Isabelle Raspo

Sofiane Meradji

Bernard Zappoli

Heterogeneous Reaction induced by the Piston Effect in Supercritical Binary Mixtures

Keywords: supercritical fluids, multiphase reactions, hydrodynamics, solubility, piston effect, numerical analysis

It is now well established that the large compressibility of supercritical fluids is responsible for the strong enhancement of the thermo-acoustic heating, leading to the speeding up of the heat transport thanks to the piston effect instead of the expected slowing down. We show in this paper, through numerical simulations, that the hydrodynamics behavior of supercritical fluids also couples with the critical behavior of the solubility of solids to cause the release of a heterogeneous reaction at solid surfaces in dilute binary supercritical mixtures.

Introduction

A supercritical fluid is any substance set above the temperature and pressure of its gas-liquid critical point. In this quite special state, the physical properties of the fluid are intermediate between those of liquids and gases. It exhibits a liquid-like density but mass transport properties, such as viscosity, are close to those of gases improving its mass transport rate. Moreover, as the critical point is approached, the isothermal compressibility and the thermal expansion coefficient diverge, while the thermal diffusivity tends to zero. As a consequence, density changes significantly for minor variations of pressure or temperature. More precisely, if we note

( ) c c T T T - = ε (with
T c the critical temperature) the dimensionless distance to the critical point on the critical isochore, relative density perturbations are 24 . 1 ε times larger than the temperature ones in thermal diffusion layers, especially near isothermal boundaries [START_REF] Zappoli | Anomalous heat transport by the piston effect in supercritical fluids under zero gravity[END_REF]. The strong compressibility and dilatability of supercritical fluids induce several specific behaviors, in particular a very fast heat transport, due to thermo-acoustic effects (known as Piston Effect), whereas heat diffusion is very slow. Indeed, the heating of a confined supercritical fluid provokes a very thin diffusion boundary layer which strongly expands and compresses adiabatically the rest of fluid. This process results in a very fast and homogeneous increase of the temperature and pressure in the whole fluid.

The piston effect was theoretically predicted for a pure supercritical fluid in the early nineties by several teams [START_REF] Boukari | Critical speeding up in pure fluids[END_REF][START_REF] Onuki | A. Fast adiabatic equilibration in a singlecomponent fluid near the liquid-vapor critical point[END_REF][START_REF] Zappoli | Anomalous heat transport by the piston effect in supercritical fluids under zero gravity[END_REF], and then it was extensively studied in microgravity and normal gravity conditions [START_REF] Garrabos | Relaxation of a supercritical fluid after a heat pulse in the absence of gravity effects: theory and experiments[END_REF][START_REF] Guenoun | Thermal cycle around the critical point of carbon dioxide under reduced gravity[END_REF][START_REF] Kogan | Density response and equilibration in a pure fluid near the liquid-vapor critical point[END_REF][START_REF] Straub | The process of heat and mass transport at the critical point of pure fluids[END_REF][START_REF] Zhong | Thermal response of a fluid near its critical point: 3He at T>T c[END_REF]. Since the strong compressibility of supercritical fluids deeply Published in Chemical Engineering Science, 62, pp. 4182-4192, 2007 changes the leading mechanism of heat transfer, it was most likely that the induced pressure and density variations might also have consequences on the behavior of an interface reaction, and associated mass transfers, in supercritical mixtures and our aim was to check this assumption. More precisely, we were interested in the case of an adsorption-desorption reaction taking place between a solid interface and an active compound in a dilute binary supercritical mixture. Now, the solubility of solids in supercritical fluids is very sensitive to small changes of temperature and pressure near the solvent's critical point. Indeed, the phase diagram of binary mixtures of interest to explore this kind of heterogeneous reaction in supercritical mixtures exhibits a crossing point (named the Lower Critical EndPoint (LCEP)) between the solid-liquid-vapor coexistence line and the critical line of the mixture starting at the solvent's critical point (Fig. 1). Above this crossing point and up to the upper critical endpoint (UCEP), there exists a phase equilibrium, in the whole range of pressure and composition, between a pure solid phase of one of the compounds and the supercritical mixture. At the LCEP, the solubility of the solid changes with an infinite slope as a function of temperature on the isobar and as a function of pressure on the isotherm. So, just above this critical endpoint, small changes of temperature or pressure cause large variations of the solubility, and the LCEP temperature is usually a few Kelvin above the critical temperature of the solvent. Therefore, in the vicinity of the LCEP, the change of pressure induced by the piston effect should influence the reaction at the solidsupercritical fluid interface because it will cause a strong change of the solubility of the solid. Moreover, the behavior of the solubility of solids in supercritical fluids is made more complex by the existence of the retrograde phenomenon, namely the decrease of the solubility with an isobaric increase of temperature for a certain range of pressures including the LCEP pressure. Since the beginning of the eighties, the solubility of solids Published in Chemical Engineering Science, 62, pp. 4182-4192, 2007 in supercritical fluids was extensively studied, but these studies were essentially devoted to measurements of the solubility of various compounds and to the development of thermodynamic models allowing to obtain a good correlation of experimental data (see the review paper of [START_REF] Higashi | Solubilities and diffusion coefficients of high boiling compounds in supercritical carbon dioxide[END_REF]. Several papers were also published concerning measurements or modeling of adsorption equilibrium of various solutes in supercritical fluids [START_REF] Afrane | Adsorption in near critical binary solvent mixtures: thermodynamic analysis and data[END_REF][START_REF] Uchida | Adsorption behaviors of 2,6 and 2,7dimethylnaphthalene in supercritical carbon dioxide using NaY-type zeolite[END_REF][START_REF] Tapia-Corzo | Experimental study of retrograde adsorption in supercritical fluids[END_REF][START_REF] Jha | Modeling of adsorption equilibria in supercritical fluids[END_REF], but these models did not take into account hydrodynamics. To the authors' knowledge, no investigation was previously performed on heterogeneous reactions in supercritical mixtures, taking into account at the same time the specific supercritical hydrodynamics and the critical behavior of the solubility of solids in supercritical fluids. A first attempt using asymptotic analysis was published by [START_REF] Zappoli | Speeding up of heterogeneous reactions in near-critical phases[END_REF] but the very simple model, based on the van der Waals equation of state with linear mixing rules, did not take into account the critical behavior of the solubility with respect to pressure and neglected the solubility variations with temperature. In this paper, by means of numerical simulations based on the solution of the complete Navier-Stokes equations, we show that, close to the LCEP, the piston effect, coupled with the critical behavior of the solubility, can be responsible for the release of a heterogeneous reaction at a chemically active surface in a dilute binary supercritical mixture.

Physical problem and modeling

Problem under study

The Naphthalene-CO 2 mixture was chosen as a reference system with the abovementioned phase diagram because of the wide experimental data on solubility which can Published in Chemical Engineering Science, 62, pp. 4182-4192, 2007 be found in literature, allowing an easier validation of our thermodynamic model [START_REF] Nicolas | The Sanchez-Lacombe Lattice fluid model for the modeling of solids in supercritical fluids[END_REF]. The physical properties of each pure compound are given in table 1. We consider a dilute mixture of Naphthalene (the minority species, named species 2) and supercritical CO 2 (named species 1), confined between two infinite solid plates that are spaced by a distance L=1cm and made of pure solid Naphthalene (Fig. 2).

Naphthalene can be adsorbed or desorbed at the two plates following a diffusion-limited reaction. The fluid is initially at rest, in thermodynamic equilibrium at a uniform temperature T i , slightly above the LCEP temperature T LCEP =307.65 K [START_REF] Tsekhanskaya | Solubility of diphenylamine and naphthalene in carbon dioxide under pressure[END_REF], (

( ) LCEP T T i ε + = 1 with ( ) 1 << - = LCEP LCEP i T T T ε the dimensionless distance to the LCEP on the isochore LCEP ρ ρ =
), at the LCEP density ρ LCEP =470 kg.m -3 (which is not very different from the critical density of CO 2 ) and in equilibrium with the solid plates. We note that T LCEP is only a few Kelvin above the critical temperature of CO 2 .

Then, the temperature of the solid plate at x=0 is gradually increased up to T i +∆T (with ∆T about some mK) whereas the other one (at x=L) is kept at the initial temperature. In order to suppress the buoyant convective transport, we assume that the system is under zero gravity conditions. In such a case, the system essentially evolves in the direction of the temperature gradient and a one-dimensional model can be rightfully considered.

Governing equations

The evolution of the system is governed by the one-dimensional time-dependent and compressible Navier-Stokes equations, coupled with energy and mass diffusion

equations. An equation of state completes the model. Since we are interested in phenomena occurring at the time scale of the piston effect which is much longer than the acoustic one, the Low Mach number approximation [START_REF] Paolucci | On the filtering of sound from the Navier-Stokes equations[END_REF] is used to filter Published in Chemical Engineering Science, 62, pp. 4182-4192, 2007 acoustic waves, allowing to strongly reduce computational costs. This approximation leads to a splitting of the pressure P into a homogeneous thermodynamic part P th (t), appearing in the equation of state and in the energy equation, and a non-homogeneous dynamic part p(x,t), appearing in the momentum equation. For the problem treated here, the dynamic pressure is extremely small compared to the thermodynamic part (about 10 5 times smaller). As a consequence, the total pressure P is almost equal to the thermodynamic pressure P th . This latter is computed through the global mass conservation. The governing equations are thus:

( ) 0 = ∂ ∂ + ∂ ∂ u x t ρ ρ (1) ( ) ( )       ∂ ∂ ∂ ∂ + ∂ ∂ - = ∂ ∂ + ∂ ∂ x u x x p u u x u t η ρ ρ (2) ( ) ( )       ∂ ∂ ∂ ∂ = ∂ ∂ + ∂ ∂ x w D x w u x w t 21 ρ ρ ρ (3) ( ) ( )       ∂ ∂ ∂ ∂ Γ -       ∂ ∂ ∂ ∂ + ∂ ∂ - = Λ ∂ ∂ + Λ ∂ ∂ x w D x x T x x u P T u x T t th 21 ρ λ ρ ρ (4)
where u is the velocity, w is the mass fraction of Naphthalene, η is the dynamic viscosity, λ is the thermal conductivity, D 21 is the diffusion coefficient of Naphthalene in supercritical CO 2 , and Λ and Γ are respectively a function of (T,ρ,w) and a constant that are given in appendix A. We supposed that the transport properties of the mixture are those of pure CO 2 , which is a valid assumption for a dilute mixture (w<<1). The following correlations are then used for the thermal conductivity λ (W.m -1 .K -1 ) and the dynamic viscosity η (Pa.s) of CO 2 [START_REF] Arai | Supercritical fluids: molecular interactions, physical properties and new applications[END_REF]:

( ) ( ) ( ) ( ) ρ λ ρ λ λ ρ λ , , 0 T T T c e ∆ + + = , ( ) ( ) ρ η η ρ η e + = 0 ,
where the first term corresponds to the limit of small densities:

( )

c c C T T T ρ ρ ρ λ - -       × + - - = ∆ - with:    > × = < × = - - 1 3 1 3 if 10 9818 . 6 if 10 7112 . 6 c c C C ρ ρ ρ ρ .
The diffusion coefficient of Naphthalene in supercritical CO 2 , D 21 (m 2 .s -1 ), is calculated with the Wilke-Chang equation [START_REF] Higashi | Solubilities and diffusion coefficients of high boiling compounds in supercritical carbon dioxide[END_REF]:

6 . 0 2 1 15 21 10 4 . 7 b v M T D η φ - × = , (5)
with v b2 (cm 3 .mol -1 ) the molar volume of Naphthalene at boiling point, φ the association factor (φ=1 for CO 2 ) and M 1 (expressed here in g. mol -1 ) the molecular weight of CO 2 .

The thermodynamic state of the mixture is described by the Peng-Robinson equation of state in the framework of the one-fluid theory:

( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) , 2 1 , 1 2 2 2 2 1 w w b w w b w T a w w b w T M R P th θ ρ θ ρ ρ θ ρ θ ρ - + - - = ( 6
)
where R is the perfect gas constant (R=8.3145 J.mol -1 .K -1 ) and:

( ) ( ) ( ) ( )( ) ( ) ( ) ( ) w T a w w T a w T a w T a w M M w , 1 2 1 , , 1 1 2 2 12 2 1 2 1 + - + - = - - = θ ( ) ( ) ( ) ( ) , 1 2 1 2 2 1 2 12 2 1 w M M b w w b w b w b + - + - = ( ) , 1 25307 . 0 , 1 1 48742 . 1 2 cj j cj j cj cj j j b T T T M R T a ρ β ρ =                 - + = ( ), 2 , 1 , 26992 0 54226 . 1 37464 . 0 2 = - + = j . j j j ω ω β ( ) ( ) ( ) ( ) ( ), 1 2 1 , 1 12 2 2 1 1 12 12 2 1 12 l b M M b b k T a T a T a -         + = - =
with M 2 (kg.mol -1 ) the molecular weight of Naphthalene and ω the acentric factor. The binary interaction parameters k 12 and l 12 are determined so as to minimize the error between the solubility (or equilibrium mole fraction e y 2 ) calculated by the model and the experimental data, using a least squares method. More precisely, the minimized objective function is: shown that temperature dependent binary parameters are necessary to obtain a reliable correlation of solubility data in a wide range of temperatures. Therefore, the following formula was considered for k 12 and l 12 :

( ) ( ) ( ) , 1 2 exp 2 2 exp 2 ∑ =           - = N i
      - ′ ′ + ′ =       - ′ ′ + ′ = 1 15
T l l l T k k k (8)
Published in Chemical Engineering Science, 62, pp. 4182-4192, 2007 The model consists in determining the solubility e y 2 of Naphthalene in supercritical CO 2 from the equilibrium condition, namely the equality of Naphthalene fugacity in the solid and supercritical phases. This equilibrium condition leads to the following equation for the solubility: 6) written for the total pressure P (instead of P th ) and for the molar variables, y 2 and v, which are linked to specific variables by the formula:

        - =
( ) ( )ρ θ θ w M v w w M M y 1 2 1 2 , = = (10)
In fact, since the fugacity coefficient ) , ( 22 sub P T ϕ differs little from 1, it was approximated by 1 in Eq. ( 9). A mean relative deviation of 8.4% between experimental values and those calculated with Eq. ( 9) was obtained in the temperature range [308.15K, 328.15K] in which solubility data are available. The values of the binary interaction parameters are then: k' 12 =0.0395, k" 12 =0.0114, l' 12 =-0.1136 and l" 12 =-0.3103. The solubility of Naphthalene, predicted by the model with these values, versus pressure is drawn for three isotherms in Fig. 3. This figure clearly shows the zone of strong variation of the solubility around the LCEP pressure. Since the heating of Published in Chemical Engineering Science, 62, pp. 4182-4192, 2007 the plate is very small (some mK), the binary parameters k 12 and l 12 are calculated for the initial temperature using Eq. ( 8) and, then, they are kept constant during the simulation.

Solid-fluid interface reaction

The reaction at interfaces x=0 and x=L is supposed to be diffusion-limited. Thus, the fluid phase is assumed to be locally in equilibrium with the solid interface at any time, namely the partial density of Naphthalene is equal to its equilibrium value

) ( ) ( ) t L x n w t L x w D t L x u , or 0 1 , or 0 , or 0 21 = ∂ ∂ - = - = = (12)
with n the outer normal to the boundary.

Published in Chemical Engineering Science, 62, pp. 4182-4192, 2007

Numerical method

The governing equations ( 1)-( 4) and ( 6) are solved in a dimensionless form using The numerical method, which is based on finite volume approximations, is very similar to that developed by [START_REF] Amiroudine | Numerical solutions of 1D unsteady near-critical fluid flows using finite volume methods[END_REF]. The time scheme is fully implicit with the power-law scheme used for the discretization of the convective terms. On the other hand, the unsteady terms are approached by a 2 nd order backward Euler scheme whereas a 1 st order scheme was used by Amiroudine et al. We used a non-uniform staggered mesh: the scalar variables are computed at the cells centre while the velocity is defined at the cells faces. The mesh, which includes 201 points, is refined near the boundaries so as to accurately represent thin boundary layers (about 20% of grid points inside the boundary layers). The velocity-pressure coupling of the Navier-Stokes equations is treated by the SIMPLER (SIMPLE-Revised) algorithm and numerical stability in terms of time steps and grid sizes has been carefully checked.

Published in Chemical Engineering Science, 62, pp. 4182-4192, 2007

Results and discussion

3.1-General description

Simulations were performed for initial temperatures T i ranging from 307.75K to 323.15K (i.e. a dimensionless distance to the LCEP, ε, ranging from

4 10 2 . 3 - × to 2 10 5 - ×
) and a temperature increase ∆T=2mK, 5mK and 10mK at the plate x=0. It must be noted that T i =307.75K (that is only 0.1K above the LCEP temperature) is out of the temperature range for which the model parameters k 12 , l 12 were fitted on solubility data.

The values of the binary parameters k 12 , l 12 are computed from Eq. ( 8) but the accuracy of the model at this temperature cannot be evaluated since there is no available solubility data. The values of the initial pressure P i and mass fraction of Naphthalene w i , respectively obtained through the equation of state and the equilibrium condition (9), are given in table 2 for each initial temperature. Table 2 also presents, for each initial states, the values of the thermal diffusivity α of CO 2 calculated from the model as explained in appendix B, of the diffusion coefficient D 21 of Naphthalene in supercritical CO 2 calculated with Eq. ( 5), and various characteristic times, namely the thermal diffusion time t d , the mass diffusion time t Md and the time scale of the piston effect t PE .

It can be noted that the diffusion coefficient D 21 , as the thermal diffusivity α, is very small (about 10 -8 ) but, unlike α, it little changes in the temperature range that we considered. In fact, some measurements revealed that the binary diffusion coefficient in supercritical fluids strongly decreases in the vicinity of the LCEP [START_REF] Higashi | Diffusion coefficients of naphthalene and dimethylnaphthalene in supercritical carbon dioxide[END_REF][START_REF] Nishiumi | Diffusion of acetone in supercritical carbon dioxide[END_REF][START_REF] Tsekhanskaya | Diffusion of naphthalene in carbon dioxide near the liquid-gas critical point[END_REF]) but the Wilke-Chang equation does not represent this decrease. As a consequence, the characteristic time of mass diffusion t Md remains almost constant for all the values of T i and it is about 1h10min. Table 2 also shows that the time scale of the piston effect is very much smaller than the two other Published in Chemical Engineering Science, 62, pp. 4182-4192, 2007 characteristic times. In particular, t PE is about 45 thousand times smaller than the characteristic time of mass diffusion at T i =308.15K and it remains 252 times smaller than t Md at T i =323.15K, i.e. at 15.5K above the LCEP.

In all the cases, the same phenomenon was observed. We discuss here the case T i =308.15K and ∆T=10mK. The temperature of the plate x=0 is gradually increased using a cosine law during t heat =10t PE , i.e. approximately 1s. More precisely, the temperature increase is defined by: ( )

     > ≤                 - = heat heat heat t t ∆T t t t t ∆T . t T if if cos 1 5 0 π δ
As in the case of a pure supercritical fluid, the boundary heating induces a thin thermal diffusion layer, associated with a large density gradient, near the heated plate.

Due to the high dilatability of the fluid, this boundary layer strongly expands and compresses adiabatically the rest of the fluid leading to a fast and homogeneous increase of its temperature by the piston effect (Fig. 4). The fluid near the isothermal boundary is then warmer than the solid plate and a second thermal boundary layer forms near this latter where diffusion cools the fluid. The fluid layer is therefore divided into three zones: two thermal boundary layers, associated with large density gradients (Fig. 5), and the isothermal bulk of the fluid (Fig. 4). We noted that the velocity profiles are also similar to those obtained in a pure supercritical fluid. Therefore, the thermomechanical response of the mixture to the heating of the plate is the same as that of the pure supercritical fluid.

Published in Chemical Engineering Science, 62, pp. 4182-4192, 2007 We consider now the chemical response of the system to the heating of the plate.

Since the temperature increase at x=0 is very small, the boundary condition (11) for w can be written at first order as:

( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) (
).

, 0 , 0 , , 0 , , 0

i i i i i i T e c i i i P e c i t x w P t x P P T P w T t x T P T T w w t x w ρ ρ ρ - = - - =                 ∂ ∂ + - =                 ∂ ∂ + = = (13) with: ( ) ( ) ( )
( ) i i y P i i i P e i i i i i i i P e c P T w P T T y P T v M w M M w M M P T T w , , , 1 1 , 2 , 2 2 1 1 2 2 1 β ρ -         ∂ ∂       -                 - - =         ∂ ∂ (14) ( ) ( ) ( )
( ) i i y T i i i T e i i i i i i i T e c P T w P T P y P T v M w M M w M M P T P w , , , 1 1 , 2 , 2 2 1 1 2 2 1 χ ρ +         ∂ ∂       -                 - - =         ∂ ∂ (15)
where ( ) ( ) ( )

i i P T i i P T y v P T v , , , 2 2 ∂ ∂ =
is the partial molar volume of Naphthalene (which is negative), An equation similar to Eq. ( 13) (without the term due to the temperature perturbation) can be written for w (x=L, t). Equations ( 14) and ( 15 Published in Chemical Engineering Science, 62, pp. 4182-4192, 2007 So, the term due to the temperature perturbation results from the coupling between the critical behavior of the solute solubility with respect to temperature and the strong dilatability of the supercritical solvent, and the term due to the pressure perturbation results from the coupling between the critical behavior of the solute solubility with respect to pressure and the strong compressibility of the supercritical solvent, especially the piston effect that it causes. It can be noted that the term due to the density . As a consequence, the term due the temperature perturbation is the largest one in Eq. ( 13). Thus, the temperature increase of the plate x=0 leads to a strong decrease of w(x=0, t) as it is shown in Fig. 6. At the same time, the piston effect, generated by the boundary heating, provokes a fast, strong and homogeneous increase of the pressure in the whole cell (Fig. 7), leading to a strong increase of the term due to the pressure perturbation in Eq. ( 13).

( ) ( ) ( ) i i i y P i i y P v P T T v P T , , 2 2 , , ∂ ∂ = β is the thermal expansion coefficient at constant composition for ( ) i i e P T y y , 2 2 = , and
( ) ( ) ( ) i i i y T i i y T v P T P v P T , , 2 2 ,
Since the increase of the plate temperature is done using a cosine law, this temperature increase is weaker near the end of the heating phase (i.e. for t approaching t heat ) and the terms due to the pressure and density perturbations become then the largest ones so that w(x=0, t) begins to increase again before the end of the heating phase (t=t heat ) (Fig. 6).

From t≈10s, w(x=0, t) slowly keeps on increasing towards an asymptotic value. The maximum perturbation of w at the heated plate,

( ) ( ) i w t x w t x w - = = = , 0 , 0 δ
, which is obtained about 0.05s before the end of the heating phase, is equal to -7.6973×10 -5 , i.e. it corresponds to a decrease about 1% of the mass fraction of Naphthalene for an increase of 0.0032% of the plate temperature. The perturbation of the mass fraction of Published in Chemical Engineering Science, 62, pp. 4182-4192, 2007 Naphthalene is therefore 312 times larger than that of temperature. Since the diffusion coefficient of Naphthalene in supercritical CO 2 is very small, the strong decrease of the mass fraction at the plate x=0 due to the heating induces a thin diffusion boundary layer that grows very slowly, at the mass diffusion speed (see Fig. 8). To summarize, the leading mechanism governing the evolution of the mass fraction of Naphthalene at the heated plate is the critical behavior of the solubility with respect to temperature. We have just induced a change of the mass fraction of Naphthalene at the plate x=0 by modifying the parameters of the interface reaction (i.e. by increasing the plate temperature), exactly as in the case of a perfect gas mixture. However, the response of the system is very different from that of a perfect gas because of the retrograde solubility phenomenon that is specific to supercritical fluids [START_REF] Raspo | Fast mass transfer at a solid-supercritical fluid interface by piston effect[END_REF].

Moreover, the perturbation of the mass fraction resulting from the temperature perturbation is much larger than that obtained in a perfect gas mixture.

On the other hand, the piston effect, coupled with the critical behavior of the solute solubility with respect to pressure, is actually the leading mechanism governing the evolution of the mass fraction of Naphthalene at the isothermal plate x=L. Indeed, the term due to the temperature perturbation disappears from Eq. ( 13) since temperature remains unchanged. The piston effect induced by the heating of the opposite plate provokes a strong and homogeneous increase of the pressure in the whole cell, and therefore at the isothermal interface. Thus, due to the critical behavior of the solubility with respect to pressure, this strong increase of pressure induces a fast and strong increase of w at the plate x=L through the term due to the pressure perturbation in Eq. ( 13) (Fig. 6). At the end of the heating phase (t=t heat ), the perturbation of w at the isothermal plate,

( ) ( ) i w t L x w t L x w - = = = , ,

δ

, is equal to 5×10 -5 . Therefore, it corresponds to an increase about 0.65% of the mass fraction of Naphthalene, i.e. a Published in Chemical Engineering Science, 62, pp. 4182-4192, 2007 perturbation 203 times larger than that of temperature. So, we induced a change of the mass fraction at the isothermal plate x=L by modifying the parameters of the interface reaction (namely the pressure) but this modification is generated by a perturbation that is applied at the opposite side of the system and that affects the plate x=L thanks to the non-local character of the piston effect. Therefore, the phenomenon occurring at the isothermal plate is specific to supercritical fluids. The behavior of the system is completely different from that of a perfect gas mixture [START_REF] Raspo | Fast mass transfer at a solid-supercritical fluid interface by piston effect[END_REF]. As at the heated plate, the increase of w(x=L, t) induces, since the beginning of the simulation, a thin diffusion layer which grows at the mass diffusion speed (Fig. 8). There is a very fast mass transfer at the isothermal plate, as at the heated plate, which is illustrated by the evolution of the mass flux

n w D ∂ ∂ = Φ 21 ρ
(with n the outward normal to the boundary) at the two reactive interfaces in Fig. 9. Indeed, the mass flux exhibits a very large peak at the end of the heating phase (t=t heat ). On the other hand, the mass transfer from the plate towards the bulk of the fluid is very slow due to the very small value of the binary diffusion coefficient of Naphthalene in supercritical CO 2 .

To summarize, thanks to the piston effect (affecting very rapidly the whole system since its characteristic time is less than 0.1s), the heating of the plate x=0 causes the fast release of the heterogeneous reaction at the opposite plate, kept at the initial temperature, because of the critical behavior of the solute solubility. The behavior of the interface reactions at the two plates results, from the very beginning of the heating phase, in a mass fraction field divided into three zones just like the temperature field: two thin diffusion boundary layers near the solid plates and the bulk of the cell where the mass fraction of Naphthalene remains unchanged (Fig. 8). After the heating phase (i.e. for t>t heat ), the mass fraction of Naphthalene at the reactive plates slightly evolves and Naphthalene slowly diffuses in the bulk of the cell.

Published in Chemical Engineering Science, 62, pp. 4182-4192, 2007

3.2-Proximity of the LCEP

The above-described behavior of the mass fraction of Naphthalene at the two reactive interfaces was observed for all the considered initial temperatures T i and temperature increases ∆T. However, the proximity of the LCEP influences the value of the mass fraction at the reactive interfaces, as the maximum of the mass flux, since the derivatives of the solubility become smaller and the piston effect becomes less effective as one moves away from the LCEP. Distances to the LCEP ranging from 0.1K to 15.5K were considered. For T i =307.75K, 308.15K and 309.15K, the heating phase lasts about 1s (more precisely 120t PE , 10t PE and 2t PE respectively). On the other hand, for T i =311K, 318.15K and 323.15K, the characteristic time of the piston effect is larger than 1s (see table 2) and the plate temperature is gradually increased during 1t PE , since the behavior of the reaction at the isothermal plate is governed by the piston effect. Figure 10 shows the ratio between the relative perturbation of the mass fraction at the two reactive interfaces (i.e. ) as a function of the dimensionless distance to the LCEP, ε, for ∆T=5mK. We noted that, for a given value of ε, this ratio little changes with ∆T; its variation from the value obtained for ∆T=5mK does not exceed 0.2% for the smallest ε. Very close to the LCEP (for T i =307.75K, i.e. Published in Chemical Engineering Science, 62, pp. 4182-4192, 2007 inverted for larger values of ε. This behavior change is due to the ratio between the derivative of the solubility with respect to temperature and that with respect to pressure. Indeed, if the contribution of the term due to the density perturbation is neglected, we can write: , the ratio between the relative perturbation of pressure and that of temperature is about 3, and the ratio of the derivatives of e c w , little different from that of the derivatives of the solubility, is larger that 6 so that the relation ( 16) is satisfied. 16) is no longer satisfied and, consequently, the perturbation of the mass fraction becomes larger at the isothermal interface than that at the heated plate. Thus, far enough from the LCEP, the heating of the plate at x=0 induces a quantity of matter involved in the reaction which is larger at the opposite isothermal plate than that at the heated plate, and the relative perturbation of the mass fraction at x=L nevertheless remains about 20 times larger than that of temperature due to the heating.

( ) ( ) ( ) ( ) ( )         ∆         >                 ∂ ∂                 ∂ ∂ - ⇔ = > = i i i i T e c i i i P e c i
We noticed that the ratio ( ) ( )

Conclusion

Therefore, we have shown that, by heating a supercritical dilute mixture at one side, the piston effect (through the strong and homogeneous increase of pressure that it provokes in the whole fluid) causes the release of a heterogeneous reaction at the opposite plate which was kept at the initial temperature. This phenomenon is possible because of the non-local character of the piston effect and because of the critical behavior of the solubility of a solid in a supercritical fluid with respect to pressure.

Simulations in the case of a cooling of the plate x=0 were also carried out and a similar phenomenon was observed: the strong decrease of the pressure caused by the piston effect induces a fast and strong decrease of the mass fraction at the isothermal plate, leading to similar but reversed profiles of the mass fraction (Fig. 12). The route is now opened towards the exploration of the coupling between convection and heterogeneous reactions in supercritical mixtures, including hydrothermal synthesis and crystal growth.

      + + - +          ∂ ∂ -    + Ω +    +    + + + = Λ ρ θ ρ θ θ 2 1 1 2 1 1 , 2 2 5 4 3 2 4 1 1 3 1 1 2 1 1 1 1 1 1 w w b w w b Ln T a T w T a w b w T w T M E T M D T M C T M B M A w ( ) ( ) ( ) 1 0 1 2 0 2 0 0 M T H M T H T T - + Ω = Γ with T 0 =298.15K, ( ) 0 2 T H and ( ) 0
1 T H the perfect gas enthalpy of Naphthalene and CO 2 respectively at temperature T 0 , and:

Published in Chemical Engineering Science, 62, pp. 4182-4192, 2007 ( )

5 4 3 2 4 1 1 2 2 3 1 1 2 2 2 1 1 2 2 1 1 2 2 1 1 2 2 T M E M E T M D M D T M C M C T M B M B M A M A T         - +         - +     -     +         - +         - = Ω
where the coefficients

j j j j j E D C B A , , , ,
come from the expression of the heat capacity at constant pressure for the perfect gas [START_REF] Poling | The properties of gases and liquids[END_REF]:

( ) . 4 3 2 T E T D T C T B A T C j j j j j j p + + + + =

Appendix B

The thermal diffusivity α of CO 2 at initial conditions is defined by the formula:

P i C ρ λ α =
The specific heat at constant pressure Cp is computed from the specific heat at constant volume C V , that is fixed at its value for the perfect gas ( .K J.kg 04 . 648

1 0 - = = V V C C ),
using the thermodynamic equation:

T i i V P P T P T C C       ∂ ∂       ∂ ∂ + = ρ ρ ρ 2 2
in which the derivatives are calculated using the Peng-Robinson equation for pure CO 2 . Published in Chemical Engineering Science, 62, pp. 4182-4192, 2007 Fig. 11. Maximum of the mass flux at the two plates versus the dimensionless distance to the LCEP, ε, for ∆T=5mK.

FIGURES

  and those estimated by the model.[START_REF] Nicolas | The Sanchez-Lacombe Lattice fluid model for the modeling of solids in supercritical fluids[END_REF] have

.

  the following boundary condition for w: fraction for the critical density ρ LCEP . The equilibrium partial density of Naphthalene is computed from its solubility v the molar volume of the mixture given by the Peng-Robinson equation for T, P, and That is why it is necessary to accurately represent the solubility behavior. The boundary condition for the velocity is obtained through the Fick's law:

(

  ) show that the derivatives of e c w depends not only on the solubility derivatives (which are very large Published in Chemical EngineeringScience, 62, pp. 4182-4192, 2007 near the LCEP) but also on the thermal expansion coefficient and on the isothermal compressibility that are also very large since the solvent is supercritical. The contribution of these two coefficients increases the term due to the solubility derivatives, so that the derivatives of e c w are about 4 times larger than the solubility ones. For T i =308.15K, we have ( temperature change of 1K (namely about 0.3%) induces a solubility variation equal to 3.78×10 -3 (namely about 142% !) and a pressure change of 1bar (namely about 1%) induces a solubility variation equal to 2.21×10 -3 (namely about 83% !). Equation (13) shows that the evolution of the mass fraction of Naphthalene at the solid interfaces is governed by three mechanisms:1. The critical behavior of the solute solubility with respect to temperature and pressure through the very large values of the derivatives ( ) the terms due to the temperature and pressure perturbations; of course, the critical behavior of the solubility with respect to temperature acts only at the heated plate;2. The strong compressibility of the supercritical fluid through the derivative piston effect, that results from it, through the pressure variation;3. The strong dilatability of the supercritical fluid which plays through the derivative

  t=t heat and that of the temperature at the heated plate (i.e.

  the mass fraction at the heated plate and that at the isothermal interface are respectively about 400 and 300 times larger than the temperature one which causes it. Figure10also shows that the ratio (

  that the relation (

.

  However, as the LCEP is approached, this law is no longer true andplate does not evolve as the inverse of a power of ε.Finally, figure11shows the maximum of the mass flux at the two reactive interfaces as a function of the dimensionless distance to the LCEP, ε, for ∆T=5mK. As for the relative perturbation of the mass fraction, the mass flux at the heated plate is larger than that at the isothermal interface tendency is inverted for larger values of ε.

Fig. 1 .

 1 Fig. 1. Phase diagram of binary mixtures of interest. (CP: Critical Point; LCEP: Lower

Fig. 2 .

 2 Fig. 2. Physical configuration.

Fig. 4 .

 4 Fig. 4. Profiles of the temperature perturbation at several times for T i =308.15K and ∆T=10mK.

Fig. 5 .

 5 Fig. 5. Profiles of the density perturbation at several times for T i =308.15K and ∆T=10mK.

Fig. 6 .

 6 Fig. 6. Evolution of the mass fraction of Naphthalene w at the heated plate x=0 and at the isothermal plate x=L for T i =308.15K and ∆T=10mK. The small figure shows the

Fig. 8 .

 8 Fig. 8. Profiles of the perturbation of the mass fraction at several times for T i =308.15K and ∆T=10mK.

Fig. 9 .

 9 Fig. 9. Evolution of the mass flux at the heated plate x=0 and at the isothermal plate x=L for T i =308.15K and ∆T=10mK.

Fig. 10 .

 10 Fig. 10. Ratio of the relative perturbation of the mass fraction of Naphthalene at the two plates, ()i i heat w w t L x w -= , or 0, at the end of the heating phase and that of the

Fig. 12 .

 12 Fig. 12. Profiles of the perturbation of the mass fraction at several times for T i =308.15K in the case of a cooling of the plate x=0 for ∆T=-10mK.

  Fig.1

Fig. 4

 4 Fig. 4

Fig. 5

 5 Fig. 5

Fig. 6

 6 Fig. 6

Fig. 7

 7 Fig. 7

Fig. 8

 8 Fig. 8

  Fig. 9

  T LCEP as reference temperature, ρ LCEP as reference density,

											ρ	LCEP	(	M R 1	) LCEP T	as
	reference pressure, L as reference length, the time scale of the piston effect as reference
	time (	PE t =	( t	d	) 2 3 Pr ε	, where	Pr	=	η p C	λ	is the Prandtl number for CO 2 , with C p the
	specific heat at constant pressure that is calculated from the Peng-Robinson equation of
	state (see appendix B), and t

d is the characteristic time of thermal diffusion) and the corresponding velocity (i.e. L/t PE ) as reference velocity.

  perturbation tends to increase w at the plate x=0 and to decrease w at the plate x=L.

													Due
	to the retrograde phenomenon, the derivative ( ∂ w ∂ e c T	) P	, that is very large, is negative.
	Moreover, it is much larger than ( ∂ w ∂ e c P	) T		(in dimensionless form,
	i T	( ∂ w c e	∂ T	) ( ) 96 . 4 , -= i i P P T	whereas ( ∂ i w c e P	∂ P	) ( ) , i i T P T	=	0	.	175	)

  Published in Chemical EngineeringScience, 62, pp. 4182-4192, 2007

		Published in Chemical Engineering Science, 62, pp. 4182-4192, 2007 Published in Chemical Engineering Science, 62, pp. 4182-4192, 2007
	y 2 Superscript mole fraction of Naphthalene.
	Notation cal calculated value.
	a Greek letters energy parameter in the Peng-Robinson equation. e value for solid-supercritical fluid equilibrium.
	b α exp experimental value. covolume in the Peng-Robinson equation. thermal diffusivity.
	D 21 β s	diffusion coefficient. thermal expansion coefficient. value in the solid phase.
	k 12 , l 12 χ isothermal compressibility. binary interaction parameters in the mixing rules. sub value at sublimation.
	L ε	distance between the two reactive plates. dimensionless distance to the gas-liquid critical point for the pure compound or to
	M	molecular weight. the Lower Critical EndPoint (LCEP) for the mixture.
	P φ	pressure. association factor.
	δP Φ	perturbation of the pressure. mass flux at the solid interfaces.
	Pr η	Prandtl number. viscosity.
	P th ϕ	thermodynamic pressure in the low Mach number approximation. fugacity coefficient.
	p ρ	dynamic pressure in the low Mach number approximation. density.
	R ω	perfect gas constant. acentric factor.
	t		time.
	t d		characteristic time of thermal diffusion.
	Subscript
	t Md		characteristic time of mass diffusion.
	b		value at boiling point.
	t PE		characteristic time of the piston effect.
	c		value at the gas-liquid critical point.
	T		temperature.
	∆T i		temperature increase applied at the plate x=0. value at the initial state.
	j		refers to component j of the mixture.
	u		velocity.
	LCEP	value at the LCEP.
	v		molar volume.
	1		refers to CO 2 .
	w		mass fraction of Naphthalene.
	2		refers to Naphthalene.
	δw		perturbation of the mass fraction at the reactive plates.
	x		space coordinate.

Table 1 :

 1 TABLES Pure component properties.

Table 2 :

 2 Initial states (T i , P i , w i ) considered in the study and the corresponding values of the dimensionless distance to the LCEP ε, of the thermal diffusivity α, of the diffusion coefficient D 21 and of the various characteristic times (thermal diffusion t d , piston effect t PE , mass diffusion t Md ).

		T c	ρ c	M	ω	v s	v b
		(K)	(kg.m -3 )	(kg.mol -1 )		(m 3 .mol -1 )	(cm 3 .mol -1 )
	CO 2 (1)	304.21	467.8	4.401×10 -2	0.225		
	Naphthalene (2) 748.40	314.9	1.282×10 -1	0.302	1.1×10 -4	155

Table 1 .

 1

	T i (K)	ε	P i (MPa)	w i	(m 2 .s -1 )	D 21 (m 2 .s -1 )	t d (s)	t Md (s) t PE (s)
	307.75 3.25×10 -4	7.836815 7.5216×10 -3	8.8941×10 -9	2.2298×10 -8	11243 4485 0.0084
	308.15 1.62×10 -3	7.909092 7.6751×10 -3	9.6465×10 -9	2.2327×10 -8	10366 4479	0.094
	309.15 4.88×10 -3	8.089188 8.0713×10 -3	1.1375×10 -8	2.2399×10 -8	8791	4464	0.49
	311	1.09×10 -2	8.420040 8.8531×10 -3	1.4112×10 -8	2.2533×10 -8	7086	4438	1.62
	318.15 3.41×10 -2	9.666847 1.2557×10 -2	2.1632×10 -8	2.3051×10 -8	4623	4338	9.02
	323.15 5.04×10 -2 10.503144 1.5929×10 -2	2.5724×10 -8	2.3413×10 -8	3887	4271	16.2

Table 2 .

 2

Acknowledgements

The authors acknowledge the financial support from the CNES (Centre National d'Etudes Spatiales). They gratefully thank Pr. Evelyne Neau and Dr. Christophe Nicolas (MSNM-GP) and Dr. Roger Prud'homme (LMM, UMR 7607, Paris, France) for their friendly help and fruitful discussions.

Appendix A

Coefficients Λ and Γ appearing in Eq. ( 4) are defined by: