Mouna Lazrag

Cecile Lemaitre

Christophe Castel

Ahmed Hannachi

Danielle Barth
email: danielle.barth@univ-lorraine.fr

Cécile Lemaitre

Aerogel production by supercritical drying of organogels: experimental study and modelling investigation of drying kinetics

Keywords: Organogel, supercritical drying, aerogel, diffusion -convection, CFD

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Introduction

1.1.Processes used to dry gels

The drying of a gel containing a solvent is operated by extracting the interstitial solvent. This process allows to produce a dry material with a highly porous structure and so a low thermal conductivity. Because of the risk of degrading the product solid structure, this operation is delicate. Different drying processes have been developed and used for inorganic gels such as silica gels: freeze drying, evaporative drying and supercritical drying (Fig. 1) [START_REF] Bisson | Drying of Silica Gels to Obtain Aerogels: Phenomenology and Basic Techniques[END_REF]. Freeze drying applied to nanostructured silica gels leads to cryogels that are usually cracked and show obvious damages in the pore structure [START_REF] Scherer | Freezing gels[END_REF]. Evaporative drying is performed at atmospheric pressure and the obtained material is generally called xerogel [START_REF] Durães | Effect of the Drying Conditions on the Microstructure of Silica Based Xerogels and Aerogels[END_REF]. Supercritical drying was developped in the 1930's by Kistler [START_REF] Kistler | Coherent Expanded-Aerogels[END_REF]. Above the critical point, the sharp distinction between liquid and gas phases disappears, inhibiting surface tension effects which are held responsible for the gel degradation. This technique has proved to be far more efficient and to produce monolithic aerogel materials [START_REF] Bisson | Drying of Silica Gels to Obtain Aerogels: Phenomenology and Basic Techniques[END_REF]. Fig. 1. Methods of gel drying and their outcome [START_REF] Bisson | Drying of Silica Gels to Obtain Aerogels: Phenomenology and Basic Techniques[END_REF].

1.2.Structure of the resulting aerogels, xerogels and cryogels

Aerogels are derived from the supercritical drying of two main categories of gels (Fig. 1): inorganic gels such as silica aerogels and organic gels, such as the organogels studied in this paper. SEM (scanning electron microscopy) images of xerogels, cryogels and aerogels obtained by the evaporative drying, freeze drying and the supercritical drying techniques respectively were studied in previous works [START_REF] Durães | Effect of the Drying Conditions on the Microstructure of Silica Based Xerogels and Aerogels[END_REF][START_REF] Su | Low-cost and fast synthesis of nanoporous silica cryogels for thermal insulation applications[END_REF][START_REF] Jamart-Grégoire | Monolithic organic aerogels derived from single amino-acid based supramolecular gels: physical and thermal properties[END_REF][START_REF] Ganesan | Design of aerogels, cryogels and xerogels of cellulose with hierarchical porous structures[END_REF][START_REF] Reichelt | Biocompatible polysaccharide-based cryogels[END_REF]. The structural patterns are very ramified which allows a low bulk density and a large porous volume in the materials. All aerogels are less dense than xerogels and cryogels. Indeed, aerogel samples display the least shrinkage compared to cryogels and xerogels. Among these materials, aerogels are generally the most porous media and possess the largest surface area [START_REF] Durães | Effect of the Drying Conditions on the Microstructure of Silica Based Xerogels and Aerogels[END_REF][START_REF] Ganesan | Design of aerogels, cryogels and xerogels of cellulose with hierarchical porous structures[END_REF]. Xerogels, cryogels and aerogels produced from silica gels have similar cloudy and diffuse structures, with very small interconnected units [START_REF] Durães | Effect of the Drying Conditions on the Microstructure of Silica Based Xerogels and Aerogels[END_REF][START_REF] Su | Low-cost and fast synthesis of nanoporous silica cryogels for thermal insulation applications[END_REF] . The pores have similar nanometric sizes (generally 1-100 nm) [START_REF] Shi | Synthesis and characterization of silica aerogels by a novel fast ambient pressure drying process[END_REF][START_REF] Masmoudi | Supercritical CO2 drying of silica aerogels synthesized in 2-propanol[END_REF]. On the contrary, the materials obtained from organogels display highly entangled fibrillar structures, presenting a discontinuous skeleton of fibers with random distribution [START_REF] Jamart-Grégoire | Monolithic organic aerogels derived from single amino-acid based supramolecular gels: physical and thermal properties[END_REF][START_REF] Ganesan | Design of aerogels, cryogels and xerogels of cellulose with hierarchical porous structures[END_REF]. The aerogel fibers are less thick than the fibers of the materials obtained from the same organic gels but with other drying techniques [START_REF] Ganesan | Design of aerogels, cryogels and xerogels of cellulose with hierarchical porous structures[END_REF]. Organic aerogels are less fragile than inorganic aerogels [START_REF] Novak | Low-Density, Mutually Interpenetrating Organic-Inorganic Composite Materials via Supercritical Drying Techniques[END_REF][START_REF] Saliger | Carbon aerogels from dilute catalysis of resorcinol with formaldehyde[END_REF]. It was shown that they can be converted into carbon aerogels through pyrolysis [START_REF] Tamon | SAXS Study on Gelation Process in Preparation of Resorcinol-Formaldehyde Aerogel[END_REF].

The first organic aerogels of natural origin and their derivatives were produced by Kistler [START_REF] Kistler | Coherent Expanded-Aerogels[END_REF][START_REF] Kistler | Coherent Expanded Aerogels and Jellies[END_REF]. Resorcinol/formaldehyde (RF) and melamine/formaldehyde (MF) are two of the most common precursor mixtures to form the organic polymer network [START_REF] Bock | Influence of monomer and catalyst concentration on RF and carbon aerogel structure[END_REF][START_REF] Nguyen | Effects of processing variable on melamine-formaldehyde aerogel formation[END_REF]. Organic aerogels are also prepared from cellulose, the most abundant polymer in nature [START_REF] Ganesan | Design of aerogels, cryogels and xerogels of cellulose with hierarchical porous structures[END_REF][START_REF] Tan | Organic Aerogels with Very High Impact Strength[END_REF]. The pore volume fraction of nanofibers at the cell wall of cellulose aerogels has an important influence on the mechanical properties [START_REF] Ganesan | Design of aerogels, cryogels and xerogels of cellulose with hierarchical porous structures[END_REF]. However, cellulose aerogels display an impact strength much larger than that of RF aerogels and thousand times larger than for inorganic aerogels [START_REF] Tan | Organic Aerogels with Very High Impact Strength[END_REF].

The aerogels studied here are obtained from organogels, which can be prepared with different types of solvents. For aerogels produced from toluene and tetralin organogels, with a weight percentage of gelator between 0.5 and 1 mass % introduced in the solvent, the fiber diameters are nanometric in both cases.

1.3.Properties and applications of aerogels produced by supercritical drying

Silica aerogels, the most common and extensively studied aerogels, show outstanding properties such as high porosity (80-99.8%), high specific surface area (500-1200m 2 /g), very low conductivity ( 0.02W.m -1 K -1 ) [START_REF] Stolarski | Synthesis and characteristic of silica aerogels[END_REF], low density (∼0.003 g/cm 3 ), low index of refraction (∼1.05) and ultralow dielectric constant (k = 1.0-2.0) [START_REF] Soleimani Dorcheh | Silica aerogel; synthesis, properties and characterization[END_REF]. Owing to these properties, the aerogels open numerous application perspectives such as acoustic and thermal insulation and to a lesser extent, catalysis, electric devices, filtration and collection devices [START_REF] Hrubesh | Aerogel applications[END_REF]. The industrial production of aerogels is mainly limited to silica-based systems, carbon, and some small amounts of organic aerogels [START_REF] Smirnova | Aerogels in Chemical Engineering: Strategies Toward Tailor-Made Aerogels[END_REF].

Organogels used to produce aerogels

Organogels are physical organic gels made of a three-dimensional network of gelator molecules, which trap the solvent and prevent its flow. Weak bonds ensure the cohesion of this supramolecular network. The gelator molecules, also called organogelators, have a low molecular weight (<3000 g.mol -1 ) and can be amino acid derivatives such as phenylalanine, as in the present study. The bonds responsible for the gelation phenomenon in this case are π-π stacking interactions, hydrogen bonding and van der Waals interactions [START_REF] Jamart-Grégoire | Monolithic organic aerogels derived from single amino-acid based supramolecular gels: physical and thermal properties[END_REF][START_REF] Brosse | A family of strong low-molecular-weight organogelators based on aminoacid derivatives[END_REF][START_REF] Pham | Influence of the gelator structure and solvent on the organisation and chirality of selfassembling fibrillar networks[END_REF][START_REF] Yemloul | Solvent Dynamical Behavior in an Organogel Phase As Studied by NMR Relaxation and Diffusion Experiments[END_REF]. The organogelators show gelation capacity in various solvents such as aromatic solvents (toluene, benzene, tetralin...), alcohols, and ketones. The organogel studied in this paper are produced with tetralin as a solvent.

1.5.Some lessons from previous supercritical gel drying practices

The gel supercritical drying to produce aerogel is based on two crucial steps, the solvent removal from the organogel by supercritical CO 2 extraction and the separation of the solvent-CO 2 mixture. Extraction of the liquid solvent by preserving the gel texture and in particular its original porous network is an important issue in aerogel production. In experimental studies, this step needs to be conducted without creating a liquid-vapor interface, since the resulting capillary forces in the pores cause shrinkage and cracking of the gel samples due to internal stress [START_REF] Van Bommel | Drying of silica aerogel with supercritical carbon dioxide[END_REF]. According to Novak and Knez [START_REF] Novak | Diffusion of methanol-liquid CO2 and methanol-supercritical CO2 in silica aerogels[END_REF], the quality and the shape of this kind of aerogels, monolithic cylindrical silica gels formed with methanol, were governed by diffusion of methanol solvent in the CO 2 phase. Furthermore, cracking of samples resulted from rapid drying while transparent crack-free aerogels were obtained for slow drying. Non-transparent areas were observed inside the silica aerogel with intermediate drying times [START_REF] Novak | Diffusion of methanol-liquid CO2 and methanol-supercritical CO2 in silica aerogels[END_REF].

In order to understand the phenomena occurring within the gels during supercritical drying, the most important parameter to follow is the solvent transport into the porous medium. In the earliest studies, researchers considered that mass transfer was purely governed by diffusion phenomena. For this reason, they focused on the experimental determination of the effective solvent-CO 2 diffusion coefficient through gel structure. In the work of Wawrzyniak [START_REF] Wawrzyniak | Diffusion of ethanol-carbon dioxide in silica gel[END_REF], this parameter was determined for ethanol-CO 2 at different temperatures and pressures, by combining the diffusional resistances in the fluid phase and structural gel parameters, porosity and tortuosity [START_REF] Wawrzyniak | Diffusion of ethanol-carbon dioxide in silica gel[END_REF]. The effective methanol-CO 2 diffusion coefficients through cylindrical gels at two different temperatures were also determined experimentally by Novak and Knez [START_REF] Novak | Diffusion of methanol-liquid CO2 and methanol-supercritical CO2 in silica aerogels[END_REF].

A recent experimental research showed the reliability of NMR (Nuclear Magnetic Resonance) technique in-situ to estimate this parameters at ambient temperature for different methanol-CO 2 and ethanol-CO 2 mixtures within silica gels as a function of the solvent concentration and pressure [START_REF] Behr | Self diffusion coefficients of organic solvents and their binary mixtures with CO2 in silica alcogels at pressures up to 6MPa derived by NMR pulsed gradient spin echo[END_REF]. Sanz-Moral et al. [START_REF] Sanz-Moral | View cell investigation of silica aerogels during supercritical drying: Analysis of size variation and mass transfer mechanisms[END_REF] studied experimentally the supercritical CO 2 aerogel drying by image analysis of video captures. Cylindrical silica alcogels produced with methanol were followed in a high pressure view cell. The mass transfer mechanisms were investigated through analysis of the aerogels transparency evolution during drying. In the early stages of drying, the mass transfer inside the gel was found to be dominated by convection, the drying rate evolution directly depending on the CO 2 flow rate. On the contrary, diffusion governed the transfer of methanol through the alcogel in the later stages [START_REF] Sanz-Moral | View cell investigation of silica aerogels during supercritical drying: Analysis of size variation and mass transfer mechanisms[END_REF].

Quiño et al. [START_REF] Quiño | Supercritical drying of aerogel: In situ analysis of concentration profiles inside the gel and derivation of the effective binary diffusion coefficient using Raman spectroscopy[END_REF] used Raman Spectroscopy to determine the composition of the CO 2 /ethanol mixture inside gels during supercritical drying, as a function of time and position. From the obtained CO 2 and ethanol concentration fields, the authors concluded that the ethanol is permanently removed from the structure of the gel while CO 2 travels into and out of the sample at the beginning and at the end of the drying operation, respectively. The diffusion coefficient has been determined as function of CO 2 concentration by fitting the experimental data with the analytical model of Crank [START_REF] Crank | The mathematics of diffusion[END_REF].

State of the art on modelling and simulations of gel supercritical drying

In order to model and optimize the supercritical drying process, a good understanding of the phenomena that occur in the complex system of solvent and carbon dioxide is necessary.

Process optimization leads to an overall cost reduction and modelling is the key to estimate Orlovic et al. [START_REF] Orlović | Mathematical modeling and simulation of gel drying with supercritical carbon dioxide[END_REF] studied the case of alumina/silica gels containing 1-butanol solvent. They simulated the transfer of the solvent in the gel by pure diffusion. An effective diffusivity was determined with four different models. The two first models were shrinking core models in which a unique effective diffusivity was taken for the whole porous medium. One used Knudsen diffusivity while the other used tortuosity and porosity in the calculation of the diffusivity. The two last models were based on particular pore arrangements. Parallel pore arrangement modeled the simultaneous drying of all pores. Pores in series modelled the successive drying of the pores from the largest to the smallest. A different effective diffusivity was used for each pore size by considering Knudsen diffusion for the micropores.

Better fitting of experimental data was obtained for the two last models.

In the theoretical approach of Griffin et al. [START_REF] Griffin | Continuous extraction rate measurements during supercritical CO2 drying of silica alcogel[END_REF], gels of annular geometry and blown around by SCCO 2 were modeled with 2D axisymmetric COMSOL model. The mass transfer model considered pure diffusion within the gel. Only molecular diffusion was taken into account and Knudsen diffusion terms were not considered. The gel structure, porosity and tortuosity were assumed not to affect diffusion. The model included a diffusion coefficient dependent on the solvent composition. The results were compared with experimental data obtained from a pilot unit that ensured a continuous measurement of the solvent (ethanol) extraction for different gel thickness and CO 2 mass flow rate. A good agreement was found between theoretical and experimental results. It was also concluded from experiments that the ethanol extraction process is due to sole diffusion phenomena, a significant effect on drying time of both gel thickness and CO 2 mass flow rate was shown.

In the study of Ozbakir et al. [START_REF] Özbakır | Experimental and theoretical investigation of supercritical drying of silica alcogels[END_REF] the mass transfer within the cylindrical gel and in the flowing SCCO 2 was modelled. The mass transfer inside the gel was assumed to be diffusive.

Mass transfer occurred by convection from the gel surface to the flowing SCCO 2 stream. The effective diffusion coefficient for the porous gel was estimated proportional to both the CO 2ethanol diffusion coefficient and gel porosity and inversely proportional to the gel tortuosity.

The binary diffusion coefficient was considered a function of the CO 2 mole fraction. To describe mass transfer along the tubular extraction vessel, a convective mass term was taken into account. Experiments were carried out with a SCCO 2 continuous flow surrounding a cylindrical silica gel in a tubular extraction vessel allowing to measure the mass of ethanol removed as a function of time. The gel experimental extraction profiles were compared to the model predictions. A good agreement was reached using mass transfer coefficients fitted from the experiments that varied from an experiment to other. The effect of the CO 2 flow rate on drying kinetic was studied both experimentally and theoretically. It was concluded that the increase of CO 2 flow rate led to a decrease of effluent concentration at a specific time.

However, the effect on the rate of extraction of ethanol was not significant. The simulations were used to investigate the effect of the effective diffusion coefficient and gel thickness. It was found that the drying time of the silica gel decreased with increasing effective diffusion coefficient and decreasing gel thickness. For instance, the solvent concentration was divided by two after 10 min for a diffusion coefficient of 5.10 -9 m 2 /s and after120 min for a diffusion coefficient ten times lower (0.5.10 -9 m 2 /s). Also, the drying time increased from 50 min to 300 min when the diameter was increased from 0.5 cm to 2 cm. Supercritical solvent extraction process was also simulated using CFD (Computational Fluid Dynamics) techniques. Lebedev et al. [START_REF] Lebedev | Modeling and scale-up of supercritical fluid processes. Part I: Supercritical drying[END_REF] performed simulations to describe the supercritical drying process of silica gels. They developed a mathematical model describing the flow in the reactor of a supercritical alcohol-CO 2 mixture, assumed a compressible viscous fluid. In their approach, heat and mass transfer were also accounted for. ANSYS-Fluent software was used for simulations. Two flow domains were considered in the system: the volume of the porous body (gel) and the free volume of the reactor. In the porous medium, mass transfer was In almost all the previous researches, the dried gels were silica gels. Very few papers treated the case of organic gels. Garcia-Gonzalez et al. [START_REF] García-González | Supercritical drying of aerogels using CO2: Effect of extraction time on the end material textural properties[END_REF] have studied the supercritical drying of gels from different precursors (inorganic silica gel and organic starch gels) in order to show the precursor effect on the kinetics profile. The solvent (ethanol) removal process was followed experimentally. The solvent extraction was studied experimentally by measuring its amount in the CO 2 gaseous stream, using an alcoholmeter device. It was concluded that the gel supercritical drying is governed by a mass transfer mechanism based on a combination of both convection and diffusion inside the gel. In the first drying step (0-60% of recovery rate), convective mass transfer was shown to play a significant role while the last drying step is controlled by diffusion. The experimental drying profiles were compared to theoretical profiles calculated considering only Fickian diffusion-based model without convective transport. This model failed to predict correctly the experimental data. The effective diffusion coefficient was determined by using the curve fitting Matlab tool. The obtained values were slightly higher than those found in the previous literature for the silica aerogel ( .

The effective diffusion coefficient of organic starch aerogel were almost ten times lower than that found for the monolithic silica aerogel ( ). This last value is the only value found in the literature for organic gels. The drying profiles of inorganic silica gel and organic starch gels were compared to investigate the effect of precursor nature of the gel. It was found that both drying profiles followed a similar two-step drying rate behavior. However, the drying rate was lower in the case of organic gel. The authors attributed this to the difference between the gels textural properties which could affect the diffusion process. Moreover, they assumed that the interaction of ethanol with the starch components of the organic gel might hinder and delay the solvent extraction. The effect of morphology in the drying process was also studied by comparing the experimental drying profiles of cylindrical monoliths and microspheres gels having the same target densities. It was observed that these profiles were similar in the first drying step (0-60% of recovery rate) then the drying process of microspheres was faster than for the cylindrical monoliths. This was explained by the threshold point at which the drying process became more affected by the diffusion phenomena. The minimum diffusion path length was shown to have a significant effect on the drying process by comparing organic gels profiles with different geometrical shapes (cube and cylinder) and sizes.

So far, most models reported in the literature describe the supercritical drying of inorganic silica gels (Table 1). The present study focuses on the supercritical drying of organic gels, derived from a low molecular weight molecule, which have been received little attention. The structure and properties of aerogels derived from organogels are very different from those derived from silica gels. Furthermore, the solvents used to produce silica gel were often alcohols like ethanol, methanol and isopropanol. The solvent chosen here is tetralin, which is aromatic and not volatile. At the best of our knowledge, it was never used in the past to produce gels. A new theoretical approach is developed in this paper in order to model the supercritical drying of organogels accounting for diffusion as well as convective mass transfer. In fact, two different approaches are assumed and compared, depending on the penetrable or impenetrable nature of the organogel.

Experimental drying of cylindrical organogels

In this section, a drying experiment of organogels is described and kinetic results are presented.

2.1.Organogel preparation

The process to obtain organogel is based on two main steps. The first step is the synthesis of low molecular weight organogelator molecules derived from amino acids (Phenylalanine Phe) which was described by Brosse et al. [START_REF] Brosse | A family of strong low-molecular-weight organogelators based on aminoacid derivatives[END_REF]. It consists of an esterification of the Z group (benzyloxycarbonyl) protected amino acid, called Z-Phe-OMe, followed by a hydrazide synthesis (Z-Phe-NH-NH 2 ) by reacting the ester with hydrazine hydrate. Then, the final compound, the organogelator (Z-Phe-NH-Napht) is obtained by condensation of naphthalic anhydride with the hydrazide. The second step consists in the gel formation by dissolving and stirring a desired amount of the organogelator into a desired amount of organic solvent (here tetralin) in a round bottomed flask connected to a reflux condenser at T=80°C. The hot solution is then transferred into a cylindrical mold and cooled down to T= 4°C [START_REF] Jamart-Grégoire | Monolithic organic aerogels derived from single amino-acid based supramolecular gels: physical and thermal properties[END_REF]. Four cylindrical organogel samples with 30 mm diameter and 6.7 mm, 10.5 mm, 10.5 mm and 12.9 mm height respectively were prepared following this experimental protocol.

2.2.Experimental setup for organogel drying

The organogels drying is carried out in a home-made supercritical laboratory equipment represented in Fig. 2. This unit is mainly composed of a high pressure cylindrical extractor or autoclave 9 of 250 ml volume where the organogel is dried. Gaseous CO 2 (50-60 bar) supplied by a CO 2 bottle 1 is liquefied by cooling in a cold exchanger 2 at 4°C. A pump 4 manufactured by LEWA (EK1) pushes the liquid CO 2 into the back pressure regulator 6 . The pressure of the CO 2 leaving the pressure regulator is 180 bar. Then, it is heated in a hot exchanger 7 up to 45°C. Under these conditions (45°C and 180 bar), the CO 2 flow is under supercritical state.

The autoclave containing the gel is fed with the supercritical CO 2 . The organogel sample is placed on a bed of glass beads (Fig. 3). The CO 2 entering from the bottom of the autoclave first flows through this bed before it is blown around the sample. The cumulative mass and the flow rate of CO 2 are measured using a Micromotion 8 Coriolis mass flowmeter placed upstream from the autoclave with accuracy. The pressure inside the autoclave is controlled by a pressure gauge (PI1).The supercritical CO 2 allows organogel drying by extracting the tetralin from its porous structure. The CO 2 -tetralin mixture leaving the autoclave is then separated in a cascade of three successive separators 10 operating at 50 bar, 30 bar and atmospheric pressure respectively and at 20°C. Further details about the drying process and the separation are provided in [START_REF] Lazrag | Thermodynamic and hydrodynamic study of a gas-liquid flow in a cyclone separator downstream supercritical drying[END_REF]. The organogel drying procedure in order to obtain the aerogel follows several steps, during which the temperature and pressure in the autoclave evolve with time, as described in Fig. 4 and detailed hereafter:

- The sample thickness and the CO 2 flow rate differed for every experiment. This is why it was not possible to perform replication tests.

2.4.Drying experiments results

The four cylindrical samples with different size described in section 2.1 are dried according to this experimental protocol. The experimental data and results, gel thickness, organogelator weight fraction, tetralin mass in the gel, mean CO 2 flow rate, drying time, initial tetralin concentration, and tetralin degree of recovery, are reported in Table 2. The initial tetralin concentration is calculated as the ratio of the tetralin mass in the gel to the pore volume of the sample, multiplied by the tetralin molar mass. The experimental tetralin degree of recovery is the ratio of the total quantity of tetralin recovered from the autoclave to the initial total quantity of tetralin in the gel. The final recovery is satisfactory, around 90% but is not complete (Table 2). A thermodynamic modeling of CO 2 /tetralin separation has been carried out and the tetralin degree of recovery has been computed in a former work [START_REF] Lazrag | Experimental and thermodynamic comparison of the separation of CO2/toluene and CO2/tetralin mixtures in the process of organogel supercritical drying for aerogels production[END_REF] .

The computed recovery (theoretical tetralin recovery) were in agreement with the experimental values (experimental tetralin recovery). The discrepancies may be attributed to experimental errors and to variations of CO 2 flow rate and pressures during experiments. In the thermodynamic model, on the contrary, the flow rate and the pressures are assumed constant all along the experiments.

The unrecovered amount is due to thermodynamic reasons: about 10% of the tetralin stays in the gas phase and leaves the separators with the CO 2 through the upper gas exits, as explained by Lazrag et al. [START_REF] Lazrag | Experimental and thermodynamic comparison of the separation of CO2/toluene and CO2/tetralin mixtures in the process of organogel supercritical drying for aerogels production[END_REF]. To compare the experimental results with the numerical calculations presented later, the instantaneous recovery rate is considered relatively to the recovered value at long times, and is called the normalized tetralin recovery rate ( ). In Fig. 5, the experimental is presented as a function of time.

Mathematical models and simulation results

3.1.Thermodynamics Data

As explained in section 1, supercritical monophasic conditions are enforced in order to obtain a well-dried and non-cracked aerogel. In all studied experiments (Table 1), the tetralin concentration does not exceed 1% (mol). In Fig. 6, the equilibrium diagram of the CO 2tetralin binary mixture is presented at two temperatures (40°C and 71°C), between which lies the actual autoclave temperature of the drying process (45°C). The experimental bubbles and dew points were taken from the literature and were summarized in a previous work [START_REF] Pham | Influence of the gelator structure and solvent on the organisation and chirality of selfassembling fibrillar networks[END_REF]. The theoretical curves were determined from a thermodynamic model detailed in the same paper.

This diagram shows that the selected experimental conditions, 45°C and 180 bar with a tetralin molar fraction inferior to (CO 2 molar fraction produce a supercritical Solid line: predicted curves with the PPR model [START_REF] Pham | Influence of the gelator structure and solvent on the organisation and chirality of selfassembling fibrillar networks[END_REF] for the system CO 2 -tetralin at two temperatures : 40°C and 71°C

3.2.Mathematical models

In this modeling study, the system is the organogel sample and the autoclave. The mathematical models are developed to describe the flow hydrodynamics of the supercritical fluid (SCCO 2 ) in the autoclave and the mass transfer of tetralin during the organogel drying process. This model is based on equations of continuum mechanics. The system is divided into three zones: the upper autoclave volume surrounding the sample, the gel volume, and the inferior autoclave volume containing the glass beads.

The produced aerogels samples possess a very high porosity (void fraction) with an average value of 0.97, determined from SEM images [START_REF] Allix | Etude Physico-Chimique d'Organogels et d' Aérogels de Faible Poids Moléculaire Dérivés d'Acides Aminés[END_REF]. For this reason, an important issue is discussed here, which is to investigate whether the gel sample is penetrable to the CO 2 flow or not. Two approaches are therefore considered here. In the first approach, the organogel is assumed to be an impenetrable solid. In the second one, the organogel is regarded as penetrable. Both approaches are implemented in Computational fluid dynamics (CFD) software ANSYS Fluent. In this approach, the organogel is considered as an impenetrable solid meaning that the surrounding fluid cannot pass through it. The considered assumptions are as follows:

-The flow of the supercritical CO 2 in the free volume is considered laminar ( ) and is described by Navier-Stokes equations. Due to the low velocities involved, the Mach number is inferior to 0.3 and the flow may be considered incompressible.

-The inferior volume of glass beads is described as a homogeneous porous medium and the CO 2 flow is governed by Darcy law.

-Tetralin mass transfer in the porous medium of the gel volume is described by Fick's diffusion equation without convective transport. An effective diffusion coefficient is considered, taking into account the porous structure of the gel.

-The system is assumed to be isothermal and isobaric, since it is the case in the experiments, which leads to a constant fluid density. where is the CO 2 velocity, is the CO 2 density, is the pressure in the autoclave, is the gravity constant, is the mass fraction of tetralin and is the CO 2 dynamic viscosity.

The fluid velocity at the walls is zero, i.e. no-slip boundary conditions. Tetralin flux at the walls is also set to zero. There is no tetralin in the CO 2 flow entering the autoclave.

At initial time, the CO 2 fluid around the gel is free of tetralin. The sample contains only tetralin with a uniform molar concentration as described in Table 2.

The permeability of the glass beads bed is estimated with Kozeny-Carman equation [START_REF] Bear | Dynamics of Fluids in Porous Media[END_REF]:

where is the glass beads diameter here equal to 10 mm, is the porosity of random stacking of monodisperse spheres, is the Kozeny coefficient that is equal to 4.17, is the form factor which is equal to for a spherical packing yielding where is the tortuosity of the glass beads bed [START_REF] Dias | Permeability and effective thermal conductivity of bisized porous media[END_REF][START_REF] Dias | Permeability analysis in bisized porous media: Wall effect between particles of different size[END_REF][START_REF] Mota | Binary spherical particle mixed beds : porosity and permeability relationship measurement[END_REF].

The molecular diffusion coefficient is the diffusion coefficient of tetralin (A) in supercritical CO 2 (B). In the literature [START_REF] Zhou | Catalytic oxidation of toluene and tetralin in supercritical Carbon Dioxide[END_REF], this coefficient was determined at temperature and pressure conditions different from supercritical drying conditions in this process. It is equal to 1.741 x 10 -3 cm 2 /s at 573 K and 8.97 MPa, and 2.221 x l0 -3 cm 2 /s at 648 K and 8.97 MPa. The hydrodynamic theory of Stokes-Einstein was used by Medina [START_REF] Medina | Determination of diffusion coefficients for supercritical fluids[END_REF] to calculate the diffusivity in supercritical fluids and gave acceptable agreements with experiments. This model assumes the diffusion of large molecules of solute through a continuum of smaller molecules of solvent which seems relevant here considering that the CO 2 molecules (the solvent) are smaller than the solute molecules (tetralin). For this reason, this model is adopted here to estimate the diffusion coefficient :

Where T is the temperature, k is the Boltzmann constant, is the viscosity of the solvent, here CO 2 , is the hydrodynamic radius of solute A, is the van der Waals volume parameter and is the van der Waals volume of solute A (tetralin).

The organogel sample is considered as homogenous solid within which tetralin diffuses. The free mean path of the tetralin molecules is increased since they must circumvent the gel fibers, which contributes to slow down their motion and is taken into account in the effective coefficient .

Several models have been established for a long time in order to predict the diffusion of solutes in liquid fluids and supercritical fluids [START_REF] Medina | Determination of diffusion coefficients for supercritical fluids[END_REF]. The determination of the diffusion coefficient of a solute in a solid matrix such as solid polymers and gels was also addressed in the literature in order to understand these complex systems from the point of view of mass transfer. Different mathematical models describing the retardation of solute diffusion within gels have been developed which were based on different theoretical approaches. A very good overview of these different theories is given in the literature review of Amsden [START_REF] Amsden | Solute diffusion in hydrogels: an examination of the retardation effect[END_REF] and Masaro et al. [START_REF] Masaro | Physical models of diffusion for polymer solutions, gels and solids[END_REF]. In the present paper, we are concerned by the theoretical approach of obstruction that focuses on gels of fibrillar structure. The presence of impenetrable fibers leads to an increase in the path length for diffusive transport. The fibers, in models based on this theory, are assumed to be at rest and randomly embedded. In this theoretical approach, the greatest importance is given to the structural description of the system. For this reason, the decrease of the diffusion coefficient is correlated with the solute size and the volume fraction of the fibers [START_REF] Sanz-Moral | View cell investigation of silica aerogels during supercritical drying: Analysis of size variation and mass transfer mechanisms[END_REF]. Among these models, the one proposed by Ogston [START_REF] Ogston | On the Transport of Compact Particles Through Solutions of Chain-Polymers[END_REF][START_REF] Ogston | The spaces in a uniform random suspension of fibres[END_REF] is selected to predict the effective diffusivity , diffusion coefficient of the solvent in the supercritical fluid through the matrix of the gel. It allows to predict the ratio from the structural data on the solute and the gel matrix for the fibrous medium. The Ogston model is usually adopted for fibrous matrix and for random distribution of fibers inside the gel [START_REF] Ogston | On the Transport of Compact Particles Through Solutions of Chain-Polymers[END_REF][START_REF] Ogston | The spaces in a uniform random suspension of fibres[END_REF]. This model describes the repartition of spaces inside a fiber suspension which structure can be compared to that of organogels. The assumptions of this model were the following: the fibers were considered rectilinear and the solute was considered rigid and spherical. The Ogston model is a phenomenological approach where the solute diffusion is the result of random successive movements. This model is chosen in this study to predict the ratio and the free mean path is taken as the average distance between the organogel fibers. It is expressed in equation ( 7) as a function of the fiber radius ( ), the solute hydrodynamic radius ( ) and the fiber volume fraction ( ), which is equal to , with the gel porosity.

The calculated effective diffusion coefficient is equal to 4.8×10 -9 m 2 /s (Table 2). This value is close to that found by fitting in the work of Garcia-Gonzalez [START_REF] García-González | Supercritical drying of aerogels using CO2: Effect of extraction time on the end material textural properties[END_REF] for organic aerogels which equals to 1×10 -9 m 2 /s.

3.2.1.2.Approach 2:

Organogel sample as a penetrable solid (diffusionconvection transfer inside the gel)

In this approach, the organogel sample is assumed to be a penetrable solid allowing the CO 2 to flow through its pores. In this approach the free volume surrounding the gel and the glass beads volume are treated the same manner as in the 1 st approach. But the organogel sample is treated as a volume of porous medium. The convective transport is added to molecular diffusion in this medium. The CO 2 flow is governed by Darcy law. In this approach, equation ( 2) is kept for Volume 1 and equation ( 4) for Volume 3. But in Volume 2, the organogel sample, equation ( 3) is replaced by [START_REF] Reichelt | Biocompatible polysaccharide-based cryogels[END_REF] The permeability of the fibers bed is calculated using the expression proposed by Davies [START_REF] Davies | The Mechanics of Large Bubbles Rising through Extended Liquids and through Liquids in Tubes[END_REF] :

where is the fibers diameter and is the organogel porosity, which are measured through SEM images. The values of the different parameters used in these models are reported in Table 3. Since the tetralin molar fractions are very low (below 0.13%), the considered density and viscosity are those of pure CO 2 . Furthermore, the diffusivities were assumed to be independent of tetralin concentration. (CO 2 ) (Pa.s) 6.95×10 -5 [START_REF] Fenghour | The Viscosity of Carbon Dioxide[END_REF] (cm 3 /s) 80.98

(m 2 /s) 4.8×10 -9 -(m 2 ) 0.39 [START_REF] Rumpf | The influence of porosity and grain size distribution on the permeability equation of porous flow[END_REF] (m 2 ) 10 -7 -0.97 [START_REF] Allix | Etude Physico-Chimique d'Organogels et d' Aérogels de Faible Poids Moléculaire Dérivés d'Acides Aminés[END_REF] α (m 2 ) 1.29×10 -13 -3.2.2. Simulation methods

3.2.2.1.Numerical schemes

In ANSYS-Fluent, the convective terms are discretized using the second order upwind approach. The interpolation model is maintained for the pressure term. In order to solve the transport equation of tetralin, the first order upwind scheme is selected.

3.2.2.2.Computational mesh

The software Gambit is used to create the computational domain and the mesh shown in Fig.

8.

The mesh is three-dimensional (3D). The domain is divided into three volumes:

 Volume 1, the upper free volume surrounding the sample is discretized with tetrahedral elements  Volume 2, the organogel sample is meshed with tetrahedral elements.

 Volume 3, the lower autoclave volume that contains glass beads is discretized with hexahedral elements. The CO 2 velocity is imposed at the autoclave inlet and no-slip conditions are applied at the solid walls.

3.2.2.4.Mesh independence test

The first experiment corresponding to sample 1, of 6.7mm thickness, is chosen as a reference to build the mathematical models. The other experiments (sample 2, 3, and 4) will be compared to theory in a second step. In order to perform mesh independence tests, two meshes are created, a coarse and a refined mesh composed of 3.3 million and 6.5 million cells respectively. In Fig. 9, tetralin mass fraction profiles in volume 2, the organogel sample, are compared for both meshes with the first approach. These profiles are plotted along two segments, I and II, indicated in Fig. 8 The calculations provide the tetralin mass fraction inside the organogel sample as a function of time. The theoretical tetralin recovery rate extracted from the pores at the drying conditions mentioned in section 2 is also predicted using the following equation where is the total density and is the tetralin mass fraction Both simulations are carried out with a time step which is equal approximately to a thousandth of the characteristic diffusion time computed from approach 1.

The theoretical tetralin recovery rate is computed with the two approaches described in sections 3.1.1. In Fig. 10, the models results are compared to the experimental extraction profiles given in Fig. 5. A large amount of tetralin is collected in the initial period of the experiment. An extraction rate (amount of tetralin extracted per unit time) of 75% is reached between 25 and 30 min for all experiments, Fig. 10(a-d). The extraction rate decreases with drying progress. The two theoretical drying kinetics are close but the kinetics predicted by approach 2 is closer to the experiment. The discrepancies between theory and experiment are quantified by the least square method and are equal to and 9.4% and 5.6% for approaches 1 and 2 respectively, in the case of the first experiment, Fig. 10(a). For the other experiments 2, 3, and 4, the discrepancies between approach 2 and the experiments are equal to 16%, 19.6% and 16.6% respectively. While, the discrepancies between approach 1 and the experiments are equal to 31.1%, 34.8% and 30.1% respectively. So, the second approach can be considered the closest approach to the experimental reality. The gel appears to behave as a penetrable porous medium. The fibers are not an obstacle against the penetration of fluid. Accordingly, the fibrous matrix of the gel can be assimilated to a sieve in which the solute can diffuse freely.

In this fibrous organogel, it is possible that penetration velocity is not sufficient which can explain the difficulty to discriminate the drying kinetics given by both approaches. But approach 2, which considers the organogel as a penetrable solid allows to represent in a satisfactory manner all the carried out experiments. The deviations, not exceeding 20%, can be explained by the following facts:

-The CO 2 flow rate was not perfectly stable during the experiments while an average constant flow rate was used in the simulations.

-In the theoretical approach, the organogel is considered as a penetrable porous medium. The porosity and permeability values of each organogel sample were unknown and estimated mean values were used for all calculations and simulations.

-In the literature, experimental data corresponding to the molecular diffusion coefficient of tetralin in CO 2 could not be found for the actual operating conditions.

The Stokes-Einstein equation used for supercritical fluids was selected to predict the binary diffusivity.

A parametric study on the operating conditions of the supercritical drying, like CO 2 flow rate and organogel thickness, is necessary to show the sensitivity of the drying kinetic with these parameters and may also allow to differentiate the approaches.

Parametric study

A parametric study is also performed to determine the sensitivity of the drying kinetic to variables such as SCCO 2 flow rate and gel thickness using the second approach.

The effect of SCCO 2 flow rate on the drying kinetics of a 6.7mm-height, 30mm-diameter organogel sample at 45°C and 180 bar is examined using a wide range of CO 2 flow rate between 100g/h and 1700 g/h, that includes the experimental flow rates. The tetralin mass fraction in the organogel as a function of time is converted into recovery rate of removed from the pores with drying time (Eq 10). Fig. 12 indicates how the tetralin recovery rate evolves with time for various flow rates. An increasing flow rate of SCCO 2 leads to a decrease of the effluent concentration and consequently to a decrease of the drying time. The different simulations demonstrate that a significant effect is observed on drying for flow rates in the range 100g/h and 1700g/h. A similar conclusion was also reached by a similar study by Lebedev et al. [START_REF] Lebedev | Modeling and scale-up of supercritical fluid processes. Part I: Supercritical drying[END_REF] which was recently published. These results should be confirmed with experiments to conclude that for industrial scale, it is necessary to run at a high CO 2 flow rate in order to minimize drying time. But also, an optimal CO 2 flow rate should be evaluated because a very high CO 2 flow rate can lead to cracking of the organogel sample as it is observed experimentally. Organogel thickness is one of the most important parameters influencing drying time. Effect of gel thickness on drying kinetic is evaluated by performing simulations for organogel samples with 30mm diameter and with heights of 6.7 mm, 10.5 mm and 12.9 mm. The drying conditions are 45°C and 180 bar with a SCCO 2 flow rate of 729g/h. The simulations results are plotted in Fig. 13. As expected and as is concluded in a similar study [START_REF] Özbakır | Experimental and theoretical investigation of supercritical drying of silica alcogels[END_REF] thicker organogel samples need longer drying times. The CO 2 which penetrates into the gel follows a longer path as the gel sample thickness increases. The drying time increases from 135 to 210 min as the height increases from 6.7mm to 12.9 mm. Both the SCCO 2 flow rate and the gel thickness have important effects on the drying kinetics.

Conclusion

In order to investigate the supercritical drying process of organogel for aerogels production, both experimental and theoretical studies have been carried out. The experiments have been performed in a supercritical pilot unit. Different samples of 30 mm diameter and 6.7mm to 12.9 mm thickness were dried with a supercritical CO 2 stream in an autoclave at 45°C and 180 bar. Based on the previsions of continuum mechanics, two mathematical models have been established to describe the hydrodynamics of the CO 2 flow and mass transfer of solvent during drying process. In a first approach, the organogel was considered as an impenetrable sample. Mass transfer in the gel was described by Fick diffusion, with an effective diffusivity to account for the gel porosity. In a second approach, the organogel was regarded as penetrable by the CO 2 flow. In this case, convective transport occurred together with diffusion. For both models, the molecular diffusion coefficient, the diffusivities, gel and glass beads permeabilities were estimated from literature correlations. The numerical results, tetralin recovery rate as function of time, of both models were compared to experimental data.

The second model, in which the gel was considered as penetrable, yielded results closer to experiments with relative discrepancies below 20%. This showed that the approach considering the organogel as a penetrable sample better represents the experimental reality.

Using the second approach, many simulations were performed in order to study the effect of CO 2 flow rate and gel thickness, showing a significant effect of both parameters. Likewise, a parametric study with simulations on structural parameters like fibers diameters and solvent hydrodynamic radius would be interesting. It could allow to discriminate the two approaches and investigate the sensitivity of drying kinetics to these parameters.

  the drying time and to contribute to the process development. With this respect, different studies can be found in the literature. The systems studied are generally silica gel and an alcohol. A mechanistic model was proposed by Mukhopadhyay et al. on supercritical drying of silica aerogel by focusing on mass transfer modeling of CO 2 and ethanol to and from the gel [32]. It was based on the assumption that the pores are parallel arrangements of cylinders. The pore structure was considered open on one or both ends, with direct contact with supercritical CO 2 (SCCO 2 ) at the open end. The diffusive CO 2 flux and the convective mass ethanol flux to and out of the open end of the pores were calculated. The effective diffusion coefficient is introduced as a function of molecular and Knudsen diffusion. The C++ language was used to solve classical mass balance equations of the 1D model. It was concluded that the supercritical extraction of ethanol occurred by spillage due to the supercritical CO 2 dissolution in the pore liquid rather than by the convective evaporation. The drying time was found theoretically to depend strongly on different parameters such as flow velocity of SCCO 2 , gel thickness, temperature, and pressure. Only the effect of the gel thickness was validated experimentally.

  described by Fick diffusion equation accounting for molecular diffusion only. The CO 2alcohol mixture inside the reactor was described by Peng-Robinson equation of state. The experimental study of the supercritical drying kinetics was performed with ten cylindrical monolithic gels. The developed model represented adequately the experiments. The influence of flow rate and gel thickness on kinetic curves was investigated showing a significant effect.

dependent

  of one pore, -Diffusion only -Boundary condition: convection at the end of the pore -Effective diffusivity function of molecular and Knudsen diffusion -Molecular diffusivity in CO 2

Fig. 2 .

 2 Fig.2. Scheme of the supercritical drying set-up: 1-CO 2 bottle; 2-cold exchanger; 3-CO 2 tank 4-pump; 5-Damper, 6-back pressure regulator; 7-hot exchanger; 8-Coriolis mass flowmeter; 9-autoclave; 10-three cyclone separators; 11-Flowmeter, PI-pressure gauge, TI-thermostatic bath.

Fig. 3 .

 3 Fig. 3.Organogel sample placed in the autoclave

Step 1 :Fig. 4 .

 14 Fig. 4.Steps of drying in the autoclave: temperature and pressure enforced in time.

Fig. 5 .

 5 Fig. 5. Experimental supercritical drying kinetics of the four organogel samples

Fig. 6 :

 6 Fig. 6: × experimental bubble points, ♦: experimental dew points.

3. 2 . 1 . Model equations 3 . 2 . 1 . 1 .

 213211 Approach 1: Organogel sample as an impenetrable solid ( Pure diffusion transfer inside the gel)

Fig. 7 . 3 )

 73 Fig. 7. Sketch of the modeled flow and transport domain. Volume 1: free volume above the sample. Volume 2: organogel sample. Volume 3: Glass beads supporting the sample.

Fig. 8 .

 8 Fig. 8. (a) System dimensions (mm) and segments I and II used for plotting concentration profiles, (b) Mesh

Fig. 9 . 8 . 3 . 3 .

 9833 Fig.9. Mesh independence tests: Comparison of the tetralin mass fraction profiles in the organogel sample, obtained with the first approach for two meshes (coarse and refined).(a) In the radial direction of the gel, along segment I (b) in the axial direction of the gel, along segment II. Segments I and II are defined in Fig. 8. 3.3.Supercritical drying simulation results 3.3.1. Comparison of the models results

Fig. 10 .Fig. 11 .

 1011 Fig. 10. Comparison of experimental drying kinetics to model results at 180 bar and 45°C (approach 1: impenetrable solid, approach 2: penetrable solid) (a) Experiment 1: Q=729g/h and e=6.7mm. (b) Experiment 2: Q=733g/h and e=10.5mm. (c) Experiment 3:Q=820g/h and e=10.5mm. (d) Experiment 4:Q=540g/h and e=12.9mm

Fig. 12 .

 12 Fig.12. Variation of tetralin recovery rate as a function of time at varying SCCO 2 mass flow rate at 180 bar and 45°C, with a 30 mm sample diameter and a 6.7 mm sample height.

Fig. 13 .

 13 Fig. 13. Variation of tetralin recovery rate as a function of time at varying organogel thickness (6.9mm, 10.5mm and 12.9mm) at 180 bar and 45°C and SC-CO2 flow rate of 729 g/h

Table 1 .

 1 Models describing supercritical drying of gels in literature and in the present paper

Table 2 .

 2 Experimental data and results of the four organogel samples

	Sample	Exp.1	Exp.2	Exp.3	Exp.4
	Gel thickness (mm)	6.7	10.5	10.5	12.9
	Organogelator weight fraction (%)	1	1	1	0.5
	Tetralin mass in the gel (g)	4.500	7.086	7.075	8.735
	Initial tetralin concentration (mol.L -1 )	7.409	7.445	7.443	7.470
	CO2 mean flow rate (g/h)	729	733	820	540
	Drying time (min.)	150	240	240	240
	Experimental tetralin degree of recovery (%)	81	91	88	87
	Theoretical tetralin degree of recovery (%)	89.0	85.9	85.9	92.3
	Mean tetralin molar fraction (%)	0.09	0.07	0.07	0.13

Table 3 .

 3 Parameter values used in the simulations

	Parameter	Value	Reference
	(CO 2 ) (kg.m -3 )	772.86	[52]

Acknowledgements

This work was partially supported by the Institut Carnot ICEEL.