Prof Ahmadi Azita

Prof Juan Senichault

researcher-HDR Martinez

Dominique

Pro.f Fekri Meftah

Mcf Khaled

Dr Mohamed Bourbatache

Guilherme Fiorot

Jia Fu

Ilham Benhsine

Saly Serhal

Sambath Ky

Tran Van

Hieu Nelson

Alvarado Patino

Ramez Baydoun

Chuan Chen

Boyuan Yang

Qingxiang Xiong

Yongbo Deng

Dr Yi Liu

Tian Xia

Jiali Xu

Hengyang Wei

Gang Shibo Liu

Huang Shijiang Wang

Hua Lu

Xiao Song

Yong Huang

Shijian Tian

Haiyun Yao

Qiong Wang

Ang Zhou

Yang Zhang

Ye Tian

Jinzeng Wang

Limiao Shi

Jiabin Liang

Gao, for their encouragement and friend ship letting me feel happy and warm in the foreign country.

More important, I would like to

Introduction

Fluid-solid two phase flow widely exists in nature and engineering such as mining and cement industries, particle transport in porous media, industry of power generation as well as environmental engineering [START_REF] Mcdowell-Boyer | Particle transport through porous media[END_REF][START_REF] Tehrani | Filtration of aerosol particles by cylindrical fibers within a parallel and staggered array[END_REF]3]. The response of solid phase can be strongly affected by the action of the fluid phase, i.e. fluidized bed reactor [4], pipe leakage [START_REF] Cui | Coupled DEM-LBM simulation of internal fluidisation induced by a leaking pipe[END_REF], poured concrete for construction [START_REF] Edouard | Modélisation numérique des écoulements granulaires denses immergés dans un fluide[END_REF]. The fluid phase is characterized by its velocity and pressure and exerts hydrodynamic force on the interface of the solid, which leads to movement of the solid under its action. In turn, the distribution of flow velocity field varies due to the presence of the solid. This retroaction of the interaction between the fluid and the solid makes the problem of fluid-solid far more complex than its single-phase counterpart.

There are a variety of phenomena involving both fluids, mainly water or air, and solid particles. These phenomena raise a great number of scientific problems in terms of the complexity of physical mechanisms. The transport of solid particles suspended in fluid takes place in many physical processes [START_REF] Ding | Extension of the lattice-Boltzmann method for direct simulation of suspended particles near contact[END_REF], such as sediment transport, debris flow, soil erosion, filtration etc. To gain fundamental insight in fluid-solid systems, some examples of problems involving fluid-solid interactions are presented in Fig. 1 and Fig. 2 Particles involve in industrial emissions, which leads to contamination that directly bring about an antagonistic effect on human environment. It is essential to remove particles from their surrounding sources. The problem of removing particles from fluid has become increasingly significant from the point of reducing health hazard [START_REF] Yeh | Aerosol filtration by fibrous filters -I. Theoretical[END_REF]. Thus an efficient fluid-solid separation system is required to deal with the problem of fluid purification.

Fluid-particle separation refers also to a process for filtrating particles from fluid. Filtration, one of the particle separation processes, is a very important technology [START_REF] Wang | Filtration of aerosol particles by elliptical fibers: A numerical study[END_REF] and Figure 3: Filtration application, top: filtering stage and clean stage, bottom: particle separation process, n upstream and n downstream represent the number of particle, at the upstream and downstream respectively [START_REF] Ringholm | Understanding filter beta ratios[END_REF].

numerically investigated. This study reveals very interesting points that can be helpful to optimize performance of filter. Chapter 4 focus on the influence of arrangement of fiber on filtration process. The circular, diamond and square fiber are staggered arranged in a filter. The effects of Reynolds number, particle-to-fiber diameter ratio and particle-to-fluid density ratio on pressure drop and filtration efficiency of filter are investigated. The simulation results are compared to explore the fiber's cross-sectional shape effect on quality factor. Finally, we summarize the general conclusions of the thesis and make recommendations for future work.

Chapter 1

Literature review on hydrodynamic and mechanical interaction of fluid and solid particles

Introduction

Particle transport in fluid flows has a wide range of applications. This phenomenon generally refers to fluid-particle and particle-particle interactions. These interactions lead to the complexity of flow field. For this reason, a flow field is initially needed. Then suspended particles flows in porous media can be described. Thus the objective of this chapter is to introduce the physical phenomena involved in particle transport and deposition in porous media at the particle scale. In this chapter, we first introduce the governing equation used for flow field and describe the porous media and fluid flow in porous media. Since the particle filtration is commonly used to describe the fluid-particle problems. Then the filtration parameters and capture mechanisms as well as the numerical methods are introduced.

Finally some previous results about particle filtration in literature are presented.

Flow of suspensions through porous medium 1.2.1 Governing equations in fluid flow

In order to study the transport and deposition of particles flow through porous media, it is first of all necessary to analyze velocity flow field through porous media under steady state.

CHAPTER 1. LITERATURE REVIEW ON HYDRODYNAMIC AND MECHANICAL INTERACTION OF FLUID AND SOLID PARTICLES

Generally, for the flow field, the classical approach is based on the numerical solution of Navier-Stokes equations [START_REF] Wang | Hydraulic fracture simulation based on coupled discrete element method and lattice Boltzmann[END_REF]. Navier-Stokes equations are first numerically solved by using the finite volume method developed by Patankar [START_REF] Patankar | Numerical heat transfer and fluid flow[END_REF]. The fluid phase is often viewed as a continuum for practical purposes in Navier-Stokes equations.

Continuity equation

The mass conversation law is known as the equation of continuity. In fluid dynamics, the rate of fluid mass enters a small stationary volume element is equal to the fluid mass leaving the volume element. It can be written as [START_REF] Zou | On pressure and velocity boundary conditions for the lattice Boltzmann BGK model[END_REF]:

∂ρ ∂t + ∇ • (ρ u) = 0 (1.1)
where ρ is fluid density and u is fluid velocity. The behavior of a fluid in porous medium should conforms to this equation. For incompressible fluids with constant density, then ρ is constant in Eq.(1.1) and the continuity equation can reduce to:

∇ • u = 0 (1.2)

Navier-Stokes equations

The Navier-Stokes equations for flow of fluids of single phase are well accepted and closedform solution for specific cases are well documented [START_REF] Crowe Clayton | Multiphase flows with droplets particles[END_REF]. For constant fluid density ρ and constant viscosity µ, one form of the Navier-Stokes equations is expressed as:

ρ ∂ u ∂t + u • ∇ u = -∇p + µ∇ 2 u + ρ f b (1.3)
Navier-Stokes equations describe the momentum and mass conservation of fluid phase at time t with velocity u, pressure p, dynamic viscosity µ and applied body force per unit mass f b . It is used for calculating flow in incompressible fluids.

Reynolds number (Re) is an important dimensionless quantity in fluid mechanics. Re is a criterion to determine the relative importance of inertial and viscous effects and defined as:

Re = ρU D f µ (1.4)
where D f is the characteristic length of porous media. Reynolds number is often used to 1.2. FLOW OF SUSPENSIONS THROUGH POROUS MEDIUM predict flow patterns. For general flow such as piping flow, when the Re is lower a critical value 2000 which is validated by experimental data [START_REF] Happel | Low Reynolds number hydrodynamics[END_REF], the flow is called laminar flow. However, for Re > 2000, the flow is called turbulent flow. For porous media, this limit is equal to 160 for Reynolds number. In laminar flow regime, if the Reynolds number is very small, much less than 1, the fluid is called creeping flow, where the viscous forces of the fluid dominate the inertial forces.

Porous media 1.2.2.1 Porosity of porous media

A porous medium is constituted by a solid matrix and voids called pores. These pores can be partially or completely filled with water, air and other fluid. Generally the fraction of volume of pores over the total volume in porous media is called porosity φ varying between 0 and 1. It is defined as the ratio of the volume of pores to the total sample volume.

φ = volume of pore total sample volume (1.5) In a general way, there are three types of porosity, intergranular porosity, fracture porosity and cavernous porosity as illustrated in Fig. 1. 1 [22]. Intergranular porosity is encountered between the grains of soil, sediment, or incompletely cemented sedimentary rock. As shown in Fig. 1.1 (a), the circles represent grains and cross-hatching represents the pore space. Fracture porosity is produced by the tectonic fracturing of rock. Crosses represent rock and black is fractures in Fig. 1.1 (b). Cavernous porosity only occurs in rocks that can be chemically weathered by solution. The black region has been dissolved and filled with air as shown in Fig. 1.1 (c), the blue region is almost filled with groundwater.

The porosity has the relationship with the volume occupied by the solid termed as solid volume fraction (or packing density) and can be given by:

Ψ = 1 -φ (1.6)

Characteristic of stack for sphere porous media

Consider the porous medium composed of sphere grains on periodic cubic lattice. The sphere grains touched each other and they are not compacted any more. The size of all the sphere are identical. The proportion of space filled by the spheres is called the packing INTERACTION OF FLUID AND SOLID PARTICLES In three dimensions, the most four common types of unit cell are the simple cubic (SC), body-centered cubic (BCC), face-centered cubic (FCC) and hexagonal close packing (HCP) [START_REF] Roberts | Grain consolidation and electrical conductivity in porous media[END_REF][START_REF] Higdon | Permeability of three-dimensional models of fibrous porous media[END_REF][START_REF] Lorenz | Similarity of percolation thresholds on the HCP and FCC lattices[END_REF][START_REF] Larson | A periodic grain consolidation model of porous media[END_REF] as presented in Fig. 1.2. In Fig. 1.2, the neighbouring sphere grain should contact each other, here we just show the location of each sphere for the sake of simplicity. Then the packing density of each cubic are presented in Table 1.1: For type of SC, BCC, FCC and HCP, the maximum porosity of porous media is SC which has the minimum packing density. The porosity of SC is equal to 1 -π/6 = 47.6%.

Another two simple arrangements within the close-packed packing correspond to regular INTERACTION OF FLUID AND SOLID PARTICLES lattices. One is FCC and the bottom of FCC is triangle. Whereas, another is HCP and the bottom of HCP is hexagon. For FCC and HCP, each sphere is surrounded by 12 other spheres. Then the porosity of BCC and HCP is 1 -

√ 2π 6 = 25.9%.
When the sphere grains are randomly arranged, the packing density varies between the random loose packing (RLP) and the random close packing(RCP) [START_REF] Valverde | Random loose packing of cohesive granular materials[END_REF]. The packing density for upper limit is referred to RCP Ψ RCP = 64% and φ RCP = 36%, the lower limit is RLP Ψ RLP = 56% and φ RLP = 34% [START_REF] Valverde | Random loose packing of cohesive granular materials[END_REF][START_REF] Onoda | Random loose packings of uniform spheres and the dilatancy onset[END_REF].

Particles deposition in porous media

Particle size distribution

In the study of particles transport and deposition in porous media, particle size is a very important parameter which governs the flow of a dispersed two-phase mixture. Porous medium such as soils and sands are often described by their particle size distribution.

Particle size distribution is often a critical parameter in the manufacture of many products [START_REF] Araya | A physicoempirical model to predict the soil moisture characteristic from particle-size distribution and bulk density data[END_REF][START_REF] Fredlund | Prediction of the soil-water characteristic curve from grain-size distribution and volume-mass properties[END_REF]. It can be used to predict physical properties such as permeability and porosity [31].

The most general definition of the spread of the particle size distribution is monodisperse or polydisperse. A monodisperse distribution is one in which the particles are all of the same size, whereas polydisperse suggests a wide range of particle sizes.

A typical example of particle size distribution of sand is given in Fig. 1.3 [START_REF] Fredlund | Equations for the soil-water characteristic curve[END_REF]. It shows the percentage passing of a particular particle as a function of particle size. From Fig. 1.3, for a certain size of particle, we can easily know the percentage passing of a particular particle size. For example in Fig. 1.3, the particle size distribution is nearly 10% for 10 µm. [START_REF] Fredlund | Prediction of the soil-water characteristic curve from grain-size distribution and volume-mass properties[END_REF].

Particle straining in constriction

The clogging of a porous media is characterized by the ratio between diameter of particles d p and diameter of grains D f . A particle penetrates a triangular constriction formed by three tangent spherical grains as shown in Fig. 1.4. There exist a limit for particles, the limit can be determined by the following relation [START_REF] Herzig | Flow of suspensions through porous media-application to deep filtration[END_REF]:

d p D f limit = 2 √ 3 -1 = 0.154 (1.7)
Above this limit, the pore size is smaller than diameter of particle, no particle can penetrate through the constriction and all particles are captured by the grains. If dp D f < 0.154, the clogging of porous medium is weak and particle can easily be transported from the pore.

It is also possible to determine the straining by several particles. For three particles and four particles, the limit are respectively: The threshold of particles trapped in a triangular construction [START_REF] Herzig | Flow of suspensions through porous media-application to deep filtration[END_REF].

d p D f limit = 0.1 (1.8)
d p D f limit = 0.082 (1.9)
It is noticed that simultaneous arrival of particle is not necessary for straining. However, straining may occur by the successive stoppages of particles. As the constriction is blocked, all particles entrained by fluid are retained at this place.

Fluid flow in porous media

The pore structure in porous media is often complex and complicated flow patterns exist within the pores and between the grains. Permeability is a measure of the ability of a porous medium to be traversed by an interstitial fluid flow. It is commonly used to investigate the effect of pore structure on the properties of porous media [START_REF] Nabovati | A general model for the permeability of fibrous porous media based on fluid flow simulations using the lattice Boltzmann method[END_REF]. In general, the porous media are characterized by porosity and permeability. These two intrinsic magnitudes are macroscopic variables estimated on a representative elementary volume containing a large number of microscopic entities composing the material.

Darcy's law

Fluid flow through porous media plays an important role in a variety of engineering system. Here we consider the saturated porous media for which the volume of the pores is completely filled with fluid. Permeability and pressure drop are two important parameters in porous media. Darcy's law which regards the medium as continuum is satisfactory for the prediction of flow through porous media [START_REF] Hellou | Stokes flow through microstructural model of fibrous media[END_REF]. Here the considered porous medium is homogeneous and isotropic. Thus, the pressure drop corresponding to a desired flow rate through the porous medium can be obtained by Darcy's law:

FLOW OF SUSPENSIONS THROUGH POROUS MEDIUM

Q = -κA µ ∆p L (1.10)
Where Q is the volume flow rate, µ is the dynamic viscosity of fluid. κ and A are respectively permeability of the medium and a cross-sectional area perpendicular to the fluid flow direction. ∆p is the total pressure drop between inlet and outlet, L is total length on which the pressure difference acts.

Q = V D A (1.11) V D = -κ µ ∇p (1.12)
where V D is the flow rate per unit area of the cross section, ∇p is pressure gradient. We can get the fluid velocity V P through the pores by the following:

V P = V D φ (1.13)

Kozeny-Carman equation

Kozeny-Carman equation is used to predict the flow rate through porous media such as filter beds and fluidized beds [START_REF] Tien | Granular filtration of aerosols and hydrosols[END_REF]. It is obtained on the basis that flow is laminar and expressed as:

∆p L = 180µ Φ 2 s D 2 p (1 -φ) 2 φ 3 V D (1.14)
where Φ s is the sphericity of the particles and Φ s = 1 for a sphere, which is the measure of how closely the shape of a particle approaches a perfect sphere. D p is the diameter of the related spherical particle. We product A at both sides of Eq 1.14 and get: 2 (1. [START_REF] Ringholm | Understanding filter beta ratios[END_REF] for the fixed fluid and porous media, Γ =

∆P L A = 180µ Φ 2 s D 2 p (1 -φ) 2 φ 3 Q (1.
Q = Γ • ∆P L A • φ 3 (1 -φ)
Φ 2 s D 2 p
180µ is constant. By combining with the Eq 1.10, the link between porosity and permeability is created as follows:

Γ • φ 3 (1 -φ) 2 = -κ µ (1.17)
Darcy's law and Kozeny-Carman equation are used to describe the flow of fluid through porous media. In order to obtain detailed information about the flow field associated with the porous medium, the solution of flow field in porous media need to be determined.

Kuwabara solution

In porous media, the flow field determination is generally difficult and problem must be solved numerically. Nevertheless, in some relatively simple configurations, it can be solved analytically. For example, we can introduce an analytical solution of Stokes equation based on stream function expressed as:

∇ 4 Ψ(r, θ) = 0 (1.18)
and the general solution of Stokes equation finally can be given as follows [START_REF] Hellou | Etude numerique and experimentale de l'écoulement à structure cellulaire, engendre par la rotation d'un cylindre dans un canal[END_REF]:

Ψ(r, θ) = a 0 + b 0 ln(r) + c 0 r 2 + d 0 r 2 ln(r) + (a 1 r -1 + b 1 r ln(r) + c 1 r + d 1 r 3 ) sin(θ) + (a ′ 1 r -1 + b ′ 1 r ln(r) + c ′ 1 r + d ′ 1 r 3 ) cos(θ) + ∞ n=2 (a n r -n + b n r -n+2 + c n r n + d n r n+2 ) sin(nθ) + ∞ n=2 (a ′ n r -n + b ′ n r -n+2 + c ′ n r n + d ′ n r n+2 ) cos(nθ) (1.19)
where a 0 , b 0 , c 0 , d 0 , a

1 , b 1 , c 1 , d 1 , a ′ 1 , b ′ 1 , c ′ 1 , d ′ 1 , a n , b n , c n , d n , a ′ n , b ′ n , c ′ n , d
′ n are coefficients. The components of velocity are satisfied and calculated using the following relationships:

V r = ∂Ψ r∂θ (1.20) V θ = - ∂Ψ ∂r (1.21)
The symmetry condition:

   Ψ(r, θ) = -Ψ(r, -θ) Ψ(r, θ) = Ψ(r, π -θ) (1.22)
These conditions lead to the following relations: [(a 2n-1 r -2n + b 2n-1 r -2n+2 + c 2n-1 r 2n-2 + d 2n-1 r 2n )(2n -1)] cos((2n -1)θ)

           a 0 = b 0 = c 0 = d 0 = 0 a ′ 1 = b ′ 1 = c ′ 1 = d ′ 1 = 0 a ′ n = b ′ n = c ′ n = d
V θ = (-a 1 r -2 + b 1 ln(r) + b 1 + c 1 + 3d 1 r 2 ) sin(θ)- ∞ n=2,3...
[(-2n + 1)a 2n-1 r -2n + (-2n + 3)b 2n-1 r -2n+2 + (2n -1)c 2n-1 r 2n-2 + (2n + 1)d 2n-1 r 2n ] sin((2n -1)θ) (1.26) For the boundary condition of the streamfucntion, we use the Kuwabara model [START_REF] Kuwabara | The forces experienced by randomly distributed parallel circular cylinders or spheres in a viscous flow at small Reynolds numbers[END_REF][START_REF] Marshall | An improved analytic solution for analysis of particle trajectories in fibrous, two-dimensional filters[END_REF] in which the boundary is a cylindrical outer boundary as shown in Fig. 1.5 (b). A series of periodic arrangement of cylinders with a square unit is shown in Fig. 1.5 (a). At the surface of the fiber, we have:

v r = v θ = 0, at r = R f (1.27)
For the outer boundary condition, Kuwabara uses v r = cos θ and ξ = 0, at r = R f + ∆R (1.28) where ξ is the vorticity, then Kuwabara's solution is

1.3. FILTRATION 19
Ψ K (r, θ) = R f [C 1 ( r R f ) 3 + C 2 r R f + C 3 R f r + C 4 r ln( r R f )] sin θ (1.29)
where C 1 , C 2 , C 3 , C 4 are the coefficients, which are found using the boundary conditions (1.27)-(1.28) 2 (1.30)

C 4 = - 4 2 ln(1 -φ) -1 + 4φ + (1 -φ)
C 1 = - C 4 (1 -φ) 4
(1.31)

C 3 = C 1 + C 4 2 (1.32) C 2 = -C 1 -C 3 (1.33)
where φ is the porosity, for the circular unit it is given by:

φ = 1 -( R f (R + ∆R) ) 2 (1.34)
The Kuwabara's solution is known, then the fluid velocity field can be calculated in such simple configurations.

Filtration

Filtration type

Filtration is commonly used to describe the deposition and transport of particles from fluid suspension through a porous medium. Particle separation from the fluid requires a high efficiency filtration system. In terms of filtration process, variety of filtration type exists, they can be classified into cake filtration, granular filtration, fibrous filtration [START_REF] Tien | Introduction to the cake filtration: Analyses, Experiments and Applications[END_REF].

Cake filtration

Cake filtration for particle separation is often carried out in several stages such as cake formation and growth [START_REF] Tien | Introduction to the cake filtration: Analyses, Experiments and Applications[END_REF]. A filter cake is formed by the solid particles that are retained on the surface of filter medium as shown in Fig. 1.6. At the initial stage of cake filtration, the smaller particles can penetrate into the filter layer. In the subsequent filtration process, INTERACTION OF FLUID AND SOLID PARTICLES the accumulation of these particles gradually leads to the increase of the thickness of the cake, then smaller particles deposit within the porous cake formed by the large particles [START_REF] Tien | Analysis of cake growth in cake filtration: Effect of fine particle retention[END_REF][START_REF] Civan | Practical model for compressive cake filtration including fine particle invasion[END_REF]. Finally it allows only the passage of the suspending fluid.

Flow direction

Filter cake Porous media

Granular filtration

Granular filtration is a process that primarily clarifies dilute suspensions using granular media as collected bodies. In granular filtration, two solid phases are included: one is the granular media through which the fluid flows, and the other solid phase comprises particles that are suspended in the fluid phase [START_REF] Kolakaluri | Filtration model for polydisperse aerosols in gas-solid flow using granule-resolved direct numerical simulation[END_REF]. Granular porous media initially are free of particles, they can accumulate deposits by removing particles from suspension (see Fig. 1.7) [START_REF] Mays | Contrasting clogging in granular media filters, soils, and dead-end membranes[END_REF]. For granular filtration, particle deposition takes place throughout the entire filter medium.

Fibrous filtration

Fibrous filtration refers to the process in which the removal of particles from fluid pass through fibrous media. In terms of the constituted filter media, particle retention occurs in the form of filter cake either throughout the medium or at the surface of the medium. If the medium is formed by packing fibers loosely such as those applied in air-condition, particle retention mostly happens within the interior of the medium [START_REF] Tien | Granular filtration of aerosols and hydrosols[END_REF]. This kind of fibrous filtration is similar to granular filtration. Generally speaking, a fibrous filter medium comprise nature and synthetic materials for use in both gas and liquid application.

Quality factor

In several decade years, many researchers have focused on the study of particle filtration process [START_REF] Marshall | An improved analytic solution for analysis of particle trajectories in fibrous, two-dimensional filters[END_REF][START_REF] Schweers | Experimental investigation of particle collection on single fibres of different configurations[END_REF][START_REF] Rembor | Measurements of single fibre effciencies at critical values of the Stokes number[END_REF][START_REF] Haugen | Particle impaction on a cylinder in a crossflow as function of Stokes and Reynolds numbers[END_REF]. The filter composed of fibers is used to remove the particle from the fluid, thus the filter performance need to be estimated in order to ensure the long term use of filter. A best-performing filter first of all not only assures a high filtration efficiency of filter η but also maintains a low pressure drop ∆p. The detailed of the parameters is presented in the next section. In order to judge the performance of filter, the definition of quality factor (or the figure of merit), which is an indicator of the ratio of penetration rate P (P = 1 -η) to the pressure drop, is expressed as [START_REF] Wang | Filtration of aerosol particles by elliptical fibers: A numerical study[END_REF]:

QF = -ln(P ) ∆p (1.35)
From the filter quality factor standpoint, the larger value of quality factor QF indicates a better quality of fibrous filter.

Filtration efficiency

The deposition of particles on the surface of filter medium can modify the pore morphology. The accumulation of the particles leads to the filter clogging. As a consequence, the permeability declines and pressure drop increases which damage filter life. Thus, it is essential to quantitatively estimate a filter performance. A filter is generally composed of a number of unit bed elements (UBE) connected in series. The UBE consists of several individual collectors with some simple geometry such as a capillary, sphere or constricted tube as depicted in Fig. 1.9. Generally, a filter performance is described by its overall efficiency, which is deduced from the efficiency of UBE by simple association formula, that will be introduced hereafter.

Here we use circular circle to represent fiber in filter as shown in Fig. 1.10. The filtration efficiency of a filter η is depicted in Fig. 1.10 (a) and defined as [START_REF] Tien | Granular filtration of aerosols and hydrosols[END_REF]:

η = c in -c ef f c in (1.36)
where c in and c ef f denote the influent and effluent particle concentrations, respectively. A filter generally includes a number of UBEs, the efficiency of the ith UBE is illustrated in

e i = c i-1 -c i c i-1 (1.37)
where c i represents the particle concentration of the suspension exiting the ith unit collector. The efficiency of UBE is dependent on both individual collector efficiency and arrangement of porous media. The individual collector efficiency corresponds to the fraction of particles encountering the collector. It is not only relative to the nature and mechanism of the transport of particle from the suspension to the collector surface and adhesion of the particles to the collector surface, but also to the flow field around the individual collector.

Finally, the filtration efficiency can be expressed in terms of efficiency of UBE :

η = 1 - N i=1 (1 -e i ) (1.38)
where N is total number of UBEs connected in series, is the symbol of product.

Single fiber capture efficiency

The flow of particles through porous media is a very complex phenomenon due to the diversity of the mechanisms involved [START_REF] Herzig | Flow of suspensions through porous media-application to deep filtration[END_REF]. A number of factors are responsible for particle 1.3. FILTRATION transport. As a transported solid particle approaches a fiber, the particle deposits on the surface of fiber by simultaneous action of several mechanisms, such as Brownian diffusion, interception and inertial impaction [START_REF] Wang | Numerical simulation of particle capture process of fibrous filters using Lattice Boltzmann two-phase flow model[END_REF]. The filtration efficiency η of clean filter medium is closely linked to single fiber capture efficiency η s . Single fiber capture efficiency is defined as [START_REF] Hinds | Aerosol technology[END_REF]:

η s =
number of particles captured by fiber number of particles in the volume geometrically swept out by fiber (1.39) It is based on mass balance on a filter element of thickness dl and integrated on the total filter thickness L. Then the relationship of total filtration efficiency and single fiber capture efficiency is [START_REF] Flagan | Fundamentals of air polllution engineering[END_REF]:

η = 1 -exp - 4βη s L π(1 -β)D f (1.40)
where β is the solid volume fraction occupied by the fibers. D f is the fiber diameter.

Single fiber capture efficiency is related to collision efficiency and adhesion probability [START_REF] Schweers | Experimental investigation of particle collection on single fibres of different configurations[END_REF]. Collision efficiency of fiber η c is the ratio of number of particle collided with fiber to the number of particles entering the project area of fiber. The relation ship between the single fiber capture efficiency η s and collision efficiency η c can be expressed:

η s = η c j (1.41)
where j is adhesion probability specifying the fraction of particles adhered on the fiber with respect to the total number of particles contacting with it. In this work, the adhesion probability j = 1, so the single fiber capture efficiency is simply taken equal to the collision efficiency.

The single fiber capture efficiency η s depends on the capture mechanism involved in filtration process. The mechanical single fiber capture efficiency is expressed as [START_REF] Qian | Retraction: Effects of the operating conditions and geometry parameter on the filtration performance of the fibrous filter[END_REF]:

η s = 1 -(1 -η D )(1 -η R )(1 -η I ) (1.42)
where η D is capture efficiency due to Brownian diffusion, η R is efficiency due to interception, η I is capture efficiency due to inertial impaction. For a particular range of particle size, generally one mechanism is dominant, the terms in Eq (1.42)

η D η R , η D η I , η R η I and η R η D η I
are neglected. Thus the single fiber capture efficiency can be calculated as: INTERACTION OF FLUID AND SOLID PARTICLES

η s = η D + η R + η I (1.43)
For the purpose of better understanding the meaning and physical significance of mechanisms, we will now examine the particle behavior individually and present the three mechanisms.

Capture mechanism 1.3.5.1 Brownian diffusion

In diffusion, particles transport from a higher concentration to a lower concentration, whereas in Brownian diffusion, particle suspended in a fluid transports irregularly due to the collisions with the surrounding particles. Brownian diffusion is one of the basic particle capture mechanisms [START_REF] Hosseini | Modeling particle filtration in disordered 2-D domains: A comparison with cell models[END_REF]. The influence of the Brownian diffusion force on particle deposition is significant for the submicrometer particles [START_REF] Tien | Granular filtration of aerosols and hydrosols[END_REF]. Brownian diffusion is the dominant factor when particle size is generally smaller than 0.1 µm [START_REF] Wang | Numerical simulation of particle capture process of fibrous filters using Lattice Boltzmann two-phase flow model[END_REF] for the deposition in a filter medium. The Brownian force is usually taken into account for analyzing the trajectories of small particles. Fig. 1.11 illustrates the particle trajectory subject to capture by Brownian diffusion. The Brownian motion is strong enough to move particle from the original streamlines to the surface of a fiber. This contribution increases as the particle size decreases. The particle capture corresponding to Brownian diffusion is related to the dimensionless number called Péclet number P e, which is the ratio of advective transport of particles to their Brownian transport (or random transport of particles) near a fiber surface [START_REF] Mays | Hydrodynamic aspects of particle clogging in porous media[END_REF]. The Péclet number provides an indication of the relative importance of diffusion and convection 1.3. FILTRATION 27 [START_REF] Wang | Modeling of filtration efficiency of nanoparticles in standard filter media[END_REF]. The Péclet number is:

P e = U D f D (1.44)
where U is flow velocity, D f is fiber diameter and D is Brownian diffusion coefficient defined as:

D = k B T C u 3πµd p (1.45)
where k B is the Boltzmann constant and T is the temperature. µ and d p are the fluid dynamic viscosity and particle diameter. C u is the Cunningham coefficient (or slip correction factor) related to the particle diameter d p and mean free path of gas molecules λ given by [START_REF] Li | Dispersion and deposition of spherical particles from point sources in a turbulent channel flow[END_REF]:

C u = 1 + 2λ d p 1.257 + 0.4 exp -0.55 d p λ (1.46)
When the Brownian diffusion is the dominant capture mechanism for small particles, the single fiber capture efficiency is approximately equal to diffusion capture efficiency η D .

In some articles, a number of theoretical formulas for capture efficiency due to Brownian diffusion are developed. [START_REF] Stechkina | Studies on fibrous aerosol filters -I. Calculation of diffusional deposition of aerosols in fibrous filters[END_REF] [START_REF] Stechkina | Studies on fibrous aerosol filters -I. Calculation of diffusional deposition of aerosols in fibrous filters[END_REF] dealt with filter model consisting of a system of parallel circular cylinders and proposed a diffusion equation based on the boundary layer analysis:

η D = 2.9K -1/3 u P -2/3 e + 0.624P e -1 (1.47)
where K u is the hydrodynamic factor of Kuwabara's flow [START_REF] Yeh | Aerosol filtration by fibrous filters -I. Theoretical[END_REF][START_REF] Kuwabara | The forces experienced by randomly distributed parallel circular cylinders or spheres in a viscous flow at small Reynolds numbers[END_REF] and is equal to:

K u = - 1 2 ln β -0.75 + β - 1 4 β 2 (1.48)
where β is the volume fraction occupied by the solid.

For submicron particles, which have a low diffusion coefficient and a large Péclet number, Lee and Liu (1982) [START_REF] Lee | Theoretical study of aerosol filtration by fibrous filters[END_REF] presented an analysis of the convective Brownian diffusion equation to determine the capture efficiency:

η D = 2.6 1 -β K u 1/3 P e -2/3 (1.49)
Eq 1.49 used the Kuwabara's flow field to consider the interference effects between the INTERACTION OF FLUID AND SOLID PARTICLES neighboring fibers, however, this equation does not account for the effect of slip at the gas-fiber interface [START_REF] Wang | Removal of nanoparticles from gas streams by fibrous filters: A review[END_REF].

For nanoparticles, which have a small Péclet number (P e is of the order of unit),

Wang et al. [START_REF] Wang | Modeling of filtration efficiency of nanoparticles in standard filter media[END_REF] developed an empirical power law model for diffusion capture efficiency depending on P e, they gave the following empirical equation:

η D = 0.84P e -0.43 (1.50)
Eq 1.50 agreed well with experimental data. It is noticed that the power of P e is -0.43 not -2/3 as in eq 1. [START_REF] Flagan | Fundamentals of air polllution engineering[END_REF], indicating lower dependence of capture efficiency on P e.

Interception

The second basic capture mechanism is interception in the filtration process. Some particles deposited on the surface of a fiber are the result of interception. Particle size is generally larger than 0.1 µm for interception mechanism [START_REF] Wang | Numerical simulation of particle capture process of fibrous filters using Lattice Boltzmann two-phase flow model[END_REF]. In this size range, particles are assumed to have the same mass as fluid particles and follow the fluid streamlines [START_REF] Liu | Pressure drop and interception efficiency of multifiber filters[END_REF]. Interception takes place when the distance between the center of a particle and surface of a fiber is equal to or less than the particle radius, even if particle remains on the corresponding streamline [START_REF] Wang | Removal of nanoparticles from gas streams by fibrous filters: A review[END_REF]. Fig. 1.12 illustrates particle capture by the fiber due to interception effect. For each size fiber, there exists a specified region as shown in Fig. 1.12 (capture zone) , within which all the particles are captured by the fiber in the fluid flow. Outside of this region, particles are not deposited on the fiber and flow out from the filter. The interception efficiency of single fiber due to the interception mechanism depends on intercept coefficient R, which is defined as the ratio of particle diameter to fiber diameter as follows:

Capture zone

1.3. FILTRATION 29 R = d p D f (1.51)
Some investigators provided a number of empirical relations to calculate the interception efficiency. A complete expression for the interception efficiency of single fiber η R is obtained from the Kuwabara's solution [START_REF] Kuwabara | The forces experienced by randomly distributed parallel circular cylinders or spheres in a viscous flow at small Reynolds numbers[END_REF]. This complete expression is [START_REF] Lee | Theoretical study of aerosol filtration by fibrous filters[END_REF]:

η R = 1 + R 2K u 2 ln(1 + R) -1 + 1 (1 + R) 2 1 - β 2 - β 2 (1 + R) 2 (1.52)
Interception is a capture mechanism between diffusion and inertial impaction regime.

However, it is common to observe that interception exists with other capture mechanisms at the same time in fluid flow. For the capture efficiency via simultaneous action of diffusion and interception, one of the simplest way is considering these two mechanisms together to obtain single fiber capture efficiency. Lee and Liu [START_REF] Lee | Theoretical study of aerosol filtration by fibrous filters[END_REF] provided a final equation to calculate capture efficiency for diffusion and interception.

η DR = 2.6 1 -β K u 1/3 P e -2/3 + 1 -β K u R 2 1 + R (1.53)
When using the Eq.(1.53), one assumption is that particle is mainly dominated by one mechanism (interception or diffusion), the contribution from the other mechanisms is small.

Inertial impaction

One of the major capture mechanisms in filtration process is inertial impaction in the absence of external force such as electrostatic force and gravitational force [START_REF] Zhu | Inertial impaction-dominated fibrous filtration with rectangular or cylindrical fibers[END_REF]. Particle departs from streamlines when it is transported in fluid. When it is close to the fiber, fluid streamlines start to change direction. However, particle trajectory is different from fluid streamlines near to a fiber. Due to inertia, it is unable to adjust to the abrupt changes from corresponding streamlines, then inertial impaction occurs. This mechanism can be observed for particle size greater than 1 µm [START_REF] Tien | Granular filtration of aerosols and hydrosols[END_REF][START_REF] Payen | Influence of fiber diameter, fiber combinations and solid volume fraction on air filtration properties in nonwovens[END_REF]. Fig. 1.13 illustrates that the particle is captured due to inertial impaction. Some of particle trajectories deviate from the fluid streamlines due to their inertia and may intersect with the fiber surface. This intersection leads to the particle deposition. around the fiber [START_REF] Israel | Use of a generalized Stokes number to determine the aerodynamic capture efficiency of Non-Stokesian particles from a compressible gas Flow[END_REF] and it is given by:

St = ρ p d 2 p U 18µD f (1.54)
where ρ p is the particle density. This mechanism plays an important role for capture efficiency when the effect of inertial impaction in the filtration regime is significant. Generally, large particle with large St number is easy to be captured by the inertial impaction mechanism [START_REF] Wang | Numerical simulation of particle capture process of fibrous filters using Lattice Boltzmann two-phase flow model[END_REF].

A series of equations used to compute the capture efficiency due to inertial impaction are proposed by some researchers. [START_REF] Landahl | Sampling of liquid aerosols by wires, cylinders, and slides, and the efficiency of impaction of the droplets[END_REF] [START_REF] Landahl | Sampling of liquid aerosols by wires, cylinders, and slides, and the efficiency of impaction of the droplets[END_REF] experimentally determined the efficiency due to inertial impaction. Their work resulted in the following correlation for fiber Reynolds number equal to 10:

η I = St 3 St 3 + 0.77St 2 + 0.22 (1.55)
The limitation of eq 1.55 is that it does not predict the capture efficiency for small St.

For small Stokes number, the capture efficiency is given by [START_REF] Stechkina | Studies in fibrous aerosol filters -IV. Calculation of aerosol depositionion in model filters in the range of maximum penetration[END_REF] η

I = StJ 2K u (1.56)
where J is a parameter derived in the Kuwabara's flow field, J = (29.6 -28β 0.62 )R 2 -27.5R 2.8 (1.57)

The Eq 1.56 is valid for 0.01 ≤ R ≤ 0.4 and 0.0035 ≤ β ≤ 0.111.

Brown (1993) [START_REF] Brown | Air Filtration[END_REF] considered inertial impaction for circular fiber at large Stokes number 1.3. FILTRATION and gave

η I = 1 -ζ St (1.58)
where ζ is a constant depending on the flow field. Brown gave ζ = 0.805 for the Kuwabara's flow field with β = 0.05. It notices that the Eq 1.58 does not take the particle size into account.

In some cases, the inertial impaction and interception exit in the same flow regime.

Both of them need to be considered at the same time. Different authors, Stechkina [START_REF] Stechkina | Studies in fibrous aerosol filters -IV. Calculation of aerosol depositionion in model filters in the range of maximum penetration[END_REF] and Davies [START_REF] Davies | The separation of airborne dust and particles[END_REF] also proposed different equations to compute the capture efficiency combining the interception and inertial impaction. [START_REF] Schweers | Experimental investigation of particle collection on single fibres of different configurations[END_REF] [START_REF] Schweers | Experimental investigation of particle collection on single fibres of different configurations[END_REF] improved the approximation proposed by Suneja and Lee [START_REF] Suneja | Aerosol filtration by fibrous filters at intermediate Reynolds numbers (<100)[END_REF] who solved the Navier-Stokes equation to obtain the capture efficiency. The correlation obtained by Schweers et al. is:

η IR = St St + 0.8 - 2.56 -log Re -3.2R 10 √ St (1 + R) (1.59)
Eq 1.59 is used under the following conditions: 1 < Re < 60 and 0 < R < 0.15.

Retention sites

The elementary mechanisms of particles deposition in porous media is described in the following section. Delanchambre [START_REF] Delanchambr | Contriution á l'étude de l'écoulement d'une suspension á travers un milieu poreux et du mécanisme de la filtration[END_REF] defined four retention sites that can limit the particle migration as illustrated in Fig. 1.16.

-Surface sites: The retention of the particle occurs on the surface of the porous medium (Fig. 1.16 a).

-Crevice sites: The particle in the suspension is located at the contact line between two convex surface of two grains. The particle can be retained by this mechanical wedging in the crevice (Fig. 1.16 b).

-Constriction sites: Some pores formed by three convex grains in contact are very frequent in porous medium. When these narrow pores whose dimensions are smaller than those of the suspended particles, the particle stops (Fig. 1.16 c).

-Cavern sites: Several grains formed a small pocket like the elbows of the pores, the fluid can not penetrate through this cavern so that the transported particle is retained (Fig. 1.16 d).

Pressure drop across a filter

As explained before, prediction of pressure drop across a filter is a very important part during filtration process. Pressure drop or flow resistance is a critical factor. Large flow resistance can cause the failure of a filter. Pressure drop is used to evaluate the filtration performance. It can be directly evaluated from the fluid flow. It can be expressed in terms of the fluid dynamic viscosity µ, filter thickness L, filtration velocity U and fiber diameter D f [START_REF] Wang | Removal of nanoparticles from gas streams by fibrous filters: A review[END_REF] as:

∆p L = f µU D 2 f (1.60)
where f stands for the pressure drop factor. When the particles accumulates inside a filter, the adhesion of particles to the surface of fiber cause the increase in pressure drop. The problem associated with the pressure drop can be analyzed by estimating the drag force of fiber. A number of theoretical works considering the calculation of pressure drop have been presented. Kuwabara [START_REF] Kuwabara | The forces experienced by randomly distributed parallel circular cylinders or spheres in a viscous flow at small Reynolds numbers[END_REF] solved Stokes equations to propose a mathematical model for computing the pressure drop factor. The expression is given:

f (β) = 4π K u (1.61)
Davies [START_REF] Davies | Air filtration[END_REF] developed an empirical correlation to estimate the pressure drop factor which is based on a large number of experimental data. This common relation is given as:

f (β) = 64β 3/2 (1 + 56β 3 ) (1.62)
Another empirical expression for predicting pressure drop factor is given by Jackson and James [START_REF] Jackson | The permeability of fibrous porous media[END_REF]. This expression is based on the three-dimensional media and presented as follows:

f (β) = 20β 3(-ln β -0.931) (1.63)
The above mentioned correlations demonstrate that pressure drop factor just depend on the solid volume fraction β. Liu and Wang [START_REF] Liu | Pressure drop and interception efficiency of multifiber filters[END_REF] presented a relation where pressure drop factor can be expressed as a function of Reynolds number Re, solid volume fraction β, Knudsen number Kn and ratio of horizontal to vertical distance between fibers l h . The relation of pressure drop factor in multi-fibrous filters can be mathematically sumerized as:

1.4. NUMERICAL MODELING FOR FLUID-PARTICLE FLOWS 35 f = W Re, β, Kn, l h (1.64)
Knudsen number is used to measure the degree of rarefaction in fluid flow and defined as Kn = 2λ/D f . It is noticed that the effect of Knudsen number on the pressure drop factor can be neglected for small value of Kn according to literature [START_REF] Davies | Air filtration[END_REF]. Based on the filtration efficiency of filter and pressure drop presented above, then we can calculate the quality factor defined in section 1.3.2.

Numerical modeling for fluid-particle flows

The fluid-particle flow plays an important role in experiment and industrial process of particle filtration, it is essential to understand the phenomenon of particle transport in fluid flow and the relative physical problems. Because the cost of experiments can be excessive and some information can not be easy to obtain experimentally, numerical simulation of fluid-particle flow provides a promising way to deal with this kind of issues. In recent years, researchers of this field have developed fluid-particle flow models such as one-way coupling, two-way coupling and four-way coupling [START_REF] Elghobashi | On predicting particle-laden turbulent flows[END_REF][START_REF] Filippova | Lattice-Boltzmann simulation of gas-particle flow in filters[END_REF][START_REF] Wang | Lattice Boltzmann method for simulations of gas-particle flows over a backward-facing step[END_REF]. One-way coupling means the fluid phase and solid phase are computed separately, only considering the drag force acting on the particles. Two-way coupling means that the back effect of solid phase on the fluid phase is taken into account. Four-way coupling considers the fluid-particle and particle-particle interactions.

The fluid-particle flow models have gradually popularized from one-way coupling to two-way coupling and four-way coupling. In terms of volume fraction of particles Φ p , Fig. 1.17 displays the range of applicability of the different computational approach [START_REF] Elghobashi | On predicting particle-laden turbulent flows[END_REF].

For a low value of Φ p ≤ 10 -6 , the effect of particle on the fluid can be neglected, thus one-way coupling could be sufficient. In the second regime 10 -6 < Φ p ≤ 10 -3 , the mutual interaction between the fluid and particle need to be considered: two-way coupling. As increased in the volume fraction of particle Φ p > 10 -3 , both the fluid-particle interaction and particle-particle interaction should be included, four-way coupling is needed.

In this section, different types of numerical models for simulating the particle transport in fluid flow are outlined. It includes the numerical methods for fluid phase and particle phase respectively. The problem of fluid-solid firstly requires the selection of a velocity field. Computational Fluid Dynamics (CFD) uses the numerical methods and algorithms to model and analyze the problem of fluid mechanics. In order to solve this kind of problem, Navier-Stokes equation is usually solved using different numerical methods such as Finite Element Method (FEM) [START_REF] Shin | Finite element simulation of deep bed filtration[END_REF][START_REF] Sun | 3D finite element model of semi-solid permeability in an equiaxed granular structure[END_REF], Finite Volume Method (FVM) [START_REF] Breuer | Accurate computations of the laminar flow past a square cylinder based on two different methods: Lattice-Boltzmann and finite-volume[END_REF][START_REF] Chen | Coupling between finite volume method and lattice Boltzmann method and its application to fluid flow and mass transport in proton exchange membrane fuel cell[END_REF], Finite Difference Method (FDM) [START_REF] Sangani | Slow flow past periodic arrays of cylinders with applicaon to heat transfer[END_REF], Smoothed Particle Hydrodynamics (SPH) [START_REF] Morris | Modeling low Reynolds number incompressible flows using SPH[END_REF][START_REF] Swegle | Smoothed particle hydrodynamics stability analysis[END_REF][START_REF] Valizadeh | SPH simulation of 2D turbulence driven by a cylindrical stirrer[END_REF] and Lattice Boltzmann Method (LBM) [START_REF] Ladd | Lattice-Boltzmann simulations of particle-fluid suspensions[END_REF][START_REF] Guo | Lattice Boltzmann model for incompressible flows through porous media[END_REF][START_REF] Przekop | Lattice-Boltzmann approach for description of the structure of deposited particulate matter in fibrous filters[END_REF][START_REF] Grucelski | Lattice Boltzmann simulations of flow past a circular cylinder and in simple porous media[END_REF].

One-way coupling

Two-way coupling

The fluid dynamics problems are solved by traditional numerical method FEM, FVM, FDM. The common feature of these methods is that they divide a domain into several elements using structured or unstructured mesh. Then it reconnects elements at nodes as if nodes were pins or drops of glue that hold elements together. They are stable and give very good precision to the problems of the fluid dynamics . However, for some types of problems, these methods are suited poorly. Problems in question are especially those with an extreme deformation as a moving boundary. The complications arising while solving these problems result from the use of a mesh [START_REF] Niedoba | Meshfree methods for computational fluid dynamics[END_REF].

Smoothed particle hydrodynamic is a Lagrangian method where the numerical results can be achieved without mesh [START_REF] Morris | Modeling low Reynolds number incompressible flows using SPH[END_REF]. It is originally designed to use in astrophysical applications. In SPH, the studied domain for fluid field is discretized in particles. These particles interact with each other and move while carrying a set of physical properties of the medium. An advantage of SPH is that it allows boundaries to move or deform as modeling the interaction of several fluid phases bound by a free surface. Therefore, SPH is commonly used to analyze high deformation events and to involve the bodies moving problems in fluids [START_REF] Valizadeh | SPH simulation of 2D turbulence driven by a cylindrical stirrer[END_REF]. According to literature [START_REF] Morris | Modeling low Reynolds number incompressible flows using SPH[END_REF], errors obtained by SPH can sometimes be larger than those obtained using grid-based methods. Moreover, SPH can be more time consuming than alternative techniques for a given application.

The LBM method is an approach based on the Boltzmann equation which describes the evolution of a fluid. LBM is different from the conventional numerical method, i.e. FEM and FVM, which solve the macroscopic Navier-Stokes equation. It uses a kinetic equation model and corresponding relations between the real simulated statistical dynamic at microscopic level and transport equation at the macroscopic level [START_REF] Ansari | Numerical study of solid particles motion and deposition in a filter with regular and irregular arrangement of blocks with using Lattice Boltzmann method[END_REF]. It describes fluid phase from the mesoscopic perspective in principle. The detail of principle of LBM will be introduced in chapter 2. Breuer [START_REF] Breuer | Accurate computations of the laminar flow past a square cylinder based on two different methods: Lattice-Boltzmann and finite-volume[END_REF] makes a comparison of results obtained using LBM and FVM by simulating confined flow around a square mounted inside a plane channel. The INTERACTION OF FLUID AND SOLID PARTICLES agreement between LBM and FVM simulation results is excellent by comparing the profiles of streamwise velocity and cross-stream velocity. As literature stated [START_REF] Breuer | Accurate computations of the laminar flow past a square cylinder based on two different methods: Lattice-Boltzmann and finite-volume[END_REF], FVM is more efficient for simple geometries whereas LBM is advantageous for highly complex geometries such as those happening in porous media. Beside this, LBM has also its advantage in efficiency, parallel scalability [START_REF] Ladd | Lattice-Boltzmann simulations of particle-fluid suspensions[END_REF].

Numerical methods for solid phase

An analysis of transport of particles in fluid flow needs a through understanding of both flows and particles. It is important to know the information of trajectory and velocity of each individual particle in fluid flow. Particle-tracking method has been applied to determine trajectory and velocity of particle by solving particle equations of motion. Numerical method such as FEM [START_REF] Cheng | A particle tracking technique for the Lagrangian-Eulerian finite element method in multi-dimensions[END_REF], FVM [START_REF] Chen | Numerical simulation of aerosol collection in filters with staggered parallel rectangular fibres[END_REF] and Discrete Element Method (DEM) [START_REF] Cundall | A discrete numerical model for granular assemblies[END_REF][START_REF] Luding | Introduction to discrete element methods. Basic of contact force models and how to perform the micro-macro transition to continuum theory[END_REF] are used to model the particle transport under the effect of various forces.

Conventional continuum methods, such as the FEM and FVM, may be appropriate to describe the behavior of small particles, which the size of particle is much smaller than the mesh size. FEM is flexible enough to treat homogeneous materials and non-linear boundary conditions. Therefore, FEM is mostly used to study mechanics problems of structures and solids. FEM and FVM are not suitable to handle the interface tracking problems, i.e., loss of solids involving the use of very small meshes and requiring a remesh process which takes much computational time. FEM and FVM have limited success in the simulations of particulate flows with a high number of particles, especially in threedimensional simulations [START_REF] Feng | The immersed boundary-lattice Boltzmann method for solving fluid-particles interaction problems[END_REF].

DEM is one of the generally suitable method for modeling the mechanical behavior of particles. It has grown considerably to became a reliable and powerful research tool for the modeling of grain issue. DEM is based on the concept that individual material elements are considered to be separate and are connected only along their boundaries by appropriate physically based interaction laws [START_REF] Feng | Coupled lattice Boltzmann method and discrete element modelling of particle transport in turbulent fluid flows : Computational issues[END_REF]. It allows us to study these discrete systems in the presence of physical phenomena at the particle scale. Since DEM can treat the spontaneous particle-particle interactions, it has a broad variety of applications in soil mechanics, granular materials and fluid mechanics [START_REF] Jing | Numerical methods in rock mechanics[END_REF]. The basic difference between the DEM and continuum-based methods is that the contact patterns between components of the system are continuously changing with the deformation process for the former, but are fixed for the latter [START_REF] Jing | Numerical methods in rock mechanics[END_REF]. An example of particle contact with DEM is performed by Lominé and Oger [START_REF] Lominé | Transport of small particles through a 3D packing of spheres: experimental and numerical approaches[END_REF]. They present experiments and numerical model based on DEM to study 1.5. SUMMARY OF EARLIER RESEARCHERS' RESULTS FOR PARTICLE FILTRATION 39 transport properties of particles through porous structure, which access the internal motion of particles and examine the influence of physical parameters on the relevant percolation behavior. The detail of the DEM will be presented in chapter 2.

Choice of numerical method

The numerical treatment of the particle transport problems depend on the size of particle in relation with the grid or mesh size [START_REF] Feng | Coupled lattice Boltzmann method and discrete element modelling of particle transport in turbulent fluid flows : Computational issues[END_REF]. For small particle size within a grid cell, fluid phase is generally described by the Navier-Stokes equations where FEV and FVM are often used. However for the problems in which the particle size is large more than one grid, LBM can make it particularly suitable for larger sized particles from the computational point of view. Large particles can be accommodated within LBM framework. Another advantage of LBM is the simplicity in the implementation of the algorithm and its capability of modeling problems with complex interior fluid domains. For fluid-particle interactions, DEM has capability to extend to deal with multiphysic issues. Thus, a coupled LBM and particle methods, LBM-DEM, can be implemented. The combined LBM-DEM solution strategy is chosen to simulate the fluid-particle systems with volume fraction of particles in the framework of this thesis.

Summary of earlier researchers' results for particle filtration 1.5.1 Particle capture on a single fiber

A real filter consists of many individual fibers and its filtration efficiency is heavily dependent on the one of single fiber [START_REF] Zhu | Inertial impaction-dominated fibrous filtration with rectangular or cylindrical fibers[END_REF]. Hence, the comprehensive investigation of particle captured by the single fiber is necessary. In order to achieve this, studying the particle deposit on the surface of single fiber is an effective way [START_REF] Han | Experimental and numerical studies of aerosol penetration through screens[END_REF]. In this part, we present particle capture on circular and non-circular fiber.

Particle capture on circular fiber

Cylindrical fiber with circular cross-section, as one of the most attractive and common shapes, has been widely used to study the filtration process. In early studies, many researches employed theoretical analyses [START_REF] Harrop | The theoretical prediction of inertial impaction efficiencies in fibrous filters[END_REF], numerical simulations [START_REF] Michael | Particle collision efficiencies for a sphere[END_REF][START_REF] Wang | Deposition of solid particles on a collector: Formulation of a new theory[END_REF] and exper-INTERACTION OF FLUID AND SOLID PARTICLES imental measurements [START_REF] Wong | Inertial impaction of aerosol particles on cylinders[END_REF] to examine particle deposition over different obstacles such as cylinders, spheres and disks [START_REF] May | The impaction of aerosol particles on cylinders, spheres, ribbons and discs[END_REF]. Lee and Liu [START_REF] Lee | Theoretical study of aerosol filtration by fibrous filters[END_REF] considered a theoretical approach, based on Kuwabara's flow field [START_REF] Kuwabara | The forces experienced by randomly distributed parallel circular cylinders or spheres in a viscous flow at small Reynolds numbers[END_REF], to predict the filtration of aerosol particles where diffusion and interception are dominant, the fiber capture efficiency calculated from the proposed expression agrees with that of experiment [START_REF] Lee | Theoretical Study of Aerosol Filtration by Fibrous Filters[END_REF]. Schweers et al. [START_REF] Schweers | Experimental investigation of particle collection on single fibres of different configurations[END_REF] determined experimentally the single fiber capture efficiency where the configuration of neighboring fibers was considered.

Particle capture on non-circular fiber

With the fast development of new instruments for manufacturing technology, the production of non-circular fiber cross-sectional shapes is becoming viable. Researches show great interest for the study of filtration by non-circular fibers [START_REF] Wang | Filtration of aerosol particles by elliptical fibers: A numerical study[END_REF][START_REF] Salmanzadeh | Particle transport and deposition in a duct flow with a rectangular obstruction[END_REF][START_REF] Jafari | Investigation of particle dispersion and deposition in a channel with a square cylinder obstruction using the lattice Boltzmann method[END_REF][START_REF] Hosseini | On the importance of fibers' cross-sectional shape for air filters operating in the slip flow regime[END_REF][START_REF] Kirsh | Stokes flow and deposition of aerosol nanoparticles in model filters composed of elliptic fibers[END_REF]. Non-circular fiber have larger specific surface area compared to the circular cross-section fiber with the same volume fraction. Due to this, it is easy to perform higher capture efficiency of particles and to increase mechanical strength, porosity, particle loading capacity for fibers.

SUMMARY OF EARLIER RESEARCHERS' RESULTS FOR PARTICLE FILTRATION 41

Several researches have been dedicated to square or rectangular fibers. Jafari et al. [START_REF] Jafari | Investigation of particle dispersion and deposition in a channel with a square cylinder obstruction using the lattice Boltzmann method[END_REF] used the square cylinder to numerically investigate the particle deposition and dispersion.

The simulation results show that the Brownian diffusion affects the deposition rate of the ultrafine particle on both the front and back sides of the square cylinder.

Brandon and Aggarwal [START_REF] Brandon | A numerical investigation of particle deposition on a square cylinder placed in a channel flow[END_REF] numerically modeled the particle deposition on a square prism in unsteady vertical flows and investigated the effect of Reynolds number on particle trajectory and deposition. They concluded that the Reynolds number has very little effect on the deposition. However, Re has more discernible influence on particle distribution. Zhu et al. [START_REF] Zhu | Inertial impaction-dominated fibrous filtration with rectangular or cylindrical fibers[END_REF] numerically and analytically investigated the impaction-dominated filtration process with rectangular fiber for particulate size much smaller than the one of fiber.

Influence of fiber aspect ratio, filter volume fraction, particulate size and Stokes number on collection efficiency of a rectangular fiber have been determined for impaction-dominated filtration. The results showed that the efficiency of combined interception and impaction is a parabolic function of Stokes number, the overall size-averaged filtration efficiency is higher than that of median sized particles. Hosseini and Tafreshi [START_REF] Hosseini | On the importance of fibers' cross-sectional shape for air filters operating in the slip flow regime[END_REF] investigated the effect of a fiber's cross-section shape on pressure drop and collection efficiency with nanofibers and microfibers. They found that the fiber geometry is more important for pressure drop prediction rather than single fiber capture efficiency estimation. Wang et al. [START_REF] Wang | Simulating and modeling particulate removal processes by elliptical fibers[END_REF] simulated the particle capture process of elliptical fiber. They gave the fitted correction factors which are used to calculate the fiber capture efficiency and pressure drop dominated by Brownian diffusion, interception and inertial impaction respectively. These fit correction factors exhibit reasonable predictions and can be applied in engineering. Huang et al. [START_REF] Huang | Numerical study of pressure drop and diffusional collection efficiency of several typical noncircular fibers in filtration[END_REF] numerically studied the pressure drop and capture efficiency for diffusional dominant mechanism with rectangular, trilobal, quatrefoil and triangular cross-sections at low Reynolds number. They concluded that the capture efficiency due to diffusion of noncircular fibers is higher than that of the circular fiber with the same volume fraction. The dimensionless force on the noncircular fibers is independent of Reynolds number when Re < 1. Wang [START_REF] Wang | Stokes flow through an array of rectangular fibers[END_REF] solved the Stokes flow across an array of rectangular fibers and proposed an empirical formula to calculate the drag of a single square fiber. Fardi and Liu [START_REF] Fardi | Efficiency of fibrous filters with rectangular fibers[END_REF] studied the capture efficiency of rectangular fiber dominated by the Brownian diffusion and interception mechanisms. The single fiber capture efficiency due to the pure diffusion and pure interception was found to be proportional to powers of the Peclet number and the interception number, respectively.

Particle transport in filter with multi-fiber

The influence of neighboring filters can affect the deposition pattern on a certain filter.

Hence, the filtration process is more complicated than the deposition on a single fiber.

The capture efficiency and pressure drop are influenced by the structure of the filter [START_REF] Wang | Simulation of filtration process for multi-fiber filter using the Lattice-Boltzmann two-phase flow model[END_REF].

Understanding particle behavior in filter with multi-fiber is of significant importance for improving the filter performance. Liu and Wang [START_REF] Liu | Pressure drop and interception efficiency of multifiber filters[END_REF] analyzed several parameters including volume fraction and fiber

SUMMARY OF EARLIER RESEARCHERS' RESULTS FOR PARTICLE FILTRATION 43

separation ratio during the filtration process. It was found that both the pressure drop and interception efficiency increase with the fiber separation ratio at a given volume fraction. The optimal fiber separation ratio can be used to improve the efficiency and reduce the pressure drop. Wang [START_REF] Wang | Simulation of filtration process for multi-fiber filter using the Lattice-Boltzmann two-phase flow model[END_REF] numerically investigated the pressure drop and filtration efficiency of filter for clean multi-fiber and particle loading multi-fiber cases. It has been found that the staggered model is better than the parallel model from the view of the filtration efficiency and the pressure drop. Lin et al. [START_REF] Lin | An early stage of aerosol particle transport in flows past periodic arrays of clear staggered obstructions: A computational study[END_REF] examined how the obstruction morphology (circular, square and diamond staggered arrayed) affects particle-surface collision efficiency in a free-stream flow.

They concluded that the collision of particles on diamond-shaped obstruction is more favorable than that on cylindrical and square obstruction with the presence of vortex-shedding.

Fardi and Liu [START_REF] Fardi | Flow field and pressure drop of filters with rectangular fibers[END_REF] investigated the pressure drop with a staggered array of rectangular INTERACTION OF FLUID AND SOLID PARTICLES fibers in a viscous fluid flow. The dimensionless drag force on rectangular fiber with aspect ratio > 1 is independent of aspect ratio. However, for fiber aspect ratio ≤ 1, the dimensionless drag force is a function of both the aspect ratio and fiber solid volume fraction.

Conclusion

In this chapter, we posed the problem of the particle transport driven by fluid in porous media, which is one of phenomena existing in fluid-solid two-phase flow. This phenomenon involving fluid-particle interactions has been the subject of fluid-solid separation.

Firstly, flow of suspensions through porous medium is presented. Continuity equation and Navier-Stokes equation are often used in fluid flow. The porous media is described such as porosity and characteristic of porous media. For simple configurations of porous media, Kuwabara solution can be used to determine the flow field. Secondly, the knowledge of filtration including filtration type, quality factor, filtration efficiency and pressure drop are reviewed. These important parameters are commonly used in particle filtration process. The relation between filtration efficiency of filter and single fiber capture efficiency is introduced due to three mechanisms (Brownian diffusion, interception and inertial impaction) involved in filtration process. Thirdly, for the purpose of understanding the particle behavior as their transport in porous media, several numerical methods are presented here to simulate fluid and solid phases. LBM has the advantage of simplicity in dealing with the complex boundary condition and in the implementation of the algorithm. DEM has the capability to model problems with a large number of particles. Thus, the coupled LBM-DEM is chosen to investigate the particle transport and deposition in filtration process. Finally, we conclude the earlier researchers' results for particle filtration. Since the single fiber is the fundamental element to constitute a filter, particle capture on circular and non-circular single fiber is reviewed. Then particle transport in filter with multi-fiber is described, where different fiber arrangements are considered.

For particle transport and deposition, it generally refers to fluid-particle and particleparticle interactions. Due to these interactions and different capture mechanisms, they lead to the complexity of particle filtration process. Some details of this process are not well understood. Thus, in order to consider the influence of particle motion on the flow field and particle-particle interactions, we use the coupled LBM-DEM to study the particle filtration process. In the following chapter, we present the detail of the LBM and DEM respectively and a coupling between LBM and DEM. A series of examples are given to valid our code. In this work, we use the LBM-DEM to investigate the single fiber capture efficiency, filtration efficiency, pressure drop and quality factory of filter. The effect of fiber's cross-sectional shape, fiber arrangement and relative parameters on filtration performance are studied.

Chapter 2 Lattice Boltzmann and Discrete Element methods

Introduction

Lattice Boltzmann method (LBM) was originally developed from the lattice gas cellular automata (LGCA) [START_REF] Frisch | Lattice as automata for the Navier-Stokes equation[END_REF][START_REF] Frisch | Lattice gas hydrodynamics in two and three dimensions[END_REF] used for modeling physics and solving a series of fluid dynamics problems. LBM is based on Boltzmann's equation proposed by Ludwig Boltzmann in 1872. The Boltzmann equation makes it possible to describe the evolution of the distribution function f ( x, c, t), defined as the probability density associated with a particle in the position x with a velocity c at time t.

∂f ( x, c, t) ∂t + c • ∇ x f ( x, c, t) + F ext m • ∇ c f ( x, c, t) = Ω(f ) (2.1)
where F ext is the external force acting on the particle in the fluid. The term Ω(f ) represents the variation of the number of particles due to collision in a volume element of the phase space. Chapman and Enskog [START_REF] Enskog | Kinetische theorie der vorgänge in mässig verdünnten gasen[END_REF][START_REF] Tg | The mathematical theory of Non-uniform gases: An account of the kinetic theory[END_REF] used a complex collision operator to derive the Navier-Stokes equation from the Boltzmann equation.

The first lattice gas cellular automata model was introduced by Hardy, Pomeau and de Pazzis (called HPP) [START_REF] Hardy | Time evolution of a two-dimensional model system. I. Invariant states and time correlation functions[END_REF][START_REF] Hardy | Molecular dynamics of a classical lattice gas: Transport properties and time correlation functions[END_REF] in 1973. HPP is a two-dimensional model where fluid is located at each lattice node. However, it uses a square lattice with insufficient symmetries to derive Navier-Stokes equation [START_REF] D'humières | Lattice gas models for 3D hydrodynamics[END_REF]. In 1986, Frisch, Hasslacher and Pomeau (named FHP) [START_REF] Frisch | Lattice as automata for the Navier-Stokes equation[END_REF] proposed a hexagonal lattice. This model based on hexagonal lattice instead of square lattice studied transport properties of fluids. The correct Navier-Stokes

CHAPTER 2. LATTICE BOLTZMANN AND DISCRETE ELEMENT METHODS

equation was obtained through the lattice gas automata on a hexagonal lattice for first time. FHP creates a new situation for LGCA, which makes it develop rapidly. However, despite the advantages provided by this model, the biggest problems of FHP is noise.

By the ends of 1980s, McNamara and Zanetti [START_REF] Mcnamara | Use of the boltzmann equation to simulate latticegas automata[END_REF] provided a method to replace the Boolean variables. This method governs the evolution of particle occupancies, in LGCA by calculating the time evolution of a probability density distribution of the particles, the time evolution is controlled by Boltzmann equation. This model, commonly referred to as 'Lattice-Boltzmann', completely eliminated the statistical noise and saved much computer time [START_REF] Bernsdorf | Simulation of Complex Flows and Multi-Physics with the Lattice-Boltzmann Method[END_REF]. The kinetic form of Lattice Boltzmann is still the same as the LGCA and the collision operator has the drawback of complexity. Therefore a lot of work has been investigated to find ways to reduce this complexity. In 1989, Higuere and Jinenez [START_REF] Higuera | Boltzmann approach to lattice gas simulations[END_REF] who linearized the collision operator by introducing the equilibrium distribution function simplify the model. Several research groups independently proposed single relaxation time for collision operator making use of a relaxation time towards the local equilibrium [START_REF] Koelman | A simple Lattice Boltzmann scheme for Navier-Stokes fluid flow[END_REF][START_REF] Chen | Lattice Boltzmann model for simulation of magnetohydrodynamics[END_REF][START_REF] Chen | Recovery of the Navier-Stokes equations using a lattice-gas Boltzmann method[END_REF]. This linear collision operator was introduced by Bhatnagar-Gross-Krook (BGK) [START_REF] Bhatnagar | A model for collision processes in gases. I. Small amplitude processes in charged and neutral one-component systems[END_REF]. The BGK dramatically reduces the computation cost and gives pretty good results in various applications [START_REF] Guo | Lattice Boltzmann model for incompressible flows through porous media[END_REF][START_REF] Feng | Coupled lattice Boltzmann method and discrete element modelling of particle transport in turbulent fluid flows : Computational issues[END_REF][START_REF] Yu | Viscous flow computations with the method of lattice Boltzmann equation[END_REF][START_REF] Boek | Lattice-Boltzmann studies of fluid flow in porous media with realistic rock geometries[END_REF][START_REF] Wang | Evaluation of three lattice Boltzmann models for particulate flows[END_REF].

Lattice Boltzmann method has a number of characteristics that are different from those of traditional fluid dynamics methods [START_REF] Pan | An evaluation of lattice Boltzmann schemes for porous medium flow simulation[END_REF]. The conventional numerical methods, such as finite element and finite volume methods, compute flow field by solving the incompressible Navier-Stokes equation. LBM, as an efficient alternative for fluid flow, solves fluid field in terms of density distribution function from the perspective of mesoscopic level. The fluid phase in LBM is considered as fictitious fluid particles which can collide and propagate on a lattice. The macroscopic properties (i.e., pressure, fluid density and velocity) can be obtained through the density distribution functions of fluid particles. Compared with conventional methods, LBM demonstrates significant advantages due to its parallel computation and capability to deal with complex boundary conditions [START_REF] Wang | Lattice Boltzmann method for simulations of gas-particle flows over a backward-facing step[END_REF]. In the following section, we describe the details for LBM including the principle and implementation of boundary conditions.

LATTICE BOLTZMANN METHOD

2.2 Lattice Boltzmann method 2.2.1 Principle of lattice Boltzmann method

The computational domain in LBM is first divided into a regular lattice with spacing h. The fluid phase in LBM is then assumed as a group of fluid particles which are located at each node of the lattice. They are constrained to migrate along the links between the nearest regular lattices and collide with each other at the grid points. Here, we employ the classical two dimensional D2Q9 model proposed by D'Humières et al. [START_REF] D'humières | Lattice gas models for 3D hydrodynamics[END_REF]. In the D2Q9 model, a computational domain is discretized using square lattice with nine discrete velocities e i (shown in Fig. 2.1). Discrete velocities in the D2Q9 model are defined as:

e i =                (0, 0) (i = 0) C(cos π(i -1) 2 , sin π(i -1) 2 ) (i = 1, ....4) √ 2C(cos π(2i -9) 4 , sin π(2i -9) 4 ) (i = 5, ....8) (2.2)
in which C means the lattice speed given by:

C = h/dt (2.3)
dt is the time step.

ρ = 8 i=0 f i (2.4) u = 1 ρ 8 i=0 f i e i (2.5)
LBM algorithm is an iterative process over time. At each time step, the fluid particles located on regular lattice have to experience an essential sequence of two mechanisms, called collision and propagation. By adopting a linearized collision operator, BGK (Bhatnagar-Gross-Krook) [START_REF] Bhatnagar | A model for collision processes in gases. I. Small amplitude processes in charged and neutral one-component systems[END_REF], to deal with fluid particle collision [START_REF] Qian | Lattice BGK models for Navier-Stokes equation[END_REF], the evolution equation can be given:

Collision step:

f i ( x, t + ) = f i ( x, t) - 1 τ [f i ( x, t) -f eq i ( x, t)] (i = 0, ..., 8) (2.6)
Propagation step:

f i ( x + e i dt, t + dt) = f i ( x, t + ) (2.7)
where the term f i ( x, t + ) in Eq (2.6) and (2.7) is called post-collision term, t + is an intermediate time t < t + < t + dt and follows the collision process. τ is a dimensionless parameter named relaxation time. For any grid node at position x, x + e i dt is its nearest node along the direction i. f eq i is the ith equilibrium distribution functions defined as:

f eq i ( x, t) = w i ρ 1 + 3 C 2 e i • u + 9 2C 4 ( e i • u) 2 - 3 2C 2 u • u (2.8)
where w i is the weight coefficient which is dependent on the model. In the D2Q9 model,

w 0 = 4 9 , w 1 , 2 , 3 , 4 = 1 9 , w 5 , 6 , 7 , 8 = 1 36 .
The kinematic viscosity of fluid ν is related to the lattice velocity C, relaxation time τ and the lattice spacing h through following relation:

ν = 1 3 (τ - 1 2 )Ch (2.9)
It is noted that the restriction of τ in Eq (2.9) satisfies τ > 0.5. Smaller values of τ

LATTICE BOLTZMANN METHOD

51

implie smaller time steps and therefore increase computational cost. Generally, results are acceptable for τ < 1.5 [START_REF] Gallivan | An evaluation of the bounce-back boundary condition for lattice Boltzmann simulations[END_REF]. According to Eq (2.

3) and Eq (2.9), the time step of LBM is:

dt = 1 3ν (τ - 1 2 )h 2 (2.10)
The fluid is considered to be slightly compressible. The value of Mach number is limited to M = umax C ≤ 0.1 in order to insure the compressibility error M 2 less than 1%, where u max is the highest velocity in the flow. The fluid pressure is determined by:

p = C 2 s ρ (2.11)
where C s is the speed of sound, defined as

C s = C √ 3 [8].

Boundary conditions on obstacles and systems

Boundary conditions on obstacles

Solid particles or any other kinds of obstacles are handled with the LB method by positioning them on the lattice. The simplest scheme to calculate momentum transfer between a SB and a FB node is the "mid-plane bounce-back" boundary condition on the boundary links [START_REF] He | Analytic solutions of simple flows and analysis of nonslip boundary conditions for the lattice Boltzmann BGK model[END_REF]. In this method, the solid surface is assumed to be located halfway between the fluid node and solid node. Since fluid particles cannot cross the solid border line that limits the outer solid domain, particle distribution functions coming from SB or FB nodes are reflected along the boundary link σi. For a stationary solid surface, the propagation step, given by Eq 2.7 is then replaced by:

CHAPTER 2. LATTICE BOLTZMANN AND DISCRETE ELEMENT METHODS

SB node FB node

LATTICE BOLTZMANN METHOD

53 f -σi ( x F B , t + dt) = f σi ( x F B , t + ) f σi ( x SB , t + dt) = f -σi ( x SB , t + ) (2.12)
For a moving boundary, the solid surface velocity need to be considered in the bounceback rule in order to ensure no-slip condition on the solid surface. This is achieved by using the modified bounce-back rule proposed by Ladd [START_REF] Ladd | Numerical simulations of particulate suspensions via a discretized Boltzmann equation. Part 1. Theoretical foundation[END_REF]:

f -σi ( x F B , t + dt) = f σi ( x F B , t + ) -2α i u s . e i f σi ( x SB , t + dt) = f -σi ( x SB , t + ) + 2α i u s . e i (2.13) where x SB = x F B + e σi dt, α i = 3w i ρ/C 2 .
u s is the boundary velocity at the middle of the boundary link σi. It can be expressed as

u s = u p + ω p ∧ r c (2.14)
where u p and ω p are the translational and angular velocities respectively at the center of mass of the solid obstacle. r c is the vector joining the center of mass x c of the solid obstacle to the middle of the boundary link.

r c = x F B + e σi dt 2 -x c (2.15)

Boundary conditions on systems

As presented in part 2.2.1, we use the distribution function f i to deduce the pressure and velocity through Eqs 2.4 and 2.5. Thus the pressure or velocity condition on the boundary cannot be imposed directly. For the boundary condition on the system, we adopt the boundary conditions proposed by Zou and He [START_REF] Zou | On pressure and velocity boundary conditions for the lattice Boltzmann BGK model[END_REF] as it is specified below.

(a) Velocity boundary condition

The rectangular simulation domain presented in Fig. 2.4 needs to define eight sets of distribution functions denoted from a) to g). The velocity boundary conditions

U 0 =   u x u y  
are specified to the inlet of the system (case a)). The key step is to determine all the distribution functions at the boundary nodes. After the propagation step, distribution functions f 2 , f 3 , f 4 , f 6 , f 7 are already known and f 1 , f 5 , f 8 are unknown. We need to solve

CHAPTER 2. LATTICE BOLTZMANN AND DISCRETE ELEMENT METHODS

for ρ, f 1 , f 5 and f 8 , which means that four equations are needed. According to the Eqs 2.4 and 2.5, we can get the following form:

f 1 + f 5 + f 8 = ρ -(f 0 + f 2 + f 3 + f 4 + f 6 + f 7 ) (2.16) f 5 -f 8 = ρu y -f 2 + f 4 -f 6 + f 7
(2.17)

f 1 + f 5 + f 8 = ρu x + f 3 + f 6 + f 7 (2.18)
A fourth equation can be obtained from the non-equilibrium part of distribution functions normal to the boundary. We assume that the bounce-back rule is still correct for the non-equilibrium part of distribution functions. Then it follows that:

f 1 -f eq 1 = f 3 -f eq 3 (2.19)
Then the unknown distribution functions f 1 , f 5 , f 8 and fluid density ρ can be obtained by solving Eqs 2.16 -2.19

ρ = 1 1 -u x [f 0 + f 2 + f 4 + 2(f 3 + f 6 + f 7 )] (2.20
)

f 1 = f 3 + 2 3 ρu x (2.21) f 5 = f 7 - 1 2 (f 2 -f 4 ) + 1 6 ρu x + 1 2 ρu y (2.22) f 8 = f 6 + 1 2 (f 2 -f 4 ) + 1 6 ρu x - 1 2 ρu y (2.23)
The distribution functions and density in cases b), c), d) of Fig. 2.4 can be solved in a similar procedure. However for the corner, the above formulations are not applicable due to different number of unknown distribution functions at the corner nodes ( case e), f), g) and h)). A specific procedure is needed. Consider the bottom left corner node (case e)) as an example. After the propagation step the distribution functions f 3 , f 4 and f 7 are known, f 1 , f 2 , f 5 , f 6 and f 8 need to be solved. Again using the bounce-back rule of the non-equilibrium part, we can find that:

CHAPTER 2. LATTICE BOLTZMANN AND DISCRETE ELEMENT METHODS

f 1 = f 3 + 2 3 ρu x (2.24)
f 2 = f 4 + 2 3 ρu y (2.25)
In addition, by combining Eqs 2.22 and 2.23 which gives

f 5 -f 6 + f 8 = ρu x -(f 1 -f 3 -f 7 )
(2.26)

f 5 + f 6 -f 8 = ρu y -(f 2 -f 4 -f 7 ) (2.27)
Then based on the Eqs 2.5 and 2.24-2.27, f 5 , f 6 and f 8 are obtained as:

f 5 = f 7 + 1 6 ρ(u x + u y ) (2.28) f 6 = 1 2 [ρ(1 -u x - 2 3 u y ) -f 0 -2(f 3 + f 4 + f 7 )] (2.29
)

f 8 = 1 2 [ρ(1 - 2 3 u x -u y ) -f 0 -2(f 3 + f 4 + f 7 )] (2.30)
Finally, all the relationships between distribution functions, densities and velocities can be solved with the same procedure for case f), g) and h).

(b) Pressure boundary condition

If pressure condition p is imposed at the inlet of channel (case a)), we specify the fluid densities at the flow boundaries because density is deduced from the formula p = c 2 s ρ. For convenience, we define the vertical velocity u y = 0. ρ is fixed and velocity u x needs to be determined first. Following the procedure for the velocity boundary conditions, the solutions are given by:

f 1 = f 3 + 2 3 ρu x (2.31) f 5 = f 7 - 1 2 (f 2 -f 4 ) + 1 6 ρu x (2.32
)

f 8 = f 6 + 1 2 (f 2 -f 4 ) + 1 6 ρu x (2.33)

LATTICE BOLTZMANN METHOD

57

The similar way for treating the corner for the velocity flow boundaries can be used to deal with the pressure boundary condition as described in the above part.

(c) Periodic boundary condition

Periodic boundary condition is one of the mostly used boundary conditions. The system becomes closed by attaching the edges such as connecting the outlet and inlet. Fig. 2.5

presents the schematic view of periodic boundary condition. The fluid particle distribution functions such as f 1 , f 5 , f 8 leaving from the outlet are assigned to the inlet. The distribution functions f 3 , f 6 , f 7 at the inlet will transfer to the outlet. If (x max + e i dt) or (x min + e -i dt) is out of system, then the periodic boundary condition is realized by:

f i ( x min , t + dt) = f i ( x max , t + ) (2.34) f i ( x max , t + dt) = f -i ( x min , t + ) (2.35)
where x max = (x max , y) and x min = (x min , y)

(d) Neumann's boundary condition on velocity

Neumann's boundary condition on velocity is an effective approach to simulate problems with an infinitely fluid domain [START_REF] Aidun | Direct analysis of particulate suspensions with inertia using the discrete Boltzmann equation[END_REF]. In some particle transport problems, a constant

CHAPTER 2. LATTICE BOLTZMANN AND DISCRETE ELEMENT METHODS

velocity is imposed at the upstream and the flow field is considered to be fully developed at the outlet. Thus, Neumann's boundary condition is used without the need to exceed the computational domain.

The downstream of flow such as outlet is generally set to:

∂u x ∂x = 0 at x = x max (2.36)
In order to achieve zero velocity gradient, if j is a unit vector normal to the boundary pointing outside, the distribution functions at the outlet should be defined as:

f i ( x max , t + dt) = f i ( x max -j, t + dt) (2.37)
where x max = (x max , y).

Discrete element method 2.3.1 Contact law

The discrete element method (DEM), introduced by Cundall and Strack [START_REF] Cundall | A discrete numerical model for granular assemblies[END_REF], is a very popular numerical approach to deal with solid particle contact interactions. The interactions between particles respect contact law. In DE method, particles move according to Newton's law and their contact follows a "soft spheres" approach. Particles are allowed to slightly overlap amongst each other during contact. We consider two particles a and b located respectively at positions r a and r b with radius R a and R b as shown in Fig. 2.6.

Their velocities are V a and V b respectively. The angular velocities of particle a and b are ω a and ω b . A collision between two particles causes a deformation, then the overlap χ n is defined as

χ n = R a + R b -( r a -r b ) • n (2.38)
where n is unit normal vector as seen in Fig. 2.6 and calculated as

n = r a -r b r a -r b (2.39)

DISCRETE ELEMENT METHOD

59

The relative velocity V rab of the particle a relatively to the particle b at the contact point is written:

V rab = ( V a -ω a ∧ R a n) -( V b + ω b ∧ R b n) (2.40) = V ab -(R a ω a + R b ω b ) ∧ n (2.41) with V ab = V a -V b .
The relative velocities in the normal and tangential directions are equal to:

V n rab = V ab • n n (2.42) V t rab = -n ∧ n ∧ V rab (2.43)
According to the particle velocity, tangential vector t is defined:

t = V t rab V t rab (2.44)
The overlap during a contact is kept relatively small compared to the particle size, in addition, in order to keep valid the rigid body assumption of particles, this overlap has to stay negligible. In these conditions, the contact force can be divided into two components:

F c ab = F n n + F s t (2.45)
where F n and F s are normal contact force and shear force respectively.

Normal contact force

The calculation of the normal contact force F n from particle a to particle b is directly linked to the value of χ n . If the overlap χ n < 0 as illustrated in Fig. 2.6, then an interaction force for the two colliding particles is computed:

χn = - F a n m a + F b n m b (2.
= - F n m p (2.47)
in which m p = mam b ma+m b . Several models of contact law exist in literature [START_REF] Kruggel-Emden | Review and extension of normal force models for the Discrete Element Method[END_REF]. Here we will use the widely sprayed "linear spring dashpot" (LSD) model [START_REF] Lominé | Dispersion of particles by spontaneous interparticle percolation through unconsolidated porous media[END_REF][START_REF] Dippel | Force schemes in simulations of granular materials[END_REF][START_REF] Schwager | Coefficient of restitution and linear-dashpot model revisited[END_REF][START_REF] Lominé | Transit time during the interparticle percolation process[END_REF] for the normal component of the contact force.

F n = K n χ n + γ n χn (2.48)
where K n is the normal stiffness coefficient of the interaction and γ n is a damping coefficient.

χn is the time derivative of χ n . We can get the following equation by combining the Eqs. 2.47 and 2.48,

m p χn + γ n χn + K n χ n = 0 (2.49)
With initial conditions χ n (t = 0) = 0 and χn (t = 0) = u 0 p , we can get the solution of the Eq.2.49:

χ n = u 0 p t c π exp(- γ n 2m p t) sin( π t c t) (2.50)

DISCRETE ELEMENT METHOD

61

where t c is the contact duration which can be expressed as [START_REF] Lominé | Dispersion of particles by spontaneous interparticle percolation through unconsolidated porous media[END_REF][START_REF] Luding | Introduction to discrete element methods. Basic of contact force models and how to perform the micro-macro transition to continuum theory[END_REF]

t c = π k n m p -( γ n 2m p ) 2 -1 2 (2.51)
The stability of the explicit scheme for the time integration holds if the time step of the DE method dt DE has to be sufficiently small to describe the wave propagation in the massspring system constituted by the solid granular assembly. With the LSD model, equation 2.51 is of practical importance, since the integration of the motion equations is stable only if the integration time-step dt DE is much smaller than t c . A ratio of 10 between dt DE and t c is usual to avoid instability of the numerical method [START_REF] Lominé | Modeling of fluid-solid interaction in granular media with coupled lattice Boltzmann/discrete element methods: application to piping erosion[END_REF]. Collision inelasticity is taken into account through the damping γ n previously introduced. The LSD force model leads to a constant restitution coefficient defined as:

κ = u 1 p u 0 p = exp(- π 4k n m/γ n 2 -1 ) (2.52)
in which u 0 p and u 1 p are the velocities of the particles before and after collision, respectively.

Shear contact force

The shear force F s is calculated using an extension of the method proposed by Cundall-Strack [START_REF] Cundall | A discrete numerical model for granular assemblies[END_REF]. When the contact is formed, F s is initialized to zero. F s is computed incrementally with an increment ∆F s calculated by accumulating the path of moving contact point between the two particles in contact [START_REF] Herrmann | Micro-mechanical investigation of the granular ratcheting[END_REF].

F s = F s + ∆F s (2.53) ∆F s = -K s ∆u s (2.54)
K s represents the interaction shear stiffness and ∆u s the increment of the tangential relative displacement of the contact point. Tangential sliding occurs if:

|F s | > µF n (2.55)
with µ = min(µ a , µ b ) and µ a and µ b are particle friction coefficients. Contact parameters K n and K s can be derived from particle properties E, ν, which are respectively Young's

CHAPTER 2. LATTICE BOLTZMANN AND DISCRETE ELEMENT METHODS

modulus and Poisson's coefficient [START_REF] Potyondy | A bonded-particle model for rock[END_REF]:

K n = 2 E a R a E b R b E a R a + E b R b and K s = 2 E a ν a R a E b ν b R b E a ν a R a + E b ν b R b (2.56)
For each particle, total contact force F c is computed with summation of contact force over all interacting partners.

Particle dynamics

The particle is subject to a contact force and external forces at the same time. Position and velocity of each solid particle are deduced from the explicit time integration of the Newton's second law:

m p d u p dt = F c + F ext (2.57)
where m p , x p and u p = d xp dt are respectively particle mass, position and velocity of particle. F c is contact force acted from other particles and F ext is the external force. Rotations of particles are computed in the same way by integrating the equation:

J d ω p dt = T c (2.58)
where T c is the torque resulting from the inter-particle forces, J the moment of inertia of particles and ω p the angular velocity. Moment of inertia of particles can be expressed as:

J =        m p r 2 p for circle 2m p r 2 p 5 for sphere (2.59)
where r p is the particle radius. At each time step of the DEM, the time integration is performed by Verlet's algorithm [START_REF] Verlet | Computer " experiments " on classical fluids I. Thermodynamical properties of Lennard-Jones molecules[END_REF]. The variation of the position of the particle subject to the force f can be derived from the Taylor development as:

x(t + dt) = x(t) + ∂x(t) ∂t dt + f m p dt 2 2 + ∂ 3 x(t) ∂t 3 dt 3 3! + o(dt 4 ) (2.60) x(t -dt) = x(t) - ∂x(t) ∂t dt + f m p dt 2 2 - ∂ 3 x(t) ∂t 3 dt 3 3! + o(dt 4 ) (2.61)

COUPLING OF LBM-DEM

63

We can express a new one by merging these two equations:

x(t + dt) = 2x(t) -x(t -dt) + f m p dt 2 (2.62)
and then, we are able to calculate new positions at time t + dt base on position at time t and t -dt and force at time t. The particle velocity can be calculated:

u x (t) = x(t + dt) -x(t -dt) 2dt (2.63)

Coupling of LBM-DEM

The exchange of information is required when combining LBM with DEM to model the particle interactions. Hydrodynamic forces and torques are deduced from the LBM computation and used in the DEM cycle to compute new positions and velocities of solid particles, which in turn define the new boundary conditions for the fluid dynamic problem [START_REF] Lominé | Modeling of fluid-solid interaction in granular media with coupled lattice Boltzmann/discrete element methods: application to piping erosion[END_REF]. The achievement of the coupled LBM-DEM has to be done by considering the fact that each method is characterized by a different time discretization. Thus in the following, we will present the calculation of hydrodynamic force and numerical time step adjustment for the coupling.

Subcycling time integration

The basic idea for the coupling of the DEM and LBM is a subcycling scheme. An overview of flowchart of the coupled LBM-DEM is illustrated in Fig. 2

Forces and torques acting on the particles

The fluid force and torque acting on an obstacle can be evaluated from the momentum exchange between fluid and solid particle surface over all boundary links of the obstacle [START_REF] Lominé | Modeling of fluid-solid interaction in granular media with coupled lattice Boltzmann/discrete element methods: application to piping erosion[END_REF]:

F h (t + 1 2 dt) = σi 2 Ω dt f σi ( x, t + ) -α i u s . e i e σi
(2.65)

T h (t + 1 2 dt) = σi T σ ( x, t + 1 2 dt) (2.66)
where Ω is the cell lattice surface in 2D ( cell lattice volume for 3D case). To be time consistent, hydrodynamic forces and torques applied on obstacles are obtained by an average over two consecutive LBM time-steps:

F h (t) = 1 2 F h (t + 1 2 dt) + F h (t - 1 2 dt) (2.67) T h (t) = 1 2 T h (t + 1 2 dt) + T h (t - 1 2
m p d u p dt = F c + F h (2.69) T c + T h = J d ω p dt (2.70)
Then, the DEM cycle computes new positions and velocities of solid particles, which in turn define the new boundaries for the fluid dynamic problem. During the subcycling, the hydrodynamic forces and torques remain unchanged, this is why n in Eq.(2.64) should stay small. Finally the coupling between LBM and DEM is established. The coupled LBM-DEM are implemented in a "house" code developed at Laboratoire de Génie Civil et Génie Mécanique (LGCGM) named "Albadec".

CHAPTER 2. LATTICE BOLTZMANN AND DISCRETE ELEMENT METHODS

Validations of LBM-DEM

In this section, the study of classical fluid mechanics problems are presented in order to validate the 2D computation code based on the LBM algorithm and on the coupling of the LB-DE methods respectively.

2D Poiseuille flow

We Theoretical stationary solution of Poiseuille flow can be solved analytically. Velocity of Poiseuille flow reaches its maximum in the middle of the channel, the relationship between average velocity U aver and maximum velocity is given as:

U aver = 2 3 u max with u max = ∆pL 2 y 8µL x (2.71)
Then Reynolds number of the Poiseuille flow is used here:

Re = U aver L y ν (2.72)
where ν is kinematic viscosity.

VALIDATIONS OF LBM-DEM

67

The LBM is validated not only in stationary state by comparing with the series solution for the transient behavior of Poiseuille which is expressed as [START_REF] Morris | Modeling low Reynolds number incompressible flows using SPH[END_REF]

u x (y, t) = ∆p 2L x νρ (y(y -L y )) + ∞ n=0 4∆pL 2 y L x νρπ 3 (2n + 1) 3 sin( πy(2n + 1) L y ) exp( -1(2n + 1) 2 π 2 νt L 2 y ) (2.73)
A series of important parameters used in the simulation are given in Table 2.1. Fig. 2.9

shows a comparison between numerical simulation results and series solution. The velocity profiles agree closely with series analytical solution at different time in Fig. 2.9. A peak velocity of Poiseuille flow is u peak = 9.8 × 10 -4 m/s for the steady state solution. The maximum velocity calculated from LBM is u max = 9.8307×10 -4 m/s. The discrepancy between the maximum velocity can be calculated:

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W W W W W V <MlPu 0 4î50 í- 50 í1 5m4î50 í1 ×î50 í1 X [ MMlP)Vu 0 ×î50 í--î50 í-3î50 í-2î50 í-50 í1
err peak = u peak -u max u peak × 100% = 0.313% (2.74)
In order to obtain more precise results, the effect of relaxation time τ and lattice spacing h on numerical solution is studied. The discrepancy between LBM and analytical solution is computed:

err = Σ i,j |u analy (i, j) -u LBM (i, j)| Σ i,j |u analy (i, j)| × 100% (2.75)
where u LBM (i, j) is the fluid velocity of a node indexed by i and j in LBM and u analy (i, j) is the analytical solution. The results of discrepancy between LBM and analytical solution of Poiseuille flow are shown in

Couette flow

Couette flow is usually used as a mean to study the suspension flow behavior in simple The series solution for the time-dependent behavior of the Couette flow is [START_REF] Morris | Modeling low Reynolds number incompressible flows using SPH[END_REF]:

u x (y, t) = U 0 L y y + ∞ n=1 2U 0 nπ (-1) n sin( nπ L y y) exp(-ν n 2 π 2 L 2 y t) (2.76)
Some simulation parameters are listed in Table 2.4 for the simulation. The velocity profiles across the flow obtained using LBM are compared with the series solution at several time steps including the steady state solution. The numerical results are presented in Fig. 2.11. The predicted LBM results agree well with the analytical solution. The discrepancy between LBM and analytical solution of Couette flow is calculated.

The results are presented in Table 2.5. From these results, we find that the impact of relaxation time τ and lattice spacing h on numerical solution of Couette flow have the same tendency as the one of Poiseuille flow. A small τ or small h can obtain more accurate solution and decrease the discrepancy compared to analytical solution. However, it leads to the increase of the computational cost.

Flow around a fixed cylinder in 2D

The single obstacle flow case is mostly used as the validation example. The problem is shown in Fig. 2.12. The circle obstacle is set in a rectangular channel on the middle place along y and asymmetrically along x. The channel length and height are L x and L y respectively. The obstacle represents a cylinder with diameter d. The computational domain and specified boundary conditions are shown in Fig. 2.12.

wall wall

err = Σ i,j |u xCF D (i, j) -u xLBM (i, j)| Σ i,j |u xCF D (i, j)| × 100% (2.77)
where u xLBM (i, j) is the fluid velocity of a node indexed by i and j in LBM. u xCF D (i, j) is the fluid velocity obtained with Comsol and its position is the same the u xLBM (i, j). Then the discrepancy for d/h = 8, 10, 12 and CFD software is obtained. The computational The drag force F D applied on the cylinder is also calculated. The drag coefficients C D which depends on fluid density, cross-sectional area S and fluid average velocity U 0 is expressed as:

&RPVRO G-K G-K 56 G-K 5 8[ PV 6 6×65 6×6 6×6 6×6 < P 6 î56 í î56 í î56 í î56 í &RPVRO G-K G-K 56 G-K 5
C D = 2 F D ρU 2 0 S (2.78)
In 2D case, the cross-sectional area S is equal to d.

Flow in porous media

In many engineering problems such as filtration [START_REF] Hellou | Stokes flow through microstructural model of fibrous media[END_REF], geotechnics problems [START_REF] Boek | Lattice-Boltzmann studies of fluid flow in porous media with realistic rock geometries[END_REF], a fundamental understanding of the flow through porous media is of important interest. In the following, Darcy's law is used to test the program through simulating the fluid flow in the porous media. A series of parameters used in the simulations are shown in Table .2.9.

Pressure drop is applied between the inlet and outlet of the channel. The velocity field in an example of porous media is shown in Fig. 2.14. The flow rate Q can be computed at the outlet of the channel. According to the Eq.(1.10), the term -κ µ is constant for fixed fluid and porous media. Fig. 2.15 presents the relationship between Q and A |∇p|. κ µ represents the slope of the line in Fig. 2.15. As a consequence, the relationship of discharge and the product of cross-sectional area and the pressure gradient(A • ∇p) obtained with our code is tested to be linear κ µ = Q A|∇p| = 4.5625 × 10 -5 . It is observed that the simulation results follows the Darcy's law. (1-φ) 2 should be linear. Fig. 2.16 presents dependancy of Q with (φ) 3

(1-φ) 2 obtained with our LB code. It can be clearly observed that the linear relation is retrieved which validates our program.

Moving cylinder in channel flow

In this section, we validate the coupled LBM-DEM. First case, we consider a fixed cylinder placed in a channel as Fig. 2.12. A series of pressure drop is applied between the inlet and outlet respectively. The values of pressure drop is shown in Table 2.10. The same boundary conditions are used with Fig. 2.12. Then the drag coefficient of cylinder is calculated with different Reynolds number. It is noted that the grid resolution used is equal to d/h = 20. This is because larger pressure drop leads to the increase of the maximum fluid velocity and Mach number. Mach number is related to lattice spacing. Then the decrease of the lattice spacing is to reduce the Mach number and to assure the fluid incompressible. The same simulations are also investigated with the CFD software. Second case, we impose a constant velocity on the cylinder as illustrated in Fig. 2.17. The velocity condition is imposed on the wall u x = u y = 0. The system is initially at rest. Periodic condition for fluid and cylinder are used. At t = 0, cylinder moves with a constant velocity. Then drag coefficient with different velocities is computed.

wall wall periodic periodic

Conclusion

This chapter presented the numerical method used in this work to simulate fluid and particle flow in porous media. The fluid flow is calculated using LBM and the particle transported by fluid is simulated by DEM. A series of classical fluid mechanics examples are presented to validate the computation code, such as Poiseuille flow, Couette flow, flow around a fixed cylinder in 2D and in porous media. These simulations results agree well with the analytical solutions. The validation of filtration problems will be presented in chapter 3 and 4. After validating the computation code, the particle transport and deposition in a filter is investigated using LBM-DEM in following chapter.

Chapter 3

Numerical simulation of particle capture on single fiber

Introduction

The knowledge of behavior of particles during their transport and deposition, on various fibers placed in a fluid flow, can provide important information for many environmental and industrial applications. The underlying physics of particle filtration process in filter is important [START_REF] Yang | Mechanistic studies of initial deposition of fine adhesive particles on a fiber using discrete-element methods[END_REF]. Since single fiber is the basic element to constitute a filter, it is necessary to understand particle behavior in filter with single fiber during filtration process. Actually, single fiber can be made of a variety of shapes such as circular, diamond and square etc. The shape of fiber considerably affects the distribution of fluid field. The velocity and position of solid particle in filter rely evidently on the fluid velocity field. Thus, it is essential to acquire an accurate velocity field for particle trajectory analysis which is appropriately described by the Lagrangian approach. Conventional CFD model, based on the incompressible Navier-Stokes equation, combined with Lagrangian approach are usually used to simulate the fluid-solid flow within porous media such as filters [START_REF] Marshall | An improved analytic solution for analysis of particle trajectories in fibrous, two-dimensional filters[END_REF][START_REF] Ouyang | Analytical solution of flow field and pressure drop for filters with noncircular fibers[END_REF][START_REF] Araújo | Critical role of gravity in filters[END_REF]. For dilute suspension, the solid particle is viewed as mass point without volume (point-like approach), therefore the trajectory and velocity of the particle are calculated from the equation of the motion using Lagrangian approach by consideration of the different external forces such as drag force and Brownian force [START_REF] Wang | Numerical simulation of particle capture process of fibrous filters using Lattice Boltzmann two-phase flow model[END_REF]. As presented in part 1.4 of chapter 1, it is reasonable to neglect the effect of particle on the fluid flow when the particle-laden flow are dilute enough (Φ p ≤ 10 -6 ) [START_REF] Elghobashi | On predicting particle-laden turbulent flows[END_REF], the error caused by this assumption is believed to be limited since the results in the validation are acceptable [START_REF] Luo | Effects of preferential concentration on collision and erosion between solid particles and tube bank in a duct flow[END_REF]. Based on these information, obviously,

80

CHAPTER 3. NUMERICAL SIMULATION OF PARTICLE CAPTURE ON SINGLE FIBER point-like approach can estimate the actual displacement of particle within a time step.

However, particle with large size (d p ≥ 10 µm) injected in fluid channel can change and affect significantly the flow field. In this case the volume of the particle and its influence on the flow field should be considered.

In this chapter, in a first part, we investigate the effect of particle size d p and Stokes number St on the particle capture efficiency using two methods. The first one is one-way coupling combining lattice Boltzmann method (LBM) with Lagrangian point-like approach.

The second one is two-way coupling based on the coupling between lattice Boltzmann method and discrete element method (DEM), which consider the particle influence on the fluid. Due to this consideration, the two-way coupling approach may be more suitable to study the transport problems for relative large particle. Therefore, we focus on the study to find the critical particle size between one-way and two-way coupling methods. In a second part, the single fiber capture efficiency characterized by Stokes number (St) and quality factor defined in chapter 1 part 1.3.2 are simulated by LBM-DEM. The effect of geometries of fiber on the filtration performance is also considered such as diamond, square and rectangular.

One particle injection

In this part of the work, the particles are injected one by one into the system, which means that the particle can be injected as soon as the previous one captured by the fiber or flows out from the filter. Particles are removed out from the flow field once they are captured by the fiber or flow out from the filter. Therefore, each particle moves alone, there is no inter-particle interaction in the fluid flow and the fiber stays clean .

Model description

The schematic view of computational domain of simulation is illustrated in Fig.

err = Σ i,j |u i,j (t + dt) -u i,j (t)| Σ i,j |u i,j (t + dt)| (3.1)
where u i,j is the fluid velocity of node. We investigate particle capture efficiency with first a one-way coupling method, that will be described hereafter. The limit trajectory of the particle deposited on the fiber surface is sketched as mentioned in Fig. 3.1. The particle released distance to the lattice axis is called δ. Thus, since the flow is symmetric, we define the fiber capture efficiency according to definition given in part 1.3.5 of chapter 1 as:

η s = 2δ D f (3.2)

One-way coupling

With one-way coupling method a solid particle is simply viewed as a point. In other words, the fluid does not see the particle and is therefore not affected by its presence. For this reason, when the fluid velocity field reaches a steady state, it remains unchanged until the end of the simulation. The velocity and trajectory of a particle is directly calculated by the traditional point-like approach tracing individual particle in the Lagrangian frame [START_REF] Luo | Effects of preferential concentration on collision and erosion between solid particles and tube bank in a duct flow[END_REF][START_REF] Filippova | Lattice-Boltzmann simulation of gas-particle flow in filters[END_REF]. Moreover, with this method there is no particle-particle interaction. Each particle is treated as if it were an isolated one. This assumption is reasonable when the volume fraction of solid particles is less than 10 -6 [START_REF] Elghobashi | On predicting particle-laden turbulent flows[END_REF][START_REF] Elghobashi | Particle-laden turbulent flows: direct simulation and closure models[END_REF] as presented in Fig. 1.17 in Chapter 1.

The one-way coupling algorithm can be described as follows. Firstly, the stationary FIBER solution of fluid flow velocity field is computed. Then, each solid particle is injected at the inlet of the domain. Its initial velocity along the flow direction is set to be equal to the fluid velocity at that position. In this subsection, the fluid velocity field is computed with the LBM presented in chapter 2 and streamfunction presented in part 1.2.3.3 of chapter 1 respectively, and coupled with the particle motion dynamics.

Particle motion of equation

Inertial impaction is one of the major mechanisms in the particle-laden flow. The inertial effect in particle capture, shown in Fig. 3.1, can be quantified by the dimensionless Stokes number [START_REF] Araújo | Critical role of gravity in filters[END_REF][START_REF] Wang | Numerical simulation of particle capture process of fibrous filters using Lattice Boltzmann two-phase flow model[END_REF]. Stokes number has crucial effects on inertial-interception-dominated capture efficiency which has been described in chapter 1. Generally speaking, the particle with large size have a larger St number, (i.e., dp >10 µm [START_REF] Wang | Numerical simulation of particle capture process of fibrous filters using Lattice Boltzmann two-phase flow model[END_REF][START_REF] Wang | Simulation of filtration process for multi-fiber filter using the Lattice-Boltzmann two-phase flow model[END_REF] and the inertial impaction dominates).

When a spherical particle moves with velocity u p in a fluid stream moving with the velocity u, in the absence of any other external force, just by considering the drag force, given by the Stokes law, the equation of particle motion is expressed as:

4 3 πρ p R 3 p d u p dt = 6πµR p ( u -u p ) (3.3)
We use the dimensionless form by using U 0 :

u * p = u p U 0 (3.4) u * = u U 0 (3.5) t * = t D f U 0 (3.6)
where the superscript * denotes the dimensionless quantity. Using these dimensionless quantities lead to follwing equations [START_REF] Tien | Granular filtration of aerosols and hydrosols[END_REF]:

St d u * p dt * = u * -u * p (3.7) d x * p dt * = u * p (3.8)
A fourth order Runge-Kutta (RK) has been used to numerically solve the equations

ONE PARTICLE INJECTION

83

(3.7) and (3.8). When new particle position is found, we compute the fluid velocity u * at this new position and again we solve equations (3.7) and (3.8). Therefore, it is possible to compute all injected particle trajectories.

Particle trajectory analysis

The transport of particle in fluid and its trajectory are analyzed. In 2D case, particles that come closer than distance (D f + d p )/2 from the fiber center are considered as trapped by the fiber and are then removed from the simulation. With this consideration, we estimated particle capture efficiency using equation (3.2).

The first one-way coupling method coupled the analytical solution (Kuwabara solution) for fluid velocity field and particle motion of equation (3.7) and (3.8) solved by Runge-Kutta, we call it Kuwabara-RK, whereas the second one is the LB-RK, in which the fluid velocity field is obtained with LBM. The results obtained by the Kuwabara-RK and LB-RK are compared each other. When the flow field is solved using LBM, it is worth to note that the solid particle is not constrained to locate only at the same lattice nodes as the fluid particles of the LBM. Because the velocity field of fluid in LBM is discrete, only the lattice nodes have the value information of the fluid velocity. Thus the fluid velocity u * at the location of the solid particle r p is solved using bilinear interpolation among the four closest nodes of the lattice. As seen in Fig. 3.2, the blue and red circles represent the positions of fluid particles and the solid particle, respectively. The expression of the fluid velocity inside the lattice is given by:

u * (x p , y p ) = y j+1 -y p y j+1 -y j u * (x p , y j ) + y p -y j y j+1 -y j u * (x p , y j+1 ) (3.9)
where u * (x p , y j ) and u * (x p , y j+1 ) are:

u * (x p , y j ) = x k+1 -x p x k+1 -x k u * (x k , y j ) + x p -x k x k+1 -x k u * (x k+1 , y j ) (3.10) u * (x p , y j+1 ) = x k+1 -x p x k+1 -x k u * (x k , y j+1 ) + x p -x k x k+1 -x k u * (x k+1 , y j+1 ) (3.11)
The trajectory of particle during their transport in fluid are calculated. closely. As a consequence, particle is not captured by the fiber. For St = 0.03, the trajectories of particle still follow the streamlines at upstream of fiber. However, the particle starts to derives from streamlines after passing the fiber. This is because the inertia of particle increased at top of fiber as shown in Fig. 3.3b. When St increases to 0.3, particle transports near to the fiber, but it cannot be captured by the fiber. As St increases (St = 0.5), particle follows slightly the fluid streamlines at several time steps and then derives from it. The reason is that the inertia of particle with St = 0.5 is not intense enough at initial, but it is strong enough to be captured by fiber. When St is sufficiently large (i.e. St = 1, 1.5), inertial impaction mechanism becomes more and more stronger. Particle deviates from the corresponding streamlines during they approach the fiber and finally deposits on the surface of the fiber.

The numerical results are compared with the work of Araujo et al. [START_REF] Araújo | Critical role of gravity in filters[END_REF] who obtained these results using the Kuwabara-RK. The fiber capture efficiency obtained by LB-RK and Kuwabara-RK is presented in Fig. 3.4. From Fig. 3.4 we can conclude that fluid velocity field obtained using LBM combined with bilinear interpolation is good enough to reproduce previous results presented in literature. These series of simulations validate our LBM code and the one-way coupling approach.

Obviously, as stated in references [START_REF] Yao | Inertial particle resuspension in a turbulent, square duct flow[END_REF][START_REF] Rabiee | Investigation of the characteristics of particulate flows through fibrous filters using the lattice Boltzmann method[END_REF][START_REF] Tehrani | Filtration of aerosol particles by cylindrical fibers within a parallel and staggered array[END_REF] and presented in section 1.4, the use of one-way coupling method to investigate particle capture efficiency is reasonable only if particle effect on fluid flow can be neglected.

Two-way coupling

In this part, we use the two-way coupling method (LBM-DEM) to examine the particle transport and deposition. The idea is to investigate if the effect of the particle on fluid flow can still be neglected for relatively large particle. In this method, the particle is viewed as an object with its own boundary, which is seen by the fluid as a no-slip boundary condition.

The fluid force and torque acting on the particle are directly computed through summation of momentum transfer over the particle surface (equations 2.65 and 2.66) and applied on it with equations 2.67 and 2.69.

The effect of particle size

To investigate particle size on particle capture efficiency, various simulations with different particle diameters (8 µm, 10 µm, 12 µm, 14 µm, 16 µm and 18 µm,) are performed.

The results of particle capture efficiency are compared with those obtained with one-way coupling method (Fig. 3.5). Figure 3.5: Single fiber capture efficiency as a function of ratio between the particle diameters d p and fiber diameter D f obtained from one-way and two-way coupling methods, respectively. FIBER When the solid particle size is small (d p < 8 µm for our model), volume of particles have little effect on single fiber capture efficiency. However as the particle size increases, the effect of particle volume on the capture efficiency becomes much more obvious. Therefore, it can be stated that one-way coupling method is not appropriate in our case for d p > 8 µm, and corresponding Stokes number of the particle is St = 0.9.

ONE PARTICLE INJECTION

The effect of fiber shape

Three types of fiber with different cross-section, circular, diamond and square, are considered to study the capture efficiency for one particle injection. The solid particle size used in the simulation is d p = 10 µm. In order to compare the capture efficiency for circular, diamond and square fiber respectively, the projected area of three fibers perpendicular to the flow direction are selected as characteristic length which are all equal to D f = 1.6 × 10 -4 m.

They keep constant during the simulation. For diamond fiber, it is a square rotated by π/4 with a diagonal D f . The circular, diamond and square fiber are located at the middle of the computational domain with length L and width H as shown in Fig. The capture efficiency of circular, diamond and square fiber as a function of St is calculated with Eq.(3.2) and results are plotted in Fig. 3.7 for isolated particle injection. We vary the particle density to change Stokes number in simulation. Fig. 3.7 displays that the capture efficiency of diamond is closer to that of circle for St = 0.5. As St increases, the difference varies obviously. However, the capture efficiency of square fiber is more important than those of circular and diamond fiber for 0.5 ≤ St ≤ 3. This is because the streamline close to the square are more curved as shown in Fig. 3.8. The trajectories of the particle with corresponding streamlines around the three shapes of fiber are also presented for St = 1. The streamlines approach to the fiber are different for circular, diamond and square cases, which shows that the flow field is affected by fiber shape. Generally speaking, we can conclude that the square case have more chance to capture the particle.

Quality factor of filter

The capture efficiency of circular, diamond and square fiber have been analyzed. In order to judge the performance of filter with single fiber, we use the quality factor (or the figure of merit) which is presented in chapter 1:

QF = -ln(1 -η) ∆p ( 3

ONE PARTICLE INJECTION

91

According to references [START_REF] Wang | Filtration of aerosol particles by elliptical fibers: A numerical study[END_REF][START_REF] Wang | The influence of fiber geometry and orientation angle on filtration performance[END_REF], it could be derived as follows:

QF = η s (1 -β) • (F * /Re) • (ρU 2 0 ) (3.13)
β is fiber volume fraction which is calculated as the ratio of volume occupied by fiber to the volume of total computational region. F * is dimensionless drag force applied on the fiber, the value is presented in Table 3.1. Fig. 3.9 presents the quality factor of filter with circular, diamond and square fiber with various Stokes number for isolated particle injected. It demonstrates that St affect the quality factor. Quality factor for circular, diamond and square fiber increases with St. Square fiber has better performance than the cases of circular and diamond fiber for St < 1.

As St increases, the diamond fiber display higher quality factor for St > 1. Obviously, FIBER although the square fiber exhibits the advantages in capture efficiency, a larger pressure drop for square fiber leads to its lower quality factor.

In the next section, we are interested in the study of sing fiber capture efficiency and filtration efficiency of filter when several particles are present simultaneously in the fluid flow. The single fiber capture efficiency has been widely studied using one-way coupling method, even when relatively large number of particles were injected [START_REF] Yao | Inertial particle resuspension in a turbulent, square duct flow[END_REF][START_REF] Rabiee | Investigation of the characteristics of particulate flows through fibrous filters using the lattice Boltzmann method[END_REF][START_REF] Lin | An early stage of aerosol particle transport in flows past periodic arrays of clear staggered obstructions: A computational study[END_REF]. Since results presented in Fig. 3.5 illustrate that the effect of an isolated particle on fluid flow cannot be neglected for d p > 8 µm in our case, we can assume that it cannot be neglected when several particles are transported at the same time with such particle diameter. For this reason, LBM-DEM will be used to study the capture efficiency with d p > 8 µm in the rest of this chapter. Nevertheless, even if our coupled method can deal with inter-particle interaction, solid volume fraction used here is less than 10 -3 . Therefore inter-particle interactions do not play an important role in this case.

Several particles injection

In this section we focus on the study of the capture efficiency with a number of solid particles injected (d p ≥ 10 µm) into the system. First, we consider single fiber capture efficiency. The injected particles are located in the projected area of single fiber at inlet. Then we investigate the filtration efficiency of filter which particles are injected at inlet of computational domain. Since the particle size used here is d p ≥ 10 µm, the Brownian diffusion is relatively weak, and the Brownian force on a particle is far less than the drag force exerted by the fluid [START_REF] Wang | Numerical simulation of particle capture process of fibrous filters using Lattice Boltzmann two-phase flow model[END_REF]. For this reason, the Brownian diffusion is not considered in the simulation.

Particles injected at projected area of fiber 3.3.1.1 Model description

The two-dimensional computational domain, shown in Fig. 3.10, is a rectangular one of length L and width H in which a circular fiber of diameter D f is set on the center. The grid resolution for LBM, fixed here as d p /h = 10, has been proved to be good enough to compute a reliable flow field and hydrodynamic force in chapter 2. A periodic boundary condition is applied on the left and right sides, both the fluid and solid particles flowing out from one side will enter into the domain from the other side. No-slip boundary condition

SEVERAL PARTICLES INJECTION

93

is imposed at the surface of the fiber. The condition for fluid is given by Eq (3.14). The

constant velocity U 0 =   0 u 0   (u 0 = 0.67m/s
) is applied at the inlet and outlet.

u y (x, 0) = u 0 u y (x, L) = u 0 (3.14)
The fluid density and kinematic viscosity used here are 1.2 kg/m 3 and 1.57 × 10 -5 m/s 2 respectively. After the flow achieves a steady state (velocity vectors are shown in Fig. 3.10), 500 particles are injected randomly in the projected area of fiber at the inlet of computational domain. The initial velocity of the injected particle is assumed to be equal to fluid velocity.

Capture efficiency of circular fiber

As in section 3.2, the particles in the flow are assumed to be captured once they collide on the surface of the circular fiber. Then, collided particles are removed from the flow. The single fiber capture efficiency is defined as [START_REF] Jafari | Investigation of particle dispersion and deposition in a channel with a square cylinder obstruction using the lattice Boltzmann method[END_REF]:

η s = N inject -N escape N total (3.15)

CHAPTER 3. NUMERICAL SIMULATION OF PARTICLE CAPTURE ON SINGLE FIBER

where N inject and N escape represent the total number of solid particles injected in the projected area of the surface and removed from the outlet respectively.

Series of snapshots from one simulation of particle transport and deposition with coupled LBM-DEM are shown in Fig. 3.11. It is seen that after injection of particles, the velocity field of fluid changes remarkably which, in turn, leads to the variation of the particle velocity and trajectory. Single fiber capture efficiency is also influenced significantly by the Stokes number. Several studies have been performed to predict single fiber efficiency resulting from the inertial impaction. Israel and Rosner [START_REF] Israel | Use of a generalized Stokes number to determine the aerodynamic capture efficiency of Non-Stokesian particles from a compressible gas Flow[END_REF] investigated the aerodynamic capture efficiency of non-Stokesian particles and determined the following expression for St ef f > 0.14:

η s = [1 + 1.25(St ef f -0.125) -1 -1.4 × 10 -2 (St ef f -0.125) -2 + 0.508 × 10 -4 (St ef f -0.125) -3 ] -1 (3.16)
where St ef f = G(Re p )St, Re p is particle Reynolds number defined as Re p = U 0 dp ν , G is a function of Re p . In our simulation, Re p = 0.425 and G(Re p ) = 0.95. Eq (1.59) for single fiber efficiency which has been described in chapter 1 is also used here.

In order to examine the effect of St on the single fiber capture efficiency, we add another two cases for injecting 1000 and 1500 particles respectively to calculate the single fiber capture efficiency. A comparison between the single fiber capture efficiency obtained with LBM-DEM and the one obtained from the empirical relationships calculated according to

SEVERAL PARTICLES INJECTION 95

Eqs (3.16) and (1.59) has been made and presented in Fig. 3.12. In our model, since the particle diameter is d p = 10 µm, the fiber Reynolds number and intercept coefficient are

Re = U 0 D f ν = 6.
8 and R = dp D f = 0.0625 respectively. The single fiber capture efficiency is plotted as a function of St ranging from 0.5 < St < 3. The results obtained with 500, 1000 and 1500 particles are nearly the same. Thus number of particle for 500 is enough to repeat the simulation results. Agreement between the simulation results and empirical formula Eq (1.59) is observed. However, with respect to Eq (3.16), we use G(Re p )St instead of St ef f in Fig. 3.12, a small difference is found for St > 1 and significant difference is observed for

St < 1. 6FKZHHUVhHWhDO ,VUDHOhHWhDO )BBhSDUWLFOHVhhhl/%0i'(0I MBBBhSDUWLFOHVhhl/%0i'(0I M)BBhSDUWLFOHVhhl/%0i'(0I Ș V B B(- B(E B(1 B(. M 6W B B() M M() - -() D D() Figure 3
.12: Comparison between capture efficiency of single circular fiber obtained from numerical method (LBM-DEM) and from the empirical correlations. Re = 6.8, R = 0.0625.

Capture efficiency of diamond and square fiber

We investigate now the influence of the fiber shape on the singe fiber capture efficiency. The simulations are conducted with two types of fibers, square and diamond respectively.

In order to compare the capture efficiency with the circular fiber results, characteristic FIBER length is selected as in section 3.2.3.2 (projected area perpendicularly to the flow direction).

Except for the variation in fiber shapes, we use exactly the same operating conditions and values that the ones used in the circular case described in section 3.3.1.2. The projected area of diamond and square fiber is equal to that of the circular fiber. On the other hand, particles with large Stokes number ( St > 1) exhibit a strong inertial behavior, they are essentially unaffected by the flow. As a consequence, a large number of particles are collected on the front side of the fiber.

To investigate the distribution of captured particles on the surface of fiber. The front side of fiber is divided by 9 zones (1-9) shown in Fig. 3.14, since the projected area of circular, diamond and square fiber are the same, here we take the diamond as example. Then the number of particles captured on the front side of fiber is counted. Fig. 3.15 displays the results of single fiber capture efficiency in each zone for St = 1.97. For these three types of fibers, the single fiber capture efficiency at the middle of the fiber is nearly the same (part 5). The number of captured particles are approximately symmetrically distributed on the front side of the fiber.

We also examine the single fiber capture efficiency for different fiber shapes. A comparison among circle, diamond and square is presented in Fig. 3.16. For the single fiber capture efficiency, there is no significant differences among these three fiber shapes when St = 0.5. This is because the single fiber capture efficiency is very weak. Araujo et al. [START_REF] Araújo | Critical role of gravity in filters[END_REF] showed this existence of critical Stokes number, below which particles cannot be captured. The capture efficiency of diamond fiber is close to that of circular fiber as St is near to unity. As Stokes number increases, the square fiber has predominant preferences in particle collection. Generally, the number of collected particles on the front surface of square is greater than those of circle and diamond for different Stokes number. We also plot the single fiber capture efficiency for isolated particle in Fig. 3.16 to compare with the one for several particles. The two cases are different as shown in Fig. 3.16. For the same fiber shape and fixed St, the single fiber capture efficiency for several particles are smaller than the one for isolated particle. This is due to the fact that when several particles are injected,

Quality factor of filter for serveral particles injected

The quality factor is calculated for several particles injected. Fig. 3.17 presents the performance of circular, diamond and square fiber with different Stokes number values. It demonstrates that the St affects the quality factor of these three fibers. Although the square fiber has a high capture efficiency, the pressure drop of circular and diamond fiber is lower than square case for St > 1.5. When considering these two parameters together, diamond fiber is at advantage of quality factor over the square and circular fiber. The quality factor for isolated particle injected is also presented in Fig. 3.17. For the same fiber shape, the case of isolated particle generally keeps higher quality factor than the case of several particles for a fixed St. This is due to higher capture efficiency for the case of isolated particle while the same pressure drop.

Particles injected along whole inlet

In this section, the filtration efficiency of filter is considered, thus the particles are injected at whole inlet of computational domain. From the simulations presented above, we know that the geometries of fiber have considerable influence on the filtration performance. More specifically, diamond cross-section shape displays a better performance by comparison of square fiber, which promote us to investigate how the orientation angle and aspect ratio of rectangular fiber affect the filtration performance. Thus, in this section, we focus on the study of the effect of orientation angle and aspect ratio of rectangular fiber on pressure drop, filtration efficiency of filter and quality factor for various Reynolds numbers.

Model description

A schematic view of a periodic cell of the filter is illustrated in Fig. 3.18. It also represents the computational domain of width H and length L. The block located at the center of the unite cell represents the cross-section of a rectangular fiber with orientation angle θ relatively to x axis. w and b are the width and length of fiber respectively.

A constant velocity

U 0 =   u 0x u 0y 
 is applied at the inlet. At the outlet, we have imposed ∂u 0x (x, 0)/∂y = 0 and ∂u 0y (x, 0)/∂y = 0, where u 0x and u 0y are components of velocity As in section 3.3.1.2, particles that are not captured are removed when they escape from the outlet. As defined previously in section 1.3.3 of chapter 1, the filtration efficiency of filter can be obtained as follows:

η = N total -N escape N total (3.17)
where N total and N escape represent the number of solid particles injected at the inlet and removed from the outlet.

Reynolds number is defined as Re = U 0 D h /ν, where ν is the fluid kinematic viscosity

Pressure drop of clean filter

As expressed in chapter 1 part 1.3.7, pressure drop ∆p between inlet and outlet of the periodic cell, which is related to drag force F D on a unit length of single fiber, is a key parameter to evaluate filtration performance [START_REF] Liu | Pressure drop and interception efficiency of multifiber filters[END_REF]. The determination of the dimensionless drag force F * = F D /(µU 0 ) permits to calculate pressure drop, where µ is the dynamic viscosity, with the following relation:

∆p L = F D β bw (3.18)
Ouyang and Liu [START_REF] Ouyang | Analytical solution of flow field and pressure drop for filters with rectangular fibers[END_REF] determined dimensionless drag force for rectangular fiber with θ = 0 in Stokes flow regime. It is worth noting that the volume fraction has also influence on the dimensionless drag force for rectangular fiber [START_REF] Fardi | Flow field and pressure drop of filters with rectangular fibers[END_REF]. In order to compare with these results and to validate our model, we have plotted in Fig. 3.19 the dependency of dimensionless drag force, obtained with our program, on volume fraction of fiber for different aspect ratios w/b and Re = 0.2 and more generally Re < 1 [START_REF] Ouyang | Analytical solution of flow field and pressure drop for filters with rectangular fibers[END_REF]. Fig. 3.19 shows a good agreement that demonstrates that our model can be used to simulate the pressure drop of filter with rectangular fiber for low Reynolds number.

In previous works, many researchers choose generally the fiber Reynolds number less than 1 for fluid flow [START_REF] Rabiee | Investigation of the characteristics of particulate flows through fibrous filters using the lattice Boltzmann method[END_REF][START_REF] Huang | Numerical study of pressure drop and diffusional collection efficiency of several typical noncircular fibers in filtration[END_REF]. As Re < 1, Liu and Wang [START_REF] Liu | Pressure drop and interception efficiency of multifiber filters[END_REF] found that the dimensionless drag force is independent of Reynolds number for circular fiber. Furthermore, Fardi and Liu [START_REF] Fardi | Flow field and pressure drop of filters with rectangular fibers[END_REF] concluded also that dimensionless drag force is relatively independent of Re for rectangular fiber. In this section, we focus on the study of the relationship between dimensionless force and Reynolds number for the case of Re > 1. To vary the Reynolds number, we have chosen to vary the fluid velocity at the inlet. Fig. 3.20 shows the dimensionless drag force F * as a function of Reynolds number, varying from 2 to 10, with four typical aspect ratios and θ = 0. As presented in Fig. 3.20, F * is influenced by Reynolds number and increased with it in the region of 2 ≤ Re ≤ 10. When Re increases, the growth rate of F * gradually enhanced for aspect ratio (w/b) from 1 to 4. Thus, Reynolds number has significant effect on the pressure drop for large value of aspect ratio. We know that a lower pressure drop across a filter is more favorable. Here at a fixed Reynolds number, it is obtained for rectangular fiber with a small aspect ratio (i.e., w/b = 1). shows evolution of windward area l c , defined as:

The streamlines around rectangular fiber with aspect ratio of w/b = 2 and different

l c = w cos θ + b sin θ (3.19)
which is represented in Fig. 3.23. The maximum value of l c for each aspect ratio is also plotted. Figures 3.24 shows the relationship between dimensionless drag force and orientation angle with different aspect ratios. It illustrates that when aspect ratio w/b = 1 (i.e., square), F * is larger for θ = π/4 than those of other angles due to the largest windward area in comparison with other angle values.

The maximum windward area l c is arctan b/w. However, F * decreases with θ for other aspect ratios w/b = 2, 3, 4. The larger w/b is, the faster F * decreases. It can also be noted that F * at θ = π/2 is lower than that at θ = 0 for w/b = 2, 3, 4. The reason is that windward area of θ = π/2 is much smaller than that of θ = 0 as observed in Fig. 3.22. Therefore, as θ increases, the windward area of rectangular fiber decreases, which allows the fluid to pass through filter more easily. It can also be noted, in Fig. 3.22 that values of l c are quite similar for θ = 3π/8 and π/2 leading to quite similar values of drag force as observed in Fig. 3.22. Based on this analysis, it is concluded that a larger θ with a smaller FIBER windward area of rectangular fiber is more stable to obtain a lower pressure drop.

Filtration efficiency of filter

Filtration efficiency, defined in Eq.3.17, is another important parameter used to assess the filter performance. As presented in 3. illustrates that the larger cross-section area is, the higher filtration efficiency can be obtained. The filtration efficiency almost linearly increased with the cross-section area, which results from the strong inertial force. Fig. 3.30 presents the same results for different Reynolds number. Same behavior can be observed, with a small dependency on Re, also indicating clearly that filtration efficiency is mainly governed by the windward area l c in this flow regime.

Quality factor of filter with rectangular fiber

The quality factor of rectangular fiber are calculated. Fig. 3.31 depicts the relationship between quality factor and Reynolds number with different aspect ratios. It can be observed Finally, for fixed Re = 8, we investigate which orientation angle and aspect ratio are favourable for a better performance of filter. Fig. 3.32 shows the effect of orientation angle on the quality factor for Re = 8. As Fig. 3.32 illustrates, quality factor for rectangular fiber increases with the orientation angle at first and then decreases. The quality factor is dependent on the ratio of η to F * . Due to this value, we find that a higher quality factor can be obtained when θ = π/4 for aspect ratio w/b = 1. According to these analysis, the rectangular fiber with aspect ratio w/b = 1 and θ = π/4 (or diamond fiber) exhibits a better performance than the rectangular fiber and circular fiber, which agrees well with the result in section 3.2.3.3.

Moreover for θ = 3π/8, values of quality factors are quite similar for the different aspect ratio investigated due to the fact that dimensionless drag force and filtration efficiency are not very dependent on w/b for this value of θ. The previous analysis of dependency of drag force of filtration efficiency on cross section l c show clearly that both of them increase with l c . In term of quality factor, it exists an optimal value of l c /H (around 0.3), as illustrated in Fig. 3.33, that maximize quality factor QF . In Fig. 3.34, we have plotted the QF as a function of l c /H. QF is strongly influenced by the variation of l c . It is good for low Reynolds number. Although increasing Reynolds number induces performance drop, a more uniform performance with l c can be obtained.

Conclusion

In this chapter, we studied the effect of the particle size on the capture efficiency obtained with one-way coupling and two-way coupling respectively. For these small particles (d p ≤

8 µm), one-way coupling in our model is sufficiently accurate to estimate the particle transport and deposition. The capture efficiency from our simulation agrees well with the results of Araujo et al. [START_REF] Araújo | Critical role of gravity in filters[END_REF]. However, with respect to the larger particles, two-way coupling is much more practical owing to consideration of the influence of particle motion on the flow.

Single fiber capture efficiency is also investigated with two-way coupling for isolated particle and several particles cases. The influence of fiber shape on the single fiber capture 3.4. CONCLUSION

115

efficiency is analyzed for different Stokes number. The particle deposition on the fiber surface is essentially characterized by the particle Stokes number and intercept coefficient.

It is found that the rate of increase of capture efficiency for single fiber is rapid at small Stokes number and then gradually becomes smaller at large Stokes number. For several particles case, a good agreement with empirical correlation of Schweers et al. [START_REF] Schweers | Experimental investigation of particle collection on single fibres of different configurations[END_REF] is observed. The impaction of particles on the front side of square-shaped fiber is more favorable than those on circular and diamond cases. With the same fiber shape, capture efficiency for isolated particle case is higher than the one for several particles case. The quality factor has been used to characterize the filtration performance in order to obtain a high filtration efficiency and low pressure drop. It is found that a filter with diamond fiber exhibits a better performance in terms of the quality factor for both isolated particle and several particles cases.

Based on these simulations, the diamond fiber has a good performance. It is one of types of rectangular shape and can be viewed as the rotation of the rectangular fiber. Thus, The effect of aspect ratio and orientation angle on filtration performance parameters, pressure drop and filtration efficiency of filter, have been investigated for rectangular fiber. The following conclusions can be drawn based on the simulation results obtained in this work.

Firstly, the dimensionless drag force is dependent on not only the orientation angle but also on Reynolds number when 2 ≤ Re ≤ 10. For θ = 0 and large value of aspect ratio (i.e., w/b = 4), the dimensionless drag force increases rapidly with Re. Due to the decrease of windward area of rectangular fiber, the dimensionless drag force decreased greatly with the orientation angle at this aspect ratio. Indeed, it has been illustrate that drag force increase greatly with l c .

Secondly, the filtration efficiency is dependent on both orientation angle and aspect ratio during the filtration process. A larger aspect ratio (i.e., w/b = 2, 3, 4) at θ = 0, π/8

generally lead to a higher filtration efficiency. For square fiber (w/b = 1) at θ = π/4, filtration efficiency is higher than that of other orientation angles due to its larger crosssection area. Indeed, we found that for Re > 1, the filtration efficiency increases almost linearly with windward area. However, it can be noted that filtration efficiency increases slightly with Re.

Finally, in terms of quality factor, it is valuable to measure the filtration performance concerning the filtration efficiency and pressure drop of filter. For these different aspect ratios, the quality factor increases with orientation angle and reaches its maximum at a certain value, then decreases. For example, the largest quality factor is obtained for square fiber at θ = π/4. Filter performance, through quality factor, has also been analysed in term of windward area. This study shows that it exists an optimal value l c that guarantee the minimization of pressure drop and maximization of filtration efficiency.

Generally, through the analysis of pressure drop, filtration efficiency and quality factor, the current study provide a theoretical indication for filter design and optimization with single fiber. Since the filtration performance such as quality factor corresponds to filter made of several identical fibers separated by a distance, in the next chapter, we will focus on the investigation of the filtration performance of filter with staggered fiber.

Chapter 4

Numerical simulation of filtration process for filter with staggered fibers

Introduction

The particle capture process in filter depends on particle and fiber diameters, particle and fluid densities as well as fiber arrangement. Since fiber arrangement, such as parallel and staggered configurations, influences significantly the filtration performance including filtration efficiency and pressure drop of filter [START_REF] Wang | Simulation of filtration process for multi-fiber filter using the Lattice-Boltzmann two-phase flow model[END_REF], thus more studies on modeling the particle process in filter with different fiber arrangement are needed. Wang et al. [START_REF] Wang | Simulation of filtration process for multi-fiber filter using the Lattice-Boltzmann two-phase flow model[END_REF] consider a series of circular fibers with parallel and staggered models to investigate the pressure drop and filtration efficiency. They found that the staggered model is better than the parallel model in terms of quality factor at the same conditions, because the staggered model exhibits higher filtration efficiency while nearly same pressure drop. Rabiee et al. [START_REF] Rabiee | Investigation of the characteristics of particulate flows through fibrous filters using the lattice Boltzmann method[END_REF] study the particulate flows through filter with three different circular fiber arrangements (parallel, staggered and random). By comparing the filtration efficiency, it is found that the staggered configuration is more efficient to capture particles. Ansari et al. [START_REF] Ansari | Numerical study of solid particles motion and deposition in a filter with regular and irregular arrangement of blocks with using Lattice Boltzmann method[END_REF] realized numerical study of solid particles motion and deposition in a filter with parallel and staggered arrangement of square fibers. They obtained that the first column fiber has a more importance in the particle deposition for both arrangements. Most particle deposition occurs on the first fiber for large St and other fibers have a negligible effect.

Other researchers also investigated the effect of fiber arrangement on filtration efficiency including Tehrani et al. [START_REF] Tehrani | Filtration of aerosol particles by cylindrical fibers within a parallel and staggered array[END_REF], Schweers et al. [START_REF] Schweers | Experimental investigation of particle collection on single fibres of different configurations[END_REF], Hosseini and Tafreshi [START_REF] Hosseini | Modeling particle filtration in disordered 2-D domains: A comparison with cell models[END_REF], Li et al. [START_REF] Li | Study and optimization of the filtration performance of multi-fiber filter[END_REF].

Since the staggered arrangement is much more effective compared with parallel model, the In chapter 3, the filtration performance of filter with a row of fibers in a laminar flow to their axes is investigated. Thus, in this chapter, we use the coupled LBM-DEM to simulate the filtration process of filter with staggered arrays of circular, diamond and square fibers. The effects of Reynolds number Re, intercept coefficient R and ratio of particle density to fluid density ε on pressure drop and filtration efficiency are investigated. Then the simulation results are used to explore the fiber's cross-sectional shape effect on filtration factors.

Circular fiber arrangements

Different fiber configuration can lead to the variation of the single fiber capture efficiency.

Here we present three types of fiber arrangements, one beside the other, one underneath the other and staggered as shown in Fig. Our fiber arrangement in part 3.3.1 of chapter 3 can be viewed as one beside the other due to the using of periodic boundary conditions (Fig. 4.1a). To investigate the effect of fiber arrangement on single fiber capture efficiency, we compare our simulation results with the experimental work of Schweers et al. [START_REF] Schweers | Experimental investigation of particle collection on single fibres of different configurations[END_REF], although the fiber size used in chapter 3 is not the same with the literature, where the fiber size in literature is equal to 40 µm. For several fibers existing in filter, the single fiber capture efficiency is obtained by calculating the mean particle fluxes in section upstream and downstream of fiber and the fiber projection area.

CIRCULAR FIBER ARRANGEMENTS 119

Single fiber capture efficiency as a function of St for different fiber arrangements is presented in Fig. 4.2. Our simulation results obtained in chapter 3 for 500 particles injected are plotted in Fig. 4.2 for the configuration one beside the other. For two fibers one beside the other, single fiber capture efficiency increases with the Stokes number for results obtained by both LBM-DEM and experiment. The differences between simulation and experiment is due to the different fiber sizes used in the two cases. Another reason is that the rebound effect of particle is not considered in the simulation. For fibers one underneath the other, the shading effects reduces the number of captured particles. With decreasing distance between fibers w 0 , the first fiber plays an important role concerning capture efficiency and no particles contact with the second fiber. For the case of staggered fibers, since the displacement of left fiber in Fig. 4.1c exerts a substantial effect, the single fiber capture efficiency is higher in contrast to other two cases. From Fig. 4.2, it is found that the single capture efficiency for fibers one underneath the other is generally lower than the staggered fibers and fibers one behind the other. As staggered configurations are more efficient to capture particles, in the following, we focus on the study of these configurations. To valid our code, we first performed the simulations of the flow through the staggered circular fibers as depicted in Fig. 4.3, where we only consider a strip of domain between two lines. Then the simulation results are compared FILTER WITH STAGGERED FIBERS with previous work published in the literature. A series of staggered arranged circular fibers of diameter D f are placed in a rectangular domain with length L and width H. w 0 and l 0 are the horizontal and vertical separation distance between two neighboring fibers, where

H = 2w 0 . A constant velocity U 0 =   0 u 0y 
 is applied at the inlet. ∂u 0x (x, 0)/∂y = 0 and ∂u 0y (x, 0)/∂y = 0 are imposed at the outlet. We use the periodic boundary at both left side and right sides.

The numerical results for the staggered configuration (Fig. 4.3) are compared with those of Liu and Wang [START_REF] Liu | Pressure drop and interception efficiency of multifiber filters[END_REF] and Wang et al. [START_REF] Wang | Simulation of filtration process for multi-fiber filter using the Lattice-Boltzmann two-phase flow model[END_REF], and also compared with the dimensionless drag force expression for the staggered model proposed by Kuwabara [START_REF] Kuwabara | The forces experienced by randomly distributed parallel circular cylinders or spheres in a viscous flow at small Reynolds numbers[END_REF]. The analytical expression of dimensionless drag force resulted from Kuwabara flow is expressed as:

F * = 4π -0.5 ln β -0.75 -0.25β 2 + β (4.1)
where β = πD 2 f 8l 0 w 0 is the fiber volume fraction. According to this expression, the value of drag force applied on fibers is only dependent of fiber volume fraction. Re in the case of β = 0.056 and l 0 /2w 0 = 0.875. Here we use the l 0 /2w 0 = 0.875 in order to be consistent with the literature. It is found that the drag force obtained in the literature [START_REF] Liu | Pressure drop and interception efficiency of multifiber filters[END_REF][START_REF] Wang | Simulation of filtration process for multi-fiber filter using the Lattice-Boltzmann two-phase flow model[END_REF] is nearly constant for Re < 1. However, the value of drag force increased with

Periodic Periodic

Reynolds number for 1 < Re < 20. Our simulation results obtained by LBM are in good agreement with the literature [START_REF] Liu | Pressure drop and interception efficiency of multifiber filters[END_REF][START_REF] Wang | Simulation of filtration process for multi-fiber filter using the Lattice-Boltzmann two-phase flow model[END_REF] in Fig. 4.4(a). Compared with Kuwabara model, it should be noted that Re = 1 is the turning point for the pressure, which indicates that expression (4.1) is not valid for Re > 1.

Dimensionless drag force as a function of fiber volume fraction is also studied. The Reynolds number and separation ratio are fixed, F * is calculated for several values of volume fraction. The value of dimensionless drag force corresponding to Re = 0.2 and l 0 /2w 0 = 0.875 are obtained with different β, as shown in Fig. 4.4(b). It is obvious that the value of dimensionless force increases rapidly as β increases. Our numerical results are in very good agreement with Kuwabara's expression and with the prediction of Wang et al. [START_REF] Wang | Simulation of filtration process for multi-fiber filter using the Lattice-Boltzmann two-phase flow model[END_REF] at Re = 0.2 and l 0 /2w 0 = 0.875.

The effect of aspect ratio of the array on the dimensionless drag force is determined at Re = 0.2 and β = 0.056, the results are shown in Fig. 4.4(c). This validation also approves the results of Liu and Wang [START_REF] Liu | Pressure drop and interception efficiency of multifiber filters[END_REF] and Wang et al. [START_REF] Wang | Simulation of filtration process for multi-fiber filter using the Lattice-Boltzmann two-phase flow model[END_REF].

Numerical study of particles motion and deposition on staggered arrayed fibers with various shapes

In this section, we consider the effect of the vertical neighboring fiber on the filtration performance. The filter for fibers arranged with staggered configurations is presented in Fig. 4.5. Since the fiber arrangement is not regular, the particles are injected along the whole inlet, here we still focus on the filtration efficiency, pressure drop and quality factor of filter.

Model description and filtration parameters

A schematic view of a periodic cell of the filter is illustrated in Fig. 4.5. Our simulations are conducted for two rows of fibers staggered in a rectangular domain, of width H = 0.001 m and length L = 0.0017 m, bounded by velocity inlet and periodic sides. Fibers with three cross-sectional shapes which are circular, diamond and square are considered respectively.

NUMERICAL STUDY OF PARTICLES MOTION AND DEPOSITION ON STAGGERED ARRAYED FIBERS WITH VARIOUS SHAPES 123

The horizontal and vertical separation distance between two neighboring fibers are w 0 and l 0 respectively, where H = 2w 0 . The projected area of each fiber perpendicular to the flow direction is D f /H = 1/2.5. For diamond fiber, it is a square rotated by π/4 with a

diagonal D f . A constant velocity U 0 =   0 u 0y 
 is applied at the inlet. At the outlet, we have imposed ∂u 0x (x, 0)/∂y = 0 and ∂u 0y (x, 0)/∂y = 0. Both left side and right side boundaries are periodic for fluid and solid particles. The fluid density and kinematic viscosity are 1.2 kg/m 3 and 4.4 × 10 -5 m/s 2 respectively. As for the case of previous simulations, once the flow field reaches a steady state, particles are randomly injected on a line at the inlet of the channel (see Fig. Following previous study presented in part 3.3.2.2, pressure drop ∆p is related to drag force F D per unit length of a single fiber oriented transverse to the flow as [START_REF] Liu | Pressure drop and interception efficiency of multifiber filters[END_REF]:

∆p = F D L 2A (4.2)
where the cross-section is here equal to A = l 0 w 0 . Dimensionless drag force F * is:

F * = F D µU 0 (4.3) FILTER WITH STAGGERED FIBERS
where µ is fluid dynamic viscosity and U 0 is the inlet velocity.

The filtration process investigated in this chapter is an initial stage of the process, in a sense that the fiber shape does not change with time and the filter is regarded as a clean one, because the deposition of particles inside the filter is relatively small in the initial stage and its effect on the filter is negligible [START_REF] Duval | la nature et le domaine de l'ingénierie. On peut par example citer les industries minières et cimentières, le transport[END_REF]. Thus, it is reasonable to assume that particles are removed from the flow if they deposit on the surface of fibers or escape from the channel outlet [START_REF] Lin | An early stage of aerosol particle transport in flows past periodic arrays of clear staggered obstructions: A computational study[END_REF]. We recall that the definition of the filtration efficiency of filter is given by Eq (3.17) of chapter 3.

In the following parts, we use the staggered configurations in Fig. 4.5 to investigate the effect of Reynolds number Re, intercept coefficient R = dp D f and density ratio ε = ρp ρ on pressure drop and filtration efficiency of filter with circular, diamond and square crosssectional shapes. For R, various particle diameters are used in the simulations (20 µm, 30 µm, 40 µm, 50 µm, 60 µm). For ε, the particle densities are 720 kg/m 3 , 1200 kg/m 3 , 1680 kg/m 3 , 2160 kg/m 3 , and 2430 kg/m 3 . The volume fraction of circular, diamond and square are respectively

β c = πD 2 f 8l 0 w 0 = 0.157, β d = D 2 f 4l 0 w 0 = 0.1 and β s = D 2 f 2l 0 w 0 = 0.2.
The projected area of fiber D f = 0.0004 m, fiber volume fraction and separation ratio l 0 /w 0 = 1.6 are kept fixed for all the simulations.

Pressure drop of clean filter with staggered fibers

When Re ≤ 1, Huang et al. [START_REF] Huang | Numerical study of pressure drop and diffusional collection efficiency of several typical noncircular fibers in filtration[END_REF] found that the dimensionless drag force is independent on the Reynolds number for both circular and noncircular fiber. In order to complete the study, in this section we investigate the relationship between dimensionless drag force and Reynolds number for circular, diamond and square fiber for Re > 1 with Fig. 4.5. We vary the Reynolds number by changing the fluid velocity in the following simulations. Fig. 4.6 presents the flow field around the fiber with different shapes for Re = 5. We focus on the streamlines near to the fiber, part of the streamlines in the computational domain are showed. It is noticed that the streamlines close to circular, diamond and square fiber are different. Obviously, the cross-sectional shape of fiber modifies the structure of fluid field which leads to the variation of the pressure drop across the filter. Fig. 4.7 shows the dimensionless drag force versus the Reynolds number for circular, diamond and square fiber. As illustrated in Fig. 4.7, F * is influenced by Re and linearly increased with it in the region of 2 ≤ Re ≤ 10. The dimensionless drag force for the diamond fiber is smaller than the cases of circular and square fiber. It indicates that filter with diamond fiber has a lower pressure drop. This can be ascribed to a better streamlined shape of diamond fiber.

CHAPTER 4. NUMERICAL SIMULATION OF FILTRATION PROCESS FOR FILTER WITH STAGGERED FIBERS

A better streamlined shape can reduce the fiber's drag force [START_REF] Wang | The influence of fiber geometry and orientation angle on filtration performance[END_REF].

Filtration efficiency of filter with staggered fibers

The filtration efficiency of filter with circular, diamond and square fiber is calculated by Eq.(3.17). In present simulations, 500 particles are randomly injected at the inlet (see Fig. 4.5a) once the flow field reaches the steady state. The velocities and trajectories of solid particles depend on the fluid velocity field, whose the regime is characterized by the Reynolds number. Therefore, the relationship between filtration efficiency and Reynolds number is investigated and the numerical results are plotted for small particle size (R = 0.05) in Fig. 4.8 and for large particle size (R = 0.15) in Fig. 4.9. Fig. 4.8 demonstrates that the filtration efficiency η increases with an increase in Re for small R (i.e. R = 0.05).

Then η reaches a certain value and does not change with Re anymore for Re < 10. As seen in Fig. 4.9 for large particle size, the filtration efficiency of filter with three cross-sectional fibers reaches their maximum value at Re = 2 and then slightly varied with Re for large intercept coefficient (R = 0.15). By comparison of Figs.4.8 and 4.9, Reynolds number has an obvious effect on filtration efficiency for small R. When the Reynolds number increases, particle performs strong inertia and deviate from the streamlines. As particles approach the fiber, they do not change their trajectories and finally deposit on the surface of the fiber. The effect of Reynolds number can be negligible as the inertia of particle increases to the value that most particles at the upstream of the system deviate from streamlines and move straightforward. The filtration efficiency of filter with square fiber has a higher value than that of circular and diamond fiber for both small and large R.

In order to understand the particle behavior in fluids more clearly, Fig. 4.10 presents instantaneous velocity field of fluid and particle distribution for filter with circular, diamond and square fiber for Re = 5. Particle process including particles injected, particle transport and deposition on the fiber is shown at different time steps. We can recognize that particles move toward the front of fiber which is due to the inertia effect. As seen in Fig. 4.10, particles transport affects the fluid velocity field compared with fluid streamlines presented in Fig. 4.6.

As mentioned above, R is also an important parameter that determines the filtration efficiency. The effect of this parameter on the filtration efficiency is presented for low density ratio (ε = 600) in Fig. 4.11 and for high density ratio ε = 2025 in Fig. 4.12. For Fig. 4.11, it is observed that the filtration efficiency for circular and diamond fiber is more sensitive to R than square case. As R is increased, the number of particles captured by three types of fibers improves. For Fig. 4.12, large particles with high density are easier to deposit on the surface of fiber than small particles. The gap of filtration efficiency for large and small particles is gradually reduced. This is because particles with high density have a higher inertia. From Fig. 4.11 and Fig. 4.12, we can see that square fiber is more favorable to capture the particles compared with the circular and diamond fiber, which accords with the previously results presented in chapter 3. The filtration efficiency of filter as a function of density ratio is depicted in Fig. 4.13 for small R = 0.05 and in Fig. 4.14 for large R = 0.15. In the simulations, a series of particle densities are used to change the density ratio ε. Fig. 4.13 indicates that for small particles the number of deposited particles on the surface of fiber first increases rapidly with ε, then increases at a relatively lower slope for three cases. It is because the particle deposition is characterized by the Stokes number (St) and the fact that the inertia for small particles in the region of density ratio 600 ≤ ε ≤ 2200 grows as the St increases from a low value to a large one. This effect is gradually reduced with the increases in St. Fig. 4.14 for large particles illustrates that the filtration efficiency for circular, diamond and square fiber is independent on the density ratio. The inertia of large particle is intense enough and the fluid field has little effect on the particle deposition. As a consequence, the majority of particles are captured on the front surface of the fiber. The number of captured particle

Capture contribution of each fiber to the filtration efficiency

The filtration efficiency of filter with different fibers were analysed in section 4.3.3. In order to know more about the information inside the filter, here we focus on the study of contribution of each fiber to the filtration efficiency. The concept of capture contribution η i proposed by Wang et al. [START_REF] Wang | Simulation of filtration process for multi-fiber filter using the Lattice-Boltzmann two-phase flow model[END_REF] is used in this section. It is defined as the ratio of captured particle number by ith fiber N capture,i to the total number N capture,tot of captured particles: We consider the effect of Re, R and ε together on the capture contribution, then capture contribution for circular, diamond and square fiber as a function of St is presented in Fig. 4.25. The contribution of each bottom fiber decreases with St and then is kept constant for St > 5. This is contrary for top fiber, η i increases with St and then arrive a constant value which is close to the one of bottom fiber for St > 5. It is noted that Stokes number has obvious effect on capture contribution of each fiber for St < 5. As Stokes number increases ( 0 < St < 5), the gap of contribution of each fiber is gradually reduced due to the effect of inertia. respectively. Fig. 4.26 shows that the quality factor QF increases with R for high and low particle density. For high particle densities, diamond case exhibits a better quality than the circular and square cases. The reason is that filter with diamond fiber has the lowest pressure drop among three cases although filtration efficiency for square fiber is the largest one. However, for low particle densities, square fiber has the greatest quality factor concerning R = 0.05, and ε = 600. Then the QF of diamond and circular fiber increases remarkably compared with the square case, which indicates a better performance for 0.075 ≤ R ≤ 0.15. Indeed, the filtration efficiency increases faster than pressure drop, which leads to a rise in quality factor. Fig. 4.27 depicts the quality factor QF as a function of particle-to-fluid density ratio. It can be observed that quality factor of three fiber configurations is independent on particleto-fluid density ratio for R = 0.15. However, for R = 0.05 the quality factor of three fiber increase with the particle-to-fluid density ratio. Generally, diamond case still indicates a better filtration performance at the same condition. Although square case has a higher filtration efficiency than diamond case as seen in Fig. 4.13 and Fig. 4.14, it also leads to a higher pressure drop. diamond and square fibers is small. In the case of St > 4, this gap becomes obviously, especially for the case of the diamond and circular fibers. The higher quality factor of staggered diamond fibers is due to higher filtration efficiency and lower pressure drop.

η i = N capture,i N capture,tot (4.4) It is noted that the i η i = 1. 53iTiC 30ܭoioC 6TXDUH0%) 6TXDUH07) 'LDPRQG0%) 'LDPRQG07) &LUFXODU0% )

Conclusions

In this chapter, firstly, three kinds of fiber arrangements, one beside the other, one underneath the other and staggered, are presented. The single fiber capture efficiency for different fiber arrangements is compared, from which the staggered configuration shows higher capture efficiency. Then, staggered circular fibers are used to validate our code, the simulation results agree well with the previous work. Finally, the filtration performance (pressure drop and filtration efficiency) of filter with three types of fibers (circular, diamond and square fibers) is evaluated using a two-way coupling method in staggered arrangement. The effect of pertinent parameters such as Reynolds number Re, R, ε, St on pressure drop and filtration efficiency were investigated at the initial stage of filtration. To obtain a high filtration efficiency and low pressure drop, the quality factor is used to characterize the filtration performance. The following conclusion can be drawn from this study:

Dimensionless drag force for three types of fibers is dependent on Reynolds number and linearly increased with it in the region of 2 ≤ Re ≤ 10. Filter with staggered arranged diamond fiber has a lower pressure drop compared with circular and diamond cases, this accord with the results in chapter 3 when concerning single fiber or for arrangement of one beside the other.

The number of captured particles by three type of fibers increases for small particles when Reynolds number varies from 2 to 10. However the filtration efficiency for circular, diamond and square fibers is independent on Re for large particles.

The increases in parameter R leads to a higher filtration efficiency for three types of fibers. The filtration efficiency for circular and diamond fibers is more sensitive to the R than the square fiber.

The filtration efficiency for three type of fibers increases with the ε for small particle, however it is independent on ε for large particle due to the strong inertial force. The injected particles are more easily captured by the square fiber than circular and diamond fibers.

Filtration efficiency of filter for circular, diamond and square fibers increases with the Stokes number and that this trend diminishes for larger Stokes numbers. The results also reveal that η for square fiber is larger than the one for circular and diamond cases when it arrives a constant value for a fixed St.

In terms of QF , filter with staggered arranged diamond fiber exhibits a better filtration performance for higher density ratio, large particle size and Stokes number. This result is the same for single fiber or for arrangement of one beside the other in chapter 3 when isolated particle and several particles are injected.

Conclusion and perspectives

General conclusions

In this thesis, numerical researches is performed to study to investigate the particle transport and deposition in porous media. A coupled lattice Boltzmann (LBM) and discrete element (DEM) methods to deal with fluid-particle interaction are presented. Filtration performance of filter including capture efficiency and pressure drop are analysed, where single fiber and multi-fiber with different cross-section shape are considered.

We introduced the principle of LBM, DEM as well as coupled of these two methods, where the LBM is employed to describe the fluid flow and the DEM is s used to deal with the particle dynamics. With a series of model validations including the Poiseuille flow, Couette flow, flow around a fixed cylinder, Darcy law and Kozeny-Carmann equation, it has been demonstrated that the coupled LBM-DEM have the capability to deal with fluid-particle problems at the particle scale with quite good accuracy and that its implementation in the code used in this work is correct. Moreover, the code has been also validated with more complex validation test cases, by comparison of literature results on filtration experiments in chapter 3 and 4.

The single fiber capture efficiency for circular, diamond and square cross-section shape are numerically investigated for various Stokes number. The capture efficiency of these single fibers all increase with Stokes number. For capture efficiency of circular fiber, a good agreement with existing empirical correction proposed by Schweers at al. are observed for 0.5 ≤ St ≤ 3. The square fiber is more favorable to capture the particle on the front side than the circular and diamond cases. However, diamond fiber exhibits a better filtration performance in terms of quality factor for both cases of one particle and several particles injected, which takes the pressure drop into account.

The influence of aspect ratio and orientation angle of rectangular fiber on filtration performance are also studied for various Reynolds number. It has been shown by Fardi

144

Conclusion and perspectives

and Liu that pressure drop is independent of Reynolds number for Re < 1. With the complementary study conducted during this work, we put in evidence that the pressure drop for filter with rectangular fiber is dependent both the orientation angle and Reynolds number for 2 ≤ Re ≤ 10. It increases with the Re for θ = 0 whereas it decreases with the orientation angle for large windward area.

With respect to filtration efficiency, it linearly increased with windward area for 2.08 ≤ St ≤ 10.4 due to the strong inertia. Through these analysis, the largest quality factor is obtained for square fiber at θ = 0 or diamond fiber, which leads us to the same conclusion that filter with diamond fiber exhibits better filtration performance when several particles are injected.

In addition to the fiber geometry, the fiber arrangement has effect on the filtration performance. The staggered arranged fibers with circular, diamond and square crosssection are considered to determine the filtration efficiency and pressure drop of filter. The influence of parameters such as R = dp D f , ratio of particle density to fluid density ε = ρp ρ and St on filtration performance are investigated. The increases in parameter R leads to a higher filtration efficiency for circular, diamond and square fiber. The filtration efficiency of filter with circular and diamond fibers is more sensitive to R than the square case. Filtration efficiency increases with ε for small particle, however it is independent on ε for large particle due to the strong inertial force. For small value of Stokes number, the filtration efficiency first increases rapidly and then increases at a relatively slower rate for filter with circular, diamond and square fibers. For large St, filtration efficiency arrives its maximum and becomes essentially independent of St. For filter with circular, diamond and square fibers staggered arranged, a higher quality factor is observed for staggered arranged diamond fibers for high density ratio, large particle size and Stokes number.

Globally, this work presents the numerical study on the particles transport and deposition in filter. Our investigation helps to analyse how particle and fluid properties determine the filtration performance and affect the particle filtration process. However, as a first step, to better understand such phenomenon, following aspects are recommended for further work. This work investigates the particles transport at the initial stage of filtration, where the fiber shape is not effected by injected particles. This assumption is reasonable while these particles are very small compared to fiber or when system is quite dilute without a large number of particles. However, particle loading on the surface of fibers may not be neglected when filtration process is located at aging stage which corresponds to the clogging of filter and the changes of fiber shape. With the particles attached on the surface of fiber, the pressure drop in aging stage will increase compared with initial stage. However, the filtration efficiency is not easy to predict due to the variation of fiber shape.

Nevertheless the moving boundary can be implemented by LBM-DEM developed in this work, LBM-DEM has the capability to deal with such unsteady filtration process.

The study of particle dense case could be added. For particle-particle interactions, lubrication forces on the sliding particles may be calculated when two particles come to near contact in further research. The extend of this study to more dense cases can be performed with the model developed and used here. Indeed, using definitions presented in chapter 1, our model can be qualified as a four way coupling in the sense that interparticle interactions are already dealt by our program.

The coupled LBM-DEM model has been validated in this work in 2D. Extension of the LBM-DEM model for 3D could be considered for further simulations. The result is compared with the work of Araujo et al [START_REF] Araújo | Critical role of gravity in filters[END_REF] En termes de processus de filtration, différents de types de filtration existent, ils peuvent être classés en filtration de gâteau, filtration granulaire, filtration fibreuse, etc. Le transport de particules à travers les milieux poreux est un phénomène très complexe dû à la diversité des mécanismes impliqués. Dans la filtration, le facteur de qualité, y compris l'efficacité de filtration et la chute de pression, est utilisé pour évaluer les qualité de filtration. L'efficacité de filtration du milieu filtrant propre est liée à l'efficacité de la capture de fibre isolé. La relation entre eux peut être créée en raison de différents mécanismes de capture, tels que la diffusion brownienne, l'interception et l'impaction inertielle. La capture de particules via la diffusion brownienne est liée au nombre adimensionnel appelé nombre de Péclet (P e). Le coefficient d'interception, un paramètre adimensionné, (R) est utilisé pour décrire l'effet de l'interception. L'impaction inertielle est quant à elle régie par le paramètre adimensionnel appelé nombre de Stokes (St). La séparation des particules du fluide nécessite un système de filtration à haute efficacité avec une faible chute de pression. La prédiction de la chute de pression à travers un filtre est donc très importante. Il peut être directement évalué à partir du débit de fluide. L'arrangement des fibres et la géométrie des fibres ont également une influence significative sur l'efficacité de la capture et la chute de pression. Ces dernière années, le processus de filtration de particule a été étudié par de nombreux chercheurs.

List of symbols

Plusieurs modèles mathématiques ont été proposés et des investigations expérimentales ont été effectuées pour prédire et améliorer la chute de pression et l'efficacité de la capture. -L'impaction des particules sur la face avant de la fibre de forme carrée est plus important que sur les fibres circulaires et losanges.

-La fibre de losange présente une meilleure performance en termes de facteur de qualité.

-La chute de pression causée par la fibre dépend non seulement de l'angle d'orientation, mais aussi du nombre de Reynolds lorsque 2 ≤ Re ≤ 10.

-En ce qui concerne l'efficacité de la filtration avec une fibre rectangulaire, elle dépend à la fois de l'angle d'orientation et du rapport d'aspect. Cependant, l'efficacité de la filtration augmente légèrement avec Re.

-En termes de qualité de filtration en prenant en compte l'efficacité de la capture et la chute de pression, le facteur de qualité diminue avec le nombre de Reynolds pour θ = 0. Cependant, le facteur de qualité augmente avec l'angle d'orientation et atteint son maximum à une certaine valeur, puis diminue. On trouve que le facteur de qualité le plus grand est obtenu pour la fibre carrée à θ = π/4 Chapitre 4 Simulation numérique du processus de filtration pour filtre à fibre disposées en quinconce L'agencement des fibres, tel que le mode parallèle et le mode en quinconce, influence de manière significative les performances de filtration, y compris l'efficacité de la capture et la chute de pression. Généralement, dans les mêmes conditions, le modèle en quinconce est meilleur que le modèle en parallèle en termes de qualité de filtration, car le modèle en quinconce présente un rendement de la capture plus élevé alors que la chute de pression est quasiment identique.

Dans ce chapitre, d'abord, nous présentons trois types d'arrangements de fibres, l'un à côté de l'autre, l'un en dessous de l'autre et en quinconce. L'efficacité de capture de fibre unique obtenue à partir du travail expérimental de Schweers et al et nos résultats de simulation sont comparés pour trois configurations de fibres. Ensuite, le processus de filtration du filtre avec des fibres circulaires, losange et carrées disposées en quinconce ont été étudié numériquement. Les effets du nombre de Reynolds, du rapport entre le diamètre des particules et les fibres (R) et du rapport entre la masse volumique des particules et le fluide (ε) sur la chute de pression et l'efficacité de la capture sont étudiés. Pour R on fait varier le diamètre des particules et pour ε on fait varier la masse volumique des particules. Les résultats des simulations de circulaire, losange et carée sont comparés pour explorer l'effet de la forme transversale de la fibre sur les facteur de qualité. La Fig. 4.34 présente les champs d'écoulement autour de la fibre avec différentes formes pour un nombre de Reynolds Re = 5. Une partie des lignes de courant dans le domaine de calcul sont représentées. On remarque que les lignes de courant proches de la fibre carrée changent remarquablement. De toute évidence, la forme de la section transversale de la fibre modifie la structure du champ de fluide qui conduit à la variation de la chute de pression à travers le filtre. Cependant, pour R = 0.05, le facteur de qualité de trois fibres augmente avec ε.

Sur la base de l'étude présentée dans ce chapitre, les conclusions suivantes peuvent être formulées :

-Pour trois types d'arrangements de fibres, les configurations en quinconce sont plus efficaces pour capturer les particules.

-Le filtre à fibre de section losange a une chute de pression inférieure à celle des cas circulaire et de carée.

-Pour l'efficacité de la capture, lorsque le nombre de Reynolds augmente, le nombre de particules capturées par les fibres augmente pour les particules de petite taille. Cependant, l'efficacité de la capture des fibres circulaires, losanges et carrées est indépendante de Re pour les grosses particules.

-Une augmentation du rapport R entre le diamètre de la particule et celui de la fibre conduit à une augmentation de la capture des particules. Ceci est d'autant plus vrai pour les fibres de formes circulaires et losanges comaprativement à celles de formes carrées.

-L'efficacité des la capture des trois types de fibres augmente avec ε pour les petites particules, mais elle est indépendante de ε pour les grosses particules en raison de la forte force d'inertie.

-En termes de facteur de qualité, combinant l'effet de la capture et la chute de pression, le filtre à fibre de losange présente une meilleure qualité de filtration pour deux cas

Abstract

The objective of the present research was to numerically investigate the transport and deposition of particles in porous media at the pore scale.

Firstly, a developed coupled lattice Boltzmann method (LBM) and discrete element method (DEM) is used to simulate the fluid-particle flow. LBM is employed to describe the fluid flow around fibers whereas DEM is used to deal with the particle dynamics. The corresponding method is two-way coupling in the sense that particle motion affects the fluid flow and reciprocally. It allowed us to predict the capture efficiency and pressure drop at the initial stage of filtration process. The quality factor is also calculated for determining the filtration performance.

Secondly, we focus on the study the capture efficiency of single fiber with circular, diamond and square cross-section, respectively. The results of LBM-DEM for filtration process of single circular fiber agree well with the empirical correlation. The impaction of particles on the front side of square-shaped fiber is more favorable than those on circular and diamond cases. However, diamond fiber exhibits a good filtration performance. Then the variations of quality factor due to the different orientation angle and aspect ratio of rectangular fiber were studied using LBM-DEM. For each case, we have found the optimal value of the windward area to which corresponds a maximum value of the quality factor. The comparison of the performance of the different forms of fibers shows that the largest quality factor is obtained for square fiber oriented with angle π/4.

Finally, the influence of the arrangement of fiber on filtration performance is analyzed by considering the staggered configuration.

Simulations conducted for several particle size and density show that the diamond with staggered array performs better for large particles and high particle-to-fluid density ratio in terms of quality factor. The present study provide an insight to optimize the filtration process and predict filtration performance.

Keywords: Lattice Boltzmann method, discrete element method, particle transport and deposition, capture efficiency, particle filtration, filter.

Figure 1 :

 1 Figure 1: Example of problem involving fluid-solid interactions: snapshots from the numerical simulation of piping erosion [8]. Particles are detached under the hydraulic loading. Then, detached particles are transported by the flow toward the channel outlet.

Figure 2 :

 2 Figure 2: Example of problem fluid-solid involving interactions: gravity-driven flow [9], particles are represented in light gray and walls in dark gray. The yielded region of the fluid grows from a small portion to the whole sample. When a new equilibrium is reached, the yielded region reduces and the flow is slowed.

Figure 1 . 1 :

 11 Figure 1.1: Porosity type (a) intergranular porosity, (b) fracture porosity, (c) cavernous porosity [22].

Table 1 . 1 :Figure 1 . 2 :

 1112 Figure 1.2: The structures for stacks of monotonous spheres, a) SC , b) BCC, c) FCC, d) HCP.

Figure 1 . 3 :

 13 Figure 1.3: Example of particle size distribution curve [30].

14 CHAPTER 1 .

 141 Figure 1.4: The threshold of particles trapped in a triangular construction[START_REF] Herzig | Flow of suspensions through porous media-application to deep filtration[END_REF].

  of the even term of the sin(nθ) are zero, we use 2n instead of na 2n = b 2n = c 2n = d 2n = 0 (1.24)Therefore, the expression of the streamfuction and the velocity are:Ψ(r, θ) = (a 1 r -1 + b 1 r ln(r) + c 1 r + d 1 r 3 ) sin(θ)+ ∞ n=2,3... (a 2n-1 r -2n+1 + b 2n-1 r -2n+3 + c 2n-1 r 2n-1 + d 2n-1 r 2n+1 ) sin((2n-1)θ)(1.25) INTERACTION OF FLUID AND SOLID PARTICLES V r = (a 1 r -2 + b 1 ln(r) + c 1 + d 1 r 2 ) cos(θ)+ ∞ n=2,3...

Figure 1 . 5 :

 15 Figure 1.5: (a) Cylinders with square unit cell. (b) Solution domain for exact solutions of streamfunction.

Figure 1 . 6 :

 16 Figure 1.6: Cake filtration

Fig. 1 . 8 presentsFigure 1 . 7 :Figure 1 . 8 :

 181718 Figure 1.7: Schematic view of granular filtration[START_REF] Mays | Contrasting clogging in granular media filters, soils, and dead-end membranes[END_REF]

Fig. 1 . 23 Figure 1 . 9 :

 12319 Fig.1.10 (b) and given:

Figure 1 . 10 :

 110 Figure 1.10: Filtration efficiency of filter (a) and efficiency of UBE (b)

Figure 1 . 11 :

 111 Figure 1.11: Particle capture by Brownian diffusion

Figure 1 . 12 :

 112 Figure 1.12: Particle capture by interception

Figure 1 . 13 :

 113 Figure 1.13: Schematic representation of particle capture by inertial impaction

Figure 1 . 14 :Figure 1 . 15 :

 114115 Figure 1.14: Capture efficiency of square block as a function of St

Figure 1 . 16 :

 116 Figure 1.16: Retention sites (a) surface sites, (b) crevice sites, (c) constriction sites (d) cavern sites[START_REF] Delanchambr | Contriution á l'étude de l'écoulement d'une suspension á travers un milieu poreux et du mécanisme de la filtration[END_REF]

Figure 1 . 17 : 37 1. 4 . 1

 1173741 Figure 1.17: Map of flow regimes in particle-laden flows [74], Φ p : volume fraction of particles, Φ p = N V p /V , N : number of particles, V p : volume of a single particles, V : volume occupied by particles and fluid, d p : diameter of particle, τ p : particle response time = ρ p d 2 p /(18ρ f ν) for Stokes flow, τ k : Kolmogorov time scale = (ν/ε) 1/2 , where ρ p and ρ f are particle and fluid density, ǫ is the dissipation rate of turbulence kinetic energy, Stokes number is St = τ p /τ k .

Fig. 1 .

 1 18 displays the particles captured by two consecutively aligned fiber. The most of particle are almost captured by the upper fiber.

Figure 1 . 18 :

 118 Figure 1.18: Scanning electron micrograph (SEM) of particles captured by the fibers [44].

Figure 1 . 19 :

 119 Figure 1.19: Particles captured by different capture mechanisms, (a) diffusion (P e = 102), (b) inertial impaction (St = 1.61), (c) interception (R = 0.0357) [112].

Fig. 1 .

 1 [START_REF] Zou | On pressure and velocity boundary conditions for the lattice Boltzmann BGK model[END_REF] presents particle loading staggered fiber cases with different capture mechanisms.Ansari et al.[START_REF] Ansari | Numerical study of solid particles motion and deposition in a filter with regular and irregular arrangement of blocks with using Lattice Boltzmann method[END_REF] investigated and the motion and deposition of particles in the regular and irregular structure filters using the LBM. They found that most of the particles deposited on the first block in the filter for both structures. Fig.1.20 illustrates an example of particle trajectories in the regular and irregular structure filter for Re = 5 .

Figure 1 . 20 :

 120 Figure 1.20: Particles trajectories in the regular and irregular structure filter for Re = 5 (a) regular structure filter, (b) irregular structure filter, [90].

Figure 2 . 1 :

 21 Figure 2.1: Discrete velocities in the D2Q9 model.

Fig. 2 .

 2 [START_REF] Tehrani | Filtration of aerosol particles by cylindrical fibers within a parallel and staggered array[END_REF] shows for example a circular solid obstacle mapped on the LB lattice. The nodes inside an obstacle which are the closest ones to its external surface are called "Solid Boundary" (SB) nodes (red). All the fluid nodes located outside the obstacle, with at least one direction pointing towards a SB node, are called "Fluid Boundary" (FB) nodes (green). All other nodes are fluid nodes (blue). We call a link between a SB and a FB node as a boundary link σi as illustrated in Fig.2.3. Boundary seen by LBM and real solid boundary are shown respectively. A boundary link σi means the direction i pointing from fluid to solid. -σi simply denotes the opposite direction of σi.

Figure 2 . 2 :

 22 Figure 2.2: A circular solid obstacle mapped on the LB lattice.

Figure 2 . 3 :

 23 Figure 2.3: A boundary link.

Figure 2 . 4 :

 24 Figure 2.4: LBM boundary conditions.

Figure 2 . 5 :

 25 Figure 2.5: Schmeatic view of streaming step when using periodic boundary conditions.

  Figure 2.6: Illustration of a collision between two particles with overlap χ n .

Figure 2 . 8 :

 28 Figure 2.8: Schematic view of 2D Poiseuille flow and boundary condition.

Figure 2 . 9 :

 29 Figure 2.9: Comparison of velocity profile obtained with LBM and series solutions for Poiseuille flow at different times.

  shear flow. It is another channel flow similar to the Poiseulle flow, but the flow is driven by a moving plate. The fluid is confined between two infinite walls at y = 0 and y = L y . The schematic view of Couette flow and boundary condition are shown in Fig.2.10. The lower wall is fixed and the upper wall is moving with a constant velocity U 0 . Periodic boundary condition is applied on the left side and right side.

Figure 2 . 10 :

 210 Figure 2.10: Schematic view of Couette flow in 2D and boundary condition.

Figure 2 . 11 :

 211 Figure 2.11: Comparison of velocity profile obtained with LBM and series solutions for Couette flow at different times.

Figure 2 . 12 :

 212 Figure 2.12: Schematic view of the problem of the flow around a circle between two parallel plats.

Table 2 . 9 : 3 [

 293 Parameters used in the simulation of the fluid through porous media Channel length, L x , 0.1, 0.3, 0.4, 0.5, 0.6 [Pa] Fluid kinetic viscosity, ν 10 -6 [m 2 /s] Fluid density, ρ 10

Figure 2 . 14 :

 214 Figure 2.14: Velocity field in porous media for ∆p = 0.4 Pa .

Figure 2 . 15 :Figure 2 . 16 : 2 s D 2 s

 21521622 Figure 2.15: The relationship between discharge Q calculated at the outlet and the product of cross-sectional area and the pressure gradient(A • ∇p) follows Darcy's law.

Figure 2 . 17 :

 217 Figure 2.17: Schematic view of cylinder moving with a constant velocity applied and boundary condition.

  u 0x 0 

 0 3.1. A circular fiber of diameter D f is introduced in computational domain. The fluid flows in x direction from left (inlet) to right (outlet). A constant velocity U 0 = is imposed at inlet and outlet. When steady state of fluid velocity field is reached, small particles with diameter d p are injected at the inlet. Eq.(3.1) is used to calculated the error of the fluid 3.2. ONE PARTICLE INJECTION 81 velocity field. We considered that a steady state is established when err < 10 -5 .

Figure 3 . 1 :

 31 Figure 3.1: Particle trajectory and deposition on a single fiber.

Fig. 3 Figure 3 . 2 :

 332 Figure 3.2: Schematic drawing of fluid velocities at one lattice in LBM.

Figure 3 . 3 :Figure 3 . 4 :

 3334 Figure 3.3: Particle trajectories with the corresponding streamlines around the fiber for (a) St = 0.001, (b) St = 0.03 (c) St = 0.3, (d) St = 0.5, (e) St = 1, (f) St = 1.5.

Figure 3 . 6 :Figure 3 . 7 :

 3637 Figure 3.6: Schematic view of structure of (a) circular fibers (b) diamond fibers (c) square fibers.

.12) 90 CHAPTER 3 .

 903 NUMERICAL SIMULATION OF PARTICLE CAPTURE ON SINGLE FIBER

Figure 3 . 8 :

 38 Figure 3.8: Trajectories of particle with streamlines for circular, diamond and square fiber

Figure 3 . 9 :

 39 Figure 3.9: Quality factor as function of St for isolated particle injected.

Figure 3 .

 3 Figure 3.10: A steady state velocity vector of flow around a circular fiber calculated with LB method

Figure 3 . 11 :

 311 Figure 3.11: Velocity field of fluid and particle distribution for St=0.5.

Fig. 3 .

 3 Fig.3.13 presents the velocity field of fluid and particle distribution with diamond and square fibers. Particles are almost impacted on the front side of the fiber during their transport. This is because fiber Reynolds number Re = 6.8 is not large enough and the flow remains smooth with very little vortices observed behind the fiber. For the intermediate Stokes number (0.5 < St < 1), solid particles do not behave like the fluid particles any more. They are dominated by both the interception and inertial impaction mechanisms.

Figure 3 . 14 :

 314 Figure 3.14: Front side of fiber is divided by 9 parts from 1 to 9.

Figure 3 . 15 :

 315 Figure 3.15: Particles distribution on the front side of circular, diamond and square fiber for St = 1.97.

Figure 3 . 16 :

 316 Figure 3.16: Single fiber capture efficiency as a function of Stokes number St with different fiber shapes.

Figure 3 . 18 :

 318 Figure 3.18: Schematic view of computational domain for simulating flow field around rectangular fiber.

1. 57 ×

 57 10 -5 m/s 2 , D h is the hydraulic diameter of the fiber equal to 2wb/(w + b) = 2 × 10 -4 . Four different values of aspect ratio (w/b = 1, 2, 3, 4) are used in the simulations while keeping D h constant. Five orientation angles (θ = 0, π/8, π/4, 3π/8, π/2) are studied for FIBER each aspect ratio. The solid volume fraction of the fibers is computed as β = bw/(LH).

Figure 3 . 19 :Figure 3 . 20 : 104 CHAPTER 3 .Figure 3 . 21 : 8 3Figure 3 .

 319320104332183 Figure 3.19: Dimensionless drag force as a function of volume fraction, LB simulation results are presented and compared with the work of Ouyang and Liu [154].

Figure 3 . 23 :Figure 3 . 24 :Figure 3 . 25 :

 323324325 Figure 3.23: The cross-section area of rectangular fiber

Figure 3 .

 3 Figure 3.25 presents the same simulations results of Fig. 3.24 in terms of cross section l c for all aspect ratios investigated. For l c , we vary w and b in order to keep D h fixed in the simulation. Figure 3.25 illustrates that drag force increase with l c . In this analysis, we present the relationship between F * and θ (or l c ) in the case where b ≤ w. However, for the case of b > w (i.e. w/b = 1/2, 1/3, 1/4.), symmetrical results about θ = π/4 with those of Fig.3.24 can be obtained, which are not demonstrated in this work.

3 . 2 . 1 ,Figure 3 . 26 :Figure 3 . 27 : 110 CHAPTER 3 .Figure 3 . 28 :Fig. 3 .

 32132632711033283 Figure 3.26: Filtration efficiency of filter as a function of Reynolds number for θ = 0.

Figure 3 . 29 :Figure 3 . 30 :Figure 3 . 31 :

 329330331 Figure 3.29: Filtration efficiency as a function of cross-section area of rectangular fiber for Re=8 and different orientation angles, w/b = 1, 2, 3, 4.

Figure 3 . 32 :Figure 3 . 33 : 114 CHAPTER 3 .Figure 3 . 34 :

 3323331143334 Figure 3.32: Quality factor as a function of orientation angle.

118 CHAPTER 4 .

 1184 NUMERICAL SIMULATION OF FILTRATION PROCESS FOR FILTER WITH STAGGERED FIBERS filtration performance of filter with staggered configurations should be focused and studied.

  4.1.

Figure 4 . 1 :

 41 Figure 4.1: Fiber arrangements (a) one beside the other, (b) one underneath the other, (c) staggered.

Figure 4 . 2 :

 42 Figure 4.2: Single fiber capture efficiency as a function of Stokes number with different fiber arrangements.

Figure 4 . 3 :Figure 4 . 4 :

 4344 Figure 4.3: A schematic diagram of the staggered structure of circular fibers

  4.5a). The initial velocity of injected particles is the same as the fluid velocity.

Figure 4 . 5 :

 45 Figure 4.5: Schematic view of structure of staggered model of filters with (a) circular fibers (b) diamond fibers (c) square fibers.

4. 3 .Figure 4 . 6 :Figure 4 . 7 :

 34647 Figure 4.6: Streamlines around fibers with (a) circular (b) diamond and (c) square crosssectional shape for Re = 5, l 0 = 0.0008 m, w 0 = 0.0005 m, D f = 0.0004 m.

4. 3 .

 3 NUMERICAL STUDY OF PARTICLES MOTION AND DEPOSITION ON STAGGERED ARRAYED FIBERS WITH VARIOUS SHAPES 127

Figure 4 . 8 :

 48 Figure 4.8: Filtration efficiency versus Reynolds number for R = 0.05 and density ratio ε = 2025.

Figure 4 . 9 :

 49 Figure 4.9: Filtration efficiency versus Reynolds number for R = 0.15 and density ratio ε = 2025.

Figure 4 . 10 :

 410 Figure 4.10: Instantaneous velocity field and particle distribution for filter with circular, diamond and square fibers for Re = 5, R = 0.05 and ε = 2025.

Figure 4 . 11 :

 411 Figure 4.11: Filtration efficiency versus intercept coefficient R, density ratio ε = 600 and Re = 5.

Figure 4 . 12 :

 412 Figure 4.12: Filtration efficiency versus intercept coefficient R, density ratio ε = 2025 and Re = 5.

4. 3 .Figure 4 . 13 :Figure 4 . 14 :

 3413414 Figure 4.13: Filtration efficiency versus density ratio ε = ρ p /ρ for R = 0.05 and Re = 5.

Figure 4 . 18 :

 418 Figure 4.18: Filtration efficiency of filter with circular, diamond and square fiber as a function of St.

Figure 4 . 19 :Fig. 4 .

 4194 Figure 4.19: Capture contribution of fiber as a function of Re for R = 0.05 and ε = 2025.

Figure 4 . 20 :

 420 Figure 4.20: Capture contribution of fiber as a function of Re for R = 0.15 and ε = 2025.

Fig. 4 .

 4 Fig.4.21 and Fig.4.22 present the effect of R on capture contribution of circular, diamond and square fibers. Capture contribution as a function of R for particles with low density is shown in Fig.4.21.For bottom fiber, it is found that the capture contribution of circular, diamond and square fibers decreases remarkably with R for 0.05 ≤ R ≤ 0.1 and then reaches a constant value when 0.125 ≤ R ≤ 0.15. For the top fiber, the capture contribution of circular, diamond and square fibers increases with R for 0.05 ≤ R ≤ 0.1 and is close to η i = 0.5. For diamond fiber, the capture contribution of bottom fiber and top fiber are nearly the same when 0.1 ≤ R ≤ 0.15. Fig.4.22 demonstrates the capture

Figure 4 . 21 :

 421 Figure 4.21: Capture contribution of fiber as a function of R for Re = 5 and ε = 600.

4. 3 .

 3 NUMERICAL STUDY OF PARTICLES MOTION AND DEPOSITION ON STAGGERED ARRAYED FIBERS WITH VARIOUS SHAPES 137

Figure 4 .

 4 Figure 4.22: Capture contribution of fiber as a function of R for Re = 5 and ε = 2025.

Figure 4 . 23 :

 423 Figure 4.23: Capture contribution of fiber as a function of ε for Re = 5 and R = 0.05.

Figure 4 . 24 :

 424 Figure 4.24: Capture contribution of fiber as a function of ε for Re = 5 and R = 0.15.

Figure 4 . 25 : 139 4. 3 . 5

 42513935 Figure 4.25: Capture contribution of circular, diamond and square fibers as a function of St.

Figure 4 . 26 :

 426 Figure 4.26: Quality factor QF as a function of R.

Figure 4 . 27 :Figure 4 . 28 :

 427428 Figure 4.27: Quality factor QF as a function of ε.

Fig. 4 .

 4 Fig.4.28 presents the relationship between quality factor and St for circular, diamond and square fiber. It demonstrates that St has the influence on quality factor for three types of fibers. Quality factor increases with St. For St < 4, the gap of QF for circular,

List of Figures 1 150 LIST OF FIGURES 1 .

 11501 Example of problem involving fluid-solid interactions: snapshots from the numerical simulation of piping erosion [8]. Particles are detached under the hydraulic loading. Then, detached particles are transported by the flow toward the channel outlet. . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 Example of problem fluid-solid involving interactions: gravity-driven flow [9], particles are represented in light gray and walls in dark gray. The yielded region of the fluid grows from a small portion to the whole sample. When a new equilibrium is reached, the yielded region reduces and the flow is slowed. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 Filtration application, top: filtering stage and clean stage, bottom: particle separation process, n upstream and n downstream represent the number of particle, at the upstream and downstream respectively [16]. . . . . . . . . . . . . . 1.1 Porosity type (a) intergranular porosity, (b) fracture porosity, (c) cavernous porosity [22]. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.2 The structures for stacks of monotonous spheres, a) SC , b) BCC, c) FCC, d) HCP. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.3 Example of particle size distribution curve [30]. . . . . . . . . . . . . . . . 1.4 The threshold of particles trapped in a triangular construction [33]. . . . . 1.5 (a) Cylinders with square unit cell. (b) Solution domain for exact solutions of streamfunction. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.6 Cake filtration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.7 Schematic view of granular filtration [43] . . . . . . . . . . . . . . . . . . . 1.8 Schematic view of fibrous filter . . . . . . . . . . . . . . . . . . . . . . . . . 1.9 Schematic representation of filter [14] . . . . . . . . . . . . . . . . . . . . . 1.10 Filtration efficiency of filter (a) and efficiency of UBE (b) . . . . . . . . . . 1.11 Particle capture by Brownian diffusion . . . . . . . . . . . . . . . . . . . . 12 Particle capture by interception . . . . . . . . . . . . . . . . . . . . . . . . 1.13 Schematic representation of particle capture by inertial impaction . . . . . 1.14 Capture efficiency of square block as a function of St . . . . . . . . . . . . 1.15 Filtration efficiency of filter versus particle diameter illustrating different capture mechanisms [70] . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.16 Retention sites (a) surface sites, (b) crevice sites, (c) constriction sites (d) cavern sites [71] . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.17 Map of flow regimes in particle-laden flows [74], Φ p : volume fraction of particles, Φ p = N V p /V , N : number of particles, V p : volume of a single particles, V : volume occupied by particles and fluid, d p : diameter of particle, τ p : particle response time = ρ p d 2 p /(18ρ f ν) for Stokes flow, τ k : Kolmogorov time scale = (ν/ε) 1/2 , where ρ p and ρ f are particle and fluid density, ǫ is the dissipation rate of turbulence kinetic energy, Stokes number is St = τ p /τ k . . 1.18 Scanning electron micrograph (SEM) of particles captured by the fibers [44]. 1.19 Particles captured by different capture mechanisms, (a) diffusion (P e = 102), (b) inertial impaction (St = 1.61), (c) interception (R = 0.0357) [112]. 1.20 Particles trajectories in the regular and irregular structure filter for Re = 5 (a) regular structure filter, (b) irregular structure filter, [90]. . . . . . . . .

3. 1

 1 Particle trajectory and deposition on a single fiber. . . . . . . . . . . . . . 3.2 Schematic drawing of fluid velocities at one lattice in LBM. . . . . . . . . . 3.3 Particle trajectories with the corresponding streamlines around the fiber for (a) St = 0.001, (b) St = 0.03 (c) St = 0.3, (d) St = 0.5, (e) St = 1, (f) St = 1.5. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.4 Particle capture efficiency on fiber as a function of the Stokes number St.

Chapitre 2 Chapitre 3

 23 Méthods numérique LBM et DEM Dans ce chapitre, nous présentons une méthode numérique couplant la méthode lattice Boltzmann (LBM) et la méthode des élements discrets (DEM). L'écoulement du fluide est calculé par la LBM tandis que la dynamique des particules est simulée via la DEM. Nous utilisons le modèle classique D2Q9. Le domaine de calcul dans la méthode LBM est d'abord divisé en un maillage carré régulier. La phase liquide dans LBM est alors supposée comme un groupe de particules de fluide se trouvant à chaque noeud du réseau. Elles sont contraintes de migrer le long des liens vers le noeuds les plus proches et d'entrer en collision les unes avec les autres aux niveau de noeuds. Les variables microscopiques des fluides comme la densité du fluide et la vitesse du fluide peuvent être déduites des moments de la fonction de distribution. une approche numérique très populaire pour traiter les interactions de contact entre particules solides. Les particules se déplacent selon la loi de Newton et leurs contacts sont modélisés par un loi de contact basée sur une approache de sphères légerement déformables. Nous utilisons le modèle LSD (linear spring dashpot) et une extension de la méthode proposée par Cundall-Strack, pour modéliser respectivement la force de contact normale et la force de contact de cisaillement. Les forces hydrodynamiques et les moments résultants sont déduits du calcul LBM et utilisés dans le cycle DEM pour calculer de nouvelles positions et vitesses de particules solides, qui à leur tour définissent les nouvelles conditions aux limites pour le problème de la dynamique des fluides. La réalisation du couplage LBM-DEM est faite en considérant que chaque méthode est caractérisée par une discrétisation temporelle différente. Les méthodes LBM-DEM couplées sont implémentées dans un code "maison" développé au Laboratoire de Génie Civil et Génie Mécanique (LGCGM) dénommé "Albadec". Les problèmes classiques de mécanique des fluides sont présentées afin de valider le code de calcul 2D basé sur l'algorithme LBM et sur le couplage des LBM-DEM respectivement. Les résultats de la simulation de l'écoulement de Poiseuille, l'ecoulement de Couette et de l'écoulement autour d'un cylindre circulaire sont présentés. Simulation numérique de la capture de particules sur une seule fibre Dans ce chapitre, nous étudions l'effet de la taille des particules d p et du nombre de Stokes St sur l'efficacité de la capture des particules en utilisant deux méthodes. La premiére est un couplage unidirectionnel combinant la méthode Lattice Boltzmann (LBM) avec une approche ponctuelle Lagrangienne. La seconde est un couplage bidirectionnel basé Résumé en français

Figure 4 . 29 :

 429 Figure 4.29: Comparaison du profil de vitesse obtenu avec la LBM et des solutions en série pour l'écoulement de Poiseuille et l'écoulement de Couette.

Figure 4 . 30 :

 430 Figure 4.30: Profils de vitesse obtenus avec la méthode LBM et avec un logiciel CFD pour un écoulement de fluide circulant autour d'un cylindre circulaire

Figure 4 . 31 :

 431 Figure 4.31: Champ de vitesse du fluide et position des particules dans les cas d'écoulements autour de fibres de sections différentes (circulaire, carrée et diamantée), pour St = 0.5.

Figure 4 . 32 :-Figure 4 . 33 :

 432433 Figure 4.32: Champ de vitesse du fluide et distribution des particules pour St = 8.33, Re = 8, (a) θ = 0, (b) θ = 1/8π, (c) θ = 1/4π, (d) θ = 3/8π, (e) θ = 1/2π.

Fig. 4 .

 4 35 a) présente le facteur de qualité en fonction de R pour les masses volumiques des particules élevées et faibles. Pour les particules de masse volumique élevées, les fibres losanges présente une meilleure qualité que les fibres circulaires et carrés. Cependant, pour les masses volumiques faibles, les fibres carrées ont le facteur de qualité le plus élevé en ce qui concerne R = 0, 05 et ε = 600. Alors le facteur de qualité du losange et circulaire augmente remarquablement par rapport au carré, ce qui indique une meilleure performance pour 0.075 ≤ R ≤ 0.15. La Fig.4.35 b) présente le facteur de qualité en fonction de ε. On peut observer que le facteur de qualité de trois fibres est indépendant de ε pour R = 0.15.

Figure 4 . 34 :

 434 Figure 4.34: Représentation des lignes de courant autour des fibres de sections transversales différentes (circulaire, losange et carré).

Figure 4 . 35 :

 435 Figure 4.35: Facteur de qualité QF en fonction de a) R et b) ε.

  .7. Generally, the DEM time step dt DE is smaller than the LBM time step dt. Consequently an integer number of DEM loops should be included in a single LBM cycle. In order to achieve the correspondence between the two time steps used by LBM and DEM, the DEM time step dt DE is reduced to a new value.

	2.4. COUPLING OF LBM-DEM	65
	dt ′ DE = Hydrodynamics dt n force and torque Update particle position and velocity DEM loop Calculate contact force and torque Final time ? End Yes No where n is the nearest integer greater or equal to dt/dt Compute LBM loop Figure 2.7: Flowchart of the coupled LBM and DEM. Save data	(2.64)

DE . Subcycling time integration is created between LBM and DEM. Eq 2.64 indicates that one computational time step of the LBM includes n substeps of time integration with the DEM.

64 CHAPTER 2. LATTICE BOLTZMANN AND DISCRETE ELEMENT METHODS

Table 2 .

 2 1: Parameters used in the simulation of Poiseuille fow.

	Channel length, L x	0.01	[m]
	Channel height, L y	0.002	[m]
	Pressure drop, ∆p	0.2	[Pa]
	Fluid kinetic viscosity, ν 10 -5	[m 2 /s]
	Fluid density, ρ	10 3	[kg/m 3 ]
	Reynolds number, Re	0.13	[-]
	Relaxation time, τ	1	[-]
	Lattice spacing, h	2 × 10 -5 [m]

Table 2

 2

	Table 2.2: Discrepancy of velocity profile between LBM and series solution of Poiseuille
	flow for different τ .
	Relaxation time, τ 0.6	0.8	1	1.2
	Discrepancy, err	0.023% 0.07% 0.14% 0.15%

.2 and Table

2

.3. Based on Eq 2.10, LBM comes with the definition of three parameters: h, τ and dt. It is noted that the relation time should τ > 0.5. For a fixed lattice spacing and a given fluid viscosity, smaller τ implies smaller time step, as a result, the discrepancy between the LBM and analytical solution decreases as presented in Table

2

.2. As discussed in Ref.

[START_REF] Gallivan | An evaluation of the bounce-back boundary condition for lattice Boltzmann simulations[END_REF]

, results are acceptable for τ < 1.5.

The lattice spacing h has a direct impact on the discrepancy. Smaller h lead to more precious results, however, it increases the computation cost. As shown in Table

2

.3, the discrepancy between LBM and analytical solution decreases with the decrease of lattice spacing, where N x is node number. Larger node number means smaller h. Thus, the 2.5. VALIDATIONS OF LBM-DEM 69 parameter h, τ and time step should be chosen in balance in LBM.

Table 2 .

 2 4: Parameters used in the simulation of Couette flow.

	Channel length, L x	0.01	[m]
	Channel height, L y	0.002	[m]
	Wall velocity U 0		0.0001	[m/s]
	Fluid kinetic viscosity, ν	10 -6	[m 2 /s]
	Fluid density, ρ		10 3	[kg/m 3 ]
	Reynolds number, Re = U 0 Ly ν Relaxation time, τ	0.2 0.8	[-] [-]
	Lattice spacing, h	2 × 10 -5 [m]
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Table 2 .

 2 5: Discrepancy of velocity profile between LBM and series solution of Couette flow.

	Relaxation time, τ 0.6	0.8	1
	Discrepancy, err	0.4% 0.5% 0.7%
	Node number, N x 400	500	600
	Discrepancy, err	0.71% 0.5% 0.3%

Table 2 .

 2 Velocity profiles obtained with LB methods and with CFD software for a fluid flowing around a circular cylinder, different values d/h of lattice are studied, where d is the cylinder diameter and h is the lattice spacing. time t end and time step are given in Table2.7. The reason leads to this discrepancy is that the solid boundary in LBM is dealt with by half-way bounce-back rule and it is located between fluid boundary and solid boundary and just likes steps. For the resolutions of d/h = 8, the discrepancy reached as much as 7.26%, this accuracy is not sufficient whereas for d/h = 12 the resolution takes much computation time. Thus, a resolution of d/h = 10 is a good compromise between accuracy and computation time. 8: Parameters used in the simulation of the fluid flowing around a cylinder

		6×6
	8[ PV	6×65 6×6 6×6
		6
		6	6×65	6×6	6×6	6×6	6×6
				; P
	Figure 2.13:

  Table 2.8 compares C D obtained from LBM with different resolutions d/h of lattice and CFD software. The value of C D can be accepted for d/h ≥ 10.

Table 2 .

 2 10: Parameters used in the simulation of the cylinder moving with constant velocity

	Channel length, L x	0.05	[m]
	Channel height, L y	0.008	[m]
	Cylinder diameter, d	0.001	[m]
	Pressure drop, ∆p	0.001, 0.01, 0.02, 0.03, 0.05, 0.1,
		0.2, 0.3, 0.4, 0.5, 1	[Pa]
	Fluid kinetic viscosity, ν 10 -6	[m 2 /s]
	Fluid density, ρ	10 3	[kg/m 3 ]
	Relaxation time, τ	0.6	[-]
	Lattice spacing, h	5 × 10 -5	[m]

number is used. The Fig.

2

.18 presents the numerical results obtained with LBM-DEM and CFD software respectively for C D = f (Re). It is observed that the drag coefficient of cylinder obtained using LBM-DEM agrees well with the CFD software for fixed cylinder. For the moving cylinder, the drag coefficient can be satisfied for LBM-DEM even though there exists small difference. Figure 2.18: Drag coefficient as a function of Reynolds number of cylinder 78 CHAPTER 2. LATTICE BOLTZMANN AND DISCRETE ELEMENT METHODS

Table 3

 3

	.1: Dimensionless force applied on the fiber
	F * Circular Diamond Square
	29.2	24.9	38.5

  Figure 4.15: Filtration efficiency of filter with circular fiber as a function of St.

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  Figure 4.17: Filtration efficiency of filter with square fiber as a function of St.

	4.3. NUMERICAL STUDY OF PARTICLES MOTION AND DEPOSITION ON
	STAGGERED ARRAYED FIBERS WITH VARIOUS SHAPES	133
	,4
	,4
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			6W 15H5 5 ,4, ܭ ,
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	Figure 4.16: Filtration efficiency of filter with diamond fiber as a function of St.

  Flowchart of the coupled LBM and DEM. . . . . . . . . . . . . . . . . . . 2.8 Schematic view of 2D Poiseuille flow and boundary condition. . . . . . . . 2.9 Comparison of velocity profile obtained with LBM and series solutions for Poiseuille flow at different times. . . . . . . . . . . . . . . . . . . . . . . . . 2.10 Schematic view of Couette flow in 2D and boundary condition. . . . . . . . 2.11 Comparison of velocity profile obtained with LBM and series solutions for Couette flow at different times. . . . . . . . . . . . . . . . . . . . . . . . . Velocity profiles obtained with LB methods and with CFD software for a fluid flowing around a circular cylinder, different values d/h of lattice are studied, where d is the cylinder diameter and h is the lattice spacing. . . . 2.14 Velocity field in porous media for ∆p = 0.4 Pa . . . . . . . . . . . . . . . . 2.15 The relationship between discharge Q calculated at the outlet and the product of cross-sectional area and the pressure gradient(A • ∇p) follows Darcy's law. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.16 Discharge Q at the outlet is proportional to φ 3 (1-φ) 2 as in Kozeny-Carman relation. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.17 Schematic view of cylinder moving with a constant velocity applied and boundary condition. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.18 Drag coefficient as a function of Reynolds number of cylinder . . . . . . . .

	LIST OF FIGURES
	2.13

2.1 Discrete velocities in the D2Q9 model. . . . . . . . . . . . . . . . . . . . . 2.2 A circular solid obstacle mapped on the LB lattice. . . . . . . . . . . . . . 2.3 A boundary link. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.4 LBM boundary conditions. . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.5 Schmeatic view of streaming step when using periodic boundary conditions. 2.6 Illustration of a collision between two particles with overlap χ n . . . . . . . 2.7 2.12 Schematic view of the problem of the flow around a circle between two parallel plats. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

  Single fiber capture efficiency of isolated particle for fiber with three different shapes. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.8 Trajectories of particle with streamlines for circular, diamond and square fiber 3.9 Quality factor as function of St for isolated particle injected. . . . . . . . . 3.10 A steady state velocity vector of flow around a circular fiber calculated with LB method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .3.11Velocity field of fluid and particle distribution for St=0.5. . . . . . . . . . . qui concerne le transport et le dépôt de particules dans des milieux poreux, le dépôt de particules à la surface des milieux poreux peut modifier la morphologie des pores. L'accumulation de particules entraîne un colmatage progressif du filtre. En conséquence, la perméabilité diminue et la chute de pression augmente, ce qui détériore la durée de vie du filtre. Ainsi, il est essentiel d'estimer quantitativement une performance de filtre.

. . . . . . . . . . 3.5 Single fiber capture efficiency as a function of ratio between the particle diameters d p and fiber diameter D f obtained from one-way and two-way coupling methods, respectively. . . . . . . . . . . . . . . . . . . . . . . . . 3.6 Schematic view of structure of (a) circular fibers (b) diamond fibers (c) square fibers. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.7 3.12 Comparison between capture efficiency of single circular fiber obtained from numerical method (LBM-DEM) and from the empirical correlations. Re = 6.8, R = 0.0625. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ce

Acknowledgements

I would like to express my sincere gratitude to my advisors Prof. Mustapha Hellou and MCF Franck Lominé for the continuous support and valuable discussion during my Ph.D studies. Great thanks should be given to MCF Franck Lominé who devoted great time and efforts to the code used in my thesis. These two advisors helped me not only in research work but also in daily life. I learned a lot from their serious attitude and passion for life and work.

This work has been supported by China Scholarship Council (CSC) and INSA de Rennes. I would like to thank CSC for the financial support and the opportunity to study in France.