower changes in the dissipation in the flow field. If Pe < 1 then: (Ce1 P − Ce2 ρe) becomes less
negative for increasing values of Ce1 . This in principle would point out in the direction of increasing the
vapour penetration distance as the dissipation does not feel fast enough the perturbation caused by the
spray. The same argument for lower values than the standard ones of Ce2 are also valid. This approach
was also suggested by S.B Pope [16]. In this publication Pope modifies8 Ce1 such that Ce1 =1.6 in order
to solve the turbulent jet penetration under prediction It is possible to think of this coefficients as "flow
damping" coefficients.
Based on the previous analysis, the following experiments will be carried out:
Design Point
Ce1
Ce2
Vapour [mm]
Liquid [mm]
1
1.2
1.2
50.93
10.0
2
1.2
1.5
40.84
7.22
3
1.2
1.8
35.89
5.63
4
1.5
1.2
94.36
24.6
5
1.5
1.5
55.37
11.51
6
1.5
1.8
43.36
9.09
Table 2.2: RNG K − e Calibration Domain
100
Vapour Penetration [mm]
80
C 1 = 1.2 C2 = 1.2
C 1 = 1.2 C2 = 1.5
60
C 1 = 1.2 C2 = 1.8
40
C 1 = 1.5 C2 = 1.2
C 1 = 1.5 C2 = 1.5
20
C 1 = 1.5 C2 = 1.8
Std Values
0
-20
0
0.2
0.4
0.6
0.8
Time [ms]
1
1.2
1.4
1.6
Liquid Penetration [mm]
30
25
C 1 = 1.2 C2 = 1.2
20
C 1 = 1.2 C2 = 1.5
C 1 = 1.2 C2 = 1.8
15
C 1 = 1.5 C2 = 1.2
10
C 1 = 1.5 C2 = 1.5
C 1 = 1.5 C2 = 1.8
5
Std Values
0
-5
-0.2
0
0.2
0.4
0.6
0.8
Time [ms]
1
1.2
1.4
1.6
Figure 2.12: Turbulence Model Analysis
The time evolution of the spray given the previous constants values can be visualized in Fig.2.12. Mainly
at this step, the goal is to match as good as possible the vapour penetration curve. The values corresponding to (Ce1 , Ce2 ) = (1.5, 1.5) seem to match the best the vapour penetration without strongly
compromising the liquid penetration curve.
The liquid penetration peak at approximately 0.1 ms seems to be independent of the turbulence model
8 This
paper is related to the standard k − e model
22
(at least on the range considered). It will always overshoot for all the turbulence model combinations
studied. The magnitude of the peak can be correlated with increasing turbulence production (which can
be identified by lowering the ce2 or increasing Ce1 ).
One must note that the combination of (Ce1 , Ce2 ) = (1.5, 1.2) that is, increasing the production rate
while decreasing the dissipation rate constants critically increases the vapour penetration and liquid
penetration 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 Ce1 and Ce2 (left most column) has only a scaling effect
on the turbulence penetration. The structure of the spray remains similar to the one computed using
standard values. It is possible to conclude that by increasing the Ce2 constant, the destruction of dissipation is increased and therefore the turbulent kinetic energy is larger mixing more the flow with its
surroundings. As a consequence, a wider spray appears.
At the right most column, the value of Ce1 is then increased while keeping Ce2 at its lowest value consid23
ered. This modification leads to a flow field more turbulent where the flow is relatively slow to adapt to
the changes produced by the fuel spray. Besides, the higher levels of turbulence and therefore smaller
scales present lead to believe that the structure of the flow field is different enough to believe this might
be 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 penetration 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 the
time. This helps identifying the asymptotic slope of the penetration as shown below:
120
EXP
C 1 = 1.2 C2 = 1.2
C 1 = 1.2 C2 = 1.5
C 1 = 1.2 C2 = 1.8
100
C 1 = 1.5 C2 = 1.2
C 1 = 1.5 C2 = 1.5
C 1 = 1.5 C2 = 1.8
Std Values
Vapour Penetration [mm]
80
60
40
20
0
0
0.2
0.4
0.6
0.8
√
1
1.2
1.4
t [ms0.5 ]
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 the
spray should be matched as good as possible but on the other hand, the tendency of the curve at large
times 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 matching
the penetration as good as possible during the simulated time. In practise, the fuel sprays deal with
high temperature combustion and the distance that they traverse before the fuel ignites is relatively
lower 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 of
view, 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]
1.8
1.7
1.6
90
1.8
80
1.7
20
1.6
[]
70
[]
ǫ2
1.5
C
ǫ2
C
Liquid Penetration [mm]
60
1.4
15
1.5
1.4
50
10
1.3
1.3
40
1.2
1.2
1.25
1.3
1.35
Cǫ1 []
1.4
1.45
1.2
1.2
1.5
1.25
1.3
1.35
Cǫ1 []
1.4
1.45
1.5
1.8
1.7
s l = 10.1 [mm]
Std. RNG
C
ǫ2
[]
1.6
s v = 56.7 [mm]
1.5
1.4
1.3
1.2
1.2
1.25
1.3
1.35
Cǫ1 []
1.4
1.45
1.5
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 would
ideally give us a set of values for the turbulence model that could be used for both liquid and vapour
phases calibration. It is advised to the reader that the previous response surfaces originates from relatively 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 calibration
constants. 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
45
40
35
error
30
25
20
15
10
5
1
2
3
4
Design Point
5
6
STD
Figure 2.18: Error Metric in order to choose Calibration Constants
From now on, the set of constants (Ce1 , Ce2 ) = (1.5, 1.5) will be chosen in our simulations as they are
the ones to predict the best the vapour penetration curve in general. The comparison with experimental
data might be visualized in a better way below:
Vapour Penetration [mm]
60
40
20
C 1 = 1.5 C2 = 1.5
0
Std. RNG
Experimental Data
-20
0
0.2
0.4
0.6
0.8
Time [ms]
1
1.2
1.4
1.6
Liquid Penetration [mm]
20
15
10
C 1 = 1.5 C2 = 1.5
5
Std. RNG
Experimental Data
0
-0.2
0
0.2
0.4
0.6
0.8
1
1.2
1.4
1.6
Time [ms]
The same misprediction of the spray penetration using the standard RNG K − e model is shown in
[17]. This seems to be a problem related to the RANS models. In [19] LES Simulations resolving smaller
turbulent scales seem to be the only possibility to accurately predict the vapour penetration of a fuel
spray [19]. The computational cost is extremely high (between 25 to 30 million cells are needed just
for the spray). The alternative is to use a calibrated RANS Model. It predicts more accurately the
vapour penetration (even though the matching is not perfect), but the set-up of the turbulence model
will be valid only for the current operating conditions (chamber pressure, temperature and background
turbulence levels mainly).
Argonne National Laboratories (USA) and ConvergentScience (USA) show in [11] the deficits of RANS
modelling 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 calibration
is to perform an LES simulation.
By using a more accurate turbulence model (LES) a better prediction of the turbulence levels is obtained
and therefore a much better spray penetration prediction. The cost of an LES is extremely high and
typically 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 to
calibrate the mean liquid penetration, the size and time constants affecting the primary break up are
here studied using grid #4 as described Table2.1.
Let us recall here the equations describing the characteristic break time and droplet radius:
rc = B0 ΛKH
τKH =
(2.4)
3.726B1 r
3.726B1 r
=
ΩKH ΛKH
UKH
(2.6)
dr
r − rc
=−
dt
τKH
(2.5)
The standard break up model according to [9] dictates that B0 = 0.68, B1 = 40. However the range of
values that B1 can take depends on the fuel as well as operating conditions. According to [13], the time
constant 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 therefore
will affect to the size more than the speed at which the break up occurs. However one can rearrange the
previous equations to lead to:
r − B0 ΛKH
r̄ − B0
1 dr̄
1
dr
=−
= {r̄ = r/ΛKH } = −
=
ΛKH ΩKH dt
3.726B1 r
3.726B1 r̄
ΩKH dt
(2.40)
In principle, one would be interested in noting the most sensitive parameter. In this sense one can define
a coordinate system based on the values of B0 and B1 . The biggest changes, that is, the steepest slope of
the response surface to Ω1KH dr̄
dt can be found by computing the gradient over this coordinates as:
∂
∂
~u0 +
~u
∂B0
∂B1 1
1 dr̄
1
~u0 +
=
ΩKH dt
3.726B1 r̄
1
B0
−
2
3.726B1
3.726B12 r̄
!
~u1
(2.41)
Using the previous equation, one must first identify the order of magnitude of r̄ = r/ΛKH . For that
purpose, the results from Eq.2.2 are here retrieved:
1 + 0.45Zl0.5 1 + 0.4T 0.7
ΛKH
r̄ =
= 9.02
(2.2)
r
1 + 0.87We0.87
g
In principle, according to the previous equation r̄ ≈ 10−4 . Based on this number it would be possible to
design a set of experiments following the gradient vector equation in Eq.2.41 as:
28
Max Slope From Std. Values
7
18
Steepest Path
Std. Values (Start Point)
Considered Combinations
6
16
14
Liquid Penetration [mm]
5
B1
4
3
12
10
8
6
2
4
[B 0 ,B 1 ] = [0.68,7.00]
[B 0 ,B 1 ] = [2.62,6.53]
1
[B 0 ,B 1 ] = [4.56,5.36]
2
[B 0 ,B 1 ] = [6.50,2.70]
0
0
0
1
2
3
4
B0
5
6
7
8
0
0.05
0.1
0.15
time [ms]
0.2
0.25
0.3
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 maximum penetration achieved by the liquid spray. Increasing the constant B0 while decreasing the constant
B1 increases the size of the particles along the jet. This is mainly due to Eq. 2.4 which determines the
size of children parcels after break up. The jet breaks but the new particles are still quite big. Recall
that bigger spheres suffer less drag9 and therefore a higher overshoot is achieved. This is however just
an academical exercise since in our aim to match the experimental data, the interest is to decrease such
overshoot.
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 than
the one with standard values shown in the upper figure. This seems to be in agreement with B0 value
which 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 opposite
direction (that is decreasing B0 and increasing B1 ). The size constant B0 cannot be reduced much more,
9 Drag
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 the
overshooting. Below an analysis of this two constants is presented:
Varying B0
15
Varying B1
15
sl [mm]
10
sl [mm]
10
B0 = 0.10
5
B1 = 4.00
5
B0 = 0.68
B1 = 7.00
B0 = 2.05
B1 = 12.00
B0 = 4.00
B1 = 20.00
0
0
0
0.02
0.04
0.06
0.08
0.1
0.12
Time [ms]
0.14
0.16
0.18
0.2
0
0.02
0.04
0.06
0.08
0.1
0.12
Time [ms]
0.14
0.16
0.18
0.2
B1
4
6
8
10
12
14
16
18
20
12.5
Max Liquid Penetration [mm]
12.5
12
12
11.5
11.5
11
11
10.5
10.5
0
0.5
1
1.5
2
B0
2.5
3
3.5
4
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 B1
and B0 leads to a smaller overshoots (around t = 0.07 [ms]) due to a slower decrease in the droplet
size. The larger the surface area of the droplets the faster the evaporation and therefore the mean liquid
penetration is achieved in a smoother way and earlier time. At the same time, this is reflected in the
vapour penetration curve as:
Varying B0
50
40
30
20
sv [mm]
sv [mm]
Varying B1
50
40
B0 = 0.10
30
20
B1 = 4.00
B0 = 0.68
B1 = 7.00
B0 = 2.05
10
B1 = 12.00
10
B0 = 4.00
B1 = 20.00
0
0
0
0.5
1
1.5
0
0.5
1
Time [ms]
1.5
Time [ms]
B1
4
6
8
10
12
14
16
18
Max Vapour Penetration [mm]
45
20
45
44.5
44.5
44
44
43.5
43.5
43
43
42.5
42.5
42
42
0
0.5
1
1.5
2
B0
2.5
3
3.5
4
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 the
constant B0 achieves the creation of much smaller particles with respect to the standard model. More
interestingly is the physical process involving high B1 values. This constant does not generate so many
child parcels and empties the core of the spray thus leading to also less overshooting. The increase in
the characteristic time maintains big parcels with high number of droplets within them without actually
colliding 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 size
constant B0 = 0.1 and keeps constant the standard time constant. Correcting the overshoot was not
possible.
Vapour Penetration [mm]
60
40
20
0
Experimental
Calibrated RNG and KHRT
-20
0
0.2
0.4
0.6
0.8
Time [ms]
1
1.2
1.4
1.6
Liquid Penetration [mm]
15
10
5
0
Experimental
Calibrated RNG and KHRT
-5
-0.2
0
0.2
0.4
0.6
0.8
1
1.2
1.4
1.6
Time [ms]
Figure 2.24: Calibrated Turbulence and Break Up Model. [ce1 , ce2 , 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-similar
jet penetration can be obtained following the derivation of [6]. This self-similar approach is dependent
on the liquid properties during the injection.




Uv ( x, r ) = min Uinj ,

2 d2

3Uinj
eq

2 
2 d2 r 2

3Uinj
eq
32νt x 1 + 256ν2 x2
(2.42)
t
Eq. 2.42 depends on the equivalent diameter which is a function of the liquid density which is ill defined
as the temperature changes with the penetration distance. For the next plot a value of fuel liquid density
of ρl = 425Kg/m3 will be used in accordance to the material properties at 600 Kelvin. This temperature was chosen as the average between the injection temperature and the chamber temperature. The
equivalent diameter and the turbulent viscosity are defined as:
r
ρl
deq = dnozz
(2.43)
ρg
νt =
1 √
Ct πUinj deq
2
(2.44)
Where Ct was reported by Abraham [7] to be Ct = 0.0161. For the present case (Sandia Spray A) it has
been found that a value of Ct = 0.0191 gives the best possible match with the experimental data for the
self similar jet.
The density of the fuel changes dramatically due to the fuel thermal expansion. The density changes are
related to the coefficient of thermal expansion as:
β=−
1 ∂ρ
ρ ∂T
(2.45)
C12H26 Material Properties
1000
Liquid Density [Kg/m3]
900
800
700
600
500
400
300
0
100
200
300
400
Temperature [K]
500
600
Figure 2.25: Fuel Material Properties
32
700
Finally, using Eq.2.42, the vapour penetration of the spray tip can be computed as:
t( x ) =
Z x
x =0
dx
U ( x, r = 0)
(2.46)
Besides, Hiroyasu and Arai [8] worked on the experimental correlation of fuel sprays discharging in a
hot, inert and stagnant chamber. They found the following correlations which serve to appreciate the
importance of some typical spray parameters as the discharging pressure and the chamber temperature
(gas density).
q

0.39 2∆P
0 ≤ t ≤ tb
ρ t
l0.25 √
s=
(2.47)
2.95 ∆P
d
t
t
≤
t
nozz
b
ρg
The vapour penetration prediction through the different models are plotted in Fig. 2.26 together with
the penetration prediction given by the calibrated CFD Model.
60
Experimental Penetration
Self-Similar Jet
Hiroyasu and Arai
CFD Calibrated
50
Vapour Penetration [mm]
40
30
20
10
0
0
0.5
1
Time [ms]
Figure 2.26: Self-Similar Jet, Statistical Correlation and CFD Prediction
33
1.5
Real Injection Profiles
Real Injection Profiles are non constant. In order to handle them it is possible to use the Duhamel
superposition 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 correct
Eq. 2.42. In this fashion, it is possible to express the equation as:
Uv =
3Ue f f deq
if x > xb
K
(2.48)
At this point, it is interesting to insert a momentum balance to account for the drag of the droplets as:
m
1 πD2
dv
= ρg
CD (u − v)|u − v|
dt
2
4
(2.49)
Where v represents the velocity of the gas and u the velocity of the droplet. Under the assumption of
low Reynolds Number, the drag of a sphere can be approximated as:
CD =
24
Re
(2.50)
Being possible now to rewrite Eq. 2.49 as:
18µ CD Re
dv
1
=
(u − v) =
(u − v)
dt
τv
ρl D2 24
(2.51)
Since the ratio CD24Re 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 shape
of Eq. 2.52. According to [6], the variations in the injection profile can be accounted by Duhamel superposition as:
n
∑ A(x, t − tk )(∆Uinj )k
Ue f f = Uinj (t = 0) +
(2.53)
k =1
t − tk
A( x, t − tk ) = 1 − exp −
τv,k
(2.54)
At the same time, in [6] it is proposed to generalize the response time as τvk = St Ux . St stands for the
k
Stokes number.
Now it is possible to rewrite the previous equations in integral for by:
(∆Uinj )k ≡
(∆Uinj )k
∆tk
∆tk
34
(2.55)
Meaning that:
Ue f f ( x, t) = Uinj ( x, t = Break) +
Z τ =t
τ =Break
t−τ
dU
1 − exp −
dτ
τav
dτ
(2.56)
In [6] it is proposed to use an average response time based on:
τav =
St deq
Uav
(2.57)
Where deq is given by Eq. 2.43 and the average velocity Uav = U inj . The Stokes Number is a constant that
requires some fine tuning. Here a value of St = 50 as employed in [6] has been used. Error minimization
by 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 injection profile as it really happens in fuel injectors.
35
Chapter 3
Case Study: Diesel ULPC Piston Bowl
Optimization
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, fuel
spray orientation, and piston bowl geometry using Converge CFD and CAESES, for the flow simulation
and automated optimization respectively. The goal of the optimization is to reduce NOx and SOOT
emissions.
The Ultra Low Particulate Combustion (ULPC) Piston Bowl is analyzed in this study. These types of geometries increase the local air to fuel ratio, improving the mixing and finally leading to lower pollutant
emissions. For the analysis, a variable parametric geometry of the piston bowl with volume control is
generated in CAESES. CAESES drives the automated execution of Converge CFD computations including 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 effects
of spray orientation are then not well considered for different types of geometries. In [21], it is analysed
the effects of the bowl radius (in plan view) and depth on the emissions obtaining a reduction of 60% in
PM. In [22] the effects of the lobe diameter with respect to the depth are analysed. The results conclude
that a higher temperature in the cylinder reduces the final emissions of Soot, however it increases the
amount of NOx. The injection strategy and fuel spray orientation have a strong impact on the pollutants
considering that this are ultimately determined by the thermodynamic state of the piston and therefore
the 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 most
relevant parameters are the diameter of the bowl, the spray angle and start of injection. The number of
nozzles in the injector has almost no effect on the emissions.
Summarizing, there is evidence enough in the literature to substantiate an optimization that considers
the injection timing, cone angle and geometrical parameters together.
36
3.2
3.2.1
ULPC Piston Bowl
Pilot and Post Injection
The pilot injection is meant to produce and initial flame ahead of the injector in order to heat the surrounding air. The goal with the pilot injection is to increase the chamber temperature at TDC beyond the
auto-ignition temperature of Diesel1 . At the same time, a uniform increase of the chamber temperature
will help to avoid engine "knock"2 as well as reduce the evaporation time reducing wall films. This in
the 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. This
triggers another design variable in the optimization process of ICE. It is reported that late injections
together with high dwell times reduce the formation of NOx. Soot and unburned hydrocarbons seem to
be larger in this case therefore making one of the optimization objectives be agains the other one. This
will 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 exhaust gases to displace part of the air entering the chamber during the intake stroke. Diesel running
always lean achieves higher temperatures than petrol engines. The main consequence is a higher NOx
production. In order to reduce the emissions, the exhaust gases reduce the amount of fresh air thus slowing 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 combustion
chemical reaction which provides sufficiently accuracy for this calculation.
79
C7 H16 + ξ O2 + N2 −→ α1 CO2 + α2 H2 O + α3 N2
(3.1)
21
The solution leads to ξ = 11, α1 = 7, α2 = 8 and α3 = 82.761. This is the so called stoichiometric ratio
between fuel and fresh air (here consider to be a unique composition of oxygen and nitrogen). In order
to calculate the species mass fractions χi , the next step is to calculate the weight of each of the products
mi of the previous equations.
1 Diesel
Auto Ignition temperature, depending on the exact composition and thermodynamic state is approximately 220◦
uncontrolled fuel ignition
2 localized
37
CO2
44
H2 0
18
N2
28
Table 3.1: Molecular Weights in [g/mol]
The mass fraction of each species under stochiometric conditions is given by:
χi =
αi mi
∑ mi
(3.2)
For a given EGR mass fraction ψ, one defines the mass fraction of exhaust gases within the chamber by
computing:
χi = ψχi
(3.3)
And the mass fraction of pure air will be given by:
χO2 = (1 − ψ) × 0.21
(3.4)
χ N2 = (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. The
Intake stroke is the first one occurring in the cycle. Fresh Air enters the chamber when the piston is at
TDC (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 fully
close. 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 close
the valves again at TDC (top dead center), the injector releases the spray of fuel and the diesel will auto
ignite due to the chamber pressure and temperature. The release of energy will push the piston down
producing mechanical work.
Slightly before getting to BDC (bottom dead center), the exhaust valves will open again allowing to
release 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 process
of the internal combustion engine. The analysis of the power stroke can be strongly simplified without decreasing strongly the accuracy of the simulation by employing a periodic control volume. The
simplification 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 motion
within the cylinder.3 . At this point it is not the scope of the project to asses the differences or the error
due to simplifying the simulation in this way. Petersen and Miles in [24] performed a set of experimental
measurements at Sandia National Laboratories confirming the swirling nature of the flow in the chamber
prior to combustion.
For the simulation, the flow will be initialize using the so called wheel-paddle method. The flow initialization might be part of the validation process since a perfect description of the flow at the beginning of the
simulation might not be available. Reuss et Al. [25] investigated the importance of the flow initialization
in combustion simulations. It has a strong effect on the combustion efficiency and fuel mixing with the
surrounding 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 flow
near the walls of the cylinder. Petersen et Al [24] solve5 the angular momentum of the Navier-Stokes
equations to give:
"
#
Ωrb Rs α
r
α2
vθ (r, t) =
· J1 (α ) · exp − 2 νt (t − t IVC )
(3.7)
4J2 (α)
rb
rb
3 In
petrol engines the typical motion goal is "tumble" motion
the z-coordinate is always aligned with the axis of the cylinder
5 This solution is subject to assumptions such as axisymmetric flow or axial uniformity (∂/∂z = 0) among others
4 Assume
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 the
bore radius, Rs is the swirl ratio, νt is a viscosity value that whose calibration allows to partially consider
turbulent 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 behaviours. Typically the swirl profile value used in flow initialization is approximately α = 3.11 (at least
according to [13])
Angular Speed Profiles
25
α = 2.8
α = 3.11
α = α n.s
Cylinder Wall
20
vθ [m/s]
15
10
5
0
0
10
20
30
40
Radial Location [mm]
50
60
Figure 3.5: Initial Velocity Profiles at t = t IVC
40
70
3.4.2
CFD Set Up
For our computational model, following the recommendations of [13], the swirl profile at initialization
is set to α = 3.11. The turbulence will be modelled using RNG k − e model. The fuel spray is modelled
following 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 correlation
will 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. and
Argonne National Laboratories in [26]. The number of parcels at a given minimum grid size is given by
the publication and the following scaling law is used when correcting the mesh size:
Np = 50 × 104 × 2
dx
b
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 (Shell
auto-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 the
combustion 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 number
of chemical paths to produce a given chemical specie is enormous. The Zeldovich Model for example
accounts 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, for
example these reactions:
N2 O + O
N2 + O2
(3.13)
N2 O + O
2NO
(3.14)
N2 O + H
N2 + OHNNH
NNH + O
N2 + H
NH + NO
(3.15)
(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 allow
to reproduce the results with very few unknowns which can be derived from the values present in the
publication. The engine specifications are:
Stroke
Bore Diameter
Engine Speed
Swirl Ratio
Compression Ratio
Intake Manifold Pressure
Temperature at Manifold
Air Fuel Ratio
100 mm
125 mm
1900 RPM
1.4
17.4
2.8 bar
356 K
21.3
Table 3.2: Engine Characteristics
The current validation case has an EGR value of 25% thus providing the following mass fraction composition of each specie at initial time:
CO2
2.47%
H2 0
1.15%
O2
18.57%
N2
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 three
different steps: Pilot, Main and Post injection.
1
0.9
Normalized Inj. Rate
0.8
0.7
0.6
0.5
0.4
0.3
0.2
0.1
0
-20
-10
0
10
CAD [deg]
20
30
40
Figure 3.6: Fuel Injection History
The validation will focus in matching the pressure and the heat release rate traces. The experimental
value of such is also published in [21]. This two ultimately determine the emission values as all chemical
reactions 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-Eulerian
spray models. The grid dependency problems have been previously discussed in the Part.2 of the current document.
The grid convergence was previously achieved on grids strongly refined around the nozzle. The size
of the cells around the injector had to be approximately the same as the diameter of the nozzle. This
imposes a strong CPU cost on the simulations as the CFL condition increases dramatically the number
of time steps to perform on a given time span. It must be noted that not only the CFL condition limits
the 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 tune
the spray models and grid sizes in order to achieve a realistic flow description without achieving grid
convergence. 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 detailed
chemistry (200 species and about 2000 chemical reactions). In order to achieve grid convergence, both
the chemical reactions and the Lagrangian phase must be able to converge at some point and produce
the same set of chemical species after the combustion. This increases the predictive capabilities of the
models and decrease the number of tuning factors.
In [26] the cell count in a single sector cylinder simulation with fully detailed chemistry is around 10
million cells. The piston bowl of this case study shares some of the engine specifications in [26], however
the simulation time reported is around 90h to complete on a 64 core workstation.The time requirements
strongly limit our capacity to perform an engine optimization, which in the end is our future goal. The
goal will be to match the heat release rate as well as the pressure trace curves varying the mesh on a
range which is known to be grid dependent.
For this project, the CTC Model will be used to calculate the characteristic time required to achieve
chemical equilibrium. The chemical equilibrium is dependent on both chemical kinetics and local turbulence levels (i.e RANS modelling). The CTC model is used together with the Shell model. This last
model is employed to predict the fuel ignition.
The heat release rate ultimately determines the chemical reactions rate and therefore the production of
pollutants. The following sections will study the combustion process from −147◦ < CAD < 60◦ which
spans 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 and
heat release rate are plotted. At the same time it is of interest to monitor the changes in SOOT and NOx
which 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 front
and therefore on the chemical reaction rates which ultimately determine the emissions amount. The
initial flame front (first row) in the finest case is more elongated and does not interact with the piston
lower 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 above
the lip. It is possible to appreaciate a much smaller flame zone compared to coarser meshes. A faster
combustion in the fine grid reduces the amount of oxygen therefore lowering the local equivalence ratio
and 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 the
grid size can be found:
Base Grid
Min Cell Size
Peak Cell Count (millions)
A
2.8 mm
0.7 mm
0.11
B
2.23 mm
0.55 mm
0.13
C
2.23
0.28 mm
0.45
Table 3.4: Engine Characteristics
The grid affects to the flame structure, evolution and temperature. This is clearly evidenced when looking at the pressure trace and heat release rates:
44
Grid A
Grid B
Grid C
14
12
10
8
6
4
2
-20
-10
0
10
20
CAD [deg]
30
40
50
Heat Realease Rate [J/deg]
150
60
4000
3000
100
2000
50
1000
0
-20
-10
0
10
20
CAD [deg]
30
40
50
Cumulative Realeased [J]
Mean Cylinder Pressure [MPa]
16
0
60
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 combustion that releases more energy than on medium or coarser grid. In the sector simulations performed
in [21] and [9], constant grid sizes were approximately 1mm. This has served as a basis for the current
simulation.
It is interesting to identify a higher pressure trace with a fine grid which is of course correlated with a
much faster heat release rate (HRR). The total heat released (HR) is however approximately the same in
all cases. The differences are mainly due to combustion efficiency differences. Besides, the HRR peak
during the pilot injection is lower in the fine grid case and higher for coarser grids. This evidences again
the dependency of the combustion model on the grid sizes.
The grid dependency can also be observed in the pollutant formation. Below the accumulated value of
NOx and SOOT is plotted:
8
×10-7
7
SOOT Grid A
NOx Grid A
SOOT Grid B
NOx Grid B
SOOT Grid C
NOx Grid C
6
Mass [Kg]
5
4
3
2
1
0
-20
-10
0
10
20
CAD [deg]
30
40
50
60
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 SOOT
however is approximately equal for the medium (B) and fine grids (C). It is possible to observe that
there 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 experimental data and our CFD calculations.
160
Experimental
Grid A
Grid B
Grid C
Pressure [bar]
140
120
100
80
60
40
-30
-20
-10
0
10
20
30
40
50
60
Time CAD
HRR [J/deg]
150
Experimental
Grid A
Grid B
Grid C
100
50
0
-30
-20
-10
0
10
20
30
40
50
60
Time CAD
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 standard
break up models seems to provide an almost perfect match with the pressure curves. The heat release
rate is however more difficult to match as it strongly depends on the chemical reactions. Using the
current 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 a
decrease in the pressure trace of the experimental data likely due to the fact that the combustion was
not able to maintain the pressure level. Later the combustion becomes more intense and the pressure
rises again. The behaviour of the pressure trace in the very end of the time spanned by the simulation is
almost fixed by thermodynamics, therefore it is basically grid independent.
Regarding the heat release rate, the pilot injection is over predicted. However the post injection barely
increases the pressure. This might be due difficulties of modelling the re-ignition process. Using fully
detailed chemistry it is expected that this problem could be easily solved. Also it must be noted that
Grid 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 chemical
path what is modified.
It is concluded that the set up involved using Grid B is accurate enough for the calculations that are of
our 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 was
performed. Variation of the turbulence constants Ce1 and Ce2 between 1.3 and 1.8 (both of them) is performed. The effect on the pressure trace and heat release rate is shown below
Varying Cǫ2
Varying Cǫ1
Experimental Data
150
Experimental Data
150
C ǫ2 = 1.3
C ǫ1 = 1.55
Pressure [Bar]
Pressure [Bar]
C ǫ1 = 1.3
Standard
100
50
0
-30
C ǫ2 = 1.55
C ǫ2 = 1.8
100
Standard
50
0
-20
-10
0
10
20
CAD [deg]
30
40
50
60
-30
150
-20
-10
0
10
20
CAD [deg]
30
60
Experimental Data
C ǫ1 = 1.3
C ǫ2 = 1.3
C ǫ1 = 1.55
100
HRR [J/deg]
HRR [J/deg]
50
150
Experimental Data
Standard
50
0
-30
40
C ǫ2 = 1.55
100
C ǫ2 = 1.8
Standard
50
0
-20
-10
0
10
20
CAD [deg]
30
40
50
60
-30
-20
-10
0
10
20
CAD [deg]
30
40
50
60
At the same time, it would be interesting to monitor what is the effect on the production of pollutants,
mainly NOx and Soot.
47
6
Varying C
×10-6
ǫ2
6
5
C ǫ1 = 1.55
Standard
4
3
2
1
C ǫ2 = 1.55
C ǫ2 = 1.8
4
Standard
3
2
1
0
-30
6
-20
-10
0
10
20
CAD [deg]
30
40
50
0
-30
60
×10-7
6
-20
Standard
NOx [Kg]
2
1
1
-10
30
40
50
60
0
10
20
CAD [deg]
30
40
50
60
C ǫ2 = 1.8
0
10
20
CAD [deg]
30
40
50
0
-30
60
Standard
3
2
-20
10
20
CAD [deg]
C ǫ2 = 1.55
4
3
0
-30
0
C ǫ2 = 1.3
5
C ǫ1 = 1.55
4
-10
×10-7
C ǫ1 = 1.3
5
NOx [Kg]
ǫ1
C ǫ2 = 1.3
SOOT [Kg]
SOOT [Kg]
5
Varying C
×10-6
C ǫ1 = 1.3
-20
-10
It is not possible to find a much better combination of Ce1 and Ce2 than the standard one for the heat
release rate evolution. The pressure trace seems to be however almost independent of the turbulence
model. This is probably due to the fact that no extremely big changes in the combustion and therefore in
the Heat Release Rate occur. However one can note that higher values of Ce2 seem to have a stabilizing
effect over the combustion, therefore contributing to raise the maximum temperature and it is thought
that because of that, there is a higher NOx prediction.
There exists many chemical "paths" through which NOx can be produced. According to [30], the NOx
production will always be increased by an increase in the temperature of the flame, independently of the
dominant 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 modification) in the NOx leads to a decrease in Soot and vice-versa. This is a first indicative of the difficulty to
minimize 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 academically
speaking to analyse the effect of the break up model in the combustion process. Let us perform a study
of the size and time constants analogously to the previous one performed in the Sandia Spray A Section.
48
Experimental
150
Pressure [Bar]
(B0 ,B 1 )=(0.4,7)
(B0 ,B 1 )=(2,7)
(B0 ,B 1 )=(0.68,20)
100
Standard
50
0
-30
-20
-10
0
10
20
30
40
50
60
Time CAD
150
Experimental
HRR [J/deg]
(B0 ,B 1 )=(0.4,7)
(B0 ,B 1 )=(2,7)
100
(B0 ,B 1 )=(0.68,20)
Standard
50
0
-30
-20
-10
0
10
20
30
40
50
60
Time CAD
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 happening due to larger drops in the case of increase the time constant B1 . The particles reduce their size in
a slower way therefore given up higher amount of concentrated fuel vapour thus increasing the violence
of 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 and
ignition process take longer to occur. Based only on the previous results one could in principle expect
that an intermediate value of the size and time constant could help to obtain a better correspondence
between CFD and experimental results. Besides, an increase in the constants leads to much longer
penetration 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 combustion
efficiency. The improvement is given by modifying the geometrical parameters and the spray orientation.
However the designs variations can vary the volume of the bowl. This affects to the compression ratio
which 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 Candidate (red)
A variation of the "Bowl Radius" is represented in the left most column of the previous figure. The
second picture from the left represents a variation in the depth of the bowl. The third one from the left
increases the entrainment of the "secondary lip" typical from ULPC piston bowls whereas the last one is
a variation of the so called "inner diameter".
Additionally, a variation of the spray direction is also taken into account. This parameter is shown in
Fig.3.14
Figure 3.14: Spray Angle Parametrization
The spray direction it is weakly dependent on other parameters. The direction is measured from the
injector position at SOI (start of Injection) towards the secondary lip. This direction is thus altered by
parametric variation of the geometry. The spray direction parameter is only an offset of with respect to
this direction. Before, the domain of study here proposed is shown:
Parameter
Bowl Radius
Depth
Secondary Lip
Inner Diameter Scale
Spray Offset Angle
Lower Bound
4
4
0
0.9
0
Baseline Value
6
6
3
1
0
Upper Bound
8
8
6
1.1
30
However this changes almost always have to be corrected so that the compression ratio is fulfilled. The
parameters chosen to produce such change are:
50
The left most picture represents a variation in the arc composing the bowl. This is accomplished by
varying the tension of the tangency between the arc of the bowl and the guide line from the centre of
the piston. The centre figure represents a vertical scaling. This parameter is only allowed to make
small variations so that it does not affect to the depth design parameter. Finally the right most column
represents a displacement of the bowl arc centre horizontally.
Parameter
Tangency Tension
Vertical Scaling
Horizontal Offset
3.5.2
Lower Bound
-3
0.95
3
Baseline Value
0
1
3
Upper Bound
2
1.05
10
Optimization Algorithm
The optimization algorithm is a surrogate multi-objective genetic algorithm. This algorithm is specially well
suited 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 the
square of the number of parameters (In this case N = 52 = 25). However this is know to be a
lower 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 a
solution 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 based
on the Kriging method.
3. Based on this response surface, new global minimum candidates are computed, 4 in this case. The
algorithm is also tuned to select design points which are at least 10% far from each other in the
design space.
4. The new computed designs allow to improve the quality of the response surface and next 4 design
candidates 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 a
curve 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. The
evolution of the optimization algorithm is shown below:
Pareto Front After 4 Generations
1
Baseline
Initial Sampling
Gen 1
Gen 2
Gen 3
Gen 4
Pareto Front
0.9
0.8
0.7
SOOT []
0.6
0.5
0.4
0.3
0.2
0.1
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
NOx []
Figure 3.15: Pareto Front after 4 Generations
The squares in blue represent the initial pool of design found while sampling the domain whereas the
dashed (- -) lines represents the actual evolution of the algorithm. The response surface of the algorithm
estimates the location of the minimum. However the actual value of NOx and Soot is corrected by
running 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 is
evidenced 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 pareto
front algorithm. This indicates that more generations should be run in order to obtain a smoother and
better posed pareto front.
Designs in the Pareto Front
In order to understand why this improvement has taken place, an analysis starting from the geometry
variations should be done.
The left most design is the best performing design in order to reduce NOx. The main characteristic of
this design is its more pronounced secondary lip with a smaller radius. The so called corner design is the
closest 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, Corner
Value, Minimum Soot
with a slightly more pronounced angle from the bowl axis to the tangency point. Finally the best Soot
performing design features a deeper bowl with a reduced inner diameter which pushes the secondary
lip 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 spray
is 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 NOx sprays
towards the lower side of the bowl. The best design for Soot reduction however sprays towards the liner
upper part of the bowl. The corner design, that is the one that reduces both the Soot and the NOx chooses
to spray towards the secondary lip splitting the flame front between an upper and lower side.
54
Parameter
Bowl Radius
Depth
Secondary Lip
Inner Diameter Scale
Spray Offset Angle
Best NOx
5.99
6.22
5.45
1.04
16.77
Corner
4.41
6.12
2.48
1.01
25.64
Best Soot
5.66
4.05
0.7
1.06
4.3
Table 3.5: Parameter Specification for Pareto Designs
The flame front chooses to generate a vortical motion in the best for NOx design (left most column). This
flame front keeps a larger volume of air at around 1600 K. This is not the extreme value at the flame
front but hot enough to stop soon enough the generation of nitrous oxides. The nitrous oxides seem
to be generated at the hottest portions of air. Relatively fine flame fronts seem to help reducing the
generation of NOx .
The right most case has a more chaotic behaviour. The flame is more turbulent and generates a thicker
flame that increases the average temperature in the burning zone, this hotter region enhances the speed
at which chemical reactions occurred leading to higher NOx amounts.
However, the best for NOx design seals the burning zone from the surrounding air due to its more
laminar alike structure (not so much chaotic). The sealing reduces the amount of oxygen (equivalence
ratio) 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 of
Soot. This is strongly evidenced for the best NOx design which accumulates almost 5 times the stoichiometric fuel to air ration near the axis of the cylinder.
Below the temporal evolution of NOx and Soot is shown.
Metric
NOx
Soot
Baseline
1
1
Best NOx
0.16
1.95
Corner
0.34
0.22
Best Soot
1.49
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 quantities
such as carbon monoxide (CO), carbon dioxide (CO2 ) or hydrocarbons (HC). For this reason one would
55
1.5
Baseline
Best NO X
Soot [mg]
1
Corner Value
Best Soot
0.5
0
-150
-100
-50
0
CAD
50
100
150
-100
-50
0
CAD
50
100
150
0.6
NOx [mg]
Baseline
0.5
Best NO X
0.4
Corner Value
Best Soot
0.3
0.2
0.1
0
-150
Figure 3.18: Pareto Front Emissions
like to analyse the time evolution of not only the goal value (NOx and Soot) but also other pollutants.
At the same time the temporal evolution can provide information about the combustion and oxidation
processes. For example in Fig.3.18, the combustion process for the best soot design finish earlier burning
all the fuel and therefore the soot curve becomes flat earlier than the rest. The best for NOx however
keeps on decreasing at a steady slow rate which leads to think that incomplete combustion might be
happening.
CO [mg]
200
150
100
Baseline
Best NO X
Corner Value
Best Soot
50
0
-150
-100
-50
0
CAD
50
100
150
-100
-50
0
CAD
50
100
150
-100
-50
0
CAD
50
100
150
CO2 [mg]
400
300
200
100
0
-150
HC [mg]
40
30
20
Baseline
Best NO X
Corner Value
Best Soot
10
0
-150
Figure 3.19: Effect on other pollutants
Similarly, the best for NOx or soot designs have some relations that are worth remarking. The best for
soot design leads to almost zero carbon monoxide emissions. This is a measure of the completeness of
the combustion process. This is also why a higher amount of CO2 is produced. The corner design also
achieves complete combustion and exactly same amount of CO2 emissions, whereas the baseline and
56
the best NOx design seem to reach the same amount of CO and CO2 .
Finally the amount of hydro carbons being released from the best NOx design leads to believe that this
design is not worth the reduction of NOx .
3.5.4
Conclusions
The validation of diesel combustion simulation can be done without requiring an excessive amount
computational power but at the cost of requiring experimental data from a test engine. Therefore, there
are advantages and disadvantages to this method.
In order to have simulation that cover all the chemical reaction and the turbulence modelling one can
always keep increasing the accuracy of the models up to LES and fully detailed chemistry. Mainly, the
cost of LES restricts the optimization possibilities and number of iterations required. It is the opinion
of this author that RANS modelling can be sufficient for piston bowl design. Besides detailed detailed
chemistry 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 allows to choose the most relevant chemical paths decreasing the cost while maintaining the accuracy of
this type of simulations. Modelling emissions strongly depends on models which require of some fine
tuning. Detailed chemistry removes such tuning tasks increasing the predictive capabilities of the CFD
simulations.
The optimization algorithm relies on purely automatic analysis. It is believed that the best for NOx design
is not a reliable design. The strong deviation of the equivalence ratio from its stoichiometric value near
the axis zone reveals a strong risk of knocking. This design needs to be tested under slightly different
operating 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 strong
value which is most likely related to the fact that not so much fuel has been burned. Leading to higher
amounts of unburned fuel (Soot) and lower NOx amounts. Optimization in order to minimize NOx has
shown to be much more difficult than expected at the beginning.
The corner design shares many geometrical similarities together with the baseline design, specially the
spray being directed towards the bowl instead of the secondary lip. This seems to be one of the most
relevant 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 ultimately
determines 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 on
the power and pollutant emission.
The heat release rate can be estimated on hand by the combustion efficiency (η) which in turn is a
function of the local equivalence ratio6 and the adiabatic index γ = c p /cv
∂Qch
≈ η (φ( ~X, t), γ(t)) ṁ ρ E
∂t
(3.17)
On the other hand, the first Law of Thermodynamics allow us to express [1]:
δQch = dU + δW + δQht + ∑ hi dmi
(3.18)
The main difficulty of the previous equation is usually related to the accuracy to which we can determine the parameters entering into the previous equation. U represents the internal energy, W the work
performed by the piston, Qht the heat transfer across the walls and ∑ hi dmi the energy leakage across the
crevice and valves. The unusual terms in the first law of thermodynamics are due to the "open" nature
of 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)
∑ hi dmi = hcr dmcr + h f dm f
(3.20)
∆W = pdV
(3.21)
∆Qht ≈ Ahc ( T − Tw ) = Ahc (
6 The
local equivalence ratio is defined as φ ≡
metric ratio of fuel to oxidizer agent.
χ f uel /χox
(χ f uel /χox )st
PV
− Tw )
Rg
(3.22)
, that is the ratio of fuel to oxidizer agent with respect to the stoichio-
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 the
temperature. This would exacerbate the heat losses due to flow leakage in the crevice region. However
the increase in pressure would improve the sealing in the valve reducing the leakages and therefore one
can usually neglect leakages in the valve region:
∑ hi dmi = hcr mcr + h f dm f
≈ hcr dmcr
(3.23)
Under this assumption, in the changes of internal energy:udm = −udmcr . Introducing at the same time
the ideal gas law, one obtains:
c
cv
v
+ 1 pdV + (hcr − u)dmcr + ∆Qht
(3.24)
∆Qch = Vdp +
R
R
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 and
time is given by the simple relationship expressed below.
∂Qch
1 ∂Qch
= {t = ωθ } =
∂t
ω ∂θ
(3.25)
The crank angle degree is measured (θ = 0) from Top Dead Center (that is, when the cylinder is compressed the most)
In [1], the author uses a empirical correlation in order to evaluate the term: (hcr − u)dmcr . Also Eq.3.24
will be differentiated with respect to time in order to obtain the heat release rate. In this fashion the time
derivative is given by:
Qch
γ
∂V
1
∂P Qleak
∂Qht
=
p
+
V
+
+
∂θ
γ − 1 ∂θ
γ − 1 ∂θ
∂θ
∂θ
Where Qleak is given by the empirical relation in [1]
0
∂Qleak
T
T
1
γ−1
∂p
≈ Vcr
+
+
ln
0
∂θ
Tw
Tw (γ − 1) bTw
γ −1
∂θ
59
(3.26)
(3.27)
In principle one could solve for the rate of fuel mass being injected however the approximation turns
out to be really crude given the difficulty of estimating the previous equations. That is:
η (φ( ~
X, t), γ(t)) ṁ ρ E ≈
γ
∂V
1
∂P Qleak
∂Qht
p
+
V
+
+
γ − 1 ∂θ
γ − 1 ∂θ
∂θ
∂θ
(3.28)
The previous equation is difficult to model due to the variation in time of the pressure. The heat transfer
through the walls is also another difficult term to estimate. Mainly the convection and radiation will
dominate the heat transfer through the walls. Possible analytical solutions can exist neglecting the heat
transfer and assuming either a constant pressure (which is somehow true in some cases for an interval of
time spanning a few crank degrees from TDC). Another possibility
is to assume
an evolution described
VTDC γ̃
∂
∂P
in a way analogous to an isentropic expansion, that is ∂θ = ∂θ PTDC V
.
θ
In all of the previous cases, the heat transfer plays a very important role which can overcome the importance of the pressure and volume evolution. This sits out of the scope of the current report and can be
used as a basis for a future study.
60
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