François Xavier

Nouha

Henrietta Siavoch

Khalisa

Emannuel, Hiba, Chantal Ali

Keywords: Air filtration, Filter pleats, Airflow pattern, CFD calculation, Hot Wire Anemometry, Clogging

This work would not have been possible without the support of a lot of people. I'd

both the difficult and rewarding aspects of this thesis. To Christelle and Sarah for their help and, for organizing the convivial Lebanese gathering that made me feel at home. I thank also my office colleagues Shawki and Corentin for all of the fruitful discussions and unending support.

Many thanks to my closest friends in Nantes, who became like family to me: Roy, Rita, Yara, Hiba, and Victor. I recognize their availability, encouragement, and unconditional support.

Finally, I'd like to express my heartfelt appreciation to my family for their love and sacrifices.

Without their support, I would not have been able to finish my thesis.

I dedicate this thesis to you, Wael, my little angel and blessing...

Résumé

Le défi de la conception des filtres fibreux utilisés dans les CTA, est de proposer des recommandations qui rendent les filtres compétitifs en termes de performances (efficacité de filtration et consommation d'énergie tout au long du colmatage), tout en respectant les contraintes de surfaces disponibles dans les systèmes de ventilation. Ceci permet de diminuer les pertes de charge et de maintenir un haut niveau d'efficacité pendant le colmatage. De nombreux paramètres influencent les performances des filtres fibreux, notamment les caractéristiques des particules (masse volumique, distribution granulométrique, morphologie), les conditions de fonctionnement (vitesse de filtration, température et humidité relative de l'air) et les caractéristiques du filtre. Ces dernières sont classées en trois échelles : l'échelle des fibres (matériau et morphologie), l'échelle du média fibreux (diamètre et longueur des fibres, porosité, taille des pores, épaisseur, perméabilité), et l'échelle du filtre (dimensions caractéristiques de la géométrie selon laquelle le média fibreux est mis en oeuvre : plissé, poches, dièdre, ...). Dans le cas d'un filtre plissé, la dernière échelle correspond à l'échelle du pli.

La conception des filtres nécessite la sélection du média fibreux pour atteindre l'efficacité souhaitée ainsi que la détermination des propriétés géométriques pour minimiser la perte de charge. En effet, la mise en oeuvre d'un média fibreux à travers une géométrie plus complexe que la géométrie plane, comme la géométrie plissée, permet d'augmenter la surface offerte à la filtration, ce qui peut entraîner des gradients de vitesse locaux responsables d'une augmentation significative de la perte de charge et éventuellement de variations locales d'efficacité. En fait, la vitesse à laquelle les particules atteignent le média fibreux est nommée, dans la littérature, la vitesse de filtration, qui est toujours calculée comme le rapport entre le débit d'air approchant le filtre et la surface développée du média filtrant. Cette vitesse est l'un des paramètres critiques qui affectent la performance de la filtration, et en particulier l'efficacité de la collecte des particules. Une vitesse de filtration élevée en amont du média tend à diminuer la collecte des particules par diffusion brownienne et à augmenter la collecte par le mécanisme d'impaction inertielle. C'est pourquoi, l'étude de l'écoulement d'air en amont des plis du filtre est cruciale pour évaluer précisément son éventuelle influence sur les performances de filtration. De plus, l'étude de la configuration de l'écoulement de l'air en aval du filtre est également importante, notamment dans les applications d'air intérieur où les filtres sont implantés en série. En effet, la distribution non-uniforme de la vitesse générée par un pré-filtre peut affecter l'efficacité et la durée de vie du filtre. De plus, les variations de vitesse peuvent induire des phénomènes physiques non-stationnaires lors du colmatage, tels que le rebond des particules et leur ré-entraînement en aval du filtre.

Ce projet vise à mieux comprendre le comportement du flux d'air au voisinage des plis d'un filtre afin d'identifier les tendances locales de vitesse qui pourraient affecter les performances de la filtration. Pour cela, l'objectif est de coupler les caractérisations des champs de vitesse à proximité du filtre aux mesures de performance de filtration, c'est-à-dire l'évolution de la perte de charge du filtre et l'efficacité de collecte des particules pendant le colmatage. Les mesures de vitesse sont effectuées à proximité du filtre vierge et à un niveau de colmatage avancé, afin de pouvoir détecter l'influence des changements dans les propriétés structurelles du média sur les profils de vitesse en aval du filtre. La caractérisation du profil de vitesse au voisinage du pli du filtre est réalisée à l'aide de moyens expérimentaux (anémométrie à fil chaud) et de simulations numériques (utilisation du code ANSYS Fluent). Les mesures expérimentales permettent de prendre en compte l'influence de la géométrie des plis sur les valeurs locales de vitesse auxquelles les particules sont exposées pendant le colmatage. L'étude de l'influence des propriétés géométriques du filtre plissé propre (hauteur et largeur des plis) sur sa perte de charge et les variations de vitesse au voisinage du filtre est réalisée à l'aide de simulations numériques. De plus, la comparaison des profils de vitesse numériques avec les valeurs expérimentales permet de valider les modèles d'écoulement, ce qui n'est pas possible en comparant simplement les valeurs de perte de charge.

Enfin, les connaissances acquises sur les propriétés structurelles du média fibreux, d'une part, et les variations locales de vitesse au voisinage des plis vierges et colmatés du filtre, d'autre part, fournissent des données expérimentales importantes qui peuvent être utilisées pour améliorer le modèle d'efficacité de collecte existant notamment, en prenant en compte les gradients de paramètres spécifiques qui peuvent induire des gradients locaux d'efficacité.

L'étude bibliographique présentée dans le premier chapitre, a montré que la conception de filtres fibreux nécessite la sélection d'un média fibreux pertinent pour atteindre l'efficacité visée, et la détermination de propriétés géométriques pertinentes permettant de minimiser la perte de charge.

De nombreuses études rapportées dans la littérature traitent de l'amélioration des performances de filtration (i.e. diminuer la perte de charge tout en assurant l'efficacité de filtration visée) des filtres plissés en combinant des modèles semi-empiriques avec des simulations numériques. À cette fin, le principal défi rencontré par les auteurs était de prendre en considération la répartition de l'écoulement de l'air à l'approche des filtres plissés. Même si certaines études présentées dans la littérature ont détecté des gradients de vitesse dans les plis, elles ne les ont pas intégrés dans le calcul de l'efficacité de collecte. Elles considèrent plutôt la vitesse de filtration, ce qui ne reflète pas les conditions locales subies par les particules et affecte donc la précision des modèles. De plus, la littérature présente de nombreuses études numériques traitant de l'effet des propriétés géométriques du filtre sur le modèle de vitesse à proximité du filtre [START_REF] Fotovati | Modeling instantaneous pressure drop of pleated thin filter media during dust loading[END_REF][START_REF] Feng | Modeling filtration performance of pleated fibrous filters by Eulerian-Markov method[END_REF][START_REF] Théron | Numerical and experimental investigations of the influence of the pleat geometry on the pressure drop and velocity field of a pleated fibrous filter[END_REF][START_REF] Kang | Characterization of pleated filter media using particle image velocimetry[END_REF][START_REF] Alilou | Impact sur le colmatage en régimes transitoire et permanent des écoulements d'air induits par le plissage des médias filtrants THE[END_REF][START_REF] Maddineni | Numerical investigation of pressure and flow characteristics of pleated air filter system for automotive engine intake application[END_REF]. La plupart de ces études se sont concentrées sur le comportement de l'écoulement de l'air sur le côté amont des plis puisqu'il s'agit du principal intérêt concernant la collecte des particules.

Pour tous les calculs numériques mentionnés dans ce chapitre, la difficulté réside dans la sélection d'un modèle d'écoulement d'air approprié pour simuler l'écoulement d'air à l'approche du filtre.

Dans la littérature, la validation du modèle d'écoulement d'air est presque toujours basée sur la perte de charge qui est un paramètre macroscopique [START_REF] Tronville | Minimization of Resistance in Pleated-Media Air Filter Designs: Empirical and CFD Approaches[END_REF][START_REF] Théron | Numerical and experimental investigations of the influence of the pleat geometry on the pressure drop and velocity field of a pleated fibrous filter[END_REF]. Peu d'études ont validé leur approche numérique grâce à des paramètres locaux tels que les valeurs de vitesse par exemple. Ceci est probablement dû au fait que la mesure des valeurs de vitesse locale à proximité des filtres représente un défi expérimental.

Sur la base de ces arguments, il a été décidé de développer une méthodologie pour la caractérisation des profils de vitesse au voisinage des filtres fibreux plissés qui combine des mesures expérimentales et des simulations numériques. En raison de la complexité de la géométrie des plis, le principal défi consiste à mesurer les profils de vitesse locale à l'intérieur des plis du filtre et jusqu'à une courte distance de la surface du média filtrant. Les matériaux utilisés, ainsi que les stratégies expérimentales et numériques, sont décrits en détail dans le chapitre 2. Parce qu'elle ne nécessite pas de traceurs de particules, l'anémométrie à fil chaud (HWA) a été choisie de préférence aux autres techniques (PIV et LDA) pour mesurer les profils de vitesse expérimentaux en amont et en aval du filtre. Le banc expérimental dans lequel ont été menées ces mesures consiste en une boucle ouverte fonctionnant en aspiration. La conduite principale dans laquelle est situé le filtre possède une longueur totale de 332 cm, et est de section carrée de 15 cm de côté. Cette conduite en inox comporte une section contenant le filtre appelée section de mesure, située à 167 cm par rapport à l'entrée de la veine, et de 41 cm de long. Cette zone de mesure est en PPMA antistatique, et donc transparente pour permettre de visualiser le positionnement du fil chaud. Les profils de vitesse transversale ont été acquis en aval à cinq distances différentes du sommet du pli (à l'intérieur du pli à x = 1,5 et 2 cm ; à l'extérieur du pli à x = 3,0 -3,5 et 5 cm), ainsi qu'en amont du pli à (à l'intérieur du pli à x = 0,2 cm -1,0 et 1,3 cm). Le média fibreux choisi pour cette étude est un média commercial en fibre de verre de type ISO ePM1 50-50 pour cent, qui est couramment utilisé dans le deuxième étage des CTA pour les applications d'air intérieur. Deux géométries de filtre ont été testées (H = 28 mm, L = 25 mm) et (H = 27 mm, L = 10 mm) à la même vitesse de filtration Vfiltration = 0,27 m.s -1 . Les simulations de dynamique des fluides par ordinateur (CFD) ont été réalisées à l'aide du code Ansys Fluent basé sur un domaine numérique de demi-plis.

L'étude dans le chapitre 3 est menée l'échelle du média. Le but est de caractériser les propriétés structurelles du média. Les filtres fibreux sont couramment utilisés dans les systèmes de ventilation pour l'élimination des particules. Les médias fibreux utilisés pour leur fabrication sont généralement des tissus non tissés, souvent constitués de fibres de verre ou de polymère, ou de mélanges de fibres de différents matériaux. En raison de la variété des procédés de fabrication utilisés pour produire les réseaux de fibres (soufflage à l'état fondu, collage par filage, pose humide, pose à l'air, pose à sec, électrofilage), ainsi que des méthodes utilisées pour assurer leur résistance mécanique (laçage par filage, aiguilletage, collage thermique ou chimique), les matériaux non tissés présentent une large gamme de propriétés structurelles : distribution du diamètre des fibres, morphologie des fibres, épaisseur du support, porosité moyenne, distribution de la taille des pores, etc. L'un des grands défis de la communauté de la filtration est d'optimiser les performances des filtres, c'est-à-dire de réduire la perte de charge et de maintenir l'efficacité d'élimination des particules visée. Les variations locales dans le plan des propriétés structurelles du média citées précédemment entraînent un degré d'hétérogénéité plus ou moins important qui peut conduire à des lignes de cheminement préférentielles concernant le flux d'air. Ce phénomène peut être responsable d'un colmatage hétérogène, et d'un vieillissement accéléré du filtre. Il est donc important d'aborder la question de la prédiction des propriétés applicatives, et principalement de la perméabilité du média qui représente la résistance du média au flux d'air, avec une description précise des propriétés structurelles du média et de leurs variations potentielles dans le plan. Ce chapitre consiste en la description de la caractérisation des propriétés structurelles du média à deux échelles différentes : (i) les propriétés "en moyenne" en déterminant les valeurs moyennes de la porosité, du diamètre des fibres, de l'épaisseur et de la perméabilité, (ii) les variations locales des propriétés structurelles (porosité et orientation des fibres). Les images MEB des deux faces du support ont révélé une distribution bimodale de la taille des fibres. La porosimétrie au mercure et la micro-tomographie à rayons X ont été utilisées pour déterminer la porosité du média. Malgré le fait que ces stratégies soient basées sur des principes différents, les résultats étaient très similaires. L'étude par microtomographie à rayons X des variations de la porosité dans le plan indique que le média a une structure assez homogène. La nature anisotrope de l'arrangement des fibres a été révélée en calculant le tenseur d'orientation des fibres à l'aide des données de la micro-tomographie à rayons X. La plupart des fibres sont orientées dans des plans perpendiculaires à l'écoulement. Les caractéristiques structurelles du média ont été utilisées pour évaluer la capacité de différents modèles rapportés dans la littérature pour prédire sa perméabilité. Parmi les modèles testés, le modèle isotrope RUC a prouvé son efficacité à reproduire la perméabilité expérimentale du média bimodal étudié. Alors que le tenseur d'orientation des fibres a révélé que le média est plutôt anisotrope en 3D, ce résultat pourrait être attribuable à la capacité du modèle RUC isotrope à décrire la légère déviation de la structure fibreuse par rapport aux fibres orientées strictement en 2D. De plus, au lieu d'utiliser le diamètre moyen des fibres pour représenter la distribution bimodale du diamètre des fibres, des diamètres équivalents ont été utilisés pour représenter la distribution bimodale du diamètre des fibres. Les modèles de perméabilité étant très sensibles aux variations de porosité, même mineures, des cartographies de perméabilité dans le plan ont été présentées. Cela permet d'étudier l'effet des variations de porosité sur les valeurs de perméabilité locales. Comme les médias fibreux destinés à la filtration de l'air sont très délicats, très poreux et généralement hétérogènes, il est plutôt problématique de prédire correctement leur perméabilité. Avoir accès à de telles variations de perméabilité dans le plan ouvre de nouvelles possibilités pour prédire les performances locales des filtres sur la base de leurs propriétés réelles.

Selon la revue de la littérature (chapitre 1), il n'y a que quelques études qui ont traité de l'influence du colmatage sur le profil de vitesse en aval des filtres fibreux. Ces recherches, cependant, étaient uniquement numériques, sans validation expérimentale. La recherche menée au cours de cette thèse est différente dans la mesure où elle propose des mesures de vitesse locale en aval des plis du filtre à un niveau de colmatage défini, pour deux aérosols solides et géométries de filtre différents. Ces caractérisations permettent d'évaluer l'influence du colmatage sur les propriétés de l'écoulement à proximité du filtre. À notre connaissance, le travail présenté dans cette étude est l'une des premières investigations expérimentales de l'effet du colmatage sur les profils de vitesse en aval d'un filtre fibreux plissé. Le principal avantage d'un tel travail est la compréhension des phénomènes physiques se produisant en aval des plis du filtre, tels que le rebond des particules et le réentraînement. Les résultats peuvent également être utilisés comme base de données pour la modélisation de l'écoulement de l'air et l'optimisation de la conception des filtres, en particulier lorsque les filtres sont mis en oeuvre en série.

Trois types de particules ont été utilisés pour ce travail :  Deux types de particules ont été utilisés séparément pour colmater le filtre : des particules test de silice microniques et des particules de chlorure de sodium submicroniques.

 Des particules de fluorescéine de sodium ont été générées à la fin du processus de colmatage pour identifier tout dépôt préférentiel de particules sur le filtre plissé.

Les particules de silice et de chlorure de sodium ont été choisies pour le colmatage car elles ont des diamètres aérodynamiques moyens distincts, à savoir dae, 50, Nacl = 0,70 µm et dae, 50, Silica = 1,00 µm. Les particules NaCl sont classées comme submicroniques (d50≤ 1,00 µm), tandis que les particules Silica sont classées comme microniques (d50 ≥1,00 µm). Le diamètre des particules influence la morphologie du gâteau pendant le colmatage : pour les particules submicroniques, le gâteau présente une forme de type dendritique avec une forte porosité du gâteau.

Dans le cas de particules microniques, le gâteau est plus compact et se présente sous forme d'agrégats. Cependant, comme les particules submicroniques ont un petit diamètre, leur surface spécifique est encore plus grande, ce qui entraîne une perte de charge plus importante du filtre par rapport aux particules microniques. La sélection de particules submicroniques et microniques vise à étudier l'influence potentielle de la géométrie du gâteau sur les profils de vitesse en aval des plis.

Le colmatage a été arrêté à ΔP ~ 400 Pa, représentatif d'un stade peu avancé du colmatage de surface. L'efficacité fractionnelle du filtre est déterminée expérimentalement. Ainsi, les propriétés structurelles moyennes et locales du média sont utilisées pour tester la capacité des modèles d'efficacité mono-fibre présentés dans la littérature à prédire l'efficacité expérimentale de collecte des particules. Les résultats indiquent qu'une description précise des caractéristiques structurelles du média améliore l'adéquation entre le modèle et les résultats expérimentaux. En effet, la caractérisation de la distribution bi-modale des fibres par un diamètre bi-modal équivalent est plus précise que l'utilisation du diamètre moyen des fibres. Pour la gamme de taille de particules étudiée, le mécanisme d'interception domine le mécanisme de collecte des particules en raison des conditions d'écoulement de l'air testées et des propriétés structurelles du filtre. La vitesse au voisinage du filtre est très sensible aux changements des propriétés structurelles et géométriques du filtre plissé. En cas de colmatage, la géométrie du plissage ainsi que les propriétés structurelles du média sont modifiées en raison du dépôt de particules. L'impact du colmatage sur le profil de vitesse n'est pas abordé dans la littérature. De telles observations sont importantes pour l'optimisation de la conception, en particulier lorsque les filtres sont placés en série. Les mesures de vitesse locale fournissent également des données qui peuvent être utilisées pour valider les choix de modèles d'écoulement d'air dans les simulations numériques qui prédisent les performances des filtres pendant le colmatage. Dans la deuxième partie du chapitre 5, l'influence du colmatage sur les champs de vitesse est évaluée en représentant les mesures de vitesse locale en aval des plis du filtre pour les deux géométries. Pour la même géométrie (H = 28 mm, W = 25 mm) et deux essais de colmatage séparés avec des particules de NaCl submicroniques et des particules de silice microniques, une tendance similaire du profil de vitesse est trouvée à des positions à l'intérieur (x = 1,5 cm) et à l'extérieur du pli (x = 3,5 cm). L'amplitude de la vitesse augmente en aval du pic du pli et diminue vers le média. L'injection d'un traceur à base de fluorescéine et l'analyse MEB ont confirmé ces résultats, révélant une augmentation du dépôt de particules et, par conséquent, du passage du flux d'air près du pic du pli. Il a également été démontré que la sélection des particules n'avait aucun effet sur les profils de vitesse en aval du filtre. Cela pourrait être dû au fait que la différence de diamètre médian des deux aérosols de test n'est pas significative, et que les colmatages ont tous été arrêtés au même niveau de perte de charge. Dans le cas d'une géométrie " étroite " (H = 27 mm, W = 10 mm) et de la même vitesse de filtration (Vfiltration = 0,27 m.s -1 ), les profils de vitesse en aval du filtre étaient différents de ceux de l'autre géométrie. Malgré le fait qu'un seul prototype ait été testé, ce résultat est important car il pourrait impliquer un colmatage non-homogène en amont du filtre. En fait, le chapitre 4 discute l'initiation d'un gradient de vitesse qui est particulièrement sensible à la forme des plis. Par conséquent, des incohérences dans les propriétés géométriques au niveau des plis pourraient entraîner un gradient de vitesse près des plis, conduisant à un dépôt de particules qui diffère d'un pli à l'autre. Il a également été révélé qu'en approchant le média à la même position étudiée à l'intérieur et à l'extérieur du pli, le profil de vitesse diminue plus rapidement lorsque la largeur du pli est plus étroite.

Mots clés : Filtration de l'air, Filtre plissé, Profil d'écoulement, CFD, Anémométrie à fil chaud, Colmatage.

List of Figures

List of Tables

General introduction

For several years, and more recently due to the recent COVID-19 crises, air quality has gained the interest of the general public, as well as the scientific, industrial, and political communities.

Because of its direct impact on comfort and health, indoor air treatment is a major focus for researchers, building professionals, automotive, and industrial sectors where contamination control is critical for products (agri-food, medical, microelectronics) or for worker protection. Pollutants in confined spaces are recognized to have serious effects on human health.

There are several processes for removing particles from the air. These processes are selected and sized based on the targeted applications, the desired air quality, and the quality of the air entering the system. According to the literature, the air-particle matrix can be separated using centrifugation, gravitational sedimentation, washing, electrostatic precipitation, and filtration [START_REF] Ali | élimination des particules[END_REF]. Particle filtration by fibrous filter is the most common process implemented in air handling units (AHUs).

It has an interesting efficiency, maintenance and operation are relatively simple, and several configurations and dimensions of filters can be adapted to air conditioning installations. The role of air filtration in AHUs is to improve air quality and reduce the transport of pollutants from the outside to the inside, particularly particles.

The filters used in AHU systems are classified according to the ISO 16890 standard [START_REF]Air filters for general ventilation -Part 1:Technical specifications, requirements and classification system based upon particulate matter efficiency (ePM)[END_REF]. This classification system reports a percentage of total collection efficiency according to several particle size ranges. Filtration in AHUs is often performed in two stages in non-residential buildings [START_REF] Araujo | Air filtration systems and restrictive access conditions improve indoor air quality in clinical units: Penicillium as a general indicator of hospital indoor fungal levels[END_REF]. A first stage of coarse filtration for the removal of large particles (ISO ePM10 and ISO coarse filters are used) and a second stage of fine filtration for the removal of small particles. The efficiency of the filters implemented in the second stage according to the ISO 16890 standard [START_REF]Air filters for general ventilation -Part 1:Technical specifications, requirements and classification system based upon particulate matter efficiency (ePM)[END_REF] is ePM1 ≥ 50%

i.e. more than 50% total efficiency for particle ranging from 0.3 to 1 μm or ePM2.5 ≥ 65% i.e. more than 65% total efficiency for particle ranging from 0.3 to 2.5 μm. A commercial glass fibrous medium, widely implemented in the filters in the second stage of filtration, is chosen for this study.

This filtering medium efficiency is ISO ePM1 50 -55 % (ISO 16890).

The challenge in designing the fibrous filters used in AHUs, is to make recommendations that make them competitive in terms of performance (filtration efficiency and energy consumption during clogging), while respecting the constraints of available surfaces in ventilation systems. This helps in lowering pressure drop and maintaining a high level of efficiency throughout clogging. Many parameters influence the performance of fibrous filters, including particle characteristics (density, size distribution, morphology), operating conditions (filtration velocity, air temperature and relative humidity), and filter features. The latter are classified into three scales: the fiber scale (material and morphology), the fibrous medium scale (diameter and length of fibers, porosity, pore size, thickness, permeability), and the filter scale (characteristic dimensions of the geometry according to which the fibrous medium is implemented: pleated, pockets, dihedral, ...). In case of pleated filter, the last scale corresponds to the scale of the pleat.

The design of filters requires the selection of the fibrous medium to achieve the desired efficiency as well as the determination of the geometric properties to minimize the pressure drop. Indeed, implementing a fibrous medium through a more complex geometry than the flat geometry, like the pleated geometry, allows for an increase in the surface offered for filtration, this may result in local velocity gradients responsible for a significant increase in pressure drop and possibly local efficiency variations. In fact, the velocity at which particles reach the fibrous medium is referred to, in the literature, as the filtration velocity, which is always calculated as the ratio between the air flowrate approaching the filter and the developed filter medium surface. This velocity is one of the critical parameters that affect the filtration performance, and especially the particle collection efficiency. A high filtration velocity at the upstream side of the medium tends to decrease particle collection by Brownian diffusion and to increase the collection by the inertial impaction mechanism. This is why, the study of the airflow upstream of filter pleats is crucial to precisely evaluate its possible influence on the filtration performance. Moreover, the study of the airflow pattern downstream of the filter is also important, particularly in indoor air applications where filters are implemented in series. Indeed, the non-uniform velocity distribution generated by a prefilter can affect the filter's efficiency and life span. Furthermore, velocity variations may induce non-stationary physical phenomena during clogging, such as particle bounce and re-entrainment downstream of the filter.

This project aims to better understand the behavior of the airflow in the vicinity of filter pleats to identify local velocity trends that could affect the filtration performance. For that purpose, the objective is to couple the characterizations of the velocity pattern near the filter to the filtration performance measurements i.e. evolution of the filter pressure drop and particle collection efficiency during the clogging. The velocity measurements are carried out in the proximity of the clean filter and at an advanced clogging level, to be able to detect the influence of the changes in the medium structural properties on the velocity profiles downstream of the filter. The characterization of the velocity pattern in the vicinity of the filter pleat is carried out using experimental means (hot-wire anemometry) and numerical simulations (use of the ANSYS Fluent code). Experimental measurements allow to take into account the influence of the pleat geometry on local velocity values to which the particles are exposed during the clogging. The influence of the geometric properties of the clean pleated filter (pleat height and width) on its pressure drop and velocity variations at the vicinity of the filter is performed with numerical simulations. Moreover, comparing numerical velocity profiles with experimental values allows validating the flow models, which is not possible by merely comparing pressure drop values.

Finally, the knowledge gained about the structural properties of the fibrous medium, on the one hand, and the local velocity variations at the vicinity of the clean and clogged filter pleats, on the other hand, provide important experimental data that can be used to improve the existing collection efficiency model notably, by taking into account the gradients of specific parameters that can induce local gradients of efficiency. The main difficulty encountered in this thesis is proposing an efficient methodology for measuring the local velocity profile within the filter pleats and up to a close distance from the filter surface. In accordance with the previously stated objectives the manuscript is organized as follows:

Chapter 1 proposes a literature review on the characterization of the air flow in the vicinity of the pleated filter. A review on the filtration theory, including common models used to predict the medium permeability and particle collection efficiency of a fibrous filter are also presented.

Chapter 2 presents the methodology for the characterization of velocity patterns in the vicinity of filter pleats that was developed throughout this thesis. Two pleated fibrous filters geometries are tested. The goal of the proposed experimental and numerical approaches is to be able to validate numerical simulations using local experimental velocity values. In this Chapter, the experimental setup, materials, and experimental and numerical strategies used to measure velocity near filter pleats are described in detail.

Chapter 3 is dedicated to the presentation of the studied medium, and to the description of the experimental techniques used to characterize its structural properties. Among these techniques Xray micro-tomography has been used as it enables to determine local values of some properties like the porosity for example, which allows the evaluation of the heterogeneous character of the medium. The results obtained in terms of structural properties are used to predict permeability, with a focus on the evaluation of the heterogeneous nature of the medium thanks to the analysis of fiber orientation and medium in-plane porosity variations.

Chapter 4 presents the local experimental velocity profiles upstream and downstream of the clean filter pleats and compares them to the numerical simulation results. After having validated the CFD simulations, further numerical investigations of the velocity at the air/filter interface are conducted.

Finally, the experimental local turbulence intensity values acquired by the HWA technique in the vicinity of filter pleats are presented.

Chapter 5 presents the performance of the filter along with clogging evolution, i.e., efficiency and pressure drop. Local experimental velocity profiles are also presented in this Chapter to evaluate the impact of clogging on the flow properties downstream of the filter pleats.

Finally, a general conclusion is drawn, and new perspectives for the continuation of the work are suggested.

Evolution of the filtration performance of fibrous filters during clogging

For a homogeneous fibrous filter, the evolution of the pressure drop (ΔP) and the particle collection efficiency during the clogging with solid particles can be described in detail by identifying three filtration stages for a flat filter as described in Figure 1 [4]:

Depth filtration

The change in pressure drop and filtration efficiency of the filter is linear at the start of the filtration process. The ΔP gradually increases while the total efficiency rapidly increases until the filter cake appears. During this phase, the particles are mostly deposited within the filtering medium, which has no effect on the fluid flow.

Transition phase

This corresponds to the beginning of a flow disturbance. The depth of the medium is densely packed with particles, and the surface is partially covered with particles. ΔP changes in an exponential way, while efficiency changes more slowly.

Surface or cake filtration

This refers to the formation of the filter cake, which causes a significant disturbance in the flow.

The evolution of ΔP and efficiency is linear. The ΔP increases much faster while the total efficiency remains constant.

For a pleated filter, two additional phases are observed:

Transition phase

The evolution of the pressure drop is no more linear, due to contentious reduction of the filtration surface due to the cake deposit.

Reduction in filtration area

The reduction of the filtration area becomes very significant, resulting in an exponential increase in the pressure drop. This phase is rarely observed under normal operating conditions, but it is frequently observed under degraded-mode operation, i.e. clogging caused by combustion aerosols [START_REF] Thomas | Clogging of ÿbrous ÿlters by solid aerosol particles Experimental and modelling study[END_REF]. The reduction of the effective surface area of the filter during clogging is highly dependent on the filter geometrical properties. Teng et al. [START_REF] Teng | Research on the influence of pleat structure on effective filtration area during dust loading[END_REF] recently demonstrated experimentally that the reduction of effective surface filtration is more significant for mini-pleats (Height H = 20.0 mm, width W = 1.3 -5.0 mm). The filter clogged with polydispersed ash particle (dp = 3 -100 µm) at a low filtration velocity of 0.04 m.s -1 .

Prediction of the filtering medium permeability

One of the most important factors in filter selection is the pressure drop. In general, the goal of any new filter design is to minimize it and, as a result, to reduce the energy expenditure. The total pressure drop produced by a filter includes two contributions: the pressure drop produced by the medium itself (the resistance to the flow) and the pressure drop produced by the singularities represented by the geometrical properties of the pleated medium. Over the last 50 years, scientists have worked to develop models that can predict the pressure drop of fibrous medium or filters, and more specifically the medium permeability, which is directly related to the pressure drop according to Darcy's Law.

In the literature, the permeability is generally assumed as being function of the fiber diameter 𝑑 𝑓 and the solid volume fraction 𝜑 (i.e. 1 -ε), i.e. the ratio of the total volume of fibers to bulk volume of the medium. According to Jackson and James [START_REF] Jackson | The permeability of fibrous porous media[END_REF] a permeability 𝐵 can be defined as follows:

To predict the permeability, the literature exhibits a variety of strategies and approaches, which are summarized in the following section.

Modeling strategies

Many permeability models are reported in the literature. These models are categorized into empirical [START_REF] Jackson | The permeability of fibrous porous media[END_REF][START_REF] Doormaal | Determination of permeability in fibrous porous media using the lattice Boltzmann method with application to PEM fuel cells[END_REF][START_REF] Davies | The Separation of Airborne Dust and Particles[END_REF], analytical [START_REF] Tomadakis | Viscous Permeability of Random Fiber Structures: Comparison of Electrical and Diffusional Estimates with Experimental and Analytical Results[END_REF][START_REF] Happel | Viscous flow relative to arrays of cylinders[END_REF][START_REF] Woudberg | Permeability prediction of an analytical pore-scale model for layered and isotropic fibrous porous media[END_REF], and numerical models, or can be combinations of different types of model [START_REF] Jackson | The permeability of fibrous porous media[END_REF][START_REF] Doormaal | Determination of permeability in fibrous porous media using the lattice Boltzmann method with application to PEM fuel cells[END_REF]. The cylindrical unit cell approach used by Happel and Kuwabara [START_REF] Happel | Viscous flow relative to arrays of cylinders[END_REF] was the first analytical attempt to predict the permeability of fibrous medium. The authors assumed that the fibrous medium is made of a set of unit cells, each containing a cylindrical fiber. As a result, rather than studying the flow in the whole fibrous medium, it is sufficient to solve Stokes flow equations in a unit cell. This approach does not take into account the effect of the environment (rest of fibers) on the airflow near the fiber. This effect was covered by Spielman and Goren [START_REF] Spielman | Model for predicting pressure drop and filtration efficiency in fibrous media[END_REF] who added a term to the Stokes equation resulting from the superimposition of Stokes equation and Darcy's law. Another analytical strategy, proposed by Hasimoto [START_REF] Hasimoto | On the Fundamental Solutions of the Stokes Equations and Their Application to Viscous Flow Past a Cubic Array of Spheres[END_REF], involved Fourier series applied to square arrays of simple, body-centered and face-centered lattices. A series of authors like Jackson and James [START_REF] Jackson | The permeability of fibrous porous media[END_REF], Tomadakis et al. [START_REF] Tomadakis | Viscous Permeability of Random Fiber Structures: Comparison of Electrical and Diffusional Estimates with Experimental and Analytical Results[END_REF], Woudberg et al. [START_REF] Woudberg | Permeability prediction of an analytical pore-scale model for layered and isotropic fibrous porous media[END_REF][START_REF] Woudberg | Investigating the influence of local porosity variations and anisotropy effects on the permeability of fibrous media for air filtration[END_REF] have reported detailed summaries of these strategies.

The number of numerical studies on flow through fibrous porous medium has increased significantly in recent years due to the development of powerful numerical tools that allows reducing the computational time [START_REF] Tahir | Influence of fiber orientation on the transverse permeability of fibrous media[END_REF][START_REF] Soltani | 3D Fiber Orientation Characterization of Nonwoven Fabrics using X-ray Micro-Computed Tomography[END_REF][START_REF] Jaganathan | a realistic approach for modeling permeability of fibrous media: 3-D imaging coupled with CFD simulation[END_REF]. This method entails numerically constructing the medium and coupling it to CFD simulations in order to solve the flow equations. Many authors have used techniques such as X-ray micro-computed tomography (e.g. Soltani et al. [START_REF] Soltani | 3D Fiber Orientation Characterization of Nonwoven Fabrics using X-ray Micro-Computed Tomography[END_REF]) digital volumetric imaging to build artificial fiber networks (e.g. magnetic resonance imaging). The flow is recreated by using numerical methods such as the Lattice Boltzmann method, the spectral boundary element method, the finite element method, to solve the Stokes equations. The advantage

𝐵 𝑑 𝑓 2 = 𝑓(𝜑) (1)
of these methods is that it takes into consideration the impact of the fiber orientation and medium structural properties on the flow approaching the filter.

Despite the fact that experimental investigations are generally time-consuming and costly (Soltani et al. [START_REF] Soltani | 3D Fiber Orientation Characterization of Nonwoven Fabrics using X-ray Micro-Computed Tomography[END_REF]), the data offered invaluable information for the validation of numerical and analytical models. Van Doormal and Pharaoh [START_REF] Doormaal | Determination of permeability in fibrous porous media using the lattice Boltzmann method with application to PEM fuel cells[END_REF] emphasize the importance of permeability predictions generated from analytical models in computational fluid dynamics models. Several writers have integrated numerical and analytical or empirical modeling techniques in recent years. In order to calculate the permeability for their drag model, Stylianopoulos et al. [START_REF] Stylianopoulos | Permeability calculations in three-dimensional isotropic and oriented fiber networks[END_REF], coupled their numerical technique with the volume averaging method. Nabovati et al. [START_REF] Nabovati | A general model for the permeability of fibrous porous media based on fluid flow simulations using the lattice Boltzmann method[END_REF] adjusted the coefficient values of integrated theoretical, numerical, and empirical model of Gebart et al. [START_REF] Gebart | Permeability of Unidirectional Reinforcements for RTM[END_REF], using their numerical data. In order to examine the influence of fiber orientation on permeability, Tahir and Tafreshi [START_REF] Tahir | Influence of fiber orientation on the transverse permeability of fibrous media[END_REF] employed the empirical permeability of Davies [START_REF] Davies | The Separation of Airborne Dust and Particles[END_REF] in their numerical simulations.

Following a review of the strategies used by the authors to model permeability, we will present a classification of the existing permeability models based on the orientation of the fibers with respect to the airflow. The fact that fiber orientation is taken into account in permeability is crucial since it takes into account the real anisotropic/isotropic nature of the filter tested.

Permeability model classifications according to the fiber orientation regarding the airflow direction

The analytical and empirical models presented in the literature are classified by fiber orientation regarding the airflow direction, which can be 1D with flow parallel or perpendicular to the unidirectional fibers axis, 2D with flow parallel (in-plane) or perpendicular (through-plane) to the medium plane, or 3D fiber arrangement randomly dispersed.

2D layered fibrous medium, in which the filter paper is made up of fibers that have been deposited onto a flat surface so that the fibers' axes are randomly distributed into planes parallel to the surface [START_REF] Spielman | Model for predicting pressure drop and filtration efficiency in fibrous media[END_REF], became very important in the filtration industry. As a result, more researchers have turned their focus to evaluating the permeability of these medium [START_REF] Tahir | Influence of fiber orientation on the transverse permeability of fibrous media[END_REF][START_REF] Soltani | 3D Fiber Orientation Characterization of Nonwoven Fabrics using X-ray Micro-Computed Tomography[END_REF][START_REF] Tamayol | Transverse permeability of fibrous porous media[END_REF][START_REF] Tamayol | Parallel Flow Through Ordered Fibers: An Analytical Approach[END_REF][START_REF] Gostick | In-plane and through-plane gas permeability of carbon fiber electrode backing layers[END_REF]. Some authors like Tamayol and Bahrami [START_REF] Tamayol | Parallel Flow Through Ordered Fibers: An Analytical Approach[END_REF] presented equations for estimating permeability for all categories (1D, 2D and 3D), but added empirical coefficients to each of them in order to get correlation with experimental results. Woudberg [START_REF] Woudberg | Permeability prediction of an analytical pore-scale model for layered and isotropic fibrous porous media[END_REF], on the other hand, offered an interesting analytical approach for predicting the permeability of 1D, 2D, and 3D fiber arrangements based on physical principles without the use of empirical coefficients. In contrast to the other authors, the geometry of the fibers is considered to be rectangular. For anisotropic medium, the author introduced RUC models, which are 2D geometrical models made up of rectangular fibers in a cubic representative unit cell.

Following that, further information is presented.

RUC models

Figure 2 (a) depicts the RUC model adopted by Woudberg [START_REF] Woudberg | Permeability prediction of an analytical pore-scale model for layered and isotropic fibrous porous media[END_REF] to predict the permeability of anisotropic fibrous medium, so called RUC TP. d is the cubic cell dimension. ds is the rectangular solid width of the fiber assumed to be equal to the cylindrical fiber diameter df. The average geometry of real fibers with random orientations in planes parallel to each other is represented by two solid fibers oriented in the two major 2D directions of a rectangular Cartesian coordinate system. The average fiber geometry surfaces next to the RUC in the through plane flow direction are also shown. The surrounding solid fiber layer is not included in the RUC, but its impact is taken into consideration when building the piece-wise parallel plate geometry between which flows occurs in the limit of low Reynolds number flow. Only the Through plane direction, i.e. the direction perpendicular to the plane of the medium, is studied in the RUC TP model. it corresponds to the direction of the flow regarding the fibrous medium considered experimentally to determine the permeability by permeametry. The RUC TP model equation is given in Table 1. The RUC model used by Woudberg [START_REF] Woudberg | Permeability prediction of an analytical pore-scale model for layered and isotropic fibrous porous media[END_REF] to predict the permeability of isotropic fibrous medium (RUC iso) is depicted in Figure 2 Woudberg [START_REF] Woudberg | Permeability prediction of an analytical pore-scale model for layered and isotropic fibrous porous media[END_REF] suggested an alternate analytical methodology to produce 3D isotropic permeability models by assigning a weight of 2/3 to the permeability for flow perpendicular to 2D fibrous medium and a weight of 1/3 to the permeability for flow parallel to 1D fibrous medium. By applying this approach to the RUC transverse and parallel analytical flow model as well as on

Happel [START_REF] Happel | Viscous flow relative to arrays of cylinders[END_REF] and Tomadakis and Robertson [START_REF] Tomadakis | Viscous Permeability of Random Fiber Structures: Comparison of Electrical and Diffusional Estimates with Experimental and Analytical Results[END_REF] models the RUC iso [START_REF] Woudberg | Permeability prediction of an analytical pore-scale model for layered and isotropic fibrous porous media[END_REF], Happel iso [START_REF] Happel | Viscous flow relative to arrays of cylinders[END_REF][START_REF] Woudberg | Permeability prediction of an analytical pore-scale model for layered and isotropic fibrous porous media[END_REF] and the alternative Tomadakis and Robertson iso models [START_REF] Tomadakis | Viscous Permeability of Random Fiber Structures: Comparison of Electrical and Diffusional Estimates with Experimental and Analytical Results[END_REF][START_REF] Woudberg | Permeability prediction of an analytical pore-scale model for layered and isotropic fibrous porous media[END_REF] respectively (see Table 1) were obtained.

1 4 [ 1 3 [ 53 8 𝜑 2 (1 -𝜑) 3 ] + 2 3 [ 9𝜋√6 16 ( √ 𝜋 2√3𝜑 -1) -5/2 ] ] -1 [[ 1 3 (1 -𝜑) 3 8𝜑 2 ] -1 + 2 3 [ (1 -𝜑) 3 exp(-𝛼𝜑) 8𝜑 2 (1 + 𝛼(1 -𝜑)) 2 ] -1 ] -1
Analytically weighted In general, the models under predict the experimentally measured permeability, which could be due to preferential passages of airflow through the real medium. Indeed, the models do not account for the heterogeneities of the structural properties of the fibrous medium, which can result in preferential airflow passages. Almost all previous models relied on average input data of structural properties such as fiber diameters and solid volume percentages. Permeability modeling gets more challenging for medium comprised of fiber mixtures, which typically have multimodal fiber diameter distributions and are commonly utilized in indoor air filtration applications. Few authors report encountering bimodal fiber distributions, owing to the scarcity of graphs displaying the distribution in the literature. In most of studies, authors did not take into account the bimodal aspect in the permeability prediction and assumed a unimodal fiber distribution.

Permeability of bimodal fiber diameter

Commonly, a mixture of fibers may be considered bimodal if the ratio of the modal diameters of both populations is between 2 and 5 [START_REF] Gervais | Permeability modeling of fibrous media with bimodal fiber size distribution[END_REF]. The coarse fibers improve the medium mechanical strength, and the fine fibers ensure the filtration efficiency [START_REF] Woudberg | Permeability prediction of an analytical pore-scale model for layered and isotropic fibrous porous media[END_REF][START_REF] Gervais | Permeability modeling of fibrous media with bimodal fiber size distribution[END_REF]. Mattern and Deen [START_REF] Mattern | Mixing rules" for estimating the hydraulic permeability of fiber mixtures[END_REF] demonstrated that the fundamental challenge in estimating the permeability of a fiber mixture is that the hydraulic resistances of fiber populations are interconnected. In other words, it is extremely difficult to obtain a realistic correlation for permeability in the case of a binary arrangement if the model is built separately for the force operating on thin fibers and the force acting on thick fibers.

The authors opted to compare four mixing methods from the literature with the results of simulations created with the code COMSOL Multiphysics TM based on a periodic arrangement of two types of fibers in two dimensions.

Starting from there, some authors calculated a unimodal equivalent diameter for the bimodal distribution using numerical simulation in order to incorporate it into the dimensionless permeability calculation [START_REF] Jaganathan | a realistic approach for modeling permeability of fibrous media: 3-D imaging coupled with CFD simulation[END_REF][START_REF] Fotovati | Analytical expressions for predicting performance of aerosol filtration media made up of trilobal fibers[END_REF]. These works, however, are limited to 2D representations of fibrous medium. From an analytical point of view, Tafreshi et al. [START_REF] Vahedi Tafreshi | Analytical expressions for predicting permeability of bimodal fibrous porous media[END_REF] compared several equivalent diameters for fibrous medium. They showed that the volume weighted diameter of Clague and Philips [START_REF] Clague | A numerical calculation of the hydraulic permeability of three-dimensional disordered fibrous media[END_REF], the area average weighted diameter of Brown and Thorpe [START_REF] Brown | Glass-fibre filters with bimodal fibre size distributions[END_REF] and their own equivalent diameter relationship give the best predictions for 0.05 ≤ φ ≤ 0.15 (i.e. for porosity between 85 and 95%) and coarse to fine fiber ratio between 1 and 5. The three afore-mentioned Chapter 1 Literature review 41 bimodal equivalent diameter formulas are resumed in Table 2. dc and df are the coarse and fine fiber diameters respectively, and nc and nf their corresponding number fraction in the fiber mixture.

The choice of the most suitable equivalent diameter is barely tackled in the literature. According to Jaganathan et al. [START_REF] Jaganathan | a realistic approach for modeling permeability of fibrous media: 3-D imaging coupled with CFD simulation[END_REF] and Mattern and Dean [START_REF] Mattern | Mixing rules" for estimating the hydraulic permeability of fiber mixtures[END_REF] the area-weighted average diameter proposed by Brown and Thorpe [START_REF] Brown | Glass-fibre filters with bimodal fibre size distributions[END_REF] yield the highest permeability values compared to the other methods (see Table 2). As the literature offers a broad variety of permeability models as well as equations to calculate equivalent diameters for cases of bimodal fiber diameter distribution, the question arises of the most accurate model to predict the permeability of a fiber mixture. The models of Davies [START_REF] Davies | The Separation of Airborne Dust and Particles[END_REF] and Jackson and James [START_REF] Jackson | The permeability of fibrous porous media[END_REF] are often used by the filtration community. These are empirical models, and as such, they are valid for the df and porosity ranges for which they were created, and especially for the medium for which they were developed. For example, Woudberg et al. [START_REF] Woudberg | Investigating the influence of local porosity variations and anisotropy effects on the permeability of fibrous media for air filtration[END_REF] revealed the limitations of these models in predicting the experimental permeability of fibrous medium with different structural properties (df, mean = 6.5 -13.8 m, 0.92 ≤εmean≤0.99, Bexp order of 10 -9 m 2 ). The isotropic Davies [START_REF] Davies | The Separation of Airborne Dust and Particles[END_REF] and isotropic Jackson and James [START_REF] Jackson | The permeability of fibrous porous media[END_REF] underestimated the permeability, resulting in a Bexp to Bmodel ratio ranging between 14.2 and 16.7.

Prediction of the particle collection efficiency of fibrous filters

In addition to pressure drop, efficiency is the most important criterion to consider when evaluating filter performance. Various studies in the literature address how to determine the efficiency of a fibrous filter based solely on knowledge of the structural properties of the filter medium, particles, The filtration efficiency of a fibrous filtering medium is determined by the capacity of its structural fibers to collect particles with defined diameters; thus, efficiency for a single fiber can be defined as "total unit efficiency of a fiber" (η). This single-fiber efficiency can be calculated using the unit efficiencies of particle capture by the several mechanisms derived from semi-empirical and theoretical models.

For calculating the unit efficiency of filtration, two approaches stand out. Both are based on the flow of fluid around an obstacle. The first is to use the best empirical or theoretical efficiency model that adjusts to experimental results, and the second is to numerically calculate the aerosol trajectories around the fibers.

The first approach is much more common in the literature to evaluate the unit efficiency of fiber collection; thus, many expressions are used to calculate the overall and unit efficiency for each mechanism, primarily Brownian diffusion, inertial impaction, interception, and the combined effect of diffusion and interception.

This section focuses on the particle capture mechanisms that occur during the fibrous filtering process, as well as the unit efficiencies that result from them.

Particle capture mechanisms

Brownian diffusion

To understand the diffusion mechanism of aerosols captured on fiber filters, the kinetic theory of gas has to be reminded. Each particle in a gas stream is modeled as a hard sphere moving in a straight line as long as they do not collide with one another; the particle size is small in comparison to the intermolecular distance. Actually, these particles are moving at random (Brownian motion)

and colliding with each other as much as they are moving at random along a zigzagging path [START_REF] Macdougall | An Introduction to the Kinetic Theory of Gases. By Sir James Jeans[END_REF].

This Brownian movement is the driving force behind the diffusion mechanism, which is extremely effective for small particles with diameters less than 0.1 µm. In fact, small particles have more time to zigzag, and thus have more chance of sticking to a fibrous filter [START_REF] Lee | Theoretical Study of Aerosol Filtration by Fibrous Filters[END_REF]. The diffusion mechanism is illustrated in Figure 3. When the small particle size value is close to their mean free path λ, the Brownian diffusion motion of the particles is defined by the diffusion coefficient D in function of the Cunningham coefficient (Cu). Cu is a correction factor that takes into account the non-continuous aspect of the gaseous medium λ (6.4 x 10 -8 at normal pressure and temperature). The diffusion coefficient is calculated as follows:

Where k corresponds to the Boltzmann constant (k = 1.38.10 -23 J.K -1 ), T is the air temperature (K), μ is the dynamic viscosity of the fluid (Pa.s) and dp is the particle diameter (m).

Cu is the Cunningham correction coefficient that depends on Knudsen number and takes into account the discontinuities of the medium. It can be expressed as follow:

The theoretical diffusion filtration efficiency is calculated in function of the dimensionless Peclet number Pe:

D = kTCu 3πµd p (2)
Cu = 1 + kn p [1.155 + 0.471 exp (- 0.596 Kn p )] (3)
Kn p = 2λ d p (4)
Chapter 1 Literature review 44

Where df is the fiber diameter (µm) and Vfiltration is the filtration velocity (m.s -1 ). The Peclet number describes the relationship between particle transportation via convection and diffusion; note that particle diffusion becomes more important as particle diameter decreases. Many models that define the theoretical diffusion efficiencies for fibers can be found in the literature; the most used models in the literature, are listed in Table 3. Most of them was reviewed by Thomas et al. [START_REF] Thomas | Aerosol Filtration[END_REF] and completed throughout this study.

Interception

Capture by interception happens when a particle that follows a streamline, comes within one molecule radius on the fibers' surface. The particle is captured and removed from the gas once it Pe = d f V filtration D

(5)

Chapter 1 Literature review 45 touches the fiber. This phenomenon is independent of the velocity, and the efficiency of capture increases whenever the particle size increase or the fiber diameter decrease. The interception mechanism is illustrated in Figure 4.

Figure 4 Illustration of particle capture on fibers by the interception mechanism

The interception is characterized by the interception number R in function of the fiber df and the particle diameter dp, There are numerous models in the literature that define theoretical interception efficiencies for fibers; the most common models reviewed by Thomas et al. [START_REF] Thomas | Aerosol Filtration[END_REF], are listed in Table 4.

Cai cited by Miecret [START_REF] Thomas | Aerosol Filtration[END_REF][START_REF] Miecret | Mathematic Expression of HEPA and ULPA Filters Efficiency, Experimental Verification, Practical Alliance to New Efficiency Test Method[END_REF] 1989 2.4𝛼

Inertial impaction

When a large particle with a finite mass is displaced from the air flow line near the fiber because it cannot adjust quickly to changes in streamline direction, particle collection by inertia mechanism occurs.

Figure 5 Illustration of particle capture on fibers by the inertial impaction mechanism

The dimensionless Stokes number (St) is used to quantify the inertial impaction mechanism:

Where ρp is the density of the particles (kg.m -3 ), and Vfiltration is the filtration velocity (m.s -1 )

Table 5 lists the main models available in the literature that define the particle collection efficiency by impaction for fibers. It is more common for large particles (dp > 1 µm) to occur depending on the filter medium, and it becomes noticeable for high fiber density [START_REF] Hakobyan | Introduction to Basics of Submicron Aerosol Particles Filtration Theory via Ultrafine Fiber Media[END_REF]. An illustration of the sieving mechanism is presented in Figure 6.

Electrostatic forces

When aerosol particles and fibers are charged, this mechanism occurs. The use of electrostatic forces can improve the collection efficiency of a fibrous filter, particularly for the capture of medium-sized particles that are difficult to collect using other mechanisms. When electrostatic forces interact, there are three possible outcomes [START_REF] Renoux | Les aérosols : physique et métrologie[END_REF]:  Charge-Dipole: either the fiber or the particle is charged; when they interact, the charged body induces a dipolar moment in the uncharged one.

 Dipole-Dipole: both uncharged particles and fiber are located in an electrical field that induces dipolar moments in both bodies, causing them to attract each other.

An illustration of the electrostatic attraction mechanism is presented in Figure 7.

Other mechanisms

While the fundamental mechanisms for submicron particle filtration are dominant, there are a couple of others that may be critical under certain conditions. These various mechanisms include the Van der Waals forces [START_REF] Kirsch | The Effect of van der Waals' Forces on Aerosol Filtration with Fibrous Filters[END_REF] and thermophoretic drive [START_REF] Tan | Anisotropic thermophoresis[END_REF], among others.

Because fine particle removal occurs on a thin gas layer attached to the fine fiber, a precise understanding of Van der Waals forces and aerosol close to the ultrafine fiber surface is required to model the particle capture process. The following is an example of the capture coefficient upon particle deposition caused solely by Van der Waals forces [START_REF] Kirsch | The Effect of van der Waals' Forces on Aerosol Filtration with Fibrous Filters[END_REF]:

η w = 0.143( A. Cu. d p 2 K. µ . V filtration 5/2 ) 2/7 (8)
Chapter 1 Literature review 49

Where A is the constant of the Van der Waal's interactions and Vfiltration the filtration velocity (m.s -1 ), Cu is the Cunningham factor (-), K is the Boltzmann constant (J.K -1 ), µ is the dynamic viscosity and dp is the aerodynamic particle diameter (m).

The thermophoresis mechanism can also be cited among the different mechanisms [START_REF] Tan | Anisotropic thermophoresis[END_REF]. It occurs when a small particle is suspended in a gas with a temperature gradient, the particle is subjected to a force directed in the opposite direction of the temperature gradient.

Some authors such as Stechkina et al. [START_REF] Stechkina | Studies of fibrous aersol filters-IV Calculation of aerosol deposition in model filters in the range of maximum penetration[END_REF], Mirecret and Gustavson [START_REF] Miecret | Mathematic Expression of HEPA and ULPA Filters Efficiency, Experimental Verification, Practical Alliance to New Efficiency Test Method[END_REF], and Kirsh et al. [START_REF] Kirsch | The Deposition of Aerosol Submicron Particles on Ultrafine Fiber Filters[END_REF] consider the interaction between the diffusion and interception mechanism ηdr . Their equations include in addition to Pe number and R number, the Hydrodynamic factor H associated to the flow field around the cylindical fiber. The expression of ηdr and H are resumed in Table 6.

The total unit efficiency of a fiber

The unit filter efficiency η of a fiber depends on the unitary fractional efficiencies related to the mechanisms of particle captures, for example ηd for the diffusion, ηr for the intreception, ηI for the impaction, ηdr for the combined diffusion-interception, ηel for the electrostatic interactions.

The simplest approach is to assume that the various mechanisms (diffusion, impaction, interception, etc.) interact independently, and thus the overall unitary efficiency can be expressed using the expression of Kasper et al. [START_REF] Kasper | The collection efficiency of a particleloaded single filter fiber[END_REF]:

(1 -η) = ∏ (1 -η i ) i (9)
Chapter 1 Literature review 50 For the interception, impaction, and diffusion, the equation can be written as follow: Table 7 lists the prediction models for total unit filter efficiencies found in the litterature.

Influence of various parameters on the filtration performance

This section investigates the effect of various parameters on the clogging of flat or pleated filters by a solid aerosol: the particle size, the filtration velocity and the medium structural properties.

Influence of the particle size on the structure of the cake deposit

The structure of the deposit depends on the collection mechanism. Thus, submicron (dp < 1 µm) particles that are collected by interception, diffusion or electrostatic mechanisms tend to form a dendrite structure. However, micron particles mainly collected by inertial mechanism tend to form aggregates. The forces applied on the collected particles i.e. adhesion forces, drag forces and gravity, influence also the structure of the deposit.

η = 1 -(1 -η r )(1 -η I )(1 -η d ) (10)
Chapter 1 Literature review 51

As shown in Figure 8, if Brownian diffusion is dominant (at low Pe) and there is no interception, the deposit around the fiber is dendritic-like and homogeneous. However, when inertial impaction is dominant (at high St) and there is no interception, the deposition is compact and oriented to face the flow. The formation of dendrite structure is favored by additional interception [START_REF] Kanaoka | The Morphology of Particle Agglomerates on a fiber when inertia and interception are predominant collection mechanisms[END_REF].

In the first case (dendrites), the deposit is more aerated and thus has a higher porosity. However, the smaller the particle diameter, the greater the specific surface area, and thus the greater the pressure drop of the filter. For larger particles, the deposit is more compact, but the specific surface area is lower than for submicron particles, resulting in a lower pressure drop for the same mass of particles deposited. Similarly, the transition zone appears for a surface density of collected particles as the particle diameter decreases. In terms of filtration efficiency, Thomas et al. [START_REF] Thomas | Clogging of ÿbrous ÿlters by solid aerosol particles Experimental and modelling study[END_REF] demonstrate that particles of 0.18 µm in diameter penetrate deeper into the medium than particles of 0.31 µm in diameter for flat filters. The observations for pleated filters are the same as for flat filters.

Influence of the filtration velocity

The velocity at which particles reach the fibrous medium is referred in the literature as the filtration velocity, which is calculated as the ratio between the air flow rate approaching the filter and the developed filter medium surface. This velocity is one of the critical parameters that affect the filtration performance: the initial pressure drop and the particle collection efficiency.

Pressure drop

The filter pressure drop is proportional to the filtration velocity (Darcy's law), thus in order to assess the effect of the filtration velocity on the pressure drop evolution during clogging, the authors define a ratio ΔP/Vfiltration that varies in function of the collected mass of particle. In case of flat filter (dp = 0.18 -0.31 µm, z = 370 -575 mm) and for a tested filtration velocity of 0.01 -0.50 m.s -1 , Thomas et al. [START_REF] Thomas | Clogging of ÿbrous ÿlters by solid aerosol particles Experimental and modelling study[END_REF] found that all of the curves describing the evolution of the ratio are ΔP/Vfiltration in function of the collected mass are superimposed. This means that the structure of the deposit formed by the particles within the medium and at its surface is independent of the filtration rate over the entire filtration rate range.

In case of pleated filter, the observations are not the same. The pleating process leads to contract the air at the entry of the pleat and by consequence to an additional pressure drop that increases with filtration velocity [START_REF] Fotovati | A macroscale model for simulating pressure drop and collection efficiency of pleated filters over time[END_REF][START_REF] Fotovati | Modeling instantaneous pressure drop of pleated thin filter media during dust loading[END_REF][START_REF] Saleh | Modeling service life of pleated filters exposed to poly-dispersed aerosols[END_REF][START_REF] Feng | Modeling filtration performance of pleated fibrous filters by Eulerian-Markov method[END_REF]. As a result, the effect of increasing the velocity upstream due to the pleating on particle deposition and, as a result, pressure drop has been numerically investigated in a series of studies presented in section 6.3.

Filtration efficiency

The dependence of collection efficiency on velocity has been well investigated based on the models of particle collection by single fiber, as demonstrated in the previous section. In general, filtration modelling can predict the most penetrating particle size for specific filtration velocity and filter medium values.

Indeed, the Peclet number Pe ( 5) is used in the models that predict particle collection with diffusion (see Table 3). Furthermore, the Stokes number St [START_REF] Jackson | The permeability of fibrous porous media[END_REF] is used in the models that predict particle collection by impaction. These dimensionless numbers are directly proportional to the filtration velocity. The single fiber efficiency models that predict impaction are proportional to St number, whereas the diffusion models are inversely proportional to Pe number. As a result, a high filtration velocity value tends to degrade Brownian diffusion collection and promote inertial impaction collection. In case of submicron particles (5-40 nm), Charvet et al. [START_REF] Charvet | Association of fibrous filters for aerosol filtration in predominant Brownian diffusion conditions[END_REF], demonstrated that increasing the filtration velocity from 1.3 cm.s -1 to 3.8 cm.s -1 reduced the efficiency of HEPA and Chapter 1 Literature review 53 medium efficiency filters. These findings are consistent with the filtration velocity effect on Brownian diffusion. Numerical simulation using CFD has also demonstrated the significant influence of the filtration velocity on the particle collection efficiency. Lehmann et al. cited by Kasper et al. [START_REF] Kasper | The collection efficiency of a particleloaded single filter fiber[END_REF], calculated the flow field after particle deposition in their 2-D simulations. A circle with a diameter of 2 µm was used to model single fiber efficiency. They demonstrated that collection efficiency and pressure drop are strongly related to clogging packing density, which is strongly affected by the flow velocity field. Based on these findings, Kasper et al. [START_REF] Kasper | The collection efficiency of a particleloaded single filter fiber[END_REF] added a sticking probability of the particle to the single fiber to their model. Indeed, not every particle that comes into contact with the fiber is collected. "Bouncing -effect" occurs when some particles bounce back once they hit the fiber, this effect is amplified for micron particles at high filtration velocities [START_REF] Maddineni | Air-borne particle capture by fibrous filter media under collision effect: A CFD-based approach[END_REF]. Accordingly, the characterization of the airflow downstream of filter pleats, based on local parameters such as velocity measurements could identify possible trends that help in understanding non-stationary phenomenon like particle bouncing and re-entrainment

Influence of filter structural parameter and operating conditions on particle collection efficiency

Thomas et al. [START_REF] Thomas | Aerosol Filtration[END_REF] investigated the effect of filter structural characteristics and operating conditions on particle collection efficiency. Table 8 summarizes the parameters that affect the particle efficiency of filtering mechanisms. These results were obtained from clogging using fibrous filters in flat configuration in the literature [START_REF] Payet | Penetration and pressure drop of a HEPA filter during loading with submicron liquid particles[END_REF][START_REF] Gougeon | Filtration des aérosols liquides par les filtres à fibres en régimes d'interception et d'inertie[END_REF]. Diethyl sebacat and NaCl particles are used, with a range of aerodynamic particle size diameters of 0.01-1 µm and 0.02 and 0.5 µm, respectively.

Average structural parameters are examined. The average fiber diameter is between 1.1 and 2.7 µm. The average filtration velocity of the investigated medium is between 0.02 and 0.05 m.s -1 .

Despite the fact that none of the models provided a satisfactory agreement with the experimental results, it was discovered that the structural characteristics of the medium, as well as the filtration velocity, have a significant influence on collection through diffusion, interception, and inertia.

Chapter 1 Literature review 54

↘ ↘ ↗ ↘ ↗ ↘ Impaction ↗ ↗ ↗ ↘ ↗ ↗ Interception - ↗ ↗ ↘ ↗ - ↗: Increase ↘: Decrease

Theoretical prediction of the initial pressure drop of pleated filters

Many models derived from experimental data have been reported in the literature [START_REF] Del Fabbro | Air flows and pressure drop modelling for different pleated industrial filters[END_REF][START_REF] Chen | Optimization of Pleated Filter Designs Using a Finite-Element Numerical Model[END_REF], but practically all of them were restricted to the range of medium characteristics and filtration velocity of that they modeled. Indeed, the authors emphasize that their proposed models underestimate the experimental pressure drop at high filtration velocity. In 2007, Callé-Chazelet cited by Thomas et al. [START_REF] Thomas | Aerosol Filtration[END_REF], proposed an analytical approach based on energy balances to predict the initial pressure drop of the pleated filter, which has more physical meaning than the other empirical models.

Based on the work of Del-Fabbro et al. [START_REF] Del Fabbro | Air flows and pressure drop modelling for different pleated industrial filters[END_REF], Callé-Chazelet cited by Thomas et al. [START_REF] Thomas | Aerosol Filtration[END_REF], determines the pressure drop across a clean pleated filter ΔPpleated,0 as the sum of the initial pressure drop of the flat filter ΔPflat,0 (determined by Darcy's law) and the pressure drop at the singularities ΔPS, i.e. that of the geometrical effect such as contraction and widening.

The pressure drop associated to the change in the geometry of the pleat, is proportional to the kinetic energy of the air Where ζ is the pressure drop coefficient across the singularities, ρ the air density (kg.m -3 ), Vmean is the mean air duct velocity approaching the pleat (m.s -1 ). By adapting the model to the Del-Fabro [START_REF] Del Fabbro | Air flows and pressure drop modelling for different pleated industrial filters[END_REF] experimental results, the pressure drop coefficient ζ can be attributed to the pleat width W, yielding in equation

∆𝑃 𝑝𝑙𝑒𝑎𝑡𝑒𝑑,0 = ∆𝑃 𝑓𝑙𝑎𝑡,0 + ∆𝑃 𝑠 ( 11
)
∆𝑃 𝑠 = 1 2 𝜁𝜌𝑉 𝑚𝑒𝑎𝑛 2 (12)
Chapter 1 Literature review 55

Where H is the height of the pleat and Vfiltration is the filtration velocity (m.s -1 ). The models succeed in predicting the experimental pressure drop for filtration velocity ranging between 0-0.12 m.s -1 ; however, this model takes into consideration the kinetic energy of the fluid at the entrance of the pleat and neglects the airflow behavior within the pleats that influences the pressure drop. Given the difficulty of assessing the impact of numerous parameters experimentally on the pressure drop of a pleated filter, the authors used numerical methods such as CFD to predict the airflow within the pleat and, as a result, the initial pressure drop and/or particle collection efficiency of the filter.

Numerical Airflow study of at the vicinity of pleated filter in clean and clogged state

Many studies reported in the literature dealt with the improvement of the filtration performance (i.e. lowering the pressure drop while ensuring the targeted filtration efficiency) of pleated filters by combining semi-empirical models to numerical simulations. These approaches cover various filter types (and thus their properties), operating conditions, and applications. As a result, it is difficult to draw broad conclusions about the filter design of a pleat filter because each case must be considered separately. It appears that the pressure drop of the filter is generally taken into account because it is directly related to the filter's energy consumption when the fan in the HVAC system is rotating at a fixed velocity. To this end, the main challenge encountered by the authors was to take into consideration the airflow pattern approaching pleated filters. CFD was an effective tool to study the air flow at the vicinity of the pleated filter. In this section, after presenting the principle of numerical simulations, we will review the numerical studies that dealt with air flow in the vicinity of fibrous pleated filter in its clean state and also during clogging.

Generalities about numerical simulations

The flow of air through the filters is governed by the Navier-Stokes equations (NS). The general form is given in equation [START_REF] Tahir | Influence of fiber orientation on the transverse permeability of fibrous media[END_REF]. Flow numerical simulation is based on the solution of these equations using digital strategies and algorithms that can be implemented and solved on computing stations. Indeed, CFD is defined as "the art of replacing the integrals or partial derivatives (as

∆𝑃 𝑝𝑙𝑒𝑎𝑡𝑒𝑑,0 = ∆𝑃 𝑓𝑙𝑎𝑡,0 + 0.278 𝑊 ( 𝐻 𝑊 )𝜌𝑉 𝑓𝑖𝑙𝑡𝑟𝑎𝑡𝑖𝑜𝑛 2 (13)
Chapter 1 Literature review 56 applicable) in the Navier-Stokes equations with discretized algebraic forms, which are then solved to obtain numbers for the flow field values at discrete points in time and/or space" [START_REF] Dick | Introduction to Finite Volume Methods in Computational Fluid Dynamics[END_REF] With V is the fluid velocity (m.s -1 ), P is the Pressure (Pa) μ the dynamic viscosity (Pa.s), 𝜌 fluid density (kg/m3), and F the sum of the external forces applied on the fluid (N)

To carry out a numerical simulation, the first step is to mesh the computation domain (discretization in space), and then the equation is solved at the scale of each cell, which can be done in one of three ways:

Direct numerical simulations (DNS)

Direct numerical simulation is one in which the Navier-Stokes equations are numerically solved without the use of a turbulence model. This means that the turbulence's entire range of spatial and temporal scales must be resolved. This method requires a very large number of cells and a lot of computation time, thus it is rarely used in the industrial domain.

Large eddy simulations (LES)

The basic principle underlying LES is to reduce computing cost by filtering the NS equations. Only the "large" structures of the flow are explicitly solved, while the smallest are taken into account using so-called "Under mesh" models.

Reynolds Averaged Navier-Stokes Simulation (RANS)

RANS is an attempt to calculate average velocity fields and pressure by averaging the NS equations through statistical models. This method is substantially more efficient in terms of calculation time, but it introduces a new term to the equation: The Reynolds tensor. The latter describes the fluid's velocity fluctuations around its mean velocity. In order to overcome this issue, turbulence models are used. The most common models are listed below:

 The k-ε is a widely used model with two equations (the turbulent kinetic energy k and its dissipation rate ε). This model best suits to free shear (away from walls) areas with low pressure gradients. Regarding numerical simulations of the airflow in the vicinity of pleated

𝜌𝑉(V. ∇) = -∇𝑃 + 𝜇∆𝑉 + 𝐹 ⃗ (14)
Chapter 1 Literature review 57 filters, this model was commonly used to model the airflow for the turbulent flow regime [START_REF] Tronville | Minimization of Resistance in Pleated-Media Air Filter Designs: Empirical and CFD Approaches[END_REF][START_REF] Théron | Numerical and experimental investigations of the influence of the pleat geometry on the pressure drop and velocity field of a pleated fibrous filter[END_REF][START_REF] Feng | Assessment of various CFD models for predicting airflow and pressure drop through pleated filter system[END_REF]  The 𝑘-𝜔 model introduces two equations for k and ω that can be considered as the ratio k / ε. Thus, ω represents the reciprocal of the characteristic time of dissipation of turbulent kinetic energy. This model appears to be more suitable for near-wall treatment [START_REF] Wilcox | Formulation of the k-w Turbulence Model Revisited[END_REF]  The 𝑘-𝜔 SST model is a combination of both previous models; the model 𝑘-𝜔 is used in close wall zones, while the model k-ε is used in internal shear zones [START_REF] Sunny | A Comparison of Predictions by SST and Wilcox k-w Models for a Mach 0.9 Jet[END_REF].

 The transition-SST model consists of coupling the 𝑘-𝜔 SST transport equations with two other transport equations: one for transition onset criteria and one for intermittency, regarding the momentum-thickness Reynolds number.

 RSM This method does not rely on turbulent viscosity (Boussinesq's hypothesis [START_REF] Schmitt | About Boussinesq's turbulent viscosity hypothesis: historical remarks and a direct evaluation of its validity[END_REF]), but rather on a complete resolution of the Reynolds tensor. This is a six equations model, as opposed to previous models. Like the k-ε model, this model has been used for predicting the pressure drop of a pleated fibrous filter [START_REF] Tronville | Minimization of Resistance in Pleated-Media Air Filter Designs: Empirical and CFD Approaches[END_REF][START_REF] Théron | Numerical and experimental investigations of the influence of the pleat geometry on the pressure drop and velocity field of a pleated fibrous filter[END_REF][START_REF] Feng | Assessment of various CFD models for predicting airflow and pressure drop through pleated filter system[END_REF].

There are also hybrid methods for solving the NS equations. For example, the DES (Detached Eddy Simulation) approach combines the RANS and LES models. The flow is resolved using a RANS approach close to the wall, and the LES method is used internally. [START_REF] Feng | Assessment of various CFD models for predicting airflow and pressure drop through pleated filter system[END_REF][START_REF] Kang | Characterization of pleated filter media using particle image velocimetry[END_REF].

The RANS approaches is adopted in this study; all the mathematical equations are provided by ANSYS fluent code.

Airflow study at the vicinity of clean pleated filter

The main challenge in the filtration domain is to optimize the design of a filter, i.e. to minimize pressure drop while increasing filter efficiency. The pleating process causes the air at the pleat's entrance to contract, resulting in an additional pressure drop that increases with filtration velocity.

The authors' main challenge has been to account for velocity variations near the filter pleat, which can affect filtration performance. To this end, numerical simulations such as computational fluid dynamics were broadly used in the literature to study the air flow at the vicinity of filter pleats.

Chapter 1 Literature review 58

The first calculations to simulate the airflow in the vicinity of a pleated filter goes back to the earlies nighties, and aimed at studying the effect of the pleat geometry on the pressure drop [START_REF] Chen | Optimization of Pleated Filter Designs Using a Finite-Element Numerical Model[END_REF][START_REF] Raber | Predicting dust filter performance: pressure drop optimisation and dust capacity estimation for a deep-pleated industrial air filter using small sample data, predicting dust filter performance: pressure drop optimisation and dust capacity estimation for a deep-pleated industrial air filter using small sample data[END_REF][START_REF] Yu | Optimized Ultra High Efficiency Filter for High-Efficiency Industrial Combustion Turbines[END_REF]. The common point between these studies is to create a simplified model that can describe the flow at the vicinity of the pleated medium to avoid the extensive computational time of direct CFD calculation. All of these models have one thing in common: they simplify the NS equations to produce a one-dimensional system of equations. These models differ based on the assumptions and simplifications used. These studies have been limited to rectangular pleats with uniform width, Among these studies, Chen et al. [START_REF] Chen | Optimization of Pleated Filter Designs Using a Finite-Element Numerical Model[END_REF] created a numerical model to optimize the design of a pleated filter in order to discover the ideal pleat number with the minimum pressure loss. The upstream flow in this model is normal to the filter plane, parallel to the pleating direction, and flows through the medium until it reaches the downstream zone of the pleat. The pressure drop is caused by the contraction of the upstream flow, the expansion of the downstream flow, and the viscous drag, in contrast to previous works [START_REF] Raber | Predicting dust filter performance: pressure drop optimisation and dust capacity estimation for a deep-pleated industrial air filter using small sample data, predicting dust filter performance: pressure drop optimisation and dust capacity estimation for a deep-pleated industrial air filter using small sample data[END_REF][START_REF] Yu | Optimized Ultra High Efficiency Filter for High-Efficiency Industrial Combustion Turbines[END_REF]. The airflow is modeled in the upstream and downstream zones by a steady laminar flow. The permeability of the porous zone is predicted by use the Darcy-Lapwood-Brinkman equations. Figure 9 depicts the numerical domain adopted by the authors. It was limited to a 2D -rectangular half-pleat inserted between symmetry conditions. Following the same approach as previously mentioned authors [START_REF] Chen | Optimization of Pleated Filter Designs Using a Finite-Element Numerical Model[END_REF][START_REF] Raber | Predicting dust filter performance: pressure drop optimisation and dust capacity estimation for a deep-pleated industrial air filter using small sample data, predicting dust filter performance: pressure drop optimisation and dust capacity estimation for a deep-pleated industrial air filter using small sample data[END_REF][START_REF] Yu | Optimized Ultra High Efficiency Filter for High-Efficiency Industrial Combustion Turbines[END_REF], Rebaϊ et al. [START_REF] Rebaï | A semi-analytical model for gas flow in pleated filters[END_REF],

develop a semi-analytical model for air flow within a pleated filter to predict pressure drop across a filter panel used in automotive air applications. Their approach is based on a special averaging of the NS equations over the pleat's entrance and takes advantage of the pressure drop and velocity profile similarities within the pleat. The study area is reduced to one rectangular or triangular halfpleat as shown in Figure 10.

Figure 10 Calculation domain

Darcy's law predicts the permeability of the porous zone. Based on the permeability of the pleat curved zone, two sub-models were proposed: fully permeable and impermeable pleat zone. This approach refines the work of Yu and Goulding [START_REF] Yu | Optimized Ultra High Efficiency Filter for High-Efficiency Industrial Combustion Turbines[END_REF] by taking into account, in addition to a uniform width pleated filter i.e. rectangular geometry, a variable width of the pleat when resolving equations i.e. a triangular pleated geometry. This study was inspired by the study carried out by Oxarango et al. [START_REF] Oxarango | Laminar flow in channels with wall suction or injection: a new model to study multi-channel filtration systems[END_REF] in liquid filtration application. According to Rebaϊ 's model [START_REF] Rebaï | A semi-analytical model for gas flow in pleated filters[END_REF], the velocity of filtration does not remain constant with pleat depth. This is likely to have an effect on particle deposition in the pleat, resulting in a preferential deposit at the bottom. By comparing the model to experimental pressure drop data, it was discovered that this model is only suitable for high filtration velocities and fail to predict the pressure drop at low filtration velocities. The model is also sensitive to the Chapter 1 Literature review 60 medium's low permeability at the curved zone of the pleat, i.e. the pleat bottoms and peak. This is attributable to the fact that these areas exhibit changes in medium structural properties that are difficult to reproduce numerically. The resolution of the developed set of equations thus represents a good approach to estimating the pressure drop of the clean filter, but further work on the pleat bottom is required to complete this work. Similarly, to the previous work, the authors confirmed the existence of an optimal pleat geometry for a given inlet velocity in the tested filtering panel, namely (H = 51 mm, W = 25 mm) for a filtration velocity of 0.31 m. s -1 .

Fotovati et al. [START_REF] Fotovati | Modeling instantaneous pressure drop of pleated thin filter media during dust loading[END_REF] performed numerical simulations with the same goal as the previous studies, namely determining the optimal number of pleats in a high-efficiency HEPA filter. Rectangular pleats of varying widths (W = 2.1 -6.4 mm) and triangular pleats of varying angles (angle = 2 o -20 o ) were simulated. The flow equations are the same as those used by Chen et al. [START_REF] Chen | Optimization of Pleated Filter Designs Using a Finite-Element Numerical Model[END_REF]. What is interesting about this work is that it takes the anisotropic nature of the medium into account. In contrast to previous works, the fibrous medium is modeled as a multilayered porous structure with a tensor of permeability (in-plane and through-plane) rather than a mean isotropic permeability value (see section 2.2). In their previous work [START_REF] Fotovati | Analytical expressions for predicting performance of aerosol filtration media made up of trilobal fibers[END_REF], the authors demonstrated that the fibers in the majority of HEPA are randomly oriented in directions parallel to the plane of the pleat, and thus the medium is not considered isotropic. Indeed, the authors used Spielmen and Goren's [START_REF] Spielman | Model for predicting pressure drop and filtration efficiency in fibrous media[END_REF] expression to calculate the in plane and through plane permeability components. The computational fluid mechanics code ANSYS Fluent® was used for the simulations. According to their numerical simulations, the author concludes that neither the inlet velocity of the filter nor the fiber orientation have any effect on the optimal number of pleats required for a minimum pressure drop.

In all the numerical approaches mentioned in this the section, the pressure drop was the only parameter to look out for the optimization of the filter design. All the authors agreed that there is an optimal pleat geometry at the which the pressure drop is minimal. However, none of the previous work included the effect of the particle deposition on the particle collection efficiency, in addition to the pressure drop.

Airflow studies at the vicinity of a pleated filter during clogging

As it was discussed previously, numerical simulations are an effective tool to optimize the filter design in clean conditions. However, only few studies investigated numerically the effect of the Chapter 1 Literature review 61 particle clogging on the filtration performance. As it was described in section 6.2, the non-linear increase of the pressure drop is related, in addition to the reduction in filter surface, to a heterogeneous structure within the pleats. The heterogeneity of the particles deposit depends on the airflow conditions as explained in section 4. The numerical simulation of particle deposition within the pleat consists of solving the NS's equations of the airflow, and also simulating the particle motion. Both Lagrangian and Eulerian numerical approaches were adopted to describe the particle motion.

 The Lagrangian method consist of injecting a group of particle called batch and then calculating the trajectory of each article individually by using Newton's second law. [START_REF] Fotovati | A macroscale model for simulating pressure drop and collection efficiency of pleated filters over time[END_REF][START_REF] Fotovati | Modeling instantaneous pressure drop of pleated thin filter media during dust loading[END_REF][START_REF] Saleh | Modeling service life of pleated filters exposed to poly-dispersed aerosols[END_REF][START_REF] Saleh | A simple semi-numerical model for designing pleated air filters under dust loading[END_REF]  The Eulerian methods is based on continuum phase particle concentration and consist of solving an additional transport equation. [START_REF] Feng | Modeling filtration performance of pleated fibrous filters by Eulerian-Markov method[END_REF] The limitation of these methods is the large computational time. Indeed, many iterations are required at each particle injection to calculate the instantaneous variation of pressure drop and velocity gradient within the pleat. In the literature, the previously described methods have been broadly used to predict the performance of pleated filter.

Fotovati et al. [START_REF] Fotovati | Modeling instantaneous pressure drop of pleated thin filter media during dust loading[END_REF], adopted a laminar airflow model to solve the NS's equations and a Lagrangian approach to simulate the particle motion. The aim of their work was to predict the evolution of the pressure drop and collection efficiency of a pleated filter during clogging with micron particles (dp = 0.3 and dp = 10 µm) and at a mean duct velocity ranging between 0.2 and 1 m.s -1 . Triangular and rectangular pleat shapes are considered and two geometries have been tested (H = 25.4 mm -W = 8.5 mm) and a narrow width geometry (H = 25.4 mm -W = 1.3 mm). They observed that in case of the narrow width geometry, the velocity gradient upstream of the filter increased, resulting in a non-uniform deposit within the pleats. The particle tends to deposit deeper inside the pleat, i.e.

pleat hollow, for large particles (dp = 10 µm). The shape of the filter pleats plays a major role in this phenomenon. It appears that this effect was less pronounced in case of triangular pleats compared to triangular pleats. However, this study is only numerical and considers only the surface filtration, thus the penetration of the particle inside the medium is not taken into consideration. It also lacks of experimental validation such as cake layer distribution and pressure drop. To complete the previous work, Fotovati et al [START_REF] Fotovati | A macroscale model for simulating pressure drop and collection efficiency of pleated filters over time[END_REF] considered the depth filtration into consideration for Vshaped pleated filters. They used similarly to the previous work, a laminar airflow model and a

Lagrangian based model to simulate the particle deposition. Their model was validated by taking into account the flat filter configuration based on a semi-empirical model developed by Thomas.

et al [START_REF] Thomas | Clogging of ÿbrous ÿlters by solid aerosol particles Experimental and modelling study[END_REF]. Based on Fotovati et al. previous works [START_REF] Fotovati | A macroscale model for simulating pressure drop and collection efficiency of pleated filters over time[END_REF][START_REF] Fotovati | Modeling instantaneous pressure drop of pleated thin filter media during dust loading[END_REF], Saleh et al [START_REF] Saleh | Modeling service life of pleated filters exposed to poly-dispersed aerosols[END_REF] developed a Lagrangianbased macroscale model to predict the life of a filter exposed to polydisperse micron aerosols (dp = 1 to 10 µm). Their simulation revealed that exposure to polydisperse aerosols reduces the service life of the filter when compared to exposure to monodisperse aerosols with the same diameter range; however, this study is purely numerical and has not been validated experimentally.

Saleh et al [START_REF] Saleh | A simple semi-numerical model for designing pleated air filters under dust loading[END_REF] developed a semi-numerical Lagrangianbased model to study the effects of particle size, pleat shape, pleat count, velocity and the instantaneous performance of the HEPA filter. The average velocity field was predicted by empirical expressions instead of CFD simulations. This work was completed two years later by developing a new set of analytical equations [START_REF] Saleh | An analytical approach to predict pressure drop and collection efficiency of dust-load pleated filters[END_REF]. Their results were validated on the base of experimental pressure drop data provided by Gervais et al. [START_REF] Gervais | Combination of Single-Photon Emission and X-Ray Computed Tomography to visualize aerosol deposition in pleated filter[END_REF]. Nevertheless, their work was limited to the prediction of HEPA filter performance at early clogging stages only, and remain non-precise at higher clogging stages.

All the previous work predicted the filtration performance of a filter in laminar airflow regime and followed a Lagrangian approaches. However, Feng et al [START_REF] Feng | Modeling filtration performance of pleated fibrous filters by Eulerian-Markov method[END_REF] simulated the instantaneous evolution of the pressure drop and collection efficiency of a mini-pleated rectangular shape using ANSYS Fluent. Due to the turbulent airflow regime, the airflow is modeled using the DNS model. A modified Eulerian approach is used to model particle motion. Because it considers depth filtration and a turbulent air flow regime, this work extends the work of Fotovati [START_REF] Fotovati | Modeling instantaneous pressure drop of pleated thin filter media during dust loading[END_REF]. The findings were validated using experimental data on collection efficiency. At high filtration velocities, the results also confirm a preferential deposition at the pleat hollow.

Due to the wide range of structural parameters, geometrical properties, and airflow conditions covered by numerical simulation in this section (clean and clogged), it is impossible to draw firm conclusions about the filter design of a pleat filter. Indeed, each case must be addressed separately based on its particular set of conditions and parameters. As a result, it was decided to summarize the structural parameters, pleat geometrical properties, and air flow conditions studied throughout this review in Table 9.

Chapter 1 Literature review 63 B, ε, and z are the medium, permeability, porosity, and thickness, respectively. The pleat height and width are denoted by the letters H and W. The fiber mean diameter is denoted by df. Pleated filters can be triangular (V shape) or rectangular in shape (U shape).

The majorities of authors assume isotropic medium and use B as the mean permeability. Only Fotovati et al. [START_REF] Fotovati | Modeling instantaneous pressure drop of pleated thin filter media during dust loading[END_REF], took the medium anisotropic character into account by introducing in plane and through plane permeability components. However, they conclude that fact of considering two permeability components has no impact on the flow behavior.

Discussion

The medium can be isotropic in-plane but anisotropic globally, resulting in different in-plane and through-plane permeability. This is the assumption Fotovati et al. [START_REF] Fotovati | Modeling instantaneous pressure drop of pleated thin filter media during dust loading[END_REF] had to make in order to consider only two permeability components; otherwise, the third would have been introduced.

Because the fibers were randomly arranged in the plane, there is only one permeability value for the two components of permeability in the plane of the medium, which is the in-plane permeability.

Accordingly, their conclusion applies only to their case and cannot be general. An experimental characterization of the fiber orientation is necessary to choose the right permeability components

that should be considered.

In terms of flow conditions, the authors generally specified the mean filtration velocity Vfiltration, but they also provide sometimes the mean inlet duct velocity Vmean. In order to describe the airflow regime, some authors considered the hydraulic Reynolds number Reh which takes into account the mean duct velocity and the hydraulic diameter of the duct Dh as the characteristic length, this Reynolds describe the airflow field at the filter scale i.e. the airflow regime in the duct [START_REF] Tronville | Minimization of Resistance in Pleated-Media Air Filter Designs: Empirical and CFD Approaches[END_REF][START_REF] Théron | Numerical and experimental investigations of the influence of the pleat geometry on the pressure drop and velocity field of a pleated fibrous filter[END_REF][START_REF] Feng | Assessment of various CFD models for predicting airflow and pressure drop through pleated filter system[END_REF].

According to Rebaï et al. [START_REF] Rebaï | A semi-analytical model for gas flow in pleated filters[END_REF], the flow inside the pleat channel can be characterized by the pleat Reynolds number, However, at a lower scale, i.e. that of a half-pleat, Rebaï et al. [START_REF] Rebaï | A semi-analytical model for gas flow in pleated filters[END_REF], characterized the flow inside the pleat by the pleat Reynolds number, which takes into account the mean duct velocity and halfpleat width as the characteristic length:

Re h = ρV mean D ℎ µ (15)
Chapter 1 Literature review 64

They consider a laminar regime for Re w = 1000. Table 9 presents the Rew calculated by the equation for all the authors who reported information about the mean duct velocity Vmean and their pleat width. For the studies who considered a turbulent flow [START_REF] Tronville | Minimization of Resistance in Pleated-Media Air Filter Designs: Empirical and CFD Approaches[END_REF][START_REF] Théron | Numerical and experimental investigations of the influence of the pleat geometry on the pressure drop and velocity field of a pleated fibrous filter[END_REF][START_REF] Feng | Assessment of various CFD models for predicting airflow and pressure drop through pleated filter system[END_REF][START_REF] Alilou | Impact sur le colmatage en régimes transitoire et permanent des écoulements d'air induits par le plissage des médias filtrants THE[END_REF][START_REF] Maddineni | Numerical investigation of pressure and flow characteristics of pleated air filter system for automotive engine intake application[END_REF], however, it was found that Rew < 1000.

Discussion

As a result of these observations, the question of the Reynolds characteristics that could be used to predict the air flow within the pleat arises. Due to the developed surface of the medium, a turbulent flow approaching the filter may become laminar in the pleats. A critical Re of 2000 allowing to identify the passage of a laminar flow to turbulent is valid only in rectilinear ducts, but it is unknown from which Rew a flow will become turbulent in the vicinity of a pleat.

The selection of the appropriate model becomes that can predict reproduce the transition between turbulent and laminar model, as well as the choice of the suitable Reynolds number to characterize the flow at the vicinity of the pleat are a real challenge in turbulent and transitory airflow regime.

To sum up, the selection of the appropriate model becomes a real challenge in turbulent and airflow regime. Indeed, selecting an appropriate airflow model capable of simulating the flow field in the pleat system is required for design optimization and performance analysis should be well addressed based on experimental validation, and notably on the basis of local parameters such as local velocity measurements. This aspect is highlighted in the next section.

Re w = ρV mean 0.5W µ

Chapter 1 Literature review 65

Validation of the airflow on the basis of experimental data

With all the numerical calculations mentioned above, the difficulty arises in selecting an appropriate airflow model to simulate the airflow approaching the filter. In the literature, validation of the airflow model is almost always based on the pressure drop, which is a macroscopic parameter as it will be explained in the next section.

Validation of the airflow model based on experimental pressure drop data

Tronville and Sala [START_REF] Tronville | Minimization of Resistance in Pleated-Media Air Filter Designs: Empirical and CFD Approaches[END_REF] investigated the ability of the standard-k-ε and RSM turbulence models to predict the pressure drop of a mini-pleated (H = 25.0 mm, W = 3.2 mm) filter panel in a turbulent air flow regime approaching the filter 1.8 × 10 4 <Reh < 7.1 × 10 4 ). (see Table 9) for the geometrical m.s -1 ) and for a low isotropic permeability of B ~ 10 -12 m 2 .

However, the simulated pressure drop with these two models agreed well with the experimental pressure drop provided by Theron et al. [START_REF] Théron | Numerical and experimental investigations of the influence of the pleat geometry on the pressure drop and velocity field of a pleated fibrous filter[END_REF], for a series of clean pleated filters with geometrical properties (H = 20.0 -40.0 mm, W = 11.3 -19.8 mm) and a higher isotropic permeability than

Tronville and Sala's [START_REF] Tronville | Minimization of Resistance in Pleated-Media Air Filter Designs: Empirical and CFD Approaches[END_REF] filter permeability (i.e. B ~ 10 -9 m 2 ).

Furthermore, the numerical simulations of Feng et al. [START_REF] Feng | Assessment of various CFD models for predicting airflow and pressure drop through pleated filter system[END_REF] with standard k-ε, Detached eddy simulation models, and V2-f models (models close to the k-ε developed for low bounded flow and low velocities) were in good agreement with the experimental pressure drop of a rectangular pleated filter (H = 50 mm, W = 50 mm) with low isotropic permeability B ~ 10 -12 m 2 . However, they also demonstrated that only V2-f models and DES models provided a good prediction of the pressure drop for the filter tested by Rebaϊ et al [START_REF] Rebaï | A semi-analytical model for gas flow in pleated filters[END_REF] (H = 51 mm, W = 12.5 mm) with a higher permeability, i.e. B ~ 10 -10 m 2 .

These results show that the selection of turbulence model is critical for each filter configuration (mean velocity range, filter geometry, medium permeability, etc.). Choosing the appropriate airflow model based on pressure drop values, on the other hand, is inaccurate. Indeed, the airflow model selected affects not only the pressure drop, but also the velocity profile near the filter pleats.

As a result, the researchers recently became aware of these issues and began validating their numerical code using local experimental parameters such as velocity values rather than pressure drop values, which are a macroscopic value. Following that, studies that validate their code using velocity measurements are presented.

Validation of the airflow model based on local velocity measurements

Few studies have validated their numerical approach using local parameters such as velocity values.

This is probably due to the fact that measuring local velocity values in close proximity to filters represents a challenging issue for the experiment. Table 10 summarizes the most recent studies that validated their numerical code using local velocity measurements near filter pleats. The most commonly-used techniques for this purpose are Hot Wire Anemometry (HWA) [START_REF] Feng | Assessment of various CFD models for predicting airflow and pressure drop through pleated filter system[END_REF][START_REF] Maddineni | Numerical investigation of pressure and flow characteristics of pleated air filter system for automotive engine intake application[END_REF] and Particle Image Velocimetry (PIV) [START_REF] Kang | Characterization of pleated filter media using particle image velocimetry[END_REF][START_REF] Alilou | Impact sur le colmatage en régimes transitoire et permanent des écoulements d'air induits par le plissage des médias filtrants THE[END_REF].

Chapter 1 Literature review 68

Feng et al. [START_REF] Feng | Assessment of various CFD models for predicting airflow and pressure drop through pleated filter system[END_REF], for example, measured the one-dimensional longitudinal velocity component using HWA downstream of a rectangular-shaped filter (height H = 50 mm, width W = 50 mm). They measured transversal velocity profiles inside and outside the pleat. They were able to get as close as 3 mm from the filter surface. He concludes that the DES (Detached Eddy Simulation) approach is more suitable for the prediction of the pressure drop and the flow within the filter.

Figure 11. Pleat geometry, different velocity measurement zones and the velocity profile inside the pleat (adapted from Feng et al. [59])

Maddineni et al. [START_REF] Maddineni | Numerical investigation of pressure and flow characteristics of pleated air filter system for automotive engine intake application[END_REF] used the same technique to measure transversal velocity profiles downstream, on the outside of a mini-pleated geometry filter panel (H = 26 mm, W = 4.5 mm) within that is typically used for automotive engine intake application. Since they were not able to go inside the filter due to the restricted space available, the HWA probe was 3.1 mm away from the pleat entrance. The velocity measurement was performed longitudinally along five axes across the pleat length. The expermental velocity profiles confirmed that the standard k-ε model was the suitable turbulence model for their study.

PIV was used by Alilou [START_REF] Alilou | Impact sur le colmatage en régimes transitoire et permanent des écoulements d'air induits par le plissage des médias filtrants THE[END_REF] to measure the velocity within mini-pleats of a HEPA filter used in nuclear facilities (H = 20 mm, W = 2.5 mm). Transversal velocity profiles were obtained upstream of the mini-pleats at different distances from the pleat entrance, ranging from x = 0.9 mm to x = Kang et al. [START_REF] Kang | Characterization of pleated filter media using particle image velocimetry[END_REF] used the same technique to measure the velocity downstream of a custom-built rectangular pleat (H = 51 mm, W = 51 mm). Similarly, to Feng et al. [START_REF] Feng | Assessment of various CFD models for predicting airflow and pressure drop through pleated filter system[END_REF], the authors concludes that the DES (Detached Eddy Simulation) approach is more suitable for the prediction of the pressure drop and the flow within their rectangular-shape filter. Their results showed significant velocity gradients downstream of the pleat. Feng also showed a non-uniform bimodal local velocity gradient downstream the rectangular shaped geometry.

Discussion

With PIV, the tracer particles can deviate from the air streamline and be collected by the filter. So they are no longer representative of the airflow in the vicinity of the medium and can increase the pressure drop. Unlike PIV, HWA does not require tracer particles. Another advantage of HWA is its ability to measure velocity fluctuations in addition to local velocity magnitude, and also turbulent characteristics like Reynolds stresses and vorticity. The importance of the velocity gradient downstream of the pleat on filtration performance is barely tackled in the literature, despite the potential importance of optimizing filtration performance, especially for indoor air applications where filters are installed in series. The non-uniform velocity distribution generated by a pre-filter can actually affect the efficiency and life span of the filter. A recent numerical study of Maddineni et al. [START_REF] Maddineni | Air-borne particle capture by fibrous filter media under collision effect: A CFD-based approach[END_REF] showed that a high filtration velocity could amplify the penetration of particles through the filter by causing particle bounce and re-entrainment downstream of the filter.

Graphic removed to respect copyright

Chapter 1 Literature review 70

To ensure better optimization of filter performance during clogging a complete characterization of the velocity profile on the upstream and downstream side of the pleat should be addressed. To summarize, studying the airflow near the filter pleats is crucial in order to accurately predict the performance of a pleated filter and optimize its design. As a result, selecting an appropriate air flow model that accurately predicts pressure drop and velocity variations near the pleat is important. However, the capacity of a turbulence model to accurately estimate velocity patterns varies significantly depending on the filter, medium properties, and flow conditions. As a consequence, validation of the airflow model based on local parameters such as local velocity parameters is a necessary step before proceeding with any further assessment

Conclusion

The literature review showed that the design of fibrous filters requires the selection of a relevant fibrous medium to achieve the targeted efficiency, and the determination of relevant geometrical properties that enable to minimize the pressure drop.

Indeed, implementing a fibrous medium through a more complex geometry than the planar one such as the pleated geometry (which is the object of the present investigations) enables to increase the surface offered for filtration but causes significant local velocity gradients and possibly local efficiency gradients. In fact, the velocity at which particles reach the fibrous medium is referred in the literature as the filtration velocity, which is calculated as the ratio between the air flow rate approaching the filter and the developed filter medium surface. This velocity is one of the critical parameters that affect the filtration performance: the initial pressure drop and the particle collection efficiency.

The pleating process lead to contract the air at the entry of the pleat and by consequence to an additional pressure drop that increases with filtration velocity. Thus, the global pressure drop is considered to be the sum of the pressure drop of the medium and that of the pleating.

Chapter 1 Literature review 72

Concerning the particle collection efficiency of the filter, a high filtration velocity value tends to demote collection by Brownian diffusion, and to promote the collection by the inertial impaction mechanism.

Accordingly, the study of the airflow at the vicinity of fibrous filters of complex geometrical properties like pleated filters is crucial to precisely evaluate its possible influence on the filtration performance that serve as a basis for the design of filters.

Many studies reported in the literature dealt with the improvement of the filtration performance (i.e. lowering the pressure drop while ensuring the targeted filtration efficiency) of pleated filters by combining semi-empirical models with numerical simulations. To this end, the main challenge encountered by the authors was to take into consideration the airflow pattern approaching pleated filters. Even if some studies presented in the literature have detected velocity gradients in the pleat channel, they did not integrate them into the calculation of the collection efficiency. They rather consider the filtration velocity, which does not reflect the local conditions experienced by the particles and thus affects the accuracy of the models. Moreover, the literature exhibits many numerical studies dealing with the effect of the filter geometrical properties on the velocity pattern at the vicinity of the filter [START_REF] Fotovati | Modeling instantaneous pressure drop of pleated thin filter media during dust loading[END_REF][START_REF] Feng | Modeling filtration performance of pleated fibrous filters by Eulerian-Markov method[END_REF][START_REF] Théron | Numerical and experimental investigations of the influence of the pleat geometry on the pressure drop and velocity field of a pleated fibrous filter[END_REF][START_REF] Kang | Characterization of pleated filter media using particle image velocimetry[END_REF][START_REF] Alilou | Impact sur le colmatage en régimes transitoire et permanent des écoulements d'air induits par le plissage des médias filtrants THE[END_REF][START_REF] Maddineni | Numerical investigation of pressure and flow characteristics of pleated air filter system for automotive engine intake application[END_REF]. Most of these studies focused on the airflow behavior on the upstream side of the pleat channel since it is the primary interest regarding particle collection.

For all the numerical calculations mentioned in this Chapter, the difficulty arises in the selection of an appropriate airflow model to simulate the airflow approaching the filter. In the literature, the validation of the airflow model is almost always based on the pressure drop which is a macroscopic parameter [START_REF] Tronville | Minimization of Resistance in Pleated-Media Air Filter Designs: Empirical and CFD Approaches[END_REF][START_REF] Théron | Numerical and experimental investigations of the influence of the pleat geometry on the pressure drop and velocity field of a pleated fibrous filter[END_REF]. Few studies validated their numerical approach thanks to local parameters such as velocity values for example. This is probably due to the fact that measuring local velocity values at the close vicinity of filters represents a challenging experimental issue.

This study combines experimental velocity measurements using HWA and numerical simulations in the vicinity of filters pleat, in order to investigate the local velocity gradients upstream and downstream of a clean and clogged pleated filter having geometrical properties representative of those used in air handling unit. The methodology adopted for this purpose is detailed in the next Chapter.

Chapter 2 Experimental and numerical strategies for air flow pattern study in the vicinity of filter pleats 1. Introduction

In this Chapter a methodology for the characterization of velocity patterns in the vicinity of pleated fibrous filters by combining experimental and numerical methods has been developed. The main aim is to validate the numerical simulations by local experimental velocity values. Once the numerical conditions are validated, these simulations allow a fine analysis of the local values of the velocity to which the particles are exposed as they come into contact with the medium at the beginning of the clogging process (in so far as the simulations are carried out with a clean filter).

These data are compared to the filtration velocity, which is currently the mean velocity value taken into account in filtration efficiency models to reflect the airflow. This numerical tool makes it also possible to test different geometrical configurations in a short time, in order to determine the influence of the pleat width and height on the initial pressure drop and on the velocity behavior at the vicinity of the filter.

This methodology is developed based on a fibrous filtering medium commonly used in the second filtration stage in air handling units of buildings, which is implemented as a flat pleated geometry.

Two pleated geometries were tested: one with a wide pleat entrance width (distance between two pleat peaks) and one with a narrow width. The geometric properties of the pleated filters tested, i.e., their width and depth, are also representative of those found for commercial filters. The filtration velocities tested are also in the range of velocities encountered in ventilation ducts. The technique chosen to perform experimental air velocity measurements near to the filter is the Hot Wire Anemometry (HWA). This technique, which is used to measure the velocity of a fluid at a single point, enables high-frequency acquisitions and does not require the use of particulate tracers like other laser techniques. The materials used as well as the experimental and numerical strategies used for the measurement of velocity at the vicinity of filter pleats are detailed in this Chapter 5.

Experimental bench

The experimental bench designed for this analysis is shown in Figure 13 The experimental bench consists of an open loop operating in aspiration mode: the fan is located at the outlet of the loop to suck air through the system. The test bench is mainly composed of a straight stainless steel duct with a length of 3 m and a cross-section of 150 × 150 mm representing the order of magnitude of duct dimensions used in the ventilation systems i.e., between 125 and 150 mm square section on average. The measurement zone (see Figure 13), where the filter is located, is made of transparent antistatic PMMA to allow visibility and control of the HWA probe position in relation to the filter. The filter is located at a distance of x = 167 cm from the duct entrance. High-efficiency filters H14 (EN 1822:2019) are located at the inlet and outlet of the duct to ensure particle-free air conditions inside it. The pressure drop generated by the tested filter is measured with a differential pressure sensor operating between 0 and 700 Pa (see Table 11) for specifications). The mean duct velocity that can be reached by the fan ranges from 0 to 1.72 m.s -1

(Qbench from 0 to 3.87 m 3 .h -1 ), which corresponds to the hydraulic Reynolds range of 0-17,060. The mean duct velocity is monitored by two flow rate sensors whose characteristics are in given Table 11. The calculation of the mean duct velocity Vmean and the filtration velocity Vfiltration are detailed as follows:

Where Qbench (m 3 .s -1 ) is the volumetric air flow rate in the bench, Sbench (m 2 ) is the duct cross-section and Sfiltration (m 2 ) is the developed medium surface. Flow rate, pressure drop, temperature and relative humidity are recorded over time (every second), and the flow rate is controlled by connecting the fan to the flow meter. The relative humidity and the temperature are measured using sensors which characteristics are given in Table 11.

V mean = Q bench S bench (17)
V filtration = Q bench S filtration (18)

Characteristics of the filter

In order to carry out the experimental measurements, pleated-filter prototypes with a 15 × 15 cm cross section area were developed in the laboratory. The prototype filters were constituted by a fibrous medium developed by the Lydall© Company, commonly used for indoor air treatment applications. The structural properties of the medium and the experimental techniques used to characterize them are presented in Table 12. Refer to Chapter 3 for more details. The shape and dimensions of the prototypes are shown in Figure 15. The design of the filter is a compromise between reproducing the filtration velocity of the commercial filter, i.e., between 0.20 and 0.30 m.s -1 , and providing a relative appropriate width of the pleats that allows the HWA to be inserted while remaining representative to the real dimensions of the filter. Two pleated geometries were tested: one with a wide pleat entrance width (distance between two pleat peaks) and one with a narrow width. Three prototypes are created for the "wide" geometry. These prototypes consist in six pleats of 28 ± 1 mm height (H) and 25 ± 1 mm width (W), and a filtration surface of 55,200 ± 2,000 mm² which is more than twice the filter surface when in flat configuration. The nominal filtration velocity given by this geometry is Vfiltration = 0.27 m.s -1 achievable for a mean duct Vmean = 0.67 m.s -1 . Only one prototype is made for the narrow geometry.

79

This prototype has fifteen pleats with a height of 27 ±1 mm and a width of 10 ± 1 mm, and a filtration surface of 123,600 ± 14,000 mm 2 , which is more than five times the filter surface when flat and twice the surface of the wider width geometry. This geometry yields the same nominal filtration velocity of Vfiltration = 0.27 m.s -1 as the first geometry for a mean duct Vmean = 1.48 m.s -1 .

For simplifications, the first geometry is written as (H = 28 mm, W = 25 mm), and the second geometry as (H = 27 mm, W = 10 mm).

Experimental Strategy for velocity measurement

In this section, descriptions are given of the experimental technique and the different materials used for the acquisition of velocity in the vicinity of the filter pleats.

Choice of the HWA technique

Hot wire anemometry (HWA) is an effective method for experimental airflow studies, and is relevant for the analysis of the airflow in the vicinity of filter pleats. HWA has various advantages as compared to other airflow velocity measurement techniques such as LDA (Laser Doppler Anemometry) and PIV (Particle Image Velocimetry). Unlike PIV and LDA [START_REF] Kang | Characterization of pleated filter media using particle image velocimetry[END_REF], HWA is a direct measurement tool that does not require particle injection to measure the airflow velocity. With PIV, the tracer particles can deviate from the air streamline and be collected by the filter, so they are no longer representative of the airflow in the vicinity of the medium and can increase the pressure drop in the filter. HWA, like LDA, can measure velocity fluctuations in addition to local velocity magnitude, but it is less expensive and does not require a particle tracer that can be captured by the filter and then increase its pressure drop.

Principle of the HWA technique

The HWA method is based on calculating the cooling effect of the surrounding flow on a heated thin wire. The hot wire is heated by an electrical current, and then cooled by the fluid. The power carried by the fluid gives an indirect measurement of the flow velocity. Basically there are two modes of operation of hot wire anemometers Constant Current Anemometer (CCA) and Constant Temperature Anemometer (CTA) [START_REF] Bruun | Hot-wire anemometry: principles and signal analysis[END_REF]. CTA is the most known HWA technique in the literature and is also used in this study. Figure 16 depicts the principle diagram of the CTA HWA technique.

A Wheatstone bridge is used to overcome the cooling effect of the surrounding flow on the wire while keeping the temperature steady. The electric current passing through the wire generates heat, which once in equilibrium, is equal to the heat losses of the fluid. The current intensity is controlled through the wire sensor by a servo amplifier to keep the bridge in balance, hence the wire has constant resistance and, therefore, constant temperature. The bridge voltage, E, is thus a direct measurement of the fluid velocity as it represents the heat loss. The combination of the high servo loop amplifier gain and the low thermal inertia of the wire provide a very fast response to the fluid velocity fluctuations. As the velocity is not directly proportional to voltage output, the HWA has to be calibrated prior to the measurement. The velocity of the fluid flow, V, is proportional to the voltage drop across the HWA, E, according to an exponential fitting function known as King's Law [START_REF] Jaganathan | a realistic approach for modeling permeability of fibrous media: 3-D imaging coupled with CFD simulation[END_REF]

Materials for velocity measurements

Miniature hot wire anemometer probes (Dantec) are used for velocity measurements. They consist of tungsten wires with a length of 1.25 mm and a diameter of 5 μm, fixed between two pins. In this analysis, three probe configurations are used: a probe with straight pins and a wire perpendicular to the probe axis (55P11), a probe with right-angle pins and a wire parallel to the probe axis (55P13) and a probe with right-angle pins and a wire perpendicular to the probe axis (55P14). The 55P14 was used instead of the 55P13 to take a single measurement inside the filter upstream (H = 27 mm, W = 10 mm). It should be noted that using a parallel or perpendicular probe to the flow has no effect on measurement accuracy; the only disadvantage is that fewer points can be acquired. The specifications of the probe and area of application for this research are detailed in Table 13.

The CTA (Constant Temperature Anemometry) module is used to measure the voltage across the hot wire and the resistance of the circuit assembly of the acquisition chain. The uncertainty of

V = 𝐶 0 + C 1 E + C 2 E 2 + C 3 E 3 + C 4 E 4 , (20)
Chapter 2 Experimental and numerical strategies for air flow pattern study in the vicinity of filter pleats.

82 velocity measurement of a HWA CTA is 3%. The major contribution in the error comes from the calibrator (2%). Variations of the airflow temperature, atmospheric pressure and curve fittings can also affect the measurement uncertainty. The precision of this method is strongly dependent on the accuracy of the calibration, which relates the measured resistance of the hot wire to the velocity.

The calibration blower is used to establish the calibration curve connecting the voltage measured by the CTA module to the air velocity. It measures the voltage across the hot wire at known air velocities. A temperature sensor is used to measure the ambient temperature to perform the temperature correction when measuring the voltage across the hot wire. The displacement mechanism used to control the position of the probe is manual, made of two perpendicular calipers with a precision of 10 μm, enabling the probe to be moved transversely in x direction and lengthwise in z directions (see Figure 13). The two calipers are mounted on the support where they can be moved vertically along the y direction by adjusting the corresponding screw. In order to make measurements at the desired z positions, PMMA plates corresponding to the vertical duct wall portion through which the hot wire probe is inserted, shown in Figure 20, are manufactured with a slot drilled at the desired z position. In order to avoid air leakage, the slots are sealed by rubber joints. More information on the methodology of velocity measurements is provided in the following section. 11). The calibration velocity range is adapted to the flowrate implemented in the experimental bench, and therefore in particular to the maximum velocity reached in the center of the duct. As a result, the probe calibration involved 30 points between 0.10 and 1.60 m.s -1 for the characterizations carried out at the Vmean = 0.67 m.s -1 , and 45 points between 0.10 and 3.0 m.s -1 for the characterizations carried out at the Vmean = 1.48 m.s -1 . A logarithmic distribution of the calibration velocity range has been adopted to obtain the most precision at low velocities. Once calibrated, the probe is placed in the measurement zone at the desired position as it is shown in Figure 18. The acquisition can then start, and in addition to velocity measurements, the bench supervision software saves other parameters like air flowrate, temperature, and relative humidity. 85

Figure 19 displays a 3D diagram of the displacement of the probe in the duct for the characterizations conducted at the Vmean = 0.67 m.s -1 . Transversal velocity profiles i.e., along the y-axis are recorded at two x distances from the entrance in the bench (see Figure 14):

-at x = 167.5 cm which corresponds to 5 mm downstream of the filter when it is implemented in the duct;

-at x = 150.5 cm to evaluate whether the flow is established in the measurement zone by comparison to profiles acquired at x = 167.5 cm.

As shown in Figure 19, measurements for both lengthwise positions (x = 150.5 cm and x = 167.5 cm) are made at three z positions: in the middle of the duct, i.e., z = 7.5 cm; at z = 9.5 cm; and at z = 5.5 cm. In order to preform measurements at the desired z positions, PMMA plates corresponding to the vertical duct wall portion through which the hot wire probe is inserted, shown in Figure 20, are designed with a slot drilled at the desired z position. In order to avoid air leakage, the slots are sealed by rubber joints.

Twenty-five acquisition points are taken between y = 0.8 and 14.0 cm for each transversal profile (0.8 cm and 1 cm respectively far from the vertical duct walls). The probe displacement begins at y = 0.8 cm and increases with a step of Δy = 0.4 cm, close to the entrance vertical wall of the ducts;

when the velocity begins to be constant, i.e. reaching the bulk flow zone, we increase the step to Δy = 1.0 cm; and when the velocity begins to decrease again, the probe displacement returns to Δy = 0.4 cm. The displacement steps near the walls are thus smaller than those in the central zone where the velocity is constant, in order to have enough measurements to determine the thickness of the boundary layer. For each velocity measurement point, the frequency and the sampling time are 200 Hz and 60 s respectively. Each measurement is repeated three times, resulting in a total of 75 measurements. This has enabled the determination of the boundary layer thickness of the horizontal duct walls and the mean velocity at the duct center.

The flow in the duct is more turbulent at Vmean = 1.48 m.s -1 (Reh = 14,920) than at Vmean = 0.67 m.s - 1 (Reh = 6,760). According to fluid mechanic's theory, the thickness of the boundary layer decreases as the turbulence in the duct increases, and thus the bulk flow region expands. To be certain, the boundary layer thickness for V = 1.48 m. s -1 was determined only at x = 167.5 cm and y = 7.5 cm, i.e. 5 mm downstream the filter when installed in the duct. These measurements include 27 points Chapter 2 Experimental and numerical strategies for air flow pattern study in the vicinity of filter pleats.

86

ranging from y = 1 cm to y = 14 cm, with a simplified probe displacement step of Δy = 0.5 cm.

Each measurement is taken three times, for a total of 81 measurements. From the results described above, in order to avoid boundary layer effects on the accuracy of measurements in the vicinity of the filter, a bulk flow zone in which the velocity is homogeneous is determined and corresponds to 4.5 cm < y < 10.5 cm and 5.5 cm < z < 9.5 cm for Vmean = 0.67 m.s -1 (see Figure 23 (a)). The bulk flow region determined experimentally for Vmean = 1.48 m.s -1 , corresponds to 3.5 cm < y < 11.5 cm at z = 7.5 cm (see Figure 23 (b)).

For each geometry tested, the studied half-pleats are chosen in the bulk flow region as illustrated in Figure 23. The turbulent intensity profile at x = 167.5 cm and z = 7.5 cm is represented in Figure 24.

V = V ̅ + v rms ′ ( 21
)
TI = v rms ′ V ̅ (22)
Chapter 2 Experimental and numerical strategies for air flow pattern study in the vicinity of filter pleats.

91

increases when approaching the duct wall i.e., for y > 11.5 cm and y < 3.5 cm. It reaches a maximum value of 12 % at the nearest measurement point to the vertical wall i.e., at y = 1.0 cm.

The TI is slightly higher in the first case due to possible leakage, particularly near the wall, which can cause more fluctuations of the measured velocity around the mean value. These fluctuations have no effect on the central region, where measurements are taken downstream of the pleats.

Additional sealing is performed on the duct areas that can cause leakage, particularly at the level of the PMMA sheets, to reduce air leakage during velocity measurement. (see Figure 20)

Velocity measurements in the vicinity of filter pleats

Measuring local velocity values in close proximity to filters represents a challenging issue. After having presented the HWA technique and determined the boundary layer thickness for the tested mean velocity, the experimental methodology used to measure velocity profiles in the vicinity of clean and clogged filters is detailed in this section.

Velocity measurements were performed upstream and downstream of the filter. These For these measurements, a straight probe (55P11, Dantec ©) mounted on a straight support was used (see Table 13) The calibration of the anemometry probe involved 30 points between 0.1 and 4.0 m.s -1 for Vmean = 0.67 m.s -1 (see Figure 28 (a)) and 45 point between 0.1 -7.0 m.s -1 (Figure 28 (a)) For each measurement point, the frequency and the sampling time were 200 Hz and 60 s respectively. For each prototype, these transverse velocity profiles were recorded downstream of the four half pleats located in the bulk flow, i.e., outside the boundary layer, as shown in Figure 25 (a). Since these measurements were carried out for the three filter prototypes, a total of twelve halfpleats were characterized. For each half pleat, transversal velocity profiles were acquired at five x distances downstream of the pleat peak (for this point we considered a longitudinal coordinate x = 0 cm), as shown in Figure 25 (b): x = 1.5; 2.0; 3.0; 3.5 and 5.0 cm. These measurements were performed at the height z = 7.5 cm, i.e., half the height of the duct. The y step of the probe displacements was 1 mm. For the two profiles acquired inside the pleat (x = 1.5 and 2.0 cm), the minimum distance to the medium surface at which the measurements were carried out was 1 mm.

HWA probe Straight support

Displacement system Filter

Half-pleats x y Chapter 2 Experimental and numerical strategies for air flow pattern study in the vicinity of filter pleats.

94

Transverse velocity profiles were also acquired upstream of the filter (H = 28 mm, W = 25 mm) as shown in Figure 25 (c). These measurements were carried out inside one half pleat (x = 0.2 cm; 1.0 and 1.3 cm) with a right-angled hot wire probe (55P13, Dantec©) mounted on a straight support.

The probe was inserted inside the pleat through a hole drilled on the filter prototype frame. Pictures from different perspectives of the probe position upstream of the filters are shown in Figure 29.

In case of the narrow pleats (H = 27 mm, W =10 mm), only one measurement point has been done upstream inside one -pleat using a right-angled hot wire probe (55P14, Dantec©). The probe was also inserted delicately inside the pleat through a hole drilled on the filter prototype frame.

For each measurement point, the frequency and sampling time were 200 Hz and 60 s respectively.

For the first case i.e. at Vmean = 0.67 m.s -1 , the calibration of the anemometry probe involved 30 points between 0.1 and 4.0 m.s -1 (see Figure 28 (a)). The y step of the probe displacements was 1 mm. The minimum distance to the medium surface at which these measurements were carried out was 5 mm. For the second case, i.e. at Vmean = 1.48 m.s -1 , the calibration of the anemometry probe involved 45 points between 0.1 and 7.0 m.s -1 (see Figure 28(b)). The minimum distance to the medium surface at which these measurements were carried out was 4 mm.

Numerical strategy for velocity measurements

The numerical simulations were carried out with the commercial code ANSYS Fluent©. It is a commercial software based on the finite volume method to solve the Navier-Stokes equations. In this section, the computational domain, the boundary conditions and the choice of the models tested to represent the airflow and the porous medium are detailed.

Computational domain and boundary conditions

A numerical domain consisting of a two-dimensional mesh was generated to simulate the flow through one half pleat. The purpose is not to predict precisely the pressure drop of the pleated medium, but only to check the consistency between the experimental results and those of Fluent.

That's why we deliberately used simplified 2D half-pleat geometry of the pleated medium under Fluent that has been already tested in the literature [START_REF] Théron | Numerical and experimental investigations of the influence of the pleat geometry on the pressure drop and velocity field of a pleated fibrous filter[END_REF][START_REF] Rebaï | A semi-analytical model for gas flow in pleated filters[END_REF] The numerical domain was divided into two: a fluid representing air and its thermodynamical characteristics at 20°C, and a porous material representing the fibrous medium. The half-pleat was positioned after a straight length of 251 mm, with a total length of 550 mm corresponding to the 30). Since these modifications are difficult to measure, especially those affecting the thickness and permeability of the medium, we assumed that the local permeability in this zone ranged between a low value, relating to the extreme case of non-permeability, and the same permeability as elsewhere along the pleat. To evaluate the sensitivity of the pressure drop to the local permeability of the pleat curved zones to, as suggested by Rebaï et al [START_REF] Rebaï | A semi-analytical model for gas flow in pleated filters[END_REF], two simulation series were carried out, considering the zones as permeable or non-permeable (see Chapter 4).

CFD models and equations

The 9). Regarding the Reynolds number range mentioned above, therefore, it is useful to test the ability of a transition model such as transition SST to predict the pressure drop and airflow in the vicinity of the tested pleated filters.

For the filtration velocity range for which the pressure drop was measured (0 -0.65 m.s -1 ), the pore Reynolds number Rep was calculated using the following equation:

Where µ and ρ are the air dynamic viscosity and density, Vfiltration is the filtration velocity and dpore is the modal pore diameter of the fibrous medium.

According to the literature, the laminar linear regime occurs for Rep < 1. The laminar inertial regime initiates at a Rep between 1 and 10 and persists at Rep~150 [START_REF] Dybbs | A New Look at Porous Media Fluid Mechanics -Darcy to Turbulent[END_REF]. Deviation from the linearity is relatively weak for 1 ≤ Rep ≤ 3.

The porous medium presents two pores populations, characterized by modal diameters dp1 = 33 µm and dp2 = 183 µm (see Table 12). For the velocity range tested for pressure drop characterization, and by considering dp1 in the calculations (which is the most important population), Rep1 = 0 -1.44.

For the largest pore diameter dp2 =183 µm (which is the less important population), Rep2 = 0 -8.01.

This indicates that for the highest velocities tested the initiation of the inertial regime is reached.

Re p = ρV filtration d pores µ (23)
Chapter 2 Experimental and numerical strategies for air flow pattern study in the vicinity of filter pleats.

98

The inertial resistance to the flow of the medium is therefore assumed to be negligible and the pressure drop that it generates is predicted using the Darcy equation:

Where z is the thickness of the medium (mm) and B its permeability (m 2 ). The medium is assumed to be isotropic.

Sensitivity of the solution convergence to the meshing size

A mesh convergence study was carried out on the geometry (H = 28 mm, W = 25 mm) to determine the optimal mesh sizes to ensure an independent solution. A non-structured meshing has been adopted with a refinement in the air layer of 0.3 mm thickness close to the medium surface since significant velocity gradients are expected in this zone according to the literature [START_REF] Ali | élimination des particules[END_REF][START_REF] Teng | Research on the influence of pleat structure on effective filtration area during dust loading[END_REF]. To be consistent with the exit condition, we have taken care to systematically check that the flow is well re-homogenized at the outlet. All the results are analyzed only when the residuals of each simulation parameter are constants and lower than 10 -9 .

First of all, we did a sensitivity analysis of the pressure drop to the meshing as it is a macroscopic parameter. The airflow model used for this analysis is the SST transition model, as it includes the largest number of parameters that could be sensitive to meshing compared to the standard 𝑘-𝜀 and the laminar models (see section 8.2). To do so, we have determined a percentage difference of the obtained pressure drop, ∆𝑃, from a reference pressure drop, ∆𝑃 𝑟𝑒𝑓 , as a function of the number of meshes used according to equation [START_REF] Gostick | In-plane and through-plane gas permeability of carbon fiber electrode backing layers[END_REF]. A highly-refined meshing of 350,000 cells has been used as a reference. The results are presented in Figure 31.

The results show that the error percentage starts to stabilize at 0 for around 80,000 cells. The velocity is highly influenced by the mesh size. For this reason, the velocity profiles was compared at three positions from the pleat peak (see Figure 32) for three meshing configurations: 80,000, 200,000, and 350,000 cells (i.e. the finest meshing adopted):

 x= -2.6 mm i.e., 2 mm upstream of the pleat entrance (zmedium = 0.6 mm);  x= 30.0 mm i.e., 2 mm downstream of the pleat exit;

ΔP = µz B V filtration (24)
Error Percentage (%) = |∆𝑃 𝑟𝑒𝑓 -∆𝑃| ∆𝑃 × 100

 0.1 mm above the medium surface i.e., in the thin air layer of 3 mm thickness upstream of the medium surface. It is not the case for the longitudinal velocity profile at 0.1 mm above the filter surface as shown in Figure 33 (c). The profiles obtained with 210,000 and 300,000 mesh cells are superimposed, but it is not the case with 80,000 cells. a mesh with 210,000 cells was used to make a reasonable balance between mesh numbers and computation time. The final meshing for the geometry (H = 28 mm, W = 25 mm) is presented in Figure 34 (a). In all the numerical domain zones, quadrilateral mesh morphology is used. The mesh size in the thin air layer close to the medium surface is 0.1 mm 100 while it is 0.2 mm for the rest of the domain. In the case of narrow geometry (H = 27 mm, W = 10 mm), the mesh size is chosen to be representative of the reduction in the pleat width size. More mesh refinement is thus performed, particularly near surface medium. Quadrilateral mesh morphology is also used in all numerical domain zones. The cell size in the thin air layer near the medium surface is 0.05 mm, while the rest of the domain has a mesh size of 0.1 mm, yielding a cell number of around 200, 000. The final meshing is presented in Figure 34 (b).

Optimization

Filter Geometry

Meshing size

• Optimal size that Ensure independent solution • Provides reasonable balance between the mesh number and the computational time.

Upstream filter pleats

Propose new filter geometry

Local velocity simulation

Experimental approach Numerical approach Chapter 3 Experimental characterization of the structural properties of the fibrous medium for an accurate prediction of its permeability

Introduction

The methodology used to study airflow at the scale of the filter pleat was previously presented.

Following the multiscale investigation of the various parameters that could influence the filtration performance, the investigation in this Chapter is conducted at a lower scale, i.e. that of the medium. Its purpose is to characterize the medium's structural properties. Fibrous filters are commonly used in ventilation systems for particle removal. The fibrous medium used to manufacture them are generally nonwoven fabrics, often made of glass or polymer fibers, or blends of fibers of different materials. Owing to the variety of manufacturing processes employed to produce the fiber networks (melt blowing, spun bonding, wet laying, air laying, dry laying, electro spinning), as well as methods used to ensure their mechanical strength (spun lacing, needle-felting, thermal or chemical bonding), nonwoven materials display a broad range of structural properties: fiber diameter distribution, fiber morphology, medium thickness, mean porosity, pore size distribution, etc. One of the filtration community's big challenges is to optimize the performance of filters, i.e. to reduce the pressure drop and maintain the targeted particle removal efficiency. Local in-plane variations of the previously cited structural properties of the medium result in a more or less significant degree of heterogeneity that can lead to preferential path lines regarding the airflow. This phenomenon can be responsible for a heterogeneous clogging, and to an accelerated ageing of the filter. It is therefore important to address the issue of the prediction of applicative properties; and mainly of the medium permeability which depicts the medium resistance to the airflow; with an accurate description of the medium structural properties and their potential in-plane variations.

The first part of this Chapter is dedicated to the presentation of the studied medium, and to the description of the experimental techniques used to characterize it. Among those techniques Xray micro-tomography has been used as it enables to determine local values of some properties like the porosity for example, which allows the evaluation of the heterogeneous character of the medium. The results obtained in terms of structural and porous properties used to predict permeability are then exposed, with a focus on the evaluation of the heterogeneous nature of the medium thanks to the analysis of fiber orientation and medium in-plane porosity variations.

The final section is dedicated to the prediction of the permeability. Different models of the literature are compared, and the influence of the fiber diameter formulation is assessed. The sensitivity of the permeability to the porosity is also analyzed, and on the basis on in-plane porosity maps obtained by X-ray micro-tomography permeability maps are presented.

Materials and methods

Studied medium

The fibrous medium studied involves 94 w% of glass fibers and 2 w% of synthetic fibers. The rest consists in acrylic fiber binders and hydrophobic additives. The medium is manufactured by the Lydall© Company and is widely used for indoor air treatment applications. It is produced by a wet-laid process It has an efficiency expressed as ISO ePM1 50-55% according to the ISO16890 Standards [START_REF]Air filters for general ventilation -Part 1:Technical specifications, requirements and classification system based upon particulate matter efficiency (ePM)[END_REF]. Some of the medium mechanical properties provided by the supplier are resumed in Table 14. To complement the information provided by the supplier, the fibrous/porous structure of the medium is characterized in terms of:

-Thickness based on low pressure measurements -Air permeability determined by permeametry (pressure drop in the function of air velocity).

-Fiber diameter based on SEM images -Mean porosity and pore size distribution thanks to mercury intrusion porosimetry -Through-plane porosity thanks to X-ray microtomography

Finally, the heterogeneous nature of the medium is assessed using X-ray microtomography images to analyze fiber orientation and medium in-plane porosity variations.

The images at x500 magnification were treated using the Image J software in order to characterize the fiber diameter distribution. A total of 951 distinct fibers; only located in the foreground of the images; were analyzed.

Mercury Porosimetry

Three samples of the medium of approximately 1 cm² area were analyzed with the Autoport IV Micrometrics device. The penetrometer used for the measurements is the 14-0806_05-

X-ray micro-tomography

Experimental conditions

Two circular samples of the medium were analyzed by X-ray micro-tomography on the ID19 beamline of the European Synchrotron Radiation Facility (ESRF, Grenoble, France). These measurements were carried out with a spatial resolution of 0.16 μm/pixel to be relevant with the characteristic length scales of the solid structure of the medium, and especially with the smallest fiber diameter determined thanks to SEM analysis (see results presented in Section 3.1.3). The beamline energy was 19 keV. In order to achieve a pixel size of 0.16 μm, a camera of 2560x2160 pixel² equipped with a x40 zoom lens was used.

d pores = -4γcosθ P (27)
The counterpart of a high resolution for X-ray micro-tomography observations is that a small zone of the sample can be observed. To reach the resolution of 16 µm/pixel the diameter of the analyzed zone was thus about 500 µm. This is the reason why; in order to assess the representativeness of the observed surface area; two analysis were carried out. The diameter of the two samples was 5 mm. It is higher than the diameter of the observed zone in order to avoid the potential modifications of the medium properties due to the sample preparation. The two analysis led to 3611 and 3286 tomograms, corresponding to 2D images of medium "slices" respectively.

Image treatment

The thickness of the analyzed zone was calculated by multiplying the number of tomograms Having applied an image mask to avoid processing out of field pixels, to reduce background noise and preserve fibers' edges without blurring effects, a non-linear anisotropic diffusion filter was applied [START_REF] Perona | Scale-space and edge detection using anisotropic diffusion[END_REF]. It smoothes homogeneous zones on the image, i.e. fibers, and reduces unwanted discontinuities, i.e. background noise. Next, binary images were obtained applying an adaptive local threshold approach [START_REF] Bradley | Adaptive Thresholding using the Integral Image[END_REF], using a sliding window (1/8 of the original image size) for which is calculated the corresponding local threshold [START_REF] Otsu | A threshold selection method from gray-level histograms[END_REF].

After having separated the solid fraction (fibers) from void by the segmentation step, the porosity is calculated for each 2D image. The evolution of the porosity through the medium thickness, also called porosity gradient, can thus be represented with a thickness step size equal to the resolution, i.e. 0.16 µm. For each sample, the mean porosity value is calculated by averaging the porosity over the complete stack of 2D images.

In order to detect some potential in-plane porosity variations, the porosity over the entire medium thickness was also calculated on small zones of 155x155 pixels² area (corresponding to 24.8x24.8 µm²) for each sample. These calculations enabled to represent in-plane porosity mappings.

Data obtained from micro-tomography analyses also enable the determination of the 3D fiber orientation. The orientation of an individual fiber in the 3D space is described by the orientation tensor. The structural tensor method is used to quantify fibers locally in each voxel. It is based on the assumption that the direction through which the gray level exhibits least variations corresponds to the fiber's main direction (see Appendix 3.1 for more details). By applying the Folgar-Tucker [START_REF] Folgar | Orientation Behavior of Fibers in Concentrated Suspensions[END_REF] model the orientation tensor A ̿ is obtained, with:

Where the values of the diagonal components XX, YY, and ZZ represent the fibers alignment within the coordinate directions. The off-diagonal components XY, XZ and YZ depict the amplitude of alignments that vary from the coordinate axes. They are thus equal to zero when fiber axes overlap with the principal directions of the orientation tensor. Subsequently, nonnegligible off-diagonal components mean that X, Y and Z axes are not the principal fiber axes.

The sum of the amplitudes of the three diagonal components is equal to one at each position along the medium thickness, and it is the same for the three off-diagonal components. If XX = YY = ZZ = 1/3 [START_REF] Tomadakis | Viscous Permeability of Random Fiber Structures: Comparison of Electrical and Diffusional Estimates with Experimental and Analytical Results[END_REF][START_REF] Woudberg | Permeability prediction of an analytical pore-scale model for layered and isotropic fibrous porous media[END_REF][START_REF] Tamayol | Transverse permeability of fibrous porous media[END_REF], the fibers arrangement is isotropic. Fibers are parallel to the X-axis for XX = 1. In such case the fiber network exhibits an unidirectional structure [START_REF] Happel | Viscous flow relative to arrays of cylinders[END_REF]. Fibers are perpendicular to the X-axis for XX = 0, and in such case the fiber network is characterized by a 2D layered structure [START_REF] Tomadakis | Viscous Permeability of Random Fiber Structures: Comparison of Electrical and Diffusional Estimates with Experimental and Analytical Results[END_REF][START_REF] Woudberg | Permeability prediction of an analytical pore-scale model for layered and isotropic fibrous porous media[END_REF].

Permeability

The permeability of the medium was measured by permeametry, i.e., by measuring the pressure drop as a function of the mean duct velocity by implementing the medium as a flat sheet in a ventilated duct. The experiments were conducted in a lab-scale air handling unit of 15x15 cm² square cross section (see Chapter 2section 2). The air velocity ranged between 0 and 0.70 m.s -1 , and the pressure drop was measured with a differential pressure sensor. Pressure drop measurements were conducted for three medium samples. The permeability of the medium B

A ̿ = [ XX XY XZ -YY YZ --ZZ ] (28)
Chapter 3 Experimental characterization of the structural properties of the fibrous medium for an accurate prediction of its permeability 110 is calculated from the slope of the linear part of the pressure drop curve as a function of the air velocity according to the Darcy's law (equation ( 26)).

Results and discussions

Medium thickness and permeability

The The thickness of sample 2 is slightly lower than both other values. This may indicate that medium zones of 500 µm diameter are not strictly representative of the medium thickness "in average".

For the determination of the medium permeability on the basis of pressure measurements and by using the Darcy's equation (equation ( 26)), the mean thickness value of 601 ± 19 µm is used.

The permeability obtained is Bexp = (3.10 ± 0.10).10 -11 m 2 .

Porosity

Mean porosity and pore size distribution obtained by porosimetry

The mean porosity obtained by mercury porosimetry for the three replicates is 92.0 ± 1.6%. The differential pore size distributions corresponding to the three measurements are presented in A model, proposed by Le Coq [START_REF] Coq | Influence on Permeability of the Structural Parameters of Heterogeneous Porous Media[END_REF] has been fitted to the experimental distributions. This model, which consists in a probability distribution of porosity as a function of the pore size, represents each pore population "i" by a maximum porosity ε i max , a modal pore size ϕ i mod , and a slope attenuation coefficient at the inflection point of the cumulative porosity curves αi. The model formulation is:

The two populations are characterized by porosities of 71.0± 1.6% and 21.0± 1.6% for the peaks corresponding to modal diameters of 27 and 330 µm respectively. The smallest pore population represents the internal porosity of the medium, and the other one probably corresponds to the medium surface roughness.

`

Through-plane porosity gradient obtained by X-ray micro-tomography

Figure 37 presents the evolution of the porosity through the medium thickness for the two samples analyzed by X-ray micro-tomography. Figure 37 shows that for both curves the porosity increases on both sample sides up to 100%. This is due to the fact that the images corresponding to these two portions of each sample contain the last few fibers of each sample surface. In the medium core the porosity fluctuates between 90 and 96 %. For the sample 1 minimum porosity values of 88% are reached. The sample 2 exhibits a high porosity peak (up to 98.5 %) between 200 and 300 µm that may be explained by local arrangement of fibers.

The mean porosity of the samples 1 and 2 are 92.7% and 94.3% respectively. The fact that the mean porosity of the sample 2 is slightly higher than the one of the sample 1 is probably due to the aforementioned porosity peak. Nevertheless, these two values are in good agreement with

27 µm 330µm ε(ϕ j ) = P(ϕ < ϕ j ) = ∑ ε i max [( ϕ j ϕ i mod ) α i + 1] k i=1 (29)
Chapter 3 Experimental characterization of the structural properties of the fibrous medium for an accurate prediction of its permeability 112 the 92.0 ± 1.6% obtained by mercury porosimetry. The small area of the analyzed zones involved in these X-ray analyses; which are approximately 500 times lower than the area of samples analyzed by mercury porosimetry; thus seems well representative of the medium porous structure "in average", which indicates that the medium is rather homogeneous.

Aside from the fact that values obtained from the two techniques, that involve far different sample area, enable to address the issue of fibrous medium homogeneity, it is also very interesting to compare porosity values obtained by different methods. If some authors compared porosity values obtained by micro-tomography to these determined thanks to gravimetric measurements [START_REF] Tsarouchas | Extraction of fibre network architecture by X-ray tomography and prediction of elastic properties using an affine analytical model[END_REF][START_REF] Huang | Morphology and transport properties of fibrous porous media[END_REF][START_REF] Lam | Multi-Scale Microstructure Investigation for a PM2.5 Air-Filter Efficiency Study of Non-Woven Polypropylene[END_REF][START_REF] Manickam | Characterization of polymeric nonwovens using porosimetry, porometry and X-ray computed tomography[END_REF], there are very few comparisons to values obtained by more precise techniques like liquid extrusion porosimetry [START_REF] Manickam | Characterization of polymeric nonwovens using porosimetry, porometry and X-ray computed tomography[END_REF] or mercury intrusion porosimetery [START_REF] Manickam | Characterization of polymeric nonwovens using porosimetry, porometry and X-ray computed tomography[END_REF][START_REF] Théron | Characterization of the porous structure of a non-woven fibrous medium for air filtration at local and global scales using porosimetry and X-ray micro-tomography[END_REF].

The aforementioned results show; as reported by Theron et al. [START_REF] Théron | Characterization of the porous structure of a non-woven fibrous medium for air filtration at local and global scales using porosimetry and X-ray micro-tomography[END_REF] that if analyzed zones are representative of the medium structural properties, the results obtained by X-ray microtomography and mercury porosimetry are in good agreement with each other for lowest pore size of the order of several microns.

In-plane porosity variations

Figure 38 shows in-plane porosity mappings for samples 1 and 2. To ensure the statistical reliability of the fiber diameter distribution, the average fiber diameter is plotted versus the cumulative number of fibers in Figure 41. The mean fiber diameter reaches an asymptotic value after approximately 880 fibers, and is df,mean = 3.14 ± 2.09 µm.

Fibers orientation

Figure 43 shows the 3D fiber organization obtained by microtomography along the medium thickness for samples (a) and (b). The images clearly demonstrate a multiple fiber population and random fiber arrangement through the medium thickness, indicating a global 3D anisotropic character. This finding needs to be confirmed by quantifying the amplitude of the orientation tensor, which is presented in the following section.

Figure 44 and Figure 45 show the results of the fiber orientation analysis realized thanks to Xray micro-tomography for the two medium samples. Figure 44 presents the evolution of the amplitude of the diagonal components of the orientation tensor through the medium thickness. All the components of the orientation tensor are quite similar for both samples. These results

show, as mentioned concerning the porosity and through-plane porosity evolution (Section 3.2.2), that the area of the analyzed zones involved in these X-ray micro-tomography analyses is well representative of the medium fibrous structure. This also confirms that the medium is rather homogeneous. Chapter 3 Experimental characterization of the structural properties of the fibrous medium for an accurate prediction of its permeability 118 Indeed, the organization of fibers in 3D is dependent on the way of fabrication of the medium, and this is crucial to be aware of since it can lead the selection of permeability models, some of which are produced by making assumptions about fiber orientation.

However, few studies publish results of this sort, despite the fact that it appears interesting to characterize fiber orientation in space. Among the few published studies, Soltani et al. [START_REF] Soltani | 3D Fiber Orientation Characterization of Nonwoven Fabrics using X-ray Micro-Computed Tomography[END_REF],

revealed findings similar to the those identified in this research, namely that the XX and YY components are nearly identical, while the ZZ component is negligible for four nearly layered fibrous networks. Furthermore, for all networks, the off-diagonal components of the orientation tensor were minimal.

Permeability modeling

As a reminder, the medium permeability that has been determined by permeametry (Section 3.1.1) is Bexp = (3.10 ± 0.10).10 -11 m 2 . In this section it is compared to values predicted by several models of the literature (presented in Chapter 1 -Table 1), with a special focus on the influence of the parameters taken into account to represent the fiber diameter distribution.

Concerning the porosity, the medium in-plane porosity variations described previously showed that the medium is rather homogeneous. The mean porosity value is thus firstly used for the calculations. On the basis of the medium in plane porosity variations obtained thanks to X-ray micro-tomography data, a permeability mapping is then presented. All of the experimental values used for these calculations are recapitulated in Table 15.

Table 15 Characteristics of the medium used to test the permeability models As shown in Figure 46, all the models under predict the experimental permeability. The most commonly used models of Davies [START_REF] Davies | The Separation of Airborne Dust and Particles[END_REF] and Jackson and James [START_REF] Jackson | The permeability of fibrous porous media[END_REF] probably due to the slight deviation to purely 2D oriented fibers exhibited by the fibrous structure that has been observed thanks to the analysis of the fiber orientation tensor (see Section 3.4).

Permeability

These results are in good agreement with the analysis of Woudberg et al [START_REF] Woudberg | Investigating the influence of local porosity variations and anisotropy effects on the permeability of fibrous media for air filtration[END_REF]. They showed that the RUC models (isotropic and TP versions) were the most successful to predict the permeability of three real fibrous mediums with mean fiber diameter ranging from 6.5 ± 0.1 to 24.0 ± 0.1 µm and mean porosity ranging from 85.0 ± 1.0 to 98.6 ± 0.7%. For these mediums the RUC iso model yielded Bexp / Bmod ratios of 7.3 to 1.6. The more or less significant deviation of the modeled permeability compared to the experimental value was explained by the rather heterogeneous nature of the medium. These results thus strengthen the fact that RUC models;

and especially the isotropic RUC model; are quite successful in predicting the permeability of fibrous medium with structural properties typically representative of these used for air filtration, i.e. fiber diameters of several microns, and rather high porosities from 85 % up to almost 100%.

Moreover, these models have the advantage of being purely analytical.

Equivalent fiber diameter

Figure 47 compares the experimental permeability to these predicted by the RUC iso model by taking into account the mean fiber diameter (as presented previsouly) or the equivalent diameter to depict the bimodal fiber diameter distribution calculated thanks to the three formulations presented in Table 2. The three values obtained are df,eq,1 = 1.92 µm, df,eq,2 = 2.17 µm and df,eq,3 = 2.12 µm respectively for the formulations of Clague and Phillips [START_REF] Clague | A numerical calculation of the hydraulic permeability of three-dimensional disordered fibrous media[END_REF], Brown and Thorpe [START_REF] Brown | Glass-fibre filters with bimodal fibre size distributions[END_REF] and Tafreshi et al. [START_REF] Vahedi Tafreshi | Analytical expressions for predicting permeability of bimodal fibrous porous media[END_REF]. As in Figure 46, for the permeability predicted by models, the error bars correspond to the deviation due to the mean porosity deviation.

Figure 47 Comparison of the experimental permeability to values predicted by the RUC iso model and considering equivalent diameters to represent the bimodal fiber distribution calculated by different approaches.

Figure 47 shows that considering an equivalent diameter to represent the bimodal fiber diameter distribution (whatever the equivalent diameter formulation considered) instead of the mean fiber diameter significantly improves the prediction of the experimental permeability. For the RUC iso model the Bexp / Bmod ratio is reduced from 2.1 using the mean fiber diameter to 1.4, 1.1, and 1.2 using the equivalent diameter df,eq,1, df,eq,2 and df,eq,3 respectively. The best prediction is obtained with the equivalent diameter formulation of Brown and Thorpe [START_REF] Brown | Glass-fibre filters with bimodal fibre size distributions[END_REF] that leads to a permeability of 2.75.10 -11 m².

This result enables to extend to fibrous medium of high porosity the assessments of Gervais et al. [START_REF] Gervais | Permeability modeling of fibrous media with bimodal fiber size distribution[END_REF]. They showed that for five bimodal fibrous medium of porosity ranging from 81% to 86%, and the fibers with 2.6 < df/dc < 4.0, the Brown and Thrope [START_REF] Brown | Glass-fibre filters with bimodal fibre size distributions[END_REF] equivalent diameter formulation leads to the most successful permeability predictions.

In Figure 46 and Figure 47 the deviations of the modeled permeability are rather significant. This is due to the fact that permeability models are highly influenced by the porosity for high porosity values close to 100% like the one of the studied medium, i.e. ε = 92.0 ± 1.6%. Even if this medium is rather homogeneous, it has been observed in Section 3.2.2 that local porosity variations can reach minimum and maximum values of 86% and 98% respectively. In the next section the sensitivity of the modeled permeability calculated by the RUC iso model to the porosity is thus analyzed. Chapter 3 Experimental characterization of the structural properties of the fibrous medium for an accurate prediction of its permeability 122

Sensitivity of the permeability to the porosity and in-plane permeability variations

Figure 48 depicts the sensitivity of the permeability calculated by the RUC iso model to the porosity with the equivalent fiber diameter calculated with the Brown and Thorpe formulation.

This figure shows that the modeled permeability corresponding to the entire range of local porosity values that have been detected by X-ray micro-tomography on the two medium samples (ε = 87 -98%) ranges from 7.78.10 -12 to 4.10.10 -10 m². This significant permeability range, characterized by a 51 factor between extreme values, reflects the high sensitivity of the modeled permeability to the porosity for such high porosity values.

Figure 48 Sensitivity of the medium predicted permeability to the mean medium porosity

For the medium investigated in the present study, the experimental permeability is accurately predicted by the RUC iso model by using the mean porosity and taking into account the bimodal fiber diameter distribution thanks to an appropriate equivalent diameter. This is due to the fact that the medium is rather homogeneous compared for example to these reported in Woudberg et al. [START_REF] Woudberg | Investigating the influence of local porosity variations and anisotropy effects on the permeability of fibrous media for air filtration[END_REF]. Even if this medium is rather homogeneous, every fibrous medium exhibits more or less significant local porosity deviations, as it has been shown from Figure 38 The in-plane porosity mappings of both samples (section 3.2.3 of this Chapter) analyzed by Xray micro-tomography have been converted into permeability mappings. For that purpose, for each patch (of 155x155 pixel² area) the permeability is calculated thanks to the RUC iso model by using the local porosity value obtained for the considered zone and the equivalent fiber diameter df,eq,2 = 2.17 µm. The permeability mappings for both samples are presented in Figure 49. The minimum and maximum permeability values detected for both samples are 6.

Conclusion

In this Chapter, the characterization of the structural properties of a commercial glass fiber medium was carried out thanks to complementary experimental methods. The properties of the medium and the experiment techniques used to characterize them are summarized in Table 16.

Table 16 Properties of the medium and the experiment techniques used to characterize it The analysis of SEM images of both medium sides showed a bimodal fiber diameters distribution. The medium porosity was measured by two techniques: mercury porosimetry and X-ray micro-tomography. It is noteworthy that despite the fact that these techniques are based on different principles, the results were in quite good agreement. Moreover, as both measurement techniques involve sample area of different orders of magnitude, it enabled to point out that the medium is rather homogeneous. The analysis of in-plane porosity variations at the scale of medium samples analyzed by X-ray micro-tomography confirmed that the medium exhibit a rather homogeneous structure. The calculation of the orientation tensor on the basis of X-ray micro-tomography data revealed the anisotropic character of the fibers arrangement. The amplitude of the diagonal components showed that most of fibers are randomly oriented in planes perpendicular to the flow (i.e. parallel to the medium plane).

The structural characteristics of the medium were used to evaluate the capacity of different models reported in the literature to predict its permeability. Among the models tested, the RUC models proved their efficiency to reproduce the experimental permeability of the bimodal studied medium. Even if both RUC models proposed by Woudberg [START_REF] Woudberg | Permeability prediction of an analytical pore-scale model for layered and isotropic fibrous porous media[END_REF] (through-plane and isotropic versions) performed well, the isotropic RUC model led to the best prediction of the medium permeability. Although the analysis of the fibers orientation tensor showed that the medium is rather 3D anisotropic, this results may be due to the fact that the isotropic RUC model enables to depict the slight deviation to purely 2D oriented fibers exhibited by the fibrous structure. Moreover, using an equivalent diameter to represent the bimodal fiber diameter

X-ray microtomography

Chapter 3 Experimental characterization of the structural properties of the fibrous medium for an accurate prediction of its permeability 125 distribution instead of the mean fiber diameter; especially the one used by Brown and Thorpe [START_REF] Brown | Glass-fibre filters with bimodal fibre size distributions[END_REF]; improved the prediction of the permeability. These results highlight the importance of a precise and reliable characterization of fibrous medium structural properties for the selection of an appropriate permeability model. As permeability models are highly sensitive even to slight porosity variations in-plane permeability mappings have been presented. For that purpose, porosity mappings obtained from X-ray micro-tomography, equivalent fiber diameter representing the bimodal distribution, and the isotropic RUC model were used. This data representation enables the observation of the effect of structural variations (for the present case in terms of porosity) on local permeability values. Due to the particularly soft, highly porous, and generally heterogeneous nature of fibrous medium for air filtration, the issue of predicting accurately their permeability is really challenging. Having access to such in-plane permeability variations offers interesting outlooks for the prediction of filters' performance by taking into account their real properties.

Chapter 4 Local variations of air velocity in the vicinity of clean filter pleats -experimental and numerical approaches 1. Introduction

In Chapter 2, the methodology to investigate the local velocity gradients upstream and downstream of a pleated filter with geometrical properties similar to those found in air handling units was detailed. It combines experimental velocity measurements using HWA and numerical simulations in the vicinity of filter pleats. This methodology is implemented in this Chapter on the basis of two pleated filter geometries: one with a relatively wide width (H = 28 ± 1 mm, W = 25 ± 1 mm) and another with narrow pleats (H = 28 ± 1 mm, W = 10 ± 1 mm). Because these prototypes are being implemented in the same duct, the second has more pleats and a larger developed surface.

The first section of the Chapter investigates the effect of filter geometry and medium permeability on the simulated pressure drop and compares it to the experimental results.

Following that, experimental velocity profiles upstream and downstream of the filter pleats are presented and compared to the numerical simulation results. After having validated the CFD simulations, further numerical investigations of the velocity at the air/filter interface are conducted. Finally, the experimental local turbulence intensity values acquired by the HWA technique in the vicinity of filter pleats are presented.

Prediction of filter pressure drop

In this section, the results of the experimental and numerical pressure drop, which is a macroscopic parameter, are presented in function of the filtration velocity for the two tested geometries for: "wide-width geometry" (H = 28 ± 1 mm, W = 25 ± 1 mm) and the "narrowwidth geometry" (H = 27± 1, W =10 ± 1 mm). The influence of the airflow model, pleat geometry and permeability of the pleat curved zones, i.e. pleat peak and hollow, are also studied.

Influence of airflow model on pressure drop

Influence of pleat geometry and curved pleat zone permeability on pressure drop

The results presented in Figure 50 and Figure 52 show that the CFD calculations reproduce the order of magnitude of the experimental pressure drop quite well, even if they tend to under- Figure 50 (a) shows that the pressure drop is very sensitive to pleat geometric properties. This aspect has also been pointed out by Theron et al. [START_REF] Théron | Numerical and experimental investigations of the influence of the pleat geometry on the pressure drop and velocity field of a pleated fibrous filter[END_REF]. The pressure drop increases with lower pleat width and higher pleat height, which concurs largely with tendencies reported in the literature for different filter properties and flow conditions [START_REF] Del Fabbro | Air flows and pressure drop modelling for different pleated industrial filters[END_REF][START_REF] Théron | Numerical and experimental investigations of the influence of the pleat geometry on the pressure drop and velocity field of a pleated fibrous filter[END_REF][START_REF] Rebaï | A semi-analytical model for gas flow in pleated filters[END_REF][START_REF] Maddineni | Numerical investigation of pressure and flow characteristics of pleated air filter system for automotive engine intake application[END_REF]. Consequently, the maximum deviation between experimental and numerical pressure drop for the filtration velocity range studied is reduced from 20% with mean pleat height and width (H = 28 mm, W = 25 mm) to 15% with the set of parameters H = 29 mm and W = 24 mm. were obtained with the set of parameters H = 29 mm and W = 24 mm; the deviation between numerical and experimental pressure drop was also reduced to 12%. These results are in accordance with the CFD simulation results of Rebaï et al. [START_REF] Rebaï | A semi-analytical model for gas flow in pleated filters[END_REF]. This can be explained by the fact that the total available filtration area is reduced, causing the pressure drop to increase. The air streamlines approaching the impermeable curved zone are deviated, which leads to increased air drag due to the higher velocity in the pleat channel and consequently to increased pressure drop across the filter. For the same filtration velocity, the pressure drop for is slightly higher for (H = 27 mm, W = 10 mm). This result is consistent with the literature [START_REF] Del Fabbro | Air flows and pressure drop modelling for different pleated industrial filters[END_REF][START_REF] Théron | Numerical and experimental investigations of the influence of the pleat geometry on the pressure drop and velocity field of a pleated fibrous filter[END_REF][START_REF] Rebaï | A semi-analytical model for gas flow in pleated filters[END_REF][START_REF] Maddineni | Numerical investigation of pressure and flow characteristics of pleated air filter system for automotive engine intake application[END_REF].

As previously stated, pressure drops increase as width decreases. However, in numerical simulations for the tested filtration velocity range (0-0.31 m.s -1 ), this effect is less pronounced.

It could be related to the fact that pressure is sensitive to the pleat geometry at higher filtration velocities. The irregularities of the experimental pleats can also amplify this effect.

To summarize, the simulated pressure drop obtained with the highest pleat height and lowest pleat width (H = 29 ± 1 mm, W = 24 ± 1 mm) is closest to the experimental pressure drop of the geometry (H =28 ± 1 mm, W = 25 ± 1 mm) for the permeable curved zone. In addition, the

Velocity patterns

Following the presentation of the pressure drop results, the experimental velocity profiles upstream and downstream of the filter pleats of both tested geometries are presented in this

ΔP(Pa)

V filtration (m.s -1 ) W=25 H=28 Darcy W=25 H=28 Darcy.Fsh Experimental data section, and compared to the numerical simulation results. Furthermore, the effect of the pleat curved-zone permeability on numerical velocity profiles downstream the filter pleat (case of H = 28 mm and W = 25 mm) is investigated.

Numerical velocity fields in the vicinity of the filter

Figure 54 shows the velocity field in the vicinity of the half-pleat (H =28 mm, W = 25 mm) obtained by CFD for the three airflow models tested (laminar, standard 𝑘-𝜀 and transition SST)

for Vmean = 0.67 m.s -1 (corresponding to Vfiltration = 0.27 m.s -1 ). The scale adopted to represent the velocity contours is adapted to the maximum velocity obtained by the models i.e. that of the laminar in this case.

Figure 55 shows the velocity field in the vicinity of the "narrow width" half-pleat (H =27 mm, W = 10 mm) obtained by CFD for the same three airflow models tested for Vmean = 1.48 m.s -1

(corresponding to same filtration velocity as the first geometry, i.e. Vfiltration = 0.27 m.s -1 ).

For both geometries the three velocity fields show that no significant velocity gradients are detected upstream of the pleat for this filtration velocity, while significant gradients are observed downstream of the pleat whatever the airflow model, with the exception of the 𝑘-𝜀 model in the case of (H = 27 mm, W = 10 mm). The intensity of the velocity gradient obtained with the 𝑘-𝜀 model upstream of the filter is similar to that observed downstream of the filter (gradient of velocity between 1.8 and 2.4 m.s -1 ). In order to visualize this gradient, Figure 55 depicts the velocity gradient at a velocity scale adapted to the maximum velocity reached by the 𝑘-𝜀 model downstream of the pleat i.e. 2.4 m.s -1 .

Upstream of the filter pleat (H= 28 mm, W = 25 mm), the velocity tends to be homogeneous and close to the mean duct velocity (0.67 m.s -1 ) approaching the filter for the three models tested. In the case of "the narrow width" geometry (H = 27 mm, W = 10 mm), a slight increase of the velocity is detected up to 2.00 m.s -1, i.e. 1.40 times higher than the mean duct velocity (1.48 m.s -1 ) for the 𝑘-𝜀 model (see Figure 55).

Experimental velocity profiles downstream of the filter

Two representations are given of the experimental transversal velocity profiles downstream of the filter (see Figure 58). The data in Figure 58 (a) represent the evolution of the velocity along the y coordinate for different longitudinal x positions. In this graph, y = 0 cm represents the bottom symmetry line, i.e. the symmetry axis passing through the pleat peak, as depicted in Fig. 6. In Figure 58 (b) these profiles are presented as dimensionless values: the velocity is normalized by the mean duct velocity: Vmean = 0.67 m.s -1 for (H = 28 mm, W =25 mm) and Vmean = 1.48 m.s -1 for (H = 27 mm, W =10 mm) and the vertical coordinate y is normalized by y0, defined as the distance between the symmetry axis passing through the pleat peak and the medium surface at the x coordinate in question. The ratio y/y0 is thus equal to 0 at the bottom symmetry line, and to 1 at the medium surface, as illustrated in Figure 57. that the results are highly reproducible and that the Hot Wire Anemometry technique is appropriate for these measurements, even inside the pleats and in close proximity to the filter medium (1 mm in the case of (H = 28 mm, W = 25 mm) and 0.5 mm in the case of (H = 27 mm, W = 10 mm)). For the both cases, the most significant deviations are recorded downstream of the pleat peak (i.e. for y values of 0 to 2 mm). This can be explained by the fact that this zone is located directly downstream of a curved portion of the pleat that might exhibit local deviations of the structural properties of the medium (porosity, thickness, permeability, etc.), and/or deviations in the shape of the curved zones due to the manual pleating process. For both geometries, these velocity deviations are less pronounced downstream of the pleat hollow. This may be because the measurement points in this zone are very close to the medium and the air velocity is therefore lower than elsewhere in the pleat channel. Some small irregularities in terms of pleat height and width can also lead to non-negligible deviations of the velocity profiles, which may also explain the general amplitude of the error bars. Due to the relative small width of the geometry (H = 27 mm, W = 10 mm), the velocity is more sensitive to these deviations compared to the wide width geometry (H = 28 mm, W = 25 mm). the pleat at x = 2.0 cm and (b) outside the pleat at x = 3.5 cm. The velocity profiles inside the pleats, i.e. at x = 2.0 cm, are fairly similar. However, at x=3.5 cm downstream, the narrowwidth filter pleats (H = 27 mm, W = 10 mm) exhibit a lower V/Vmean ratio than the geometry pleats (H = 28 mm, W = 25 mm) for y/y0>0.3 and y/y0<0.1. For both geometries, the velocity reaches a maximum at y = 0 mm. At this point the ratio V/Vmean is close to 3 in case of the geometry (H = 28 mm, W = 25 mm), whatever the x coordinate concerned, as observed in Figure 58 (b). However, in the case of the geometry (H = 28 mm, W = 10 mm), a slightly smaller ratio V/Vmean of 2.5 is observed, as shown in Figure 59 (b). For both cases, the velocity then decreases as y increases. For the two profiles inside the pleat (x = 1.5 and 2.0 cm), the minimum recorded velocity Vmin, i.e. in closest proximity to the medium surface, is near to the Chapter 4 Local variations of air velocity in the vicinity of clean filter pleats -experimental and numerical approaches 137 filtration velocity of 0.27 m.s -1 . The minimum velocity recorded for the three velocity profiles outside the pleat (x = 3 -5 cm) is 0.14 m.s -1 for (H = 28 mm, W = 25 mm) (see Figure 58 (a)) and 0.18 m.s -1 for (H = 27 mm, W = 10 mm) (see Figure 59 (a)), respectively. This velocity is detected downstream of the pleat hollow where low velocity variations are observed (see Figure 54). As mentioned previously, this can be explained by the fact that this zone of the filter is located directly downstream of a curved pleat zone. In addition, Figure 58 (a) and Figure 59 (a) show that at the three positions outside the filter (x = 3 -5 cm) the profiles are identical. It therefore reveals that the geometric properties of the filter and airflow conditions concerned lead to significant velocity gradients downstream of the filter and persist for at least 2.5 cm downstream of the filter (2.5 cm in height). Figure 58 (a) Figure 59 (a) also shows that the velocity profiles are sharper going deeper into the pleat (i.e. x decreases). This is basically due to the continuous reduction of available space for flow along the pleat length in the case of triangular pleats.

When the velocity profiles are presented as dimensionless parameters they are identical for the geometry under consideration, as observed in Figure 58 For y ≤ 2 mm in the case of (H = 28 mm, W = 25 mm) and y ≤ 1 mm in the case of (H = 27 mm, W = 10 mm), the discrepancies between the numerical velocity values are more significant.

Numerical versus experimental velocity profiles downstream of the filter

This can be explained by the fact that this zone is located downstream of a curved zone of the pleat (the pleat peak) which gives rise to high-velocity gradients difficult to capture numerically. Also, as mentioned previously (see sections 2.1 and 2.2), in this zone the filter medium may be subject to modifications of its structural properties (porosity, thickness, permeability) which are difficult to reproduce numerically and may influence airflow behavior.

These observations concur closely with the numerical results produced by Rebaï et al. [START_REF] Rebaï | A semi-analytical model for gas flow in pleated filters[END_REF], who showed that the structural properties of the medium in the pleat curved zones are modified by the pleating process and that resistance to flow increases in this area, which may affect the airflow pattern locally. Similar deviations have also been reported by Kang et al. [START_REF] Kang | Characterization of pleated filter media using particle image velocimetry[END_REF], who To summarize, the overall trends for velocity profiles obtained using HWA and CFD calculations concurred closely for the three numerical airflow models tested, except near the pleat curved zone. The laminar model was revealed as the one best able to reproduce the velocity distribution downstream of these zones. The next section addresses the issue of sensitivity of velocity profiles downstream of the filter to the permeability of pleat curved zones. In respect of the considerations mentioned above, only data obtained with the laminar model will be given for the analysis of numerical velocity profiles.

Influence of pleat curve-zone permeability on numerical velocity profiles downstream of the filter

As mentioned previously (see section 2.1), considering the pleat curved zone as impermeable significantly influences the pressure drop. The effect of such an assumption on the velocity The subject of the following section is thus a comparison of experimental and numerical velocity profiles downstream of the filter in order to assess the effect of pleat curved-zone permeability on velocity profiles. the numerical velocity profile obtained with impermeable curved zones is identical to that with completely permeable ones, except for y ≥ 9 mm. From this position the velocity starts to decrease and reaches a null value at y = 10.5 mm, i.e. downstream of the pleat hollow. It then increases to become identical with the numerical value obtained for the completely permeable ones and the experimental value at the end of the curve. Similar curve shapes were obtained for all x coordinates studied downstream, outside of the filter (x = 3.0 and 5.0 cm). These results

show that assuming the curved pleat zones to be non-permeable has a significant effect on velocity distribution downstream of this zone outside the pleat, more precisely downstream of the pleat hollow, but becomes less pronounced elsewhere along y, or inside the pleat. Figure 66 (a) compares the experimental velocity profile upstream of the filter at x = 0.5 cm, to the one obtained numerically using the laminar airflow model. As previously stated in Chapter 3, the HWA probe was used to take a single measurement inside one pleat, with the HWA placed horizontally, i.e. perpendicular to the air flow in the x-y plane (see Chapter 2 -Section 5). As a result, the experimental measurement corresponds to the average value over the entire 1.5 mm wire length and we represented the average experimental value as a 0.75 mm line.

Numerical versus experimental velocity profiles upstream of the filter

The minimum distance between the middle of the horizontal wire and the filter medium surface at which the velocity was measured was 4 mm. Re w = ρV mean 0.5W µ [START_REF] Vahedi Tafreshi | Analytical expressions for predicting permeability of bimodal fibrous porous media[END_REF] considerably higher than the mean duct velocity approaching the filter. These studies, however, are characterized by a turbulent airflow regime approaching the filter. For example, the CFD simulation of Theron et al. [START_REF] Théron | Numerical and experimental investigations of the influence of the pleat geometry on the pressure drop and velocity field of a pleated fibrous filter[END_REF] showed that for Vmean =2.8 m.s -1 , corresponding to Reh = 22,480 and Rew = 2150 and 1080 (for W = 23 and 11.5 mm respectively), the velocity was over 5 times the mean duct velocity at the pleat center upstream of the filter. These considerations might indicate that in terms of the airflow inside the pleat, the onset of turbulence appears at a channel Reynolds number between 560 and 1080. However, the numerical simulation of Rebaï et al. [START_REF] Rebaï | A semi-analytical model for gas flow in pleated filters[END_REF] for Vfiltration = 0.39 m.s -1 and Rew = 1097 (for W = 4 mm) showed that the velocity varied slightly at the pleat center upstream of the mini-pleated filter and barely reached 1.1 times the mean duct velocity, which is close to the results obtained in the present study. The appearance of turbulence on the upstream side of the pleat therefore depends on a number of parameters (filtration velocity, filter geometry and structural properties), and further assessment is required to determine the transition channel Reynolds number above which the turbulence starts to occur.

The numerical velocity profiles in Figure 65 and Figure 66 (b) enable to evaluate the order of magnitude of local velocities at which particles might reach the filter at the air/filter interface (0 mm from the medium surface) and two distances from the medium surface: 0.05 and 0.10 mm for both geometries (H = 28 mm, W =25 mm) and (H =27 mm, W =10 mm). For a same distance from the surface, the trend of the numerical velocity in the case of the geometry (H=28 mm, W = 25 mm) is different from that of the geometry (H = 27 mm, W =10 mm). In the first case the numerical velocity decreases when x increases i.e. towards the pleat hollow (see Figure 65 (b)). However, when the pleat width is reduced, the numerical velocity profiles slightly increases when approaching the hollow of the pleat. Rebaï et al. [START_REF] Rebaï | A semi-analytical model for gas flow in pleated filters[END_REF] reported similar findings, demonstrating numerically that for Vfiltration = 0.39 m.s -1 and a pleat width of 4 mm, the velocity tends to increase towards the bottom of the pleat, possibly causing preferential particle deposition in this zone during clogging. However, their hypotheses were not validated through experimental observations and were entirely based on numerical simulations.

For the straight part of the pleat (H =28 mm, W =25 mm) located at 0.10 mm above the surface of the medium, i.e. for x coordinates between 1.0 and 27.0 mm as shown in Figure 65 The order of magnitude of local velocity decreases significantly when approaching the surface of the medium. In the case of the geometry (H = 28 mm, W = 25 mm), the velocity ranges between 0.60 m.s -1 and 0.45 m.s -1 along the straight line 0.05 mm above the surface of the medium, and between 0.98 m.s -1 and 1.27 m.s -1 in the case of the geometry (H = 27 mm, W = 28 mm), representing 2.2 -1.5 and 3.6 -4.7 times the theoretical filtration velocity respectively.

In the case of the geometry (H = 28 mm, W = 25 mm), the local velocity ranges between 0.46 m.s -1 and 0.38 m.s -1 along the straight line on the medium surface, and between 0.50 m.s -1 and 0.54 m.s -1 in the case of the geometry (H = 27 mm, W = 10 mm), representing 1.9 -1.6 and 1.8 -2.0 times the theoretical filtration velocity, respectively.

These results show that even though the airflow upstream of the pleat is mostly laminar, some velocity gradients may appear close to the surface of the medium. These variations may generate some local particle collection efficiency gradients. Accordingly, these observations raise the question of the relevance of adopting the filtration velocity to describe the velocity at which the particles are exposed at the close proximity of the filter when predicting the particle collection efficiency.

Turbulence intensity profiles

Of the few studies in the literature that have reported experimental velocity recordings in the vicinity of the filter [START_REF] Feng | Assessment of various CFD models for predicting airflow and pressure drop through pleated filter system[END_REF][START_REF] Kang | Characterization of pleated filter media using particle image velocimetry[END_REF][START_REF] Maddineni | Numerical investigation of pressure and flow characteristics of pleated air filter system for automotive engine intake application[END_REF] none has shown turbulence intensity values, to our knowledge.

The turbulence intensity TI represents the ratio of the root-mean-square fluctuating velocity v rms ′ to the average velocity V ̅ :

For both the mean duct velocity in question of 0.67 m.s -1 and 1.48 m.s -1 , the velocity recordings made in the empty duct led to mean TI values at the duct center of 7.0% and 6.7%, respectively. 68). This increase may be due to the complex shape of the pleat and the locally-modified structural properties of the medium. It appears also that considering a narrow width, increases the turbulent intensity 2.5 cm downstream the curved pleat zone compared to the wide geometry.

Additional characterization of the change in the structural properties of the curved zone due to the pleating is thus, necessary for a better understanding of such phenomena.

To summarize, the turbulence intensity profiles downstream of the filter showed a very low gradient inside the pleat and an increase outside the pleat, especially downstream of the pleat hollow. At the filtration velocity tested (Vfiltration = 0.27 m.s -1 ), no significant turbulence intensities were detected close to the medium surface. However, it would be useful to measure turbulence intensities for higher mean duct velocity values, especially downstream of the filter, since as reported by Maddineni et al. [START_REF] Maddineni | Numerical investigation of pressure and flow characteristics of pleated air filter system for automotive engine intake application[END_REF], significant velocity fluctuations beyond a critical value during clogging may cause particle bouncing and increase particle penetration by particle re-entrainment.

Conclusion

In this study, the methodology for the characterization of velocity patterns in the vicinity of pleated fibrous filters combining experimental and numerical methods that have been developed in Chapter 2, in the main aim of being able to validate the numerical simulations by local experimental velocity values. The comparison of experimental to numerical velocity values was carried out for a transitional airflow regime approaching the filter, for which few studies have been reported in the literature. For the CFD calculations the ability of the laminar, standard 𝑘-𝜀 and transition SST airflow models to predict the pressure drop and airflow pattern in the vicinity of the filter was thus tested.

If the numerical pressure drop was not significantly affected by the airflow model, the experimental pressure drop was under-predicted by the numerical simulations. It was shown that even slight variations of the pleat dimensions, and modifications of the medium properties;

especially due to the pleating process in the pleat curved zones; led to non-negligible pressure drop variations. The numerical pressure drop was thus increased by considering the pleats slightly higher and narrow than the mean experimental prototypes dimensions, and by making the extreme hypothesis of impermeable curved pleat zones.

These investigations indicate that for pleat dimensions like those that have been studied (H = 28 mm and W = 25 mm) and (H = 27 mm, W = 10 mm), despite its intrusive character the Hot Wire Anemometry technique is appropriate for velocity acquisitions inside the pleat downstream of the filter, up to a very short distance from the medium surface, and for low velocity values (up to 0.14 m.s -1 ). The comparison of numerical to experimental velocity profiles in the close vicinity of the filter enabled to show that the laminar model was the best for the prediction of the overall airflow pattern. For a transitional airflow regime, and quite "wide" pleats, the air velocity was quite homogeneous upstream of the filter, and very close to the mean duct velocity value. For a turbulent air flow regime approaching the filter and a narrow width geometry, the air velocity upstream the filter was also quite homogeneous, but higher than the mean duct value, indicating the initiation of a velocity gradient. The CFD simulations enabled to point out that velocity variations, and more precisely sharp decreases, appeared only in a thin air layer along the medium surface, to reach values close to the theoretical filtration velocity. Significant velocity gradients were observed downstream of the filter and persisted outside the filter.

Another advantage of the HWA anemometry technique is that it also enables to derive turbulence intensity profiles from velocity profiles. The turbulence intensity profiles downstream of the filter showed low variations inside the pleat, and increased outside the pleat.

It would be interesting to measure velocity fluctuations for higher filtration velocity values in order to quantify possible fluctuations leading to non-stationary physical phenomena during clogging, such as particle bouncing and re-entrainment.

Chapter 5 Influence of the clogging on the filtration performance and on the velocity profiles downstream of the filter pleats 1. Introduction

According to the literature review (see Chapter 1), there are just a few studies that dealt with the influence of clogging on the velocity profile downstream of fibrous filters. These investigations, however, were solely numerical, with no experimental validation (see Chapter 1section 6.3). Our research is different in that it offers local velocity measurements downstream of filter pleats at a defined clogging level, for two different solid aerosols and filter geometries. These characterizations make it possible to evaluate the influence of the clogging on the properties of the flow near the filter. To our knowledge, the work presented in this study is one of the first experimental investigations of the effect of clogging on velocity profiles downstream of pleated fibrous filter. The main benefit of such work is the understanding of physical phenomena occurring downstream of filter pleats such as particle bouncing and reentrainment. The results can be also used as a database for airflow modeling and filter design optimization, particularly when filters are implemented in series.

Three types of particles have been used for this work:

 Two types of particles were used separately to clog the filter: micron Silica test particles and submicron sodium chloride particles.

 Soda fluorescein particles were generated at the end of the clogging process to identify any preferential particle deposition on the pleated filter.

Silica particles and sodium chloride particles were chosen for clogging because they have distinct average aerodynamic diameters i.e. dae, 50, Nacl = 0.70 µm and dae, 50, Silica = 1.00 µm. Nacl particles are classified as submicron (d50≤ 1.00 µm), whereas Silica particles are classified as micron (d50 ≥1.00 µm).

As discussed in Chapter 1, particle diameter influences the morphology of the cake during clogging: for submicron particles, the cake exhibits a dendritic-like form with high cake porosity. In the case of micron particles, the cake is more compact and displayed in the form of aggregates. However, since submicron particles have a small diameter, their specific surface area is even greater, which leads to a greater pressure drop in the filter compared to micron particles. The selection of submicron and micron particles aims at studying the potential influence of the cake geometry on velocity profiles downstream of pleats.

Chapter 5 Influence of the clogging on the filtration performance and on the velocity profiles downstream of the filter pleats 154

The materials used to monitor the filtration performance during clogging (i.e. pressure drop and particle filtration efficiency) are presented first in this Chapter, followed by the strategy employed to carry out velocity measurements downstream of the filter. Secondly, the results of the filtration performance and the local velocity profile are presented and discussed. Finally conclusions and perspectives are proposed.

Materials used for clogging

The clogging experiments were carried out using the experimental bench described in Chapter 2. The bench was designed to measure air velocity profiles at the vicinity of clean and clogged filters. The bench and the methodology used for velocity measurements have been broadly

discussed in Chapter 2. The materials used to clog the filter are explained in this section.

Description of the particles used in this study

Sodium chloride particles

Sodium chloride (NaCl) particles are produced from crystals. The structure of NaCl corresponds to two cubic sub-networks with a centered face. It has a solubility of 359 g.dm -3 in water and a density of 2165 kg.m -3 . The sodium chloride tends to absorb moisture and is thus hygroscopic. According to Joubert et al [START_REF] Joubert | Influence of Humidity on Clogging of Flat and Pleated HEPA Filters[END_REF]. when relative humidity reaches a certain level, known as Relative Deliquescence Humidity (RDH), crystalline particles spontaneously capture water to create bigger salt droplets. As the relative humidity falls, the aqueous saline particles begin to evaporate and crystallize at a relative humidity value known as Relative Efflorescence Humidity (REH) or delitescence. For NaCl particles having diameter larger than 0.10 µm and at a temperature of 25 o C, RDH NaCl is around 75% and REH NaCl is around 45% [START_REF] Joubert | Influence of Humidity on Clogging of Flat and Pleated HEPA Filters[END_REF]. The hygroscopic nature of NaCl particles was an additional challenge during the experiments as the particle morphology changes when the humidity increases. SEM pictures at a x500 magnification show particle deposition on the surface of a flat filter at (a) RH = 61% and (b) RH = 70%. First, since the particles are submicron, the deposit is dendritic-like [START_REF] Joubert | Influence of Humidity on Clogging of Flat and Pleated HEPA Filters[END_REF], and it can be seen as particles adhering one to the other.. However, at an increased humidity i.e. at 70% more clusters are visible. Indeed, due to the hygroscopic nature of the NaCl, the morphology Despite the interesting effect of relative humidity observed on the filtration performance and discussed later in section 4, the main aim of this study is to evaluate the effect of solid particle deposition on velocity profiles downstream of filter pleats. The effect of humidity on velocity profiles is not broadly discussed in this study but opens an interesting perspective for future investigations. As a result, we only evaluate the effect of clogging with hygroscopic NaCl particles on velocity profiles downstream of the pleats of a filter with geometrical dimensions (H = 28 mm, W = 25 mm) in dry conditions (RH = 40 -50 %). Concerning the study of the influence of particle deposition on velocity profiles downstream of "the narrow-width" geometry (H = 27 mm, W = 10 mm), as it was carried out only with non-hygroscopic particles, the humidity constraint was avoided.

Silica test particles

ISO 12103-1, A2 FINE TEST DUST commercially known as "Silica test particles" is used in many applications such as air filter performance testing. It is a complex mixture of many particles, of which the chemical composition is summarized in Table 17. The mixture has an average density of 2600 kg.m -3 . Silica particles are non-hygroscopic and hence less sensitive to humidity than NaCl particles. As a result, humidity has no impact on particles morphology.

Fluorescein particles

Fluorescein is a chemical substance having the formula C20H10O5. It is a dark orange/red powder that is somewhat soluble in water (solubility = 500 g.dm -3 at 20 o C). It emits a fluorescence-reflective light when exposed to UV radiation. Its density is 1500 kg.m -3 .

Depending on it's generation mode, it is commonly used as a fluorescent tracer in a variety of applications, including air filtration. In this study, it was generated at the end of a clogging experiment of a pleated filter (H = 28 ± 1 mm, W = 25 ± 1 mm) with NaCl particles to determine whether or not the clogging was homogeneous over the surface of the filter medium.

Particle generation technologies

Two technologies have been used to generate solid particles:

 AGK 2000 atomizer from PALAS for Fluorescein and NaCl particles generation  Ring dust particle disperser 3410U from TSI for Silica test particles generation

Following is an explanation of their specifications and operation.

AGK 2000 Palas

The An adjustable pressure regulator drives pressurized air to the binary atomizer nozzle. The pressure, adjustable between 0 and 4 bars, was set to 2 bars for the experiments. The liquid is drawn from the liquid reservoir by the vacuum generated in the nozzle and pushed out of the liquid reservoir by the vacuum generated inside the nozzle. The nozzle produces a particle cloud (liquid droplets) that travels tangentially into a cyclone. Large droplets are caught and returned to the liquid reservoir due to centrifugal forces. The others leave the cyclone through the exit tube. The droplets that exit the generator are then dried before passing through the filter via the TR2000 drying stage and the airflow in the test stream.The size distribution of these droplets is slightly influenced by the nozzle-generated main droplet distribution, while the cyclone separation characteristics have a far higher impact. As a result, the size distribution of many Chapter 5 Influence of the clogging on the filtration performance and on the velocity profiles downstream of the filter pleats 158 substances is unaffected by the mass flowrate of the liquid in the nozzle. The cyclone separation properties are influenced by the air volume flowrate and, as a result, by the binary pre-pressure atomizer of the nozzle. As the volume flowrate or pre-pressure increases, the separation limit changes to smaller particle sizes. Pure liquids, solvents, and suspensions in general are dispersed by the AGK 2000. The volatile components of solvents and suspensions evaporate, whereas pure liquid particles with low vapor pressure seldom vary as they exit the cyclone.

Ring TSI dust 3410U disperser

Silica test dust particles are generated by the TSI dust 3410U disperser. A photograph of the generator is shown in Error! Reference source not found.. The Dust Particle type 3410U

provides precise continuous and constant generation (output particle concentration range between 0.05 g.m -3 -20 g.m -3 ). The powder is poured constantly around a metal ring. Any powder in excess that falls down the side is collected and stored in a reservoir. Metal rings with different thickness are used to monitor particle concentrations generated. The thicker the ring, the more surface for poured powder is offered, and hence the higher the output concentrations.

The powder is inhaled through a ceramic tube, and transferred to an ejector nozzle. Shear forces in the ejector nozzle disperse and de-agglomerate particles. The reservoir can be refilled while in operation to accommodate long particle generation duration.

Particle counting technologies

In the following sections, we will briefly describe the operation of the two devices used: the aerodynamic particle counter APS and the optical particle counter WELAS.

Aerodynamic particle sizer (APS)

The Aerodynamic Particle Sizer® 3321 (APS ™ spectrometes) was used in this study to The device is made of two main components: a particle nozzle and a laser detector. The operation mode of the APS is described by Renoux and Boulaud [START_REF] Renoux | Les aérosols : physique et métrologie[END_REF].. The particle-laden air is Chapter 5 Influence of the clogging on the filtration performance and on the velocity profiles downstream of the filter pleats 159 forced through a tiny orifice, increasing its acceleration. Due to their inertia, particles need time to adapt their velocity to that of air. As the aerodynamic diameter is proportional to this adjustment delay, the aerodynamic diameter is calculated simply by measuring the velocity of each particle downstream of the nozzle. Two light beams generated by a laser diode pass through each particle at the exit orifice, producing two pulses separated by a specific delay termed "time of flight" (TOF) recorded using a clock. A photomultiplier detects the light pulses and transforms them into an electrical response. Larger TOFs represent large particles moving at a slower rate than air. Time-of-flight analyzers are devices that measure the aerodynamic diameter of particles for a given flow domain and at a constant acceleration. These devices enable to classify thousands of particles in a matter of seconds (nearly a second), allowing to get the sample aerodynamic diameter distributions. The "time-of-flight" analyzer used in this study is the Aerodynamic Particle Sizer (APS). This device measures both the particle aerodynamic diameter based on particle time of flight and optical diameter based on scattered light intensity. The APS specifies the particle size distribution for particles with aerodynamic diameters ranging from 0.5 to 20 micrometers and optical diameters ranging from 0.3 to 20 micrometers. Moreover, in order to widen the range of diameter studied we used the optical counter WELAS from Palas© which measures optical diameters ranging from 0.2 µm to 40 µm.

Optical counter WELAS Palas ©

The WELAS DIGITAL particle size analyzer is an optical spectrometer that counts and measures particle size, providing a high resolution response in a quick and reproducible way. It The WELAS instrument measures the optical diameter (in the 0.2 -40 µm range) and concentration of particles using white light dispersed at 90 o . The counting method is based on the fact that particles travel through a well-defined region with a consistent and intense light intensity. As a result, the particles reflect rays in all directions, which are detected at an angle of 90 o . This detection provides an accurate measurement of particles in the optical diameter range of 0.2 to 40 µm. The signal generated by particles is concentrated in an optical fiber, which transfers it to a photomultiplier in real time to measure its light intensity. Furthermore, particle concentration may be calculated as a function of the number of light signals detected by the photomultiplier (Nparticle) and the sampling rate (Qsampling) which was fixed at 5 L.min -1 .

The concentration of the particles is thus calculated as follows:

Particles Sampling

Particles are sampled using nozzles made in the laboratory to ensure isokinetic sampling conditions. The aim of isokinetic sampling is to sample a quantity of particles that is as close to that found in a volume of flow within the setup as possible; in terms of concentration and size distribution. The key requirement in isokinetic sampling is the absence of flow line divergence around the sampler input. The input sampling nozzle must face the flow lines to prevent or minimize the divergence of the measured stream. As a result, the sample and sampling streams must have the same velocity. The airflow inside the sampling probe nozzle must, in reality, be equal to the flow at the sampling point. Thus, the nozzle diameter depends on the inlet velocity, i.e. the sampling stream velocity inside the bench. The diameter of the nozzle can be calculated as follows:

Where Qsampling is the sampling rate of the WELAS fixed at 5 L.min -1 and Vmean the sampling stream velocity inside the bench. The Vmean values were calculated to achieve the same filtration Vfiltration = 0.27 m.s -1 , i.e. the filtration velocity recommended by the supplier, for all geometries tested, as described in Chapter 4.

C particle = N particle Q sampling (32)
d nozzle (m) = √ Q sampling (m 3 . s -1 ) V mean (m. s -1 ) (33)
Chapter 5 Influence of the clogging on the filtration performance and on the velocity profiles downstream of the filter pleats 161 Table 18 summarizes the sampling nozzle interior diameters designed for particle sampling upstream and downstream of the three filter geometries used in this study. The error between the measured and theoretical diameter is less than 10%. A photograph of the nozzles is shown in Figure 71. The sampling nozzles are located 25 cm upstream of the filter and 155 cm downstream of the filter. Prior to each clogging experiments, and in absence of filter in the test bench, the placement of the nozzles was validated by comparing the particle size distribution and the concentration of NaCl particles or Silica test particles measured by the WELAS. The results confirmed that there is no variation in the particle size distribution and concentration between the two sampling points.

General methodology of the experiments with clogged filters

Strategy

The major goal of the clogging tests in this study is to evaluate how the particle deposition affects the velocity profiles downstream of pleated filters. Following the characterization of the airflow downstream of filter pleats in their clean state, the flat and pleated filters were evaluated in the laboratory in terms of collection efficiency (E) and pressure drop (∆P) at a given level of clogging.

For pleated filters, and in order to be representative of the recommended filter final pressure in HVAC systems i.e. 450 Pa it was decided to stop the clogging at a pressure drop ranging from 350 to 400 Pa, i.e. 3.5∆P0 regardless the particle used and the pleats geometrical properties. In fact, In the case of flat filters, clogging stopped when the pressure drop exceeds 400 Pa. It is well known that the pressure drop generated by a pleated filter is greater than the pressure drop generated by a flat filter for a similar collected amount of particle [START_REF] Thomas | Aerosol Filtration[END_REF][START_REF] Del Fabbro | Air flows and pressure drop modelling for different pleated industrial filters[END_REF], due to the increased pressure drop caused by the pleating. Thus we can express the pressure drop of the clogged filter as follow:

(a) (b) (c)
∆P clogged pleated filter = ∆P particle deposit + ∆P pleating [START_REF] Lee | Theoretical Study of Aerosol Filtration by Fibrous Filters[END_REF] Chapter 5 Influence of the clogging on the filtration performance and on the velocity profiles downstream of the filter pleats 163

The pressure drop caused by particle deposition in the pleated filter, i.e. ΔPparticle deposit, is graphically estimated by comparing the evolution of a pleated filter pressure drop to that of a flat filter, as shown in Figure 72.

Knowing the filtration velocity (Vfiltration = 0.27 m.s -1 ) and applying Darcy's law, a new permeability of the filter that represents the level of clogging, calling it "apparent permeability" Bapp, can be calculated. The estimated apparent permeability may be used as parameter in numerical simulations to represent the level of clogging, for example.

Clogging experiments with NaCl and Silica particles were carried out on flat filters and pleated filters (H = 28 ± 1 mm, W = 25 ± 1 mm). After having clogged pleated filters, transversal velocity measurements were performed downstream of filter pleats at the same locations as for clean filters (see Chapter 3). This allows assessing the effect of clogging on velocity profiles inside and outside the filter pleats. A single clogging (N=1) was performed with the non-hygroscopic Silica particle and a narrowwidth geometry pleated filter (H=27 ± 1 mm, W = 10 ± 1 mm) to examine experimentally the effect of geometry on the velocity profile downstream of the filter pleats for the same level of clogging, by comparing the results to those obtained on the "wide-width" geometry. To complement the results obtained with fluorescein injection and to observe the structure of the cake deposit, SEM observations were made at different locations of the filter, such as pleats peaks, hollows, and straight zones of pleats. The experimental characterizations performed on the clogged pleated filters are summarized Table 19.

Experimental protocol

The clogging experiments were carried out on the same experiment bench in which velocity measurements at the vicinity of the filter were performed (see Chapter 2 -Section 2). The experimental methodology for clogging with NaCl and Silica particles is detailed in the logic diagram presented in Figure 73. The investigations were carried out with a filtration velocity Vfiltration = 0.27 m.s -1 . The filter is first weighed using a digital balance (METLER TOLEDO ME4001, precision of 0.1 g) and then placed in the measuring zone of the bench. The initial pressure drop of the clean filter is measured, as well as the mean duct velocity, the relative humidity, and temperature. When the inlet velocity is stable, the particle generation process starts. Every 10 min, the particle removal efficiency is evaluated by particle sampling and counting with the WELAS counter upstream and downstream of the filter in the following order: Upstream -Downstream -Upstream. The twofold samplings upstream allow verifying that the generation is stable, what is required for precise efficiency measurements. Each efficiency sampling cycle takes 5 min, including 2 min of "break time," in which the sampled gas passes through the optical counter WELAS without data acquisition to steady the flow of the sampled gas and prevent measuring the particle concentration recorded in the previous gas sample. Each measurement point requires 1 min of acquisition with the WELAS. When the pressure drop is ~ 3.5 times the initial pressure drop in the case of a pleated filter, particle generation is stopped, and we proceed to measure the velocity profiles downstream of the filter (see section 6). After velocity measurements, the filter is removed from the bench and reweighed.

Chapter 5 Influence of the clogging on the filtration performance and on the velocity profiles downstream of the filter pleats 166

Characteristic values used to express the results

This section describes the methods and mathematical equations used to present the results.

Characteristic diameter of the particles

In reality, aerosols do not have an spherical shape. Therefore, equivalent diameters must be used to characterize them:

 The equivalent diameter in volume dev, which is the diameter of the sphere with the same total volume as the particle. If the particle is not porous, the equivalent diameter in volume is the same as the equivalent diameter in mass. The optical diameter doptical measured by the WELAS can be assimilated to a dev. Chapter 5 Influence of the clogging on the filtration performance and on the velocity profiles downstream of the filter pleats 167  The aerodynamic diameter dae is defined as the physical diameter of a spherical unit density particle (ρ = 1000 kg/m 3 ) that settles in air at the same velocity as the studied particle.

 The diameter of Stokes dSt, which is the diameter of a sphere with the same density as the studied particle and settles in air at the same velocity. The relationship between the Stokes diameter and aerodynamic diameter for particles can be written as follow [START_REF] Thomas | Aerosol Filtration[END_REF]:

Particles with the same airborne behavior have the same aerodynamic diameter regardless of size, shape, density, or composition. Thus, in order to compare the particle distributions obtained by NaCl and Silica particles, the aerodynamic diameter is considered.

The filtration efficiency was determined by the particle distribution measured by the WELAS upstream and downstream of the filter, which is represented in terms of optical diameter.

However, in order to represent the distributions in terms of aerodynamic diameter, a single measurement with the APS prior to and in the absence of the filter was performed. Due to particles non-sphericity and particles density, there is a sliding of curves for all particles with an aerodynamic modal diameter (measured with the APS) higher than the optical modal diameter (measured with the WELAS) as shown in Figure 74.

For example, 0.66 vs. 0.27 µm (shift ratio of 2.45) for NaCl and 0.89 vs. 0.31 µm (shift ratio of 2.80) for Silica powder. Other parameters, such as particle diffusivity and the index of refraction used for optical measurement and calibration, can also contribute to this shift. The WELAS distributions are displaced by a factor equal to the shift ratio when presenting the results, which allows covering a broader range of aerodynamic diameter.

d St = d aerodynamic √ 1000(kg. m 3 ) ρ particle (kg. m 3 ) (35)

Particle number distribution

The particle distribution of a polydispersed aerosol composed of N particles is usually lognormal. In terms of the number of particles N, the following parameters were used to characterize the particle size distributions of the NaCl and Silica particles used for this study.

The following equations can be used to compute the statistical distribution parameters [START_REF] Renoux | Les aérosols : physique et métrologie[END_REF]:


log d g = ∑ n i . log d i i N (36)
F(d 50 ) = ∫ f(d p ). dd p d 50 0 = ∫ f(d p ). dd p = 1 2 ∞ d 50 (37)
Chapter 5 Influence of the clogging on the filtration performance and on the velocity profiles downstream of the filter pleats 169 With f representing the frequency or probability distribution function and F representing the cumulative probability distribution function.

 geometric standard deviation σ:

Since NaCl is hygroscopic, particle distributions were examined in terms of relative humidity (RH %) to out whether humid conditions impact the distribution. In the case of non-hygroscopic Silica particles, however, we ignore this impact. The number particle distribution dN is normalized by the total number of particles N and represented as dN N for each sampling. It allows comparing particle distributions at various concentrations and inlet velocity, as well as filtering geometries. It also assumed that the NaCl and Silica particles are spherical, thus the aerodynamic diameter is taken into account.

Filtration performance

To characterize the performance of a fibrous filter medium, the evolution of its efficiency and pressure drop during particle clogging must be evaluated as a function of the total collected mass of particles per unit area (g/m 2 ). It corresponds to the difference between the weight of particles generated upstream of the filter wup(t) and of these left downstream of the filter wdown(t):

In order to calculate the weight of particles upstream (equation ( 40)) or downstream (equation ( 41)) of the filter for the sampling time t, it is assumed that particles are spherical.

log σ = √ ∑ n i . (log d i -log d g ) 2 i N (38)
w collcted (t) = w up (t) -w down (t)

w up (t) = ∑ N up,i (t). π. d p,i 3 6 n class i . ρ p (39)
With i the the particle size class, dp,i the count median diameter of the class, Nup,i(t) and Ndown,i (t) the number of particles per particle size class collected upstream and downstream of the filter during a time interval t (t = 1 min with the particle counter WELAS) respectively.

The total collected mass per surface wexp (g/m 2 ) is measured experimentally for the clogging with NaCl and Silica particles by weighing the filter before and after the clogging and dividing by the total surface of the filter. Because of possible fluctuations in the generated particle concentration and the efficiency of the filter, the evolution of the collected mass per surface area is not linear during clogging. As a result, the collected mass of particles by the filter at time t is linearly corrected as follows:

The total number efficiency EN (-) of the filter is calculated from the number of particles N collected downstream (Ndown) and upstream (Nup) of the filter during the clogging. Each sampling cycle consists of two measurements upstream and one measurement downstream of the filter, thus the Nup is the average number concentration of particles of the two sampling upstream of the filter.

The evolution of the fractional efficiency is expressed as a function of the aerodynamic particle diameter dae. The fractional efficiency evolution at the beginning of clogging allows determining the initial MPPS value specific to the filter, as well as the corresponding minimum filtration efficiency. The following equation is used to calculate the filtration efficiency for a given particle diameter:

With Nup(dp) and Ndown(dp) the number concentration of particles having median aerodynamic diameter dp upstream and downstream of the filter, respectively. Nup(dp) is the average number

w down (t) = ∑ N down,i (t). π. d p,i 3 6 n class i . ρ p (41)
w corrected (t) = ∑(w up (t) t 0 -w down (t)) × w exp ∑ (w up (t) -w down (t)) t final 0 (42)
E N (t) = N up (t) -N down (t) N up (t) (43)
concentration of particles with diameter dp of the two measurements upstream of the filter.

During the clogging, the pressure drop of the tested filter is monitored. To be able to compare the filters, the pressure drop values are normalized by the initial pressure drop ∆P0. The evolution of the ∆P/∆P0 ratio is expressed as a function of the particle collected mass. The evolution of the normalized pressure drop during clogging allows the determination of the different stages of filtration i.e. from depth filtration to surface filtration (see Chapter 1section 1).

Filtration performance

The results of the filtration performance, i.e. the evolution of the pressure drop and particle collection efficiency during the clogging experiments, are presented in this section. First, the experimental conditions during the clogging are presented.

Experimental conditions investigated

The conditions of the clogging experiments with NaCl and Silica particles are resumed in Table 20 and Table 21 respectively. The filtration performance of the fibrous filters were studied in respect to the NaCl and the Silica particles in flat and pleated configurations at the filtration velocity recommended by the supplier: Vfiltration = 0.27 m.s -1 .

Size distribution of the particles

Following the general methodology and experimental conditions, we describe the particle size distributions before presenting the results in terms of filtration performance.

NaCl particles

Figure 75 shows the log-normal number particle size distributions of NaCl particles from sampling and counting with the WELAS counter upstream of (a) flat filters and (b) pleated filters at different RH values. The results are the average distribution of total measurements upstream of the filter during clogging. The results show that the humidity slightly affects the particle size distribution in the range studied (dae = 0.43 -15.66 µm i.e. the range in which at least 1 particle is counted).

Regardless the humidity, the mode diameter of the distributions is nearly the same i.e. around 0.66 µm. However, when humidity increases, the distribution widens; for example, results obtained with flat filters (Figure 75(a)) demonstrates that the geometrical standard deviation of the distribution is σ = 1.67 µm for RH = 70.1 % and decreases to σ = 1.39 µm for RH = 44.0 %.

The particle size number distribution at RH = 70% and RH = 61% are very close. The mean geometrical aerodynamic diameter dae,g is around 0.90 µm and the median aerodynamic diameter dae,50 is around 0.8 µm. The particle size number distribution for dryer conditions remains the same for RH= 44% and RH = 48%. dae,g is around 0.80 µm which is 11% smaller, while the median aerodynamic diameter dae,50 is around 0.70 µm which is 13% smaller. This effect is barely visible in the case of the pleated filter (Figure 75 (b)) as the studies were conducted under similar hygroscopic conditions ranging from RH = 45 to 52 %. As a result, no noticeable change in particle size distribution is detected. The results are the average distribution of total measurements upstream of the filter during clogging. They reveal that the number distribution profiles are similar and, as a result, do not differ considerably regarding the range of mean duct velocities investigated (Vmean = 0.27 -1.48 m.s -1 ) (see Table 21). As Silica test particles are non-hygroscopic, humidity had no effect on particle size distribution, despite the wide humidity range covered by these trials: RH = 31-67 % (see Table 21). The aerodynamic particle size ranges from 0.48 to 15.9 µm with a geometric mean diameter dae,g = 1.18 µm, a count median diameter dae,50 =0.89 µm, and a standard geometric deviation σ of 1.65 µm, which is 1.16 times greater than for the NaCl particle size distributions in dry conditions. Even though only one prototype was tested, and further reproducibility of results would be more accurate, these findings are intriguing as they suggest a possible effect of the pleat geometry on particle collection efficiency that begins only at a threshold pleat width at the scale of the pleat (or pleat count at the scale of the filter) that would induce velocity gradient upstream the filter.

As a result, the change in the trend of the initial fractional filter efficiency can be linked to a possible induced efficiency gradient. Indeed, an initiation of a velocity gradient upstream of the pleats was observed experimentally and numerically with (H = 28 ± 1 mm, W = 10 ± 1 mm), in contrast to what was observed with (H = 28 ± 1 mm, W = 25 ± 1 mm). In both cases, the predominant filtration mechanisms are the inertial impaction and interception.

The MPPS of NaCl is lower than that of Silica: it is around 0.57 µm versus 0.63 µm. With NaCl particles, the lowest efficiency is around 0.50, while with Silica particles, it is around 0.6. For the diameter range 0.57 -4.70 µm, the efficiency of Silica is always greater than that of NaCl.

The difference in efficiencies could be explained by differences in the charge of the particles.

In fact, the particle generation modes differ: Silica is produced by a dry mean with frictional forces that charge the particles, whereas NaCl is produced by a wet mean with a very light aerosol charge. The difference in charge can also be explained by the two particles' different chemical surfaces. Even when the filter is initially uncharged, electrostatic forces contribute to to the humidity, as stated previously in section 4.2.. indeed, at an increase humidity, many clusters of particles are formed, in addition to the dendritic-like network (see Figure 69).

Indeed, as humidity rises, multiple clusters of particles grow in addition to the dendritic-like network (see Figure 69). As a result, as indicated by Joubert et al. [START_REF] Joubert | Influence of Humidity on Clogging of Flat and Pleated HEPA Filters[END_REF], the overall surface area of the numerous agglomerated particles is less than the sum of their individual surface areas.

This could explain why a cake deposit built in humid conditions has a lower pressure drop than a cake formed in dry conditions for the same collected mass.

 For dryer conditions, i.e. RH = 44 -48%, the evolution of the pressure drop is similar and closer to classical filtration theory of clogging with sub-micron particles. There are two distinct phases: deep filtration, which is defined by a slow increase in pressure drop, and surface filtration, which is characterized by a linear evolution of pressure drop. An inflection point at about wcollected = 6 g.m -2 separates both stages (Figure 81 (b)).

 For air humidity values ranging from 60 to 70%, the evolution of the pressure drop appears to be in a transition zone. This trend is similar to what has been observed in liquid particle filtration [START_REF] Joubert | Influence of Humidity on Clogging of Flat and Pleated HEPA Filters[END_REF]. As a result, drying is required to carry out the tests. 8) is used to compute the approximate average apparent permeability of a clean flat filter that generates a pressure drop of 222 Pa at a filtration velocity of 0.27 m.s -1 . It is found to be Bapp,NaCl = 1.31×10 -11 m 2 , which is 2.4 more resistant than the clean filter.

Note that in numerical simulations, the apparent permeability value will be employed as an input parameter to represent the clogging level. 21).

Silica particles

The three curves behave similarly in terms of ΔP/∆P0. When the pleat density increases, i.e.

pleat width decreases, the pressure drop increases more quickly: the increase of the ΔP/∆P0 in the case of (H = 27 ± 1 mm, W = 10 ± 1) geometry is faster than the increase in the case of (H= 28 ± 1 mm, W = 25 ± 1 mm) geometry and flat filter. For all three geometries, the particle generation was stopped at nearly the same pressure drop level, namely ΔP ~ 3.5×ΔP0 ~ 400 Pa.

It occurs at wcollected = 12.5 g/m 2 in case of (H = 27 ± 1 mm, W = 10 ± 1) geometry, at wcollected Using the same methodology as previously described for estimating "apparent permeability"

for an equivalent flat filter, it is found that Bapp,iso is 9.02 × 10 -12 m 2 for (H = 27 ± 1 mm, W = 10 ± 1 mm) and 8.27 × 10 -12 m 2 for (H = 28 ± 1 mm, W = 25 ± 1 mm) which is 3.7 and 3.4 more resistant than the clean filter respectively. The clogging was stopped at the same pressure drop level, i.e ΔP/ΔP0 = 3.2 and 3.5 in the case of NaCl and Silica, respectively. The results show that the particle diameter has a significant impact on clogging. For the same collected particles weight, the increase in pressure drop is faster in the case of NaCl than in the case of Silica particles. The observations are in accordance with the results of Del Fabbro et al. [START_REF] Del Fabbro | Air flows and pressure drop modelling for different pleated industrial filters[END_REF] who demonstrated that when working at the same According to Thomas et al. [START_REF] Thomas | Aerosol Filtration[END_REF], in the case of submicron particles, the cake of particle deposit is more aerated and thus has a lower compactness. In fact, lower particles diameter results in increased specific surface area, and in higher pressure drop. Larger particles, on the other hand, exhibit a higher deposit compactness, but their specific surface area is lower than that of submicron particles, resulting in a lower pressure drop for a similar deposited particles weight. The results show that the humidity has a significant impact on the evolution of total efficiency.

High humidity values, causes a slower evolution of total efficiency for a similar collected mass compared to that observed for lower humidities: for example, at wcollected 16 g/m2 EN is about As previously mentioned regarding change in pressure drop (see section 182, the wide range of humidity (RH = 31 -67 %, see Table 21) during the tests has no effect on the filtration performance with the non-hygroscopic Silica particles. The total efficiency increases progressively for the three geometries tested. The trends are similar in case of flat and pleated geometry (H= 28 ± 1 mm, W = 25 ± 1 mm), and they slightly differ for the geometry (H = 27 ± 1 mm, W = 10 ± 1 mm). The initial efficiency in the first case is around 0.8, while it is around 0.7 in the case of the "narrow-width" geometry, which is 12.5 % than the other geometries.

Despite the fact that only one prototype was tested with geometry (H = 27 ± 1 mm, W = 10 ±1 mm), this result is interesting in terms of the effect of geometry on particle collection at the which may be potentially related to the flow behavior upstream of the filter. In Chapter 4section 3.5, it was shown that decreasing the pleat width from 25 mm to 10 mm resulted in a more significant velocity gradient . Moreover, it was observed thanks to numerical simulations that the filtration velocity is not uniform all over the filter for both tested geometries, and that this effect is more pronounced in the "narrow-width" geometry i.e. (H = 27 mm, W = 10 mm).

This hypothesis needs to be validated by comparing the fractional efficiencies at the beginning of the clogging. The total efficiency of the filter for Silica particles is significantly higher than the efficiency for NaCl particles. In fact, the initial efficiency is around 0.6 in case of NaCl while it is 0.8 in case of Silica. However, at wcollected = 10.5 g/m 2 the filters is nearly 90% efficient for both particles (E = 0.9). As previously explained in section (4.4.1), the electrostatic forces that may be induced during the generation of Silica particles in dry conditions could be responsible for increasing collection efficiency regardless of particle size.

Silica v.s NaCl particles

The performance of the pleated filter during clogging with submicron NaCl particles and Silica particles has been presented in this section. Two pleated geometries (H = 28 ± 1mm, W = 25 ± 1 mm) and (H = 27 ± 1 mm, W = 10 ± 1 mm) have been tested. The clogging stopped when the pressure drop reached around 400 Pa. For the same filtration velocity and collected particles weight, the increase in pressure drop is faster in the case of submicron NaCl than in the case of micron Silica particles. The evolution of total efficiency of the filter for Silica particles is significantly higher than the efficiency for NaCl particles. The electrostatic forces that may be induced during the generation of Silica particles in dry conditions could be responsible for increasing collection efficiency regardless of particle size. For the same reasons, the initial fractional efficiency of the filter for Silica particles is higher than the efficiency for NaCl particles.

Chapter 5 Influence of the clogging on the filtration performance and on the velocity profiles downstream of the filter pleats 188

In the following section, the experimentally determined initial fraction efficiencies of the filter for NaCl particles are compared to various collection efficiency models from the literature. The previously determined structural characteristics (see Chapter 3) are used to evaluate the ability of these models to predict the filter's collection efficiency. A sensitivity analysis is also carried out to investigate the effect of collection efficiency on the air velocity approaching the filter medium surface, taking into account the velocity variations determined upstream of the filter (see Chapter 4).

Collection efficiency modeling

Various studies in the literature address how to determine the initial efficiency of a fibrous filter based solely on knowledge of the structural properties of the filter medium, particles, and filtration velocity (see Chapter 1). Almost all previous studies used average parameters to estimate efficiency, such as average fiber diameter, global porosity (or solid volume fraction), average filtration velocity, etc. As a result, the heterogeneous structure of the medium is not taken into account, which affects the accuracy of the predictions. In addition, it was demonstrated in Chapter 4 that the pleated geometry of the filters induces a velocity gradient [START_REF] Miecret | Mathematic Expression of HEPA and ULPA Filters Efficiency, Experimental Verification, Practical Alliance to New Efficiency Test Method[END_REF] in this sensitivity study. Gustavson model [START_REF] Miecret | Mathematic Expression of HEPA and ULPA Filters Efficiency, Experimental Verification, Practical Alliance to New Efficiency Test Method[END_REF] by taking into account the mean fiber diameter (as presented previously df,mean = 3.14 µm) or the equivalent diameter to depict the bimodal fiber diameter distribution calculated thanks to the three formulations presented in Chapter 1 -Table 2. The three values obtained are df, eq,1 = 1.92 µm, df, eq,2 = 2.17 µm and df, eq,3 = 2.12 µm respectively for the formulations of Clague and Phillips [START_REF] Clague | A numerical calculation of the hydraulic permeability of three-dimensional disordered fibrous media[END_REF], Brown and Thorpe [START_REF] Brown | Glass-fibre filters with bimodal fibre size distributions[END_REF] and Tafreshi et al. [START_REF] Vahedi Tafreshi | Analytical expressions for predicting permeability of bimodal fibrous porous media[END_REF]. When a smaller fiber diameter is considered, the total efficiency increases. The equivalent bimodal diameter of Brown and Thrope [START_REF] Brown | Glass-fibre filters with bimodal fibre size distributions[END_REF] is found to translate the experimental results correctly, even It demonstrates that for all the fiber diameters considered, εmean = 0.92 and Vfiltration = 0.27 m.s -1 , the interception dominate the impaction. The consideration of an equivalent diameter increases the collection efficiency by interception while having a minor effect on particle collection by impaction. Indeed, this result is consistent with the literature, as the equivalent diameter in this case study is smaller than the mean fiber diameter, and thus all mechanisms, but particularly interception, are promoted under the actual conditions. to sum up, considering a bimodal equivalent radius improved the prediction of collection efficiency, just as it did for the prediction of permeability. based on the obtained results and to be representative of the actual fiber distribution of the medium, it was decided to use the equivalent fiber diameter calculated by Brown and Thrope [START_REF] Brown | Glass-fibre filters with bimodal fibre size distributions[END_REF] for the remainder of the study.

Next, the influence of the local in-plane porosity variations on the predicted collection efficiency is studied. Figure 90 (a) shows that the models are extremely sensitive to the porosity of the medium.

When the porosity is reduced from 0.98 to 0.87, the total particle efficiency at the MPPS increases from 10% to 90%. Figure 90 (b) shows that the medium porosity has the greatest impact on the interception mechanism, which takes precedence over the impaction mechanism as porosity decreases. For example, in the case of ε = 0.87, the single fiber efficiency was nearly 100 percent for the experimental aerodynamic particle diameter range tested, i.e. 0.6 µm < dp <7.0 µm, whereas the single fiber efficiency of the impaction mechanism ranges between 50 and 100 % for the considered particle diameter range. However, for the highest porosity considered, i.e. ε = 0.98, the single fiber efficiencies of both mechanisms are close, ranging between 20% and 100% for the tested experimental particle diameter range. These findings confirm that the heterogeneities of the medium have a significant impact on particle collection efficiency, and thus a local assessment of the in-plane porosity variations is required to improve efficiency prediction.

Sensitivity to the velocity variation at the vicinity of filter pleats

As a reminder, for a pleated filter geometry of H = 28 mm, W = 25 mm, the local velocity observation upstream of the filter and in close proximity to the filter, i.e. up to 1 mm from the surface, presented a local velocity gradient ranging between 0.44 m.s -1 and 0.74 m.s -1 . When the pleat width is reduced to 10 mm, the intensity of the gradient upstream of the filter increases to a maximum of 2.00 m.s -1 for the same average filtration velocity. As a result, it is crucial to investigate the impact of velocity gradients on collection efficiency, given that the average filtration velocity is the only velocity parameter considered in the models in the literature.

Figure 91 (a) depicts the effect of velocity variation on the total collection efficiency predicted by the Miecret and Gustavson model [START_REF] Miecret | Mathematic Expression of HEPA and ULPA Filters Efficiency, Experimental Verification, Practical Alliance to New Efficiency Test Method[END_REF]. In this study, an equivalent bimodal diameter calculated by Brown and Thrope df, eq,2 = 2.17 µm is considered, as well as a mean porosity value of 0.92. A very slight increase in efficiency for particles with dp > 0.1 µm is observed for a velocity ranging between the average filtration velocity Vfiltration =0.27 m.s -1 and a velocity of 0.74 m.s -1 and get more pronounced at Vfiltration = 2.00 m.s -1 . Figure 91 (b) depicts the impact of velocity on single collection efficiencies.

In accordance with the literature, the interception mechanism is unaffected by velocity because it is primarily determined by the structural parameters of the medium; however, the impaction mechanism is affected by velocity because it depends on the St number, which is related to the velocity. The results confirm that increasing the filtration velocity promotes particle collection by impaction. The collection by interception is dominant for velocity ranges between 0.27 and 0.74 m.s -1 , which means that for the considered structural properties of the medium (df,eq = 2.17 ε, = 0.92) and the tested geometry (H = 28 mm, W = 25 mm), the effect of velocity is negligible between V = 0.27 and V = 0.74 m.s -1 . However, it appears that there is a velocity threshold above which impaction takes over and the effect of velocity becomes more pronounced. That is, the geometry of the filer, which is responsible for the velocity gradient upstream of the filter, should be taken into account to improve the prediction of the existing model, i.e. by integrating a velocity gradient rather than a single value. Furthermore, this result is interesting because it velocity is remarkably sensitive to changes in the structural and geometrical properties of the filter. These properties are altered as a result of clogging. Thus, a thorough understanding of the impact of particle deposit on the velocity profile is required for better filter design optimization.

Local velocity measurements downstream the filter pleats of both geometries are performed in the following section to evaluate the effect of clogging on the velocity patterns. These findings can be used to validate the airflow model for CFD calculations, as well as to provide information about possible clogging heterogeneities at the pleat level. The results are then compared to the local velocity profiles downstream the clean filter pleats (see Chapter 4) to see if the particle diameter, and thus the deposit structure, has an effect on the change in the velocity pattern. The effect of reducing the pleat width on the velocity profile downstream the clogged filter is also investigated experimentally, by comparing the results obtained downstream of the pleats (H = 28 ± 1 mm, W = 25 ± 1 mm) to those obtained downstream of the narrowed pleats (H = 27 ± 1 mm, W = 10 ± 1 mm)

Experimental characterization of the local velocity gradients downstream of the clogged pleated filters 6.1 Material, method and methodology

The materials and methods used to measure velocity profiles downstream of the filter when it is clogged are identical to these used to measure the velocity profiles downstream of the filter when it is clean, as detailed in Chapter 2.

The airflow is characterized downstream of two pleated filter geometries with similar height but different width: the "wide-width" geometry (H = 28 ± 1 mm, W = 25 ± 1 mm) and the The effect of particle size distribution on velocity profiles is investigated at the same clogging level (ΔP ~ 400Pa) for both filter geometries. Transversal velocity profiles obtained at different positions downstream of the pleated filters (H = 28 ± 1 mm, W = 25 ± 1 mm) clogged with submicron NaCl particles (dae,50 = 0.7 µm) (see Table 20) are compared to those obtained with micron Silica particles (dae,50 = 1.0 µm) (see Table 21). These transverse velocity profiles were recorded downstream of the four half-pleats located in the bulk flow, i.e., outside the boundary layers, for each prototype (see Chapter 2 -Figure 14 (a)). As these tests were performed for three filter prototypes for each aerosol (NaCl or Silica), a total of twelve half-were characterized for each aerosol.

Furthermore, the effect of geometry on velocity profiles is tested. To do so, transversal velocity profiles downstream of filters with geometry (H = 27 ± 1 mm, W = 10 ± 1 mm) were compared For each half-pleat, transversal velocity profiles were acquired at two x distances downstream of the pleat peak (the peak corresponds to a longitudinal coordinate x = 0 cm), as shown in Figure 93: x = 1.5 and 3.5 cm. These measurements were taken at the height z = 7.5 cm, i.e. at the half duct height. The y step of the probe displacements was 1 mm in case of (H = 28 ± 1 mm, W = 25 ± 1 mm) and 0.5 mm in case of (H = 27± 1 mm, W = 10 ± 1 mm). For the profiles acquired inside the pleat (x = 1.5), the minimum distance to the medium surface estimated numerically at which the measurements were carried out was 1 mm in case of (H = 28 ± 1 mm, W = 25 ± 1 mm) and 0.7 mm in case of (H = 27± 1 mm, W = 10 ± 1 mm). The effect of clogging on the velocity profile is more significant at y=0 i.e. downstream of the symmetry axis passing through the pleat peak in both cases (NaCl and Silica) and at the two distances. For the considered clogging state, the maximum velocity at y = 0 is 2.51 m. s -1 in case of clogging with NaCl and 2.45 m. s -1 in case of clogging with Silica at x = 1.5 cm, which is 125% and 122% higher than the values obtained at the same position for the clean filter (V = if the increase of the velocity is slightly less pronounced, i.e. 110% higher than the value obtained at the same position for the clean filter (V= 1.94 m.s -1 ). As it was previously mentioned in Chapter 4 (See Chapter 4section 3.2), this increase is observed downstream the curved zone of the pleats, that might exhibit changes in structural and geometrical properties due to the pleating process. This increase in velocity amplitude can be explained by a decrease in the pleat width of the filter caused by particle deposition on the filter as a result of clogging, which results in a decrease in velocity amplitude toward the medium and an increase toward the pleat peak.

Similarly, particle depositions within the porous medium, make the filter pleat less permeable and thus more resistant to flow, which can converge the air stream line towards the zone downstream the pleat peak where the maximum velocity is detected. To our knowledge, the literature doesn't exhibit experimental velocity profile acquired locally at the vicinity of a clogged filter. some authors were interested in studying the airflow within the pleated filter during clogging, in order to predict its pressure drop and particle collection efficiency (see Chapter 1 -Section 6.3), however, their work was solely numerical and lacks of validation on the basis of local parameters such as local velocity measurements.

In order to detect any preferential path or dead zone to confirm these observations and understand this behavior, fluorescein particles were injected at the end of the clogging with NaCl particles. Section 7 is dedicated to presentation of the methodology and results of these tests. SEM images were also taken at various zones of the clogged filter, such as pleat peaks, hollows and straight zones, to evaluate whether any preferential cake deposition occurred. is not the case. Most of the transverse velocity profiles follow a similar trend when comparing clogged to clean state, i.e. an increase in velocity amplitude downstream of the pleat peak and a decrease as approaching the pleat bottom for x = 3.5cm and medium surface for x = 1.5 cm, but some clogged half-pleats, such as half-pleats 4, 7, and 10, show significant deviations from the others. Despite the fact that only one prototype was tested, these results suggest a possible non-homogeneous clogging upstream of the filter. In fact, a slight velocity gradient is observed and discussed in Chapter 4 for this geometry. Geometrical irregularities at the pleats level could thus induce a velocity gradient near the pleats, and thus at the medium surface, resulting in heterogeneities in terms of particle deposition from one pleat to the other. However, more testing (conducted in more filter prototypes) is required to validate this assumption. . the distance between the symmetry axis passing through the pleat peak and the medium surface at the x coordinate considered. The ratio y/y0 is thus equal to 0 at the hollow symmetry line, and to 1 at the medium surface as illustrated in Figure 97. These effects are more pronounced outside the pleat at x = 3.5 cm (Figure 98 (a)). When y/y0 > 0.4, the velocity decreases sharply from V/Vmean = 0.7 to 0.2 in the case of (H = 27 1 mm, W = 10 1 mm), but more gradually from V/Vmean = 0.9 to 0.3 in the case of (H = 28 mm, W = 25 mm). For y/y0 < 0.5 i.e. towards the center of the pleat downstream of the pleat peak, the velocity increases from V/Vmean = 0.7 to 3.2 in case of (H = 27 mm, W = 10 mm) and presents an inflection point at y/y0 = 0.1, while it increases gradually from V/Vmean = 0.9 to 2.7 in case of (H = 28 mm, W = 25 mm).

Effect of the filter geometry on the velocity profiles downstream of filters

The experimental results resented in this section revealed that at the conditions investigated and at the same level of clogging with Silica particles, the geometry has an effect on the velocity profiles downstream of the pleat. It was discovered that when the pleat width is narrow, the velocity profile decreases faster when approaching the medium at the same position studied inside and outside the pleat. Spite of the fact that only one prototype was tested, this finding is intriguing because it may indicate non-homogeneous clogging within the filter pleats. In fact, in Chapter 4, an initiation of a velocity gradient that is highly sensitive to the pleat geometry is observed and discussed. As a result, irregularities in geometrical properties at the pleat level could cause a velocity gradient near the pleats, resulting in particle deposition that differs from one pleat to the next. The effect of geometry on the particle deposition homogeneity within the filter has been barely discussed in the literature [START_REF] Teng | Research on the influence of pleat structure on effective filtration area during dust loading[END_REF][START_REF] Fotovati | Modeling instantaneous pressure drop of pleated thin filter media during dust loading[END_REF][START_REF] Saleh | Modeling service life of pleated filters exposed to poly-dispersed aerosols[END_REF][START_REF] Feng | Modeling filtration performance of pleated fibrous filters by Eulerian-Markov method[END_REF][START_REF] Alilou | Impact sur le colmatage en régimes transitoire et permanent des écoulements d'air induits par le plissage des médias filtrants THE[END_REF]. Despite differences in the studied pleat geometrical dimensions and airflow conditions, the authors agreed that narrowing the pleat would increase velocity on the upstream side of the pleat, resulting in more heterogeneous particle deposition. However, the observations were theoretical, and none of them were validated by local parameters like local velocity values. This study is innovative in that it provides a local investigation of the effect of clogging on the velocity pattern downstream the filter under real conditions rather than relying on virtual injection of a batch of particles.

Evaluation of the particle deposit homogeneity at the pleat scale

When the pleated filters (H = 28 ± 1 mm, W = 25 ± 1 mm) were clogged with NaCl (dae,50 = 0.7 µm) and Silica particles (dae,50 = 1.0 µm) at the same pressure drop level, i.e. ΔP ≈ 400 Pa, the velocity profiles downstream of the filters revealed an increase in the velocity amplitude downstream of the pleat peak and a slight decrease towards the pleat hollow. In the clogged case, the error bars broaden, indicating that the clogging is heterogeneous, possibly between plies. These clogging-induced heterogeneities may result in new preferential flow paths at the pleat scale. The aim of injecting fluorescein at the end of the clogging was to evaluate whether some preferential flow paths or dead zone happened during clogging, and that would result in an heterogeneous particles deposit. These tests were carried out for one pleated filter (H = 28 ± 1 mm, W = 25 ± 1 mm) clogged with NaCl particles. Despite the fact that the presented methodology is implemented on a single prototype, it enables to better explain the velocity profiles downstream of filters. Finally, SEM observations are carried out to visualize the cake structure and particle deposition in various zones of the pleat. The observations help to improve our understanding of the effect of potential heterogeneities on the local airflow velocity pattern downstream of the filter pleats.

Characterization using fluorescein particles as tracer

Fluorescein particles are used to quantify possible deposit heterogeneity, which could be a reflection of preferential passages. It is a non-toxic organic with a high water solubility and one of the highest fluorescence levels, making it easily detectable at low concentration levels [START_REF] Gutowski | Photolytic transformation products and biological stability of the hydrological tracer Uranine[END_REF].

As a result, it is an effective method for detecting and quantifying the presence of preferential particle deposition zones within filter pleats. According to the literature, this technique is usually applied to configuration where air channels are in contact with a porous medium [START_REF] Terzis | A method to visualise near wall fluid flow patterns using locally resolved heat transfer experiments[END_REF][START_REF] Hurel | Innovative use of fluorescein for the air path study within light-weight wall assemblies[END_REF]. Fluorescein particle injection is used to quantify deposit heterogeneity, which could be due to preferential passages. As a result, it's a useful technique to observe and quantify the presence of preferential particle deposition zones within filter pleats [START_REF] Hurel | Innovative use of fluorescein for the air path study within light-weight wall assemblies[END_REF]. Due to its ease and efficiency, this technique is used in this study.

Materials and methods

The experiment is carried out with the same experimental setup as for the clogging and velocity measurements (see Chapter 2section 2). One prototype of (H = 28 ± 1 mm, W = 25 ± 1 mm) clogged with NaCl particles has been tested. In fact, once the clogging was stopped at the desired pressure drop, the filter is carefully removed from the bench so as not to disturb the Chapter 5 Influence of the clogging on the filtration performance and on the velocity profiles downstream of the filter pleats 209 cake deposit, re-weighed, and then carefully re-inserted in the bench. It was checked that no significant change in the pressure drop has occurred.

A 50 g.L -1 fluorescein solution was nebulized at 167 cm distance from the filter. The nebulizer used is the AGK 2000 model of the PALAS® brand (see section 2.2.1), operating under compressed air at 2 bars of pressure. The generation was stopped after 5 min, once the "color" of the filter changes from white to orange, and the filter is removed to be analyzed in the laboratory. The filter is then cut and removed from its frame. Eleven discs of 1 cm diameter are sampled with a hydraulic press as shown in Figure 99. The concentration of the samples Cfluo,sample was calculated using equation [START_REF] Kasper | The collection efficiency of a particleloaded single filter fiber[END_REF], which was derived from the Beer-Lambert relationship and the linear equation of the calibration curve.

The calculated concentrations in mg. L -1 were then multiplied by the extraction volume of 4 mL to get the weight of fluorescein collected on the sample wfluo, collected. Then, by dividing by the area of a sample, they were converted to µg.cm -2 of filter medium.

Results

Chapter 5 Influence of the clogging on the filtration performance and on the velocity profiles downstream of the filter pleats 211

Chapter. Table 23 presents the fluorescein concentrations for all the samples. Figure 102 represents the fluorescein distribution on the filter based on values presented in Table 23. To simplify the interpretation, a color gradient from yellow (the lowest value) to dark red (the highest value) is used. First, the fluorescein concentrations at the extreme half pleats L and R, are close and within a 6% difference range. This indicated that the clogging at these zones is somehow symmetrical.

The results also show that the fluorescein concentration per surface at hollows is the lowest compared to the straight parts and the peaks. When average values throughout different pleat zones are considered, it is revealed that the maximum values are on straight areas, and it is lower for pleat hollow and peaks, with a little less for peaks. By assuming that fluorescein follows the air streamlines, these results may indicate that the airflow preferentially passes through the straight zones of the pleat, and near the peak rather than near the hollow. Even though the results presented in this section are approximations that would require more data to be statistically representative, they are consistent with the conclusions made on the basis of the velocity profiles downstream of the clogged filter (see section 6.2). The fluorescein observations can explain the previously described increase in velocity amplitude downstream of the filter peak pleats and decrease towards the hollow. The SEM pictures were obtained thanks to a JEOL JSM 7600F apparatus. Photographs have been taken from samples of 1.5 cm². Three samples were taken from three pleat peaks and three samples were taken from three pleat hollows. All of them were located on the central pleats where the airflow had been studied. Images at x25 magnification were taken for each sample in order to observe the general particle repartition over the analyzed zone and the cake structure.

Results

Figure 103 shows the SEM observations of Silica deposit for a (a) pleat peak and a (b) pleat hollow on the upstream side of the filter, for the case of the geometry (H =28 mm, W = 25 mm).

Figure 104 shows the SEM observations at similar types of locations for the geometry (H = 27 mm, W = 10 mm).

First, some particle deposition is observed over the pleat peak and hollow, which confirms that this zone cannot be considered completely impermeable, as stated by Rebaϊ et al. [START_REF] Rebaï | A semi-analytical model for gas flow in pleated filters[END_REF].

However, a dark strip can be seen in all of the images, which corresponds to a less concentrated deposit in comparison to the rest of the medium. The shape and thickness of this strip differs from one curved zone to another, which is most likely due to the pleating process, and thus to the local modification of the medium shape. This causes irregularities and changes in the structural properties of the filter in this zone, making it difficult to draw general conclusions about the shape and properties of the curved area of the pleats. The fact that fewer particles are observed in this region might indicate that this region is less permeable. This less-permeable strip was also seen on the filter after fluorescein injection, and it appeared as a white strip (see Figure 99)

with

Conclusion

The methodology presented in this Chapter is a first attempt to study experimentally the effects of clogging on the behavior of velocity profiles downstream of pleated filters. The HWA has been shown as being an effective technique for measuring velocity values downstream of clogged pleats, even at a close distance to the surface.

The experimental strategy for measuring velocity inside filter pleats using HWA techniques, which was developed throughout this thesis and detailed in Chapter 2, was successful in providing data that exhibit the influence of clogging on velocity profiles downstream of filter pleats at a level of clogging ΔP ~ 400 Pa, representative of an early stage of the surface filtration.

A similar trend of velocity profile is observed at positions inside (x = 1.5 cm) and outside the pleat (x =3.5 cm) for the same geometry (H = 28 mm, W = 25 mm) and for two different cloggings with submicron NaCl particles and micron Silica test particles. It shows that the velocity amplitude increases downstream of the pleat peak and decreases towards the medium.

These findings were confirmed by the injection of a fluorescein tracer and SEM analysis, which revealed more particle deposition, and as a result airflow passage, close to the pleat peak. It was shown also that the choice of the particle did not affect the velocity profiles downstream of the filter. This could be related to the fact the difference between count median diameter is not that significant, and also that clogging experiments were stopped at a similar pressure drop level.

In the case of a "narrow-width" geometry (H = 27 mm, W = 10 mm) and the same filtration velocity Vfiltration = 0.27 m. s -1 , the velocity profiles downstream of the filter were different from the other geometry. Despite the fact that only one prototype was tested, this result is intriguing because it could point out a non-homogeneous clogging upstream of the filter. In fact, an initiation of a velocity gradient that is highly sensitive to the pleat geometry is observed and discussed in Chapter 4. As a result, irregularities of geometrical proprieties at the level of the pleats could cause velocity gradient near the pleats, resulting in particle deposition that would differ from one pleat to the other.

The experimental results also revealed that at the conditions investigated and at the same level of clogging with Silica particles, the geometry has an effect on the velocity profiles downstream

General conclusion and perspectives

One of the issues of researches in the indoor air filtration domain is to be able to study the filter's performance in conditions as close to its real-world application as possible. Implementing a fibrous medium through a more complex geometry than the flat one, such as the pleated geometry, increases the surface area available for filtration but results in significant local velocity gradients. Because airflow velocity is one of the critical parameters that affects filtration performance, particularly particle collection efficiency, the results presented in this manuscript could serve as some optimization criteria for newly designed pleated filters. The main objective of this study were:

 The development of a methodology to characterize the velocity pattern in the vicinity of commercial pleated fibrous filters that combines experimental and numerical methods, by validating numerical simulations with local experimental velocity values.

 The experimental characterization of the structural properties of the fibrous medium for an accurate prediction of its permeability, and by consequence its pressure drop.  The experimental and numerical studies of the effect of the geometrical and structural properties of the pleated filter on the velocity profiles upstream and downstream of the filter pleats. Two pleated geometries were tested at the average filtration velocity recommended by the filter medium supplier.

 The experimental investigation of the effect of clogging on the airflow velocity pattern downstream of the filter pleats. For that purpose, the experimental characterization of the filtration performance of two pleated filter geometries with micron and submicron particles was carried out.

In order to respond to these objectives, a literature review was first carried out and presented in variations in medium permeability in the pleat curved zones, resulted in non-negligible pressure drop variations.

The comparison of numerical to experimental velocity profiles in the vicinity of the filter revealed that the laminar model was the best to predict the overall airflow pattern. the Reynolds pleat number Rew, which considers the half pleat width as a characteristic diameter and the mean duct velocity approaching the filter is Rew ~ 500 in both cases), which is quite low and may justify the choice of the laminar airflow model. For some cases reported in the literature, significant velocity gradients were detected upstream of the pleat, with maximum velocity values in the central pleat zone considerably higher than the mean duct velocity approaching the filter. These studies, however, are characterized by a turbulent airflow regime approaching the filter. A critical Re of 2000 allowing to identify the passage of a laminar flow to turbulent is valid only in rectilinear ducts, but it is unknown from which Rew a flow will become turbulent in the vicinity of a pleat. Further investigations are thus required to determine the threshold Reynolds number, above which the turbulence occurs in the vicinity of the pleat.

The air velocity upstream of the filter was homogeneous for a transitional airflow regime approaching the filter and wide pleats (H = 28 mm, W = 25 mm), and very close to the mean duct velocity value (Vmean = 0.67 m.s -1 ). The air velocity upstream of the filter was also homogeneous for a turbulent air flow regime approaching the filter and the narrow width geometry, but 20% higher than the mean duct value (Vmean = 1.48 m. s -1 ), indicating the initiation of a velocity gradient. These investigations also show that, despite its intrusive nature, the Hot Wire Anemometry technique is appropriate for velocity acquisitions inside the pleat downstream of the filter, up to a 1 mm distance from the medium surface, and for low velocity values (up to 0.14 m.s -1 ) for the pleat dimensions like those studied. The CFD simulations enabled to point out that velocity variations, and more precisely sharp decreases, appeared only in a thin air layer along the medium surface, to reach values close to the theoretical filtration velocity. Significant velocity gradients were observed downstream of the filter and persisted up to 2.5 cm outside the filter. Due to the high acquisition frequency of the HWA technique, turbulent intensity profiles have been derived from velocity profiles. The turbulence intensity profiles downstream of the filter showed low variations inside the pleat, and increased outside the pleat. Which also reveal a transition between a laminar regime within the pleat and a more turbulent regime in the duct.

Chapter 5 investigates experimentally the influence of the clogging on velocity profiles downstream of the filter. In the first part of this chapter, the filter's filtration performance were evaluated for the two pleated geometries tested. The filters were clogged with submicron NaCl particles and micron Silica particles, resulting in a dendritic-like cake deposit in the first case and a more compact cake deposit in the second case. The clogging stopped at ΔP ~ 400 Pa, representative of an early stage of surface clogging. The fractional efficiency of the filter is determined experimentally. As a result, average and local structural properties of the medium are used to test the capacity of the single fiber efficiency models presented in the literature to predict the experimental particle collection efficiency. The results indicate that a precise description of the medium structural features improve fittings between the model and the experimental results. Indeed, characterizing the bi-modal fiber distribution with an equivalent bi-modal diameter is more accurate than using the mean fiber diameter. For the particle size range studied, the interception mechanism dominates the particle collection mechanism due to the tested airflow conditions and filter structural properties. The velocity at the vicinity of the filter is highly sensitive to changes in its the structural and geometrical properties of the pleated filter. As a result of clogging, the geometry of the pleat as well as the structural properties of the medium are altered due to particle deposition. The impact of the clogging on the velocity profile is not tackled in the literature. Such observations are important for design optimization, particularly when the filters are placed in series. Local velocity measurements also provide data that can be used to validate the airflow model choices in numerical simulations that predict filter performance during clogging.

In the second part of this chapter, the influence of clogging on the velocity pattern is evaluated by representing local velocity measurements downstream of the filter pleats for both geometries. For the same geometry (H = 28 mm, W = 25 mm) and two separate clogging tests with submicron NaCl particles and micron Silica particles, a similar trend of velocity profile is found at positions within (x = 1.5 cm) and outside the pleat (x = 3.5 cm). The velocity amplitude increases downstream of the pleat peak and drops towards the medium. The injection of a fluorescein tracer and SEM analysis confirmed these findings, revealing increased particle deposition and, as a result, airflow passage near the pleat peak. It was also demonstrated that the particle selection had no effect on the velocity profiles downstream of the filter. This could be due to the fact that the difference in count median diameter is not significant, and clogging were all stopped at the same pressure drop level. In the case of a "narrow-width" geometry (H = 27 mm, W = 10 mm) and the same filtration velocity (Vfiltration = 0.27 m.s -1 ), the velocity profiles downstream of the filter were different from the other geometry. Despite the fact that only one prototype was tested, this result is important because it could imply non-homogeneous clogging in the filter upstream side. In fact, Chapter 4 observes and discusses the initiation of a velocity gradient that is particularly sensitive to the pleat shape. As a result, inconsistencies in geometrical properties at the pleat level could result in a velocity gradient near the pleats, leading in particle deposition that differs from one pleat to another. It was also revealed that while approaching the medium at the same position investigated within and outside the pleat, the velocity profile reduces faster when the pleat width is narrower.

The obtained results are promising in terms of the study of airflow in the vicinity of clean and clogged pleated filters, and they open up possible directions for future research. As a result, it would be interesting to:

 Extend the velocity measurements to cover a broader range of pleat geometries and filtration velocities in order to determine the conditions under which an important velocity gradient occurs upstream of the filter, and thus determining the threshold pleat

Reynolds number Rew at which the transition between the laminar and turbulence regimes occurs within the pleat.

 Adjust the velocity measurement strategy to take advantage of the most recent innovative nano-scale HWA technique [START_REF] Kokmanian | Development of a nanoscale hot-wire probe for supersonic flow applications[END_REF]. This enables investigation of velocity at closer distances to the filter surface, as well as for narrower geometries to which the current technique is limited.

 Investigate experimentally the structural properties modifications caused by pleating in order to accurately estimate the reduction in permeability. This could improve numerical simulations and, as a result, the prediction of the pressure drop and the velocity variations in the vicinity of the pleat curved zones.

 Study the effect of duct obstacles (such as Pre-filters), and elbows located on the upstream side of the filter, on the uniformity of the velocity profile approaching the filter.

Annex 3-1: Calculation of differential and cumulative pore size using mercury prosimetry data

The AutoPore IV generates tabular and graphical reports of percentage pore volume vs.

diameter, and a summary report of percentage porosity in the defined size ranges. First, we start to cumulate the incremental mercury volume registered for each pressure; then we calculate the cumulative porosity as follows:

With ρapp is the apparent density of the sample (g/cm 3 ) and VHg cum is the cumulative volume of mercury (cm 3 ). the apparent density ρapp is given by the following detailed equations:

The weight of the mercury m Hg (g) is obtained by withdrawing the weight measured initially of the penetrometer containing the sample, from the total weight of the penetrometer at the end of the analysis. Thus the equation (4) becomes:

The cumulative porosity calculated using equation (3), is plotted in the function of the pores diameter registered by the system. In order to provide a more tangible overview of the pore sizes present in the medium, a differential logarithmic porosity distribution is required, this type of distribution shows equally well the distribution of large and small pore sizes. To proceed, the differential porosity is calculated as follow:

ε cum = ρ app × V Hg cum (47
Annexes 227

The experimental data of the porosity versus pore size (both differential and cumulative) are then modeled using equation ( 29) by taking into consideration the different level of pore sizes in the medium under study.

x -axis and x-y plane. Even if the way of presenting the results is different, the approach adopted in this study remain close to what was discribed elsewhere.

A group of N fibers can be described by the orientation tensor A, By averaging the tensors of each fiber over a "window" of several voxels. The second order orientation tensor 𝐴 ̿ is the mostly used to quentify the fiber orientation. the algotithm of calculation Fogar-Tucker [START_REF] Folgar | Orientation Behavior of Fibers in Concentrated Suspensions[END_REF] and It is calculated as follow:

N corresponds to the number of voxels belonging to the fiber phase in the window. The trace the laminar model was used for this purpose.

The numerical velocity fields presented in Figure 106 show that, similar to what was observed at a lower filtration velocity, Vfiltration = 0.27 m.s -1 (see Figure 54), no significant velocity gradients are detected upstream of the pleat for this filtration velocity, whereas significant gradients are detected downstream of the pleat. To validate the numerical code, experimental velocity data have been acquired upstream and downstream one half-pleat chose in the central zone of the duct. Abstract: The optimization of the pressure drop of a pleated filter is an economic and energy issue. Computational Fluid Dynamics (CFD) is a powerful tool for optimizing the design of pleated filters. However, the validation of CFD models reported in the literature relies only on experimental values of pressure drop, without taking into account significant local velocity variations in the vicinity of the medium surface that may affect particle collection mechanisms. The pleated filters investigated were made of a glass fiber filter medium, and two geometries (height -pleat width) were tested: (28 mm -25 mm) and (28 mm -25 mm) (27 mm -10 mm). Hot wire anemometry (HWA) was used to measure the experimental velocity profiles upstream and downstream of the filters. Transversal velocity profiles were acquired downstream at five different distances from the pleat peak (inside the pleat at x = 1.5 and 2 cm; outside the pleat at x = 3.0, 3.5, and 5 cm), as well as upstream of the pleat at (inside the pleat at x = 0.2 cm; 1.0, and 1.3 cm). downstream of the filter, and especially at the close vicinity of the filter medium.

For the tested filtration velocity, no The numerical velocity data were in quite good agreement with the experimental velocity profiles upstream and significant turbulent intensities were detected close to the medium surface. Clogging has an effect on the experimental airflow profile downstream of the filter, causing the velocity amplitude to increase downstream of the pleat peak and decrease toward the hollow. A preferential passage of airflow near the peak of the pleat was confirmed by fluorescein injection and microscopic analysis. When the width of the pleat was reduced, the velocity decreased more rapidly as it approached the surface of the medium. The experimental and numerical investigations upstream of the pleat revealed a slight increase in velocity when the pleat width is reduced, which could affect the homogeneity of particle deposition within the pleat. The possibility of clogging heterogeneity can explain the differences in velocity trends observed downstream of the pleat.

4. 5

 5 Change in the filter total efficiency ........................................................................... 184 5. Collection efficiency modeling ......................................................................................... 188 5.1 Methodology .............................................................................................................. 189 5.2 Influence of the equivalent particle diameter to express the experimental fractional efficiency ............................................................................................................................... 191 5.3 Influence of the parameters used to represent the fiber diameter distribution ........... 192 5.4 Sensitivity to the in-plane porosity variations ........................................................... 194 5.5 Sensitivity to the velocity variation at the vicinity of filter pleats ............................. 195 6. Experimental characterization of the local velocity gradients downstream of the clogged pleated filters ............................................................................................................................ 197 6.1 Material, method and methodology ........................................................................... 197 6.2 Effect of the size of the collected particles on the velocity profile downstream of the filters ................................................................................................................................... 200 6.3 Effect of the filter geometry on the velocity profiles downstream of filters.............. 202 7. Evaluation of the particle deposit homogeneity at the pleat scale .................................... 207 7.1 Characterization using fluorescein particles as tracer ................................................ 208 7.2 Characterization using SEM images .......................................................................... 213 8. Conclusion ......................................................................................................................... 216 General conclusion and perspectives ........................................................................................ 218 Annexes ....................................................................................................................................... 225 Annex 3-1: Calculation of differential and cumulative pore size using mercury prosimetry data ............................................................................................................................................... 226 Annex 3-2: Calculation of the fiber orientation tensor based on X-ray microtomography data ............................................................................................................................................... 228 Annex 3-3: SEM images at different magnifications ........................................................... 230 Annex 4-1: Comparison of numerical and experimental transversal velocity profiles downstream of the filter (H = 28 mm, W = 25 mm) for Vmean = 0.67 m.s -1 (Vfiltration = 0.27 m.s - 1 ): at various distances inside and outside the pleat. ............................................................. 231 Annex 4-2: Comparison of numerical and experimental transversal velocity profiles downstream of the filter (H = 27 mm, W = 10 mm) for Vmean = 1.48 m.s -1 (Vfiltration = 0.27 m.s - 1 ): at various distances inside and outside the pleat .............................................................. 233 Annex 4-3: Influence of the increase of the filtration velocity from Vfiltration = 0.27 m.s -1 to Vfiltration = 0.69 m.s -1 on the intensity of the velocity gradient upstream the filter (H = 28 mm, W = 25 mm) .......................................................................................................................... 235

Figure 1 37 Figure 3 43 Figure 4 45 Figure 5 46 Figure 6 47 Figure 7 48 Figure 8 51 Figure 9 90 Figure 25

 13734344554664774885199025 Figure 1 Evolution of the filtration performance during clogging by a solid aerosol .................... 33 Figure 2 RUC models: (a) TP flow (b) flow thorugh isotropic medium (Removed to respect copyright) ....................................................................................................................................... 37 Figure 3 Illustration of particle capture on fibers by the Brownian diffusion mechanism. ........... 43 Figure 4 Illustration of particle capture on fibers by the interception mechanism......................... 45 Figure 5 Illustration of particle capture on fibers by the inertial impaction mechanism ............... 46 Figure 6 Illustration of particle capture on fibers by the sieving mechanism ................................ 47 Figure 7 Illustration of particle capture on fibers by the electrostatic mechanism ........................ 48 Figure 8 Relationship between the structure of the particle deposit and the particle collection mechanism ...................................................................................................................................... 51 Figure 9 Numerical domain ............................................................................................................ 58 Figure 10 Calculation domain ........................................................................................................ 59 Figure 11. Pleat geometry, different velocity measurement zones and the velocity profile inside the pleat (adapted from Feng et al. [59]) .............................................................................................. 68 Figure 12 experimental velocity probe location (MP1 -MP5) downstream the filter element (adapted from Maddineni et al [73]) (Removed to respect copyright)........................................... 69 Figure 13 Transparent measurement zone of the experimental bench (Removed to respect copyright). ...................................................................................................................................... 76 Figure 14 Diagram of experimental bench (HWA probe for velocity measurements located downstream of the filter ................................................................................................................. 77 Figure 15 Pleated filter: (a) Example of prototypes, (b) geometrical specifications. .................... 79 Figure 16 CTA HWA principle diagram [74] ................................................................................ 81 Figure 17 Calibration curve of the straight HWA probe used for airflow characterization in the bench in the absence of a filter at a mean duct velocity (a) Vmean = 0.67 m.s -1 , (b) V mean = 1.48 m.s -1 . ............................................................................................................................................... 84 Figure 18 Photo of the HWA probe placement during the air-flow characterization measurement in the empty duct. ........................................................................................................................... 84 Figure 19 3D-Diagram of the displacements of the probe in the duct (for Vmean = 0.67 m.s -1 ). .... 86 Figure 20 Custom made PMMA sheets with slots allowing measurements at desired z positions 86 Figure 21 Comparison between the velocity profile for Vmean = 0.67 m.s -1 at two different x positions and at three z positions: (a) z = 5.5 cm, (b) z = 7.5 cm, (c) z= 9.5 cm and (d) all superimposed (average N=3 measurements; error bars = minimum and maximum values). ........ 88 Figure 22 Velocity profile for Vmean = 1.48 m.s -1 at x=167.5 cm and z = 7.5 cm (average N=3 measurements; error bars = minimum and maximum values). ...................................................... 89 Figure 23 Bulk-flow measurement zone (a) for Vmean = 0.67 m.s -1 , (b) Vmean = 1.48 m.s -1 ........... 89 Figure 24 Turbulent intensity profile at x = 167.5 cm, z = 7.5 cm and for (a) Vmean = 0.67 m.s -1 ; (b) Vmean = 1.48 m.s -1 (average N=3 measurements; error bars = standard deviation) ................. 90

Figure 31

 31 Figure 25 Diagram of hot wire probe displacements: (a) top view of the filter (H = 28 mm, W = mm); (b) transverse displacements downstream of one half pleat; (c) transverse displacements upstream of a one half pleat. .......................................................................................................... Figure 26 Diagram of hot wire probe displacements: (a) top view of the filter (H = 28 mm, W = mm); (b) transverse displacements downstream of one half pleat; (c) transverse displacements upstream of a one half pleat. .......................................................................................................... Figure 27 Pictures of the probe position downstream of the filter pleats (H = 28 mm, W = 25 mm) (top view) ....................................................................................................................................... Figure 28 Calibration curve for straight and right angle probes used for velocity measurements downstream and upstream of filter pleats, respectively (a) for Vmean = 0.67 m.s -1 ; (b) for Vmean = 1.48 m.s -1 . ....................................................................................................................................... Figure 29 Pictures taken from two angles of a right-angled probe mounted on a straight support and inserted in the pleat upstream of the filter. .............................................................................. Figure 30 Simulation domain and boundary conditions. ...............................................................Figure 31 Error percentage compared to the finest mesh number according to the number of total meshes of the numerical domain. ................................................................................................... Figure 32 Diagram of the velocity profiles position used for the meshing convergence study. .... Figure 33 Comparison of numerical transversal velocity profiles obtained with three configurations involving 80 000; 210 000 and 350 000 elements: (a) at x = -2.6 mm; (b) at x= 30.0 mm; and (c) longitudinal velocity profiles at 0.1 mm above the filter surface ................................................. Figure 34 Meshing used for CFD calculations (a) (H = 28 mm, 1/2W = 12.5 mm); (b) (H = 27 mm, 1/2W = 5 mm). ............................................................................................................................. Figure 35 Diagram showing the methodology developed in this study ....................................... Figure 36 Pore size: (a) differential distribution (b) cumulative distribution. ............................. Figure 37 Evolution of the medium porosity through its thickness. ............................................ Figure 38 In-plane porosity mapping of : (a) sample 1, (b) sample 2. ........................................ Figure 39 Histogram of in-plane porosity distribution in : (a) sample 1, (b) sample 2. .............. Figure 40 SEM Images of the medium at: (a) × 250 magnification; (b) × 500 magnification. .. Figure 41 SEM Evolution of the mean fiber diameter with the number of fibers. ...................... Figure 42 Fiber diameter: (a) Differential distribution; (b) Cumulative distribution. ................. Figure 43 3D fiber distributions images obtained by micro tomography analysis: (a) sample 1; (b) sample 2. ....................................................................................................................................... Figure 44 Evolution of the amplitude of the diagonal components of the orientation tensor through the medium thickness: (a) Sample 1; (b) Sample 2. ..................................................................... Figure 45 Evolution of the amplitude of the off-diagonal components of the orientation tensor through the medium thickness: (a) Sample 1; (b) Sample 2. ....................................................... Figure 46 Comparison of the experimental permeability to modeled values by taking into account the mean fiber diameter. ............................................................................................................... Figure 47 Comparison of the experimental permeability to values predicted by the RUC iso model and considering equivalent diameters to represent the bimodal fiber distribution calculated by different approaches. ....................................................................................................................

Figure 48

 48 Figure 48 Sensitivity of the medium predicted permeability to the mean medium porosity ....... Figure 49 In-plane permeability mapping of: (a) sample 1; (b) sample 2. ................................... Figure 50 Comparison of experimental and numerical filter pressure drop: sensitivity analysis of (a) pleat geometric properties (b) pleat curved zone permeability (with H = 28 ± 1 mm and W = 25 ± 1 mm). ................................................................................................................................. Figure 51 Influence of airflow models on the pressure drop ....................................................... Figure 52 Comparison of experimental and numerical filter pressure drop obtained for both geometries tested H = 28 mm, W= 25 mm) and (H = 27 mm, W= 10 mm). ............................... Figure 53 Comparison of experimental and numerical filter pressure drop using Darcy and Darcy Darcy-Forchheimer. ..................................................................................................................... Figure 54 Numerical velocity fields obtained for three airflow models at the vicinity of 2D halfpleat (H = 28 mm, W =25 mm): (a) laminar, (b) transition SST, (c) standard 𝑘-𝜀. ..................... Figure 55 Numerical velocity fields obtained for three airflow models at the vicinity of 2D halfpleat (H = 27 mm, W =10 mm): (a) laminar, (b) transition SST, (c) standard 𝑘-𝜀. ..................... Figure 56 Numerical velocity fields obtained for the standard 𝑘-𝜀.models at the vicinity of 2D halfpleat (H = 27 mm, W =10 mm) by reducing the scale to 0 -2.44 m.s -1 . ..................................... Figure 57 Diagram of the dimensionless coordinate y/y0 adopted to represent the transversal velocity profiles downstream of the filter .................................................................................... Figure 58 Experimental transversal velocity profiles downstream of the filter (H =28 mm, W = mm) for Vmean = 0.67 m.s -1 (Vfiltration = 0.27 m.s -1 ). ...................................................................... Figure 59 Experimental transversal velocity profiles downstream of the filter (H =27 mm, W = mm) for Vmean = 1.48 m.s -1 (Vfiltration = 0.27 m.s -1 ). ...................................................................... Figure 60 comparison of the experimental transversal velocity profiles downstream of the filter (H =28 mm, W = 25mm) for Vmean = 0.67 m.s -1 to those of the filter (H =27 mm, W = 10 mm) for Vmean = 1.48 m.s -1 (Vfiltration = 0.27 m.s -1 ) at (a) x = 2.0 cm, (b) x = 3.5 cm. .............................. Figure 61 Comparison of numerical and experimental transversal velocity profiles downstream of the filter (H = 28 mm, W = 25 mm) for Vmean = 0.67 m.s -1 (Vfiltration = 0.27 m.s -1 ): at (a) x = 2.0 cm; and at (b) x = 3.5 cm. ................................................................................................................... Figure 62 Comparison of numerical and experimental transversal velocity profiles downstream of the filter (H = 27 mm, W = 10 mm) for Vmean = 1.48 m.s -1 (Vfiltration = 0.27 m.s -1 ): at (a) x = 2.0 cm; and at (b) x = 3.5 cm. ................................................................................................................... Figure 63 Influence of pleat curved-zone permeability on numerical airflow streamlines (calculated with the laminar airflow model) for Vmean = 0.67 m.s -1 (Vfiltration = 0.27 m.s -1 ) by considering (a) permeable curved zone; (b) impermeable curved zone. ............................................................... Figure 64 Influence of pleat curved-zone permeability on numerical velocity profiles (calculated with the laminar airflow model) downstream of the filter for Vmean = 0.67 m.s -1 (Vfiltration = 0.27 m.s - 1 ): at (a) x = 2.0 cm; and at (b) x = 3.5 cm. .................................................................................. Figure 65 (a) Numerical (calculated with the laminar airflow model) and experimental transversal velocity profiles upstream of the filter pleats (H = 28 mm, W = 25 mm) at x = 0.2 cm, x = 1.0 cm and x = 1.3 cm for Vmean = 0.67 m.s -1 (Vfiltration = 0.27 m.s -1 ). (b) Numerical longitudinal velocity profiles at the air/filter interface and at 0.05 mm and 0.10 mm from the medium surface. .........

Figure 66 (Figure 80 Figure 92

 668092 Figure 66 (a) Numerical (calculated with the laminar airflow model) and experimental transversal velocity profiles upstream of the filter pleats (H =27 mm, W = 10 mm) at x = 0.5 cm, for Vmean = 1.48 m.s -1 (Vfiltration = 0.27 m.s -1 ).(b) Numerical longitudinal velocity profiles at the air/filter interface and at 0.05 mm and 0.10 mm from the medium surface. .............................................. 147 Figure 67 Experimental transversal turbulence intensity profiles for Vmean = 0.67 m.s -1 (Vfiltration = 0.27 m.s -1 ) (a) upstream of the filter at x = 0.2 cm, x = 1.0 cm and x = 1.3 cm (single values over 1 half pleat); (b) downstream of the filter at x = 1.5 cm, x = 2.0 cm, x = 3.0 cm, x = 3.5 cm and x = 5.0 cm (average values over 12 half pleat). .................................................................................. 148 Figure 68 Experimental transversal turbulence intensity profiles for Vmean = 1.48 m.s -1 (Vfiltration = 0.27 m.s -1 ) downstream of the filter (H = 27 mm, W = 10 mm) at x = 1.5 cm, x = 2.0 cm, x = 3.0 cm, x = 3.5 cm and x = 5.0 cm. (Average values over 10 half pleats) ......................................... 148 Figure 69 SEM images at × 500 magnification of NaCl particles deposited at the surface of flat filter under relative humidity conditions of (a) RH = 61% and (b) RH = 70% ........................... 155 Figure 70 SEM images of Silica particles depostion at the surface of a flat filter at a magnification of (a) × 500 and (b) × 2000. ........................................................................................................ 156 Figure 71 Nozzles used for particle sampling (a) for flat filter at Vmean = 0.27 m.s -1 , (b) for the pleated filter (H = 28 ± 1 mm, W = 25 ± 1 mm) at V inlet = 0.67 m.s -1 and (c) for the pleated filter (H = 27 ± 1 mm, W = 10 ± 1 mm) at Vmean = 1.48 m.s -1 . ............................................................. 162 Figure 72 Estimation of ΔPparticle deposit, by comparing the evolution of the pressure drop of a clogged pleated filter to that of a clogged flat filter. .................................................................................. 163 Figure 73 Logic diagram of the experimental protocol ................................................................ 166 Figure 74 Particle size distributions expressed in aerodynamic diameters from APS measurements and in optical diameters from Welas measurement (a) case of NaCL; (b) case of Silica (average values for N = 3 for Welas measurements -SD error bars; N = 1 for APS measurement) ........ 168 Figure 75 Log-normal number particle size distribution of NaCl particles at different RH levels, as well as the statistical parameters associated, from sampling upstream of (a) Flat filters , (b) Pleated filter with geometry (H= 28±1 mm , W =25±1mm) (Average values for N = 3 and more -SD error bars. .............................................................................................................................................. 173 Figure 76 Log-normal number particle size distribution of NaCl particles at RH around 45% from sampling upstream of flat and pleated filters (H= 28±1 mm , W =25±1mm) (Average values for N = 3 and more -SD error bars) ...................................................................................................... 174 Figure 77 Log-normal number particle size distribution of Silica particles from sampling upstream of different filter geometries as well as the statistical parameters associated (Average values for N = 3 and more -SD error bars) ...................................................................................................... 175 Figure 78 Evolution of the initial fractional number efficiency of the filters clogged by submicron NaCl particles at Vfiltration = 0.27 m.s -1 (a) Flat filter for RH ranging from 44 to 70% (b) Pleated filters (H = 28 ± 1 mm, W = 25 ± 1 mm) and flat filters for RH between 44 and 56 %. ............. 176 Figure 79 Evolution of the initial fractional number efficiency of the flat and pleated filters clogged by Silica test particles at Vf = 0.27 m.s -1 (Average values, N = 3 for flat and pleated filters (H = 28 mm, W =25mm); N = 1 for pleated filter (H = 27 mm, W = 10 mm) -Error bars representing min and max values) ............................................................................................................................ 178

Figure 1

 1 Figure 1 Evolution of the filtration performance during clogging by a solid aerosol

Figure 2

 2 Figure 2 RUC models: (a) TP flow (b) flow thorugh isotropic medium

  (b). Three solid fibers from the 1D fiber RUC model are placed Graphic removed to respect copyright Chapter 1 Literature review 38 in the same cubic RUC to create the isotropic RUC model. The three fibers are orthogonal in relation to one another. Two of the fibers run perpendicular to the flow direction, while the third runs parallel. Due to the random 3D arrangement of the actual fiber medium, the rectangular fiber oriented in each of the three major directions represents the degree of solid contribution in each direction.

Chapter 1 Literature review 42 and

 42 filtration velocity. The authors are interested in predicting the filter collection efficiency by studying the physical phenomenon of particle capture and deposition at the fiber scale. Fine particle capture and deposition are caused by electrical attraction or/and mechanical means. There are numerous capture mechanisms known, at least five of which are: diffusion, interception, impaction, as well as sieving and electric forces.

Figure 3

 3 Figure 3 Illustration of particle capture on fibers by the Brownian diffusion mechanism.

SievingFigure 6

 6 Figure 6 depicts the sieving mechanism. Particles with a diameter greater than or equal to the characteristic dimension of the pores of the fibrous medium are captured by sieving mechanisms.

Figure 6

 6 Figure 6 Illustration of particle capture on fibers by the sieving mechanism

Chapter 1 Literature review 48 

 48 Charge-Charge: the attraction or repulsion of unipolar or heteropolar fibers and particles by Coulomb forces.

Figure 7

 7 Figure 7 Illustration of particle capture on fibers by the electrostatic mechanism

Figure 8

 8 Figure 8 Relationship between the structure of the particle deposit and the particle collection mechanism

Figure 9

 9 Figure 9 Numerical domain

Chapter 1

 1 Literature review 67 and structural parameters of the pleat). The standard-k-ε predicts pressure poorly, whereas the RSM turbulence model predicts pressure well for the tested mean velocity range. (Vmean = 0.5 -2.0.

15. 3 Literature review 69 Figure 12

 36912 Figure 12 experimental velocity probe location (MP1 -MP5) downstream the filter element (adapted from Maddineni et al[START_REF] Maddineni | Numerical investigation of pressure and flow characteristics of pleated air filter system for automotive engine intake application[END_REF])

  and Figure 14. It was entirely developed and built by the technical team of the GEPEA laboratory at IMT Atlantique (Nantes). It enables measuring the airflow velocity in the vicinity of the pleated filters as well as to Chapter 2 Experimental and numerical strategies for air flow pattern study in the vicinity of filter pleats. 75 carry out clogging experiments at nominal airflow conditions. It should be noted that the materials used for clogging experiments will be detailed in Chapter 5.

Figure 13

 13 Figure 13 Transparent measurement zone of the experimental bench.

Figure 14

 14 Figure 14 Diagram of experimental bench (HWA probe for velocity measurements located downstream of the filter

Chapter 2

 2 Experimental and numerical strategies for air flow pattern study in the vicinity of filter pleats.

Figure 15

 15 Figure 15 Pleated filter: (a) Example of prototypes, (b) geometrical specifications.

  or a 4 th order Polynomial fitting function (20): Chapter 2 Experimental and numerical strategies for air flow pattern study in the vicinity of filter pleats.81And thus the terms A, B, n, and C0 to C4, are constants deduced from the calibration curve. The polynomial curve fit is used in this analysis, as linearization errors are often less than 1%.

Figure 16 CTA

 16 Figure16CTA HWA principle diagram[START_REF] Bruun | Hot-wire anemometry: principles and signal analysis[END_REF]



  Velocity measurements without the filter to characterize the airflow in the measurement zone, at the mean duct velocity.  Transversal velocity acquisitions downstream the filter pleats.

Chapter 2

 2 Experimental and numerical strategies for air flow pattern study in the vicinity of filter pleats. 83 55P13 3.5 Ω 0.05 -500 m.s -1 400 kHz  Transversal velocity acquisitions upstream the filter pleats. 55P14 3.5 Ω 0.05 -500 m.s -1 400 kHz  Transversal velocity acquisitions upstream the filter pleats 6. Airflow characterization in the bench 6.3 Methodology Before measuring the airflow in the vicinity of the filter pleats, a preliminary set of experiments was performed without the filter to characterize the airflow in the measurement zone. It allows determining whether or not the flow is established in the measurement zone, as well as the thickness of the airflow boundary layer. This characterization was carried out at the mean duct velocity corresponding to the nominal filtration velocity of the fibrous medium recommended by the supplier (Vfiltration = 0.27 m.s -1 ), i.e. at Vmean= 0.67 m.s -1 , for (H = 28 mm, W = 25 mm) and at Vmean= 1.48 m.s -1 for (H = 27 mm, W = 10 mm). The corresponding hydraulic Reynolds number are 6,760 and 14,590 respectively. hence the flow regime in the duct is transitional in the first case, and turbulent in the second. For this purpose, a straight probe (55P11, Dantec ©) mounted on a straight support was used (see Table

Chapter 2 84 Figure 17

 28417 Figure 17 Calibration curve of the straight HWA probe used for airflow characterization in the bench in the absence of a filter at a mean duct velocity (a) Vmean = 0.67 m.s -1 , (b) V mean = 1.48 m.s -1 .

Figure 18

 18 Figure 18 Photo of the HWA probe placement during the air-flow characterization measurement in the empty duct.

Figure 19

 19 Figure 19 3D-Diagram of the displacements of the probe in the duct (for Vmean = 0.67 m.s -1 ).

Figure 20 Chapter 2

 202 Figure 20 Custom made PMMA sheets with slots allowing measurements at desired z positions

Figure 22

 22 Figure 22 compares the velocity profiles for the mean duct velocity Vmean = 1.48 m.s -1 for x =167.5 cm and z = 7.5 cm. The horizontal error bars represent the minimum and maximum local velocity values obtained from three different trials. The velocity profiles are well superimposed and symmetrical depending the vertical plane of the duct. The velocity in the central zone is constant at about 1.62 m.s -1 , i.e. 1.1 times higher than Vmean = 1.48 m.s -1 , for y values between 3.5 cm and 11.5 cm. The boundary layer thickness for this mean duct velocity is thus δ = 3.5 cm, which is 1 cm less than the thickness obtained for Vmean = 0.67 m.s -1 .

Figure 21

 21 Figure 21 Comparison between the velocity profile for Vmean = 0.67 m.s -1 at two different x positions and at three z positions: (a) z = 5.5 cm, (b) z = 7.5 cm, (c) z= 9.5 cm and (d) all superimposed (average N=3 measurements; error bars = minimum and maximum values).

Chapter 2 89 Figure 22

 28922 Figure 22 Velocity profile for Vmean = 1.48 m.s -1 at x=167.5 cm and z = 7.5 cm (average N=3 measurements; error bars = minimum and maximum values).

Figure 23 δChapter 2 90 6. 5

 232905 Figure 23 Bulk-flow measurement zone (a) for Vmean = 0.67 m.s -1 , (b) Vmean = 1.48 m.s -1

Figure 24

 24 Figure 24 Turbulent intensity profile at x = 167.5 cm, z = 7.5 cm and for (a) Vmean = 0.67 m.s -1 ; (b) Vmean = 1.48 m.s -1 (average N=3 measurements; error bars = standard deviation)

  measurements were carried out at the nominal filtration velocity of the fibrous medium recommended by the supplier, i.e., Vfiltration = 0.27 m.s -1 . The mean duct velocity required to achieve this value is Vmean = 0.67 m.s -1 in the case of the geometry (H = 28 mm, W = 25 mm) and Vmean = 1.48 m.s -1 in the case of the narrow geometry (H = 27 mm, W = 10 mm). The corresponding hydraulic Reynolds number is 6,760 and 14,920, respectively. The flow regime in the duct is thus transitional in the first case, and turbulent in the second case. The airflow characterization consists of the acquisition of transversal velocity profiles (i.e., through the y axis) (see Figure25and Figure 26), at different x positions upstream and downstream of the filter. The displacements through the y axis corresponding to the velocity measurements downstream of the filter (H = 28 mm, W = 25 mm) and (H = 27 mm, W = 10 mm) are presented in Figure 25 (a) and Figure 26 (a), respectively. An upper view photograph of the probe position downstream the pleats (H= 28 mm, W = 25 mm) is shown in Figure 27.

Figure 25

 25 Figure 25 Diagram of hot wire probe displacements: (a) top view of the filter (H = 28 mm, W = 25 mm); (b) transverse displacements downstream of one half pleat; (c) transverse displacements upstream of a one half pleat.

Figure 26 Chapter 2 Figure 27

 26227 Figure 26 Diagram of hot wire probe displacements: (a) top view of the filter (H = 28 mm, W = 10 mm); (b) transverse displacements downstream of one half pleat; (c) transverse displacements upstream of a one half pleat.

Figure 28 Figure 29

 2829 Figure 28 Calibration curve for straight and right angle probes used for velocity measurements downstream and upstream of filter pleats, respectively (a) for Vmean = 0.67 m.s -1 ; (b) for Vmean = 1.48 m.s -1 .

2

 2 Experimental and numerical strategies for air flow pattern study in the vicinity of filter pleats. 96 real distance between pressure samplers located upstream and downstream of the tested filter in the set-up to determine the pressure drop. The boundary conditions used are presented in Figure 30. The half pleat was inserted between two symmetry conditions. The transverse component of the velocity, v = 0, as well as the gradient of the longitudinal component 𝜕𝑢 𝜕𝑥 , are zero in the plane of symmetry. A constant velocity inlet value was fixed at the domain inlet. At the domain outlet, a pressure outlet of 101,325 Pa was fixed. The shaded areas in Figure 30 correspond to the porous medium. The manufacturing process for the pleated filter may have produced local modifications of the structural properties of the fibrous medium in the curved zones (represented in grey in Figure

Figure 30

 30 Figure 30 Simulation domain and boundary conditions.

2

 2 airflow was modeled by Navier-Stokes equations at a steady state. The range of inlet air velocity (or mean duct velocity for the purposes of experiments) for which the numerical and experimental pressure drop were compared was 0 -1.56 m.s -1 . The corresponding hydraulic Reynolds number range on the bench was Reh = 0 -15,750. This range of airflow conditions therefore covers laminar and turbulent flow regimes, including the transitional flow regimes. This is why three models: the laminar model, the standard 𝑘-𝜀 turbulence model and the transition SST Experimental and numerical strategies for air flow pattern study in the vicinity of filter pleats. 97 model; were tested to evaluate their ability to predict the filter pressure drop and local velocity patterns for the inlet velocity value of Vmean = 0.67 m.s -1 (Reh = 6,760) for the geometry (H = 28 mm, W = 25 mm) and Vmean = 1.48 m.s -1 (Reh = 14,920) for the geometry (H = 27 mm, W =10 mm). The standard 𝑘-𝜀 turbulence model is widely used to solve the NS equations, using two equations (turbulent kinetic energy k and its dissipation rate ε). This model better suits free shear areas (away from walls) with low pressure gradients. According to the literature, this model succeeded in predicting the pressure drop for unused pleated filters with low permeability[START_REF] Tronville | Minimization of Resistance in Pleated-Media Air Filter Designs: Empirical and CFD Approaches[END_REF][START_REF] Théron | Numerical and experimental investigations of the influence of the pleat geometry on the pressure drop and velocity field of a pleated fibrous filter[END_REF]. The transition SST model consists of coupling the 𝑘-𝜔 SST transport equations (combination of the 𝑘-𝜔 model used in near-wall areas and the 𝑘-𝜀 model applied in internal shears areas) with two other transport equations: one for transition onset criteria and one for intermittency, regarding momentum-thickness Reynolds number. According to the literature, none of the studies dealing with numerical simulations of airflow in the vicinity of pleated filters concerned the transitional flow regime (see Chapter 1 -Table

Figure 31

 31 Figure[START_REF] Clague | A numerical calculation of the hydraulic permeability of three-dimensional disordered fibrous media[END_REF] Error percentage compared to the finest mesh number according to the number of total meshes of the numerical domain.

Figure 32

 32 Figure 32 Diagram of the velocity profiles position used for the meshing convergence study.

2

 2 Experimental and numerical strategies for air flow pattern study in the vicinity of filter pleats.

Figure 33

 33 Figure 33 Comparison of numerical transversal velocity profiles obtained with three configurations involving 80 000; 210 000 and 350 000 elements: (a) at x = -2.6 mm; (b) at x= 30.0 mm; and (c) longitudinal velocity profiles at 0.1 mm above the filter surface

Chapter 2 9 .

 29 Experimental and numerical strategies for air flow pattern study in the vicinity of filter pleats. 101 Conclusion of the Chapter with diagram of the methodology In this Chapter a methodology for the characterization of velocity patterns in the vicinity of pleated fibrous filters combining experimental and numerical methods has been developed, in the main aim of being able to validate the numerical simulations by local experimental velocity values. The methodology can be resumed in the diagram shown in Figure 35.

Figure 35

 35 Figure 35 Diagram showing the methodology developed in this study

17 , 3

 173 Bulb, 0.41 Stem, Powder type, with a volume of 3.1058 mL. It consists of a sample cup connected to a metal-clad, precision-bore and glass capillary stem. The mercury volume penetrating the porous medium was measured at different pressures to relate the porosity to the pore size by the Washburn equation: Where P is the applied pressure (Pa), γ the mercury superficial tension (N.m -1 ), and θ ( o ) the contact angle between mercury and the solid material. The values of θ and γ were 130° and 0.48 N•m -1 respectively. The analyses involved 50 pressure points ranging from 1.85 to 172.08 kPa which correspond to a pore size range of 7 µm -680 μm. The equilibration time at each pressure point was of 10 seconds. The equations used for calculation and steps follow to plot the pore size distributions are presented in Appendix 3.1.

  used in the analysis by the resolution. By processing the stacks of 2D images obtained from these micro-tomography analyses, different characteristics of the porous (mean porosity, through-plane porosity gradient and in-plane porosity variations) and fibrous structure of the medium were determined.The characterization of the porous structure is based on image segmentation, in order to differentiate fibers and image background. This work was carried out in the frame of the FIFT Project (Fibrous Filter and Tomography, Carnot TSN project, 2020) in partnership with researchers (Fatma Haouas, post-doctoral student, and John Puentes, Professor) of the ITI Department of IMT Atlantique and LabSTIC laboratory.

  mean medium thickness obtained by the low-pressure measurement technique is z = 601 ± 19 µm. The values estimated by the two micro-tomography analysis are 577.6 and 525.6 µm for the samples 1 and 2 respectively. The results obtained by the two methods are thus in quite good agreement, especially if one compares the thickness obtained by the low-pressure measurement technique and the one estimated for the sample 1 analyzed by micro-tomography.

Figure 36 (

 36 Figure 36 (a) and the cumulative pore size distribution in Figure 36 (b). These distributions are well superimposed, which proves the reproducibility of the measurement. They exhibit two interpenetrated populations characterized by modes of 27 µm and 350 µm respectively, and a minimum of approximately 7 µm.

Figure 36

 36 Figure 36 Pore size: (a) differential distribution (b) cumulative distribution.

Figure 37

 37 Figure 37 Evolution of the medium porosity through its thickness.

Figure 4

 4 Figure38shows in-plane porosity mappings for samples 1 and 2. Figure4presents their corresponding porosity distribution. The minimum and maximum porosity values detected for both samples are 88% and 98% respectively, but most of zones (of 24.8x24.8 µm² area) exhibit

Figure 38

 38 Figure 38 In-plane porosity mapping of : (a) sample 1, (b) sample 2.

Figure 39 Figure 40

 3940 Figure 39 Histogram of in-plane porosity distribution in : (a) sample 1, (b) sample 2.

Figure 40 SEM

 40 Figure 40 SEM Images of the medium at: (a) × 250 magnification; (b) × 500 magnification.

Figure 41 SEM

 41 Figure 41 SEM Evolution of the mean fiber diameter with the number of fibers.

Chapter 3

 3 Experimental characterization of the structural properties of the fibrous medium for an accurate prediction of its permeability 115The fiber diameter distribution is presented in Figure42. The error bars represent the widths of the diameter intervals. As shown in Figure42(a), the distribution involves two interpenetrated populations characterized by modal diameters which represent the "fine" fibers and the "coarse" fibers of df = 2.2 µm dc = 6.0 μm respectively. In order to estimate the fraction of fibers of each population, the area under the curve is divided at its inflection point, i.e. at d =4.2 μm. The fraction of fine and coarse fibers are nf = 0.68 and nc = 0.32 respectively. The cumulative distribution in Figure 42 (b) enables to determine the characteristic diameters of the distribution: d20 = 1.2 µm; d50 = 2.0 µm; and d90 = 5.1 µm.

Figure 42

 42 Figure 42 Fiber diameter: (a) Differential distribution; (b) Cumulative distribution.

Chapter 3 Figure 44 (

 344 Figure44 (a) shows that for sample 1 the XX and YY components both fluctuate around 0.4 -0.5. For sample 2 (Figure44 (b)), the XX and YY components range between 0.3 -0.4 and 0.5 -0.6 respectively. For both samples, the ZZ component is significantly lower through the whole medium thickness: it slightly fluctuates around 0.10. As the XX and YY components are the most important ones through the entire medium thickness, these results reveal that the medium is globally 3D anisotropic, having a quasi-layered structure with fibers randomly oriented in X-Y planes. And as the XX and YY components are of similar amplitude, the medium can be considered as quasi 2D isotropic.

Figure 45 depicts

 45 Figure 45 depicts the evolution of the amplitude of the off-diagonal components through the medium thickness for the two samples. As it is shown in Figure 45 (a) and Figure 45 (b), the components XZ and YZ are almost null for both samples, while the XY component slightly varies between -0.1 and 0.1. These results are complementary to previously described ones. It confirms that fibers are randomly oriented in the X-Y plane of the medium and rarely oriented through its thickness.

Figure 43 3D

 43 Figure 43 3D fiber distributions images obtained by micro tomography analysis: (a) sample 1; (b) sample 2.

Figure 44

 44 Figure 44 Evolution of the amplitude of the diagonal components of the orientation tensor through the medium thickness: (a) Sample 1; (b) Sample 2.

Figure 45

 45 Figure 45 Evolution of the amplitude of the off-diagonal components of the orientation tensor through the medium thickness: (a) Sample 1; (b) Sample 2.

32 Chapter 3 4 . 1 Influence of the parameters used to represent the fiber diameter distribution on the predicted permeability 4 . 1 . 1

 32341411 Figure 46 compares the experimental permeability to these predicted by different models of the literature (see Chapter 1 -Table1), by using the mean fiber diameter to represent the fiber diameter distribution. The error bars for the experimental permeability correspond to the deviation due to the thickness (see Section 3.1: z = 601 ± 19 µm). For the permeability predicted by models, the error bars correspond to the highest deviation due to the respective deviations of each of the two parameters of the model (ε and df) considered separately, which is here the porosity.

Figure 46

 46 Figure[START_REF] Kasper | The collection efficiency of a particleloaded single filter fiber[END_REF] Comparison of the experimental permeability to modeled values by taking into account the mean fiber diameter.

2 . 10 -12 m 2 and 2 .

 21022 67.10 -12 respectively, but most of zones (of 24.8x24.8 µm² area) exhibit values close to average predict value of permeability calculated for εmean = 0.92, i.e. 1.84.10 -11 .The medium is thus rather homogeneous. Moreover, these values are in good agreement with the mean permeability of (3.10±0.10).10 -11 m 2 obtained by permeametry.

Figure 49

 49 Figure 49 In-plane permeability mapping of: (a) sample 1; (b) sample 2.

Figure 50 and

 50 Figure 50 and Figure 52 compare the experimental and numerical filter pressure drop for the filtration velocity range 0 -0.65 m.s -1 (case of H = 28 mm, W = 25 mm) and 0 -0.31 m.s -1 (case of H = 27 mm, W = 10 mm). The experimental data are the mean pressure drop obtained from the measurements carried out on three prototypes (H = 28 mm, W = 25 mm) and two prototypes

Figure 50

 50 Figure 50 Comparison of experimental and numerical filter pressure drop: sensitivity analysis of (a) pleat geometric properties (b) pleat curved zone permeability (with H = 28 ± 1 mm and W = 25 ± 1 mm).

Figure 51

 51 Figure 51 Influence of airflow models on the pressure drop

1 )

 1 Figure 50 (a) represent the mean height and width (H = 28 mm, W = 25 mm) and the value obtained with the sets of parameters representing the maximum (H = 29 mm, W = 24 mm) and minimum (H = 27 mm, W = 26 mm) pressure drop.

Figure 50 (

 50 Figure 50 (b) represents the sensitivity of the numerical pressure drop to the pleat curved zone permeability. The results show that the simulated pressure drop (calculated with mean pleat width and height) increases when the pleat curved zones are considered impermeable. The deviation between numerical and experimental pressure drop is reduced to 16%. Similar results

Figure 52 compare

 52 Figure 52 compare the experimental and numerical pressure drop (obtained with the laminar flow model) of the geometry (H = 28 mm, W = 25 mm) to these of the narrow width geometry (H = 27 mm, W = 10 mm). For the same filtration velocity, the pressure drop for is slightly

Figure 52

 52 Figure 52 Comparison of experimental and numerical filter pressure drop obtained for both geometries tested H = 28 mm, W= 25 mm) and (H = 27 mm, W= 10 mm).

Figure 53

 53 Figure 53 Comparison of experimental and numerical filter pressure drop using Darcy and Darcy Darcy-Forchheimer.

1 )

 1 Experimental data (H = 28 mm, W =25 mm) CFD ( H = 28 mm, W = 25 mm) Experimental data (H = 27 mm, W = 10 mm) CFD (H = 28 mm, W =10 mm)

Downstream of the filter

  pleat (H= 28 mm, W = 25 mm) velocity values of up to 1.70 m.s -1 , i.e. 2.50 times higher than the mean duct velocity (0.67 m.s -1 ), are reached. Similar results were observed downstream of the filter pleat (H= 27 mm, W = 25 mm), where velocity values of up to 4.00 m.s -1 , i.e. 2.70 times higher than the mean duct velocity (1.48 m.s -1 ), are reported. These velocity patterns also show that local velocity values downstream of the filter are greatly influenced by the airflow model. The subject of the next section is therefore a comparison between experimental and numerical velocity profiles downstream of the filter.

Figure 54

 54 Figure 54 Numerical velocity fields obtained for three airflow models at the vicinity of 2D half-pleat (H = 28 mm, W =25 mm): (a) laminar, (b) transition SST, (c) standard 𝑘-𝜀.

Figure 55

 55 Figure 55 Numerical velocity fields obtained for three airflow models at the vicinity of 2D half-pleat (H = 27 mm, W =10 mm): (a) laminar, (b) transition SST, (c) standard 𝑘-𝜀.

Figure 56

 56 Figure 56 Numerical velocity fields obtained for the standard 𝑘-𝜀.models at the vicinity of 2D half-pleat (H = 27 mm, W =10 mm) by reducing the scale to 0 -2.44 m.s -1 .

Figure 57 1 Chapter 4

 5714 Figure[START_REF] Tronville | Minimization of Resistance in Pleated-Media Air Filter Designs: Empirical and CFD Approaches[END_REF] Diagram of the dimensionless coordinate y/y0 adopted to represent the transversal velocity profiles downstream of the filter

Figure 58 (

 58 Figure 58 (a) and Figure 59 (a) show that all the velocity profiles for the geometry (H = 28 mm, W = 25 mm) and (H = 27 mm, W = 10 mm) are similar in shape. Figure 60 compares the experimental velocity profiles of both geometries (H = 28 mm, W = 25 mm) and (H = 27 mm, W = 10 mm) normalized by the mean velocity value downstream of the filter pleats (a) inside

  (b) andFigure 59 (b). A similar result was obtained by Rebaï et al.[START_REF] Rebaï | A semi-analytical model for gas flow in pleated filters[END_REF]: the normalized velocity profiles (using dimensionless parameters similar to those previously described) obtained numerically at eight positions inside the pleat channel (x = 0.5 -4.5 cm) were fairly similar for the filtration velocity Vfiltration = 0.39 m.s -1 .

Figure 58

 58 Figure 58 Experimental transversal velocity profiles downstream of the filter (H =28 mm, W = 25 mm) for Vmean = 0.67 m.s -1 (Vfiltration = 0.27 m.s -1 ).

Figure 59 Figure 60

 5960 Figure 59 Experimental transversal velocity profiles downstream of the filter (H =27 mm, W = 10 mm) for Vmean = 1.48 m.s -1 (Vfiltration = 0.27 m.s -1 ).

Figure 61 compares

 61 Figure61compares experimental and numerical velocity profiles obtained with the three airflow models downstream of the filter pleats (H = 28 mm, W = 25 mm), at two x positions: x = 2.0 cm, i.e. inside the pleat, and x = 3.5 cm i.e. outside the pleat (Figure57). Figure62

Figure 62 compares

 62 experimental and numerical velocity profiles downstream " the narrow-width" filter pleats (H = 27 mm, W = 10 mm) obtained with the same airflow models and at the same x positions as for the previous filter geometry.For the filter pleats geometry (H = 28 mm, W = 25 mm), the three numerical profiles agree quite well with the experimental ones for y ≥ 2 mm as shown in Figure61. For this y coordinate range, the data obtained with the laminar and transition SST models, in particular, are identical and almost all appear inside the error bars of the experimental values. However, in the case of the geometry (H = 27 mm, W = 10 mm), only the laminar and transitional model agree quite closely with experimental data for y ≥ 1 mm as shown in Figure62. The results obtained with the k-ε model do not reproduce at all the experimental results.

  compared their experimental velocity profiles obtained by Particle Image Velocimetry (PIV) downstream of a custom-built rectangular pleated filter (H = 50.4 mm, W = 50.4 mm) with numerical values obtained by CFD. They showed that the DES model proposed by Feng et al.[59], succeeded in reproducing the experimental results except in close proximity to the pleat curved zone. Similar curve shapes were obtained for the other x coordinates downstream of the filter and for both geometries (x = 1.5, 3.0 and 5.0 cm), whether located inside or outside the filter, and the same conclusions can be drawn (see Annex 4-1 and Annex 4-2).

Figure 61

 61 Figure 61 Comparison of numerical and experimental transversal velocity profiles downstream of the filter (H = 28 mm, W = 25 mm) for Vmean = 0.67 m.s -1 (Vfiltration = 0.27 m.s -1 ): at (a) x = 2.0 cm; and at (b) x = 3.5 cm.

Figure 62

 62 Figure 62 Comparison of numerical and experimental transversal velocity profiles downstream of the filter (H = 27 mm, W = 10 mm) for Vmean = 1.48 m.s -1 (Vfiltration = 0.27 m.s -1 ): at (a) x = 2.0 cm; and at (b) x = 3.5 cm.

Chapter 4

 4 Local variations of air velocity in the vicinity of clean filter pleats -experimental and numerical approaches 142 distribution downstream of the filter is addressed in this section. Only the geometry (H = 28 mm, W = 25 mm) is considered for this study.

Figure 63

 63 Figure63depicts the airflow streamlines near the filter pleats when a permeable and impermeable curved zone are considered. Because the medium is isotropic, the flow tends to minimize the path inside the medium by entering perpendicular to it. By considering an impermeable curved zone, the streamline upstream the pleat deviates at the proximity of the curved impermeable zone and enters the pleat through its straight part. In this case, clear dead zones are observed downstream of the pleat hollow where no airflow streamline is observed.

Figure 63

 63 Figure 63 Influence of pleat curved-zone permeability on numerical airflow streamlines (calculated with the laminar airflow model) for Vmean = 0.67 m.s -1 (Vfiltration = 0.27 m.s -1 ) by considering (a) permeable curved zone; (b) impermeable curved zone.

Figure 64 Chapter 4

 644 Figure 64 compares the numerical velocity profiles (calculated with the laminar airflow model)for permeable and impermeable curved zones of the pleat at two x coordinates downstream of the filter: x = 2.0 cm (i.e. inside the pleat, Figure64 (a)), and x = 3.5 cm (i.e. outside the pleat, Figure64 (b)). At x = 2.0 cm, the numerical velocity profiles are perfectly identical, except for y ≤ 1 mm, i.e. downstream of the pleat peak, where the velocity for the impermeable pleat peak slightly decreases by 4% compared to that for the permeable one. Similar results were obtained

Figure 64

 64 Figure[START_REF] Raber | Predicting dust filter performance: pressure drop optimisation and dust capacity estimation for a deep-pleated industrial air filter using small sample data, predicting dust filter performance: pressure drop optimisation and dust capacity estimation for a deep-pleated industrial air filter using small sample data[END_REF] Influence of pleat curved-zone permeability on numerical velocity profiles (calculated with the laminar airflow model) downstream of the filter for Vmean = 0.67 m.s -1 (Vfiltration = 0.27 m.s -1 ): at (a) x = 2.0 cm; and at (b) x = 3.5 cm.

Figure 65 (Chapter 4

 654 Figure65(a) compares the experimental velocity profiles upstream of the filter (H = 28 mm, W =25 mm) at three x coordinates x = 0.2 cm, x = 1.0 cm and x = 1.3 cm (the coordinate x = 0 cm corresponds to the pleat peak, as shown in Figure57) to those obtained numerically using the laminar airflow model. The experimental velocity profiles involve y coordinate ranges mostly located in the center of the pleat. The minimum distance from the filter medium at which the velocity was recorded was 2.0 mm for the profile at x = 1.3 cm. Figure65shows that the numerical velocity values in this pleat zone are equal to the experimental ones, confirming that

Figure 66 (

 66 b) shows that the numerical velocity values in this pleat zone are equal to the average experimental one, confirming that the laminar flow model is relevant for predicting the velocity pattern for the airflow conditions tested.In addition, Figure65and Figure66show that despite the mean airflow approaching the filter being transitional for the geometry (H = 28 mm, W = 25 mm) (Reh = 6,760) or even turbulent for the geometry (H = 27 mm, W = 10 mm) (Reh = 14,920), at the pleat center, the velocity is homogeneous and almost equal to the mean duct velocity (Vmean = 0.67 m.s -1 and 1.48 m.s -1 respectively). The numerical simulations that produce velocity data up to the medium surface indicate that velocity decreases sharply within a very thin air film, approximately 0.3 mm thick for both geometries, at the medium surface, with values in the order of Vfiltration (0.27 m.s -1 ). For the airflow in question, this result supports the decision to use the laminar flow model. More generally it raises the issue of the relevant Reynolds number to be considered when selecting the appropriate airflow model for air velocity patterns inside the filter pleats. According to Rebaï et al.[START_REF] Rebaï | A semi-analytical model for gas flow in pleated filters[END_REF], the flow inside the pleat channel can be characterized by the channel Reynolds number, which takes into account the mean duct velocity and half-pleat width as the characteristic length:For the present cases, Rew is equal to 560 for a W = 25 mm (Vmean = 0.67 m.s -1 ) and 496 for W = 10 mm (Vmean = 1.48 m.s -1 ), which is quite low and may justify the choice of the laminar airflow model. For some cases reported in the literature, significant velocity gradients were detected upstream of the pleat, with maximum velocity values in the central pleat zone

  (b), the velocity ranges between 0.74 m.s -1 and 0.44 m.s -1 , which represents 2.7 to 1.6 times the theoretical filtration velocity (Vfiltration = 0.27 m.s -1 ). In the case of the narrow width geometry (H = 27 mm, W =10 mm), and at 0.10 mm above the surface, i.e. x coordinates between 0.5 and 26.5 mm as shown in Figure66(b), the velocity reaches ranges between 1.40 m.s -1 and Chapter 4 Local variations of air velocity in the vicinity of clean filter pleats -experimental and numerical approaches 146 1.61 m.s -1 which is close to the mean velocity (Vmean = 1.48 m.s -1 ), and 5.8 to 6.0 times the theoretical filtration velocity (Vfiltration = 0.27 m.s -1 ).

Figure 65 (Chapter 4

 654 Figure 65 (a) Numerical (calculated with the laminar airflow model) and experimental transversal velocity profiles upstream of the filter pleats (H = 28 mm, W = 25 mm) at x = 0.2 cm, x = 1.0 cm and x

Figure 66 (

 66 Figure 66 (a) Numerical (calculated with the laminar airflow model) and experimental transversal velocity profiles upstream of the filter pleats (H =27 mm, W = 10 mm) at x = 0.5 cm, for Vmean = 1.48 m.s -1 (Vfiltration = 0.27 m.s -1 ).(b) Numerical longitudinal velocity profiles at the air/filter interface and at 0.05 mm and 0.10 mm from the medium surface.

Figure 67 ( 1 V 1 VmeanChapter 4

 67114 Figure67 (a) shows TI profiles upstream of the pleat (H = 28 mm, W =25 mm) at the x positions considered in the previous section. Each point corresponds to one measurement taken in a single half-pleat. The turbulence intensity varies slightly along y for the three profiles, ranging from 6% to 7.0% at x = 0.2 cm, and from 5.0% to 6.0% at x = 1.0 cm and 1.3 cm. For the pleat (H

Figure

  Figure Experimental transversal turbulence intensity profiles for Vmean = 0.67 m.s -1 (Vfiltration = 0.27 m.s -1 ) (a) upstream of the filter at x = 0.2 cm, x = 1.0 cm and x = 1.3 cm (single values over 1 half pleat); (b) downstream of the filter at x = 1.5 cm, x = 2.0 cm, x = 3.0 cm, x = 3.5 cm and x = 5.0 cm (average values over 12 half pleat).

FigureFigure 67 (

 67 Figure Experimental transversal turbulence intensity profiles for Vmean = 1.48 m.s -1 (Vfiltration = 0.27 m.s -1 ) downstream of the filter (H = 27 mm, W = 10 mm) at x = 1.5 cm, x = 2.0 cm, x = 3.0 cm, x = 3.5 cm and x = 5.0 cm. (Average values over 10 half pleats)

Chapter 5

 5 Influence of the clogging on the filtration performance and on the velocity profiles downstream of the filter pleats 155 of cake has changed as the humidity increase leads particles to collapse and grow especially close to the RDH NaCl i.e. in the case of RH 70%.

Figure 69 SEM

 69 Figure 69 SEM images at × 500 magnification of NaCl particles deposited at the surface of flat filter under relative humidity conditions of (a) RH = 61% and (b) RH = 70%

Chapter 5 156 Figure 70

 515670 Figure 70 presents SEM images of Silica particles at the surface of a flat filter at a magnification of (a) × 500 and (b) × 2000. It depicts a dense cake deposition in aggregate form. It also shows that particles are rather polydispersed.

Figure 70 SEM

 70 Figure 70 SEM images of Silica particles depostion at the surface of a flat filter at a magnification of (a) × 500 and (b) × 2000.

  AGK 2000 Palas particle generator is made of a cyclone, a liquid reservoir, and a binary nozzle. Error! Reference source not found. (a) displays a diagram of the different components of the generator. The reservoir is filled with NaCl solution at a 100g/L concentration for NaCl solid particles generation and at a 10g/L concentration for Fluorescein particles generation. The solution (NaCl or Fluorescein solution) is nebulized by the AGK 2000 particle generator using a binary atomizer nozzle. A TR 2000 drying stage, fed by dry pressurized air, can be added at the atomizer aerosol output ( see Error! Reference source not found. (b)).

  measure the particle distribution of the Silica particles and NaCl prior to clogging experiments. It provides high-resolution real-time aerodynamic measurements of particles from 0.5 to 20 microns. The input particle concentration should not exceed 10 4 particles.cm -3 . A photograph of the device is shown in Error! Reference source not found. (b).

  was used in this work to assess collection efficiency during clogging with Silica test dust and NaCl particles by measuring particle distribution upstream and downstream of the filter (whether it was implemented in pleated or flat geometry). Error! Reference source not found. (a) shows a photograph of the device and Error! Reference source not found. (b) an operating mode diagram.

Figure 71

 71 Figure 71 Nozzles used for particle sampling (a) for flat filter at Vmean = 0.27 m.s -1 , (b) for the pleated filter (H = 28 ± 1 mm, W = 25 ± 1 mm) at V inlet = 0.67 m.s -1 and (c) for the pleated filter (H = 27 ± 1 mm, W = 10 ± 1 mm) at Vmean = 1.48 m.s -1 .

Figure 72

 72 Figure 72 Estimation of ΔPparticle deposit, by comparing the evolution of the pressure drop of a clogged pleated filter to that of a clogged flat filter.

5

 5 Influence of the clogging on the filtration performance and on the velocity profiles downstream of the filter pleats 164 Fluorescein particles were generated at the end of a single experiment with a pleated filter (H = 28 ± 1 mm, W = 25 ± 1 mm) clogged with NaCl particles, to evaluate if preferential deposition occurred. It helped in the understanding and interpretation of local velocity profiles, as well as the verification of clogging homogeneity at the surface of the pleated medium.

Figure 73

 73 Figure 73 Logic diagram of the experimental protocol

  parameters V mean , ΔP 0 , RH, T Start of the NaCl particle generation using AGK2000 nebuliser Particle counting upstream and downstream the filter using WELAS optical counter Stop of particle generation at ΔP≥400 Pa Measurment of the parameters V mean , ΔP 0 , RH, T Start of the isofine particle generation using TSI ring dust disperser Particle counting upstream and downstream the filter using WELAS optcical counter Stop of particle generation at ΔP= 350 -

Figure 74

 74 Figure 74 Particle size distributions expressed in aerodynamic diameters from APS measurements and in optical diameters from Welas measurement (a) case of NaCL; (b) case of Silica (average values for N = 3 for Welas measurements -SD error bars; N = 1 for APS measurement)

  geometric mean diameter dg:  count median diameter d50 :

Figure 75

 75 Figure 75 Log-normal number particle size distribution of NaCl particles at different RH levels, as well as the statistical parameters associated, from sampling upstream of (a) Flat filters , (b) Pleated filter with geometry (H= 28±1 mm , W =25±1mm) (Average values for N = 3 and more -SD error bars.

Figure 76 compares

 76 Figure76compares the particle size number distribution collected upstream of the flat and pleated filters at the same humidity level (RH =45%). The drying column was added to the generator in the case of the flat filter, but not in the case of the pleated filter. The results reveal that the particle size distribution is relatively similar and that the drying column has no effect on the distribution in this case. It should be noted that sampling takes place under different airflow conditions depending on the filter geometry used. In the case of flat and pleated filters, the mean duct velocity is Vmean = 0.27 m.s -1 and Vmean = 0.67 m.s -1 , respectively. It is equivalent to Vfiltration = 0.27 m.s -1 .

Figure 76 4 . 2 . 2

 76422 Figure 76 Log-normal number particle size distribution of NaCl particles at RH around 45% from sampling upstream of flat and pleated filters (H= 28±1 mm , W =25±1mm) (Average values for N = 3 and more -SD error bars)

Figure 77 4 . 3 Initial fractional efficiency of the filters 4 . 3 . 1

 7743431 Figure 77 Log-normal number particle size distribution of Silica particles from sampling upstream of different filter geometries as well as the statistical parameters associated (Average values for N = 3 and more -SD error bars)

Figure 78 (Chapter 5

 785 Figure78(a) depicts a clear impact of the humidity on the initial fractional efficiency of the filter, which is consistent with the previously discussed pressure drop and total efficiency evolutions. The lowest efficiency zone is found around dae = 0.57 µm (doptical = 0.22 µm). At

Figure 78

 78 Figure 78 Evolution of the initial fractional number efficiency of the filters clogged by submicron NaCl particles at Vfiltration = 0.27 m.s -1 (a) Flat filter for RH ranging from 44 to 70% (b) Pleated filters (H = 28 ± 1 mm, W = 25 ± 1 mm) and flat filters for RH between 44 and 56 %.

4. 3 . 2

 32 Figure79shows the evolution of the initial fractional efficiency of the filters in at the beginning of the clogging (ΔP/ΔP0~1) with Silica particles (dae,50 =1.0 µm) for three filter configurations:

Figure 79 4 . 3 . 3

 79433 Figure 79 Evolution of the initial fractional number efficiency of the flat and pleated filters clogged by Silica test particles at Vf = 0.27 m.s -1 (Average values, N = 3 for flat and pleated filters (H = 28 mm, W =25mm); N = 1 for pleated filter (H = 27 mm, W = 10 mm) -Error bars representing min and max values)

FlatChapter 5

 5 Influence of the clogging on the filtration performance and on the velocity profiles downstream of the filter pleats 179 particle collection; this is known as image force, and it occurs when particles but not fibers are charged.

Figure 80 4 . 4 Change in the filter pressure drop and determination of the apparent permeability 4 . 4 . 1

 8044441 Figure 80 Comparison of the evolution of the initial fractional number efficiency of the pleated filters at the beginning of clogging with submicron NaCl (d50 = 0.7 µm) vs. micron Silica (d50 = 1.0 µm) for the same geometry (H = 28 mm, W =25 mm) and at Vf = 0.27 m.s -1 . (Average values, N = 3 -Error bars representing min and max values)

Figure 81 (

 81 Figure81 (b) shows the evolution of the pressure drop of the pleated filters (H = 28 ± 1 mm, W = 25 ± 1 mm) compared to the two tests performed with flat filters and drying column under similar air humidity conditions ranging from 44 to 56 %. The results show that in accordance with filtration theory, the increase in pressure drop in the case of a pleated filter is faster than in the case of a flat filter: the depth filtration phase is shorter, and the inflecting point occurs at lower collected mass i.e. about wcollected = 4 g/m 2 . The pressure drop induced by the pleating causes the pressure drop to evolve faster than in the flat configuration.

Figure 81

 81 Figure 81 Evolution of the pressure drop of filters clogged by submicron NaCl particles at a filtration velocity of 0.27 m.s -1 (a) Flat filters for RH ranging from 44 to 70% (b) Pleated filters (H = 28 ± 1 mm, W = 25 ± 1 mm) and flat filters for RH between 44 and 56 %.

Figure 82 depicts

 82 Figure82depicts the evolution of flat and pleated filter pressure drop through an average curve of duplicated samples with the same filtration velocity of 0.27 m.s -1 and humidity range (RH = 44-56 %). The dotted lines represent the pressure drop range that varies due to humidity and possible heterogeneity of the medium. Such a representation allows for an approximation of the "apparent permeability" of the pleated filter as follows:

Figure 82

 82 Figure 82 Comparison of the pressure drop evolution of flat and pleated filters clogged by NaCl particles at a filtration velocity of 0.27 m.s -1 .

Figure 83 depicts

 83 Figure 83 depicts the pressure evolution of flat and pleated filters clogged by micron nonhygroscopic Silica test particles at 0.27 m.s -1 filtration velocity.

  9 g/m 2 in case of (H = 27 ± 1 mm, W = 10 ± 1) geometry and at wcollected = 15.2 g/m 2 in case of flat filter.

Figure 83 4 . 4 . 3

 83443 Figure 83 Evolution of the pressure drop of the flat and pleated filters clogged by Silica test particles at a filtration velocity of 0.27 m.s -1 (Average values, N = 3 for flat and pleated filters (H = 28 mm, W =25mm); N = 1 for pleated filter (H = 27 mm, W = 10 mm)min and max values error bars)

Chapter 5

 5 Influence of the clogging on the filtration performance and on the velocity profiles downstream of the filter pleats 184 filtration velocity, and with the same weight of deposited particles, the increase in the pressure drop is more important for smaller particles.

Figure 84 4 . 5 Change in the filter total efficiency 4 . 5 . 1 Figure 85

 844545185 Figure 84 Comparison of the evolution of the filter pressure drop during clogging with submicron NaCl (dae,50 = 0.7 µm) vs. micron Silica (dae,50 = 1.0 µm) particles for the same geometry (H = 28 mm, W =25 mm) and at Vfiltration = 0.27 m.s -1 . (Average values -N = 3)

0Figure 86

 86 Figure 86 shows the total efficiency evolution of the flat and pleated filters clogged by micron non-hygroscopic Silica test particles at Vfiltration = 0.27 m.s -1 . The curves represent the average total efficiency values determined from three filter clogging experiments in the case of flat and pleated geometry (H= 28 ± 1 mm, W = 25 ± 1 mm), with error bars representing the maximum and minimum values. Only one pleated geometry prototype (H = 27 ± 1 mm, W = 10 ± 1 mm) has been tested.

Figure 86

 86 Figure 86 Evolution of the total number efficiency of the flat and pleated filters clogged by Silica test particles at a filtration velocity of 0.27 m.s -1 (Average values, N = 3 for flat and pleated filters (H = 28 mm, W =25mm); N = 1 for pleated filter (H = 27 mm, W = 10 mm) -Error bars representing min and max values)

  mm, W = 10 mm beginning of clogging. In fact, it exhibits a distinct behavior compared to the other geometries,

Figure 87

 87 Figure 87 compares the total number efficiency of the pleated filter during clogging with submicron NaCl particles (dae,50 = 0.7 µm) to that with micron Silica particles (dae,50 = 1.0 µm) for the same geometry (H = 28 mm, W =25 mm) and Vfiltration = 0.27 m.s -1 .

Figure 87

 87 Figure 87 Comparison of the evolution of the total number efficiency of the pleated filters during clogging with submicron NaCl particles (dae,50 = 0.7 µm) vs. micron Silica particles (dae,50 = 1.0 µm) for the same geometry (H = 28 mm, W =25 mm) and at Vfiltration = 0.27 m.s -1 . (Average values, N = 3 -Error bars representing min and max values)

5 . 2

 52 Figure 88 (a) compare the experimental initial fractional collection efficiency of the pleated filter (H =28 mm, W =25 mm) to those obtained by different models of the literature by considering a mean fiber diameter dmean = 3.14 µm, mean porosity ε = 0.92 (𝜑 =0.08) and a mean filtration velocity (Vfiltration = 0.27 m.s -1 ) as model input parameters. The models fairly reproduce the efficiency for optical and Stokes particle diameters in the range of 0.2 and 0.6 µm. They depict a MPPS diameter range between 0.1 and 0.3 µm. The wide range of expressions found in the literature makes it impossible to favor any particular model. The reasonfor this is that most collection efficiencies have been experimentally determined or simulated on a single fiber, or are based on empirical determinations of the efficiency of the fibrous medium from which the single fiber collection efficiency is deduced. The trends of the three analytical models are close on one another in the range of diameter studied, so it was decided deliberately to use the Miecret and Gustavson model[START_REF] Miecret | Mathematic Expression of HEPA and ULPA Filters Efficiency, Experimental Verification, Practical Alliance to New Efficiency Test Method[END_REF] in this sensitivity study.

Figure 88 5 . 3

 8853 Figure 88 Comparison of the experimental efficiency values of the filter (H = 28 mm, W = 25 mm) and those for different models from the literature (df,mean = 3.14 µm, εmean = 0.92, Vfiltration = 0.27 m.s -1 )

Chapter 5

 5 Influence of the clogging on the filtration performance and on the velocity profiles downstream of the filter pleats 193 if somewhat slightly overestimated. Moreover, using Brown and Thrope[START_REF] Brown | Glass-fibre filters with bimodal fibre size distributions[END_REF] diameter brings the slope of the predicted efficiency closer to the slope of the experimental data in the particle diameter range 0.2<dp<3.0 µm.

Figure 89 (

 89 Figure 89 (b) depicts the evolution of single fiber interception and inertial impaction efficiency.

Figure 89

 89 Figure 89 Influence of considering equivalent diameters to represent the bimodal fiber distribution on the estimation of (a) the total fractional efficiency calculated by Miecret and Gustavson model (b) the corresponding single fiber efficiency by interception and inertial impaction. (εmean = 0.92, Vfiltration = 0.27 m.s -1 )

5 . 4

 54 Figure 90 depicts the sensitivity of the in-plane porosity variations determined by X-ray microtomography analysis (a) on the total fiber collection efficiency calculated by Miecret and Gustavson and (b) on the corresponding single fiber collection efficiency by interception and impaction. The values of porosity considered represent the minimum and maximum observed by microtomography i.e. ε = 0.87 and ε = 0.98, as well as the mean porosity values obtained by mercury porosimetery i.e εmean = 0.92 ± 0.016.

Figure 90

 90 Figure 90 Influence of considering the in-plane porosity variations on: (a) the total fractional efficiency calculated by Miecret and Gustavson model; (b) the corresponding single fiber efficiency by interception and inertial impaction. (df ,eq,2 = 2.17 µm, Vfiltration= 0.27 m.s -1 )

  Impaction -ε = 0.90 Interception -ε = 0.90 Impaction -ε = 0.92 Interception -ε = 0.92 Impaction -ε = 0.98 Interception -ε = 0.98 Impaction -ε = 0.94 Interception -ε = 0.94

"

  narrow-width" geometry (H = 27± 1 mm, W = 10 ± 1 mm). The experiments were conducted at the same nominal filtration velocity Vfiltration = 0.27 m. s -1 which corresponds to a mean duct velocity Vmean = 0.67 m. s -1 and to a hydraulic Reynolds number Reh = 6760 (transitional airflow regime) in case of (H = 28 ± 1 mm, W = 25 ± 1 mm) and to Vmean = 1.48 m.s -1 and Reh = 14919 (Turbulent airflow regime) with the "narrow-width" geometry.For each measurement point, the acquisition frequency and the sampling time were 200 Hz and 60 s respectively. The airflow characterizations required determination of transversal velocity Chapter 5 Influence of the clogging on the filtration performance and on the velocity profiles downstream of the filter pleats 198 profiles (i.e., through the y-axis) (see Chapter 2 -Figure25), at different x positions downstream of the filter. For these measurements, a straight probe (55P11, Dantec ©) mounted on a straight support was used (see Chapter 2 -Table13). The calibration of the anemometry probe involved 45 points between 0.1 and 4.0 m.s -1 in the case of (H = 28 ± 1 mm, W = 25 ± 1 mm) (Figure92 (a)) and 60 points between 0.1 and 7.0 m.s -1 in the case of (H = 27± 1 mm, W = 10 ± 1 mm) (Figure92 (b)).

Figure 92

 92 Figure 92 Calibration curves of the straight HWA probe used for airflow characterization downstream of pleated filters (a) case of (H = 28 ± 1 mm, W = 25 ± 1 mm), (b) case of (H = 27± 1 mm, W = 10 ± 1 mm).

Chapter 5

 5 Influence of the clogging on the filtration performance and on the velocity profiles downstream of the filter pleats 199 to these with filter geometry (H = 28 ± 1 mm, W = 25 ± 1 mm) at the same clogging level with Silica particles (ΔP ~ 400 Pa). A single pleated prototype (H = 27 ± 1 mm, W = 10 ± 1 mm) is tested, and transverse velocity profiles are recorded downstream of ten half-pleats in the bulk flow.

Figure 93 5 6 . 2 Chapter 5

 935625 Figure 93 Transverse displacement downstream of clogged filter

Figure 94

 94 Figure 94 Transverse velocity profiles downstream of the filters at Vfiltration = 0 27 m.s -1 (Vmean = 0.67 m.s -1 ) for the filter (H = 28 ± 1 mm, W = 25 ± 1 mm) at its clean and clogged state by Silica and NaCl particles at (a) x = 1.5 cm and (b) x = 3.5 cm. (Average values N = 12 half pleats for clogged filters and N = 24 half pleats for clean filters, SD errors)

Chapter 5

 5 1 -clean Half-pleat 1 -Clogged hlaf-pleat 2 -clean Half-pleat 2 -clogged half -pleat 3 -clean half-pleat 3 -clogged half-pleat 4 -clean half-pleat 4 -clogged half-pleat 5 -clean half-pleat 5 -clogged half-pleat 6 -clean half-pleat 6 -clogged half-pleat 7 -clean half -pleat 7 -clogged half-pleat 8 -clean half-pleat 8 -clogged half-pleat 9 -clean half-Influence of the clogging on the filtration performance and on the velocity profiles downstream of the filter pleats 202 2.00 m. s -1 ) respectively. The same phenomenon is observed at x=3.5 cm (Figure 94 (b)) even

Figure 95 and

 95 Figure95and Figure96show the evolution of transverse velocity profiles acquired downstream of the filter with "narrow-width" geometry (H = 27 ± 1 mm, W = 10 ± 1 mm) clogged with Silica particles, for the same filtration velocity as for the geometry (H = 28 ± 1 mm, W = 25 ± 1 mm), i.e. at Vfiltration = 0.27 m.s -1 (which corresponds to Vmean = 1.48 m.s -1 ) at x = 1.5 cm and x = 3.5 cm, respectively. Only one prototype is tested in this case.

Figure 95 (

 95 Figure 95 (b) and Figure 96 (b) show the evolution of the transverse velocity profiles downstream of the filter pleats in their clean and clogged states, respectively, using average local velocity values measured at x = 1.5 cm and x = 3.5 cm downstream of the 10 half pleats studied. For the clean filters, the error bars indicate the standard deviations, and in the case of the clogged filters, the dashed green lines cover the minimum and maximum velocities recorded. This representation is chosen to facilitate the results interpretation.

Figure 95 Figure 96

 9596 Figure 95 Transverse velocity profiles downstream of the filter at Vfiltration = 0.27 m.s -1 (corresponding to Vmean = 1.48 m.s -1 ) for the filter (H = 27 ± 1 mm, W = 10 ± 1 mm) at its clean and clogged state by Silica particles at x = 1.5 cm (a) local measurements per pleat and (b) average values (N = 10 half pleats/1 prototype ; SD errors for clean filter; min and max cover range in case of clogged filter)

Figure 98 Figure 98 Chapter 5

 98985 Figure 98 presents the dimensionless velocity profiles downstream of filter pleats in clean and clogged states (ΔP ~ 400 Pa) for both geometries (H = 28 ± 1 mm, W = 25 ± 1 mm) and (H = 27 mm, W = 10 mm) and at Vfiltration = 0.27 m.s -1 for two positions (a) x = 1.5 cm and (b) x = 3.5 cm. The mean duct velocity corresponding to this filtration velocity is Vmean = 0.67 m. s -1for the case of (H = 28 ± 1 mm, W = 25 ± 1 mm) and 1.48 m. s -1 for the case of (H = 27 mm, W = 10 mm).

Figure 99 5

 995 Figure 99 Photograph of the filter with the position of the samples

Figure 100

 100 Figure 100 Calibration curve of the fluorescein absorbance

Figure 101

 101 Figure101depicts the non-homogeneous deposition of fluorescein over the filter. At the pleat level, a pale orange hue can be seen close to the pleat hollow that becomes darker towards the pleat peak. It implies that the fluorescein is most likely concentrated at the level of the pleat peak and less concentrated at the level of the peak hollow. A white strip is seen at on peaks and hollows, which could indicate airflow dead zones caused by the modifications in structural properties of the filter medium due to pleating process, as mentioned in section 6.2 of this

Chapter 5 212 Figure 101 Chapter 5

 52121015 Figure 101 Photograph of the clogged filter with the deposit of fluorescein particles

7. 2 7 . 2 . 1

 2721 Characterization using SEM images Material and methodsSEM pictures have been taken at different pleat hollows and peaks to observe if there is any preferential deposition that could explain the previously mentioned velocity trends downstream of the filter. These analyses are carried out on two prototypes of the filter clogged with Silica particles: one prototype of configuration (H = 28 ± 1 mm, W = 25 ± 1 mm) and one prototype of configuration (H = 27 ± 1 mm, W = 10 ± 1 mm). It was decided to conduct the analysis on the filter clogged with Silica particles in order to avoid humidity constraints during storage and transport, that could affect the observations.

  few particle traces, indicating potential airflow dead zone areas. The effect of the structural modifications due to the pleating process on particle deposit within pleated filters was recently investigated. Teng et al.[START_REF] Teng | Research on the influence of pleat structure on effective filtration area during dust loading[END_REF] demonstrated experimentally that the curved dead zone caused by pleating increases the structural resistance of the medium and decreases the effective filtration area (EFA) during clogging. Pleated filters geometries with same height (H = 20 mm) and different widths (W = 5 -27 mm) were clogged with polydispersed ash particle (dp = 3 -100 m) at a low filtration velocity of 0.04 m. s -1 . They conclude that the influence of these curved dead zones is less significant in the width range W = 14 -27 mm, which is close to the one used in this study.Except for the dead zone strip part, it appears that more particles are deposited in the pleat peak than in the pleat hollow for both geometry cases (H = 28 ± 1 mm, W = 25 ± 1 mm) and (H = 27 ± 1 mm, W = 10 ± 1 mm). The images taken at the pleat peak (Figure103 (a)) and (Figure104(a)) reveal particle layers on the surface that appear to be thicker than those taken at the pleat hollow. The cake deposit in the first case looks dense and covers the surface of the filer while the fibers are still visible in the second case. This observation, in conjunction with the fluorescein observations and local velocity measurements downstream the clogged pleats, indicates that the deposition is not homogeneous within the pleats. The particles are more likely to be collected close to pleat peaks rather than pleat hollows. Few studies in the literature investigated the non-homogeneous deposition within the pleat but most of them concerned rectangular pleat shapes or triangular mini-pleats. Fotovati et al[START_REF] Fotovati | Modeling instantaneous pressure drop of pleated thin filter media during dust loading[END_REF] numerically studied the influence of the particle deposition on the surface of a pleated filter. Triangular and rectangular pleat shape are considered. two geometries have been tested (H = 25.4 mm -W = 8.5 mm) and Chapter 5 Influence of the clogging on the filtration performance and on the velocity profiles downstream of the filter pleats 215 a narrow width geometry (H = 25.4 mm -W = 1.3 mm). They observed that in case of the narrow width geometry, the velocity gradient upstream the filter increased, resulting in a not uniform deposit within the pleats. The particle tends to deposit deeper inside the rectangular pleat i.e. pleat hollow, for large particles (dp = 10 µm). this effect was less pronounced in case of triangular pleats where more homogeneous deposit. However, this work was purely numerical and lack of experimental validation.

Figure 103

 103 Figure 103 SEM observations of the Silica particle deposit at x25 magnification for the case of the geometry (H = 28 mm, W =25 mm) (a) at a pleat peak (b) at a of pleat hollow.

Figure 104

 104 Figure 104 SEM observations of the Silica particle deposit at x25 magnification for the case of the geometry (H = 27 ± 1 mm, W = 10 ± 1 mm) (a) at a pleat pleak (b) at a pleat hollow.

Chapter 1 .

 1 CFD simulations are a powerful tool to optimize the design of a pleated filter.However, the authors' main challenge remains in selecting an appropriate airflow model to predict the airflow approaching the filter, depending on the airflow conditions investigated. The validation of the airflow model in the literature is almost always based on the filter pressure drop, which is a macroscopic parameter. Few studies validate the choice of the airflow model based on local parameters such as velocity values. Furthermore, although several studies described in the literature discovered velocity gradients in the pleat channel, they did not incorporate them into the collection efficiency calculation. They instead consider the average filtration velocity, which does not reflect the local conditions encountered by the particles and hence reduces the model's accuracy. The significance of the velocity gradient downstream of the pleat on filtration performance has received little attention in the literature, despite the fact that it might be significant to improve filtration performance, particularly in interior air applications where filters are installed in series. Similarly, the non-uniform velocity distribution produced by a pre-filter can have an impact on the filter's efficiency and life span. Moreover, despite the fact that particle clogging modifies the medium's structural characteristics and reduces the filter geometry due to surface particle deposition, the literature did not provide experimental research of the clogging on the velocity profile near filter pleats. The investigations were purely computational, with no experimental validation based on local velocity values. As a result, to provide better optimization of the filtration efficiency during clogging, special attention should be paid to the flow behavior on both sides of the pleat.Based on these arguments, it was decided to develop a methodology for the characterization of velocity patterns in the vicinity of pleated fibrous filters that combines experimental measurements and numerical simulations. Due to the complexity of the pleating geometry, the main challenge is to measure local velocity profiles within the filter pleats and up to a short distance from the filter medium surface. The materials used, as well as the experimental and numerical strategies, are described in detail in Chapter 2. Because it does not require particle tracers, Hot Wire Anemometry (HWA) was chosen over other techniques (PIV and LDA) to measure experimental velocity profiles upstream and downstream of the filter. Transversal velocity profiles were acquired downstream at five different distances from the pleat peak (inside the pleat at x = 1.5 and 2 cm; outside the pleat at x = 3.0, 3.5, and 5 cm), as well as upstream of the pleat at (inside the pleat at x = 0.2 cm; 1.0, and 1.3 cm). The fibrous medium chosen for this study is a commercial glass fiber medium type ISO ePM1 50-50 percent, which is commonly used in the second stage of AHUs for indoor air applications. Two filter geometries have been tested (H = 28 mm, W = 25 mm) and (H = 27 mm, W = 10 mm) at the same filtration velocity Vfiltration = 0.27 m. s -1 . The Computational Fluid Dynamics (CFD) simulations were carried out using the Ansys Fluent code based on a half-pleat numerical domain.

of A is always 1 .Annex 3 - 3 :

 133 Figure 105 SEM images of the medium at (a) × 25, (b) ×500, (c) × 1000.

Annex 4 - 1 :Annex 4 - 2 :Annex 4 - 3 :

 414243 Comparison of numerical and experimental transversal velocity profiles downstream of the filter (H = 28 mm, W = 25 mm) for Vmean = 0.67 m.s -1 (Vfiltration = 0.27 m.s -1 ): at various distances inside and outside the pleat. Comparison of numerical and experimental transversal velocity profiles downstream of the filter (H = 27 mm, W = 10 mm) for Vmean = 1.48 m.s -1 (Vfiltration = 0.27 m.s -1 ): at various distances inside and outside the pleat Influence of the increase of the filtration velocity from Vfiltration = 0.27 m.s -1 to Vfiltration = 0.69 m.s -1 on the intensity of the velocity gradient upstream the filter (H = 28 mm, W = 25 mm)The purpose of this section is to determine whether increasing the mean duct velocity from 0.67 m.s -1 to the ventilator's maximum achievable velocity, i.e. = 1.70 m.s -1 , will result in an important gradient upstream the filter (H = 28 mm, W = 25 mm), contrary to what was previously discussed in section 1.7.)The corresponding filtration velocity for a Vmean = 1.70 m.s -1 , i.e. the highest velocity achievable by the ventilator, is Vfiltration 0.69 m.s -1 , which is 2.5 higher than the filtration velocity tested in this study (Vfiltration = 0.27 m.s -1 ). The hydraulic Reynolds number in the duct is Reh = 17120, indicating a turbulent regime. The pleat Reynolds number is Rew = 1425 indicating a laminar flow within the pleat. As demonstrated throughout this study (see sections 3.3 and 3.5), the Rew is the parameter to look out when predicting the airflow behavior within a pleat channel;

Figure 107 Figure 106

 107106 Figure 107 depicts numerical and experimental transversal velocity profiles (a) upstream and (b) downstream of the filter pleats at x = 1.3 cm and x = 3.5 cm, respectively. The laminar model fairly well reproduces the experimental results upstream and downstream of the filter,similarly to be previously discussed in section 3.3. Figure19 (a)shows that for the tested geometry (H =28 mm, W =25 mm) and filtration velocity of 0.67 m.s -1 , the velocity is rather homogeneous and close to the mean duct velocity (Vmean = 1.70 m.s -1 ), similar to what was observed at lower filtration velocity i.e Vfiltration = 0.27 m.s -1 (Vmean = 0.67 m.s -1 )(see sections 3.5) thus no influence of the increase of the filtration velocity on the intensity of the gradients upstream the filter (H = 28 mm, W =25 mm).

Figure 107

 107 Figure 107 Numerical (calculated with the laminar airflow model) and experimental transversal velocity profiles (a) upstream of the filter pleats at x = 1.3 cm; (b) Downstream the filter pleats at x = 3.5 cm (H = 28 mm, W = 25 mm) for Vmean = 1.70 m.s -1 (V filtration = 0.69 m.s -1 )

Table 1

 1 Permeability models of the literature for fibrous medium ................................................ Table 2 Expressions of unimodal equivalent diameter .................................................................. Table 3 Prediction models for filtration efficiencies by diffusion .................................................Table 4 Prediction models for interception filtration efficiencies .................................................. Table 5 Prediction models for inertial impaction filtration efficiencies. ....................................... Table 6 Prediction of interaction between diffusion and interception efficiency .......................... Table 7 Prediction models for total unit filter efficiencies. ............................................................ Table 8 Influence of filter structural parameter and operating conditions on particle collection efficiency ........................................................................................................................................ Table 9 Studies of the literature dealing with numerical simulations of air flow through the pleated filter ................................................................................................................................................ Table 10 Studies dealing with experimental velocity measurements at the vicinity of pleated filter ........................................................................................................................................................ Table 11 Characteristics of sensors ................................................................................................ Table 12 Structural properties of the medium studied. .................................................................. Table 13 Miniature HWA probes specifications and applications ................................................. Table 14 Technical data sheet of the medium AX1691HD provided by Lydall .......................... Table 15 Characteristics of the medium used to test the permeability models ............................ Table 16 Properties of the medium and the experiment techniques used to characterize it......... Table 17 Provided data on the chemical composition w% of the Silica test particles from the supplier (Powder Technology Inc.) .............................................................................................. Table 18 Sampling nozzle diameters designed for particle sampling upstream and downstream of the different filter geometries .......................................................................................................Table 19 Synthesis of the experimental characteristion performed ............................................. Table 20 Conditions of clogging experiments with NaCl particles ............................................. Table 21 Conditions of clogging experiments with Silica particles ............................................. Table 22 Parameters used to test the efficiency models (case of H =28 mm, W =25 mm) ......... Table 23 Distribution of surface weight of fluorescein concentrations captured by filter ...........

	Nomenclatures

Table 1

 1 Permeability models of the literature for fibrous medium

	Authors	B/df 2

Table 2

 2 Expressions of unimodal equivalent diameter

	Authors	deq equations
	Clague and Phillips (1997) [31]	1 2	√𝑛 𝑐 𝑑 𝑐 2 + 𝑛 𝑓 𝑑 𝑓 2
	Brown and Thrope. (2001) [32]	1 2	( 𝑛 𝑐 𝑑 𝑐 2 + 𝑛 𝑓 𝑑 𝑓 2 𝑛 𝑐 𝑑 𝑐 + 𝑛 𝑓 𝑑 𝑓	)
	Tafreshi et al. (2009) [30]	1 2	√𝑛 𝑐 𝑑 𝑐 3 + 𝑛 𝑓 𝑑 𝑓 3 3

Table 3

 3 Prediction models for filtration efficiencies by diffusion

	Authors	Year Theoretical prediction	Flow field used	Remarks
	Langmuir [35]	1942	1.7 k 0 -1/3 Pe -2/3	Lamb's Flow		k 0 = 2 -2 ln Re
	Natanson [35]	1957	2.9 k 0 -1/3 Pe -2/3	Lamb's Flow		k 0 = 2 -2 ln Re
	Friedlander [35]	1957	3.25𝐴 1/3 𝑃𝑒 -2/3	Tomotika and	A is a constant function of φ and Re
						Aoi
	Stechkina	1966	2.9 𝑘 0 -1/3 𝑃𝑒 -2/3	Lamb's flow		𝑘 0 = 2 -2 𝑙𝑛 𝑅𝑒
	[35,36]		+ 0.62𝑃𝑒 -1
	Davies [35]	1973	1.5𝑃𝑒 -2/3		0.02< φ<0.35
	Liu. Rubow [35,37]	1990	1.6( 1 -𝜑 𝐻 𝐾𝑢	)𝑃𝑒 -2/3 𝐶𝑑	Kuwabara flow	𝐻 𝑘𝑢 = -	1 2	𝑙𝑛 𝜑 -075 + 𝜑 -	1 4	𝜑 2
							0.5<Vfiltration<100 cm.s -1 ; 0.005<dp<1 µm
							𝐶𝑑 = 1 + 0.388k𝑛 𝑓 [	𝐻 𝑘𝑢 (1 -φ)𝑃𝑒	] 1/3
	Payet et al. [35,38]	1992	1.6 ( 1 -𝜑 𝐻 𝐾𝑢	) 𝑃𝑒 -2 3 𝐶𝑑. 𝐶𝑑′	Kuwabara flow			𝐶𝑑 ′ =	1 1 + 𝜂 𝑑
							𝜂 𝑑 is the theoretical expression of Liu.Rubow
							(1990)

Table 4 Prediction models for interception filtration efficiencies

 4

				𝑅 =	𝑑 𝑝 𝑑 𝑓
	Authors	year	Theoretical prediction	Flow field	Remarks
	Stechkina, Fuchs [35]	1966	2(1 + 𝑅)𝑙𝑛(1 + 𝑅) -(1 + 𝑅) + 2𝐻 𝐾𝑢	1 1 + 𝑅	Kuwabara flow
	Yeh,Liu [35]	1974
								Kuwabara
	Kirsch [35]	1978 2(1 + 𝑅)𝑙𝑛(1 + 𝑅) -(1 + 𝑅) +	1 + 𝑅 1 + 2.86(2 + 𝑅)𝑅𝐾𝑛 𝑓	flow
			2𝐻 𝐾𝑢
	Lee,Liu [35]	1982	0.6 ( 1 -𝛼 𝐻 𝐾𝑢	) ( 1 + 𝑅 𝑅 2	)	Kuwabara flow	1<Vf <30 cm.s -1 0.05<dp<1.3µm
								0.0045<R<0.12
								0.0086<𝜑<0.151

Table 5

 5 Prediction models for inertial impaction filtration efficiencies.

			𝑆𝑡 =	18µ𝑑 𝑓 𝐶𝑢. 𝜌 𝑝 . 𝑉 𝑓𝑖𝑙𝑡𝑟𝑎𝑡𝑖𝑜𝑛 . 𝑑 𝑝 2		(7)
	Authors	Year	Theoretical prediction	Flow field	Remarks
	Langmuir [35]	1948		St 2	Transitory	Ref<1
			(1 + 0.55St) 2	flow

Table 6

 6 Prediction of interaction between diffusion and interception efficiency

	Author	Expression of ηdr			Remarks
	Stechina [36]	1.24H ku -1 2 R	2 3 Pe -1 2	𝐻 𝑘𝑢 = 𝜑 -	1 2	𝑙𝑛𝜑 -	1 4	𝜑 2 -	3 4
	Miecret and Gustavson [39]
	Kirsh et al. [45]	1.24H Fan -1 2 R	2 3 Pe -1 2	𝐻 𝑓𝑎𝑛 = -1.43 (1 -𝜑)𝑘𝑛 𝑓 1 2 𝑙𝑛𝜑 -0.52 + 0.64𝜑 +

Table 7

 7 Prediction models for total unit filter efficiencies.

	Authors	Year	Expression of η	Expression of ηd Expression of ηr	Expression of Expression ηdr of ηI
	Stechkina et al [36] 1969	ηd+ ηr+ ηdr+ ηI	Stechkina	Yeh et al.	Stechkina	Stechkina
	Kirsh et al. [45]	1978	ηd+ ηr+ ηdr	Kirsch	Kirsch	Fuchs	-
	Kasper et al.[46]	1978	1-(1-ηd) (1-ηr) (1-ηI)	Davies	Kuwabara	-		Langmuir
	Lee.Liu [34]	1982	ηd+ ηr	Lee. Liu	Lee. Liu	-		-
	Miecret.Gustavson [39]	1989	ηd+ ηr+ ηdr+ ηI	Stechkina	Kuwabara modified by Cai	Stechkina. Fuchs Suneja. Lee
	Liu. Rubow [37]	1990	ηd+ ηr	Liu. Rubow Liu. Rubow	-		-
	Payet [38]	1992	ηd+ ηr	Payet	Liu. Rubow	-		-
	Gougeon [40]	1994	ηd+ ηr+ηI	Payet	Liu. Rubow	-		Gougeon

Table 8

 8 Influence of filter structural parameter and operating conditions on particle collection efficiency

	Filtration mechanism	Parameters of the particles	Structural parameters of the filter	Operating conditions
		ρp ↗	dp↗	z↗	df↗	𝜑↗	Vfiltration↗
	Diffusion

Table 9

 9 Studies of the literature dealing with numerical simulations of air flow through the pleated filter

	Authors	Medium type and	Filter type and	Flow regime / model	Pleat	Simulation
		properties	geometrical properties	and Velocity range (m.s -1 )	number Reynolds	domain
	Fotovati et al.

Table 10

 10 Studies dealing with experimental velocity measurements at the vicinity of pleated filter

	Authors	Medium
		permeability

Table 11

 11 Characteristics of sensors

	Parameter	Sensor Type	Working range	Precision
	Relative Humidity (RH)	EE08	0 -100 %	±3%
		(Electronik)
	Temperature (T)	Pt1000	-40 -80 °C	±0.2°C
		(Electronik)
	Flow sensor	SS 20.600	0 -220 m.s -1	± 3%
		(SCHMIDT®90
	Differential pressure sensor	SETRA ® FC	0 -700 Pa	± 1%

Table 12

 12 Structural properties of the medium studied.

	Property	Values	Measurement technique
	Thickness	601 ± 19 µm (11 samples)	Low-pressure measurement
	Mean fiber diameter	3.14 ± 2.09 µm (950 fibers counted)	SEM (JEOL JSM 7600F) and picture analysis using Image J
	Permeability	(3.10 ± 0.10).10 -11 m² (3 samples)	Permeametry
	Mean porosity	92 ± 2 % (3 samples)
		2 populations :	Mercury porosimetry (Micrometrics Autoport
	Mean pore diameter	-dpores,1 = 33 µm -dpores,2 = 183µm	IV)
		(3 samples)

Table 13 Miniature

 13

HWA probes specifications and applications

  Experimental and numerical strategies for air flow pattern study in the vicinity of filter pleats.

	Chapter 2
		1.60									1.80
											1.75
		1.55
											1.70
		1.50									1.65
	E (V)	1.45								E (V)	1.55 1.60
		1.40									1.50
											1.45
		1.35
				y = -0.0328x 4 + 0.1604x 3 -0.3107x 2 + 0.3988x + 1.2987	1.40	y = -0.0059x 4 + 0.0501x 3 -0.1664x 2 + 0.3419x + 1.364
						R² = 0.9998							R² = 0.9998
		1.30									1.35
		0.0	0.2	0.4	0.6	0.8	1.0	1.2	1.4	1.6	0.0	0.5	1.0	1.5	2.0	2.5	3.0
						V (m.s -1 )								V (m.s -1 )
						(a)								(b)
											HWA probe
											Displacement system Displacement system

Table 14

 14 Technical data sheet of the medium AX1691HD provided by Lydall

	Properties	values
	Basic weight (g/m 2 )	68
	Tensile, MD*(Pa)	3000
	Tensile, CD**(Pa)	1100
	Stiffness, MD (Pa)	1100
	% Combustibles	6
	Water Repellency (mm)	300

*Machine direction. ** Cross direction

Table 17

 17 Provided data on the chemical composition w% of the Silica test particles from the supplier (Powder Technology Inc.)

	Component	Weight Fraction (%)
	Silica (fine dust)	69 -77
	Aluminum oxide	8-14
	Calcium oxide	2.5-5.5
	Potassium oxide	2-5
	Sodium oxide	1-4
	Iron (III) oxide	4-7
	Magnesium oxide	1-2
	Titanium dioxide	0-1

Table 18

 18 Sampling nozzle diameters designed for particle sampling upstream and downstream of the different filter geometries

	Filter geometry	Vmean (m.s -1 )	d nozzle, theoretical (mm)	d nozzle, measured (mm)
	Flat	0.27	17.56	18.00
	H = 28 mm, W = 25 mm	0.67	11.20	12.00
	H = 27 mm, W = 10 mm	1.48	7.50	8.00

Table 19

 19 Synthesis of the experimental characteristion performed

	H = 28 ± 1 mm, W =25 ± 1 mm	H = 27±1 mm, W
			=10±1 mm
	Clogging with	Clogging with	Cloggeing with
	NaCL	Silica	Silica
	Solution	Powder	Powder dispersion
	atomization	dispersion	(Ring TSI dust
	(AGK 2000	(Ring TSI dust	disperser)
	Palas)	disperser)

Table 20

 20 Conditions of clogging experiments with NaCl particles

	Filter geometry	Sampl e	RH (%) a	T ( o C) b	Vmean (m.s -1 )	Duration (min)	Cupstream c (P/cm 3 )	Number of efficiency cycles	∆P0 (Pa)	Drying stage condition
		1	70.1±0.4	18.3±1.2	0.27	203	94298±3131	10	113	no
	Flat	2 3	61.5±0.6 48.6±0.5	22.6±1.6 23.3±0.2	0.27 0.27	186 84	89866±3568 39641±1266	13 5	116 105	no yes
		4	44.0±0.5	22.5±0.1	0.27	130	40100±1511	5	104	yes
		1	45.1±0.5	21.3±0.1	0.67	67	33750±1600	6	110	no
	H= 28 mm, W = 25 mm	2	56.2±0.3	19.4±0.1	0.67	63	27344±1334	6	114	no
		3	47.3±0.5	24.5±0.1	0.67	66	24428±1581	6	112	no

a,b: Average values over the entire clogging duration time; standard deviation errors for N = 3 and greater. c: Average values based on the total number of efficiency cycles; standard deviation errors for N = 3 and greater.

Table 21

 21 Conditions of clogging experiments with Silica particlesAverage values over the entire clogging duration time; standard deviation errors for N = 3 and greater. c: Average values based on the total number of particle sampling; standard deviation errors for N = 3 and greater.

	Filter geometry	Sample	RH (%) a	T ( o C) b	Vmean (m.s -1 )	Duration (min)	Cupstream (P/cm 3 ) c	Number of efficiency cycles	∆P0 (Pa)
		1	52.7±0.7	26.1±0.5	0.27	310	2138±290	5	113
	Flat	2	63.7±0.9	23.5±0.3	0.27	107	9855±1439	5	107
		3	67.2±0.5	22.5±0.2	0.27	58	10770±1458	5	103
		1	48.4±0.4	18.3±0.6	0.67	61	8221±991	6	116
	H= 28 mm, W = 25 mm	2	58.2±0.2	19.0±0.1	0.67	52	8354±1331	6	115
		3	40.0±0.8	19.1±0.2	0.67	66	8449±1607	6	112
	H = 27 mm, W = 10 mm	1	31.3±0.3	18.3±0.7	1.48	141	2576±433	11	120
	a,b:

Table 23

 23 Distribution of surface weight of fluorescein concentrations captured by filter

	Pleat Zone	Sample	A	A corrected	C (mg.L -1 )	w(mg)	w(µg)	wfluo, corrected (µg.cm -2 )	waverage (µg.cm -2 )
		a	0.092	0.038	0.905	0.004	3.62	4.61
	Peak	bc	0.097	0.043	1.026	0.004	4.10	5.23	4.86
		d	0.093	0.039	0.929	0.004	3.72	4.74
		a	0.103	0.049	1.170	0.005	4.68	5.96
		b	0.104	0.050	1.194	0.005	4.78	6.08
	Straight	c d	0.101 0.101	0.047 0.047	1.122 1.122	0.004 0.004	4.49 4.49	5.72 5.72	5.84
		L	0.103	0.049	1.170	0.005	4.68	5.96
		R	0.100	0.046	1.098	0.004	4.39	5.59
	Hollow	ab cd	0.088 0.09	0.034 0.036	0.809 0.857	0.003 0.003	3.24 3.43	4.12 4.37	4.25

  )

			𝜌 𝑎𝑝𝑝 =	𝑚 𝑆𝑎𝑚𝑝𝑙𝑒 𝑉 𝑎𝑝𝑝	(48)
		𝜌 𝑎𝑝𝑝 =	𝑚 𝑠𝑎𝑚𝑝𝑙𝑒 𝑉 𝑝𝑒𝑛 -𝑉 𝐻𝑔
		𝜌 𝑎𝑝𝑝 =	𝑚 𝑠𝑎𝑚𝑝𝑙𝑒 𝑚 𝐻𝑔 𝑉 𝑝𝑒𝑛 -𝜌 𝐻𝑔
	𝜌 𝑎𝑝𝑝 =	𝑉 𝑝𝑒𝑛 -	𝑚 𝑠𝑎𝑚𝑝𝑙𝑒 𝜌 𝐻𝑔 𝑚 𝑝𝑒𝑛+𝑠𝑎𝑚𝑝𝑙𝑒+𝐻𝑔 -𝑚 𝑝𝑒𝑛+𝑠𝑎𝑚𝑝𝑙𝑒	(49)
			𝑑𝜀 𝑖 =	𝑑 𝜀 𝑐𝑢𝑚,𝑖 𝑑 log 𝐷 𝑝,𝑖

Acknowledgments

Characterization methods

The characterization methods, as well as the equations used to determine the structural properties, are presented in this section.

Thickness measurements

The measurement of the medium permeability by permeametry (detailed further in Section2.2.5) requires the use of its thickness to derive the permeability value thanks to the Darcy's equation:

Where ΔP is the pressure drop (Pa), µ is the air viscosity (Pa.s), V is the air velocity (m.s -1 ) and z is the medium thickness (m).

To be able to determine the permeability with as much precision as possible and to compare it to predicted values, it is important to determine the medium thickness accurately, under pressure conditions relevant with these to which fibrous medium are typically exposed. The thickness of the medium was thus measured by the low pressure technique developed by Zerrouati et al. [START_REF] Zerrouati | Study of Paper Transverse Shrinkage During Thermal Drying[END_REF]. Eleven samples of the medium were analyzed. These measurements were carried out on three randomly distributed zones per sample of 100 cm 2 area. The pressure exerted on the sample by the plate and the probe is fixed at 2.1 mbar; i.e. the lowest achievable value; which is relevant with the order of magnitude of pressure drop to which medium are exposed in ventilation systems. Such measurements also enable comparisons with local thickness values obtained by X-ray tomography which involved relaxed medium samples.

Scanning Electron Microscopy (SEM)

SEM pictures of the medium were obtained thanks to a JEOL JSM 7600F apparatus of the IMN Laboratory -Nantes, France. Six samples of 1.5 cm² area were analyzed. For each sample images of x250 and x500 magnification were taken in order to observe the general fiber network organization and to measure the fiber diameters. A total of 60 images were taken, involving both sides of the medium. ). Therefore, to analyze the sensitivity of the numerical pressure drop to the geometrical properties of the pleat and the permeability of its curved zone, presented in Figure 50, the numerical pressure-drop results shown are those obtained with the laminar model.

The fact that the pressure drop is not influenced by the airflow model has already been reported in the literature. For example, Feng et al. [START_REF] Feng | Assessment of various CFD models for predicting airflow and pressure drop through pleated filter system[END_REF] showed that the large eddy simulation, the detached eddy simulation, the low Re 𝑘-𝜀, the standard 𝑘-𝜀 and the v2f models can predict the pressure drop at low filtration velocity (Vfiltration = 0.072 m.s -1 ) for rectangular pleats (H = 50 mm, W = 50 mm) and low-permeability medium (B = 8.33.10 -12 m 2 ). Theron et al. [START_REF] Théron | Numerical and experimental investigations of the influence of the pleat geometry on the pressure drop and velocity field of a pleated fibrous filter[END_REF] showed that the Reynolds stress and standard 𝑘-𝜀 models give similar pressure drop to that measured experimentally for the filtration velocity range 0.14 -0.79 m.s -1 with a filter with triangular pleats (H = 40 mm, W = 23 mm) and medium permeability of 2.0.10 -9 m 2 . On the other hand, Rebaï et al. [START_REF] Rebaï | A semi-analytical model for gas flow in pleated filters[END_REF] tested the same turbulence models as Feng et al. [START_REF] Feng | Assessment of various CFD models for predicting airflow and pressure drop through pleated filter system[END_REF] and did not obtain a similar pressure drop for triangular pleats (H = 51 mm, W = 12.5 mm) and medium permeability of 3.57. 10 -10 m 2 , for Vfiltration = 0.39 m.s -1 . Tronville et Sala [START_REF] Tronville | Minimization of Resistance in Pleated-Media Air Filter Designs: Empirical and CFD Approaches[END_REF] showed that for the inlet velocity range 0.2 -1.0 m.s -1 , the Reynolds stress model (RSM) was the best of three models (RSM, standard 𝑘-𝜀 and RNG 𝑘-𝜀) for predicting the pressure drop of a mini-pleated filter (H = 25 mm, W = 3.2 mm) and a low permeability medium (B = 7.5.10 -12 m 2 ). In conclusion, it is not possible to give general recommendations for selecting a relevant airflow model, even to predict the pressure drop, as it depends on the airflow conditions and the properties of the filter and medium.

upstream of the filter, the intensity of which is dependent mostly on structural properties of the medium and geometrical properties of the filter. Thus, it is proposed in this study to evaluate the sensitivity of existing models to the structural parameters of the studied commercial fibrous medium by taking into account its bimodal fiber size distribution and in-plane porosity variation. The sensitivity of the efficiency models to velocity variations upstream of the filter pleats is also investigated considering flow field results of numerical simulation. These findings broaden the experimental range of model validation and propose recommendations for improving collection efficiency predictions.

Methodology

Most of the models that are widely used to predict the filter collection efficiency do not take into account the full range of potential collection mechanisms. Electrostatic effects, in particular, are frequently neglected due to the difficulty in estimating some of the parameters involved in these mechanisms. As a result, it was decided to use the experimental fractional efficiency data obtained with NaCl particles rather than those obtained with Silica particles.

Indeed, as previously explained in section 4. Because of the experimental particle diameter ranges tested (see Table 9), impaction and interception are supposed to be the predominant collection mechanisms. As a result, among the various models in the literature, it is decided to test the models that take into account, in addition to the diffusion mechanism, interception, and impaction. As a result, it was decided to take into account the models proposed by Stechkina et al. [START_REF] Stechkina | Studies of fibrous aersol filters-IV Calculation of aerosol deposition in model filters in the range of maximum penetration[END_REF], Kasper et al. [START_REF] Kasper | The collection efficiency of a particleloaded single filter fiber[END_REF], and Miecret and Gustavson [START_REF] Miecret | Mathematic Expression of HEPA and ULPA Filters Efficiency, Experimental Verification, Practical Alliance to New Efficiency Test Method[END_REF](see Chapter 1 -Table 7 for the equations). Table 22 summarizes the parameters used to implement the models. These models are analytical and thus doesn't contain experimental fitting coefficients like other empirical models [START_REF] Gougeon | Filtration des aérosols liquides par les filtres à fibres en régimes d'interception et d'inertie[END_REF].

Chapter 5 Influence of the clogging on the filtration performance and on the velocity profiles downstream of the filter pleats 190 The NaCl particles are not spherical; instead, they have a faced cubic center structure. However, because it is difficult to characterize the geometry of the particles finely, an equivalent spherical diameter is considered, i.e. aerodynamic, Stokes, optical, and so on (see section 3.3.1). The selection of an appropriate equivalent diameter to describe the initial fractional efficiency is determined by the physical mechanism that governs the interaction between the fibers and the particles. In the case of nanometric particles, the experimental fractional collection efficiency is frequently expressed in terms of an equivalent electrical mobility diameter since collection by diffusion mechanism is dominant [START_REF] Boudhan | Influence of air humidity on particle filtration performance of a pulse-jet bag filter[END_REF]. Impaction and interception mechanisms are intended to be the predominant collection mechanisms. The optical diameter is more representative of interception, which is a steric effect mechanism, than the other equivalent diameter, whereas a Stokes or aerodynamic diameter is more representative of impaction which is an aerodynamic mechanism. confirms the complementarily of studies at the medium scale, i.e. determining the structural properties of the medium while taking heterogeneity and fiber distribution into account, and studies of airflow at the scale of the pleat, in order to improve particle efficiencies prediction.

Summary of the first part of the Chapter

The filtration performance of the filter were evaluated in the first part of this Chapter for the two tested geometry. Submicron monodispersed NaCl articles and micron iosfine articles were used to clog the filters, resulting in a dendritic-like cake deposit in the first case and a more compact cake deposit in the second. The clogging is stopped at a pressure drop, around 400 pa, which corresponds to an early stage of surface filtration. The filter's initial fractional efficiency is determined experimentally. As a result, the capacity of the single fiber efficiency models presented in the literature is tested using the filter's average and local structural properties. The results demonstrate that an accurate description of the structural properties of the medium improves prediction of collection efficiency. Indeed, using an equivalent bi-modal diameter to describe the bi-modal fiber distribution is more accurate than the mean fiber diameter. Due to the tested airflow conditions and the filter structural properties, the interception mechanism dominate the particle collection mechanism. However, the models' sensitivity to velocity revealed that there is a threshold velocity above which impaction dominates interception and the models become sensitive to velocity. As previously demonstrated in the Chapter 4, the of the pleat. It was discovered that when the pleat width is narrow, the velocity profile decreases

The obtained results are promising in terms of the study of the airflow in the vicinity of a clogged pleated filter, as they confirm an important influence of the clogging on the velocity profiles downstream of the pleats. These findings can be used to validate the airflow model for CFD calculations, as well as provide useful data for modeling the filtration performance of a medium-efficiency pleated filter. These studies can also improve performance prediction during clogging and thus the optimization of the filter design in series configuration.

The non-uniform velocity produced by a pre-filter can have an effect on the filter's performance.

As a result, the incorporation of velocity gradients, Furthermore, local investigation of the velocity pattern downstream the clogged filter is critical for understanding and modeling nonstationary physical phenomena like particle bouncing and re-entrainment.

Chapter 3 consists in the description of the characterization of the structural properties of the medium at two different scales: (i) the properties "in average" by determining mean values of porosity, fiber diameter, thickness, and permeability, (ii) the local variations of the structural properties (porosity and fibers orientation). SEM images of both medium sides revealed a bimodal fiber sizes distribution. Mercury porosimetry and X-ray micro-tomography were used to determine the porosity of the medium. Despite the fact that these strategies are based on different principles, the results were very similar. The X-ray micro-tomography investigation of in-plane porosity variations indicated that the medium has a fairly homogeneous structure.

The anisotropic nature of the fiber arrangement was revealed by calculating the fiber orientation tensor using X-ray micro-tomography data. Most of fibers are oriented in planes perpendicular to the flow. The structural characteristics of the medium were used to evaluate the capacity of different models reported in the literature to predict its permeability. Among the models tested, the isotropic RUC model proved its efficiency to reproduce the experimental permeability of the bimodal studied medium. Whereas the fibers orientation tensor revealed that the medium is rather 3D anisotropic, this finding could be attributable to the isotropic RUC model's ability to describe the fibrous structure's slight deviation from strictly 2D oriented fibers. Furthermore, instead of using the mean fiber diameter to represent the bimodal fiber diameter distribution, equivalent diameters have been used to represent the bimodal fiber diameter distribution. Since permeability models are highly sensitive to even minor porosity variations, in-plane permeability mappings have been presented. This allows for the investigation of the effect of porosity variations on local permeability values. As fibrous medium for air filtration are very delicate, highly porous, and generally heterogeneous, properly predicting their permeability is rather problematic. Having access to such in-plane permeability variations opens up new possibilities for predicting local filter performance based on their actual properties.

In Chapter 4, investigations of local velocity variations were conducted at the scale of a clean pleats. In the case of the "wide" pleated geometry i.e. (H = 28 mm, W = 25 mm), the comparison of experimental to numerical velocity values was carried out for a transitional airflow regime approaching the filter (Reh = 6760), for which few studies have been reported in the literature.

In the case of the "narrow" pleated geometry, the airflow approaching the filter was turbulent (Reh = 14,920). Accordingly, the ability of the laminar, standard 𝑘-𝜀 and transition SST airflow CFD models to predict the pressure drop and airflow pattern in the vicinity of the filter was tested and compared. It was demonstrated that variations in pleat dimensions, as well as  Evaluate the influence of the velocity variations upstream of the filter on particle collection efficiency, for example, by incorporating velocity profiles into efficiency models.

 Extend the airflow study in the vicinity of the clogged filter to cover: various clogging levels, larger particle diameter ranges (up to nanoparticles) and flow conditions.

 Extend the field of validity of current single fiber collection efficiency model to include non-spherical particles. This method entails finely characterizing the particles so that we do not rely on a single characteristic measure (e.g., aerodynamic diameter, equivalent volume diameter, etc.).

 Improve the prediction of particle collection efficiency by using high-performance numerical codes (GeoDict for example) in combination with the real structure of fibrous medium, as determined by x-ray micro-tomography, which takes into account air flow variations in the vicinity of the pleated filter, as well as the true characteristics of the aerosols.

Annexes

Annexes

Annexes

Annex 3-2: Calculation of the fiber orientation tensor based on X-ray microtomography data

The algortihm applied by NOVITOM for the fiber orientation tensor calculation is divided into the following steps:

 Gaussian image filtering to eliminate the finest structures that are not fibers and that can distort the results.

 Calculation of the gray level gradients ∆I in each voxel in the directions X, Y and Z.

 Calculation of the structural tensor in each voxel.

 Gaussian filtering to average the structure tensor over a neighborhood around each voxel.

 Diagonalisation of 𝑇(𝑋, 𝑌, 𝑍) in each voxel.

The eigenvector associated with the lowest eigenvalue corresponds to the direction in which the gray levels vary the least, i.e. the main direction of the fiber to which the voxel belongs.

This vector is called the 3D orientation vector 𝑝 ⃗ ⃗⃗. It is a unity vector parallel to the fiber and defined in the cartisian coordinate system as follow: Some authors adopted an angular coordinated system to define the 3D fiber orientation vector p [START_REF] Soltani | 3D Fiber Orientation Characterization of Nonwoven Fabrics using X-ray Micro-Computed Tomography[END_REF]. two angles (θ, ∅) were defined to describe the special orientation. θ is the angle formed between the fiber axis and the z-axis, ∅ is the angle formed between the fiber projection on the