Marie Grégoire

Pierre Ouagne
email: pierre.ouagne@enit.fr

Benjamin Barthod-Malat

Philippe Evon

Laurent Labonne

Vincent Placet

Pierre And Barthod-Malat

Granulomé Trique

Les Proprié Té S Mé

M Gré Goire

Extraction de fibres de lin oléagineux pour des applications textiles techniques : influence des paramètres de pré-traitement sur le rendement en fibres, la répartition granulométrique et les propriétés mécaniques

Keywords: lin olé agineux, extraction de fibres, rendement d'extraction, répartition granulomé trique, proprié té s mé caniques

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INTRODUCTION

Le lin olé agineux est principalement cultivé pour ses graines qui contiennent jusqu'à 45 % d'huile végétale. L'huile de lin olé agineux est utilisé e dans de nombreux secteurs industriels. Tout d'abord, les principaux acides gras de l'huile de lin oléagineux sont l'acide α-linolénique (C18:3 ω-3) (56 %), l'acide linoléique (C18:2 n-6) (16 %) et l'acide olé ique (C18:1 n-9) (18 %). Il s'agit donc d'une importante source d'acides gras oméga-3 et omé ga-6 qui sont fortement recommandés pour la consommation humaine. D'autre part, grâce à ses qualités nourrissantes, l'huile de lin oléagineux est é galement utilisé e pour la formulation de nombreux produits cosmé tiques. Pour finir, l'huile de lin oléagineux est une huile siccative qui durcit spontanément à l'air. Elle est donc utilisé e en tant que siccatif pour les encres vé gé tales ainsi que pour la formulation des peintures à l'huile, les nettoyants pour planchers et les produits de pré servation du bois.

Les anas, la partie ligneuse des tiges de lin, peuvent ê tre extraits. Ils sont principalement utilisé s en tant que litiè res animales à cause de leur haute capacité d'absorption de l'eau. Cependant, il est raisonnable de supposer que les anas de lin peuvent é galement ê tre utilisé s en tant que charges renforcé es pour la production de composites bois-polymè re extrudables et/ou moulables par injection [START_REF] Gamon | Twin-screw extrusion impact on natural fibre morphology and material properties in poly (lactic acid) based composites[END_REF], pour la fabrication d'isolants thermiques renouvelables ou de panneaux de fibres à haute densité par pressage à chaud [START_REF] Evon | New thermal insulation fibreboards from cake generated during biorefinery of sunflower whole plant in a twin-screw extruder[END_REF][START_REF] Evon | Influence of thermo-pressing conditions on mechanical properties of biodegradable fibreboards made from a deoiled sunflower cake[END_REF], pour la production de blocs d'isolation thermique par moulage par compression [START_REF] Evon | New insulation fibreboards from sunflower cake with improved thermal and mechanical properties[END_REF] ou en tant que bio-agré gats à mé langer avec des liants miné raux pour la conception d'agrobétons [START_REF] Gazagnes | Maté riau composite de construction incorporant de la chè nevotte de chanvre[END_REF]. D'autre part, les poussières vé gé tales, une autre partie des tiges de lin, peuvent é galement ê tre potentiellement utilisé es pour le renforcement mé canique des polymères thermoplastiques dans l'industrie plastique.

La derniè re fraction vé gé tale inté ressante dans les tiges de lin olé agineux concerne les fibres. Dans une é tude publié e en 2011 par l'Agence de l'Environnement et de la Maitrise de l'Énergie [START_REF]Assessment of natural fibre availability in France[END_REF], il apparait que 11 000 ha de lin olé agineux sont cultivés en France chaque année. Il s'agit d'une petite fraction de la production mondiale qui a é té identifié e en 2009 comme étant d'environ 1,7 Mha dont 650 000 ha pour le Canada qui est le principal producteur. La quantité de paille est gé né ralement considérée comme étant d'environ 2 tonnes/ha pour le lin olé agineux. Cependant, il faut rappeler que les tiges de lin sont stoppées dans leur croissance afin d'éviter leur verse dans le champ, ce qui serait un problè me lors de la ré colte des graines. Dans leur étude rapportant l'influence du gé notype de 11 varié té s, [START_REF] Rennebaum | Fibre quality of linseed (Linum usitatissimum L.) and the assessment of genotypes for use of fibres as a byproduct[END_REF] [START_REF] Rennebaum | Fibre quality of linseed (Linum usitatissimum L.) and the assessment of genotypes for use of fibres as a byproduct[END_REF] ont montré que jusqu'à 5,5 t/ha de paille peuvent être récoltées sans compromettre la quantité de graines. Si les graines de lin olé agineux sont bien valorisé es, le reste de la plante et en particulier la paille n'est pas du tout utilisé. Il est dé chiqueté en mê me que les graines sont ré colté es. Cela explique principalement pourquoi les textiles techniques à base de lin olé agineux ne sont pas disponibles sur le marché même si les proprié té s mé caniques des fibres individuelles de lin olé agineux reporté es dans quelques é tudes [START_REF] Rennebaum | Fibre quality of linseed (Linum usitatissimum L.) and the assessment of genotypes for use of fibres as a byproduct[END_REF][START_REF] Tomljenovic | Characterisation of textile and oleaginous flax fibrous and shives material as potential reinforcement for polymer composites[END_REF] montrent qu'elles sont situées dans la gamme basse des propriétés mé caniques des fibres de lin textile. Pillin et al. [START_REF] Pillin | Could oleaginous flax fibers be used as reinforcement for polymers[END_REF] ont montré que les fibres individuelles de lin olé agineux extraites manuellement avec soin pré sentes des proprié té s en traction qui sont situé es dans la gamme des proprié té s mesuré es pour des fibres de lin textile é galement extraites manuellement, ces derniè res é tant couramment utilisé es pour la production de renforts textiles pour des applications composites structuraux ou semi-structuraux. Cependant, cette derniè re affirmation pourrait ne plus ê tre vraie en considé rant les proprié té s des fibres après la phase d'extraction des fibres des tiges.

La récolte du lin oléagineux ne permet pas à la paille d'être traité e avec les mê mes techniques que celles utilisé es pour le lin textile. Les pailles sont fauché es et directement absorbé es par la moissonneuse-batteuse qui sé pare les graines des pailles avec ses batteuses inté gré es. Les pailles sont donc soumises à des sollicitations mé caniques pendant la phase de battage. A la fin de cette derniè re, les pailles tombent de faç on ré guliè re de la moissonneuse-batteuse et forment un andain de tiges orienté es alé atoirement. Les pailles peuvent ensuite ê tre laissées dans le champ afin qu'un rouissage réalisé par les micro-organismes du sol puisse ê tre mis en place. Cette é tape de pré traitement, qui est bien connue et documenté e pour le lin textile, ne peut pas ê tre ré alisé e pour le lin olé agineux en suivant le mê me protocole puisque les pailles ne sont pas aligné es ni bien distribué es sur le sol. Pendant le rouissage, le contact avec le sol et donc les micro-organismes n'est pas le mê me pour toutes les pailles. Même s'il pourrait être possible de retourner l'andain pendant la période de rouissage, l'homogénéité du rouissage pourrait être remise en question. De plus, puisque les fibres ne sont pas alignées dans l'andain, les tiges de lin olé agineux sont conditionné es avec des orientations aléatoires en de larges balles d'environ 200 kg, et les tiges ne peuvent pas être alignées comme c'est nécessaire pour le teillage et le peignage, employé s habituellement pour sé parer les diffé rentes fractions vé gé tales de la plante pour le lin textile. Par consé quent, un dispositif «toutes fibres» doit être utilisé. Différents dispositifs, inspirés de l'industrie papetiè re, sont gé né ralement utilisé s. Cependant, ces dispositifs sont souvent trè s agressifs pour les fibres et ils peuvent mener à l'apparition de défauts tels que des dislocations dans les fibres comme c'est le cas durant l'extraction des fibres de chanvre [START_REF] Hernandez-Estrada | Assessing the susceptibility of hemp fibre to the formation of dislocations during processing[END_REF].

Le but de ce travail est d'étudier l'impact de deux paramè tres de pré traitement (le degré de rouissage et le taux d'humidité des tiges à l'entrée de la machine d'extraction des fibres) sur le rendement en fibres extraites de la paille de lin olé agineux soumise à une ré colte et à une extraction mé canisé es ainsi que sur les longueurs et les proprié té s en traction de ces fibres.

MATÉRIELS ET MÉTHODES

Maté riels

Tous les essais sont mené s en utilisant deux lots diffé rents de tiges de lin olé agineux (Linum usitatissimum L.) de varié té Everest, cultivé s dans le Sud-Ouest de la France et fournis par Ovalie Innovation (Auch, France). Le premier lot consiste en des pailles collecté es et mises sous forme de balles de 200 kg immédiatement après la récolte des graines, c'est-à -dire au dé but du mois de juillet, ce qui correspond au lot non roui (NR). Le second lot (R) consiste en des pailles rouies pendant 3 semaines aprè s la ré colte des graines. Les deux lots viennent du mê me champ.

De plus, les deux lots dé crits pré cé demment sont réhumidifiés par aspersion d'eau liquide avant l'extraction des fibres. Un troisiè me et un quatriè me lot sont alors cré é s. Il s'agit du lot non roui et humide (NRH) et du lot roui et humide (RH).

Mé thodes analytiques

Puisque le comportement mé canique des fibres lignocellulosiques et de leurs composites associé s change en fonction de la teneur en humidité [START_REF] Placet | Influence of environmental relative humidity on the tensile and rotational behaviour of hemp fibres[END_REF][START_REF] Moothoo | Vegetal fibre composites for semi-structural applications in the medical environment[END_REF], il est important de déterminer le niveau d'eau absorbé e en entré e du dispositif d'extraction des fibres pour tous les lots considérés. Il est ainsi possible de déterminer si l'humidité a une influence sur les proprié té s physiques et mé caniques é tudié es ici. Les teneurs en humidité sont dé terminé es selon la norme ISO 665:2000 [START_REF]Oilseeds -Determination of moisture and volatile matter content[END_REF].

Le dispositive d'extraction de fibres

Un équipement d'extraction «toutes fibres» Laroche

Cadette 1000 (France), situé sur la plateforme AGROMAT (Tarbes, France), la halle de transfert technologique du Laboratoire de Chimie Agro-industrielle, est utilisé pour sé parer les diffé rentes fractions vé gé tales pour tous les lots de pailles de lin olé agineux é tudié s dans cette é tude. Un sché ma du dispositif est pré senté sur la Figure 1. A la fin de chaque module, il y a é galement un cylindre perforé où la ventilation est appliqué e. Le cylindre perforé a trois fonctions: extraire la poussiè re vé gé tale du maté riau, former le mat et le transfé rer vers le module suivant ou à la sortie. Chaque ventilateur de dé poussié rage est é quipé d'un moteur ayant une vitesse de rotation maximale de 2865 rpm.

Pendant les expériences, les débits d'entrée en paille de lin olé agineux sont approximativement de 175 kg/h, ce qui correspond à une vitesse d'avancement de 3,5 m/min. La vitesse de transmission du mat du module 1 au module 2 é tait de 2,2 m/min et elle é tait de 1,5 m/min du module 2 au module 3. Pour finir, la vitesse du tapis de sortie é tait de 1,8 m/min. La vitesse de rotation des rouleaux d'extraction était de 725 rpm pour chacun des 3 modules. La vitesse de rotation des moteurs des ventilateurs de dé poussié rage é tait de 1500 rpm pour le module 1 et de 2000 rpm pour les deux modules suivants.

Les fractions vé gé tales

Tous les essais sont mené s sur les lots non rouis (NR) et rouis (R). Pour ces lots, deux expé riences sont ré alisé es, une sans ré -humidification (RS et NRS) et une autre avec réhumidification avec de l'eau liquide (RH et NRH). Chaque pré lè vement a é té effectué pendant 10 min. Ensuite, les trois fractions obtenues (mat, anas et poussiè res collecté es dans les trois modules et mé langé es en une seule fraction de poussiè res) sont pesé es. Le bilan de matiè re peut ensuite ê tre dé terminé pour chaque é chantillon. Le mat est principalement constitué de fibres mais aussi d'anas de lin piégés dans le mat. Pour chaque mat, un é chantillon de 20 g a é té pré levé pour déterminer le taux d'anas dans le mat et ainsi le taux réel de fibres. Un tamisage manuel de 2 min a é té ré alisé sur le mat afin de faire tomber les anas de lin. Ensuite, les anas ré siduels, toujours pié gé s dans le mat, sont collecté s manuellement. La figure 2 montre des photographies des trois fractions vé gé tales obtenues.

Longueur des fibres techniques

Une poigné e de faisceaux de fibres a é té extraite des diffé rents mats desquels les anas ré siduels avaient é té enlevé s et 500 faisceaux ont é té pré levé s. Chaque faisceau a é té mesuré en longueur. Pour cela, une extré mité du faisceau é tait maintenue pendant que l'autre était tirée de manière à déployer le faisceau de fibres. La longueur a é té mesuré e entre ces deux extré mité s. La moyenne arithmé tique de la longueur a ensuite ét é calculé e ainsi que l'é cart type correspondant.

Tests de traction sur fibres é lé mentaires

Pour chaque lot, entre 24 et 30 fibres é lé mentaires sont extraites manuellement des faisceaux. Dans le but de faciliter l'extraction des fibres élémentaires, les faisceaux sont plongé s pendant 20 min dans de l'eau chaude. Selon Charlet [START_REF] Charlet | Contribution à l'étude de composites unidirectionnels renforcé s par fibres de lin: relations entre la microstructure de la fibre et ses proprié té s mé caniques[END_REF], ce prétraitement n'a pas d'influence sur les proprié té s en traction (contrainte en traction et module d'Young) des fibres de lin. Les fibres é lé mentaires sont collé es sur des cadres en papier ayant une longueur de jauge de 10 mm à cause de la faible longueur des fibres é lé mentaires (entre 10 et 50 mm). À la suite de cette é tape, les fibres sont stocké es. Elles é changent de l'humidité avec l'air ambiant et ainsi sèchent jusqu'à atteindre un taux d'humidité correspondant à leur équilibre avec l'atmosphère environnant (T = 21 °C, RH = 65 %). Avant les tests en traction, le diamè tre apparent moyen de chaque fibre a é té dé terminé . Il s'agit d'une valeur moyenne de 5 mesures ré alisé es le long de la fibre é lé mentaire en utilisant un microscope optique (Olympus PMG3-F3, France). Pour calculer les contraintes en traction et les modules d'Young, les aires transversales ont é té dé terminé es pour ces diamè tres apparents moyens en considé rant que les fibres sont parfaitement cylindriques. Ces cadre's papier sont monté s dans les mors d'une machine de traction Bose (USA) Electroforce 3230 équipée d'une cellule de force de 22 N. Les proprié té s en traction ont ensuite é té dé terminé es selon la norme approprié e [START_REF]Standard Test Method for Tensile Properties of Single Textile Fibres[END_REF].

RÉSULTATS

Analyse des fractions vé gé tales

Comme expliqué dans les sections 2.3 et 2.4, le dispositif d'extraction de fibres Laroche Cadette 1000 offre la possibilité de sé parer (tout du moins en partie) les diffé rentes fractions de plantes fibreuses telles que les tiges de lin olé agineux. Les poussiè res vé gé tales et les anas sont envoyé s vers des sacs de ré cupé ration via les systè mes pneumatiques et le mat de fibres va vers le tapis de sortie du dispositif. Les quantité s des trois fractions vé gé tales extraites sont pré senté es aprè s un cycle d'extraction à travers les trois modules du dispositif dans le Table 1. Les ré sultats expé rimentaux donné s correspondent aux diffé rentes matiè res collecté es à la sortie de la machine d'extraction pour 100 kg de paille sèche de lin oléagineux en entrée. La quantité d'humidité est pour cela soustraite des masses des différents constituants. Les taux d'humidité des quatre lots considé ré s (RS, NRS, RH et NRH) sont respectivement de 8,7 %, 9,1 %, 16 % et 18,1 %. Après l'élimination complète des anas et autant que possible des poussières végétales, les quantités finales de fibres, d'anas et de poussiè res sont pré senté es dans le Table 3. Le Table 3 montre les quantité s respectives des constituants extraits des tiges de lin olé agineux pour les quatre lots é tudié s. Plus de la moitié de la masse des tiges est composée d'anas. Les valeurs sont situé es dans une gamme comprises entre 52 et 58 %, en fonction du pré traitement effectué sur les tiges. On peut considé rer que la dispersion des ré sultats peut ê tre relativement grande puisqu'ils peuvent dépendre de la façon de sé lectionner les tiges dans les balles et du pourcentage exact d'anas extraits du mat. Puisque le critère de tri est visuel, des variations peuvent survenir et quelques petits morceaux d'anas et de la poussiè re peuvent rester dans le mat, et ce mê me si c'est en faible proportion.

La proportion de fibres extraites de la tige est situé e entre 38 et 40 %. L'influence du rouissage et de l'humidité n'a pas d'influence particulière sur le rendement en fibres de lin oléagineux. Cela suggère donc que les prétraitements n'ont pas d'influence sur le degré d'extraction des fibres. Pour le lot RS, la quantité de poussiè res est plus importante que pour les autres lots. Cela peut être dû au fait que, durant l'extraction des fibres, quelques fibres et/ou anas sont coupé s et transformé s en toutes petites particules (poussiè res) alors que, pour les autres pré traitements, il reste plus de poussiè res à l'intérieur du mat de fibres ou autour des anas.

La proportion de fibres extraites des tiges de lin olé agineux (38-40 %) est beaucoup plus importante que celles indiqué es dans les quelques é tudes trouvé es dans la litté rature. Dans une étude de l'Agence de l'Environnement et de la Maitrise de l'Énergie [START_REF]Assessment of natural fibre availability in France[END_REF], les quantités respectives de fibres, d'anas et de poussiè res é taient de 25 %, 65 % et 10 %. Dans leur é tude, Rennebaum et al. [START_REF] Rennebaum | Fibre quality of linseed (Linum usitatissimum L.) and the assessment of genotypes for use of fibres as a byproduct[END_REF] ont trouvé un rendement moyen en fibres de 23 % pour onze varié té s diffé rentes avec une valeur maximale à 29 %. Nos résultats montrent donc qu'une plus grande proportion de fibres a é té extraite avec une plus faible quantité d'anas (la proportion de poussières est équivalente dans toutes les é tudes). On pourrait dire que le mat é tudié ici contient toujours quelques anas ou de la poussiè re. Or ce n'est pas le cas puisqu'à la fin du procédé les anas sont supprimés par tamisage et que les derniers anas restants sont enlevé s manuellement avec soin avec une pince à é piler. La quantité globale de fibres extraites des tiges de lin olé agineux est comparable au pourcentage de fibres extraites du lin textile (25 % de fibres longues plus 15 % de fibres courtes (é toupes), ce qui correspond à une quantité totale de fibres de 40 %) [START_REF]Assessment of natural fibre availability in France[END_REF].

Les résultats présentés dans cette étude indiquent qu'avec des tiges de varié té Everest cultivé es et ré colté es par l'entreprise Ovalie Innovation pour ses graines, le pourcentage relatif de fibres pouvant ê tre extraites du lin olé agineux est plus é levé que pour les é tudes pré cé dentes. Cette large proportion de fibres peut augmenter l'intérêt économique de la ré colte des tiges de lin au lieu de les dé chiqueter pendant la ré colte des graines.

Proprié té s des fibres

Les longueurs moyennes des fibres des quatre lots é tudié s dans ce travail sont pré senté es dans le Table 4. Les ré sultats pré senté s dans le Table 4 montrent que le pré traitement de ré -humidification a pour consé quence de mieux pré server la longueur des faisceaux de fibres. Le rouissage n'a pas d'influence sur la longueur des faisceaux de fibres. Quand les fibres sont ré -humidifié es en entré e du dispositif d'extraction, leurs longueurs demeurent plus grandes (5 cm au lieu de 4 cm sans ré-humidification). Cela est dû à la diminution de la rigidité de la fibre, ce qui entraine moins de ruptures. La limitation des défauts par l'optimisation des paramètres d'extraction ou par l'utilisation de techniques d'extraction plus douces pourrait ê tre é tudié e dans le futur. Cependant, mê me si les contraintes en traction sont affecté es par les procé dé s d'extraction, réduisant probablement les propriétés potentielles des fibres par deux, cela montre que les valeurs obtenues sont largement suffisantes pour des applications textiles techniques, pour des composites semi-structuraux ou des gé otextiles.

CONCLUSION

L'extraction des fibres des tiges de lin oléagineux et particulièrement l'influence des prétraitements tels que le rouissage ou la ré -humidification des tiges avant l'extraction sont é tudié s dans ce travail. Les ré sultats montrent que le rouissage considé ré dans cette étude n'a globalement pas d'influence sur le rendement en fibres et les proportions d'anas extraits. D'autres études pourraient être réalisées pour confirmer ces résultats avec d'autres conditions de rouissage. La ré -humidification des tiges avant l'extraction des fibres a tendance à favoriser leur extraction. La quantité de fibres extraites des tiges de lin é tudié es dans ce travail est plus importante que celles trouvé es dans des é tudes pré cé dentes. Il est difficile d'expliquer ce résultat très sé duisant (obtenu pour tous les lots é tudié s) à ce stade mais cela pourrait ê tre attribué à de trè s bonnes conditions de croissance associé es à des procédés d'extraction bien adaptés, qui maintiennent bien les fibres ensemble et ne cassent pas une grande part de celles-ci. Des é tudes supplé mentaires pourraient ê tre ré alisé es avec d'autres lots de tiges de variétés différentes car, si les résultats pré senté s dans cette é tude é taient confirmé s, cela augmenterait certainement le potentiel é conomique du lin olé agineux. Les tiges pourraient alors ê tre davantage valorisé es qu'actuellement. Le potentiel mécanique des fibres est affecté lors de leur extraction, étant réduit d'environ 50 % de son potentiel initial é valué pour des fibres extraites manuellement et avec le plus grand soin [START_REF] Pillin | Could oleaginous flax fibers be used as reinforcement for polymers[END_REF]. Né anmoins, ces proprié té s restent à un niveau qui est parfaitement acceptable pour des applications techniques telles que des composites semistructuraux ou des gé otextiles.

Figure 1 .

 1 Figure 1. Dispositif Laroche Cadette 1000 d'extraction des fibres (à partir du site de l'entreprise Laroche) Avec une largeur de 1 m, cette machine de dé chiquetage a la capacité de ré aliser à la fois le dé fibrage et le nettoyage des fibres naturelles, ainsi que la ré alisation de mats. Elle est équipée de 3 modules. A l'entrée de chacun d'entre eux, l'alimentation en matière brute est assurée par une paire de rouleaux en caoutchouc, l'un lisse et l'autre rainuré. Ensuite, chaque module dispose d'un cylindre équipé de pointes, le cylindre d'extraction (ou rouleau d'extraction de fibres). Sa vitesse de rotation est ajustable (de 750 à 1800 rpm). Sous le cylindre d'extraction, une trappe permet d'évacuer les anas de lin par gravité .A la fin de chaque module, il y a é galement un cylindre perforé où la ventilation est appliqué e. Le cylindre perforé a trois fonctions: extraire la poussiè re vé gé tale du maté riau, former le mat et le transfé rer vers le module suivant ou à la sortie. Chaque ventilateur de dé poussié rage est é quipé d'un moteur ayant une vitesse de rotation maximale de 2865 rpm.Pendant les expériences, les débits d'entrée en paille de lin olé agineux sont approximativement de 175 kg/h, ce qui correspond à une vitesse d'avancement de 3,5 m/min. La vitesse de transmission du mat du module 1 au module 2 é tait de 2,2 m/min et elle é tait de 1,5 m/min du module 2 au module

Figure 2 .

 2 Figure 2. Diffé rentes vues de fractions vé gé tales obtenues après les étapes d'extraction des fibres

Table 1 .

 1 Matiè res ré colté es en sortie de dispositifLes ré sultats pré senté s dans le Table1montrent que les quantité s d'anas extraits des lots ré-humidifié s sont plus importantes que celles des lots simplement stocké s à température ambiante. L'extraction des anas est ainsi favorisée depuis les lots ré -humidifié s. Pour les lots à humidité naturelle, l'extraction des poussiè res est favorisé e mais pas celle des anas.Sur les quatre lots, des quantités importantes d'anas font toujours partie du mat de fibres. Un tamisage manuel supplémentaire, suivi d'une récupération manuelle des derniers anas, ont é té alors ré alisé s pour quantifier les quantités d'anas restantes dans le mat. Les résultats sont pré senté s dans le Table2.Le Table2montre que pour trois des quatre lots, à savoir les lots ré -humidifié s (RH et NRH) et le lot non roui à condition de stockage ambiant (NRS), les quantité s de fibres dans le mat sont d'environ 70 %. Pour le lot RS, plus d'anas sont pié gé s dans le mat puisque 43 % de la masse du mat est constituée d'anas. Ces résultats indiquent clairement que le mat doit ê tre traité ulté rieurement pour enlever les anas ré siduels, soit en utilisant un tamisage soit en soumettant le mat à un second passage dans le dispositif d'extraction de fibres.

	RH	100	54	40,8	5,2
	NRH	100	56	38,3	5,7
	RS	100	66,6	23,6	9,8
	NRS	100	57,3	37,2	5,5

d'extraction (masses sèches données pour 100 kg de matiè re sè che en entré e) Lots Tiges (kg) Mat (kg) Anas (kg) Poussiè res (kg)

Table 2 .

 2 Fractions massiques d'anas dans le mat après une étape d'extraction

	Lots	Taux de fibres dans le mat (%)	Taux d'anas dans le mat (%)
	RH	69,3	30,7
	NRH	72,1	27,9
	RS	56,7	43,3
	NRS	68,4	31,6

Table 3 .

 3 Fractions vé gé tales aprè s la sé paration totale des constituants (ré sultats donné s en pourcentages massiques)

	Lots	Taux de fibres total (%)	Taux d'anas total (%)	Taux de poussiè res total (%)
	RH	37,5	57,6	4,9
	NRH	40,4	53,9	5,7
	RS	37,8	52,4	9,8
	NRS	39,2	55,3	5,5

Table 4 .

 4 Longueurs des faisceaux de fibres

	Lots	Longueur des faisceaux de fibres (cm)
	RH	5,3 ± 2,9
	NRH	5,1 ± 2,9
	RS	3,9 ± 2,2
	NRS	3,7 ± 1,9

Table 5 .

 5 Diamè tres et ré sistance à la traction des fibres individuellesLes ré sultats pré senté s dans le Table5montrent que les diamè tres des fibres individuelles (20-23 µm) sont bien situé s dans la gamme des valeurs observé es dans des é tudes pré cé dentes pour les lins textile et olé agineux[START_REF] Rennebaum | Fibre quality of linseed (Linum usitatissimum L.) and the assessment of genotypes for use of fibres as a byproduct[END_REF][START_REF] Tomljenovic | Characterisation of textile and oleaginous flax fibrous and shives material as potential reinforcement for polymer composites[END_REF]. Les valeurs des contraintes à la rupture sont dans une gamme comprise entre 333 et 377 MPa pour les 4 lots. S'il peut être observé que les valeurs des lots ré -humidifié s sont plus importantes, elles ne sont pas diffé rentes statistiquement (test de Student) de celles des 2 autres lots. Il est par consé quent difficile de conclure sur l'intérêt d'un prétraitement concernant les proprié té s mé caniques. Les valeurs pré senté es dans le Table5sont comparables avec celles pré senté es par Rennebaum et al. ou par Tomljenovic et Erceg[START_REF] Rennebaum | Fibre quality of linseed (Linum usitatissimum L.) and the assessment of genotypes for use of fibres as a byproduct[END_REF][START_REF] Tomljenovic | Characterisation of textile and oleaginous flax fibrous and shives material as potential reinforcement for polymer composites[END_REF] (valeurs moyennes d'environ 420 MPa). Dans ces études, ils utilisent des procédés d'extraction mécaniques pour extraire les fibres des tiges aprè s un mois de rouissage. Au contraire, Pillin et al.[START_REF] Pillin | Could oleaginous flax fibers be used as reinforcement for polymers[END_REF] extraient directement et manuellement les fibres des tiges.Leurs ré sultats montrent des contraintes à la rupture plus é levé es (863 ± 444 MPa). La dispersion des ré sultats observé s dans cette é tude est aussi importante que celle montré e dans le

	Lots	Diamè tre (µm)	Contrainte (MPa)	Contrainte maximale (MPa)	Contrainte minimale (MPa)
	RH	22,2 ± 3,7	377 ± 189	980	153
	NRH 21,3 ± 3,9	371 ± 160	774	105
	RS	23,7 ± 5,2	324 ± 110	578	115
	NRS 20,2 ± 4,3	333 ± 108	635	167

Table 5 .

 5 Les ré sultats les plus é levé s sont bien situé s dans la gamme des valeurs pré senté es par Pillin et al.[START_REF] Pillin | Could oleaginous flax fibers be used as reinforcement for polymers[END_REF] alors que les valeurs les plus faibles sont relativement basses. Cela est probablement dû au fait que les fibres qu'ils avaient utilisées n'avaient pas été soumises à une quelconque sollicitation mé canique pour les sé parer des anas et des poussiè res. Des dé fauts tels que des dislocations sont probablement introduits durant l'extraction des fibres comme cela a déjà été observé pour l'extraction des fibres de chanvre[START_REF] Hernandez-Estrada | Assessing the susceptibility of hemp fibre to the formation of dislocations during processing[END_REF] avec des mé thodes d'extraction agressives similaires. Certaines fibres sont probablement plus affectées par le processus d'extraction que d'autres. Le coefficient de variation pour les fibres réhumidifié es est de 0,50 pour le lot RH et de 0,43 pour le lot NRH. Ces coefficients sont plus importants que pour les deux autres lots (0,33 et 0,32 pour RS et NRS). Pour les lots RS et NRS, les coefficients de variation sont plus faibles que pour les lots ré -humidifié s. Cela peut suggé rer que les dé fauts d'extraction apparaissent systé matiquement pour les lots RS et NRS tandis que pour les lots RH et NRH de plus hautes valeurs de contraintes sont observé es pour certaines fibres (980 MPa pour le lot RH), suggé rant que certaines des fibres sont extraites sans cré er de dé fauts. Dans ce cas, les valeurs des contraintes maximales en traction observé es pour les lots RH et NRH sont dans la gamme des proprié té s des fibres extraites manuellement [9,16] sans le moindre dé faut.


A Berth

D Lecouturier

K Loubiere

P Dhulster

G Delaplace

Modelling and optimisation of gas-liquid mass transfer in a microporous hollow fiber membrane aerated bioreactor used to produce surfactin

Keywords:

An approach coupling DA and DOE is proposed to mode! membrane,aerated bioreactors. • Effects of liquid flow rate, volume, gas pressure and surface tension are explored. • Process relationships to predict aeration performances are proposed. • Scaling-up of such bioreactor performances are discussed, guidelines are proposed.

Introduction ABSTR ACT

Aeration by a membrane contactor is a convenient method to produce surfactin, a bacterial surfactant com pound, while avoiding foam to overflow as it is the case with most of aerated bioreactors equipped with gas sparger. This work helps improving knowledge on oxygen transfer in membrane-aerated bioreactors and opti mizing the adjustment of culture aeration performances. In this work, oxy genation of a surfactin solution was studied in a bioreactor aerated by a microporous hollow fi ber membrane contactor. First, a dimensional analysis was coupled in an innovative way with a fractional design of experiments, thus reducing gready the number of experiments. Theo, the analysis of the mode! helped to understand thoroughly the influence of the four main parameters, namely the liquid flow rate inside the fibers, the gas pressure outside the fibers, the liquid volume in the tank and the amount of surfactant in the bulk. Empirical process relationships were proposed to predict either the volumetric oxygen transfer coefficient (k L a) or the liquid-side oxygen transfer coefficient (kJ (with an average standard deviation < 11 %). The liquid flow rate, the liquid volume and the gas pressure were found to be signifi cantly influencing unlike the surface tension. The validity of the relationships with surfactin fermen tations obtained at a larger scale was demonstrated.

Membrane aerated bioreactors have been developed to run aerobic cultures in some specific processes in which air bubble sparging is in appropriate. They offer a very interesting alternative as they allow good mass transfer capacities, due to a great surface exchange, white keeping each phase separated from either si<le of the membrane, thus avoiding both mixing and foam formation. Applications have been developed in mammalian cell culture (1,2) or blood [START_REF] Catapano | Turbulent flow technique for the estimation of oxygen diffusive permeability of membranes for the oxygenation of blood and other cell suspensions[END_REF][START_REF] Shah | Bloodless evaluation of blood oxygenators[END_REF] oxygenation to prevent cell damage and gas embolism risks. More reoently, this technology has been used in wastewater treatment [START_REF] Martin | The membrane biofilm reactor (MBfR) for water and wastewater treatment: principles, applications, and recent developments[END_REF][START_REF] Kim | Preparation and application of patterned hollow-fiber membranes to membrane bioreactor for wastewater treatment[END_REF], in methane biohydroxylation [START_REF] Pen | An innovative membrane bioreactor for methane biohydroxylation[END_REF] and in surfactant production (8 10) where foam formation is not desired. For this latter application, Goutte et al. [START_REF] Coutte | Production of surfactin and fengycin by Bacillus subtilis in a bubbleless membrane bioreactor[END_REF] have tested various membranes and selected a microporous hollow fiber membrane for the production of surfactin, a powerful lipopeptidic surfactant synthetized by the aerobic bacteria Bacillus subtilis. Oxygen transfer in the culture medium during the fermentation was found to greatly impact the me tabolism of the bacterium and surfactin synthesis, but no comprehen sive description of oxygen mass transfer has been reached since, which at present prevents to improve culture oxygen feeding and to develop Abbreviati.ons: DOE, design of experiment; DO, dissolved oxygen; CMC, critical micellar concentration; OFAT, one factor at a time; MSD, mean standard deviation; TPIFB, three phase inversed fluidised bed bioreactor; CFD, computational fluid dynamics • Corresponding author. E-mail address: alexandre.berth@univ-lille.fr (A. Berth).

https://doi.org/10.1016/j.bej.2018.10.029 consistent strategies to scale up this process. Over the last years, oxygen transfer in bubbling bioreactors has been widely investigated revealing different transfer phenomena (bubble size dependency, mass transfer film theories, bubble flow regimes, local hydrodynamics,…) exhaustively described in Garcia Ochoa et al. [START_REF] Garcia-Ochoa | Bioreactor scale-up and oxygen transfer rate in microbial processes: an overview[END_REF]. These phenomena affect the efficiency of bubbling bioreactors as sparged bioreactors, bubble columns [START_REF] Merchuk | A novel approach to the correlation of mass transfer rates in bubble columns with non-Newtonian liquids[END_REF] or vertical gas liquid jets [START_REF] Lima | Effect of nozzle design on bubbly jet entrainment and oxygen transfer efficiency[END_REF]. These bioreactors are commonly evaluated and compared using the volumetric oxygen transfer coefficient, noted k L a [START_REF] Paul | Handbook of Industrial Mixing: Science and Practice[END_REF]. Such global parameter, which is the inverse of a mass transfer characteristic time, does not need a priori the knowledge of the interfacial area.

Unfortunately, much less research has been undertaken about gas transfer in membrane aerated bioreactors. The mass transfer through membranes is preferentially described using the liquid side mass transfer coefficient, noted k L , the exchange surface being in general known [START_REF] Côté | Bubble-free aeration using membranes : mass transfer analysis[END_REF]. In the case of a membrane aerated bioreactor, both parameters, k L a and k L , can be considered as relevant target variables to evaluate the aeration performances, but only k L is found in the litera ture. Hence, the comparison of the performances between a sparged bioreactor and a membrane aerated bioreactor is uneasy.

Many relations have been developed to attempt to model target variables governing oxygen transfer in membrane aerated bioreactors,

Nomenclature

C

dissolved Oxygen concentration (g.L Fig. 1. Schematic representation of bioreactor used in this study with an external hollow fiber membrane contactor.

Aeration was carried out with a 24 cm² external polysulfone hollow fiber membrane (CFP 6 D MM01 A, GE Healthcare, MA, USA) chosen from the work of Coutte et al. [START_REF] Coutte | Production of surfactin and fengycin by Bacillus subtilis in a bubbleless membrane bioreactor[END_REF] running in flow through mode to avoid water vapour accumulation [START_REF] Côté | Bubble-free aeration using membranes : mass transfer analysis[END_REF]. The membrane aerated bior eactor module contained four fibers. Each fiber length was 0.308 m long with 0.75 mm internal diameter and was composed of 0.65 μm pores diameter.

Air was used as gas phase; the associated flow rate was maintained constant (Q G = 4.2 10 -6 m 3 .s 1 ) for all experiments. Gas inlet pressure was controlled by a restriction and measured with an analogical man ometer (7833500, WIKA). The liquid flow rate was pumped with a peristaltic pump from the bioreactor and was limited below 20 mL.min 1 because equilibrium between fluids from either side of the membrane became instable above this value. A restriction controlled the liquid pressure to avoid mixing of liquid and gas phases. The liquid volume in the tank varied from 0.15 L to 0.45 L.

Model fluid used

Surfactin solutions in distilled water were used as model fluids. Surfactin is a surface active molecule that modifies surface tension. The CMC measured in our laboratory and equally found in Slivinski et al. [START_REF] Slivinski | Production of surfactin by Bacillus pumilus UFPEDA 448 in solid-state fermentation using a medium based on okara with sugarcane bagasse as a bulking agent[END_REF] was 0.12 g.L -1 .

The range of surfactin concentration was chosen to be under and above the CMC (0 1 g.L -1 ). This range of surfactin concentration al lowed us to explore a large surface tension range (from 0.072 N.m -1 to 0.027 N.m -1 ) to measure its impact on oxygen transfer. The surface tension was measured by a tensiometer Lauda TD1.

Density and viscosity were not significantly impacted by addition of surfactin when compared to pure distilled water in the range of con centrations used in this study. Zdziennicka et al. [START_REF] Zdziennicka | Volumetric properties of rhamnolipid and surfactin at different temperatures[END_REF] demonstrated that in the range of 0 50 mg.dm -3 of surfactin in distilled water, density and viscosity variation were smaller than 2% at 40 °C. Only surface tension properties of the fluid were modified.

Measurements of aeration performances

The aeration performances of membrane aerated bioreactor were evaluated in terms of k L a and k L and measured for different operating conditions

Determination of volumetric oxygen mass transfer coefficient k L a

The volumetric mass transfer coefficient k L a (where "a", the specific surface area, refers to the ratio between the membrane inner surface area to total volume of liquid) was determined by the dynamic method similarly reported in Garcia Ochoa et al. [START_REF] Garcia-Ochoa | Bioreactor scale-up and oxygen transfer rate in microbial processes: an overview[END_REF]. Briefly, after removing oxygen from the medium by flushing nitrogen (the initial level of DO was thus controlled), air was turned back on and the change in the DO concentration was monitored with an 8 mm classic polarographic DO sensor (Z010017080, MiniBio 500, Applikon). Once the concentration signal was stabilised (t = t1), oxygenation was followed for enough data to be collected (t = t 2 ) (Fig. 2).

Considering a perfectly mixed behaviour of the liquid bulk in the bioreactor, the mass balance could be written, leading to deduce the coefficient k L a by Eq. (1):

⎛ ⎝ ⎜ - - ⎞ ⎠ ⎟ = - C C C C k a t t ln • ( ) p p L ( ) * 2 ( ) * 1 2 1 G G (1)
Fig. 2. Flowchart of the experimental procedure to measure the oxygen transfer coefficients.

both in dimensional and dimensionless forms. Empirical relations linking k L or k L a to operating parameters are rarely complete and ex haustive in the literature. In particular, it appears that the impact of some key process parameters encountered in membrane contactors, such as gas pressure outside the fibers, surface tension or liquid volume in the tank are rarely considered altogether [START_REF] Côté | Bubble-free aeration using membranes : mass transfer analysis[END_REF]. Dimensionless rela tions linking the Sherwood number (in which the oxygen transfer coefficient is embedded) to dimensionless operating parameters are still rare in the literature, and are limited to the adaptation of already ex isting general relations, that are not well adapted to membrane aerated bioreactors [ 16 18]. Such relationships are not enough in the per spective of intensifying the process of surfactin production with a membrane aerated bioreactor as they do not allow to deeply char acterise the aeration performances. There is thus an obvious need to establish consistent process relationship, enabling to describe the de pendence of the oxygen transfer coefficient with the whole set of the influencing parameters.

In this respect, this paper aims to model for the first time oxygen transfer performances in a bioreactor aerated by a microporous hollow fiber membrane used to produce a biosurfactant as surfactin. The novel strategy implemented consists in coupling dimensional analysis, an ef ficient method for defining dimensionless numbers affecting a process [START_REF] Delaplace | Dimensional Analysis of Food Processes[END_REF], with ad hoc experimental programs such as a fractional factorial DOE. By this combination never used before, it is expected to quickly assess the process relationship while reducing the number of experi ments.

Materials and methods

Membrane aerated bioreactor set up

Experiments were carried out with 0.5 L bench top fermenter Minibio (Applikon Biotechnology, Schiedam, the Netherlands) at 37 °C. This fermenter consisted of a rounded bottom tank agitated with Rushton Turbine at 150 rpm with an external loop for aeration by membrane (Fig. 1). This configuration was chosen to be the closer possible as a scale down of the work of Coutte et al. [START_REF] Coutte | Production of surfactin and fengycin by Bacillus subtilis in a bubbleless membrane bioreactor[END_REF].

In Fig. 1, V L was the liquid volume inside the tank; d t , d , d if and d p were respectively tank, impeller, fiber and membrane pore diameter; C b was the clearance bottom between agitator and tank. L was the length of the fiber and n the number of fibers; g was the gravity acceleration, ρ L and μ L were respectively the liquid density and dynamic viscosity whereas ρ G , μ G were the gas density and dynamic viscosity; σ was the surface tension; D was the oxygen diffusivity in the liquid phase; N was the rotational impeller speed; Q G was the gas flow rate outside the fi bers, Q L was the total liquid flow rate inside the fibers and p G was the gas pressure outside the fibers; C * was the DO concentration and T° the temperature.

p geo {} 1 and p geo {} 2 were sets of geometrical parameters which were not explicitly listed but were required geometrical parameters for a complete description of the mixing system and membrane. Some of them were respectively length dimension (m) such as height of blades of the Rushton turbines, while others were dimensionless ( ) such as number of blades of Rushton turbines.

In the experiments, impeller speed was voluntarily maintained constant as the oxygen mass transfer is controlled by the exchange across the membrane and not by the hydrodynamics in the tank. Note that tank top was closed as in sterile culture conditions.

where C p ( ) * G is the DO concentration at saturation at 37 °C (liquid tem perature) at a gas inlet pressure = + p p p G atm relative , C 1 is the DO con centration measured at time t 1 and C 2 is the DO measured at time t 2 . C p ( ) * G was obtained by flushing air into the stirred tank until reaching the equilibrium.

The two film theory of Lewis and Whitman [START_REF] Lewis | Principles of gas absorption[END_REF] assumes that (i) the phase equilibrium is achieved at the interface, that (ii) the main re sistances are situated on each diffusion film (liquid and gas sides) and that (iii) the overall liquid side mass transfer coefficient, K L , is the result of two local mass transfer coefficients (k L and k G ). Nevertheless, in the present cases, due to the low solubility of oxygen in surfactin solutions, one can consider that all the resistance to oxygen mass transfer is in the liquid film [START_REF] Garcia-Ochoa | Bioreactor scale-up and oxygen transfer rate in microbial processes: an overview[END_REF], leading to K L ≈ k L .

According to the Henry's law (Eq. ( 2) where H is the Henry's con stant, at T = 37 °C, H = 1.1 10 -8 mol. kg 1 . Pa 1 [START_REF] Chaix | Oxygen and Carbon Dioxide Solubility and Diffusivity in Solid Food Matrices: A Review of Past and Current Knowledge[END_REF]), the amount of a gas dissolved in a liquid is proportional to the partial pressure of the gas over the liquid; C p ( ) * G was thus directly deduced from the ratio of applied gas inlet pressure p G to p atm and C p ( )

* atm = => = + C p H C C p p p • p G p p atm relative atm ( ) * ( ) * ( ) * G G atm (2)

Determination of oxygen transfer coefficient k L

The liquid side mass transfer coefficient k L was estimated as ac cording to the method suggested in Ahmed et al. [START_REF] Ahmed | Use of sealed end hollow fibers for bubbleless membrane aeration: experimental studies[END_REF]. These authors determined k L by plotting after combining and deriving mainly two equations. The first one described the oxygen transfer across hollow fiber (Eq. ( 3)) and the second one (Eq. ( 5)) dealt with oxygen mass balance in the tank (model fluid reservoir). For sake of clarity, it was chosen to briefly remember these equations and the associated main steps below. For more details concerning the assumption in the bellow derivation, readers can refer directly to Ahmed et al. [START_REF] Ahmed | Use of sealed end hollow fibers for bubbleless membrane aeration: experimental studies[END_REF].

Assuming a plug flow behaviour, the oxygen transfer across hollow fiber could be described by Eq. (3).

= - v dC dz k S C C • • ( ) L z L p z ( ) * G (3)
Where v L is the liquid superficial velocity inside one fiber

( = v Q π d 4. / L L
2 ), C z the DO concentration at a given axial position in the fiber, referenced by z, and S the exchange surface given by the manufacturer (24 cm²).

By integrating Eq. ( 3) with the boundary conditions

= C C z i n at = z i n l e t 0 (
) and = C C z o u t at = z L (outlet), Eq. ( 4) could be deduced.

= - - - C C C C exp k S L v ( ) • ( • • ) out p p in L L ( ) * ( ) * G G (4)
Where C out is the DO concentration at the end of the fiber, C in the DO concentration measured in the bulk of the tank (before the inlet) and L the fiber length (0.308 m).

The oxygen mass balance in the tank (reservoir containing liquids) could also be written by Eq. ( 5)

= - v dC dt Q C Q C • • • L in L out L in (5)
Where Q L is the liquid flow rate. By substituting C out from Eq. ( 4) in Eq. ( 5), and then integrating with the boundary condition = C C in 1 at t = t 1 and = C C in 2 at t = t 2 , Eq. ( 6) was then expressed.

⎛ ⎝ ⎜ - - ⎞ ⎠ ⎟ = ⎡ ⎣ ⎢ ⎛ ⎝ - - ⎞ ⎠ ⎤ ⎦ ⎥ - C C C C Q V exp k S L v t t ln 1 ( • • ) ( ) p p L f L L ( ) * 1 ( ) * 2 2 1 G G (6)
Where V f is the liquid volume inside one fiber.

Eq. ( 6) showed that plotting

⎜ ⎟ ⎛ ⎝ ⎞ ⎠ - - ln C C C C p G p G ( ) * 1 ( ) * 2 vs. - t t ( ) 2
1 and measuring the slope of the straight line, noted x, allowed to deduce the mass transfer coefficient k L according to Eq. ( 7)

= - ⎡ ⎣ ⎢ - ⎤ ⎦ ⎥ k v S L ln x V Q • • 1 • L L f L (7)

Set of dimensionless numbers governing the aeration performances

Dimensional analysis leads to establish the set of dimensionless numbers governing the aeration performances in a membrane aerated bioreactor. The method of modelling process by dimensional analysis has been deeply described in [START_REF] Delaplace | Dimensional Analysis of Food Processes[END_REF], and consequently will not be de tailed in the present work. Only some short comments on the six main steps will be reported. These six main steps are namely the following:

• defining the target variable, • listing the relevant independent physical quantities potentially in fluencing the target variable,

• determining the dimension of the physical quantities, • applying the Vaschy Buckingham theorem and building of di mensionless numbers,

• rearranging the dimensionless numbers,

• establishing the process relationship by making experiments.

All the relevant physical quantities involved in aerated bioreactor system operating in isothermal conditions were listed as in Hassan et al. [START_REF] Hassan | A consistent dimensional analysis of gas-liquid mass transfer in an aerated stirred tank containing purely viscous fluids with shear-thinning properties[END_REF]. This list should be completed to integrate the specificity of the aeration device here under study, i.e. the membrane. These physical quantities included:

• Boundary and initial conditions: Tank and impeller:

V d d C , , , L t i b Membrane : d L d n , , , f p
Other geometrical parameters:

p p { } , { } geo geo 1 2
Field of gravity: g

• Material parameters: ρ μ ρ μ σ D , , , , , L L G G • Process parameters: N Q Q p , , , G L G
The full list of parameters considered in the dimensional analysis was therefore reported in Eq. ( 8): ,,,,,,,,,,, { } { } , , , , , ,

= ⎛ ⎝ ⎜ ⎞ ⎠ k or k a f ρ μ d Q V p σ D g d d C p p d L n ρ μ N Q 1 ,
L L k L L f L L G t i b geo geo p G G G 1 2 , (8)
With k an integer varying from 1 to 2, depending on the target di mensional parameter chosen (k L or k L a respectively) and f k a mathe matical function.

As shown in Table 1, where the dimensional matrix associated with these physical quantities was reported, three different dimensions (namely length, mass, and time) were required to describe the dimen sions of all physical quantities. Consequently, three independent phy sical variables (named base) should be used as repeated variables [START_REF] Delaplace | Dimensional Analysis of Food Processes[END_REF].

Table 1

Dimensional matrix of physical quantities. The core matrix appeared in dark grey while the residual matrix was in light grey.

Here ρ μ , L L and d f were chosen as base. As indicated in [START_REF] Delaplace | Dimensional Analysis of Food Processes[END_REF], once the repeated variables were chosen, some matrix operations should be applied to the dimensional matrix to transform the core matrix into an identity matrix.

During this transformation, the residual matrix (in light grey in Table 1) was modified and became modified residual matrix (in light grey Table 2). The dimensionless numbers characterizing the physical phenomena could be written using the coefficients contained in the columns of the modified residual matrix.

Consequently, the determined dimensionless numbers appearing from the modified residual matrix were given in Eq. ( 9):

= ⎛ ⎝ ⎜ ⎜ ⎜ ⎞ ⎠ ⎟ ⎟ ⎟ k ρ d μ or k a ρ d μ f Q p σ D g p n N Q • • • • 2 • ; ; • ; • ; • ; • ; ; ; ; { } ; ; ; ; ; ; ; • ; • L L f L L L f L k L ρ μ d V d G ρ d μ ρ d μ ρ μ ρ d μ d d d d C d geo p d d d L d ρ ρ μ μ ρ d μ G ρ μ d 2 • • • • 1 { } • • L L f L f L f L L f L L L L f L i f t f b f geo f p f f G L G L L f L L L f 3 2 2 2 2 3 2 2 2 (9)
With f2 a mathematical function.

After rearranging initial dimensionless numbers, Eq. ( 9) could be rewritten and gave rise to certain common dimensionless numbers whose physical meaning is well established.

= = = = ⋅ = ⎛ ⎝ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ = = ⋅ = = = = ⋅ = = ⋅ = = = ⋅ = = = = = = = = = = = = ⋅ = ⋅ ⎞ ⎠ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ π Sh or π k a k a f π Q π V π p p π σ σ π π g π π π π p π π π S* n π π π π π Q * 3 Re , *
target k •d D target L L ρ •d μ k L ρ μ d n L V d G G ρ d
L f L f L L L f L f L f L L f L L L L f L i f t f b f geo f p f f G L G L L f L L L f 2 3 2 2 2 2 3 2 2 2 ( 10
)
With f3 k a mathematical function The classical dimensionless correlations describing Sherwood number in a tube side flow system involve systematically Reynolds and Schmidt numbers [START_REF] Wickramasinghe | Mass transfer in various hollow fiber geometries[END_REF]. As there is two components in the system (the reactor and the membrane), one should add the dimensionless numbers that concern a phenomenon related to the membrane. For example, the Reynolds number used in this study represents the liquid flow patterns in the fibers, that should be differentiated from the Reynolds number that represents liquid flow patterns in the tank (π ) [START_REF] Kavousi | Hydrodynamics and gas transfer performance of confined hollow fibre membrane modules with the aid of computational fluid dynamics[END_REF] . In agreement with these correlations, the present results give a more exhaustive view of the others potentially influencing causal dimensionless numbers, thus leading to a more complete description of the process.

In the experimental program investigated in this work, a unique system of membrane aerated reactor was tested as well as a unique gas phase. Moreover, N was fixed to 150 rpm. Q G at 250 L.min -1 . Consequently, some dimensionless numbers remained unchanged (π 5 to π 18 ). So, their influence of some dimensionless numbers appearing in Eq. ( 22) on target dimensional numbers could not be quantified.

From the application of the dimensional analysis to the set up stu died and for a given fluid to be aerated, four dimensionless numbers will govern the evolution of target dimensionless numbers Sh ( or k a L * ), namely:

= ⎛ ⎝ ⎜ ⎜ ⎜ = = = = = = = = ⎞ ⎠ ⎟ ⎟ ⎟ Sh or k a F π Re Q π V π p p π σ σ • , , • , • L k L ρ μ d n L V d G G ρ d μ ρ d μ * 1 • • 2 * 3 * • 4 * • L L f L f L f L L f L 3 2 2 2 (11)
With F k a mathematical function. These four dimensionless numbers revealed to be independent from each other and were the internal measures of four independent di mensional parameters (Q V p σ , , , ). 4 and 2 levels (low coded X i = 1 and high coded X i =+1). These 4 factors π π π π ( , , , )

1 2 3
4 gave rise to 4 variables X X X X ( , , , )

1 2 3
4 using Eq. ( 12).

= - X π π π Δ i i i i 0 ( 12
)
With X i the coded value of the factor i, π i the causal dimensionless number (indicated in Table 3 and established in Section 3.1). π i0 is the value of dimensionless number i at the center point and π Δ i is the step change value for dimensionless number i ( = -

+ - π π π Δ i i i 1 1
). From Table 3, one could note that the sign of σ * (also named π 4 , corresponding to the dimensionless number for surface tension) was inversed. This was voluntary done to make match low value with pure water and high value with addition of surfactant. π 4 factor was expected to influence the response factor in a binary way, meaning either nude membrane or membrane saturated with surfactant.

A fractional factorial experiment offers the advantage to reduce the number of experiments by carefully choosing the subset fraction of the experimental runs of a full factorial design; for that, it is assumed that an interaction could be replaced by an alias structure (also called gen erators). The alias structure determines which effects are confounded with each other [ 27 29]. Here the third degree interaction X X X

1 2 3
supposed negligible [START_REF] Pierlot | Design of experiments in thermal spraying: a review[END_REF][START_REF] Montgomery | Design and Analysis of Experiments[END_REF] was chosen as the generator and replaced by the fourth variable X 4 (Table 4).

The presence of a binary factor (σ ) * led to the impossibility to make central point's experiments. Central points were therefore replaced by the duplication of each experiment.

Regression models extracted of fractional factorial experiment

The classical multi regression method associated to current 2 4-1 fractional factorial design allowed to obtain a model describing the evolution of the π target in terms of coded variables as indicated in Eq. ( 13): A, α i and α ij were respectively a constant, the main variables for effects and for interaction effects, which were estimated using least

∑ ∑ = + + + = = π A αX α XX ε target i i i i j ij i j 1

Table 2

Modified residual matrix (in light grey) obtained after transformation of core matrix (in dark grey) into identity matrix. square regression on experimental data. ε was the negligible error, ac counting for interactions of three or more variables.

Once experimental results were led (presented in Table 6, exp. n°1 8), they were processed using the software Modde Pro 12 (Ume trics, Umea, Sweden) to obtain effects and statistical information (Q² and R²) that indicated the robustness of the factorial design. Q² showed an estimate of the future prediction precision: Q² should be greater than 0.5 to obtain accurate predicted values. R² highlighted the proportion of variation in the data described by the fit model and was varied be tween 0 and 1; 1 standing for a perfect fit.

2.5.2.1. Polynomial mathematical form for process relationship. By introducing Eq. ( 12) in Eq. ( 13), the coded variables X i were transformed back (Eq. ( 14) and rearranged (Eq. ( 15)) to obtain a process relationship linking causal dimensionless numbers π i (factors) to π target numbers (responses)

∑ ∑ = + - + - - + = = π A α π π π α π π π π π π ε Δ ( ) Δ •Δ ( ) ( ) target i i i i i i j ij i j i i j j 1 4 0 , 1 4 0 0 (14)
∑ ∑ = + + - - ′ = = π A α π π α π π π π π π π π Δ • Δ •Δ ( • • • ) target i i i i i j ij i j i j i j j io 1 4 , 1 4 0 (15) Where = -∑ + ∑ + ′ = = A A π π π ε • • • i α π i i j α π π i j 1 4 Δ 0 , 1 4 Δ •Δ 0 0 i i ij i j
was a constant term, in which ε the negligible error was included.

2.5.2.2. Monomial mathematical form for process relationship. As remembered above, the use of fractional factorial DOE gives traditionally rise to a polynomial relationship (Eq. ( 15)) between target numbers and causal dimensionless numbers, π i .This mathematical form of the equation is not so usual in chemical engineering. Indeed, a monomial process relationship, such as given in Eq. ( 16), is more commonly encountered for describing empirical process relationship.

∏ = = π E π • ( ) target i i γ 1 4 i ( 16
)
with E a constant and γ i the exponent coefficients.

For providing such a monomial form of process relationship, the results from the factorial DOE were examined again considering ln π ( target ) and ln π ( i ) instead of respectively π target as responses and π i as factors.

In this case, Eq. ( 13) is transformed into Eq. ( 17). The same meth odology was applied to transform back variables into factors (Eq. ( 18) [START_REF] Kavousi | Hydrodynamics and gas transfer performance of confined hollow fibre membrane modules with the aid of computational fluid dynamics[END_REF] to ( 20)).

∑ ∑

= + + + = = ln π ln B β ln X β X ln X ε ( ) ( ) ( ) ln( ) ( ) target i i i i j ij i j 1 4 , 1 4 (17)
( ) ( ) Δ ( ) ( ( ) ( )) Δ ( ). Δ ( ) ( ( ) ( ))( ( ) ( )) ( ( )) target i i i i i i j ij i j i i j j 1 4 0 , 1 4 0 0 (18) ∏ ∏ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ = ⎛ ⎝ ⎞ ⎠ ⎡ ⎣ ⎢ ⎛ ⎝ ⎞ ⎠ ⎛ ⎝ ⎞ ⎠ ⎤ ⎦ ⎥ = = π B π π β π π π π π π ε • • e x p Δ ln ( )•Δ ln ( ) •ln • ln •exp ( ) target i i io i j ij i j i io j jo 1 4 , 1 4
β i π i Δ ln ( ) (19) ∏ ∏ ⎜ ⎟ ⎜ ⎟ = ⎡ ⎣ ⎢ ⎛ ⎝ ⎞ ⎠ ⎛ ⎝ ⎞ ⎠ ⎤ ⎦ ⎥ ′ = = π B π e x p β π π π π π π • ( ) Δ ln ( )•Δ ln ( ) •ln •ln target i i i j ij i j i io j jo 1 4 , 1 4
β i π i Δ ln ( ) (20)
with Experimental run

∏ = ′ = - B B π ε • ( ) •exp( ) i i β π 1 4 0 Δ ln ( ) i i (21)
X 1: Re X V : L * 2 X p : G * 3 = X X X X 1 2 3 4 : σ * = X X X X 1 2 3 4 = X X X X 1 3 2 4 = X X X X 1 4 2 3 1 - - - - + + + 2 + - - + - - + 3 - + - + - + - 4 + + - - + - - 5 - - + + + - - 6 + - + - - + - 7 - + + - - - + 8 + + + + + + + Table 3
Variables, Factors and Levels used in the fractional factorial design.

∑ = - MSD i abs Y Y Y 1 ( ) i i predicted i exp i exp ( 22
)
Where i was the number of experiments, Y i predicted the predicted values and Y i exp the experimental values. The effects obtained from the fractional factorial design (Eq. ( 12) or Eq. ( 17), depending of the desired form of mathematical equation (polynomial or monomial) were used as first initial set of coefficients in the Microsoft Excel 2016 solver.

Results and discussion

Experimental database

The experimental dimensional results for k L a and k L obtained for the dimensional experimental conditions tested are given in Table 5. Table 6 reports the corresponding dimensionless experimental condi tions and dimensionless experimental data calculated from Eq. [START_REF] Béchet | Production of a novel mixture of mycosubtilins by mutants of Bacillus subtilis[END_REF]. Data from the DOE are in grey and numbered from 1 to 8 whereas data from the OFAT are in white and numbered from 9 to 18. Each data is the mean result of at least 3 repetitions.

It can be observed that k L a values are significantly smaller than the values usually encountered in bioreactor studies (k L a of 10 -2 s -1 [START_REF] Hassan | A consistent dimensional analysis of gas-liquid mass transfer in an aerated stirred tank containing purely viscous fluids with shear-thinning properties[END_REF][START_REF] Fahim | Oxygen transfer in three phase inverse fluidized bed bioreactor during biosurfactant production by Bacillus subtilis[END_REF]). This is due to the scale of the membrane (24 cm²) that was voluntarily chosen the smallest possible to make the research easier and initiate the scale up. Values of k L a are expected to be proportional to the exchange surface and consequently will be greater with systems in cluding larger membranes (k L a of 10 -2 s -1 for a 2.5 m² membrane [START_REF] Coutte | New integrated bioprocess for the continuous production, extraction and purification of lipopeptides produced by Bacillus subtilis in membrane bioreactor[END_REF]).

Concerning the values of k L , a comparison is made difficult because the latter parameter represents the liquid side mass transfer resistance linked to (i) liquid phase physicochemical properties and (ii) liquid side local hydrodynamic inside the fibers dependant of membrane geo metry. A large range of k L values can be found in the literature, for example : 10 -4 m.s -1 for a laboratory scale bubble column [START_REF] Painmanakul | Effect of surfactants on liquid-side mass transfer coefficients[END_REF] and 10 -2 m.s -1 in a 0.0258 m 2 dense silicone rubber hollow fiber [START_REF] Côté | Bubble-free aeration using membranes : mass transfer analysis[END_REF]. The use of fiber membrane induce a laminar flow (Re < 50) due to the thin size of the fibers.

Statistical analysis of the fractional factorial design

The statistical results of the factorial design, obtained with the Modde Pro 12 software, lead to a value of R² of 0.95 for Eq. ( 15), thus indicating that the regression model obtained by fractional factorial design describes well the data. The value of Q² is found to be 0.89 for Eq. ( 15), thus showing that the model has an accurate predictive ca pacity. The same observation can be made for Eq. ( 20) (respectively 0.95 for R² and 0.89 for Q²). These findings demonstrate that the fractional factorial design coupled with dimensional analysis is fully relevant to give process relationships.

The effects calculated using the DOE are presented in Table 7. Each parameter seems to be independent as no interaction is significantly Table 5 Experimental dimensional conditions tested, and dimensional aeration performances measured (presented in the last two columns in bold). Data provided by DOE appear in grey zone (# 1 to 8). The others (# 9 to 19) are obtained by OFAT experimental program.

In Eq. ( 20), B′ was also constant integrating the negligible error ε. β i and β ij were respectively the main variables effects and interaction effects which were estimated using least square regression on experimental data.

One factor at a time experiments and optimisation of the process relationships

Experiments with OFAT variation were also carried out to con solidate the process relationship obtained by fractional factorial design and to enlarge the validity domain of the final equation modelling oxygen transfer. For OFAT experiments, all the parameters (Q LL ,,Vp g ,σ) were kept fixed except one at a time.

A set of dimensional physical quantities used to obtain OFAT data are presented in Table 5 (# 9 19). The values of dimensionless numbers obtained with OFAT data are reported in Table 6 (# 9 19).

Afterwards, the solver of Microsoft Excel 2016 was used to fit the whole experimental data (factorial design + OFAT # 1 19) and to obtain monomial and polynomial equations describing the evolution of experimental data (π target ) with causal dimensionless numbers (optimi sation of the process relationships).

The coefficients of the optimized process relationship were achieved by minimising the MSD (Eq. ( 22)), as in Hassan et al. [START_REF] Hassan | A consistent dimensional analysis of gas-liquid mass transfer in an aerated stirred tank containing purely viscous fluids with shear-thinning properties[END_REF]: influent when comparing each effect with the confidence interval, re ported in Table 7 legend.

The comparison between the effects and confidence intervals in dicates that only three of four factors studied have a significant influ ence over the Sherwood number and on k a L * . The most influencing factors are clearly : (i) the membrane Reynolds number, (ii) the dimensionless volume of liquid in the tank and (iii) the dimensionless gas pressure because their effects are greater than the confidence intervals, and this both for monomial and polynomial re lations. The Reynolds number is commonly found in process relation ships encountered in the literature [START_REF] Catapano | Turbulent flow technique for the estimation of oxygen diffusive permeability of membranes for the oxygenation of blood and other cell suspensions[END_REF][START_REF] Gabelman | Hollow fiber membrane contactors[END_REF] whereas volume and gas pressure are usually neglected.

All these observations allow to rewrite Eqs. ( 15) and ( 20) respec tively in Eqs. ( 23) and [START_REF] Ahmed | Use of sealed end hollow fibers for bubbleless membrane aeration: experimental studies[END_REF], getting rid of poorly influent factors.

= + + + ′ π A α Re Re α V V α p p Δ Δ Δ target L L G G 1 2 * * 3 * * (23) = ′ π B R e V p • • • target β Re L G Δ ln ( ) * * β V L β p G 1 2 Δ ln ( * ) 3 Δ ln ( * ) (24)
Note that Eqs. ( 23) and ( 24) contain only three main effects sig nificantly influent and no interaction.

Replacing each constant and effect by their values in Eqs. ( 23) and ( 24) lead to Eqs. ( 25) to ( 28).

= - - Sh Re V p 1.6• •( ) •( ) L G 0.29 * 0.25 * 0.17 (26)
= - - k a Re V p 0.01• •( ) •( ) L L G * 0 . 2 9 * 0.25 * 0.17 (27)
Eqs. ( 25) to (28) are preliminary process relationships for Sherwood number and k a L * , adopting different mathematical equations (poly nomial or monomial forms).

The comparisons between experimental and predicted values are good since MSD are respectively equal to 26%, 17%, 16% and 15% for Eqs. (25) [START_REF] Net | Experimental design approach to the optimisation of hydrocarbons extraction from the sediment: method development and application[END_REF].

The methodology developed in this work has proved that the combination of dimensional analysis and DOE is an efficient method to obtain rapidly mathematical equations for process relationships (de termined by multi linear regression). Here only 8 experiments arising from a fractional factorial design have been used to obtain it.

In addition, it was established that DOE could also provide mono mial process relationships (see Section 2.5.2) without requiring any additional experimental efforts. This finding is very interesting since a lot of empirical correlations (i.e. process relationships) are in general provided in chemical engineering under this mathematical form. As underlined in [START_REF] Delaplace | Dimensional Analysis of Food Processes[END_REF], even if nothing guarantees that the process re lationship is able to adjust the "true" physical law (which is theoretical but analytically inaccessible), the monomial form is generally im plemented. This because it minimizes the number of unknown model parameters to be identified, and also because of its ability to approx imate various families of mathematical functions.

Table 6

Experimental dimensionless conditions tested, and dimensionless aeration performances obtained (presented in the last two columns in bold). Data provided by DOE appear in grey zone (# 1 to 8). The others (# 9 to 19) are obtained by OFAT experimental program.

Table 7

Effects resulting from DOE for Sh and k a L * dimensionless numbers using monomial or polynomial forms for process relationship. Confidence intervals are for (i) Sherwood polynomial ± 0.8 10 3 , (ii) Sherwood mononomial ± 0.9, (iii) k a L * polynomial ± 0. To improve the accuracy of the process relationship established by using the experimental data contained in the DOE (#1 to #8), all ex perimental results obtained in this work (#1 to #19) were used. Fitting and multi linear regression were used for this purpose (see Section 2.5) Table 8 reports the different predictive equations (Eqs. ( 29) to ( 32)) for Sherwood number and k L a* found in this study. These predictive equations, monomials and polynomials, consider the significantly in fluent causal dimensionless numbers. The MSD associated to the pre dictive equations is also given.

Each relation is valid for the domain 11 < Re < 111, 3.6 10 5 < V L * < 1. The process relationship with a monomial form describing the aeration performances often consider the Schmidt number with a fixed exponent of 0.33 [START_REF] Kavousi | Hydrodynamics and gas transfer performance of confined hollow fibre membrane modules with the aid of computational fluid dynamics[END_REF]. The Schmidt number, μ/ρD, is a pure material number, that depends only on the agitated liquid properties and on the oxygen diffusivity, D, all considered constant in this work.

= = = = = = = = = = - - - - π π π π π π π π π π 4.
Côté et al. [START_REF] Côté | Bubble-free aeration using membranes : mass transfer analysis[END_REF] obtained experimentally the correlation = Sh Re Sc 0.61 0.363 0.333 (Re varying from 0.6 to 49) to describe mass transfer of membranes using bubble free aeration. More recently, Ka vousi et al. 2016 [START_REF] Kavousi | Hydrodynamics and gas transfer performance of confined hollow fibre membrane modules with the aid of computational fluid dynamics[END_REF] proposed = Sh Re Sc 1,7 0.42 0.33 to describe, by CFD approach, gas transfer performance of confined hollow fibre membrane modules (Re varying from 1100 and 5500). So, we can assume that the Reynolds coefficient of 0.47 in Eq. ( 30) of monomial form is consistent with literature.

As expected, the analysis of Table 8 revealed that the accuracy of process relationships increases when experimental data are more nu merous. However, it could be noted that the accuracy of preliminary process relationships (respectively Eqs. ( 25) to [START_REF] Net | Experimental design approach to the optimisation of hydrocarbons extraction from the sediment: method development and application[END_REF] obtained from DOE) are very close to the "optimised" process relationships obtained by fitting all data (respectively Eqs. [START_REF] Box | Statistics for Experimenters: Design, Innovation, and Discovery[END_REF] to [START_REF] Fahim | Oxygen transfer in three phase inverse fluidized bed bioreactor during biosurfactant production by Bacillus subtilis[END_REF]), specifically for equations including k L a*. Fig. 3 compares, for monomial forms that offered best MSD, the predicted and experimental values of Sh (respectively k L a*) obtained by Eqs. ( 27) and (30) (respectively Eqs. ( 28) and ( 32)). It also illustrates the fact that a weak improvement of accuracy is obtained by optimi sation with k L a* equations whereas Sherwood equations are sig nificantly improved. The trends found at the end of the DOE were confirmed by integrating additional experiments (the OFAT values).

At last, one can conclude that Eqs. [START_REF] Pierlot | Design of experiments in thermal spraying: a review[END_REF] to [START_REF] Fahim | Oxygen transfer in three phase inverse fluidized bed bioreactor during biosurfactant production by Bacillus subtilis[END_REF] show the likeness of Sh and k L a* behaviours toward causal dimensionless numbers. Indeed, coefficients observe the same trends and are for each parameter rela tively close one of the another.

An enlarging study towards a larger amount of data with the OFAT study will allow to expand the validity domain of the relation and therefore the field of potential applications. Results confirm that monomial form often proposed in literature, as Eq. ( 16), best suits to

Analysis of each preliminary process relationships effects

The analysis of empirical correlations (Eqs. ( 25) to ( 28)) shows that the same trends were observed for the variations of k L a* and Sherwood number with causal dimensionless numbers. Dimensionless liquid flow rate and volume of liquid in the tank were found to have the greatest impact on oxygen transfer, respectively in a positive and negative way. Dimensionless gas pressure had a moderate impact and dimensionless surface tension was not significantly influent.

The liquid flow rate clearly enhance oxygen transfer as already noted by Ahmed et al. [START_REF] Ahmed | Use of sealed end hollow fibers for bubbleless membrane aeration: experimental studies[END_REF]. This can be explained by the fact that an increase in liquid velocity leads to reduce the diffusion boundary layer thickness [START_REF] De Napoli | Transport modeling of convection-enhanced hollow fiber membrane bioreactors for therapeutic applications[END_REF][START_REF] Casey | Biofilm development in a membrane-aerated biofilm reactor : effect of flow velocity on performance[END_REF]. A similar study, based on the use of a dense silicone membrane and liquid on the shell side, has reported a rise in Sherwood number from 4.7 to 19.3 when varying Reynolds from 0.6 to 49 [START_REF] Côté | Bubble-free aeration using membranes : mass transfer analysis[END_REF]. The oxygen transfer can be therefore optimised by choosing wisely the greatest liquid flow rate available with the system studied (pump ca pacities or loss of charges).

The effect of pressure was expected to be smoothed by correcting C p with each pressure, but a decrease of oxygen transfer rate is still ob served with pressure. This observation means that the latter parameter does not only influence the driving force of the mass transfer process but equally the transfer resistance. Ahmed et al. [START_REF] Ahmed | Use of sealed end hollow fibers for bubbleless membrane aeration: experimental studies[END_REF] observed an im proved transfer with high pressure in a similar system with sealed end hollow fiber but this enhancement was supposed to be an artefact due to back diffusion of nitrogen into the gas phase. In the present case, there is a priori no back diffusion as a gas flow is maintained during all the experiments. Another study with dense membrane and gas circu lated inside the fiber showed that oxygen partial pressure had no effect up to 3 bars [START_REF] Côté | Bubble-free aeration using membranes : mass transfer analysis[END_REF]. In the present study, the decrease of oxygen transfer observed could be explained by a membrane deformation when in creasing gas pressure (which circulate outside the fibers); this de formation would reduce pore size and therefore contact surface be tween gas and liquid.

On the contrary, the dimensionless surface tension does not seem to have a significant impact. The influence of this parameter has been investigated with other gas liquid contactors such as bubble columns, stirred tank bioreactors or TPIFB This result is quite uncommon in the field of gas liuid mass transfer in bubbly reactors. Indeed, when the aeration is performed by bubbling (sparging), the surface tension affects (i.e. reduces) strongly the bubble size (leading to an increase of inter facial area), and to a lesser extent, modifies the liquid side mas transfer (k L ) by changing the renewal characteristic time at the bubble interface [START_REF] Painmanakul | Effect of surfactants on liquid-side mass transfer coefficients[END_REF]. In the present work, there are no bubbles, so surfactant could only have an effect since they adhere to the membrane. A study which used the same materials except for the membrane size (2.5 m² vs 24 cm²) found a k L a decrease of 35% with a saturated membrane [START_REF] Coutte | Production of surfactin and fengycin by Bacillus subtilis in a bubbleless membrane bioreactor[END_REF]. Another study with a silicone membrane observed a 20% increase in mass transfer coefficient attributed to the membrane properties when adding the equivalent of 6 mg/L of surfactant [START_REF] Côté | Bubble-free aeration using membranes : mass transfer analysis[END_REF]. These findings would suggest that the effect of surface tension is greatly dependant of the membrane type used and thus on the interaction between surfactant and membrane material. Here, the oxygen transfer through the micro porous polysulfone membrane does not seem to be modified by the presence of surfactant. predict the oxygen transfer across a membrane.

Validity of empirical correlations for k L a and k L at larger scale

A previous work using the same bioreactor set up to produce sur factin studied the oxygen mass transfer with the same polysulfone membrane, but having a greater exchange surface of 2.5 m² (Table 9) [START_REF] Coutte | Production of surfactin and fengycin by Bacillus subtilis in a bubbleless membrane bioreactor[END_REF]. It was found a k L a of 4.3 10 -3 s 1 on used membranes (i.e. after fermentation and washing) but no study of influencing parameters has been performed.

When using the relationships found in this study (Eq. ( 32)) to pre dict the k L a, a different result was found (4.3 10 -5 s 1 instead of 4.3 10 3 s 1 ). This would mean that some parameters that have not been varied in this study and which effect are therefore included in the constant of Eqs. [START_REF] Hassan | A consistent dimensional analysis of gas-liquid mass transfer in an aerated stirred tank containing purely viscous fluids with shear-thinning properties[END_REF] to [START_REF] Fahim | Oxygen transfer in three phase inverse fluidized bed bioreactor during biosurfactant production by Bacillus subtilis[END_REF] have a non negligible influence. The dimensionless ratios integrated in the constant which are affected by the membrane scale change (24 cm² to 2,5 m²) are ratio π 7 to π 11 , π 13 , π 14 and π 18 as shown in Table 9. So, it can be supposed that the constant of process

With A'' and B'' constants for a different membrane scale, f4 and f5 two mathematical function and r the ratio of surface to volume ratios (r = 133). Then, it allows to evaluate k L a for other membrane sizes by multi plying constant of Eqs. ( 25) to [START_REF] Fahim | Oxygen transfer in three phase inverse fluidized bed bioreactor during biosurfactant production by Bacillus subtilis[END_REF] by the ratios of surface between small to larger scales. With such hypothesis, the established relation ship seems valid for the set up of Coutte et al. [START_REF] Coutte | Production of surfactin and fengycin by Bacillus subtilis in a bubbleless membrane bioreactor[END_REF], confirming the predictive interest of relations [START_REF] Hassan | A consistent dimensional analysis of gas-liquid mass transfer in an aerated stirred tank containing purely viscous fluids with shear-thinning properties[END_REF] to [START_REF] Fahim | Oxygen transfer in three phase inverse fluidized bed bioreactor during biosurfactant production by Bacillus subtilis[END_REF].

Conclusions

The aeration performances of a microporous hollow fiber membrane aerated bioreactor producing biosurfactant were investigated by cou pling dimensional analysis and DOE. Coupling fractional factorial DOE with DA led to the use of dimensionless numbers as factors (i.e. varia tion of liquid flow rate in the fiber, gas pressure outside the fiber, liquid volume and surface tension). Such a combination, not reported until now, appeared to be an efficient method to rapidly identify the most influencing parameters and to obtain, with a minimum experimental effort, empirical correlations describing the evolution of the di mensionless volumetric oxygen transfer k L a* and the Sherwood number.

The DOE allowed to obtain good preliminary process relationships, of both monomial and polynomial forms, with a good level of accuracy. Additional OFAT experiments were performed to enlarge the domain of validity leading to two empirical equations Eqs. [START_REF] Pierlot | Design of experiments in thermal spraying: a review[END_REF] and [START_REF] Fahim | Oxygen transfer in three phase inverse fluidized bed bioreactor during biosurfactant production by Bacillus subtilis[END_REF] (MSD less than 11%) describing the oxygen transfer performances. It was also proved that k L a correlation at a smaller membrane aerated bioreactor scale could be used to predict experiments at a larger scale. From these equations, it could be deduced that to improve aeration performances in a microporous hollow fiber membrane aerated bioreactor, the set of parameters should be chosen as follows: (i) use the maximum liquid flow technically possible with pump and membrane, (ii) choose the minimum gas pressure enabling to avoid membrane surface deforma tion and (iii) minimize the liquid volume in the tank. Dimensionless relations also guide the choice of the right membrane size, liquid flow rate and tank geometry when designing a membrane aerated bioreactor at a different scale for a wanted value of k L a.

Finally, beyond this study which is devoted to the study of the aeration performance in a membrane aerated bioreactor, this work provides guidelines on the way of coupling dimensional analysis and 27); full square, Eq. ( 30), empty circle [START_REF] Net | Experimental design approach to the optimisation of hydrocarbons extraction from the sediment: method development and application[END_REF], empty square [START_REF] Fahim | Oxygen transfer in three phase inverse fluidized bed bioreactor during biosurfactant production by Bacillus subtilis[END_REF]. Dotted line predicted values with ± 15% of standard deviation.

Table 9

Comparison of parameters from Coutte et al. [START_REF] Coutte | Production of surfactin and fengycin by Bacillus subtilis in a bubbleless membrane bioreactor[END_REF] and this study.

2 . 5 .

 25 Experimental program to study key factors and identify process relationship2.5.1. Fractional factorial design of experimentThe goal was here to select the important causal dimensionless numbers (in this study the factors in the factorial DOE) for the re sponses (target dimensionless numbers, Sh or k a L * ) while minimizing the number of experiments. For this purpose, a study was carried out using a fractional factorial design 2 4-1 with 4 factors π π π π (

  π target the target dimensionless number, either for Sh or k a L * .

)

  relationship for this larger scale is different. Nevertheless, let's assume that the dimensionless exchange surface ( is the main geometrical dimensionless ratio including the effect of the membrane exchange surface and that the value of the constant in Eqs.[START_REF] Hassan | A consistent dimensional analysis of gas-liquid mass transfer in an aerated stirred tank containing purely viscous fluids with shear-thinning properties[END_REF] to[START_REF] Fahim | Oxygen transfer in three phase inverse fluidized bed bioreactor during biosurfactant production by Bacillus subtilis[END_REF] is proportional to the dimensionless exchange surface.

Fig. 3 .

 3 Fig. 3. Comparison between experimental and predicted values for Sherwood number and k a L * . Predicted values have been computed by different equations. Full circle, Eq. (27); full square, Eq. (30), empty circle[START_REF] Net | Experimental design approach to the optimisation of hydrocarbons extraction from the sediment: method development and application[END_REF], empty square[START_REF] Fahim | Oxygen transfer in three phase inverse fluidized bed bioreactor during biosurfactant production by Bacillus subtilis[END_REF]. Dotted line predicted values with ± 15% of standard deviation.

Table 4

 4 Experimental runs of the fractional factorial design with Re: Reynolds number, V L * dimensionless liquid volume, p G

	Levels
	Dimensional value	Dimensionless value

* dimensionless gas pressure, σ * dimensionless surface tension.

Table 8

 8 Comparison of mathematical equations and MSD for predictive values of the Sherwood number and.k a

	L	* .

Acknowledgments

The authors thank the REALCAT platform for the use of micro bioreactors MiniBio (Applikon Biotechnology, Schiedam, the Netherlands) in this work. The REALCAT platform is benefiting from a state subsidy administrated by the French National Research Agency (ANR) within the frame of the 'Future Investments' program (PIA), with the contractual reference 'ANR 11 EQPX 0037′. The European Union, through the ERDF funding administered by the Hauts de France Region, has co financed the platform. Centrale Lille, the CNRS, and Lille 1 University as well as the Centrale Initiatives Foundation, are thanked for their financial contributions to the acquisition and implementation of the equipment of the REALCAT platform.

This work has been carried out in the framework of Alibiotech project which is financed by European Union, French State, and the French Region of Hauts de France.

This work was supported by the European Funds of INTERREG V FWVL Bioprod Project.