Numerical modeling of two-phase reactive flows

7/29/2019 Numerical modeling of two-phase reactive flows

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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

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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

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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

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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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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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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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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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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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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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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


28/124


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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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 multicomponent 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


30/124


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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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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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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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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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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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


36/124


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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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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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


Z

p

p fp


+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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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


Z

p

p fp


+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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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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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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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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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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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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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


46/124


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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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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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


49/124


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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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

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Numerical modeling of two-phase reactive flows