A Busnel

K Samhat

E Gérard

A Kazbar

H Marec

E Dechandol

Le Gouic

J.-L Hauser

Jeremy Pruvost
email: jeremy.pruvost@univ-nantes.fr

B Le Gouic

Development and validation of a screening system for characterizing and modeling biomass production from cyanobacteria and microalgae: Application to Arthrospira platensis and Haematococcus pluvialis

Keywords:

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

Introduction

Numerous solutions are currently being investigated to substitute chemical synthesis with biosynthesis, using plants and microorganisms to produce food, animal feed, chemicals, etc., and also to reduce the environmental impact of various processes. Microalgae, including cyanobacteria, represent a potential source of biomolecules for many different applications [START_REF] Muller-Feuga | Microalgues marines: les enjeux de la recherche[END_REF][START_REF] Vonshak | Spirulina Platensis Arthrospira: Physiology, Cell-Biology and Biotechnology[END_REF][START_REF] Romay | Antioxidant and anti-inflammatory properties of C-phycocyanin from blue-green algae[END_REF][START_REF] Romay | Further studies on anti-inflammatory activity of phycocyanin in some animal models of inflammation[END_REF][START_REF] Řezanka | Natural microbial UV radiation filtersmycosporine-like amino acids[END_REF][START_REF] Oren | Mycosporines and mycosporine-like amino acids: UV protectants or multipurpose secondary metabolites?[END_REF]. They can be found almost in every biotope of the globe, representing around 300,000 different genera. This adaptation to different environmental conditions results in the production of huge biodiversity of biomolecules (pigments, fatty acids, proteins, polysaccharides, vitamins, etc.). However, the lack of knowledge about many species and hindsight on their cultivation methods, strongly restrict the industrial potential of microalgae. Today, only around 20 microalgae species are used and commercialized at the industrial level, for the cosmetics, animal feed and food supplement markets. A wide discrepancy exists between the number of species cultivated at an industrial scale and the great potential for microalgal biodiversity.

The production of biomass and/or molecules of interest is closely related to culture conditions and strain choice. Some species, Arthrospira platensis for example, will have better growth at a high pH [START_REF] Soni | Spirulina -from growth to nutritional product: a review[END_REF]. For others, metabolites of interest will be produced in stress conditions, as with the production of beta-carotene by Dunaliella salina [START_REF] Nguyen | High light stress regimen on Dunaliella salina strains for carotenoids induction[END_REF] and astaxanthin by Haematococcus pluvialis [START_REF] Boussiba | Carotenogenesis in the green alga Haematococcus pluvialis: cellular physiology and stress response[END_REF]. Thus, to better understand the parameters that affect biomass and metabolite production before taking them to an industrial scale, important characterization work is required at the lab scale. This is particularly the case for poorly studied and documented species, for which different culture parameters such as culture medium, light, temperature, pH, etc., must be first determined. This step is crucial in an industrial context, to determine the best cultural conditions to achieve the best productivity and to evaluate the technoeconomic and scale of production. However, this characterization step can be time-consuming [START_REF] Loubière | A new photobioreactor for continuous microalgal production in hatcheries based on external-loop airlift and swirling flow[END_REF]. In general, equipment and methods which could reduce the delay between laboratory to industrial scale would be extremely valuable.

It is well-known that there is a strong link between photoautotrophic growth and light absorption conditions, which are related to culture system geometry [START_REF] Takache | Experimental and theoretical assessment of maximum productivities for the microalgae Chlamydomonas reinhardtii in two different geometries of photobioreactors[END_REF]. For example, geometry responding to a onedimensional hypothesis such as a flat panel system, or cylindrical geometry presenting cylindrical symmetry, allows simple analytical equations to be used to represent light transfer in culture systems [START_REF] Pruvost | Modeling dynamic functioning of rectangular photobioreactors in solar conditions[END_REF][START_REF] Pruvost | Knowledge models for the engineering and optimization of photobioreactors[END_REF][START_REF] Takache | Kinetic modeling of the photosynthetic growth of Chlamydomonas reinhardtii in a photobioreactor[END_REF][START_REF] Pruvost | Microalgae culture in building-integrated photobioreactors: biomass production modelling and energetic analysis[END_REF]. Of the numerous works using PBR geometries corresponding to the one-dimensional hypothesis for an accurate study of photosynthetic microorganism growth, we can cite Kandilian [START_REF] Kandilian | Influence of light absorption rate by Nannochloropsis oculata on triglyceride production during nitrogen starvation[END_REF] and Soulies et al. [START_REF] Soulies | Investigation and modeling of the effects of light spectrum and incident angle on the growth of Chlorella vulgaris in photobioreactors[END_REF], who investigated lipid accumulation and light spectrum effects respectively. Investigations including the modeling approach were therefore facilitated.

Screening studies to determine growth conditions are usually conducted on microplates or in Erlenmeyer flasks or serum bottles to compare medium or strain potential, with system volumes ranging from 5 mL to 4 L [START_REF] Kurano | Fixation and utilization of carbon dioxide by microalgal photosynthesis[END_REF][START_REF] Yang | Isolation and characterization of Taiwanese heterotrophic microalgae: screening of strains for docosahexaenoic acid (DHA) production[END_REF][START_REF] Sydney | Screening of microalgae with potential for biodiesel production and nutrient removal from treated domestic sewage[END_REF][START_REF] Van Wagenen | Microplate-based method for high-throughput screening of microalgae growth potential[END_REF][START_REF] Radzun | Automated nutrient screening system enables high-throughput optimisation of microalgae production conditions[END_REF]. These systems, however, do not provide strict control of culture conditions, especially regarding light absorption. Recently, screening systems dedicated to photosynthetic microorganisms have been proposed, including microPHAROS™ from Xanthella and Multi-Cultivator® (MC 1000-OD) developed by Photon Systems Instruments. This latter PBR consists of eight 85 mL test tubes, can maintain an ambient temperature of up to 60 • C for all the tubes, and a sensor to monitor growth in each tube is provided. CO 2 is constantly injected by a CO 2 /airflow, and a specific spectral LED light flux can be applied from 405 to 730 nm. A turbidostat mode can be held on each cultivation PBR cell. Similarly, a screening PBR named EOSS was proposed [START_REF] Taleb | Development and validation of a screening procedure of microalgae for biodiesel production: application to the genus of marine microalgae Nannochloropsis[END_REF]. As for the Multi-Cultivator system, continuous and semicontinuous cultures are possible, but a cylindrical geometry was used in both systems with frontal illumination. As a consequence, the light transfer is not one-dimensional, which makes it difficult to determine the light absorption conditions [START_REF] Loubière | A new photobioreactor for continuous microalgal production in hatcheries based on external-loop airlift and swirling flow[END_REF].

This work presents the design of a new screening photobioreactor named EOSS2-PBR. The main interest is to characterize and model the pairing of light absorption conditions and photosynthetic growth. The cyanobacterium Arthrospira platensis and the green microalga Haematococcus pluvialis were used in this study to demonstrate the efficiency of EOSS2-PBR. Biomass productivities obtained were compared to the predicted productivities with the kinetic growth model for Arthrospira platensis and experimental data from the 1 L Airlift flat-panel PBR for Haematococcus pluvialis.

Materials and methods

Experimental setup

Two types of PBR were used during this study: the EOSS2-PBR that is the basis of this work and the 1L Airlift flat-panel PBR used to validate the results obtained in EOSS2-PBR.

EOSS2-PBR description

EOSS2-PBR is the second version of the first screening PBR, EOSS1-PBR, which is fully described in Taleb et al. [START_REF] Taleb | Development and validation of a screening procedure of microalgae for biodiesel production: application to the genus of marine microalgae Nannochloropsis[END_REF]. It permits screening of (i) microalgae species performance; (ii) medium composition; (iii) growth temperature; (iv) illumination conditions (continuous or day/ night cycles) in reliable conditions (Fig. 1). EOSS2-PBR consists of 6 mini flat PBRs operated independently in parallel, their geometry has been modified to provide accurate control of light absorption conditions in the culture volume than on EOSS1-PBR. Each PBR is therefore constructed as a flat-panel PBR to respond to the one-dimensional hypothesis.

Each PBR has a volume of V r = 75 mL (100 × 40 × 20 mm with a rounded shape at the bottom of R20 diameter) and a specific illuminated surface a light = 50 m -1 (depth of 2 cm). It is fully automated in terms of light illumination and thermoregulation, with a possibility of continuous culture. This system is made in polymethyl methacrylate Fig. 1. Schematic representation (A) and pictures of the front (B), the backplate with the heating/cooling system (C) and the gassing system (D) of EOSS2-PBR.

(PMMA) for the transparent part (Fig. 1B) and the stainless steel backplate (Fig. 1C). A PT100 temperature sensor is used to measure and pilot a heating/cooling system composed of a Peltier system, a copper radiator, and a ventilator (Fig. 1C). Then, each dilution rate is imposed by a pump (ISMATEC® reglo ICC) to inflow fresh medium and a calibrated solenoid valve (SMC® VDW22UAH) to harvest. The calibration of the harvest solenoid valve has been done by mass of liquid measurement as a function of the opening time. A maximal error of 8% has been observed with the given harvest method. Airlift agitation is obtained by injection of air enriched in CO 2 with continuous flow maintained at 100 mL⋅min -1 by mass flow meter/controller (EL-FLOW® Select, Bronkhorst). The gas flow has been homogeneously distributed to the 6 PBRs by 6 independent valves (Fig. 2,V1) via mixing time measurement (see Section 3.1). For optimal mixing, a 3 ways solenoid valve permits direct flux in two of four gas injection points (SMC® VDW2505G201FHQ) (Fig. 2,V2). All injection points are located at the bottom of the PBR (Fig. 1D). Two injection stainless steel needles (intern diameter 0.8 mm) are located on each side at the front part of the used volume and the two others at the background (Fig. 1D). The incident photons flux density (PFD) was provided by a set of 6 independent small white LED flat panels with light intensity ranging from 40 to 400 μmol hν ⋅m -2 ⋅s -1 which allows either a constant or a cycled illumination intensity. Each electronic device is piloted by a program build on LabVIEW software (National Instruments, USA).

Airlift flat-panel PBR description

The airlift flat-panel PBR (AL-PBR) was used, in this study, to validate the performance of EOSS2-PBR in characterizing microalgae strains performances, as represented here by the areal biomass productivity as a meaningful quantity when scaling production facilities and culture systems [START_REF] Pruvost | Knowledge models for the engineering and optimization of photobioreactors[END_REF]. This AL-PBR, described in more details by Pruvost et al. [START_REF] Pruvost | Investigation of biomass and lipids production with Neochloris oleoabundans in photobioreactor[END_REF], has a volume of V r = 1 L (16 × 24.5 × 3 cm) and a specific illuminated surface a light = 33.33 m -1 (depth of 3 cm). Its non-optical rear face is made of stainless steel while its front face is constructed with transparent and non-toxic PMMA ensuring the passage of light into the PBR. PBR was set with a complete loop of common sensors and automation (Mettler Toledo M300) for microalga culture, namely pH, temperature, and gas injections (CO 2 and air). pH was regulated by automatic injection of CO 2 and temperature by ambient air blowing [START_REF] Pruvost | Investigation of biomass and lipids production with Neochloris oleoabundans in photobioreactor[END_REF]. The incident PFD was generated by a panel of white LED, previously calibrated, which was placed vertically in front of the PBR, parallel to the optical face, to provide one-dimensional light attenuation to the PBR.

Algal strains and pre-cultivation in shake flasks

Two strains were used in this work, namely the cyanobacterium Arthrospira platensis (PCC8005) and the green microalga Haematococcus pluvialis (SAG 34-7).

Arthrospira platensis was maintained in Zarrouk medium [START_REF] Zarrouk | Contribution à l'étude d'une Cyanophycée, influence de divers facteurs physiques et chimiques sur la croissance et la photosynthèse de "Spirulina maxima[END_REF] while Haematococcus pluvialis was maintained in a modified Bold Basal Medium (BBM) [START_REF] Bold | The cultivation of algae[END_REF]. Both species were cultivated in an Erlenmeyer glass vessel and were thermo-regulated at 21 • C, stirred at 70 rpm and illuminated at 30 μmol hν ⋅m -2 ⋅s -1 by an incubation chamber (INNOVA44®

Fig. 2. Schematic of EOSS2-PBR set-up.
Incubator shaker series from New Brunswick Scientific).

Operating conditions 2.3.1. Arthrospira platensis cultivation in EOSS2-PBR

For A. platensis, EOSS2-PBR was operated at a constant temperature of 35 • C. An incident PFD of 300 μmol hν ⋅m -2 ⋅s -1 was fixed and three dilution rates of 0.011, 0.033 and 0.058 h -1 were applied to obtain different biomass concentrations and cover different light attenuation conditions. Air injection was enriched at 1% CO 2 to ensure a constant pH of 9.5 ± 0.3 while providing sufficient carbon to avoid growth limitation conditions.

Haematococcus pluvialis cultivation in EOSS2-PBR and airlift-PBR

H. pluvialis was cultivated in both EOSS2-PBR and AL-PBR under the same culture conditions. Cultures were run in continuous (chemostat) mode with optimal growth conditions (i.e., green stage). The fresh medium was injected using a dosing pump (Ismatec®), and harvesting was by overflowing into a sterilized bottle, making the reactor volume V r constant. The temperature was maintained at 25 • C using a heating resistor connected to the transmitter probe and which is located on the rear face of the reactor. Thus, the pH is regulated by injecting CO 2 gas (2.5% CO 2 ) to maintain a constant pH of 7.5 ± 0.5, the optimal pH for the growth of H. pluvialis, and to meet the needs in dissolved carbon to avoid growth limitation conditions. Two incident PFDs of 75 and 200 μmol hν ⋅m -2 ⋅s -1 were fixed and three dilution rates of 0.01, 0.02 and 0.03 h -1 were applied to investigate various light absorption conditions.

Before starting each experiment, the PBRs were sterilized for 30 min using a 5 mM peroxyacetic acid solution and rinsed twice with sterile deionized water.

Analytical methods

Measurement of mixing time

Determination of mixing time (t m ) in the EOSS2-PBR was performed by monitoring the conductivity of the liquid phase. The conductivity meter used was a conductometer (CD810 Tacussel, France) with a selfmade probe made of two nickel wires (0.5 mm diameter 95% purity) spaced at 5 mm with an insert made by a milling machine. The end of the tube was sanded to obtain two circular wires with no glue on them. The analog output (0 V-7 V) of the conductometer was plugged into a card (USB-6009 National Instrument, USA) in the differential input.

For all experiments for mixing time measurement, distilled water was used as the liquid phase in the EOSS2-PBR, maintained at ambient temperature (21 ± 1 • C). Mixing was provided by air injection. The air was introduced at the bottom of the reactor through four (1 mm diameter) needles (two on each side, one close to the front wall, and one close to the back wall) (Fig. 1D). Air was entered from the gas circuit at a flow rate of 25 mL⋅min -1 and a pressure of 3.0 × 10 4 Pa at the bottom of each PBR (Fig. 2).

When used for cultivation, gas can be introduced alternately by the left and right needles (Fig. 1D). This guarantees good mixing on both sides of the PBR to avoid biofilm appearing on the optical surface. Since the mixing-time experiments were not conducted with microorganisms, the air was only injected on the right side. Due to the design symmetry of the PBR, the same results were achieved when using the left-side injection (data not shown).

A conductivity probe was placed in the EOSS2-PBR cell, passing by the event position (on the left of the reactor), 2.5 cm from the top of the PBR cell. The conductivity tracer consisting of 800 μL saturated salt solution (NaCl) was injected into the reactor (at time t 0 ), passing by the medium inlet position on the left of the reactor, also 2.5 cm from the top. Thus, NaCl solution was introduced on the opposite side to the bubbling. The injection was performed as quickly as possible to constitute a punctual perturbation. Conductivity was monitored by the conductivity meter for 6 min, the time needed for stabilization, at the rate of one measure per second (1 Hz).

The obtained values (conductivity as a function of time) were normalized by the maximum value obtained for the experiment. Mixing time is defined as when the conductivity value exceeds 95% of the maximum value. All analyses were done in triplicate.

Measurement of gas-liquid mass transfer

The volumetric oxygen transfer coefficient (k L a) was determined. Distilled water at ambient temperature (21 ± 1 • C) was used as the liquid phase. The dissolved oxygen concentration was monitored by an optical oxygen sensor (Pyroscience®, OXROB10) connected to Pyro oxygen logger (Pyroscience®) software. This probe took one measurement per second (1 Hz).

The medium was first deoxygenated by replacing the airflow containing oxygen with gaseous nitrogen. This step was performed until a dissolved oxygen concentration of less than 2% saturation was obtained in experimental conditions. The nitrogen flow was then turned off and replaced with airflow at the rate of 25 mL⋅min -1 . The oxygenation step was maintained until thermodynamic equilibrium was achieved (i.e., constant dissolved oxygen concentration).

In the absence of microorganisms, the oxygen uptake rate (OUR) is equal to 0. The oxygen mass balance in the liquid phase, therefore, leads directly to the oxygen transfer rate (OTR), and the kinetics of the reoxygenation phase is governed by the following differential equation:

dC L dt = k L a ( C * L -C L ) = OTR (1)
where C L and C L * are the dissolved oxygen concentration at a given time and at equilibrium respectively. Volumetric oxygen transfer was determined by plotting ln

( C * L -CL C * L )
versus time, resulting in a straight line with a slope corresponding to the k L a value.

Characterization of thermal behavior of EOSS2-PBR

When full light attenuation occurs, microalgal culture can be considered as a black body, with a light absorption capacity close to 100% [START_REF] Goetz | Photon statistics of propagating thermal microwaves[END_REF]. Black Indian ink was therefore used to simulate a microalgal culture for thermal behavior characterization. The aim was to determine the maximum and minimum temperatures applicable at each incident PFD by determining the corresponding cooling and heating powers. Six incident PFDs were applied to EOSS2-PBR (50, 150, 200, 300 and 400 μmol hν ⋅m -2 ⋅s -1 ) with a temperature set point between 5 and 50 • C for all minis PBRs. Minimal and maximal temperatures were noted when the thermic equilibrium was obtained.

Reproducibility evaluation

Since EOSS2-PBR consists of 6 mini PBR (named PBR cells), it was important to test their reproducibility to verify there is no significant effect in the operation of each PBR cell on the growth of microalgae. Therefore, H. pluvialis was cultivated in 3 PBR cells simultaneously under the same culture conditions (PFD = 100 μmol hν ⋅m -2 ⋅s -1 and D = 0.02 h -1 ). The biomass productivity and the pigment contents were measured during cultivation.

Dry weight and biomass productivity

For both strains, the microalgal dry weight concentration C x was determined by filtration through a pre-dried and pre-weighed glass-fiber filter (Whatman GF/F). The filters were dried for 24 h at 105 • C then weighed in triplicate. The volumetric biomass productivity P v in the continuous cultures at dilution rate D was calculated from biomass concentration C x measured at steady state 3 days in a row [START_REF] Pruvost | Hydrodynamics influence on light conversion in photobioreactors: an energetically consistent analysis[END_REF]. Measurements of productivities were taken in triplicate on three consecutive days. The areal biomass productivity S x was then deduced from the specific illuminated surface a light :

P v = C x D (2)
S x = P v V r S light = P v a light (3)
Note that the use of a geometry responding to the one-dimensional hypothesis facilitates the determination of areal productivity as a light for flat panel PBR, obtained by:

a light = S light V r = 1 L ( 4
)
where L is the depth of culture. Considering that areal biomass productivity in a light-limited condition is not dependent on the PBR geometry [START_REF] Pruvost | Knowledge models for the engineering and optimization of photobioreactors[END_REF], this introduces another benefit to using such geometry to obtain meaningful quantities.

Pigment contents

According to Qiu et al. [START_REF] Qiu | A new method for fast extraction and determination of chlorophylls in natural water[END_REF], a volume V 1 (mL) of H. pluvialis culture was first centrifuged at 13400 rpm for 15 min. The pellet was then suspended in a volume V (mL) of dimethyl sulfoxide (DMSO) and then incubated at 50 • C for 1-2 h in the dark until the sample turned white. After cooling, the pigment extract was diluted with a volume V 2 (mL) of 90% acetone (the ratio of DMSO: 90% acetone should be equal to 1:4) before centrifugation at 13400 rpm for 15 min. The optical density OD λ of the supernatant was measured at 480, 630, 645, 665 and 750 nm with a UV-vis spectrophotometer (JASCO V-630). All extractions were performed in triplicates. Pigment concentrations (chlorophyll a (Chl-a), b (Chl-b) and photoprotective carotenoids (PPC)) were calculated as follows [START_REF] Strickland | A Practical Handbook of Seawater Analysis[END_REF]:

where l is the cell path length (1 cm). The corresponding mass fraction of pigment "i" per dry weight of biomass can be estimated as w i = C i / C x .

Theoretical considerations 2.5.1. Overview of modeling of Arthrospira platensis growth

If light source and PBR are planar systems, the one-dimensional hypothesis can be applied for light transfer modeling [START_REF] Pottier | A fully predictive model for one-dimensional light attenuation by Chlamydomonas reinhardtii in a torus photobioreactor[END_REF]. Equations are a function of the depth of the PBR (L), so the fluence rate G(z) can be determined at each point of culture depth (z), allowing calculation of the local rate of photon absorption A(z). This can therefore be linked to photosynthetic growth (r x ) by introducing the kinetic growth model. For the cyanobacterium Arthrospira platensis, the model of Cornet et al. [START_REF] Cornet | A structured model for simulation of cultures of the cyanobacterium Spirulina platensis in photobioreactors: II. Identification of kinetic parameters under light and mineral limitations[END_REF] was used:

r x (z) = ρ ϕ A(z) (6) ρ = ρ M K K + G(z) (7) ϕ = M X ϕ ′ ( 8
)
where ρ is the energy yield for photon conversion, ρ M the maximum energy yield for photon conversion, K the half-saturation constant of photosynthesis, ϕ and ϕ ′ mass and molar quantum yields for the Zscheme of photosynthesis respectively, and M X the C-molar mass of biomass. For Arthrospira platensis, ρ M = 0.8 and ϕ = 1.85 × 10 -9 kg x ⋅μmol hν -1 (with NO 3 2-as N-source) and K = 90 ± 5 μmol hν ⋅m -2 ⋅s -1 [START_REF] Cornet | A simple and reliable formula for assessment of maximum volumetric productivities in photobioreactors[END_REF].

An average biomass volumetric growth rate can then be calculated by integration over the depth of culture [START_REF] Pruvost | Knowledge models for the engineering and optimization of photobioreactors[END_REF]:

< r x >= 1 L ∫ L l=0 r x (z)dz (9)
At steady state in continuous culture, <r x > is equal to the volumetric biomass productivity of the PBR, P v : < r x >= P v [START_REF] Loubière | A new photobioreactor for continuous microalgal production in hatcheries based on external-loop airlift and swirling flow[END_REF] Note that the areal biomass productivity S x can be deduced from Eq. (3).

Modeling of light attenuation and light absorption conditions

Light attenuation through PBR (G(z)) is a function of (i) the technological parameters of PBR geometry and PFD and (ii) the biological parameters of biomass concentration and corresponding radiative properties as mass absorption cross-section (Ea λ in m 2 ⋅kg -1 ), mass scattering cross-section (Es λ in m 2 ⋅kg -1 ) and backward scattering fraction (b 2λ , dimensionless).

For geometries responding to the one-dimensional hypothesis, the two-flux model proved effective to properly describe light diffusion and absorption phenomena in algal suspensions. This leads to the analytical solution, as represented by [START_REF] Pottier | A fully predictive model for one-dimensional light attenuation by Chlamydomonas reinhardtii in a torus photobioreactor[END_REF]:where R d represents the diffuse reflectance of the PBR back wall (R d = 0 for a transparent wall) while α λ and δ λ are 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 = ̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅ ̅ Ea λ Ea λ + 2b λ Es λ √ (12) δ λ = C x ̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅ ̅ Ea λ ( Ea λ + 2b λ Es λ ) √ (13) G λ (z) q λ,0 = 2 [R d (1 + α λ )e -δλL -(1 -α λ )e -δλL ]e δλL + [(1 + α λ )e δλL -R d (1 -α λ )e δλL ]e -δλL (1 + α λ )2e δλL -(1 -α λ ) 2 e -δλL -R d (1 -α λ 2 )e δλL + R d (1 -α λ 2 )e -δλL
(OD 630 -OD 750 ) } V 2 V 1 l C PPC [mg/L] = {4.0 (OD 480 -OD 750 ) } V 2 V 1 l (5)
The field of fluence rate can be used to determine the local rate of photon absorption A(z) which is obtained by:

A(z) = 1 L ∫ L 0 ∫ λ=700 λ=400 A λ (z)dzdλ = 1 L ∫ L 0 ∫ λ=700 λ=400 Ea λ G λ (z)dzdλ (14)
Values can be then integrated over the PAR region and related to the growth kinetics (Eq. ( 6)). Radiative properties have already been determined for Arthrospira platensis in [START_REF] Cornet | A simple and reliable formula for assessment of maximum volumetric productivities in photobioreactors[END_REF] (Ea = 162 ± 8 m 2⋅ kg -1 ; Es = 640 ± 30 m 2⋅ kg -1 ; b 2 = 0.03 ± 0.002).

Three light regimes are usually described by light attenuation conditions, as represented by fluence rates field G(z): (i) full light absorption regime corresponding to "G(z = L) < G C ", (ii) luminostat regime with "G(z = L) = G C " and (iii) kinetic regime corresponding to "G (z = L) > G C ". G C here is the fluence rate at the compensation point. It represents the minimum fluence rate with positive growth [START_REF] Pruvost | Knowledge models for the engineering and optimization of photobioreactors[END_REF]. Cornet et al. [START_REF] Cornet | A simple and reliable formula for assessment of maximum volumetric productivities in photobioreactors[END_REF] have shown a compensation point at 1.5 μmol hν ⋅m -2 ⋅s -1 for Arthrospira platensis.

Prediction of maximal biomass productivity

Maximal biomass productivity can be predicted either by solving the overall equation model including the radiative (Eq. ( 10)) and kinetic growth (Eq. ( 5)) models or by using the simplified engineering relation proposed by Cornet et al. [START_REF] Cornet | A simple and reliable formula for assessment of maximum volumetric productivities in photobioreactors[END_REF]. Maximal biomass productivity is obtained from cyanobacteria when full light absorption occurs (G (z = L) < G c ). The following analytical equation can be derived to obtain maximal areal biomass productivity S x,max [START_REF] Cornet | A simple and reliable formula for assessment of maximum volumetric productivities in photobioreactors[END_REF]:

S x,max = ρ M ϕ O2 ' . 2α 1 + α . K Ea .ln [ 1 + q 0 Ea K ] (15)
This formula, avoiding a complete resolution of the kinetic and radiative transfer models, was validated by Cornet et al. [START_REF] Cornet | A simple and reliable formula for assessment of maximum volumetric productivities in photobioreactors[END_REF] with an overall accuracy of around 15%.

Statistical tests performed

Regarding the measurement replicates, all the experiences have been done in triplicates. The error has been estimated through standard deviation calculation.

All the productivities have been calculated on steady states. Dry matter and dilution rates have been measured in triplicates three days in a row after reaching a steady state.

Results and discussion

Mixing efficiency in EOSS2-PBR

Airflow was fixed at 100 mL⋅min -1 in EOSS2-PBR and distributed using a valve control in each PBR cell. Mixing time measurements were taken for each PBR cell for the first homogenized mixing at each valve control position on 6 PBR cells (Fig. 2,V1), to verify that the same mixing conditions occur in each EOSS2-PBR cell. Triplicate assays on one PBR cell highlighted a standard deviation of 15%. This standard deviation has been estimated small enough regarding the appreciation of the opening capacity of valves (manual valves with no graduation). Homogenization of mixing time was therefore carried out, accepting an error percentage of 15% between the six EOSS2-PBR cells.

As shown in Fig. 3, a mixing time of around 3.6 min was obtained for all PBR cells. Regarding biological kinetics such as doubling time (function of strains, but roughly from hours to days [START_REF] Sakai | Chlorella strains from hot springs tolerant to high temperature and high CO 2[END_REF][START_REF] De Morais | Isolation and selection of microalgae from coal fired thermoelectric power plant for biofixation of carbon dioxide[END_REF], this mixing time was considered sufficiently low to assume perfectly mixed conditions. Also, the mixing time value was used as a reference to define the alternation time of lateral air injection through the 3-way solenoid valve (Fig. 2,V3). The 3-way valve permits to orientate the gas flow on the left side pipes then, a timer set at 3.6 min, switch and orientate the gas flow on the right-side pipes.

Gas-liquid mass transfer

After deoxygenation of the medium with gaseous nitrogen, airflow with oxygen was reintroduced in the EOSS2-PBR cell. The dissolved oxygen concentration increased for around 100 s before stabilizing. The corresponding k L a value was equal to 0.028 s -1 at 21 ± 1 • C for a flow rate of 25 mL⋅min -1 . This was higher than the k L a value reported in 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] for an intensified PBR technology presenting high volumetric productivity and its associated high oxygen production (k L a = 0.01 s -1 ), so it was assumed that gas-liquid mass transfer performance was sufficient to consider that EOSS2-PBR has sufficient capacity to remove oxygen produced by photosynthesis and avoid its over-accumulation.

Temperature regulation efficiency

The temperature efficiency of EOSS2-PBR was tested at each PFD value to find out the limits of the system's thermoregulation. It was Fig. 3. Mixing time distribution across six EOSS2-PBR cells (n = 3).

found that the temperature could be stabilized at the target set points in all cases. A minimum and maximum temperature set point of respectively 5 • C and 50 • C can be maintained for a PFD from 40 to 400 μmol hν ⋅m -2 ⋅s -1 in a room thermo-regulated at 20 • C.

Reproducibility evaluation

Fig. 4 shows the areal biomass productivities and the pigment contents for Haematococcus pluvialis cultures in three EOSS2-PBR cells. In the same operating conditions, the difference between areal biomass productivity in the three EOSS2 cells did not exceed 5%; it was around 14.01 ± 0.1 g⋅m -2 ⋅d -1 . Similar observations can be drawn for pigment contents; in the three cells, Chl a and b content was around 1.22 ± 0.025% DW with 0.4 ± 0.015% DW of carotenoids. These results confirm that all EOSS2-PBR cells have the same performance, hence confirming the reliability of the EOSS2-PBR to screen various conditions by conducting parallel experiments in each PBR cell.

Characterization of A. platensis growth and comparison with growth model

Biomass productivities obtained on Arthrospira platensis at 300 μmol hν ⋅m -2 ⋅s -1 and dilution rates of 0.011, 0.033 and 0.058 h -1 are 12.2, 15.8 and 5.0 g⋅m -2 ⋅d -1 respectively (Fig. 5). A comparison with the model prediction was added. Note that uncertainty regarding the dilution rate as fixed by the feeding pump (i.e., 10%) was considered in simulations. A correct agreement was observed. In addition to validating the model described in Cornet et al. [START_REF] Cornet | A structured model for simulation of cultures of the cyanobacterium Spirulina platensis in photobioreactors: II. Identification of kinetic parameters under light and mineral limitations[END_REF] to predict the growth of A. platensis in PBRs, this result also demonstrates the interest of the EOSS2-PBR system for determining the kinetic constants of a growth model. For example, in the present case, the constant K which represents the half-saturation constant of photosynthesis, therefore directly related to the photosynthetic response of the strain to received light (G(z)), may have been obtained by adjusting the model parameter K to the experimental productivity data (using a simple regression tool such as fmin type in Matlab software).

Our maximal biomass productivity (S x,max = 15.84 g.m -2 .d -1 ) was compared to the one predicted by the engineering formula (Eq. ( 15)) proposed by Cornet et al. [START_REF] Cornet | A simple and reliable formula for assessment of maximum volumetric productivities in photobioreactors[END_REF]. With this equation, a theoretical value of S x,max of 15.93 g⋅m -2 ⋅d -1 was obtained. The discrepancy between these two results is therefore only about 3%. This confirms the interest of EOSS2-PBR as a reliable investigator of light-limited growth in a PBR.

As shown in Fig. 5, an optimal productivity of 15.84 g⋅m -2 ⋅d -1 at 300 μmol hν ⋅m -2 ⋅s -1 is reached at D = 0.033 ± 0.003 h -1 . At higher dilution rate values, productivity decreased as far as culture washout. This has already been seen in several other works [START_REF] Pruvost | Knowledge models for the engineering and optimization of photobioreactors[END_REF][START_REF] Takache | Kinetic modeling of the photosynthetic growth of Chlamydomonas reinhardtii in a photobioreactor[END_REF]. More surprisingly, for lower dilution rates, biomass productivity should have been found close to the maximal due to the negligible respiration activity of cyanobacteria in light [START_REF] De | Participation of plastoquinone, cytochrome c 553 and ferrodoxin-NADP (+) oxido-reductase in both photosynthesis and respiration in Spirulina maxima[END_REF]. As a consequence, the dark volume as obtained for low dilution rates (i.e., large biomass concentrations) has a negligible effect on the resulting PBR biomass productivity [START_REF] Cornet | A structured model for simulation of cultures of the cyanobacterium Spirulina platensis in photobioreactors: II. Identification of kinetic parameters under light and mineral limitations[END_REF]. This is contradictory to our results at D = 0.011 ± 0.0009 h -1 , with productivity 16% lower than predicted compared to the maximal value achieved at D = 0.033 ± 0.003 h -1 . Such decrease could be attributed to a possible deviation in the PBR operation, as lowest dilution rate corresponds also to larger biomass concentration and longer residence time. The dilution rate D = 0.011 h -1 corresponds to a residence time (=1/D) of around 91 h. Considering the steady-state is achieved after at least 3 times the residence time value, more than 12 days were necessary to obtain the steady-state for biomass productivity measurement (25 days for D = 0.005 h -1 ). Such a long duration increases the risk of biofilm formation or operation deviation such as a drift in the feeding pump flowrate (and then dilution rate). However, such deviation could be considered acceptable (16%).

Validation of EOSS2-PBR for the determination of growth performance of Haematococcus pluvialis in photobioreactors

To validate the interest of using EOSS2-PBR for the determination of growth performance of strains in photobioreactors (as here represented by the areal biomass productivity), the photosynthetic growth of H. pluvialis was determined experimentally in both EOSS2 and AL-PBR. Cultures were conducted in chemostat mode, under 2 constant illuminations of 75 and 200 μmol hν ⋅m -2⋅ s -1 and 3 dilution rates of 0.01, 0.02 and 0.03 h -1 .

According to Fig. 6, in AL-PBR, the dilution rate that maximized the productivity was 0.02 h -1 , yielding the highest values of areal biomass productivity of 10.87 ± 0.18 and 24.91 ± 0.15 g⋅m -2 ⋅d -1 for 75 and 200 μmol hν ⋅m -2⋅ s -1 respectively. Similar results were obtained with EOSS2-PBR, with maximal biomass productivities of 10.75 ± 0.22 and 25.92 ± 0.18 g⋅m -2 ⋅d -1 reached with D = 0.02 h -1 for 75 and 200 μmol hν ⋅m -2 ⋅s -1 respectively. Fig. 6 summarizes the experimental values of areal biomass productivity S x for all studied conditions in both PRBs. This confirms that, considering the standard deviations below 5% between all results, there are no significant differences in the productivities obtained for given culture conditions between both culture systems. Similar areal biomass productivities are obtained for same culture conditions (i.e., PFD and dilution rate), indicating that EOSS2-PBR can be used for the rapid determination of microalgae strain kinetics performances. For instance, using a single 1L Airlift PBR would request around one week per condition investigated, leading to around one month of culture to retrieve data to determine optimal dilution rate for a given PFD (i.e., at least 3-4 values of dilution rate are needed). With EOSS2-PBR, more data will be obtained in one week (6 dilution rates can be applied in parallel).

Conclusion

This work presents, for the first time, the design of a new screening photobioreactor named EOSS2-PBR. The main interest is the control of light absorption conditions (i.e., flat panel system) facilitating the characterization and modeling of the photosynthetic growth of microalgae.

EOSS2-PBR was characterized in terms of mixing and gas-liquid mass transfer performance. Both were found efficient enough to consider homogeneous nutrient concentrations in the culture and avoid dissolved oxygen accumulation, which could impair photosynthetic growth. EOSS2-PBR allows screening of culture temperature from 5 to 50 • C.

EOSS2-PBR was proven suitable for investigating light-limited growth in photosynthetic microorganisms. This was demonstrated on the cyanobacterium Arthrospira platensis. Its kinetic growth model was validated through experiments. It allowed also generating quantitative data to characterize strain performances such as biomass productivity, as shown in the case of the green microalga Haematococcus pluvialis.

( 11 )

 11 C Chl-a [mg/L] = {11.6 (OD 665 -OD 750 ) -1.31 (OD6 645 -OD 750 ) -0.14 (OD 630 -OD 750 ) } V 2 V 1 l C Chl-b [mg/L] = {20.7 (OD 645 -OD 750 ) -4.34 (OD 665 -OD 750 ) -4.42

Fig. 4 .Fig. 5 .

 45 Fig. 4. Areal biomass productivities S x , chlorophyll a concentration C Chl-a , chlorophyll b concentration C Chl-b and carotenoid concentration C PPC obtained in three EOSS2-PBR cells cultivating H. pluvialis at PFD = 100 μmol hν ⋅m -2⋅ s -1 and D = 0.02 h -1 .

Fig. 6 .α′

 6 Fig. 6. Comparison of the areal biomass productivities S x of H. pluvialis grown in Airlift-PBR and EOSS2-PBR at 75 and 200 μmol hν ⋅m -2⋅ s -1 as a function of dilution rates D.

Acknowledgments

Special thanks to Atlantic Microalgae (AMI), a scientific consortium of all the establishments in the Pays de la Loire region working on microalgae and the Lebanese University (LU) that funded this study.

Declaration of competing interest

The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.

No conflicts, informed consent, human or animal rights applicable.


Kashif Hussain Mangi

Zakaria Larbi

Jack Legrand

Jérémy Pruvost

El-Khider Si-Ahmed
email: el-khider.si-ahmed@univ-nantes.fr

Passive thermal regulation approach for Algofilm © photobioreactor through phase change

Keywords:

This study deals with the development of a self-sustained passive thermal regulation technique in Algofilm © solar photobioreactor. The reported experimental investigation includes, the evaluation of an average equivalent condensate film thickness, in indoor operating conditions, as function of the flowing liquid T l and ambient T amb temperatures, inclination angle as well as air flowrate. For instance, an average condensate film thickness of 2.2 ± 0.2 mm at T l = 50 • C, for liquid flow rate of 0.08 l/s and = 4 • in steady state conditions was recorded. Furthermore, a correlation estimating the condensation rate as function of the flowing liquid and ambient temperatures, was developed. A theoretical, heat and mass transfer model was proposed and validated with the indoor experimental results. The numerical predictions of the model adapted to a simulated sunny day were in good agreement with the experimental recordings. This validation evidenced the model applicability to real solar conditions, where an accurate evaluation of the condensed water is required for predicting the infrared filtration.

Introduction

Microalgae are unicellular microorganisms that can convert sunlight into biochemical energy via the process of photosynthesis. It is a promising feedstock for bulk commodities like chemicals, food, nutraceutical, pharmaceutical industries and bio-fuels [START_REF] Litchman | Competition of phytoplankton under fluctuating light[END_REF][START_REF] Spolaore | Commercial applications of microalgae[END_REF][START_REF] Axelsson | Perspective: Jatropha cultivation in southern India: Assessing farmers' experiences[END_REF][START_REF] Draaisma | Food commodities from microalgae[END_REF]. Microalgae is becoming a prominent technology for wastewater treatment [START_REF] Derakhshan | A comparative study of hybrid membrane photobioreactor and membrane photobioreactor for simultaneous biological removal of atrazine and CNP from wastewater: A performance analysis and modeling[END_REF][START_REF] Xie | Blending high concentration of anaerobic digestion effluent and rainwater for cost-effective Chlorella vulgaris cultivation in the photobioreactor[END_REF][START_REF] Yang | Coupling a photosynthetic microbial fuel cell (PMFC) with photobioreactors (PBRs) for pollutant removal and bioenergy recovery from anaerobically digested effluent[END_REF][START_REF] Luo | Evaluating the resilience of photobioreactors in response to hazardous chemicals[END_REF], and has been considered as one of the sustainable solutions towards CO 2 control and capture [START_REF] Al Ketife | A mathematical model for carbon fixation and nutrient removal by an algal photobioreactor[END_REF] to counter the global warming and climatic change [START_REF] Zhao | Nature inspired fractal tree-like photobioreactor via 3D printing for CO2 capture by microaglae[END_REF]. [START_REF] Pruvost | Industrial photobioreactors and scale-up concepts[END_REF], and gas hold up [START_REF] Sabri | Investigating the cross-sectional gas holdup distribution in a split internal-loop photobioreactor during microalgae culturing using a sophisticated computed tomography (ct) technique[END_REF]. Temperature and light are important factors to control the biomass and lipid production in algae [START_REF] Ugwu | Photobioreactors for mass cultivation of algae[END_REF][START_REF] Pruvost | Theoretical investigation of biomass productivities achievable in solar rectangular photobioreactors for the cyanobacterium arthrospira platensis[END_REF][START_REF] Xu | Microalgal bioreactors: Challenges and opportunities[END_REF][START_REF] Solimeno | Mechanistic model for design, analysis, operation and control of microalgae cultures: calibration and application to tubular photobioreactors[END_REF][START_REF] Shuba | Microalgae to biofuels: 'Promising' alternative and renewable energy, review[END_REF]. Algal growth rates increases with temperature up to an optimum range Muller-Feuga et al. (1998). The optimum light [START_REF] Cheng | Numerical simulation on optimizing flow field and flashing-light effect in jet-aerated tangential swirling-flow plate photobioreactor to improve microalgal growth[END_REF] and temperature [START_REF] Singh | Effect of temperature and light on the growth of algae species: a review[END_REF] were studied for the maximum growth of different species of algae. [START_REF] Ras | Temperature effect on microalgae: a crucial factor for outdoor production[END_REF] reported the opti-mum temperature ranges, to achieve maximum microalgae productivity, for different types of microalgae were 25, 25, 40 • C respectively for psychrophilic (Asterionella formosa), mesophilic and thermophilic (Chaetoceros, Anacystis nidulans) species. The estimation of microalgae production is quite difficult in outdoor systems [START_REF] Mata | Microalgae for biodiesel production and other applications: a review[END_REF][START_REF] Béchet | Full-scale validation of a model of algal productivity[END_REF]. For instance, the temperature cannot be easily controlled at outdoor scale, because algae species experience significant temperature change during varied weather conditions such as cloudy conditions, Day/Night cycle and seasonal variations [START_REF] Béchet | Mechanistic modeling of broth temperature in outdoor photobioreactors[END_REF][START_REF] Pruvost | Thermal modeling and optimization of microalgal biomass production in the harsh desert conditions of State of Qatar[END_REF].

Nomenclature

Overheating of photobioreactors is an unavoidable challenge for microalgal mass production, resulting in low photosynthetic efficiency and low biomass productivity [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] Pruvost | Thermal modeling and optimization of microalgal biomass production in the harsh desert conditions of State of Qatar[END_REF]. Microalgae production requires thermal regulations in outdoor conditions [START_REF] Bechet | Modelling the impact of temperature on microalgae productivity during outdoor cultivation[END_REF][START_REF] Nwoba | Can solar control infrared blocking films be used to replace evaporative cooling for growth of Nannochloropsis sp. in plate photobioreactors?[END_REF][START_REF] Pruvost | Thermal modeling and optimization of microalgal biomass production in the harsh desert conditions of State of Qatar[END_REF]. Microalgae needs the photo-synthetically active spectrum i.e. 400-700 nm for biomass growth [START_REF] Wondraczek | Shifting the Sun: solar spectral conversion and extrinsic sensitization in natural and artificial photosynthesis[END_REF][START_REF] Baer | Optimization of spectral light quality for growth and product formation in different microalgae using a continuous photobioreactor[END_REF]. Therefore a substantial portion of the solar spectrum is unwanted for these solar PBRs which includes infrared and ultraviolet ranges which are the basic cause of overheating and cell-damaging on microalgae culture [START_REF] Nwoba | Can solar control infrared blocking films be used to replace evaporative cooling for growth of Nannochloropsis sp. in plate photobioreactors?[END_REF]. Some recent researches have witnessed that microalgae culture temperature can be regulated through passive evaporating cooling (PEC) and glazing which can filter infrared spectrum [START_REF] Béchet | Mechanistic modeling of broth temperature in outdoor photobioreactors[END_REF][START_REF] Nwoba | Growth comparison of microalgae in tubular photobioreactor and open pond for treating anaerobic digestion piggery effluent[END_REF]. Although PEC is an active thermal regulation approach but still due to cost and performance limitation, the use of PEC systems to maintain the optimum culture temperatures is not economically or environmentally sustainable. However, the glazing affects the transmission of the visible spectrum. Therefore, an innovative technology for microalgal PBR that requires less, or no, freshwater to filter the infrared spectrum will be a game-changer [START_REF] Nwoba | Can solar control infrared blocking films be used to replace evaporative cooling for growth of Nannochloropsis sp. in plate photobioreactors?[END_REF].

On the passive thermal regulation for a semi-buried open raceway, [START_REF] Pruvost | Thermal modeling and optimization of microalgal biomass production in the harsh desert conditions of State of Qatar[END_REF] has developed a thermal model to cultivate microalgae and achieved very good agreement with experimental results for temperatures. However, the simultaneous heat and mass transfer study for any open or closed PBR system, up to our knowledge, has not been reported yet. In solar photobioreactors, evaporation is often observed during microalgae production [START_REF] Torzillo | Production of spirulina biomass in closed photobioreactors[END_REF][START_REF] Moheimani | Coccolithophorid algae culture in closed photobioreactors[END_REF][START_REF] Hindersin | Irradiance optimization of outdoor microalgal cultures using solar tracked photobioreactors[END_REF], also in Algofilm © photobioreactor [START_REF] Goetz | A generic temperature model for solar photobioreactors[END_REF], which can be driven to develop condensate droplets or a film at the inner surface of the top cover. The developed condensate film can absorb infrared radiations reducing then the overheating in the Algofilm © PBR. Subsequently, water (condensate) is a promising filter which can transmit the visible light, required for microalgae growth, and absorb the infrared the major cause of overheating in outdoor PBRs [START_REF] Nwoba | Can solar control infrared blocking films be used to replace evaporative cooling for growth of Nannochloropsis sp. in plate photobioreactors?[END_REF]. For instance, [START_REF] Krauter | Increased electrical yield via water flow over the front of photovoltaic panels[END_REF] used a thin water film over a photovoltaic panel and achieved a huge reduction of 22 • C in the panel temperature.

The absorption of infrared radiation is a volumetric process [START_REF] Kirk | Solar heating of water bodies as influenced by their inherent optical properties[END_REF], therefore, it is imperative to quantify the amount of condensate which will be developed in outdoor conditions. The estimation of condensate film is extremely complicated in outdoor condition, due to the involvement of complex phenomena such as temperature variations with simultaneous heat and mass transfer with phase change (evaporation and condensation), in addition to solar radiation, ambient air velocity, relative humidity and the day/night cycle [START_REF] Apel | Engineering solutions for open microalgae mass cultivation and realistic indoor simulation of outdoor environments[END_REF][START_REF] Apel | Open thin-layer cascade reactors for saline microalgae production evaluated in a physically simulated mediterranean summer climate[END_REF][START_REF] Huesemann | Simulation of outdoor pond cultures using indoor led-lighted and temperature-controlled raceway ponds and phenometrics photobioreactors[END_REF]. Therefore, an optimized study cannot be carried out to establish a robust basis for modelling purposes.

Understanding and quantifying the phase change phenomena at controlled indoor conditions will constitute the basis for the estimation of condensate in outdoor conditions. For that purpose, this paper aims to build up a theoretical and experimental study which can be further used for outdoor conditions. The experimental study is conducted on indoor lab-scale Algofilm © prototype with the constant operating parameters to investigate the influence of flowing liquid temperature, air injection, and the inclination angle of the cover on the hourly condensation mass flux. A theoretical study was also carried out by considering the global simultaneous heat and mass transfer model with phase change.

Experimental setup and model considerations

Experimental setup

The lab-scale Algofilm © photobioreactor of the GEPEA laboratory, as shown in Fig. 1a, has a 0.33 m 2 surface area and it is mainly made up of stainless steel and aluminum with a transparent cover of Plexiglas. This prototype has an ability to be inclined up to 65 • , which made possible the collection of condensed mass that could be compared with the numerical predictions. The Algofilm © PBR was divided into four subsystem including bottom plate (i.e stainless steel), flowing liquid on bottom plate, condensate as well as the top cover plate (i.e. plexiglas), and humid air in between flowing liquid and condensate film as shown in (see Fig. 1a). All experiments were conducted in fully controlled indoor conditions such as regulated room temperature, the flowing liquid temperature was maintained constant through the external water heater, and was continuously re-circulated, at a constant volume flow rate, through an integrated pump. Liquid film was injected uniformly through a flow stabilizer at the inlet point (see Fig. 1c), which is quite effective to make a smoother and uniform flow of liquid on the bottom of Algofilm © PBR. The size and distribution of holes were already optimized on the same experimental setup in previous studies by Le Borgne (2014), [START_REF] Pruvost | Development of a thin-film solar photobioreactor with high biomass volumetric productivity (AlgoFilm©) based on process intensification principles[END_REF].

Experimental data acquisition

Indoor experimental setup is mainly based on the precise data acquisition regarding mass quantification and temperature distribution in every subsystem of Algofilm © PBR. A dedicated LabVIEW program was established to record the data (i.e. temperature, airflow rate etc) of each subsystem (glasscondensate film, humid air medium (inside air), flowing liquid and the bottom) as shown in Fig. 1b. The thermocouples for each sub-system are uniformly distributed on the centre in every zone (i.e., top, middle, and lower zones) see Fig. 1c (showing the thermocouples distribution for the flowing liquid sub-system). The horizontal surface of exchange is 30 × 100 cm, i.e. 0.3 m 2 . The interest of the indoor setup was to measure accurately the condensate mass flux. After reaching steady state the condensate mass was collected every hour, and every experiment was at least triplicated on same operating conditions (Inclination angle, liquid flow rate and temperature) and the mean data is considered for the averaged experimental condensation rate for each experiment.

Theoretical modeling

The mathematical model was developed for the prediction of heat and mass transfers involved in the passive thermal regulation technique of the Algofilm © photobioreactor. All heat and mass fluxes are briefly illustrated in Fig. 2.

The following assumptions hold:

(i) The film flow is assumed to be at a constant velocity. The residence time of liquid is very short with respect to characteristic evaporation time, therefore the variation in the flowing film thickness is assumed to be negligible. (ii) The temperature of the film is assumed to be constant throughout the process. (iii) The overall system of the photobioreactor is subdivided into subsystems (bottom, liquid, humid air, condensate, glass), where each subsystem is characterized by its averaged temperature T l , T b , T ha , T w/g . It was assumed that the condensate and the glass are thermally thin body and characterized by the temperature T w/g (iv) The heat and mass conservation in each subsystem is assumed to be modeled by global balances, based on the variation in enthalpy of each subsystem. Therefore, the spatial average values were considered that is:

∂ H i ∂t + U i ∂ H i ∂x = Q in - Q out (1)
where H i = H i,ref + m i cp i T i denotes the enthalpy of the subsystem i = b, l, ha, g/w. m i , cp i were respectively the mass and specific heat capacity of the subsystem i. Q in , Q out refers to input and output heat fluxes, respectively. U i is the averaged velocity over thickness of each participating subsystem.

U i = 1 ı i ı i 0 U i (y, t)dy and H i = 1 ı i ı i 0 H i (x, y, t)dy
(2)

Heat balance equations

It should be observed that the momentum Equations are not included since it is assumed a constant mass flow rate and the coupling with the heat balance is achieved through the residence time.

(3) Flowing Liquid:

The liquid film thickness ı l can be considered as constant over the length L of the PBR (smooth interface). The local energy Equation for a fully developped flow with an average velocity U l , is expressed as follows:

l cp l ∂T l ∂t + l cp l U l ∂T l ∂x = k l ∂ 2 T l ∂y 2 0 < x < L, 0 < y < ı l (3)
where k l is the thermal conductivity of the liquid, T l (x, y, t) is the temperature field of the liquid film. The transport mechanism with respect to the direction x is assumed to be purely advective, and purely diffusive in the y direction. Eq. ( 3) is subjected to the following initial and boundary conditions:

⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩ T l (x, y, t = 0) = T l,0 T l (x = 0, y, t) = T l,i k l ∂T l ∂y y=0 = q conv,l-b k l ∂T l ∂y y=ı l = -q conv,l-ha -q evap (4) S = L × W is the surface of exchange.
T l,i is the inlet temperature. q conv,l-b and q conv,l-ha are density of convective heat fluxes (W/m 2 ) between bottom-liquid, and liquid-humid air, respectively, and q evap is the evaporation heat flux density.

The integration of Eq. ( 3) over the liquid film thickness leads to:

l cp l ∂ ∂t ı l y=0 T l dy + l cp l U l ı l y=0 ∂T l ∂x dy = k l ∂T l ∂y y=ı l - ∂T l ∂y y=0 (5)
The averaged temperature over the thickness of the liquid film reads as follows:

T l = 1 ı l ı l 0 T l (x, y, t) dy (6)
The substitution of the boundary equations ( 4) and the averaged temperature over the thickness (6) into Eq. ( 5) gives:

l cp l ı l ∂ T l ∂t + l cp l ı l U l ∂ T l ∂x = -q conv,l-b -q conv,l-ha -q evap (7)
The integration of Eq. ( 5) over the length L gives the following equation:

l ı l cp l d dt L x=0 T l dx + l ı l cp l U l ( T l | x=L -T l | x=0 ) = L x=0 (-q conv,l-b -q conv,l-ha -q evap ) dx (8)
The spatial average temperature of the flowing liquid, and the averaged heat fluxes over the surface of exchange are defined as follows:

T l = 1 L ı l L x=0 ı l y=0 T l dxdy (9)
Q i = W L x=0 q i dx (10)
where i = conv, lb;conv, lha;evap. The substitution of the averaged temperature and fluxes ((9, 10)) in Eq. ( 8) give the global energy equation of the flowing liquid.

m l cp l dT l dt + ṁl cp l ( T l | x=L -T l | x=0 ) = (-Q conv,l-b -Q conv,l-ha -Q evap ) (11)
where ṁl = l ı l W U l is the mass flowrate of the liquid and ṁl = l ı l WL.

The characteristic time for heat exchange between the liquid and participating sub-systems i.e. (humid air and bottom) is defined as follows:

t c = m l cp l S(h fc + h cm + h evap ) (12)
The residence time can be defined as function of the average velocity U l and the length of PBR as follows:

t r = L U l (13)
When the residence time is smaller than the heat transfer characteristic time t r < t c , the heat flow is approximately equal to the heat exchanged between the liquid and the participating subsystems ṁl cp l (

T l | x=L -T l | x=0 ) ≈ -Q conv,l-b -Q conv,l-ha -Q evap .
In such case the accumulation of the averaged sensible heat over the residence time is negligible, which allow to write dT l /dt ≈ 0. Such approximation was experimentally verified (see Fig. 7b), since the liquid film residence time was short and the theoretical endorsement of t r < t c , the global modeling approach can be comfortably applied in the lab-scale Algofilm © . Therefore the Equation ( 11) can be re-written as:

dT l dt = 0 (14)
(4) Humid air: The overall heat and mass transfer of the system are highly dependent on humid air as it is the main transporting medium between the flowing liquid and condensate film. The development of condensate film is highly dependent on this subsystem as it receives convective heat flux and mass from the flowing liquid and transports them to the condensate film. The global heat balance Eq. ( 1) in the case of humid air read as:

d H ha dt + U ha d H ha dx = Q evap + Q conv,l-ha -Q conv,ha-w -Q cond (15)
where,

• <U ha > is the average velocity of the injection air (see Fig. 1a). • H ha = H ha,ref + m ha cp ha T ha is the enthalpy of the humid air. • Q evap is the evaporation heat flux.

• Q conv,l-ha is the convective heat flux between liquid and humid air. • Q conv,ha-w is the convective heat flux between humid air and condensate film.

• Q cond is the condensation heat flux.
One should note that some experiments were conducted without air injection which leads to reduction of Eq. ( 15) to:

d H ha dt = Q evap + Q conv,l-ha -Q conv,ha-w -Q cond (16)
(5) Glass-condensate: The condensate film development is mainly a function of vapour transported by humid air and condensed at the undersurface of the glass cover. The heat balance for condensate film is a function of heat and vapour transported from humid air to the glass and heat evacuated from the glass to the ambient air. In addition, an assumption of the thermally thin body was considered for the glass and condensate due to the small thickness of condensate film, accordingly, the global heat balance equation for this subsystem is written as follows:

d H w/g dt = Q cond + Q conv,ha-w -Q conv,w-amb (17)
where,

• H w/g = H w/g,ref + (m w cp w + m g cp g )T w/g is the enthalpy of the glass-condensate. • Q conv,ha-w is the convective heat flux between condensate-glass and ambient air. (6) Bottom: In the case of indoor Algofilm © bottom is exposed to the ambient atmosphere without having any insulation, therefore, it receives heat from flowing liquid in the form of forced convection and transfer heat to ambient atmosphere in the form of natural convection. The global heat balance equation (1) for the bottom will be rearranged as:

d H b dt = Q conv,l-b -Q conv,b-amb (18)
where

• H b = H b,ref + m b cp b T b
is the enthalpy of the bottom.

• Q conv,l-b is the forced convective heat flux between liquid and bottom. • Q conv,b-amb is the convective heat flux between bottom and ambient air.

Mass balance equation

The mass transfer is mainly happening due to evaporation, and condensation. It is assumed saturated humid air and all the vapour produced at the flowing liquid interface is condensed at the undersurface of the glass cover. The mass balance equation expressing the variation of m w reads as follows:

ṁha out -ṁha in = ṁevap -ṁw (19)
where ṁevap is the evaporation rate, ṁha is the mass flow rate of the injected air. Without air injection Eq. ( 19) becomes: ṁw = ṁevap (20)

Heat and mass transfer fluxes

The heat and mass transfer model was developed by considering all significant transport fluxes including convective and latent heat transfers. The heat transfer in Algofilm © photobioreactor can be divided into two main categories internal and external heat transfers as shown in Fig. 1a. Internal heat transfer includes latent heat transfer (i.e. evaporation and condensation), and free/forced convection heat transfer between flowing liquid to humid air and bottom surface, and humid air to glass cover of Algofilm © . However, the external heat transfer is considered for the heat evacuated from the top cover and bottom surface to the ambient atmosphere by natural convection.

Convective heat transfer fluxes

The convective heat fluxes are described by Newton's law of cooling. The internal heat transfer coefficients are dependent on temperature and partial pressure of the vapour, due to the presence of simultaneous heat and mass transfer. However, the external heat transfer coefficients do not involve any mass transfer. The internal heat transfer fluxes are given by:

Q conv,l-ha = Sh cm (T l -T ha ) (21) Q conv,ha-w = Sh cha (T ha -T w ) ( 22
)
where S is the surface of the Algofilm © PBR and h cm , h cha are the heat transfer coefficients (from flowing liquid to humid air, humid air to condensate) and are theoretically derived by a heat and mass transfer analogy by [START_REF] Dunkle | Solar water distillation: the roof type still and a multiple effect diffusion still[END_REF], applied on solar photobioreactor [START_REF] Artu | Etude et optimisation de la culture de microalgues en photobioréacteurs solaires[END_REF] to quantify the free convective heat transfer in closed geometries. These coefficients are:

h cm = 0.884 (T l -T w ) + p v,l -p v,ha 2016 -p v,l T l 1 3 (23) h cha = 0.884 (T l -T w ) + p v,ha -p v,w 2016 -p v,l T l 1 3 (24)
The heat transferred between the flowing liquid and the bottom is given by:

Q conv,l-b = S h fc (T l -T b ) (25)
where h fc is the forced convective heat transfer coefficient and expressed as:

h fc = Nuk l L ( 26
)
where k l , L is the thermal conductivity of liquid and the length of the Algofilm © PBR, and Nu is the Nusselt number expressed as function of Re and Pr as follows [START_REF] Welty | Fundamentals of Momentum, Heat, and Mass Transfer[END_REF]:

Nu = 0.664Re 1 2 Pr 1 3 (27)
The amount of heat evacuated through external surfaces of Algofilm © PBR read as follows:

Q conv,b-amb = S h b-amb (T b -T amb ) (28) Q conv,w-amb = S h w/g-amb (T w/g -T amb ) ( 29
)
where h b-amb , h w/g-amb are the heat transfer coefficients for the heat flow in upward and downward direction.

h b-amb = Nu b k air L c ( 30
)
h w/g-amb = Nu w/g k air L c ( 31
)
where k air is the thermal conductivity of ambient air,L c is the characteristic length of the heat exchanging surface and Nu b , Nu w/g are the Nusselt numbers for natural convection for downward and upward heat flow respectively, which are the given as a function of Rayleigh number Ra.

Nu b = 0.27Ra 1 4 (32)
Nu w/g = 0.54Ra 1 4

(33)

Latent heat transfer fluxes

The latent heat fluxes of evaporation and condensation are given by the analogical expression of Newton's law of cooling but in terms of partial vapour pressures instead of temperatures of the participating subsystems. These fluxes are expressed as:

Q evap = L evap ṁevap (34) Q cond = L evap ṁcond (35)
where L evap is the latent heat of evaporation, and ṁevap , ṁcond are the evaporation and condensation mass fluxes. In this case the humid air was assumed to be saturated with vapour, allowing then to write ṁevap = ṁcond and Q evap = Q cond [START_REF] Yu | A molecular dynamics investigation on evaporation of thin liquid films[END_REF].

Mass transfer fluxes

The evaporation mass flux ṁevap is computed from:

ṁevap = S h evap (p v,l -p v,ha ) (36)
where p v,i is the partial vapour pressure of subsystem i = l, w/g, and the mass transfer coefficient h evap is obtained from the heat transfer coefficient h cm (Eq. ( 23)) with a correlation factor of 9.15 × 10 -7 , and is theoretically derived by a heat and mass transfer analogy by [START_REF] Dunkle | Solar water distillation: the roof type still and a multiple effect diffusion still[END_REF], applied on solar photobioreactor by [START_REF] Artu | Etude et optimisation de la culture de microalgues en photobioréacteurs solaires[END_REF] to quantify the evaporative heat transfer as follows:

h evap = 9.15 × 10 -7 h cm (37)

Numerical solution for heat and mass transfer model equations

The substitution of all the above defined heat and mass fluxes and enthalpy expressions in equations ((14, 16),(17, 18),( 20)) lead to system of five ODEs to be solved simultaneously as given in the following for the case of zero injected air flow:

⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩ dT l dt = 0 m air cp air dT ha dt = Q evap + Q conv,l-ha -Q conv,ha-w -Q cond (m w cp w + m g cp g ) dT w/g dt = Q cond + Q conv,ha-w -Q conv,w-amb m b cp b dT b dt = Q conv,l-b -Q conv,b-amb dm w dt = S h evap (p v,l -p v,ha ) (38)
with the following initial conditions:

⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩ T l (t = 0) = T l,0 T ha (t = 0) = T ha,0 T w/g (t = 0) = T w/g,0 T b (t = 0) = T b,0 m w (t = 0) = 0 (39)
The solution of heat and mass transfer model was obtained by integration with respect to time of the system of ordinary differential equations (38) taking into account the initial conditions (39). The integration of these ODEs was carried out by the MATLAB predefined function.

Results and discussions

The experimental and theoretical results are presented with different environmental and controlled operating conditions. The experimental results were also compared with the theoretical model in terms of condensed mass at the undersurface of the glass cover, and the temperatures of each subsystem.

Experimental results

The experiments were conducted by varying the inclination angle, flowing liquid temperature, and the air injection to evaluate the effect on hourly condensation rate, and the temperatures of each subsystem. The effect of inclination angle is a very important factor [START_REF] Goshayeshi | Effect of absorber plate surface shape and glass cover inclination angle on the performance of a passive solar still[END_REF]) not only to see its impact but also to quantify the maximum experimental condensation rate. Fig. 3 shows the results for hourly condensation rate on a wide range of inclination angles 4-65

• , on fixed parameters such as liquid temperature (35, 40 and 50 • C) and flow rate of 0.08 l/s in the Algofilm © PBR.

The experimental results showed that the collection rate is increased with increasing up to a critical value = 35 • ), where the collection rate reached the asymptotic value and revealed the experimental quantification of condensation rate. This fact was caused due to the increased gravitational force which drives condensate to flow into the condensate collector [START_REF] Goshayeshi | Effect of absorber plate surface shape and glass cover inclination angle on the performance of a passive solar still[END_REF].

The influence of the flowing liquid temperature on the condensation rate is depicted in Fig. 4. The results showed that the condensation rate increased with the increasing flowing liquid temperature, where a large increase was observed from 9.5 g/h to 49.5 g/h on 35 and 50 • C respectively. This effect can be explained by the increase of transport of the vapour through buoyancy force which highly depends on the temperature gradient between flowing liquid and the glass cover. Moreover, the increase of the temperature of the flowing liquid also increased the partial pressure of the vapour at the interface, which results in the increase in evaporation as well as the condensation rate. This can help to correlate the indoor study with the outdoor atmosphere while dealing with the relevant temperature range.

Fig. 5 shows the photos taken from the top of the Algofilm © PBR for inclination angle of 6 • at the temperature of 50 • C of the flowing liquid. These photos show the existence of polydisperse and connected droplets for all the cases considered. This observation made it possible to consider an irregular film of condensate at the undersurface of the glass cover.

The existence of irregular film made possible the estimation of an averaged condensate film on different inclination angles as shown in Fig. 5, calculated from the data of condensation rate experiments on different . The averaged condensate film thickness was calculated as follows:

ı cond = ṁw t d1 l S (40)
where ı cond in the averaged film thickness, ṁw is the maximum condensation rate (maximum condensation rate was collected by varying the inclination angles up to 65 • , however after 35 • the experimental condensation rate was constant, which allowed to assume that the condensation rate achieved at the critical inclination angle 35 • is equivalent to the maximum experimental condensation rate and independent of ), t d1 is the time taken by the first droplet to slide into the condensate collector, and S is the glass surface area. Furthermore, to estimate the averaged condensate film thickness the time between the initial time t 0 and the time when the first droplet to start slide t d1 was recorded, and the condensation rate was taken into account for the observed duration to compute the amount of condensate stocked at the under the surface of the top cover. The computed volume of condensate per surface area of glass cover gave the approximate averaged condensate film thickness. Fig. 6 represents the averaged condensate film thickness estimated on the range of =4-35 • . The maximum film thickness 2.2 mm was achieved on lowest of 4 • , however, the film thickness remained minimum 0.1 mm on higher of 35 • . The averaged condensate film thickness is highly dependent on inclination angle . If inclination angle is higher the condensate will continue to slide down into the condensate collector and less amount of condensate will remain on the undersurface of glass cover which leads the reduction in condensate film thickness. Nevertheless, when is lower as 4 • the condensate could not slide into the collector and remained at the glass cover due to the least gravitational effect which results a significant condensate thickness as presented in Fig. 6, however sometimes droplets coalescence was observed due to short sliding of droplets, but that coalescence could not drive the condensate droplets into the collector. Such results showed the importance of the indoor experiments, which made possible, to estimate this condensate film thickness which was very difficult to obtain in outdoor conditions, where this infor-

Experimental validation of the theoretical assumptions

The major assumptions made in the theoretical modeling include (i) natural convection in the humid air and (ii) uniform temperature within the subsystems. Accordingly, these assumptions were endorsed by the experimental results discussed hereafter.

It is well known that the influence of air injection in PBR is important for the microalgae as they require CO 2 injection for pH regulation and growth. From the point of view of heat and mass transfer, the air flow rate can influence the transport mechanism in the humid air, which can impact the condensation rate. To understand the impact of air injection on the transport behavior, experiments with different flow rates (i.e. 2.5-7.5 l/min) were conducted, while considering the airflow rate range of outdoor Algofilm © PBR i.e. 2-4 l/min. The vapour can be transported by air outside by forced convection reducing the amount of condensate on the undersurface glass cover. Fig. 7a represents the effect on condensation rate for different airflow rates form 2.5 to 7.5 l/min, a case without air injection was included as base of comparison.

Fig. 7a showed that natural convection dominated in Algofilm © PBR, therefore no any significant impact on the condensation rate was observed. Furthermore, it was also shown that the Richardson Number Ri for the air flow rates considered (2.5, 5, 7.5 l/min), had respectively the values (8440,2110,940), showing that the natural convection was the main heat transfer process. These results endorsed the assumption considered in the theoretical modeling that the dominated transport mechanism in humid air is natural convection.

Regarding the assumption of uniform temperature within each subsystem was also verified by measuring the temperature at entrance/upper zone and exit/ lower zone of the Algofilm © (see Fig. 1a), the comparison for each subsystem was depicted in Fig. 7b.

The results on Fig. 7b show that the gradient between the upper and lower zone is negligible. These results endorsed the global modeling approach for predicating heat and mass transfer in Algofilm © PBR. The assumption of uniform temperature within each subsystem was also discussed and validated experimentally by [START_REF] Goetz | A generic temperature model for solar photobioreactors[END_REF]. However in the large-scale outdoor PBR the temperature gradient may not be negligible, in such situation the local modeling approach is recommended. In the case of lab-scale Algofilm PBR the length and the height of flowing liquid are small enough to acquire the temperature gradient within the flowing liquid film as it is controlled through and external water heater, and the residence time is very short.

Theoretical results and comparison with experiments

The validation of the simulated results consists of the comparison with the experimental results for temperature, averaged mass transfer flux and the condensed mass. The comparison was made for the temperature of humid air, condensate/glass, and the bottom of Algofilm © PBR.

The solution of Eqs. ( 38) for indoor conditions allowed to predict the temperatures of each subsystem and condensate amount at the undersurface of the glass cover. The simulations were carried out on three liquid temperatures 35, 40, and 50 • C as reported in Fig. 8. For all the cases considered, the evolution of the temperature of each subsystem showed transient behaviour in the first hour before reaching steady-state. The simulated results were in a good agreement with the experimental results for all the investigated temperatures. Whereas the maximum deviation at the steady-state was observed at most 1 • C. However, considering the complexities of the simultaneous heat and mass transfer with phase change phenomena and the global model this deviation is acceptable. Furthermore, from the point of view of microalgae growth and temperature modeling studies this deviation will not have a significant effect [START_REF] Pruvost | Thermal modeling and optimization of microalgal biomass production in the harsh desert conditions of State of Qatar[END_REF].

Fig. 10a represents the evolution of experimental and numerical condensed mass for different temperature of flowing liquid i.e (T l = 35, 40, 50 • C). The evolution of m w over the time exhibited linear behaviour after achieving steady-state for all the temperatures of flowing liquid considered. The comparison of results is in acceptable range with a maximum deviation of 2 g/h for 50 • C, but for lower temperatures which represent the actual range for microalgae production about 20-30 • C, the modeling results are in very good agreement with a maximum deviation 0.8 g/h.

The linear behaviour of condensed mass over time allowed us to calculate the averaged condensation rate with respect to the liquid temperature. Fig. 10b represents the comparison of experimental and simulation results of averaged condensation rate for all the investigated temperatures. The mass flux is seeming to be a highly dependent on the flowing liquid temperature as it increased from 10 g/h to 49.5 g/h by varying the temperature from 35 to 50 • C. Fig. 9 shows the evolution of heat fluxes i.e. evaporation and condensation over the range of studied flowing liquid temperature (35, 40, and 50 • C) with respect to time. The humid air between flowing liquid (evaporating surface) and top cover (condensing surface) was assumed to be instantaneously saturated with vapour in this case, therefore all the vapour produced at the liquid/air interface will be condensed at under surface of the top cover, since the PBR is closed. This assumption was justified for a similar case through lattice Boltzmann simulations, accordingly the latent heat of evaporation and condensation are same, since m evap = m cond [START_REF] Yu | A molecular dynamics investigation on evaporation of thin liquid films[END_REF]. The maximum value for evaporation and condensation heat fluxes after reaching fully developed condition is about 21.9, 39.19, 89.65 w/m 2 for (35, 40, 50 • C), respectively. The driven force for evaporation and condensation is the temperature difference between the flowing liquid (which is controlled through water heater) and the ambient temperature (regulated indoor temperature). Therefore, at the start of simulation the magnitude of these fluxes is higher because of the large temperature gradient as the initial conditions for liquid temperature are (35, 40, 50 • C), however the indoor ambient temperature is almost constant at 23 • C which also corresponds to the humid air temperature experimentally as well as numerically at initial conditions. The transient time for simulation or experiments is maximumly about one hour for studied range of temperature as shown in Fig. 8, which caused higher values for evaporation and condensation fluxes for first hour but declining continuously. This decline in the heat fluxes is due to decrease in the temperature difference (between flowing liquid and humid air) which was maximum at initial conditions, however after reaching fully developed conditions (after one hour) these fluxes became constant as the temperature gradient between humid air and flowing liquid reached at a constant value (numerically as well as experimentally).

The evolution of averaged condensation rate over the range of liquid temperature is presented in Fig. 11 which shows the same trend of as Fig. 10b, where the increase in liquid temperature results an increase in averaged condensation rate. This figure also shows a critical temperature for which no condensation was observed, that corresponds perfectly to the ambient temperature. As expected, this result seems to be logical while there is no temperature gradient which is driving force for evaporation, therefore the system has achieved thermal equilibrium. Such ascertainment shows the impact of the ambient temperature on evaporation which was not so expected even when the liquid temperature is automatically regulated. This result was further studied to see the impact of ambient temperature on the condensation rate and some simulations were carried out by considering the ambient indoor temperature variation. The results of these simulations exhibited that the ambient temperature can also be an influencing parameter for condensation rate, as it was significantly influenced by these small temperature deviations i.e. (23 ± 2 • C). The influence of flowing liquid and ambient temperatures can be stretched to develop a correlation to estimate the condensation rate for closed PBRs in realistic conditions. [START_REF] Liu | An analytical model of the heating and evaporation of bi-component wall film[END_REF] showed that the mass flux of an evaporating liquid film can be proportional to the mass flux derived from the analytical solution of Stefan flow transport Equation in binary gas mixture, which is expressed as follows:

ṁ = ˇ × S air D v ı a ir ln 1 -Y v,amb 1 -Y v,l (41)
where:

• Y v,l , Y v,∞ are the mass fraction of vapor in liquid and gas respectively, ˇ is the factor of proportionality, and ı air is the thickness air gap. • D v is the binary diffusion coefficient, which can be expressed in this case as a function of the flowing liquid temperature by the relation of Fuller-Schettler-Giddings [START_REF] Green | Perry's Chemical Engineers' Handbook/edición Don W[END_REF]:

D v = 10 -7 T 1.75 l 1 Ma + 1 Mv P amb 1/3 a + 1/3 v ( 42
)
where M i , i = a, v is the molecular mass for air and vapor and i is the diffusion volumes for air and vapor molecules. • air is the density of vapor evaluated at the liquid temperature, which can be expressed by the perfect gas law as follows:

air = P amb M air RT amb (43)
The influencing parameters of the evaporation flux [START_REF] Liu | An analytical model of the heating and evaporation of bi-component wall film[END_REF] are the temperatures of the evaporating liquid film and the gas. However in current study the driving force of evaporation is the temperature gradient between the flowing liquid film and ambient air, accordingly the mass flux in current study can be expressed in same way as Eq. ( 41), therefore a simulation was carried out by considering ambient air temperature as 23 • C and results are plotted in Fig. 12, where the mass flux given by Eq. ( 41) showed the same trend with simulation results, while considering a constant of proportionality ˇ = 98. However, the constant of proportionality ˇ appeared as a function of ambient air temperature when some simulations were carried out by considering the wide ange of ambient temperature with respect to seasonal variations. Therefore, to extend the applicability the developed correlation ˇ was fitted as func- tion of ambient air temperature, which leads to the following expression:

ˇ = 2.011 T amb T K 2 -5.168 T amb T K + 3.335 (44)
T K = 273.15 K is the absolute temperature. This correlation takes into account the temperature of the flowing liquid and the ambient air. These temperatures, in outdoor conditions, are always varying with respect to the weather conditions, which may allow to predict the mass of condensed water on the undersurface of the glass cover of Algofilm © PBR, which is very important to quantify the infrared filtration. Fig. 12 showed the comparison of results of simulation and Eq. ( 41) for average condensation rate with respect to T l . The comparison showed that the correlation can be applied to predict the condensation rate over the wide range of ambient temperature, which will be of great interest in outdoor conditions.

Validation of the indoor model in a simulated sunny day

Fig. 13a andb are showing the evolution in simulated sunny day experiment and simulation for the temperatures of different sub-systems and the condensed mass, respectively.

In order to test the response of the developed heat and mass transfer model on dynamic conditions, a mimic experiment was conducted to validate the model on a representative condition of a typical outdoor day, by varying liquid temperature using an external water heater. Flowing liquid temperature was increased from 35 • C up to 50 • C and then decreased inversely with a time interval of one hour by considering the thermal inertia of solar radiation in a typical sunny day. This approach of simulated sunny day with artificial solar light was also considered by [START_REF] Goetz | A generic temperature model for solar photobioreactors[END_REF] in controlled indoor conditions for the validation of heat transfer model to predict the temperatures of the different subsystems.

The simulated results for temperature predictions of each subsystem (without any fitting or adjustment) were compared with the experimental ones and showed a good agreement (see Fig. 13a). In addition to temperature validation, the condensed mass prediction was also in good agreement (Fig. 13b) over the period of mimic outdoor day. The experimental mass collected over the period of whole experiment was 95 g, however the deviation of 15 g was observed at the end of experiment but that corresponds to the mass remained at the glass cover before the sliding of the first droplet as discussed in Fig. 6. These results showed the capability of developed model to be applied on real outdoor solar conditions. [START_REF] Goetz | A generic temperature model for solar photobioreactors[END_REF] carried out some outdoor experiments and the mean temperature of flowing liquid reached up to 55 • C in a typical sunny day in the month of July, but in the current study it is experimentally and numerically revealed that evaporation rate at this temperature can easily reached around 65 g/h (see Fig. 11), which is not a negligible value. The condensed water mass can serve the purpose of passive evaporating cooling and filtration of infrared spectrum to avoid overheating of microalgae culture with a passive approach. Indeed, [START_REF] Nwoba | Can solar control infrared blocking films be used to replace evaporative cooling for growth of Nannochloropsis sp. in plate photobioreactors?[END_REF] have shown the influence of the infrared filtration on microalgae culture temperature, where passive evaporating cooling was used as a filter for infrared radiation to avoid the overheating of microalgae and temperature reduction of 8 • C was observed when compared with and without PEC. This fact endorsed the need of incorporation of simultaneous heat and mass transfer model for thermal regulation of Algofilm © PBR.

Conclusion

The present work was aimed to develop a simultaneous heat and mass transfer model to predict the condensate film thickness which will absorb the Infrared spectrum (main source of overheating). The interest of the indoor study was to validate the model with controlled operating parameters (not possible in outdoor conditions) and experimentally investigating all the influencing parameters for evaporation and condensation. Accordingly, a detailed experimental and theoretical study on lab-scale has been conducted. The experimental study focused on the evolution of temperature and condensed mass as a function of inclination angle, air injection, flowing liquid and ambient temperature. The results showed that the temperature of flowing liquid is the most important factor on condensation rate, whereas the air injection has not shown any significant effect for the investigated range.

These results revealed that air injection in outdoor condition will not affect the transport of heat and mass by natural convection. The numerical results were carried out for the prediction of temperature and condensed for the wide range of flowing liquid temperatures i.e. 30-50 • C. The experimental and numerical results were found in a good agreement for the whole range of flowing liquid temperature in steady state conditions. Besides, the model has the ability to predict the temperature and condensed mass on artificial dynamic condition. The good agreement between numerical and experimental results made possible to develop a correlation to estimate the condensation rate as a function the flowing liquid and ambient temperature; the latter appeared to be important in the evaporation process.

The water is the potential filter for the infrared spectrum (main source of overheating the microalgae). Active cooling systems such as passive evaporating cooling system is not viable due to the water constraints, therefore the developed passive thermal regulation approach will serve both purposes: (i) do not need any fresh water to filter infrared spectrum as reported by [START_REF] Krauter | Increased electrical yield via water flow over the front of photovoltaic panels[END_REF], [START_REF] Nwoba | Can solar control infrared blocking films be used to replace evaporative cooling for growth of Nannochloropsis sp. in plate photobioreactors?[END_REF], [START_REF] Ghosal | Performance of solar photovoltaic module through combined air and water cooling in warm and humid climatic condition of India[END_REF]; (ii) condensate film development will be a function of solar irradiation intensity (condensation would be maximum at the peak hours of the day which will filter infrared radiations since the present indoor study showed that the evaporation rate is a function of flowing liquid and ambient temperatures). The development of current simultaneous heat and mass transfer model with a good agreement on a simulated sunny will be a prime step to apply this model on real solar conditions, where the estimation of condensed mass is essential to quantify the amount of filtered infrared radiations.

Fig

  Fig. 1 -(a) Schematic description of indoor AlgoFilm ©. (b) Experimental setup for AlgoFilm © indoor prototype. (c) Flow for uniform film distribution and (d) Distribution of the thermocouples on sub-system "Flowing liquid".

Fig. 3 -

 3 Fig. 3 -Effect of inclination angle on condensation rate.

Fig. 5 -

 5 Fig. 5 -Development of irregular condensate film on T l = 50 • C and = 6 • .

Fig. 6 -

 6 Fig. 6 -Estimation of averaged condensate film thickness over the range of inclination angles .

Fig. 7 -

 7 Fig. 7 -Experimental validation of theoretical assumptions: (a) Effect of air injection on condensation rate; (b) Temperature gradient of each subsystem for T l = 50 • C (without air injection).

Fig. 8 -

 8 Fig. 8 -Comparaison between simulated and experimental temperature of each subsystem for T l = 35, 40, 50 • C.

Fig. 9 -

 9 Fig. 9 -Evolution of evaporation and condensation latent heat fluxes for T l = 35, 40, 50 • C.

Fig. 10 -

 10 Fig. 10 -Comparison of experimental and simulated results for T l = 35, 40, 50 • C: (a) Condensed mass (b) Mass flux.

Fig. 11 -

 11 Fig. 11 -Evolution of condensation rate as a function of flowing liquid and ambient temperature.

Fig. 12 -

 12 Fig. 12 -Evolution average condensation rate as a function of ambient temperature (comparison of developed correlation with simulations).

Fig. 13 -

 13 Fig. 13 -Evolution of condensed mass and temperatures of each sub-system in simulated sunny day.

Acknowledgment

The Higher Education Commission, Pakistan is acknowledged for the financial support to the PhD student under their project HEC-UESTP, Pakistan.

Declaration of Competing Interest

The authors report no declarations of interest.