Jean-François Cornet

Claude-Gilles Dussap

A Simple and Reliable Formula for Assessment of Maximum Volumetric Productivities in Photobioreactors

This paper establishes and discusses the consistency and the range of applicability of a simple, but general and predictive analytical formula, enabling to easily assess the maximum volumetric biomass growth rates (the productivities) in several kinds of photobioreactors with more or less 15% of deviation. Experimental validations are performed on photobioreactors of very different conceptions and designs, cultivating the cyanobacterium Arthrospira platensis, on a wide range of volumes and hemispherical incident light fluxes. The practical usefulness of the proposed formula is demonstrated by the fact that it appears completely independent of the characteristics of the material phase (as the type of reactor, the kind of mixing, the biomass concentration…), according to the first principle of thermodynamics and to the Gauss-Ostrogradsky theorem. Its ability to give the maximum (only) kinetic performance of photobioreactors cultivating many different photoautotrophic strains (cyanobacteria, green algae, eukaryotic microalgae) is theoretically discussed but experimental results are reported to a future work of the authors or to any other contribution arising from the scientific community working in the field of photobioreactor engineering and potentially interested by this approach.

1-Introduction and Objectives

Modeling, design and scale-up of photobioreactors (PBR) is a research field of fast increasing importance because the consciousness that photosynthesis will play a crucial role in the future, both in the mitigation of the CO 2 emissions and in the sustainable and renewable synthesis of chemicals (biofuels, raw molecules for chemistry or high valuable products), is progressively imposed to the population all around the world [START_REF] Banerjee | Botryococcus braunii: a renewable source of hydrocarbons and other chemicals[END_REF][START_REF] Spolaore | Commercial applications of microalgae[END_REF][START_REF] Chisti | Biodiesel from microalgae[END_REF]. In this prospect, the cultivation of photosynthetic microorganisms in PBR enabling higher productivities, a rigorous control of minerals and an optimized management of water fluxes is the only way to avoid an unacceptable competition with plant crop cultivation for food.

Because of the specific requirement of light (sunlight or artificial light) as a main energy source for the PBR functioning, a great variety of concepts exists in the literature [START_REF] Pultz | Photobioreactors: production systems for phototrophic microorganisms[END_REF][START_REF] Carvalho | Microalgal reactors: a review of enclosed system designs and performances[END_REF], even if, from a chemical engineering point of view, they can just be subdivided in completely stirred tank reactors (CSTR) and plug flow tubular reactors (PFTR). The former are generally used for applications with artificial light and present a strong advantage to operate with a spatial homogeneous biomass concentration, enabling a fine control of the field of radiation inside the reactor. At the opposite, PFTR are often preferred for sunlight applications as they seem more adapted to large reactors designed for outdoor cultivation, except if the recent concept of internally lighting PBR is envisaged [START_REF] Ogbonna | An integrated solar and artificial light system for illumination of photobioreactors[END_REF][START_REF] Cornet | Procédés limités par le transfert de rayonnement en milieu hétérogène. Etude des couplages cinétiques et énergétiques dans les photobioréacteurs par une approche thermodynamique[END_REF][START_REF] Zijffers | Design process of an area-efficient photobioreactor[END_REF].

Whatever the variety of existing concepts and their intrinsic complexity, engineering purposes require to establish simple but predictive formula for the assessment of kinetic and energetic performances of PBR, mainly useful for design and scale-up operations, or for comparison of different systems between them, on a Cartesian basis, at the opposite of a discourse on the famous effect of the "biological diversity". Moreover, the idea that more complicated is the theoretical demonstration of a result, less is its practical usefulness, or at the opposite, that only simple demonstrations and formula are of any practical usefulness for potential applications, is known as the Hartmanis' principle and has been partly 4 demonstrated recently using the Kolmogorov complexity theory [START_REF] Kreinovich | How important is theory for practical problems? A partial explanation of Hartmanis' observation[END_REF]. The aim of the present paper is to demonstrate that, even if a complete PBR understanding and modeling remains rigorously a very complex problem with numerous intricate levels of description (see hereafter), it is however possible, according to the Hartmanis' principle, to easily establish a simple but general and reliable formula for the assessment of the maximum volumetric biomass productivities. The fundamental interest of such a predictive analytical formula, if it can be established, is that it gives always definitive and general responses and solutions whereas complex and full numerical simulations only depict in more details a particular situation. As stated by the Hartmanis' principle, the authors hope that the simple and easy-touse relation proposed in this paper will be of great practical interest to clarify and drive the future developments in the field of PBR engineering at two levels. First, the rigorous demonstration itself gives enlightenments on the key variables enabling to properly formulate the coupling between light transfer and kinetic rates in PBR, clarifying by the way many important points which often lead to misinterpretations in the specialized literature. Second, the final concise formula establishes the respective importance of each main variable in the assessment of kinetic performances of PBR with associated lumped parameters, which is of considerable interest in the perspective of designing new efficient PBR for future applications.

Clearly, it is not envisaged in this paper to take under consideration all the factors influencing the biomass growth rate in a PBR [START_REF] Carvalho | Microalgal reactors: a review of enclosed system designs and performances[END_REF]. Stating that the main limiting engineering factor is the light distribution and availability in the reactor [START_REF] Aiba | Growth kinetics of photosynthetic microorganisms[END_REF][START_REF] Cornet | A structured model for simulations of cultures of the cyanobacterium Spirulina platensis in photobioreactors -I. Coupling between light transfer and growth kinetics[END_REF][START_REF] Cornet | Kinetics and energetics of photosynthetic micro-organisms in photobioreactors. Application to Spirulina growth[END_REF][START_REF] Cassano | Photoreactor analysis and design: fundamentals and applications[END_REF][START_REF] Csogör | Light distribution in a novel photobioreactor -Modelling for optimization[END_REF][START_REF] Yun | Kinetic modelling of the light-dependent photosynthetic activity of the green microalga Chlorella vulgaris[END_REF][START_REF] Pruvost | Hydrodynamics influence on light conversion in photobioreactors: An energetically consistent analysis[END_REF], only the effect of the radiant light transfer on modeling and its consequences on the mean biomass volumetric productivities observed in PBR will be investigated in the following. All the other important parameters (culture medium, temperature, pH, dissolved gases…) will be assumed to be controlled in optimal conditions along this work. The useful and reliable formula will be first established in the framework of photoautotrophic CSTR photobioreactors, but a possible extension of its domain of validity in calculating maximum kinetic performances for PFTR photobioreactors will be then discussed. In all cases, a special attention will be paid on the energetic consistency of the obtained theoretical result, as a proof of the global coherence of the proposed approach. The experimental validations of the final formula will be envisaged on eight artificially lightened PBR of very different conceptions and designs with volumes varying on three order of magnitude and hemispherical incident photon flux densities (PFD) varying on two orders of magnitude. At this stage however, only one model of photoautotrophic microorganism, the cyanobacterium Arthrospira platensis, will be used to prove the validity of the theoretical approach developed in the paper.

2-Materials and Methods

For all the experiments, the microorganism used was Arthrospira (Spirulina) platensis PCC 8005 (Institut Pasteur, Paris, France). It was grown axenically in Zarrouk medium modified by Cornet (17) or by Cogne (18) for trace elements, depending of the experiment (but always using nitrate as nitrogen source). The pH (between 8 and 10) and the temperature (35-36°C) were controlled at their optimal values for growth. Results for eight different kinds of artificially lightened PBR (CSTR) were reported in this study with working volumes varying between 0.1 and 77 L and incident hemispherical photon fluxes ranging between 30 and 1600 µmol.m -2 .s -1 (PAR, using multi-points measurements with a LI-COR cosine quantum sensor LI-190SA and confirmed by actinometry in case of PBR marked with a star in the following). The main parameters influencing the biomass productivities (geometry, characteristics of the lighting system) and the culture conditions (mixing, pH and temperature control) are depicted hereafter for each kind of PBR.

-PBR 1: rectangular PBR of 4 L working volume illuminated by one side with 5 fluorescent tubes (Philips, white industry, 20W in the spectral range [380-750 nm]) and agitated by bubbling air-CO 2 6 (2%) at a flow rate of 0.02 vvm (enabling to control the pH) plus magnetic stirring. Temperature was controlled at the level of the room (culture chamber). The path length of the reactor was 8 cm giving a specific illuminated area a light of 12.5 m -1 without dark fraction (f d = 0).

-PBR 2: cylindrical stirred tank reactor (Applikon) of 5 L working volume illuminated by one side with two halogen projectors (Philips, 500 W in the spectral range [350-1100 nm]) and agitated by two rushton turbines. Temperature was controlled with an internal exchanger fed with an external thermal bath and pH was regulated using 1 mol/L H 2 SO 4 . The diameter of the reactor was 16 cm giving a specific illuminated area a light of 12.5 m -1 without dark fraction (f d = 0). Special attention must be paid in this case that the mean incident hemispherical PFD appearing in Table 2 is given in front of the reactor and must be cosine averaged on half of the cylinder (divided by 2) in order to use in the final eq. ( 22). -PBR 7: cylindrical reactor with a centrifugation field generated by 4 paddles (300 rpm) and radially illuminated by 44 fluorescent white tubes (Philips, 18 W, in the spectral range [380-750 nm]). The total volume of the reactor was 13.2 L (total diameter of 22 cm) but working volume was only 6 L (biofilm thickness of 3 cm at the wall). Temperature was regulated using fans and convective air in order to remove the heat of the lamps and pH was controlled with H 2 SO 4 , 1 mol/L. The corresponding specific illuminated area a light was 40 m -1 without dark volume fraction f d = 0.

-PBR 8*: rectangular PBR of 0.5 L working volume illuminated by one side with a panel of 180 red LED (Agilent, 38mW -16000 mcd -8°, in the spectral range [635-655 nm]) and agitated by bubbling air-CO 2 at a flow rate of 0.02 vvm (with a regulated CO 2 mole fraction enabling to control the pH) plus magnetic stirring. Temperature was controlled using a thermostatic bath and external heat exchanger.

The path length of the reactor was 4 cm giving a specific illuminated area a light of 25 m -1 without dark fraction (f d = 0). The dry biomass concentration (DM) was systematically determined by weighting filtered samples (Millipore, 0.45 µm) preliminary dried at 110°C during 24 hours. The dry mass productivities were determined according to the material balance for biomass (eq. 24), i.e. measuring the output flow rate and (light-limited) biomass concentration over at least six stable residence times for continuous cultures, or determining the slope of the (light-limited) biomass concentration time course for batch cultures.

3-Theoretical Considerations

3.1-The Complexity of a Predictive Knowledge Model for PBR

As already explained by the authors [START_REF] Cornet | Modeling stability of photoheterotrophic continuous cultures in photobioreactors[END_REF], a general thorough understanding and knowledge modeling of PBR is a very difficult task requiring to manage a seven dimensional Euclidean space in which the relaxation times of the involved kinetic phenomena overlap twelve order of magnitude. This relies first on a fine description of the spectral radiant light transfer inside the reactor for any geometry and boundaries conditions, because in the general case, formulating the local kinetic coupling requires to accurately determine the local radiant power density, considering the PBR as a scattering and participative medium. This work is only rigorously feasible using the spectral (Boltzmann-type) quasi steady-state radiative transfer equation (RTE) inside the PBR, leading to solve [START_REF] Cassano | Photoreactor analysis and design: fundamentals and applications[END_REF][START_REF] Cornet | Conversion of radiant light energy in photobioreactors[END_REF]:

( ) ( ) ( ) Ω d p I C Es I C Es Ea I X X ′ ′ + + - = ⋅ ∇  Ω Ω Ω , 4 4π λ λ λ λ λ λ λ π (1)
The spectral absorption Ea λ and scattering Es λ coefficients with the phase function for scattering ( )

Ω Ω ′ , λ p (i.
e. the so-called radiative properties for the medium) appearing in eq. ( 1) can be obtained using sophisticated experimental methods [START_REF] Berberoglu | Experimental measurements of the radiation characteristics of Anabaena variabilis ATCC 29413-U and Rhodobacter sphaeroides ATCC 49419[END_REF], but recently, the authors have proposed an entirely predictive methodology, using the generalized Lorenz-Mie theory and requiring only to know some basic information as a model of shape, a size distribution and the pigment content for the microorganism [START_REF] Cornet | Procédés limités par le transfert de rayonnement en milieu hétérogène. Etude des couplages cinétiques et énergétiques dans les photobioréacteurs par une approche thermodynamique[END_REF][START_REF] Pottier | A fully predictive model for one-dimensional light attenuation by Chlamydomonas reinhardtii in a torus photobioreactor[END_REF][START_REF] Farges | Spectral kinetic modeling and long-term behavior assessment of Arthrospira platensis growth in photobioreactor under red (620 nm) light illumination[END_REF]. Finally, eq. ( 1) must be solved for a given geometry of PBR and boundary illuminating conditions, using one-dimensional methods in simple cases like modulated two-flux and discrete ordinate methods [START_REF] Cornet | Procédés limités par le transfert de rayonnement en milieu hétérogène. Etude des couplages cinétiques et énergétiques dans les photobioréacteurs par une approche thermodynamique[END_REF], or more sophisticated Monte Carlo [START_REF] Cassano | Photoreactor analysis and design: fundamentals and applications[END_REF][START_REF] Csogör | Light distribution in a novel photobioreactor -Modelling for optimization[END_REF] and finite element methods [START_REF] Cornet | Conversion of radiant light energy in photobioreactors[END_REF] in the general case. The spectral field of irradiance G λ is then known everywhere inside the reactor as a preliminary stage in formulating the kinetic coupling, by averaging over all the wavelength in the PAR both the field of irradiance

  = λ π λ Ω λ d d I G
  = λ π λ λ Ω λ d d I C Ea X 4 A .
The former quantity enables to define the working illuminated fraction V V ℓ = γ inside the reactor (from the localization of the compensation point for photosynthesis in relation to the irradiance G C ), corresponding to a volume in which photosynthetic growth occurs, and then to write the mean volumetric growth rate in the completely stirred PBR as a result of two different averaged metabolic zones [START_REF] Cornet | A structured model for simulations of cultures of the cyanobacterium Spirulina platensis in photobioreactors -I. Coupling between light transfer and growth kinetics[END_REF][START_REF] Cornet | Kinetics and energetics of photosynthetic micro-organisms in photobioreactors. Application to Spirulina growth[END_REF][START_REF] Cornet | A simplified monodimensional approach for modelling coupling between radiant light transfer and growth kinetics in photobioreactors[END_REF]:

dV r V dV r V r d V d X d V X X   - + = > < , , 1 ) 1 ( 1 γ γ ℓ ℓ ℓ (2)
For cyanobacteria, and specially for A. platensis, having common electron carriers chains for photosynthesis and respiration, it is well established [START_REF] Cornet | Procédés limités par le transfert de rayonnement en milieu hétérogène. Etude des couplages cinétiques et énergétiques dans les photobioréacteurs par une approche thermodynamique[END_REF] that a first analysis of the relaxation time necessary to switch the metabolism from photosynthesis to respiration leads to consider in any case that 0 , = d X r in the dark volume V d , even in partially lightened PBR with low values of the γ fraction. As a consequence, the maximum and constant volumetric growth rate for cyanobacteria is reached as soon as the γ fraction is lower or equaled to 1, corresponding to the well-known physical limitation by light transfer in PBR (all the incident photons are absorbed and used in the vessel). Evidently, as it will be discussed later in the paper, this assumption is not true for eukaryotic microorganisms as microalgae for which photosynthesis and respiration operate separately in chloroplasts and mitochondria and nor for photoheterotrophic bacteria [START_REF] Cornet | Modeling stability of photoheterotrophic continuous cultures in photobioreactors[END_REF].

10

The latter quantity A, enables to formulate the local kinetic coupling in the illuminated volume, leading to the final knowledge expression for ℓ , X r . It is well-known indeed that this coupling rigorously relies on the local radiant light power density A (13), defining properly the energetic yield ρ and the mass quantum yield φ of conversion, as already proposed by the authors [START_REF] Cornet | Modeling stability of photoheterotrophic continuous cultures in photobioreactors[END_REF][START_REF] Cornet | Simulation, design and model based predictive control of photobioreactors[END_REF]:

G C Ea r X X φ ρ φ ρ = = A ℓ , (3)
As recently explained [START_REF] Cornet | Procédés limités par le transfert de rayonnement en milieu hétérogène. Etude des couplages cinétiques et énergétiques dans les photobioréacteurs par une approche thermodynamique[END_REF][START_REF] Farges | Spectral kinetic modeling and long-term behavior assessment of Arthrospira platensis growth in photobioreactor under red (620 nm) light illumination[END_REF], this partition is a consequence of first principle analyses of the excitation energy transfer in the photosynthetic antenna toward the reaction center and then of the socalled Z-scheme for photosynthesis leading to the synthesis of ATP and reducing power NADPH 2 . The energetic yield ρ represents the dissipative part of the photonic energy absorbed by the antenna. This local value decreases with the local irradiance from a convenient relationship obtained on the basis of an approximation of a theoretical quantum physics study of the excitation transfer mechanisms in antenna [START_REF] Bejan | Unification of three different theories concerning the ideal conversion of enclosed radiation[END_REF]:

G K K K G M M + = + ≅ ρ ρ ρ 1 1 (4)
in which ρ M is the maximum value of the yield, obtained when the system operates in the thermodynamic optimal conditions, at the irradiance of compensation G C (the irradiance G becomes negligible in regard to the half-saturation constant K). It must be pointed out at this stage that, even if eq. ( 4) remains a representation model, it gives some theoretical background to the well-known but sometimes discussed hyperbolic behavior of the photosynthesis in regard to the local irradiance.

At the opposite, the mass quantum yield φ is a mean value in time (as indicated by the bar), at a time scale corresponding to the pseudo-steady state hypothesis for the products of the Z-scheme for classical photosynthesis like ATP and NADPH 2 (one to some minutes). It corresponds to the conservative part of the photonic energy absorbed in the reactor (the part of the photonic processes leading to a charge separation at the reaction center), and has been demonstrated to be consequently a mean value in space [START_REF] Cornet | Procédés limités par le transfert de rayonnement en milieu hétérogène. Etude des couplages cinétiques et énergétiques dans les photobioréacteurs par une approche thermodynamique[END_REF] for the general case of well-mixed photobioreactors. It can then be easily deduced from the true mole quantum yield φ ′ by just using the mean C-molar mass of the produced biomass in the reactor M X :

φ φ ′ = X M (5)
As discussed hereafter, this true mole quantum yield φ ′ may be obtained by a stoichiometric analysis of the Z-scheme, taking into account the cyclic photophosphorylation from the well-known P/2e -ratio [START_REF] Cornet | Kinetics and energetics of photosynthetic micro-organisms in photobioreactors. Application to Spirulina growth[END_REF][START_REF] Farges | Spectral kinetic modeling and long-term behavior assessment of Arthrospira platensis growth in photobioreactor under red (620 nm) light illumination[END_REF], completing then the proposed knowledge model for the kinetic coupling, using finally from eqs. [START_REF] Chisti | Biodiesel from microalgae[END_REF][START_REF] Pultz | Photobioreactors: production systems for phototrophic microorganisms[END_REF][START_REF] Carvalho | Microalgal reactors: a review of enclosed system designs and performances[END_REF] the local equation:

G K G C Ea K r X M X + = φ ρ ℓ , (6)

3.2-Reification for the Kinetic Parameters Calculation

Clearly, the proposed approach for the local kinetic coupling is completely predictive, only if all the parameters and variables involved in eq. ( 6) can be determined a priori by a theoretical analysis. As already explained above with the help of the relevant literature, it is the case for all the data involved in the physical phenomenon of radiant light transfer in scattering and participating dense media, as the spectral or mean field of irradiance G, and the mean absorption coefficient for the biomass Ea.

Among the three other kinetic parameters requiring a reification ρ M , φ and K, the maximum energetic yield ρ M will be first discussed in the following. It corresponds at the maximum thermodynamic efficiency, or the minimum losses, for the photonic conversion and excitation transfers in the photosynthetic units or photosynthetic domains (26) towards the reaction centers, leading finally to a charge separation process. Even if the maximum efficiency for radiant energy conversion processes has been strongly debated in the past and clarified by Bejan [START_REF] Bejan | Unification of three different theories concerning the ideal conversion of enclosed radiation[END_REF][START_REF] Bejan | Advanced Engineering thermodynamics[END_REF], we can retain here the most simple 12 approach proposed by Jeter and used for photosynthesis from a long time [START_REF] Duysens | [END_REF]. These authors just consider an extension of the Carnot formula for the maximum radiation conversion [START_REF] Bejan | Unification of three different theories concerning the ideal conversion of enclosed radiation[END_REF]:

R M T T - = 1 ρ (7)
in which the temperature for the radiation is given by the ideal black body formula of Planck, considering for the spectral intensity

λ , C I
in thermodynamic optimal conditions, the value at the compensation point for the mean considered wavelength λ in the reactor:

        + = 5 , 2 2 c h 2 1 ln k c h λ λ λ C m m R I n n T ( 8
)
It is obvious that The true mole quantum yield φ ′ , enabling to easily derive the mass quantum yield φ (eq. 5) can be also determined from a stoichiometric analysis of the Z-scheme for photosynthesis [START_REF] Farges | Spectral kinetic modeling and long-term behavior assessment of Arthrospira platensis growth in photobioreactor under red (620 nm) light illumination[END_REF]. This requires establishing a structured stoichiometry for a given photoautotrophic microorganism [START_REF] Roels | Energetics and kinetics in biotechnology[END_REF] as it has been done for A. platensis for reasonably low hemispherical incident photon fluxes [START_REF] Cornet | Kinetics and energetics of photosynthetic micro-organisms in photobioreactors. Application to Spirulina growth[END_REF] [START_REF] Aiba | Growth kinetics of photosynthetic microorganisms[END_REF] eliminating by summation the cofactors and the ATP syntheses and demonstrating that, if the Nsource for photosynthesis was nitrate, the resulting photosynthetic quotient

+   →  + +   →  + + + ATP COF J J
Q P ( 2 2 CO O P J J Q =
) is roughly 1.4. Obviously, eq. ( 10) provides the mean value of the P/2e -ratio for the considered conditions:

275 . 1 e 2 / = = - COF ATP J J P (11)
Finally, the calculation of the true mole quantum yield for the Z-scheme, corresponding to the conservative part of the photonic energy conversion, and only P/2e -dependent is straightforward (from eqs. 9-11): [START_REF] Cornet | Kinetics and energetics of photosynthetic micro-organisms in photobioreactors. Application to Spirulina growth[END_REF] Of course, this value seems linked to the macromolecular composition of A. platensis via eq. ( 9) and restricted to the conditions of low incident PFD (eqs. [START_REF] Aiba | Growth kinetics of photosynthetic microorganisms[END_REF][START_REF] Cornet | A structured model for simulations of cultures of the cyanobacterium Spirulina platensis in photobioreactors -I. Coupling between light transfer and growth kinetics[END_REF], because it has been demonstrated that the P/2e -ratio varied significantly with the hemispherical incident photon flux [START_REF] Cornet | Procédés limités par le transfert de rayonnement en milieu hétérogène. Etude des couplages cinétiques et énergétiques dans les photobioréacteurs par une approche thermodynamique[END_REF][START_REF] Cornet | Kinetics and energetics of photosynthetic micro-organisms in photobioreactors. Application to Spirulina growth[END_REF]. Nevertheless, from a detailed theoretical analysis of the Z-scheme functioning, using the linear thermodynamics of irreversible processes (requiring lengthy and tedious developments remaining out of the scope of this paper), the authors proved (7) that the numerical value (eq. 12) appeared quasi-constant in all the domain of incoming PFD onto the PBR, because the increase of the P/2e -ratio was counterbalanced by a decrease in the stoichiometric coefficient

X H NADPH - + , υ
related to the global formula of the produced biomass quality. Additionally, the theoretical approach used being general (at the thylacoïd level) and completely independent of any given microorganism, the numerical value determined by eq. ( 12) can thus be considered as a quasi universal result (within more or less 10% of deviation) for photosynthesis as long as nitrate ions are used for the N-source and if all the physiological conditions are optimal for growth (pH, temperature). Conversely, the conversion from mole to mass quantum yield φ from eq. ( 5) can be questionable if the quality of the biomass varies with lighting conditions of the PBR and with the considered microorganism. If it is rigorously true that the macromolecular composition is different for each microorganism and culture conditions, except for severe substrate limitations which are not considered in this paper, it has been well-documented again (30) that the resulting C-molar mass M X for the biomass rarely exceed 10% of deviation. Taking for photosynthetic microorganisms the mean value

1 kg.mol 024 . 0 - = X M
leads definitively to the general value:

%) 15 ( μmol . kg 10 85 . 1 -1 h X 9 ± × ≅ - υ φ (13)
As briefly sketched above, in the optimal conditions for growth (only light-limited cultures, no mineral limitations) this quantum yield is mainly affected by the corresponding N-source. For example, the previous global analysis could be perform again considering ammonia (

+ 4
NH ) as unique N-source for growth (an interesting case for many photosynthetic microorganisms). In this case, the photosynthetic quotient of the stoichiometric equations (9-10) becomes roughly

Q P = 1.1, that is 2 . 2 , ≅ - + X H NADPH υ
, and using eq. ( 12) with the same C-molar mass M X leads to the quantum yields values:

1 - h X 9 -1 h X 7 μmol . kg 10 4 . 2 μmol . mol 10 0 . 1 υ υ φ φ - - × ≅ × ≅ ′ (14)
This interesting result demonstrates that the efficiency of photosynthesis is 25% higher if ammonia is used instead of nitrate as nitrogen source, which was a well-known phenomenon on aerobic microorganisms as previously calculated by Roels [START_REF] Roels | Energetics and kinetics in biotechnology[END_REF]. Finally, it is clear again that these proposed optimal values (eqs. 12-14) of quantum yields do not take into account specific conditions of some extremophilic microorganisms cultivated in (for example) highly saline media or at the opposite very low CO 2 partial pressures and needing to dissipate important amount of ATP (leading to considerably decrease the previous quantum yields) to maintain a given level of intracellular concentrations or osmotic pressure. This is in any case a limitation of this paper, the aim of which being to calculate the maximum productivities of photobioreactors only.

The last kinetic parameter, the half-saturation constant K must be discussed now. Because eq. ( 4) is only an approximated relation, there is today no mean to have a predictive method giving a theoretical value to K as a function of the molecular architecture of the photosynthetic antenna. K appears then as the only parameter of the model to be identified and this is indeed a specific value for a given microorganism. Its calculation may be easily done however using for example independent measurements of O 2 evolution as a function of the irradiance (if this latter can be rigorously quantified).

Once a time, if conducting such experiments on different kinds of photosynthetic microorganisms (prokaryotic or eukaryotic), leads to the determination of surprisingly close values within a range of +/-30%. Nevertheless, such variations indicate that an experimental determination for each considered microorganism can significantly increase the accuracy of the proposed approach. The corresponding value determined for A. platensis used in this study has been already widely reported [START_REF] Cornet | Kinetics and energetics of photosynthetic micro-organisms in photobioreactors. Application to Spirulina growth[END_REF][START_REF] Farges | Spectral kinetic modeling and long-term behavior assessment of Arthrospira platensis growth in photobioreactor under red (620 nm) light illumination[END_REF] to be:

1 2 hυ s . m . μmol 90 - - = K (15)

3.3-Problem Reduction in a Particularly Simple Case

In order to establish and discuss the physical meaning of the final formula, a very simple case (but often encountered in the field of artificially lighting PBR) is first examined. We consider a completely stirred and sufficiently thin rectangular photobioreactor artificially illuminated by one side with a quasi-homogeneous and collimated hemispherical incident photon flux density ∩ q . Such geometry enables to make the assumption of one-dimensional light transfer attenuation along a given z-axis, and then to use the simplified form of the RTE in the calculation of the field of irradiance [START_REF] Berberoglu | Experimental measurements of the radiation characteristics of Anabaena variabilis ATCC 29413-U and Rhodobacter sphaeroides ATCC 49419[END_REF][START_REF] Pottier | A fully predictive model for one-dimensional light attenuation by Chlamydomonas reinhardtii in a torus photobioreactor[END_REF], with mean spectrally averaged data on the PAR (absorption and scattering coefficients, phase function, irradiances and incident hemispherical PFD). The possible deviations linked to this assumption have been already

discussed [START_REF] Pottier | A fully predictive model for one-dimensional light attenuation by Chlamydomonas reinhardtii in a torus photobioreactor[END_REF] and do not exceed the degree of approximation envisaged in the following.

Unfortunately, the corresponding analytical solution already proposed by the authors [START_REF] Pottier | A fully predictive model for one-dimensional light attenuation by Chlamydomonas reinhardtii in a torus photobioreactor[END_REF][START_REF] Farges | Spectral kinetic modeling and long-term behavior assessment of Arthrospira platensis growth in photobioreactor under red (620 nm) light illumination[END_REF] is still much more complicated to have an analytical treatment of the kinetics coupling using the integral eq.

(2). A new physically consistent approximation of the one-dimensional irradiance profile, deduced from the rigorous two-flux approach and using only one attenuation term is needed. It has been previously

established [START_REF] Cornet | A structured model for simulations of cultures of the cyanobacterium Spirulina platensis in photobioreactors -I. Coupling between light transfer and growth kinetics[END_REF] that, for a collimated incident hemispherical PFD, the best energetically-consistent approximation was in the form:

      + - ≅ ∩ z C Ea q G X z α α 2 ) 1 ( exp (16)
In this equation, Assuming then a coupling formulation for cyanobacteria (a new restriction) leading to take (as

previously discussed) 0 , = d X r
in eq. ( 2) and with the help of eq. ( 6) for ℓ , X r , it becomes easy to perform the resulting spatial integral (eq. 2) over the illuminated volume ℓ V of the CSTR to obtain (assuming that the irradiance of compensation G C was always sufficiently low to verify

1 1 ≅ + K G C ):       + + = > < ∩ K q L K r M X 1 ln 1 1 2 max α α φ ρ (17)
This surprisingly simple formula gives a very good approximation of the maximum volumetric growth rate max > < X r for any cyanobacteria cultivated in one-dimensional rectangular PBR illuminated by one side. As explained above in this case, it appears completely independent of the illuminated fraction γ, because the maximum kinetic performances for cyanobacteria (for which respiration is inhibited by light) are reached as soon as the physical limitation by light is established (γ is just equaled to one), leading to constant volumetric growth rates for any lower values of 1 ≤ γ , and then to the observed linear time courses for biomass concentration in batch experiments. To understand the physical meaning of this formula, it must be first noticed that the term 1/L corresponds, for the considered case of rectangular reactor illuminated by one side, to the specific illuminated area a light defined from the total working volume of the reactor V by:

V S a light light = (18)
Thus, eq. ( 17) demonstrates that the maximum volumetric biomass growth rate which can be reached in the PBR is mainly dependent on both this specific illuminated area a light and the hemispherical incident PFD ∩ q , once the kinetic parameters for the coupling are fixed ( K M , ,φ ρ ). This is the consequence of the physical light transfer limitation, which is (by analogy with the well-known physical limitation by the dissolved oxygen transfer rate) characterized mainly by a geometrical factor (a light ) and a "force" factor ( ∩ q ). At the opposite, this result shows that maximum kinetic rates are not dependent of the radiation field inside the reactor (the radiative properties being only involved in the definition of radiation losses at the boundaries), or of any state variable for the material phase as the biomass concentration or the kind of mixing (as long as turbulent conditions are guaranteed). Finally, the form of eq. ( 17) may be easily understood from the consistent energetic analysis presented in appendix of this paper. This appendix indeed established in the general three-dimensional case, and especially in the simplified one-dimensional case, that the maximum spatial radiant light power density absorbed in a photobioreactor (physically light-limited) is given by:

∩ + ≅ > < q a light α α 1 2 max A (19)
i.e. only depends on a surface balance in photonic energy onto the reactor and is completely independent of the irradiance profile inside this reactor. So, comparison of eqs. ( 17) and [START_REF] Cornet | Modeling stability of photoheterotrophic continuous cultures in photobioreactors[END_REF] demonstrates that it is possible to assess the maximum productivities of a PBR just by multiplying <A> max with properly defined energetic and quantum yields φ ρ M . This is only true at the limit because eq. ( 17) consistently integrate the fact (in the term [ ]

K q K ∩ + 1 ln
instead of ∩ q ) that the energetic yield of photosynthesis strongly decreases when the hemispherical incident PFD ∩ q increases. Taking the Taylor expansion of eq. ( 17) when K q / ∩ approaches 0 (or more rigorously the irradiance of compensation G C ) is then demonstrative as we obtain directly:

max max 1 2 > < = + = > < ∩ A φ ρ α α φ ρ M light M X q a r ( 20
)
This confirms that a PBR only operates at its maximum thermodynamic efficiency (ρ M ) for very low values of the hemispherical incident PFD ∩ q , completing the energetically consistent analysis of biomass growth rates from radiant light conversion in PBR as sketched in appendix. Conversely, for higher incident photon flux inputs, i.e. in the general case, eq. ( 20) must be multiplied by an efficiency factor for photosynthesis:

[ ] K q q K E ∩ ∩ + = 1 ln ϕ (21)
only related to the hemispherical incident PFD and to the half saturation constant, and limiting the maximum potential biomass productivity whereas the mean radiant light power density increases, according to eq. ( 19).

3.4-Restriction Removal, Generalization and Final Formula

The previous useful formula (eq. 17) has been established with strong restrictive hypotheses both on the geometry and lighting conditions for the PBR and on the considered microorganism; nevertheless, as examined in this paragraph, it can be generalized to almost any generally encountered actual situations of PBR.

3.4.1-Other geometries and lighting conditions of PBR

From the general demonstration given in appendix, it is first obvious that the mean spatial radiant light power density absorbed < A > (eqs 19 and 20) can be easily defined in any given geometry characterized by a specific illuminated area a light . For example (see appendix), in case of rectangular PBR illuminated by two opposite sides a light = 2/L; in the same way, for a cylindrical vessel radially illuminated, a light = 4/D. Moreover, for more complicated situations, it is always feasible to define geometrically a light from eq. ( 18) with the associated definition of a mean hemispherical incident PFD on the surface S light (see the next paragraph).

On the other hand, the mathematical form of the efficiency factor (eq. 20) comes from the integration of the local kinetic rate (eq. 6) with a simplification for the one-dimensional attenuation for the field of irradiance in the particular case of a rectangular reactor illuminated by one side (eq. 16). If it is easy to demonstrate that it can be generalized to any problem in Cartesian coordinates, its validity is less obvious in other geometries. It is however possible, from a rigorous treatment of the RTE in curvilinear coordinates with an appropriate redefinition of the moving frame of reference (the unit vector Ω Ω Ω Ω) to prove that the form of eq. ( 21) applies in any geometry and lighting situation, giving then a general predictive character to eq. ( 17). This step requiring lengthy and tedious developments remains indeed out of the scope of this paper but an extensive treatment of this problem can be found in other author's publication [START_REF] Cornet | Procédés limités par le transfert de rayonnement en milieu hétérogène. Etude des couplages cinétiques et énergétiques dans les photobioréacteurs par une approche thermodynamique[END_REF].

Finally, the domain of validity of eq. ( 17) must be still extended to match with many different original and more or less complex concepts of PBR existing in the literature ( 5) such as partially lighting vessels, annular reactors and/or internally lighting PBR. In the former case, it is necessary to introduce a dark volume fraction f d , corresponding to the volume of the PBR which is not illuminated by construction (i.e. a dead volume for the photosensitized reactions). In other cases, the problem is just to properly calculate the specific illuminated area a light as in eq. ( 18), defining a productivity relative to the working liquid volume (annular) or the total volume (taking into account the internal lighting devices for example) of the reactor and then modifying the final value of max > < X r accordingly.

3.4.2-Other assumptions on the nature of the hemispherical incident photon flux

Eq. ( 17) has been established from the assumption of an homogeneous and quasi-collimated hemispherical incident PFD ∩ q . Nevertheless, these restricted conditions can be easily enlarged to the more general situations encountered in practice. First, if the incident PFD is not homogeneous on the input surface S light , it is always possible to average this flux on the considered surface. This can be done both experimentally, using light cosine sensors with multiple points of measurement or using a chemical actinometer [START_REF] Cornet | Procédés limités par le transfert de rayonnement en milieu hétérogène. Etude des couplages cinétiques et énergétiques dans les photobioréacteurs par une approche thermodynamique[END_REF], or theoretically, from emission model of the artificial sources (13), or for example, averaging the diurnal or seasonal variations of the zenith ϕ and azimuth θ angles in case of solar irradiation.

Many practical cases also deal with non-collimated incident photon fluxes. Here again, a general approach can be proposed, introducing the degree of collimation for the radiation field inside the PBR. It relies on an extension of the two-flux method (historically restricted to collimated or diffuse radiation fields, i.e. intensities independent of the polar angle) recently proposed by the authors [START_REF] Cornet | Procédés limités par le transfert de rayonnement en milieu hétérogène. Etude des couplages cinétiques et énergétiques dans les photobioréacteurs par une approche thermodynamique[END_REF], and consisting in a modulation of the radiant intensities by the n th moment, before integrating over each hemisphere.

This degree of collimation n enables to take into account all the actual situations with polar dependent intensities between the two limiting previous cases, the collimated field with ∞ = n , and the diffuse field with n = 0. It can be demonstrated [START_REF] Cornet | Procédés limités par le transfert de rayonnement en milieu hétérogène. Etude des couplages cinétiques et énergétiques dans les photobioréacteurs par une approche thermodynamique[END_REF], after a rigorous reformulation of the two-flux method and of the assumption given by eq. ( 16), or its equivalent in curvilinear coordinates, that a simple correction

term 1 2 + + n n
then appears in the general improved eq. ( 17). This extension is particularly useful, for example, to take in consideration both the direct and diffuse components of any solar irradiation.

3.4.3-Eukaryotic microorganisms

For eukaryotic microorganisms of course, the previous assumption used in formulating the kinetic coupling for the establishment of eq. ( 17) that 0

, = d X r
is not verified, because the compartmentation (chloroplasts, mitochondria) authorizes the respiration both in darkness or in light. This evidences then, from eq. ( 2), that a second integral term must be rigorously evaluated for the correct assessment of the mean volumetric growth rate in the PBR. Nevertheless, if the respiration in light remains negligible (as it is often the case), it must be emphasized that eq. ( 2), or its resulting integral form eq. ( 17), give again the maximum kinetic performance of any considered PBR cultivating eukaryotic microalgae, because the existence of a negative respiratory term (if the working illuminated fraction γ is lower than 1) would lead to a decrease of the volumetric biomass productivity. This demonstrates, as previously stated by the authors with the introduction of the concept of luminostat [START_REF] Cornet | Modeling stability of photoheterotrophic continuous cultures in photobioreactors[END_REF], the crucial importance of the radiation field control inside the PBR for optimization purposes. This is particularly true if cultivating eukaryotic microalgae, because in this case, the working illuminated fraction γ must be rigorously equal to 1 in order to reach the maximum volumetric productivity for the PBR max > < X r , as the second integral term in eq. ( 2) then always vanishes, resulting finally in a perfect validity of eq. ( 17).

3.4.4-Final general formula

Summarizing here all the extensions of the previous restrictions used in building eq. ( 17) in a final general and useful formula as discussed in the preceding paragraphs leads finally to write the maximum biomass volumetric growth rate in any PBR in the form:

( ) ( )       + + - = > < ∩ + + + + K q K a f r n n n n light M d X 1 2 1 2 max 1 ln 1 2 ) 1 ( α α φ ρ (22)
or indifferently, with the help of eqs. [START_REF] Cornet | Conversion of radiant light energy in photobioreactors[END_REF][START_REF] Berberoglu | Experimental measurements of the radiation characteristics of Anabaena variabilis ATCC 29413-U and Rhodobacter sphaeroides ATCC 49419[END_REF] to:

( ) ( )       + > < - = > < ∩ + + ∩ + + K q q K f r n n n n M d X 1 2 1 2 max max 1 ln ) 1 ( A φ ρ (23)
These simple relations are clearly of considerable interest, taking into account their degree of generality and their predictive character in the sense where all the parameters may be known a priori, either from the design of the PBR or from the previous theoretical work of the authors, as explained in this paper. Additionally, even if they have been established in the restricted framework of CSTR, as already discussed above, they appear completely independent of the material phase characteristics (according to the theoretical analysis given in appendix), and particularly of the biomass concentration.

They can thus also be used for PFTR as long as their conception has been optimized with a recycle to ensure an optimal functioning in physical light limitation in all their volume. If it is not the case, eqs. [START_REF] Pottier | A fully predictive model for one-dimensional light attenuation by Chlamydomonas reinhardtii in a torus photobioreactor[END_REF][START_REF] Farges | Spectral kinetic modeling and long-term behavior assessment of Arthrospira platensis growth in photobioreactor under red (620 nm) light illumination[END_REF] can be used to assess the theoretical maximum productivity in a given PFTR for its possible optimization. Moreover, their general form is also available for photoheterotrophic microorganisms, even if the value of φ is more difficult to obtain in this case taking into account the complexity of the photosystem functioning [START_REF] Cornet | Modeling stability of photoheterotrophic continuous cultures in photobioreactors[END_REF].

4-Experimental Validations

The experimental validations have been performed in eight (very) different, completely stirred PBR, the working liquid volume of which varying between 0.1 and 77 L with dark volume fractions f d ranging between 0 and 0.48 as described in the M&M section. Numerous kinds of geometries have been investigated with different artificial lighting systems and different ways in mixing the culture medium.

For each kind of PBR, the experimental values of productivities have been reported in Table 2. Both batch and continuous cultures have been investigated, and in all cases, the experimental mean volumetric growth rate has been obtained according to the biomass balance onto the well-mixed PBR (defining the so-called residence time for continuous cultures

L L Q V = τ ): dt C d C r X X X + = > < τ (24)
For calculations of the theoretical maximum productivities with the proposed simple formula, we used the general predictive parameters determined in this paper for ρ M , φ ′ and φ , and specific predictive parameters adapted to A. platensis for Ea, Es, b and K as determined elsewhere [START_REF] Pottier | A fully predictive model for one-dimensional light attenuation by Chlamydomonas reinhardtii in a torus photobioreactor[END_REF][START_REF] Farges | Spectral kinetic modeling and long-term behavior assessment of Arthrospira platensis growth in photobioreactor under red (620 nm) light illumination[END_REF]. All their values are summarized in Table 1. It must be noticed here that if the radiative properties Ea, Es and b are specific data of a given microorganism, they are only used in eq. ( 22) for the calculation of the linear scattering modulus α. However, calculations of the authors on many different photosynthetic microorganisms experiences that the value of α often ranges between 0.85 and 0.95. Consequently, and considering that the exact calculation of radiative properties is a complicated problem, a mean value of α = 0.9 can be used by default with only ± 5% of deviation on the final result.

Table 1:

Values of all the predictive parameters used (general and specific) for applying the simple formula (eq. 22) giving the maximum biomass productivities in PBR. (eq. 15)

General predictive parameters

Specific

The comparison between the experimental < r X > and theoretical max > < X r biomass productivities calculated by eq. ( 22) for each kind of PBR and for hemispherical incident PFD ∩ q ranging between 30 and 1600 µmol.m -2 .s -1 (PAR) are given in Table 2.

Table 2 : Comparison between experimental productivities obtained in very different kinds of photobioreactors cultivating Arthrospira platensis and the simple formula (eq. 22). The reactors main characteristics and the experimental conditions are given in the M&M section. (2.6 ± 0.2) ×10 -3 (4.7 ± 0.4) × 10 -3 (5.0 ± 0.5) ×10 -3 (5.3 ± 0.5) ×10 -3 (7.1 ± 0.7) ×10 -3 (7.2 ± 0.7) ×10 -3 (9.5 ± 0.8) × 10 -3 (1.1 ± 0.1) ×10 -2 (8.0 ± 0.7) ×10 -3 (1.2 ± 0.1) ×10 -2 (1.3 ± 0.1) ×10 -2

2.8 ×10 -3 4.9 × 10 -3 5.4 ×10 -3 5.9 ×10 -3 7.4 × 10 -3 7.8 ×10 -3 8.9 ×10 -3 9.6 ×10 -3 8.3 ×10 -3 1.1 × 10 -2 1.3 ×10 -2 (1.9 ± 0.2) ×10 -2 (2.7 ± 0.1) × 10 -2 (3.3 ± 0.5) ×10 -2

1.4 ×10 -2 2.2 ×10 -2 2.8 ×10 -2 3. (1.0 ± 0.1) ×10 -2 (1.3 ± 0.1) ×10 -2 (1.7 ± 0.2) ×10 -2 (1.9 ± 0.2) × 10 -2

1.1 ×10 -2 1.4 ×10 -2 1.9 ×10 -2 (1.1 ± 0.1) ×10 -2

3.5 ×10 -3

1.0 ×10 -2 + 6

5-Discussion and Concluding Remarks

The analysis of the experimental results given in Table 2 and their comparison with the theoretical maximum biomass productivities given by the general formula (eq. ). The maximum deviation observed, as stated in the core of the paper, is ± 15% demonstrating the considerable capability of the proposed simple formula (eq. 22) in the particular case of the cyanobacterium A. platensis. This confirms the idea developed in introduction that although the inherent complexity of PBR modeling in the general case, it is possible to easily establish a simple and then useful formula, available in any geometry and operating condition, for the assessment of maximum volumetric productivities in PBR.

The involvements of this formula are of considerable interest for the field of PBR engineering because it rigorously demonstrates the well-known but intuitive idea that the only design parameters in conceiving a good reactor are the specific illuminated area a light (which must be increased as possible) and the dark volume fraction f d (which must be decreased as possible). The key role played by the hemispherical incident photon flux ∩ q in the PBR performances is also demonstrated: eq. ( 22) enables to calculate with confidence the possible increase in productivity by increasing ∩ q for a given fixed geometry, but conversely, eq. ( 21) or its generalization obtained from eq. ( 23) enables to assess the corresponding loss in thermodynamic efficiency. As illustrative example, it is very simple to calculate that increasing a quasi-collimated incident flux ∩ q from 20 µmol.m -2 .s -1 to 200 µmol.m -2 .s -1 and then to 2000 µmol.m -2 .s -1 (full sun at zenith condition), leads to increase the PBR productivity roughly by 6 and then by 16, but with respective efficiencies (relative to the maximum thermodynamic efficiency for photosynthesis) equal to 0.9 (at 20 µmol.m -2 .s -1 ), 0.53 and finally 0.14. If the value of the maximum thermodynamic efficiency η th at the compensation point for cyanobacteria photosynthesis is taken at 18% ( 7), these values give respectively η th = 16%, 9% and 3% for the same respective incident PFD.

These estimations are in good agreement with the previous calculations of thermodynamic efficiencies in PBR obtained by the authors with full modeling of light transfer using a sophisticated numerical finite element method (20) which confirms, according to the demonstration given in appendix, the energetic consistency of eq. ( 22). Besides, it must be pointed out also that eq. ( 22) enables easily to calculate the required increase of specific illuminated area a light if the incident PFD ∩ q is decreased in order to keep constant the kinetic performance of the PBR (and reciprocally). Finally, it must be kept in mind that the proposed formula (eq. 22) gives only the maximum performances of the PBR (CSTR or PFTR operating in condition of physical limitation by light, i.e. using all the incident PFD) but can be thus very useful to discuss losses in volumetric biomass growth rates which could be obtained in case of photoinhibition It is well documented in the relevant literature [START_REF] Chandrasekhar | Radiative transfer[END_REF][START_REF] Siegel | Thermal radiation heat transfer[END_REF] that the integration of the RTE (eq. 1 in the paper) over any solid angle Ω: Ω Ω Ω leads to the radiant energy balance for the photonic phase of any radiative process. In the particular case of PBR applications, the turbid and participative medium can be considered as non-emitting and this integration gives the following balance at pseudo steady state [START_REF] Cornet | Kinetics and energetics of photosynthetic micro-organisms in photobioreactors. Application to Spirulina growth[END_REF][START_REF] Cassano | Photoreactor analysis and design: fundamentals and applications[END_REF][START_REF] Cornet | A simplified monodimensional approach for modelling coupling between radiant light transfer and growth kinetics in photobioreactors[END_REF]: for the integral (A7) in a form corresponding to that obtained in the paper by a different approach:

A = ⋅ ∇ - q (
( )

∩ + + = - = = = > <  q a q q a q a dz z G C Ea L light ref light light X L α α 1 2 ) ( 1 0 0 0 0 max A (A8)
This result rigorously establishes the energetic consistency of the proposed kinetic relations (eq. 17 or 22), if they are interpreted through eq. [START_REF] Cornet | Modeling stability of photoheterotrophic continuous cultures in photobioreactors[END_REF][START_REF] Cornet | Conversion of radiant light energy in photobioreactors[END_REF][START_REF] Berberoglu | Experimental measurements of the radiation characteristics of Anabaena variabilis ATCC 29413-U and Rhodobacter sphaeroides ATCC 49419[END_REF]. This demonstration is clearly not limited to this particular simple case because of the degree of generality of eq. (A6) already pointed out. If we restricted the field of application of eq. (A6) to PBR functioning in condition of physical limitation by 40 light which is the only considered case in this paper for the assessment of maximum volumetric biomass productivities, it is evident that any geometry obeys to the more general equation, available in any situation as discussed in the core of the paper:

∩ + =         ⋅ - = > <  q a d V light S α α 1 2 1 max max S q A (A9)

-PBR 3 *

 3 : cylindrical stirred tank reactor (Applikon) of 5 L working volume radially illuminated by 55 halogen lamps (Claude BAB 38°, 20 W, 12 V in the spectral range [350-1100 nm]) and agitated by two rushton turbines. Temperature was controlled with an internal exchanger fed with an external thermal bath and pH was regulated using 1 mol/L H 2 SO 4 . The diameter of the reactor was 16 cm giving a specific illuminated area a light of 25 m -1 without dark fraction (f d = 0). -PBR 4: cylindrical air-lift reactor (Bioengineering) of 7 L working volume radially illuminated by 48 halogen lamps (Sylvania BAB 38°, 20W, 12 V in the spectral range [350-1100 nm]) and agitated by bubbling air-CO 2 at a flow rate of 1 vvm. Temperature was controlled with an internal exchanger fed with an external thermal bath and pH was regulated by adjusting automatically the CO 2 mole fraction in the input gas. The diameter of the reactor was 10 cm giving a specific illuminated area a light of 40 m -1 with a dark volume fraction f d = 0.48. -PBR 5*: Oblate cylindrical membrane reactor of 0.106 L working volume lightened by the height with one halogen lamp (Claude BAB 38°, 20 W, 12 V in the spectral range [350-1100 nm]) and magnetically stirred. Temperature was controlled with an exchanger fed with an external thermal bath and pH was regulated by CO 2 pulses in the gas phase when necessary. The path length of the reactor was 2.3 cm giving a specific illuminated area a light of 43.5 m -1 without dark fraction (f d = 0). -PBR 6: double cylindrical (riser and down comer) air-lift reactor (Bioengineering) of 77 L working volume radially illuminated by 350 halogen lamps (Sylvania BAB 38°, 20W, 12 V in the spectral range [350-1100 nm]) and agitated by bubbling air-CO 2 at a flow rate of 1 vvm. Temperature was controlled with an internal exchanger fed with an external thermal bath and pH was regulated by adjusting automatically the CO 2 mole fraction in the input gas. The diameter of each cylinder was 15 cm giving a specific illuminated area a light of 26.7 m -1 with a dark volume fraction f d = 0.33.

4 and

 4 the field of radiant light power density

  value of the irradiance of compensation G C , on the angular nature of the radiation field for the radiant intensity calculation, and on the spectral nature of the field for the calculation of λ . Nevertheless, evaluating ρ M from eqs.[START_REF] Cornet | Procédés limités par le transfert de rayonnement en milieu hétérogène. Etude des couplages cinétiques et énergétiques dans les photobioréacteurs par une approche thermodynamique[END_REF][START_REF] Zijffers | Design process of an area-efficient photobioreactor[END_REF] in a wide range of previous conditions, both for prokaryotic and eukaryotic microorganisms having different optimal temperatures of functioning, leads generally to numerical values ranging between 0.76 and 0.82 (in the PAR). This enables to retain in first approximation the constant value 8 . 0 = M ρ with less than 10% deviation for any photosynthetic microorganism considered in this study. It should be mentioned here that this value is in very good agreement with the theoretical ones determined by a thorough analysis of the excitations transport in antenna on a quantum physics basis[START_REF] Paillotin | In: Etude théorique des modes de création, de transport et d'utilisation de l'énergie d'excitation électronique chez les plantes supérieures[END_REF], demonstrating that, around the compensation point, photosynthesis operates close to the optimum thermodynamic conditions.

  correct in a very first approximation the Bouguer's law from scattering. Clearly, this equation is too simple to accurately describe the radiant light profile inside the culture medium and then to give actual local volumetric growth rates in the general case; nevertheless, as it will be demonstrated in the following, it just contains the physical information needed for the objectives of the paper (not more and not less).

7 )

 7 Mass absorption coefficientEa = 8 162 ± m 2 .kg -1Mass quantum yield (nitrate as N-source):Mass scattering coefficient Es = light sources used at any PFD)Half saturation constant K = 5 90 ± µmol.m -2 .s -1

2 ) 3 1

 23 sidea light = 12.5 m -1 (f d = 0) ×10 -3 (2.1 ± 0.2) ×10 -3 (3.2 ± 0.2) × 10 -

  light = 40 m -1 (f d = 0.48)

(E 1 Kqqαρ

 1 too high PFD), in kinetic regime (excess of photons in the PBR) or in light-limited cultures with low values of γ for eukaryotic microorganisms (microalgae). This simple and reliable formula appears then as a robust and practical tool for sizing new original PBR (as internally lighting reactors) or for the comparison of different designs and technologies, independently of the considered microorganism. This last point of course would rigorously required that the experimental results presented and discussed above concern many different photoautotrophic microorganisms, as a proof of the supposed degree of generality claimed along with the paper (specially for the values of the general kinetic parameters ρ M , φ ′ and φ ). Clearly, the results presented in this NOMENCLATURE a light Specific illuminated area for any given photobioreactor [m -1 ] A Local volumetric radiant power density absorbed [µmol.s -1 .m -3 ] b Back-scattered fraction for radiation [dimensionless] c Speed of light in vacuum, c = 299792458 m.s -1 C X Biomass concentration [kg.m -3 or g.L -1 ] D Total diameter of a cylindrical photobioreactor [m] Ea Mass absorption coefficient [m 2 .kg -1 ] Es Mass scattering coefficient [m 2 .kg -1 ] ϕ Efficiency factor [dimensionless] f d Design dark volume fraction of any photobioreactor [dimensionless] G Local spherical irradiance [µmol.s -1 .m -2 ] h Planck constant, h = 6.626× 10 -34 J.s I Specific radiant intensity [µmol.s -1 .m -2 ] J i Molar specific rate for species i [mol.kg X -1 .s -1 ] k Boltzmann constant, k = 1.381 ×10 -23 J.K -Half saturation constant for photosynthesis [µmol.s -1 .m -2 ] L Total length of a rectangular photobioreactor [m] M X C-molar mass for the biomass [kg.mol -1 ] n Degree of collimation for the radiation field [dimensionless] n m Refractive index of the liquid medium in the photobioreactor [dimensionless] Photon flux density [µmol.s -1 .m -2 ] ∩ Hemispherical incident photon flux density (PFD) in the PAR [µmol.s -1 .m -2 ] Q L Volume liquid flow rate [m 3 .s -1 ]Q P Photosynthetic quotient [dimensionless]r X Biomass volumetric growth rate (productivity) [kg.m -3 .s -1 or kg.m -3 .h -1 ] S Surface [m 2 ] S light Illuminated surface of any photobioreactor [m 2 ] Linear scattering modulus [dimensionless] γ Fraction for working illuminated volume in the photobioreactor [dimensionless] η th Thermodynamic efficiency of the photobioreactor [dimensionless] Energetic yield for photon conversion [dimensionless] ρ M Maximum energetic yield for photon conversion [dimensionless] τ Hydraulic residence time [h] ij υ Stoichiometric coefficient [dimensionless] φ Mass quantum yield for the Z-scheme of photosynthesis [kg X .µmol hν -1 ] φ' Mole quantum yield for the Z-scheme of photosynthesis [mol X .µmol hν -input surface of a rectangular photobioreactor c Relative to the compensation point for photosynthesis L Relative to the output surface of a rectangular photobioreactor λ Relative to a spectral quantity for the wavelength λ APPENDIX Calculation of the maximum spatial radiant light power density < A A A A > max in a photobioreactor as a proof of the energetic consistency of the proposed relation (eq. 17 or 22)

  photonic phase, i.e. just balancing the input and output mean flux densities at the boundaries. In case of

  :

	→  + 1.463 CH H CO 2 1.575 + O 0.173 N 0.459 HNO S 0.006 + 0.173 O 3 2 X J	+ H 3.538 SO 0.006 P 0.006 4 P + i 2	+ 3.544 3.544 ATP ADP +	+ (NADPH, NADP 2.779 2.779 +	H	+	)	(9)
	Eq. (9) is associated to the corresponding Z-scheme stoichiometric equation:
	2.779	NADP	2.779	H	2	O	2.779	(NADPH,	H	)	1.389	O	2
		3.544	(ADP		P i	)	3	544 .	ATP	3.544	H	2	O

Table 2 (

 2 continued) : Comparison between experimental productivities obtained in very different kinds of photobioreactors cultivatingArthrospira platensis and the simple formula (eq. 22). The reactors main characteristics and the experimental conditions are given in the M&M section.

30

study have been focused on the diversity of the PBR technology and size (3 orders of magnitude in volume), and cultivating different microorganisms in these eight different PBR would have represented a considerable amount of work, exceeding the framework of a scientific paper. Another possibility could have been to try a comparison with the numerous experimental results of the literature, but this requires again to know with some confidence the geometrical details of the PBR design (a light and f d ) with the corresponding mean values of the PFD ∩ q and to be sure that the data were obtained in conditions of physical limitation by light. This complete information is rarely available in the literature, and except very good results obtained with Porphyridium cruentum cultivated in the 100 L ALP2 PBR (33, data not shown), it has not been possible to perform this analysis. Nevertheless, a secret wish of the authors would be that the scientific community working in the field of photobioreactor engineering tries to use the proposed simple formula on many different kinds of artificial or solar reactors and different photoautotrophic microorganisms in order to specify its range of validity for the assessment of maximum potential biomass productivities.

ACKNOWLEDGMENT

The authors thank the European Space Agency (ESA/ESTEC) which partly supported this work through the MELiSSA project.

max Maximum value for a radiant power density A or a volumetric productivity r X In eq. (A1) the local radiant density absorbed A may be considered as a sink term for the photonic phase, but eq. (A3) clearly establishes that it is also a source term for the material phase (the volumetric absorption coefficient EaC X establishing then the link between the photonic field of irradiance G or q and the actual absorbed energy by the material phase A). As already explained elsewhere [START_REF] Pruvost | Hydrodynamics influence on light conversion in photobioreactors: An energetically consistent analysis[END_REF]39), the conservation of the total energy, i.e. the first principle of thermodynamics applied on the PBR imposes that this terms be equaled, whatever the method involved for their calculation, from the photonic or the material phases. This conclusion also holds for the mean spatial volumetric radiant light power density absorbed in any PBR and just defined by the volume integral:

Superscripts

As explained in the core of this paper, this term, corresponding to the total spatial radiant light power density available for the photosensitized reactions can serve as a basis for the assessment of any PBR kinetic performances. Using the definition (A4) together with eq. (A1) and applying the Gauss-Ostrogradsky theorem enables then to write:

On the other hand, eq. (A4) may be combined with eq. (A3) on the material phase to give finally, with eq. (A5) the powerful relation:

This equation of considerable interest is just what we need to prove the consistency of the useful formula (eq. 17 and 22) developed in this paper. First it represents the unique condition for the


Documentatioiu

Xavier Irz

Professor Terry Roe

lNùiltut NATl()t{At 0Ê tA ft[tllÊHuHt Abhurrrrurruu' &stlm d'Ëconomh et Sociologio Hualae

Keywords: structural transformations, economic growth and agriculture, French agriculture November, 1996

A general equilibrium model ofstructural change featuring three sectors (agriculture, manufacturing, services) and endogenous growth is presenûed to explain the evolution of agriculture in a growing market economy. Non-homothetic preferences, necessary to reproduce the Engel effect, as well as technological change occurring at different rates in the three sectors and originating in learning-by-doing constitute the main originalities of the model. The analytical properties of the model are then shown to be consistent with one of the main regularities of the process of economic development: the continuous flow of labor out of agriculture and into the services seAEtor.

The model is then calibrated to France and the results of the simulations are compared to the pattern of structural transformations which has characterized the evolution of the French economy over the 1950.'2000 period. The drastic reduction in agricultural worldorce is reproduced as well as the phenomenon of de-industrialization (decrease in the relative importance of manufacturing ) after 1970. The model is then used to analyze the evolution of the domestic competitiveness of the agricultural sector over the period considered and the negative effect on the farm sector of deterioratng domestic terms of trade is highlighted.

I Introduction

Ftench agriculture has lnd.ergone tremendous changes over the post wa,r period as highlighted by the following facts: in 1950, the farm sectoi employed the largest share of the country's work force, and. it was a net importer of food (Eck). By 1992, agriculture accounted for only 6 % (USDA ) of the work force and produced a $10 bittion trade surplus, which ofiset by one third the French industrial trade deficit (USDA). In 40 years, an apparently very traditional a,nd relatively backward sector has become a powerful, dynamic and modern industry, thanks to a level of performances that Ruttan (19?8) qualifies as "little short of spectacular"l .

This study develops an analytical model which helps to provide key insights erçlaining the pattern of structural transformation which has cha,racterized the evolution of the Flench economy over the last five d.ecades, with a special attention given to the farm sector. We identify a^nd explain the interplay of the main forces which concurred to shape Flench agriculture into its current form a.s described. by macroeconomic data on output, employment in the context of the entire Fbench economy. This same approach should also be applicable for other OEDC countries.

In order to define an appropriate approach, it must be noticed that the pattern of structural tran^sformations followed by Ftance does not constitute in a.ny vray an isolated phenomenon. The decrease in the importance of agriculture both in terms of value added sha,re of GDP and in terms of the sha.re of work force employed actually constitutes one of the most robust stylized fact of economic d,evelopment. This can be ehecked empirically not only from historical individual experiences of countries2 but also from cross section data of countries in the world (Sundrum).

These empirical findings lead us to the general view that changes a.ffecting the fa,rm sector can_only be understood in the context of a growing economy taken as a whole. Unfortunateh the literature in the field of economic growth, which presents numerous models derived from the seminal work of Solow, has ignored almost completely the issue of structural transformation. The neo-classicat growth model as well as the more recent endogenous growth models only consider one final output a^nd are by nature unable to e>çlain the reallocation of resources across sectors of a given economy.

The literature in the freld of economic development, although usually of a descriptive natrue as pointed out by Shin(1990), provides more insight into what really constitutes the engine of lRuttan actually refers to the period covering the late 1950s and the 1960s 2Shin(1990) reports that the US agriculture was employingTg% of the workforce in 1820, 53% ia 1860, 40% in 1900, 20% in 1940 and currently less than 3%. structural transformations, although no analytical framework of a general equilibrium nature is provid.ed. In particular, the Engel effect, i.e. the fact that the income elasticity for agricultural output is typically less than one, while the income elasticities of demand. for industrial output an6 services are typically greater than one, is widely recognized as one of the main forces driving structural transformations of a developing economy' Technological progress, describing the outward shift in the production possibility frontier of a cormtry over time, seems to play also a firnd.a,mental role in explaining structural transformations.

First, it lead.s to higher per capita income which is a necessary condition for the Engel effect to apply. Second, it is not uniform across sectors of the economy [START_REF] Pasinetti | structural change and economic growth -A theoritical essay on the dynamics of the wealth of nations[END_REF]. Therefore, productivity of economy wid.e resources3 in the different sectors of the economy varies over time which lead.s to a reallocation from low productivity activities to high productivity onae.

A few authors have tried to cast these elements into an analytical frarnework based on the theory of economic growth. For exarnple, there wa^s arr ea,rly attempt (1972) by Henrichsmeyer who d.eveioped a two-sector (manufactudng and agriculture) model of exogenous growth and cal- ibrated it to Germany. [START_REF] Pasinetti | structural change and economic growth -A theoritical essay on the dynamics of the wealth of nations[END_REF], motivated. by the inability of growth mod.els to e>çlain ,,uneven d.evelopmentfrom sector to sector and region to region" of post-war Europe tried to integrate the input-output analysis and the macro-dyna.rnic growth models but at a purely theo- retical level. [START_REF] Shin | Economic growth, structural transformation and agriculture: the case of U.S. ans S. Korea[END_REF] studied the structural tra''sformations of the US and Korean economies and eiaborated an end.ogenous growth model including two distinct sectors (marrufacturing and agriculture). However, the fact that his model ignored the services sector, which in most OECD countries constitutes a large share of value added GDP and is large in terrns of the share of work force employed., constitutes a major drawback. Finally, Eschevarria (1992) built a threesector model ( agriculture-ma"nufacturing-services) where growth originates in capital accumulation a-s well as exogenous technological progress occurring at a different rate in each sector. Her cal- ibration, based on data for the US and other OECD countries, reproduces the typical logistic growth patterna but leads to counter-factual results with regard to structural tra.nsformations in the sense that her model simulations predict a decrease in the importance of the services sector for high levels of per capita income while the manufacturing sector expands steadily. This paper intends to show that a simple multi-sector growth model incorporating non- 3By economy wide resource we understand a factor of production which can be used indifierently in all sectors of the economy, one example of which is labor aOr as stated by [START_REF] Lucas | On the mecha,nics of economic development[END_REF]: "The poorest countries tend to have the lowest growth;.the wealthiest next; the "middle income' countries highest".

of structural changes and help us understand how agricultwe evolves as an economy grov,,s . Sec- tion 2 presents the model and its dynamic properties. A calibration procedure is then developed in section 3 and the outcome of the simulations are analyzed. Section 4 concludes the paper.

2 The Model

A. The environment

The economy is supposed autarchic and features three sectors: agricultue, manufacturing and services. Although trade has been evidently important in determining the evolution of the French economy, the present model must be understood as a fust step towa.rd a more appropriate approach. F\:.rthermore, the analysis of a closed economy can be justified by making two simple observations. First, trade still remains small as compaxed to total GDP even for economies usually corsidered opens. Second, as pointed out by Ventura(1994), a world. of open economies is a closed.

economy so that our analysis can also give some insight into the functioning of agriculture in the context of the world economy.

s side Eadr sector is composed of a finite mrmber of identical producers. All produc- ers in a sector j : a, nz, s make use of the sa,me neo-classical, constant returns to scale technology which employes two inputs to generate a single output.. One of these inputs, referred to as labor, is "economy wid.e" in the sense that it can be used. in all three sectors while the other input is sector speciûc. Such a formulation was adopted because of our focus on agriculture where land plays a major role but has little a.lternative use in other indraetriesd. Producers behave competi- tively a.nd treat prices parametrically. Under these conditions, it can be shown that the individual production functions can be aggregated and the resulting sectorial production functions can be written a.s?:

X', : SlH(n2), XL : Mf (ny), Xl: A$(L -rLLt -n r)

(2.1)

sFor iastance, trade related activities avnonnt to less tha l.b% of GDP in the US. 6Fbr the manufacturing and services sector, the formulation is somewhat less appealing. However, some kind.

of managerial capital can be thought of as playing the role of a sector specific factor in these two industries.

TRrnctions 4 G and ff are stricly increasing and strictly concave.

where n11 and n21 denote the shares of labor force employed in the manufacturing and services sectors respectively as to time t. Mr, Sr, As designate productivity indices in manufacturing, services and agriculture and Xj ls the output of sector j'

The engine of growth in this economy lies in technological change in the three sectors origi- nating in learning-by-doing: there is a positive link between the aggregate output of a sector and its rate of total factor productivity growth. Learning-by-doing can be justified by considering that as a sector attracts more resources and increases its output, more knowledge relating to the technology is accumulated and such knowledge is then used to improve the production process.

It is important to note, however, that learning-by-doing effects are purely external to the firms that generate them: knowledge only accumulates as a by-product of the activity of each' ûrm.

Formally, prod.uctivity indices change over time accord.ing to the following laws of motion 8:

Mt:6X!*, 3t: pXto, A1: p,Xt"

(2.2)
where a dot designates a variables derivative with respect to time.

Parameters 6, p md p, strictly positive, are exogenous and represent the speed of the learning process which can vary across sectors. ll Demand side The d.emand side of the economy results from constrained utility ma>cimiza- tion of a representative consumer who owns the sector specific factors as well as one unit of labor services.e The agent marcimizes a discounted flow of instantaneous utility of the StoneGeary type subject to a sequential budget constraint : lo** lu"log(ci -t")+ log(cf,) + É,loe(ci + %)] e-ptdt s.t. c! + prci,, + pt ct, s at * rt cL>0, cL> o, c: > o where C'", CL, Cj denote aggregate consumption of each good as to time t , P ) 0 is the discount rate, pt^ and p3 correspond, to the prices of manufactures and services in terms of food, a'rs is the wage rate and zq the returns to sector specific factors sThis formulation is similar to those of [START_REF] Lucas | On the mecha,nics of economic development[END_REF], [START_REF] Shin | Economic growth, structural transformation and agriculture: the case of U.S. ans S. Korea[END_REF]Matsuyama(1992). eAggregate demands derived from this set up are the s4me as those generated by a conthuum of consumers with identical preferences as long as income inequalities a,re not too "high" (a.s long as all the agents consume services).

Mar

cL,cl^,c! (2.3) (2.4) (2.5)
There is no asset in the economy allowing the consumer to smooth consumption over the entire time horizon. Therefore, this dynarnic problem can actually be solved as a succession of static problems where each agent spends his full income on current consumption of the three goods.

Parameters 9o, Fr, -ya,.ys are strictly positive. .yo is usually interpreted as a subsistence level and we impose the condition that the economy has reached a stage of development where it can actually feed its population : A.G(L) t ''lo .

In addition to the fact that it allows for a simple aggregation, the utility function was chosen because it features non-homotheticity which is believed to be a fundamental component of any model of structural change. Computation of the income elasticities for the three goods e'o, e'^, ej shows that the choice of preferences implies food to be a necessary (but not inferior) good (0 < ul < L) while services present the characteristics of a lunrry good (1 < €i). The case of manufactures is more ambiguous but it is possible to establish that eI" < e{" < ej. Therefore, the d.emand for services can be expected to increase and the demand for food to decrea.se (at least in relative terms) as the economy grows due to technological progress in the three productive sectors.

(iii) Equilibrium path The evolution of the economy over time is now computed as the com- petitive outcome of the environment specified above. In each sector, a representative producer minimizes cost subject to an aggregate production function given by equations 2.1. Prices as well as productivity indices are treated parametrically by this representative producer. Equilibrium on the market for the economy wide factor requires the marginal productivity of labor in the three sectors to be equal to the wage rate at any point in time : u)t : AtG' (1 -rr,nrr) -pt^M1F'(nrr) : f,S6H'(n2t)

(2.6)

with the price of food normalized to one.

Aggregate demands solve problem 2.3. The first order conditions to the malcimization allow us to establish an expression for the product mix of final demand :

C'": Êae'^C!^+ n

(2.7) c:: B"!*cL-t,

(2.8)

to the following equilibrium conditions:

ArlG(L -rl1--n2)F'(nrr) -Ê"G'(L -nu -n22)F(ntt)l : 1"F'(nu)

(2.e)

-SlH(n2,)F'(nr,) -p,H'(n r)F(n")l -'y,F'(nt 1 (Z't0)

These equations d.efine an allocation of labor across sectors (nninzù given any set of pro- ductivity indices Ar, Mr, St. Once the proportions of the work force employed in each sector are determined, it is straightforward to compute all the variables of the economy: equations 2.9-2-L0 completely characterize the static equilibrium.

The evolution of the economy over time is then given by the a^ssociation of.2.9-2-L0 with the laws of motion for productivity indices 2.2 as weli as iniiiat conditions Ao, Mo, ^9s. However, it is possible to reduce the equilibrium conditions to a system of two differential equations and two initiat conditions whictr helps clarify the dynarnic properties of the model. Taking the logarithm of 2.10 a^nd differentiating with respect to time we obtain :

,r,:orffi*nt,u'ffi (2'11) where the arguments of all the production firnctions have been dropped for clarity.

proceeding similarly with equation 2.9 and using the expression for n26 derived above, vye can derive the following expression :

Ft Ft H'F_H

(2.r2) fLLt:

Equations 2.11 and 2.L2 form a system of two differentia,l equations in two variables rù16 ârrd n21. The initial values n1s and n2s cà\be computed from equations 2.9-2.L0 and initial conditions Ao, So. Therefore, this dyna,mic system is completely defined and we can derive the whole pattern of resource reallocation simultaneous to the growth of the economy.

B. A few important properties

It should be clear at this point that the structural transformations affecting the economy can be characterized. by solving numerically the system of differential equations 2.11-2.12. However, F' F, + before proceeding to the next section, 'we can derive a few general properties that are imposed by the structure of the modello.

(i) Lons-run state of the economy The first question that arises concerns the long-run state of the economy: will economic growth evolve in such a way that the economy reaches some form of stability? It turns out that the reallocation of economy wide resources (labor) is going to grind to a halt so that the economy will eventually reach a steady state as claimed in the following proposition, Proposition 111: The econorny, oaer time, conuerges to the unique steady state (ni;ni) defined by the following system of equations:

p,Ht(ni)F("i) -H(ni)F'(ni) : G(l -ni -ni)F'("ï -P"G'(1-ni-ni)F(ni) : 0 0 (2.13) (2.t4)
It must first be pointed out that these equations are identical to system 2.9-2.L0 characterizing the static equilibrium for values of 7o and n equal to zero. 'I:r other words, had the preferences been of the Cobb-Douglas type (hence homothetic) U : *c*4', the economy would have "jumped" directly to the steady state without any resouce reallocation. This result confirms that the non-homotheticity of preferences constitutes a firnda,mental drive of structural transformation.

It can be shown that at the steady state, ctranges in relative prices exactly offset differences in rates of tecbaological progress across sectors so that the marginal product of labor evolves at the sa.rre speed in the three industries, preventing any fiuther reallocation of the work force. rii) Characteristics of the oattern of structural transformations Proposition2r2: The proportion of the work force employed, in the serv'ices sector monotically increases along the growth path of the economy. Simultaneously, labor steadily fl,ows out of agriculture.

It can not be determined in which way the manufacturing sector is going to evolve over time. These properties appear highly desirable for any modei of structural change. They fit nicely loDemonstrations of aII the results a^re not presented. here due to the lack of space. They are however available from the author upon request rrThe steady state is determined by ûnding cond.itions ensuring ô11 a^nd ,it21to be simultaneously equal to 0.

The convergence property can be established by drawing the equivalent of a phase diagram in plane (nr;nz). r2Theproof comesfrs6flsnsfsrmationsof ocpressions 2.lL-2.tzandfromtheobservationthat HF'-&HtF <0

and GF' -\oG'F > 0 along any equilibrium path (see eq. 2.9-2.10) the typical pattern of resource reallocation that an economy experiences as it grows richer a,s described in the introduction.

3 An example of calibration A. Method of calibration

The sectorial production fi-rnctions axe chosen of the Cobb-Douglas form so that they can be written as: X!: At(Irltr -n2t)oo, X!*: Mr(nr,)o^, X! : S'(nz')o' '

Therefore, calibrating the model entails defining 10 parameter values:

-ea, dm, a, for the sectorial production functions -Êo, Ar,'ya,'fs which deûne the Stone-Gary utility function lt, p,,6 which determine the speed of learning-by-doing in each sector

We also need to define three initial conditions As, Ms, Ss'

The proced,ure of calibration can be briefly summarized as follows'

The pa,rameters of the production firnctions oo, drn, ds a.re straightforward to ca.librate since they represent the share of labor cost in total production cost in each sector.

Coefficients Bo and B, from the utility function can be computed from equations 2.13-2.14 given a steady state allocation of labor ("î;"î). It is assumed at that level thad the US have reached the steady state which is therefore defined. by the following allocation of the work force across sectors: ni: .32, ni: .65.

The system of d,ifferential equations 2.11-2.12 shows that the pattern of resource reallocation d,oes not depend. upon parameters As, S9, Mo, ^lo, ?s once the initia'I values for n1 and n2 have been established. F\rrthermore, it appears from equations 2.9-2.L0 that these initial values n1s and n26 only depend. upon two ratioé : f; ana fl. Therefore, we decide to set all three productivity indices equal to one at t :0. Pararneters 7o and 7s ca,n then be deduced from an initial allocation of labor (nro;rzzo) by solving equations 2.g-2.L0. An initial allocation rLLo : .34, n2s: .33 , corresponding to the state of the French economy in 1950 as documented by sundrum (1991), was chosen.

At this point, only values for the three pa,rameters defining the speed of learning-by-doing in each sector need. to be determined. Gopinath and aI.(1996) obtain a measure of the rate of TFP change in the FYench agriculture over the 19741993 period equal to 6To while Martin and Mitra(1996) report a 2Yo rate of growth in TFP in the Ftenctr manufacturing sector. While no estimate of the rate of TFP dtange in the services sector was found, there seenË to be a consensus in the literature that this rate is lower than in other sectors of the economy (Sundrum p.144) and we therefore set ittoLTo. Ultimately, values for pt,, p,6were set to.15,.04 and.02 respectively.

The following (i) Pattern of resource reallocation Figure 1 presents the result of the numerical simulation of the model (that is, the solution to the system of non-linear equations (2.LL-2.12)). As claimed by proposition 2, labor is continuously drawn out of agricuiture and into the services sector. But the main support to the model comes from the magnitude of these changes: while the parameters determining the speed of the dynamicsl3 have been defined from estimates of rates of technological change in each sector, they lead to a very plausible profrle of labor reallocation across sectors. In particular, the French agriculture still employes 6% of the work force which corresponds precisely to the result of the simulation.

The simulated evolution of the manufacturing sector lends some more credibility to the ap proadr followed. From 1950 to 1970, labor is attracted to the manufactuing sector but this trend grinds to a halt and is eventually reversed from 1970 to year 2000. This result reproduces the phe- nomenon of "deindustrialization" of developed economies (Sundrum p. 144) that is documented for the European Community as a whole by [START_REF] Stoeckel | Intersectoral effects of the CAP: growbh, trade and unemployment[END_REF] for the period. 1971-1982. Simulated values of other economic variables are now presented to better understand how this pattern of resource reallocation was obtained and to show that the model provides a very coherent picture of the development of the Fbench economy over the last 50 years.

(ii) Overall srowth oerformance of the economv The rate of growth in real GDP wa.s computed and figure 8 presents the result. Starting fuom 4% in 1950, the growth rate diminishes sharply to eventually stabilize a^round. L.5%. Although this profile appears too steep, the model reproduces the fact that the Flench economy was growing quickly after the second world war while the 70s and 80s were cha.racterized by low rates of economic growth. In the model, the reduction in l3That is, parameters p,,6, p.

the rate of growth in GDP can be associated with the transfer of resources from sectors undergoing fast technological change (agriculture and manufacturing) to a sector cha^racterized by a slow rate of technological progress (services).

(iii) of sectorial competitiveness Although the concept of competitiveness is rather elgsive, we give it a clear meaning following the definition of Gopinath and Roera: if within an economy, the rate of growth in sector j's real GDP exceeds that of the economy, i.e., d(ln G D Pj) I dt > d,(lnGDP)f dt then sector j is increasing its competitiveness relative to the other sectors of the economy.

Figures 2 and 7 present respectively the sectorial composition of GDP and the rates of growth in sectorial GDP. According to the definition adopted, agriculture loses its domestic competitive' ness over the entire period while the services sector gains competitiveness. The manufacturing sector, after strenghtening its position from 1950 to 1970, eventually undergoes a slow erosion of its competitiveness. This evolution can be better understood by decomposing the increase in value-added output of each sector into different sources.

(iii).1 Evolution of relative prices. Figure 3 presents the evolution of the sectorial terms of tradels. As expected from the combination of a low income elasticity of demand and a high rate of technological progress, the real price of agricultural output is dwindling steadily. In the same time, the real price of services increases while the manufacturing sector faces slightly decreasing terms of trade after 1955. Atthough data for the Fbench economy were not found, one can notice that this evolution is very similar (both in shape and in magnitude) to the one reported by Gopinath (1995) for the US.

(iii).2 Determinants of the domestic competitiveness of the agricultural sector. The increase in sectorial GDP can be d.ecomposed into a price effect, capturing the changes in the valuation of final output, an input effect (here increase in the production due to an increase in the labor input) and a technological rate effect (denoted TFP)16. Figure 4 presents this decomposition l4This is the domestice dimènsion of competitiraeness. The authors also define the concept of international competitiveness, which is not considered here since the ecouomy is firnctionning in autarky.

rsReal prices were computed by deflating prices by the following price indoc Pl -tt"p', + st"e',.+ s3p3 where the weights sj correspond to the sectoral value'added sha,res of GDP.

16For instance, the rate of increase in agricultural GDP can be decomposed as follows: W : -+ + * -%(n'r, + n2ù i for the agricultural sector and gives some insight into why the farm sector has lost its domestic competitiveness over the last five decades. In spite of a fast rate of technological progress, agri- cultural GDP d.ecreases in real terms from 1950 to 2000. This is due to the conjunction of two very negative effects: first, the real price of food decreases quickly (from 6To a year in 1950 to 2.2% a year in 2000). Second, the agricultual work force has diminished at a rate comprised between 6% ayear and lTo aYear- Correspond.ing results are presented in figures 5 and 6 for manufacturing and services.

Conclusion

This paper shows that a simple thre+sector model of endogenous growth featuring non-ho- mothetic preferences can replicate one of the major regularities of economic development: the steady expansion in the services sector and the simultaneous decrease in the fa^rm sector in terms of work force. The numerical simulation of the model, calibrated to post war Flance' gave a very coherent an{ plausible picture of the growth pattern of a developed economy. ln pa,rticular, the phenomenon of de-industrialization of a mature economy u/as reproduced and the evolution of the sectorial terms of trade obtained appears consistent with what is reported in the literature.

It is also believed that the paper hightights the close link between structural transformations a,nd economic growth which have usually been analyzed in two different fields of economics; on one hand, economic growth results in higher per capita income and hence in a change in the composition of final demand if the income elasticities for final goods differ. On the other hand, the reallocation of resources across sectors characterized by different rates of technological progress influences the overall growth rate of the economy.

Finally, the model gives some insight into what constitutes the major determinants of the evolution of each sector. The work is seen as complementary to classical growth accounting exercises which have been used to analyze the competitiveness of the farm sector [START_REF] Gopinath | General equilibrium analysis of growth and competitiveness of U.S. agriculture[END_REF] but whose results remain of a descriptive nature. By contrast, the model presented gives a fully consistent representation of an economy and can therefore be used as a tool for policy analysis.

The next step will consist in the development of an open-economy and more applied version of the model to assess the impact of the Common Agricultural Policy on the pattern of structural -fi t the real price efiect, fr tne TFP change efiect and -ft1":r, + nïr) the input effect.

transformations followed by the Flench economy. It is believed that the formulation of techno- logical progress chosen, whictr creates a link between economic policies and the evolution of the comparative advantage of a country will have some interesting implications in the analysis of the competitiveness of the Flench agricultrue. 1950 1960 1970 1980 1990 2000 Figure 4 : Sources of growth in ag GDP t increase in serv -0.

-0.

Figure

  Figure 2: Sectoral composition of GDP

Figure

  Figure 3: Evolution of sectorial terms of trade t Lncrease in uarruf GDP 0.08 0.06 0.0{ 0.02

Figure 5 :

 5 Figure5: Sources of growth in man. GDP @P CrorÈh

Figure

  Figure 6: Sources of growth in serv GDP crortlr latè

Figure

  Figure 8: GDP growth rate -0 -0 -0

  table summarizes the choice of parameters:

	Cobb-Douglas production functions	Ston+Gary Utility	Learning-by-doing
	ao Q,M Qs Ao Mo	^90	Ê"	Ê"	'Yo	'Ys	l-L	p	lr.
	.6	7	.8	1	1	1	109 1.78 .461 .450	.15 .04	02
	B. Results