Dimitri Mutschler

Mohamed Aziz Larabi

Abdelkader Mojtabi

Theoretical models for the thermo-gravitational separation process in porous media filled by N-component mixtures ⋆

The aim of this work is to present a theoretical analysis of the separation of an N -component mixture. In this study, two analytical models explaining the thermo-gravitational separation of components in N -component mixtures for vertical cavity filled by a porous medium are presented and assessed. The basic state and the separation are expressed in terms of the separation ratio, and the Lewis, crossdiffusion and Rayleigh numbers. Our computational analysis confirms that, for the given values of the mass fractions, thermodiffusion can be measured with a thermo-gravitational column, strongly supporting the experimentally determined transport coefficients.

Introduction

Transport properties of fluid mixtures have attracted much interest because of their numerous applications. For instance, the double-diffusion process plays an important role in the transport of contaminants in soil, in crystal growth and in the separation of components in a mixture [1][2][3]. More details about the fundamental and industrial applications are presented by Legros et al. [4], Vafai [START_REF] Vafai | Handbook of Porous Media[END_REF], Nield and Bejan [START_REF] Nield | Convection in Porous Media[END_REF], Baytaş and Pop [START_REF] Baytaş | [END_REF]. Many works have been devoted to improving the thermogravitational separation in a mixture. In 1939, Furry, Jones and Onsager [8] developed a theory that allowed the separation of binary gases in a thermo-gravitational column to be predicted. A thermo-gravitational column is a rectangular vertical cavity that is differentially heated and filled with a mixture. The difference in temperature produces a unicellular flow which separates the components between the bottom and the top of the column. The theory developed in [8] is called the FJO theory. In their model the influence of the mass fraction in the buoyancy term is neglected (the "forgotten effect"). In order to improve the separation in thermo-gravitational columns, Lorenz and Emery [9] introduced a porous medium in the cavity and Bennacer et al. [10] suggested splitting the thermo-gravitational column into three sub-domains filled with a porous medium. Nowadays, thermo-gravitational columns are used to determine the transport properties ⋆ Contribution to the Topical Issue "Non-isothermal transport in complex fluids", edited by Rafael Delgado-Buscalioni, Mohamed Khayet, José María Ortiz de Zárate and Fabrizio Croccolo.

a e-mail: abdelkader.mojtabi@imft.fr of binary and ternary mixtures. Several approaches have been developed for the experimental determination of the diffusion, thermal diffusion, and Soret coefficients for binary and ternary mixtures. Blanco et al. [11] measured the thermodiffusion coefficients of binary and ternary mixtures using a thermo-gravitational column, while Kolodner et al. [12] and Gebhardt and Kölher [13] used optical techniques to determine the transport coefficients of ternary mixtures. If the heaviest component of the mixture has a negative Soret coefficient, it will be transported to the warmer part of the column by the thermodiffusion effect and to the bottom of the cavity by the gravitational field. These phenomena prevent a negative thermodiffusion coefficient from being measured in ground conditions and for this reason Bou-Ali et al. [14] have made measurement of the thermodiffusion coefficients in the International space station. The mixture of 1,2,3,4-tetrahydronaphthalene (THN), isobutylbenzene (IBB) and n-dodecane (nC12) at mass fraction 0.8-0.1-0.1 were chosen as a benchmark to validate space measurements since, at this composition it allows for ground measurements [14]. Nowadays, binary mixtures are widely studied and their transport coefficients measured accurately. This is however not the case for ternary mixtures, as the cross-diffusion effect is still poorly understood and the measured cross-diffusion coefficients are highly dispersed [14,15]. In order to improve our understanding of multi-component mixtures numerical and theoretical studies have been performed. Larre et al. [16] have studied the onset of convection in infinite horizontal layers filled with a ternary mixture without taking the cross-diffusion effect into account. Haugen and Firoozabadi [17,18] have developed a model for measuring the thermodiffusion and diffusive coefficients in N -components mixture. Theirs model is based on the transient duration to the steady state. The authors obtained good agreement when comparing the transports coefficients determined with their theoretical model with the literature for seven binary mixtures [19]. Ryzhkov et al. [20] obtained an analytical expression of the basic flow for an N -component mixture in a thermo-gravitational column. Their dimensionless formulation introduced the matrix of Schmidt numbers Sc with (N -1) 2 elements. Their theoretical models are based on the parallel flow approximation and the FJO theory. The stability of the unicellular flow in an infinite or finite thermo-gravitational column is studied in [21,22]. Larabi et al. [23] proposed a method to indirectly validate the measured transport coefficients of ternary mixtures in space conditions. Shevtsova et al. [24] and Lyubimova and Zubova [25] performed three-and two-dimensional numerical simulations of the Soret-driven convection of a cell filled by binary or ternary mixtures. Ghorayeb and Firoozabadi [26] numerically predicted the mass fraction of ternary hydrocarbon mixtures in a two-dimensional reservoir. The aim of the present study is to extend Larabi et al.'s work [23] and the FJO theory [8] to an N -component mixture. A two-dimensional vertical cavity filled with a porous medium saturated by an N -component mixture is considered. The governing equations are obtained and solved using the parallel flow approximation and the "forgotten effect" for a multicomponent mixture. The mass fraction, temperature, velocity and separation are analytically expressed as functions of the filtration Rayleigh number, Lewis number, separation ratio and cross-diffusion numbers. All the theoretical results are assessed by numerical simulations and the difference between the two theoretical studies and the numerical results is quantified.

2 Problem statement

Mathematical formulation

This study focuses on the thermo-gravitational separation process in a vertical cavity filled with a porous medium saturated by an N -component mixture. L is the height of the cavity along the y-axis and h the thickness along the x-axis. The porous medium is considered homogeneous and isotropic. Figure 1 illustrates the system under consideration along with the thermal boundary conditions. The vertical walls are maintained at uniform temperature while the horizontal walls are thermally insulated. All boundaries are assumed stiff and impermeable. The problem is described by Darcy's law, the mass conservation of each component in the mixture and the energy conservation equation. The Oberbeck-Boussinesq approximation is considered and all the thermophysical coefficients of the fluid are assumed to be constant except for the density in the buoyancy term

ρ d = ρ d 0 1 -β T (T d -T d 0 ) - n-1 i=1 β ci (C d i -C d i0 ) , (1)
T 1 T 2 L J T • n =0 J T • n =0 g x y h
Fig. 1. Vertical thermo-gravitational column and thermal boundary conditions. g is the gravity vector.

where the superscript d stands for the dimensional formulation, N is the number of components in the mixture, ρ d is the density of the mixture and ρ d 0 is the density at the reference temperature and mass fraction. β T and β Ci are the thermal expansion coefficient and the mass fraction expansion coefficient, respectively, of component i at the reference temperature and mass fraction. T , C i and C i0 are the temperature, the mass fraction of component i and the initial or reference mass fraction of component i, respectively. The fluid is assumed to be incompressible. The convective flow and the heat and mass transfer are governed by the following dimensionless equations [23]:

∇ • V =0, (2)
V = -∇P + Ra T + ψ T • C e y , (3)
ε∂ t C +(V • ∇) C = Le -1 •∇ 2 (Cr • C -1 • T ) , (4)
∂ t T + V • ∇T = ∇ 2 T, (5)
where the reference scales are h for the geometric parameter and h 2 κ/a for the time (κ =( ρc p ) p /(ρc p ) f ), with (ρc p ) p the volumetric heat capacity of the saturated porous medium and (ρc p ) f the mixture heat capacity. Furthermore

T =(T d -T 2 )/∆T , where T d is the dimensional temperature, ∆T = T 1 -T 2 .
The mass fraction is scaled by

C i =(C d i -C 0 )/∆C i . ∆C i = -∆T D ′ Ti /D ii , where D ii is the mass diffusion coefficient of the component i. C is a vector of length (N -1) and D is the diffusion matrix of size (N -1) × (N -1). D ′ Ti is the thermal diffusion coef- ficient of the component i. 1 is a vector of length (N -1) equal to 1 =[1...1]
. P is the pressure and V the filtration velocity vector. All vectors are column and the superscript T is the transposed vector. The filtration velocity represents the mean fluid velocity taken over a representative elementary volume and is equal to V = εV f , where ε is the normalized porosity ε = ε * (ρc) f /(ρc p ) * and ε * is the cell porosity. The thermal filtration Rayleigh number

Ra = Khgβ T ∆T νa , (6)
where a = λ/(ρc p ) f is the equivalent thermal diffusivity, λ is the thermal conductivity of the mixture, K is the permeability of the porous medium and (ρc p ) f is the specific heat capacity of the mixture. The separation ratio vector (length: N -1) is given by

ψ i = - β ci • D ′ Ti β T • D ii . ( 7
)
Note that the definition of the separation ratio is differente than in Ryzhkov et al. [21,22]. The matrix of Lewis numbers Le ii = a/D ii and the cross-diffusion numbers

Cr ij =(D ij /D ii )(∆C j /∆C i ) are of size (N -1) × (N -1).
Note that the matrix Le is diagonal and the matrix Cr has the main diagonal elements equal to unity, hence, they both comprise only (N -1) × (N -1) independent coefficients. In conclusion, the problem under consideration, accounting for separation ratios, Rayleigh number and porosity, depends on (N -1) 2 + N + 1 dimensionless parameters. The dimensionless boundary conditions for x =0andx =1are

V • n =0 , J i • n =0, (8) T (x =0,y)=1 ,T (x =1,y)=0, ( 9
)
where n is the unit vector and J i is the dimensionless mass flow of the component i,

J i = -Le -1 •∇(Cr • C i -1 • T ). ( 10
)
The dimensionless boundary conditions for y =0,A are

V • n =0 , J i • n =0, ( 11
) ∂ y T (x, y =0)=0,∂ y T (x, y = A)=0, (12)
where A = L/h is the aspect ratio of the column. Ryzhkov et al. [21,22] or Lyubimova and Zubova [25] reduced the number of dimensionless parameters of this problem by working in the standard basis of the matrix of Schmidt numbers. This method, based on the matrix diagonalization technique, uncouples the equation of conservation of each mass fraction. This transformation would be useful when studying the stability of flows in multi-component mixtures. We intend to implement it in our forthcoming work on such stability. We used the data of the benchmark DCMIX1 [14] to provide the values of the diffusion coefficients so that our results would apply to a real-world ternary mixture. This benchmark was composed of 1,2,3,4-tetrahydronaphthalene (THN), isobutylbenzene (IBB) and ndodecane (nC12) with mass fractions 0.8-0.1-0.1. All the transport coefficients were taken from [14]. Note that in

D ′ T 1 = D 11 S T 1 + D 12 S T 2 , (13)
D ′ T 2 = D 22 S T 2 + D 21 S T 1 . (14)
The results obtained were D 12 =0 .15 • 10 -10 m 2 /sa n d D 21 =1.1 • 10 -10 m 2 /s. In porous media, tortuosity affects all the transport coefficients, a tortuosity equal to τ =1.35 has been determined by Costesèque et al. [27]. The thermodynamic coefficients β T =8.48•10 -4 /K, β c1 = -0.136, β c2 =0 .120, µ =1 .719 • 10 -3 kg/(m • s) and the density ρ = 925.3kg/m 3 were taken from [28]. For the thermal conductivity of the mixture inside the pores, an average value was considered based on the thermal diffusivity of each pure component found in [29], λ =0.128 W/(m • K). The dimensionless numbers obtained using these thermodynamic coefficients are listed in table 1.

3 Parallel Flow Approximation

Basic flow

For binary mixtures, a previous study [30] showed that the Parallel Flow Approximation (PFA) was valid in shallow cavities. With the PFA, the streamlines are assumed to be parallel to the vertical walls of the thermo-gravitational column, thus the horizontal velocity is taken equal to zero. The PFA is valid far from the horizontal walls. The basic state is approximated by

V = v(x) e y , (15)
C(x, y)=m y + f (x), ( 16
) T (x, y)=m T y + g(x). (17)
Equations ( 15)-( 17) describe a unicellular flow for which there is only one convective roll in the thermogravitational column. By inserting eqs. ( 15)-(17) into the system of eqs. ( 2)-( 5), the following system for v(x), T (x) and f (x) is obtained:

∂ 2 x v(x) -ω 2 v(x)=0 ,∂ 2 x T (x)=0, (18)
Le -1 • Cr • ∂ 2 x C(x, y)=m v(x), (19)
where

ω 2 = Ra ψ T • Le • Cr -1 • m.
The PFA is not valid close to the horizontal walls, thus, the boundary condition on the latter must be modified. In order to provide for the lack of boundary condition, the following hypotheses are considered:

-the mass flow rate is assumed to be equal to zero through any horizontal cross-section

1 0 v(x)dx =0, (20)
-the conservation of each component in the cavity is assumed

1 0 A 0 C(x, y)dx dy =0, (21)
-the thermal flow rate through any horizontal crosssection is equal to zero

1 0 (V • T + J T ) • n dx =0. ( 22
)
J T is the thermal flow equal to J T = -∇T . A last boundary condition is required to solve the system of eqs. ( 18)- (19). In order to obtain another boundary condition, the curl operator was applied to eq. ( 3) and evaluated for x =0, 1:

∂ x v = Ra ∂ x (T + ψ T • C). (23)
The temperature was

T =1-x. (24)
The solution for the velocity and the mass fraction depend on the sign of ω,i fω 2 > 0 then

v(x)=-ψ 0 1 -cosh(ω) sinh(ω) cosh(ωx) + sinh(ωx) , (25)
C(x, y)= Le • Cr -1 • m ω 2 v(x)+Ra x - Ra 2 +m y - A 2 . ( 26
)
If

ω 2 = 0, then v(x)=Ra ψ T • (Cr • 1)+1 1 2 -x , (27)
C(x, y)=RaLe • Cr -1 • m x 2 4 - x 3 6 - 1 24 +Cr • 1 1 2 -x + m y - A 2 . ( 28
)
On the other hand, if ω 2 < 0 then v(x)=-ψ 0 cos(ω) -1 sin(ω) cos(ωx)+sin(ωx) , (29)

C(x, y)=- Le • Cr -1 • m ω 2 v(x)+Ra x - Ra 2 -2 x Cr -1 • 1 + m y - A 2 , (30)
where

ψ 0 = Ra ω (1 + ψ T • Cr -1 • 1).

Vertical Mass fraction Gradient and separation

For vertical cavities, a unicellular flow separates each component i between the top and bottom of the cavity. Thus the separation of the component i is equal to S = A • m, m is a vector of length N -1 and is called the Vertical Mass fraction Gradient (VMG). In order to determine the VMG of each component, the mass flow rate of each component through a horizontal cross-section is assumed equal to zero:

1 0 (V C i + J i ) • n dx =0. (31)
Solving this coupled system of equations gives the separation of each component of an N -component mixture. If ω = 0, an analytical expression can be obtained:

m =10Ra ψ T • (Cr • 1)+1 (RaLe) 2 +120Cr 2 -1 Le • 1.
(32) For a mixture, the two experimental control parameters of a thermo-gravitational column are the column thickness and the difference in temperature. Figure 2 shows the influence of the control parameters of a thermo-gravitational column on the separation component for a ternary mixture. The separation is strongly influenced by the thickness of the cavity.

Furry, Jones and Onsager model 4.1 Basic flow

In 1939 [8] the thermo-gravitational separation process was analytically studied for vertical cavities filled with a binary mixture of gases. In [8], the mass fraction influence on the density was neglected as opposed to the temperature influence on the term of the gravitational force. This model is called the FJO model. Applying the curl operator and the FJO model to eq. ( 2), the system of eqs. ( 2)-( 5) leads to

∇ • V =0, (33) ∂ x v(x)=Ra∂ x T (x), ( 34
)
ε∂ t C +(V • ∇) C = Le -1 •∇ 2 (Cr • C -1 • T ) , ( 35
) ∂ t T + V • ∇T = ∇ 2 T. ( 36
)
-0.01 -0.005 0 0.005 0.01 The basic state under the parallel flow approximation and the FJO model is obtained from the system of eqs. ( 33)-(36) using the boundary conditions ( 8), ( 9) and the three hypotheses:

0 1 2 3 4 5 (a) m d [/m] ∆ T [K] m d 1 , h =0.03 [m] m d 1 , h =0.05 [m] m d 1 , h =0.10 [m] m d 2 , h =0.03 [m] m d 2 , h =0.05 [m] m d 2 , h =0.
m d [/m] h[ m ] m d 1 ,∆T =0.14 [K] m d 1 ,∆T =1.00 [K] m d 1 ,∆T =10.0[ K ] m d 2 ,∆T =0.14 [K] m d 2 ,∆T =1.00 [K] m d 2 ,∆T =10.0[ K ]
-the conservation of each component in the cavity, -the thermal flow rate through any horizontal crosssections is equal to zero, -the mass flow rate of each component through a horizontal cross-section is equal to zero,

v(x)= 1 2 Ra (1 -2 x), (37)
C(x, y)= Ra 2 Le • Cr -1 • m x 2 2 - x 3 3 - 1 12 (38) +Cr -1 • 1 1 2 -x + m y - A 2 , T (x)=1-x. ( 39
)

Vertical Mass fraction Gradient and separation

In order to determine an analytical expression for the VMG, the mass flow rate of each component through a horizontal cross-section is assumed to be equal to zero. This assumption allows an analytical expression for the separation to be obtained:

m =10Ra Cr -1 •LeRa 2 +120Le -1 •Cr -1 •Cr -1 •1 (40) =10Ra (Ra Le) 2 + 120Cr 2 -1 • Le • 1. ( 41
)
The VMGs for the N -1 components are expressed in terms of the Lewis number, the cross-diffusion numbers and the thermal Rayleigh number in eq. ( 40). For the component N , the VMG can be obtained using the relation

N i=1 C i =0. (42)
The same results were obtained for a binary mixture (N = 2) by Furry, Jones and Onsager [8] and for a ternary mixture (N = 3) by Larabi et al. [23]. For the FJO model, the separation ratio (ψ) was neglected. Figure 3 according to the separation ratio. For the same dimensionless parameters the VMG obtained by the FJO model is m d 1fjo = -8.80 • 10 -3 /m. However, the PFA predicts a VMG between -8.80 • 10 -3 /m and -9.15 • 10 -3 /m for (ψ 1 ,ψ 2 ) ∈ [0, 0.5] 2 (fig. 3). If the influence of the crossdiffusion can be neglected, then the analytical expression for the VMG leads to

m i = 10 Le i Ra (Le i Ra) 2 + 120 . ( 43
)
Larabi et al. [23] studied the influence of the crossdiffusion on the VMG for the case of ternary mixture. Equation (40) in the dimensional form leads to a relation between

D ′ Ti , D ij : 10 Kg∆T 2 D ′ Ti β T D 2 ii ν N -1 j=1 Khgβ T ∆T νD ii 2 + 120 N -1 k=1 D ik D kj D ′ Tj D kk D jj D ′ Ti -1 + m d i =0. ( 44
)
If the cross-diffusion effect can be neglected then eq. ( 44) leads to

D ′ Ti = - m d i 10Kg∆T 2 νD 2 ii β T Khg∆T β T D ii ν 2 + 120 . (45)
5 Direct simulation

. 1F l o wi nt h ec a v i t y

In order to assess the hypothesis considered for the theoretical models (PFA and FJO), a direct numerical simulation was performed using the Comsol Multiphysics software. The system of eqs. ( 2)-( 5) and the boundary conditions (8) were solved numerically for a ternary mixture using a finite elements method. The aspect ratio of the thermo-gravitational column was A = 20. A rectangular mesh was used and the spatial resolution was 50 × 200:

y i = A 2 cos πi N y -1 +1 for i =0,...,N y -1.
(46) The grid was uniform in the x-direction and distributed as y i in eq. ( 46) (with N y = 200) for the y-direction. The order of convergence with this distribution was equal to 2.5. The streamlines and the temperature are presented for the stationary state in fig. 4. As expected, there was a unicellular flow in the cavity, the streamlines were parallel to the vertical walls and the temperature depended linearly on the thickness of the column.

Assessment of the theoretical models

In order to quantify the error, two norms were used:

E 2 = 100 • x the -x num 2 x num 2 , ( 47
)
E ∞ = 100 • x the -x num ∞ x num ∞ , (48)
where x the and x num are the theoretical and numerical values, respectively, ... 2 is the 2-norm and ... ∞ is the infinity norm. E 2 and E ∞ are the mean error and the worst case error. These norms were calculated for all the numerical values on figs. 5, 6, 7 and 8 and are reported in table 2. The horizontal velocity was neglected for the two theoretical models, while for the PFA, the vertical velocity was as given by eq. ( 25) if ω 2 > 0 and by eq. ( 29) if ω 2 < 0. For the FJO model the vertical velocity was a linear function depending only on the Rayleigh number.

The velocity profiles obtained in the two cases (ω 2 > 0 and ω 2 < 0) are illustrated in fig. 5. The results using the two theoretical models are compared to the direct simulation for a difference in temperature of ∆T = 10 K. For the PFA, the two errors are lower than 1%, so the PFA is in very good agreement with the numerical results. The FJO model based on physical considerations leads to an error of 5% for the velocity. In fig. 6 the VMG of components 1 and 2 are presented according to the temperature using the three methods. As table 2 shows, the two theoretical models are in good agreement with the numerical results. in temperature of 6 K and a thickness of 3 • 10 -2 mu s i n g the two theoretical models and the numerical simulations. The THN is the heaviest component of the ternary mixture THN(0.8)-IBB(0.1)-nC12(0.1) and nC12 is the lightest component. In fig. 7 the THN is more concentrated at

m d [/m] ∆ T [K] m d 1 ,F J O m d 1 ,P F A m d 1 , DNS m d 2 ,F J O m d 2 ,P F A m d 2 , DNS

Conclusion

In this paper a theoretical analysis for the separation of an N -component mixture in a vertical porous thermogravitational column is proposed. Two theoretical models for the separation of N -component mixtures are presented. These models, based on the parallel flow approximation allow the separation to be determined for each component of an N -component mixture. The mass fraction, velocity and temperature at the stationary state were obtained analytically. These models indirectly validate the measured transport coefficients. The Furry-Jones-Onsager theory was extended to the N -component mixture and the separation was expressed as a function of the Rayleigh number, the matrix of Lewis numbers and the crossdiffusion numbers. In order to assess the theoretical work, direct numerical simulations were performed for a cavity with an aspect ratio of 20. A good match was obtained for the mass fraction in 80% of the cavity. The theoretical and numerical results for the separation in the cavity were in good agreement, thus validating the assumptions made in the theoretical models. A quantitative analysis of the errors for these models was performed.

2. 2

 2 Ternary mixture THN(0.8)-IBB(0.1)-nC12(0.1)

Fig. 2 .

 2 Fig. 2. Influence of the control parameters (the temperature difference (a) and the cavity thickness (b)) on the vertical mass fraction gradient of components 1 and 2 for a ternary mixture (Le1 = 613, Le2 = 538, ψ1 =0 .17, ψ2 =0 .10, Cr12 = -0.016, Cr21 = -0.24).

Fig. 3 .

 3 Fig. 3. Influence of the separation ratio on the VMG obtained from the PFA for a ternary mixture (Le1 = 613, Le2 = 538, Cr12 = -0.016, Cr21 = -0.24) for a temperature difference equal to ∆T = 5 K and a thickness h =3• 10 -2 m.

Fig. 4 .

 4 Fig. 4. (a) Streamlines and (b) temperature in a thermogravitational column filled with a ternary mixture (Le1 = 613, Le2 = 538, ψ1 =0 .17, ψ2 =0 .10, Cr12 = -0.016, Cr21 = -0.24) at the stationary state for a temperature difference equal to ∆T = 6 K and a thickness h =3• 10 -2 m (Ra =0.4013).

Figures 7 and 8 Fig. 5 .

 85 Fig. 5. Numerical (DNS) and theoretical (PFA and FJO) horizontal profiles of the vertical velocity in the thermogravitational column at the stationary state for ternary mixtures (a) ψ1 =0 .17, ψ2 =0 .10, (b) ψ1 = -0.17, ψ2 = -0.10 and Le1 = 613, Le2 = 538, Cr12 = -0.016, Cr21 = -0.24 for ∆T = 10 K and h =3• 10 -2 m( Ra =0.6688).

Fig. 6 .Fig. 7 .

 67 Fig.[START_REF] Nield | Convection in Porous Media[END_REF]. Theoretical (PFA and FJO) and numerical (DNS) Vertical Mass fraction Gradient according to the temperature difference for a ternary mixture (Le1 = 613, Le2 = 538, ψ1 =0 .17, ψ2 =0 .10, Cr12 = -0.016, Cr21 = -0.24) for a thickness equal to (a) h =3• 10 -2 ma n d( b )5• 10 -3 m.

Fig. 8 .

 8 Fig. 8. Theoretical (PFA and FJO) and numerical (DNS) mass fraction of component 2 (Le1 = 613, Le2 = 538, ψ1 =0 .17, ψ2 =0 .10, Cr12 = -0.016, Cr21 = -0.24) in the cavity at the stationary state for a temperature difference of 6 K and a thickness of 3 • 10 -2 m( Ra =0.4013).

Table 1 .

 1 Dimensionless numbers for the ternary mixture THN-IBB-nC12 with mass fractions 0.8-0.1-0.1 in a porous medium of permeability K =1 0 -10 m 2 , normalized porosity ε =0.25 and thermal diffusivity a =2.0 • 10 -7 m 2 /s. Consequently the numerical values of diffusion and thermodiffusion coefficients need to be changed: the thermodiffusion coefficients were equal to D ′ T 1 =0 .65 • 10 -12 m 2 /(s K) and D ′ T 2 = -0.49 • 10 -12 m 2 /(s K). The mean values were used for the pure diffusive coefficients D 11 =5 .96 • 10 -10 m 2 /s, D 22 =6 .79 • 10 -10 m 2 /s. The measured values for the cross-diffusion coefficients were dispersed, while the Soret coefficients S T 1 and S T 2 were in good agreement. The value of the cross-diffusion coefficients were determined by using the equations from[13]

	Dimensionless number	THN	nC12
	Lewis number (Le)	613	538
	Separation factor (ψ)	0.17	0.10
	Cross-diffusion number (Cr) Cr12 = -0.016 Cr21 = -0.24
	this work we are adopting a component order different
	from the one in ref. [14], component 1 being THN and
	2 nC12.

Table 2 .

 2 Mean and worst case error between the numerical and theoretical results (PFA and FJO). bottom of the cavity and in fig.8nC12 is at the top of the thermo-gravitational column, which is consistent with physics and the sign of the VMG observed in fig.6. The white zones observed in figs. 7 and 8 are the zones in the thermo-gravitational column where the streamlines are not parallel to the vertical walls so neither the PFA nor FJO results are valid. The white zones represent 20% of the area of the column and the coloured zone 80%. The norms reported in table 2 are calculated for 80% of the cavity. As can be observed in the table the error for the mass fraction in 80% of the column is less than 0.1%, the two theoretical models are a perfect match for the simulations.

	Figure	PFA		FJO
		E2	E∞	E2	E∞
	5(a)	0.16%	0.21%	3.47%	4.64%
	5(b)	0.17%	0.15%	3.72%	5.41%
	6(a)	0.92%	2.16%	1.15%	2.24%
	6(b)	0.62%	1.10%	1.73%	2.06%
	71 0 -3 %1 0 -3 %1 0 -3 %1 0 -3 %
	8	0.01%	0.05%	0.04%	0.06%

the

This research is supported by the CNES (Centre National Études Spatiales).

Author contribution statement

DM and AM conceived the original idea and developed the theory. MAL and DM planned and carried out the simulations. DM wrote the manuscript in consultation with MAL and AM.


Oussama Abahri

Djamel Sadaoui

Kacem Mansouri

Abdelkader Mojtabi

Marie Catherine Mojtabi

Thermogravitational separation in horizontal annular porous cell

Keywords: Thermodiffusion, convection, perturbation method, porous medium

Thermogravitational separation has until now, been used in different heated vertical cells called thermogravitational columns. The cell can be an annular cavity with two isothermal faces maintained at different temperatures. The main objective of this paper is to study the two dimensional coupled convection with thermodiffusion process. It concerns a theoretical and numerical investigation of species separation in a binary liquid mixture saturating a horizontal porous annulus space where the inner cylinder is heated isothermally. This kind of geometry is used instead of the annular vertical cell, hence the novelty of this technique. Analytical resolution is performed using the perturbation method function of time versus the corresponding physics (Raleigh and Lewis numbers. . . ). Results reveal that the separation can be increased with an optimum for small values of Rayleigh number. Further, these values are less important than the critical value of Raleigh leading to the loss of unicellular flow stability found in literature.

Introduction

Thermogravitational diffusion is the combination of two phenomena; convection and thermodiffusion. Coupling these two phenomena leads to species separation constituting a mixture of gas or liquid unfilled a welldefined geometrical space.

A literature survey related to this topic revealed that previous studies are substantially orientated toward the study of concave geometries and only few works deal with this type of geometry in the presence of porous media. Most of them are numerical studies. Clusius and Dickel [START_REF] Clusius | Neues Verfahren zur Gasenmischung und Isotroprennung[END_REF] are among the first in the field: they have successfully carried out the separation of gas mixtures in a vertical cavity heated from one side (thermogravitational column, TGC). Soon thereafter Furry et al. [START_REF] Furry | On the theory of isotope separation by thermal diffusion[END_REF] have developed a theoretical and experimental study on thermogravitation species separation in binary mixtures (FJO Theory) in order to interpret the experimental processes of isotope separation. Subsequently, many works appeared, in order to justify or extend the FJO theory and results on the binary liquid case [START_REF] De Groot | Théorie Phénoménologique du procédé thermogravitationnel de séparation dans un liquide[END_REF]. Furthermore, other research a Corresponding author: oussama.abahri@gmail.com on the improvement of experimental devices to increase separation can be found in literature. In this regard, Lorenz et al. [START_REF] Lorenz | The packed thermal diffusion column[END_REF] proposed the introduction of a porous medium into the cavity. Later, still regarding this topic, Caltagirone [START_REF] Caltagirone | Thermoconvective instabilities in a porous medium bounded by two concentric horizontal cylinders[END_REF] used the perturbation method by developing the temperature and stream function in terms of Rayleigh number until order 3, to provide solution for low flow regimes. Thereafter, Burns and Yien [START_REF] Burns | Natural convection in porous medium bounded by concentric spheres and horizontal cylinders[END_REF] treated a steady flow with other types of boundary conditions for the inner cylinder (i.e. uniform temperature or imposed flow). They used the finite difference method for solving the linear system based on the S.O.R procedure and found the Caltagirone results for Dirichlet boundary conditions for the temperature. Later, Bau [START_REF] Bau | Thermal Convection in a horizontal eccentric annulus containing a saturated porous medium: an extended perturbation expansion[END_REF] investigated the effect of the cylindrical eccentricity on heat transfer, using a perturbation method with Rayleigh number as a development parameter. Indeed, Himasekhar [START_REF] Himasekhar | Two-dimensional bifurcation phenomena in thermal convection in horizontal concentric annuli containing saturated porous media[END_REF] used the Galerkin method developed at higher levels to analyze the stability of two-dimensional unicellular flow with two different approaches. The first approach estimates the convergence of the obtained solution in terms of entire series of Rayleigh number, while the second one focuses on the linear stability of the obtained solution for radius ratios ranging between 2 1/8 and 2. Thus, they determine the critical Rayleigh numbers of transition between two-dimensional systems based on the radius ratio R. Dyko et al. [START_REF] Dyko | A numerical and experimental investigation of stability of natural convective flows within a horizontal annulus[END_REF] considered both linear and nonlinear stability of two-dimensional flow, using linear theory and the energy method. This study allows defining the criteria for the onset of the secondary flows observed experimentally. In fact, they obtained a critical Rayleigh number that delimited the flow stability from the linear stability analysis. Indeed, a subcritical Rayleigh number that defines a necessary condition for global flow stability was determined using the energy method. Therefore, Desrayaud et al. [START_REF] Desrayaud | An analytical solution for the stationary behavior of binary mixtures and pure fluids in a horizontal annular cavity[END_REF] consider thermogravitational separation of binary mixture and pure fluid in a horizontal annulus space delimited by two concentric and isothermal cylinders. They developed an analytical model able to describe the phenomenological equation of heat and mass transfer occurring in this space. Also, Mojtabi et al. [START_REF] Mojtabi | Transferts de chaleur dans les milieux poreux, Techniques de l'Ingénieur[END_REF] obtained the following correlation for average Nusselt number in an annular space as a function of the Rayleigh number based on the layer thickness N u = 1 + 17Ra (r e /r i )

2 -((r e /r i ) -1) 3 /40 320.

This correlation is still valid for an inner temperature higher than the outer one (T i > T e ). It is sufficient to describe heat transfer inside the annular space for all values of Ra lower than 200. The difference compared with the numerical results is even smaller than the inner radius is less than √ 2. Under such conditions, an experimental visualization by Christiansen effect highlighted the existence ofunicellular and bi-cellular flows for different aspect ratios.

Recently works on the annular configuration, take into account changes in fluids properties (i.e. the Boussinesq assumption is no valid) thus the local thermal equilibrium between phases is not retained; T s = T F . Otherwise, preliminary studies concerning unicellular steady state flow characterized by a single convective cell in the annular layer was performed by [START_REF] Burns | Natural convection in porous medium bounded by concentric spheres and horizontal cylinders[END_REF]. More recently, Hadidi et al. [START_REF] Hadidi | Bi-layered and inclined porous collector: Optimum heat and mass transfer[END_REF] studied the double-diffusive free convective heat and mass transfer phenomena in a rectangular inclined cavity bounded by two adiabatic and impermeable horizontal walls and two isothermal vertical sides filled with two parallel porous layers. Results indicate that the permeability of the two porous layers has a significant effect on the flow behavior and heat transfer. According to the numerical analysis three regimes has been distinguished; a diffusive regime for low values of R K (permeability ratio), a transition regime where the mean Nusselt and Sherwood numbers increase with an increase of R K and an asymptotic regime where Nu and Sh become independent of R K .

Thorough analytical and numerical analysis conducted recently by Khouzam et al. [START_REF] Khouzam | Species separation of a binary mixture in the presence of mixed convection[END_REF] on species separation process in a binary fluid mixture within a horizontal rectangular cavity heated either from the top or the bottom, it was shown that the separation is controlled by horizontal temperature gradient and the upper moving wall. Further, in addition to five parameters setting the general problem, namely Ra, Pe, Pr, Le and ψ; the optimal separation solution is also influenced by the mass Peclet and Rayleigh numbers.

The present paper investigates analytically and numerically the separation of a binary mixture and pure fluid within a porous horizontal annulus. Both the inner and outer cylinders are heated isothermally, respectively T i and T o with T i > T o . In other words, we propose a new procedure coupling the convection and the thermodiffusion to obtain an important separation between the top and the bottom of the annular horizontal porous layer. This kind of geometry is used instead of the annular vertical cell and the classical rectangular one studied by Elhajjar et al. [START_REF] Elhajjar | Separation in an inclined porous thermogravitational cell[END_REF] or that of Oueslati et al. [START_REF] Oueslati | Analytical and numerical solution to the convection problem in a shallow cavity filled with two immiscible superposed fluids[END_REF], hence the novelty of this technique.

Mathematical formulation

It should be recalled that thermodiffusion effect induces a mass fraction gradient in binary fluid mixtures subjected to a temperature gradient. In addition to the usual isothermal contribution, there is an additional contribution proportional to the concentration, so that:

J = -ρ D ∇ C -ρC (1 -C) D T ∇T ( 1
)
where D is the mass diffusion coefficient, D T the thermodiffusion coefficient, ρ the density, and C the mass fraction of the denser component.

The investigated problem is illustrated schematically in Figure 1, which consists of an annular porous layer between two coaxial horizontal cylinders of length L. The inner cylinder of radius r i is at fixed warm temperature T i while the outer cylinder of radius r e is maintained at constant and uniform temperature T e with T e < T i .

Darcy's law is assumed valid where fluid and solid phases are in local thermal equilibrium. It is also assumed that viscous dissipation, compressibility and Dufour effects are neglected [START_REF] Elhajjar | Separation in an inclined porous thermogravitational cell[END_REF] because of their minor influence on liquid mixture. Also, the Boussinesq approximation is invoked for binary fluid properties to relate the density changes to temperature and concentration and to couple in this way the temperature and concentration fields to the flow field: Here, β T and β c are the coefficients of thermal and solutal expansion, respectively, while ρ 0 is the fluid mixture reference density at temperature T 0 and mass fraction c 0 . Under these assumptions, the equations of continuity, momentum, energy and chemical species are cast in their dimensional form as follow:

ρ = ρ 0 [1 -β T (T -T 0 ) -β C (C -C 0 )] (2)
⎧ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎩ ∇ • (V ) = 0 V = -K μ (∇P -ρg) λ * ∇ 2 T = (ρc) * ∂T ∂t + (ρc) f V • ∇T ε ∂C ∂t + V • ∇C = D * ∇ 2 C + D * T C 0 (1 -C 0 ) ∇ 2 T (3)
Here it was assumed that there is little variation in the term C (1 -C) of the species equation, so that it can be replaced by C 0 (1 -C 0 ); with C 0 the initial mass fraction. Equations (3) were transformed in the nondimensionalized form by scaling length, time and temperature by r i ; (ρc) * r 2 i /λ * and [T i -T e ], and the velocity, pressure and mass fraction by [λ * /(r i (ρc

) f )]; [λ * μ/(k(ρc) f )] and ΔC = -ΔT c 0 (1 -c 0 )D * T /D * , respectively.
Thus, the dimensionless governing conservation equations for mass, momentum, energy and chemical species are:

⎧ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎩ ∇ • (V ) = 0 V = Ra (T + spC) e z ∂T ∂t + V • ∇T = ∇ 2 T ε ∂C ∂t + V • ∇C = 1 Le (∇ 2 C -∇ 2 T ) (4)
The studied problem depends on five non-dimensional parameters; the thermal filtration Rayleigh number Ra = gβ

(T i -T e ) (ρc) f K r i /λ * ν; the Lewis number Le = a * /D * ; the normalized porosity έ = ε/σ; the separation ratio sp = -(β C /β T ) (D * T /D * ) C 0 (1 -C 0 )
and the aspect ratio R = r e /r i .

Here σ = (ρc) * /(ρc) f where (ρc) * is the effective volumetric heat capacity of the porous medium, and a * the effective thermal diffusivity of the porous medium.

Using polar coordinates (r, ϕ) and the associated local database (e r , e ϕ ), the velocity can be defined by:

V = V r e r + V ϕ e ϕ ( 5
)
The vector e Z takes the following form: e z = cos (ϕ) e rsin(ϕ)e ϕ . With V r and V ϕ are the radial and tangential velocity component, respectively.

Using stream function formalism, the velocity components can be expressed as:

V r = 1 r ∂ψ ∂ϕ and V ϕ = - ∂ψ ∂r
The continuity equation is identically satisfied, by taking the rotational of Darcy's equation. Studying the steady state we have to solve the following system:

⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩ ∇ 2 ψ = Ra sin (ϕ) ∂ ∂r (T + sp C) + cos (ϕ) r ∂ ∂ϕ (T + sp C ) ∇ 2 T = 1 r ∂ψ ∂ϕ ∂T ∂r - ∂ψ ∂r ∂T ∂ϕ ∇ 2 C -∇ 2 T = Le r ∂ψ ∂ϕ ∂C ∂r - ∂ψ ∂r ∂C ∂ϕ ∇ 2 . = ∂ 2 . ∂r 2 + 1 r ∂. ∂r + ∂ 2 . r 2 ∂ϕ 2 (6)
The corresponding dimensionless boundary conditions are:

⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩ T = 1 for r = 1, ψ= ∂ 2 ψ ∂r 2 = 0, ∂C ∂r = ∂T ∂r , ∀ϕ ∈ [0, 2π] T = 0 for r = R, ψ = ∂ 2 ψ ∂r 2 = 0, ∂C ∂r = ∂T ∂r ,
∀ϕ ∈ [0, 2π]

ϕ = 0, π ∂T ∂ϕ = ∂C ∂ϕ = 0, ψ= 0, ∀r (7)
For symmetric regime and single-cell system r varies in the range [1-R], with R = r e /r i . Moreover, a new variable can be introduced in function of temperature and the concentration η = C -T .Thus, the coupled system takes the form:

∇ 2 ψ = Ra sin (ϕ) ∂ ∂r ((1 + sp) T + sp η) + cos (ϕ) r ∂ ∂ϕ ((1 + sp)T + sp η ) ( 8
)
∇ 2 T = 1 r ∂ψ ∂ϕ ∂T ∂r - ∂ψ ∂r ∂T ∂ϕ (9) ∇ 2 η = Le r ∂ψ ∂ϕ ∂ ∂r (T + η) - ∂ψ ∂r ∂ ∂ϕ (T + η) ( 10
)
3 Results and discussion

Analytical results

An analytical resolution, of the coupled mass, energy and momentum equations (Eqs. ( 8) to ( 10)) using the perturbation method is conducted in this section. The development is performed till second order.

Perturbation method is an approximation technique based on the principle of asymptotic expansion; the proposed solution is generally represented by the first developed terms. This development may involve a parameter (large or small) that appears in the equations, or that artificially introduced. It is called disturbance parameter.

In the present study, this method leads to express temperature, stream function and concentration as an expansion of Rayleigh number:

ψ = n i=1 Ra i ψ i (r, ϕ) T = n i=0 Ra i T i (r, ϕ) η = n i=0 Ra i η i (r, ϕ) (11)
By introducing these three expressions in Equations ( 8) to [START_REF] Desrayaud | An analytical solution for the stationary behavior of binary mixtures and pure fluids in a horizontal annular cavity[END_REF], we obtain an infinite sequence of coupled partial differential equations that can be solved analytically by recurrence.

The zero order is given by:

∇ 2 T 0 = 0 ( 12
)
∇ 2 η 0 = 0 (13)
The following equation system is resulting from the development at order 1:

∇ 2 ψ 1 = sin (ϕ) (1 + sp) ∂T 0 ∂r + sp ∂η 0 ∂r + cos(ϕ) r (1 + sp) ∂T 0 ∂ϕ + sp ∂η 0 ∂ϕ ( 14
)
∇ 2 T 1 = 1 r ∂ψ 1 ∂ϕ ∂T 0 ∂r - ∂ψ 1 ∂r ∂T 0 ∂ϕ ( 15
)
∇ 2 η 1 = Le r ∂ψ 1 ∂ϕ ∂ ∂r (T 0 + η 0 ) - ∂ψ 1 ∂r ∂ ∂ϕ (T 0 + η 0 ) (16)
The system developed at order 2 is given by:

∇ 2 ψ 2 = sin (ϕ) (1 + sp) ∂T 1 ∂r + sp ∂η 1 ∂r + cos(ϕ) r (1 + sp) ∂T 1 ∂ϕ + sp ∂η 1 ∂ϕ ( 17
)
∇ 2 T 2 = 1 r ∂ψ 2 ∂ϕ ∂T 0 ∂r + ∂ψ 1 ∂ϕ ∂T 1 ∂r - ∂ψ 2 ∂r ∂T 0 ∂ϕ - ∂ψ 1 ∂r ∂T 1 ∂ϕ ( 18
)
106-page 4

∇ 2 η 2 = Le r ⎡ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎣ ∂ψ 1 ∂ϕ ∂ ∂r (T 1 + η 1 ) + ∂ψ 2 ∂ϕ ∂ ∂r (T 0 + η 0 ) - ∂ψ 1 ∂r ∂ ∂ϕ (T 1 + η 1 ) - ∂ψ 2 ∂r ∂ ∂ϕ (T 0 + η 0 ) ⎤ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎦ (19)
At order N the equations that can be solved analytically take the following general form:

⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩ ∇ 2 ψ (i) = sin (ϕ) (1 + sp) ∂T (i-1) ∂r + sp ∂η (i-1) ∂r + cos(ϕ) r (1 + sp) ∂T (i-1) ∂ϕ + sp ∂η (i-1) ∂ϕ ∇ 2 T (i) = n j=1 1 r ∂ψ j ∂ϕ ∂T (i-j) ∂r - ∂ψ j ∂r ∂T (i-j) ∂ϕ ∇ 2 η (i) = n j=1 Le r ∂ψ j ∂ϕ ∂ ∂r T (i-j) + η (i-j) - ∂ψ j ∂r ∂ ∂ϕ (T (i-j) + η (i-j) ) (20)
The associated boundary conditions are:

T 0 = 0 for r = 1 ∂η ∂r = 0, ψ (i) = 0, for r = 1, R for any value of i. T 0 = 0 for r = R, ∂η ∂ϕ = ∂T ∂ϕ = 0, ψ (i) = 0 for ϕ = 0, π for any value of i.

T (i) = 0 for r = 1, R, i
Taking into account the boundary conditions of temperature and concentration and after direct integration, the following expressions are obtained for zero order:

T 0 = 1 - ln(r) ln(R) ( 21
)
η 0 = cte (22)
By recurrence and after replacing T 0 and η 0 by their expressions in Equation ( 14) the following partial differential equation is obtained:

∂ 2 ψ 1 (r, ϕ) ∂r 2 + 1 r ∂ψ 1 ∂r + 1 r 2 ∂ 2 ψ 1 (r, ϕ) ∂ϕ 2 + (1 + sp) rln(R) sin (ϕ) = 0 (23)
Using the separation of variables method, the solution at order 1 of Equation ( 23) is:

ψ 1 = ⎛ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎝ 1 4 r (1 + sp) (2R 2 ln (R) -R 2 + 1) ln (R) (R 2 -1) - R 2 (1 + sp) 2r (R 2 -1) - r (2 ln (R) -1) (1 + sp) 4 ln (R) ⎞ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎠ sin(ϕ) (24)
The same procedure is applied to the temperature where the following equation should be solved:

∂ 2 T 1 (r, ϕ) ∂r 2 + 1 r ∂T 1 ∂r + 1 r 2 ∂ 2 T 1 (r, ϕ) ∂ϕ 2 + Q (r) cos (ϕ) = 0 (25) with, Q (r) = 1 r 2 ln(R) ⎛ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎝ r (1 + sp) (2R 2 ln (R) -R 2 + 1) 4 ln (R) (R 2 -1) - R 2 (1 + sp) 2 r(R 2 -1) - r (2 ln (r) -1) (1 + sp) 4 ln(R) ⎞ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎠
(26) In this case, the corresponding solution of Equation (25) takes the following form:

T 1 = a 1 r ln 2 (r) + a 2 r ln ( r) + a 3 r -1 ln (r) +a 4 r -1 + a 5 r cos(ϕ) (27)
where, a i are coefficients that depend only on the ratio R and the separation factor sp.

Similarly η takes the form partial differential equation form as the temperature:

∂ 2 η 1 (r, ϕ) ∂r 2 + 1 r ∂η 1 ∂r + 1 r 2 ∂ 2 η 1 (r, ϕ) ∂ϕ 2 + g (r) cos (ϕ) = 0 (28) with, g (r) = Le r 2 ln(R) ⎛ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎝ r (1 + sp) (2R 2 ln (R) -R 2 + 1) 4 ln (R) (R 2 -1) - R 2 (1 + sp) 2 r(R 2 -1) - r (2 ln (r) -1) (1 + sp) 4 ln(R) ⎞ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎠
(29) Then, the solution of this equation is:

η 1 = Le b 1 r + b 2 r -1 + b 3 r -1 ln (r) + b 4 r ln (r) +b 5 r ln(r) 2 . cos (ϕ) (30)
where, b i are coefficients that depend only on the ratio R and the factor sp.

Further, the concentration c (i) are identified using the relationship between:

c (i) = η (i) + T (i) (31)
After injecting the temperature, the concentration and the current function found previously at order 1 and order in Equation ( 17), and after using the separation variables method, the current function at order 2 (ψ 2 ) takes the following form:

ψ 2 = c 1 r 2 ln 2 (r) + c 2 r 2 ln ( r) + c 3 ln (r) + c 4 r 2 +c 5 + c 6 r -2 sin (2ϕ) (32)
Concerning the temperature at order 2, this variable assumes the general following form:

T 2 = f T (r) cos(2ϕ) + g T (r) (33)
And also for the concentration:

η 2 = f η (r) cos(2ϕ) + g η (r) ( 34
)
where c i , f T (r), f η (r), g T (r), and g η (r) are function of R and the separation factor Expressions of temperature, stream function and concentration at a given point will be approximated by the three equations developed earlier:

⎧ ⎪ ⎨ ⎪ ⎩ ψ = Raψ 1 + Ra 2 ψ 2 T = T 0 + RaT 1 + Ra 2 T 2 η = η 0 + Raη 1 + Ra 2 η 2 (35)
Here, the perturbation method is applicable only for small values of Rayleigh number.

Numerical results

To highlight the developed analytical method, analytical results are compared against those obtained by solving the basic equations of the coupled model. It is undertaken using COMSOL Multiphysics code [START_REF] Femlab | Comsol 33 Multiphysics FEM Software package[END_REF] which is a powerful environment for modeling and solving a variety of research and engineering problems based on the finite element method (FE). Moreover, this software is particularly adapted for the treatment of Multiphysical problems where several phenomena are simultaneously studied. Thus, the assessment of the incidence of heat and mass transfer on other phenomena such as the ingress of aggressive agents (chlorides, ions) and the ingress of pollutants (Volatile Organic Compounds) becomes possible. Also, different kind of meshes are easily generated which is more convenient for 3D studies. It is obvious that the accuracy of numerical results is function of the mesh size.

To find a proper grid size, a grid testing is performed using various grid combinations (90 × 10 to 180 × 20) of control volumes. For each grid size, separation (S) and Nusselt number (Nu) are presented in Figure 2 for Ra = 10, sp = 0.1, R = 35/32 and Le = 100. Throughout this investigation, grid sizes greater or equal than 150 × 20 are sufficient to have a solution independent of the mesh. Hence, considering both the accuracy and the computational costs, most computations reported in the current work were performed with a multiple grid system of 150 × 20, leading to good agreement between the numerical and analytical results.

Figure 3 represents the isoconcentration lines within an annular space of aspect ratio r = 35/32, obtained for different Ra (0.1, 10 and 100, respectively). In order to highlight the effect of Ra alone, both Lewis number, R and sp were kept constant at values. As seen, the form of the isoconcentration line is affected by the increasing value of Ra. Increasing the Rayleigh number intensifies distortions of the concentration profiles, particularly in the vicinity of the two poles. In fact, for a low value of Ra (0.1), it can be seen that these isoconcentrations have quasi-uniform and homogeneous radial distribution along the horizontal cell. This means that the separation is mainly due to the thermodiffusion where the role of the convection is almost negligible. When increasing Ra, convection velocity rises; this affects the isoconcentrations curvatures; so that concentration plumes emerge above the poles of the inner cylinder, which signal an increase in the mass transfer rate, as is indicated in Figure 3b. Therefore, we approach increasingly from horizontal separation, in which case the coupling convection/ thermodiffusion is optimal. For Ra = 100, the convection develops compared to the thermodiffusion, this promotes instability of the isoconcentrations repartition and therefore the separation decreases significantly.

Comparison between analytical and numerical results

In this section, we compared results of a numerical solution obtained by the finite element method (COM-SOL) to those given by an analytical method developed above. The analytical method used to solve the coupled Equations ( 8)-( 10) with the associated boundary conditions ( 7) is a perturbation technique based on the principle of asymptotic expansion with a development till second order. The developed perturbation method is applicable only for low Rayleigh numbers.

The comparison is based on the main variables of the problem; namely the stream function (ψ); temperature (T ), mean Nusselt number (Nu) and separation (S). The global Nusselt number characterizing the rate of total heat transfer is defined as the ratio of the total heat flux to the conductive heat flux through the same surface. Thus, the internal and external Nusselt can be defined as follow:

N u gi = - ln(R) π π ∫ 0 ∂T ∂r r=1 dϕ N u ge = -R ln(R) π π ∫ 0 ∂T ∂r r=R dϕ
The separation factor (S) is defined as the difference of the mass fractions of the heavier component between two points [(R + 1)/2] φ=0 and [(R+1)/2] φ=π as shown in Figure 4. Thus, separation is: S = C max -C min . Maximum separation means an optimal coupling between the thermodiffusion and convection. According to the reference variable adopted for the mass fraction, separation is always greater than 1.

The comparison of numerical and analytical stream function, separation and temperature results is illustrated in Table 1 and Figure 5. Upon inspection of Table 1 it is observed that the comparison of temperature behavior is globally in good agreement. It is also noted a good agreement for the values of Nusselt number for a wide range of Ra, the maximum deviation was within one percent.

While a slight difference not more than 6% between analytical and numerical results of the stream function values is observed. We noted that the stream function vanishes on the walls (r = 1 and r = R = 35/32), this is due to the condition of adhesion to the surface; then it present an extremum for r = 1.06 (-0.123814 and -0.127929 for numerical and analytical results, respectively).

For the Separation the maximum deviation increases with the Rayleigh number. Analytical values for the separation are slightly lower than that calculated numerically. The results are very close at low Ra number, the maximum deviation increases with Ra (i.e. the error passes from 0.1% for Ra = 0.1 to 5.4% for Ra = 0.5).

The various results presented allow us to conclude that the analytical perturbation method developed till second order is more appropriate for small values of Ra.

In the following, to better understand the considered phenomenon it is instructive to study the effects of several parameters on the separation "S". The invoked parameters are the Rayleigh and Lewis numbers with the annulus dimension.

Table 2 highlights the evolving separation within the porous annular space according to Ra for three main values of the separation factor (sp). These values are: sp = 0.1305; 0.12 and 0.1. Results indicate that the separation factor sp does not entail any significant change in the separation (S) for all Ra values. Subsequently, the value of sp will be set equal to 1.

In order to highlight the effect of Ra alone, both Lewis number, separation factor (sp) and gap ratio (R) were kept constant at values equal to 100, 0.1 and 35/32, respectively. Figure 6 shows the separation factor versus Ra in the range Ra = 0 to 50. A detailed investigation of the result show that the optimal separation is at Ra = 1.75. This maximum corresponds to the optimal coupling between convection and thermal diffusion. By raising Ra, the separation predictions show a gradual dip and then assume an asymptotic trend so that the effect of Ra on the separation seems to be insignificant. For slightly high Ra values, the flow is intense and it does not provide sufficient separation. In contrast, for low Ra numbers, convection is smaller compared with the thermal-diffusion which controls separation time. Furthermore, numerical calculations have been performed for different Lewis numbers. This number provides a measure of thermal diffusivity of a fluid to its mass diffusivity. Thus when Lewis number equals one, the thermal diffusivity equals to the mass diffusivity. This means that heat and species diffuse with the same characteristic time and the isothermal lines are congruent with the isoconcentration lines. Lewis number in the ongoing investigation was varied in the range from 0 to 100 as depicted in Figure 7 showing separation versus Ra.

Results shown in this figure indicate that the maximum separation value is practically independent on the Lewis number. Furthermore, the results also point out direct impact of Lewis number on the optimum Ra; they are inversely proportional. In fact, the optimal Rayleigh is even small when Lewis number increases (i.e : Ra opt = 1.75; 3.75 for Le = 100 and 50, respectively).

To highlight the effects caused by changing the radius ratio (R) on the Separation, Figure 8 depict the separation following different annulus dimension and three aspect ratios; R = 35/32 (narrow gap), R and R = 2 1/2 (wide gap). The ratio (R) characterizing the annulus dimension is defined as a ratio between the outer and the inner radii of the crown (R = r e /r i ). As can be seen, the ratio R has an influence on both the maximum separation and the optimum Rayleigh number. Indeed, reducing the dimensionless ratio (R), improve both the maximal separation (S max ) and Ra opt (i.e. R = 35/32: Ra opt = 1.75 and S max = 13.75; R = 2 1/2 : Ra opt = 0.7 and S max = 3). The evolution of S also shows a rapid tendency toward an asymptotic behavior as Ra grows (i.e. R = 2 1/2 : Ra asym = 10 and S asym = 0.45; R = 35/32: Ra asym = 30 and S asym = 2).

To achieve the effect of dimensionless parameter on the separation we focused on the influence of the crown section characterized by the gap width e = (r er i ). The dependence of the separation on the gap (e) is displayed in Table 2 showing the optimal separation for Le = 100, sp = 0.1 and R =1.09375. Note that the annular space width (e) vary in multiples of the lowest value (e = 3) corresponding to r e = 35 and r i = 32. It follows that the optimal separation remains substantially constant around 13.5 regardless of the gap width e. In contrast, the optimal Rayleigh (Ra opt ) decreases when increasing diameters, therefore the gap (e).

In the general formalism the radii ratio is involved, in reality separation also depends on other quantities (dimension). Therefore, it is interesting to consider the same thickness (e) case and vary the mean radius of crown; e = (r e + r i )/2, just like R. Thus; Table 3 also illustrates separation (S) according to the average radius. The increase in the prediction separation is vivid as (e ) is increased; this is due to the behavior of the mixture inside the cell. Increasing the mean radius will raise the workspace with considerable concentration distribution at the vicinity of the outer cylinder leading to a migration of the heavier component to the coldest side by simple diffusion. This component will be advected to the left side of the cell by convection so that the accumulation of the concentration difference between the top and bottom of the cell concentration creates great separation. In contrast to the previous case, the prediction of the optimal Rayleigh shows a growth as (e ) increases.

Conclusion

This paper focuses on the characterization of binary fluid separation process in annular porous layer A horizontal annular cell is used instead of the rectangular cell, hence the novelty of this technique.

In a first part, an analytical resolution, of the coupled mass, energy and momentum equations, using the perturbation method was conducted. The development was performed till second order. Regarding the encouraging results obtained for temperature and stream function solutions, it was observed that for high values of the Rayleigh number (Ra above the optimal) concentration diverges (concentration varies proportionally with the Rayleigh number), although the value of Nusselt are in very good agreement with the numerical results.

In a second part, several numerical simulations, using the finite element method were performed to corroborate the results obtained analytically. These simulations allow a good agreement with the analytical temperature fields, the stream function and the separation parameter especially for Ra values less than the optimum.

Then, a parametric study based on the Rayleigh number Lewis number, separation factor porosity and characteristic size of the annular configuration was performed. It shown that for low ratios of R, the separation is optimal and varies proportionally with the increase of the mean radius.

Further experimental works are required as perspective to check the real functionality of the proposed annular geometry.

Fig. 1 .

 1 Fig. 1. Geometry of the physical problem.

Fig. 2 .

 2 Fig. 2. Detail of the grid dependency.

Fig. 3 .

 3 Fig. 3. Iso-concentration for Le = 100, sp = 0.1 and R = 35/32 for Ra = 0.1, Ra = 10 and Ra = 100 from the left to the right side.

Fig. 4 .

 4 Fig. 4. Separation concept.

Fig. 5 .

 5 Fig. 5. Analytical and numerical temperature for different Ra and for Le = 100, sp = 0.1, φ = 0 and R = 35/32.

Fig. 6 .

 6 Fig. 6. Effect of Rayleigh number on separation curves for Le = 100, sp = 0.1 and R = 35/32.

Fig. 7 .

 7 Fig. 7. Effect of Lewis number on separation curves for sp = 0.1 and R = 35/32.

Fig. 8 .

 8 Fig. 8. Separation curves for R = 2 1/2 , R = 35/32, Le = 4 and sp = 0.1.

Table 1 .

 1 Comparison between analytical and numerical solution for Le = 100, sp = 0.1; and R = 35/32 for: Stream function (Ra = 1, φ = π/2); mean Nusselt number (inner cylinder) and Separation.

	Stream function (Ra = 1, ϕ = π/2)		Nusselt (r = 1)	Separation "S"
	r	Num. sol	Anal. sol	Ra	Num. sol	Analy. sol	Num. sol Anal. Sol.
	1.00			0.05 1.000000 1.000000001	0.957	0.957
	1.01	-0.04817	-0.05026	0.1	1.000000 1.000000006	1.904	1.915
	1.02	-0.08543	-0.08794	0.2	1.000000 1.000000561	3.738	3.831
	1.03	-0.10994	-0.11329	0.3	1.000000 1.000000989	5.442	5.747
	1.04	-0.12116	-0.12654	0.4	1.000000 1.000001520	7.259	7.663
	1.05	-0.12381	-0.12792	0.5	1.000000 1.000002018	8.312	9.579
	1.06	-0.11071	-0.11768	1.	1.000000 1.000004379
	1.07	-0.09238	-0.09601	10	1.000002 1.000072115
	1.08	-0.06119	-0.06313	20	1.000011 1.000287791
	1.09	-0.01881	-0.01925	30	1.000023 1.000649663
	1.0937			40	1.000047 1.001148666

Table 2 .

 2 Separation for Le = 204 and R = 2 (1/2) . Separation for sp = 0.1305 3.092 3.335 2.250 1.903 1.597 1.224 0.790 0.462 0.309 Separation for sp = 0.1 3.064 3.336 2.251 1.903 1.334 1.222 0.788 0.461 0.308

	Ra	0.1	0.3	0.75	1	1.75	2	4	10	20

Table 3 .

 3 Effect of the gap width and mean radius on the separation (Le = 100, sp = 0.1).

			R = 1.09375					e = 3
	re	ri	E Raopt	S	r e	ri	e	R	Raopt	S
	35	32	3	1.750 13.609	35	32	33.5	1.09375	1.75	13.609
	70	64	6	1.000 13.595	40	37	38.5	1.08108	2.00	15.644
	105	96	9	0.600 13.598	50	47	48.5	1.06382	2.75	19.506
	140 128 12 0.500 13.500	55	52	53.5	1.05769	2.77	21.741
						60	57	58.5	1.05263	3.00	23.793
						70	67	68.5	1.04477	4.00	27.718
						102 99 100.5 1.03030	6.00	40.618