Audrey Common
email: acommon@mines-albi.fr

Elisabeth Rodier

Martial Sauceau

Jacques Fages

Biosourced polymer foam production using a (SC CO2)assisted extrusion process

A process based on extrusion coupled with supercritical carbon dioxide (scCO 2 ) was implemented. scCO 2 modifies the rheological properties of the material in the barrel of the extruder and acts as a blowing agent during the relaxation at the passage through the die. An experimental device based on a single-screw extruder allows the injection of scCO 2 into the melt, the mixing of both components and the creation of porosity into the extruded polymer. In this work, it was used to produce foams of two bio sourced, semi-crystalline polymers The effects on material porosity of different parameters such as, temperature before and after the die, pressure drop or CO 2 concentration, were studied. Different configurations on the extruder were tested.

INTRODUCTION

Polymers are widely used in several areas. However, due to their slow degradation and the predicted exhaustion of the world petroleum reserves, significant environmental problems have arisen. Therefore, it is necessary to replace them with biosourced polymers with equivalent properties. It is also important to find new way to process them, which are more environmental friendly.

Extrusion is a process converting a raw material into a product of uniform shape and density by forcing it through a die under controlled conditions [START_REF] Rauwendaal | Polymer Extrusion[END_REF]. It has extensively been applied in the plastic and rubber industries, where it is the most important manufacturing process. A particular application concerns the generation of polymeric foams. Polymeric foams are expanded materials with large applications in the packaging, insulating, pharmaceutical and car industries because of their high strength/weight ratio or their controlled release properties.

Conventional foams are produced using either chemical or physical blowing agents. Various chemical blowing agents, which are generally low molecular weight organic compounds, are mixed with a polymer matrix and decompose when heated beyond a threshold temperature. This results in the release of a gas, and thus the nucleation of bubbles. This implies however the presence of residues in the porous material and the need for an additional stage to eliminate them. Injection of scCO 2 in extrusion process modifies the rheological properties of the polymer in the barrel of the extruder and scCO 2 acts as a blowing agent during the relaxation when flowing through the die [START_REF] Sauceau | Improvement of extrusion processes using supercritical carbon dioxide[END_REF]. The pressure drop induces a thermodynamic instability in the polymer matrix, generating a large number of bubbles. The growth of cells continues until the foam is rigidified (when T<T g or T f ). Moreover, its relatively high solubilisation in the polymer results in extensive expansion at the die. The reduction of viscosity decreases the mechanical constraints and the operating temperature within the extruder. Thus, coupling extrusion and scCO 2 would allow the use of fragile or thermolabile molecules, like pharmaceutical molecules. The absence of residues in the final material is also an advantage for a pharmaceutical application.

Our lab has developed a scCO 2 -assisted extrusion process that leads to the manufacturing of microcellular polymeric foams and already elaborated microcellular foams using a biocompatible amorphous polymer [START_REF] Nikitine | Residence time distribution of a pharmaceutical grade polymer/supercritical CO2 melt in a single screw extrusion process[END_REF][START_REF] Nikitine | Controlling the structure of a porous polymer by coupling supercritical CO2 and single screw extrusion process[END_REF]. In this work, this process has been applied to two semi-crystalline biosourced polymers. Foam production of semi-crystalline polymer is less frequent in the literature. Crystallinity hinders the solubility and diffusion of CO 2 into the polymer and leads consequently to less uniform porous structure [START_REF] Doroudiani | Processing and characterization of microcellular foamed high-density polyethylene/isotactic polypropylene blends[END_REF]. Moreover, it has been shown that a large volume expansion ratio could be achieved by freezing the extrudate surface of the polymer melt at a reasonably low temperature [START_REF] Park | Low density microcellular foam processing in extrusion using CO2[END_REF]. Thus, in this work, we investigate the feasibility of this process with biosourced semicrystalline polymers. We first try to control and improve the porous structure, studying the influence of melt and die temperatures on PHBV, which a copolymer of poly3hydroxybutyrate and hydroxyvalerate. Then, we modify the experimental device in order to experiment the mixing of carbon dioxide into the polymer. We used a different polymer for this study, called BP here, and we looked at the influence of the mixing device and of the carbon dioxide content.

MATERIALS AND METHODS

PHBV (M w =600 kDa), with a HV content of 13 % and plasticized with 10 % of a copolyester was purchased from Biomer (Germany). Melting onset temperature was measured at 159°C by DSC (ATG DSC 111, Setaram) and no glass transition temperature was detected at temperatures as low as -20°C. The solid density € ρ P , determined by helium pycnometry (Micromeretics, AccuPYC 1330) was found to be 1216 kg.m -3 .

Melting onset temperature of the BP polymer was measured at 184°C by DSC (ATG DSC 111, Setaram) and the glass transition temperature was detected at 44°C. The solid density € ρ P , determined by helium pycnometry (Micromeretics, AccuPYC 1330) was found to be 1027± 5 kg.m -3 .

Figure 1 shows the experimental set up, which has previously been detailed elsewhere [START_REF] Nikitine | Residence time distribution of a pharmaceutical grade polymer/supercritical CO2 melt in a single screw extrusion process[END_REF][START_REF] Nikitine | Controlling the structure of a porous polymer by coupling supercritical CO2 and single screw extrusion process[END_REF]. The single-screw extruder has a 30 mm-screw diameter and a length to diameter ratio (L/D) of 35 (Rheoscam, SCAMEX). A great L/D ratio generally indicates a good capacity of mixing and melting but important energy consumption. The screw is divided into three parts. The first one has a length to diameter ratio of 20 and the two others have a length to diameter ratio of 7.5. Between each part, a restriction ring has been fitted out in order to obtain a dynamic gastight which prevents scCO 2 from backflowing. The first conical part allows the transport of solid polymers and then, their melting and plasticizing. Then, the screw has a cylindrical geometry from the first gastight ring to the die. This die has a diameter of 1 mm and a length of 11.5 mm. The temperature inside the barrel is regulated at five locations: T a and T b before the CO 2 injection, T c and T d after the injection and T e in the die.

There are three pressure and two temperature sensors: P 1 after the CO 2 injector, P 2 and T 1 before the second gastight ring and P 3 and T 2 by the die. This allows measuring the temperature and the pressure of the polymer inside the extruder. Errors associated to pressure and temperature measurements were about 2 bars and 3.3°C respectively. CO 2 (N45, Air liquide) is pumped from a cylinder by a syringe pump (260D, ISCO) and then introduced at constant volumetric flow rate. The pressure in the CO 2 pump is kept slightly higher than the pressure P 1 . The CO 2 injector is positioned at a length to diameter ratio of 20 from the feed hopper. It corresponds to the beginning of the metering zone, that is to say the part where the channel depth is constant and equal to 1.5 mm. The pressure, the temperature and the volumetric CO 2 flow rate are measured within the syringe pump. CO 2 density, obtained on NIST website by Span and Wagner equation of state [START_REF] Span R | A New Equation of State for Carbon Dioxide Covering the Fluid Region from the Triple-Point Temperature to 1100 K at Pressures up to 800 MPa[END_REF], is used to calculate mass flow rate and thus the CO 2 mass fraction € w CO 2 . In the second part of the work, this device was modified, replacing the last part of the screw (7.5 L/D) by a mixing element, that is a pineapple mixer with diamond-shaped pins. This one can be seen on Figure 2. Dynamic mixers are often used to improve mixing in single screw extruder. Distributive mixing and dispersing one are commonly distinguished even if they are not physically separated. The component to be mixed here is CO 2 and the first mixing to ensure is the distributive one. Hence, the aim here is to go further in this aspect. We chose a distributive mixer, like the pineapple one, which distributes spatially the phase to be mixed acting on the shear flow. The pineapple has advantages in that the flow turns to be streamlined, the barrel is well wiped by the pins and rather frequent flow splitting occurs [START_REF] Rauwendaal | Mixing in polymer processing[END_REF]. The other type of mixers, the dispersive ones, generate high enough shear to allow breaking agglomerates. There are rather used to disperse a solid phase into a polymer melt. Experimental conditions have to ensure that the polymer flows through the barrel without being thermally degraded. Temperatures have been adapted to each polymer to obtain rapid enough solidification after the die to avoid coalescence of the bubbles and a collapse of the foam structure but also to avoid the blocking of the screw.

As for PHBV, influence of the die temperature was tested. For each experiment, only the temperature of the metering zone T d and of the die T e were changed. The three other temperatures T a , T b and T c were kept constant at 160°C. CO 2 mass fraction € w CO 2 was also kept constant at 1.3 %, which might be less than solubility: in [START_REF] Cravo | Solubility of carbon dioxide in a natural biodegradable polymer: determination of diffusion coefficients[END_REF], for a similar polymer with a HV content of 12%, at 313K and 15 MPa, the solubility of CO 2 was found to be of 55% in weight. The rotation speed of the screw was fixed at 30 rpm. Three series of experiments were carried out. T d was fixed at 140°C, 135°C and 130°C respectively and T e varied from 140 down to 110°C. At lower values of T e , the extruder stopped due to over-limited pressure P 3 according to the established alarm value.

Then we tested the distributive mixing level in the metering zone using another biosourced polymer (BP). The introduction of a mixing element was tested in order to know its influence on the mixing efficiency. In each configuration, CO 2 flowrate was varied and we studied its influence on the porosity of the foams.

Once steady state conditions were reached, extrudates were collected and flowrates were measured. Several samples were collected during the experiment in order to check the homogeneous structure of the extrudates. To calculate the apparent porosity € ρ app , samples were weighed and their volumes were evaluated by a water pycnometer. Porosity, defined as the ratio of the pore volume to total volume is calculated by equation 1:

€ ε = 1- ρ app ρ P (1)
€ ρ P is the polymer density and € ρ app the apparent density of the extrudates. The theoretical maximum porosity € ε max is obtained considering that all the dissolved CO 2 becomes gaseous inside the extrudate at ambient conditions and thus create porosity. It is then calculated by the following equation:

€ ε max = w CO 2 ρ P w CO 2 ρ P + ρ CO2 (atm) (2)
€ w CO 2 is the CO 2 mass fraction and € ρ CO2 (atm) is the CO 2 density at ambient conditions.

Samples were also analysed with an helium pycnometer (Micrometerics AccuPyc 1330).

Helium can reach open pores, so the density measurements allow to evaluate the closed porosity with the same equation 1.

To complete the characterization of the porosity structure, samples were examined by scanning electron microscopy (ESEM, FEG, Philips).

RESULTS

Results obtained with PHBV when testing the die and melt temperatures.

T . The varying operating conditions are given, which are temperature T e in the die and T d before the die, together with the resulting pressure evolution, torque, temperature T 2 measured before the die, and polymer flowrate measured at the die outlet. Rotation speed of the screw is fixed together with CO 2 concentration. In order to investigate the temperature profiles, the Brinkman number, Br, was evaluated: it compares the extent of viscous heat generation relative to the heat conduction resulting from the imposed temperature difference (T Barrel -T product ), T product being the temperature of the product measured before the die, that is T 2 . It is calculated according to

€ Br = η v B 2 λ P T B -T p (3)
where v B is the relative velocity of the barrel, η the molten polymer viscosity, λ P , the molten polymer conductivity. A rheological study, which was previously investigated using a rheometer MARS, Thermo Scientific, showed a pseudo plastic behaviour and the viscosity was taken to be 1630 Pa.s for an average shear rate of 31 s -1 in the pumping zone. λ P of the solid polymer was measured at 150°C and found to be 0.205 W/m/K. in function of temperature of the barrel T d in the pumping zone. Brinkman number is found to be larger than 2, which means that there is a maximum temperature, over the barrel one, at a position interm ediate between the barrel wall and the screw surface. Viscous dissipation is then important in our case. Considering temperature effects at fixed die temperature, porosity decreases when melt temperature increases. This last one influences the CO 2 solubility, which decreases as melt temperature increases. It was previously observed for polystyrene that, at a reasonably low temperature of polymer melt, there exists an optimal die temperature for which large volume expansion ratio are achieved by freezing the extrudate surface [START_REF] Park | Low density microcellular foam processing in extrusion using CO2[END_REF]. This effect is explained because more gas is retained in the foam at lower temperature and used for cell nucleation and growth. However, when the nozzle temperature was further decreased, the volume expansion ratio decreased because of the increased stiffness of the frozen skin layer. In our case, we didn't observe this phenomenon since we couldn't decrease more the temperature because of torque limitation. At fixed melt temperature and as die temperature increases, porosity decreases or even remains constant at melt temperature of 140°C. Note that the highest die temperatures correspond to the lowest pressures before die P 3 , which are around 100±7 bars. Indeed, when depressurization range decreases, nucleation rate decreases [START_REF] Han | Effect of die temperature on the morphology of microcellular foams[END_REF]. These explanations could be satisfying considering the porosity as a global characteristic but they may be incomplete considering the pore size and number.

Going further into the pore structure analysis, it appears that the pore structure differs greatly according to the pressure drop through the metering zone, P 3 -P 2 , as it is shown on figure 5. As P3 is greater than P2 and (P 3 -P 2 ) is positive, CO 2 remains dissolved into the melt until the flow reaches the die where nucleation occurs. This corresponds to the case when melt temperature is 140°C, which is then equal or higher than the die temperature. In addition, no significative effect of the die temperature is then observed. As P3 is lower than P2 and (P 3 -P 2 ) is negative, the mixture may desaturate, nucleation begin in the melt zone and then only growth take place in the die. This induces less pores but bigger ones, leading to a greater porosity. This last one decreases with die temperature when transfer of CO 2 out of the extrudate is enhanced. Endly, it can be noticed that, in any case, porosities measured remain below the maximal one, which was estimated at 90%.

Results obtained with BP when studying the influence of the distributive mixing and the CO 2 concentration on porous structure.

Experiments investigating the influence of carbon dioxide content and the influence of the mixing device were performed. Table 2 presents the operating conditions of each experiment. The first experiment was done with the former unmodified screw. We can see on Figure 6 that the total flowrate decreases with the increase of the CO 2 content and then reaches a constant value above 3% of CO 2 . This transition is also observed considering the structure aspect of the foams. Below 3% of CO 2 , the pores are rather small near the surface of the extrudate and are bigger in the center of the extrudate cross section. This can be explained by coalescence and growth enhanced by a slower cooling in the center than near the surface at the oulet of the die Beyond the same amount of CO 2 (3%), we can see on Figure 7 a change in the porous structure. As the global porosity decreases, the closed one increases significantly and the mean pore diameter decreases. At the highest CO 2 concentration, the whole porosity seems to be closed. Those porosity measurements are in agreement with SEM observations. The increase of CO 2 amount seems to increase the number of nucleation sites. In addition, the increasing cooling effect of the depressurisation through the die freezes the structure as soon as the outlet of the die is reached or even before. The evolution of the porosity with carbon dioxide concentration is consistent with other reported works. Comparing to PHBV results, at low concentration of CO 2 , foams obtained look rather the same. The negative pressure drop (P3-P2) might be the governing process leading to big pores. However, at high CO 2 concentration, porosity levels remain rather high and the porous structures seem to follow the classical statements from the literature: the governing mechanisms are those implied during the depressurization through the die and the cooling rate at the outlet of the die [START_REF] Han | Effect of die temperature on the morphology of microcellular foams[END_REF]. The negative pressure drop (P3-P2) through the metering zone doesn't seem to influence anymore the porosity results: mass and heat transfer kinetics may be significantly different due to the big amount of carbon dioxide.

The second experiment was conducted using the pineapple mixing element. This device greatly changes the pressure profile in the extruder. It increases the pressure gradient from P 2 to P 3 enhancing the shear flow towards the die. It seems to improve the mixing since higher contents of CO 2 can be incorporated into the melt without noticing any failure while processing the mixture. As we can see on Figure 8, the total flowrate is not disturbed by the addition of great amounts of carbon dioxide. Besides, the high porosity level is maintained until 15% of CO 2 . Above this amount, the total porosity is drastically decreased, as it is shown on Figure 9 and the whole porosity is closed. There is far too much CO 2 and the induced freezing at the exit of the die prevents the pores from growing. The pores obtained with the pineapple mixer are less homogeneous in size and are bigger than those obtained with the unmodified screw, at similar CO 2 contents. This could be explained by the pressure profile induced by the mixer compared to the unchanged screw. Pressure drop (P 3 -P 2 ) is also negative but P 2 is much higher, which may induce a significative desaturation of the mixture. Hence, nucleation may take place in the metering zone of the extruder, as it is suspected for PHBV. Therefore, the use of this device may not be adapted to the production of foams with small and numerous pores. Obviously, it allows to incorporate greater amounts of CO 2 but the porous structure obtained is not homogeneous and pores remain rather large. Besides, the high pressure drop induced in the mixer element zone enhances the shear flow towards the die together with a desaturation of the mixture. It also increases the counterpressure flow before this mixing zone forcing some carbon dioxide towards the hopper, which could explain some CO 2 leaks that we occasionally observed. Considering then the higher amounts of CO 2 and the raise in pressure P 2 at the upstream part of the mixing device, not all the carbon dioxide may be processed through the die. Therefore, if great amounts of CO 2 are required in the mixture depending on the process aim, possibly above the solubility value, an additional static mixer could be placed between the screw and the die. The CO 2 introduction could then be splitted between the actual one and another one placed just before the static mixer. This last one might improve again the mixing and allow incorporating carbon dioxide above the solubility value without flowing back through the hopper. Experiments are currently conducted to understand the impact of those mixing devices on the porosity and on the homogeneity of the extrudates.

CONCLUSION

PHBV foaming by extrusion assisted by supercritical fluid is thus feasible according to the temperature profile established. Porosity up to 70% was obtained. However, beyond the temperature profile effect, this foaming seems greatly influenced by the pressure profile, especially by the pressure drop through the metering zone just before the die. Shear flow may be enhanced together with desaturation of the mixture leading to early nucleation before the die. Comparing to the case when counterpressure flow is favoured and the mixture kept saturated, it leads to less pores but bigger ones and to a greater porosity. Acting on distributive mixing level by the introduction of a pineapple element and testing another biosourced polymer led to similar observations concerning the effect of the pressure profile. Foams with porosities up to 85% were obtained but the porous structure is not homogeneous and pores remain rather large. The high pressure drop induced in the mixer element zone enhances the shear flow towards the die together with a desaturation of the mixture. It also increases the counterpressure flow before this mixing zone forcing some carbon dioxide towards the hopper. Hence, this mixing element is not well adapted if dense and small pores are required. Endly, experiments with this biosourced polymer with the unmodified screw, so without the pineapple element, led to results consistent with literature ones. When saturation of the mixture is maintained all along the screw until the die is reached by maintaining a sufficient pressure level, the governing mechanisms are those induced by the depressurisation rate through the die and the cooling rate at the die outlet.

Figure 1 :

 1 Figure 1: Experimental device

Figure 2

 2 Figure 2 Mixing device (pineapple)

Figure 3 :

 3 Figure 3: Brinkman number in function of Td

Figure 4 :

 4 Figure 4: Porosity in function of the die temperature at fixed metering zone temperature (diamonds at T barrel =140°C, circles at T barrel =135°C, crosses at T barrel =130°C)

Figure 5 :

 5 Figure 5: SEM pictures together with porosity results in function of pressure drop between P3 and P2.

Figure 6 :

 6 Figure 6: Evolution of the total flowrate with the mass fraction of CO 2

Figure 7 :

 7 Figure 7 : Evolution of the porosity with CO 2 fraction. Exp1.

Figure 8 :

 8 Figure 8: Evolution of total flowrate with CO 2 fraction. Exp 2.

Table 1 :

 1 Operating conditions and porosity results with PHBV experiments

	Td (°C)	Te (°C)	porosity (%)	T2 (°C)	P1 (bar)	P2 (bar)	P3 (bar)	C (N.m)	Qpolym(g/mn)
	before die	die
	140	110	48.04	128	94	103	120	43	37.4
	140	120	47.36	129	91	96	106	43
	140	125	48.71	130	93	98	104	42
	140	135	45.32	131	91	94	97	43.9
	140	140	47.34	132	95	90	93	42
	135	110
	135	120	57.22	125	106	131	122	44
	135	125	55.48	126	102	122	118	42
	135	135	52.62	129	104	124	102	39	41.3
	135	140	47.21	130	96	109	98	41	44.3
	130	110	69.45	128	107	142	130	43
	130	120	61.46	125	106	131	114	44
	130	125	59.08	126	107	122	107	42
	130	135	63.35	123	107	129	115	40
	130	140	52.11	129.9	106	112	108	41	45.4

Table 2 :

 2 Operating conditions in BP experiments

	Experiment number	Mixing device	Speed Rotation	Ta/b Tc/d	Te	T1	T2	P1	P2	P3
			rpm	°C	°C	°C	°C	°C	bar	bar	bar
	1	Classical screw	40	220 180 175 174 174 65-85 65-85	50
	2	Pineapple mixer	40	240 180 180 183 176 50-70 110-140 60-70

fraction of CO 2 % total flowrate (g/min)

  Figure 9: Evolution of porosity with CO 2 fraction. Exp2.

	20,00
	18,00
	16,00
	14,00
	12,00
	10,00
	8,00
	6,00
	4,00
	0%	2%	4%	6%	8%	10%	12%	14%	16%	18%	20%
	porosity %	10% 20% 30% 40% 50% 60% 70% 80% 100% weight 0% 90%			closed porosity (helium) total porosity (water)
		0%	2%	4%	6%	8%	10%	12%	14%	16%	18%	20%
						weight fraction of CO2 %


Houcine Mhemdi
email: h.mhemdi@live.fr

Rodier Elisabeth

Fages Jacques

± Kechaou

A one-pot process for the extraction of fats and essential oil from coriander seeds by supercritical CO 2

Keywords: Coriander, Vegetal oil, Essential oil, Supercritical CO 2, Extraction

The seeds of coriander (Coriandrum sativum) are rich in vegetal oil (VO) and in essential oil (EO), which have both an interesting potential for the food, pharmaceutical and cosmetic industries. The review of the literature shows that most of the research works related to the extraction of coriander seeds are aimed at extracting either the VO or the EO. The aim of this work is to develop a process that can extract selectively the VO and EO in a consecutive way by tuning experimental conditions. A 4-step process has been set up: (i) seeds preparation by grinding and sieving, (ii) extraction by supercritical CO 2 (SC CO 2 ) and (iii; iv) selective separation in two separators with different pressure level. The first part of this study consisted in studying the effects of the operating conditions (T, P, CO 2 density, flow rate and particle size distribution of ground seeds) on the yield of extraction. The results showed that the global performance of the process depended mainly on the pressure level, on the CO 2 flow rate and on the size of ground seeds. An optimization of the parameters gave an extraction efficiency of 90 %. The second part of this study dealt with the effects of operating conditions on the quality of extracts, and more precisely on their composition in VO and EO. It was observed that the supercritical extraction had no major effect on the fatty acid composition of the VO compared to classical extraction procedures, while EO is enriched in components such as linalol and that a previous drying of the seeds is deleterious. Finally, we proposed a protocol in order to extract separately VO and EO by tuning CO 2 density and separation conditions

INTRODUCTION

Coriander (Coriandrum sativum) is an annual Apiaceae (Umbelliferae) herb, which is widely used in food, pharmaceutical and cosmetic industries [1][2][3]. Coriander fruits contain vegetal oil (VO) with a high concentration of the monounsaturated fatty acid, especially petroselinic acid. This fatty acid can be oxidatively cleaved to produce a mixture of lauric acid, a compound useful in the production of detergents, and adipic acid, a C6 dicarboxylic acid which can be utilized in the synthesis of nylon polymer [4][5][6][7]. Moreover, the essential oil of coriander has been shown to have antibacterial, antioxidant, antidiabetic, anticancerous and antimutagenic activities [8][9][10][11][12][13][14][15][16]. There are three major extraction techniques used to obtain VO and EO from coriander seeds which are steam distillation, organic solvent extraction (soxhlet), and supercritical extraction. Each technique has its benefits and drawbacks as far as operating cost, capital cost, yield and quality of the extracts are concerned. Steam distillation is by far the most widely practiced and cheapest way of extraction, but it is limited to essential oil production only, and can sometimes cause chemical changes to the extract, by oxidation of some compounds. Organic solvent extraction is intermediate in capital and operating cost, and can be used for producing VO and EO from herbs. However, concerns over the solvent residues in the oleoresin products, new regulations concerning the limits of volatile organic solvent emissions in the air, and the extent of further refining that is required after the extraction step begin to limit the use of this technology. Supercritical extraction is the most recent technology that is assuming increasing importance in the production of EO, VO, and a range of other substances from natural products. This technique has the highest capital cost of the three techniques, and moderate operating costs. However, neither solvent residues remain in the product after extraction, nor there are any chemical changes due to the processing technique, which gives extract of outstanding quality [17][START_REF] Monfort | Les applications des fluides supercritiques[END_REF][START_REF] Mira | [END_REF][20]. The review of literature shows that most of the research works related to the extraction of coriander seeds are aimed at extracting either of the VO or the EO. The aim of this work is to develop a process using supercritical carbon dioxide, which is able to extract selectively the VO and EO in a consecutive way by tuning experimental conditions.

MATERIALS AND METHODS

Raw material

Coriander that was used during this research came from Canada. It was provided by the laboratory of General Herbalism, Marseille, France. The coriander seeds are kept in a refrigerator at 4°C before use. Their mass content of VO and EO, respectively determined by Soxhlet and steam distillation are 20.8% of VO and 0.4% of EO.

Particle size reduction

To study the effect of particle size on extraction yield, coriander seeds are first crushed. The grinding is performed using a grinder type IKA M20. The particle diameter after size reduction was estimated by using sieves (200 µm, 400 µm, 630 µm, 800 µm and 1 mm) to separate a known initial sample mass into weight fractions according to their corresponding particle size range.

Supercritical carbon dioxide extraction

Extraction experiments on coriander seeds were carried out on the pilot scale extraction apparatus (Separex, France) shown schematically in Figure 1. The pilot scale tests were performed by flowing carbon dioxide at the required temperature and pressure through a bed of ground coriander fraction contained in the extractor. The pressure in the extractor was controlled by a backpressure regulator and temperature was regulated by an electrical device. The carbon dioxide and dissolved solute was then depressurized causing precipitation of VO into the first cyclonic separator S1 which was held at a pressure of 70 bar and a temperature of 30°C. Then carbon dioxide flowed through a needle valve causing precipitation of the remaining EO and water into the second cyclonic separator S2. This last vessel was maintained at a pressure of 40 bar and a temperature of 10° C. Samples of extract were withdrawn at regular intervals (15 min) from the separation vessels to give the time course of the extraction of oils

Characterization of the extract : Gas Chromatography

Fatty acid composition of Vegetal Oil

The fatty acid composition of the coriander VO was determined by analysis of their methyl esters obtained by transesterification. These esters were analyzed using a gas chromatograph type Varian CP-select CB 3900 equipped with a silica capillary column (50 m x 0.25 mm x 0.25 m). Temperature was held at 185 °C for 4 min, raised to 250 °C at 15 °C/min. The injection and the detector temperature was 250 °C. The carrier gas was H 2 (1.2 ml/min). The device was equipped with a FID detector, which gave the chromatogram for each sample. The identification of fatty acids was made by comparing their retention times with those of standard samples.

Composition of Essential Oil

The separation of volatile components of the coriander EO was performed using a gas chromatograph capillary type HP 5890 series II, equipped with a DB-5MS column (30 m x 0.25 mm x 0.25 m), a detector of flame ionization (FID) and injector EPC (Electronic Pressure Control). The tests were conducted according to the following conditions. The column was held at 40 °C for 5 min and then increased at 5 °C / min up to 280 ° C over 53 min. The solvent was dichloromethane and the flow through the column was 1.3 ml/min. The FID detector was held at 300 °C. The head pressure of the column was 1.034 bar and the amount injected was 0.5 l. The identification of different constituents of EO is made by calculating the retention index (RI) and Kovats index (KI) and comparing them with those of the literature.

RESULTS

Our process consists in four steps, which are respectively grinding, sieving, extraction and fractionation. Indeed, coriander seeds are first ground. Then the particle size distribution was measured by sieving and the fractions collected. The mass of ground and sieved coriander seeds used in each experiment was approximately 50 g. After the extraction step, a yellow vegetal oil, slightly viscous was recovered in separator 1 (70 bar and 30°C) and a fragrant essential oil is collected in separator 2 (40 bar and 10°C). Samples of extract were removed at regular intervals (each 15 min) from the separation vessels in order to follow the kinetics of extraction. The experiment was stopped once the amount extracted remained constant and a plateau was reached on the extraction curve. Different operating conditions were used to study the influence of the pressure, the temperature, the density and the flow rate of CO 2 and the mean particle size of ground seeds to be processed on the yield and the composition of the extract (Table 1 andTable 2). The VO and the EO obtained were also compared with those obtained respectively by Soxhlet and steam distillation using the same particle size fraction (Table 2).

Effect of the operating conditions on extraction yield

The results are analysed according to:

 The Extraction Rate (ER) that is the total mass of extract recovered at time t from 100 g of coriander. It is expressed in % (g of extract / 100 g coriander)  The Extraction Yield (EY) that is the percentage of oil extracted from the total quantity of oil contained in the seeds of coriander

2.1.1-Influence of pressure

The influence of two different pressures (100 and 210 bar) on the yield of extraction was studied at 35°C, a mean particle size of ground seeds of 0.5 mm and a supercritical fluid flow rate of 1 kg/h (Table 1 and figure 2). Analysis of the extraction curves (Figure 2) shows that the increase of pressure enhances the kinetics and greatly reduces the extraction time while increasing the extraction efficiency. Indeed, as the pressure increases from 100 to 210 bar, the extraction yield rises from 23% to 85% and the time needed to achieve the extraction decreases from more than 300 min to 200 min. These effects could be attributed to the increase of the CO 2 density, which improves its solvent power and then the solubility of the solute. The effect of pressure on extraction has been observed by several research teams who have worked on the extraction of oil seeds by supercritical CO 2 [8][9][10][11][12][13][14][15]. In addition, to go further in the enhancement of the extraction kinetics, diffusivities of the solute transferred into the supercritical fluid were evaluated using the mathematical model of Owen and al [7]. Indeed, the values of diffusivities at a pressure of 100 bar and 210 bar were respectively 0.5 x 10 -13 m 2 /s and 1.06 x 10 -13 m 2 /s. When the pressure increases from 100 bar to 210 bar, the amount of extracted EO remains the same while the amount of HV extracted increases significantly. Data from the literature [4] show that for a solvent density less than 500 kg / m3, only EO can be extracted while a higher density is required to extract the VO. The extraction of EO can then be performed at a moderate enough pressure of 100 bar while the VO extraction requires higher pressures.

Influence of temperature

The influence of the temperature was studied using the following extraction conditions of pressure, flow rate, and particle size: 210 bar, 1 kg/h and 0.3 mm (Table 1 and figure 4) The temperature did not have a significant effect on the extraction rate. At 35°C, the extraction yield was 88% and at 55°C it was 91%. This limited or even lacking effect of the temperature could be explained by two opposite phenomena that occur simultaneously. Indeed, the increase of temperature induces a drop of the solvent density and of the solvent power, so the rate of extraction decreases. In the same time, the rise in temperature induces an increase of the solute vapour pressure and therefore its solubility in the supercritical fluid, so the extraction efficiency increases. These two opposite effects are competing. It has been often observed that the effect of vapour pressure prevails and the result is a positive effect on the extraction rate [START_REF] Brunner | Gaz Extraction[END_REF]. In our case, the balance of both effects appears without significant impact.

Influence of density of CO 2

The study of the effect of pressure and temperature on the extraction led to study the influence of solvent density on the extraction yield. 5 extraction tests were performed for 5 different values of density and under the same conditions of particle size (0.3 mm) and flow of CO 2 (1 kg/h) (Table 1 and Figure 5). The extraction yield drastically increases with density, that is from 9% for a density of 400 kg / m 3 to 90% for a density value of about 800 kg / m 3 . This increase was due to the enlargement of the solvent power of the carbon dioxide with its density. However, increasing the density above 800 kg / m 3 improves the extraction efficiency Reducing the particle size from 0.9 mm to 0.5 mm improves the extraction yield from 23% to 90%. This can be attributed to the increase of the contacting area and to the decrease of the transfer resistance. The diffusivity is estimated to be 1.08 x 10 -13 m 2 / s for particle size of 0.3 mm while the one for a diameter of 0.9 mm is smaller and equal to 0.1 x 10 -13 m 2 / s

Influence of the particle size

Influence of flow rate

To study the effect of solvent flow rate on extraction efficiency, three extraction tests were performed under the same operating conditions with three different flow rates of CO 2 which are respectively 1 kg / h, 1.5 kg / h and 2 kg / h. (Table 1 and figure 7) Increasing the flow of CO 2 accelerates the kinetics of extraction. Indeed, the time required to reach the plateau decreased from 160 minutes for 1 kg / h to 80 minutes for 2 kg / h. However, the extraction yield decreases from 90% to 61% when the flow of CO 2 increases to 2 kg / h. This result may be explained by the reduction of the contact time between the carbon dioxide and the bed of coriander during the extraction step.

Effect of the operating conditions on the composition of VO and EO

Vegetal oil of coriander

Different samples of VO are analyzed by GC and their fatty acid composition is shown in Table 2.

Table 2. Fatty acid composition of different samples of vegetal oil

The VO of coriander is very rich in unsaturated fatty acid: about 95% are unsaturated, 80% are monounsaturated and 15% are polyunsaturated. The major fatty acid is petroselinic acid (>70%). These values are similar to the results found by Msaada and Holmes [5,6].Comparison between the composition of the VO obtained by organic solvent extraction (Soxhlet) and that obtained by supercritical CO 2 shows that both fatty acid profiles are similar. However, we note that 10 fatty acids have been identified from VO obtained by supercritical CO 2 , while only 9 have been identified from the VO extracted by Soxhlet, which does not contain arachidic acid but it is richer in gadoleic acid (0.10%) than the VO extracted by supercritical extraction (0.39%). We note also that operating conditions (T; P, Q and d) have no major effect on the fatty acid composition of VO.

Essential oil of coriander

The composition of the EO obtained after the supercritical extraction and the one obtained by steam distillation were determined. The EO extracted from seeds of coriander, which were previously dried at 70 °C and 600 mbar for 24 h was also analyzed.

Results show that EO extracted by supercritical CO 2 contains 29 compounds, whereas 35 compounds were identified in the case of extraction by steam distillation. This could be attributed to hydrolysis and / or oxidation of certain compounds of the EO during the extraction by steam distillation. We note also that EO extracted by supercritical CO 2 is richer in linalool and in limonene (66% and 3%) than that obtained by hydrodistillation (57% and 0.02%). Drying coriander seeds before extraction affects the quality of the EO. Indeed, only 16 compounds were identified after drying. This could be explained by the evaporation of some of the volatile compounds during drying.

CONCLUSION

Coriander seeds were extracted using supercritical carbon dioxide to obtain selectively vegetal oil and essential oil. A process in 4 steps has been set up. A range of pressure, temperature, particle diameters and flow rates were used to study the effect of operating conditions on the yield extraction and the composition of the extract. The results showed that pressure, particle size and flow rate have a significant effect on the extraction yield and they have no major effect on the composition of VO and EO. The VO obtained is rich in petroselinic acid, while linalool is the major compound of the EO. 210 bar and 35 ° C as extraction conditions together with a superctitical fluid flow rate of 1 kg/h and a mean size of 0.5 mm for the ground seeds were the best conditions tested to extract vegetal oil of good quality and to maximize the extraction yield (90%). In addition, 100 bar in the extractor are enough to extract selectively essential oil.

  Nomenclature t = time (s) P = Pressure (bar) T = temperature (° C) M = Mass of extract (g) d = diameter (m) ℓ = Density (kg / m³) Q = Flow rate of CO ₂ (kg / h) D = diffusion coefficient (m² / s) EY = Extraction Yield (%) SC = SuperCritical SCFE = SuperCritical Fluid Extraction VO = Vegetal Oil EO = Essential Oil FA = Fatty Acid SFA = Saturated Fatty Acid MUFA = MonoUnsaturated Fatty Acid PUFA = PolyUnsaturated Fatty Acid ER = Extraction Rate (%)

Figure 1 .

 1 Figure 1. Schematic of pilot scale extraction apparatus. 1-CO 2 cylinder; 2-condenser; 3-CO 2 pump; 4-co-solvent pump; 5-refrigerator tank; 6-refrigerator pump; 7-heaters; 8-mass flowmeter; 9-extractor; 10-heating baths; 11-pumps of heating systems; 12-recirculation pump; 13-separators (S1 and S2); 14-filter; Pr-pressostat; PT-pressure transmitter, TT-temperature transmitter.

Figure 2 .

 2 Figure 2. Effect of pressure on the extraction rate versus time for T=35°C, Q=1kg/h and d=0.5mm

  To confirm this observation, an extraction test in two steps was performed in order to extract first the EO and second the VO at two different pressures (80 bar and 210 bars) and thus at two different densities (Figure 3): -First Step: 80 bar and 35 ° C, which corresponds to a CO 2 density of 419.9 kg/m 3 -Second Step: 210 bar and 35 ° C, which corresponds to a density of 873.67 kg/m 3

Figure 3 .

 3 Figure 3. Effect of pressure on the selectivity of extraction in two steps Figure 3 shows that during the first extraction step (80 bar), only the EO was recovered while in the second stage (210 bar) only VO was recovered. These results confirm those of literature, the previous comments about the effect of pressure and density of CO 2 on the selectivity of the extraction of EO and HV. We can conclude that the extraction in two steps allows to extract selectively the VO and the EO from coriander seeds.

Figure 4 .

 4 Figure 4. Effect of temperature on the extraction rate for P=210 bar, Q=1kg/h and d=0.3mm

Figure 5 .

 5 Figure 5. Effect of the density of CO₂ on the extraction yield for Q=1kg/h and = 0.3mm

Figure 6 .

 6 Figure 6. Effect of the particle size on the extraction rate for P=210 bar,T=35°C & Q=1kg/h

Figure 7 .

 7 Figure 7. Effect of the flow rate on the extraction yield versus time for 210 bar, 35°C and 0.3 mm

Table 1 .

 1 Supercritical fluid extraction experiments : operating conditions used

Acknowledgments: this work is part of a research project financed with the help of Région Midi-Pyrénées which is gratefully acknowledged. The authors thank the team from LCA-ENSIACET Toulouse and especially Mr Thierry Talou for the chromatographic analyses.


N Kalboussi
email: nesrinekalboussi@gmail.com

A Rapaport
email: alain.rapaport@inra.fr

T Bayen

N Ben Amar

F Ellouze
email: ellouze_fatma@yahoo.fr

J Harmand
email: jerome.harmand@inra.fr

E Ellouze

Optimal control of membrane filtration systems

Keywords: Membrane filtration process, Physical backwash strategy, Optimal Control, Pontryagin Maximum Principle, Singular Arcs

This paper studies an optimal control problem related to membrane filtration processes. A generic mathematical model of membrane fouling is used to capture the dynamic behavior of the filtration process which consists in the attachment of matter onto the membrane during the filtration period and the detachment of matter during the cleaning period. We consider the maximization of the net water production of a membrane filtration system (i.e. the filtrate) over a finite time horizon, where the control variable is the sequence of filtration/backwashing cycles over the operation time of process. Based on the Pontryagin Maximum Principle, we characterize the optimal control strategy and show that it exhibits a singular arc. Moreover we prove the existence of an additional switching curve before reaching the terminal state, and also the possibility of having a dispersal curve as a locus where two different strategies are both optimal.

I. INTRODUCTION

Membrane filtration systems are widely used as physical separation techniques in different industrial fields like water desalination, wastewater treatment, food, medicine and biotechnology. The membrane provides a selective barrier that separates substances when a driving force is applied across the membrane. Different fouling mechanisms are responsible of the flux decline at constant transmembrane pressure (TMP) or the increase of the TMP at a constant flux. Hence, the operation of the membrane filtration process requires to perform regularly cleaning actions like relaxation, aeration, backwashing and chemical cleaning to limit the membrane fouling and maintain a good filtrate production. Usually, sequences of filtration and membrane cleaning are fixed according to the recommendations of the membrane suppliers or chosen according to the operator's experience. This leads to high operational cost and to performances that can be far from being optimal.

A variety of control approaches have been proposed to manage filtration processes. In practice such strategies were based on the application of a cleaning action (physical or chemical) when either the flux decline through the membrane or the TMP increase crosses predefined threshold values ( [START_REF] Ferrero | Automatic control systems for submerged membrane bioreactors: A state-of-the-art review[END_REF]). Smith et al. developed a control system that monitors the TMP evolution over time and initiates a membrane backwash when the TMP exceeds a given set-point, [START_REF] Smith | A new approach to backwash initiation in membrane systems[END_REF]. In [START_REF] Hong | A new approach to backwash initiation in membrane systems[END_REF] the TMP was also used as the monitoring variable but the control action was the increase or decrease of membrane aeration. The permeate flux was used in [START_REF] Vargas | Controlled backwashing in a membrane sequencing batch reactor used for toxic wastewater treatment[END_REF] as the control variable to optimize the membrane backwashing and prevent fouling. Moreover, knowledge-based controllers found application in the control of membrane filtration process. In [START_REF] Robles | Advanced control system for optimal filtration in submerged anaerobic MBRs (SAnMBRs)[END_REF], Robles et al. proposed an advanced control system composed of a knowledge-based controller and two classical controllers (on/off and PID) to manage the aeration and backwash sequences. The permeability was used by [START_REF] Ferrero | Automatic control system for energy optimization in membrane bioreactors[END_REF] as a monitoring variable in a knowledgebased control system to control membrane aeration flow. To date, different available control systems are able to increase significantly the membrane filtration process performances. However, more enhanced optimal control strategies are needed to cope with the dynamic operation of the purifying system and to limit membrane fouling. The majority of the control strategies previously cited address energy consumption, but regulation and control have not being proved to be optimal.

In the present work, we consider the maximization of the net production (i.e. the filtrate) per area of a membrane filtration system over a given operation duration. The control variable is the direction of the flow rate: forward for filtration through the membrane and backward for backwashing attached foulants. This problem is quite generic for various fluids to be filtered Membrane fouling is assumed to be only due to the particle deposition onto the membrane surface while pores blocking is neglected. The aim of the present work is to give a complete optimal synthesis in a quite generic way (i.e. without giving the exact expressions of the functions involved in the model) characterizing the occurrence of such singularities. The analysis of these singularities (cf. [START_REF] Bonnard | Singular Trajectories and their role in Control Theory[END_REF]) is important for practical implementation because it gives the structure of the control strategies to be applied (how many switches, where or when to switch...) and the information (i.e. which variable and when) that is needed to be measured.

In Section II, we present the model and state the optimal control problem, with preliminary results about the structure of the optimal control near to the terminal time. Section III is devoted to the analysis of singular arcs (existence and optimality). In Section IV, we show that a switching curve appears and moreover that a phenomenon of dispersion along this curve may occur. We then give a complete optimal synthesis, which is illustrated in Section V on two filtration models of the literature.

II. MODEL DESCRIPTION AND PRELIMINARY RESULTS

We consider a simple form of the model of [START_REF] Benyahia | HARMAND A simple model of anaerobic membrane bioreactor for control design: coupling the "AM2b" model with a simple membrane fouling dynamics, 13[END_REF] to describe the membrane filtration process. In a previous work, it was shown that this model is very generic in the sense that it is able to capture the dynamics of a large number of models available in the literature while simple enough to be used for optimizing and control purposes, see [START_REF] Kalboussi | A comparative study of three membrane fouling models -Towards a generic model for optimization purposes[END_REF]. In the present work, it is assumed that the membrane fouling is only due to the particle deposition onto the membrane surface. Let m be the mass of the cake layer formed during the water filtration (m ≥ 0), that is assumed to follows a dynamics ṁ = f 1 (m) with f 1 : R + → R + . We further assume that the physical cleaning of the membrane is performed by a backwashing which consists in reversing the flow. During this phase, the filtration is stopped and the mass detaches from the membrane surface with a dynamics ṁ = -f 2 (m) with f 2 : R + → R + . The system is operated by alternating the filtration and backwash modes: we consider a control u that takes values 1 during filtration and -1 during retro washing, as follows:

ṁ = 1 + u 2 f 1 (m) - 1 -u 2 f 2 (m) with m(0) = m 0 , (1)
where m 0 ≥ 0 is the initial mass. The objective is to determine an optimal switching between the two modes which maximizes the net water production of the membrane filtration process during a time interval [0, T ]. Assuming that the flux of water through the membrane during forwards and backwards operation is given by a function g : R + → R that depends on m, the net amount of water per area of membrane is

J T (m 0 , u(•)) = T 0 u(t)g(m(t))dt.
to be maximized. Let us recall that sufficient conditions for the existence of an optimal control require the extended velocity set to be convex (see for instance [START_REF] Lee | Foundations of Optimal Control Theory[END_REF]). As the dynamics and the integrand are both linear w.r.t. to the control, this amounts here to require that u(•) takes values in the interval [-1, 1], what we consider in the following. The question of practical applicability of a control that takes values different to -1 and 1 relies on approximations with chattering controls [START_REF] Zelikin | Theory of Chattering Control with applications to Astronautics[END_REF]. Hence, we will focus in this paper on the following optimal control problem:

max u(•)∈U J T (m 0 , u(•)),
where U denotes the set of measurable functions over [0, T ] taking values in [-1, 1]. We consider the following hypotheses on the model. Hypothesis 2.1: f 1 and g are smooth L-functions 1 . f 2 is a smooth K-function 1 .

Let us comment about these hypotheses. i) When a membrane operates in filtration, the resistance to flow is never zero and increases according to the mass m of the cake layer formed on the membrane surface, which subsequently decreases the permeate flux. Thus, we assume that the rate f 1 at which the mass of material adheres to the membrane surface during filtration is a positive decreasing function. ii) When starting membrane backwash, the cake layer is decomposed and the 1 We recall that

f from R + to R + if a K-function if it is increasing with f (0) = 0, and a L-function if it is decreasing with lim x→+∞ f (x) = +∞.
membrane's permeability increases again. So, the speed f 2 of the cake detachment can be described by a positive increasing function. When the membrane is clean (m = 0), there is nothing to be detached: f 2 (0) = 0. iii) At constant TMP, the permeate flux decreases as the extent of fouling gradually increases. Therefore, the variation of the permeate flux J can be described by a decreasing positive function of the mass of the fouling layer.

Under Hypothesis 2.1, one obtains straightforwardly the following property.

Lemma 2.1: The domain {m > 0} is positively invariant whatever is the control u(•).

For convenience, we define the functions f + : R + → R + and f -: R + → R defined by

f + (m) := f 1 (m) + f 2 (m) 2 , f -(m) := f 1 (m) -f 2 (m) 2 ,
thus the dynamics can be equivalently written

ṁ = f -(m) + uf + (m), u ∈ [-1, 1].
We shall use the Maximum Principle of Pontryagin (PMP) [START_REF] Pontryagin | The Mathematical Theory of Optimal Processes[END_REF] to obtain necessary conditions. Defining the Hamiltonian

H(m, λ, u) = λf -(m) + u (λf + (m) + g(m)) , (2)
the PMP states2 that if a pair (u, m) is optimal, then i. there exists an absolutely continuous adjoint variable λ solution of the adjoint equation λ = -∂H ∂m for a.e. t ∈ [0, T ], with the terminal condition λ(T ) = 0. ii. the control u satisfies the Hamiltonian condition

u(t) ∈ arg max ω∈[-1,1] H(x(t), λ(t), ω) for a.e. t ∈ [0, T ]. iii. t → H(m(t), λ(t), u(t) is constant A triplet (x(•), λ(•), u(•)
) that verifies i-ii-iii is called an extremal, and the corresponding x(•) an extremal trajectory.

Here, the adjoint equation is

λ = -λf -(m) -u λf + (m) + g (m) , a.e. t ∈ [0, T ] (3) with    u = +1 when φ(m, λ) > 0, u = -1 when φ(m, λ) < 0, u ∈ [-1, 1] when φ(m, λ) = 0, ( 4
)
where φ is the switching function

φ(m, λ) := λf + (m) + g(m).
Proposition 2.1: An extremal satisfies λ(t) < 0 for any t ∈ [0, T [. Moreover, for any initial m 0 there exists t < T such that the control u(t) = 1 is optimal for t ∈ [ t, T ].

Proof: λ = 0 implies φ(m, 0) = g(m) > 0, u = 1 and λ = -g (m) > 0. To have λ(T ) = 0, λ has to be negative for t < T . As t → φ(m(t), λ(t)) is positive at T , it remains positive on an interval [ t, T ] and u = 1 on this interval.

III. SINGULAR ARC AND FIRST OPTIMALITY RESULTS

For convenience, we define the functions

ψ(m) := g(m) f -(m)f + (m) -f -(m)f + (m) +g (m)f + (m)f -(m) γ(m) := - g(m)f -(m) f + (m)
We now consider the following hypothesis:

(m(•), λ(•), u(•)) satisfies m(t) = m and u(t) = ū, t ∈ [t 1 , t 2 ], where ū := - f -( m) f + ( m) . ( 5
) Moreover, λ(•) is constant equal to λ where λ ∈ R is defined by λ = - g( m) f + ( m) . (6)
Proof: One can easily check that it is possible to factorize φ and ψ in the expression of φ as follows:

φ = ψ f + + φ f + f --f -f + f + , (7)
where for simplicity we wrote φ for the derivative of t → φ(m(t), λ(t)) and we dropped the m dependency of functions f -, f + and g. The conclusions follow from [START_REF] Cogan | A method for determining the optimal back-washing frequency and duration for dead-end microfiltration[END_REF].

We show now that for certain initial conditions the optimal solution consists in a most rapid approach path to m. This property is also known as exact turnpike in the literature [START_REF] Rapaport | Turnpike theorems by a value function approach[END_REF], [START_REF] Élat | The turnpike property in finite-dimensional nonlinear optimal control[END_REF], [START_REF] Faulwasser | Exact turnpike properties and economic NMPC[END_REF]. However, Lemma 2.1 shows that an optimal solution that reaches m has to leave it before the terminal time, at a time T that we make explicit in the following proposition. Proposition 3.1: Suppose that Hypothesis 3.1 hold true and let m 0 > 0 be an initial condition. Then, the following properties are satisfied:

i. When m 0 < m, the control u = 1 is optimal as long as the corresponding trajectory satisfies m(t) < m, ii. When m 0 > m, either the control u = 1 is optimal until t = T , or the control u = -1 is optimal until a time

t < T with m( t) ≥ m. If m( t) > m then u = 1 is optimal on [ t, T ]. iii. Suppose that f -( m) ≥ 0.
Then, for any initial condition m 0 ≥ m, an optimal control satisfies u = -1 over some time interval [0, t] with t ∈ [0, T ] and u = 1 over [ t, T ]. iv. Suppose that f -( m) < 0 and let T ∈ R be defined by

T := T - mT m dm f 1 (m) with mT := g -1 (γ( m)). (8)
Then, if T > 0, any singular trajectory is optimal until t = T . Proof: From Hypothesis 3.1 and ( 7), we can deduce that when φ(m) = 0 with m < m then φ < 0. This implies that φ can change its sign only when decreasing. Therefore only a switching from u = 1 to u = -1 can be optimal in the domain {m < m}. When φ(m) = 0 with m > m then φ > 0. This implies that φ can change its sign only when increasing. Therefore, only a switching point from u = -1 to u = 1 can be optimal in the domain {m > m}.

Let us prove i. Take m 0 < m, and suppose that the control satisfies u = -1. It follows that the trajectory remains in the domain {m < m}. From Proposition 2.1, the trajectory necessarily has a switching at a time t c (otherwise, we would have u = -1 until the terminal time t = T and a contradiction) implying φ(t c ) ≥ 0. On the other hand, we deduce from [START_REF] Cogan | A method for determining the optimal back-washing frequency and duration for dead-end microfiltration[END_REF] that φ(t c ) = ψ(m(tc)) f+(m(tc)) < 0 which is a contradiction. Hence, we must have u = 1 in the domain {m < m}.

The proof of ii is similar as in the domain {m > m}, any optimal trajectory has at most one switching from u = -1 to u = 1. It follows that only three cases may occur: either u = 1 is optimal over [0, T ], or the trajectory reaches m = m at some instant t < T , or it has exactly one switching in the domain {m > m} from u = -1 to u = 1.

Let us prove iii. If one has u = 1 at time 0, then the result is proved with t = 0. Suppose now that one has u = -1 at time 0. We know that if the trajectory switches at some time t ∈ [0, T ] before reaching m = m, then one has u = 1 for t > t and the result is proved. Suppose now that an optimal trajectory reaches the singular arc before t = T and that one has m(t) = m on a time interval of non-null length. Since the Hamiltonian is constant along any extremal, one must have H = λf -( m). Moreover, as the Hamiltonian at time T is given by H = g(m(T )), one should have λf -( m) = g(m(T )) > 0. As λ < 0, we conclude that when f -( m) ≥ 0, this situation cannot occur. Hence, a singular arc is not optimal.

Finally, let us prove iv and suppose that f -( m) < 0. Accordingly to Propositions 2.1 and 3.1, an optimal control satisfies u = 1 in a left neighborhood of t = T . Let us compute the last instant T < T (if it exists) until a singular arc is possible. From the previous analysis, we necessarily have u = 1 on [ T , T ]. This imposes (using that the Hamiltonian is constant) the final state to be mT = m(T ) as a solution of

g( mT ) = λf -( m) = - g( m)f -( m) f + ( m) = γ( m), (9)
which is uniquely defined as g is decreasing, lim m→+∞ g(m) = 0, and

-f-( m) f+( m) ∈ [0, 1]
. This also imposes that the switching time T can be determined integrating backward the Cauchy problem ṁ = f 1 (m), m(T ) = mT until m( T ) = m, which amounts to have the expression [START_REF] Cogan | Optimal backwashing in dead-end bacterial microfiltration with irreversible attachment[END_REF].

We now show that an extremal trajectory leaving the singular arc m = m at a time t < T cannot be optimal. To do so, consider a trajectory m(•) leaving the singular arc at a time t < T (necessarily with u = 1 until the terminal time T ).

In particular, we have m(T ) > mT . Since the dynamics is ṁ = f 1 (m) with u = 1, the corresponding cost from time t can be written as follows:

J 1 (t) := m(T ) m g(m) f 1 (m) dm = mT m g(m) f 1 (m) + m(T ) mT g(m) f 1 (m) ,
to be compared with the cost J s (t) of the singular arc strategy from time t (i.e. u = ū over [t, T ] and then u = 1 over [ T , T ]), which is equal to

J s (t) := - g( m)f -( m) f + ( m) ( T -t) + mT m g(m) f 1 (m) dm.
Thanks to [START_REF] Cogan | Optimal backwashing in dead-end bacterial microfiltration with irreversible attachment[END_REF] and

T -t = m(T ) m dm f1(m) , we get T -t = (T -t) - mT m dm f 1 (m) = m(T ) m dm f 1 (m) - mT m dm f 1 (m) = m(T ) mT dm f 1 (m)
.

The difference of costs δ(m(T )) := J 1 (t) -J s (t) can be then written as:

δ(m(T )) = m(T ) mT g(m) + g( m)f -( m) f + ( m) dm f 1 (m)
Let us now study the behavior of δ as a function of m(T ).

For convenience, we write m in place of m(T ) and recall that m ≥ mT since m(T ) ≥ mT . By a direct computation, one has:

δ (m) = g(m) + ᾱ f 1 (m) , δ (m) = g (m)f 1 (m) -(g(m) + ᾱ)f 1 (m) f 2 1 (m)
,

where ᾱ := g( m)f-( m) f+( m)
. From this last expression, since g < 0, one has at each m > 0:

δ (m) = 0 =⇒ δ (m) < 0.
Now, it is to be observed that δ( mT ) = 0 and that δ ( mT ) = 0 (from (9))). The previous analysis then shows that δ < 0 on ( mT , +∞). It follows that δ is decreasing over [ mT , +∞). Hence, we obtain that δ(m) < 0 for any m > mT . To conclude, we have proved that J 1 (t) < J s (t) at any t ∈ [0, T ), thus any singular trajectory is such that it is optimal to stay on the singular arc until T (and then use u = 1 up to T ).

Remark 3.1: One can easily check that Hypothesis 3.1 implies that ( m, ū) is the unique steady state that maximizes the integrand g(m )u among steady states (m , u ), and that the dynamics is strictly dissipative for the supply rate w(m, u) = g( m)ū-g(m)u (with the storage function

S(m) = m m -g(ξ)
f + (ξ) dξ. The turnpike property of the optimal solution can be deduced from [10, Th. 2], but here we characterize explicitly the last switching time T . Considering the following disjoint sub-domains:

D -:= {(t, m) ∈ [0, T ] × [0, m]}, D + := {(t, m) ∈ [0, T ] × [ m, +∞)}.
one deduces straightforwardly the following result.

Corollary 3.1: Under Hypothesis 3.1, are verified: i. When f -( m) ≥ 0, u = 1 is optimal at any (t, x) ∈ D -. ii. When f -( m) < 0 with T ≤ 0, where T is defined in [START_REF] Cogan | Optimal backwashing in dead-end bacterial microfiltration with irreversible attachment[END_REF], u = 1 is optimal at any (t, x) ∈ D -. iii. When f -( m) < 0 and T ∈ (0, T ) with T defined in [START_REF] Cogan | Optimal backwashing in dead-end bacterial microfiltration with irreversible attachment[END_REF],

u = 1 if m < m or t ≥ T , ū if m = m and t < T ,
is optimal at any (t, x) ∈ D -. iv. D + is optimally invariant (i.e for any initial condition in D + , an optimal trajectory stays in D + at any time).

IV. SWITCHING LOCUS AND FULL SYNTHESIS

A. Study of the switching locus in D + Accordingly to Proposition 3.1, a switching outside the singular arc can occur only in D + and from u = -1 to u = 1. When f -( m) < 0, consider the (possibly empty) curve in D + :

C := ( T ( m), m) | m ≥ m and T ( m) > 0 , (10)
where T : [ m, +∞) → R is the function defined by

T ( m) := T - g -1 (γ( m)) m dm f 1 (m) , m ≥ m. ( 11
)
Proposition 4.1: Assume that Hypothesis 3.

1 is fulfilled. i. When f -( m) ≥ 0, u = 1 is optimal for (t, m) ∈ D + . ii. When f -( m) < 0, one as: a. If C is empty, u = 1 is optimal for (t, m) ∈ D + . b. When C is non-empty, u = -1 if (t, m) ∈ W, ū if m = m and t < T , 1 otherwise. (12)
is optimal, where the domain W is defined as

W := (t, m) ∈ [0, T )×] m, +∞) | t < T (m) . (13)
Furthermore, C is tangent to the trajectory that leaves the singular arc at ( T , m) with the control u = 1. Proof: Suppose that f -( m) ≥ 0 and let us prove i. We only have to show that any optimal control satisfies u = 1 in D + . In this case, we know that no singular arc occurs, therefore it is enough to exclude switching from u = -1 to u = 1 in D + . Also, since one has u = 1 in a neighborhood of t = T , it is enough to consider terminal states m T ≥ m. By integrating backward the dynamics with u = 1, one has H = g(m T ) = g(m(t)) + λ(t)f 1 (m(t)) for t < T as long as the switching function

φ(m, λ) = g(m) + λf + (m) = g(m) + (g(m T ) -g(m)) f + (m) f 1 (m) = f + (m) f 1 (m) (g(m T ) -γ(m)) , (14)
remains positive. As f -(m) ≥ 0, one has γ( m) ≤ 0. Notice also that for m ≥ 0, one has

γ (m) = - ψ(m) f + (m) 2 , ( 15
)
so that γ is increasing over [0, m] and decreasing over [ m, +∞). We deduce that γ(m(t)) ≤ 0 for any t ∈ [0, T ]. Consequently, φ cannot change its sign, and u = 1 is optimal at any time. Suppose now f -(m) < 0 and let us prove ii. Again, we consider terminal states m T ≥ m. Note that when m T = mT , one has g( mT ) = γ( m) by conservation of the Hamiltonian.

Consider now an initial state m T > mT and the system backward in time with u = 1. If an optimal control is such that u = 1 until reaching the singular arc, we deduce (by ( 14)) g(m T ) -γ( m) < g( mT ) -γ( m) = 0, (since g is decreasing). Thus, the switching function is negative when m is reached backward in time. By the mean value Theorem, we conclude that there necessarily exists a switching at some value m > m such that γ( m) = g(m T ), and accordingly to Proposition 3.1 this switching (from u = -1 to u = 1) is unique. From the monotonicity of γ on [ m, +∞), for each m T > m, m is uniquely defined by m = γ -1 (g(m T )), or reciprocally, for any m ≥ m, m T is uniquely defined as a function of m: m T ( m) = g -1 (γ( m)) (as g is also a decreasing invertible function), with

m T ( m) = γ ( m) g (m T ( m)) ≥ 0. ( 16
)
Then, the corresponding switching time T ( m) satisfies

T -T ( m) = m T ( m) m d f 1 (m) . ( 17
)
If T ( m) ≤ 0 then no switch occurs at m i.e. the constant control u = 1 is optimal from 0 to T . It follows that if C is empty, then u = 1 is optimal in D + as was to be proved. When a switching occurs, that is when C is non-empty, the previous analysis shows that it occurs on the curve of D + given by ( 10) and the corresponding switching times are given by ( 11) as was to be proved. The optimality of the feedback control [START_REF] Ferrero | Automatic control systems for submerged membrane bioreactors: A state-of-the-art review[END_REF] follows by noting that in D + , optimal trajectories have at most one switching point u = -1 to u = 1 or from u = -1 to ū.

Finally, the derivative of T with respect to m can be determined from expressions [START_REF] Kalboussi | Optimal control of physical backwash strategy -towards the enhancement of membrane filtration process performance[END_REF] and [START_REF] Kalboussi | Optimal control of a membrane filtration system[END_REF] as

T ( m) = 1 f 1 ( m) - m T ( m) f 1 (m T ( m)) = 1 f 1 ( m) - γ ( m) g (m T ( m))f 1 (m T ( m))
.

At m = m, one has T ( m) = T and γ ( m) = 0 (since ψ( m) = 0), which gives T ( m) = 1/f 1 ( m) > 0. Thus, the parameterized curve C is indeed tangent to the trajectory that leaves the singular arc with u = 1 at ( T , m). Remark 4.1: When f -( m) < 0 and T > 0, ( T , m) belongs to C which is then non-empty. C could be a set of disjoint curves (for instance if the function T has several changes of sign). However, in the examples we met, it is always a single curve (bounded or not), see Section V. Notice also that the map m → T ( m) has no a priori reason to be monotonic, as it is shown in examples in Section V.

B. Dispersal locus

When the set C is non-empty (under the condition f -( m) < 0), consider the partition:

C = C s C d with C s := (t, m) ∈ C ; 1 + T (m)f 2 (m) > 0 , C d := (t, m) ∈ C ; 1 + T (m)f 2 (m) ≤ 0 .
Corollary 4.1: Assume that Hypothesis 3.1 is fulfilled with f -( m) < 0 and C defined in Proposition 3.1 non empty.

• C s is not reduced to {( T , m)} and is a switching locus.

• C d (when it is non-empty) is a dispersion locus i.e. from any state in C d the two trajectories 1) with u = 1 up to the terminal time, 2) with u = -1 up to reaching m = m or C s , are both optimal. Proof: The domain W (when it is not empty) is exactly the set of points (t, m) ∈ D + for which the optimal control satisfies u = -1 (see Proposition 3.1). From such a state, the optimal trajectory has to leave the domain W (as ṁ is bounded from above by -f 2 ( m) < 0 in this set) reaching either the singular arc or the set C. At some point (t, m) in C, an outward normal n to W is then given by

n(t, m) = 1 -T (m) ,
and the velocity vectors v -1 , v 1 for the control u = -1 and u = 1 respectively are

v -1 (t, m) = 1 -f 2 (m) , v 1 (t, m) = 1 f 1 (m)
.

Notice that by construction of the set C, the velocity vector v 1 points outward of W at any point (t, m) ∈ C. Hence, the velocity vector v -1 points outward when the scalar product n • v -1 is positive, that is when (t, m) belongs to C s .

We consider now optimal trajectories that reach C from W and distinguish two cases. 1) At states in C s , the velocity vectors v -1 , v 1 both point outward of the set W. Therefore an optimal trajectory reaching C s with u = -1 leaves it with u = 1. Then, accordingly to Proposition 2.1, the optimal control stays equal to 1 up to the terminal time.

2) At states in C d , v -1 points inward of W while v 1 points outward. Therefore an optimal trajectory cannot reach a point located on C d . From states in C d , there exist two extremal trajectories: one with u = 1 up to the terminal time, and another one with u = -1 up to the singular arc or to the curve C s (accordingly to Propositions 2.1 and 3.1) and then u = 1 up to the terminal time. At these points, the Hamiltonian dynamics of (m, λ) is an upper semi-continuous multi-valued map with convex compact values, and therefore its solutions are continuous with respect to the initial condition (see for instance [START_REF] Filippov | Differential Equatiions with Discontinuous Righthand Sides[END_REF]). Moreover, the value function of a Lagrange problem with smooth data is everywhere Lipschitz continuous (see for instance [START_REF] Bardi | Optimal Control and Viscosity Solutions of Hamilton-Jacobi-Bellman Equations[END_REF]). For initial condition on each side of C d , we have shown that the trajectory with u = -1 or u = 1, depending on the side, are optimal. Therefore, by continuity w.r.t. to the initial condition, the two extremal trajectories leaving the set C d have the same (optimal) cost (and are thus optimal). Finally, let us show that C s is not reduced to a singleton. The state ( m, T ) belongs to C (as it is indeed a point where the switching function vanishes) but it also belongs to the singular arc m = m. So, there exists a trajectory with u = -1 that crosses C transversely at this point. By continuity of the solutions with u = -1 w.r.t. the initial condition, we deduce that there exist locally other trajectories that cross the nonempty curve C transversely with the control u = -1.

Figure 1 illustrates the two kinds of points in C. Remark 4.2: For a problem with free terminal state and smooth data, the value function V is Lipschitz continuous (see for instance [START_REF] Bardi | Optimal Control and Viscosity Solutions of Hamilton-Jacobi-Bellman Equations[END_REF]) but not necessarily differentiable. When there is no dispersion locus i.e. C d = ∅, the extremals, or equivalently the characteristics of the HJB equation, do not intersect in the (t, m) plane. Then the relation between the adjoint variable and the sub-and super-differentials of V (see [START_REF] Yong | Stochastic controls, Hamiltonian Systems and HJB equations[END_REF]) allows to show that V is indeed differentiable. However, in presence of dispersion locus, the lack of uniqueness of optimal trajectories leads to the non differentiability of V [START_REF] Frankowska | On singularities of value function of Bolza optimal control problems[END_REF].

C. Feedback synthesis

We reformulate Corollary 3.

u[t, m] = -1 if (t, m) ∈ W, ū if m = m and t < T , 1 otherwise. ( 18
)
is an optimal feedback, where W is defined in [START_REF] Filippov | Differential Equatiions with Discontinuous Righthand Sides[END_REF].

V. TWO NUMERICAL CASE STUDIES A. The Benyahia et al model

The next functions have been experimentally validated [START_REF] Benyahia | HARMAND A simple model of anaerobic membrane bioreactor for control design: coupling the "AM2b" model with a simple membrane fouling dynamics, 13[END_REF]:

f 1 (m) = b e + m , f 2 (m) = am, g(m) = 1 e + m ,
d( m) = f -( m)f + ( m)-f -( m)f + ( m) = -g ( m)f + ( m) g( m) f -( m).

B. The Cogan-Chellam model

The following functions have been proposed [START_REF] Cogan | A method for determining the optimal back-washing frequency and duration for dead-end microfiltration[END_REF], [START_REF] Cogan | Optimal backwashing in dead-end bacterial microfiltration with irreversible attachment[END_REF]

f 1 (m) = b e + m , f 2 (m) = am e + m , g(m) = 1 e + m ,
where a, b and e are positive numbers. Clearly Hypothesis 2.1 is fulfilled. Moreover, one has

ψ(m) = (am -b) 2 -2abe -2b 2 4(e + m) 4
Therefore, the function ψ can have two changes of sign at

m-= b - √ 2b 2 + 2abe a , m+ = b + √ 2b 2 + 2abe a ,
where mand m+ are respectively negative and positive numbers. One has also

ψ (m) = a 2 em + abe + abm 2 (e + m) 5
, which is positive. Therefore ψ is an increasing function and Hypothesis 3.1 is fulfilled with m = m+ . Moreover one can write

f -( m) = - √ b 2 + 2abe e + m < 0.
Then, as for the previous model, Proposition 3.1 and Corollary 4.1 allow to conclude that there exists a singular arc when T > 0 and a switching locus when T ( m) > 0. Figure 3 gives the optimal feedback synthesis, where C is spitted into two non-empty subsets C s and C d .

C. Discussion

Although the two models are very close and have similar optimal syntheses, a main difference occurs on the size and on the shape of the domain W where backwash has to be applied (see Figures 2,3). In particular, its boundary C is entirely a switching curve in one case while most of it is a dispersal curve in the second case. This should give valuable information to the practitioners about when and how long backwashing (i.e. u = -1) has to be applied out of the singular arc. Notice that the duration T -T in (8) is independent of T . For arbitrary large horizon T , the simpler strategy which consists in reaching as fast as possible the singular arc and stay on it for ever gives a good approximation of the optimal average cost, which moreover converges to to the optimal value in infinite horizon. For the practical implementation of the optimal control (where only the values u = -1 and u = 1 can be applied) it is not possible to stay exactly on the singular arc m = m, but an approximation by a sequence of filtration/backwashing can be applied to stay on the vicinity of the singular arc. This sequence can be chosen so that the average value of m is m, which provides a good approximation of the optimal cost as it has been tested in [START_REF] Kalboussi | Optimal control of a membrane filtration system[END_REF], [START_REF] Kalboussi | Optimal control of physical backwash strategy -towards the enhancement of membrane filtration process performance[END_REF]. One may argue that the problem could be reformulated in discrete time where the time step is the smallest period of switching between filtration and backwashing that could be applied in practice. We believe that this approach gives less geometric insights of the nature of the optimal control than the continuous formulation. Moreover, computing the optimal cost in continuous time gives an upper bound of what could be intrinsically expected from the process, independently of its practical implementations.

VI. CONCLUSION

The Pontryagin Maximum Principle has been applied to a membrane filtration model and shows interesting results for maximizing the net water production (per filtrate). The optimal synthesis exhibits bang-bang controls with a most rapid approach to a singular arc and a switching curve before reaching the final time. We have shown that a dispersal locus may occur, leading to the non-uniqueness of optimal trajectories. Practically, the determination of the singular arc allows to compute a sequence of filtration/backwashing to stay about the singular arc, and the determination of the curve C provides the information about the domain where backwashing has to be applied. The synthesis also reveals that if one wants to implement a feedback controller, which is more robust than an open-loop one, the on-line measurement of the mass deposit m or of any invertible function of m, such as the water flow rate, is crucial. The main advantage of the present analysis is to describe an optimal synthesis for a very large class of models relying on simple qualitative properties of the functions f 1 , f 2 and g.

Perspectives of this work are first to implement the optimal synthesis with real process constraints, and compare the water production (per filtrate) of the membrane filtration process with the classical operating strategies that are proposed in the literature and currently used. Extensions to other fluids or non constant TMP and consideration of multiple objectives (production and energy consumption) could be also the matter of future works, as well as possibilities of multiple singular arcs.

Hypothesis 3 . 1 : 3 . 1 :

 3131 The function ψ admits an unique positive root m and is such that ψ(m)(m -m) > 0 for any m = m. Under Hypothesis 3.1, one can characterize m = m as the unique candidate singular arc.Lemma Consider a singular arc defined over a time interval [t 1 , t 2 ]. Then the corresponding singular extremal

Figure 1 .

 1 Figure 1. Switching point (left) versus dispersion point (right) on the set C.

4 . 1 :

 41 1 and Proposition 4.1 in terms of feedback u[ ] : (t, m) ∈ [0, T ] × [0, +∞) → u[t, m]. Theorem Assume that Hypothesis 3.1 is fulfilled. i. If f -( m) ≥ 0 or f -( m) < 0 and C empty, then u[t, m] = 1 is optimal for any (t, m) ∈ [0, T ]×(0, +∞). ii. If f -( m) < 0 and C non-empty, then

  where a, b and e are positive numbers. One can check that Hypothesis 2.1 is fulfilled. A straightforward computation of the function ψ gives the following expressiona 2 e 2 m 2 +2a 2 em 3 +a 2 m 4 -2abe 2 -6abem -4 abm 2 -b 2 4 (e+m)4 and a further computation of its derivative a 2 e 3 m+2a 2 e 2 m 2 +a 2 em 3 +abe 2 +5abem + 4abm 2 +2b 2 2 (e+m) 5 which allows to conclude that ψ is increasing on R + . As one has ψ(0) = -(2abe 2 + b 2 )/(4e 4 ) < 0 and lim m→+∞ ψ(m) = +∞, we deduce that Hypothesis 3.1 is fulfilled. When ψ is null for m = m, one has

  Therefore f -( m) and d( m) have the same sign. A straightforward computation gives d(m) = -ab (e + 2 m) 2 (e + m) 2 < 0, and thus one has d( m) < 0. Therefore, from Proposition 3.1 and Corollary 4.1, there exists a singular arc when T > 0 and a switching locus when T ( m) > 0. Figure 2 gives the optimal synthesis, where C is entirely a switching locus (i.e. C = C s ).

Figure 2 .

 2 Figure 2. Optimal synthesis for model V-A with a = b = e = 1 and T = 10 hours. The set W is depicted in blue and the switching locus in yellow.

Figure 3 .

 3 Figure 3. Optimal synthesis for model V-B with a = b = e = 1 and T = 40 hours. The set W is depicted in blue, the switching locus in yellow and the dispersion locus in gray.

As the terminal state is free, the abnormal multiplier λ 0 is not considered.

http://www.inra.fr/treasure

Acknowledgments. The authors thank the TREASURE research network 1 for its financial support. The third author would like to thank the MIA Division of INRA and the MISTEA lab, Montpellier for providing him a half year delegation during the academic year 2017-2018.


published or not. The documents may come

Introduction

This book looks at the French language through corpora, and comprises four parts dealing respectively with diachrony, syntax, sociolinguistics and issues arising in the learning and teaching of French. Each part is headed by a chapter that provides an overview of the given field in relation to the themes running through the book. Other than contributing to our general understanding of the French language today, this book specifically addresses the use of corpora for the study of language and the links between tools, methods and analyses. How do we use corpora? What are the underlying theoretical and/or methodological considerations? How have these changed our way of formulating linguistic descriptions? What are the implications for descriptive accounts of French? What are the applications of corpus studies?

Each chapter focuses on specific aspects of French and addresses (often indirectly) issues such as the ways in which researchers use existing resources, the reasons for producing new resources, or questions arising from different types of data use. One aim is to challenge or complete existing work, not least in relation to the possibilities that are now made available through corpus use.

Corpora provide data , and a common theme throughout the book is one of empiricism. A distinction is sometimes drawn between corpus-based and corpus-driven approaches, and this is apparent here: a corpus-based study looks to gather findings that test a certain idea or model, whatever its scope or ambition; a corpus-driven study is more ecological insofar as it aims to build conclusions on the sole basis of the findings (Tognini-Bonelli 2001). These are perhaps two ends of a continuum, and in many cases there will be a continuous interaction between data consultation and the questions one has, each influencing the other at all stages, from corpus compilation through to final outcomes. But in all cases it is important not to lose sight of the fact that language is not just a neutral collection of data, hence the inclusion of the concept ecology in the title.

Ecology has become something of a buzz-word of late in many spheres. The rationale for the ecological leitmotif here derives from a shared interest in language and environment: ecology is to do with understanding language not as an abstract system but as an integral part of human existence (Haugen 1972). Ecology also refers to the authentic nature of the data and issues to do with their collection, transcription and editing, mark-up and tagging, or any other treatment resulting from human intervention whether deliberate or not.

Corpora are used nowadays in many areas of linguistics (and beyond), often for purposes not anticipated by the original designers. Burnard (2002: 67), for example, readily admits that the linguists behind the landmark BRITISH NATIONAL CORPUS (BNC) never imagined that it would be bought by individuals, nor that it could be of any direct use for teachers and learners. Nonetheless, where large corpora are now available freely on line, teachers and learners tend to make up the majority of users, as is the case for the BYU or Lextutor interfaces to several large corpora in different languages. In some cases, corpora have been used as simple repositories of linguistic 'facts', almost as archives or databases to dip in to when needed and in whatever way required. This approach can be useful in philology, for example, where the corpora of available texts are often limited and difficult to gather together into a single coherent corpus. Such activity may be considered peripheral to corpus linguistics as an identifiable discipline -or even by some as not true corpus linguistics at all. It does, however, have the virtue of quietly filtering corpus linguistics into 'the rest of linguistics', thereby making it more accessible to a wide research community and extending the influence of its methods and results among the general field. This has also given rise to a debate about whether corpus linguistics represents 'merely' a methodology precisely because it is open to so many uses in so many fields (McEnery et al . 2006: 8), or something rather more (Tognini-Bonelli 2001), given its tremendous impact on all that we know about language use -not for nothing do McCarthy (2008) and others talk of a corpus revolution.

Turning specifically to French, Gadet (2009: 115) points out that while "for many contemporary linguists, 'doing linguistics' means working with corpora […] the study of French distinguishes itself in this respect from that of other languages". She gives two reasons for the comparative underuse of corpora in French studies: the fact that sizeable corpora for French have been late in coming in comparison to other languages (in Europe, at least); and the noncentrality of many existing French corpora (it is of note that there is no French national corpus). Gadet does concede, though, that the "broadening of the data" over the last few decades has brought about a number of interesting changes, as different research methodologies and theoretical backgrounds find common ground. This is particularly true perhaps for the study of spoken grammar (Blanche-Benveniste 2007: 129), an area where French corpus work has proved particularly innovative through the work of the GARS (Groupe aixois de recherches en syntaxe) school of linguists. Blanche-Benveniste notes that this is beneficial for grammatical description of the language as a whole, both in terms of suggesting new methods and for discovering new phenomena.

Though we might optimistically imagine the future will hold greater collaboration between sectors of linguistics, positions can become entrenched and exchanges rather heated, as in the "bootcamp debate" (see Worlock Pope 2010) between two camps of corpus linguiststhose who are keen to see interaction with cognitive linguistics, and those who are sceptical of any compromise on what may be perceived as an epistemological or even ethical divide between empiricism and intuition/introspection. Corpus linguists may be understandably wary, having been branded as "butterfly collectors" (Chomsky 1979) lacking theory -corpus linguistics as a field simply "doesn't exist" claims Chomsky in an interview reported in Aarts (2001: 5). On the other hand, it is perhaps worth remembering that linguistics has been through a number of pendulum swings over the last century or so (for reasons of technical and methodological possibilities and indeed for cultural or political reasons as much as for theoretical reasons), and that it may be more profitable to seek common ground and collaboration between different approaches (e.g. Fillmore 1992). In other words, should butterfly collectors hunt in a protected area or should they open up to different areas and points of view? Should linguists who do not go in for butterfly collecting become interested in the findings of their collector colleagues, and vice versa? Answers to these questions (any many more) represent a major challenge for the ecology of the linguistics world. As Labov (1971: 119) pointed out several decades ago now, "it is not necessary for everyone to use the same methods -indeed it is far better if we do not […] Data from a variety of distinct sources and methods, properly interpreted, can be used to converge on right answers to hard questions". Labov also draws our attention to the "cumulative principle" of linguistic research, whereby "the more that is known about language, the more we can find out about it" (p. 98). And so it is that this book offers a series of studies that come together in their concern for furthering our understanding of the French language, of its uses, its forms, its variation, its acquisition, etc., as well as in promoting ecological approaches to using corpora for studying these questions.

Bibliography

  Henry Tyne, Virginie André, Alex Boulton, Christophe Benzitoun, Yan Greub To cite this version: Henry Tyne, Virginie André, Alex Boulton, Christophe Benzitoun & Yan Greub. (2014). Introduction to In H. Tyne, V. André, A. Boulton, C. Benzitoun & Y. Greub (eds), Ecological and Data-Driven Perspectives in French Language Studies. Newcastle: Cambridge Scholars, p. 250-256.

  Aarts, B. 2001, "Corpus linguistics, Chomsky and fuzzy tree fragments", In C. Mair & M. Hundt (eds.), Corpus Linguistics and Linguistic Theory, Amsterdam: Rodopi, 5-13 Blanche-Benveniste, C. 2007, "Corpus de langue parlée et description grammaticale de la langue", Langage et société 121-122(3), 129-141 Burnard, L. 2002, "Where did we go wrong? A retrospective look at the British National Corpus", In B. Kettemann & G. Marko (eds.), Teaching and Learning by Doing Corpus Analysis, Amsterdam: Rodopi, 51-70 Chomsky, N. 1979, Language and Responsibility (based on conversations with Mitsou Ronat), New York, Pantheon Fillmore, C. 1992, "Corpus linguistics or computer-aided armchair linguistics", In J. Svartik (ed.), Directions in Corpus Linguistics, Berlin: Mouton de Gruyter, 35-60 Gadet, F. 2009 "Stylistic and syntactic variation: introduction", In K. Beeching, N. Armstrong & F. Gadet (eds.), Sociolinguistic Variation in Contemporary French, Amsterdam: John Benjamins, 115-120 Haugen, E. 1972, The Ecology of Language: Essays by Einar Haugen (ed. A. S. Dil), California: Stanford University Press Labov, W. 1971, "Some principles of linguistic methodology", Language in Society 1, 97-120 McCarthy, M. 2008, "Accessing and interpreting corpus information in the teacher education context", Language Teaching 41(4), 563-574 McEnery, T., R. Xiao & Y. Tono 2006, Corpus-Based Language Studies: An Advanced Resource Book, London: Routledge Tognini-Bonelli, E. 2001, Corpus Linguistics at Work, Amsterdam: John Benjamins Worlock Pope, C. (ed.) 2010, "The bootcamp discourse and beyond", International Journal of Corpus Linguistics 15(3)

This is a pre-publication version. For the version of record or full book, please see http://www.cambridgescholars.com/french-through-corpora (or email me at alex.boulton@atilf.fr)