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University Sapienza of RomeDepartment of Electric Engineering
Numerical modeling of two-phase
reactive flows
PhD in Energetics - XXII Cycle 2006/2009
Sector ING-IND 06
Author: F. Donato, Aerospace Engineer
Tutor: Prof. B. Favini, "Sapienza" University of Rome
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Contents
Nomenclature iii
Introduction 1
1 Mathematical model 12
1.1 Continuous phase model . . . . . . . . . . . . . . . . . . . . . . . . 12
1.1.1 Filtered conservation equations . . . . . . . . . . . . . . . . 14
1.1.2 The Constitutive Equations . . . . . . . . . . . . . . . . . . . 18
1.1.3 Some Thermodynamics Definitions . . . . . . . . . . . . . . 23
1.2 Dispersed phase model . . . . . . . . . . . . . . . . . . . . . . . . . 27
1.2.1 Average operators definition . . . . . . . . . . . . . . . . . . 28
1.2.2 Conservation equations . . . . . . . . . . . . . . . . . . . . . 30
1.2.3 Model closures . . . . . . . . . . . . . . . . . . . . . . . . . 41
1.2.4 Cylindrical formulation . . . . . . . . . . . . . . . . . . . . . 47
1.2.5 Finite volume formulation . . . . . . . . . . . . . . . . . . . 49
1.2.6 Subgridscale closure model . . . . . . . . . . . . . . . . . . 51
2 Numerical approach 59
2.1 Numerical scheme for the gas phase equations . . . . . . . . . . . . . 59
2.1.1 Treatment of variables on the axis . . . . . . . . . . . . . . . 62
2.1.2 Metric correction . . . . . . . . . . . . . . . . . . . . . . . . 62
2.1.3 Artificial viscosity . . . . . . . . . . . . . . . . . . . . . . . 63
2.2 Dispersed phase . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64
2.2.1 Reconstruction method . . . . . . . . . . . . . . . . . . . . . 652.3 Time evolution scheme . . . . . . . . . . . . . . . . . . . . . . . . . 69
i
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3 Model validation 72
3.1 Sommerfeld and Qiu experiment . . . . . . . . . . . . . . . . . . . . 73
3.1.1 Computational grid . . . . . . . . . . . . . . . . . . . . . . . 74
3.1.2 Boundary conditions . . . . . . . . . . . . . . . . . . . . . . 76
3.1.3 Code settings . . . . . . . . . . . . . . . . . . . . . . . . . . 78
3.2 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 79
3.2.1 Gas phase . . . . . . . . . . . . . . . . . . . . . . . . . . . . 79
3.2.2 Dispersed phase . . . . . . . . . . . . . . . . . . . . . . . . 93
3.3 Chapter conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . 107
Conclusion 109
References 116
ii
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Nomenclature
Symbols
fp Dispersed phase probability density function conditioned by the flow realization
cp, Vp Single particle velocity
fi Body force acting on species i
Ji Species i mass diffusion flux
q Thermal flux
qD Thermal flux due to Dufour effect
uf Gas phase velocity
up Dispersed phase velocity
Vc Correction velocity to impose mass conservation due to approximated diffusion
fluxes for chemical species
Vi Species i mass diffusion velocity
E Gas mixture internal energy
Hf Gas mixture enthalpy
Hf,i Enthalpy of the ith species in the gas mixture
K Gas mixture kinetic energy
Ru Universal gas constant
Independent coordinate along the tangential direction
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Cpi Species i specific heat at constant pressure
Cp Gas mixture specific heat at constant pressure
Cvi Species i specific heat at constant volume
Cv Gas mixture specific heat at constant volume
Di j Binary diffusion coefficient of species i into species j
Di Diffusion coefficient of species i into the gas mixture
dp Particle diameter
Ek Turbulent kinetic energy associated to the lengthscale k
Ep f Work done by aerodynamic force on the continuous phase in a time unit
fp Dispersed phase probability density function
fi j jth component of the body force acting on species i
h0fi Enthalpy of formation at reference state for the i
th
species
Hp Dispersed phase enthalpy
hsi Sensible enthalpy for the ith species
hs Sensible enthalpy for the gas mixture
kf Fourier coefficient for heat conduction
Lv
Phase change heat
mp Particle mass
np Particle number density
Ns Number of chemical species in the gas mixture
p Pressure
Qloss Energy loss from the continuous phases (e.g. radiation)
Qp f Energy dissipation due to the aerodynamic interaction between phases
iv
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r Independent coordinate along the radial direction
Rg Gas constant
Re Reynolds number
Rep Particle Reynolds number
St Stokes number
t Time
Tf Gas phase temperature
Tp Dispersed phase temperature
Vi j jth component of species i mass diffusion velocity
Wi Species i molecular weight
Wmix Gas mixture molecular weight
WRU M Specific energy flux from the condensed to the gas phase due to the uncorrelatedmotion and to the aerodynamic interaction
Xi Species i mole fraction
Yi Species i mass fraction
z Independent coordinate along axis direction
Qp Dispersed phase heat flux vector
Vp Particle uncorrelated velocity
Hf Gas phase flow realization
i Species i thermodiffusion coefficient
p Dispersed phase volumetric fraction
p Dispersed phase uncorrelated energy
Filter lengthscale
v
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p, p Single particle diameter
Kolmogorovs dissipative lengthscale
p Specific mass flow rate from the condensed to the gas phase
p,i Specific mass flow rate of species i from the condensed to the gas phase
RU M Specific energy flux from the condensed to the gas phase due to the evaporation
and the uncorrelated motion
Cinematic viscosity
i Species i mass production due to chemical reactions
f Dissipation function
p Specific energy flux from the condensed to the gas phase due to the evaporation
Generic particle property
f Gas phase density
p Density of the particle material
tf Turbulence characteristic timescale
p Particle relaxation time
p, hp Single particle enthalpy
p, p Single particle temperature
Operators
[] Total derivative with respect time
[] Reynolds average
[] Fvre average[]
p Mass average
{[]}p Ensemble average
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Superscripts
[] Fluctuation with respect to Reynolds average
[] Fluctuation with respect to Fvre average
Subscripts
[]f,j jth component of a vector in the continuous phase
[]f Continuous phase
[]p,j jth component of a vector in the dispersed phase
[] Dissipative lengthscale
Tensors
R Dispersed phase generalized stress tensor
E Strain tensor
S Stress tensor
Viscous stress tensor
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Introduction
Research scope
Much attention has been payed to multiphase flows modelling in the last decades. Two
concurrent reasons can justify such an interest in the research community. The first mo-
tivation is that multiphase flows can be found in a variety of industrial processes and
components (e.g. fluidized beds, gas turbine burners, etc.) as well as in many every
day life devices (e.g. computer printers); the second reason can be addressed to the in-
creasing computational capabilities, which is making Computational Fluid Dynamics
(CFD) to be a more and more useful tool in the design and optimization process of
such devices and components. In the CFD framework, Large Eddy Simulation (LES)
approach is gaining in importance as a tool for simulating turbulent combustion pro-
cesses. Such a technique is nowadays a standard as far as single phase phenomena are
considered, while much work is being done to improve its performance in multiphase
flow applications.
The present work is intended to be a contribution to the development of reliable
models for multiphase dispersed reactive flow simulations within LES framework. The
aim of this research project is to provide an improvement to the capability of exist-
ing particle transport models in predicting the dispersed phase evolution under dilute
condition and for inertial particles. The main applications towards which this work is
oriented are coal powder burners and spray combustors. The coupling of the LES accu-
racy in predicting gas phase turbulent combustion and an improved model for particledispersion in the carrier gas could help in the design process of such components.
1
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Introduction 2
Physics of turbulent flows
Turbulent flows are one of the physical phenomena that are easiest to find in nature.
They characterize the evolution of the oceans as well as of the atmosphere. They are
often involved in the operation of man-made devices. In fact, turbulent flows represent
the natural status of fluid motion while laminar flows are the exceptions, even though
the sequence of their study in mechanical engineering has been inverted. Differently
from the laminar case, in turbulent flows the fluid variables at a given point are func-
tions not only of the position but also of time and the instantaneous velocities present
components normal to their averaged values.
In Figure 1 a typical example of a two-phase turbulent flow is shown. By paying
attention to the multitude of structures that can be seen in the plume of the pyroclastic
eruption of St. Helenes mountain, it may be clearer what is meant when turbulence is
said to be an example of deterministic chaos. A turbulent field is in fact characterized
by the presence of organized structures (eddies), presenting a finite dimension in space
(lengthscale) and time (timescale). The Navier-Stokes equations, used as a model for
fluid dynamics, are able to describe any turbulent field evolution (hence the "deter-
ministic" connotation) for common fluids, but no general solution is available. Thestrong nonlinearity of the Navier-Stokes equations makes it impossible to give an a
priori estimate of the evolution of perturbations in a turbulent flow. Hence the chaotic
nature of turbulence. Figure 1 also shows how the size of the involved structures may
differ by many order of magnitude. Although the energy distribution over these scales
Figure 1: St. Helenes eruption
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Introduction 3
Figure 2: Turbulence Energy Spectrum
may appear at first sight completely unorganized, experimental data as well as Kol-
mogorovs theory [1] demonstrate that the turbulent kinetic energy distribution over
the scales can be represented by the energy spectrum shown in Figure 2. Different
characteristic lengthscale can be here identified through the corresponding wavenum-
bers. The integral lengthscale lI (kI) identifies the energy containing scales. It can be
expressed bylI =
1
u2L2
Z
u(x, t)u(x + r, t)dr (1)
where u is a fluctuation with respect to the averaged velocity and is the L2-norm. From the integral lengthscale, energy is transfered to smaller scales down to
Kolmogorovs scale (kK) that is given by
Re =u
= 1 (2)
where Re is the Reynolds non dimensional number, is the kinematic viscosity and
u is the characteristic velocity of Kolmogorovs scale. The Re number can be seen
as the ratio of diffusion and convection characteristic times. The fact that Re = 1
says that is the dissipative scale where kinetic energy is transformed into heat. As
reported in Figure 2 the slope of the spectrum plot between kI and kK is 5/3 (whencompressible flow effects are weak). This is a result of Kolmogorovs theory [1] that
has been experimentally validated. This range of wavenumbers is called inertial range.
The fact that this feature of turbulence is statistically reproduced by all the turbulentflows confirms their deterministic nature and provides a link for modelling strategies.
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Introduction 4
Another important lengthscale reported in Figure 2 is the Taylor scale T (kT). This is
an index of the position in the wave number space of the inertial range.
In order to study turbulent flows, due to the lack of a general analytical solution,
scientists must numerically solve the Navier-Stokes equations. A complete and reli-
able solution should resolve all the wavenumbers down to Kolmogorovs scale . This
approach does not require any modelling and it is called Direct Numerical Simulation
(DNS). Nevertheless, performing a DNS is usually not possible for practical applica-
tions, where the Reynolds number Re based on the domain characteristic dimension L
is too high. It can be shown for the lengthscale separation that
lI
Re 34 (3)
thus in 3D calculations one will need more than Re94 grid points to perform a DNS
calculation. Since in practical applications it is easy to find ReL O(106) or higher,one can understand how the computational effort necessary to perform a DNS is not
affordable by any present computer. To given an idea, the physical time required to
perform a time step on a grid with 4.5 106 nodes, with a single core of a core2 duo
Intel processor P8700 and the HeaRT code used in this work, is approximately 25 s.The simulated time during a numerical time step is O(108) s.
In order to overcome the computational cost of a DNS, two modelling strategies
are available:
Large Eddy Simulation (LES): only a part of the spectrum is directly simulatedwhile the highest frequencies are modelled. This approach allows to retain the
unstationary features of turbulence and the results have an high level of reliabi-
lity. The computational effort is still quite heavy.
Reynolds Average Navier Stokes (RANS): all the turbulent spectrum is modelledand the time dependence is removed from the solution. It is the most widely
spread technique because of its low computational cost. Nevertheless RANS
models are complex and their applicability not certain. Calibration constants
may change from a configuration to another and the final results are thus not so
reliable.
The LES approach is thus growing in importance since, when a sufficient portion ofthe energy spectrum is resolved, it is the only available and reliable tool with prediction
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Introduction 5
capability for complex flows. In the present work this features will be coupled with
new techniques developed for the dispersed phase in order to obtain a predictive tool
for dispersed multiphase flows under dilute conditions.
Physics of the dispersed phase
Dispersed multiphase flows involve different physical phenomena each of them cha-
racterized by a proper time and length scale. For example, a particle moving in a
carrier phase is subjected to the effects of several forces (drag, lift, added mass...etc);
when the considered particle is not rigid, his shape and integrity depend on the ba-lance of the pressure, viscous stresses and surface tension. Furthermore, when mass
exchanges take place at the particle interface they introduce a continuous variation of
the considered forces, caused by the change of particle dimensions. This large variety
of phenomena and associated scales imposes to carefully choose the modelling stra-
tegy looking for the phenomena that are negligible or that can be evaluated by simple
relations, in order to develop a reliable but also usable model.
Following the above considerations, the first thing to be done is to provide a classi-
fication of the different possible regimes for dispersed multiphase flows, and to select
then the conditions the model to be developed is supposed to be applied to. The first
classification that can be easily found in the literature is related to the different levels of
coupling between the carrier phase and the dispersed ones. This classification is based
on the dispersed phase volume fraction ; another useful parameter to understand the
kind of interaction between particles and turbulence is the local Stokes number St,
which is the ratio between a particle response time to the aerodynamic forces and a
characteristic time of turbulence. The level of coupling is generally classified into four
degrees [3]; in Figure 3 the classification for particle-laden flows is reported. Coupling
regimes are classified into
one-way: whereby the particle motion is affected by the continuous phase butnot vice-versa. This is the case of dilute dispersions of small particles that do
not exchange mass with the carrier flow;
two-way: whereby the dispersed phase affects the continuous phase through the
inter-phase coupling, e.g. (mass, momentum and energy exchange). These arethe conditions typically met inside fuel spray combustors as well as in powder
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Introduction 6
Figure 3: Turbulent particle laden flow regimes. Adapted from reference [2]
burners, far enough from the injector, where the fuel volume fraction is smallenough to neglect particle-particle interactions. Under these conditions small
particles (i.e. small St) will subtract energy to the turbulent scales while larger
particles (higher St) will transfer energy to the scales comparable to their wake
dimension;
three-way: whereby individual particle flow disturbances locally affect the mo-tion of the other nearby particles, i.e. fluid-dynamic interactions between parti-
cles;
four-way: whereby particle collisions are present and have an influence on theoverall particle motion.
Note that in Figure 3 the range for three-way coupling is not shown explicitly
because, for particle laden flows, it overlaps the four-way coupling one: when particles
are so close to each other to feel the interaction through the aerodynamic disturbancesinduced by the particles themselves, it is likely they will also collide.
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Introduction 7
In the present work dilute conditions for inertial particles are assumed. This as-
sumption allows not to model particle-particle interactions and to focus on the dis-
persed phase evolution under conditions that are of interest in real combustor appli-
cations. Consider, for example, a spray injector in a gas-turbine combustor: few mil-
limeters after the fuel injection the spray is completely developed and the fuel volume
fraction is sufficiently small to justify the use of a two-way coupling model. It seems
therefore reasonable to separately model the spray injection, instead of conditioning
the model for the all chamber to account for conditions that will only occur in a small
portion of the domain. Similar considerations hold for coal powder entrained flow
reactors or slurry reactors. Since the present work is oriented towards this kind ofapplications, only two-way coupling effects will be accounted for in the model and
particle-particle interactions will be neglected. Furthermore the assumption of dilute
conditions allows to estimate the needed source terms in the carrier phase equations
by a proper averaging procedure of the exchange terms calculated for a single particle,
while under three-way coupling assumption this would not be strictly possible.
The second step is to determine the mass, momentum and energy exchange me-
chanism for the isolated particle. The momentum transfer between particle and carrier
phase depend on the force Fp acting on the particle; this force can be splitted in differ-
ent contributions [4]
Fp = FD + Fg + FL + FS+ FH + FW (4)
where FD is the aerodynamic drag force, Fg is the gravity force, FL is the lift force,
FS is the Tchen force, FH is the Basset History force and FW is the wall interaction
force.F
S takes into account the acceleration of the carrier flow at the position occupiedby the particle while the history force accounts for its wake development. The above
separation is not always valid as there can be non linear interactions between various
forces but typically they are small enough to be neglected. A preliminary estimation
of the weight of each force shows (see equation 1.116) that for heavy particles the
prominent forces acting are the gravity force, the drag force and the Basset force. The
history force scales like the inverse of the diameter while the drag force scales as the
square of inverse diameter [2]; the drag force is thus the dominant force, together with
the gravity, for small particles. In our model the drag force for the single particle ismodeled by an expression based on the Stokes drag (strictly valid only for creeping
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Introduction 8
flows) corrected by a factor ffunction of the particle Reynolds number Rep
Rep = |uf@p cp|dpf
(5)
where uf@p is the carrier phase velocity at the particle location, cp is the particle
velocity, dp is the particle diameter. With this assumption the drag force acting on
a particle can be expressed by FD =(cpuf@p)
p. Here p is the particle relaxation
time that can be interpreted as the time taken by the particle to reach the 63 % of
the carrier phase velocity, when the particle is initially at rest and the carrier phase
velocity is constant. This parameter is used, together with a characteristic time f of
the carrier phase (e.g. a turbulent time scale tf), to build the non dimensional Stokes
number St = pf . The Stokes number is a very important parameter since it measures
the possibility for the particle to follow the velocity fluctuations corresponding to a
given scale of the fluid velocity field.
Regarding the mass and energy exchange, they will not be an object of the present
work since the latter will be focused on the transport model improvement. In addition,
the mass exchange model depends on the particular fuel and operational range and will
therefore not be treated in this work.
As to particle heating, when practical applications are considered, the models
adopted in the literature are usually rather simple. This is due to the fact that particle
heating should be modelled taking into account the effects of turbulence, combustion
and all related phenomena in realistic 3D enclosures. Hence, a compromise between
the complexity of the involved phenomena and the computational efficiency of the
adopted models is an essential precondition for a CFD tool. A complete model should
account for:
the effect of convection;
the effect of the flow around the particle (e.g. different mass flow between up-stream and downstream stagnation point);
fractional mass exchange of multicomponent fuels;
temperature distribution inside the particle;
the effect of internal recirculation caused by the frictional stresses on the surface(for liquid fuels);
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Introduction 9
other minor phenomena.
Several models can be found in the literature [5]; the so called "Infinite conductivity
model" is usually selected because of its simplicity. In this model the particle inner
temperature is radially constant but variable with time. Since this model is developed
for a particle in a stagnant environment an empirical correlation is used to account
for convection (Ranz-Marshall). This model has been implemented in the CFD code
used for the validation of the transport model but, since no reactive validation could be
performed during this work, it has not been validated.
From the brief review of the phenomena concerning an isolated particle in an hot
convective environment, one can easily see the importance of the particle diameter.
In real devices a spectrum of particle sizes is usually found. These dimensions range
from below the micron up to the hundred of microns, depending on the kind of reactor
that is being considered. It is worth to say that even if it were possible to inject only
particles of a given size, the cloud becomes polydispersed [6]. This is due to the
different mass exchange rates experienced by the particles in different zones of the
combustor. From experiments and theoretical observation many researchers developed
different size distribution functions, but, since many of the models are developed formonodisperse distributions, it is useful to define some representative diameters that
can reproduce some aspect of the polydispersed nature of the spray. They are usually
defined as
Di j =
RDi f(D)dDRDj f(D)dD (6)
The most known are the Sauter diameter D32 and the surface diameter D20. The first
one has the same volume to surface ratio of the whole particle cloud and for this reason
is frequently used in combustion; the latter has the same surface of the whole particle
system and is sometimes preferred to the Sauter diameter for problems in which the
mass exchange rate is particularly important (e.g. ignition).
State of the art
In scientific literature several modeling strategies for the simulation of multiphase dis-
persed flows are present; they can be roughly divided in two classes, Lagrangian andEulerian, with respect to the framework in which the secondary phase is described. In
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Introduction 10
the past, LES technique has been used together with a Lagrangian description for the
dispersed phase [7]. This is due to the easiness in modeling a single particle behavior
with respect to model the behavior of a group of particles, present in a given control
volume at a given time instant. This historical trend makes it easy today to find LES
simulations of reacting two-phase flows in literature [8], performed with this approach.
Greater attention has been more recently gained by the Euler-Euler formulation.
This is mainly due to the massive diffusion of parallel computation techniques as a
standard to increase computational capacity. The parallelization of a numerical code
for two-phase dispersed flow application with the Eulerian-Lagrangian (EL) approach
is a difficult task. Workload distribution among processors is not straightforward. Inaddition the large amount of informations that have to be exchanged among the CPUs,
and the time needed to do it, tend to decrease parallelization efficiency. This is espe-
cially true when particle distribution is not uniform throughout the domain [9].
A two-fluid Eulerian model was first proposed by Druzhinin and Elghobashi [10].
This model laid on the assumption that particle equations were obtained by filtering
on a length scale smaller than the smallest characteristic scale lp of particle velocity
field. This hypothesis ensures the unicity ofup at the scale lp.
Simonin et al. [11] proposed a model for dispersed two-phase flows, based on
the separation of particle velocities into a "mesoscopic" correlated part, representative
of a group of particles, and an uncorrelated part proper of each single particle. They
also proposed a correlation, strictly holding for Homogeneous Isotropic Turbulence
(HIT), for the evaluation of the turbulent kinetic energy part due to uncorrelated mo-
tion. Kaufmann [2] proposed models for the second order velocity moments and the
terms appearing after filtering the equations of Simonin et al. [11] model, under the
assumption of non-colliding particles. Moreau [12] made an a priori evaluation of the
closures proposed by Kaufmann [2] by applying them to Direct Particle Simulation
(DPS) results. Selected models were then applied in LES simulations of particle laden
flows [13] and confined bluff-body gas-solid flows [9].
All the cited models are strictly developed for monodisperse sprays but extension
to polydispersion are also present in literature. The classical way to extend a model to
polydispersion is called Sectional Method. The size PDF is splitted in n classes, each
of them representing a monodisperse distribution. For each class it is possible to apply
one of the cited model adding proper relations to define the migration of particlesbetween different classes. Another way is the so called "Presumed Shape PDF"[14]
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Introduction 11
that calculates the deviation from monodispersion as moments of the PDF.
Summary of this work
In Chapter 1 the model equation for both continuous and dispersed phase will be pre-
sented. For the latter in particular the model discussed by Moreau [12] will be extended
in order to partially keep into account effects that are second order in the relaxation
time. The latters become relevant when inertial particles within dilute conditions are
considered. These are the conditions typically met in industrial as well as in aeroengine
burners, with the exception of the injector region, where the diluition assumption is of-ten not met. The coupling between continuous and dispersed phase will be treated by
classic empiric correlations. A revised Fractal Model (FM) for the SGS terms closure
is also presented.
In Chapter 2 the adopted numerical treatment for both phases is described. A robust
numerical treatment for the dispersed phase, based on a Finite Volume ENO (Essen-
tially Non Oscilatory) scheme is here proposed.
In Chapter 3 the validation of a reference model, that does not contain the additional
terms proposed in Chapter 1 is presented. This solution was necessary in order to:
a) validate the numerics and the revised SGS model; b) obtain a reference solution.
The obtained results are described and criticized. Expected improvements due to the
application of the proposed model are identified.
Unfortunately, the numerical solver for the gas-phase implemented in the HeaRT
CFD code, which has been selected for the validation against a literature test case
[15], did not prove to be robust under these conditions. The stability problems caused
a delay on the research project that made it impossible to validate the whole model
within this PhD period. The final validation will thus have to be performed in a future
activity.
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Chapter 1
Mathematical model
1.1 Continuous phase model
When two-phase flows are considered, great attention must be paid to the assumptions
under which the adopted model is developed, since these assumptions will affect the
transport equations that govern the evolution of the two-phase system. The model de-
veloped in the present work is based on two main hypothesis: a) a condensed phase
is dispersed in a continuous gaseous phase; b) dilute conditions are assumed. Underthese hypothesis two way coupling between phases can be assumed, meaning that the
equations governing each phase evolution present terms that account for the interaction
with the other phase. Under dilute conditions it is also possible to model this interac-
tion in the continuous phase by simply adding source terms to the single phase balance
equations.
Let a mixture of Ns ideal gases in local thermodynamic equilibrium and chemi-
cal non-equilibrium be considered. The complete set of transport equations for the
gas phase, that expresses the conservation of mass, momentum, energy and chemical
species mass fractions, together with the thermodynamic state equation, is
Conservation of Mass
f
t+ fuf= p (1.1)
Conservation of Momentum
fuft
+ fufuf= S + f Nsi=1
Yifi pup + ppup ufp
(1.2)12
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Mathematical model 13
Conservation of Energy (internal + kinetic)
tf (E+K) + [fuf (E+K)] = (Suf) q (1.3)
+fNs
i=1
Yifi (uf + Vi) + ppp
(up uf) uf Ep f
+pp
p(up uf)2
Qp f
p (Hp +Lv)
h
p
1
2up up
RUMWRUM p
Conservation of Species Mass Fraction
fYit
+ fufYi= Ji + fi p,i (1.4) Thermodynamic State Equation
p = fNs
i=1
Yi
WiRuTf (1.5)
The terms Ep f and Qp f in equation (1.3) represent the work done by the aerodynamicforce on the continuous phase and the energy dissipation into heat during aerodynamic
interaction respectively. When larger particles are considered (large Rep) part ofQp f
should account for the energy transferred to the turbulent structures in the particle
wake. However, in the limit of small particles, the energy injection will occur at dissi-
pative scales and the assumption that all Qp f is dissipated into heat is acceptable.
The physical meaning of the other source terms that appear at the right hand side
of the equations and account for two-way coupling, as well as their genesis, will be
presented in section 1.2.2.
In equations (1.3) and (1.4) the relation between the mass flux Ji of the ith species
due to diffusion and the corresponding diffusion velocity Vi has been used
Ji = fYiVi (1.6)
Equations (1.1)-(1.5) must be coupled with the constitutive equations which de-
scribe the type of flow, and in particular its behavior in relation to molecular properties.
It should be noted that summation of all species conservation equations in (1.4)yields total mass conservation equation (1.1), so that these Ns +1 equations are linearly
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Mathematical model 14
dependent and one of them is redundant. Furthermore, to be consistent with mass
conservation, the diffusion fluxes and source terms due to chemical reactions and mass
exchange between the two phases must satisfy
Ns
i=1
Ji = 0Ns
i=1
i = 0Ns
i=1
p,i = p (1.7)
where p is the overall mass flux from the dispersed towards the gas phase while p,i
is the fraction of such mass flux involving the ith species.
After subtracting from (1.3) the conservation equation for the kinetic energy[16],
the energy equation can be written in terms of enthalpyH
f as:
t
fHf
+ fHfuf = Dp
Dt q + f Qloss + f
Ns
i=1
Yifi Vi (1.8)
p (Hp +Lv) h
RUMWRUM p + Qp f
Here q is the heat flux; f = : u is the dissipation function, being the viscous part
of the stress tensor; Qloss the heat loss (e.g., by radiation), fi the body force per unit of
mass acting on the ith chemical species that diffuses at velocity Vi.The heat flux q is given by three contributions, Fourier, Dufour and that associated
to the diffusion of each species transporting its own enthalpy:
q = qF + qD + qVi = kfTf + qD + fNs
i=1
YiHf,i(Tf)Vi (1.9)
1.1.1 Filtered conservation equations
It is common practice, while studying turbulent flows, to treat velocities and scalarswith classical Reynolds averaging, where the quantity q is split into a mean q and a
deviation from the mean denoted by q. Nevertheless, in turbulent flames, fluctuationsof density are observed because of the thermal heat release and classical Reynolds ave-
raging induces some additional difficulties. For example, averaging the mass balance
equation leads to:
f
t
+
xi(fui +
fu
i) = 0 (1.10)
where the velocity/density fluctuations correlation fui appears.
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Mathematical model 15
To avoid the explicit modeling of such correlations, a Fvre (mass weighted) average
[] is introduced and the generic quantity q is then decomposed into q = q + q whereq = f q
f(1.11)
The objective of Large Eddy Simulation is to explicitly compute the largest structures
of the flow (typically the structures larger than the computational mesh size), while the
effects of the smaller ones are modeled. In LES, the relevant quantities q are filtered in
the spectral space (components greater than a given cut-off frequency are suppressed)
or in the physical space (e.g. weighted averaging in a given volume). The filter opera-
tion is defined by:
f(x) =Z
Dfx
Gx, x
dx (1.12)
where G is the filter function. The latter must have the following properties:
1. G (x) = G (x);
2. RD
G
(x, x) dx
= 1;
3. G (x) small outside the compact domainx 2 ,x + 2
.
Standard filters are:
A cut-off filter in the spectral space:
G(k) =
1 ifk 0 otherwise
(1.13)
where k is the spatial wave number. This filter preserves the length scales
greater than the cut-off length scale 2.
A box filter in the physical space:
G(x1,x2,x3) =
1 if|xi| 20 otherwise
where (x1,x2,x3) are the spatial coordinates of the location x. This filtercorresponds to an averaging of the quantity q over a box of size .
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Mathematical model 16
A Gaussian filter in the physical space:
G(x1,x2,x3) = 6
23/2
exp 6
2
x21 +x22 +x
22
By applying the filter operator to the system equations, balance equations for the fil-
tered quantities q and q are obtained. In this work, the filter operation is implicitlydefined by the mesh size. The uncertainties related to the procedure of exchanging
the order of the filter and differential operators (commutation errors), are neglected
and assumed to be incorporated in the sub-grid scale modeling. It has however been
demonstrated that the commutation error is O(2
) [17]. Fvre filtering leads to a set ofequations formally similar to the Reynolds averaged balance equations:
Mass:f
t+
xj(fuf,j) = p (1.14)
Momentum:
t(fuf,i) +
xj(fuf,iuf,j) =
p
xi+
i jxj
SGSi j
xj(1.15)
pup,i + pp
up,i uf,ip
+ f
Ns
s=1
Ysfs,i
i j =
uf,ixj
+uf,j
xi
2
3
uf,kk
i j (1.16)
SGSi j = fuf,iuf,j f
uf,i
uf,j (1.17)
Species equations:
t(fYi) + xj
fYiuf,j =
xj
fYiVi j+ fi JSGSi jxj p,i (1.18)
JSGSi j = fYiuf,j fYiuf,j (1.19)where the assumption has been made, and will be used in the further development,
that the subgridscale effects due to diffusion, arising from
Ji j, may be neglected with
respect to those due to the SGS species transport JSGSi .
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Mathematical model 17
As to the energy equation:
tf Hf+ xj f Hfuf,j= DpDt xj qj + qSGSj + f Qloss (1.20)
+f
Ns
i=1
Yi fi jVi j h RUMWRUM p + Qp fqSGSh,j = f
Hfuf,j f Hfuf,j (1.21)Again the subgridscale heat flux due to diffusion effects has been considered negligible
with respect to SGS heat transport qhSGS.
Finally, the filtered equation of state is
p = f
Ns
i=1
RuYi
WiTf f
Ns
i=1
RuYi
WiTf (1.22)
Another important assumption that has been adopted in the development of the
present model is that particles present a high value of their inertia. The meaning of this
sentence and its importance in the development of the model will be cleared in section
1.2.3. Among other implications, the restriction to highly inertial particles allows to
assume that particle motion is not influenced by the unresolved scales of turbulence.
A first exploitation of this statement leads to the absence of SGS modeling in the dis-
persed phase balance equations. Another turnaround is the possibility to model the
filtered source terms that account for phase interaction, by means of their unfiltered
form, where gas-phase filtered variables take the place of their unfiltered values. This
possibility is also granted by the fact that most of the source terms accounting for
phase interaction are modeled after semi-empirical correlations (see for example [2])that implicitly take into account the turbulence effects. Nevertheless, these correla-
tions are based on the gas-particle relative velocity and correction may be necessary
when this parameter is small but the turbulence intensity is high. In such a situation
the correlations may in fact predict a laminar behavior. These effects are probably
more important in the evaluation of the mass (p) and heat (p) exchange than in
aerodynamics forces, since when the gas-particle relative velocity is low the particle
momentum is close to its equilibrium, while temperature and species concentrations
may be far from their own. Nevertheless, the improvement of the mass and heat ex-change source terms goes beyond the objectives of the present work, being the latter
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Mathematical model 18
oriented to improve the transport model and numerics. Therefore, these topics will not
be addressed here.
The quantities to be modelled are:
the unresolved Reynolds stresses SGS, requiring a subgrid scale turbulencemodel;
the unresolved molecular transport fluxes JSGSi ;
the unresolved heat transport fluxes qSGS;
the filtered chemical reaction rate i; all the source terms accounting for phase interactions.
The filtered balance equations presented in this section, coupled with subgrid scale
models, may be numerically solved to simulate the unsteady behavior of the filtered
fields.
1.1.2 The Constitutive Equations
Each material has a different response to an external force, depending on the properties
of the material itself. The constitutive equations describe this behavior. In particular,
for a gas mixture they should model the stress-strain relation S E, the heat flux qand the species mass flux Ji. In the preceding section the hypothesis has been made
that SGS effects other than those accounted for in the source terms and those due
to small scale transport are negligible with respect to these contributions. Given this
assumptions, all the quantities that will appear in the following development must beconsidered as filtered values. The average signs [] and [] are thus dropped.The Diffusive Momentum Flux
For all gases that can be treated as a continuum, and most liquids, it has been observed
that the stress at a point is linearly dependent on the rates of strain (deformation) of
the fluid. A fluid that behaves in this manner is called a Newtonian fluid. With this
assumption, it is possible to derive a general deformation law that relates the stresstensor S to the pressure and velocity components:
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Mathematical model 19
S = p + ufI + 2E = pI + (1.23)where E is the strain rate, is the coefficient of viscosity (dynamic viscosity) and
is the second coefficient of viscosity. The two coefficients of viscosity are related to
the coefficient of bulk viscosity b by the expression b = 2/3 + . In general, it
is believed that b is negligible except in the study of the structure of shock waves
and in the absorption and attenuation of acoustic waves. With this assumption (Stokes
hypothesis), is equal to 2/3, and the viscous stress tensor becomes:
i j = uf,ixi
+ 212uf,i
xj+ uf,j
xi . (1.24)
Pressure at the macroscopic level corresponds to the microscopic transport of momen-
tum by means of molecular collisions in the direction of molecules motion. Instead,
molecular momentum transport in other directions is what at macroscopic level is
called viscosity. They are of different nature. In terms of work done, when continuous
distribution are considered, pressure produces reversible transformations (changes of
volume), while viscous stresses produce irreversible transformations where dissipation
of energy into heat occurs.
The Diffusive Heat Flux
The heat flux q for a gaseous mixture ofNs chemical species consists of three different
transport contributions.
The first is the heat transfer by conduction, modeled by the Fouriers law. At the mi-
croscopic level it is due to molecular collisions: since kinetic energy and temperature
are equivalent, molecules with higher kinetic energy (at higher temperature) "energize"
collisionally the ones with less kinetic energy (at lower temperature); in the continuum
view, heat is transfered by means of temperature gradients.
The second heat transport contribution is due to molecular diffusion, acting in mul-
ticomponent mixtures and driven by concentration gradients: where Yi = 0, eachspecies diffuses with its own velocity Vi. In this way each molecule transports its own
enthalpy contribution; this means that there is energy transfer even in a gas at uniform
temperature, or in a rarefied gas (with negligible conduction).
The third heat transport mechanism is the so called Dufour effect. The Onsagerprinciple of microscopic reversibility in the thermodynamics of irreversible processes
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Mathematical model 20
implies that if temperature gradients cause species diffusion (thermo-diffusive or Soret
effect), concentration gradients must cause a reciprocal (Dufour effect) heat flux. The
Dufour effect is neglected. [18].
The total energy flux q is finally modeled:
q = kfTf + fNs
i=1
Hf,iYiVi +RuTfNs
i=1
Ns
j=1
XjiWiDi j
Vi Vj
. (1.25)
where i is the thermodiffusion coefficient of the ith species
The Diffusive Species Mass Flux
To be useful, equation (1.4) requires the knowledge of diffusive species mass flux,
Ji, that expresses the relative motion of chemical species with respect to the motion
of their (moving) center of mass. Within the continuum mechanics this motion can
be expressed by a constitutive law rather than additional momentum equations for
chemical species. Both modelling and calculation of individual species diffusive mass
fluxes is not easy. The distribution ofNs chemical species in a multicomponent gaseous
mixture, at low density, is rigorously obtained by means of kinetic theory [18]
Xi =Ns
i=1
XiXj
Di j
Vj Vi
+ (Yi Xi)
p
p +DV PG
+fp
Ns
j=1
YiYj
fi fj
+
Ns
j=1
XiXj
fDi j
jYj
iYi
TfTf
BF SE
(1.26)
where Di j is the binary diffusion coefficient of species i into the species j, Xj and
Yj are the molar and the mass fraction of the jth species respectively, fj the body
force per unit mass, acting on species j, j the thermodiffusion coeffcient of species
j. Equations (1.26) are referred to as the Maxwell-Stefan equations, since Maxwell
[19, 20] suggested them for binary mixtures on the basis of kinetic theory, and Stefan
[21, 22] generalized them to describe the diffusion in a gas mixture with Ns species.
The main feature of (1.26) is that they couple inextricably all diffusion velocities Vj,
and thus all fluxes to all concentrations Xj and Yj and their gradients. According to(1.26), concentrations gradients (e.g., X i ) can be physically created by:
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Mathematical model 21
differences in Diffusion Velocities (DV)
Pressure Gradients (PG) ("pressure diffusion")
differences in Body Forces (BF) per unit mass acting on molecules of differentspecies;
thermo-diffusion, or Soret Effect (SE), i.e., mass diffusion due to temperaturegradients, driving light species towards hot regions of the flow.
This last effect, often neglected, is nevertheless known to be important, in particular
for hydrogen combustion, and in general when very light species play an important
role. The Soret effect has the Dufour effect as reciprocal, but is more important than
this. The linear system (1.26) for the Vj has size Ns x Ns and requires knowledge of
Ns(Ns 1)/2 diffusivities. Only Ns 1 equations are independent, since the sum ofall diffusion fluxes must be zero. This system must be solved in each direction of the
frame of reference (coordinate system), at every computational node and, for unsteady
flows, at each time step. Extracting the diffusion velocities is a mathematically difficult
task, therefore, simplified models, such as the Ficks law and the Hirschfelder and Cur-
tiss law, are preferred in most CFD (Computational Fluid Dynamics) computations.
These simplified models still involve the estimation of individual chemical species dif-
fusion coefficients into the rest of the mixture; also at this step some simplifications
are usually assumed. These will be analyzed in the hereafter.
Many combustion codes use a simplified model for the diffusion velocities, the
Ficks law approximation, assuming
binary mixture (two species A and B),
thermo-diffusion negligible,
fA = fB
This law is usually adopted for the sake of simplicity also for multi- component mix-
tures (more than binary):
Ji = fYiVi = fDYi (1.27)
A more accurate (but still simple) approximate formula for diffusion velocities ina multicomponent mixture is that of Hirschfelder and Curtiss, which has been used in
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Mathematical model 22
this work.
Vi =
DiXi
Xi(1.28)
with :
Di =1Yi
Nsj=1, j=i
XjDji
(1.29)
The coefficient Di is not a binary diffusion but an equivalent diffusion coefficient
of species i into the rest of the mixture. Mass conservation problem arises in cal-
culations when inexact expressions for diffusion velocities are used (as when using
Hirschfelders or Ficks laws), and in general when differential diffusion effects areconsidered, i.e., the species diffusion coefficients are different. In fact, the diffusion
velocities do not necessarily satisfy the constrain Nsi=1 Ji =
Nsi=1 fYiVi = 0. A sim-
ple empirical remedy to impose global mass conservation consists in subtracting any
residual artificial diffusional velocity from the flow velocity in the species transport
equations. In fact, summing all species transport equations, the mass conservation
equation must be obtained, while it is found:
f
t + fuf= (fNs
i=1YiVi) (1.30)Thus, in order for the conservation of mass to be respected, a term fVc involving
a correction velocity Vc must be introduced. Vc is defined as
Vc = Ns
i=1
YiVi (1.31)
and assuming Hirschfelders law holds, it becomes
Vc =
Ns
i=1Wi
WmixDiXi (1.32)
The correction velocity must be computed at each time step and added to the flow
velocity in the species convective term. The corrected convective term of species trans-
port equations must then become
(fufYi) (f(uf + Vc)Yi) (1.33)With this "trick", any artificial flow due to the nonzero diffusional mass flux is thereby
cancelled, and solving for Ns 1 species and global mass, results into a "correct"concentration for the last Ns species (the last species can be obtained as 1 Ns1i=1 Yi).
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Mathematical model 23
1.1.3 Some Thermodynamics Definitions
In a multi-species system the enthalpy, Hf(Yi,Tf), is given by two contributions: oneis the potential energy of the molecular force field (expressed in terms of formation
energies that must depend on temperature because the molecular force field changes
when temperature changes), and the other is the kinetic energy of molecules (sensible
enthalpy), obtained considering all their degrees of freedom (expressed in terms of
specific heat) and not only the effect of temperature Tf. The enthalpy Hf(Yi,Tf) is
defined as
Hf(Yi,Tf) =Ns
i=1
YiHf,i(Tf) (1.34)
and therefore
Hf(Yi,Tf) =Ns
i=1
Yi
h0fi (Tf) + hsi (Tf)
=
Ns
i=1
Yih0fi
(Tf) + hs(Yi,Tf) (1.35)
where Yi is the mass fraction of the ith chemical species, h0fi (Tf) and hsi (Tf) are respec-
tively the formation and the sensible enthalpies of the ith species.
The sensible enthalpy is defined thermodynamically:
dhs = CpdTf (1.36)
and therefore
hs(Yi,Tf) =ZTf
Tf,r
Cp(Yi,Tf)dT + hs(Yi,Tf,r) (1.37)
where Tf,r is a reference temperature and Cp(Yi,Tf) is the specific heat at constant
pressure given by
Cp(Yi,Tf) =Ns
i=1
YiCpi (Tf) . (1.38)
Also the internal energy is defined thermodynamically:
des = CvdTf (1.39)
and therefore
es(Yi,Tf) =ZTf
Tf,r
Cv(Yi,Tf)dT + es(Yi,Tf,r) (1.40)
where Cv(Yi,Tf) is the specific heat at constant volume given by
Cv(Yi,Tf) =Ns
i=1
YiCvi (Tf) . (1.41)
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Mathematical model 24
The relation between the sensible enthalpy and the internal energy is obtained by
subtracting (1.40) from (1.37):
dhs = des + (Cp Cv)dTf = des +RgdTf (1.42)
having used
Cp Cv = Rg . (1.43)
The gas constant Rg is defined as
Rg =Ru
Wmix=
Ru
1/N
si=1 Yi/Wi= Ru
Ns
i=1
Yi
Wi(1.44)
where Ru is the universal gas constant, and Wmix and Wi are respectively the mixture
and the single species molecular weight.
Energy Equation in Terms of Tf and Cp
The aim of this subsection is to derive the energy equation written in terms of the fil-
tered temperature and specific heat at constant pressure starting from equation (1.20),
since this is the form used in the HeaRT code used for validation in the present work.When termodynamic relations are applied to filtered quantities, terms accounting for
subgrid scale effects should appear. This terms will be omitted in the following deriva-
tion and their effect will be thought as modelled in the SGS heat flux qSGS together
with qhSGS.
The material derivative of enthalpy Hf(Yi,Tf) can be calculated as:
DHf(Yi,Tf)
Dt=
DHf(Yi,Tf)
DTf
DTf
Dt+
Ns
i=1
DHf(Yi,Tf)
DYi
DYi
Dt=
= CpDTf
Dt+
Ns
i=1
Hf,i(Tf)DYi
Dt(1.45)
having used equations (1.35) and (1.36) for working out DHf(Yi,Tf)/DTf and equa-
tion (1.34) for working out DHf(Yi,Tf)/DYi. Using the species mass fraction transport
equation for DYi/Dt yields
DHf(Yi,Tf)
Dt= Cp
DTf
Dt+
Ns
i=1
Hf,i(Tf)pYi fYiVc
(1.46) fYiVi JSGSi + fi p,i
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Mathematical model 25
where i the production / destruction rate of species i and an approximated form for
the ith species diffusion velocity Vi, with the correction for mass conservation Vc, has
been assumed.
The divergence of the diffusive heat flux qVi , due to single species diffusion velocity
(second term in (1.9)), can be written as
qVi =Ns
i=1
fYi (Vi + V
c) Hf,i(Tf) +Hf,i(Tf)
fYi (Vi + Vc)
(1.47)
Considering that
H
f,i(T
f) = h0
fi (T
f) + h
si (T
f) =C
pi T
f (1.48)equation (1.47) can be written as:
qVi =Ns
i=1
fYiCpi (Vi + V
c) Tf +Hf,i(Tf)
fYi (Vi + Vc)
(1.49)
Substituting (1.46), (1.9) and (1.49) into equation (1.20), and taking into account
the correction due to the approximation in the species diffusion, it is finally found:
fCpDTf
Dt
=Dp
Dt
+
kfTf qD + qSGS+Ns
i=1
Hf,i
JSGSi (1.50)
+f Qloss + fNs
i=1
Yi
fi Cpi Tf (Vi + Vc) Ns
i=1
fHf,ii
p (Hp +Lv) RU MWRU M p + Qp f +Ns
i=1
Hf,ip,i
It has to be observed that the last term depending on the formation enthalpies changes
with temperature is erroneously neglected in many books and numerical codes. The
formation enthalpies are usually calculated at a reference temperature, neglecting the
dependence of the molecular force field with temperature.
Energy Equation in Terms of Tf and Cv
The aim of this subsection is to write equation (1.50) in terms of temperature and
specific heat at constant volume. It is observed that the sum of the two terms containing
the material derivative of temperature and pressure in (1.50) can be written as
fCpDTf
Dt Dp
Dt
= fCpDTf
Dt fRg
DTf
Dt fTf
DRg
Dt RgTf
Df
Dt
(1.51)
= fCvDTf
Dt fTfDRg
Dt+ p uf +RgTfp
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Mathematical model 26
after having used the equation of state (p = fRgTf), the thermodynamic relation Cp Rg = Cv, and the continuity equation
Substituting (1.51) into (1.50) leads to
fCvDTf
Dt= p uf +
kfTf
qD + q
SGS
+Ns
i=1
Hf,i JSGSi (1.52)
+fTfDRg
Dt+ f Qloss + f
Ns
i=1
Yi
fi Cpi Tf (Vi + Vc) Ns
i=1
fHf,ii
p
Hp +Lv +RgTf
RU MWRU M p + Qp f +
Ns
i=1
Hf,ip,i
Using equation (1.44) the following relation is obtained
fTfDRg
Dt= fRuTf
Ns
i=1
1
Wi
DYi
Dt
When the expression for the mass fraction material derivative for the ith chemi-
cal species is used in the above relation and it is substituted in ( 1.52) together with
equation (1.28), the following form for the energy equation is finally obtained
fCvDTf
Dt= p uf +
kfTf
qD + q
SGS
+Ns
i=1
Hf,i JSGSi (1.53)
+f Qloss Ns
i=1
fi Cpi Tf
f WiWmix
DiXi + fYiVc
+RuTf
Ns
i=1
1
Wi
fWi
WmixDiXi
fYiV
c + JSGSi + fi p,i Ns
i=1
Hf,i
fi p,i p (Hp +Lv) RUMWRUM p + Qp f
This form is useful in numerical codes because it contains the material derivative ofTf
only. It is implemented in the HeaRT code which has been used for the validation of the
models proposed in this work, even though no reactive simulation has been performed.
The results that will be presented have been obtained neglecting the Dufour effect,
the heat loss, the kinetic energy dissipation during aerodynamic interaction between
phases and the dissipation function. The latters are usually small at the resolved scales,when low Mach number conditions are considered.
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Mathematical model 27
1.2 Dispersed phase model
The model that will be presented hereafter is based on the Mesoscopic Formalism that
was first introduced by Fvrier et al.[23] and takes advantage from the results of the
Kinetic Theory for gases [24].
When particles are assumed to behave like rigid spheres in an empty medium du-
ring their motion, they follow the rules of Hamiltonian systems. Let the function
fp (cp,p,p; xp, t) bedefinedasa Probability Density Function (PDF) giving the num-
ber of particles that at the time instant t are in the volume x + dx, with a velocity Vp
within the range cp +
dcp
, temperature p
within the range p +
dp
and diameter p
within the range p +dp. The evolution of fp is described by the Maxwell-Boltzmann
equation given by
fpt
+ cp,jfpxj
+ cp,jfp
cp,j+ p
fpp
+ pfpp
=
fpt
coll
(1.54)
where the term at the right hand side takes into account particle collisions effects, and
the velocities cp,j, p, p in the phase subspace (cp,p,p) are assumed to depends
only on (t,x). When the latter assumption is not valid anymore (e.g. the drag forceacting on a particle depends on the particle velocity itself), the evolution of the PDF is
described by the generalized form of the Maxwell-Boltzmann equation as follows
fp
t+
cp,j fp
xj+
cp,j fp
cp,j+
pfp
p+
pfp
p=
fp
t
coll
(1.55)
When multiphase flows are considered, particles do not move in an empty space but
interact with a surrounding medium. This interaction will affect particle PDF, that willdepend on the continuous phase flow variables as well. In order to overcome the diffi-
culties arising from this coupling Fvrier et al.[23] introduced, for the flow of inertial
particles under dilute conditions, a PDF conditioned by the continuous phase flow re-
alization Hf, that is fp
cp,p,p; xp, tHf. The reasons underlaying this choice may
be searched in the fact that, when dilute conditions are considered, the effects of inter-
particle collisions may be neglected[25]. When such an assumption holds, there is no
reason for two different particles to share the same velocity at the same moment and
place, with the exception of the action done by the continuous phase, and the influenceof the initial and boundary conditions. This means that the particle system does not
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Mathematical model 28
tend towards a solution where particle share a component of their velocity, since there
is not a mechanism for momentum redistribution among particles.
Thus, let the mass average of the particle velocity cp be taken, where cp is weighted
by the particle mass and conditioned by the new PDF fp,
up (xp, t) =1
pp
Zmp(p)cp fp
cp,p,p; xp, tHf
dcpdpdp (1.56)
where p is the dispersed phase volume fraction, p is the particle material density
assumed constant and mp is the particle mass. The assumption that a sufficient number
of particles is involved in the average process is implicit for the average to have anysense. The resulting velocity up is called mass weighted mesoscopic velocity or cor-
related velocity, meaning that it is due to the correlation brought to particle velocities
by the action done by the surrounding fluid. Should this action be neglected (ballistic
trajectory limit) and the average be taken at a position and time such that the influence
of initial and boundary conditions could be neglected as well, the up resulting from
(1.56) would be zero.
1.2.1 Average operators definition
Two different average operators will be used in this work: an ensemble average and a
mass average.
Ensemble average
The ensemble average
{}p of a generic property (cp,p,p) of the dispersed phase
is defined as
{}p =1
np
Z(cp,p,p) fp
cp,p,p; xp, tHf
dcpdpdp (1.57)
If(cp,p,p) = 1 is set in (1.57) the definition for the particle number density np is
obtained
np = Z(cp,p,p) fp cp,p,p; xp, tHfdcpdpdp (1.58)where np is the particle number per unit volume.
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Mathematical model 29
Mass average
The mass average of the generic property (cp,p,p) of the dispersed phase, that hasalready been used in the definition of the correlated velocity up, is defined as
p = 1pp
Zmp (p) (cp,p,p) fp
cp,p,p; xp, tHf
dcpdpdp (1.59)
where the particle mass mp is expressed in terms of the particle diameter
mp (p) = p3p/6 (1.60)
Both average operators are linear and have the property of being commutative with
respect to derivation in space and time, e.g.
xj
p
={}p
xj(1.61)
The two averages differ from each other with the exception of the case where p and
dp are constant. For the general case the following relation holds (see [14])
ppp = npmpp (1.62)Correlated and uncorrelated quantities for the dispersed phase
The mass weighted correlated velocity up has already been defined in (1.56). Let now
the k-th particle be considered, which finds itself at the time t in the position Xkp and
whose velocity is Vkp. It is possible to split such a velocity into two components: the
correlated up part shared by all the particles involved in the average operation and the
uncorrelated part Vkp which is characteristic of the k-th particle
Vkp
Xkp, t
= up
Xkp, t
+ Vkp
Xkp, t
(1.63)
A similar decomposition could be introduced for the particle temperature kp, this be-
ing considered unique within each particle (infinite thermal conductivity assumption).
After introducing the mesoscopic temperature Tp
Tp (xp, t) =
1
ppZ
mp (p) p (cp,p,p) fp cp,p,p; xp, tHfdcpdpdp(1.64)
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Mathematical model 30
the following decomposition can be adopted
kp
Xkp, t
= Tp
Xkp, t
+ kp
Xkp, t
(1.65)
as well as for the particle enthalpy kp
kp
Xkp, t
= Hp
Xkp, t
+ kp
Xkp, t
(1.66)
where Hp =p
p.
1.2.2 Conservation equations
Now that mesoscopic quantities have been introduced it is possible to obtain the con-
servation equations describing their evolution. Consider again the generic particle
property (cp,p,p). The conservation equation for the corresponding mesoscopic
quantity can be obtained by simply multiplying by (cp,p,p) equation (1.55), con-
ditioned by the flow realization Hf, and taking the mass average of the product. Thefollowing equation can thus be obtained
Z fpt
mp (p) (cp,p,p) dcpdpdp +
+Z
xj
cp,j fp
mp (p) (cp,p,p) dcpdpdp =
Z
cp,j cp,j fp
mp (p) (cp,p,p) dcpdpdp + (1.67)
Z
p
p fp
mp (p) (cp,p,p) dcpdpdp +
Z
p
p fp
mp (p) (cp,p,p) dcpdpdp +
+Z
fpt
coll
mp (p) (cp,p,p) dcpdpdp
By taking advantage from the commutation properties of the mass average recalled in
section 1.2.1, it is now possible to take the derivative operators with respect to x and toutside from the integrals, thus obtaining
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Mathematical model 31
t
Zfpmp (p) (cp,p,p) dcpdpdp +
+ xj
Zcp,j fpmp (p) (cp,p,p) dcpdpdp =Z
cp,j
cp,j fp
mp (p) (cp,p,p) dcpdpdp + (1.68)
Z
p
p fp
mp (p) (cp,p,p) dcpdpdp +
Z
p
p fp
mp (p) (cp,p,p) dcpdpdp +
+Z fpt
collmp (p) (cp,p,p) dcpdpdp
Taking now advantage of the average definitions and omitting for seek of simplicity
to write the independent variables inside the parenthesis, the equation (1.68) can be
written as
tppp +
xjppcp,jp
A
= (1.69)
Z cp,j
cp,j fpmpdcpdpdp B
Z p
p fpmpdcpdpdp C
+
Z
p
p fp
mpdcpdpdp
D
+Z
fpt
coll
mp (p) dcpdpdp C()
The terms B, Cand D at the right hand side can be developed following the procedure
described in [14] which is here recalled. For the term B it is possible to write
B = Z
cp,jcp,j
mpfpdcpdpdp B1
Z
fpcp,j
cp,jmpdcpdpdp B2
(1.70)
The term B2 can be integrated by parts yielding to
B2 =
cp,j fpmp
Z
cp,j
cp,jmp
fpdcpdpdp (1.71)
By developing the second term in the right hand side of eq. (1.71) and substituting inequation (1.70) the following relation can be written
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Mathematical model 32
B = Z
cp,j
cp,jmpfpdcpdpdp
B1
cp,j fpmp
B21+Z
cp,j
cp,jmpfpdcpdpdp
B22=B1
+Z
cp,jmp
cp,jfpdcpdpdp
B23
(1.72)
The symbol represent the boundary of the particle phase space, that is the do-
main containing all the possible particle states. This domain is in theory unbounded,
meaning that the variables can retain any value. However, all the realistic PDF tend
to zero for infinite values (positive or negative) of a variable, representing the physical
constraint that it is impossible to find a particle with infinite velocity, dimensions ortemperature. Nothing can be said a priori on the behavior of
cp,jcp,j
for this boundary.
The same problem is present for the part of the domain boundary corresponding to
particles of null diameters p. For both cases the assumption is made that the inverse
of fp tends to an infinite value while approaching the particle phase space boundary
with an higher order with respect tocp,jcp,j
mp. Under this assumption it is possible
to write
B21 = 0 (1.73)yielding to
B = B23 = ppcp,j cp,j
p (1.74)A similar development can be applied to term C in equation (1.69), leading to
C= ppp p
p (1.75)
Let the term D in equation (1.69) be now considered. It should be recalled here that,
since spherical shape and constant density have been assumed for particles, their mass
can be simply expressed by
mp = p
63p (1.76)
that after substitution in D leads to
D = p 6Z
p
p fp
3pdcpdpdp (1.77)
After developing the product inside the squared brackets the following relation is ob-tained
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Mathematical model 33
D = p 6Z
p
p3pfpdcpdpdp D1
p 6Z
fpp
p3pdcpdpdp D2
(1.78)
Let now the integration by parts with respect to p be applied to D2. The following
relation is obtained
D2 =
p3pfp
Z
p
p
3p
fpdcpdpdp (1.79)
where is the boundary for p domain. The development of the product in squared
brackets in the last equation, after substitution in equation (1.77) yields to
D = p 6
Z
p
p3pfpdcpdpdp
D1
+
+
p3pfp
D21
Z
p
p3pfpdcpdpdp
D1
+ (1.80)
Z p3p
p fpdcpdpdp
D23
Z p3p
p fpdcpdpdp
D24
For the same kind of assumptions that allowed to write equation (1.73) follows that
D21 = 0 (1.81)
Term D24 can be developed as
D24 = 3Z
p3p
p
fpdcpdpdp (1.82)
After the substitution of equations (1.81) and (1.82) into equation (1.80), the following
expression for term D is finally obtained
D = p
6(D23 +D24)
= p
6
Zp
3p
fp
3
p+
p
dcpdpdp (1.83)
=
Zpmp fp
3
p+
pdcpdpdp
= ppp3 p
+
p
p
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Mathematical model 34
The expressions found for B, C and D can now be substituted into equation (1.69).
Enskog general equation is thus obtained
tppp +
xjppcp,jp = C()
+ ppcp,j cp,j
p
+ ppp p
p (1.84)
+ ppp3
p+
ppwhere
C() =Z
fpt
coll
mp (p) (cp,p,p) dcpdpdp (1.85)
Note that at the RHS of equation (1.84), besides the collisional term, the variable pdependence on the velocities in the phase space is present. These velocities are the
particle acceleration cp and the time variation of particle temperature p and of the
particle dimension p. All these phenomena depend on the particle interaction with
the surrounding fluid and the respective terms in (1.84) will generate the source terms
responsible for the two-way coupling in the equation system.
Since cp represents the general value that can be assumed by k-th particle velocity
Vkp in the position x, the decomposition introduced in equation (1.63) can be substituted
in the second term at the left hand side of equation (1.84), yielding to
tppp +
xjppup,jp = T() +C()
+ ppcp,j cp,j
p
+ ppp p
p (1.86)
+ ppp
3
p+
p
p
where
T() =
xjpp
Vkp,j
p (1.87)
or, by taking advantage of the average properties (1.62), in terms of ensemble average
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Mathematical model 35
T() = xj
np
mpV
kp,j
p(1.88)
It is important not to confuse T with a SGS term. No filter operation in space
or time has been applied to the generic conservation equation of the dispersed phase
up to this point. T only expresses the fact that at a given point and time one could
find many particles each having a different velocity. Note that if the assumptions of
monodispersion [14] or that the temperature is unique within a particle cloud [26] are
removed, additional terms will appear as well in the equations. The information in T
is necessary to build a energy conserving system for the dispersed phase. It will be
shown in Chapter 3 how the lack of this term in the equations may have an influenceon the computed solution.
Starting from equation (1.86) it is possible to write the conservation equations for
the particle number density np, the dispersed phase mass pp, momentum ppup,
enthalpy ppHp and uncorrelated energy ppp
Particle number density
When = 1mp
is substituted in equation (1.86), the equation for the particle number
density np is obtained as follows
tpp 1
mpp +
xjppup,j 1
mpp = T
1
mp
+C
1
mp
+ ppp
3
1mp
p+
1mp
p
p (1.89)
By taking the derivative of the inverse of equation (1.76) with respect to p
1mp
p=
p
p
63p
1=
p
63p
23p
62p (1.90)
= 3p
6
3p
1
p= 3
mpp
and making use of (1.62), which for = 1mp
gives
pp
1
mp p = np (1.91)
and after substitution in (1.89) of (1.90) and (1.91), the following equation is obtained
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Mathematical model 36
npt
+
xjnpup,j = T
1
mp+C
1
mp(1.92)
The termC
1mp
represents droplet number increase or decrease because of collisions,
break-up or coalescence. In this work the absence of such physical aspect is assumed,
following from the dilute conditions hypothesis, which imply C
1mp
= 0. Note that,
when the only transport effect due to collisions of rigid spheres is considered, since the
particle number can not change because of collisions, it would be C
1mp
= 0 anyway.
T
1
mp
represents the change in particle number density because of the uncorre-
lated motion. When
f is a regular function, the simple substitution of=1
mp in (1.88)shows that T
1
mp
= 0. When f is not a regular function, which means that discon-
tinuities may arise, Maxwell-Boltzmann equation is not valid anymore throughout the
domain and it can not be integrated through the discontinuity. Proper jump conditions
must be adopted in that case.
Dispersed phase volume fraction
When = 1 is chosen, the equation for the dispersed phase volume fraction is obtained
tpp +
xjppup,j = T(1) +C(1) + pp 3
ppp
p
(1.93)
where the term p takes into account the mass exchange between phases. If= 1 is
set in the T definition (1.87), and remembering that for constant mp within the particle
cloud
Vp,jp = 0 (1.94)
it is possible to recognize that
T
1
mp
= 0 (1.95)
Again this is valid only where the function f is regular. C(1) = 0 can be set for the
same considerations done in the preceding paragraph. The term p will be modelledin the next sections.
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Mathematical model 37
Momentum
In order to obtain the momentum conservation equation it is necessary to substitute= cp,i into equation (1.69), which leads to
tppcp,ip +
xjppcp,jcp,ip = C(cp,i)
+ ppcp,j cp,icp,j
p
+ ppp cp,ip
p (1.96)
+ ppp3cp,ip
+cp,i
p
pIf the mesoscopic velocity definition (1.56) is substituted in the second term in the left
hand side of (1.96) one obtains
xjppcp,jcp,ip =
xjpp
up,j + Vp,j
(up,i + Vp,i)p
=
xjpp up,jup,i + up,iVp,jp (1.97)
+up,jVp,ip + Vp,jVp,ip
After applying the definitions (1.56) and (1.59), and performing some simplifications,
the following form for the momentum equation is obtained
tppup,i +
xjppup,jup,i =
xjppRp,i j +C(cp,i)
+ pp
Fp,i
mp p (1.98)
+ pp 3cp,ip
pp u,j
where Fp,i is the overall force acting on a particle in the i-th direction and
Rp,i j = Vp,jVp,ip = 1pp
ZVp,iVp,j fpdcpdpdp (1.99)
This second order moment of the particle velocity can be seen as a generalized stresstensor, even though it is very different in nature from the "conventional" one. While the
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Mathematical model 38
latter is due to the molecule momentum exchange through collisions, Rp,i j expresses
how the uncorrelated motion of the particles induces a variation in the correlated mo-
tion. Consider for example an isolated particle cloud at a given position x in space,
in a still medium. Let the particles belonging to this cloud have different velocities,
such that the correlated velocity results to be zero. At a successive time instant each
particle will have travelled in the direction of its initial velocity. If the average velocity
is now taken on the particles sharing the same new position, a correlated velocity will
be found having the direction of the line where both the initial and actual position lay.
The uncorrelated motion has thus induced a correlated velocity, on a smaller group of
particles.As to the collisional term, since the sum of the momentum within the particle cloud
can not change during collisions C(cp,i) = 0 can be set.
Finally, when use is made of the average property (1.62) the term u,j can be de-
veloped as follows
u,j = pp
3
cp,i
pp
p = npmpp3
cp,i
p = npup,idmp
dt (1.100)from which it is apparent that the term u,j expresses the effect of the particle mass
exchange on the dispersed phase momentum. It will have to be modelled as well as
Fp,i.
Uncorrelated energy
By setting = 12 Vp,iVp,i it is possible to obtain the conservation equation for the
uncorrelated energy p, which is defined as
p =1
pp12
Vp,iVp,ip (1.101)
From equation (1.84), by attending the procedure described in [24], the following con-servation law is found
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Mathematical model 39
tpp12Vp,iVp,ip +
xjppup,j12Vp,iVp,ip =
ppRp,i j uixj
xj
12
Qp,ii j +C1
2Vp,iVp,i
+ ppcp,j
12 Vp,iVp,i
cp,jp
+ ppp 12 Vp,iVp,i
pp (1.102)
+ ppp
312 Vp,iVp,i
p+
12 Vp,iVp,i
p
p
where Qp,i jk is the third order moment of particle velocity given by
Qp,i jk = ppVp,iVp,jVp,kp =Z
Vp,iVp,jVp,k fpdcpdpdp (1.103)
It can be observed that the sum of particle kinetic energies within the particle cloud
does not change during collisions (C
12 Vp,iVp,i
= 0) and
12 Vp,iVp,i
cp,j = Vp,iVp,icp,j = Vp,i
cp,j up,jcp,j = Vp,i (1.104)thus leading to
tppp +
xjppup,jp = ppRp,i j ui
xj
xj
1
2Qp,ii j
+ ppVp,i Fimp
p
WRUM(1.105)
+ pp32pp
Vp,iVp,ip RUM
where WRU M is the work done by the forces on particles and finally RU M is the rate of
change ofp due to mass exchange.
Enthalpy
When = hp is set in (1.86), hp being the particle enthalpy, the following conservationequation for the correlated enthalpy Hp is obtained
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Mathematical model 40
tpphpp +
xjppup,jhpp = T(hp) +C(hp)
+ ppcp,j hpcp,j
p
+ ppp hpp
p (1.106)
+ ppp
3hp
p+
hp
p
p
and after the substitution of
Hp =
hp
p (1.107)
and some simplification, the final expression for the correlated enthalpy conservation
equation is obtained
tppHp +
xjppup,jHp = T(hp) +C(hp)
+ ppp hpp
p
p
(1.108)
+ nphp dmpdt
h
where p represents the effect of the heat exchange between phases due to convection.
It is assumed that no enthalpy flux occurs during collisions thus leading to C(hp) = 0
In this work the temperature Tp is considered constant within the particle, and it
will then be possible to obtain it from the equation
fHT(Tp) = Hp Ho
p (1.109)
where Hop is the formation enthalpy. Since fHT(Tp) may be complex the solution of
the last equation may require to be solved by iteration or tabulated.
The terms at the right hand side, whose meaning should now be clear to the reader,
will be modelled in the following sections.
Final remarks
In the preceding section a model for the evolution of a dispersed phase, has been ob-tained starting from the equation of Maxwell-Boltzmann. The corresponding set of
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Mathematical model 41
equations may be used to describe the evolution of a monodispersed phase; in alterna-
tive different sets of equations may be used to simulate different classes of particles.
Each class is to be considered homogeneous in terms of size, temperature and physical
properties.
In order to obtain the model conservation equations, the following assumptions
have been made:
1. particles are supposed to be rigid and spherical;
2. particle dimensions should be small compared to the gas phase integral length
scale of turbulence (i.e. small compared to the discretization cell volume);
3. dilute conditions have been assumed;
4. the temperature is homogeneous within the particle;
5. particle temperatures and dimensions are considered unique within a particle
cloud.
When the equations will be numerically solved, no explicit filtering or SGS mo-
delling will be applied. A further assumption is then made: all the scales of the par-ticle motion are resolved, when a LES resolution for the gas phase is used in the dis-
cretization of the dispersed phase conservation equation. A DNS like approach will
then be used. This can be physically justified only for sufficiently inertial particles
St =pf
>> 1 where f is the smallest characteristic time scale of the resolved tur-
bulent structures. It should be clear to the reader now that, due to the lack of particle-
particle collisions, the dispersed phase flow does not create structures by its own. If
these are present, they must be the result of the interaction with the carrier phase. In
order for this to happen the particle relaxation time and the characteristic time of thegas-phase structure must be at least comparable.
It is here important to underline that the only differences in the present work with
respect to former works (see in particular [14] [12]) will be in the models for Rp,i j
and Qp,ii j.
1.2.3 Model closures
In the present section closures will be provided for the terms that came out from theequation development presented in 1.2.2. These are both the particle velocity moments
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Mathematical model 42
and the terms that express mass, momentum and energy exchanges between phases.
Closures for particle velocity moments
Let equation (1.55) be recalled here and the contributions due to mass and heat ex-
change be temporary neglected while searching for a solution for the transport equa-
tions
fpt
+ cp,j fpxj
+ cp,j fp
cp,j=
fpt
coll
fp cp,jcp,j
(1.110)
At the right hand side it is possible to recognize the co