Siddharth Rajupet

Soleiman Bourrous

Francois Gensdarmes

Mamadou Sow
email: mamadou.sow@irsn.fr

Coulombic force contribution to nano scale aerosol capture by a wire grid: Quantitative comparison of experiments and simulations

Fibers of filter media and aerosol particles both typically have some distributions of electric charge. Attractive coulombic interactions between charged particles and fibers enhance aerosol filtration efficiency; however, these coulombic interactions are difficult to characterize due to the complex, nonuniform fiber charge, diameter and spacing of typical filter media. We develop an experimental filter assembly to control these coulombic interactions. The filter assembly consists of three sequential electrically isolated metallic grids-high voltage is applied to the middle grid and the adjacent outer grids are electrically grounded such that the electric potential distribution in the filter assembly is well-defined. We test this filter assembly with aerosols of controlled diameter (ranging between ~70 -500 nm) and electric charge (1e -), and systematically increase the applied voltage to the middle grid to enhance coulombic interactions and consequently the filter efficiency. We develop particle trajectory simulations to model our experimental filter assembly such that all parameters in the simulation have direct basis from the experimental system. We find excellent quantitative agreement between the filter efficiency measured in experiments and predicted from simulations without any fitting parameters. The precise control and accurate simulation of coulombic interactions demonstrated in this study indicate that the experimental and simulation methodologies developed here may be applied to uncover fundamental insights into the role of coulombic forces in more complex phenomena such as filter clogging.

I. INTRODUCTION

The filtration of aerosols is of significant interest in a number of environmental, health, and safety applications including mitigating indoor and outdoor air pollution [START_REF] Vohra | Global mortality from outdoor fine particle pollution generated by fossil fuel combustion : Results from GEOS-Chem[END_REF], reducing the transmission of respiratory viruses [START_REF] Anderson | Consideration of the Aerosol Transmission for COVID-19 and Public Health[END_REF][START_REF] Cowling | Aerosol transmission is an important mode of influenza A virus spread[END_REF], and preventing dissemination of airborne hazardous materials [START_REF] Kaneyasu | Sulfate aerosol as a potential transport medium of radiocesium from the fukushima nuclear accident[END_REF][START_REF] Malá | Particle size distribution of radioactive aerosols after the Fukushima and the Chernobyl accidents[END_REF][START_REF] Povinec | Aerosol radioactivity record in Bratislava/Slovakia following the Fukushima accident -A comparison with global fallout and the Chernobyl accident[END_REF]. We are particularly interested in the filtration of radioactive aerosols, which may be produced from fires, liquid leaks, or general dismantling processes of nuclear plants and laboratories [START_REF] Halverson | Combustion aerosols formed during burning of radioactively contaminated materials: Experimental results[END_REF][START_REF] Sow | Aerosol release fraction by concrete scarifying operations and its implications on the dismantling of nuclear facilities[END_REF][START_REF] Sow | Experimental study of aerosol release following liquid leaks of fission products concentrates simulants[END_REF].The efficiency of filters at capturing aerosols depends on the aerosol size and typically, filters are least effective at capturing aerosols with diameters between ~100-300 nmthis size range is termed the maximum penetrating particle size (MPPS). Aerosol particles within this size range are particularly hazardous to health since they penetrate deep into human respiratory pathways and disseminate far from their source.

Aerosols and fibers of filters typically have a distribution of charge, and attractive electrostatic interactions between particles and fibers are well-known to increase filtration efficiency, particularly within the MPPS range [START_REF] Alonso | Deposition of ultrafine aerosol particles on wire screens by simultaneous diffusion and image force[END_REF][START_REF] Kerner | Evaluation of electrostatic properties of electret fi lters for aerosol deposition[END_REF][START_REF] Thakur | Electret Air Filters[END_REF]D. C. Walsh & Stenhouse, 1997a, 1997b, 1998;[START_REF] Walsh | Experimental Studies of Electrically Active Fibrous Filter Loading[END_REF][START_REF] Zhao | Household Materials Selection for Homemade Cloth Face Coverings and Their Filtration Efficiency Enhancement with Triboelectric Charging[END_REF] where other aerosol capture mechanisms like diffusion and impaction are less effective. These interactions include coulombic interactions, where charged fibers attract charged particles, dielectrophoric interactions, where charged fibers polarize particles, and image forces, where charged particles polarize fibers. Improving filter efficiency by incorporating electrostatic interactions is particularly advantageous since the efficiency is improved without increasing the pressure drop of the filter. As a result, the fabrication of filters typically includes steps to impart electrostatic charge and several recent studies have sought to enhance electrostatic interactions by applying external electric fields during filtration to directly charge, or polarize fibrous filter media [START_REF] Choi | Al-Coated Conductive Fiber Filters for High-Efficiency Electrostatic Filtration: Effects of Electrical and Fiber Structural Properties[END_REF][START_REF] Choi | Al-Coated Conductive Fibrous Filter with Low Pressure Drop for Efficient Electrostatic Capture of Ultrafine Particulate Pollutants[END_REF][START_REF] Han | Electrically Activated Ultrathin PVDF-TrFE Air Filter for High-Efficiency PM1.0 Filtration[END_REF][START_REF] Han | Advances in air filtration technologies: structure-based and interaction-based approaches[END_REF][START_REF] Jeong | High Efficiency, Transparent, Reusable, and Active PM2.5 Filters by Hierarchical Ag Nanowire Percolation Network[END_REF][START_REF] Tian | Toward energy saving and high efficiency through an optimized use of a PET coarse filter: The development of a new electrostatically assisted air filter[END_REF][START_REF] Tian | Ultralow Resistance Two-Stage Electrostatically Assisted Air Filtration by Polydopamine Coated PET Coarse Filter[END_REF][START_REF] Zhang | High-performance particulate matter including nanoscale particle removal by a selfpowered air filter[END_REF][START_REF] Zhang | Electrostatic polyester air filter composed of conductive nanowires and photocatalytic nanoparticles for particulate matter removal and formaldehyde decomposition[END_REF][START_REF] Zhao | Charged graphene aerogel filter enabled superior particulate matter removal efficiency in harsh environment[END_REF].

The role of electrostatic effects on the efficiency of a clean filter has been evaluated theoretically by analytically determining the trajectories of particles around a single fiber while accounting for electrostatic forces [START_REF] Kraemer | Collection of Aerosol Particles in Presence of Electrostatic Fields[END_REF][START_REF] Natanson | Deposition of aerosol particles by electrostatic attraction upon a cylinder around which they are flowing[END_REF][START_REF] Zebel | Deposition of aersol flowing past a cylindrical fiber in a uniform electric field[END_REF]. However, rigorous quantitative comparison of the electrostatic filtration efficiency between theory and experiments has never been achieved due to the complex structure of industrially manufactured filters. Traditional non-woven filters have a distribution of fiber diameters and inter-fiber spacings. Furthermore, experimentally characterizing the charge on fibers of filters through direct methods is challenging-this charge is non-uniform with different fibers having different polarities of charge [START_REF] Kilic | Measuring electrostatic properties of fi brous materials : A review and a modi fi ed surface potential decay technique[END_REF][START_REF] Thakur | Spatial charge distribution in fi brous dielectrics[END_REF]. Instead, a more common method to estimate the charge on fibers is to measure the initial efficiency of a filter and fit the fiber charge based on analytic theory [START_REF] Baumgartner | The collection performance of electret filters in the particle size range 10 nm -10 um[END_REF][START_REF] Kim | Initial Collection Performance of Resin Wool Filters and Estimation of Charge Density[END_REF][START_REF] Lathrache | Fractional Penetrations for Electrostatically Charged Fibrous Filters in the Submicron Particle Size Range[END_REF][START_REF] Lathrachi | Deposition of submicron particles on electrically charged fibers[END_REF][START_REF] Lee | Prediction of collecion efficiency of highperfromance electret filters[END_REF][START_REF] Romay | Experimental Study of Electrostatic Capture Mechanisms in Commercial Electret Filters[END_REF][START_REF] Thomas | An improved model for the penetration of charged and neutral aerosols in the 4 to 80nm range through stainless steel and dielectric meshes[END_REF]D. C. Walsh & Stenhouse, 1997a, 1997b, 1998;[START_REF] Walsh | Experimental Studies of Electrically Active Fibrous Filter Loading[END_REF][START_REF] Wu | Deposition of Particles on a Single Cylinder by a Coulombic Force and Direct Interception[END_REF] or particle trajectory simulations [START_REF] Kerner | Numerical and experimental study of submicron aerosol deposition in electret microfiber nonwovens[END_REF][START_REF] Kerner | Evaluation of electrostatic properties of electret fi lters for aerosol deposition[END_REF]. This method, however, precludes the rigorous experimental validation of theory describing electrostatic effects in filtration.

A method to circumvent the difficulty in characterizing filter charge is to directly apply a potential to a conductive filter. In prior studies, traditional nonwoven filters were coated with aluminum and the filtration efficiency was enhanced by increasing the voltage applied to the filter [START_REF] Choi | Al-Coated Conductive Fiber Filters for High-Efficiency Electrostatic Filtration: Effects of Electrical and Fiber Structural Properties[END_REF][START_REF] Choi | Al-Coated Conductive Fibrous Filter with Low Pressure Drop for Efficient Electrostatic Capture of Ultrafine Particulate Pollutants[END_REF]. However, even when the potential of the fibers is controlled, the electric field distribution (and consequently the coulombic forces acting on the particles) near the filter is difficult to characterize due to non-uniformity in fiber size, orientation, and spacing in the filterthese factors strongly influence the electric field distribution when constant potential is applied. Such complexities make it difficult to quantitatively predict the contribution of electrostatic mechanisms to the global efficiencies of real fibrous filters using simulations.

Here we develop novel experimental and simulation methodologies through which we can quantitatively-and non-empirically-compare electrostatic filtration efficiencies in experiments and fundamental particle trajectory simulations, enabling rigorous understanding of the basis of electrostatic mechanisms on filtration performance. In our experiments, we carefully control the properties of both the aerosol particles and the filter media. To control the filter properties, we use a metallic grid with uniform fiber diameter and spacing as the filter and we control the fiber charge by applying a potential to this metallic grid such that the electric field distribution in our system is always well-defined. We then develop particle trajectory simulations based on the fundamental forces acting on particles as they flow past an individual fiber representative of a fiber in the highvoltage metallic grid. All parameters in our simulation have a direct basis from our experimental system i.e. no fitting parameters are used. Finally, we compare the electrostatic filtration efficiency measured in our experimental bench to that determined from the simulations.

II. METHODS

A. Experiments

To rigorously investigate the effect of electrostatic forces in aerosol filtration, we develop an experimental system in which we control the size and charge of aerosol particles as well as the size, spacing, and charge of fibers in our filter media. We then evaluate how the aerosol penetration through the filter (i.e. the fraction of aerosols passing through and not captured by the filter) varies with the electrostatic properties of the aerosol particles and filter.

We generate a monodisperse aerosol stream with a controlled charge, by first aerosolizing a solution of monodisperse polystyrene latex (PSL) particles (unless otherwise specified) using a constant output collision type atomizer (TSI Model 3076). After generation, the aerosol droplets pass through a diffusion dryer to evaporate the liquid to form a dry aerosol. The dried aerosol stream has a stable number concentration and size distribution. Then, we use a differential mobility analyzer (DMA) (TSI Electrostatic Classifier Model 3082) to select only the monodisperse PSL particles carrying one elementary charge in the aerosol stream by precipitating other particles based on their electrical mobility. Only particles within a narrow range of electrical mobilities pass through the DMA such that the particles exiting the DMA are monodisperse and are charged with 1e -(the particles have negative polarity). Due to the narrow size distribution of particles generated in our system and the relatively small particle sizes considered in this study, the fraction of particles exiting the DMA with multiple elementary charges is negligible. The schematic of the experimental system is depicted in Fig. 1.

We develop a filter with well-characterized geometry and electrostatic potential, as depicted in Fig. 2. The filter assembly consists of three identical metallic grids in series, which are electrically insulated from each other using polyether ether ketone (PEEK) spacers. The fibers of the metallic grids have a diameter of Df = 100 µm and edge-to-edge spacing of h = 250 µm. The PEEK spacers are annuli with inner diameter of 35 mm, outer diameter of 47 mm and thickness of 11 mm such that the metallic grids are separated from each other by d = 11 mm and the filter cross sectional diameter exposed to the aerosol stream is 35 mm. The middle grid is connected to a positive highvoltage supply and the outer two grids are connected to electrical ground such that the electric potential in the filter assembly has well-defined boundaries. The electric field within the assembly is oriented such that particles are attracted to fibers of the center grid both when particles are upstream and downstream of the center grid. The filter assembly is placed inside a standard conical metallic 47 mm filter holder which is connected to electrical ground. The monodisperse, singly charged aerosol flows through the filter assembly and the aerosol concentration after the filter assembly is measured by a Condensation Particle Counter (CPC) (TSI Model 3775). The average airflow face velocities examined ranged between 0.6 -5 cm/s and in our analysis was considered to be uniform over the face of the grids. Thus, we do not account for nonuniform inlet fluid velocities which may arise in experiments due to undeveloped flow profiles or edge effects at the perimeter of the filter holder. However, due to the small fluid velocities tested (<5 cm/s) we do not expect these effects to be significant. We measure the aerosol concentration when no voltage difference is applied between the center and outer grids of the filter assembly, CV=0. Then, we apply a voltage difference between the center and outer grids of the filter assembly and measure the aerosol concentration, CV>0.

We test experimental conditions which help isolate coulombic effects. In particular, we choose particle sizes within the MPPS regime (~70 -500 nm) and large fiber diameters (100 µm) such that mechanical capture mechanisms, i.e., diffusion, interception, and inertia, are less significant and the primary contributor to the aerosol capture efficiency is attractive coulombic forces between the negatively charged particles and positively charged metallic grid. To further distinguish the role of electrostatic forces from mechanical capture, we assume mechanical and coulombic mechanisms of particle capture are entirely independent (i.e. CV=0 = 𝑃 𝑚𝑒𝑐ℎ 𝑒𝑥𝑝 Co and CV>0 = 𝑃 𝑚𝑒𝑐ℎ 𝑒𝑥𝑝 𝑃 𝑐𝑜𝑢𝑙 𝑒𝑥𝑝 Co where Co is the constant aerosol concentration upstream the grid, and 𝑃 𝑚𝑒𝑐ℎ 𝑒𝑥𝑝 and 𝑃 𝑐𝑜𝑢𝑙 𝑒𝑥𝑝 are the penetration of particles through the metallic grids due to mechanical and coulombic interactions respectively). If 𝑃 𝑚𝑒𝑐ℎ 𝑒𝑥𝑝 is the same with and without voltage applied, the ratio 𝑃 𝑐𝑜𝑢𝑙 𝑒𝑥𝑝 =CV>0/CV=0 represents the penetration of particles through the metallic grids only due to coulombic interactions. The physical interpretation of 𝑃 𝑐𝑜𝑢𝑙 𝑒𝑥𝑝 assumes that mechanical and coulombic mechanisms of particle capture are independent; however, given the little prevalence of mechanical capture in our system, we expect error due to this assumption to be insignificant.

B. Simulation

We simulate the trajectories of particles based on the fundamental forces acting on the particles as they flow past a single fiber and relate the fraction of particles captured by the fiber in our simulation to the total aerosol penetration through a full metallic grid. Our simulation cell is characteristic of a fiber within the center, high-voltage, grid of our experimental filter assembly.

A circular fiber of diameter Df = 100 µm is positioned in the center of a rectangular simulation cell depicted in Fig. 3. The width of the simulation cell is 2d = 22 mm and the height of the simulation cell is Df + h = 350 µm. The simulation is carried out in two dimensions such that the simulation represents particles flowing around infinitely long cylindrical fibers oriented parallel to each other. In simulating the particle trajectories, we consider forces due to the airflow as well as coulombic forces between the particle and fiber. First, we solve the Navier-Stokes equations to determine the laminar fluid flow field and Poisson's equation to determine electrostatic potential field in the system. We model the particles as spheres with diameter Dp with a single elementary charge. The forces due to airflow are then calculated using Stokes law together with the fluid flow field, and the coulombic forces are calculated from the gradient of the electrostatic potential field. We assume the particles have no effect on the fluid flow and potential field and the fluid velocity and electric field at the center of the particle are used to calculate the drag force and coulombic force respectively. Note, random fluctuations in particle position due to Brownian motion are not considered in the simulation.

The Navier-Stokes equations are solved using finite element methods. The fluid is modeled using properties of air and enters through the left boundary of the simulation cell with a uniform velocity profile u. The surface of the fiber has a no-slip boundary condition and the top and bottom of the rectangular cell have periodic boundary conditions.

Poisson's equation is also solved using finite element methods. We model the fiber with a constant surface potential ΔV and the left and right boundaries of the simulation with V = 0, characteristic of the two grounded metallic grids in our filter assembly. The top and bottom of the rectangular cell have periodic boundary conditions and the cell is modeled with dielectric properties characteristic of air. We emphasize that unlike the potential distribution in the constant charged fiber case, the potential distribution in the present system (with constant potential fibers) does not have an analytic solution preventing the use of more classical single fiber theory [START_REF] Natanson | Deposition of aerosol particles by electrostatic attraction upon a cylinder around which they are flowing[END_REF].

We simulate the particle trajectories by integrating Newton's law of motion. We simulate the trajectories of no particles, with no = 10,000. The initial positions of the particles are evenly distributed on the left boundary of the simulation cell and with initial velocities equal to the airflow velocity, u. As the particles move, if the boundary of the particle contacts the fiber surface, the particle is considered captured by the fiber; the number of particles captured is denoted nc. The fraction of particles collected, nc/no, can be compared with the fraction of cross-sectional area that is blocked by the fiber, Df/(Df+h), to give the collection efficiency of the single fiber due to coulombic interactions, ηC,

𝜂 𝐶 = 𝑛 𝑐 𝑛 𝑜 ( 𝐷 𝑓 +ℎ 𝐷 𝑓 ) (1)
Physically, all particles with initial positions within ηC(Df+Dp)/2 of the center of a fiber are captured. From ηC in the simulation, we can estimate a total coulombic penetration by considering fibers arranged in a grid pattern as shown in Fig. 4, similar to that in the experiments. We assume ηC, which is calculated for fibers oriented parallel to each other, does not change for fibers in the grid orientation. Then, the total penetration is the ratio of particles initially positioned further than ηC(Df+Dp)/2 from a fiber center, 𝜌 𝑝 (ℎ + 𝐷 𝑓 -𝜂 𝐶 (𝐷 𝑓 + 𝐷 𝑝 ))

2

, to the total number of particles approaching the filter, 𝜌 𝑝 (ℎ + 𝐷 𝑓 )

2 , where ρp is the number of particles per cross sectional area of the grid. We assume particles approaching the grid have uniformly distributed initial positions such that the penetration is given by,

𝑃 𝑐𝑜𝑢𝑙 𝑠𝑖𝑚 = 𝐶 V>0 𝐶 V=0 = (ℎ+𝐷 𝑓 -𝜂 𝐶 (𝐷 𝑓 +𝐷 𝑝 )) 2 (ℎ+𝐷 𝑓 ) 2 (2)
This simulated columbic penetration can be directly compared to that obtained from experiments.

The COMSOL Multiphysics software was used to solve the Poisson and Navier-Stokes equations as well as to integrate the equations of motion for particle trajectories. All parameters used in the simulation are listed in Table 1.

III. RESULTS AND DISCUSSION

Figure 5 shows the experimental results for Pcoul through the filter assembly as a function of the applied potential difference between the high-voltage grid and grounded grids, the airflow velocity at the face of the grid, and the particle diameter. We note that in all data presented, aerosol particles have one negative elementary charge, the middle metallic grid has a positive applied potential, and the metallic grid has fiber diameter of 100 µm and spacing of 250 µm. We emphasize that all parameters used in our simulation have values taken from our experimental system i.e. no fitting parameters are used.

As shown in Figure 5a, Pcoul decreases as the applied potential difference increases. This effect occurs because the attractive forces between the particles and fibers increase with greater applied potential differences.

As shown in Figure 5b, Pcoul increases as the airflow velocity increases. With higher airflow velocity, the particles spend less time in proximity of the high-voltage grid. Thus, electrostatic forces do not displace particles as significantly toward the fibers and the penetration increases.

As shown in Figure 5c, Pcoul increases as the particle diameter increases. As the particle diameter increases, the drag force on the particle increases, which decreases the drift velocity of the particles towards the fiber. Meanwhile the coulombic attractive force between the particles and fibers remain the same since the particle charge is constant.

The coulombic penetration predicted from simulations agrees with experiments very well at large particle sizes but overestimates the penetration as the particles size decreases. Our analysis below suggests a reason for this overestimation at small particle sizes. Our simulation methodology employs some simplifying assumptions to model the experimental system. In particular, we do not account for nonuniform airflow velocities. Indeed, the average airflow face velocities examined ranged between 0.6 -5 cm/s and in our analysis was considered to be uniform over the face of the grids. Thus, we do not account for nonuniform inlet fluid velocities which may arise in experiments due to undeveloped flow profiles or edge effects at the perimeter of the filter holder. This is justified by that fact that, due to the small Reynolds number (fluid velocities tested <5 cm/s), the regime is laminar with low inertial effects, thus we do not expect these effects to be significant. Instead, we hypothesize that the discrepancy between experiments and simulation at small particle sizes is a result of molecular slip which is more significant for small particles and is not accounted for in the simulation.

To characterize the significance of molecular slip in our system, we examine the single fiber efficiency since it is a more general quantity that is normalized with respect to the filter thickness. We estimate the experimental coulombic single fiber efficiency by rearranging Eq. 2. In Fig. 6, we compare the experimental and simulated single fiber efficiencies for all of the data shown in Fig. 5. In this Figure, we distinguish conditions where the Knudsen number is "high" and "low" by grouping conditions where the Knudsen number is greater than or less than 1 respectively. The Knudsen number,

𝐾𝑛 = 2𝜆/𝐷 𝑝 (3)
where λ, is the molecular mean free path of air (~65 nm), relates the degree to which fluid flow around aerosol particles can be treated as a continuum flow. At larger Knudsen numbers where the particle size is smaller than or on the same order as λ, the continuum flow assumption breaks down as air molecules "slip" past the aerosol particles, decreasing the fluid drag. The experiments and simulation agree more closely at small Knudsen numbers, where flow around particles is better approximated by continuum flow, and Stokes law applies for calculation of the drag force. At larger Knudsen numbers, the simulation, which does not account for molecular slip, overestimates drag forces on aerosol particles causing the drift velocity of particles toward fibers, and consequently the filtration efficiency, to be underestimated in the simulation.

We develop a dimensionless coulomb number, KC, which is the ratio of the drift velocity due to coulomb attraction to the airflow velocity. The drift velocity, vd, is obtained by setting the coulombic force equal to the Stokes drag force 3πµDpvd/C (µ is the viscosity of air) and solving for vd. To account for molecular slip, we include the Cunningham slip correction factor, C. The coulomb force on a particle far from the fibers and between the high-voltage and grounded grids, is qΔV/d (q is the charge on each aerosol particle); since the electric field nearby the fiber is enhanced at smaller Df and larger h, we scale the characteristic coulomb force accordingly: qΔVh/dDf. Therefore, we obtain the coulomb number as:

𝐾 𝐶 = 𝑞∆𝑉ℎ𝐶 3𝜋𝜇𝐷 𝑝 𝑑𝐷 𝑓 𝑢 (4)
Physically, small values of KC signify that particle drift due to electrostatic forces is small relative to fluid velocities causing electrostatic forces to have little effect on the efficiency. Large values of KC indicate that attractive electrostatic forces dictate the trajectories of particles as opposed to fluid drag causing efficiency to be significantly enhanced due to attractive electrostatic effects. A similar coulomb number has previously been developed for the case of a fiber with constant charge density, rather than the present case of a fiber with constant electrostatic potential [START_REF] Natanson | Deposition of aerosol particles by electrostatic attraction upon a cylinder around which they are flowing[END_REF].

Figure 7 depicts the experimental and simulated coulombic single fiber efficiencies as a function of the dimensionless coulomb number described from Eq. 4. Since molecular slip is not accounted for in the simulation, we consider KC with C = 1 for the simulated results (Fig. 7b). On the other hand, since molecular slip affects the experimental results, we account for slip in KC in the experiments (Fig. 7a) using the Cunningham correction,

𝐶 = 1 + 𝐾𝑛(𝛼 + 𝛽 exp (- 𝛾 𝐾𝑛 ) (5)
Where α, β, and γ are 1.257, 0.4, and 1.1 respectively, as is commonly used for spheres in air (C N [START_REF] Davies | Definitive equations for the fluid resistance of spheres[END_REF]. For both experimental and simulation results, we fit ηC to KC using a power-law expression. We find that in both the experiments and simulation, ηC scales very well with KC as indicated by the R 2 values close to 1. Furthermore, we find excellent agreement in the scaling of ηC with KC between the experiments and simulation (power-law exponents of 0.914 and 0.911 respectively) and very good agreement in the magnitude coefficient as well (12.7 and 16.1 respectively). The effect of electrostatic interactions on aerosol filtration performance has long been the subject of study, but rigorous experimental control over electrostatic interactions has been elusive due to difficulty controlling the charge and potential distribution in filter media. More recently, progress has been made in controlling electrostatic properties of filter media, both for the purpose of fundamental investigation of electrostatic filtration effects [START_REF] Alonso | Deposition of ultrafine aerosol particles on wire screens by simultaneous diffusion and image force[END_REF], and for application in active-electrostatically assisted filtration [START_REF] Choi | Al-Coated Conductive Fiber Filters for High-Efficiency Electrostatic Filtration: Effects of Electrical and Fiber Structural Properties[END_REF][START_REF] Choi | Al-Coated Conductive Fibrous Filter with Low Pressure Drop for Efficient Electrostatic Capture of Ultrafine Particulate Pollutants[END_REF][START_REF] Han | Electrically Activated Ultrathin PVDF-TrFE Air Filter for High-Efficiency PM1.0 Filtration[END_REF][START_REF] Han | Advances in air filtration technologies: structure-based and interaction-based approaches[END_REF][START_REF] Jeong | High Efficiency, Transparent, Reusable, and Active PM2.5 Filters by Hierarchical Ag Nanowire Percolation Network[END_REF][START_REF] Tian | Toward energy saving and high efficiency through an optimized use of a PET coarse filter: The development of a new electrostatically assisted air filter[END_REF][START_REF] Tian | Ultralow Resistance Two-Stage Electrostatically Assisted Air Filtration by Polydopamine Coated PET Coarse Filter[END_REF][START_REF] Zhang | High-performance particulate matter including nanoscale particle removal by a selfpowered air filter[END_REF][START_REF] Zhang | Electrostatic polyester air filter composed of conductive nanowires and photocatalytic nanoparticles for particulate matter removal and formaldehyde decomposition[END_REF][START_REF] Zhao | Charged graphene aerogel filter enabled superior particulate matter removal efficiency in harsh environment[END_REF].

Alonso et al. probed the role of electrostatic image forces (where charged particles polarize fibers creating an attractive force) in filtration of nano-scale charged aerosols by using grounded metallic grids as the filter media [START_REF] Alonso | Deposition of ultrafine aerosol particles on wire screens by simultaneous diffusion and image force[END_REF]. This filter media was advantageous since the image force between a charged particle and a grounded wire is well-defined and the uniform size, spacing, and orientation of wires in the metallic grid facilitated consistent image forces throughout the filter media. Similarly, coulomb forces have previously been controlled to a certain extent by applying potentials to complex, nonuniform conductive filter media, yet the complexity of these systems precluded rigorous evaluation these electrostatic interactions.

The filter assembly developed here integrates ideas from these prior works to precisely control coulombic forces-we apply a potential to the filter media and ensure the potential distribution is uniform and well-defined by using a metallic grid with adjacent electrically grounded grids on either side of the high-voltage grid. Our experiments clearly demonstrate that coulombic forces substantially enhance filtration efficiency particularly within the typical MPPS regimes tested in this study.

The simulations of this well-controlled experimental system elucidate key physics governing particle capture within the filter. In particular, the comparison between experiments and simulation at small particle sizes indicate the importance of non-continuum, "slip" effects at high Knudsen numbers. This slip, weakens drag forces on aerosol particles, thereby increasing displacement of particles toward fibers due to coulombic forces and consequently improves the filter efficiency.

Prior works have developed corrections to the drag forces at high Knudsen numbers in effort to address these slip effects [START_REF] Allen | Slip Correction Measurements of Spherical Solid Aerosol Particles in an Improved Millikan Apparatus[END_REF][START_REF] Davies | Definitive equations for the fluid resistance of spheres[END_REF][START_REF] Cunningham | On the velocity of steady fall of spherical particles through fluid medium[END_REF][START_REF] Millikan | The Isolation of an Ion, a Precision Measurement of Its Charge, and the Correction of Stokes's Law[END_REF].

Both experiments and simulations scale well with the coulomb number developed here which is applicable to the constant electrical potential limit of charged fibers (in contrast to the constant charge limit developed in prior work [START_REF] Kraemer | Collection of Aerosol Particles in Presence of Electrostatic Fields[END_REF][START_REF] Natanson | Deposition of aerosol particles by electrostatic attraction upon a cylinder around which they are flowing[END_REF][START_REF] Zebel | Deposition of aersol flowing past a cylindrical fiber in a uniform electric field[END_REF]). Further, we emphasize that this coulomb number is developed for the case where each charged fiber is adjacent to equally spaced electrical grounds, and thus may not be applicable to more general, complex conductive filter media used in prior studies [START_REF] Choi | Al-Coated Conductive Fiber Filters for High-Efficiency Electrostatic Filtration: Effects of Electrical and Fiber Structural Properties[END_REF][START_REF] Choi | Al-Coated Conductive Fibrous Filter with Low Pressure Drop for Efficient Electrostatic Capture of Ultrafine Particulate Pollutants[END_REF].

Though electrostatic interactions between particles and fibers are well-known to increase the efficiency of a clean filter, many open questions still remain regarding, in particular, how electrostatic interactions impact the long-term performance of a filter. As filters are used, particles accumulate on the fibers of the filter changing both the fluid flow, as well as the electric field distribution around the fibers. Electrostatic interactions change the particle deposit distributions around fibers and prior work along with the present trajectory simulations suggest that electrostatic interactions cause particles to deposit more uniformly on the upstream and downstream sides of fibers [START_REF] Kanaoka | Stochastic simulation of the agglomerative deposition process of aerosol particles on an electret fiber[END_REF]D. C. Walsh & Stenhouse, 1997a, 1997b, 1998;[START_REF] Walsh | Experimental Studies of Electrically Active Fibrous Filter Loading[END_REF].

However, complexity of real filter media makes systematic investigation of how electrostatic effects change the loading of filters challenging. Furthermore, analytically determining the trajectories of particles around fibers, as has previously been done for clean fibers [START_REF] Kraemer | Collection of Aerosol Particles in Presence of Electrostatic Fields[END_REF][START_REF] Natanson | Deposition of aerosol particles by electrostatic attraction upon a cylinder around which they are flowing[END_REF][START_REF] Zebel | Deposition of aersol flowing past a cylindrical fiber in a uniform electric field[END_REF], is not possible for fibers loaded with particles. As such, experimental methodologies with more highly controlled filter properties and more robust modeling methods are required for a fundamental understanding of how electrostatic interactions affect the long-term performance of filters.

To the authors' knowledge, the experimental methodology in the present work is the first to control all aspects of the filter media and aerosol as to achieve excellent quantitative agreement in the electrostatic filtration efficiency with particle trajectory simulations without any fitting parameters.

Experiments with this filter assembly can be adjusted to systematically test complexities in real electret filter media. For instance, inter-fiber spacing can be varied by using a finer grid to investigate the role of fiber proximity on electrostatic capture, or clogging studies may elucidate how previously deposited particles affect further electrostatic capture. As such, the experimental and simulation methodologies developed here provide valuable tools to address remaining open questions in electrostatic filtration from a fundamental, rather than empirical, perspective.

IV. CONCLUSION

We develop an aerosol filtration bench wherein electrical and geometrical properties of both aerosol particles and the filter media are precisely controlled. Using this filtration bench we demonstrate that coulombic interactions significantly enhance filtration efficiency and can be tuned in our filter assembly by varying the applied voltage. We develop particle trajectory simulations designed to model our experimental system and find excellent quantitative agreement between simulations and experiments without any fitting parameters. The combination of the experimental and simulation methodologies developed here elucidate key fundamental physics governing electrostatic filtration and are valuable to examine more complicated phenomena in electret filters such as clogging.

Figure 1 .

 1 Figure 1. Schematic of experimental system. At the outlet of the Electrostatic Classifier 3082, the aerosol is composed of monodisperse, singly charged PSL particles. The electric potential of the metallic grid is controlled using a high-voltage supply.

Figure 2 .

 2 Figure 2. a) Schematic of filter assembly. The three metallic grids are insulated from each other and the conical filter holder by PEEK spacers. The middle grid contacts a positive high-voltage line while the outer two grids are connected to electrical ground. The thickness of the PEEK spacers is d = 11 mm such that the metallic grids are separated by 11 mm. b) Optical microscope images of the metallic grid used as the filter media. The edge-to-edge distance between fibers is h = 250 µm and the fiber diameter is Df = 100 µm.

Figure 3 .

 3 Figure 3. Schematic of simulation cell. The simulation cell is characteristic of a single fiber within the middle, high-voltage metallic grid and all parameters are taken directly from the experimental system i.e no fitting parameters are used.

Figure 4 .

 4 Figure 4. Schematic of the grid arrangement of fibers used to calculate the total penetration through a metallic grid from the single fiber efficiency, ηC, calculated from the trajectory simulations. Particles with initial positions within the shaded red region are captured by the fiber.

Figure 5 .

 5 Figure 5. Coulombic penetration as a function of a) applied potential difference at a constant u = 0.6 cm/s at three particle diameters: Dp = 100 nm (black), Dp = 300 nm (red), Dp = 500 nm (blue); b) airflow face velocity at a constant Dp = 300 nm and ΔV = 1000 V; c) particle diameter at a constant u = 0.6 cm/s and ΔV = 1000 V. Open points are data from particle trajectory simulations and solid points are from experiments. In c) aerosol particles are generated from a potassium chloride (KCl) solution and monodisperse, singly charged KCl particles are sorted using the DMA.

Figure 6 .

 6 Figure 6. Simulated coulombic single fiber efficiency vs the experimental coulombic single fiber efficiency. The black dashed line represents where the simulation and experimental results are equal. Black data points have conditions where the Knudsen number is less than 1 and red data points have conditions where the Knudsen number is greater than 1.

Figure 7 .

 7 Figure 7. The a) experimental and b) simulated coulombic single fiber efficiency as a function of the coulomb number. In a) C in KC is calculated using Eq. 5 whereas in b) C = 1.

Table 1 .

 1 Parameters used in particle trajectory simulations. All of these parameters have a direct basis from the experimental system. Other relevant parameters not listed include the viscosity, density and permittivity of air.

	Parameter	Symbol	Value/Range
	Particle diameter	D p	70 -500 nm
	Particle charge	q	1e -
	Fiber diameter	D f	100 µm
	Fiber spacing	h	250 µm
	Grid spacing	d	11 mm
	Applied potential difference	ΔV	0 -6000 V
	Airflow face velocity	u	0.6 -5 cm/s


Thomas Gelain

Laurent Ricciardi

François Gensdarmes

Implementation of a Particle Resuspension Model in a CFD Code: Application to an Air Ingress Scenario in a Vacuum Toroidal Vessel

Keywords: aerosol, resuspension, CFD, tokamak, LOVA

de niveau recherche, publiés ou non, émanant des établissements d'enseignement et de recherche français ou étrangers, des laboratoires publics ou privés.

INTRODUCTION

As part of the construction of ITER, it is important to develop knowledge that ensures the highest level of safety in the nuclear facility [START_REF] Van Dorsselaere | Progress of IRSN R&D on ITER Safety Assessment[END_REF]. To assess the safety of nuclear facilities and the relevance of associated radiation protection measures, a key step consists in determining source terms of contamination during normal operation and for various accident scenarios. Since it concerns potential contamination in aerosol form, source terms are calculated by using resuspension coefficients to link the quantity of aerosol formed to the quantity of matter that may potentially be dispersed, depending on the scenario.

In ITER's normal operating conditions, several hundred kilograms of particles containing beryllium (Be) and tungsten (W) will be produced following erosion of the walls of the vacuum vessel (blanket and divertor) by the plasma [START_REF] Roth | Recent analysis of key plasma wall interactions issues for ITER[END_REF]. The particles are subject to conditions found in the vacuum vessel (VV) and may thus be radioactive (tritium and activation products [START_REF] Pace | Biological hazard issues from potential releases of tritiated dust from ITER[END_REF]) or exhibit acute chemical toxicity like beryllium particles [START_REF] Cortes | Optimization at the design phase of the potential impact of ITER on workers, the public and the environment[END_REF][START_REF] Rosanvallon | Dust control in tokamak environment[END_REF], or constitute, with air or water vapour, a potentially explosive mixture [START_REF] Roth | Recent analysis of key plasma wall interactions issues for ITER[END_REF].

The mobility of these airborne particles and their containment in the event of an accident such as a LOVA have become, over time, a major topic for assessing the safety of ITER [START_REF] Glor | Hazards due to electrostatic charging of powders[END_REF][START_REF] Paci | Bases for dust mobilization modelling in the light of STARDUST experiments[END_REF][START_REF] Porfiri | Dust mobilization experiments in the context of the fusion plants-STARDUST facility[END_REF][START_REF] Sharpe | A review of dust in fusion devices: Implications for safety and operational performance[END_REF]2006).

Several studies were conducted in order to understand the phenomenon of resuspension and to highlight the main parameter which is the friction velocity (Garcia-Cascales et al., 2010;[START_REF] Gelain | CFD modelling of the wall friction velocity field in the ITER tokamak resulting from airflow during loss of vacuum accident-Consequences for particle resuspension[END_REF][START_REF] Honda | Analyses of loss of vacuum accident (LOVA) in ITER[END_REF]. These studies also allowed to characterize the competition between the airflow forces and the adhesive ones, especially depending on particle diameter, and to develop resuspension models such as the Rock'n Roll one [START_REF] Reeks | Kinetic models for particle resuspension in turbulent flows: theory and measurement[END_REF][START_REF] Biasi | Use of a simple model for the interpretation of experimental data on particle resuspension in turbulent flows[END_REF].

Considering this latter, its application can be enforced in order to evaluate the risks caused by the particle resuspension in case of LOVA. For performing this step, the Rock'n'Roll model had to be implemented in a CFD code and precisely compared to experimental data.

This article proposes to develop the implementation of the Rock'n Roll model in a CFD code (ANSYS CFX) and to provide first result of its application to the case of a LOVA in a low pressure toroidal vessel.

However, even if the application proposed in this article is nuclear oriented, the issue of particle resuspension is present in numerous industrial situations in which powders are manipulated, can fall on the floor and can be resuspended by operator displacements. Hence, the present calculations could be extended to other industrial processes.

ROCK'N ROLL MODEL DESCRIPTION

The "Rock'n Roll" model, developed by [START_REF] Reeks | Kinetic models for particle resuspension in turbulent flows: theory and measurement[END_REF], on the basis of a so-called "RRH" model developed by [START_REF] Reeks | On the resuspension of small particles by a turbulent flow[END_REF], is founded on the assumption of a particle deposited on a rough surface which is resuspended by the airflow turbulence effect, such as shown in figure 1. This competition between adhesive and airflow moment forces can be expressed by the equation 1.

1 2 F L + r p 𝑎 F D > f a , (1)
where F L is the lift force, F D is the drag force, f a is the adhesive force, a is the distance between the two irregularities on which the particle rests and r p the particle radius.

According to experimental data analysed by [START_REF] Reeks | Kinetic models for particle resuspension in turbulent flows: theory and measurement[END_REF], the ratio r p a ⁄ between the particle radius and the distance between two irregularities could be usually considered equal to 100.

This observation implies that the term including the lift force can, in most cases, be neglected, even if it is maintained in our case.

The drag and lift forces can be expressed by the following expressions: 'Neill, 1968),

F D = 8.01ρν 2 (d p + ) 2 (O
F L = 4.2ρν 2 (d p + )
2.31 [START_REF] Hall | Measurements of the mean lift force on a particle near a boundary layer in turbulent flow[END_REF],

(2

)
where ν is the fluid kinematic viscosity and d p + is the non-dimensional particle diameter defined by:

d p + = d p .u * ν , (3)
where 𝑢 * is the friction velocity and 𝑑 𝑝 is the particle diameter.

Taking into account these expressions, the balance of airflow forces required to mobilize a spherical particle deposited on a flat surface gives:

F = 1 2 F L + r p 𝑎 F D = 2.1ρν 2 (d p + ) 2.31 + 801ρν 2 (d p + ) 2 . ( 4
)
Adhesive forces cannot be simply expressed due to the floor roughness, but [START_REF] Reeks | Kinetic models for particle resuspension in turbulent flows: theory and measurement[END_REF] supposed that they are distributed according to a log-normal distribution given by:

φ(f a ′ ) = 1 √2π 1 f a ′ ln(σ a ′ ) e (- 1 2 ( ln(f a ′ f a ′ ̅̅̅ ⁄ ) ln(σ a ′ ) ) 2 ) , ( 5
)
where f a ′ = f a F JKR ⁄ represents the normalised adhesive force f a (N) thanks to the JKR theoretical model. f a ′ ̅ represents the geometric mean of normalised adhesive forces distribution and σ a ′ the geometric standard deviation of the distribution. The JKR model [START_REF] Johnson | Surface energy and the contact of elastic solids[END_REF] provides a theoretical adhesive force that takes into account particle deformation for an ideal system composed of a spherical particle on a smooth surface:

F JKR = 3 4 π𝛾d p , ( 6
)
where 𝛾 represents the surface energy of the system particle/surface entering in interaction. Consequently f a ′ could be view as a reduction factor from JKR force that allows to take into account surface and particle roughnesses.

For operational Rock'n Roll model calculation, [START_REF] Biasi | Use of a simple model for the interpretation of experimental data on particle resuspension in turbulent flows[END_REF] proposed empirical correlations (eq. 7 and 8) as adhesion forces distribution parameter f a ′ ̅ and σ a ′ ̅̅̅ , for model input.

These correlations were determined by fitting the Rock'n Roll model on numerous experimental data of particles resuspension for different kinds of particles and different deposit surfaces.

f a ′ ̅ = 0.016 -0.0023 ( d p 2 ) 0.545 , ( 7
)
σ a ′ ̅̅̅ = 1.8 -0.136 ( d p 2 ) 1.4 . ( 8
)
In equations 7 and 8, the particle diameter d p is expressed in micrometer and the main data used to establish them correspond to particle-surface pairs with an adhesive surface energy approaching 0.56 J.m -2 and particle diameters between 0.1 µm and 30 µm.

As a synthesis of the model, it can be seen that it is mainly depending on the friction velocity 𝑢 * , the particle diameter d p and the adhesion surface energy 𝛾, and the most of the existing models [START_REF] Ziskind | Resuspension of particulates from surfaces to turbulent flows -review and analysis[END_REF] agree with this observation.

However, knowing the presence of the fluid density in the model and given the specific nature of the LOVA in terms of pressure change, it is necessary to take into account the variation of fluid density when calculating airflow forces affecting the resuspension and dispersion models.

The "Rock'n Roll" model aims at calculating the resuspension rate which needs to know the constant resuspension specific rate 𝑝 for an adhesive force value 𝑓 𝑎 .

{ p(f a ) = n θ (e ( -(f a -F ̅)2 2f 2 ̅̅̅ ) 1 2 [1 + erf ( (f a -F ̅) √ 2f 2 ̅ )] ⁄ ) if f a -F ̅ ≥ 0.75 √ f 2 ̅ p(f a ) = n θ , if f a -F ̅ < 0.75 √ f 2 ̅ , ( 9
)
where 𝑛 𝜃 is the surface-particle resonance frequency (s -1 ), 𝐹 ̅ and 𝑓 are respectively the mean and fluctuating component of the instantaneous airflow force F(𝑡) (N); the expression of 𝐹 ̅ is given at the equation 3, whereas the fluctuation 𝑓 and the frequency 𝑛 𝜃 can be calculated by the following equations, thanks to experimental measurements carried out by [START_REF] Hall | Measurements of the mean lift force on a particle near a boundary layer in turbulent flow[END_REF]:

𝑓 = 0.2 𝐹 ̅ , (10)
𝑛 𝜃 = 0.00658 ( 𝑢 * 2 𝜈 ) . (11)
From the constant resuspension specific rate p(f a ), the resuspension rate 𝜆(𝑡), depending essentially on the particle diameter and the shear velocity, can be calculated according to airflow conditions: constant or variable.

Model for a constant shear velocity

Knowing the expressions of the constant resuspension specific rate 𝑝 and the adhesive force distribution φ, the resuspension rate 𝜆(𝑡) for constant airflow conditions, involving a constant shear velocity, can be calculated from the following expression proposed by [START_REF] Reeks | Kinetic models for particle resuspension in turbulent flows: theory and measurement[END_REF] in the "Rock'n'Roll" model:

λ(t) = ∫ φ(f a ′ )p(f a ′ )e -p(f a ′ )t df a ′ ∞ 0 . (12)
In order to obtain p(f a ′ ) , f a must be replaced by (f a ′ . F JKR ) in the equation (9).

Model adaptation for a variable shear velocity

To take into account the transient configuration, it is necessary to discretize the time with a timestep ∆𝑡 small enough for considering the shear velocity as constant. This adaptation was proposed by [START_REF] Choi | Large eddy simulation of particle resuspension during a footstep[END_REF] in a study on the particle resuspension during a footstep. The mean value of the resuspension rate λ ̅ (t) is thus calculated for the timestep ∆𝑡 considered by the following expression:

λ ̅ (t) = 1 ∆𝑡 ∫ λ(t)𝑑𝑡 ∆𝑡 0 = 1 ∆𝑡 ∫ φ(f a ′ )(1 -e -p(f a ′ )∆t )df a ′ ∞ 0 . (13)

MODEL IMPLEMENTATION IN ANSYS CFX

The Rock'n Roll model was implemented in ANSYS CFX in order to take into account the resuspension phenomenon in aerosol transport and deposition calculations, allowing to assess the resuspended particle amount for issues related to dust explosion in nuclear reactors or of dust inhalation by operators.

For this implementation, the following particle transport equation is used:

∂ρY p ∂t + ∇. (ρ𝐔Y p ) = ∇. ((ρD + μ t Sc t ) ∇Y p ) + S p , ( 14
)
where Y p is the particle mass fraction, ρ is the carrier fluid density (kg.m -3 ), 𝐔 is the mean fluid velocity vector (m.s -1 ), D is the particle diffusion coefficient (m 2 .s -1 ), μ t is the eddy viscosity (Pa.s), Sc t is the turbulent Schmidt number and S p the global particle source term (kg.m -3 .s -1 ).

The resuspension rate λ(t) including Biasi's correlation for the adhesive forces distribution is thus implemented as a wall flux term, associated to the local particle surface concentration C s (t) (kg.m -2 ):

S p,res = λ(t)C s (t). ( 15
)
As the particles are resuspended, the local particle surface concentration has to decrease over time by this way. Hence, a variable associated to the surface concentration has also been implemented in order to take into account this evolution.

{ C s (t + ∆t) = C s (t)(1 -λ(t)∆t), if u * is constant C s (t + ∆t) = C s (t)(1 -λ ̅ (∆t)∆t), if u * is variable . ( 16
)
Both types of model, either for a constant shear velocity, or for a variable one, have thus been implemented in ANSYS CFX and their preliminary validation has been conducted thanks to experimental data acquired on experimental benches available at IRSN. Some comparisons between CFX calculations and experiments on particle resuspension for transient flow are presented in Gelain (2014).

It should be noticed that a specific model for the particle transport and deposition, developed and validated by IRSN (Nérisson, 2011), was also implemented in ANSYS CFX. The transport model is integrated to the transport equation ( 14), whereas the deposition model is implemented as a wall sink term S p,dep added to the resuspension flux term S p,res .

APPLICATION TO THE CASE OF A LOVA IN A TOROIDAL GEOMETRY

CFD simulation description

To evaluate the resuspension of different particle sizes from the airflows generated during a LOVA, numerical simulations were performed using ANSYS CFX software, into which the resuspension model with a variable shear velocity was implemented.

The first step consists in representing the geometry of the torus with the duct that simulates the breach which is the source of the loss of vacuum.

For the vessel, a toroidal geometry is retained with a D-shaped section, the dimensions of which are given in Table 1.

Table 1-overall dimensions of the torus (ITER-FEAT,2000)

To represent the breach, a circular duct with a diameter of 0.16 m and a length of 2 m is used [ITER-FEAT, 2000]. The duct (representative of the equatorial port) is located halfway up the torus and is initially closed at the torus wall. Figure 2 presents a sectional view of the shape of the torus with its dimensions.

Figure 2 -geometry used for calculations

The mesh is presented in Figure 3.

Figure 3 -geometry used for calculations

For the resuspension issues, the refinement close to the wall is important and may have consequences about the amount of resuspended particles, but currently, the resuspension model is not accuracy enough to calculate all the phenomena occurring at the vicinity of the wall and of the roughness. Hence, the aim of this study is not to validate the model or the modelling but to verify the model implementation relevance and the capabilities of this model to evaluate the resuspension in the case of low pressure. A more accurate validation will be conducted as soon as relevant experimental data will be available.

The dataset parameters for the calculations are given in Table 2. The k- SST turbulence model was chosen after a sensitivity study described by [START_REF] Gelain | CFD modelling of the wall friction velocity field in the ITER tokamak resulting from airflow during loss of vacuum accident-Consequences for particle resuspension[END_REF]. This model remains a good compromise between accuracy and calculation time. The particle surface energy of adhesion is the tabulated value for tungsten (4.4 J.m -2 ). This latter is different from the main value used by Biasi for determining his correlation (0.56 J.m -2 ). Hence, the calculation is probably not conservative regarding resuspension.

The airflow parameters used for the calculations were set for a LOVA-type accident scenario as described by [START_REF] Xiao | Numerical analysis of hydrogen riskmitigation measures for support of ITER licencing[END_REF] and already used by [START_REF] Gelain | CFD modelling of the wall friction velocity field in the ITER tokamak resulting from airflow during loss of vacuum accident-Consequences for particle resuspension[END_REF]. They are given in Table 2. It is reminded that the initial pressure in this calculation is about 800 Pa whereas, in ITER conditions, the pressure is about 10 -7 Pa. The latter is too much low to ensure the validity of Navier-Stokes equation and 800 Pa is a good compromise between low pressure conditions and calculation robustness and convergence. Furthermore, what we want also to visualize with this calculation is this impact of lack of air on the resuspension and on the aerosols transport. Before starting the resuspension calculation, the floor has to be seeded by a particle surface concentration. For that, a concentration source term was applied in the first cell of the mesh close to the floor.

During the calculation, the resuspension coefficient 𝐾 𝑟𝑒𝑠 and the resuspension rate 𝜆 𝑟𝑒𝑠 were tracked in real time as well as the other usual variables such as the friction velocity, the particle mass fraction, the pressure or the floor surface concentration.

It can be noticed that the calculation was stopped at 20 s (for an inside pressure of around 12,000 Pa) because the friction velocities on the floor became too low to induce any particle resuspension.

In the next section, the simulation results are presented and discussed in terms of airflows, particle resuspension and dispersion. Finally, Figure 6 shows the airflow streamlines inside the torus.

Figure 6 -airflow streamlines inside the torus (t = 1s)

The geometry and the high velocities induce vortices which will entrain the particles along the torus walls and disperse them inside the torus. The observation of these vortices is very important to understand how the particles are dispersed as shown in the next paragraph.

 Particle resuspension and dispersion

As an illustration of the resuspension phenomenon during the calculation, Figure 7 shows the evolution of an isosurface of the particle normalized mass fraction (mass fraction divided by the maximal mass fraction) during the pressurization. At the beginning, for the time 0.1 s, it can be seen that the resuspension of the particles is in good agreement with the friction velocity field of Figure 5. Indeed, as previously mentioned, the friction velocity is the main parameter of the resuspension where the drag force on particles has to overcome the adhesive forces, which mainly depend on the particle diameter. So there is a threshold from which the resuspension happens which explains that just a little area of the floor exhibits resuspension. At the time 1 s, the friction velocity is higher on a larger area of the floor; hence, the resuspension is more extended on this area, in agreement with the friction velocity field of Figure 5. For the following times 6 s and 20 s, the friction velocities on the floor are lower; so, the resuspension too, and what is observed in Figure 7 is mainly the particle dispersion in the torus. The particles are carried along the wall, following the airflow streamlines as shown on Figure 6. As indicated in the legend of Figure 8, the resuspension rates are very different according to the diameter. Around eleven decades are observed between resuspension rate for the 2 µm aerosol and that for 10 µm . Otherwise, the resuspension kinetics is also slightly different; for the 2 µm aerosol, the latter begins to be significantly resuspended from 8 s, then the resuspension stays constant up to 18 s, before decreasing. In contrast, for the 5 and 10 µm aerosol, the resuspension occurs almost at the beginning of the pressurization and is still on at 20 s.

These observations can be confirmed by the evolution of the resuspension coefficients in Figure 9. This latter presents the resuspension coefficient time evolution (solid lines), corresponding to the floor source term, and the "in suspension" coefficient time evolution (dash lines), corresponding to the particles effectively in suspension in the torus at the given time. The difference between both is the deposition on the torus wall. On the secondary axis is presented the time evolution of the pressure in the torus.

Figure 9 -resuspension fraction time evolution

Figure 9 shows that the biggest particles are the most resuspended but also the most deposited. However, it can be observed that for the 10 µm aerosol, the total particle resuspension coefficient can reach around 25 % for the calculation conditions.

CONCLUSIONS

This article presents the implementation of the Rock'n'Roll model in a CFD code (ANSYS CFX) and an application to the case of a LOVA in toroidal geometry.

In a first part, the Rock'n'Roll model integrating the adhesive force distribution of [START_REF] Biasi | Use of a simple model for the interpretation of experimental data on particle resuspension in turbulent flows[END_REF] was described, before detailing the model implementation in ANSYS CFX, performed by the way of a source term of resuspension in the particle transport equation. The model can be used with a constant or a variable friction velocity according to the model formulation.

Then, an application of the model to a LOVA case in a toroidal geometry was presented. This type of simulation is slightly complex due to the lower pressure close to the vacuum which evolves with the air ingress.

These simulations carried out for 2, 5 and 10 µm dust corresponding to 8.8, 22 and 44 µm aerodynamic diameters showed a relevant kinetics of resuspension according to the friction velocity fields on the floor where the particles are dropped off and to the airflows which entrain the particle into the torus. The results are also in agreement with the particle resuspension physics showing a larger resuspension amount for the 10 µm aerosol than for the 2 µm one with a difference of many decades between both. This article demonstrates thus the feasibility of particle resuspension CFD simulations with the Rock'n Roll model, allowing to evaluate quantitatively the particle resuspension. However, this model with the Biasi's formulation takes into account a one layer particle surface deposition which may not be the reality in most cases. So a prospect for improving these simulations could be to implement a multi-layer particle surface deposition or to better take into account the adhesive forces by the way of recent studies which could be applied to this case. Finally, recent numerical and experimental results showed the influence of the particle polydispersity on their resuspension.

NOMENCLATURE Latin letters

𝑎

distance between two floor irregularities (m) 𝐶 𝑠 floor surface concentration (kg.m -2 ) 𝐷 diffusion coefficient (m 2 .s -1 ) 𝑑 𝑝 particle diameter (m) 𝐹, 𝑓 force (N) 𝑝 constant resuspension specific rate (s -1 ) 𝑟 𝑝 particle radius (m) 𝑆 𝑝 particle source flux (kg.m -2 .s -1 ) 𝑆𝑐 𝑡 turbulent Schmidt number 𝑡 time (s) 𝑢 * friction velocity (m.s -1 ) 𝑈 velocity (m.s -1 ) 𝑉 𝑑 deposition velocity (m.s -1 ) 𝑦 + non-dimensional wall distance 𝑌 𝑝 particle mass fraction Greek letters 𝜆 resuspension rate (s -1 ) 𝜌 density (kg.m -3 ) 𝜈 kinematic viscosity (m 2 .s -1 ) 𝜎 geometric standard deviation 𝜇 𝑡 eddy viscosity (Pa.s)

Figure 1 -

 1 Figure 1scheme of understanding of the "Rock'n Roll" model As observed on Figure 1, lift, drag and adhesive forces will create moments on the particle according to particle radius and the roughness scale of the floor.This competition between adhesive and airflow moment forces can be expressed by the equation 1.

  particle resuspension is essentially driven by the airflows and more precisely by the friction velocity. Hence, Figure4presents the time evolution of the friction velocity on the bottom where the particles are dropped off. As seen on this figure, the maximal friction velocity reaches around 8 m/s for a mean velocity of around 4 m/s.

Figure 4 -

 4 Figure 4friction velocity on the floor and pressure inside the torus

Figure

  Figure 5 -friction velocity fields on the floor at 0.1 s, 1 s, 6 s and 20 s

Figure

  Figure 7 -time evolution of the isosurface at 0.001 of the resuspended normalized particle mass fraction at 0.1 s, 1 s, 6 s and 20 s (d p = 10 µm diameter)

Figure 8 -

 8 Figure 8 -resuspension rate time evolution for the three diameters studied