Ryma Laifa

Jérôme Morchain

Ligia Barna

Pascal Guiraud
email: pascal.guiraud@insa-toulouse.fr

A numerical framework to predict the performances of a tubular photobioreactor from operating and sunlight conditions

Keywords: Tubular photobioreactor, Growth rate, Numerical experiments, Broth flowrate, Sunlight conditions

A framework model is proposed to evaluate the actual overall growth rate of microalgae in an outdoor tubular photobioreactor. A Monte Carlo-based radiative transfer modeling approach describes the local distribution of light energy inside the broth as a function of static (reactor geometry, location) and dynamic solar radiation parameters (angle of incidence, direct and diffuse solar contribution, incident radiation intensity). The light fields are coupled to a Lagrangian discrete random walk tracking of the cells to give the light variations experienced by each microalga for different broth flow rates. The cell light experiments are combined with a dynamic biological model to statistically calculate the actual overall growth rate. Using this model, 380 numerical experiments were performed for a wide range of geographic, light, biomass concentration, and broth flow turbulence conditions. Correlations for a normalized growth rate, Γ, relating the actual overall growth rate to its asymptotic behaviors (i.e., the instantaneous response and the full integration response), are proposed. The results clearly show that, for a fixed broth flowrate, Γ does not change with cell concentration variation. Under given light conditions, the level of turbulence linearly manages Γ, and thus the efficiency of sunlight utilization by the PBR biomass can be tuned by the broth flow rate in the tubular PBR. Γ also increases linearly with the diffuse fraction of solar radiation. A simple correlation is proposed for fast calculation of the actual overall growth rate.

Introduction

One of the emerging fields of application of microalgae, which has been growing steadily over the last decade, is food and feed [START_REF] Knuckey | Production of microalgal concentrates by flocculation and their assessment as aquaculture feeds[END_REF]. Due to their nutritional qualities, some microalgae (in particular Arthrospira Platensis and Chlorella Vulgaris) are used as food and /or feed supplement, they are marketed as dried powder or capsules but also as fresh raw biomass. In addition, there are a wide variety of areas where microalgae could be a renewable bioresource, such as the cosmetics [START_REF] Del Campo | Outdoor cultivation of microalgae for carotenoid production: current state and perspectives[END_REF],

pharmaceutical [START_REF] Yan | The Potential for Microalgae as Bioreactors to Produce Pharmaceuticals[END_REF] green chemistry (biofuels in particular) and water treatment industries [START_REF] Shoener | Microalgae and cyanobacteria modeling in water resource recovery facilities: A critical review[END_REF].

Industrial production of microalgae requires a high-performance system in terms of "optimal" growth and light conversion, which remains a challenge for processes operating in real outdoor conditions with sunlight.

A wide range of technologies for outdoor microalgae production systems can be identified. They are classified and qualified according to their type either open, such as raceways, mainly used for large scale commercial production [START_REF] Posten | Microalgal Biotechnology: Potential and Production[END_REF], or closed reactors called photobioreactors (PBR) which are more attractive compared to the first ones. Indeed, they offer the advantage of a controlled reaction condition and exist in a wide variety of designs depending on the production needs. In this context, exhaustive analyses of the limitations and advantages of each PBR configuration were carried out, the challenge being to design a closed system that is cost-effective, competitive and so efficient in terms of biomass productivity [START_REF] Carvalho | Microalgal Reactors: A Review of Enclosed System Designs and Performances[END_REF]- [START_REF] Huang | Design of Photobioreactors for Mass Cultivation of Photosynthetic Organisms[END_REF]. From a performance point of view, there are more arguments in favor of flat plan and tubular photobioreactors characterized by a high surface/volume ratio and efficient sunlight harvesting that guarantee a good photosynthetic efficiency [START_REF] De Andrade | Optimization of biomass production in outdoor tubular photobioreactors[END_REF]- [START_REF] Tredici | Efficiency of sunlight utilization: Tubular versus flat photobioreactors[END_REF].

They are commonly used either outdoor in direct sunlight or in a greenhouse to possibly control temperature and light. So far, there is no consensus on the ideal type of photobioreactor.

Predicting the performance of a solar photobioreactor requires an essential analytical approach focused on radiative transfer, which allows a correct estimation of the light energy, brought by the radiation, distributed in the volume and converted into biomass using mineral nutrients and water.

For outdoor systems, when all other physiological parameters are maintained at their optimal values, the productivity of solar systems in any configuration is limited by sunlight, which varies over time and according to the geographical position and orientation of the system. Therefore, it should be possible either to account for this incoming and outgoing energy or to obtain accurate information on what is happening in the reaction volume where this energy is used. With this in mind, detailed modelling approaches have been developed to evaluate outdoor solar PBRs based on the coupling of variable irradiance conditions (i.e. sunlight fluctuations intensity and the diffuse fraction of intercepted radiation) with photosynthetic growth kinetics [START_REF] Huang | Design scenarios of outdoor arrayed cylindrical photobioreactors for microalgae cultivation considering solar radiation and temperature[END_REF]- [START_REF] Slegers | Design scenarios for flat panel photobioreactors[END_REF].

Comprehensive and detailed models can provide a good simulation tool for solar systems despite their complexity. However, to the best of our knowledge, for the evaluation of photosynthetic growth rate, the aforementioned approaches focus on the instantaneous response of microalgae to local light. The latter is previously quantified by a light attenuation model either Beer-Lambert law or a two-flow model [START_REF] Pottier | A fully predictive model for one-dimensional light attenuation by Chlamydomonas reinhardtii in a torus photobioreactor[END_REF] which considers absorption without and with diffusion by microalgae respectively. Indeed, none of these simulations include the dynamic response of the cells to the light actually seen as they move along the light gradient, which allows to account for the effect of light/dark cycles called "flashing light" [START_REF] Kok | Photosynthesis in flashing light[END_REF]- [START_REF] Terry | Photosynthesis in modulated light: Quantitative dependence of photosynthetic enhancement on flashing rate[END_REF].

This effect, which is driven by the combination of mixing (large eddy scale and local turbulence) and spatial light heterogeneity, depends on bioreactor geometry. Experimental and modelling studies of lab scale illuminated PBRs for different species have frequently shown that mixing can induce a range of fluctuating light levels, with frequencies ranging from 0.05 to 50 Hz, while frequencies above 1 Hz can improve PBR performance [START_REF] Abu-Ghosh | Flashing light in microalgae biotechnology[END_REF]- [START_REF] Toninelli | Scenedesmus dimorphus biofilm: Photoefficiency and biomass production under intermittent lighting[END_REF].

Regarding the paramount importance of this rate of change in the light signal structure on the photosynthetic yield and consequently on the algae growth rate in PBR, in the last decade several numerical simulation studies have taken an interest in modeling the effect of coupling hydrodynamic, irradiance field and kinetics using different approaches. The latter were based on either direct numeric simulations [START_REF] Marshall | Simulation of light-limited algae growth in homogeneous turbulence[END_REF], the integration of a dynamic kinetics of photosynthesis and the fluid dynamics [START_REF] Wu | Simulation of algae growth in a bench-scale bubble column reactor[END_REF], [START_REF] Wu | Simulation of algae growth in a bench scale internal loop airlift reactor[END_REF] or a stochastic Lagrangian approach for bubble column PBR [START_REF] Gernigon | How Mixing and Light Heterogeneity Impact the Overall Growth Rate in Photobioreactors[END_REF], [START_REF] Tong | A theoretical analysis of the hydrodynamic influence on the growth of microalgae in the photobioreactors with simple growth kinetics[END_REF] and for tubular PBR [START_REF] Olivieri | Photobioreactors for microalgal cultures: A Lagrangian model coupling hydrodynamics and kinetics[END_REF], [START_REF] Marshall | A stochastic Lagrangian approach for simulating the effect of turbulent mixing on algae growth rate in a photobioreactor[END_REF]. Other more or less complex approaches have been proposed [START_REF] Pruvost | Hydrodynamics influence on light conversion in photobioreactors: An energetically consistent analysis[END_REF], [START_REF] Pruvost | Simulation of microalgae growth in limiting light conditions: Flow effect[END_REF] [40], [START_REF] Bitog | Application of computational fluid dynamics for modeling and designing photobioreactors for microalgae production: A review[END_REF] to evaluate the performance of different photobioreactor configurations.

We note that a robust biological model is needed to properly describe the effect of light changes on the dynamics of photosynthesis. Some existing models involve internal metabolic processes acting at different time scales (i.e. photoinhibition and acclimatation) [START_REF] Wu | A model integrating fluid dynamics in photosynthesis and photoinhibition processes[END_REF], [START_REF] Hartmann | Growth rate estimation of algae in raceway ponds: A novel approach[END_REF]- [START_REF] Han | Development of a new method of measuring bubble size[END_REF]. These works are generally based on the concept of photosynthetic unit PSU initiated by Eilers & Peeters [START_REF] Eilers | A model for the relationship between light intensity and the rate of photosynthesis in phytoplankton[END_REF] and account for the dynamic response of photosynthetic processes with photoinhibition and recovery [START_REF] Eilers | Dynamic behaviour of a model for photosynthesis and photoinhibition[END_REF]. Despite their robustness, the application of these concepts for simulation requires the calibration of all model parameters for the species under study. Moreover, their computational accuracy is compromised by the wide range of spatio-temporal scales of kinetic and hydrodynamic phenomena involved in the process. Thus, this limits their use in large-scale modeling. Other dynamic models have also already been put forward in literature [START_REF] Bernard | Hurdles and challenges for modelling and control of microalgae for CO2 mitigation and biofuel production[END_REF]- [START_REF] Bernardi | A Framework for the Dynamic Modelling of PI Curves in Microalgae[END_REF].

In the same line, Gernigon et al., [START_REF] Gernigon | How Mixing and Light Heterogeneity Impact the Overall Growth Rate in Photobioreactors[END_REF] proposed a first-order dynamic model to relate the algal response to light fluctuations in order to estimate the actual growth rate in a PBR. It considers the light history of the cells and introduces the internal physiological characteristics of the microalgae as a state variable. Their approach led to a simple expression for the photosynthetic rate, which is easily amenable to mathematical analysis under a quasi-stationary approximation. This model using a single-parameter (i.e. characteristic time) was easier to calibrate. Furthermore, it demonstrated its predictive capabilities for our strain of interest on the basis of several data from the literature [START_REF] Xue | Growth of Spirulina platensis enhanced under intermittent illumination[END_REF], [START_REF] Graham | A penalty on photosynthetic growth in fluctuating light[END_REF]. Therefore, to contribute to a continuing modeling effort of our research team, the present work is based on this model which has been experimentally validated and whose structure and mathematical properties are well established for a PBR illuminated by constant artificial light.

The main novelty of this paper is the extension of this coupling approach including fundamental investigations on: (1) the radiative transfer process, (2) Lagrangian cell tracking and (3) biological reaction dynamics, to calculate the actual performance (expressed as growth rate) in an outdoor tubular using variable sunlight. Specifically, massive simulations of 380 possible cases were performed to consider various photo-process parameters such as biomass concentration, flow rate of the culture medium as well as the characteristics of solar energy: direct sunlight incidence angles, diffuse light fraction and their intensities. The results of these numerical experiments were correlated to the investigated macroscopic parameters in order to provide a fast estimation of the growth rate. Indeed, the progression from the microscopic scale to the process scale is the key point of this work. The ambition being later to be able to use this growth rate model correlation to foresee the productivity over a full year, at any place on earth, using geo-located weather data and operating conditions of the PBR investigated.

Modeling approaches 2.1. Methodology

The core of our approach lies in accessing the distribution of "light conversion efficiency" in the algae population. Basically, in culture volume, light is spatially heterogeneous due to absorption and scattering by microalgae. Therefore, in a stationary light field, flow-transported cells are exposed to a fluctuating light signal to which they respond dynamically. The features of this signal result from the convolution of the stationary light field and the cell's velocity and location. The latter are calculated using a Discrete Random Walk model along with the known turbulent flow velocity profile (detailed in section 2.3). To establish the light field, a rigorous light transfer simulations were carried out using a Monte Carlo homemade algorithm including relevant parameters such as the PBR geometry, its wall's characteristics (refraction, reflection, and transmission), and microalgae's radiative properties (model description in section 2.2).The combination of these two independent fields determine "the rate of light change along the cell trajectory".

Microalgae are known to adopt a growth rate which is related to the integral of the light received over a defined time scale. Thus, microalgae are not at equilibrium with their local environment. In other words, their actual growth rate % & ' is not set by the average intensity received by the cells (defined as full light intensity integration i.e. %(〈*〉 & ,), nor by the local light fields as an instantaneous response (no light intensity integration i.e. 〈%(*,〉 & ) but must include the previously encountered light signal in a recent past (defined by the above-mentioned integration time scale).

This fact has been identified and reported by Terry in 1986 [START_REF] Terry | Photosynthesis in modulated light: Quantitative dependence of photosynthetic enhancement on flashing rate[END_REF], who introduced the concept of "The proportional light integration" :

Γ = % & ' -〈%(*,〉 & %(〈*〉 & , -〈%(*,〉 & (1)
This normalized specific growth rate, Γ, is observed and measured experimentally. However, there is no predictive method to quantify the actual growth rate and its departure from asymptotic cases namely (no or full light intensity integration). We propose to predict this quantity % & ' from the local distribution of individual cell capacity to convert light. In line with previous works, we name this cell property, 0, we assume that the microalgae population is segregated according to this property and put that microalgae characterized by 0 are able to grow at %(0 ,. The relationship % = 1(0, is known as "P-I relationship" describing the inherent link between constant light intensity I and photosynthesis P (see Appendix A).

If we now define the probability density function in mass 2( , 0, 4, such that 2( , 0, 4,50 is the mass

Introducing the local concentration of algae, =( ,, as the total mass of algae in that elementary volume 5 , the gain in mass per unit time over the entire volume of the reactor is retrieved from a volume integration. The resulting growth rate at the reactor scale is

% & ' = > & > & = ? =( , ? %(0,2( , 0, 4,50 (3)
Thus, the major difficulty in calculating the actual growth rate at the reactor scale comes from determining the cell property distribution 2( , 0, 4,. Indeed, the cell spatial distribution is uniform (cell iso-density is validated in section 3.2.2) so that the actual specific growth rate becomes:

% & ' (4, = 1
In this work, the distribution 2 & (0, 4, is approximated by the probability to find a cell with the property 0, whatever its location.

2 & (0, 4, = B(0, 4, ∑ B(0, 4, (6)
Hence, the problem is solved by tracking the location of a large number of particles while adapting their internal property 0. In our previous modelling works, a relaxation model was proposed and experimentally validated for a more realistic prediction of the PBR performances [START_REF] Gernigon | How Mixing and Light Heterogeneity Impact the Overall Growth Rate in Photobioreactors[END_REF]. This nonequilibrium model reflects this dynamic biological adaptation of microalgae to surrounding light fluctuations. According to [START_REF] Gernigon | How Mixing and Light Heterogeneity Impact the Overall Growth Rate in Photobioreactors[END_REF], each cell is characterized by an internal propriety defined as its light conversion efficiency, i, that tends towards the local light experienced by cell, *(D,, and it is given by:

E0 E4 F = 1 G F . (*( , -0, (7)
with G = 0.3 K being the microalga relaxation time constant which has been calibrated on the basis of fairly recent experimental data [START_REF] Xue | Growth of Spirulina platensis enhanced under intermittent illumination[END_REF], [START_REF] Gernigon | How Mixing and Light Heterogeneity Impact the Overall Growth Rate in Photobioreactors[END_REF]. This model enables description of the effect of flashing light frequency on the global growth rate. It has already demonstrated its reproducibility and success for a wide range of light frequencies (i.e. several durations of day/night cycle).

From a calculation point of view, to compute the actual overall growth rate, the integration method chosen is to sum over a large number of Lagrangian particles B L and over B M consecutive sampling time events (Monte Carlo) until the initial condition B L (0, 0, is forgotten, i.e. the ensemble average is constant.

% & ' = 1 B L B M N N μ(0, P Q MRS P T LRS (8)
From Eq.(4) and Eq.( 7), it is straightforward to show that if the response time to light fluctuations is infinitely small 2( , 0, 4, = U(0 -*( , 4,, which means that all cells at are in the same state 0 = *( , 4,. This case is commonly referred as instantaneous response.

Similarly, if the response time to light fluctuation is infinite, 2( , 0, 4, become independent of the spatial coordinate is 2( , 0, 4, = U(0 -〈*〉 & ,. This case is commonly referred as full integration.

% &,V→; ' = 1 A : : %(0,U(0 -〈*〉 & , 50 5
;

< & = 1 A : %(〈*〉 & ,5 & = %(〈 〉 & , (10)
These two asymptotic responses patterns ( 9) and ( 10), can be obtained by a direct coupling between the irradiance field results and the growth kinetic model without considering actual cell trajectories and cell capacity to convert light.

The sketch in Fig. 1. summarizes the modeling methodology undertaken in this study to compute the overall growth rate in tubular PBR then deduce the value of normalized specific growth rate, Γ, using the two asymptotic cases mentioned above.

This approach is applied to a horizontal tube (considered as the model element of a tubular PBR)

operating under a range of dynamic functioning conditions such as biomass concentration and light energy, which vary in time and according to the plant geographic position.

The key point of this work is to describe Z as a function of the studied operating parameters and thus theoretically infer the actual overall growth rate. Thus, a burst of 380 simulations were performed to sweep a wide range of operating cases, then Z were correlated to the main parameters.

Local available light inside solar tubular PBR

As the light is the main parameter affecting the growth of autotrophic microalgae, properly quantifying the local distribution of light within the PBR is important, especially when the irradiance heterogeneities are striking. In this section, the aim is to simulate the light distribution in the suspension as a function of the variable solar radiative flux, in a defined location.

Unlike many existing models that often rely, in their calculation of the overall photosynthesis rate, on the incident or average light intensity reaching the culture [START_REF] Lee | Kinetics and bioenergetics of light-limited photoautotrophic growth of Spirulina platensis[END_REF], [START_REF] Jeon | Measurement of microalgal photosynthetic activity depending on light intensity and quality[END_REF], we rigorously describe the propagation and distribution of sunlight (composed of direct and diffuse fractions) in the whole volume.

Fig. 2. Diagram of sun's position angles and orientation of a tubular PBR

The sun's apparent position is described by two main angles zenith ([) and azimuth solar (\ ] ) as shown in Fig. 2. These angles are calculated using equations from literature [START_REF] Duffie | Solar engineering of thermal processes[END_REF], which include the geographic localization (latitude & longitude) of the site, the day in the year and the time.

For an outdoor tubular cultivation system, the maximum flux collected during the sun course is obtained when the tube is aligned on a north-south axis, this orientation was chosen.

Our model includes these particular characteristics in the representation of: (i) the non-normal incidence of direct sunlight on the PBR surface that drives the photon entrance and propagation within the reactor, and (ii) the hourly and annual variation of the levels of diffuse irradiation and the flux density of the direct incident light. An appropriate OXYZ photon moving marker was defined for characterizing the incident light direction by two new angles (Ψ) and (θ), which are expressed as a function of zenith ([) and azimuth (\ ] ) solar angles. The formulae used and the corresponding diagrams are presented in supplementary document (Appendix B.1).

Because of the complexity in solving the RTE (The integro-differential Radiative Transfer Equation)

analytically or even numerically when all phenomena are included, some approaches based on approximations can be found in the literature. For example, [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 simple and reliable formula for assessment of maximum volumetric productivities in photobioreactors[END_REF] used an analytical solution based on a one-dimensional approximation of scattering known as the Two-flux method. In this approach, only the forward and backward propagation directions are considered, which corresponds to two diffuse radiation flows propagating inwards and outwards from the medium. Among the approximate analytical models, the P1 approach (assuming that energy is distributed uniformly over all directions) was proposed to solve the RTE in solar tubular photocatalytic reactors [START_REF] Arancibia-Bulnes | Development and Modeling of Solar Photocatalytic Reactors[END_REF].

However, others studies have used numerical methods of resolution as the finite volume method [START_REF] Wheaton | Modeling radiative transfer in photobioreactors for algal growth[END_REF], the discontinuous Galerkin method [START_REF] Lee | Design tool and guidelines for outdoor photobioreactors[END_REF], and Monte Carlo method applied for a plane-slab photocatalytic reactor [START_REF] Brucato | Estimating radiant fields in flat heterogeneous photoreactors by the six-flux model[END_REF] or for different PBR geometries [START_REF] Csogör | Light distribution in a novel photobioreactor-modelling for optimization[END_REF]- [START_REF] Dauchet | The practice of recent radiative transfer Monte Carlo advances and its contribution to the field of microorganisms cultivation in photobioreactors[END_REF].

The simulation of the local irradiation field in solar PBR tubes should consider the corresponding light energy balance including: the transmission, absorption, diffusion in the algal broth, as well as the illuminated surface geometry (curvature of the walls). Previous works on the same PBR design neglected the light scattering by cells (light multiple diffusion) and used only Beer-Lambert law to estimate the internal attenuation of light [START_REF] Huang | Design scenarios of outdoor arrayed cylindrical photobioreactors for microalgae cultivation considering solar radiation and temperature[END_REF], [START_REF] Slegers | Scenario analysis of large scale algae production in tubular photobioreactors[END_REF], [START_REF] Marotta | Reflection-refraction effects on light distribution inside tubular photobioreactors[END_REF]. In contrast to these studies, here the Monte Carlo method is employed as a suitable approach for multi-phenomena modeling (details about this method in [START_REF] Hammersley | Monte Carlo Methods[END_REF]). In this technique, events are considered corresponding to the following physical phenomena: (i) the deviation of the photon path due to its reflection/refraction by the PBR wall according to Snell-Descartes laws, (ii) the energy fraction carried towards the culture by reflection or transmission according to Fresnel laws, (iii) the absorption in the culture broth according to Beer-Lambert's laws, and (iv) the anisotropic multiple diffusion (i.e. diffraction effects) due to the interaction with the cells using the phase function [START_REF] Dauchet | Analyse radiative des photobioréacteurs[END_REF] (see formula in Appendix B.2).

Since the nature of incidence between direct and diffuse light is different, the corresponding irradiance fields are calculated separately. Indeed, the PBR is considered to receive parallel direct light rays on the upper hemicylindrical wall exposed to the sun, whereas the diffuse component of sunlight is assumed to have an isotropic Lambertian incidence. The diffuse light can come from any direction over the entire pipe wall (random propagation trough space). The global irradiance field * corresponding to the sunlight is finally calculated by the sum of the contributions resulting from each spatial distribution for direct * ^ and diffuse * _ radiation:

*(D, `, a, = * < × [c × * _ (D, `, a, + (1 -c, × * ^(D, `, a,] (11)
where \ is the diffuse light contribution fraction and * < is the global incident light intensity.

To reduce the calculation cost and achieve a good simulation accuracy, it was found that a 2D matrix discretized with a square mesh of 10 d segments, and about ten million photon's trajectories, were necessary to obtain the convergence of the calculation and a satisfactory representation of the light distribution field. The model was implemented and simulations are performed using MATLAB R2017b software (Mathworks) thanks to parallelization option to span the various cases of irradiance fields simultaneously. The developed model based on Monte Carlo approach was validated using analytical models for different simplified cases (detailed results in Appendix B.3).

Lagrangian study of cell transport in the pipe

In addition to the irradiance field determination, it is important to simulate the individual cell trajectories driven by the carrier fluid flow in the PBR. As the light field is identical all along the tubular reactor axis, only the radial and azimuthal microalgae movements change the light intensity they experience along their trajectory. These radial and azimuthal cell movements are generated by turbulence and wall ejections. The most complete and accurate approach for this situation would be the direct resolution of the Navier-Stokes equations DNS (Direct Numerical Simulation) or using LES (Large Eddy Simulation). However, their implementation is limited by the computational costs because it requires discretization in very small time and space steps in order to capture all the structures of the turbulence. Thus, the added value of such a complex simulation can only be justified if the fine details of the trajectories are of great importance. The particle trajectory in a RANS-based flow simulation very often refers to a discrete random walk model. In the present work, the modelling relies on a Lagrangian cell tracking method based the algal trajectories from well-known 1-D turbulent flow profiles. In this case, a discrete random walk model provides a time-saving approach. Indeed, For a flow in a tube at different Reynolds numbers, the time required to perform an advanced CFD simulation as proposed in several studies [START_REF] Pruvost | Hydrodynamics influence on light conversion in photobioreactors: An energetically consistent analysis[END_REF], [START_REF] Perner-Nochta | Simulations of light intensity variation in photobioreactors[END_REF], [START_REF] Gernigon | How Mixing and Light Heterogeneity Impact the Overall Growth Rate in Photobioreactors[END_REF], [START_REF] Luo | Verification and validation of CFD simulations for local flow dynamics in a draft tube airlift bioreactor[END_REF], [START_REF] Gao | Comprehensive computational model for combining fluid hydrodynamics, light transport and biomass growth in a Taylor vortex algal photobioreactor: Eulerian approach[END_REF], could take months, whereas the proposed approach takes only a few minutes several Re resulting from experimental data [START_REF] Toonder | Reynolds number effects in a turbulent pipe flow for low to moderate Re[END_REF] and numerical works [START_REF] Wu | A direct numerical simulation study on the mean velocity characteristics in turbulent pipe flow[END_REF].

Based on this, in our cell tracking the two transverse components of standard deviation values of the fluid velocities (RMS velocities) are supposed to be equal and constant over the section.

Furthermore, these fluctuation values can be deduced from the one obtained at the center of the tube where the turbulence is isotropic. To attain this value, calculation of turbulence in the continuous phase has to be performed.

Some hypotheses have been considered: (i) the carrier fluid is assumed to be incompressible and Newtonian, and (ii) cells have a small size and a density equivalent to that of the continuous phase.

Thus, they are considered as fluid particles that follow the flow lines without impacting the continuous phase flow.

Numerical resolution of the continuity and Navier-Stokes equations was performed using the k -l model (RANS approach).

The most faithful modeling of the irregularity and random characteristics of turbulent flow should be able to capture the details of turbulence diffusion due to vortex structures which is the main mechanism of particle dispersion in the fluid. Taylor [START_REF] Taylor | DIFFUSION BY CONTINUOUS MOVEMENTS[END_REF] introduces the concept of turbulent diffusion for the modeling of the transport of passive particles by proposing a stochastic model coupled to a Lagrangian approach for a statistically stationary isotropic homogeneous turbulent flow.

The trajectories of particles are governed by a part related to the mean flow of the continuous phase For a more realistic simulation of the fluctuating part, a stochastic approach was used here via a discrete random walk (DRW) model, initiated by Hutchinson et al., [START_REF] Hutchinson | Deposition of liquid or solid dispersions from turbulent gas streams: a stochastic model[END_REF] for the study of particle dispersion in the turbulent flow field.

The DRW model assumes that a particle interacts with a sequence of vortices that are characterized by a time interval of interaction G p and random velocity fluctuations that are kept constant during this time. The principle is that once this interaction time has elapsed, a new random fluctuation independent of the previous one is introduced to account for an interaction with a new vortex. This characteristic time is known as "The characteristic lifetime of the eddy" and is defined as a constant

given by local characteristics of the turbulent flow k and l:

G p = 2 • s t ( 13
)
s t u 0.15 • k l (14)
With s t "the fluid Lagrangian integral time"

The fluctuating velocities are randomly sampled in a normal distribution of zero mean and standard deviation equal to the RMS value of velocity fluctuations Eq. [START_REF] Lee | Design tool and guidelines for outdoor photobioreactors[END_REF]. Assuming that the turbulence is isotropic at the tube center, the RMS at any point in the flow is written as a function of the local turbulent kinetic energy as follows:

m' fgh = e' fgh = i' fgh = x(2k y /3, (16)
k y being the kinetic energy in tube center.

The time step chosen for our simulation is 54 =0.05 (s). This value is set so that it is equal or inferior to the vortices lifetime in the tube center (resulting from considerations postulated in the previous paragraph) ranging from (0.09-0.44 (s)) for the case studied. This choice of time step is also made by respecting on one hand the classical conditions allowing an acceptable numerical stability of the solution with a lower cost of calculation time, and on the other hand the frequency range of the sampled irradiance signals (i.e. 20 Hz), thus, a good integration of the events undergone by the cell over its biological relaxation time G =0.3 (s).

The particles positioning (r,{,z) is calculated as follows:

| R ~•€~= R ~+ u ' (t, • dt Θ ~•€~= Θ ~+ u † (t, • dt Z ~•€~= Z ~+ u ˆ (t, • dt (17)
Using Lagrangian approach provokes a preferential trapping of particles next to the walls of the PBR. This is due to the assumption that turbulence is isotropic everywhere in the bulk, however, in the parietal zone, the turbulence is neither homogeneous nor isotropic as observed previously in Fig.

3.

where the radial component of the velocity is smaller than the tangential one. As a result, the particles in the turbulent boundary layer are decelerated significantly and are not returned to the e ‰ ~B(0, e' fgh ,

flow. The calculated radial fluctuations in the vicinity of the wall are not sufficient to generate the return of the particles inside the flow and so the continuity equation for the particle phase is not respected. This numerical problem has also been raised by several CFD simulation studies using the Lagrangian method to track particles in different types of PBR [START_REF] Pruvost | Hydrodynamics influence on light conversion in photobioreactors: An energetically consistent analysis[END_REF], [START_REF] Gernigon | How Mixing and Light Heterogeneity Impact the Overall Growth Rate in Photobioreactors[END_REF], [START_REF] Luo | Verification and validation of CFD simulations for local flow dynamics in a draft tube airlift bioreactor[END_REF], [START_REF] Gao | Comprehensive computational model for combining fluid hydrodynamics, light transport and biomass growth in a Taylor vortex algal photobioreactor: Eulerian approach[END_REF].

As it is not possible to reproduce numerically the turbulent structures called "Bursting" present in the boundary layer using k -l, (only DNS approach could do that), we tried to overcome this problem of accumulation by managing the particle wall ejection by their rebounds. The objective is to ensure that the concentration of particles in the cross-section respects as well as the continuity equation, i.e., their concentration keeps homogeneous since there is no phenomenon concentrating them somewhere in the flow. The principle is to eject low velocity particles from the viscous sublayer into the internal flow. The particle rebound is managed by an adapted rebound coefficient. The homogeneity of the cell's concentration will be checked later (section 3.2.2).

About one hundred thousand trajectories were simulated to form a statistically representative sample of numerous paths taken by the cells. The particles are initially sampled in the entrance cross-section of tube to ensure their homogenous distribution since the PBR is considered perfectly agitated.

Results and discussions

The model is designed with the aim of simulating the dynamic evolution of the light distribution in whole-year, outdoor running, PBR. Time varying parameters are: cell concentration (in the case of batch reactor), direct light incident angles, global incident irradiance, fraction of diffuse radiation.

The study case is a horizontal tubular PBR represented by a single isolated pipe with a diameter of D=0.05 m, and the broth medium containing the strain of species Arthrospira platensis (Spirulina).

The list of physical quantities and parameters used for the simulations (model inputs) are presented in Table 1. We have tried to scan a range of representative flow velocities at typical velocities in tubular PBRs used at the technical scale [START_REF] Pulz | Photobioreactors: production systems for phototrophic microorganisms[END_REF], [START_REF] Molina | Tubular photobioreactor design for algal cultures[END_REF] to test the effect of turbulence on performance. Table 2 summarizes the different parameters of the conditions tested in this work. Azimuth angles was considered in order to represent several times of the day and in year. Therefore, using the symmetrical profile of the sun's path, it was possible to select a limited number of cases.

Ten irradiance fields are calculated for cell concentrations ranging from 0.3 to 1.5 (g/L). Hence, 280 cases were performed. Not all results for irradiance field simulation for selected cell concentrations and sun positions, are presented here.

The fields are presented in normalized form with respect to the energy received in each illuminated elementary surface (i.e. divided by the number of photons used in simulation and the area of the mesh). Due to the invariance along the horizontal axis of the tube, the geometry is reduced to crosssection. Therefore, the results are the projections of the 3D motion of the photons in the volume onto this cross-section.

Only one example is shown here, a representative case of a typical summer day in Toulouse, France As can be seen, there is a striking spatial heterogeneity of light distribution, hence, a significant difference is noticed between the regions close to the wall and those in the center of the tube Fig. 4.

This is a specific feature usually mentioned in the literature for light-limited PBRs, and which is due to the absorption and scattering of light in the medium. Furthermore, the effect of the refraction and reflection of direct collimated light on the wall is clearly seen in Fig. 4. (a). Thus, due to the incurved surface, direct sunlight passing through the PBR interface is subject to a redirecting effect and obeys

Fresnel's laws. As a result, the fraction of energy transmitted into the reactor volume is less than the solar energy arriving at the surface. This explains the non-uniform distribution of irradiation in the area near the illuminated surface (i.e. the hemisphere facing the sun). This feature has been also observed in [START_REF] Marotta | Reflection-refraction effects on light distribution inside tubular photobioreactors[END_REF]. The diffuse light irradiance field shown in Fig. 4. (b) is only concentration dependent. Since this part of radiation is isotropic, the light come from any direction and it is better distributed than direct light over the entire wall of the pipe. However, compared to the overall field, the diffuse light field seems not significant even if its contribution is \=0.38. The irradiance distributions computed will be then multiplied by the global incident irradiance value * < corresponding to the investigated case, derived from the PVGIS data. The results for the selected cell concentrations and sun positions constitute a data set to be combined with other modeling approaches in order to be exploited for photosynthetic growth calculation.

With an appropriate adaptation of the model, this approach can be applied to (i) other PBR geometries (e.g. rectangular or flat PBR), and (ii) other types of surface or volume lighting (e.g.

artificial LED

). Cell spatial distribution

Cell trajectories and their combination with the irradiance field

The goal here is to obtain data on the history of each cell exposure to light in order to further exploit use it in the estimation of overall growth rate. A collection of various trajectories was acquired using the method of sampling cell movement in the reactor described above (subsection 2.3). The interaction between hydrodynamics and light transfer is highlighted here. To ensure the statistical convergence of numerical resolution and to guarantee a meaningful representation of all possible events that microalgae undergo in PBR, a choice of 1.5 × 10 ˜ particles to be simulated, for a duration of 5s and with a time step of 0.05 (s), were deemed necessary to generate sufficient amount of information (i.e. about 10 ™ events).

An example of trajectory recorded for an individual cell is illustrated in Fig. 5 with a given irradiation field (e.g. simulated for a concentration of 0.9 (g/L) and lighting conditions of a typical summer day in Toulouse at 10h solar time where c=0.4). Hence, the temporal light regime seen by microalgae is recorded over 5 (s) with frequency of 20 (Hz) as represented in Fig.

5(c).

This procedure is performed for the whole set of simulated particles in order to represent the light regime actually seen by each particle along its travel in the reaction volume. Therefore, the interaction between turbulence and heterogeneity of illumination within a volume was highlighted.

Some microalgal trajectories relatively similar to the one presented in Fig. 5. (c) have been reported in previous studies [START_REF] Pruvost | Hydrodynamics influence on light conversion in photobioreactors: An energetically consistent analysis[END_REF], [START_REF] Perner-Nochta | Simulations of light intensity variation in photobioreactors[END_REF], [START_REF] Gernigon | How Mixing and Light Heterogeneity Impact the Overall Growth Rate in Photobioreactors[END_REF], [START_REF] Olivieri | Photobioreactors for microalgal cultures: A Lagrangian model coupling hydrodynamics and kinetics[END_REF]. However, the verification that the database of particle trajectory is correctly sampling the light volume distribution is almost never carried out. Here, we aim to validate the exploitation of light exposure database for the evaluation of growth rates for the different microalgae behaviors mentioned in section 2.1. First of all, Fig. 6. (a) shows a uniform spatial dispersion of cells along the tube radius, which is an important result in its own. Indeed, simulations reported in the literature [START_REF] Gernigon | How Mixing and Light Heterogeneity Impact the Overall Growth Rate in Photobioreactors[END_REF] performed with CFD tools by using the standard turbulence model k -l coupled to the Lagrangian tracking, overestimate the turbulent kinetic energy in this area and thus generate a numerical bias on the subsequent trajectories' calculation. These methods overestimate the particle concentration in the near walls. In our results, although this artificial buildup exists, due to the lack of the hydrodynamic boundary layer resolution and the insufficient geometric adaptation of the rebound laws, it is however reduced to less than 6% thanks to the particle wall ejection proposed here, which imposes a constraint that forces the cells back into the flow, as described in section 2.3.

Moreover, an "at the point" validation was conducted according to the method described by Gernigon et al., [START_REF] Gernigon | How Mixing and Light Heterogeneity Impact the Overall Growth Rate in Photobioreactors[END_REF] to confirm the uniformity of the distribution of cells in the volume and to analyze the acuity of the near-wall effect. This method consists in comparing the distribution of local light intensities obtained by the spatial irradiance field calculation and the database of light recorded by cells using the stochastic tracking model. This verification approach provides more sound information than an integral approach based on the comparison of the average light available in the volume and the population average obtained by the Lagrangian approach, as has been performed in other studies [START_REF] Pruvost | Hydrodynamics influence on light conversion in photobioreactors: An energetically consistent analysis[END_REF], [START_REF] Tong | A theoretical analysis of the hydrodynamic influence on the growth of microalgae in the photobioreactors with simple growth kinetics[END_REF]. In Fig. 6. (b) the probability density results show that the validation of cell iso-distribution is successful since the irradiance seen by the whole population statistically converges towards the spatial irradiance field in the PBR under given operating conditions (i.e. cell concentration of 0.9 (g/L), I < =1241(μmol. m žŸ . s žS ), diffuse coefficient α=0.4,. Furthermore, it was also verified that the total average irradiation of all the generated data (all particles and all events) corresponds to the irradiance field volume average with less than 2% of relative error between the two values.

Exposure time distribution in areas close to the wall

Despite small erroneous predictions of particle concentration on the near wall region provided by our particle tracking model, we intend to analyze its impact on the irradiation field integration by the cell population. Indeed, the question that arises here is to know if an overexposure is possible in the zones with very high light intensities causing an eventual inhibition.

To answer this, we need information on the frequency of occurrence of "viscous layer passage" events and their duration. A distribution of the exposure or residence times of all particles is calculated and normalized by the total simulation time, as shown in Fig. 7. The shape of this distribution is similar to a decreasing exponential. This latter leads to think of the residence time distribution for a perfectly agitated reactor. Once again, this finding supports the discussion mentioned previously concerning the uniform repartition of the cells in the reactor. More precisely, these data reveal the passage times of the particles through the entire viscous layer near the reactor walls (a ring of variable thickness according to flow velocity). However, this boundary layer, can be extremely heterogeneous in terms of light intensities. This heterogeneity increases as the biomass concentration increases.

To our knowledge, there is no consensus on the exact value of the light intensity and the exposure duration that actually cause inhibition of spirulina. According to the experimental data of [START_REF] Xue | Growth of Spirulina platensis enhanced under intermittent illumination[END_REF], a period of more than 10s under an irradiance of about 2600 (μ>¡¢. > žŸ . K žS ) is sufficient to have a negative growth rate leading to a total collapse of the culture. These high irradiance intensities are rarely encountered in reality except in subtropical regions. The onset of inhibition under conditions of permanent lighting was also signaled for intensities > 1300 (μ>¡¢. > žŸ . K žS ). With this in mind, we analyzed our light exposure database for whole cells tracked to check the time of cell passage through the very high illuminated zones. To do this we have chosen an extreme operating condition where microalgae could be subject to inhibition (i.e. the highest incident irradiance recorded in Toulouse, France I < = 1402 (μmol. m žŸ . s žS ) α=0.33, the lower biomass concentration considered here X=0.3 (g/L), and at the lowest flow velocity tested " ^=0.3 (m/s)). Then, we estimated for each cell the exposure time to irradiances above 1300 μ>¡¢. > žŸ . K žS . The results plotted in Fig. 8. show that the cell switching events to high lights are occasional and of short duration. Thus, this indicates that cells are rather exposed to high intensity flashes of light that are not harmful to growth as opposed to prolonged exposure as in the case of the permanent light regime which causes cell destruction.

The results confirm the harmfulness of such overexposure to light when coupled with the turbulence induced by mixing. 3.2. Analysis of growth rate results

Biological dynamic response

The database about the light history of each particle has fed the dynamic model expressed in Eq [START_REF] Lehr | Closed photo-bioreactors as tools for biofuel production[END_REF] to obtain the internal variable 0 evolution that represents the dynamic adaptation of individual cells to the detected light * L . Fig. 9 presents the internal variable signal resulting from the biological response to the recorded light fluctuations, for the same cell as in

Real growth Kinetics

The PBR overall growth rate were calculated here, according to the modelling approach described in ). In case of the dynamic biological response, the results are positioned between the two asymptotic cases. The mixing or turbulence effect becomes more remarkable with the increase of the biomass concentration since the fraction of dark volumes and light gradient increase. Indeed, as the liquid velocity increases, the growth rates tend toward the maximum limiting case. Simulation results indicate a 20% improvement in PBR performance when the velocity increases from 0.3 (m/s) to 1.68 (m/s) (i.e. Re = 15000 and Re=84000 respectively) for a cell concentration of 1.5 (g/L). This behavior can be explained by the fact that algae filter the fluctuations of light they see and respond to the captured light more slowly than the instantaneous adaptation. Thus, the greater the turbulence, the closer the growth rate is to the upper limit (i.e., the fully integrated case).This beneficial effect of turbulence has also been observed in the experimental results of [START_REF] Carlozzi | Productivity of Spirulina in a Strongly Curved Outdoor Tubular Photobioreactor[END_REF] who recorded a 29% increase in spirulina productivity when the flow regime in a straight tubular PBR switches from laminar to turbulent regime. Indeed, this confirms that the strong turbulence is responsible for the beneficial effects of the strong intermittent exposure of cells at different light intensities in highly heterogeneous spatial gradients. Nevertheless, a too strong turbulence also increases the shear stresses, which must be limited to avoid culture damages. This later effect is not considered in this study.

Towards a general correlation for the normalized specific growth rate Γ

This whole procedure of specific growth rates calculation for both asymptotic and dynamic (real) cases, was performed for a wide range of operating conditions regarding microalgae concentration, flow rate expressed in Reynolds, and received sunlight characteristics (global incident irradiance, direct/ diffuse radiation contribution) in order to cover a wide spectrum of operating possibilities for an outdoor tubular PBR. The normalized growth rate Γ was calculated by Eq. ( 1) for each case simulated.

In other words, Z translates the capacity of a PBR to reach its optimal performances according to the way the microalgae are exposed to light, which is driven by its operating conditions. Since numerical simulations require a coupling between large databases which is time-consuming, a reduction of the model seems attractive for application cases where PAR performance could be calculated for real operating conditions (i.e., biomass concentration, flow hydrodynamics, lighting quantity and quality).

Z varies between 0 and 1 indicating the efficiency of light use between instantaneous adaptation and full integration. To deeply understand the effect of different operating parameters on PBR ability to drive its performances towards the two asymptotic cases, we started to investigate whether the cell concentration =(g/L) and incident sunlight intensity * < affect the rate of light utilization (i.e. Z, and consequently the real growth rate.

At the sight of Fig. 10, it is easy to prejudge that the consequence of the biomass concentration on the kinetics is filtered out in the normalization ratio Eq.( 1). Hence, to verify this, the corresponding In order to evaluate the effect of turbulence-induced mixing on the ability to improve light utilization, Γ is plotted versus Re Fig. 12. Not surprisingly, the value of Γ increases with increasing

Re approaching the maximal asymptotic case (i.e. full integration case, when Γ=1). These results confirm again that increased turbulence improves growth rate. Experimental data leading to a To better illustrate the influence of the diffuse radiation on biological growth, the ratio Z ¥¦ F is plotted versus α. in Fig. 13. Results showed that, unlike biomass and the intensity of incident light, hydrodynamics coupled with the way in which light energy is supplied to a culture system, has a considerable effect on its ability to use available light. A mathematic expression can be proposed to couple both flow and illumination configuration as follows: Z = 6.19 10 žd ¥¦ c + 3.22 10 žd ¥¦ [START_REF] Slegers | Design scenarios for flat panel photobioreactors[END_REF] Since Re only varies, in our numerical experiments with the fluid velocity, it's more suitable to write this correlation as function of " ^ as follows as the tube diameter © =0.05 (m): Z = 0.31 10 žd " â \ + 0.16 10 žd " â [START_REF] Pottier | A fully predictive model for one-dimensional light attenuation by Chlamydomonas reinhardtii in a torus photobioreactor[END_REF] where ª being the kinematic viscosity of the fluid (m²/s). For Spirulina case, we can suppose ª = 10 žd (m²/s). So, Γ can be expressed directly as a function of the liquid flow velocity:

The values of % & ' calculated by Eq.( 8) (i.e. resulting from combination of the three information: the light distribution, the cell motion and biological response time (inertia of the internal variable 0)), are compared in Fig. 14 to the values of % & ' predicted by the expression Eq. ( 21) using Z's correlation.

Although, deviation of 20% between simulated and correlated data (probably due to the small errors made when correlating the link between Γ and turbulence and c), the linear trend is found with a coefficient of determination of ¥ Ÿ =0.98. The results obtained by Eq.( 21) can be considered in sufficient agreement with those obtained using the full modeling approach. Thus, the reliability of using these correlations for an easy estimation of the overall PBR specific growth rate is validated.

  Photobioreactor volume

Fig. 1 .

 1 Fig. 1. Diagram of the modelling approach. Green box: input parameter; Grey box: modeling method. Pink box: results database

  Only the radial and azimuthal velocity fluctuations are involved to produce the cell movement in the light field. They can be represented by two root-mean-square (RMS) components (e fgh et i fgh ) transversal to the main flow. A collection of some experimental and numerical data from the literature was used to observe and analyze these velocity fluctuations in turbulent flows in a horizontal pipe. Fig.3. presents the profiles of the ratios of the radial and azimuthal RMS variances by the friction velocity % j corresponding to different case studies[START_REF] Toonder | Reynolds number effects in a turbulent pipe flow for low to moderate Re[END_REF],[START_REF] Wu | A direct numerical simulation study on the mean velocity characteristics in turbulent pipe flow[END_REF]. This figure indicates that these two non-dimensional fluctuating velocities are almost similar regardless of the Reynolds number, namely 17800, 24600, 38000 and 44000. In addition, their values remain almost maintained in the inner region of the tube until the beginning of the viscous sub-layer where they start to progressively decrease. Indeed, the turbulence tends towards the isotropy in the pipe center and presents an increasing anisotropy towards the boundary layer whose thickness varies with Reynolds number. However, the two transverse fluctuations keep similar values. This reproduces the classical behavior of a turbulent single-phase flow in a cylindrical pipe, which has been widely studied for a long time[START_REF] Eggels | Fully developed turbulent pipe flow: a comparison between direct numerical simulation and experiment[END_REF]-[START_REF] Lawn | The determination of the rate of dissipation in turbulent pipe flow[END_REF].

Fig. 3 .

 3 Fig. 3. Profiles of radial and tangential RMS components normalized by the friction velocity for

  and another part related to the action of the turbulence structures driven by the instantaneous fluctuations. It is therefore necessary to integrate the trajectory equation by using the instantaneous velocity components seen by the particle during their movements which, in the case of a horizontal pipe, are written in cylindrical coordinates such as: m ] (4, = m n + ḿ(4, m f (4, = é(4, m o (4, = í(4, (12) With m n the average longitudinal velocity and ḿ(4,, é(4,, í(4, are the components of the instantaneous longitudinal, radial and tangential fluctuating velocities respectively.

( 43 .

 43 36° N, 1.26° E) at solar noon where the Zenith angle "= 20°, Azimuth angle \ ] =0° and diffuse fraction c=0.38. The irradiance fields with 1 (g/L) of biomass for the direct and diffuse solar radiation are shown in Fig. 4. (a) & (b) respectively and Fig. 4. (c) presents the total distribution of sunlight received on the culture.

Fig. 4 .

 4 Fig. 4. Normalized local irradiance fields at biomass concentration ==1(g/L): (a) for direct sunlight with ϑ= 20° a ˆ=0°, (b) diffuse sunlight, (c) the sum of the contribution of the two fields direct and diffuse for solar midday of summer day in Toulouse diffuse coefficient α=0.38.

  for a given hydrodynamic conditions (a flow velocity of 0.4 (m/s) corresponding to Reynolds number of 20000) over 5s of simulation. The main point of interest in this investigation is shown in Fig. 5.(a) which describes the bidimensional motion of the particle in the cross section of the tube where the light gradient occurs. These radial transitions projected in 2D plan effectively reproduce the stochastic aspect of the cell displacement predicted by the random walk model. These purely random redirections enable the capture of all light variations due to the turbulence, i.e. the local fluctuations characterizing the local eddies established in this type of flow. Fig. 5(b) illustrates the real-time tracking of the same particle where the radial movement is depicted along the longitudinal axis in 3D Cartesian coordinates. This database of the consecutive cell positions is then coupled directly

Fig. 5 .

 5 Fig. 5. Followed trajectory of an arbitrary cell for a Re= 20000 and during t= 5(s): (a) projection on an irradiation field (2D cross-section), with cell concentration of ==0.9 (g/L), I < =1241(μmol. m žŸ . s žS ), diffuse coefficient α=0.4, (b) trajectory followed along the longitudinal axis in 3D and (c) temporal light intensity recorded during its motion.

Fig. 6 .

 6 Fig. 6. (a) The radial distribution of the cell population density in the tube, (b) Comparison of the irradiance distribution recorded by the cells (using. " ^=0.4 (m/s) corresponding to a Reynold's number of 20000) with light distribution given by the spatial field for cell concentration of ==0.9 (g/L), * < =1241(μ>¡¢. > žŸ . K žS ), diffuse coefficient c=0.4.

Fig. 7 .

 7 Fig. 7. Distribution of normalized duration of cells exposure to the near-wall layer for the lowest flow velocity value tested in this study (i.e. " ^=0.3 (m/s) corresponding to Re= 15000).

Fig. 8 .

 8 Fig. 8. Light filed with * < = 1402 (μ>¡¢. > žŸ . K žS ) c=0.33, ==0.3 (g/L), (b) Distribution of cell time exposure to inhibitory zones (flow velocity " ^=0.3 (m/s) corresponding to Re= 15000).

Fig. 5 .

 5 (c). It was found that the magnitude of strong fluctuations in light exposure is smoothed by the internal variable 0 due to turbulence induced by hydrodynamic microstructures (i.e. small swirls). This could be explained by the difference in time scales of the hydrodynamic and biological processes: the relaxation time or cell response, τ=0.3 (s), used in the dynamic model, is much greater than the sampling frequency of light exposure and eddy lifetime. Therefore, this allows to filter out the high variations in light exposure, to finally represent only the pattern of fluctuations inferred by the macro-instabilities (i.e. large eddies) of the studied flow.

Fig. 9 .

 9 Fig. 9. Dynamic response (red dashed line) of an arbitrary cell to the captured light (blue dashed line) with acharacteristic time τ=0.3s using the dynamic model described by[START_REF] Gernigon | How Mixing and Light Heterogeneity Impact the Overall Growth Rate in Photobioreactors[END_REF].

Fig. 1

 1 Fig. 1 in section 2.1, for 10 concentrations between 0.3-1.5 (g/L) and for different liquid flow velocities in the tube U ¤ (see Table2) expressed in Reynolds number. Fig.10presents the results of

Fig. 10 .

 10 Fig. 10. Influence of turbulence on algae biological growth rate with constant lighting conditions at * < =1241(μ>¡¢. > žŸ . K žS ), diffuse coefficient c=0.4; response with full light integration (plain red line); instantaneous response (plain blue line); dynamic response with biological adaptation (dashed lines).

  values of Z as a function of concentration, and for several Reynolds values, were analyzed. The results in Fig.11. (a) highlights the independence of Z to variation of the biomass concentration whatever Reynolds number except at low concentrations where a very slight non-uniformity is observed. This exception could be explained by a better integration of light heterogeneity due to low spatial gradients for low biomass concentrations.

Fig. 11 .Fig. 11 .

 1111 Fig. 11. (b) shows Z values obtained from simulations of several incident light flux values (in the range of daily and seasonal irradiances) provided by PVGIS [55], for Toulouse, France. At a flow velocity of " ^=0.4 (m/s), Z is practically constant with an average value of Γ u 0.13, and similar to that shown in Fig. 11. (a) for the same ¥¦=20000. The small deviations from the mean observed especially for low irradiances may be explained by the contribution of the diffuse component of sunlight in the calculation of the light fields, which gives the PBR better integration of the light following an additional radial illumination and thus a better exposure of the microalgae to the penetrated light. Following these observations, in addition to the biomass concentration we can already free ourselves from second parameter which is the incident light intensity * < for the evaluation of this module Γ. The validation of the independency of * < (μ>¡¢. > žŸ . K žS ) in the gamma calculation based on literature is detailed in supplementary document (Appendix C)

Fig. 12 .

 12 Fig. 12. Γ vs Re for different diffuse sunlight fraction α. Round marks represent simulation results.

Fig. 14 .

 14 Fig. 14. Comparison between % & ' predicated by Eq. (21)and % & ' from numerical experiments (from the simulation approach)

Table 1 .

 1 Model inputs for the application case (spirulina culture in a tube in thin poly-methyl methacrylate

	PMMA)
	Parameter	Values				Reference
	System orientation	North-South			-
	Incidence light angles ( , ,	Variable*
	Contribution coefficient of the diffuse	Variable*			[55]
	irradiance ‹
	Global incident irradiance	Variable*			[55]
	Day of year	Variable*			-
		Variable*			-
	Mass scattering coefficient Es	640	(m²/kg	of	dry	[58]
		Spirulina)
	Mass absorption coefficient	162	(m²/kg	of	dry	[58]
		Spirulina)
	Phase function coefficient g	0.97				[67]
	Air refractive index OE	1
	Refractive index of the broth OE • Ž	1.34				[35]

*Variable value depends on location, day and time.

Table 2 .

 2 Details on the turbulent flow conditions in the pipe investigated the direct light characteristics vary over time, the range of possible values of Zenith and

	Flow Velocity		μt (friction
	" ^ (m/s)	Re (" ^.D/ν)	velocity)
	0.3	15000	0.0179
	0.4	20000	0.0231
	0.5	25000	0.028
	0.7	35000	0.0376
	1.68	84000	0.0791

Since

Table 2 )

 2 expressed in Reynolds number. This could be explained by the fact that the spatial gradient of light is very slight which provides to cells the same light exposure, therefore the responses of the microalgae become similar. However, as the biomass concentration increases, the curves become steeper and the overall growth rate diverges according to the behavior of the cells towards the captured light. A

	Fig. 10 presents the results of

'

(section 2.1). At low biomass concentrations the growth rates corresponding to the two asymptotic cases and the real case % & ' meet together, regardless of the applied flow velocity.

& ? =( ,5 &

& (4)Interestingly, the same quantity can also be expressed using the distribution of cell property at the reactor scale, 2 & (0, 4,

Abbreviations 40

Latin letters

Azimuth solar angle (°)

Tube radius (m) similar conclusion have been documented in the literature [START_REF] Molina | Tubular photobioreactor design for algal cultures[END_REF], [START_REF] Carlozzi | Productivity of Spirulina in a Strongly Curved Outdoor Tubular Photobioreactor[END_REF]: biomass productivity was improved by increasing the flow velocity. This phenomenon can be represented by a linear correlation; however, the acuity of this correlation should be validated for high flow velocities where possible cell damage could occur which implies an upper limit to the flow rate. A total collapse of the culture was also reported in the absence of mixing (low Re <11500) [START_REF] Molina | Tubular photobioreactor design for algal cultures[END_REF], which demonstrates the role of turbulence on the light perceived and used by cells.

Then, this procedure was carried out for four cases of irradiance fields which differs by the contribution of diffuse radiation in the supplied light, represented by the coefficient α. The values of α correspond to the range of the mean ratio of diffuse to global radiation given by PVGIS [START_REF] Cattaneo | Photovoltaic Geographical Information System (PVGIS)[END_REF]. Fig.

12. highlights the importance of the PBR lighting configuration on the light utilization rate. Indeed, for a given Re number, Γ values are different from one light field to another. Thus, as shown in Fig.

12. the higher the fraction of diffuse radiation, which corresponds to a radial irradiation (see section 3.1), the better is the transfer of light into the PBR volume and thus better is the light use by cells.

Indeed, in addition to the turbulence induced by the flow velocity of the fluid, the way in which light is brought to the PBR is of major importance in the orientation of the growth kinetics towards the two limit cases. From one of the two equations Eq.( 19) or Eq.( 20) it is possible to predict Γ without having to solve the coupled Monte Carlo light field -hydrodynamic -growth kinetics model. However, to avoid any confusion with the actual performance of the PBR, it is necessary to recall that Γ is an engineering module that describes the skill of the PBR to efficiently expose the cells to the various light fluctuations in its volume, or in other words, the efficiency of light utilization. Therefore, in order to predict the real global growth rate of the solar PBR, a prior estimation of the limit kinetics (instantaneous case and full integration case), relative to a given irradiance field, is required. Indeed, the actual performance is expressed, for each case, using Eq. [START_REF] Slegers | Design scenarios for flat panel photobioreactors[END_REF] or Eq. ( 20) and Eq. ( 1) determines the rate of light integration into growth kinetics. Indeed, under a given amount of radiative energy, the real overall PBR growth rate stands between two limit cases that define its extreme theoretical performance.

The normalized growth rate Γ translates the PBR's capacity in orienting its real performances towards these limit cases obtained for a given light energy. This concept defines the efficiency of light utilization/integration by biomass circulating in the PBR. Therefore, it depends on the operating conditions and the PBR design including the way the system is subjected to light energy and the quality of the turbulence.

For tubular photobioreactors exposed to the sun, the behavior of the sunlight, as well as the intensity of the turbulence induced by the flow rate (expressed by the Reynolds number), have a combined impact on the ability of the photobioreactor to expose its biomass to the energy received. In order to study the impact of each parameter, a sensitivity analysis was conducted. The results show that this rate of light integration by the PBR cells Γ is independent of biomass concentration variation however it depends entirely on turbulence (i.e., broth flow rate). With respect to light conditions, Γ is not influenced by the variation in intercepted light intensity, but is proportional to the fraction of the scattered radiation that illuminates the entire surface of the PBR.

Hence, a correlation for a fast calculation of Γ was proposed f(¥¦, c) and it highlights the effect of the PBR design and operation through the inclusion of macroscopic process variables: Reynolds (or implicitly the flow rate) and the light distribution quality at the reactor surface (diffuse light coefficient). Finally, this correlation was used to predict the real overall PBR growth rate, based on the two asymptotic growth kinetics, which were estimated directly by coupling the irradiance field with the photosynthetic growth model.

Formatting of funding sources

This work was funded by the project READYNOV INSPIRO, EU-FEDER (Fonds Europeen de Developpement Regional), Region Occitanie, ZFrance.


Léonel Philibert
email: leonel.philibert.1@ulaval.ca

David Jean Simon

+33761409336 Ayawavi

L'approche biomédicale et l'approche socioculturelle : deux perspectives aux enjeux distincts pour mieux appréhender la grossesse à l'adolescence Biomedical and sociocultural approaches: two perspectives with distinct

Keywords: Grossesse, adolescente, approche biomédicale, approche socioculturelle, homogène Adolescent, pregnancy, biomedical approach, socio-cultural approach, homogeneous

challenges to better understand teenage pregnancies

Introduction

Plus de 20 millions de filles âgées de 15 à 19 ans tombent enceintes chaque année dans le monde et environ 16 millions d'entre elles mettent au monde des enfants [START_REF] Rahmawati | Childbirth Self-Efficacy and Childbirth Expectation of Adolescent Mothers in Indonesia[END_REF][START_REF] Ayele | Determinants of teenage pregnancy in Degua Tembien District, Tigray, Northern Ethiopia: A community-based case-control study[END_REF][START_REF] Ronen | A Comparison of Childbirth Costs for Adolescents and Adults From 2001 to 2010[END_REF]. Même si le taux de fécondité des adolescentes a nettement diminué au niveau mondial, il demeure relativement élevé dans les pays en développement. En France, en Belgique et au Canada, cet indicateur est évalué à moins de 10 ‰. En revanche, dans des pays comme l'Ouganda, le Mali et le Niger, il est supérieur à 110 ‰ (World Population Prospects, 2021).

Bien que la grossesse/maternité des adolescentes soit classée comme un problème de santé publique majeur par l'Organisation mondiale de la santé (Jean [START_REF] Simon | E ̂tre me re a ̀ l'adolescence a ̀ Haı̈ti : un phe ńome ǹe persistant et multifactoriel[END_REF], sa compréhension peut varier d'une discipline à l'autre. En biomédecine, pour expliquer ce phénomène, les experts mettent très souvent en avant l'aspect accidentel (non désiré) tout en insistant sur les complications médicales, notamment les risques accrus de mortalités maternelles et infantiles (Le [START_REF] Van | La grossesse à l'adolescence : un acte socialement déviant?[END_REF]. La manière d'aborder ce phénomène en biomédecine ne fait pas l'unanimité au sein de la communauté scientifique. En effet, pour les chercheurs en sciences sociales, les grossesses ne peuvent être réduites à un « accident » ou perçues comme la conséquence d'un manque de connaissances sur les méthodes contraceptives (Le [START_REF] Van | Les Grossesses à l'adolescence normes sociales, réalités vécues[END_REF][START_REF] Mignot | La grossesse chez l'adolescente[END_REF]. Comme l'affirment Le [START_REF] Van | La grossesse à l'adolescence : un acte socialement déviant?[END_REF] et [START_REF] Delaunay | L'entrée en vie féconde : expression démographique des mutations socioéconomiques d'un milieu rural sénégalais[END_REF], elles sont aussi tributaires de facteurs sociaux, environnementaux et économiques. Ainsi, afin d'avoir une appréhension globale de ce phénomène, cet article se donne pour objectif de présenter les limites de l'approche biomédicale de la grossesse adolescente, mais aussi de montrer l'importance des facteurs socioculturels.

La construction de la grossesse à l'adolescence à l'aune de la biomédecine

Dans le domaine de la biomédecine, ce sont les experts (personnels soignants et autres) qui déterminent si un individu est en bonne santé ou pas. Pour parvenir à une telle conclusion, les critères biologiques sous-jacents sont importants pour cerner les différentes maladies [START_REF] Ankeny | Maladie et santé : concepts et représentations[END_REF]. La mesure (calculs mathématiques/statistiques) est un outil important pour déterminer le sens normal ou pathologique [START_REF] Sherlaw | La promotion de la santé : comprendre pour agir dans le monde francophone[END_REF]. Pour faire exister la santé ou la maladie, les chiffres, les images venant des examens médicaux apparaissent incontournables dans la conclusion de l'expert en biomédecine [START_REF] Sicard | La médecine sans le corps : quelques notes sur la relégation du corps[END_REF]. Ce faisant, le corps matériel devient le centre d'intérêt au détriment d'une autre pan de connaissance, à savoir le vécu expérientiel de l'individu (Gaille, 2007), tout comme les dimensions collectives, sociales ou existentielles de sa situation (Amiel-Lebigre & Gognalons-Nicolet, 1993).

Quand il s'agit de traiter la problématique de la grossesse à l'adolescence en biomédecine, deux éléments cruciaux émergent : le temps de la grossesse et le corps biologique de l'adolescente.

Pour ce qui est du premier, il sert à mesurer les risques encourus par la jeune fille et par le foetus afin de fournir des soins obstétricaux d'urgence. Quant au second, il ressort de la littérature que les jeunes filles enceintes présentent un risque élevé d'anémie, de faible poids et d'hypertension artérielle au cours de leur période de grossesse [START_REF] Alouini | Risk factors of teenage pregnancies, deliveries and post-partum in the department of Loiret[END_REF][START_REF] Wendland | Parentalité à l'adolescence : déterminants et conséquences[END_REF]. En sus, il a été observé que le taux de dépression est largement plus important chez les mères adolescentes que chez les mères plus âgées [START_REF] Noirhomme-Renard | La dépression chez les mères adolescentes : pour un repérage des situations de vulnérabilité[END_REF]. D'autres auteurs ont affirmé pour leur part, qu'il y a un risque élevé de retard de croissance intra-utérin, ce qui, par conséquent, favorise un taux élevé de naissances de faible poids chez les adolescentes [START_REF] Ngum | Early motherhood : a qualitative study exploring the experiences of African Australian teenage mothers in greater Melbourne, Australia[END_REF][START_REF] Wemaux-Denis | Grossesse et accouchement de l'adolescente : le pronostic obstétrical est-il plus défavorable entre 13 et 15 ans? Gynécologie Obstétrique[END_REF].

Outre qu'elle aborde la grossesse adolescente en considérant sa dimension pathologique [START_REF] Yakam | Santé reproductive des adolescents en Afrique : pour une approche globale[END_REF][START_REF] Van | La grossesse à l'adolescence : un acte socialement déviant?[END_REF], la biomédecine adopte une conception chronologique pour qualifier une grossesse de « grossesse adolescente » [START_REF] Côté | Brève généalogie de discours sur les mères adolescentes. Le cas de l'administration québécoise de la santé publique (note de recherche)[END_REF]. Pour les spécialistes de ce domaine, ce type de grossesse concerne toutes les femmes enceintes âgées de 10 à 19 ans [START_REF] Mchunu | Adolescent pregnancy and associated factors in South African youth[END_REF][START_REF] Self | Youth accessing reproductive health services in Malawi: drivers, barriers, and suggestions from the perspectives of youth and parents[END_REF]. De fait, toutes les jeunes filles ayant expérimenté une grossesse durant cette période sont vues comme étant plus exposées à plus de risques que les femmes plus âgées. L'adolescence devient alors un indicateur facilitant la quantification de la grossesse chez une population donnée et en même temps, elle permet d'établir la comparaison entre des catégories d'âge.

À la lecture de ce qui précède, nous pouvons déduire d'une part, que les adolescentes -quelle que soit leur appartenance sociale -représentent un groupe homogène en biomédecine, d'autre part que le temps de grossesse est indissociable à la notion de risque et que le corps de la jeune fille est essentialisé dans sa composante biologique [START_REF] Klein | La santé comme norme de soin[END_REF][START_REF] Blanc | Canguilhem et les normes[END_REF].

Il n'y a pas « une » mais « des » grossesses adolescentes

Si la grossesse adolescente a été longtemps imputée à un accident ou à un manque d'informations sur la planification familiale, aujourd'hui, il est admis et démontré que cette approche occulte les facteurs environnementaux, psycho-sociaux, économiques et institutionnels qui influencent les comportements sexuels et reproductifs des jeunes filles [START_REF] Kacou | Comportements sexuels à risque au temps du VIH/sida : le cas des jeunes en Côte d'Ivoire[END_REF][START_REF] Yakam | Santé reproductive des adolescents en Afrique : pour une approche globale[END_REF]Llomond, 2009 ;[START_REF] Dadoorian | Grossesses adolescentes[END_REF][START_REF] Emina | Situation résidentielle, scolarisation et mortalité des enfants selon la légitimité de leur naissance : Une analyse du Cameroun, de la Centrafrique et de la République Démocratique du Congo[END_REF].

Une démarche de vérification de l'intégrité du corps

La grossesse à l'adolescence peut correspondre à une démarche de vérification de l'intégrité du corps et des organes de la reproduction [START_REF] Marcelli | Grossesse et adolescence[END_REF]. Dans la société Ijo du delta du Niger (une ethnie), pour éprouver leur capacité procréative, les adolescentes cherchent à tomber enceintes de très tôt. Consciemment désirées, ces grossesses aideraient les jeunes filles à contracter un mariage [START_REF] Densen | Représentations sociales de l'adolescence. Une perspective interculturelles[END_REF]. Comme le souligne la démographe Thiriat Marie-Paule, elles (grossesses prémaritales) sont valorisées comme gage de la fécondité des jeunes filles. Les enfants appartiendront aux époux, qu'ils en soient ou non les géniteurs (Thiriat, 2000, p 84).

Une identité de femme, un statut d'adulte

Du fait que les adolescentes ne représentent pas un groupe homogène, la manière de représenter une grossesse à l'adolescence doit donc être équivoque. En occident, la grossesse adolescente est perçue comme un acte socialement « déviant » alors que dans les sociétés coutumières, elles sont tolérées (Le [START_REF] Van | La grossesse à l'adolescence : un acte socialement déviant?[END_REF]. Chez les Beti-Fang de la province du centre Cameroun, les jeunes filles procèdent à des tentatives visant à provoquer les premières règles dans le but de hâter l'entrée en vie sexuelle et féconde [START_REF] Kacou | Comportements sexuels à risque au temps du VIH/sida : le cas des jeunes en Côte d'Ivoire[END_REF]. Dans cette ethnie, la virginité prénuptiale n'a aucune importance. D'ailleurs, les parents -notamment les pères -encouragent les jeunes filles à enfanter avant de se marier [START_REF] Rwenge | Comportements sexuels parmi les adolescents et jeunes en Afrique subsaharienne Francophone et facteurs associe ś : original research article[END_REF].

L'enfantement peut être aussi perçu comme un passage à l'âge adulte. [START_REF] Yakam | Santé reproductive des adolescents en Afrique : pour une approche globale[END_REF]. Très souvent, les membres de la famille du conjoint l'incitent à se séparer de son épouse (féminisation de la stérilité), ce qui conduit à un changement de statut matrimonial. Dans un tel contexte -où la procréation revêt un caractère primordial dans la mesure où elle contribue non seulement à la stabilité du couple, mais aussi à la perpétuation du lignage ou du groupe [START_REF] Kouwonou | Fécondité et planification familiale au Sud-est Togo par une approche qualitative[END_REF][START_REF] Vignikin | Groupe internationale de partenaires population-santé. Les facteurs de la contraception au Togo au tournant du siècle : analyse des données de l'enquête démographique et de santé de[END_REF] -les jeunes filles pourraient être réfractaires à l'usage des méthodes contraceptives [START_REF] Beninguisse | Sexualité prémaritale et santé de la reproduction des adolescents et des jeunes en Afrique Subsaharienne[END_REF]; Le [START_REF] Van | La grossesse à l'adolescence : un acte socialement déviant?[END_REF]. Ce modèle culturel est particulièrement prégnant chez les Moba-Gurma de la région des Savanes et les Ewés des plateaux de Danyi au Togo, les Krou de Côte d'Ivoire, les Zandés-Nzakara de Centrafrique (Gastineau et Binet, 2006;[START_REF] Kacou | Comportements sexuels à risque au temps du VIH/sida : le cas des jeunes en Côte d'Ivoire[END_REF].

Le poids des mariages précoces

Il importe de mentionner que certains espaces sociaux traditionnels désapprouvent la sexualité et la fécondité prémaritales. Le mariage suivi de la sexualité et de la maternité constitue la trajectoire de référence pour l'accession aux statuts d' « adulte » et de « femme » [START_REF] Beninguisse | Sexualité prémaritale et santé de la reproduction des adolescents et des jeunes en Afrique Subsaharienne[END_REF]. Les jeunes filles ont pour obligation de rester vierges et de suivre des rituels afin de protéger leur virginité. Outre l'excision, le mariage précoce ferait partie des stratégies adoptées par les parents des adolescentes en vue de les empêcher de perdre leur virginité avant le mariage [START_REF] Yakam | Santé reproductive des adolescents en Afrique : pour une approche globale[END_REF]. D'après [START_REF] Delaunay | L'entrée en vie féconde : expression démographique des mutations socioéconomiques d'un milieu rural sénégalais[END_REF], ce contexte prohibitif de la sexualité et de la fécondité prénuptiale est parfois légitimé par les autorités religieuses. Ce rejet de la sexualité et de la maternité prémaritale prévaut encore chez des ethnies au Sénégal, au Congo, en Centrafrique et au Cameroun [START_REF] Kacou | Comportements sexuels à risque au temps du VIH/sida : le cas des jeunes en Côte d'Ivoire[END_REF]Adjamagbo et al. 2004;[START_REF] Diop | La dynamique de la fécondité des adolescentes au Sénégal[END_REF][START_REF] Ilinigumugabo | Causes et conséquences des grossesses chez les adolescentes au Cameroun[END_REF].

Une stratégie de survie

La pauvreté généralisée favorise le recours des jeunes filles à la sexualité comme stratégie de survie (Jean [START_REF] Simon | E ̂tre me re a ̀ l'adolescence a ̀ Haı̈ti : un phe ńome ǹe persistant et multifactoriel[END_REF][START_REF] Yakam | Santé reproductive des adolescents en Afrique : pour une approche globale[END_REF][START_REF] Beninguisse | Sexualité prémaritale et santé de la reproduction des adolescents et des jeunes en Afrique Subsaharienne[END_REF][START_REF] Vignikin | Groupe internationale de partenaires population-santé. Les facteurs de la contraception au Togo au tournant du siècle : analyse des données de l'enquête démographique et de santé de[END_REF][START_REF] Kaufman | Please:The Economies of Sex and Gifts among Adolescents in Urban South Africa[END_REF][START_REF] Nyanzi | The negotiation of sexual relationships among school pupils in South-Western Uganda[END_REF]. Très exposées aux aléas du marché du travail et au climat d'incertitude que provoque le chômage, pas mal d'adolescentes provenant des milieux défavorisés s'engagent dans des unions précoces et cherchent à satisfaire par tous les moyens (grossesse, maternité) les désirs de leur compagnon. Cette monétarisation des rapports sexuels s'effectue très souvent avec l'accord informulé (ou la pression) des parents [START_REF] Kacou | Comportements sexuels à risque au temps du VIH/sida : le cas des jeunes en Côte d'Ivoire[END_REF]. Force est de reconnaître qu'aux yeux de certaines adolescentes, les partenaires n'auraient pas le statut de « père » ou de « mari », si elles tombaient enceintes, mais plutôt celui d'un individu qui prendrait soin d'elles et de leurs enfants (Durand, 2005). Dans des zones rurales au Malawi (Ntcheu) et au Togo (Danyi, Gléï, Kpélé, Okou), par exemple, si le partenaire refuse de remplir leur devoir, l'enfant sera accueilli par la famille de la jeune fille, ce qu'on définirait par de l'entraide familiale [START_REF] Amuyunzu-Nyamongo | Qualitative Evidence on Adolescents Views of Sexual and Reproductive Health in sub-Saharan Africa[END_REF][START_REF] Clermont-Mathieu | Les tuteurs de résilience des enfants et adolescents vivant dans des camps à Port-au-Prince et dans les villes avoisinantes[END_REF]; URD/ATBEF, 2019).

Les inégalités de genre

Autres faits importants : la contraception est une responsabilité masculine dans de nombreuses sociétés traditionnelles et patriarcales. À Haïti et au Togo (pour ne citer que ceux-là), au moment des relations sexuelles, c'est à l'homme à qui il appartient de se prononcer sur le port du préservatif [START_REF] Simon | La violence subie par les adolescentes enceintes a ̀ Haı̈ti[END_REF][START_REF] Kohler | Copenhagen Consensus Center[END_REF][START_REF] Toudeka | C'est l'homme qui décide et la femme lui est soumise. » -contribution des écoles de maris pour une implication des hommes dans l'utilisation de la planification familiale (PF) au sein des couples au Togo[END_REF]. Ce qui interpelle c'est que la pratique contraceptive féminine est liée à du « vagabondage sexuel » et pas mal de jeunes filles dans ces milieux partagent cette idée (Jean [START_REF] Simon | La violence subie par les adolescentes enceintes a ̀ Haı̈ti[END_REF][START_REF] Mathieu | La grossesse survenue à l'adolescence chez les jeunes femmes vivant dans un camp d'hébergement suite au séisme du 12 janvier 2010 en Haït[END_REF].

Les femmes (jeunes ou pas) sont parfois violentées par leur partenaire lorsqu'elles exigent le port du préservatif. Selon Amuuyunzu-Nyamongo et al. ( 2005), en Ouganda, les hommes peuvent devenir suspicieux envers les femmes lorsque celles-ci veulent négocier ou recourir à un préservatif. Ce faisant, elles peuvent être frappées (violences physiques) et victimes d'injures (violences verbales et psychologiques) [START_REF] Yakam | Santé reproductive des adolescents en Afrique : pour une approche globale[END_REF][START_REF] Singer | Syndemics, sex and the city: Understanding sexually transmitted diseases in social and cultural context[END_REF][START_REF] Kouwonou | Pour une évaluation du programme de planification familiale au Togo[END_REF].

La contraception : un vice pour l'église chrétienne

Pour l'église chrétienne, l'acte du mariage est, « par sa nature même, destiné à la génération des enfants ». Toute entrave à la conception est synonyme de violation de la loi divine (Greethel, 2016). Considérant les relations extra-conjugales comme immorales, le christianisme prône un comportement sexuel « responsable ». On enseigne aux fidèles chrétiens qu'avant de débuter leur vie sexuelle, ils doivent être conscients de leurs ressources financières, de leur maturité corporelle et de l'immensité des devoirs qui se cachent derrière l'acte sexuel [START_REF] Simon | E ̂tre me re a ̀ l'adolescence a ̀ Haı̈ti : un phe ńome ǹe persistant et multifactoriel[END_REF]. Dans des assemblées protestantes à Haïti, on enseigne même aux fidèles que les préservatifs et les pilules sont des objets anticonceptionnels et démoniaques et que s'ils font usage de ces produits, ils tomberont malades ou deviendront stériles (Jean Simon, 2020).

Mythe de la « première fois », contrat de confiance

En dépit des nombreuses campagnes de sensibilisation menées par l'OMS sur le bénéfice de la planification familiale à travers le monde, il existe encore des adolescent-e-s qui croient au « mythe de la première fois ». Dans leur imagination, le premier rapport sexuel par sa spécificité et son caractère initiatique n'est pas fécondant [START_REF] Mathieu | La grossesse survenue à l'adolescence chez les jeunes femmes vivant dans un camp d'hébergement suite au séisme du 12 janvier 2010 en Haït[END_REF]. D'autres adolescentes, ellesmêmes, pour prouver leur sincérité, leur fidélité et leur loyauté, sont prêtes à jouer le jeu du sacrifice amoureux en proposant à leur partenaire du sexe sans préservatif qui est par sousentendu « le contrat de confiance » et « une preuve d'amour inconditionnel » [START_REF] Kohler | Copenhagen Consensus Center[END_REF].

La grossesse à l'adolescence : une limite des réponses institutionnelles au phénomène

La maternité adolescente est également liée à des facteurs institutionnel et organisationnel. Des études réalisées un peu partout dans le monde (Amérique, Afrique, Caraïbes) révèlent que l'on rencontre fréquemment des personnels de santé aux attitudes répressives dans les centres de planification familiale (Jean [START_REF] Simon | La violence subie par les adolescentes enceintes a ̀ Haı̈ti[END_REF][START_REF] Yakam | Santé reproductive des adolescents en Afrique : pour une approche globale[END_REF][START_REF] Olukoya | Avortement à risque chez les adolescentes[END_REF]. Jouant le rôle de moraliste, ils prédéfinissent des critères (âge, présence des parents pour être reçu, etc.) -à l'insu du Directeur de service -contrevenant au règlement de l'institution pour démotiver les adolescentes à ne plus fréquenter l'établissement [START_REF] Aujoulat | Les grossesses chez les adolescentes. Analyse des facteurs intervenant dans la survenue et leur issue. Synthèse des résultats et recommandation pour la prévention, l'éducation et l'accompagnement[END_REF]. De même, à certains centres de planification familiale, aucun traitement particulier n'est réservé aux adolescentes. Elles sont reçues au même titre qu'une femme de 30 ans ou de 40 ans et se placent comme « tout le monde » dans les files d'attente [START_REF] Kacou | Comportements sexuels à risque au temps du VIH/sida : le cas des jeunes en Côte d'Ivoire[END_REF]. Très clairement, à ce compte-là, elles pourraient se décourager et arrêter la prise de la contraception.

Quel que soit la méthode contraceptive (surtout hormonale), en plus de son côté protecteur, elles ont toutes des effets indésirables (nausées, douleurs, saignements, etc.). Et, si au bout de 2-3 mois, ils ne se sont pas estompés, il est recommandé à la patiente de prendre un rendez-vous avec son médecin traitant pour discuter des solutions envisageables (UNFPA, 2013). Pourtant, dans certaines structures de planification familiale du Togo, quand une adolescente vient s'informer sur les types de méthodes contraceptives, certains prestataires insistent plus sur les effets secondaires que sur ses bénéfices [START_REF] Vignikin | Groupe internationale de partenaires population-santé. Les facteurs de la contraception au Togo au tournant du siècle : analyse des données de l'enquête démographique et de santé de[END_REF]. Donc, par peur de perturber leur santé et par volonté de ne pas décevoir leurs parents, les adolescentes choisiraient d'encourir des risques de grossesses ou d'infections sexuellement transmissibles [START_REF] Vignikin | Groupe internationale de partenaires population-santé. Les facteurs de la contraception au Togo au tournant du siècle : analyse des données de l'enquête démographique et de santé de[END_REF].

La grossesse adolescente : une vulnérabilité psychosociale

La grossesse peut prendre « le sens d'une rupture avec la famille, le géniteur, l'enfance de l'adolescente et parfois même comme une rupture et une attaque du corps propre » (Marcelli et Alvin, 2000, p 37). Suite à des problèmes familiaux, il y a des adolescentes qui cherchent à tout prix à tomber enceintes. Ces grossesses peuvent être des quêtes pour combler certains manques (manque d'affection de la part des parents…). L'enfant constituerait le centre de gravité d'une existence nouvelle (Le [START_REF] Van | La grossesse à l'adolescence : un acte socialement déviant?[END_REF].

Dans la même veine, une grossesse adolescente peut être attribuée à une « destruction de soi » (Marcelli et Alvin 2001). Après avoir été victimes de viols, il y a des adolescentes qui mettent en danger leur corps. Voulant détruire ce dernier, elles adoptent des comportements sexuels à risques (Le [START_REF] Van | La grossesse à l'adolescence : un acte socialement déviant?[END_REF][START_REF] Yakam | Santé reproductive des adolescents en Afrique : pour une approche globale[END_REF].

Regard féministe de la grossesse à l'adolescence

Le courant féministe égalitariste voit la grossesse adolescente comme le fruit de l'influence des déterminismes sociaux qui restent hors du contrôle des jeunes filles (Charbonneau, 2002). Sous ce prisme, l'analyse de la problématique renvoie à la conception tridimensionnelle de la justice sociale [START_REF] Fraser | Justice sociale, redistribution et reconnaissance[END_REF][START_REF] Fraser | Qu'est-ce que la justice sociale? : reconnaissance et redistribution[END_REF][START_REF] Fraser | Marchandisation, protection sociale et émancipation[END_REF]. Cette approche permet d'analyser comment les injustices se renforcent mutuellement, dans la mesure où les normes culturelles sont institutionnalisées dans l'État et dans la sphère économique, et où « le handicap économique » des femmes restreint leurs possibilités à faire écouter leur voix, « ce qui entrave l'égale participation à la fabrication de la culture, dans la sphère publique et dans la vie quotidienne » (Fraser, 2005, p. 27).

Les chercheurs féministes proposent également des pistes de réflexion pour mieux répondre à cette problématique. En effet, ils suggèrent que les stratégies de prévention devraient viser les problèmes d'inégalités des conditions de vie et de pauvreté des adolescentes (Charbonneau, 2002). De surcroît, ils invitent les professionnels de santé à être davantage sensibles à l'expérience subjective de ces adolescentes et au regard qu'elles portent sur leur sexualité [START_REF] Manseau | Discours sur la grossesse et la sexualité : regards et action féministes pour l'autonomie sexuelle et reproductrice des adolescentes[END_REF].

Ainsi, comme nous pouvons le constater, les comportements sexuels et procréateurs des adolescentes résulteraient d'un processus social, culturel, économique et politique, propre à un contexte sociohistorique donné (Le [START_REF] Van | Les Grossesses à l'adolescence normes sociales, réalités vécues[END_REF].

Conclusion

La construction faite par la biomédecine de la problématique de la grossesse à l'adolescence n'est pas réellement capable de témoigner de la complexité qui lui est inhérente. Ne voir dans la grossesse des adolescentes que le résultat d'un accident ou d'un manque d'informations sur la planification familiale, comme le précisent [START_REF] Marcelli | Grossesse et adolescence[END_REF], c'est réduire singulièrement la signification d'un tel événement. Dans ce présent article, il a été bien montré que ce phénomène ne peut être traité sans une considération des réalités socioculturelles. Loin de diverger, les approches biomédicale et socioculturelle se complètent. Ainsi, pour mieux cerner la grossesse à l'adolescence dans le monde, un phénomène social complexe, il serait intéressant de privilégier à la fois les deux approches.

  , les familles nombreuses sont très valorisées, la stérilité étant socialement stigmatisée. Les femmes stériles qui vivent en couple sont marginalisées et étiquetées de « femmes maudites » (Tantchou

Dans d'autres groupes culturels de type patriarcal et fortement hiérarchisés, les adolescentes n'ont pas de statut socialement reconnu. Pour se faire valoriser, accéder aux statuts d'« adulte » et de « femme », et avoir le droit d'exploitation des terres, elles doivent prouver leur fertilité. Il convient de noter que dans ces sociétés