Carlos Montilla
email: carlos.montillaestrella@ensiacet.fr

Youssef Nasro-Allah

Renaud Ansart

R O Fox

O Simonin

Toulouse Archive Ouverte

Keywords: gas-particle flows, kinetic theory, electrostatic force, CFD simulation

Due to triboelectric charging, the solid phase in gas-particle flows can become electrically charged, which induces an electrical interaction between all the particles in the system. Because this force decays very rapidly, many current models neglect the contribution of the electrostatic interaction between the particles. Nevertheless, the impact that this force can have in many industrial configurations is well documented. In this work an Eulerian-Eulerian model for gas-particle flow is proposed in order to take into consideration the electrostatic interaction between the particles. We use the kinetic theory of granular flows to derive the transport equation for the electrical charge for dense gas-particle systems. We show that the electric charge transport equation can be derived without presuming the for the particle charge distribution. In order to close the collision integrals in the Chapman-Enskog equation, we proposed a linear model for the charge-velocity correlation. The model was tested in a 3 dimensional periodic box. The results shows that the dispersion phenomenon have two contributions: a kinetic dispersion due to the random motion of particles (predominant in dilute systems) and collisional dispersion due to the electron transfer during a particle-particle collision (predominant in dense systems). Another effect that contributes to the particle transfer is the triboconductivity effect, which is the motion of electrons following the global electric potential difference. The dispersion and triboconductivity characteristic times where calculated, and the analysis shows that, in dense systems, the two effects are comparable if the dispersion characteristic length is equal to the particle diameter.

Introduction

Nowadays, gas-particle flows play an extremely important role in many industrial technologies. Fluidized beds, cyclonic separators and the transport of air pollutants are just a few examples of this type of flow. In some configurations the particles collide with other solid materials (either another particle or a solid boundary). During these interactions the particles can get electrically charged due to the triboelectrification effect [START_REF] Matsusaka | Electrostatics of particles[END_REF]. The electrically charged particles can now interact with other charged particles via the Lorentz force. Because the particle velocity is very small compared to the speed of light, the magnetic contribution can be dropped, and only the electrostatic term is relevant.

The generation of electrical charges can be undesirable for many industrial processes. There are safety hazards such as the risk of explosions due to a spark, wall sheeting and the generation of an intense electric field. Dynamically, the presence of electrostatic forces modifies the behavior of a gasparticle flow: modifications on the minimum fluidization velocity, rise in the entrainment rate, modification on the heat transfer coefficient are just a few example of the effects of the electrostatic force [START_REF] Hendrickson | Electrostatics and gas phase fluidized bed polymerization reactor wall sheeting[END_REF][START_REF] Miller | Fluidization studies of solid particles[END_REF].

All these electrostatic effects are well documented in the literature. [START_REF] Sowinski | New technique for electrostatic charge measurement in gas-solid fluidized beds[END_REF][START_REF] Sowinski | Investigation of electrostatic charge distribution in gas-solid fluidized beds[END_REF] built a fluidized bed with a Faraday cup to measure the total particle charge after fluidization. Their results showed that the particles get charged and that the magnitude depends on the fluidization velocity. Moreover the entrained fines particles and the bed particles have an inverse polarity. [START_REF] Salama | Investigation of electrostatic charge distribution within the reactor wall fouling and bulk regions of a gas-solid fluidized bed[END_REF] extended the study to the particles inside the bed. They observed that, although globally the bed is charged negatively, there are a small percentage of particle with a positive charge. This suggests the wall sheet is formed by consecutive layers of negative and positive charged particles. Zhou, Y. et al. (2013) introduced a moving probe inside a fluidized bed to map the electric potential inside the column. Their data reveal that the bed is negative charged at the bottom and positive charged at the top. Moreover, they also found a difference in the radial profile; with the wall having a stronger potential than the center of the bed. The gas-dynamics can also be impacted; [START_REF] Dong | Experimental investigation of electrostatic effect on bub ble behaviors in gas-solid fluidized bed[END_REF] placed 4 electrostatic probes inside a fluidized bed to analyze the effect of the electrostatic force on the bubble's motion. The authors ob-served that the bubble size decrease as the electrostatic force increases. They attributed this result to the fact that most of the particles have the same charge sign, this creates a repulsive force between them, leaving less space to the bubble to grow.

The eulerian modeling is widely used to study gas-particle flow [START_REF] Hamidouche | Numerical simulation of multiphase reactive flows[END_REF]). The governing equation can be derived from the Boltzmann and Chapman-Enskog equations [START_REF] Chapman | The mathematical theory of non uniform gases[END_REF]). In dense configuration the collision term must be closed. [START_REF] Jenkins | A theory for the rapid flow of identical, smooth, nearly elastic, spherical particles[END_REF] achieved a closed form of the collision integral assuming a molecular chaos and a Maxwellian velocity distribution. If the Maxwellian hypothesis is too restrictive, a non-equilibrium distribution can be obtained following the Grad's theory [START_REF] Grad | On the kinetic theory of rarefied gases[END_REF][START_REF] Jenkins | Grad's 13-moment system for a dense gas of inelastic spheres[END_REF] or the Richmann method [START_REF] Richman | The Source of 2nd Moment in Dilute Granular Flows of Highly Inelastic Spheres[END_REF]).

Currently, some efforts have been made in order to add the electrostatic force to the Eulerian codes. Rokkam, R. G. et al. (2010) developed a model in which the electrostatic effect is added as a body force in the solid momentum equation. Later, the same authors [START_REF] Rokkam | Computational and experimental study of electrostatics in gas-solid polymerization fluidized beds[END_REF]) tested this model in a fluidized bed reactor using Fluent. Their model was in good agreement with the experimental observation, specially concerning the radial segregation of the solid phase. In this approach, however, the electrical charge is an input parameter and remain fixed throughout the simulation.

A more complex model was proposed by [START_REF] Kolehmainen | Eulerian modelling of gas-solid flows with triboelectric charging[END_REF]; they used the kinetic theory of granular flow to derive a transport equation for the particle charge. Using an uncorrelated Maxwellian and Gaussian probability density distributions for the velocity and the particle charge, they were able to close the collision integral and to derive a collisional dispersion coefficient. However, this coefficient was not enough to account for all the particle dispersion, and therefore, they decided to add a kinetic dispersion coefficient following an analogy with the heat transfer coefficient [START_REF] Hsiau | Kinetic theory analysis of flow-induced particle diffusion and thermal conduction in granular material flows[END_REF]. The results showed that this new formulation was in accordance with DEM simulations. More recently [START_REF] Ray | An Euler-Euler model for mono-dispersed gasparticle flows incorporating electrostatic charging due to particle-wall and particle-particle collisions[END_REF] extended that model and computed the charge-velocity correlation in order to derive the kinetic dispersion coefficient. The authors were also able to derive the charge variance equation in order to fully close the transport equation. They implemented their model using OpenFOAM and simulated a two-dimensional fluidized bed. The results showed that they were able to successfully predict the thickness of the particle layer formed at the wall of the reactor.

In our work, we propose a closure for the collisional and kinetic dispersion terms in the mean charge balance derived in the framework of the kinetic theory. In particular, we show that the closure assumption for the collisional contribution can be derived without presuming an uncorrelated Maxwellian probability distribution for the particle electrical charge. In addition, we derive a closure for the dispersion term due to transport by the random velocity from the transport equation governing the electrical charge due to the fluctuating particle velocity.

Particle dynamics

Following the approach described by [START_REF] Gatignol | The Faxén formulas for a rigid particle in an unsteady non-uniform Stokes-flow[END_REF] and [START_REF] Maxey | Equation of motion for a small rigid sphere in a nonuniform flow[END_REF] the motion equation for a single particle follows the Newton's second law:

m p du p,i dt = -V p ∂P g ∂x i + m p g i + F d,i + q p E i (1)
Hereinafter all the equation are presented in tensor notation using the Einstein summation convection over all indexes except p. m p is the mass of the particle, u p the particle velocity, V p the particle volume, P g is the pressure of the gas-phase, g is the gravity F d is the drag force, E in the electric field and q p is the particle electric charge.

The right hand side of the equation represents the sum of forces acting on the particles. There we found in order: the Archimedes force, the gravity force, the drag force (see [START_REF] Simonin | Drag force modelling in dilute to dense particle-laden flows with monodiperse or binary mixture of solid particles[END_REF] for a detailed description), and the last term is the electrostatic force due to the electric field generated by the presence of other charged particles.

Following the Maxwell's equation we can find the electrical field:

∇ 2 ϕ = - (2)
E i = -∇ϕ (3)
Where ϕ is the electrical potential, is the charge density and is the mixture permittivity.

Collision dynamics

One of the most important aspects of particle dynamics, are the particle-particle collisions, and the exchange of momentum and electric charge during the collision. For the sake of simplicity, we will limit our study to binary collisions of spherical particles.

Let's consider two particles p 1 and p 2 located at x p1 and x p2 . They have a given velocity c p1 and c p2 and electric charge ξ p1 and ξ p2 . We define k as the unit vector going from the center of p 1 to the center of p 2 , we also define g as the relative velocity of the particles

g i = c p1,i -c p2,i .
Using the momentum and kinetic energy conservation law, we can derive the particles velocity after the collision c + p1 and c + p2 :

c + p1,i = c p1,i - 1 2 (1 + e c ) (g j k j ) k i (4) c + p2,i = c p2,i + 1 2 (1 + e c ) (g j k j ) k i (5)
To take into account the triboelectrification phenomenon, we are going to use the model developed by [START_REF] Kolehmainen | Triboelectric charging of monodisperse particles in fluidized beds[END_REF]. They used a Hertzian collision model to calculate the overlapping area during a collision between two particles. Using the triboelectrification model proposed by [START_REF] Laurentie | Numerical modeling of triboelectric charging of granular materials in vibrated granular beds[END_REF] they were able to compute the charge transfer during the impact:

ξ + p1 = ξ p1 -ε 0 A max (g m k m ) (E * l k l ) (6) ξ + p2 = ξ p2 + ε 0 A max (g m k m ) (E * l k l ) (7)
Where E * is the total electric field, which has the contribution of the resolved electric field, plus the contribution of the electric field generated by the colliding particles:

E * i = E i - ξ p2 -ξ p1 πε 0 d 2 p (8)
And A max (g m k m ) is given by the Hertzian model:

A max (g m k m ) = 2πr * 15m * p 32Y * √ r * 2 /5 (g m k m ) 4 /5 (9)
Where r * is the effective particle radius, Y * is the effective Young's modulus and m * p is the effective particle mass:

1 r * = 1 r 1 + 1 r 2 (10) 1 Y * = 1 -ν 2 1 Y 1 + 1 -ν 2 2 Y 2 (11) 1 m * p = 1 m p1 + 1 m p2 (12)
r is the particle radius, Y is the Young's modulus, ν is the Poisson ratio, and m p is the particle mass.

Finally the charge transfer model by collision can be written as:

ξ + p1 = ξ p1 -βE i k i + β γ (ξ p2 -ξ p1 ) (g m k m ) 4 /5 (13) ξ + p2 = ξ p2 + βE i k i - β γ (ξ p2 -ξ p1 ) (g m k m ) 4 /5 (14)
With:

β = ε 0 2πr * 15m * p 32Y * √ r * 2 /5 (15) γ = πε 0 d 2 p ( 16
)

Eulerian modeling of the electrostatic phenomenon

In order to derive a continuum model for the solid phase, we will use the fact that the motion of particles in a rapid granular flow is very similar to the motion of molecules in a gas. This allows us to use the kinetic theory to obtain the governing equation of the solid phase. Let f ( x, c p , ξ p , t) be the particle number density function which gives the number of particles per unit of volume at the position x with a velocity u p = c p and a charge q p = ξ p . Using this function we have the definition for the particle number density (n p ) and the mean value for any property φ:

n p = R 3 R f dξ p d c p (17) φ = 1 n p R 3 R φf dξ p d c p (18)
This allows us to define some useful quantities such as he particle mean velocity:

U p,i = c p,i = 1 n p R 3 R c p,i f dξ p d c p (19)
The particle fluctuant velocity:

c p,i = c p,i -U p,i (20)
The particle kinetic stress tensor:

R p,ij = c p,i c p,j = 1 n p R 3 R c p,i c p,j f dξ p d c p (21)
For an isotropic flow, this tensor can be reduced to:

q 2 p = R p,ii 2 (22)
When we can assume an uncorrelated motion of particles, the granular temperature can be defined as:

Θ = R p,ii 3 = 2 3 q 2 p ( 23
)
The particle mean electric charge:

Q p = ξ p = 1 n p R 3 R ξ p f dξ p d c p (24)
The fluctuant electric charge:

ξ p = ξ p -Q p (25)
The electric charge variance:

Q = ξ p ξ p = 1 n p R 3 R ξ p ξ p f dξ p d c p (26)

Boltzmann equation

The dynamic evolution of f is given by the Boltzmann equation:

∂f ∂t + ∂ ∂x i [c pi f ] + ∂ ∂c pi du pi dt | x, c p , ξ p f + ∂ ∂ξ p dq p dt | x, c p , ξ p f = ∂f ∂t coll (27)
The notation G | x, c p , ξ p is a short form for the conditional expectation G | x, u p = c p , q p = ξ p .

The right hand side of the Boltzmann equation accounts for the variation due to particle-particle collisions.

We will consider that the particle charge only changes due to the collisions with other particle, hence dq /dt = 0.

From the Boltzmann equation, we can derive a general transport equation for any property φ:

Dn p φ Dt + n p φ ∂U pi ∂x i + ∂n p φc pi ∂x i -n p Dφ Dt -n p c pi ∂φ ∂x i -n p F i ∂φ ∂c pi + n p DU pi Dt ∂φ ∂c pi + n p c pj ∂φ ∂c pi ∂U pi ∂x j = C (φ) (28)
This is called the Chapman-Enskog equation [START_REF] Chapman | The mathematical theory of non uniform gases[END_REF] The right hand side of the equation accounts for the mean transfer rate of the property φ due to collision. Following formulation proposed by Jenkins, J. & Savage S. ( 1983) the term can be written as the contribution of a source term and a flux term:

C (φ) = χ (φ p ) - ∂ ∂x i θ i (φ p ) (29)
Where

χ = d 2 p 2 g• k>0 ∆φ (g i k i ) f (2) d kdξ p1 dξ p2 d c p1 c p2 (30)
θ i = - d 3 p 2 g• k>0 δφ (g i k i ) f (2) k i d kdξ p1 dξ p2 d c p1 c p2 (31)
Where f (2) = f (2) x p1 , c p1 , ξ p1 , x p1 + d p k, c p2 , ξ p2 , t is the particle-particle pair distribution.

∆φ accounts for the total variation of the property φ during the collision:

∆φ = φ + p1 -φ p1 + φ + p2 -φ p2 (32)
δφ is the variation of φ for the particle p 1 :

δφ = φ + p1 -φ p1 (33)
In order to close the collision integrals (equations 30 and 31), we need to give an expression for the single particle number density function f . Assuming an uncorrelated particle motion, [START_REF] Kolehmainen | Eulerian modelling of gas-solid flows with triboelectric charging[END_REF] and [START_REF] Ray | An Euler-Euler model for mono-dispersed gasparticle flows incorporating electrostatic charging due to particle-wall and particle-particle collisions[END_REF] proposed the following model:

f (2) = g 0 f ( x p1 , c p1 , ξ p1 , t) f ( x p2 , c p2 , ξ p2 , t) (34)
Where g 0 is the radial distribution function. f ( x p , c p , ξ p , t) is given by an uncorrelated Maxwellian and Gaussian distribution for the velocity and the charge respectively:

f = 1 (2πQ) 1 /2 n p (2πΘ) 3 /2 e - ξ p Q e -c 2 Θ ( 35
)
In our study, will will not assume this form for the number density function, instead we will show that the electric charge part does not have to be presumed a priori. Using the definition of the probability density function, we have:

∞ -∞ φ p1 f (2) dξ p1 dξ p2 = φ p1 | x p1 , c p1 , x p2 , c p2 f * (2) (36)
Now we are going to suppose that the property of the first particle is not correlated with the presence of the second colliding particle, therefore:

φ p1 | x p1 , c p1 , x p2 , c p2 = φ p1 | x p1 , c p1 (37)
To take into consideration the correlation between the property φ and the particle velocity, we chose a linear model of the form:

φ p1 | x p1 , c p1 = φ p1 + B j c p,j (38)
Where the coefficient B j is given, so the first two statistical moments are satisfied:

B i = R -1 p,ij φc p,j (39)
Which, in a hydrodynamic isotropic model, simplifies to:

B i = φc p,i 2 /3q 2 p (40) Now, f * (2) = f * (2) ( x p1 , c p1 , x p2 , c p2 , t
) is the particleparticle velocity distribution, which does not depend on the electric charge of the particles. This function can be modeled using the same methodology as the one shown above:

f (2) = g 0 f * ( x p1 , c p1 , t) f * ( x p2 , c p2 , t) (41)
Here, we chose to close the probability distribution using a Maxwellian distribution:

f * = n p (2πΘ) 3 /2 e -c 2 Θ ( 42
)
With this model, the collisions terms can be fully solved.

Charge transport equation

If we now use φ = ξ p in the equation 28, we find the following expression for the charge transport equation:

n p ∂Q p ∂t + n p U pi ∂Q p ∂x i + ∂n p ξ p c pi ∂x i = C (ξ p ) (43)
From this equation, two terms need to be closed: the last term on the left hand side accounts for the correlation between the charge and the velocity and the right hand side represents the mean rate of change for the charge due to collisions.

Due to the charge conservation law, it can be shown that the source term of the collision integral vanishes:

χ (ξ p ) = 0 (44)
Using the methodology described above, the flux term can be computed:

θ i (ξ p ) = d 3 p βE i g 0 n 2 p q 2 p 9 /10 Υ (1.1) -d 4 p β γ ∂Q p ∂x i g 0 n 2 p q 2 p 9 /10 Υ (2.1) + g 0 d 5 p ∂U t ∂x j β γ ∂Q p ∂x l q 2 p 2 /5 n 2 p Ψ (2.1) tlji Υ (2.2) + d 3 p β γ B i g 0 n 2 p q 2 p 7 /5 Υ (2.3) -Υ (2.4) d 4 p g 0 ∂U t ∂x j β γ B l n 2 p q 2 p 9 /10 Ψ (2.4) ltji Υ (2.4) (45)
Where Υ (•) are constants, and

Ψ (•)
abcd are 4-order constant tensors.

If we insert this, into the collision term definition, and we neglect any term proportional to ∂U /∂x, we get:

C (ξ p ) = - ∂ ∂x i (σ p E i ) + ∂ ∂x i κ coll p ∂Q p ∂x i (46) κ coll p = d 4 p β γ g 0 n 2 p q 2 p 9 /10 2 57 /10 5 3 19 /10 7 Γ 12 5 Γ 3 2 (47)
σ p = d 3 p βg 0 n 2 p q 2 p 9 /10 2 57 /10 5 3 19 /10 7 Γ 3 2 Γ 12 5 (48)
Where Γ (•) is the gamma function. κ coll p is the collisional dispersion coefficient and it accounts for the particle charge dispersion due to particle-particle collisions. σ p is the triboconductivity coefficient and it represents the charge transport generated by the electric field when two particles are in contact.

The second term in the charge transport equation (equation 43) that needs to be closed is the charge-velocity correlation c p ξ p . To accomplish this, we will write a transport equation for the correlation between the particle velocity and charge. Therefore we set φ = ξ p c p,i in the Chapman-Enskog equation (equation 28):

Charge-velocity correlation

n p D ξ p c pi Dt + ∂n p ξ p c pi c pj ∂x j + n p c pi c pj ∂Q p ∂x j -n p F i ξ p +n p c pk ξ p ∂U pi ∂x k = C ξ p c p,i -Q p C c p,i (49)
In order to find a first approximation for the chargevelocity correlation, we will make a series of assumptions to simplify the equation:

1. Steady-state ( D /Dt = 0) 2. No-velocity gradient ( ∂U /∂x = 0)
3. We neglect the third order momentum ξ p c p,i c p,j

Electric charge does not change during a collision

With this assumptions the charge-velocity transport equations reduces to:

n p c pi c pj ∂Q p ∂x j = C ξ p c p,i -Q p C c p,i (50)
Again the collision term can be fully computed:

C ξ p c p,i -Q p C c p,i = - 2 13 /2 π 1 /2 3 5 /2 d 2 p g 0 n p η q 2 p 3 /2 R -1 p,ij n p c pj ξ p (51)
Hence the final transport equation becomes:

n p c pi c pk ∂Q p ∂x k = - 2 13 /2 π 1 /2 3 5 /2 d 2 p g 0 n p η q 2 p 3 /2 R -1 p,ij n p c pj ξ p (52)
If we assume a hydrodynamic isotropic flow:

n p 2 3 q 2 p ∂Q p ∂x j = - 2 11 /2 π 1 /2 3 3 /2 d 2 p g 0 n p η q 2 p 1 /2 n p c pj ξ p ( 53
)
With this model, we can derive a gradient model for the charge-velocity correlation:

n p ξ p c p,i = -κ kin p ∂Q p ∂x i ( 54
)
κ kin p = 3 1 /2 2 9 /2 π 1 /2 ηd 2 p g 0 q 2 p 1 /2 (55)
Where

η = 1 /2 (1 + e c )
Inserting this into the charge transport equation, we obtain:

n p ∂Q p ∂t + n p U pi ∂Q p ∂x i = - ∂ ∂x i (σ p E i ) + ∂ ∂x i κ coll p + κ kin p ∂Q p ∂x i (56)
As we can observe, we have derive a transport equation for the electric charge. The effects of adding taking into consideration of the charge velocity correlation is much clearer now. The term κ kin p is the kinetic dispersion coefficient and it accounts for the dispersion phenomenon due to the random motion of particles. This terms is similar to the one found by [START_REF] Ray | An Euler-Euler model for mono-dispersed gasparticle flows incorporating electrostatic charging due to particle-wall and particle-particle collisions[END_REF]. However we have shown that it can be derived using less restrictive hypothesis on the particle density distribution.

Collisional and kinetic dispersion coefficients

The kinetic theory has shown that the particle charge disper sion can be explained as the contribution of two independent phenomena When two particles collides, there is a charge transfer that takes place. This is characterize by a collisional dispersion coefficient (,ç 0011 ). This coefficient is proportional to the radial distributio/function go, which tends to infinity as the solid fraction volume (a p = n p V p ) goes to the maxi mum compacity. This is coherent, because the more particles are in the flow, the more probable is a collision to happen, and therefore the particle charge is transported faster.

The second contribution to the dispersion coefficient is the kinetic contribution K kin _ This can be explained as the parti cle charge diffusion d�e to random motion of particles. This coefficient is important in dilute system, because gon p -+ O. This is because a particle is able to travel further where there are few other particles.

Figure 1 shows the value of the two dispersion coefficient as a function of a p , for 8 = 0.01 m 2 /s 2 • As we can see for dilute system the kinetic contribution is the most important, and for dense system the collisional terms is the dominant one. Also is worth noting that for an intermediate value of a P both terms have the same order of magnitude and therefore both have to be considered To see the effect of these dispersion coefficients, we are going to study one of the test cases proposed by [START_REF] Kolehmainen | Eulerian modelling of gas-solid flows with triboelectric charging[END_REF]. He studied a 3D peri odic box of 192d p x 8<4, x 8d p . Initially, the particles at x < 96d are charged positively Q P = Qo and the particles p at x � 96<4, are charged negatively Q P = -Qo We are gomg to neglect all the extemal forces (gravity, drag force, elec trostatic, etc.). An initial granular temperature is imposed, and it remains constant during the simulations. We are also going to drop the triboconductivity effect With these hy potheses, the charge transport equation can be simplified to a one-dimensional diffusion equation:

This equation can be solved analytically:

oo coll ün
(2 )

_ "P +"p ( 2ffn ) 2 t , 7rnX Q -� \ e np L Slll -- p -�An L n=l Àn = 2 Qo (1 -(-lf ) n7r (58) (59)
Where L = 192<4, is the box length in the x direction.

This equation allows to study the evolution of the electric charge in function of time. For the simulation, we set ail the values specified in table 1 and a P = 0.25 and we plot the particle charge spatial profile for different values of the non-dirnensional time t* = ( vB/ dp) t (Figure 2). As we can see the electric charge diffuses inside the domain as the time passes, and it tends to reach the equilibrium value Q p = O. A more interesting analysis can be performed if we sepa rate the kinetic and collisional contribution in the dispersion coefficient Figure 3, 4 and 5 show the particle charge profile at t* = 4000 for a dense system (o: p = 0.45) a dilute sys tem (o: p = 0.10) and an intermediary system (o: p = 0.25). The squares markers represent the dispersion created by the kinetic contribution, the circle markers account for the col lisional contribution, and the solid line is the sum of both contributions. As we can see, for dilute systems, the disper sion cornes almost exclusively from the kinetic contribution of the dispersion coefficient. On the contrary, for dense sys tem the collisional terms is the dominant parameter. How ever, we can see that for intermediate values, both coefficient are of the same order of magnitude, they both need to be taken into account in order to accurately predict the disper sion phenomenon. ---'-----'----'--= same parameters as before, we can determine the dynarnic evolution of the electric charge (figure 6).As we can see, the electric charge reaches the equilibrium value faster, which confirms that fact that the triboconductivity helps to the re distribution of the electric charge. However, it is worth noting that the triboconductivity effect seerns to be more important than the diffusion process, this was also reported in the liter ature [START_REF] Kolehmainen | Eulerian modelling of gas-solid flows with triboelectric charging[END_REF]. -1 '-----'-----'---'--'-----'---- In order to verify this, we are going to rewrite the equa tion 60, so we make appear the characteristic times for the dispersion (7 ,. ) and the triboconductivity Tu. Taking las the diffusion characteristic length, we have: If we chose l = L, then we can represent them as func tion of the solid volume fraction (figure 7). We remark that the triboconductivity characteristic time is much smaller than the dispersion characteristic time for alrnost ail values of a P . It is only in very dilute (where there are almost no collisions) cases where the kinetic diffusion process is more efficient This can be shown if we perform a simple analysis between the characteristic times of the dispersion and triboconductiv ity phenomena:

0.5 •••••••••••••••••••••••••••••••••••••• u ...... .
10•�-�-�-�--�-�-�-� 10 -6�-�-�-�--�-�-�-� 0.1 0.2 0

Pv T/9o a P qP

Where PP is the particle density.

(

) (68) 67
This shows, that for dense systems, the dispersion effect is only comparable to the triboconductivity when the disper sion characteristic length is of the same order of magnitude as the particle diameter. For dilute systems, the ratio between the characteristic times is more complex, because it depends on the particle physical properties (Young's module and den sity). However, we can say that when a P -+ 0 the kinetic dispersion is dominant, because there are not enough colli sion for the triboconductivity to take place.

Conclusions

An eulerian model for the electric charge transport has been derived following the kinetic theory of granular flows. As surning a correlated form between the electric charge and the particle velocity, we have been able to fully close the collision integrals without having to presume the expression for the particle electric charge distribution. This model have shown the main mechanisms involved in the electric charge dynarnics. First of ail, the dispersion phenomenon can be classified in two: a collisional dispersion that accounts for the charge transfer during a particle-particle collision, and a kinetic dispersion that due to the random motion of the par ticles. For both types of dispersion an expression was fully derived. We have shown that the collisional dispersion coef ficient is predorninant in dense regimes and that the kinetic dispersion coefficient is is the most important in dilute con figurations. There is, nevertheless, an intermediary region where both coefficient have to taken into account in order to accurately predict the dispersion effect

The second mechanism responsible for the electric charge transfer, is the triboconductivity effect. The global charge distribution can create a electric potential difference. When two particles touch each other, they behave like a conduc tor material and therefore the electric charge flows following the potential gradient. In order to characterize which effect is more important between the dispersion and the tribocon ductivity, we have derive their characteristic times. These parameters have allowed us to show that, for dense regimes, the triboconductivity and the dispersion are of the same order of magnitude if the characteristic dispersion length is compa rable to the particle diameter. For dilute regimes, the expres sion is more complicated. It depends on the particle geomet rical and physical properties, solid volume fraction and the particle agitation.

Figure 1 :

 1 Figure 1: Collisional and kinetic dispersion coefficients as a function of the solid volume fraction for 8 = 0.01 m 2 /s 2

'Figure 2 :

 2 Figure 2: Particle charge profile as a function of x/ Lat dif ferent times t* = ( ,/aja.p) t

Figure 3 :Figure 4 :8x

 34 Figure 3: Particle charge profile at t* = 4000 for aP = 0.45

Figure 5 :

 5 Figure 5: Particle charge profile at t* = 4000 for aP = 0.25

Figure 6 :

 6 Figure 6: Particle charge profile as function of x/ Lat differ ent times t* = ( ,/ej dp) t

Figure 7 :T

 7 Figure 7: Triboconductivity and dispersion characteristic times in function of solid volume fraction for e = 0.01 m 2 /s 2

Acknowledgments

This work was supported by the ANR-IPAF project, grant ANR-16-CE06-0008 of the French National Agency of Research (ANR); and by the by the program "Chaire d'attractivité AAP 2015 de l'Idex UNITI-BIREM", grant conv-ANR-l l-ldex-0002-02.


Youssef Nasro-Allah
email: youssef.nasroallah@inp-toulouse.fr

Carlos Montilla

Renaud Ansart

Rodney Fox

Olivier Simonin

Numerical and Experimental study of electrostatic charge in gas-solid fluidized beds

Keywords: fluidized beds, tribocharging, particle-fluid flow

Electrostatic charges have been one of the major issues in gas-solid fluidized beds, including wall fouling due to particle accumulation on the walls, defluidization, security issues like sparks and dust explosions, etc. At a molecular scale, the particle-wall or particle-particle contact generates electrons transfer, inducing a charge on each particle. As a result, the surrounding gas carries an electromagnetic field which results in an electromagnetic force known as Lorentz force. The aim of this work is to model this force and take its contribution into account in the momentum equation. In this work, a tribocharging model coupled with the multi-fluid model is proposed in an Eulerian approach to model the phenomenon. The numerical simulations will be carried with NEPTUNE_CFD to test the model. An experimental setup is build to form an accurate database for comparison with simulations. It will also help to adjust the model parameters.

Introduction

With polymer reactors, electrostatic charge generation can cause the formation of larger granules, as well as reactor wall fouling, which is the formation of layers of the particles on the reactor wall. [START_REF] Hendrickson | Electrostatics and gas phase fluidized bed polymerization reactor wall sheeting. Chemical Engi-(a) Diagram of the experimental setup (b) The experimental setup and the humidity generator[END_REF] reviewed electrostatic effects in polymerization fluidized-bed reactors and the causes of reactor fouling, explained the charge distribution in fluidized beds and compared the effect of electrostatic forces with other forces like drag and, gravity on particle entrainment. [START_REF] Sowinski | Effect of fluidizing particle size on electrostatic charge generation and reactor wall fouling in gas-solid fluidized beds[END_REF] studied the effect of particle size of a polyethylene resin received directly from industrial reactors on electrostatic charge generation and reactor wall fouling using a Faraday cup.

Regarding the relative humidity effect, Fotovat et al. ( 2016) performed experiments at different values of relative humidity using a mixture of glass beads as coarse particles and fine particles of different materials. These experiments aimed to study the relationship between fluidized bed electrostatic and entrainment.

On the numerical side, there was many models proposed to describe the phenomenon. [START_REF] Rokkam | Computational fluid dynamics and electrostatic modeling of polymerization fluidized-bed reactors[END_REF] developed an electrostatic model based on basic laws describing electromagnetic phenomena within an Eulerian approach. For tribocharging phenomenon, [START_REF] Lindell | Electrostatic image method for the interaction of two dielectric spheres[END_REF] analyzed the interaction of two dielectric spheres to find their reaction to external sources based on a static theory image. On the other hand, some authors proposed models based on the surface state theory [START_REF] Ali | Charge exchange model of a disperse system of spherical powder particles[END_REF], [START_REF] Laurentie | Discrete element modeling of triboelectric charging of insulating materials in vibrated granular beds[END_REF]). [START_REF] Kolehmainen | Triboelectric charging of monodisperse particles in fluidized beds[END_REF] built a Lagrangian model on the same works and tested it on a miniaturized fluidized bed in a DEM simulation. More recently, [START_REF] Kolehmainen | Eulerian modelling of gas-solid flows with triboelectric charging[END_REF] proposed the same model in an Eulerian approach.

Numerical Methods

Building on the well known Maxwell equations for electromagnetic field, a Poisson equation is solved for the electric potential :

∇ • (ε m ∇ϕ) = - N β=1 q p β α p β ε 0 (1)
where α g is the gas volume fraction, α p β is the volume fraction on the β th solid phase, N is the solid phase count in the mixture, ε m and ε 0 are the dielectric permittivity of the mixture and the vacuum respectively. ϕ is the electric potential. The walls are assumed to be grounded (ϕ = 0). Note that the effect of magnetic field is neglected here.

The gradient of the potential results in an electric field E:

E = -∇ϕ (2)
This electric field induces an electrostatic force F qp β :

F qp β = -q p β α p β ∇ϕ (3)
Numerical simulations are carried with NEPTUNE_CFD, an unstructured code using unsteady Eulerian multi-fluid approach for dilute and dense particle-laden reactive flows [START_REF] Hamidouche | Numerical Simulation of Multiphase Reactive Flows[END_REF]). The coupling between the electrostatic model and the multi-fluid was verified by simulating experimental results of [START_REF] Sowinski | Investigation of electrostatic charge distribution in gas-solid fluidized beds[END_REF]. The weakness of this model is that charges are fixed on particles, which does not take into account the tribocharging phenomenon.

The triboelectric charging mechanism assumes that the charge carried by the solid phase to the following transport equation:

α p ρ p ∂χ c p ∂t + ∂ ∂x i α p ρ p U p,i χ c p = ∂ ∂x i α p ρ p D χ ∂χ c p ∂x i +ψ p,q
(4) Where χ c p = <qp> mp is the average charge per mass unit. α p and ρ p are, respectively, the volume fraction and the density of the p th phase. U p,i is the i th component of the p th phase velocity. D χ is the diffusion coefficient. ψ p,q is a source term representing the charge generation due to collision between particles of different solid phase. The transfer between particles inside the same solid phase is included in the diffusion coefficient.

The boundary conditions are expressed by a charge flux from the wall towards the particles. This flux represents the charge generation due to the particle-wall collision.

Experimental setup

The experiments are performed on a laboratory scale pilot of 0.1 m inner diameter and 1 m height Plexiglass column shown in Fig. 1b. A diagram representing this setup is shown in Fig. 1a. The electrostatic charges are measured by means of two Faraday cups related to an electrometer. The distributor plate was welded in a valve related to an actuator to allow particles to drop into the Faraday cup. Entrainment flux is followed up via a box at the end of the cyclone. The box contains a Faraday cup placed on a balance to follow the evolution of charge and entrainment flux simultaneously. This pilot is made of the same material (Plexiglass) starting from the distributor to the end of the cyclone in order to have the same effect of walls on particles.

The humidity effects is studied by means of a humidity generator presented in Fig. 1b on the right. This generator ensure a constant value of the relative humidity in a range up to 40% without heating the system. It is connected to the compressed air supplier.

The experimental work can be divided in two main parts. The first one is the study of bubbling fluidized bed. The net charge of the particles will be measured after each experiment. This aims to compare to literature findings and to estimate the saturation charge of the particles. The second part is the study of entrainment flux. The entrained mass and charge evolution will be measured at the end of the cyclone. This allows to study the effect of electrostatic charges on entrainment flux. Both parts will be performed at different relative humidity values to highlight its effects. These experiments will provide an accurate database to adjust the tribocharging model parameters depending on relative humidity and particles properties (size, material).