Euntaek Lee

Jérémy Pruvost

Xing He

Ramakanth Munipalli

Laurent Pilon
email: pilon@seas.ucla.edu

This study provides design and operational guidelines for achieving maximum biomass productivity in outdoor photobioreactors (PBRs). Detailed simulations of coupled light transfer and growth kinetics of microalgae were performed for open ponds, vertical flat-plate, and tubular PBRs operated in batch mode and exposed to time-dependent collimated and diffuse solar irradiance. The temporal evolution of microalgae concentration was predicted by accounting for light saturation, photoinhibition, and respiration. Three-dimensional spectral light transfer simulations of collimated and diffuse solar radiation in the PBRs were performed at different times of the day. The green microalgae Chlamydomonas reinhardtii was used for illustration purposes. The study demonstrated that the daily productivity per unit of illuminated surface area for PBRs operated in batch mode was identical and depended uniquely on the ratio X 0 /a where X 0 is the initial microalgae concentration and a is the illuminated surface area per unit volume of PBR. A maximum daily productivity of about 0.045 kg/m 2 /day was achieved for X 0 /a= 0.035 kg/m 2 . Remarkably, similar results were obtained with experimental data and other simulation results based on different models reported in the literature, for different microorganisms and PBRs operated in continuous mode. The PBR optical thickness, represented by X 0 /a, constitutes a convenient parameter for designing (via a) and operating (via X 0 ) these PBRs to achieve their maximum performance.

1

INTRODUCTION

Microalgae cultivation has received significant attention in recent years as a way to fixate CO 2 generated during fossil fuel combustion and to produce liquid or gaseous biofuels [START_REF] Pilon | Radiation transfer in photobiological carbon dioxide fixation and fuel production by microalgae[END_REF] as well as food supplement [START_REF] Skjånes | BioCO 2 -a multidisciplinary, biological approach using solar energy to capture CO 2 while producing H 2 and high value products[END_REF] and protein for human or animal feed [START_REF] Yoon | High cell density culture of Anabaena variabilis using repeated injections of carbon dioxide for the production of hydrogen[END_REF]. Photosynthetic microalgae use sunlight as their energy source, water as their electron source, and CO 2 as their carbon source. They are typically grown in open ponds and photobioreactors (PBRs) of various designs where sunlight is absorbed and scattered by the microalgae kept in suspension by mechanical stirring and/or bubble sparging [START_REF] Asenjo | Bioreactor System Design[END_REF]. To be economically viable, the processes require the highest microalgae productivity and efficiency. Open ponds or PBRs can be operated in batch or in continuous mode. Batch cultures are widely used for their simplicity, flexibility, and low cost [START_REF] Barsanti | Algae: Anatomy, Biochemistry, and Biotechnology[END_REF]. Scaling-up benchtop PBRs to industrial scale remains a challenge [START_REF] Morweiser | Developments and perspectives of photobioreactors for biofuel production[END_REF]. Indeed, optimum temperature, mixing, light, and mass transfer should be maintained in photobioreactors of any sizes [START_REF] Ugwu | Photobioreactors for mass cultivation of algae[END_REF]. Current photobioreactors must be improved in order to achieve larger mass concentrations and growth rate and to minimize auxiliary energy use and capital cost [START_REF] Posten | Design principles of photo-bioreactors for cultivation of microalgae[END_REF].

The objective of this study is to develop accurate numerical simulation tools and to obtain design guidelines for the optimization and operation of efficient PBRs. To do so, light transfer in PBRs with various geometries exposed to solar radiation was analyzed on a spectral basis over the spectral region between 400 and 700 nm corresponding to the photosynthetically active radiation (PAR) region. The temporal evolution of microalgae concentration was also predicted using growth kinetics model taking into account the local available light in the PBRs at different times of the day.

BACKGROUND

Radiation Transfer in Photobioreactors

As light travels through the microalgae suspension contained in the PBR, it is absorbed by the microorganisms or by the medium and scattered by microorganisms and, possibly, by gas bubbles used to deliver CO 2 and to stir the suspension. Solar radiation intensity I λ (r, ŝ) at location r traveling along direction ŝ is governed by the radiative transfer equation (RTE) [START_REF] Pilon | Radiation transfer in photobiological carbon dioxide fixation and fuel production by microalgae[END_REF].

The local spectral fluence rate, denoted by G λ (r), and the local fluence rate average over the PAR region between 400 and 700 nm, denoted by G P AR (r), available to microalgae at location r are respectively defined as [START_REF] Pilon | Radiation transfer in photobiological carbon dioxide fixation and fuel production by microalgae[END_REF],

G λ (r) = 4π I λ (r, ŝ)dΩ and G P AR (r) = 700 400 G λ (r)dλ (1)
The average fluence rate G av over the entire PBR of volume V can be estimated from the local PAR-averaged fluence rate G P AR (r) as,

The two-flux approximation assumes one-dimensional radiation transfer and can account for in-scattering terms as well as anisotropic scattering [START_REF] Modest | Radiative Heat Transfer[END_REF] and provides a simple analytical expression for the local spectral fluence rate G λ (z) [START_REF] Cornet | A structured model for simulation of cultures of the cyanobacterium Spirulina platensis in photobioreactors: I. Coupling between light transfer and growth kinetics[END_REF][START_REF] Cornet | A simplified monodimensional approach for modeling coupling between radiant light transfer and growth kinetics in photobioreactors[END_REF]. In cases when the PBR is exposed to both collimated and diffuse irradiances G in,c,λ and G in,d,λ , the total local spectral fluence rate G λ (z) (in W/m 2 ) can be estimated by summing up its collimated and diffuse components as [START_REF] Pruvost | Theoretical investigation of biomass productivities achievable in solar rectangular photobioreactors for the cyanobacterium arthrospira platensis[END_REF],

G λ (z) = G c,λ (z) + G d,λ (z) (3)
Pottier et al. [START_REF] Pottier | A fully predictive model for one-dimensional light attenuation by Chlamydomonas reinhardtii in a torous photobioreactor[END_REF] solved the radiative transfer equation (RTE) using the two-flux approximation to model light transfer in a one-dimensional flat-plate photobioreactor with a transparent front window and a diffusely reflecting back side with spectral reflectance ρ λ . The authors derived an analytical expression for the local spectral fluence rate G c,λ (z) (in W/m 2 ) in such PBRs exposed to solar irradiance G in,c,λ incident onto the photobioreactor at an angle θ c with respect to the surface's normal direction as [START_REF] Pottier | A fully predictive model for one-dimensional light attenuation by Chlamydomonas reinhardtii in a torous photobioreactor[END_REF],

G c,λ (z) G in,c,λ = 2 sec θ c [ρ λ (1 + α λ )e -δ λ L -(1 -α λ )e -δ λ L ]e δ λ z + [(1 + α λ )e δ λ L -ρ λ (1 -α λ )e δ λ L ]e -δ λ z (1 + α λ ) 2 e δ λ L -(1 -α λ ) 2 e -δ λ L -ρ λ (1 -α 2 λ )e δ λ L + ρ λ (1 -α 2 λ )e -δ λ L (4)
where the parameters 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 torous photobioreactor[END_REF],

α λ = Āabs,λ Āabs,λ + 2b λ Ssca,λ and δ λ = X sec θ c Āabs,λ ( Āabs,λ + 2b λ Ssca,λ ) (5)
here Āabs,λ and Ssca,λ are the average mass absorption and scattering cross-sections (in m 2 /kg) while X is the microorganism mass concentration X expressed in kg dry cell weight cells per m 3 of suspension (or kg/m 3 ). The effective absorption and scattering coefficients of the suspension are given by κ λ = Āabs,λ X and σ s,λ = Ssca,λ X

For an axisymmetric phase function, the backward scattering fraction, denoted by b λ is defined as [START_REF] Pottier | A fully predictive model for one-dimensional light attenuation by Chlamydomonas reinhardtii in a torous photobioreactor[END_REF],

b λ = π π/2 Φ T,λ (θ) sin θdθ (7)
where Φ T,λ (ŝ i , ŝ) is the effective scattering phase function of the suspension. The local diffuse spectral fluence rate G d,λ (z) can be estimated from Eqs. 4 and 5 by replacing by a factor 2.

Microalgae Growth Kinetics

The time rate of change of microalgae mass concentration X can be modeled as [START_REF] Dunn | Biological Reaction Engineering: Dynamic Modelling Fundamentals with Simulation Examples[END_REF],

dX dt = μX ( 8
)
where μ is the average total specific growth rate expressed in hr -1 . Various growth kinetics models have been developed to predict the local specific growth rate µ. Fouchard et al. [START_REF] Fouchard | Kinetic modeling of light limitation and sulfur deprivation effects in the induction of hydrogen production with Chlamydomonas reinhardtii : Part I. model development and parameter identification[END_REF] expressed the photosynthetic growth rate of C. reinhardtii as a function of local fluence rate G P AR (r) at location r, according to the Haldane model [START_REF] Versyck | Practical identification of unstructured growth kinetics by application of optimal experimental design[END_REF], accounting for light limitation and inhibition as

µ p (r) = µ 0 G P AR (r) K S + G P AR (r) + G 2 P AR (r)/K I (9)
where µ 0 is the so-called maximum specific growth rate while the coefficients K S and K I are the light half-saturation and inhibition constants, respectively. The authors also considered reduction in growth rate due to cellular respiration as the catabolic process. Then, the total local specific grow rate µ(r) can be expressed as [START_REF] Fouchard | Kinetic modeling of light limitation and sulfur deprivation effects in the induction of hydrogen production with Chlamydomonas reinhardtii : Part I. model development and parameter identification[END_REF] µ(r) = µ p (r) -µ s [START_REF] Cornet | A structured model for simulation of cultures of the cyanobacterium Spirulina platensis in photobioreactors: I. Coupling between light transfer and growth kinetics[END_REF] where µ s is the respiration rate and was assumed to be constant at all times and locations [START_REF] Fouchard | Kinetic modeling of light limitation and sulfur deprivation effects in the induction of hydrogen production with Chlamydomonas reinhardtii : Part I. model development and parameter identification[END_REF].

In addition, the average total specific growth rate μ over the PBR volume can be estimated as [START_REF] Fouchard | Kinetic modeling of light limitation and sulfur deprivation effects in the induction of hydrogen production with Chlamydomonas reinhardtii : Part I. model development and parameter identification[END_REF] μ

= 1 V V µ(r)dV (11)

Photobioreactor Modeling

Modeling of PBR typically consists of solving the RTE coupled with a growth kinetics model. Aiba [START_REF] Aiba | Growth kinetics of photosynthetic microorganisms[END_REF] calculated the light intensity distribution and absorptance in one-dimensional flatplate photobioreactor with microalgae Rhodopseudomonas spheroides using the Monte Carlo method. The author considered anisotropic scattering and neglected reflection at the walls.

He compared the absorptance obtained by the Monte Carlo method with that obtained by Beer-Lambert's law as a function of cell concentration. Beer-Lambert's law was found to overestimate the absorptance in the photobioreactor because it does not consider inscattering [START_REF] Aiba | Growth kinetics of photosynthetic microorganisms[END_REF]. This was also illuestablished in Ref. [START_REF] Berberoglu | Light transfer in bubble sparged photobioreactors for H 2 production and CO 2 mitigation[END_REF].

Cornet et al. [START_REF] Cornet | A structured model for simulation of cultures of the cyanobacterium Spirulina platensis in photobioreactors: I. Coupling between light transfer and growth kinetics[END_REF], Cornet and Albiol [START_REF] Cornet | Modeling photoheterotrophic growth kinetics of Rhodospirillum rubrum in rectangular photobioreactors[END_REF], and Cornet and Dussap [START_REF] Cornet | A simple and reliable formula for assessment of maximum volumetric productivities in photobioreactors[END_REF] developed a coupled light transfer and Haldane growth kinetics model to estimate the biomass volumetric production rate of one-dimensional flat-plate PBRs. They introduced the concept of working illuminated volume which, combined with their models, can be used to retrieve the growth kinetic parameters. Based on this approach Fouchard et al. [START_REF] Fouchard | Kinetic modeling of light limitation and sulfur deprivation effects in the induction of hydrogen production with Chlamydomonas reinhardtii : Part I. model development and parameter identification[END_REF] retrieved the growth kinetics parameters of C. reinhardtii cultivated in a torus PBR with continuous injection of N 2 and CO 2 gases and illuminated with white light from fluorescent tubes considering biomass concentration, extracellular sulfur concentration, and intracellular quota. Finally, these models have been validated experimentally for different (i) PBR shapes and volumes, (ii) incident irradiance, (iii) microorganism species, and (iv) carbon sources.

Wheaton and Krishnamoorthy [START_REF] Wheaton | Modeling radiative transfer in photobioreactors for algal growth[END_REF] simulated light transfer coupled with fluid hydrodynamics within an air-lift tubular photobioreactor illuminated from inside by fluorescent lamps. The authors identified the effects of angular discretization, scattering phase function, air mass flow rate, and bubble size on the local fluence rate. They used finite volume method to solve the 3D RTE based on spectrally averaged incident radiation and radiation characteristics of cyanobacterium Synechococcus sp. over three spectral bands in the PAR region. They concluded that scattering by bubbles resulted in redistribution of the light but was negligible at high microalgae concentrations, as previously reported [START_REF] Berberoglu | Simulating light transfer in a bubble sparged photobioreactor for simultaneous hydrogen fuel production and CO 2 mitigation[END_REF].

Murphy and Berberoglu [START_REF] Murphy | Effect of algae pigmentation on photobioreactor productivity and scale-up: A light transfer perspective[END_REF] coupled light transfer with photosynthetic rate model for C. reinhardtii wild strain and transformant tla1 within plane parallel PBR considering photoinhibition. The authors solved the one-dimensional RTE using the discrete ordinates method with Gaussian quadrature to estimate the local fluence rate. They calculated the local specific and total oxygen production as function of optical thickness for different incident irradiances.

Slegers et al. [START_REF] Slegers | Design scenarios for flat panel photobioreactors[END_REF] simulated outdoor vertical flat-plate PBRs operated in continuous mode and containing Phaeodactylum tricornutum or Thalassiosira pseudonana. The authors coupled light transfer with a growth kinetics model to estimate volumetric daily and yearly biomass productivity. They treated the suspensions as gray with constant radiation characteristics averaged over the PAR region and used Beer-Lambert's law to calculate the average fluence rate within the PBR and a growth kinetic model based on pI-curves and accounting for respiration [START_REF] Geider | A dynamic model of photoadaptation in phytoplankton[END_REF]. They also defined the average total specific growth rate based on Equation [START_REF] Pottier | A fully predictive model for one-dimensional light attenuation by Chlamydomonas reinhardtii in a torous photobioreactor[END_REF]. They investigated the effect of thickness of PBRs and biomass concentration on the volumetric productivity in the Netherlands.

Huang et al. [START_REF] Huang | Simulation of the light evolution in an annular photobioreactor for the cultivation of Porphyridium cruentum[END_REF] simulated annular PBR with Porphyridium cruentum in continuous and batch cultures. The authors integrated hydrodynamics, radiation transfer, and growth kinetics models to predict biomass concentration as a function of time. The 3D RTE was solved using finite volume method and discrete ordinate method to estimate the light intensity within the annular PBR. The box model with two boxes was used to account for spectral radiation in normal diffuse incident irradiance and radiation characteristics of microalgae to estimate the light intensity within the annular PBR. Good agreement was observed between the numerically prediction of biomass concentration and experimental data reported in the literature [START_REF] Muller-Feuga | Benefits and limitations of modeling for optimization of Porphyridium cruentum cultures in an annular photobioreactor[END_REF].

Finally, Pruvost et al. [START_REF] Pruvost | Theoretical investigation of biomass productivities achievable in solar rectangular photobioreactors for the cyanobacterium arthrospira platensis[END_REF] simulated outdoor inclined rectangular PBR exposed to solar irradiance with cyanobacterium Arthrospira platensis grown in continuous culture. The authors coupled light transfer with growth kinetics model to estimate the areal biomass productivity per unit surface area illuminated. They calculated the fluence rate using the two-flux approximation [START_REF] Cornet | A simplified monodimensional approach for modeling coupling between radiant light transfer and growth kinetics in photobioreactors[END_REF] for constant biomass concentration. They investigated the effects of latitude and inclination of the PBR surface on the maximum areal biomass productivity. The location close to the earth equator had biomass productivity up to 40% larger than those at higher latitudes. In addition, the areal biomass productivity achieved with a solar tracking system was 30% larger than that obtained with constant inclination.

Previous studies often used spectrally averaged incident irradiance and radiation characteristics [START_REF] Aiba | Growth kinetics of photosynthetic microorganisms[END_REF][START_REF] Cornet | A structured model for simulation of cultures of the cyanobacterium Spirulina platensis in photobioreactors: I. Coupling between light transfer and growth kinetics[END_REF]. In addition, light transfer in PBRs has frequently been treated as onedimensional [START_REF] Cornet | A structured model for simulation of cultures of the cyanobacterium Spirulina platensis in photobioreactors: I. Coupling between light transfer and growth kinetics[END_REF][START_REF] Pottier | A fully predictive model for one-dimensional light attenuation by Chlamydomonas reinhardtii in a torous photobioreactor[END_REF][START_REF] Cornet | A simple and reliable formula for assessment of maximum volumetric productivities in photobioreactors[END_REF] and estimated using either Beer-Lambert's law [START_REF] Slegers | Design scenarios for flat panel photobioreactors[END_REF] or the two-flux approximation [START_REF] Cornet | A structured model for simulation of cultures of the cyanobacterium Spirulina platensis in photobioreactors: I. Coupling between light transfer and growth kinetics[END_REF][START_REF] Cornet | A 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][START_REF] Cornet | A simplified monodimensional approach for modeling coupling between radiant light transfer and growth kinetics in photobioreactors[END_REF]. Most studies also considered constant and normally collimated incident irradiance [START_REF] Pottier | A fully predictive model for one-dimensional light attenuation by Chlamydomonas reinhardtii in a torous photobioreactor[END_REF][START_REF] Cornet | A simple and reliable formula for assessment of maximum volumetric productivities in photobioreactors[END_REF][START_REF] Takache | Experimental and theoretical assessment of maximum productivities for the microalgae chlamydomonas reinhardtii in two different geometries of photobioreactors[END_REF]. However, solar irradiance fluctuates in direction and intensity during the day. Also, radiation characteristics of microalgae depends strongly on wavelength. In addition, actual PBRs may have complex geometries for which simplified radiation model may not be valid.

Moreover, [START_REF] Richmond | Efficient utilization of high irradiance for production of photoautotropic cell mass: a survey[END_REF], [START_REF] Richmond | Principles for efficient utilization of light for mass production of photoautotrophic microorganisms[END_REF] and [START_REF] Richmond | Principles for attaining maximal microalgal productivity in photobioreactors: an overview[END_REF] suggested the need for a unifying approach to PBR design and operation and emphasized the connection between light intensity, cell density, and the optical thickness of the PBR to maximize microalgal productivity. The authors demonstrated the need to optimize these parameters for a given microalgae species and provided qualitative guidelines obtained from experimental studies. In particular, they recommended ultra-high cell density cultures in PBRs with short path length to achieve efficient utilization of solar radiation. Under these conditions, they also discussed the importance of turbulent mixing to increase the light/dark cycle frequency. However, quantitative criteria for optimum productivity were not provided.

In this study, light transfer was accurately simulated by solving the three-dimensional RTE on a spectral basis using experimentally measured spectral radiation characteristics [START_REF] Berberoglu | Radiation characteristics of Chlamydomonas reinhardtii CC125 and its truncated chlorophyll antenna transformants tla1, tlaX, and tla1-CW +[END_REF]. Both spectral diffuse and collimated solar irradiances with different incident angle corresponding to different times of the day were considered. It was coupled with the Haldane growth kinetics model accounting for photolimitation, photoinhibition, and cellular respiration. The fluence rate, biomass concentration, and daily productivity of open ponds, vertical flat-plate, and tubular PBRs were compared and discussed with experimental and numerical results reported in the literature. Quantitative criteria to achieve maximum productivity were also derived in terms of cell density and PBR dimensions for a given microalgae species.

METHODS

Problem Statement

Algal biomass production in PBRs depends on numerous parameters including (i) the cultivation location, (ii) the day of the year and the time of the day along with (iii) the corresponding solar irradiance, (iv) the microalgae species, (v) the initial mass concentration, (vi) the PBR geometry, and (vii) its wall reflection and refraction. The present study simulates light transfer and microalgae growth in common PBRs located in Los Angeles, CA USA (34.04 • N, 118.15 • W). Simulations were performed for open pond, vertical flat-plate, and tubular PBRs. The PBRs were aligned along the north-south direction and exposed to solar irradiance comprised of both a collimated and a diffuse component. Figure 1 shows the geometries, dimensions, and boundary conditions of the PBRs simulated in this study along with the associated coordinate systems. The open pond had depth L varying from 0.05 to 1.0 m. The thickness L of the vertical flat-plate PBR ranged from 0.05 to 1.0 m while the diameter L of the tubular PBR varied from 0.1 to 1.0 m. C. reinhardtii were simulated for illustration purposes and because its growth kinetic parameters were known [START_REF] Fouchard | Kinetic modeling of light limitation and sulfur deprivation effects in the induction of hydrogen production with Chlamydomonas reinhardtii : Part I. model development and parameter identification[END_REF].

Assumptions

In order to predict light transfer and the temporal evolution of microalgae concentration in the different PBRs considered, it was assumed that: (1) the microalgae were well mixed, randomly oriented, and uniformly distributed in the PBR. In practice, this is achieved by stirring the PBR with paddle wheels or gas sparging, for example. (2) The liquid medium was non-emitting (cold) and non-scattering over the PAR region. featured interfacial area concentration smaller than 450 m -1 so their effect on light transfer could be neglected [START_REF] Berberoglu | Simulating light transfer in a bubble sparged photobioreactor for simultaneous hydrogen fuel production and CO 2 mitigation[END_REF]. [START_REF] Morweiser | Developments and perspectives of photobioreactors for biofuel production[END_REF] The PBR was neither mass transfer nor nutrient limited and operated at constant temperature. [START_REF] Ugwu | Photobioreactors for mass cultivation of algae[END_REF] The photosynthetic specific growth rate µ p was only a function of the local and average fluence rate available in the PBRs and given by Equations [START_REF] Aiba | Growth kinetics of photosynthetic microorganisms[END_REF], and (8) Finally, biomass loss at night due to respiration was ignored as PBR productivity was estimated after 12 h of exposure to sunlight. For longer duration (succession of day-night cycles), losses to the respiration at night must be considered [START_REF] Le Borgne | Investigation and modeling of biomass decay rate in the dark and its potential influence on net productivity of solar photobioreactors for microalga Chlamydomonas reinhardtii and cyanobacterium Arthrospira platensis[END_REF] 3

.3 Governing Equations

The radiation intensity I λ (r, ŝ) in direction ŝ at location r can be decomposed as the sum of a collimated I c,λ (r, ŝ) and a diffuse I d,λ (r, ŝ) component so that [START_REF] Modest | Radiative Heat Transfer[END_REF],

I λ (r, ŝ) = I c,λ (r, ŝ) + I d,λ (r, ŝ) (12)
The steady-state RTE for the collimated intensity can be written as [START_REF] Modest | Radiative Heat Transfer[END_REF],

ŝ • ∇I c,λ (r, ŝ) = -κ λ I c,λ (r, ŝ) -σ s,λ I c,λ (r, ŝ) (13)
This first and second terms on the right-hand-side of Eq. 13 account for the fact that the collimated incident radiation along the collimated direction decays as it travels through the microalgae suspension due to absorption and scattering, respectively. Similarly, the steadystate RTE for the diffuse intensity I d,λ (r, ŝ) can be written as [START_REF] Modest | Radiative Heat Transfer[END_REF],

ŝ • ∇I d,λ (r, ŝ) = -κ λ I d,λ (r, ŝ) -σ s,λ I d,λ (r, ŝ) + σ s,λ 4π 4π I d,λ (r, ŝi )Φ λ (ŝ i , ŝ)dΩ i + σ s,λ 4π 4π I c,λ (r, ŝi )Φ λ (ŝ i , ŝ)dΩ i (14)
The last two terms of Eq. 14 account for multiple scattering of the diffuse and collimated radiation intensities. In fact, the diffuse incident radiation along direction ŝi at location r is not only absorbed and scattered but also reinforced as diffuse and collimated radiations from any direction ŝi over 4π solid angle get scattered in direction ŝi .

Radiation Characteristics of C. reinhardtii

The effective absorption coefficient κ λ of the suspension can be expressed in terms of the microorganism mass concentration X as [START_REF] Berberoglu | Maximizing solar to H 2 energy conversion efficiency of outdoor photobioreactors using mixed cultures[END_REF],

κ λ = κ L,λ (1 -νX) + Āabs,λ X ( 15
)
where ν is the specific volume of microorganisms assumed to be equal to 0.001 m 3 /kg. The absorption coefficient of the liquid phase κ L,λ is expressed in m -1 and given by [START_REF] Modest | Radiative Heat Transfer[END_REF],

κ L,λ = 4πk λ λ ( 16
)
where k λ was taken as the absorption index of water reported by Hale and Querry [START_REF] Hale | Optical constants of water in the 200-nm to 200-µm wavelength region[END_REF]. On the other hand, the effective scattering coefficient σ s,λ of the suspension can be expressed as [START_REF] Berberoglu | Experimental measurement of the radiation characteristics of Anabaena variabilis ATCC 29413-U and Rhodobacter sphaeroides ATCC 49419[END_REF],

σ s,λ = Ssca,λ X (17)
The average mass absorption and scattering cross-sections Āabs,λ and Ssca,λ along with the Henyey-Greenstein asymmetry factor of C. reinhardtii between 400 and 750 nm were reported in the literature [START_REF] Berberoglu | Radiation characteristics of Chlamydomonas reinhardtii CC125 and its truncated chlorophyll antenna transformants tla1, tlaX, and tla1-CW +[END_REF]. Alternatively, they could have been predicted by Lorenz-Mie theory using the complex index of refraction retrieved by Lee et al. [START_REF] Lee | Spectral optical properties of selected photosynthetic microalgae producing biofuels[END_REF].

Boundary and Initial Conditions

The solar spectrum incident on Earth depends on the latitude, longitude, and altitude. In this study, the Simple Model of the Atmospheric Radiative Transfer of Sunshine (SMARTS) [START_REF] Gueymard | SMARTS Code, Version 2.9.2 User's direct beam spectral irradiance data for photovoltaic cell Manual[END_REF] was used to predict the incident collimated and diffuse solar irradiances at sea level in Los Angeles, CA on June 21 at different times of the day. Figure 2 shows the incident (a) collimated and (b) diffuse solar irradiances, denoted by G in,c,λ and G in,d,λ , in the PAR region for different times of the day. These irradiances were used in the boundary conditions necessary to solve Equations ( 13) and ( 14) according to I c,λ (r wall , ŝ) = G in,c,λ and I d,λ (r wall , ŝ) = G in,d,λ /2π, respectively. Here, the denominator 2π corresponds to the solid angle of the hemisphere through which the diffuse irradiance G in,d,λ is incident on the PBR [START_REF] Modest | Radiative Heat Transfer[END_REF].

The other boundary conditions depend on the PBR geometry. The open ponds were assumed to have transparent top surface, i.e., reflection by air-water interface was neglected since it does not exceed 6% over the PAR region for incident angle within 60 • of the normal to the pond surface according to Fresnels law [START_REF] Modest | Radiative Heat Transfer[END_REF]. Note that the pond surface is strongly reflecting at glazing incident angles corresponding to sunrise and sunset. Then, however, the intensity incident on the PBR was insufficient to significantly affect the biomass concentration. In addition, the bottom and side walls of the pond were treated as black (ρ λ = 0) or diffusely reflecting (ρ λ = 1). Refraction by the panels of the vertical flat-plate PBRs with constant index of refraction of 1.49 over the PAR region was also considered. The results were compared with those of simulations treating PBR walls as transparent. Based on result from the flat-plate PBRs, the wall of the tubular PBRs was assumed to be transparent and refraction by the tube wall was ignored, as discussed in Sections 4.2 and 4.3.

Two-Flux Approximation

The analytical expression derived by Cornet et al. [START_REF] Cornet | A structured model for simulation of cultures of the cyanobacterium Spirulina platensis in photobioreactors: I. Coupling between light transfer and growth kinetics[END_REF]

G λ (z, t) = G in,c,λ (t)f c (z) + G in,d,λ (t)f d (z) + G in,d,λ (t)f d (L -z) (18)
where f c (z) is defined as

f c (z) = 2 sec θ c (t) (1 + α λ )e δ λ L e -δ λ z -(1 -α λ )e -δ λ L e δ λ z (1 + α λ ) 2 e δ λ L -(1 -α λ ) 2 e -δ λ L (19)
and f d (z) is expressed as

f d (z) = 4 (1 + α λ )e δ d,λ L e -δ d,λ z -(1 -α λ )e -δ d,λ L e δ d,λ z (1 + α λ ) 2 e δ d,λ L -(1 -α λ ) 2 e -δ d,λ L (20)
hereα λ and δ λ are given by Equation ( 5) while δ d,λ = 2X Āabs,λ ( Āabs,λ + 2b λ Ssca,λ ). Moreover, Berberoglu et al. [START_REF] Berberoglu | Radiation characteristics of Chlamydomonas reinhardtii CC125 and its truncated chlorophyll antenna transformants tla1, tlaX, and tla1-CW +[END_REF] reported that the Henyey-Greenstein asymmetry factor of C. reinhardtii was 0.98 corresponding to strongly forward scattering, typical of microalgae. Then, the backward scattering fraction b λ given by Equation ( 7) can be assumed to be zero and α λ ≃ 1. Then, the fluence rate in vertical flat-plate PBR exposed to collimated and diffuse incident radiation simplifies to

G λ (z, t) = sec θ c (t)G in,c,λ (t)e -δ λ z + 2G in,d,λ (t) e -δ d,λ z + e -δ d,λ (L-z) (21)
where δ λ and δ d,λ simplify to δ λ = Āabs,λ X(t)secθ c (t) and δ d,λ = 2 Āabs,λ X(t). Similarly, the fluence rate in open ponds with reflecting back wall can be expressed as

G λ (z, t) = G in,c,λ (t)f c (z) + G in,d,λ (t)f d (z) (22)
If α λ ≈ 1, the two-flux approximation for open-ponds simplifies to

G λ (z, t) = [G in,c,λ (t) sec θ c (t) + 2G in,d,λ (t)] e -δ λ z + ρ λ e -δ λ (2L-z) (23)
These expressions apply also in the case of open ponds with black walls with ρ λ = 0.

Method of solution 3.7.1 Light transfer

The 3D RTE given by Equations ( 13) and ( 14) were solved numerically for I λ (r, ŝ) using the discontinuous Galerkin method. Detailed description of the DG method used in the present study and its validation was given in Ref. [START_REF] He | A high-order accurate GPUbased radiative transfer equation solver for combustion and propulsion applications[END_REF] and need not be repeated. Sunlight incident on the PBRs consists of a collimated and a diffuse component. The direction θ c (t) of collimated incidence changed during the course of the day. Unfortunately, conventional angular discretization methods such as discrete ordinate S N [START_REF] Fiveland | Numerical model for predicting performance of threedimensional pulverized-fuel fired furnaces[END_REF] and T N [START_REF] Chui | Computation of radiant-heat transfer on a nonorthogonal mesh using the finite-volume method[END_REF] approximations typically use fixed discrete directions. Therefore, the discretization would need to be changed for simulating different hours of the day [START_REF] Li | A new discrete ordinate algorithm for computing radiative transfer in one-dimensional atmospheres[END_REF]. Discrete ordinate scheme with infinitely small weight (DOS+ISW) [START_REF] Li | A new discrete ordinate algorithm for computing radiative transfer in one-dimensional atmospheres[END_REF] was employed in this study to simulate collimated sunlight incident on the PBRs during the course of the day. It consists of adding a discrete direction, corresponding to the direction of collimated irradiance, directly to a conventional discrete ordinate quadrature. The weight associated with this new discrete direction is set to be infinitely small [START_REF] Li | A new discrete ordinate algorithm for computing radiative transfer in one-dimensional atmospheres[END_REF]. Thus, the new discrete direction has no effect on the zeroth, first, and second order moments of the intensity [START_REF] Li | A new discrete ordinate algorithm for computing radiative transfer in one-dimensional atmospheres[END_REF]. Finally, unstructured tetrahedral elements were employed for spatial discretization. The number of elements varied depending on the size of the PBRs. The maximum number of elements in simulating open ponds, vertical flat-plate, and tubular photobioreactors was 95633, 91257, and 112464, respectively. The S 4 angular discretization, consisting of 6 discrete ordinate directions per quadrant, was used. The PAR region, defined from 400 to 700 nm, was discretized in 10 nm increments for a total of 31 wavelengths. To obtain a numerically converged solution of the RTE, the P-3 DG method was used in all simulations. with a maximum number of elements in simulating open ponds, vertical flat-plate, and tubular PBRs equal to 95,633, 91,257, and 112,464, respectively.

Growth kinetics

In the present study, the growth kinetics model described by Equations ( 8) to [START_REF] Pottier | A fully predictive model for one-dimensional light attenuation by Chlamydomonas reinhardtii in a torous photobioreactor[END_REF] was used to determine the temporal evolution of microalgae concentration in photobioreactors. Fouchard et al. [START_REF] Fouchard | Kinetic modeling of light limitation and sulfur deprivation effects in the induction of hydrogen production with Chlamydomonas reinhardtii : Part I. model development and parameter identification[END_REF] measured the average specific growth rate μ of the green algae C. reinhardtii. The authors estimated the parameters µ 0 , µ s , K S , and K I to be 0.2274 hr -1 , 0.032 hr -1 , 81.38 µmol photon m -2 • s -1 , and 2500 µmol photon m -2 • s -1 , respectively for local irradiance G P AR (r) ranging from 0 to 400 µmol photon m -2 • s -1 . These parameters resulted in prediction for µ in good agreement with experimental data reported by [START_REF] Janssen | Scale-up aspects of photobioreactors: effects of mixing-induced light/dark cycles[END_REF]. They were used in the present study after converting K S and K I , expressed in µmol photon m -2 • s -1 , in W/m 2 using the conversion factor, 1 µmol photon m -2 • s -1 ≃ 0.2174 W/m 2 over the PAR region [START_REF] Morel | Relation between total quanta and total energy for aquatic photosynthesis[END_REF].

Solution procedure

The mass concentration of microalgae as a function of time was obtained by the following procedure. First, the initial mass concentration of microalgae was set as X(t = 0) = X 0 at initial time 8:00 am. The corresponding effective absorption coefficient κ λ and the scattering coefficient σ s,λ were estimated using Equations ( 15) and ( 17), respectively. Then, Equations ( 13) and ( 14) were solved for I λ,c and for I λ,d using the DG method. Then, the local fluence rate was estimated using Equation (1). The corresponding local photosynthetic specific growth rate µ p was estimated [Equation [START_REF] Aiba | Growth kinetics of photosynthetic microorganisms[END_REF]] and used to calculate the average total specific growth rate μ [Equation [START_REF] Pottier | A fully predictive model for one-dimensional light attenuation by Chlamydomonas reinhardtii in a torous photobioreactor[END_REF]] and the microalgae mass concentration X(t) [Equation ( 8)] at subsequent time. This procedure was repeated by increment of 2 hours. During that time interval, the local fluence rate and average total specific growth rate were assumed to be constant. To assess the validity of this assumption, the temporal evolution of the concentration in open ponds was predicted for 12 hours with initial mass concentration X 0 = 0.1 kg/m 3 using the two-flux approximation and the same growth kinetics model. The local fluence rates obtained for time intervals of 30 minutes, 1 hours, and 2 hours were compared at 8:00 am, 10:00 am, 12:00 pm (noon), 2:00 pm, 4:00 pm, 6:00 pm, and 8:00 pm. The maximum relative errors in the corresponding μ(t) and X(t) throughout the day were less than 3% and 6%, respectively. Thus, a 2 hours time increment was judged appropriate for simulating coupled radiation transfer and microalgae growth kinetics in the PBR throughout the day.

RESULTS AND DISCUSSION

Open ponds

Figure 3 shows the local PAR-averaged fluence rate G P AR (r) in the east/west center plane of the open pond with either (a) black or (b) reflecting side and bottom walls at times 8:00 am, 10:00 am, 12:00 pm (noon), 2:00 pm, and 4:00 pm. Here, the pond diameter D, depth L, and initial mass concentration X 0 were taken as 2 m, 0.1 m, and 0.1 kg/m 3 , respectively. By comparing Figures 3a andb, it is evident that the reflecting walls increased the local fluence rate in the PBR, particularly before 4:00 pm. However, the relative difference, in terms of mass concentration X(t), between open ponds with black and reflecting walls was less than 1% after a day of growth. In both cases, the local fluence rate was nearly onedimensional except near the side walls where shadows were apparent in the early morning and late afternoon. Note that even though, on June 21 in Los Angeles, the sun rises at 05:42 am and sets at 8:08 pm, the average total specific growth rate μ at times earlier than 8:00 am and later than 8:00 pm was less than 0.01 hr -1 and growth was negligible.

Vertical flat-plate photobioreactor

Light transfer in a vertical flat-plate photobioreactor oriented north-south with initial mass concentration X 0 = 0.1 kg/m 3 was simulated over 12 hours. The vertical flat-plate PBR height H, width W , and thickness L were taken as 2 m, 2 m, and 0.1 m, respectively. The walls were made of 8 mm thick glass panels. The effect of refraction due to mismatch in refractive indices of the air (n = 1.0), the reactor walls (n = 1.49), and the C. reinhardtii suspension (n = 1.33) contained in the PBR was investigated. These refraction indices were assumed to be constant over the PAR region. Reflection and refraction were estimated using Fresnel's equations for optically smooth and specularly reflecting surfaces [START_REF] Modest | Radiative Heat Transfer[END_REF]. Figure 4 shows the local PAR-averaged fluence rate G P AR (r) along a vertical cross-section of a 0.1 m thick flat-plate PBR accounting for refraction, at 8:00 am, 10:00 am, 12:00 pm (noon), 2:00 pm, 4:00 pm, and 6:00 pm. Three-dimensional effects were apparent between 10:00 am and 2:00 pm when the sun was near its zenith. During this time, the flat-plate PBR intercepted a small amount of collimated solar radiation. Sunlight was also incident on the PBR vertical windows at glazing angles when reflectance is large. Overall, accounting for refraction reduced slightly the local fluence rate compared with results obtained assuming the PBR wall to be transparent. The relative difference in C. reinhardtii mass concentration after 12 h was less than 0.1% when considering or ignoring refraction. Therefore, refraction of sunlight by the front and back windows of the PBR had negligible effects on the mass concentration of microalgae and could be ignored in our simulations.

Tubular photobioreactor

A horizontal tubular photobioreactor oriented in the north-south direction with initial mass concentration X 0 = 0.1 kg/m 3 was simulated over 12 hours on June 21. Here, the tubular PBR diameter L was taken as 0.1 m. Figure 5 shows the local PAR-averaged fluence rate G P AR (r) over the cross-section of the pipe at different times of the day. It shows significant multidimensional effects. A darker region developed in the center of the tubular PBR under the combined effects of microalgae growth and the setting of the sun. Given the inherent 2D nature of this type of PBRs, the two-flux approximation could not be used.

Two-flux approximation

Figure 6a compares the PAR-averaged local fluence rate G P AR (z) at the centerline of the open pond with black walls at times 8:00 am, 12:00 pm (noon), and 4:00 pm. It also compares the numerical predictions obtained using the DG method with predictions by the two-flux approximation [Equation [START_REF] Wheaton | Modeling radiative transfer in photobioreactors for algal growth[END_REF]] and by the simplified two-flux approximation [Equation ( 23)].

The average relative difference between the numerical predictions using DG method with DOS-ISW and the two-flux approximation for G P AR (z) at the centerline of the open pond ranged between 4 and 10% depending on the time of the day and the location inside the PBR. However, prediction by the two-flux approximation was not able to predict the shadow and other multidimensional effects in the open pond. Moreover, the average total specific growth rate μ predicted based on G λ (z) and G P AR (z) predicted by the two-flux approximation fell within 1 to 7% of its numerically predicted value. These results suggest that the two-flux approximation can be used to determine G P AR (z) and the corresponding μ in open ponds or race ponds with dimensions larger than 2 m when shadow effects become less and less significant. This can be very useful in the design and real time control and operation of open ponds. 9@99 9@9A 9@9B 9@9C 9@9D 9@E9 9 Figure 6b compares numerical results with predictions from the two-flux approximation [Equation [START_REF] Modest | Radiative Heat Transfer[END_REF]] and the simplified two-flux approximation [Equations [START_REF] Fouchard | Kinetic modeling of light limitation and sulfur deprivation effects in the induction of hydrogen production with Chlamydomonas reinhardtii : Part I. model development and parameter identification[END_REF]]. The maximum relative difference between numerical results and predictions by the two-flux approximation for the local G P AR (x) without refraction ranged between 4 and 22% while the relative differences averaged over the PBR volume were about 2 to 13% depending on the time of the day. The two-flux approximation tended to overpredict G P AR (x) because it was not able to predict the shadow and other multidimensional effects. Moreover, the average total specific growth rate μ estimated using G P AR (x) predicted by the two-flux approximation [Equation [START_REF] Modest | Radiative Heat Transfer[END_REF]] fell within 2 to 10% of its numerically predicted values. Overall, the two-flux approximation predictions of G P AR (x) and the corresponding average total specific growth rate μ in vertical flat-plate PBRs were acceptable. Finally, the average total specific growth rate μ in the vertical flat-plate PBR was found to be larger than that of an open pond of identical depth with black walls at all times except at 12:00 pm because the surface area exposed to sunlight was smaller.

Comparison of daily areal biomass productivities

The daily areal biomass productivity of a open pond, expressed in kg/m 2 /day, is defined as the mass of microalgae produced after 12 hours of exposure to sunlight per unit illuminated surface area exposed to collimated solar irradiance. It is expressed as

P op = (X f -X 0 )V St f = (X f -X 0 ) Lt f (24)
where S and V are the illuminated surface area and volume of the PBR while X f is the final mass concentration and t f is equal to 1 day. The specific illuminated area of the PBRs, denoted by a (in m -1 ), is defined as a = S/V [START_REF] Cornet | A simple and reliable formula for assessment of maximum volumetric productivities in photobioreactors[END_REF]. For an open pond of diameter D, thickness L, and illuminated surface area S = πD 2 /4, the specific illuminated area is equal to a op = 1/L. In addition, the initial optical thickness of open ponds is expressed as β λ L = ( Āabs,λ + Ssca,λ )X 0 L where the average mass cross-sections Āabs,λ and Ssca,λ are intrinsic properties of the microalgae species. Thus, X 0 L = X 0 a -1 op can be considered as representative of the PBR's initial optical thickness.

The daily areal biomass productivity per unit surface area exposed to collimated solar irradiance for vertical flat-plate and tubular PBRs, respectively denoted by P f p and P tb (in kg/m 2 /day), are also defined by Equation [START_REF] Murphy | Effect of algae pigmentation on photobioreactor productivity and scale-up: A light transfer perspective[END_REF]. The specific illuminated area of flat-plate PBRs of thickness L is expressed as a f p = 1/L and that of tubular PBRs of diameter L is equal to a tb = 2/L. In other words,

P f p = (X f -X 0 )L t f and P tb = (X f -X 0 )L/2 t f (25)
Here also, X f is the final concentration after a duration t f of one day.

Figure 7 shows the daily areal biomass productivity of open ponds P op , vertical flat-plate P f p , and tubular PBRs P tb as a function of X 0 a -1 for different values of their characteristics length L (i.e., depth, thickness, or diameter) varying from 0.05 to 1.0 m and initial concentration X 0 between 0.0 and 5.0 kg/m 2 . The daily areal biomass productivity of vertical flat-plate P f p was calculated based on μ estimated with G P AR (x) predicted by the two-flux approximation, ignoring wall refraction, and accounting for respiration. On the other hand, P tb was calculated numerically based on G P AR (r) and μ. It is interesting to note that the daily biomass productivities P op , P f p , and P tb depended uniquely on the product X 0 a -1 and not on X 0 and a or L independently.

Moreover, the productivities P op , P f p , and P tb versus X 0 a -1 nearly overlapped regardless of the PBR geometry. The maximum daily areal productivity per unit illuminated surface area of these PBRs was P max =0.045 kg/m 2 /day for X 0 a -1 = 0.035 kg/m 2 . For X 0 a -1 < 0.035 kg/m 2 , the incident irradiance was not entirely absorbed by microalgae as some photons were absorbed at the bottom of the open pond or transmitted through flat-plate or tubular PBRs. Then, the biomass productivity was low and increased with increasing optical thickness. However, for X -1 0 > 0.035 kg/m 2 , dark region appeared in the PBRs thus decreasing the working illuminated volume while the effects of respiration became significant [START_REF] Takache | Kinetic modeling of the photosynthetic growth of Chlamydomonas reinhardtii in a photobioreactor[END_REF].

Note that the fact that the maximum productivity was identical for PBRs with the same specific illuminated area a irrespective of their geometry has already been predicted by Cornet and Dussap [START_REF] Cornet | A simple and reliable formula for assessment of maximum volumetric productivities in photobioreactors[END_REF] and experimentally validated by Takache et al. [START_REF] Takache | Experimental and theoretical assessment of maximum productivities for the microalgae chlamydomonas reinhardtii in two different geometries of photobioreactors[END_REF]. Here, we expanded this conclusion by demonstrating that the productivities per unit of illuminated surface area of PBRs are identical (including at their maximum) as long as they feature the same value of optical thickness represented by X 0 a -1 . These results should be evaluated in combination with the associated capital and operational costs [START_REF] Chaumont | Biotechnology of algal biomass production: a review of systems for outdoor mass culture[END_REF]. The same design tool could also be used to investigate shading between cultivation systems but this effort falls outside the scope of the present study.

Finally, the fact that daily biomass productivities P op , P f p , and P tb depend only on the product provides a simple and practical way to design (via a) and to operate (via X 0 ) these PBRs to achieve maximum productivity in batch mode. Indeed, to achieve the maximum daily production rate from the specific microalgae grown in any of the three PBRs considered, one should geometrically design the PBRs specific illuminated surface area and set the initial concentration X 0 so that X 0 a -1 = 0.035 kg/m 2 . However, one may wonder if these results are valid for (i) other types of PBRs, (ii) for continuous operation, (iii) for different light transfer and growth kinetic models, and if they are supported by experimental evidences. productivity of continuous air-lift PBRs was also a unique function of X 0 a -1 . Note that finding additional experimental data to further validate our results was made difficult by the fact that experimentally the PBR geometry and size as well as the initial concentration are arbitrarily set. Indeed, experimental parametric study similar to that performed numerically in the present study would be very time consuming and potentially costly.

Moreover, Figure 8b shows the simulation results reported by Slegers et al. [START_REF] Slegers | Design scenarios for flat panel photobioreactors[END_REF] for yearly volumetric biomass production (in kg/m 3 /year) for vertical flat-plate PBRs operated in continuous mode with P. tricornutum and T. pseudonana. Here, the PBR thickness ranged from 0.05 to 0.1 m and concentrations varied between 1.0 and 13.0 kg/m 2 . It is very interesting to observe that these data obtained independently using different light and kinetics models collapsed also on a single line when plotted as a function of X 0 a -1 , whereas they were scattered when plotted as a function of L = 1/a (Figure 3 in the manuscript by Slegers et al. [START_REF] Slegers | Design scenarios for flat panel photobioreactors[END_REF]). This may be attributed to the fact that the illuminated surface area is closely related to the solar energy input entering the PBR. In other words, the amount of biomass produced remains the same regardless of the PBR geometry for a given amount of solar energy absorbed.

CONCLUSION

This study presented accurate 3D numerical simulations for coupled light transfer and growth kinetics in the most commonly used PBRs exposed to collimated and diffuse sunlight in Los Angeles on June 21. The local fluence rate was predicted on a spectral basis by solving the 3D RTE. The temporal evolution of microalgae mass concentration was predicted by accounting for light saturation, photolimitation, and respiration. In open ponds, the reflecting walls resulted in a more uniform light distribution and increased the local fluence rate. However, the difference in overall biomass concentration after 12 hours was negligible. Similarly, refraction by the container walls, in vertical flat-plate PBRs, had no significant effect on the microalgae concentration. The study demonstrated that the two-flux approximation can be used to estimate the local fluence rate in open (or race) ponds and flat-plate PBRs for all practical purposes including designing, controlling, and operating PBRs. Finally, the daily areal biomass productivity per unit footprint was found to depend uniquely on the initial optical thickness represented by X 0 a -1 for open ponds and tubular PBRs operated in batch mode. Similar results were obtained for the daily productivity per unit surface area illuminated for flat-plate PBRs. What's more, the same conclusions were drawn be revisiting both experimental data and numerical simulations reported in the literature for similar and other PBR types cultivating other microorganisms in continuous mode. The parameter X 0 a -1 is useful and simple for designing (via a) and operating (via X 0 ) these PBRs at their maximum productivity.

( 3 )Figure 1 :

 31 Figure 1: Schematic of the (a) open pond, (b) vertical flat-plate, and (c) tubular photobioreactors simulated in this study along with coordinate systems.

Figure 2 :

 2 Figure 2: (a) collimated G in,c,λ and (b) diffuse G in,d,λ solar irradiance spectrum over the PAR region at 12:00 pm.

Figure 3 :

 3 Figure 3: Computed PAR-averaged local fluence rate G P AR (r) on June 21, in Los Angeles at 8:00 am, 10:00 am, 12:00 pm, 2:00 pm, and 4:00 pm in the midplane of an open pond having diameter D = 2 m and depth L = 0.1 m with (a) black walls or (b) reflecting walls for initial mass concentration X 0 = 0.1 kg/m 3 .

Figure 4 :

 4 Figure 4: Computed PAR-averaged local fluence rate G P AR (r) on June 21 in Los Angeles at 8:00 am, 10:00 am, 12:00 pm, 2:00 pm, 4:00 pm, and 6:00 pm with refraction by the walls in the midplane of a vertical flat-plate having height H = 2 m, width W = 2 m, and thickness L = 0.1 m with initial mass concentration X 0 = 0.1 kg/m 3 .

Figure 5 :

 5 Figure 5: Computed local PAR-averaged fluence rate G P AR (r) on June 21 in Los Angeles at 8:00 am, 10:00 am, 12:00 pm, 2:00 pm, 4:00 pm, and 6:00 pm in the cross-section of northsouth oriented tubular PBRs having diameter L = 0.1 m with initial mass concentration X 0 = 0.1 kg/m 3 .

Figure 6 :

 6 Figure 6: PAR-averaged local fluence rate G P AR (r) predicted on June 21 in Los Angeles at (a) 8:00 am, 12:00 pm, and 4:00 pm in the centerline of an open pond with black walls and (b) at 10:00 am, 2:00 pm, and 6:00 pm across a vertical flat plate PBR by (i) DG method with DOS-ISW, (ii) two-flux approximation [Equations(4)], and (iii) simplified twoflux approximation [Equations(23)]. The open pond featured had diameter D = 2 m and depth L = 0.1 m while the flat plate PBR had height H = 2 m and thickness L = 0.1 m. In both cases, the initial mass concentration was X 0 = 0.1 kg/m 3 .

Figure 7 :

 7 Figure 7: Daily areal biomass productivity per unit illuminated surface area exposed to collimated solar irradiance predicted on June 21 in Los Angeles for open ponds P op , vertical flat-plate PBRs P f p , and tubular PBRs P tb for C. reinhardtii as function of X 0 a -1 with depth or diameter L varying from 0.05 to 1.0 m and initial concentration X 0 between 0.1 and 5.0 kg/m 3 . Here, a op = 1/L, a f p = 1/L, and a tb = 2/L.

  for predicting the local fluence rate G λ (z) in vertical flat-plate PBRs exposed to collimated solar irradiance with reflecting back wall [Equation (4)] can be extended to vertical flat-plate PBRs with transparent faces, one exposed to direct collimated and diffuse incident solar irradiance G in,c,λ and G in,d,λ and the other exposed only to diffuse solar irradiance G in,d,λ . Then, the local spectral fluence rate at time t can be expressed as

 3in Ref. [START_REF] Slegers | Design scenarios for flat panel photobioreactors[END_REF] plotted as a function of X 0 a -1 for 1 m tall vertical flat-plate PBRs in continuous operation with L varying from 0.05 to 0.1 m and X 0 between 1.0 and 13.0 kg/m 3 for P. tricornutum and T. pseudonana.

Comparison with experimental data and with other models

Pruvost et al. [START_REF] Pruvost | Systematic investigation of biomass and lipid productivity by microalgae in photobioreactors for biodiesel application[END_REF] cultivated Neochloris oleoabundans in vertical flat-plate air-lift PBRs operated in continuous mode with different thicknesses L = 0.03 and 0.055 m. The authors also modeled the process using the two-flux approximation to calculate the fluence rate in the PBRs. The growth kinetics model ignored photoinhibition but accounted for respiration and for the effect of dilution to predict the volumetric and areal productivities. The parameters µ 0 , K S , and µ s were estimated from experimental data as 0.21 hr -1 , 90 µmol photon m -2 • s -1 , and 0.005 hr -1 , respectively. Figure 8a shows the experimentally measured and predicted daily areal biomass productivity of vertical flat-plate air-lift PBRs reported by Pruvost et al. [START_REF] Pruvost | Systematic investigation of biomass and lipid productivity by microalgae in photobioreactors for biodiesel application[END_REF]. First, it indicates that the productivity measured experimentally had the same order of magnitude as those predicted in this study. Figure 8a also establishes that the areal


Pascal Frucquet

Didier Chabaud

What kind of governance practices should be developed to drive "Smart City" public policies that create public value ?

❑ A great diversity of definitions : more than 100 identified in the literature (Ruhlandt 2018, p. 12-14 ;Lim and al. 2019, p. 10-12) ❑ Two main common points :

▪ The positionning of digital technologies and datas at the service of all the fields of activity and functions of urban and territorial management ▪ An ambition to create multidimensional value [START_REF] Csukas | The many faces of the smart city: Differing value propositions in the activity portfolios of nine cities[END_REF] ❑ The richness and the diversity of the expected benefits from Smart Cities led several authors, over the last 10 years, to invite researchers to address the subject of Smart Cities from the point of view of public value theory (Rodríguez Bolívar 2019)

❑ How can we define public value ?

"A multi-dimensional construct, intended for the community of citizens, reflecting their collectively expressed and politically mediated preferences, not only produced through outcomes but also through processes that generate trust or equity." [START_REF] O'flynn | A Public Value Framework for Contractual Governance[END_REF]2007) ❑ The public value creation potential of Smart Cities

Citizen (stakeholder) engagement

Activities that are intended to involve non-elected citizens directly and their groups to the operation of city systems.

Political

Social Ideological Stewardship

Dialogue

User democracy Citizen involvement Accountability

Quality of life

Activities, directly providing better conditions for the citizens to pursue happiness in their daily life, considering their needs and experience in mind, resulting in wellbeing.

Quality of life Common good

Social inclusion and equality

Activities, to realize social equality, providing better urban living conditions for all segments of the society, without discrimination.

Social Social cohesion Protection of minorities

Connectivity

Activities to create information channels in-between humans and non-humans in order to interact with each other.

Economic Social Strategic

Openness

Responsiveness Listening to public opinion

Knowledge creation and sharing

Activities that are supporting the accumulation and flow of local urban knowledge inbetween urban stakeholders, contributing to city intelligence.

Economic Strategic Cooperativeness

Cost reduction

Activities that are specifically intended to replace or substitute an existing solution, to be operated in a less resource-intensive way.

Economic Strategic

Productivity Parsimony

Scalability and transferability

Activities that incorporate features, which makes them easy to extend, replicate and adapt in different contexts.

Economic Strategic

Adaptability Innovation

Environmental impact

Activities, with objectives related to contributing to environmental sustainability. and "what do we do with it to optimize the Smart City?"; but alongside that, above all, "How do we instill, how do we spread the word internally so that everyone has the reflexes to share data?" (Deputy Mayor in charge of the "Smart City" policy)

Quality of life Sustainability

❑ Result n°3 : The need to renew the human resources management policy and to support the transformation of professions

❑

  Among the great diversity of definitions three dichotomies (Komninos and Mora 2018) emerge from the review of the Smart City literature, raising the question of governance Technology-driven approach vs. Human-driven approach Data-driven intelligence vs. Collective intelligence Top-down planning vs. Bottom-up planning « Governance can be defined as a process of coordination of actors, social groups and institutions to achieve goals that are discussed and defined collectively. » (Le Galès, 2006) ❑ The literature on « Smart Urban Governance » has also experienced a significant increase over the last years (Allam and al. 2022) Source : Allan and al. (2022, p. 8) « fragmentation » « conceptual swamp » (Meijer and Bolívar 2016) « Our current academic understanding of the management and governance of smart cities falls short since the literature is rather polarized and is dominated by success stories and best practices. For this reason, there is a need for a research-based understanding of the management and governance of smart cities. » (Grossi, 2020)

  cases : User Lab (A), e-dministration and « coordination through use » (B)❑ To create public value, "Smart City" public policies need :• To be formalized as public policies in their own right, as well as the other public policies led by local authorities• To question the objectives of public creation and not only the technological solutions to be implemented• To identify the resources to be mobilized, the partners to work with, and the KPI to evaluate their achievement• To combine proactive and open governance• To be careful to provide open governance with appropriate resources from the very beginning of the projects to fully create public value by synergies with stakeholders and inhabitants of the territoryThank you for your attentionContact : Pascal Frucquet pfrucquet@univ-pau.fr

  For the manager, it is the same. What we're asking him to do is carry out real water monitoring, to ensure that the plants perform well and to assess the results. So we're asking him to do something else." (Deputy General Manager in charge of Ecological transition and Urban planning) The development of co-creation practices (economic development, sustainable IT, new services, co-evaluation, digital twins, etc.) There is a phasing between proactive governance and open governance, with consequences on the types of public value created

	Context	Research Question	Research Framework	Methodology	Results	Conclusion
	"Our ambition is to turn the digital twin into an urban collaborative platform creating social links and economic value through various material, knowledge and space exchanges on a neighbourhood scale". (Deputy General Manager in charge of the "Smart City" policy) ❑ Result n°6 : Open governance offers greater potential for creating public value than proactive governance ❑ Result n°5 : Technological prerequisites Proactive governance Quality of life Economic Political Social Strategic Ideological	Stewardship
		Organizational and Cultural changes		Strain on ressources
	"The digital twin has three objectives : decision-making aid for elected representatives, work support for services, information and consultation tool for residents." (Deputy Mayor in charge of the "Smart City" policy) Institutional conditions Open governance Quality of life Economic Political Social Strategic Ideological Stewardship

"I always say, "Asking a gardener to manage his watering with something other than a spanner but with a telephone is not the same job at all". So that's what's happening.

❑ Result n°4 :