R ÉSUM É DE LA TH ÈSE

Étude théorique et expérimentale de l'intensification de la culture solaire de microalgues by Jack Hoeniges

Nantes Université

Laboratoire GEPEA UMR CNRS 6144 L'existence d'une crise climatique provoquée par l'homme est un fait indéniable [START_REF] Tignor | Summary for Policymakers[END_REF]. Les concentrations atmosphériques de dioxyde de carbone, de méthane et d'oxyde nitreux ont atteint des niveaux sans précédent au cours des derniers siècles en raison de l'utilisation de combustibles fossiles pour l'activité industrielle humaine [START_REF] Tignor | Summary for Policymakers[END_REF]. Il en résulte une augmentation de l'effet de serre qui a entraîné une hausse constante des températures mondiales au cours des 30 dernières années. Il en résulte une intensité et une fréquence plus élevées des phénomènes météorologiques et climatiques extrêmes, et notamment des sécheresses, des vagues de chaleur, des inondations et des incendies, ce qui a des conséquences négatives considérables sur les écosystèmes naturels, la vie humaine et les infrastructures. [START_REF] Tignor | Summary for Policymakers[END_REF]. Ces événements exacerbent également les pressions exercées sur les sources d'eau douce et les systèmes de production alimentaire [START_REF] Tignor | Summary for Policymakers[END_REF]. En outre, les conséquences de la crise climatique touchent de manière disproportionnée les populations défavorisées qui y ont le moins contribué [START_REF] Tignor | Summary for Policymakers[END_REF].

Les innovations au carrefour des systèmes alimentaires, énergétiques et hydriques seront cruciales pour faire face aux difficultés posées par la crise climatique. Dans ce contexte, les microalgues photosynthétiques ont suscité l'intérêt en tant que culture à croissance rapide pour l'alimentation humaine et animale [START_REF] Matos | Microalgae as healthy ingredients for functional food : A review[END_REF], source de biomasse pour les biocarburants neutres en carbone [START_REF] Chisti | Biodiesel from microalgae[END_REF], ingrédients dans les compléments alimentaires, les produits nutraceutiques et cosmétiques [START_REF] Jha | Microalgae-based pharmaceuticals and nutraceuticals : An emerging field with immense market potential[END_REF]. Grâce à leur croissance rapide et à leur forte teneur en nutriments et en ii protéines, la culture des microalgues utilise moins de ressources que l'agriculture conventionnelle pour une même production de biomasse [START_REF] Chew | Microalgae biorefinery : High value products perspectives[END_REF]. En outre, certaines espèces de microalgues peuvent se développer dans les eaux saumâtres ou les eaux usées [START_REF] Shen | Culture of microalga Botryococcus in livestock wastewater[END_REF][START_REF] An | Hydrocarbon production from secondarily treated piggery wastewater by the green alga Botryococcus braunii[END_REF][START_REF] Mata | Microalgae for biodiesel production and other applications : A review[END_REF]. Les microalgues métabolisent également le carbone inorganique et peuvent donc être utilisées pour fixer le CO 2 des gaz de combustion émis par les processus industriels comme les centrales électriques au charbon ou au gaz naturel [START_REF] Sydney | Potential carbon dioxide fixation by industrially important microalgae[END_REF].

Cependant, la culture de microalgues à l'échelle industrielle reste un enjeu majeur en raison, notamment, des coûts de culture élevés. Ainsi, des avancées sont nécessaires pour augmenter la productivité de la biomasse des systèmes de culture de microalgues afin de réduire le coût de la biomass produite. Cette thèse se positionne sur cet enjeu. Elle présente une évaluation théorique et expérimentale de deux méthodes d'intensification : l'intensification de la lumière reçue et l'intensification de la culture au travers de l'usage de photobioréacteur à surface spécifique éclairée élevée. La première méthode est appliquée à des bassins extérieurs de type raceway, tandis que la seconde est appliquée à des photobioréacteurs tubulaires.

Les microalgues sont souvent cultivées dans des bassins extérieurs à grande échelle, constitués d'une culture d'au moins 15 cm de profondeur, orientée selon l'axe nord/sud [START_REF] Pruvost | Large-scale production of algal biomass : photobioreactors[END_REF][START_REF] Tredici | Mass Production of Microalgae : Photobioreactors[END_REF]. Pour les bassins en plein air, de grands angles d'incidence solaire se produisent le matin et le soir, ainsi que pendant les mois d'hiver, lorsque l'angle d'élévation solaire peut être faible en fonction de la latitude à laquelle le bassin est exploité. En conséquence, le flux radiatif solaire spectral incident q in,λ (t) diminue par rapport aux situations où la lumière du soleil est presque normalement incidente sur la culture. Cela est représenté par la formule générale [START_REF] Modest | Radiative Heat Transfer[END_REF]:

q in,λ (t) = G S,λ (t) cos θ z (t) (0.1)
où G S,λ est le rayonnement solaire spectral collimaté dépendant du temps et θ z est l'angle zénithal solaire pour une heure donnée de la journée t, défini par rapport au vecteur normal pointant vers l'extérieur de la surface de culture des microalgues.

En outre, la lumière solaire délivrée à des angles obliques ne pénètre pas aussi profondément dans la culture de microalgues que la lumière normalement incidente [START_REF] Pruvost | Knowledge models for the engineering and optimization of photobioreactors[END_REF][START_REF] Souliès | Investigation and modeling of the effects of light spectrum and incident angle on the growth of Chlorella vulgaris in photobioreactors[END_REF].

iii Ainsi, une incidence non normale peut augmenter les zones sombres dans la culture où il n'y a alors pas assez de lumière pour stimuler la photosynthèse. Ce phénomène, combiné à la diminution de l'irradiation solaire le matin et le soir, a un impact négatif sur la croissance des microalgues en réduisant la quantité de lumière disponible pour les cellules en suspension.

L'objectif de la première partie de cette thèse est d'explorer l'utilisation de miroirs pour augmenter la productivité de la biomasse des raceways en augmentant le flux radiatif solaire délivré à la culture de microalgues à des moments critiques de la journée et de l'année.

La productivité quotidienne de la biomasse d'un raceway avec différentes configurations de miroirs verticaux a été prédite tout au long de l'année en utilisant des modèles validés expérimentalement, couplant le transfert de lumière et la cinétique de croissance. La performance de chaque configuration a été évaluée en termes de productivités surfacique et volumétrique et comparée au même raceway mais sans miroir. L'impact des paramètres de fonctionnement, tels que la concentration initiale de biomasse et la profondeur de culture, et des paramètres de conception, notamment la hauteur des miroirs, le rapport longueur/largeur et la localisation du bassin, a été évalué.

Quatre conceptions ont été envisagées, notamment un raceway sans miroir (Configuration A) utilisé comme référence, un bassin orienté selon l'axe nord/sud avec des miroirs sur ses côtés est et ouest (Configuration B), un bassin orienté selon l'axe est/ouest avec un seul miroir sur son côté nord (Configuration C), et un bassin rotatif à suivi solaire avec un seul miroir (Configuration D). La croissance de Chlorella vulgaris a été prédite en utilisant l'approximation à deux flux et un modèle de cinétique de croissance tenant compte de la limitation de la lumière et de l'activité respiratoire des cellules. L'utilisation de miroirs externes a permis d'améliorer les productivités volumétriques quotidiennes et les productivités de biomasse basées sur la surface de culture tout au long de l'année et jusqu'à 73 % par rapport à un raceway sans miroir. La configuration B s'est révélée plus performante que la configuration C pendant les mois d'été, alors que l'inverse était vrai pendant les mois d'hiver. En outre, la productivité de la biomasse basée sur la surface de culture des quatre configurations s'est avérée être proportionnelle au produit de la concentration initiale de la biomasse et de la profondeur de culture X 0 L. Le produit X 0 L, qui donne la productivité iv de la biomasse, dépend de la configuration et de la période de l'année. L'ajout de miroirs a permis d'obtenir la plus grande amélioration de la productivité de la biomasse pour les solaire mesuré expérimentalement était négligeable lorsque la lumière solaire directe était bloquée par des nuages ou par des arbres ou des structures à proximité. L'augmentation de la production de biomasse permise par les miroirs n'a donc pas été aussi importante que celle prévue par le modèle développé au Chapitre 2. Néanmoins, sur une période de jours clairs en août, la configuration B (orientation nord/sud du bassin avec des miroirs sur les côtés est et ouest) a augmenté le rendement total de la biomasse de 36,6 % par rapport à un raceway sans miroir. De même, sur une période de jours clairs en mars, la configuration C (orientation est/ouest du bassin avec un seul miroir sur le côté nord) a augmenté le rendement total de la biomasse de 14,9 % par rapport à un raceway sans miroir. La configuration C a également amélioré le rendement total de la biomasse de 6,1 % lors d'un essai continu en octobre. Cependant, la configuration C n'a pas permis d'améliorer le rendement de la biomasse en mai.

Dans l'ensemble, l'utilisation de miroirs s'est avérée plus efficace pour améliorer la productivité en biomasse lorsque la concentration de la culture était approximativement supérieure à 0,5 gL -1 . Les pertes par évaporation ont été observées comme étant plus importantes dans les bassins avec miroirs lorsque la culture circulait via une pompe immergée et négligeables pour la circulation via une roue à aubes. Cela peut s'expliquer par un mauvais mélange de la culture lorsque la pompe immergée, était utilisée, ce qui a conduit à une température de surface de la culture plus élevée. Ainsi, un mélange efficace des cultures peut contribuer à atténuer l'augmentation des pertes par évaporation causées par la présence des miroirs. Dans l'ensemble, il a été démontré que les bassins de type raceway équipés de miroirs augmentent la productivité de la biomasse pour des cultures suffisamment concentrées et sous un ciel clair.

En plus d'améliorer la productivité de la biomasse des bassins extérieurs par l'augmentation de la lumière incidente, nous avons cherché à améliorer la productivité de la biomasse des photobioréacteurs par des principes d'intensification reposant sur la conception. Dans des conditions de lumière limitée, la productivité volumétrique P V d'un photobioréacteur est contrôlée par la surface éclairée spécifique a light , définie par le rapport entre la surface éclairée et le volume de culture, c'est-à-dire a light = S light /V [START_REF] Pruvost | Knowledge models for the engineering and optimization of photobioreactors[END_REF]. Une augmentation de a light augvi mente la productivité volumétrique théorique maximale d'un système. Cela peut permettre de réaliser des économies d'énergie pour les processus dont la consommation d'énergie est liée au volume de culture, tels que le mélange, le pompage, le contrôle thermique et la récolte.

AlgoFilm © est un exemple d'un tel PBR intensifié qui se caractérise par une conception en couche mince tombante avec des épaisseurs de culture d'environ 2 mm (c.-à-d. a light = 500 m 2 m -3 ). Cela permet d'atteindre des productivités volumétriques élevées de 7 g L -1 day -1 avec une concentration moyenne de biomasse de 14 g L -1 en fonctionnement continu [START_REF] Pruvost | Development of a thin-film solar photobioreactor with high biomass volumetric productivity (AlgoFilm©) based on process intensification principles[END_REF].

De telles concentrations de biomasse sont nécessaires pour assurer l'atténuation totale de la lumière incidente dans le film de culture mince, et ont l'avantage supplémentaire de réduire les coûts de traitement en aval pour la centrifugation et/ou la déshydratation.

Les mêmes principes peuvent être appliqués aux photobioréacteurs tubulaires. Pour un éclairage d'un seul côté, la valeur d'a light pour un photobioréacteur tubulaire d'un hauteur H est uniquement une fonction du rayon du tube, et varie selon:

a light = πrH πr 2 H = 1 r . (0.2)
Ainsi, la productivité volumétrique maximale réalisable d'un photobioréacteur tubulaire peut être augmentée en diminuant le rayon du tube. Nous avons mené au cours de la thèse une campagne expérimentale de preuve de concept sur un photobioréacteur tubulaire baptisé ICAR et comportant des tubes minces de 4 mm de diamètre (a light = 500 m 2 m -3 ) cultivant Chlorella vulgaris. L'objectif de l'étude était d'évaluer l'adéquation de la conception des tubes minces pour la culture de microalgues à de grandes concentrations de biomasse, et d'identifier tout défi opérationnel tel que la formation de biofilms. Le coefficient de transfert de masse gaz-liquide du réacteur a également été mesuré afin d'évaluer toute limitation associée à une sur-accumulation d'oxygène. Enfin, les performances du réacteur à tubes minces ont été comparées aux performances théoriques maximales prévues par les modèles d'ingénierie.

La productivité volumétrique théorique maximale a été prédite en fonction de la densité de flux de photons incidents en utilisant les paramètres de conception du prototype actuel, c'est-à-dire la surface spécifique éclairée a light et la fraction sombre f d , ainsi que les propriétés d'absorption et les paramètres de croissance cinétique pour Chlorella vulgaris. Le vii coefficient de masse gaz-liquide (k L a) du réacteur a été mesuré en utilisant une approche de désoxygénation/ré-oxygénation pour plusieurs débits de gaz. Le plus grand k L a mesuré était de 2,2×10 -3 s -1 à un débit de gaz correspondant Q de 10 mL min -1 . Deux essais de culture en mode batch ont été réalisés avec des concentrations cibles de biomasse de 10 et 15 gL -1 .

La concentration maximale de biomasse atteinte pour chaque essai était de 9,2 et 12,0 gL -1 , respectivement. Pour les deux essais, aucun stress lumineux significatif n'a été observé. Un essai en mode semi-continu a été réalisé avec récolte de la culture et injection de milieu frais.

Le taux de croissance moyen était de 3,0 gL -1 jour -1 . avec une densité de flux photonique de 250 -2s -1 . et une concentration de biomasse variant de 9,6 à 16,3 gL -1 . Ces résultats étaient inférieurs à la productivité volumétrique théorique maximale de 5,2 gL -1 day -1 prévue à la même densité de flux photonique. Ce résultat a été attribué au faible coefficient de transfert de masse gaz-liquide, à un mauvais mélange et à des conditions d'éclairage de culture non idéales. Ainsi, les efforts pour améliorer les performances du PBR à tube mince devraient se concentrer sur l'amélioration de ces aspects du réacteur. Néanmoins, le photobioréacteur a été capable d'atteindre des concentrations de biomasse et des productivités élevées, tout en étant peu sujet à la formation de biofilms, ce qui constitue une preuve de concept réussie pour le concept ICAR de PBR intensifié à tube mince.

Après avoir démontré avec succès l'intérêt d'un PBR en tube fin, le dernier chapitre visait à développer un modèle de transfert de lumière qui permettrait d'identifier les paramètres de conception optimaux pour la distribution de la lumière dans les photobioréacteurs tubulaires. Bien que le transfert de lumière dans les photobioréacteurs tubulaires ait déjà été étudié, à notre connaissance, aucune étude n'a pris en compte l'impact de l'épaisseur de la paroi du tube sur la distribution de la lumière, et par conséquence la croissance prédite des microalgues. En effet, l'épaisseur de la paroi du tube est relativement faible pour les photobioréacteurs tubulaires typiques dont le rayon est de l'ordre du centimètre ou du décimètre.

Cependant, dans le contexte du concept de photobioréacteur à tube mince présenté au chapitre 4, l'épaisseur de la paroi du tube est beaucoup plus importante par rapport au rayon du tube extérieur. En effet, le réacteur tubulaire du chapitre 4 avait une épaisseur de paroi de 1 mm pour un rayon intérieur de 2 mm. Dans ce cas, les parois du tube interceptent un viii tiers de l'énergie radiative totale incidente sur le PBR, et la réfraction à l'intérieur du tube doit être prise en compte avec précision.

Par conséquent, l'objectif du dernière chapite était de prédire le transfert de lumière et le taux de croissance des microalgues dans un photobioréacteur tubulaire, en tenant compte de l'incidence diffuse et collimatée, des réflexions et réfractions multiples à toutes les interfaces optiques, ainsi que de l'absorption et de la diffusion anisotrope par les cellules de microalgues en suspension. L'impact de l'épaisseur de la paroi du tube, de l'angle d'incidence et de la diffusion par les microalgues sur la fraction éclairée du PBR et la croissance des microalgues a été pris en compte. Enfin, la productivité volumétrique journalière de la biomasse a été prédite pour plusieurs angles d'inclinaison et orientations différentes du PBR dans des conditions solaires correspondant à des journées ensoleillées de juin et septembre à Saint-Nazaire, en France. Le modèle et les résultats obtenus ont permis d'apporté des éléments pour optimiser la conception et le fonctionnement des PBR tubulaires.

Pour ce faire, une méthode Monte Carlo ray-tracing (MCRT) a été développée et validée par rapport à une solution analytique de l'équation de transfert radiatif utilisant la méthode à deux flux. Le modèle de transfert de lumière proposé tient compte de l'incidence diffuse et collimatée, des réflexions et réfractions multiples à toutes les interfaces optiques, ainsi que de la diffusion et de l'absorption anisotropes par les cellules de microalgues en suspension. Tout d'abord, la méthode MCRT a été utilisée pour identifier une relation de dimensionnement entre la fraction de culture éclairée γ et l'épaisseur optique, représentée par le produit de la concentration de biomasse et du rayon intérieur du tube C X r i . Ainsi, la relation déjà identifiée entre la productivité surfacique et l'épaisseur optique au Chapitre 4 a été considérée comme un résultat direct du même comportement observé pour γ. L'algorithme de ray tracing a ensuite été utilisé pour évaluer l'impact de l'épaisseur de la paroi du tube, de l'angle d'incidence et de la diffusion par les microalgues sur le taux local d'absorption des photons, la fraction éclairée et le taux de croissance volumique moyen de la culture de microalgues. On a constaté que la fraction éclairée augmentait en fonction de l'épaisseur du tube normalisée par le rayon extérieur du tube t/r o . En outre, on a constaté que γ augmentait avec l'augmentation de t/r o en raison de la réfraction par les parois du tube créant ainsi un effet de concentration ix du flux de photons incident sur la culture. Cet effet de concentration était plus prononcé à des angles d'incidence θ i plus grands. On a constaté que la diffusion par les microalgues diminuait γ. Par conséquent, le fait de négliger la diffusion dans les modèles de transfert de lumière des PBR tubulaires peut entraîner une surestimation de γ. Enfin, la productivité volumique de la biomasse a été prédite pour les conditions solaires de juin et de septembre à Saint-Nazaire pour plusieurs orientations différentes du PBR. L'intérêt de PBR à tube fin tel qu'ICAR a été démontré. En juin et en septembre, la productivité volumique de la biomasse a été de plus maximisée pour une épaisseur de tube normalisée t/r o = 0,3. En fait, l'optimisation de l'épaisseur de la paroi du tube a permis d'augmenter la productivité volumétrique des systèmes tubulaires verticaux jusqu'à 1 kg m -3 jour -1 dans les conditions solaires de septembre. Dans l'ensemble, la méthode et les résultats présentés dans la présente étude peuvent être utilisés pour optimiser le transfert de lumière dans les photobioréacteurs tubulaires par le biais de la conception (épaisseur du tube et angle d'inclinaison) et de l'exploitation (par le biais de l'épaisseur optique de la culture). En effet, l'étude a monté le rôle complexe qu'avait l'ensemble des paramètres de conception (inclinaison, diamètre et épaisseur du tube) et opérations (localisation, période de l'année) sur les performances des PBRs tubulaires.

x

To my family, and to those who I will see "later" xi The existence of a human-precipitated climate crisis is an undeniable fact [START_REF] Tignor | Summary for Policymakers[END_REF]. Atmospheric levels of carbon dioxide, methane, and nitrous oxide are at levels that are unprecedented over the past thousands of years due to the burning of fossil fuels for human industrial activity [START_REF] Tignor | Summary for Policymakers[END_REF].

This has resulted in a net positive radiative forcing effect causing global temperatures to rise consistently over the last 30 years. The result has been an increase in the intensity and frequency of extreme weather and climate events including droughts, heat waves, floods, and fire weather, yielding far-reaching negative consequences to natural ecosystems and human life and infrastructure. [START_REF] Tignor | Summary for Policymakers[END_REF]. Such events exacerbate existing strains on fresh water sources and systems of food production [START_REF] Tignor | Summary for Policymakers[END_REF]. Moreover, the consequences of the climate crisis fall disproportionately on disadvantaged populations who have done the least to contribute to it [START_REF] Tignor | Summary for Policymakers[END_REF].

Innovations at the nexus of food, energy, and water systems will be crucial to confront the difficulties posed by the climate crisis. In particular, photosynthetic microalgae have garnered interest as a fast-growing crop for human and animal feed [START_REF] Matos | Microalgae as healthy ingredients for functional food : A review[END_REF], as a source of biomass for carbon-neutral biofuels [START_REF] Chisti | Biodiesel from microalgae[END_REF], and as a valuable ingredient in food supplements, nutraceuticals, and cosmetics [START_REF] Jha | Microalgae-based pharmaceuticals and nutraceuticals : An emerging field with immense market potential[END_REF]. Thanks to their rapid growth and high nutrient and protein content, microalgae cultivation uses significantly less resources than conventional agriculture for the same quantity of biomass [START_REF] Chew | Microalgae biorefinery : High value products perspectives[END_REF]. Furthermore, a variety of microalgae species can grow in brackish or wastewater for simultaneous bioremediation and biomass production [START_REF] Shen | Culture of microalga Botryococcus in livestock wastewater[END_REF][START_REF] An | Hydrocarbon production from secondarily treated piggery wastewater by the green alga Botryococcus braunii[END_REF][START_REF] Mata | Microalgae for biodiesel production and other applications : A review[END_REF].

Moreover, microalgae can metabolize carbon in the form of CO 2 and can therefore be used 1 Table 1.1: Various microalgae species and their corresponding products (adapted from Ref.

[18]).

Algal species Products

Arthrospira [START_REF] Pulz | Valuable products from biotechnology of microalgae[END_REF]. Among the first high-value commercialized products is the carotenoid β-carotene, which is synthesized by the microalgae D. salina and sold for approximately $300-1500 per kilogram [START_REF] Borowitzka | High-value products from microalgae-their development and commercialisation[END_REF]. A wide variety of microalgae-based products are currently available for commercial use. The current worldwide market for microalgae is estimated at around $6.5 billion, 38% of which is accounted for by food supplements [START_REF] Chowdhury | Algae for Biofuels : The Third Generation of Feedstock[END_REF].

Biofuels

Biofuels are generally classified as first-, second-, or third-generation. First-generation biofuels are those derived from food crops (e.g., corn, wheat, and soybeans) [START_REF] Dahman | An Introduction to Biofuels, Foods, Livestock, and the Environment[END_REF]. Such biofuels create additional demands on resources, such as arable land, food, and water, and are therefore not considered an effective use of resources in the context of climate change [START_REF] Chowdhury | Algae for Biofuels : The Third Generation of Feedstock[END_REF].

Second-generation biofuels use non-food items, such as wood chips, waste cooking oil, and waste from food crops as feedstock [START_REF] Chowdhury | Algae for Biofuels : The Third Generation of Feedstock[END_REF]. While the use of waste material prevents additional demand for land, water, and other resources [START_REF] Dahman | An Introduction to Biofuels, Foods, Livestock, and the Environment[END_REF], no clear pathway has been established to economically produce second-generation biofuels at an industrial scale [START_REF] Sims | An overview of second generation biofuel technologies[END_REF].

Third-generation biofuels are those derived from microorganisms, the most common of which is microalgae [START_REF] Chowdhury | Algae for Biofuels : The Third Generation of Feedstock[END_REF]. Microalgae cultivation takes place in open ponds or closed photobioreactors, and does not require arable land. Furthermore, microalgae grow relatively quickly, and are vastly superior to traditional sources of biomass in terms of oil yield per acre, as indicated in Figure 1.2 [START_REF] Chisti | Biodiesel from microalgae[END_REF]. Despite the advantages afforded by third-generation biofuels, algae-based biodiesel remains prohibitively expensive at an estimated $18.63 per gallon [START_REF] Anl | Renewable diesel from algal lipids : An integrated baseline for cost, emissions, and resource potential from a harmonized model[END_REF]. Since cultivation accounts for a large portion of the production cost, improving the productivity of microalgae cultivation systems is essential to make algae-based biofuels more economically attractive [START_REF] Chowdhury | Algae for Biofuels : The Third Generation of Feedstock[END_REF][START_REF] Anl | Renewable diesel from algal lipids : An integrated baseline for cost, emissions, and resource potential from a harmonized model[END_REF].

Photosynthesis

Microalgae are photoautotrophic organisms that rely on photosynthesis for energy. Photosynthesis is the main energy vector by which light energy is harvested and stored to be used in the previously discussed applications of microalgae-based biomass and high-value molecules. Photosynthesis in green algae and other plants can be described by the following simplified formula [START_REF] Ke | Photosynthesis : Photobiochemistry and Photobiophysics[END_REF]

CO 2 + 2 H 2 O light --→ (CH 2 O) + H 2 O + O 2 . (1.1)
Photosynthesis is a biological process that takes place within the cell chloroplast wherein light-driven reactions enable the transformation of carbon dioxide and water into oxygen and organic molecules such as sugars [START_REF] Reece | [END_REF]. A simplified schematic is shown in Figure 1.3.

Light-driven reactions

The light-driven reactions take place in the thylakoid and transform solar energy into chemical energy in the form of ATP (adenosine triphosphate) and NADPH (the reduced state of nicotinamide adenine dinucleotide phosphate or NADP + ) [START_REF] Reece | [END_REF]. This process takes place via the so-called Z-scheme of photosynthesis, which is shown in Figure 1.4. Upon absorption of a photon, an electron of the pigment molecules (typically in the form of cholophyll a) jumps to an excited state before falling back to the ground state, sending the electron of an adjacent molecule to the excited state in the process. Carotenoids are also present as an accessory lightharvesting pigment [START_REF] Ke | Photosynthesis : Photobiochemistry and Photobiophysics[END_REF]. This energy propagates through the outer or peripheral antenna to the core or inner antenna before reaching the reaction center [START_REF] Ke | Photosynthesis : Photobiochemistry and Photobiophysics[END_REF]. Here, a pair of chlorophyll molecules, known as P700 and P680 for photosystems I and II, respectively, are oxidized when the excited electron reduces the primary acceptor in the reaction center. Then, in the case of photosystem II, the oxidized form of P680 becomes a powerful oxiziding "hole" which drives the splitting of water into two protons (H 2 ) and an oxygen atom [START_REF] Barber | Photosystem ii : The water-splitting enzyme of photosynthesis[END_REF]. The liberated protons are released into the the thylakoid space and contribute to a gradient in H + concentration across the thylakoid membrane [START_REF] Reece | [END_REF]. This gradient drives the synthesis of ATP [START_REF] Ke | Photosynthesis : Photobiochemistry and Photobiophysics[END_REF]. As the electron falls to a lower energy state along the electron transport chain, some of its energy is used to produce additional protons that are pumped into the thylakoid space, further contributing to the H + concentration used for ATP synthesis [START_REF] Reece | [END_REF].

Figure 1.4: A schematic illustration of the Z-scheme of photosynthesis [START_REF] Reece | [END_REF].

At the same time, light energy harvested from photosystem I's pigments molecules is used to excite an electron at the P700 chlorophyll pair in the reaction center [START_REF] Reece | [END_REF]. The oxidizing agent P700 + is then reduced by an electron arriving at the end of the electron transport chain [START_REF] Reece | [END_REF]. The electron sent to the primary acceptor from P700 is then taken along a second electron transport chain where it is used along with NADP + and H + from the stroma to produce NADPH via an NADP + reductase [START_REF] Reece | [END_REF]. This process also contributes to the H + concentration gradient across the thylakoid membrane by reducing H + concentration in the stroma [START_REF] Reece | [END_REF].

Calvin cycle

The Calvin cycle makes use of the chemical energy stored within the two compounds produced by the light-driven reactions, namely NADPH and ATP, to reduce carbon dioxide where P i denotes an inorganic phosphate. The Calvin cycle takes place in three steps (i) carbon fixation, (ii) reduction, and (iii) regeneration of the CO 2 acceptor [START_REF] Reece | [END_REF]. A schematic illustration of six turns of the Calvin cycle is shown in Figure 1.5. In the carbon fixation step, CO 2 attaches to the acceptor in the form of ribulose bisphosphate (RuBP) [START_REF] Reece | [END_REF]. The resulting molecule, referred to as Rubisco, readily splits into two molecules of 3-phosphoglycerate (PGA). These molecules then receive a phosphate group from the ATP supplied by the light-driven reactions, yielding 1,3-bisphosphoglycerate (BPG). In the reduction step, 1,3bisphosphoglycerate is reduced by NADPH, losing an inorganic phosphate in the process and becoming glyceraldehyde 3-phosphate (G3P) a three-carbon sugar. In the case of Equation (1.2), which represents six turns of the Calvin cycle, 12 molecules of G3P will be produced. Of these molecules, 10 will be used to regenerate the six molecules of RuBP needed to complete the cycle. This regeneration of RuBP consumes an additional six molecules of ATP. The remaining two molecules of G3P are then used in the cell metabolic pathways to produce glucose. Thus, six molecules of CO 2 are consumed to produce one molecule of glucose.

(

Photorespiration

Photorespiration is a competing process to photosynthesis, wherein O 2 is consumed and CO 2 is produced along with serine and ammonia [START_REF] Reece | [END_REF][START_REF] Masojídek | Photosynthesis in Microalgae[END_REF]. This process is energy-intensive Figure 1.5: Overview of the Calvin cycle for the formation of glucose [START_REF] Nave | The Calvin Cycle -HyperPhysics[END_REF].

and consumes ATP in contrast to photosynthesis where it is produced [START_REF] Reece | [END_REF]. Photorespiration is stimulated by a high dissolved oxygen concentration relative to the the dissolved CO 2 concentration [START_REF] Masojídek | Photosynthesis in Microalgae[END_REF]. Thus, to limit photorespriation and favor photosynthetic growth, microalgae cultures are cultivated at high levels of CO 2 /O 2 ratio [START_REF] Masojídek | Photosynthesis in Microalgae[END_REF]. This can be achieved by enriching the culture medium through direct injection of gaseous CO 2 . For concentrated microalgae cultures with high photosynthetic activity, high rates of oxygen production can shift the reaction equilibrium towards photorespiration, reducing the culture growth rate [START_REF] Pruvost | Development of a thin-film solar photobioreactor with high biomass volumetric productivity (AlgoFilm©) based on process intensification principles[END_REF]. In this case, strategies to degas the dissolved oxygen may be required, such as replacing the air injected for agitation with N 2 .

Specific absorption cross-section,

𝑨 𝒂𝒃𝒔, 𝝀 (m 2 /mg)
Wavelength, λ (nm)

Pigments

Solar energy required for photosynthesis is harvested by pigment molecules. In particular, Chlorophyll a is the main pigment featured in green microalgae, with accessory pigments such as chlorophyll b, c, and photosynthetic carotenoids also present to broaden the absorption spectrum of the microalgae [START_REF] Ke | Photosynthesis : Photobiochemistry and Photobiophysics[END_REF]. The absorption spectrum of a variety of pigments has been cataloged and is plotted in Figure 1.6 [START_REF] Bidigare | In-vivo absorption properties of algal pigments[END_REF]. Notably, the low absorption cross-section A abs,λ of chlorophyll a from λ = 500 -550 nm is responsible for the green color commonly seen in microalgae.

Light transfer 1.3.1 Radiative transfer equation

To fully define the local availability of photons to microalgae cells for driving photosynthesis, the intensity field within the culture must be known. For the purposes of modeling light transfer, the microalgae culture can be considered as a homogeneous absorbing, scattering, and non-emitting suspension. Then, at a given location r in the culture, the spectral radiation intensity I λ (r, ŝ), in W m -2 sr -1 nm -1 , of wavelength λ and along the direction ŝ must satisfy the integro-differential radiative transfer equation originally developed by Chandrasekhar [START_REF] Chandrasekhar | Radiative Transfer[END_REF] given by [START_REF] Pilon | Radiation transfer in photobiological carbon dioxide fixation and fuel production by microalgae[END_REF] ŝ

• ∇I λ (r, ŝ) = -κ λ I λ (r, ŝ) -σ s,λ I λ (r, ŝ) + σ s,λ 4π 4π I λ (r, ŝi )Φ λ (ŝ i , ŝ)dΩ i . (1.3)
κ λ = C abs,λ N = A abs,λ C X (1.4)
where C abs is the cell absorption cross-section in m 2 , N is the number of cells per unit volume, A abs is the mass-specific absorption cross-section in m 2 kg -1 and C X is the biomass concentration in kg m -3 .

Similarly, the second term represents the attenuation due to scattering of radiative energy out of the direction ŝ. Here, σ s,λ represents the effective spectral scattering coefficient, in m -1 of the microalgae suspension given by

σ λ = C sca,λ N = S sca,λ C X (1.5)
where C sca and S sca are the absorption cross-section and mass-specific absorption cross section, respectively.

The scattering phase function Φ λ (ŝ, ŝi ) gives the probability that light propagating along the ŝi direction in the dΩ i solid angle will be scattered into the current solid angle dΩ along Often, the Henyey-Greenstein approximation is used to express the scattering phase function Φ λ (ŝ, ŝi ) thanks to its relative simplicity. This approach is appropriate as long as the scattering particles considered are nondielectric, as is the case for microalgae. In this case the Henyey-Greenstein formulation has been shown to yield very accurate results [START_REF] Van De Hulst | Asymptotic fitting, a method for solving anisotropic transfer problems in thick layers[END_REF]. The axisymmetric Henyey-Greenstein formulation of the scattering phase function is given by [START_REF] Henyey | Diffuse radiation in the galaxy[END_REF] Φ HG,λ

(Θ) = 1 -g 2 λ [1 + g 2 λ -2g λ cos(Θ)] 3/2 .
(1.7)

Here, the scattering angle between ŝ and ŝi is represented by Θ and g λ is the so-called spectral Henyey-Greenstein asymmetry factor which is defined by the mean cosine of the scattering phase function [START_REF] Modest | Radiative Heat Transfer[END_REF][START_REF] Henyey | Diffuse radiation in the galaxy[END_REF][START_REF] Pilon | Interaction between light and photosynthetic microorganisms[END_REF] i.e.,

g λ = 1 4π 4π Φ λ (Θ) cos(Θ)dΩ = 1 2 π 0 Φ λ (Θ) cos(Θ) sin(Θ)dΘ. (1.8)
An asymmetry factor of g λ = 1.0 corresponds to pure forward scattering. In other words, a "ray" of radiative energy propagating through a suspension with g λ = 1.0 will experience no change in direction when scattered. On the other hand, an asymmetry factor of g λ = -1.0 corresponds to the limiting case of pure backward scattering and g λ = 0 corresponds to the case of isotropic scattering. Similarly, a back-scattering ratio which represents the fraction of scattered energy with a scattering angle Θ > π/2 can be expressed as [START_REF] Pottier | A fully predictive model for one-dimensional light attenuation by Chlamydomonas reinhardtii in a torus photobioreactor[END_REF] b

λ = 1 2 π π/2
Φ(Θ) sin(Θ)dΘ.

(1.9)

Here, the back-scattering ratio b λ is equal to 0 for pure forward scattering, 0.5 for isotropic scattering, and 1 for pure back-scattering. Microalgae cells tend to be strongly forward scattering in the case of visible radiation (i.e., λ = 400 -700 nm) since the cell radius is much larger than the wavelength λ [START_REF] Van De | Light Scattering by Small Particles[END_REF]. Then the values for g λ and b λ tend to be near unity and zero, respectively.

Two-flux approximation

The two-flux approximation can be used to derive an analytical solution to the onedimensional RTE for flat plate PBRs or raceway ponds [START_REF] Pottier | A fully predictive model for one-dimensional light attenuation by Chlamydomonas reinhardtii in a torus photobioreactor[END_REF][START_REF] Lee | Design tool and guidelines for outdoor photobioreactors[END_REF]. This approximation has been used extensively [START_REF] Pruvost | Knowledge models for the engineering and optimization of photobioreactors[END_REF][START_REF] Pottier | A fully predictive model for one-dimensional light attenuation by Chlamydomonas reinhardtii in a torus photobioreactor[END_REF][START_REF] Lee | Design tool and guidelines for outdoor photobioreactors[END_REF][START_REF] Cornet | A structured model for simulation of cultures of the cyanobacterium Spirulina platensis in photobioreactors : I. Coupling between light transfer and growth kinetics[END_REF][START_REF] Cornet | A simplified monodimensional approach for modeling coupling between radiant light transfer and growth kinetics in photobioreactors[END_REF][START_REF] Takache | Experimental and theoretical assessment of maximum productivities for the microalgae Chlamydomonas reinhardtii in two different geometries of photobioreactors[END_REF][START_REF] Takache | Kinetic modeling of the photosynthetic growth of Chlamydomonas reinhardtii in a photobioreactor[END_REF] and has been successfully validated against results from a 3D RTE solver for ponds and flat plate PBRs [START_REF] Lee | Design tool and guidelines for outdoor photobioreactors[END_REF]. It consists of separating the radiative intensity incident upon a microalgae cell into two parts, namely I + λ and I - λ , which represent the radiative intensity scattered into the forward (0 < Θ < π/2) and backward (π/2 < Θ < π) hemispheres, respectively, such that the local fluence rate G λ (z) is given by [START_REF] Artu | Etude et optimisation de la culture de microalgues en photobioréacteurs solaires[END_REF]

G λ (z) = 4π I λ dΩ = I + λ + I - λ .
(1.10)

Then, I + λ and I - λ can be found by resolving the one-dimensional radiative transfer equation via a system of differential equations, i.e., [START_REF] Cornet | A simplified monodimensional approach for modeling coupling between radiant light transfer and growth kinetics in photobioreactors[END_REF] dI

+ λ dz = -A abs,λ C X I + λ -S sca,λ b λ C X (I + λ + I - λ ) (1.11) dI - λ dz = -A abs,λ C X I - λ -S sca,λ b λ C X (I - λ + I + λ ).
(1.12)

In the case of a planar microalgae culture, typically found in flat-plate photobioreactors or raceway ponds, with a depth L and back wall of spectral reflectivity ρ λ subjected to a normally incident, collimated radiative flux q in,λ , the system boundary conditions are given by [START_REF] Cornet | A simplified monodimensional approach for modeling coupling between radiant light transfer and growth kinetics in photobioreactors[END_REF]

]      z = 0, I + λ = q in,λ z = L, I - λ = ρ λ I + λ .
(1.13)

Then, the local photon spectral flux G λ (z) can be expressed as [START_REF] Pottier | A fully predictive model for one-dimensional light attenuation by Chlamydomonas reinhardtii in a torus photobioreactor[END_REF] G λ (z)

q in,λ = [ρ λ (1 + α λ )e -δ λ L -(1 -α λ )e -δ λ L ]e δ λ z + [(1 + α λ )e δ λ L -ρ λ (1 -α λ )e δ λ L ]e -δ λ z (1 + α λ ) 2 e δ λ L -(1 -α λ ) 2 e -δ λ L -ρ λ (1 -α 2 λ )e δ λ L + ρ λ (1 -α 2 λ )e -δ λ L . (1.14)
Here, δ λ and α λ are given by

α λ = A abs,λ A abs,λ + 2b λ S sca,λ and δ λ = C X A abs,λ (A abs,λ + 2b λ S sca,λ ). (1.15)
For the case of strongly forward scattering microalgae, the backward scattering fraction b λ can be assumed to be zero, and α λ 1. Then, the two flux approximation for a planar microalgae culture under normal incidence with a reflecting back wall can be simplified to [START_REF] Lee | Design tool and guidelines for outdoor photobioreactors[END_REF][START_REF] Kandilian | Simple method for measuring the spectral absorption cross-section of microalgae[END_REF] G λ (z) = q in,λ e -A abs,λ C X z + ρ λ q in,λ e -A abs,λ C X (2L-z) .

(1.16)

This approach has been found to yield results that are in good agreement with the classic two-flux approximation presented in Equation (1.14) [START_REF] Lee | Design tool and guidelines for outdoor photobioreactors[END_REF].

Microalgae radiative properties

Accurate predictions of light transfer within photobioreactors requires knowledge of the radiative properties of the specific species being cultivated [START_REF] Dauchet | Photobioreactor Modeling and Radiative Transfer Analysis for Engineering Purposes[END_REF]. Due to their impressive biodiversity, microalgae feature a variety of shapes and colors which causes their radiative properties to widely depending on the species [START_REF] Pilon | Interaction between light and photosynthetic microorganisms[END_REF]. Moreover, radiative properties of a given microalgae species can depend on a variety of factors, including cell size [START_REF] Van De Hulst | Asymptotic fitting, a method for solving anisotropic transfer problems in thick layers[END_REF], pigment content [START_REF] Pottier | A fully predictive model for one-dimensional light attenuation by Chlamydomonas reinhardtii in a torus photobioreactor[END_REF], lipid content [START_REF] Kandilian | Influence of light absorption rate by Nannochloropsis oculata on triglyceride production during nitrogen starvation[END_REF], genetic mutation [START_REF] Berberoglu | Radiation characteristics of Chlamydomonas reinhardtii CC125 and its truncated chlorophyll antenna transformants tla1, tlaX and tla1-CW+[END_REF], and the arrangement of cells into colonies [START_REF] Kandilian | Absorption and scattering by fractal aggregates and by their equivalent coated spheres[END_REF][START_REF] Hoeniges | Effect of colony formation on light absorption by Botryococcus braunii[END_REF]. An example of the spectral absorption and scattering cross-sections A abs,λ and S sca,λ , respectively, and the back-scattering fraction b λ of the microalgae species Chlorella vulgaris is shown in Figure 1.8. Broadly speaking, microalgae can be treated as spherical, homogeneous [START_REF] Bhowmik | Can spherical eukaryotic microalgae cells be treated as optically homogeneous ?[END_REF], optically soft particles, with radii on the order of 1 -10 µm. As mentioned previously, microalgae can be considered much larger than the incident photosynthetically active radiation and thus tend to be strongly forwards scattering [START_REF] Van De Hulst | Asymptotic fitting, a method for solving anisotropic transfer problems in thick layers[END_REF], exhibiting back-scattering fractions b λ less than 0.01 as shown in Figure 1.8c.

The determination of the absorption and scattering cross-sections of microalgae in suspension has been performed experimentally [START_REF] Kandilian | Simple method for measuring the spectral absorption cross-section of microalgae[END_REF][START_REF] Berberoglu | Radiation characteristics of Chlamydomonas reinhardtii CC125 and its truncated chlorophyll antenna transformants tla1, tlaX and tla1-CW+[END_REF][START_REF] Jonasz | Light Scattering by Particles in Water : Theoretical and Experimental Foundations[END_REF][START_REF] Bricaud | Optical properties of diverse phytoplanktonic species : Experimental results and theoretical interpretation[END_REF][START_REF] Berberoglu | Experimental measurements of the radiation characteristics of Anabaena variabilis ATCC 29413-U and Rhodobacter sphaeroides ATCC 49419[END_REF][START_REF] Lee | Spectral optical properties of selected photosynthetic microalgae producing biofuels[END_REF][START_REF] Ma | GPU-accelerated inverse identification of radiative properties of particle suspensions in liquid by the Monte Carlo method[END_REF] and numerically [START_REF] Kandilian | Absorption and scattering by fractal aggregates and by their equivalent coated spheres[END_REF][START_REF] Hoeniges | Effect of colony formation on light absorption by Botryococcus braunii[END_REF][START_REF] Bhowmik | Can spherical eukaryotic microalgae cells be treated as optically homogeneous ?[END_REF][START_REF] Jonasz | Light Scattering by Particles in Water : Theoretical and Experimental Foundations[END_REF][START_REF] Bricaud | Light attenuation and scattering by phytoplanktonic cells : a theoretical modeling[END_REF][START_REF] Stramski | Modeling the inherent optical properties of the ocean based on the detailed composition of the planktonic community[END_REF][START_REF] Lee | Absorption and scattering by long and randomly oriented linear chains of spheres[END_REF][START_REF] Heng | Absorption and scattering by bispheres, quadspheres, and circular rings of spheres and their equivalent coated spheres[END_REF][START_REF] Charon | Monte Carlo implementation of Schiff's approximation for estimating radiative properties of homogeneous, simple-shaped and optically soft particles : Application to photosynthetic micro-organisms[END_REF][START_REF] Dauchet | Calculation of the radiative properties of photosynthetic microorganisms[END_REF].

Experimental determination of microalgae radiative properties is relatively cumbersome and expensive as it requires sophisticated equipment [START_REF] Pilon | Interaction between light and photosynthetic microorganisms[END_REF]. However, simplified methods such as that developed by Kandilian et al. [START_REF] Kandilian | Simple method for measuring the spectral absorption cross-section of microalgae[END_REF] allow for rapid estimation of the absorption crosssection using normal-hemispherical transmittance and reflectance measurements from a UV-VIS spectrophotometer.

Microalgae cultivation

The essential elements of microalgae cultivation for biomass and other useful byproducts are shown in Figure 1.9. Microalgae are composed of proteins, carbohydrates, and lipids, in varying proportions, depending on the species, growth conditions, and the final product being targeted [START_REF] Chowdhury | Algae for Biofuels : The Third Generation of Feedstock[END_REF]. Microalgae are classified as photoautotrophs, which require an inorganic carbon, typically in the form of dissolved CO 2 , and light in order to produce their own energy source in the form of various carbohydrates [START_REF] Ke | Photosynthesis : Photobiochemistry and Photobiophysics[END_REF]. The light delivered to the photosystems must be in the photosynthetically active radiation (PAR) region, defined by wavelength ranging from 400 to 700 nm, to drive photosynthesis [START_REF] Ke | Photosynthesis : Photobiochemistry and Photobiophysics[END_REF]. Artificial lighting can enable more precise control of the growth conditions, and is typically used for high-value products, while naturally occurring sunlight is used for lower-value products and/or mass scale cultivation [START_REF] Pruvost | Cultivation of Algae in Photobioreactors for Biodiesel Production[END_REF]. Nutrients are provided to the cells via a liquid media containing elements such as N, P, and S and other micronutrients such as Mg, Ca, Mn, Cu, and Fe [START_REF] Pruvost | Cultivation of Algae in Photobioreactors for Biodiesel Production[END_REF]. Such media are often neutral to slightly alkaline medium with pH from 7-9 to provide optimum growth conditions depending on the species. [START_REF] Chowdhury | Algae for Biofuels : The Third Generation of Feedstock[END_REF]. Cultivation can take place in either batch mode, where no additional nutrients are added after inoculation and harvesting takes place at the end of the growth cycle, semi-continuous mode, where nutrients are added and biomass is harvested punctually, or continuous mode, where nutrients are continually injected into the culture while biomass is simultaneously harvested.

Although microalgae grow naturally in a variety of environments such as lakes, ponds, and oceans, harnessing their potential as a sustainable resource requires industrial-scale cultivation which poses a number of technical challenges [START_REF] Tredici | Mass Production of Microalgae : Photobioreactors[END_REF]. Cultivation systems must efficiently deliver light and nutrients to the microalgae, while maintaining other parameters such as temperature, pH, and dissolved O 2 at or near their optimum operating points. Furthermore, capital expenses associated with construction and land costs must be balanced with operational expenses such as energy and water consumption and nutrient sourcing costs [START_REF] Pruvost | Large-scale production of algal biomass : photobioreactors[END_REF]. This approach features relatively low construction and operating costs and is thus the most widespread method for large-scale microalgae cultivation [START_REF] Pruvost | Large-scale production of algal biomass : photobioreactors[END_REF]. Such ponds feature culture depths ranging from 10 -30 cm and make use of exterior mounted paddle wheels to circulate the culture, prevent sedimentation, and facilitate CO 2 and nutrient mixing [START_REF] Pruvost | Large-scale production of algal biomass : photobioreactors[END_REF][START_REF] Tredici | Mass Production of Microalgae : Photobioreactors[END_REF]. Despite their relatively low cost, open raceway ponds suffer from a variety of disadvantages including contamination, lack effective temperature control, water loss by evaporation, and high harvesting costs due to their low operation at low biomass concentrations compared to closed systems [START_REF] Tredici | Mass Production of Microalgae : Photobioreactors[END_REF]. Moreover, the operation of open raceway ponds results in CO 2 degassing [START_REF] Pruvost | Development of a thin-film solar photobioreactor with high biomass volumetric productivity (AlgoFilm©) based on process intensification principles[END_REF].

Closed systems

In contrast to open systems, closed microalgae cultivation systems or photobioreactors (PBRs) feature lower risk of contamination and higher biomass productivity. Closed systems come in a wide range of geometries including closed raceway ponds, tubular, and flat-plate PBRs shown in Figures 1.10b , 1.10c, and 1.10d, respectively. Although closed systems are more expensive, they offer the possibility of total control over the physiological conditions [START_REF] Pilon | Radiation transfer in photobiological carbon dioxide fixation and fuel production by microalgae[END_REF]. This enables closed PBRs to attain higher values of biomass productivity or to be used to apply a certain set of growth conditions to favor the production a specific molecule [START_REF] Kandilian | Effect of light absorption rate and nitrate concentration on TAG accumulation and productivity of Parachlorella kessleri cultures grown in chemostat mode[END_REF].

1.4.3 PBR Engineering 1.4.3.

Solar cultivation of microalgae

Although artificial lighting enables more precise control of the radiative field within the culture, an important feature of microalgae is their ability to harvest and store sunlight directly, potentially reducing energy costs associated with their growth. Figure 1.11 schematically illustrates the parameters that define solar conditions for PBR engineering including the changing incidence angle θ, the PBR inclination angle β, and the direct I col and diffuse I dif parts of the solar incidence. The schematic also illustrates the wide variation in the solar path from summer to winter that can occur for certain locations in the northern hemisphere.

I col I col I dif I dif I dif
Indeed, sunlight is a highly variable input changing with seasons, weather patterns, and local conditions such as trees and buildings adjacent to the cultivation site [START_REF] Pruvost | Large-scale production of algal biomass : photobioreactors[END_REF]. Thus, modeling approaches which account for the dynamic solar conditions are necessary to optimize microalgae growth for solar cultivation. Pruvost et al. [START_REF] Pruvost | Modeling Dynamic Functioning of Rectangular Photobioreactors in Solar Conditions[END_REF] constructed a generic model coupling light transfer and microalgae growth kinetics to simulate the impact of solar conditions including day-night cycles, harvesting, and PBR inclination angle. The same model was used to define the theoretical maximum achievable biomass productivity for a given location, a useful benchmark for assessing PBR performance [START_REF] Pruvost | Theoretical investigation of biomass productivities achievable in solar rectangular photobioreactors for the cyanobacterium Arthrospira platensis[END_REF]. Extensive solar modeling has also been performed by Slegers et al. [START_REF] Slegers | Design scenarios for flat panel photobioreactors[END_REF][START_REF] Slegers | Scenario evaluation of open pond microalgae production[END_REF][START_REF] Slegers | Scenario analysis of large scale algae production in tubular photobioreactors[END_REF] with the goal of optimizing the design of large-scale microalgae cultivation installations using raceway ponds, flat plate PBRs, or tubular PBRs under solar conditions for various locations. Such solar PBR models can also be used to define optimum control schemes wherein the PBR biomass concentration is controlled via the dilution rate to attain optimum light transfer conditions within the reactor throughout the day [START_REF] Grognard | Optimal strategies for biomass productivity maximization in a photobioreactor using natural light[END_REF].

Illuminated fraction γ

Light attenuation conditions have a direct effect on the resulting algae growth and the culture productivity. To guide the design and control scheme for a reactor of any geometry, a simple and practical engineering parameter, the so-called illuminated fraction γ, can be used to summarize the light attenuation conditions within a microalgae culture [START_REF] Pruvost | Knowledge models for the engineering and optimization of photobioreactors[END_REF]. This parameter is defined with respect to the compensation fluence rate, denoted by G c , which represents the threshold fluence rate where the rate of oxygen production via photosynthesis is equal to the rate of oxygen consumption by cell respiration. Thus no net photosynthetic growth occurs [START_REF] Pruvost | Knowledge models for the engineering and optimization of photobioreactors[END_REF][START_REF] Torzillo | Environmental Stress Physiology with Reference to Mass Cultures[END_REF]. Then, the microalgae culture can be conveniently divided into two volumes where the local fluence rate is less than or greater than the compensation fluence rate. The illuminated fraction is defined as [START_REF] Pruvost | Knowledge models for the engineering and optimization of photobioreactors[END_REF]

γ = V c V R = A c A R = L c L R (1.17)
where the numerators represent the minimum culture volume, cross-sectional area, or depth required attenuate the local fluence rate such that G(x, y, z) ≥ G c throughout the culture.

Here, V R , A R , and L R represent the reactor volume, cross-sectional area, and depth, respectively. Thus, the illuminated fraction can be defined for cases where light attenuation is one-, two-, or three-dimensional. Clearly, knowledge of the incident irradiation conditions, microalgae cell radiative properties, i.e., A abs , S sca , g), and culture biomass concentration is necessary to predict the local fluence rate at a given culture location.

The illuminated fraction γ can then be used to define three distinct regimes of illumination conditions in PBRs namely, γ < 1, γ = 1, and γ > 1 illustrated in Figures 1.12a, 1.12b, and 1.12c, respectively. The first regime, corresponds to a biomass concentration (or reactor depth) greater than the optimum. Thus, full light attenuation occurs in the culture bulk and part of the reactor is deprived of sufficient radiation in the PAR region to drive photosynthesis. The illuminated fraction γ can be increased by increasing the photon flux on the culture surface or increasing the culture dilution rate for continuously operated cultures.

The second regime, the so-called "luminostat" regime refers to the case where γ = 1 and full light attenuation occurs and no excess reactor volume is present. This is considered to be the ideal operating point for photobioreactors for optimum biomass productivity. The final regime, wherein γ > 1 is the so-called "kinetic" regime. In this case, light that could be used for photosynthesis passes through the culture and the culture volume, area, or depth required for full light is greater than the actual culture depth, i.e.,

V c > V R , A c > A R , L c > L R .
In this case, the illuminated fraction γ may be decreased by increasing the biomass concentration or decreasing the culture dilution rate for reactors operated in continuous mode.

Maximum theoretical productivity

Previous studies have demonstrated that for the case of light-limited growth, when the condition γ = 1 is satisfied to within ± 20%, the maximum theoretical PBR productivity can be predicted by a simple engineering formula [START_REF] Pruvost | Knowledge models for the engineering and optimization of photobioreactors[END_REF][START_REF] Takache | Experimental and theoretical assessment of maximum productivities for the microalgae Chlamydomonas reinhardtii in two different geometries of photobioreactors[END_REF][START_REF] Cornet | A simple and reliable formula for assessment of maximum volumetric productivities in photobioreactors[END_REF] given by

r X max = (1 -f d )ρ M M X φ X 2α 1 + α a light K n+2 n+1 ln 1 + n+2 n+1 q in K . (1.18)
Here, ρ M is the maximum energy yield for photon conversion, M X is the microalgae species molar mass, φ X is the mass quantum yield of conversion, and K is the half saturation constant for photosynthesis [START_REF] Cornet | A simple and reliable formula for assessment of maximum volumetric productivities in photobioreactors[END_REF]. The PBR design parameters include the design dark fraction f d , the specific illuminated area a light = S light /V R , the incident photon flux q in . Finally, α is the microalgae linear scattering modulus given by Equation (1.15) [START_REF] Pottier | A fully predictive model for one-dimensional light attenuation by Chlamydomonas reinhardtii in a torus photobioreactor[END_REF] and n is the degree of collimation of the incident radiation, ranging from n = 0 for isotropic diffuse radiation to n = ∞ for collimated radiation [START_REF] Pruvost | Knowledge models for the engineering and optimization of photobioreactors[END_REF].

Intensified reactors

Photobioreactor performance can be assessed by considering the daily productivity on a volumetric P V and/or culture surface-area P A basis, in kg m -3 day -1 and kg m -2 day -1 , respectively. The two productivities can be related via the specific illuminated area a light according to

P V = P A S light V R = P A a light . (1.19)
Here, a light is defined as the ratio of the illuminated surface area S light to the reactor volume V R . Figure 1.13 plots the predicted volumetric P V and areal P A productivities obtained from a growth model of Chlamydomonas reinhardtii as a function of a light for incident photon flux q in equal to 200, 400, and 800 µmol hν m -2 s -1 . While areal productivity was independent of a light , volumetric productivity P V increased linearly with a light , demonstrating the advantage of cultivation systems with high specific illuminated area [START_REF] Pruvost | Knowledge models for the engineering and optimization of photobioreactors[END_REF]. In addition to larger volumetric productivity, such high values of a light will lead to high cell densities and more compact PBR designs, which can reduce the process energy consumption for pumping, mixing, thermal control, harvesting etc. Alternatively, Figure 1.13 indicates that both volumetric and areal productivity can be increased with increasing incident photon flux q in . Thus, both process parameters can be used to optimize the performance of cultivation systems.

Indeed, both light intensification (larger q in ) and culture intensification (larger a light )

were considered in the context of this thesis with the goal of increasing the reactor biomass productivity. Both approaches aim to increase the optimum biomass concentration, thereby increasing the system volumetric productivity. Furthermore, increasing the cultivation concentration may help avoid the preconcentration step often required before down stream processing of the biomass can occur [START_REF] Gifuni | Current bottlenecks and challenges of the microalgal biorefinery[END_REF]. This can help reduce processing costs of microalgae-based products associated with dewatering or centrifugation [START_REF] Chen | Dewatering and Drying Methods for Microalgae[END_REF]. A variety of photobioreactor configurations leveraging both approaches have been proposed by previous studies and are discussed in the following sections. an tube inner radius of 24 mm. The reactor maximum volumetric productivity obtained was 0.5 g L -1 day -1 , which the authors noted was 30% larger than previous reported values for tubular PBRs of similar dimensions cultivating Spirulina.

Cornet et al. [START_REF] Cornet | Calculation of optimal design and ideal productivities of volumetrically lightened photobioreactors using the constructal approach[END_REF] proposed a volume-illuminated photobioreactor, which features lightdiffusing fiber optic cables of a few millimeters in diameter submerged in the cylindrical culture (DiCoFluv concept). These fiber optics are attached to linear Fresnel light-harvesting surfaces, as shown in the demonstrator pictured in Figure 1.14b [START_REF] Pruvost | Industrial photobioreactors and scale-up concepts[END_REF]. This design permitted a large specific illuminated surface a light of approximately 350 m -1 . The authors estimate that such a system may provide a five-fold increase in areal productivity (based on the footprint area occupied by the unit) compared to conventional systems used for solar microalgae cultivation [START_REF] Pruvost | Industrial photobioreactors and scale-up concepts[END_REF].

Hindersin et al. [START_REF] Hindersin | Irradiance optimization of outdoor microalgal cultures using solar tracked photobioreactors[END_REF] presented a flat-panel photobioreactor mounted to a solar tracking panel with a culture thickness ranging from 15 to 22 mm cultivating Chlorella vulgaris and Scenedesmus obliquus and pictured in Figure 1.14c. The tracking system increased the daily averaged PAR photon flux by 45%. The culture was cooled actively cooled via a heat exchanger or passively cooled by turning the the panel away from the sun. The system had a specific illuminated area of 66 m -1 . To observe the effects of photoinhibition, the quantum yield of photosynthesis was measured via fluorescence. While cultures at lower biomass concentrations experienced some signs of photoinhibition, authors reported that biomass concentrations greater than 2.3 g L -1 limited the effects of photoinhibition at thanks to mutual shading between the cells. Overall, the system was found to increase the optimum biomass concentration and volumetric biomass productivity compared to a horizontal system. a few millimeters in thickness, resulting in concentrations C X ≥ 10gL -1 . Foam break-up for harvesting was performed using a packed column of hydrophobic beads. The reactor achieved a reported specific growth rate of 0.10 h -1 , with the estimated biomass concentration increasing from 2.1 g/L to 4.7 g/L over a period of 8 hours. To ensure that the cells were able to withstand the high sheer stress caused by bubble formation and collapse, the maximum quantum yield of photosystem II was measured via chlorophyll florescence. For the three runs performed, the maximum quantum yield was observed to be similar to that of microalgae cultivated in a typical flat-plate photobioreactor. The reactor was reported to attain volumetric productivities of up to 11 g L -1 day -1 for biomass concentrations up to 24 g L -1 with a relatively low light intensity of 300 µmol hν m -2 s -1 . However, the authors noted that the disadvantages of the thin flat-panel design may include overheating and increased installation costs and bubbling costs compared to classic microalgae cultivation methods.

Pruvost et al. [START_REF] Pruvost | Development of a thin-film solar photobioreactor with high biomass volumetric productivity (AlgoFilm©) based on process intensification principles[END_REF] completed an experiential campaign of a novel technology referred to as Algofilm © featuring a cascading thin-film culture with an enclosed gas head between the culture and the transparent cover. A photo and schematic representation of the Algofilm © photobioreactor are shown in Figure 1.15c. Batch cultivation of the microalgae species Chlorella vulgaris took place in a falling film with thickness ranging from 1.3 to 2.2 mm for a specific illuminated area a light ranging from 454 to 770 m -1 . The thin-film reactor achieved biomass productivity of up to 7 gL -1 day -1 for a constant photon flux density of 270 µmol hν m -2 s -1 representative of year-averaged solar conditions for Nantes, France. The gas-liquid mass transfer coefficient of the Algofilm © reactor was also measured and was considered large enough to ensure that oxygen over-accumulation did no limit the microalgae growth.

Indeed, the authors stated that the productivity of the Algofilm © reactor was limited by non-optimal light conditions, and expected that improved strategies for light management would further increase the reactor performance under solar conditions.

Objectives

Introduction

Microalgae are often cultivated in large-scale outdoor raceway ponds consisting of a culture, at least 15 cm deep, and oriented along the north/south axis [START_REF] Pruvost | Large-scale production of algal biomass : photobioreactors[END_REF][START_REF] Tredici | Mass Production of Microalgae : Photobioreactors[END_REF]. For raceway ponds in outdoor conditions, large solar incidence angles occur in the mornings and evenings and during the winter months when the solar elevation angle may be small depending on the latitude where the raceway pond is operated. As a result, the incident spectral solar radiative flux q in,λ (t) decreases compared to situations when sunlight is nearly normally incident on the culture according to [START_REF] Modest | Radiative Heat Transfer[END_REF]

q in,λ (t) = G S,λ (t) cos θ z (t) (2.1)
where G S,λ is the time-dependent spectral collimated solar irradiation and θ z is the solar zenith angle for a given time of day t, defined with respect to the outward pointing normal vector of the microalgae culture surface. Furthermore, sunlight delivered at oblique angles does not penetrate as deeply into the microalgae culture compared to normally incident light [START_REF] Pruvost | Knowledge models for the engineering and optimization of photobioreactors[END_REF][START_REF] Souliès | Investigation and modeling of the effects of light spectrum and incident angle on the growth of Chlorella vulgaris in photobioreactors[END_REF]. Thus, non-normal incidence can increase dark zones in the culture where there is not enough light to drive photosynthesis. This phenomenon, combined with the decrease in solar irradiation in the mornings and evenings, negatively impacts microalgae growth by reducing the amount of light available to the suspended cells.

The aim of this chapter is to explore the use of mirrors to increase the biomass productivity of raceway ponds by increasing the solar radiative flux delivered to the microalgae culture at critical times of the day and year. The daily biomass productivity of a raceway pond with various configurations of vertical mirrors was predicted throughout the year using experimentally-validated models coupling light transfer and growth kinetics. The performance of each configuration was assessed in terms of areal and volumetric productivities and compared to the same raceway pond but without mirrors. The impact of operating parameters, such as initial biomass concentration and culture depth, and of design parameters including mirror height, pond length-to-width ratio, and cultivation location, were assessed. (in µmol hν m -2 s -1 ) from sunrise to sunset on September 21 st as depicted in Figure 2.1A.

Methods

September 21 st was considered representative of an average day as there are approximately 12 hours of sunlight in Los Angeles, CA on this day. The daily biomass productivity was also simulated for the 21 st day of each month of the year. The raceway ponds were considered to be operated in a semi-continuous mode where harvesting took place when the maximum daily biomass concentration C X,max (in kg m -3 ) was attained, i.e., at time t(C X = C X,max ).

The solar position for a given time and day was described by the solar zenith angle θ z and the solar azimuth angle γ s defined with respect to the due south direction, as illustrated in for Configuration C. For Configuration D, the platform supporting the pond was rotated such that the mirror was always facing the sun and the rotation angle θ D was equal to the solar azimuth angle γ S . Thus, the actual and apparent solar zenith angles were equal, i.e., θ z = θ z,a . Then, the mirror height H B/C/D for each configuration was given by

H B (t) =      W/ tan θ z,a , |γ S | > tan -1 W/l l tan |γ S |/ tan θ z,a , |γ S | ≤ tan -1 W/l (2.
2)

H C (t) =              0, |γ S | > 90 • l/ tan |γ S | tan θ z,a , 90 • > |γ S | ≥ tan -1 l/W W/ tan θ z,a , |γ S | < tan -1 l/W (2.
3)

H D (t) = W/ tan θ z,a . (2.4)
To avoid unreasonably large values of mirror height when θ z,a was small, a maximum allowed height of H * = 1 m was imposed. Then, the mirror height for a given time t was the minimum value between H * and the mirror height for a given configuration from Equations (2.

Assumptions

Light transfer and microalgae growth were modeled based on the following assumptions :

(1) all mirrors were considered to be specularly-reflecting with 100% reflectivity over the photosynthetically active radiation (PAR) region from 400 to 700 nm. (2) The raceway pond was operated in the light-limited regime such that growth was only a function of the local rate of photon absorption (LRPA) within the culture. (3) Light transfer within the culture was considered to be one-dimensional along the z-axis and shading from the walls of the raceway pond was negligible, as demonstrated in Ref. [START_REF] Lee | Design tool and guidelines for outdoor photobioreactors[END_REF]. (4) Diffuse solar radiation was neglected.

(5) The culture was well-mixed with uniform biomass concentration. [START_REF] Shen | Culture of microalga Botryococcus in livestock wastewater[END_REF] The liquid medium was non-scattering and non-absorbing over the PAR region. [START_REF] An | Hydrocarbon production from secondarily treated piggery wastewater by the green alga Botryococcus braunii[END_REF] The radiative properties and kinetic growth parameters of Chlorella vulgaris were constant throughout the day. (8) The culture temperature was kept constant throughout the day. [START_REF] Sydney | Potential carbon dioxide fixation by industrially important microalgae[END_REF] The bottom of the raceway ponds were perfectly absorbing.

Reflected sunlight

The culture surface area subjected to reflected sunlight S ref for a given configuration and solar position was calculated as the difference between the total surface area of light reflected by the mirror and the area of reflected light that fell outside of the culture surface when none of the culture surface was exposed to reflected light, to 1.0 when the entire culture surface area was exposed to reflected light. Then, the incident mean spectral radiative flux q in,λ (in µmol hν m -2 s -1 ) averaged over the culture surface area at a given time t was given by q in,λ (t

(see Figures 2.1B
S ref (t) =              W ref (t)l -1 2 W ref (t) 2 /| tan γ S (t)| for Configuration B W ref (t)l -1 2 W ref (t) 2 tan γ S (t) for Configuration C W ref (t)l for Configuration D.
) = τ (θ z (t))G S,λ (t) cos θ z (t) 1 + S ref (t) S (2.6)
where τ (θ z (t)) is the transmittance of the air/microalgae culture interface predicted by Fresnel's equations for an incidence angle equal to the solar zenith angle θ z [START_REF] Modest | Radiative Heat Transfer[END_REF]. Note that since the mirror was perfectly vertical, the angle of incidence of reflected light was equal to that of light directly incident on the culture surface. The resulting incident mean photosynthetic photon flux q in,P AR (t) was obtained by integrating q in,λ (t) over the PAR region, i.e., q in,P AR (t) = P AR q in,λ (t)dλ.

(2.7)

Light transfer in microalgae culture

The two-flux approximation was used as an analytical solution to the one-dimensional radiative transfer equation governing light transfer in the microalgae culture. This method has been validated and used extensively in previous studies [START_REF] Pruvost | Knowledge models for the engineering and optimization of photobioreactors[END_REF][START_REF] Pottier | A fully predictive model for one-dimensional light attenuation by Chlamydomonas reinhardtii in a torus photobioreactor[END_REF][START_REF] Lee | Design tool and guidelines for outdoor photobioreactors[END_REF][START_REF] Cornet | A structured model for simulation of cultures of the cyanobacterium Spirulina platensis in photobioreactors : I. Coupling between light transfer and growth kinetics[END_REF][START_REF] Cornet | A simplified monodimensional approach for modeling coupling between radiant light transfer and growth kinetics in photobioreactors[END_REF][START_REF] Pruvost | Knowledge models for the engineering and optimization of photobioreactors[END_REF]. The radiation transmitted through the air/microalgae culture interface was refracted at the interface at an angle θ m = sin -1 (n a /n m sin θ z ) where n a = 1.0 and n m = 1.33 are the refractive indices of the air and culture medium, respectively. Then, for a raceway pond with a perfectly absorbing bottom wall and exposed to the mean incident spectral radiative flux q in,λ (t), the local spectral fluence rate G λ (z, t) at a given culture depth z (see Figure 2.2a) was given by

G λ (z, t) q in,λ (t) = 2 cos θ m (1 + α λ )e δ λ (L-z) -(1 -α λ )e -δ λ (D-z) (1 + α λ ) 2 e δ λ D -(1 -α λ ) 2 e -δ λ L (2.8)
where the parameters α λ and δ λ were expressed as [START_REF] Pottier | A fully predictive model for one-dimensional light attenuation by Chlamydomonas reinhardtii in a torus photobioreactor[END_REF] α λ = Āabs,λ Āabs,λ + 2b λ Ssca,λ and

δ λ = α λ C X (t) cos θ m ( Āabs,λ + 2b λ Ssca,λ ). (2.9)
Here, C X (t) is the biomass concentration (in kg m -3 ) at time t while the spectral average mass absorption Āabs,λ and scattering Ssca,λ cross-sections (in m 2 kg -1 ) and the backward scattering ratio b λ of Chlorella vulgaris were reported in Ref. [START_REF] Souliès | Investigation and modeling of the effects of light spectrum and incident angle on the growth of Chlorella vulgaris in photobioreactors[END_REF] for a pigment content of 5

wt.% (in g of total pigment per g of dry biomass) and are shown A(z, t) (in µmol hν kg -1 s -1 ), was defined as [START_REF] Souliès | Investigation and modeling of the effects of light spectrum and incident angle on the growth of Chlorella vulgaris in photobioreactors[END_REF] A(z, t)

= P AR
Āabs,λ G λ (z, t)dλ.

(2.10)

Microalgae growth kinetics

The time rate of change of the biomass concentration C X (t) in a microalgae batch culture can be expressed as [START_REF] Dunn | Biological Reaction Engineering : Dynamic Modelling Fundamentals with Simulation Examples[END_REF]

dX dt = r X (t) = µ(t) C X (t) (2.11)
where r X (t) is the average volumetric growth rate (in kg m -3 s -1 ) and µ(t) is the volumeaveraged specific growth rate (in s -1 ). The growth kinetics model and corresponding parameters reported in Refs. [START_REF] Souliès | Investigation and modeling of the effects of light spectrum and incident angle on the growth of Chlorella vulgaris in photobioreactors[END_REF][START_REF] Takache | Kinetic modeling of the photosynthetic growth of Chlamydomonas reinhardtii in a photobioreactor[END_REF] for Chlorella vulgaris are given in Table 2.1 and were used to predict the specific growth rate µ(t) of the microalgae culture as a function of time.

This model accounted for light limitation and cell respiration activity [START_REF] Takache | Kinetic modeling of the photosynthetic growth of Chlamydomonas reinhardtii in a photobioreactor[END_REF].

First, the volume-averaged specific rate of oxygen production or consumption

J O 2 (t) (in mol O 2 kg -1 X s -1
) as a function of the LRPA A(z, t) was calculated according to [14]

J O 2 (t) = 1 L L 0 ρ M K A K A + A(z, t) φ O 2 A(z, t) - J N ADH 2 ν N ADH 2 -O 2 K r K r + A(z, t)
dz.

(2.12)

Here, ρ M is the maximum energy yield for photon conversion, φ

O 2 (in mol O 2 µmol hν -1
) is the molar quantum yield of O 2 for the Z-scheme of photosynthesis, K A (in µmol hν kg -1 s -1 ) is Table 2.1: Growth kinetics parameters for Chlorella vulgaris [START_REF] Souliès | Investigation and modeling of the effects of light spectrum and incident angle on the growth of Chlorella vulgaris in photobioreactors[END_REF].

Parameter

Value Units

ρ M 0.8 - J N ADH 2 2.8×10 -3 mol NADH 2 kg -1 X s -1 ν O 2 -X 1.13 - φ O 2 1.1×10 -7 mol O 2 µmol -1 hν M X 0.024 kg X mol -1 C ν N ADH 2 -O 2 2 - K A 40,000 µmol hν kg -1 s -1 K r 556.5 µmol hν kg -1 s -1 A c 2,800 µmol hν kg -1 s -1
the half-saturation constant for photosynthesis,

J N ADH 2 (in mol NADH 2 kg -1 X s -1
) is the specific rate of cofactor regeneration on the respiratory chain related to the oxygen consumption by the stoichiometric coefficient of cofactor regeneration on the respiratory chain ν N ADH 2 -O 2 , and K r (in µmol hν kg -1 s -1 ) is a saturation constant describing the inhibition of respiration in light.

Then, the stoichiometric relationship between the production of oxygen and the production of biomass was used to predict the volume-averaged growth rate µ(t) (in s -1 ) as a function of J O 2 (t) according to [START_REF] Souliès | Investigation and modeling of the effects of light spectrum and incident angle on the growth of Chlorella vulgaris in photobioreactors[END_REF]

µ(t) = r X (t) C X (t) = J O 2 (t) M X ν O 2 -X (2.13)
where M X (in kg X mol -1 C ) is the C-molar mass in the biomass given by CH p O n and ν O 2 -X is the stoichiometric coefficient of the oxygen production.

Biomass productivity

The daily volumetric P V , culture-area-based P A,C , and land-area-based P A,L biomass productivities were considered as metrics to compare the performance of all four raceway pond configurations. The daily volumetric biomass productivity P V (in kg m -3 day -1 ) was defined as

P V = (C X,max -C X0 ) ∆t (2.14)
where C X,max is the maximum biomass concentration reached on a given day, C X0 is the initial biomass concentration, and the time increment ∆t is equal to one day. Similarly, the daily culture-area-based biomass productivity P A,C (in kg m -2 day -1 ) was defined as

P A,C = (C X,max -C X0 )V S∆t = P V L (2.15)
where S is the culture surface area. In addition, the daily land-area-based biomass productivity P A,L (in kg m -2 day -1 ) was defined as

P A,L = (C X,max -C X0 )V S L ∆t (2.16)
where S L is the land area required to accommodate both the raceway ponds and the spacing between adjacent ponds. The land area S L required for a single pond was

S L = 2S = 4 m 2
for Configurations B and C with pond width W and spacing width W sp equal to 1 m. A circular land area S L = πR 2 = π(W 2 + L 2 /4) = 6.28 m 2 was required for Configuration D.

The volumetric P V , culture-area-based P A,C , and land-area-based P A,L productivities were considered as they are related to the different costs associated with producing a kilogram of biomass. The volumetric productivity P V can be used to assess the biomass output relative to the operating costs that depend on the culture volume such as the energy required for water circulation and thermal regulation as well as downstream processing costs such as dewatering [START_REF] Pruvost | Cultivation of Algae in Photobioreactors for Biodiesel Production[END_REF]. The daily culture-area-based P A,C and land-area-based P A,L productivities can be used to assess the biomass output relative to operating and capital costs that scale with the culture area (e.g., evaporation losses, pond liners) and land area (e.g., land cost), respectively [START_REF] Gifuni | Current bottlenecks and challenges of the microalgal biorefinery[END_REF]. The land-area-based productivity P A,L can also be used to estimate the size of the facility required for a desired yield of biomass. mirrors in Configuration B was small and the average growth rate µ(t) was briefly smaller than that of Configuration A. This was caused by the decrease in light penetration due to the higher biomass concentration C X (t) in Configuration B compared to Configuration A.

Light transfer

Results and Discussion

In the morning and evening, Configuration B had a larger average growth rate µ(t) than Configuration C, while the opposite was true at midday. This was attributed to the fact that the east/west facing mirrors increased the photosynthetic photon flux q in,P AR significantly in the mornings and evenings while the south-facing mirror increased q in,P AR the most at midday (see Figure 2 Previous studies [START_REF] Lee | Design tool and guidelines for outdoor photobioreactors[END_REF][START_REF] Hoeniges | Impact of dropwise condensation on the biomass production rate in covered raceway ponds[END_REF] have demonstrated that the culture-area-based biomass productivity of photobioreactors and covered raceway ponds scales with C X0 /a where a is the specific illuminated area given by a = S/V such that a = 1/L for the present raceway ponds. Note also that the initial culture optical thickness can be expressed as β λ,0 L = ( Āabs,λ + Ssca,λ )C X0 L where β λ,0 is the initial extinction coefficient in m -1 . Thus, the product C X0 L of the initial biomass concentration C X0 and the culture thickness L is representative of the culture's initial optical thickness [START_REF] Lee | Design tool and guidelines for outdoor photobioreactors[END_REF]. for raceway ponds located in Los Angeles was greater than those located in Saint-Nazaire.

This can be attributed in part to the smaller solar zenith angles (see Figure 2.9b) experienced by the ponds in Los Angeles thanks to its lower latitude. However, Configuration C yielded a slightly larger productivity in Saint-Nazaire than in Los Angeles during the month of June despite having a significantly smaller optimum mirror height H * (see Figure 2.9a) and experiencing a larger solar zenith angle θ z . This indicates that Configuration C

was more effective at improving the biomass productivity of raceway ponds at higher lati- Overall, Configurations B-D enabled higher biomass yield per unit area and volume of culture by increasing the solar input to the microalgae culture. This would reduce the final cost per unit mass of biomass since the operating cost scales linearly with the culture surface area [START_REF] Gifuni | Current bottlenecks and challenges of the microalgal biorefinery[END_REF]. Moreover, by increasing productivity in the winter months, the growing season can be extended and yearly productivity can be improved in locations where year-long growth is currently inefficient. For example, Configuration C increased biomass production in Saint-Nazaire from September to March by 50%. Additionally, the increased incident solar flux achieved by using mirrors may decrease the energy required for thermal regulation of the culture in cooler months and/or climates where sunlight is a major source of heat for solar culture systems [START_REF] Pruvost | Thermal modeling and optimization of microalgal biomass production in the harsh desert conditions of State of Qatar[END_REF]. However, adding mirrors requires more land to prevent mutual shading between adjacent raceway ponds. Furthermore, the mirrors and control system required to implement the external reflecting pond design would increase the capital and maintenance costs compared to a standard raceway pond, particularly for Configuration B featuring two mirrors and for Configuration D featuring the rotating raceway pond. Thus, Configuration C appears to be the most practical design as it requires a single mirror but still improved the raceway pond volumetric and culture-area-based biomass productivity significantly, particularity in the winter months. The concepts explored in this study are promising and should be explored experimentally.

Conclusion

The use of mirrors to increase the daily biomass productivity of outdoor raceway ponds by reflecting additional light onto the microalgae culture was investigated theoretically. Four designs were considered including a raceway pond without mirrors (Configuration A) used 54 as a reference, a pond oriented along the north/south axis with mirrors on both its east and west sides (Configuration B), a pond oriented along the east/west axis with a single mirror on its north side (Configuration C), and a solar tracking rotating pond with a single mirror (Configuration D). The growth of Chlorella vulgaris was predicted using the twoflux approximation and a growth kinetics model accounting for light limitation and cell respiration activity. The use of external mirrors was found to improve the daily volumetric and culture-area-based biomass productivities throughout the year and by as much as 73% compared to a raceway pond without mirrors. Configuration B outperformed Configuration C in the summer months, while the opposite was true in the winter months. Furthermore, the culture-area-based biomass productivity of all four configurations was found to scale by the product of the initial biomass concentration and the culture depth C X0 L. The product C X0 L which yielded the maximum biomass productivity depended on the configuration and the time of year. The addition of mirrors was found to yield the largest improvement in biomass productivity for ponds with a pond length-to-width ratio greater than 5. Additionally, the optimum maximum mirror height was reported for Configurations C and D for both Los Angeles, CA, and Saint-Nazaire, France. Overall, Configuration C featuring a single mirror on the north side of a pond oriented along the east-west axis was considered to be the simplest and most cost-effective method for improving the biomass productivity in outdoor raceway ponds. These results provide practical guidelines for the design and operation of raceway ponds featuring mirrors for improved biomass productivity.

CHAPTER 3

Experimental assessment of the performance of a pilot-scale external reflecting raceway pond for improved biomass productivity

Introduction

In the previous chapter, we theoretically demonstrated the potential of mirrors as a novel solution to increase the solar flux incident on outdoor microalgae cultures to improve their biomass productivity. In particular, the use of an external mirror for microalgae cultivation in Saint-Nazaire, France, was found to improve the biomass productivity throughout the year without the need for excessively large mirrors. Here, we aim to experimentally assess the increase in solar flux afforded by external mirrors and the corresponding impact on the rate of biomass production. Experiments were conducted from the spring to autumn months using two raceway ponds, with and without mirrors, simultaneously cultivating Chlorella vulgaris using sunlight as their light source. Two of the configurations discussed in the previous chapter were tested, namely Configuration B featuring a north/south orientation and mirrors on its east and west sides, and Configuration C, featuring an east/west orientation and a single mirror on its north side. These experiments were intended to assess the performance of the reflecting raceway ponds under changing weather conditions and elucidate the impact of mirrors on parameters not investigated by the model such as culture temperature and evaporative losses.

Materials and methods

Experimental set-up

Two identical raceway ponds purchased from A. time interval from the last measurement.

Mirror placement

To experimentally assess the impact of reflecting additional light on the microalgae culture during solar cultivation, one of the above-described raceway ponds was equipped with an external mirror. This mirror consisted of an opaque reflecting film (MIROIR-111i) purchased from Luminis Films (Villars-les-Dombes, France) that was applied to two flat, rectangular pieces of PVC that measured 30 cm wide and 150 cm long. The reported film reflectivity was 82% in the visible part of the solar spectrum. One mirror was then placed lengthwise against the exterior side of the transparent wall, while the other was held above the raceway pond by a supporting bracket (see Figure 5.1a where the mirror is outlined in red). A top view of the two mirror configurations tested is shown in Figure 5.1b. For Configuration B, both raceway ponds were oriented along the north/south axis, and the mirrors were manually moved from the west side of the raceway pond to the east side of the raceway pond at solar noon and from east side to back to the west side after sunset and before sunrise.

For Configuration C, both raceway ponds were oriented along the east/west axis and the mirror was fixed on the north side of the raceway pond.

Inoculation and operation

The two raceway ponds, with and without a mirror, were placed side-by-side and operated simultaneously to cultivate Chlorella vulgaris in both batch and continuous modes. Here, a modified recipe for Bold's basal medium was used featuring ammonium chloride NH 4 Cl as the nitrogen source rather than sodium nitrate NaNO 3 . The medium composition is shown in Table 3.1. For each experiment, the initial biomass concentration was the same in both raceway ponds. Furthermore, for all experiments, both ponds had the same set points for pH and temperature control and paddle wheel RPM or pump flow rate depending on the method of mixing used.

Biomass concentration

The culture biomass concentration C X (in g/L) was determined gravimetrically by filtering a given volume of culture through a pre-dried, pre-weighed 0.7 µm pore size glass microfiber filter (Fisherbrand, MF 300). The filter and remaining biomass were then dried for 24 hours at 105 • C and cooled in a desiccator for 10 minutes before being weighed. Each measurement was performed in triplicates and the mean values are reported.

Pigment concentration

The extraction of the photosynthetic pigments chlorophyll a, chlorophyll b, and photoprotective carotenoids (PPC) was performed using methanol before being quantified spectrophotometrically. First samples of a given volume V 1 were taken from the raceway ponds. The samples were then centrifuged at 13,400 RPM for 10 min and the liquid medium supernatant was removed. To extract the pigments from the cells, the remaining biomass was suspended in a given volume V 2 of methanol and placed in an oven at 45

C Chla = V 2 V 1 [-8.
C Chlb = V 2 V 1 [27.
4405(OD 652 -OD 750 ) -12.1688(OD 665 -OD 750 )]

C P P C = 4 V 2 V 1 (OC 480 -OC 750 ). (3.1)
The sample V 1 and dilution V 2 volumes varied depending on the sample biomass concentration such that the optical density of the supernatant did not exceed 0.8. Then pigment content w pig can be expressed in terms of weight percent according to

w pig = C Chla + C Chlb + C P P C C X . (3.2)

Solar flux

The total and diffuse solar flux were measured and recorded continuously by a weather station located on the roof of the Algosolis facility. However, while these data provide useful information about the weather conditions throughout a given day, they may not be representative of the solar flux incident on the microalgae culture. Indeed, since the raceway ponds were located inside a greenhouse, shading from the greenhouse structure or other adjacent objects may decrease the solar flux incident on the culture compared to that measured by the weather station. Thus, punctual measurements of the total solar flux were also taken just above the culture surface using a LICOR LI-250A light meter. Here, the surface-averaged solar flux was estimated by measuring the solar flux during a period of 15 seconds while the sensor was manually moved back and forth width-wise over the surface of the culture. This method was preferable to a single-point measurement as it accounted for shadows that may be cast by the greenhouse structure. The results show that the measured increased in sunlight incident on the culture was larger in the mornings and evenings for Configuration B and in the afternoon for Configuration C.

Results and Discussion

These trends are in agreement with those predicted by the model. However, Figure 3.2c in particular highlights the difference between the predicted increase in PFD and the measured PFD. For example, in the morning and evening, shading by trees and adjacent buildings to the east and west of the raceway ponds blocked direct sunlight from reaching the culture and mirror. Thus, the mirror did not increase the incident PFD at this time of day, contrary to the model predictions where shading by adjacent objects was not taken into account. On the other hand, Figure 3.2d shows relatively good agreement throughout the day between the measured and predicted percent increase in the solar PFD since there were fewer obstructions to the south of the raceway ponds. the submerged exchanger when the culture temperature T c exceeded 20 • C. Note that the culture temperature T c for the raceway pond without a mirror is not shown in Figure 3.3d due to a malfunction in the data recording system. However, based on the results shown in Figure 3.3c it can be expected to follow the same trend as the raceway pond with a mirror.

Here the average culture temperature for the raceway pond of Configuration C was Tc = 19.8 • . Comparing Figures 3.3c and 3.3d indicates that the thermal control system reduced the maximum temperature attained by the culture. This was the case despite higher ambient temperatures and an increase in the direct solar radiation incident on the raceways in late May compared to early April (see Figures 3.3a and 3.3b).

Figures 3.3e and 3.3f plot the resulting total dry biomass m (in g) present in the ponds for the early April and late May trials, respectively. Note that biomass concentration is a function of the microalgae growth rate and the evaporation rate from the culture. Thus the total dry biomass is shown for all experiments to isolate the effect of increased microalgae growth. The results clearly demonstrate that the solar conditions later in the spring were more favorable for microalgae growth. In fact, the quantity of dry biomass produced during the trial performed in late May was larger despite the fact that the late May experiment was shorter (8 days) than that in early April (10 days). Indeed, late May featured longer days and larger solar flux as illustrated in Figures 3.3a and 3.3b, which resulted in an increased dry biomass yield. Figure 3.3e indicates that the raceway pond of Configuration C enabled the culture to produce a larger quantity of dry biomass compared to the no mirror case. For both raceway ponds, the dry biomass increased at the same rate and then stagnated slightly due to the poor solar conditions and larger biomass concentrations, around 0.9 gL -1 , in the ponds on April 5 th and 6 th . However, in the last three days of the trial, the growth rate was larger for the raceway pond of Configuration C. Indeed, at the end of the trial, the raceway pond with a mirror contained 14.9% more dry biomass than the raceway pond without a mirror thanks to the increased solar energy input. The higher growth rate for the raceway pond with a mirror in the final days of the experiment can be explained by the higher biomass concentration. At low biomass concentrations, the culture is fully illuminated and no dark zone occurs (i.e. γ = 1). Then, the additional light reflected onto the culture by the mirror will not be as effective at increasing the average photosynthetic growth rate. This was the case early in the experiment. However, as the biomass concentration increases, the sunlight does not penetrate as deeply and the culture is no longer fully illuminated (i.e. γ < 1).

Then, additional light reflected by the mirrors can enable deeper penetration of light into the culture, increasing the illuminated volume and improving the average photosynthetic growth rate. Unlike the early April trial, Figure 3.3f shows that the addition of a mirror did not increase the final dry biomass for the trial conducted in late May. This demonstrates that Configuration C is not suitable for increasing the biomass productivity of outdoor raceway ponds at this time of the year in Saint-Nazaire, France. Indeed, during the late spring and summer months the sun stays high in the southern half of the sky at midday. Thus, the south-facing mirror used in Configuration C is poorly positioned to reflect additional light on to the culture and the final dry biomass was the same for both raceway ponds. Despite this, the model developed in Chapter 2 predicted an increase in biomass productivity at this time of year. This suggests that the assumptions used in the model may cause the impact of the external mirrors to be over-estimated. However, these results confirm that Configuration C is better suited to the spring months when the the sun is low in the southern sky at midday, as predicted by the model in Chapter 2. Figure 3.4 shows a comparison of the pigment content w pig (in weight %) between the raceway ponds with and without mirrors for the spring trials in April and May. Pigment content was larger in for the April trial, when the measured direct solar radation G S was lower (see Figures 3.3a and 3.3b). This was due to pigment adaptation by the microalgae, which causes the cells to accumulate more pigments if less sunlight is available, and vice versa. However, for each trial, Figure 3.4 indicates that the difference in pigment content between the cultures in the raceway ponds with and without mirrors was slight. This suggests that significant differences in pigment adaptation did not occur between the two ponds over the course of the experiments. Since the measured difference between the two raceway ponds was slight, the pigment content is not reported for the following experiments.

Summer trials

Two trials were performed in July and August featuring a raceway pond of Configuration B (north/south pond orientation with a dual mirrors on the east and west sides). For this configuration, the mirrors were manually moved from the west side of the raceway to the east at solar noon, and then back to the west side after sunset and before sunrise. The corresponding solar conditions are shown in Figures 3.5a and 3.5b and the culture temperature T c is shown in Figures 3.5c and 3.5e. For these trials, the culture volume was increased slightly to 52L corresponding to a culture depth of 10.5 cm. This was done to increase the thermal inertia of the system in an attempt to prevent large swings in culture temperature. The data gaps in Figure 3.5c on July 6 th and July 8 th correspond to planned power outages for maintenance purposes at the Algosolis site. Additionally, there is a larger gap in data for the culture temperature of a raceway pond without a mirror in Figure 3.5c due to a malfunction in the data recording system. At several instances, both Figures 3.5c and 3.5d indicate that the culture temperature of the raceway without a mirror was higher than that with a mirror. This may be attributed to the depth of the temperature sensor in the culture and poor culture mixing. Since water has a high absorption index in the infrared part of the electromagnetic spectrum, the increased solar thermal radiation incident on the culture will be mostly absorbed within the first few centimeters of culture depth. Thus, there was not an increase in culture temperature measured since the temperature sensors were placed a depth of at least 5 cm. This effect was even more pronounced for the August trial (see Figure 3.5d) due to the higher culture biomass concentrations which decreased the penetration depth of the solar radiation. Furthermore, for both summer trials, the microalgae culture was circulated with a submerged pump. While the pump provided enough forward velocity for culture circulation in the raceway pond, it was likely less effective at mixing the culture from top to bottom. Thus, an increase in culture temperature was not measured deeper in the culture. Figure 3.5e and 3.5f plot the dry biomass in each raceway pond for the July and August trials, respectively. The intial biomass concentration was 0.17 gL -1 and 0.56 gL -1 for the July and August trials, respectively. For the July trial, Figure 3.5e indicates that the two raceway ponds, with and without mirrors, had the same biomass growth rate for the first six days.

After this point, the biomass growth rate was larger for the raceway pond of Configuration B. This was similar to the behavior observed in Figure 3.3e for a raceway pond with the singlemirror Configuration C. This indicates that for both mirror configurations, the additional light provided by the mirrors at low biomass concentrations is not effective at increasing the culture average photosynthetic growth rate. However, once higher biomass concentrations are reached, in this case beyond approximately 0.8 gL -1 , the larger incident photon flux provided by the mirrors will increase the illuminated volume of the culture which leads to a corresponding increase in growth rate. Indeed, over the course of the experiment, the raceway pond featuring mirrors produced 5.4% more biomass than the raceway pond without mirrors. To test this hypothesis, the August trial was performed over five days with a larger initial biomass concentration of 0.56 gL -1 . The results are shown in Figure 3.5f. In this case, the growth rate for the raceway pond of Configuration B was immediately larger than that of the raceway pond without a mirror thanks to the larger biomass concentration. Here, the mirrors enabled the raceway pond of Configuration B to produce 36.6% more biomass over the course of the experiment. Furthermore, the Configuration B raceway pond had a final biomass concentration that was 9.8% larger than that of the raceway pond without mirrors, 0.89 gL -1 compared to 0.81 gL -1 , respectively. This indicates that under the same solar conditions, the raceway pond featuring mirrors can achieve higher maximum biomass concentrations. This is especially interesting for raceway ponds operated in continuous mode since the productivity is directly proportional to the biomass concentration.

Evaporation

Figures 3.6a and 3.6b show the decrease in culture depth L due to evaporation for both raceway ponds as a function of time for the spring trial in April, and the summer trial in August, respectively. First, Figure 3.6 demonstrates that much more evaporative losses occurred in April compared to August. Interestingly, this was the case despite the fact that the culture temperature in April was typically lower that that in August (see Figures 3.3c and 3.5d). This result may be explained by the fact that a paddle wheel was used in the April trials for culture circulation, while a submerged pump was used in the August trial.

As the paddle wheel turns, the paddles are wetted with a thin layer of culture which can then quickly dry, increasing the evaporation rate from the culture. Figure 3.6a shows a nearly identical culture depth for the raceway ponds with and without a mirror. This indicates that the evaporation rate was the same for both cultures. This was due to the similar culture temperatures show in Figure 3.3c. On the other hand, Figure 3.6b shows a steeper decrease in culture depth for the raceway pond with mirrors compared to that without mirrors, demonstrating a higher rate of evaporation for the former. This was the case despite a similar measured culture temperature between the two raceway ponds (see Figure 3.5d). However, as discussed previously, the submerged pump may be less effective at culture mixing, leading to a vertical temperature gradient within the culture. Then, the culture temperature at the surface was likely higher for the raceway pond with a mirror, causing a higher rate of evaporation and a faster decrease in culture depth. This suggests that effective culture mixing is important to mitigate the culture evaporative losses caused by the increase in incident solar energy for raceway ponds featuring mirrors. plots the PFD q P AR incident on the culture surface for both raceway ponds measured once per day between noon and 3 :30 PM. Figure 3.7b demonstrates that the mirror increased the incident PFD only on days when there was higher direct solar radiation G S,P AR . For instance, the PFD was nearly equal for both raceway ponds on October 1 st and 13 th when the weather was overcast and no direct solar radiation was measured. The same was true for times with low direct solar radiation, such as October 2 nd and 7 st . However, for the remaining days where measurements were taken, the mirrors increased the PFD incident on the culture from 18% to 50%.

Continuous operation

The culture biomass concentration C X and volumetric productivity, given by P V = C X D in (gL -1 day -1 ), are plotted in Figures 3.7c and 3.7d, respectively. Note that here, the evaporation will not impact the biomass concentration, as long as the rate of evaporation is less than the rate at which fresh medium is added to the culture. This is because both cultures were harvested via an overflow pipe of the same height (see Figure 5.1) and thus always had the same volume. The starting biomass concentration was approximately 0.6 gL -1 for both raceway ponds. Initially, the biomass concentration decreased since the culture dilution rate was higher than the microalgae growth rate. This was likely due to the relatively cloudy conditions from October 1 st to 5 th . The decrease in biomass concentration was larger for the raceway pond with the mirror particularly between October 3 rd and 4 th . This may be the result of photoinhibition particularly on October 3 rd , where Figure 3.7b indicates that the PFD was as large as 1400 µmol hν s -1 m -2 for the raceway pond of Configuration C. Nonetheless, over the course of the experiment, the raceway pond of Configuration C produced 6.1% more biomass than the raceway pond without a mirror. Indeed, in the days following October 7 th , the microalgae growth rate was larger than the dilution rate and the biomass concentration increased. During these days, the raceway pond featuring a mirror had a volumetric productivity that ranged from 5.9% to 24.4% larger compared to the raceway pond without a mirror. These results indicate that while mirrors may be used to increase the growth rate of outdoor raceway ponds, their performance is highly dependent on the weather conditions. Thus, implementing mirrors in solar cultivation of algae would yield the best results in climates and seasons that feature mostly clear skies.

Conclusion

An experimental proof-of-concept campaign was carried out on two pilot-scale raceway ponds in both batch and continuous mode under solar conditions during different seasons and featuring two different mirror configurations. For each trial, an identical raceway pond without a mirror was operated simultaneously as a control case to assess the impact of the mirrors. For both configurations, the presence of mirrors was shown to increase the solar photon flux density (PFD) incident on the culture for clear days. For example, the total incident solar energy was increased by 52.2% and 39.6% for Configurations B and C, respectively, on a sunny day in September in Saint-Nazaire, France. However, the experimentally measured increase in solar PFD was negligible when the direct sunlight was blocked by clouds or nearby trees or structures, thus the increase in biomass production afforded by the mirrors was not as large as that predicted by the model developed in Chapter 2. Thus, the effectiveness of the mirrors was highly sensitive to the weather conditions and any shadows cast by adjacent trees or structures. Nonetheless, over a span of clear days in August, Configuration B (north/south pond orientation with mirrors on the east and west sides) increased the total biomass yield by 36.6% compared to a raceway pond without a mirror. Similarly, over a span of clear days in March, Configuration C (east/west pond orientation with a single mirror on the north side) increased the total biomass yield by 14.9% compared to a raceway pond without a mirror. Configuration C also improved the total biomass yield by 6.1% for a continuous trial in October. However, Configuration C was ineffective at improving the biomass yield in May since the south-facing mirror was poorly adapted to reflect sunlight when the solar elevation angle was large. Overall, the use of mirrors was found to be a relatively simple and effective method to improve the biomass production rate when the culture concentration was approximately greater than 0.5 gL -1 . At lower biomass concentration, full light attenuation did not occur in the culture and the additional light reflected by the mirrors did not increase the illuminated fraction. Evaporative losses were observed to be larger for the raceway ponds featuring mirrors when the culture was circulated via a submerged pump and negligible for circulation via a paddle wheel. This may be explained by poor culture mixing when the submerged pump was used, which led to a higher culture surface temperature.

Thus, effective culture mixing may help mitigate the increase in evaporative losses caused by the presence of the mirrors. Overall, the reflecting raceway ponds were shown to increase the biomass productivity for sufficiently concentrated cultures under clear skies.

CHAPTER 4

Experimental proof-of-concept of a novel, compact, thin-tube photobioreactor for high volumetric productivity

Introduction

While the previous two chapters focused on improving the biomass productivity of outdoor raceway ponds via intensification of the incident light, the following two chapters focus on improving the biomass productivity of photobioreactors via culture intensification principles. As discussed in Chapter 1, under light-limited conditions, the volumetric productivity P V of a photobioreactor is controlled by the specific illuminated area a light , defined by the ratio of the illuminated surface area to culture volume, i.e., a light = S light /V [START_REF] Pruvost | Knowledge models for the engineering and optimization of photobioreactors[END_REF]. An increase in a light increases the maximum theoretical volumetric productivity of a system. This can yield energy-savings for processes whose energy consumption is related to the culture volume, such as mixing, pumping, thermal control, and harvesting. AlgoFilm © is an example of such an intensified PBR which features a falling thin-film design with culture thicknesses of approximately 2 mm (i.e. a light = 500 m 2 m -3 ) and has achieved volumetric productivities as large as 7 g L -1 day -1 with an average biomass concentration of 14 g L -1 [START_REF] Pruvost | Development of a thin-film solar photobioreactor with high biomass volumetric productivity (AlgoFilm©) based on process intensification principles[END_REF]. Such large biomass concentrations are necessary to ensure full attenuation of the incident light in the thin culture film and have the added benefit of reducing downstream processing costs for centrifugation and/or dewatering [].

The same principles can be applied to tubular photobioreactors. For illumination on a single side, a light for a tubular photobioreactor of height H is solely a function of the tube radius, and varies according to

a light = πrH πr 2 H = 1 r . (4.1)
Thus, the maximum achievable volumetric productivity of a tubular photobioreactor can be increased by decreasing the tube radius.

Here we conduct an experimental proof-of-concept campaign on a tubular photobioreactor featuring thin tubes of 4 mm in diameter (a light = 500 m 2 m -3 ) cultivating Chlorella vulgaris (ICAR-PBR Concept). The goal of the study was to assess the suitability of the thin-tube design for growing microalgae cultures at large biomass concentrations, and identify any operational challenges such as biofilm formation. The gas-liquid mass transfer coefficient of the reactor was also measured to assess any limitations associated with oxygen over-accumulation. Finally the performance of the thin-tube PBR is compared against the maximum theoretical performance predicted from established engineering rules for maximum PBR productivity.

Materials and methods

Reactor design and operation

A schematic illustration of the prototype thin-tube photobioreactor built by Seawit Lab flow rate varied from 3 mL min -1 to 10 mL min -1 . The reactor was operated in both batch and semi-continuous modes. The semi-continuous operation consisted of harvesting a given volume of medium and injecting an equal volume of fresh medium.

The PBR was illuminated using strips of white LED lights connected to an external voltage supply and placed approximately 6 cm from the vertical tubes. Calibration of the photon flux density (PFD) was performed by taking measurements at intervals of approximately 10 cm along the height of both tubes using a LICOR LI-250A light meter for a range of applied voltages. Then, the average PFD incident on the tubes was related to the applied voltage by a calibration curve.

Medium composition

Due to their high biomass concentration, the culture medium for intensified photobioreactors must feature higher nutrient concentrations than those used in raceway ponds or traditional photobioreactors to avoid nutrient limitation. Here, a modified version of Sueoka's high salt medium was used. The concentration of a given nutrient was modified according to the desired biomass concentration and the elemental composition of Chlorella vulgaris from a stoichiometric equation of its growth [START_REF] Hadj-Romdhane | Development and validation of a minimal growth medium for recycling Chlorella vulgaris culture[END_REF] To ensure nutrient excess, an additional 10% of each nutrient was added [START_REF] Pruvost | Development of a thin-film solar photobioreactor with high biomass volumetric productivity (AlgoFilm©) based on process intensification principles[END_REF]. High concentrations of nutrients can pose problems such as the formation of salt crystals, which are unable to be metabolized by the microalgae. To avoid this problem, the ammonium chloride (NH 4 Cl) and sodium bicarbonate (NaHCO 3 ) used in the previous chapters was replaced by ammonium bicarbonate (NH 4 HCO 3 ) [START_REF] Pruvost | Development of a thin-film solar photobioreactor with high biomass volumetric productivity (AlgoFilm©) based on process intensification principles[END_REF]. The high concentration of nutrients also made it necessary to prepare the phosphates in a separate solution from the rest of the medium.

Then, the phosphate solution was injected on a daily basis to avoid large concentrations that result in precipitate formation. For all trials, an average of 1 mL per day of phosphate solution was injected. The medium recipe and the composition of the phosphate solution used is shown in Table 4.1 for target biomass concentrations of 10 and 15 gL -1 . Note that the concentration of Hutner's trace element solution was not increased with increased target biomass as the trace minerals were already expected to be in excess.

Gas-liquid mass transfer

In a closed photobioreactor, the oxygen produced by microalgae during their growth must be removed from the culture to prevent oxygen over-accumulation which can negatively impact photosynthetic growth [START_REF] Kazbar | Effect of dissolved oxygen concentration on microalgal culture in photobioreactors[END_REF]. Thus, the oxygen mass transfer rate N O 2 of the reactor must be known to avoid limiting microalgae growth due to high dissolved oxygen concentration C L . The oxygen mass transfer rate can be defined and related to the dissolved oxygen concentration according to

N O 2 = dC L dt = k L a • (C * L -C L ). (4.3)
Here, k L is the mass transfer coefficient (in m/s), a is the ratio of gas-liquid interfacial area to liquid volume (in m 2 m -3 ), and C * L is the dissolved oxygen concentration at equilibrium with the gas phase.

The volumetric gas-liquid mass transfer coefficient, given by the product k L a (in s -1 ), governs the removal of dissolved oxygen during microalgae cultivation. The value of k L a for a given reactor can be calculated by measuring the dissolved oxygen C L content throughout the course of a de-oxygenation/re-oxygenation experiment [START_REF] Loubiere | Investigations in an externalloop airlift photobioreactor with annular light chambers and swirling flow[END_REF]. This consists of removing oxygen from the culture medium by injecting gaseous nitrogen (N 2 ) instead of air until the dissolved oxygen concentration reaches a new equilibrium point. Then the injected gas is changed back to air for the re-oxygenation phase. Integrating Equation (4.3) with respect to time and assuming that k L a is time-independent yields

ln (C * L -C L ) (C * L -C L0 ) = -k L a • t (4.4)
where C L0 is the initial dissolved oxygen concentration before nitrogen injection. Then, the measured oxygen concentration C L can be used to plot the quantity ln

(C * L -C L ) (C * L -C L0 )
as a function of time t, and the volumetric gas-liquid mass transfer coefficient k L a is given by the slope of the line.

The de-oxygenation/re-oxygenation experiments were performed using culture medium with a target biomass concentration of 10 gL -1 (Section 4.2.2). The dissolved oxygen concentration was measured using an oxygen sensor (InPro6800, Mettler Toledo) positioned at the top of the reactor connected to a transmitter (M300, Mettler Toledo). The gas flow rate was controlled by a volumetric flow meter (Bronkhorst, France).

Biomass analyses

Culture biomass concentration was measured gravimetrically using the method previously described in Section 3.2.3. In brief, the biomass from a sample of a given volume was filtered and dried before being weighed to determine the dry weight of biomass present in the culture per unit volume. Pigment concentration was determined using the method previously described in Section 3.2.4. In brief, after removing the culture medium via centrifugation pigment extraction was performed in methanol. The spectral optical density OD λ of the extract was measured at 480, 652, 665, and 750 nm. Then the concentration of chlorophyll a, chlorophyll b, and photoprotective carotenoids was calculated using Equation (3.1).

Results and Discussion

Theoretical maximum productivity

The expression for the theoretical maximum productivity, given by Equation (1.18), can be simplified for ideal light conditions i.e., collimated light (n = ∞) and normal incidence (cos θ = 1) yielding :

P V,max = a light (1 -f d )ρ M φ O 2 ν O 2 -X 2α 1 + α K ln 1 + q in K (4.5)
where the energy yield for photon conversion ρ M , the molar quantum yield for the Z-scheme of photosynthesis φ O 2 , and the stoichiometric coefficient of oxygen production ν O 2 -X are given for Chlorella vulgaris in Table 2.1. Here, K represents the photon flux at the halfsaturation point for photosynthesis (in µmol hν m -2 s -1 ) and is given by the ratio of the halfsaturation constant for photosynthesis and the microalgae mass absorption cross-section, i.e, K = K A / Āabs . As in Chapter 2, the spectrally-averaged absorption cross-section of Chlorella vulgaris was taken from Ref. [START_REF] Souliès | Investigation and modeling of the effects of light spectrum and incident angle on the growth of Chlorella vulgaris in photobioreactors[END_REF] and corresponded to Āabs = 275 m 2 kg -1 for the measured total pigment content of 6 wt.% and pigment distribution of 66% Chl. a, 17% Chl. b, and 17% PPC for the current study. The corresponding surface productivity is given by

P S,max = P V,max /a light (4.6)
Then, the maximum theoretical volumetric P V,max and surface P S,max productivities can be predicted for the current photobioreactor. Figure 4.2 plots the maximum theoretical volumetric P V,max and surface P S,max productivity for a light equal to 50, 100, and 500 m 2 m -3 corresponding to a tube radius of 2 cm, 1 cm, and 2 mm, respectively. The results illustrate the increase in P V,max for decreasing tube radius, i.e, increasing a light (Eqn (4.1)). Note that the P S,max is constant for at a given PFD for all values of a light .

a li g h t = 5 0 0 m -1 a light = 100 m -1 a light = 50 m -1
A ll aligh t v a lu e s

C * L -C L C * L -C L0
used to calculate the gasliquid mass transfer coefficient k L a (Equation 4.4). Here, only the values of C L that were greater than 20% and less than 80% of the equilibrium value C * L , i.e., 0.2C * L < C L < 0.8C * L , were used to calculate k L a [START_REF] Pruvost | Development of a thin-film solar photobioreactor with high biomass volumetric productivity (AlgoFilm©) based on process intensification principles[END_REF]. These values are plotted in blue. The same data was also collected during the oxygen gassing phase. Figures 4.3a, 4.3c, and 4.3e show that the dissolved oxygen concentration C L decreased at a faster rate for increasing gas flow rates of nitrogen Q. Indeed, at higher gas flow rates, oxygen can be more quickly removed from the culture. Furthermore, at higher flow rates, more bubbles were present in the reactor at a given moment. This increased gas-liquid interfacial area which led to an increase in the rate of oxygen degassing. This is reflected in the increasingly negative slope shown in Figures 4.3b, Table 4.2: Gas flow rate Q (in mL min -1 ) and corresponding gas-liquid mass transfer coefficient k L a (s -1 ) and biomass productivity P V,crit (gL -1 day -1 ) that yields the critical dissolved oxygen concentration C L,crit = 31 mgL -1 . Q (mL min -1 ) k L a × 10 3 (s -1 ) P V,crit (gL For photobioreactors in steady state, the rate of oxygen production P O 2 (in grams of O 2 /s/L) can be linked to the biomass volumetric productivity P V via the specific yield of oxygen production Y O 2 /X according to

P O 2 = Y O 2 /X P V . (4.7)
where a conservative value of Y O 2 /X is 2 grams of oxygen per gram of biomass [START_REF] Pruvost | Development of a thin-film solar photobioreactor with high biomass volumetric productivity (AlgoFilm©) based on process intensification principles[END_REF]. We note that the rate of oxygen degassing denoted by N O2 will be equal to the rate of oxygen production for a photobioreactor in steady state with a constant value of dissolved oxygen concentration C L , i.e.,

N O2 = P O2 . (4.8)
Previous studies have shown that dissolved oxygen concentrations beyond a critical value of C L,crit = 31 mgL -1 will negatively impact the growth of Chlorella vulgaris [START_REF] Kazbar | Effect of dissolved oxygen concentration on microalgal culture in photobioreactors[END_REF]. Then, Equations (4.4) and (4.7) can be combined to estimate the critical biomass productivity for a given gas flow rate Q, beyond which oxygen over-accumulation in the culture would limit microalgae growth, i.e,

P V,crit = k L a Y O2/X (C L,crit -C * L ) (4.9)
where the equilibrium concentration C * L was measured for bubbling air.

𝑘 ! 𝑎 = 1 . 3 × 1 0 -3 s -1 𝑘! 𝑎 = 2 . 2 × 1 0 -3 s -1
Oxygen overaccumulation 4.4 illustrates the importance of gas-liquid mass transfer in photobioreactor engineering. For low gas flow rates, the gas-liquid mass transfer coefficient is lower, and the critical value of dissolved oxygen concentration is quickly reached at low biomass productivity. Table 4.2 shows the critical biomass productivity for each gas flow rate Q. Notably, the current photobioreactor configuration is expected to reach the critical value of dissolved oxygen C L,crit at a biomass productivity of 2.1 gL -1 day -1 . Then, according to Figure 4.2, this critical biomass productivity can be attained for the current PBR under light-limited conditions at incident an PFD as low as 70 µmol hν m -2 s -1 . This indicates that the low gas-liquid mass transfer in the current PBR will quickly become a factor that limits photosynthetic growth.

For example, at a photon flux density of 250 µmol hν m -2 s -1 a gas-liquid transfer coefficient k L a of 5.5×10 -3 s -1 is necessary to ensure that the culture is not limited due to oxygen overaccumulation and can approach the maximum theoretical productivity of 5.2 gL -1 day -1 . and 11.96 gL -1 were achieved after approximately 10 days for the 10 gL -1 and 15 gL -1 target biomass concentrations, respectively. Beyond this point, the biomass concentration stagnated due to nutrient limitation. Despite these high biomass concentrations, no visible sign of biofilm formation was observed after emptying and inspecting the PBR in the tube featuring bubble-driven upward flow. This was attributed to the fact that the bubble diameter was large relative to the inner diameter of the tube. This led to more effective mixing near the inner wall of the tube which prevented the formation of a biofilm [START_REF] Thobie | Different types of bubbly flows in a confined channel with the aim of limiting microalgae biofilm development ? Part II : Study of the development and removal of microalgae biofilm[END_REF]. This is a unique advantage of the thin-tube PBR design compared to traditional vertical tubular PBRs, where large tube diameters make it difficult to ensure sufficient mixing near the walls to prevent the formation of microalgae biofilms. For the tube featuring gravity-driven downward flow, some minor biofilm formation was observed as bubbles were not present to prevent biofilm formation.

≈ a light C X (q in -q o ) (4.10)
where q o is the photon flux density that exits the photobioreactor. An estimate of the MRPA Ā can be calculated by assuming full light absorption by the culture, i.e., q o = 0. For both cultures, the incident photon flux density was adjusted to maintain an MRPA between 5 and 15 µmol hν g -1 s -1 . Figures 4.5c and 4.5d show that the thin-tube design enabled microalgae growth at high concentrations without the need for high incident photon flux densities. Indeed for both trials, the photon flux density q in varied from 50 µmol hν m -2 s -1 to 300 µmol hν m -2 s -1 .

Such a PFD would be attainable in outdoor systems. For instance, the average solar photon flux density ranges from 120 µmol hν m -2 s -1 in the winter months to 500 µmol hν m -2 s -1 in the summer months in Nantes, France. for both cultures, leading to low or even negative initial growth rates. However, the average total pigment concentration was 5.2 and 6.5 wt. % for the trials with 10 and 15 gL -1 target biomass concentration, respectively. Such pigment concentrations suggest that significant light stress did not occur over the course of the experiments [START_REF] Bonnanfant | Étude des mécanismes de conversion photosynthétique de l'énergie lumineuse par Chlorella vulgaris en photobioréacteur solaire[END_REF]. ), the corresponding daily growth rate r X (in gL -1 day -1 ), and the incident PFD and MRPA, respectively, for a semi-continuous trial of the ICAR (thin-tube) photobioreactor using culture medium with a target biomass concentration of 15 gL -1 . The gas flow rate was set to Q = 10 mL min -1 . Here, the daily growth rate was calculated according to

Semi-continuous operation

r X = C X (t + ∆t) -C X (t) ∆t (4.11)
where ∆t was the time in days between biomass concentration measurements. Figure 4.6a illustrates that the ICAR (thin-tube) photobioreactor was able to sustain microalgae growth at biomass concentrations greater than 10 gL -1 for a period of approximately 7 days. The maximum biomass concentration obtained was 16.3 gL -1 . This was the case thanks to the small tube diameter which enabled full illumination of the culture even when the penetration depth of light was small. Figure 4.6b plots the biomass growth rate r X in gL -1 day -1 calculated according to Equation (4.11). The average growth rate over the course of the experiment was rX = 3.0 gL -1 day. This was a significant improvement to the average growth rate rX of 0.49 gL -1 day and 0.82 gL -1 day for the two batch trials with target biomass concentrations of 10 gL -1 and 15 gL -1 , respectively. This was due, in part, to the consistently high biomass concentration throughout the duration of the experiment. These elevated biomass concentrations and growth rates were attained without the need for excessively large incident PFD, as illustrated in Figure 4.6c. Indeed, the PFD was fixed at 250 µmol hν m -2 s -1 throughout the course of the experiment. This value is readily achievable in outdoor conditions, even on overcast or winter days.

Finally, the average growth rate of rX = 3.0 gL -1 day was significantly smaller than the maximum theoretical productivity predicted by Equation (4.5), be it 5.2 gL -1 day -1 for a PFD of 250 µmol hν m -2 s -1 . This indicates that the culture growth was not light-limited or that the dark fraction may be underestimated. Indeed, sedimentation of the the microalgae cells in the lower chamber was observed due to the relatively slow bubble driven-flow. This may have created an inhomogeneous culture mixture, with a larger concentration of biomass in the dark lower chamber. Note that the term (1-f d ) in Equation (4.5) represents the fraction of cells exposed to light to drive photosynthesis. Thus, in the case of an inhomogeneous culture with a larger biomass concentration in the lower chamber, the fraction of cells exposed to light would be overestimated by the term (1-f d ), and the actual productivity P V would be lower than that predicted by Equation (4.5). Furthermore, Equation (4.5) assumes ideal light conditions. In other words, the illuminated fraction γ, given by the fraction of culture exposed to a local fluence rate greater than the fluence rate required for positive photosynthetic growth, was assumed to be equal to 1 (with the obvious exception of the culture in the reactor dark volume). However, this may not have been the case due to factors such as the diffuse part of the incident photon flux and refraction effects between the tube wall and the culture. These phenomena will be discussed in more detail in Chapter 5.

Finally, the average growth rate attained of rX = 3.0 gL -1 day was greater than the critical value of biomass productivity P V,crit = 2.1 gL -1 day from Equation (4.9) for the gas flow rate Q = 10 mL min -1 . This indicates that the gas-liquid mass transfer coefficient of 2.2×10 -3 s -1

was too low and microalgae growth was likely limited by oxygen over-accumulation during the semi-continuous trial. As mentioned previously, a gas-liquid mass transfer coefficient of at least 5.5×10 -3 s -1 is necessary to provide sufficient oxygen de-gassing at the maximum predicted growth rate of 5.2 gL -1 day -1 . Thus, efforts to improve the performance of the ICAR thin-tube photobioreactor should first focus on improving the gas-liquid mass transfer coefficient to enable more efficient degassing of oxygen produced by the microalgae during growth. Improved mixing and optimized light transfer via detailed light transfer analysis can also be expected to enable the experiential volumetric productivity to approach the maximum achievable volumetric productivity.

Conclusion

This chapter aimed to carry out an experimental proof-of-concept campaign for an intensified photobioreactor in the form of the ICAR thin-tube PBR featuring high biomass concentrations. The species cultivated was Chlorella vulgaris and the interior tube diameter was 4 mm for a specific illuminated area a light equal to 500 m 2 m -3 . To assess the reactor performance, the maximum theoretical volumetric productivity was predicted using the design parameters of the current prototype, i.e., specific illuminated area a light and dark fraction f d , and the absorption cross-section and kinetic growth parameters for Chlorella vulgaris. First, the gas-liquid mass coefficient k L a of the reactor was determined using a de-oxygenation/reoxygenation approach for several bubbling gas flow rates. The largest measured k L a was 2.2×10 -3 s -1 at a corresponding gas flow rate Q of 10 mL min -1 . Two batch mode trials were conducted with target biomass concentrations of 10 and 15 gL -1 . The maximum biomass concentration achieved for each trial was 9.2 and 12.0 gL -1 , respectively. For both trials, pigment content was relatively stable, suggesting that no significant light stress occurred.

One semi-continuous mode trial was performed featuring culture harvesting and injection of fresh medium. The average growth rate was 3.0 gL -1 day -1 with a photon flux density of 250 µmol hν m -2 s -1 and biomass concentration ranging from 9.6 to 16.3 gL -1 . Such high productivity and biomass for a relatively low photon flux density of may enable reduce the process energy consumption required for lighting in indoor cultivation. Nonetheless, the measured productivity was lower than the maximum theoretical volumetric productivity of 5.2 gL -1 day -1 predicted at the same photon flux density and the 7 gL -1 day -1 volumetric productivity achieved by the Algofilm © concept for a similar PFD [START_REF] Pruvost | Development of a thin-film solar photobioreactor with high biomass volumetric productivity (AlgoFilm©) based on process intensification principles[END_REF]. Thus, it is expected that the productivity of the thin-tube ICAR design can be further improved. Efforts to do so should focus on improving the gas-liquid mass transfer coefficient, culture mixing, and lighting conditions. Overall, the photobioreactor was able to achieve high biomass concentrations and productivities while showing little evidence of biofilm formation which constitutes a successful proof-of-concept for the thin-tube intensified PBR design.

CHAPTER 5

Investigation of light transfer in thin-tube photobioreactors via a Monte Carlo ray-tracing modeling approach

Introduction

Light transfer through a well-mixed suspension of microalgae is governed by the radiative transfer equation (RTE) for a homogeneous absorbing, scattering, and non-emitting medium.

Frequently, for flat plate photobioreactors (PBRs) or raceway ponds, the two-flux approximation is used to obtain an analytical solution to the one-dimensional RTE [START_REF] Pottier | A fully predictive model for one-dimensional light attenuation by Chlamydomonas reinhardtii in a torus photobioreactor[END_REF][START_REF] Lee | Design tool and guidelines for outdoor photobioreactors[END_REF][START_REF] Cornet | A structured model for simulation of cultures of the cyanobacterium Spirulina platensis in photobioreactors : I. Coupling between light transfer and growth kinetics[END_REF]. This approximation has been used extensively [START_REF] Pruvost | Knowledge models for the engineering and optimization of photobioreactors[END_REF][START_REF] Pottier | A fully predictive model for one-dimensional light attenuation by Chlamydomonas reinhardtii in a torus photobioreactor[END_REF][START_REF] Lee | Design tool and guidelines for outdoor photobioreactors[END_REF][START_REF] Cornet | A structured model for simulation of cultures of the cyanobacterium Spirulina platensis in photobioreactors : I. Coupling between light transfer and growth kinetics[END_REF][START_REF] Cornet | A simplified monodimensional approach for modeling coupling between radiant light transfer and growth kinetics in photobioreactors[END_REF] and has been successfully validated against results from a 3D RTE solvers for ponds and flat plate PBRs [START_REF] Lee | Design tool and guidelines for outdoor photobioreactors[END_REF].

For tubular photobioreactors, Cornet et al. [START_REF] Cornet | A simplified monodimensional approach for modeling coupling between radiant light transfer and growth kinetics in photobioreactors[END_REF] applied the two-flux approximation to obtain a one-dimensional solution of the radiative transfer equation for a non-emitting medium and absorbing and scattering cells. The incident radiation was assumed to be radially homogeneous and scattering by the cells was assumed to be isotropic. The given analytical solution was found to lead to values of infinity at the tube center. A reformulation of this approach was presented by Takache et al. [START_REF] Takache | Experimental and theoretical assessment of maximum productivities for the microalgae Chlamydomonas reinhardtii in two different geometries of photobioreactors[END_REF] for the case of high biomass concentration.

However, in practice and particularly under solar conditions, tubular PBRs are illuminated on a single side and the case of quasi-collimated, homogeneous, radial illumination is not realistic. Thus, methods for solving the two-or three-dimensional radiative transfer equation are necessary for accurate predictions of the radiative field within the culture.

Slegers et al. [START_REF] Slegers | Scenario analysis of large scale algae production in tubular photobioreactors[END_REF] predicted the yearly biomass productivity vertical and horizontally oriented tubular photobioreactors using solar conditions for locations in The Netherlands, France, and Algeria accounting for diffuse and collimated incidence, tube reflectance, and tube mutual shading. The impact of ground reflectance, tube material and tube spacing were considered. Here, the local fluence rate within the culture was predicted assuming that the light path was normal to the reactor walls. However, previous studies have demonstrated the angle of incidence of light on a microalgae culture and have a significant impact on the local rate fluence rate and the resulting predicted growth rate [START_REF] Souliès | Investigation and modeling of the effects of light spectrum and incident angle on the growth of Chlorella vulgaris in photobioreactors[END_REF].

Lee et al. [START_REF] Lee | Design tool and guidelines for outdoor photobioreactors[END_REF] solved the three-dimensional radiative transfer equation using a discrete

Galerkin method [START_REF] He | A high-order-accurate gpubased radiative transfer equation solver for combustion and propulsion applications[END_REF] of the radiative field predicted the presence of dark zones in the tube due to refraction at the air/liquid interface. This phenomenon was more pronounced for lower biomass concentrations where the light was not fully attenuated. Indeed, at very low biomass concentration (C X = 0.02 kg m -3 ) refraction at the air/liquid interface led to a concentrating effect where the predicted local rate of photon absorption at the back of the tube was larger than that near the illuminated surface of the tube. Nonetheless, the total fraction of photons absorbed was only slightly changed when reflection and refraction at the air/liquid interface were accounted for. Finally, the photon distribution was observed to scale with the factor C X r i .

Thus, while light transfer in tubular photobioreactors has been previously investigated, to the best of our knowledge, no study has considered the impact of tube wall thickness on the light distribution and predicted microalgae growth. Indeed, the thickness of the tube wall is relatively small for typical tubular photobioreactors with a radius on the order of a centimeter or decimeter. However, in the context of the thin-tube photobioreactor concept presented in Chapter 4, tube wall thickness is much larger compared to the outer tube radius.

Indeed, the tubular reactor in Chapter 4 had a tube wall thickness of 1 mm for an inner radius of 2 mm. In this case, the tube walls intercept one third of the total radiative energy incident on the PBR, and refraction within the tube must be accurately taken into account.

Therefore, the goal of the current study was to predict light transfer and microalgae growth rate within a tubular photobioreactor, accounting for diffuse and collimated incidence, non-normal incidence, multiple reflection and refraction events at all optical interfaces, and absorption and anisotropic scattering by the microalgae cells in suspension. The impact of tube wall thickness, incidence angle, and scattering by the microalgae on the PBR illuminated fraction and microalgae growth was considered. Finally, the daily volumetric biomass productivity was predicted for several different PBR inclination angles and orientations for solar conditions corresponding to sunny days in June and September in Saint-Nazaire,

France. The current model and results can be used as practical guidelines for optimizing the design and operation of tubular PBRs, ranging from thin-culture systems of a few millimeters to typical tubular systems of several centimeters.

Methods

Problem statement

Let us consider a tubular photobioreactor of outer radius r o and inner radius r i containing a microalgae culture of biomass concentration C X and exposed on a single side to a diffuse I dif and collimated intensity I col with an incidence angle represented by θ in taken with respect to the orthogonal vector of the tube's longitudinal axis. Then, the incident hemispherical photon flux density (PFD) was taken as q in = (π/4)I col cos(θ in ) + πI dif . The tube wall thickness t was given by t = r o -r i . The PBR inclination angle β was measured with respect to the horizontal axis. The tube was considered to be non-absorbing with a refractive index representative of that of glass i.e., n t = 1.5. The culture medium was also assumed to be non-absorbing, with a refractive index representative of that of water, i.e., n c = 1.333. The culture was comprised of a suspension of cells of the microalgae Chlorella vulgaris with PARaveraged mass absorption Āabs and scattering Ssca cross-sections taken from Ref. [START_REF] Souliès | Investigation and modeling of the effects of light spectrum and incident angle on the growth of Chlorella vulgaris in photobioreactors[END_REF] as in Chapter 2. Here, the asymmetry factor g was taken as 0.974 according to the measured value reported for C. vulgaris in Ref. [START_REF] Kandilian | Comparison of experimentally and theoretically determined radiation characteristics of photosynthetic microorganisms[END_REF].

Assumptions

Light transfer within the tubular PBR was modeled using the Monte-Carlo ray-tracing method [START_REF] Dauchet | The practice of recent radiative transfer Monte Carlo advances and its contribution to the field of microorganisms cultivation in photobioreactors[END_REF][START_REF] Hoeniges | Impact of dropwise condensation on the biomass production rate in covered raceway ponds[END_REF][START_REF] Huang | Bidirectional transmittance of transparent windows with external or backside condensation of nonabsorbing capshaped droplets[END_REF][START_REF] Zhu | Transmittance of semitransparent windows with non-absorbing cap-shaped droplets condensed on their backside[END_REF][START_REF] Zhu | Light transfer through windows with external condensation[END_REF][START_REF] Yalçln | Colored Radiative Cooling Coatings with Nanoparticles[END_REF]. To do so, the following assumptions were made [START_REF] Zhu | Transmittance of semitransparent windows with non-absorbing cap-shaped droplets condensed on their backside[END_REF] : (i) the tube wall thickness was much larger than the wavelength of radiation in the PAR region such that geometric optics prevailed. (ii) Both the tube wall and the microalgae culture medium were considered to be non-absorbing over the PAR region. (iii) Refractive indices of the air, tube wall, and culture medium were taken as constant over the PAR region. Furthermore, each of these components was considered to be non-emitting and non-scattering. (iv) All interfaces were taken as optically smooth. In other words, the surface roughness was much less than the wavelength of the light in the photosynthetically active radiation (PAR) region (λ = 400 -700 nm) such that reflection and transmission were specular. Then a photon incident at an angle θ 1 , measured with respect to the interface normal vector, upon an interface between two non-absorbing media of refractive indices n 1 and n 2 , respectively, underwent refraction and was transmitted at angle θ 2 given by Snell's law, i.e., [START_REF] Modest | Radiative Heat Transfer[END_REF] θ 2 = asin sin(θ 1 ) n 1 n 2 .

(5.1)

Then, the reflectivity, which represents the probability that said photon is reflected instead of transmitted, can be given by Fresnel's equation, i.e., [START_REF] Modest | Radiative Heat Transfer[END_REF] ρ = 1 2

tan 2 (θ 1 -θ 2 ) tan 2 (θ 1 + θ 2 ) + sin 2 (θ 1 -θ 2 ) sin 2 (θ 1 + θ 2 ) . (5.2)
Predictions of the light transfer and microalgae growth within the the PBR were based on the assumptions that (v) the PBR was operated in the light-limited regime wherein growth was only a function of the amount of light available to the microalgae cells [START_REF] Souliès | Investigation and modeling of the effects of light spectrum and incident angle on the growth of Chlorella vulgaris in photobioreactors[END_REF][START_REF] Lee | Design tool and guidelines for outdoor photobioreactors[END_REF][START_REF] Pruvost | Theoretical investigation of biomass productivities achievable in solar rectangular photobioreactors for the cyanobacterium Arthrospira platensis[END_REF].

Thus the PBR was not limited by any other operating parameters such as temperature, pH, nutrient availability, or dissolved oxygen concentration. (vi) The microalgae culture was well-mixed with uniform biomass concentration C X throughout the PBR. (vii) The tube was considered to be infinitely long such that edge effects at the top and bottom of the PBR were neglected and the light transfer within the tube was two-dimensional. In other words for a given case the local fluence rate was only a function of the x and y coordinates.

Method of solution

Monte Carlo ray-tracing

In order to predict the local rate of photon absorption (LRPA) for a microalgae culture within a tubular PBR, a Monte Carlo ray-tracing (MCRT) method was used [START_REF] Hoeniges | Effect of colony formation on light absorption by Botryococcus braunii[END_REF][START_REF] Hoeniges | Impact of dropwise condensation on the biomass production rate in covered raceway ponds[END_REF][START_REF] Zhu | Transmittance of semitransparent windows with non-absorbing cap-shaped droplets condensed on their backside[END_REF][START_REF] Zhu | Light transfer through windows with external condensation[END_REF] was employed. A large number N rays of discrete collimated photon bundles or "rays"

were incident upon the tubular PBR. The incident photon bundles could undergo multiple refraction and reflection events at the air/tube, tube/culture, culture/tube, and tube/air interfaces, as well as scattering and absorption events in the microalgae culture. The rays were tracked until they were either (i) reflected away from the PBR, (ii) transmitted through the PBR, or (iii) absorbed in the microalgae culture. The step-by-step process is outlined in the flow diagram in Figure 5.2 and was as follows :

1. Randomly generate an emission location p 0 = (x 0 , y 0 , z 0 ) on the plane in front of the tube.

2. Determine ray direction vector s i based on the incidence angle θ in 3. Calculate the intersection location p between the ray and the outer air/tube interface and the reflectance ρ of the air/tube interface based on the incidence angle θ in between the ray and the outward pointing normal vector of the tube surface using the generalized Fresnel's equations (see Equation (5.2)).

4. Generate a random number R between 0 and 1 from the uniform distribution. (b) Else if l β < l geo , attenuation occurs at a location a distance of l β along the current propagation direction from its last attenuation, reflection, or refraction event at p . Use the culture single-scattering albedo ω = σ/(σ + α) and a random number R between 0 and 1 to determine if ray is scattered or absorbed.

i. If R < ω, ray is scattered. A new scattered direction s sca is defined by a randomly generated azimuthal ψ sca angle and a polar θ sca angle generated by using a new random number R and the integrated accumulated distribution of the Henyey-Greenstein function, i.e., [START_REF] Yalçln | Colored Radiative Cooling Coatings with Nanoparticles[END_REF] cos

(θ sca ) = 1 2g 1 + g 2 - 1 -g 2 1 + gs 2 (5.4)
where s = 2R -1. Return to Step 6.

ii. If R > ω, ray is absorbed and the absorption location p abs = (x abs , y abs ) is recorded.

The above process was completed for a large number of rays, until each ray was either reflected or transmitted away from the PBR or absorbed at given location p abs = (x abs , y abs ).

Recall that the radiative field within the tube was assumed to be independent of the zdirection. The cross-section of the tube was discritized into a N el × N el rectangular mesh featuring square elements of width ∆y = ∆x = 2r i /N el . A matrix [N abs ] was defined wherein each element (i, j) contained the number of rays absorbed N abs,i,j at a location p abs = (x abs , y abs ) such that x i-1 < x abs < x i and y i-1 < y abs < y i where x i and y i represent the right-hand and upper element boundaries, respectively, of element (i, j).

Each ray can be said to carry an amount of energy E ray equal to

E ray = S light q in N rays (5.5)
where S light is the illuminated surface area, given by S light = πr o H for illumination on a single side and H is the tube height considered. Then, the local rate of photon absorption A(x, y) in µmol hν kg -1 s -1 (LRPA) can be calculated from the ray absorption matrix [N abs ] according to

A(x i , y j ) = N abs,i,j E ray V el C X (5.6)
where V el is the element volume given by ∆x 2 H. Note that the tube height H is contained in both the E ray and V el terms and will thus cancel yielding results that are independent of the tube height considered. The corresponding illuminated fraction γ can then be given by In the case of solar simulations, the incident hemispherical PFD q in was composed of both collimated, direct radiation and diffuse radiation. In this case, the MCRT algorithm was executed twice, once for an incident collimated intensity of I col and once for an incident diffuse intensity of I dif yielding two separate matrices containing the local rate of photon absorption, denoted by [A col ] and [A dif ]. The two matrices were superimposed to calculate the total LRPA according to [START_REF] Lee | Design tool and guidelines for outdoor photobioreactors[END_REF] [

γ = A(A > A c ) πr 2 i (5.
A] = [A col ] + [A dif ].
(5.9)

Microalgae growth kinetics

The microalgae growth kinetics model for Chlorella vulgaris employed in Chapter 2 was used to predict the growth rate as a function of the LRPA. The relevant parameters for the growth kinetics model can be found in Table 2.1. Equations (2.12) -(2.13) were used to predict the local volumetric growth rate r X (x, y) (in kg m -3 h -1 ). Then, the average volumetric growth rate r x was calculated according to

r x = A r X (x, y)dA (5.10)
where A is the cross-sectional area of the tube given by πr 2 i .

Validation

In the case of quasi-collimated, radially incident light, the radiative field within a tubular PBR can be resolved by applying the two-flux method to obtain an analytical solution to the one-dimensional radiative transfer equation. This solution was proposed by Takache et al [START_REF] Takache | Experimental and theoretical assessment of maximum productivities for the microalgae Chlamydomonas reinhardtii in two different geometries of photobioreactors[END_REF] and is given by G(r) q in = 2 I 0 (δr) I 0 (δr i ) + αI 1 (δr i ) (5.11) where I 0 (R) and I 1 (R) represent the zero-and first-order modified Bessel functions of the first kind and δ is given by Equation (2.9). Note that the radial coordinate r is defined with respect to the tube center.

To validate the current MCRT method, the local fluence rate G(r) was predicted for an example case using PBR geometry and microalgae radiative properties from Ref. [START_REF] Takache | Experimental and theoretical assessment of maximum productivities for the microalgae Chlamydomonas reinhardtii in two different geometries of photobioreactors[END_REF]. Indeed, G(r) decreased to zero for r > 4 cm indicating complete light absorption. Here, good agreement between the two methods was found for all values of tube radius. This suggests that the predictions of G(r) given by the two methods agree when total light absorption occurs, whereas at low biomass concentrations, the MCRT predicts larger G(r) due to the rays converging at the tube center point. This suggests that the MCRT can accurately predict the illuminated fraction γ and local fluence rate G(r) in the case of full light attenuation, i.e., γ < 1.

Convergence

Since the Monte-Carlo algorithm is statistical method requiring the sampling of a large number of rays, the variability of the results will depend on the number of ray paths sampled and the corresponding mesh used to discretize the absorption field and calculate the illuminated fraction γ. For all values of N el and biomass concentration, the standard deviation in γ was less than 0.005 for N rays greater than 0.5 × 10 5 , corresponding to a 95% confidence interface of ± 1 %. However, Figure 5.4 indicates that despite this, the illuminated fractions γ predicted for different values of N el were not in agreement for a number of rays N rays as low as 0.5 × 10 5 for a given biomass concentration. Indeed, the predictions given for all values of N el converged for N rays ≥ 0.5 × 10 6 and N rays ≥ 0.5 × 10 7 for biomass concentrations of 0.5 and 1.0 kg m -3 , respectively. This indicates that the number of rays N rays required to achieve numerical convergence depended on the conditions within the microalgae culture, i.e., biomass concentration. Thus, for all simulations in the current study, a number of rays N rays was used such that the illuminated fractions γ predicted using N el equal to 100 and 200 were within ± 2% of the value predicted for N el = 50. 3) that γ is an important parameter for describing light conditions in the PBR. A value of γ > 1 indicates that some light is transmitted, while γ < 1 indicates that dark volumes occur within the culture, thus productivity is maximized for the "luminostat" regime where γ = 1.

Results and Discussion

Here, the interest of a thin-tube design is illustrated by the large illuminated fraction γ at relatively large biomass concentrations and low photon flux densities. This was especially the case for a thin tube with an inner radius r i of 2 mm where γ ranged from 0.67 to 0.12 for biomass concentrations of 5 and 30 kg m -3 , respectively.

109

C X /a light = C X r i representative of the culture optical thickness. The results demonstrate a clear scaling relationship between γ and the optical thickness. Indeed, the same scaling relationship has been demonstrated in Chapter 2 and numerous previous studies between the areal productivity P A (in kg m -2 h -1 ) and the optical thickness of a microalgae culture in raceways ponds, flat-plat PBRs, and tubular PBRs under a variety of conditions [START_REF] Lee | Design tool and guidelines for outdoor photobioreactors[END_REF][START_REF] Hoeniges | Impact of dropwise condensation on the biomass production rate in covered raceway ponds[END_REF].

Figure 5.5b demonstrates that the scaling behavior P A observed previously is a direct result of the same behavior observed for γ. Thus, despite the complexity of the underlying phenomena involved in light transfer, the results presented in the current study will be valid for any tubular PBR inner radius r i and culture biomass concentration C X such that the optical thickness C X r i and lighting conditions, i.e., incidence angle, incident diffuse and collimated photon flux, are the same.

Impact of tube wall thickness

To assess the impact of the tube on the light transfer within the photobioreactor, the flux transmitted through the tube/culture interface q cult and thus available to the microalgae cells for different tube wall thicknesses t was considered. The photon flux available to the culture q cult was calculated using the MCRT algorithm by terminating a given ray once it was transmitted through the tube/culture interface. Then q cult was taken as

q cult = E ray N cult πr i h (5.12)
where N cult is the number of rays transmitted through the tube/culture interface and available to the culture for absorption. To generalize the results, the ratio q cult /q in was considered as a function of the normalized tube wall thickness, i.e., t/r o . Note that the ratio q cult /q in is only a function of the tube geometry and is independent of the radiative properties of the microalgae cells, i.e. Āsca , Ssca , g, and the operational parameters of the PBR, i.e., C X , q in . Thus, the normalized tube wall thickness t/r o can be considered a design parameter for tubular photobioreactors.

Figure 5.6a plots the ratio q cult /q in for normalized tube wall thicknesses t/r o ranging from 0.01 to 0.9 and for a collimated photon flux with incidence angle θ in ranging from 0 • (i.e., normal incidence) to 80 • . Similarly, Figure 5.6b plots q cult /q in as a function of t/r o for the case of diffuse incidence. For collimated incidence, Figure 5.6a indicates that the ratio q cult /q in increased with increasing tube wall thickness. In fact, for all incidence angles considered, q cult /q in exceeded unity for large values of normalized tube wall thickness. This indicates that the photon flux available to the culture was larger than photon flux incident on the PBR, i.e., q cult > q in . Indeed, as tube wall thickness increased, the area of the tube increased and more light was intercepted by the PBR. This additional light was then refracted at the air/tube interface towards the tube/culture interface creating a concentrating effect and resulting in q cult /q in > 1. This suggests that a large tube wall thickness t relative to the outer radius r i of the PBR may increase the microalgae growth rate by increasing the photon flux available to the microalgae cells. For diffuse incidence, Figure 5.6b indicates that a concentrating effect was observed for t/r o > 0.2, however, it was significantly less pronounced than the direct incidence case since only a small fraction of rays were incident at a direction that would result in them being refracted towards the culture.

For lower incidence angles θ in of 0 • , 20 • , and 40 • , q cult /q in increased with increased t/r o until it reached a plateau of 1.25, 1.29, and 1.42, respectively. Once this plateau was reached, increasing the tube wall thickness did not increase q cult /q in since the additional rays captured were not incident on the tube/culture interface. For incidence angles θ in > 40 • , q cult /q in increased monotonically with increasing normalized tube wall thickness t/r o . For tube wall thickness t/r o ≥ 0.5, q cult /q in increased with increasing incidence angle θ in (with the exception of θ in = 80 • ). was equal to 0.01 kg m -2 for each inner radius r i simulated. The results indicate that the illuminated fraction γ increased to a maximum value with increasing tube wall thickness t for both incidence angles θ in and all inner radii r i considered. This can be explained by the concentrating effect observed in Figure 5.6a for thicker tube walls which sent more light to the microalgae culture, increasing the light penetration depth and the illuminated fraction γ. Figure 5.7a also indicates that γ was smaller for an incidence angle θ in equal to 45 • . This was attributed to an increase in the reflectance at the air/tube interface and a decrease total in the total energy incident upon the PBR in the case of non-normal incidence. Figures 5.7a

and 5.7b demonstrate that t/r o is a useful parameter than can be adjusted to increase γ.

Nonetheless, for the cases plotted, adjusting t/r o was not sufficient to reach γ = 1. Instead, other parameters such as optical thickness C X r i and/or incident photon flux q in would also have to be adjusted to achieve γ = 1.

Figure 5.7b plots the same predicted illuminated fraction as a function of the normalized tube wall thickness t/r o . The results for all inner radii r i collapse unto the same curve for both incidence angles. This result was also observed for diffuse incidence with I dif such that

q in =250 µmol hν m -2 s -1 (see Appendix Figure B.1).
For the current value of C X r i = 0.01 kg m -2 , the optimum normalized tube wall thickness was 0.25 and 0.4 for θ in equal to 0 • and 45 • , respectively. Figure 5.7b indicates that γ decreased steadily with increasing t/r o for both incidence angles. This was the case despite the fact that the ratio q cult /q in increased and then plateaued for values of t/r o beyond the optimum values identified here. This can be explained by the distribution of light absorption between t/r o equal to 0.5 compared to 0.9. This is illustrated in Figures 5.7c and 5.7d, respectively, which show the areas where the LRPA was greater than the compensation LRPA, i.e., A(x, y) > A c , in red and the areas where A(x, y) < A c in yellow for θ in = 45 • . Notably, at t/r o = 0.5, Figure 5.7c indicates that the culture was illuminated from the sides as well as the front due to the refractive index mismatch between the air and tube walls causing light to be refracted towards the inner cylinder (see Figure 5.6c and 5.6d). However, for t/r o = 0.9, Figure 5.7d indicates that only the front half of the culture was illuminated. Indeed, as t/r o increased, the radius r i decreased relative to the outer radius r o and the refraction effects of the tube wall were less important.

Figure 5.8a plots the resulting predicted average volumetric growth rate r X (in kg m -3 day -1 ) from the light transfer simulations presented in Figure 5.7. The average volumetric growth rate was also predicted for diffuse incidence and is plotted in Figure 5.8b. For all cases, the average growth rate is also shown for a flat panel reactor with a glass wall of the same thickness and optical properties as the tube wall. For flat panels, the optical thickness based on culture depth L was the same as that for tubular reactors, i.e., C X L = 0.01 kg m -2 . As with the illuminated fraction γ, the results demonstrate that the growth rate r X increased with increasing tube wall thickness t/r o to a maximum value for each incidence angle. A similar trend was observed for diffuse incidence in Figure 5.8b. Moreover, we note that the maximum value of γ and r X occurred at the same tube wall thickness t/r o for a given incidence angle (see Figure 5.7b). These results suggest that the biomass growth rate can be improved by optimizing the tube wall thickness. Indeed, for θ in = 45 • , increasing t/r o from 0.1 to 0.4 resulted in an increase in the predicted average growth rate of 1 kg m -3 day -1 which represents a 37% increase in growth. Similar results were observed for diffuse incidence where increasing t/r o from 0.1 to 0.4 resulted in an increase of 0.4 kg m -3 day -1 or 8%. For a flat panel reactor, the predicted growth rate r X was independent of the wall thickness since the glass was considered to be non-absorbing. Furthermore, unlike tubular reactors, flat panel PBRs did not exhibit a concentrating effect, thus the photon flux delivered to the culture was lower, and the predicted growth rate was smaller.

Impact of incidence angle

Tubular PBRs under solar conditions will can experience the whole range of possible incidence angles, i.e., θ in = 0-90 • . Thus, Figure 5.9 plots the predicted illuminated fraction γ for the case of collimated incident radiation as a function of incidence angle θ in for normalized tube wall thickness t/r o equal to 0.01, 0.30, and 0.50. The incident collimated intensity I col was 250 µmol hν m -2 s -1 and the optical thickness was C X r i = 0.02 kg m -2 . Here, the illuminated fraction decreased monotonically with increasing incidence angle θ in for all three values of t/r o . This was due to increased reflection at the air/tube interface as the incidence angle increased. For incidence angles θ in < 40 • the illuminated fraction was largest for t/r o = 0.30. This was thanks to the larger concentrating effect of the thicker tube walls compared to t/r o = 0.01 and the refraction at the air/tube interface which illuminated the front and sides of the culture as observed in Figure 5.7c. However, for incidence angles θ in > 40 • , the illuminated fraction γ was largest for t/r o = 0.50. This was due to the concentrating effect observed in Figure 5.6d which moved the focal point of the refracted rays towards the tube center of the tube as the incidence angle increased.

Impact of scattering

Figure 5.10 plots the illuminated fraction γ as a function of the optical thickness C X r i for the case of an asymmetry factor g equal to unity, i.e., negligible scattering by the microalgae, and for the case of an asymmetry factor g equal to 0.974 as reported in the literature for Chlorella vulgaris. The photon flux density q in was composed of collimated, normally incident radiation (i.e., θ in = 0 • ) and was equal to 200 µmol hν m -2 s -1 . For both cases, the illumi- nated fraction γ decreased exponentially with increasing biomass concentration (see Figure 5.5). However, Figure 5.10 indicates that γ was larger when scattering by the microalgae was neglected, i.e., g = 1.0, than when it was accounted for with an asymmetry factor of g = 0.974. Interestingly, this was the case despite the fact that an asymmetry factor g of 0.974, representing a strongly forward-scattering particle, is only slightly different than an asymmetry factor g = 1.0, representing a perfectly forward-scattering particle. This can be explained by the large mass scattering cross-section of microalgae relative to the absorption cross-section, which resulted in a large single-scattering albedo ω. Thus, photons could deviate significantly from the forward direction due to the many scattering events that took place before a photon was absorbed. Assuming negligible scattering by the microalgae could cause the illuminated fraction γ to be over estimated by 25% at low biomass concentrations where γ was the largest. Then, scattering should not be neglected for light transfer modeling in tubular photobioreactors, as has been done previously [START_REF] Marotta | Reflection-refraction effects on light distribution inside tubular photobioreactors[END_REF]. flux G(x, y)/q in for an microalgae asymmetry factor g equal to 1 and 0.974 and optical thickness C X r i of 0.05 kg m -2 . Figure 5.11b plots the area where the LRPA was greater than that of the the compensation LRPA, i.e., A(x, y) > A c . For the case of g = 1.0, the incident light was scattered in the forward direction and therefore penetrated deeper into the microalgae culture. This phenomenon explains the larger values of illuminated fraction γ achieved for the culture with negligible scattering. On the other hand, for g = 0.974, larger normalized local fluence rate G(x, y)/q in near the culture surface indicated that scattering by the algae caused light to be absorbed closer to the culture surface. Finally, 5.11b indicates that lines of larger G(x, y)/q in were present in the lower half of the tube when scattering was neglected. However, these rays were were not present for g = 0.974. This suggests that, unlike flat plate photobioreactors [START_REF] Lee | Design tool and guidelines for outdoor photobioreactors[END_REF][START_REF] Kandilian | Simple method for measuring the spectral absorption cross-section of microalgae[END_REF], the local fluence rate G(x, y) is highly sensitive to the microalgae asymmetry factor g in tubular photobioreactors.

Solar modeling

To assess the impact of tube wall thickness and PBR inclination angle β for solar microalgae cultivation, the daily volumetric productivity of a thin-tube photobioreactor was predicted for solar conditions in Saint-Nazaire, France (47.25 • N, 2.26 • ) in the months of June and September. To do so, the incident diffuse and total solar photon flux density were measured at the AlgoSolis R&D research facility in Saint-Nazaire, France (BF5 Sunshine Sensor, Delta-T Devices). The direct solar PFD was calculated by subtracting the diffuse part from the total measured solar PFD. The incident direct and diffuse PFD were then averaged over a time increment ∆t = 1 h, and the solar position was used to define the incidence angle θ in for a given PBR inclination angle β. Then, the temporal evolution of the biomass concentration was estimated by assuming the volumetric growth rate r X to be constant over the time step ∆t. Then, the current MCRT algorithm predicted the local rate of photon absorption A(x, y) according to Equations (5.6) -(5.9) and the biomass concentration at time t + ∆t was calculated according to

C X (t + ∆t) = C X (t) + r X ∆t (5.13)
and the volumetric biomass productivity could be calculated according to

P V = C X (t f ) -C X (t 0 ) ∆t (5.14)
where C X (t f ) and C X (t 0 ) represent the biomass concentration at sunset and sunrise, respectively, and ∆t was taken here as one day. all values of tube wall thickness in June. This was due to the high solar elevation angle which created large incidence angles with the tube surface and resulted in lower illuminated fraction γ as observed in Figure 5.9. For the other three PBR orientations tested in June, the normalized tube wall thickness that yielded the maximum productivity was t/r o = 0.3, this corresponds to a tube wall thickness of 1 mm for a tube with inner radius r i = 2 mm and is representative of the tube geometry used in the thin-tube demonstration experiments in Chapter 4. However, for the vertically oriented tube, the maximum biomass productivity was achieved at t/r o = 0.6. This was due to the increase in γ afforded by larger tube wall thicknesses for θ in > 40 • also observed in Figure 5.9. For the month of September, Figure 5.12b demonstrates that the optimal tube wall thickness t/r o was equal to 0.3 for all four 12f demonstrates that the predicted illuminated fraction in September was slightly larger than that predicted in June despite higher predicted volumetric productivity in June. This can be explained by the longer days and larger solar PFD in the summer, which allowed more time for the biomass to grow and increased the rate of photosynthetic growth in areas where the LRPA was greater than the compensation LRPA, i.e., A(x, y) > A c .

Conclusion

This chapter presented a detailed analysis of light transfer in tubular photobioreactors.

A Monte Carlo ray-tracing method was developed and validated against an analytical solution of the radiative transfer equation using the two-flux method. The light transfer model accounted for diffuse and collimated incidence, non-normal incidence, multiple reflection and refraction events at all optical interfaces, and anisotropic scattering and absorption by the microalgae cells in suspension. First, the MCRT method was used to demonstrate the scaling relationship between the illuminated culture fraction γ and the optical thickness, represented by the product of the biomass concentration and the tube inner radius C X r i , for a given incident photon flux density. Indeed, the scaling relationship between the areal productivity and the optical thickness demonstrated in Chapter 2 was considered to be a direct result of the same behavior observed for γ. The current ray-tracing algorithm was then used to assess the impact of tube wall thickness, angle of incidence, and scattering by the microalgae on the local rate of photon absorption, the illuminated fraction, and the average volumetric growth rate of the microalgae culture. The illuminated fraction was found to scale with the tube wall thickness normalized by the tube outer radius t/r o . Furthermore, γ was found to increase with increasing t/r o due to refraction by the tube walls which created a concentrating effect and increased the photon flux density incident on culture. This concentrating effect was more pronounced at larger incidence angles θ in . Scattering by the microalgae was found to decrease γ. Hence, neglecting scattering in light transfer models of tubular PBRs may cause γ to be overestimated. Finally, the volumetric biomass productivity was predicted for solar conditions in June and September in Saint-Nazaire for several different PBR orientations.

The predicted volumetric productivity was as large as 4.9 and 3.5 kg m -3 day -1 for June and September, respectively, demonstrating the interest of the thin-tube ICAR for solar cultivation. In both June and September, the volumetric biomass productivity was maximized for a normalized tube wall thickness t/r o = 0.3. In fact, optimizing the tube wall thickness was predicted to increase the volumetric productivity of vertical tubular PBRs by as much as 50% from 2.1 to 3.1 kg m -3 day -1 for solar conditions in September. Overall, the method and results presented in the current study can be used to optimize light transfer in a variety tubular PBR technologies through design via (tube wall thickness and inclination angle) and operation (via culture optical thickness). Finally, the thin-tube ICAR PBR concept should be operated for longer time periods in continuous mode to further test its stability and capacity to avoid biofilm formation.

Finally, Chapter 5 leveraged a Monte Carlo ray-tracing algorithm to perform a detailed investigation of light transfer in tubular photobioreactors accounting for diffuse and collimated incidence, non-normal incidence, multiple reflection and refraction events at all optical interfaces, and absorption and anisotropic scattering by the microalgae cells in suspension.

Similar to the scaling relationship observed in Chapter 2, the predicted culture illuminated fraction γ was shown to scale with the product of the biomass concentration and the tube inner radius C X r i . The spatial distribution of the local rate of photon absorption in the culture was found to be very sensitive to the cell asymmetry factor g. Scattering by the microalgae was found to favor absorption near the illuminated surface and decrease the predicted illuminated fraction γ. Furthermore, the illuminated fraction γ was shown to scale with the tube wall thickness normalized with respect to the tube outer radius, i.e., t/r o . As an original result, the illuminated fraction γ increased with increasing tube wall thickness to a maximum and then decreased. This was due to a mild concentrating effect, wherein light intercepted by the tube walls was refracted towards the inner cylindrical culture. The effects on the resulting growth were then investigated by predicting the microalgae growth as a function of tube wall thickness. The maximum growth rate and illuminated fraction γ occurred at the same value of tube wall thickness t/r o . This confirmed the relevance of γ for maximizing biomass productivity, despite the fact that light transfer was found to be far more complex in thin-tube geometries compared to that in classic PBR geometries (i.e. flat panels, raceway ponds, large diameter tubes). Finally, the productivity of a the ICAR concept thin-tube PBR was predicted for solar conditions in June and September in Saint-Nazaire, France. Optimizing mild concentrating effect of recieved light via the tube wall thickness increased the daily volumetric productivity by as much as 1 kg m -3 day -1 or 50% from 2.1 to 3.1 kg m -3 day -1 for solar conditions in September. Abstract: Microalgae cultivation costs must be reduced to be used as a renewable, lowcarbon source of food, energy, and valueadded products. To this end, two approaches are considered to increase the productivity of microalgae cultivation systems via light and culture intensification principles. First, the use of external mirrors to increase the productivity of outdoor raceway ponds was assessed theoretically and experimentally. The impact of the mirror location, solar position, and operating parameters such as biomass concentration and culture depth were considered. Indeed, up to a 37 % increase in volumetric biomass productivity was observed exper-imentally under clear skies. Then, a proof-ofconcept experimental campaign of an intensified thin-tube photobioreactor targeted for biofaçade applications was performed. The thintube design enabled high biomass productivities with minimum biofilm formation. However, culture growth was found to be limited by oxygen over-accumulation. To optimize the thintube design, a detailed study of light transfer was conducted. A Monte Carlo ray-tracing approach was used to predict the local fluence rate within the culture for a given solar PBR geometry, inclination, and operating conditions.

  bassins dont le rapport longueur/largeur était supérieur à 5. De plus, la hauteur maximale optimale des miroirs a été rapportée pour les configurations C et D pour Los Angeles, CA, et Saint-Nazaire, France. Dans l'ensemble, la configuration C, avec un seul miroir sur le côté nord d'un bassin orienté selon l'axe est-ouest, a été considérée comme la méthode la plus simple et la plus pertinente d'un point de vue coût pour améliorer la productivité de la biomasse dans les raceways extérieurs. Ces résultats fournissent des conseils pratiques pour la conception et l'exploitation de raceways munis de miroirs afin d'améliorer la productivité de la biomasse. Suite à cette démonstration théorique du potentiel des miroirs comme solution novatrice pour augmenter le flux solaire incident sur les cultures de microalgues en extérieur afin d'améliorer leur productivité en biomasse, nous avons cherché à évaluer expérimentalement l'augmentation du flux solaire permise par les miroirs externes et l'impact correspondant sur la productivité en biomasse. Des expériences ont été menées du printemps à l'automne dans deux raceways, avec et sans miroirs, cultivant simultanément des Chlorella vulgaris en utilisant la lumière du soleil comme source lumineuse. Deux des configurations précédemment discutées ont été testées, à savoir la configuration B avec une orientation nord/sud et des miroirs sur ses côtés est et ouest, et la configuration C, avec une orientation est/ouest et un seul miroir sur son côté nord. Ces expériences avaient pour but d'évaluer les performances dans des conditions météorologiques changeantes et élucider l'impact des miroirs sur des paramètres non étudiés par le modèle tels que la température de culture et les pertes par évaporation. La campagne expérimentale a été menée sur deux raceways à l'échelle pilote, en mode batch et continu de culture, dans des conditions solaires avec deux configurations de miroirs différentes. Pour chaque essai, un bassin identique sans miroir a été exploité simultanément comme référence pour évaluer l'impact des miroirs. Pour les deux configurations, la présence de miroirs a permis d'augmenter la densité du flux de photons solaires incident sur la culture par temps clair. Par exemple, l'énergie solaire incidente totale a été augmentée de 52,2 % et 39,6 % pour les configurations B et C, respectivement, lors d'une journée env soleillée de septembre à Saint-Nazaire, en France. Cependant, l'augmentation du PFD flux

3 . 7 4 . 6 5 . 7

 374657 photobioreactor considered in the current study. . . . . . . . . . . . . . . . . . .

Figure 1 . 1 :

 11 Figure 1.1: Micrographs of (a) the filamentous Arthrospira platensis (Spirulina), (b) the predominately single-cell algae Chlorella vulgaris.

Figure 1 . 2 :

 12 Figure 1.2: Oil yield of various sources of biomass adapted from data in Ref. [3].

Figure 1 . 3 :

 13 Figure 1.3: A schematic illustration of the inputs and products of photosynthesis, as well as the interaction between of the light-driven reactions in the thylakoid and the Calvin cycle [23].

Figure 1 . 6 :

 16 Figure 1.6: Spectral specific absorption cross-section A abs,λ for various photosynthetic pigments, namely chlorophyll a, b, and c, photosynthetic (PSC) and photoprotective (PPC) carotenoids, as well as phycoerythrobilin (PEB) and phycourobilin (PUB) adapted from data given by Bidigare et al. [27].

Figure 1 .

 1 Figure 1.7 depicts a schematic illustration of light transfer through a microalgae culture governed by the RTE. The first term on the right-hand of Equation (1.3) represents the attenuation of the local intensity I λ (r, ŝ) due to absorption. Here, κ λ represents the effective spectral absorption coefficients, in m -1 of the microalgae suspension given by

Figure 1 . 7 :

 17 Figure 1.7: A schematic depiction of the light propagation through a microalgae suspension featuring absorption, out-scattering, and in-scattering adapted from [30].

Figure 1 . 8 :

 18 Figure 1.8: Example of spectral radiative properties of microalgae for the species Chlorella vulgaris taken from Ref. [14].

Figure 1 . 9 :Figure 1 . 10 :

 19110 Figure 1.9: A schematic depiction of microalgae cultivation, including required inputs and various output products.

Figure 1 .

 1 Figure 1.11: A schematic illustration of solar radiation consisting of collimated direct I col radiation incident at an angle θ and diffuse I dif radiation on a PBR surface [13].

Figure 1 . 12 :

 112 Figure 1.12: Light attenuation profiles and corresponding growth rate r X and biomass concentration C X for the cases of (a) full light attenuation, i.e, γ < 1, (b) the luminostat regime, i.e., γ = 1, and (c) the kinetic regime, i.e., γ > 1 [71].

1. 4 . 4 . 1 Figure 1 .Figure 1 . 13 :

 4411113 Figure 1.14a. A tracking system maintained the tubes in the focal point of the lens throughout the day. The concentrating lenses provided a large reported incident photon flux ranging from 2,000 to 7,000 µmol hν m -2 s -1 and the biomass concentration varied from 1 to 3 g L -1 with

1. 4 . 4 . 2 Figure 1 . 14 :

 442114 Figure 1.14: Photos and schematics of (a) a tubular photobioreactor subjected to concentrated sunlight [76], (b) a prototype volume-illuminated photobioreactor featuring lightdiffusing fiber optics [77-79], and (c) a solar-tracing flat panel PBR [80].

  Gifuni et al.[START_REF] Gifuni | New ultra-flat photobioreactor for intensive microalgal production : The effect of light irradiance[END_REF] presented a thin flat-panel photobioreactor based on the same culture intensification principles. A photo and schematic representation of the thin flat-panel reactor are depicted in Figure1.15b. Here too, Chlorella sorokiniana was cultivated under artificial lighting conditions with incident photon flux ranging from 50 to 1000 µmol hν m -2 s -1 . The reactor thickness was 3 mm, corresponding to a specific illuminated area a light of 333 m -1 .

Figure 1 . 15 :

 115 Figure 1.15: Photos and schematics of (a) a liquid foam-bed photobioreactor [81], (b) a thin flat-panel photobioreactor [82], and (c) a the Algofilm © photobioreactor consisting of an enclosed thin falling-film [15].

  Based on the photobioreactor engineering principles discussed previously and within the context of this thesis, light and culture intensified reactors were considered with the goal of increasing the biomass productivity of two different solar microalgae cultivation systems, namely open raceway ponds and closed tubular photobioreactors.The first intensification approach was applied to a classic raceway pond operated under solar conditions and consisted of increasing the average illuminated fraction throughout the day by intensifying the photon flux incident on the culture. This was done by integrating external mirrors into the raceway pond design, to redirect sunlight towards the microalgae culture. Chapter 2 presents a theoretical assessment of the performance enhancement provided by this approach by coupling a model of the radiative transfer within the culture with a kinetic growth model for Chlorella vulgaris. Various mirror configurations, raceway pond locations, and operating conditions are considered. Chapter 3 presents the results of a pilot scale external reflecting raceway pond operated in Saint-Nazaire, France in both batch and continuous mode for multiple seasons, weather conditions, and mirror configurations.The second approach was applied to a tubular photobioreactor and consisted of intensifying the culture biomass concentration by increasing the culture specific illuminated area a light via decreased tube radius. A proof-of-concept experimental campaign is presented in Chapter 4 for a thin-tube PBR with an inner radius of 2 mm under artificial lighting conditions. The reactor (ICAR concept) was operated in both batch and semi-continuous modes, and the gas-liquid mass transfer coefficient was measured. Then, Chapter 5 presents a detailed model of the light transfer within the thin-tube photobioreactor to guide its design and operation. The impact of tube thickness, reactor inclination angle, and biomass concentration are investigated with the goal of maximizing the reactor illuminated fraction and volumetric productivity under solar conditions.

CHAPTER 2 A

 2 novel external reflecting raceway pond design for improved biomass productivity J. Hoeniges, W. Welch, J. Pruvost, L. Pilon, "A novel external reflecting raceway pond for improved biomass productivity", Algal Research, vol. 65, no. 102742, 2022.

2. 2 . 1

 21 Problem statement Let us consider an outdoor rectangular raceway pond of width W = 1 m and length l = 2 m located in either Los Angeles, CA, USA (34.07 • N, 118.44 • W) or Saint-Nazaire, France (47.25 • N, 2.26 • W) growing a culture of Chlorella vulgaris of depth L equal to 0.1 m, 0.2 m, or 0.3 m. These locations were chosen due to their difference in latitude in the northern hemisphere. The pond was exposed to direct, collimated solar radiation G S,λ

Figure 2 .Figure 2 . 1 :

 221 Figure 2.1A. Here, γ s = -90 • corresponded to due east and γ s = 90 • corresponded to due

Figure 2 . 2 :

 22 Figure 2.2: (a) A side view (not to scale) of Configurations B-D illustrating the reflection width W ref , pond spacing width W sp , and the apparent solar zenith angle θ z,a . (b) Mirror height H B/C/D (t) given by Equations (2.2)-(2.4) as a function of time of day on September 21 st in Los Angeles, CA for maximum allowed mirror height H * = 1 m.

(2. 5 )

 5 The amount of additional light reflected onto the culture for each pond configuration can be assessed by considering the ratio S ref /S of the culture surface area subjected to reflected sunlight S ref to the total culture surface area S = W l. The ratio of S ref /S ranged from zero,

  Figure A.1 of the Appendix A. Finally, the local rate of photon absorption (LRPA) by the microalgae cells, denoted by

Figure 2 . 3 :

 23 Figure 2.3: Block diagram illustrating the computational procedure used for predicting the temporal evolution of biomass concentration C X (t) and the daily biomass productivities for raceway Configurations A-D.

2. 3 . 1 Figure 2 .Figure 2 . 4 :

 31224 Figure 2.4a plots the fraction S ref /S of the total culture area S receiving reflected light for Configurations A-D as a function of time on September 21 st in Los Angeles, CA. Note that S ref /S was zero throughout the day for Configuration A since no reflecting mirrors were present, i.e., S ref = 0. Both Configurations B and D experienced a decrease in the reflected area S ref at midday. This was due to the small apparent solar zenith angle θ z,a at midday and the fact that the mirror height was limited to H * = 1 m (see Figure 2.2b). The ratio of S ref /S for Configuration B was nearly unity at the beginning and end of the day when the sun was positioned facing the west and east mirror, respectively. On the other hand, S ref /S = 1 for several hours in the morning and evening for Configuration D thanks to the tracking system

Figure 2 .Figure 2 . 5 :

 225 Figure 2.4b plots the incident photosynthetic photon flux q in,P AR (Equation (2.7)) averaged over the culture surface area as a function of time on September 21 st for Configurations A-D. It indicates that Configurations B-D increased the mean incident photosynthetic photon flux throughout the day compared to a raceway pond without mirrors. In the morning and evening, Configurations B and D exhibited the highest mean incident photosynthetic flux thanks to the east-and west-facing orientation of their mirrors. Nonetheless, the incident photosynthetic flux remained small in the early morning and late evening due to the weak solar irradiation G S,λ at these times. Furthermore, the mean incident photosynthetic flux of Configuration B decreased at midday to be equal to that of Configuration A as the sun aligned with the north-south axis and the reflected area S ref went to zero (see Figure 2.4a). At midday, the mean incident photosynthetic flux was higher for Configuration C than Configuration B due to the south-facing orientation of the mirror.

Figure 2 . 6 :Figure 2 .

 262 Figure 2.6: Daily culture-area-based P A,C productivity as a function of (a) initial biomass concentration C X0 and (b) the product C X0 L for culture depth L equal to 0.1 m, 0.2 m, and 0.3 m on September 21 st . (c) Volumetric P V productivity as function of C X0 for a culture depth L = 0.3 m on September 21 st . (d) Product of the initial biomass concentration and the culture depth (C X0 L) opt which maximizes biomass productivity on the 21 st day of each month of the year. All data shown is for Configurations A-D located in Los Angeles, CA.

Figure 2 .

 2 Figure 2.6b plots the same data for biomass productivity P A,C shown in Figure 2.6a but as a function of C X0 L. The results indicate that, even when using mirrors, the productivity P A,C collapsed onto a single line for all values of C X0 and L. Note that land-area-based biomass productivity P A,L = P A,C × S/S L (not pictured) also collapsed onto a single line. This indicates that the scaling relation between areal biomass productivity (in kg m -2 day -1 ) and the initial optical thickness represented by the product C X0 L holds true for raceway ponds featuring external mirrors.

Figure 2 .

 2 Figure 2.6c shows the daily volumetric biomass productivity P V for all four raceway pond configurations as a function of C X0 for a culture depth L = 0.3 m on September 21 stin Los Angeles, CA. Note that volumetric productivity is given by P V = P A,C /L and thus did not scale with the product C X0 L. Configurations B-D significantly improved the daily culture-area-based P A,C and volumetric P V biomass productivities for all values of C X0 L and C X0 , respectively. The maximum value of both P A,C and P V increased by 32%, 26%, and 45% for Configurations B, C, and D, respectively, compared to Configuration A for which P A,C,max = 0.020 kg m -2 day -1 and P V,max = 0.067 kg m -3 day -1 . The predicted productivity of Configuration A was within the typical range of productivities for well-managed open raceway ponds reported as 0.020 to 0.025 kg m -2 day -1 from Ref.[START_REF] Tredici | Mass Production of Microalgae : Photobioreactors[END_REF].

Figure 2 .

 2 Figure 2.6d plots the optimum value of the product C X0 L which yielded the maximum

Figures 2 .

 2 Figures 2.7a and 2.7b show the maximum daily culture-area-based P A,C,max , land-areabased P A,L,max , and volumetric P V,max productivities of Configurations A-D obtained from simulations of biomass concentration C X (t) from sunrise to sunset on the 21 st day of each month of the year using (C X0 L) opt reported in Figure2.6d. It is evident that adding mirrors to the raceway pond increased the maximum biomass productivities P A,C,max and P V,max throughout the year. Indeed, even the simple single-mirror design of Configuration C increased the culture-area-based P A,C,max and volumetric P V,max productivities by 52% in December.However, Configurations B-C also decreased the maximum land-area-based P A,L,max productivity throughout the year. Configuration D exhibited the largest culture-area-based P A,C,max and volumetric P V,max productivities as well as the smallest land-area-based P A,L,max productivity. This was due to the additional land area required to accommodate the rotating platform and prevent shading between adjacent ponds with external mirrors. Thus, a production facility featuring raceway ponds of Configuration D would require a larger land area to achieve the same annual yield as a raceway pond of Configuration A-C. Furthermore, Figure2.7 indicates that Configuration B had larger productivities P A,C,max , P A,L,max , and P V,max from March to September than Configuration C while the opposite was true from October

Figures 2 .

 2 Figures 2.8a and 2.8b plot the maximum daily culture-area-based biomass productivity P A,C,max as a function of the maximum allowed mirror height normalized with respect to the pond width H * /W for Configurations A-D on September 21 st in Los Angeles, CA and Saint-Nazaire, France. For the dual east/west mirror Configuration B, Figure 2.8 indicates that the productivity P A,C,max increased continuously with increasing H * /W for both locations.This was due to the fact that the apparent solar zenith angle θ z,a approached zero at midday which resulted in very large values of mirror height required to maximize the reflected area according to Equation (2.2). On the other hand, the productivity P A,C,max of Configurations C and D increased and then remained constant for H * /W ≥ 1.6 for Los Angeles, CA and

Figure 2 . 8 :

 28 Figure 2.8: Maximum daily culture-area-based biomass productivity P A,C,max as a function of normalized maximum mirror height H * /W on September 21 st for Configurations A-D in (a) Los Angeles, CA and (b) Saint-Nazaire, France.

Figure 2 . 9 :

 29 Figure 2.9: (a) Optimum mirror height normalized by the pond width (H * /W ) opt for Configurations C and D and (b) the minimum solar zenith angle θ z,min on the 21 st day of each month in Los Angeles, CA and Saint-Nazaire, France. Maximum daily culture-area-based biomass productivity P A,C,max throughout the year using (H * /W ) opt for Configurations A-D located in (c) Los Angeles, CA and (d) Saint-Nazaire, France.

Figure 3 . 1 :

 31 Figure 3.1: (a) Experimental setup of a two raceway ponds, with and without a mirror, operated simultaneously under a greenhouse at the Algosolis R&D research facility, (b) a simplified schematic illustration of the top-view of the two mirror configurations, Configuration B and C, tested in the current study, and (c) a schematic illustration of the raceway pond featuring pH, temperature control, and a paddle wheel for culture mixing. The medium feeding and culture harvesting systems are also shown for continuous operation.

3. 3 . 1

 31 Figures 3.2a and 3.2b plot the measured solar photon flux density (PFD) at the culture surface for raceway ponds of Configurations B and C, respectively, throughout a clear day in Saint-Nazaire, France (September 19 th , 2022). The solar PFD without mirrors present is also shown, along with the total, direct, and diffuse solar PFD measured by the weather station on the greenhouse roof. Figures 3.2a and 3.2b demonstrate that the solar flux incident on the raceway ponds in the greenhouse was smaller throughout the day than the total solar flux measured by the weather station. This was due to shading by the greenhouse structure and reflective losses that occurred as light passed through the greenhouse windows. For both configurations tested, Figure 3.2 indicates that the use of mirrors increased the amount of light incident on the microalgae culture throughout the day. Indeed, over the course of the day, the total solar energy incident on the culture increased by 52.2% for Configuration B and 39.6% for Configuration C.

Figures 3 .

 3 Figures 3.2c and 3.2d plot the measured and predicted percent increase in solar PFD incident on the culture for Configurations B and C, respectively. The predicted increase was based on the reflected area given by Equation (2.5) and used in the model in Chapter 2.

3. 3 . 2 Figure 3 . 2 :

 3232 Figure 3.2: Total, direct, and diffuse solar photon flux density (PFD) in Saint-Nazaire, France on September 19 th , 2022 and solar PFD measured at the culture surface for (a) a raceway pond of Configuration B (dual mirrors with north/south pond orientation) and (b) a raceway pond of Configuration C (single mirror with east/west pond orientation). The measured solar PFD is also shown for the same raceway ponds without mirrors. Measured and predicted percent increase in solar PFD for (c) Configuration B and (d) Configuration C.

Figure 3 . 3 :

 33 Figure 3.3: Data collected during two trials of raceway pond Configuration C including the temporal evolution (a,b) of the direct solar radiation G S (in W m -2 ) and (c,b) culture temperature T c and (e,f) dry biomass present in the raceway ponds with and without mirrors.

Figure 3 . 4 :

 34 Figure 3.4: Pigment content w pig as a function of time for raceway ponds with and without mirrors from experimental trials performed during April and May.

Figure 3 . 5 :

 35 Figure 3.5: Data collected during two trials of raceway pond Configuration B including the temporal evolution (a,b) of the direct solar radiation G S (in W m -2 ) and (c,b) culture temperature T c , and (e,f) dry biomass present in the raceway ponds with and without mirrors.

Figure 3 . 6 :

 36 Figure 3.6: Culture depth L as a function of time for raceway ponds with and without mirrors from experimental trials performed during (a) April and (b) August.

Figure 3 .

 3 Figure 3.7 shows the results for a continuous trial carried out in early October featuring a raceway pond without a mirror, and one of Configuration C, i.e. east/west orientation and a single mirror on the north side. The dilution rate D ranged from 0.005 to 0.007 h -1 . Here, Figure 3.7a plots the daily direct solar radiation G S,P AR in µmol hν m -2 s -1 and Figure 3.7b

Figure 3 . 7 :

 37 Figure 3.7: Experimental results for continuous operation of a raceway pond without a mirror and a raceway pond of Configuration C including (a) daily solar conditions represented by the direct PAR solar radiation G S,P AR , (b) daily incident photon flux density q P AR on the raceway pond culture, (c) evolution of the culture biomass concentration C X , and (d) volumetric productivity P V .

(Figure 4 . 1 :

 41 Figure 4.1: (a) A schematic illustration of the experimental setup of the thin-tube photobioreactor. The culture was circulated in a loop flowing upward in the left tube due to continuous air injection and downward in the right tube driven by gravity. The culture pH was monitored and controlled by injecting an air/CO 2 mixture into the reactor when the pH exceeded a given setpoint. (b) A photo of the reactor featuring ascending bubble-driven culture flow in the left tube and descending gravity driven-culture flow in the right tube and illumination via external LEDs.

Figure 4 . 2 :

 42 Figure 4.2: Maximum theoretical volumetric (black) and surface (red) productivity for tubular photobioreactors with specific illuminated area a light equal to 50, 100, and 500 m 2 m -3 .

Figure 4 .

 4 Figure 4.3 plots the results of oxygen degassing experiments performed on the reactor containing culture medium for three different gas flow rates Q equal to 3, 6, and 10 mL min -1 .Figures 4.3a, 4.3c, and 4.3e show the dissolved oxygen concentration C L as a function of time,

  Figures 4.3a, 4.3c, and 4.3e show the dissolved oxygen concentration C L as a function of time, while Figures 4.3b, 4.3d, and 4.3f show the quantity ln

  , 4.3d, and 4.3f show the quantity ln

Figure 4 . 1 -

 41 Figure 4.4 plots the dissolved oxygen concentration C L (in mgL -1 ) in the photobioreactor as a function of the biomass productivity P V for the three gas flow rates Q tested.Figure

Figure 4 . 3 :

 43 Figure 4.3: (a, b, c) Culture dissolved oxygen concentration C L (in mg/L) as a function of time t during oxygen de-gassing and (d, e, f) the corresponding term used to calculate k L a for gas flow rates Q of 3 mL min -1 , 6 mL min -1 , and 10 mL min -1 , respectively. The linear part of the data used for the calculation of k L a is shown in blue, and the corresponding linear regression is shown by the dashed line. 86

Figure 4 . 4 :

 44 Figure 4.4: Dissolved oxygen concentration C L as a function of biomass productivity P V for gas flow rates of 3 mL min -1 , 6 mL min -1 , and 10 mL min -1 .

Figure 4 . 5 :

 45 Figure 4.5: Temporal evolution of (a, b) biomass concentration C X , (c, d) mean rate of photon absorption (MRPA) Ā, and (e, f) pigment concentration w pig for Chl. a, Chl. b, and PPC for trials conducted using culture mediums with target concentrations of 10 gL -1 and 15 gL -1 , respectively. 88

Figures 4 .

 4 Figures 4.5a and 4.5b plot the temporal evolution of the biomass concentration C X (in gL -1 ) for two trials of the thin-tube ICAR-PBR operated in batch mode, using culture mediums with target concentrations of 10 gL -1 and 15 gL -1 , respectively. The error bars correspond to 95% confidence intervals. The maximum biomass concentrations of 9.16 gL -1

Figures 4 .

 4 Figures 4.5cand 4.5d plot the estimated mean rate of photon absorption (MRPA) and the corresponding incident photon flux density q in for each trial. Here the mean rate of photon absorption was estimated according to Ā ≈ a light C X (q in -q o ) (4.10)

Figures 4 .

 4 Figures 4.5e and 4.5f which plot the pigment concentration in weight percent of Chl. a, Chl. b, and photoprotective carotenoids for both trials calculated using Equations (3.1) and (3.2). Notably, Figure 4.5e shows a decrease in pigment concentration for all three pigment types during the first three days which may be indicative of pigment adaptation due to the high initial MRPA. Indeed, Figures 4.5cand 4.5d indicate that the initial MRPA was large

Figures 4 .

 4 Figures 4.6a, 4.6b, and 4.6c plot the biomass concentration C X (in gL -1 ), the correspon-

  Figures 4.6a, 4.6b, and 4.6c plot the biomass concentration C X (in gL -1 ), the correspon-

Figure 4 . 6 :

 46 Figure 4.6: Temporal evolution of (a) biomass concentration C X , (b) daily growth rate r X , and (c) the culture mean rate of photon absorption (MRPA) Ā for semi-continuous operation of the thin-tube photobioreactor using a culture medium with a target biomass concentration of 15 gL -1 .

  photobioreactor. The incident light was collimated and normally incident with respect to the longitudinal axis of the tube. The presence of the tube walls was neglected, and the reactor was modeled as a cylinder of water in air with refractive indices of 1.33 and 1.0, respectively. Reflection events at the water/air interface and scattering by the microalgae cells were neglected. Photon absorption by the microalgae cells was modeled according to Beer-Lamberts law. The culture had an effective absorption coefficient κ (in m -1 ) defined as κ = C X A abs where C X represents the culture concentration (in kg m -3 ) and A abs represents the mass absorption cross-section (in m 2 kg -1 ) of the species being cultivated. The simulated results

Figure 5 . 1 :

 51 Figure 5.1: (a) Side view and (b) section view of a simplified representation of the tubular photobioreactor considered in the current study.

  (a) If R < ρ ray is reflected, terminate ray (b) Else if R > ρ ray is refracted, calculate the new ray direction s t according to Snell's law (see Equation (5.1)) and continue. 5. Calculate new intersection location p on either (i) the tube/air interface or (ii) the tube/culture interface. Calculate the corresponding reflectance ρ and generate random number R. (a) If intersection is on the tube/air interface and ray is refracted, terminate ray. (b) If intersection is on the tube/air interface and ray is reflected, calculate the new ray direction s t according to specular reflection and return to Step 5 . (c) If intersection is on the tube/culture interface and ray is reflected, calculate new intersection location on tube/air interface p , new direction s r according to specular reflection and new reflectance ρ, return to Step 5.a or 5.b according to random number outcome.

Figure 5 . 2 : 6 .

 526 Figure 5.2: A simplified flow chart of the algorithm structure used to track a ray as it propagates through the tubular photobioreactor where it may experience multiple reflection, refraction, and scattering events before being transmitted or absorbed in the microalgae culture.

7 )

 7 where A(A > A c ) is the cross-sectional area of the tube where the local rate of photon absorption was greater than the compensation rate of photon absorption A c . The PAR-averaged local fluence rate G(x, y) in µmol hν m -2 s -1 can then be determined by simply dividing the LRPA by the PAR-averaged mass absorption cross-section, i.e., G(x, y) = A(x, y)/ Āabs(5.8)

𝐶 ! = 0.1 kg/m 3 𝒜Figure 5 . 3 :Figures 5 .

 3535 Figure 5.3: Comparison of the local fluence rate G(r) (in µmol hν m -2 s -1 ) predicted using the two-flux method and the current MCRT algorithm for a PBR with an inner radius r i = 8 cm and biomass concentration C X of (a) 0.1, (b) 0.45, and (c) 1.0 kg m -3 and (d) the corresponding illuminated fraction γ predicted by both methods as a function of biomass concentration.

Figure 5 .

 5 Figure 5.3a illustrates poor agreement in the predicted local fluence rate G(r) betweenthe two methods at biomass concentration C X = 0.1 kg m 3 . Indeed, the MCRT predicts an extremely large local fluence rate at r = 0 as the un-absorbed rays converge at the center point of the tube. This was not the case for the two-flux method which predicted a steady decay in G(r) as r decreased. At C X = 0.45 kg m -3 , Figure5.3b illustrates good agreement between the two methods for r > 1 cm. However, here too the un-absorbed rays converged at the tube center and the local fluence rate predicted by the MCRT was much larger than that predicted by the two-flux method. At larger biomass concentration, Figure5.3c illustrates a steeper decay in G(r) due to increased absorption by the microalgae cells.

Figure 5 .

 5 Figure 5.3d plots the illuminated fraction γ predicted by both methods as a function of biomass concentration. Here the compensation rate of photon absorption Ac was equal to 2800 µmol hν kg -1 s -1 [14]. Despite the disagreement observed at low biomass concentrations in Figures 5.3aand 5.3b, Figure 5.3d illustrates good agreement in the predicted illuminated fraction γ between the two methods over the range of biomass concentrations considered.

Figure 5 . 4 :

 54 Figure 5.4: Predicted illuminated fraction γ using 50, 100, and 200 mesh elements as a function of the number of rays simulated N rays for biomass concentrations of C X equal to 0.5 and 1.0 kg m -3 .

Figure 5 .

 5 4 plots the predicted illuminated fraction γ as a function of the number of rays simulated N rays for a number of elements N el equal to 50, 100, and 200 and biomass concentrations equal to 0.5 and 1.0 kg m -3 . The microalgae radiative properties Āabs and Ssca were equal to 199 m 2 kg -1 and 2873 m 2 kg -1 and were representative of those of Chlorella vulgaris for a pigment concentration of 4 wt. % according to measured pigment concentrations in Chapter 4. The photon flux density q in was composed of collimated, normally incident radiation (i.e., θ in = 0 • ) and equal to 200 µmol hν m -2 s -1 . For a given mesh and number of rays N rays , ten trials were performed to assess the standard deviation.

5. 3 . 1 Figure 5 .

 315 Figure 5.5a plots the predicted illuminated fraction γ within the culture of Chlorella vulgaris as a function of biomass concentration C X for tube inner radius r i equal to 2, 5, 10, and 20 mm. The photon flux density q in was composed of collimated, normally incident radiation (i.e., θ in = 0 • ) and was equal to 200 µmol hν m -2 s -1 . Recall from Chapter 1 (Section 1.4.3) that γ is an important parameter for describing light conditions in the PBR. A value of

Figure 5 .

 5 Figure 5.5b plots the same predicted illuminated fraction γ as a function of the quotient of the biomass concentration and the specific illuminated area of the system, i.e.,

Figure 5 . 5 :

 55 Figure 5.5: Predicted illuminated fraction γ for tube inner radius r i equal to 2, 5, 10, and 20 mm as a function of (a) the biomass concentration C X and (b) the optical thickness represented by C X r i (in kg m -2 ) for a photon flux density q in = 200 µmol hν m -2 s -1 .

Figure 5 . 6 :

 56 Figure 5.6: Flux ratio q cult /q in as a function of normalized tube wall thickness t/r o for (a) collimated radiation with incidence angle θ in ranging from 0 • to 80 • and (b) diffuse radiation and (c,d) schematic representations of the paths of refracted rays for collimated radiation and normalized tube wall thickness t/r o = 0.6 for θ in = 0 • and 60 • , respectively.

Figures 5 .

 5 6c and 5.6d show the path of rays refracted at the air/tube boundary for t/r o = 0.6 and incidence angle θ in equal to 0 • and 60 • . These figures demonstrate the increased concentrating effect at larger incidence angles θ in . As θ in increased, the focal point of refracted rays shifted closer to the tube center, increasing the intensity of the light flux on the microalgae culture.

Figure 5 .

 5 Figure 5.7a plots the predicted illuminated fraction γ as a function of tube wall thickness t for tube inner radius of r i equal to 2, 4, and 6 mm and incidence angle θ in equal to 0 • and 45 • . The incident collimated intensity I col was equal to 250 µmol hν m -2 s -1 . To isolate the impact of the tube wall thickness, the biomass concentration was taken such that C X r i

Figure 5 . 7 :

 57 Figure 5.7: Illuminated fraction γ as a function of (a) tube wall thickness for tube inner radii r i equal to 2, 4, and 6 mm and (b) normalized tube wall thickness t/r o for incidence angles θ in of 0 • and 45 • . (c,d) Map of culture locations where the local rate of photon absorption exceed the compensation rate of photon absorption A(x, y) > A c for θ in = 45 • and t/r o equal to 0.5 and 0.9, respectively.

igure 5 . 8 :

 58 Predicted average volumetric growth rate r X as a function of normalized tube wall thickness t/r o for (a) collimated incidence with incidence angles θ in equal to 0 • and 45 • and (b) diffuse incidence.

Figure 5 . 9 :

 59 Figure 5.9: Predicted illuminated fraction γ for collimated incident radiation as a function of incidence angle θ in for normalized tube wall thickness t/r o equal to 0.01, 0.30, and 0.50.

Figure 5 . 10 :

 510 Figure 5.10: Illumined fraction γ as a function of biomass concentration for a microalgae cell asymmetry factor g equal to 0.974 and 1.0 representative of negligible scattering.

Figure 5 .

 5 Figure 5.11a plots the local fluence rate normalized with respect to the incident photon

Figure 5 .

 5 Figure 5.11: (a) Normalized local fluence rate G(x, y)/q in for the case of g = 1 (i.e., negligible scattering) and g = 0.974 and (b) corresponding culture locations where the LRPA was greater than the compensation LRPA, i.e., A(x, y) > A c (red area) and less than the compensation LRPA, i.e., A(x, y) < A c (yellow).

Figures 5 .Figure 5 .

 55 Figures 5.12a and 5.12b plot the volumetric biomass productivity as a function of tube wall thickness for solar conditions in June and September, respectively, for PBR inclination angles β equal to 90 • (vertical tube), 45 • , and 0 • (horizontal tube) with the tube centerline aligned with the North/South axis. A horizontal tube aligned with the East/West axis was also considered. Recall that only a single tube is considered, thus mutual shading bet-ween tubes is neglected. The initial biomass concentration was C X (t 0 ) = 10 kg m -3 and the tube inner radius was r i = 2 mm. First, the results demonstrate the large volumetric biomass productivities that can be expected for a thin-tube photobioreactor in light-limited conditions. Indeed, the maximum biomass productivity predicted was 4.9 kg m -3 for a horizontal PBR aligned with the North/South axis in June and 3.4 kg m -3 for a PBR with an inclination angle β = 45 • aligned with the North/South axis. Similar values of productivity have been obtained for the Algofilm © concept PBR under artificial lighting and solar conditions[START_REF] Pruvost | Development of a thin-film solar photobioreactor with high biomass volumetric productivity (AlgoFilm©) based on process intensification principles[END_REF][START_REF] Louveau | Étude et optimisation de la culture solaire de microalgues en photobioréacteur intensifié[END_REF]

Figure 5 .

 5 Figure 5.12: (a,b) Predicted volumetric biomass productivity P V as a function of tube wall thickness t/r o for a vertical tubular PBR (β = 90 • ), a tilted PBR (β = 45 • ) aligned with the North/South axis and horizontal PBRs (β = 0 • ) aligned with either the North/South or East/West axis. (c,d) Temporal evolution of the biomass concentration C X and (e,f) illuminated fraction γ for the four PBR orientations and a tube wall thickness t/r o = 0.3. The results are given for solar conditions representative of a clear day in (a, c, e) June and (b, d, f) September, respectively.

CHAPTER 6 Conclusion

 6 High production cost associated with industrial cultivation of microalgae remains a major barrier to their use as a renewable resource. Thus, cost-effective strategies for improving the biomass productivity of cultivation systems must be developed to unlock the full potential of microalgae-based products. This thesis presented theoretical and experimental studies of two such strategies, light and culture intensification, applied to two different cultivation systems, namely an open raceway pond and a closed tubular photobioreactor.Chapters 2 and 3, respectively, featured a theoretical and experimental assessment of the use of external mirrors on solar raceway ponds cultivating Chlorella vulgaris with the goal of increasing the incident photon flux and increasing the raceway biomass productivity.Chapter 2 considered four designs (i) Configuration A : a raceway pond without mirrors, (ii) Configuration B : a raceway pond oriented along the north/south axis with mirrors on both its east and west sides, (iii) Configuration C : a raceway pond oriented along the east/west axis with a single mirror on its north side, and (iv) Configuration D : a solar tracking rotating pond with a single mirror. A coupled light transfer and growth kinetics model was used to predict the biomass productivity for each configuration. The areal biomass productivity was found to scale by the product of the initial biomass concentration and the culture depth C X0 L. Overall, the use of mirrors was found to increase the areal biomass productivity by as much as 73% compared to a raceway pond without mirrors, with Configuration B performing best in the summer months and Configuration C performing best in the winter months. Practical design recommendations such as the optimum mirror height and minimum length-to-width ratio were identified. Configuration C was considered to be the best balance between improved biomass productivity and increased capital expenses.Chapter 3 featured an experimental campaign of two pilot-scale raceway ponds, one with and one without mirrors, under solar conditions during different seasons and mirror Configurations B and C. The raceway ponds were operated in both batch and continuous mode. For a sunny September day in Saint-Nazaire, France, Configurations B and C were demonstrated to increase the incident solar photon flux density on the culture by 52.2% and 39.6%, respectively. However, the mirrors were ineffective when they were blocked from direct sunlight making the system performance highly sensitive to weather conditions and nearby obstacles. Nonetheless, the biomass yield was found to increase with the use of mirrors by as much as 36.6% for a span of clear days in August (Configuration B). Configuration B was more effective in the summer months while Configuration C was more effective in the spring months which was in agreement with the model predictions presented in Chapter 2.The use of mirrors was more effective for biomass concentrations greater than 0.5 gL -1 when full light attenuation occurred.Overall, the Chapters 2 and 3 demonstrated the interest of mirrors for increasing the biomass productivity of sufficiently concentrated raceway ponds under clear skies. Future studies should focus on a techno-economic analysis of a industrial-scale microalgae cultivation facility, to assess the additional capital and operational expenses required for the reflecting raceway system. Weather and climate data could be used to predict productivity and identify the locations and climates where the reflecting raceway pond would be economically viable based on a given mirror configuration. Moreover, a thermal model of the raceway pond could be integrated into the current model to predict the impact of the mirror design on the culture temperature and microalgae growth. Such a study could be complemented by a large-scale outdoor experiment of the reflecting raceway pond. Indeed, the improvement in biomass productivity may be larger in outdoor conditions where less shading would occur compared to the current experiments where the greenhouse structure cast shadows on the raceway ponds. Furthermore, on-line monitoring of chlorophyll florescence could be included to develop control strategies to detect and avoid photoinhibition due to strong photon flux densities that can occur due to the use of mirrors. The mirrors could be leveraged as a means of controlling the PFD on the culture. Finally, further research regarding the microalgae species being cultivated is needed to identify those that would benefit the most from the higher photon flux densities provided by the mirrors. Chapters 4 and 5, respectively, presented a experimental and theoretical assessment of the performance of a closed, thin-tube photobioreactor (ICAR concept). Chapter 4 featured an experimental demonstration of the ICAR thin-tube prototype cultivating Chlorella vulgaris with an inner radius of 2 mm corresponding to a specific illuminated area a light of 500 m 2 m -3 . The reactor was operated in batch and semi-continuous modes under artificial lighting. Under semi-continuous operation, the reactor achieved a daily average growth rate of 3.0 gL -1 day -1 for a relatively low photon flux density of 250 µmol hν m -2 s -1 and biomass concentrations as large as 16.3 gL -1 . Such high productivity and biomass for a relatively low photon flux density of may enable reduce the process energy consumption required for lighting in indoor cultivation. However, this growth rate was less that the predicted maximum theoretical growth rate assuming light-limited growth. Indeed, based on the measured biomass productivity and the gas-liquid mass transfer coefficient k L a of the reactor, the predicted dissolved oxygen concentration was high enough that growth limitation due to oxygen over-accumulation was expected. Overall, the ICAR concept thin-tube photobioreactor was demonstrated to achieve high biomass concentrations and productivities while showing little evidence of biofilm formation. Further experimental studies should monitor the dissolved oxygen concentration of the culture during growth to detect and avoid oxygen over-accumulation. Control and design strategies such as increased bubbling rate, nitrogen injection, and hydrodynamic design improvement should be considered for improving the gas-liquid mass transfer of the reactor. Moreover, the current reactor features a relatively large design dark fraction of f d = 50%. Thus, future design work for reactor scale-up should focus on minimizing the culture dark volumes (upper and lower chambers) and increasing the illuminated tube area to increase the maximum theoretical productivity of the reactor.

FutureAwFigure A. 2 :

 2 Figure A.2: (a) Maximum daily culture-area-based biomass productivity P A,C,max as a function of length-to-height ratio l/W on September 21 st and (b) maximum daily culture-areabased biomass productivity P A,C,max over one year for Configurations A-D with l/W equal to 5 and 20 for raceway ponds located in Los Angeles, CA.

TABLE OF CONTENTS

 OF 1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.1 Motivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.1.1 Value-added products . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.1.2 Biofuels . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .1.2 Photosynthesis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.2.1 Light-driven reactions . . . . . . . . . . . . . . . . . . . . . . . . . . Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2.1 Problem statement . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2.2 Assumptions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2.3 Reflected sunlight . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2.4 Light transfer in microalgae culture . . . . . . . . . . . . . . . . . . . 2.2.5 Microalgae growth kinetics . . . . . . . . . . . . . . . . . . . . . . . . 2.2.6 Biomass productivity . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2.7 Boundary and initial conditions . . . . . . . . . . . . . . . . . . . . . Results and Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.3.1 Reflected light from mirrors . . . . . . . . . . . . . . . . . . . . . . . 3.3.2 Batch operation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .3.3.3 Continuous operation . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.4 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 Experimental proof-of-concept of a novel, compact, thin-tube photobioreactor for high volumetric productivity . . . . . . . . . . . . . . . . . . . . . . 4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Assumptions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.2.3 Method of solution . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.3 Results and Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.3.1 Optical thickness scaling . . . . . . . . . . . . . . . . . . . . . . . . . 5.3.2 Impact of tube wall thickness . . . . . . . . . . . . . . . . . . . . . . 5.3.3 Impact of incidence angle . . . . . . . . . . . . . . . . . . . . . . . . 5.3.4 Impact of scattering . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.3.5 Solar modeling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

1.2.2 Calvin cycle . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.2.3 Photorespiration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.2.4 Pigments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.3 Light transfer . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.3.1 Radiative transfer equation . . . . . . . . . . . . . . . . . . . . . . . 1.3.2 Two-flux approximation . . . . . . . . . . . . . . . . . . . . . . . . . 1.3.3 Microalgae radiative properties . . . . . . . . . . . . . . . . . . . . . 1.4 Microalgae cultivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.4.1 Open systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.4.2 Closed systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.4.3 PBR Engineering . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.4.4 Intensified reactors . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.5 Objectives . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 A novel external reflecting raceway pond design for improved biomass productivity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xii 2.2 2.2.8 Method of solution . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.3 Results and Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.3.1 Incident radiative flux . . . . . . . . . . . . . . . . . . . . . . . . . . 2.3.2 Microalgae growth . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.3.3 Biomass productivity . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.3.4 Impact of reflecting pond dimensions and location . . . . . . . . . . . 2.4 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 Experimental assessment of the performance of a pilot-scale external reflecting raceway pond for improved biomass productivity . . . . . . . . . . . 3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2 Materials and methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2.1 Experimental set-up . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2.2 Inoculation and operation . . . . . . . . . . . . . . . . . . . . . . . . 3.2.3 Biomass concentration . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2.4 Pigment concentration . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2.5 Solar flux . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xiii 3.3 4.2 Materials and methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.2.1 Reactor design and operation . . . . . . . . . . . . . . . . . . . . . . 4.2.2 Medium composition . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.2.3 Gas-liquid mass transfer . . . . . . . . . . . . . . . . . . . . . . . . . 4.2.4 Biomass analyses . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.3 Results and Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.3.1 Theoretical maximum productivity . . . . . . . . . . . . . . . . . . . 4.3.2 Gas-liquid mass transfer . . . . . . . . . . . . . . . . . . . . . . . . . 4.3.3 Batch operation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.3.4 Semi-continuous operation . . . . . . . . . . . . . . . . . . . . . . . . 4.4 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 Investigation of light transfer in thin-tube photobioreactors via a Monte Carlo ray-tracing modeling approach . . . . . . . . . . . . . . . . . . . . . . . . 5.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.2 Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.2.1 Problem statement . . . . . . . . . . . . . . . . . . . . . . . . . . . . xiv 5.2.2 5.4 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Nomenclature . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . A Supplementary Materials for Chapter 2 . . . . . . . . . . . . . . . . . . . . B Supplementary Materials for Chapter 5 . . . . . . . . . . . . . . . . . . . .

  Note that, for Configuration B, the west mirror height H B,W (t) was given by Equation (2.2) while the east mirror was lowered in the morning when the solar azimuth angle was negative, i.e., γ S < 0. Similarly, the east mirror height H B,E (t) was given by Equation (2.2) while the west mirror was lowered in the afternoon when the solar azimuth angle was positive, i.e., γ S > 0. Note that the mirror height H C (t) was equal to zero for Configuration C when the sun was positioned behind the mirror, i.e., |γ S | > 90 • . Figure 2.2b plots the resulting mirror heights H B/C/D (t) as a function of time for Configurations B-D on September 21 st according to Equations (2.2)-(2.4) with an imposed maximum mirror height of H

2)-(2.4). * = 1 m.

  Eqs. (2.8) -(2.10)

	Growth kinetics
	Eqs. (2.12) -(2.13),
	(2.17)
	𝜃

! ≥ 90˚? 𝒜(𝑧, 𝑡) 𝐶 " (𝑡 + Δ𝑡) No Yes MICROALGAE CULTURE • Biomass concentration, 𝐶 " 𝑡 , 𝐶 ",$%& • Volumetric and areal daily biomass productivity, 𝑃 ' , 𝑃 (,) , 𝑃 (,* [Eqs. (2.14) -(2.16)]

  H * /W ≥ 1.0 for Saint-Nazaire, France. Thus, these values represented the optimum value (H * /W ) opt of the maximum mirror height H * normalized by the pond width W . Note that (H * /W ) opt for Configurations C and D on September 21 st were given by the maximum value of mirror height H C/D (t) on that day from Equations (2.3) and (2.4), respectively. Figure 2.9a plots the optimum mirror height normalized by the pond width (H * /W ) opt for the 21 st day of each month in Los Angeles, CA and Saint-Nazaire, France for Configurations C and D. Note that (H * /W ) opt was not shown for Configuration B since it was infinite at midday as the apparent solar zenith angle θ z,a approached zero [see Equation (2.2)]. Figure 2.9a indicates that (H * /W ) opt was smaller throughout the year at the higher latitude of Saint-Nazaire, France compared to that of Los Angeles, CA. This was due to the larger minimum solar zenith angle θ z,min (see Figure 2.9b) which reduced the mirror height necessary to maximize the fraction S ref /S of the culture area subjected to reflected light. Similarly, the larger solar zenith angles in the winter months resulted in smaller (H

* /W ) opt compared to the summer months for both locations. Furthermore, Figure

2

.9a demonstrates that the mirror heights required to optimize the performance of both configurations were small in the winter months, particularity for ponds located in Saint-Nazaire where (H * /W ) opt was less than unity from September to March. Figures 2.9c and 2.9d plot the predicted maximum culture-area-based productivity P A,C,max throughout the year for the optimum maximum mirror height (H * /W ) opt from Figure 2.9a for all four configurations in Los Angeles, CA and Saint-Nazaire, France, respectively. Here, the optimum maximum mirror height (H * /W ) opt for Configuration C was used for Configuration B. In general, Figures 2.9c and 2.9d indicate that the areal productivity P A,C,max

Table 3 .

 3 1: Composition of the modified Bold's Basal medium used in the current study.

		Nutrient	Concentration (mg L -1 )
		NH 4 Cl		472
		MgSO 4 ,7H 2 O		225
		CaCl 2 ,2H 2 O		25
		Additive 1		1 mL L -1
		Additive 2		1 mL L -1
		Na 2 EDTA,2H 2 O		50
		FeSO 4 7H 2 O		14
		K 2 HPO 4		150
		KH 2 PO 4		123
		NaHCO 3		1260
		Additive 1			Additive 2
	Nutrient	Concentration (mg L -1 )	Nutrient	Concentration (mg L -1 )
	ZnSO 4 ,7H 2 O	222		Na 2 MoO 4 ,2H 2 O	260
	(NO 3 )2Co,6H 2 O	44
	CuSO 4	79
	H 3 BO 3	2860
	MnCl 2 ,4H 2 O	1810

  • for at least 1 h. The samples were centrifuged a second time and the optical density OD λ of the resulting supernatant was measured using a UV-vis spectrophotometer (Jasco V-630, Lisses, France) at wavelengths λ equal to 480, 652, 665, and 750 nm. All extractions were performed in triplicates. Then, the

concentrations of chlorophyll a C Chla , chlorophyll b C Chlb , and photoprotective carotenoids C P P C in mg/L were estimated according to

[START_REF] Ritchie | Consistent sets of spectrophotometric chlorophyll equations for acetone, methanol and ethanol solvents[END_REF]

:

  CO 2 + 0.148 NH 4 + + 0.006 H 2 SO 4 + 0.010 H 3 PO 4 + 0.571 H 2 O CH 1.776 O 0.459 N 0.148 S 0.006 P 0.010 + 1.088 O 2 . (4.2)

	h ν --→

Table 4 .

 4 1: Composition of the modified Sueoka medium and phosphate solution used in the current study.

		Target biomass concentration
		C X = 10 gL -1	C X = 15 gL -1
	Nutrient	Concentration (mg L -1 )
	NH 4 HCO 3	5338	7765
	MgSO 4 ,7H 2 O	593	863
	CaCl 2 ,2H 2 O	81	118
	Hutner's Solution	1mL/L	1mL/L
	Phosphate solution
	Nutrient	Concentration (mg L -1 )
	KH 2 PO 4	2283
	K 2 HPO 4	3425

  -1 day -1 )

	3	0.75	0.71
	6	1.3	1.3
	10	2.2	2.1
	4.3d, and 4.3f, which indicate an increase in the gas-liquid mass transfer coefficient k L a with
	increasing gas flow rate Q.

Boundary and initial conditions

The incident spectral solar irradiance G S,λ (t) was determined using the Simple Model of the Atmospheric Radiative Transfer of Sunshine (SMARTS) for either Los Angeles, CA or Saint-Nazaire, France on the 21 st day of each month [START_REF] Gueymard | Simple model of the atmospheric radiative transfer of sunshine (smarts)[END_REF]. The initial biomass concentration C X0 was varied between (i) 0.03 and 0.30 kg m -3 for culture depth L = 0.3 m, (ii) 0.03 and 0.45 kg m -3 for L = 0.2 m, and (iii) 0.03 and 0.70 kg m -3 for L = 0.1 m. These ranges of culture depth and biomass concentration were found to yield positive biomass productivity on September 21 st . [START_REF] Souliès | Investigation and modeling of the effects of light spectrum and incident angle on the growth of Chlorella vulgaris in photobioreactors[END_REF]. Next, the growth kinetics model for Chlorella vulgaris was used to predict the volume-averaged specific growth rate µ(t) (Equations (2.12) -(2.13)). Then, the biomass concentration at subsequent times C X (t + ∆t) was predicted by integrating Equation (2.11) and assuming that µ(t) and C X (t) were constant over the time increment ∆t according to

Method of solution

(2.17)

where the time increment ∆t was equal to 3 minutes to obtain numerically converged results. This process was then repeated for the updated biomass concentration and sunlight parameters at t = t + ∆t until sunset, defined here as the time t where θ z ≥ 90 • .

(a) (b) tubular designs that take advantage of the concentrating effect of the tube wall could be conceived. For example, a concentric tube PBR design could be imagined featuring a center tube containing the microalgae culture and a water-jacket outer tube for simultaneously concentrating the incident visible light, and filtering the incident infrared radiation. Such a design could be particularly advantageous for thin-tube reactors, which will tend to overheat quickly due to their low thermal inertia.