Engineering and Manufacturing for Biotechnology - Marcel Hofman & Philippe Thonart

J.-F. Cornet, C.G. Dussap and J.-J. Leclercq

integration over the working illuminated volume in the reactor (Cornet et al., 1992;

1995 and 1998).

The aim of this paper is to show how, with considerable theoretical efforts, it is possible to obtain general knowledge models with a reduced number of empirical coefficients. These models can then be used in a predictive manner as powerful tools for simulation, design or control of photoreactors. The paper gives an overview of theoretical and experimental results obtained in the last decade in the field of PBR modelling.

2. Modelling photobioreactors

Modelling PBRs appears as a difficult task because of the heterogeneity of the radiation

phase function for scattering and have different values over a 4n solid angle. This leads to a complex formulation of the problem in order to calculate the local radiant light energy available from the radiative transfer theory.

It is therefore necessary to formulate the coupling between light energy transfer in

PBRs and local biomass growth rates and stoichiometries, leading to define zones in which metabolic activity occurs, and to volumetrically average local kinetic rates. This paper presents several experimental and theoretical developments obtained for two micro-organisms, Spiru/ina platens is and Rhodospirillum rubrum, cultivated in different PBRs.

2.1. RADIATIVE TRANSFER FORMULATION

Solving the general tridimensional form of the equation of radiative transfer in the given geometry of a reactor is a very difficult task. This requires Monte Carlo or finite element methods (Spadoni et 1978; Aiba, 1982; Siegel and Howell, 1992; Cornet et al., 1994) which are highly time consuming, limiting such an approach to accurate simulation of PBRs. Nevertheless, as the authors previously showed, many practical cases can be formulated with monodimensional approximation for radiative transfer (Cornet et al., 1992; 1995 and 1998). For example, a generalised two-flux method, derived from the assumptions of Schuster (1905), can be used with a sufficient accuracy if the radiative coefficients are properly determined (Brewster and Tien, 1982; Wang and Tien, 1983; Koenigsdorff et al., 1991, Cornet et al., 1995; Brucato and Rizzuti, 1997). The main advantages for using a simplified monodimensional approximation are first that analytical solutions to the radiative transfer problem exist (Cornet et al., 1995); and second that only mean values in intensities are used, corresponding to the physical quantities required in modelling the process.

Let introduce the total radiant light energy available at a point over the solid angle ω and the mean quantities over the considered visible spectrum by

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In this case, the profile of radiant energy available for the

micro-organisms in a cylindrical reactor of radius R, radially illuminated with a mean incident flux FQ is given by:

where:

We have introduced the volumetric absorption and scattering Schuster coefficients A and S, easily related to actual coefficients a and s by:

(4)

S = 2s

It is clear that these coefficients are mean coefficients in wavelength on the considered spectrum for a given micro-organism, obtained by:

Moreover, the coefficient b, appearing in equations (2-3) is the back-scattered fraction of light, obtained from the phase function of the medium and given by the Lorenz-Mie theory (Brewster and Tien, 1982; Wang and Tien, 1983; Koenigsdorff etal., 1991); that is:

where quotes indicates the scattered direction. This integral corresponds only to an incident beam parallel to the r-axis, and in fact, it is necessary to take into account all the incident directions leading, for an equivalent sphere, to the double integral (Koenigsdorff et al., 1991):

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J.-F. Cornet, C.G. Dussap and J.-J. Leclercq

Finally, in equation (1), the mean incident flux describing the boundary condition of the radiative transfer problem is clearly a key parameter. It can be obtained either by

Obviously, equation (1), which has been obtained for different geometries (rectangular, cylindrical, spherical, annular region...- see Cornet et al., 1995 -) is fully predictive if the coefficients a, s and b, called the optical properties of the medium, can be obtained theoretically. For a micro-organism, the absorption coefficient a is an intrinsic property which depends on the pigment content, and can be calculated by convolution for each wavelength from data banks, once this content is known. The Lorenz-Mie theory then provides an excellent basis to compute the wavelength dependent scattering coefficients

and the phase function

2.2. COMPUTING THE OPTICAL PROPERTIES

From the basic electromagnetic characteristics of the micro-organism, i.e. the refractive index of the medium and the complex refractive index of the particle m - n + K i, the Lorenz-Mie theory enables to calculate, with tedious computation (Van de Hulst, 1981; Bohren and Huffman, 1983), the optical properties necessary to formulate the radiative transfer model. The wavelength dependent properties are obtained from the definition of the size given by:

is the considered wavelength in the vacuum at which the computation is performed.

The details of these computations are not given in this paper, but as example, results were obtained for the calculation of the scattering efficiencies QsCA f°r tw° microorganisms, Rs. rubrum and S. platensis (Figure 1). The actual computation requires to know the size distribution f(x) for the corresponding micro-organism, enabling the assessment of the mean scattering efficiency from:

The scattering efficiency is then easily related to the scattering volumetric coefficient by:

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The calculation was performed for two extreme cases, a non-absorbed wavelength (perfect dielectric) and a wavelength at a maximum of absorption (from the highest value of K), It clearly appears that, in the range of interest for the size parameter in the visible spectrum, most of the micro-organisms can be considered as a perfect dielectric with a maximum deviation less than 10% (Figure 1). This enables to use a simplified engineering equation for the calculation of the scattering efficiency available when the ratio of refractive indexes tends to 1 (Van de Hulst, 1981, Cornet et al., 1996):

where and which is probably one of the most famous and useful relation in this field.

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J -F. Cornet, C.G. Dussap and J.-J. Leclercq

These results can be used to determine the wavelength dependent absorption and scattering mass coefficients defined by where

is the biomass concentration. These coefficients should be used in equation (4) instead of the volumetric coefficients because in batch cultivations, the biomass concentration is time dependent. An example of this determination was performed for Rs. rubrum, on which is obtained experimentally and is theoretically computed from equation (11) (Cornet et at., 1996) (Figure 2). Typically, these results enable to perform the integration of equation (5), and to determine mean coefficients in wavelength.

Finally, from the Lorenz-Mie theory (Bohren and Huffman, 1983), we have computed phase functions for the same micro-organisms and in the same extreme conditions (Figure 3). Clearly, as it is well known, the phase function for scattering of micro-organisms is strongly peaked in the forward direction From these data, relation (7) gives respectively for Rs. rubrum and S. platensis,

and

2.3. COUPLING RADIATIVE TRANSFER WITH RATES AND STOICHIOMETRY

Once a correct formulation of the radiative transfer has been done, it is necessary to

of the produced biomass. As an example, we give here theoretical results

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experimentally validated and obtained from the biochemically structured approach, using the phenomenological thermodynamics of irreversible processes (Dussap, 1988, Cornet et al., 1998). For a considered available light energy, this approach leads to the following structured stoichiometric equation for S. platensis:

in which the number of quanta is obtained from the thermodynamically calculated value of the in vivo P/2e" ratio (Cornet et al., 1998). Because this kind of equation can be theoretically established for each value of it is fully predictive for calculating the stoichiometric energetic conversion yield from the number of photons involved in the reaction. For example, we obtained from equation for S. platensis and with a similar equation, for Rs. rubrum.

Actually, photosensitised reactions does not operate at the maximum conversion yield, because, the primary quantum yield strongly decreases when increases. One can postulates thatis given by a relation of the form:

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J.-F. Cornet, C.G. Dussap and J.-J.

in which pmax is the well-known maximum quantum yield at the thermodynamics optimum. The coefficient K is a characteristic of the pigment content of the photosynthetic antenna and remains difficult to obtain theoretically. Nevertheless, this coefficient, which is the sole to obtain experimentally for the proposed model, can be easily determined (Cornet et al. 1998). Then, the local biomass volumetric reaction rate in the illuminated volume of the reactor is given by:

Because, the kinetic coefficients previously discussed are only valid in the illuminated zone of the reactor, it is obvious that the mean averaged observed volumetric rate in the reactor will be given by (Cornet et 1998 and 2000):

The working illuminated volume and the illuminated fraction are easily obtained from equation (1) (Cornet et al., 1995). The dark operative volume fraction has been introduced for photoheterotrophic micro-organisms, as Rs. rubrum, because in this case, a metabolic activity can occur for short residence time of cells in darkness, from a reverse electron transfer (Cornet et al., 2000). It is also easily determined from the appearance of a constant growth rate on batch cultures (Cornet et al., 2000). Finally, the illuminated surface fraction enables to describe cases in which only part of the photoreactor is illuminated, as it is often the case on industrial reactors.

3. Results and discussions

The formulation of the radiative transfer problem on a physical basis, together with a correct understanding of the coupling between light transfer, stoichiotnetry and rates at the level of the cell, then integrated at the scale of the whole process, leads to the proposed knowledge model. This approach presents at least the two following advantages:

•it is not specific of a given micro-organism or of a photoreactor geometry, so it is quite general;

•it is fully robust and predictive (only one coefficient of the model remains to be experimentally determined).

Consequently, this model can be successfully used to simulate experimental results in PBRs where the monodimensional approximation is justified.

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3.1. SIMULATION AND DESIGN

During the last decade, the above model has been applied on different geometries

(rectangular, cylindrical, annular region), mixing types (rushton turbine, air-lift,...) and volumes (1 to 100 litres) of artificial PBRs operating in batch and continuous mode with S. platensis and Rs. rubrum, and with incident fluxes varying by a factor 100 (4 to The standard deviation observed never exceeded 10% (Cornet, 1997; Cornet et al., 1992, 1994; 1995, 1998 and 2000). It succeeded also as a basic tool for scaling up by a factor 10 an airlift PBR with a constant productivity.

Figure 4 shows an example of comparison between model and experimental results on a 10 L cylindrical photobioreactor radially illuminated and during a step in incident light flux Clearly, the agreement between the biomass concentration and the predictive values of the model is excellent, both on steady and transient states.

It must be emphasised that more refined numerical tools exist regarding the radiative transfer formulation if a high accuracy is required for local simulation and design. They use Monte Carlo or finite elements methods with wavelength dependent coefficients

1978; Aiba, 1982, Cornet et al., 1994).

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3.2. MODEL BASED PREDICTIVE CONTROL

Model based control of processes requires predictive models with short calculation times. For this reason, the proposed model (leading to analytical solution for radiative transfer) appears as a good candidate to be used as a model based predictive control of PBRs. Very good results have then been obtained (Leclercq, 1998; Cornet et al., 1999) with the two previous micro-organisms involved in the MELiSSA project of ESA

(Figure 5).

4. Conclusions and perspectives

A model for simulation, design and model based predictive control of PBRs was presented and discussed. It relies on a monodimensional formulation of the radiative transfer problem using a generalised two-flux method, providing analytical solutions for available light energy profiles inside the reactor in a given geometry. The optical properties of the medium were shown to be theoretically obtained from the Lorenz-Mie theory, giving the model's coefficients by a predictive mean. This approach was used to define the coupling between the available light energy and both the local kinetics and stoichiometry. This led to the concept of working illuminated volume inside the reactor, allowing to calculate the mean volumetric growth rates in biomass.

Such a physically and biochemically consistent model appears fully predictive because calculations can be performed from the experimental knowledge of only one coefficient relative to the primary quantum yield of a given micro-organism.

Simulations of many experimental results proved the model very robust. Thus, good results were also obtained in model based predictive control of PBRs.

In a near future, this approach could be improved by focusing attention and developing theoretical tools in three main directions:

•further investigations about biochemically structured metabolism are necessary for different metabolic conditions (photoautotrophy with one and two photosystems, photoheterorrophy). This requires advanced formulations in the domains of

metabolic network analysis for photosynthetic micro-organisms together with energetics and irreversible thermodynamics analysis of photosynthesis;

•the formulation of the coupling between kinetic rates and radiative transfer in cases where transient states exist with short time dark efficient zones in the PBR (as it is the case for photoheterotrophic metabolisms) needs also to be further investigated. Using populations balances formalism then could be a good alternative for this;

•optical radiative properties remain difficult to assess for many photosynthetic micro-organisms, and there is a great necessity in developing data banks of in vivo pigment absorption properties, together with analytical and numerical tools for

computation of Lorenz-Mie series with actual shapes of micro-organisms

(cylinders, spheroids,...).

The elucidation of these different points is a prerequisite step for the demonstration that, besides their industrial interests, photosynthetic micro-organisms for which, at a given complexity, knowledge models can proceed with a lesser number of degree of

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freedom than for heterotrophic micro-organisms, are ideal case study for advanced modelling in the field of biochemical reactors.

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