Thermosolutal natural convection in a vertically layered fluid-porous medium heated from the side

Energy Conversion & Management 41 (2000) 1065±1090

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Thermosolutal natural convection in a vertically layered ¯uid-porous medium heated from the side Mohamed Mharzi a,*, Michel Daguenet b, Saad Daoudi a a

Universite Sidi Mohamed Ben Abdellah, Faculte des Sciences de FeÁs Dhar El Mehraz, B.P. 1796, 30000 FeÁs Atlas, Morocco b Laboratoire de Thermodynamique et EnergeÂtique, Universite de Perpignan, 52 Avenue de Villeneuve, 66860 Perpignan Cedex, France Received 19 January 1999; accepted 17 August 1999

Abstract The authors formulate the natural thermosolutal convection in an elongated enclosure of horizontal axis, partitioned by a vertical porous layer. They use the Boussinesq approximation, the stream function, the vorticity and the extended Darcy±Brinkman formulation to describe the transfer occurring in the porous layer. The chosen dimensionless parameters allow obtaining a single set of conservation equations, valid both in the two ¯uid compartments and in the porous layer. They solve numerically the set of coupling equations using the volume control approach. After recovering some literature results, they study, in an enclosure of square section which is partitioned by a porous layer into two equal ¯uid compartments, the in¯uences of the main parameters on the Nusselt and Sherwood numbers for the Rayleigh number varying from 104 to 107, the Darcy number from 10ÿ5 to 10ÿ3, the thermal conductivity ratio from 1 to 100, the solutal di€usivity ratio from 0.01 to 1, the Lewis number from 0.1 to 5 and the buoyancy ratio from ÿ5 to 5. For Rar104 , a multicellular convective ¯uid ¯ow can appear when ÿ3RNR ÿ 0:5 and if, at least, Le, Rk or Rd is di€erent from unity. # 2000 Elsevier Science Ltd. All rights reserved. Keywords: Thermosolutal natural convection; Double di€usion; Saturated porous medium; Binary ¯uid; Layered ¯uid-porous medium; Darcy±Brinkman formulation

* Corresponding author. Tel.: +212-564-1708; fax: +212-564-2500. E-mail address: [email protected] (M. Mharzi). 0196-8904/00/$ - see front matter # 2000 Elsevier Science Ltd. All rights reserved. PII: S 0 1 9 6 - 8 9 0 4 ( 9 9 ) 0 0 1 3 2 - 6

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Nomenclature A C0 C1 (C2) Da D e' E g~ H K L Le N Nu( y ) Num Pr Ra Rd Rk S Sc Sh( y ) Shm T0 T1 (T2) u (v ) x ( y) x1 a bc bt DC DT l m n y r O c

aspect ratio of cavity, H/L reference solute concentration, (C1+ C2)/2 solute concentration on left (right) vertical wall, C1 > C2 Darcy number, K=L2 solutal di€usivity (m2/s) porous layer thickness (m) dimensionless porous layer thickness, e'/L gravitational acceleration (m/s2) height of cavity (m) permeability of porous medium (m2) width of cavity (m) Lewis number, af =Df ˆ Sc=Pr buoyancy ratio, bc DC=bt DT local Nusselt number, ht y 0 =lf average Nusselt number Prandtl number, nf =af thermal ¯uid Rayleigh number, gbt DTL3 =nf af solutal di€usivity ratio, Dp =Df thermal conductivity ratio, lp =lf dimensionless solute mass fraction, …C ÿ C0 †=DC Schmidt number, nf =Df local Sherwood number, hc y 0 =Df average Sherwood number reference temperature, …T1 ‡ T2 †=2 (K) temperature on left (right) vertical wall of cavity, T1 > T2 (K) horizontal (vertical) dimensionless velocity, u 0 L=nf …v 0 L=nf † dimensionless horizontal (vertical) coordinate, x 0 =L …y 0 =L† dimensionless width of left ¯uid region, x10 =L thermal di€usivity of mixture (m2/s) solutal volumetric expansion coecient, …ÿ1=rf †…@rf =@C † thermal volumetric expansion coecient, …ÿ1=rf †…@rf =@T † (1/K) concentration di€erence, C1 ÿ C2 temperature di€erence, T1 ÿ T2 (K) thermal conductivity (w/m s) dynamic viscosity (kg m/s) kinematic viscosity (m2/s) dimensionless temperature, …T ÿ T0 †=DT density of mixture (kg/m3) dimensionless vorticity dimensionless stream function

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Subscripts c solutal parameter f refers to binary ¯uid p refers to e€ective property of porous medium t thermal parameter

1. Introduction Natural convection in closed cavities is operating in numerous industrial applications and has been the subject of several numerical and experimental studies. Generally, the ¯uid motion in natural convection phenomena is induced by density variations due to temperature and/or concentration gradients. However, the majority of the existing work has been devoted to the thermal, solutal or thermosolutal natural convection occurring in a ¯uid ®lled cavity or in a porous saturated layer. Recently, some authors have considered the case where the natural convection is occurring in a rectangular enclosure containing simultaneously a ¯uid reservoir and a porous partition saturated with the same ¯uid in order to understand the transfers in several environmental or industrial applications such as in geothermal operations, chemical catalytic reactors, thermal isolation problems, ground water seepage. Many works in this topic concern thermal natural convection in cavities where the porous layer is disposed either vertically or horizontally. Analysis was essentially focussed on the insulating e€ect of the porous partition and on the in¯uence of the ¯ow penetration through the porous layer on the thermal transfers. Tong and Subramanian [1] have studied the natural convection ¯uid ¯ow and heat transfer in a vertical enclosure divided into a ¯uid ®lled region and a porous saturated region. The interface separating the ¯uid and porous layer was supposed to be impermeable. They found that the heat transfer can be minimised under some conditions when the porous layer thickness was increased from zero to the cavity width. Campo et al. [2] have conducted a numerical study of thermal convection in an annular enclosure partially ®lled with a porous layer, and they reported an identical conclusion. Considering the ¯uid/porous layer interface as permeable and using the Darcy±Brinkman extended law for modelling the ¯uid ¯ow in the porous medium, Sathe et al. [3] showed that the thermal heat transfer is higher than in the case where the ¯uid/porous interface is impermeable. They found that increasing the thickness of the porous layer can minimise heat transfer in the cavity if the critical ¯uid/porous thermal conductivity ratio is less than one (necessary condition but not sucient). Mbaye et al. [4] are interested in a convection generated by a horizontal thermal ¯ux in a vertical ¯uid layer boarded by a ¯uid saturated porous layer with a ®nite thickness and conductivity and showed that the thermal exchange decreases as the Darcy number is made small. When the thermal conductivity of the porous layer increases, the heat exchange becomes constant for given values of the Darcy and Rayleigh numbers. The numerical and experimental studies conducted by Beckermann et al. [5,6] concerning

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thermal convection in several examples of rectangular enclosures partially ®lled with a porous layer using various glass beads, ¯uids and test cell sizes show that the penetration of ¯uid ¯ow in the porous region is strongly in¯uenced by the product of the Rayleigh and Darcy numbers. However, when the porous medium is horizontally disposed, the convective motion is localised essentially in the ¯uid layer. For given Darcy and Rayleigh numbers, the ¯uid motion in the porous medium will be suppressed when the e€ective porous layer thermal conductivity increases. Natural convection generated by combined thermal and solutal gradients, commonly named double di€usion, was extensively studied in ¯uid ®lled cavities [7±13] or in saturated porous media [14±20]. Double di€usion natural convection in rectangular enclosures partially ®lled with a porous medium disposed vertically or horizontally has received much less attention. Works in this topic have focussed essentially on the analyses of the onset of ®nger convection in a porous medium underlying a ¯uid layer [21,22]. Recently, some authors [23,24] have considered this con®guration for modelling the coupled heat, mass and ¯uid ¯ow problem in a dendritic mushy zone in the absence of phase change. However, those studies have principally analysed the in¯uence on the thermosolutal convection of the thickness of the porous layer and Darcy number when the thermal and solutal buoyancy forces are cooperating and the convection is mass dominated (N > 0). In this paper, we investigate numerically the two-dimensional cooperating and opposing steady state thermosolutal natural convection phenomenon occurring inside a square cavity that is separated in two ¯uid ®lled regions by a ¯uid saturated porous medium. The interface between the ¯uid and the porous medium is permeable, and the ¯ow in the porous layer is modelled using the Darcy±Brinkman extended law to account for no-slip on the walls and on the ¯uid/porous interfaces. The binary ¯uid in this study is air …Pr ˆ 0:71† mixed with several pollutants. Therefore, the Lewis number is varying from 0.2 to 5. To our knowledge, such con®guration has not yet been studied.

2. Physical model and governing equations The physical model which is the main concern of this study deals with the coupled thermal and solutal natural convection in a square cavity containing a binary ¯uid saturated porous layer of ®nite thickness e ' placed in the centre of the cavity …x 0 ˆ x10 ‡ e 0 =2† and bounded by two vertical binary ¯uid layers of equal width. The composite ¯uid/porous system, boundary conditions and the coordinate system for the problem under consideration are depicted in Fig. 1. The vertical side walls …x 0 ˆ 0, x 0 ˆ L† are kept at di€erent levels to a constant and uniform temperature and concentration, while on the bottom …y 0 ˆ 0† and the top …y 0 ˆ H† enclosure walls, the boundary conditions invoked are those of impermeability, non-heat and non-mass transfer. The present study has been conducted by invoking the following hypothesis: . ¯ow is assumed to be steady, laminar, incompressible and two-dimensional,

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Fig. 1. Schematic representation of the cavity.

. physical properties of the ¯uid and porous medium are treated as constants except for the buoyancy term in the momentum equation, where the Boussinesq approximation is used to account for the density variation with temperature and species concentration,   r ˆ r0 1 ÿ bt …T ÿ T0 † ÿ bc …C ÿ C0 † …1† . radiative exchange, viscous dissipation and pressure terms in the energy equation are neglected, . porous medium is assumed to be homogeneous, isotropic, saturated and in local thermal and solutal equilibrium with the binary ¯uid, . Soret and Dufour thermodynamical e€ects are neglected, . ¯ow in the porous layer is described by the Darcy±Brinkman formulation, and the porous medium e€ective dynamic viscosity mp is assumed to be equal to the ¯uid one mf [1,3,5,6,23± 26]. The governing equations expressed in conservative form and non-dimensional terms of stream function, vorticity, temperature and solutal concentration are:

Momentum equation   @uO @vO ˆ ‡ w @x @y

@ 2O @ 2O ‡ 2 @x 2 @y

!

  Ra @y @S wÿ1 ‡ ‡N ‡ O Pr @x @x Da

…2†

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Energy equation @uy @vy …1 ÿ w†…Rk ÿ 1† ‡ 1 @ 2 y @ 2 y ‡ ˆ ‡ @x @y Pr @x 2 @y2

! …3†

Species di€usion equation @uS @vS …1 ÿ w†…Rd ÿ 1† ‡ 1 @ 2 S @ 2 S ‡ ‡ ˆ @x @y Le Pr @x 2 @y2

! …4†

The above equations are written in a combined set of equations valid in the ¯uid region as well as in the porous layer. This transition from one medium to the other is assumed by the parameter w, where w ˆ 1 in the ¯uid region and w ˆ 0 in the porous layer [6]. The dimensionless stream function and vorticity are de®ned as: ! @c @c @ 2c @ 2c ‡ 2 …5† ˆu ˆ ÿv and O ˆ ÿ @y @x @x 2 @y All the above equations following de®nitions: ÿ
x, y ˆ x 0 , y 0 =L u, v ˆ …u 0 , v 0 †=…nf =L† Rk ˆ lp =lf Pr ˆ nf =af

are expressed in the non-dimensionalised variables based on the
ÿ x 1 , E ˆ x 10 , e 0 =L y ˆ …T ÿ T0 †=DT Rd ˆ Dp =Df Le ˆ af =Df

A ˆ H=L S ˆ …C ÿ C0 †=DC N ˆ bt DT=bc DC Ra ˆ gbt DTL3 =nf af

…6†

The boundary conditions on the cavity walls are: cˆ

@c ˆ0 @y

y ˆ S ˆ 0:5

cˆ

@c ˆ0 @y

y ˆ S ˆ ÿ0:5

cˆ

@c ˆ0 @x

@y @S ˆ ˆ 0 at y ˆ 0 @y @y

cˆ

@c ˆ0 @x

@y @S ˆ ˆ 0 at y ˆ A @y @y

at x ˆ 0

at x ˆ 1

…7a†

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When the vorticity is given by:   @ 2c OG ˆ ÿ at x ˆ 0, x ˆ 1, y ˆ 0 and y ˆ A @n2 G

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…7b†

where n is the normal coordinate to the cavity wall G: Modelling the composite system ¯uid/porous medium can be done by using the continuum method approach [6,23,24] or by using the two domain approach where the ¯uid and porous medium are treated separately with matching coupling conditions at the interface [3,5,25,26]. These conditions express the continuity of horizontal and vertical velocities, shear stress, normal stress, temperature, solutal concentration and, ®nally, heat and mass ¯uxes. Expressed in dimensionless parameters at interfaces x ˆ x 1 and x ˆ x 1 ‡ E …0RyRA†, these conditions are written as:     @yp @Sp @yf @Sf yf ˆ yp ˆ Rk Sf ˆ Sp ˆ Rd @x @x @x @x cf ˆ cp

@cp @cf ˆ @x @x

Of ˆ Op

  @Op @Of 1 @cp ˆ ‡ Da @x @x @x

…8†

The model using the two domain approach has the advantage of being fully predictive, so there is no need to ®t experimentally any parameter at the ¯uid/porous interface.

3. Numerical procedure The governing equations (2)±(4) are discretised using a control volume formulation in order to ensure continuity of the convective and di€usive ¯uxes, overall energy and momentum conservation over each elementary volume as well as the entire domain [27]. The convective heat and mass ¯ows through the boundary of the control volume are approximated using the upwind scheme. Note that the boundaries conditions (Eq. (7)) and the coupling ¯uid/porous interfaces conditions (Eq. (8)) are discretised using a second-order expression to maintain the same order of numerical accuracy. Since the composite ¯uid/porous system is modelled using a two domain approach where the ¯uid and porous layers are treated separately and coupled by using, at the permeable interface, the matching conditions described in Eq. (8), the simultaneous algebraic equations were solved using a line by line iterative method [27]. To accelerate convergence, it has also been necessary to use under/overrelaxation parameters (optimised relaxation coecients were determined by multiple tests). The process is repeated until the computed values …y, S, O and c† satis®ed the following condition:

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Maxj1 ÿ

fm‡1 ij j < 10ÿ4 fm ij

…9†

where m is the order of iteration and fij represent the values of the unknowns …y, S, O and c† at the nodes (i, j ). The two-dimensional domain is discretised according to a non-uniform grid [28] generated by the expression (10) in order to have a sucient number of nodes (at least three) in the thin boundary layers near the walls of the cavity and in the neighbourhood of the ¯uid/porous interfaces where Rk and Rd may present an abrupt change. x…i † ˆ

  EP 1 ÿ exp…bi=M† 2 1 ÿ exp…b †

avec: 0RiRM

…10†

In this formula, EP designates the width in the x-direction of the considered ¯uid or porous region and M the half number of nodes localised in this region, and b is the expansion coecient which allows us to control the stretching meshes in the neighbourhood of the cavity walls and ¯uid/porous interfaces. The similar relation (10) is also retained in the y-direction with EP ˆ A: Di€erent grid distributions were tested and led us to retain an expansion coecient b varying form 1.7 to 2 associated to non-uniform grid systems ranging from (60, 60) to (90, 90) depending on the Ra and Da values. The local and global heat and mass transfer rates, on the vertical walls …x ˆ 0 and 1) are expressed, respectively, by the local Nusselt Nu…y† and Sherwood Sh…y† numbers and by the averages Num and Shm which are de®ned as follows: 

@y Nu…y † ˆ ÿ @x

1 Num ˆ ÿ A



…A 0



xˆ0, 1

@y @x

@S Sh…y † ˆ ÿ @x

 xˆ0, 1

dy

 xˆ0, 1

1 Shm ˆ ÿ A

…A 0

@S @x

 xˆ0, 1

dy

…11†

The heat and mass ¯uxes on the vertical hot wall …x ˆ 0† and cold wall …x ˆ 1† are assumed to be conserved if the relative gaps between the average Nusselt number as well as the average Sherwood number verify the following conditions: j

Num …x ˆ 0 † ÿ Num …x ˆ 1 † j < 0:5% Num …x ˆ 0 †

j

Shm …x ˆ 0 † ÿ Shm …x ˆ 1 † j < 0:5% Shm …x ˆ 0 †

…12†

This relationship is used to reinforce the check on the convergence of the numerical solution. In view of condition (12), we represent only the local and the average Nusselt and Sherwood numbers evaluated on the hot wall …x ˆ 0).

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4. Results and discussions The thermosolutal natural convection under consideration is governed by 10 dimensionless parameters: Ra, Da, Pr, Le, N, Rk, Rd, A, E and x1. In the following, we restrict this study to a square cavity …A ˆ 1† ®lled with air …Pr ˆ 0:71† separated in two equal parts …x 1 ˆ 0, 4† by a saturated porous medium of dimensionless thickness E ˆ 0:2: The Rayleigh number is ranging from 104 to 107 and the Darcy number from 10ÿ5 to 10ÿ3 when Rk and Rd are varying between 1 and 100, and 0.1 to 1, respectively, the Lewis number varying from 0.1 to 5 and the buoyancy ratio N from ÿ5 to 5. 4.1. Comparisons with other results The test of the accuracy of the numerical procedure retained in the present study is accomplished by comparison with previous numerically quantitative published results concerning similar systems [3,10]. Shown in Fig. 2a is the comparison of our results obtained for 0:01RNR5 and Rat ˆ 107 with the ones obtained by the correlation Num ˆ 0:22…RajN ‡ 1j†0:27 proposed by BeÂghein et al. [10]. An excellent agreement is observed as well as that in Fig. 2(b) which compares the values of Shm given by Ref. [10] with those calculated for Rat ˆ 107 and Rac ˆ 0 as a function of Lewis number. The results of comparisons relative to the con®guration considered by Sathe et al. [3] are given in Tables 1 and 2 which regroup the values of Num as a function of A and Ra for the ®rst and as a function of E for the second. Here also, we have very good agreement.

Table 1 Variation of Num vs. the aspect ratio A and Ra A

Ra

Present work

Sathe et al. [3]

5 5 10 10

104 105 104 105

2.015 3.726 1.674 3.200

1.987 3.714 1.681 3.230

a

(1.16%)a (0.30%) (0.32%) (0.94%)

Relative error.

Table 2 Variation of Num vs. the thickness of the porous layer E for A ˆ 1, Rk ˆ 1, Ra ˆ 105 and Da ˆ 10ÿ3 E

Present work

Sathe et al. [3]

0.25 0.50 0.75

3.600 3.321 3.083

3.604 (0.11%) 3.348 (0.81%) 3.101 (0.58%)

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Fig. 2. (a) Variation of Num in function of Rat jN ‡ 1j: (b) Variation of Shm vs. Le for Rat ˆ 107 and Rac ˆ 0:

4.2. In¯uence of Ra and Da Increasing simultaneously or separately Ra and Da favours the binary ¯uid to ¯ow through the porous layer from one ¯uid region to the other, so the intensities of thermosolutal convection and, consequently, local and average Nusselt and Sherwood numbers increase. Moreover, the ¯ow in the cavity is reduced when Ra and Da are made small (Fig. 3), especially

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Fig. 3. In¯uence of Ra and Da on Num and Shm for Rk ˆ Rd ˆ N ˆ Le ˆ 1:

for moderate Rayleigh numbers …Ra < 5  105 ). Accordingly, the ¯uid ¯ow is con®ned in the ¯uid compartments, and the heat and mass transfers in the porous layer are mainly by conduction [5,6,23,24]. Fig. 5 illustrates this phenomenon and shows also the existence of an important velocity gradient at the ¯uid/porous interfaces, resulting in a strong modi®cation of the vertical velocity component (v ). Consequently, in the high part of the cavity, the ¯ow lines curve down when they leave the left ¯uid compartment and penetrate in the porous medium, while they curve up

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before leaving the porous matrix. This phenomenon is more signi®cantly accentuated in the middle part of the cavity. 4.3. In¯uence of N The thermosolutal ¯ow is driven by the net e€ect of the solutal and thermal buoyancy forces which are characterised by the parameter N (Eq. (5)). Since, in the present study, DT and DC are considered to be positive, the sign of N is depending simultaneously on the thermal expansion coecient sign, which is normally positive, and on the solutal expansion coecient sign, which can be either positive or negative. As a result, the solutal and thermal convections are cooperating when N is positive, and they are opposed when N is negative. However, the solutal convection is dominant over the thermal one for jNj > 1: It the case of N ˆ ÿ1 and Rk ˆ Rd ˆ Le ˆ 1, the buoyancy e€ects are cancelled. So, the energy and species di€usion equations become identical, independently of the values of Ra and Da [10,14]. It can be noticed here that when N is varying and Rk ˆ Rd ˆ Le ˆ 1, we have: Num …jNj† ˆ Num …ÿjNj† ˆ Shm …jNj† ˆ Shm …ÿjNj† as is shown in Fig. 4. 4.4. In¯uence of Rk Taking Rd ˆ Le ˆ N ˆ 1 and increasing Rk (porous partition more conducting of heat than the ¯uid region), we conclude, as shown in Fig. 5(a), that Shm is nearly invariant while Num decreases. Under this condition, the conductive exchange of heat in the cavity is increased. As can be seen in Fig. 6, for a high conductivity porous medium …Rk  1), the porous layer is almost isothermal, and thermal boundary layers appear at the ¯uid/porous interfaces.

Fig. 4. In¯uence of N and Ra on Num and Shm for Da ˆ 10ÿ3 and Rk ˆ Rd ˆ Le ˆ 1:

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Fig. 5. Num and Shm variation with Rk : in¯uence of Ra and Da …Rd ˆ Le ˆ N ˆ 1).

4.5. In¯uence of Rd In solutal transfer, Rd plays the same role as Rk in the thermal one, however, the solutal di€usivity ratio Rd is less than 1 (the ¯uid is generally more di€usive than the porous layer). Fig. 7 shows the variation of the average heat and mass exchange for Rd ranging from 0.01 to 0.9 and Rk ˆ Le ˆ N ˆ 1: As seen on this ®gure, Num is practically independent of the solutal

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Fig. 6. Streamlines …c), isotherms …y† and isoconcentrations lines (S ): in¯uence of Rk …Rd ˆ Le ˆ N ˆ 1†: (a) Ra ˆ 105 ; Da ˆ 10ÿ3 ; (b) Ra ˆ 106 and Da ˆ 10ÿ5 :

di€usivity ratio for a given Ra and Da, while the Shm is approximately invariant when Rd > 0:2: However, for Rd < 0:2, a substantial reduction of Shm is produced. This decrease in the average Sherwood number is more important for DaR10ÿ4 : Generally, a decrease in the molecular di€usion coecient of the porous layer is associated with an increase in the vertical concentration gradient in the porous layer (Fig. 8). 4.6. In¯uence of Le For ®xed values of the governing parameters, the solutal di€usion in the cavity is reduced, with regard to the heat conduction, for larger values of Lewis number (smaller binary di€usion coecient) [7,10,14,24]. This results in a larger solutal transfer rate (large Shm) at the vertical cavity walls and thinner boundary layers relative to the thermal one (independently of Rk and Rd). On the other hand, this increase of the solutal transfer is coupled to a smaller reduction in

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Fig. 7. Num and Shm variation with Rd: in¯uence of Ra and Da …Rk ˆ Le ˆ N ˆ 1).

Num (Fig. 9). Inversely, for Le < 1, the solutal transfer is accomplished more by di€usion than by convection, and the thermal boundary layer is thinner than the solutal one (Fig. 10). 4.7. Simultaneous in¯uence of several parameters Varying one by one the governing parameters and taking the others as constant values, we deduce that the Nusselt number grows with Ra, Da, and 1=Rk , but it has lower sensibility to

Fig. 8. Streamlines, isotherms and isoconcentrations lines for Rd ˆ 0:01 …Rk ˆ Le ˆ N ˆ 1). (a) Ra ˆ 105 and Da ˆ 10ÿ3 ; (b) Ra ˆ 106 and Da ˆ 10ÿ5 :

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Fig. 9. E€ect of Le on Num and Shm. (a) Rk ˆ Rd ˆ N ˆ 1; (b) Ra ˆ 105 , Da ˆ 10ÿ3 , N ˆ 0 and Rd ˆ 0:05:

Le and Rd. On the other hand, Shm increases with Ra, Da, Le and 1=Rd , and it is relatively insensitive to Rk. When several governing parameters are varying simultaneously, their e€ects on the thermosolutal convection are added if their actions are cooperating. However, when their actions are opposed to each other, the ¯ow ®eld becomes considerably varied and very dicult to predict. If two governing parameters are varied simultaneously, we could compare quantitatively their in¯uences. For example, as we can see in Figs. 11 and 12, the Num increases especially with Ra and Da, weakly with 1=Rk and is quasi-independent of Rd, so that the growth with

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Fig. 10. E€ects of Le and Rk: isotherms and isoconcentrations lines for Ra ˆ 105 ; Da ˆ 10ÿ3 and Rd ˆ N ˆ 1: (a) Le ˆ 0:1; (b) Le ˆ 3:

Ra, Da and Rd of Shm is noticeably of the same importance, and it is relatively independent of Rk. When N ˆ ÿ1 (the solutal and thermal buoyancy forces are opposed to each other and of equal magnitudes), the thermosolutal convection is totally suppressed if Le ˆ Rk ˆ Rd ˆ 1: Furthermore, when at least one of those parameters: Le, Rk and Rd is di€erent from unity, and according to the values of the other governing parameters, the stream function could have two extrema (the negative one indicated by cmin and the positive one indicated by cmax † indicating the existence of contra-rotating ¯uid ¯ow. In general, the stream function exhibits only one extremum, which can be either negative (clockwise: cmin † or positive (counterclockwise: cmax ). So, the simultaneous existence of negative and positive extrema is proof of the existence of contra-rotating ¯uid ¯ow. Hence, the study of the variations of the stream function extremum constitutes a simple process to describe the domains where the multi-cell structure can appear in the cavity. Figs. 13±16 illustrate well this technique which allows us to conclude that generally, for Rar104 , the multicellular ¯uid ¯ow can exist, when ÿ3RNR ÿ 0:5, only if at least one of the parameters Le, Rk and Rd is di€erent from 1. The structures of the multicellular ¯uid ¯ow observed are complex and very sensitive to the values of Le, Rk and Rd, in particular to the two ®rst ones (Figs. 13±16). In general, the more Le, Rk and 1=Rd exceed unity, the more the ¯uid ¯ow is susceptible to be multicellular. This category of the multicellular ¯uid ¯ow must not be confounded with those susceptible to appear at the high values of Rayleigh number nor with those obtained in shallow cavities. The change in the ¯uid ¯ow structure is accompanied with a change in the local Nusselt and Sherwood numbers curves. So, the study of the variations of Nu…Y † and Sh…Y † along the left (right) wall of the cavity enables us also to predict the existence of a multicellular ¯uid ¯ow. For example (Figs. 17 and 18), if we consider a running point upwards along the hot vertical left hind wall of the cavity, the existence of two (or more) minimums or maximums indicates

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Fig. 11. In¯uence of Rd, Rk and Da on Num and Shm for Le ˆ N ˆ 1: (a) Ra ˆ 105 and Da ˆ 10ÿ3 ; (b) Ra ˆ 106 and Rk ˆ 50:

clearly the existence of the contra-rotating vortices. The monotonous variation of the local Nusselt and Sherwood numbers indicates, in general, the presence of an unicellular convective ¯uid ¯ow which is clockwise if Nu…y† and Sh…y† decrease monotonously and counterclockwise if they increase monotonously. However, it should be noticed that the variations of Nu…y† and Sh…y† are not always sensible to the presence of the multicellular structure ¯uid ¯ow, especially if the contra-rotating vortex is localised in the core region of the cavity or when the intensity of the secondary cells is very low.

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Fig. 12. Variation of Num and Shm vs. Rk, Le and N for Ra ˆ 105 , Da ˆ 10ÿ3 and Rd ˆ 0:05: (a) e€ect of Rk, Le and N; (b) e€ect of Le, N for Rk ˆ 60:

5. Conclusion Using the Navier±Stokes equations coupled to the Darcy±Brinkman model, a numerical study, based on the control volume method, has been conducted to analyse the in¯uences of the governing parameters on a two-dimensional laminar and permanent, thermosolutal natural convection occurring in a square cavity containing simultaneously a binary ¯uid (air + pollutant) and a saturated vertical porous layer. The following conclusions can be drawn.

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Fig. 13. Variation of extremum of the stream function for: Da ˆ 10ÿ3 and Rk ˆ Rd ˆ 1: Ra ˆ 105 : (a) Le ˆ 0:6; (b) Le ˆ 4 ÿ Ra ˆ 106 : (c) Le ˆ 0:6; (d) Le ˆ 4:

The thermal exchange as well as the solutal one are principally sensitive to the Rayleigh and Darcy numbers, so the increase of Ra enhances the convection in the ¯uid compartments of the cavity, while the convection in the porous layer is enhanced with the increase of Da. The heat transfer is more in¯uenced by the thermal conductivity ratio Rk, while the solutal transport is essentially sensitive to the solutal di€usivity ratio Rd. The increases of Rk

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Fig. 14. Variation of extremum of the stream function for: Da ˆ 10ÿ5 and Rk ˆ Rd ˆ 1: Ra ˆ 105 : (a) Le ˆ 0:6: (b) Le ˆ 4 ÿ Ra ˆ 106 : (c) Le ˆ 0:6; (d) Le ˆ 4:

(decreases of Rd) intensify the conductive thermal exchange in the porous layer (di€usive solutal exchange). Furthermore, their global e€ects on the thermosolutal convection are lower than the ones of Ra and Da. The increase of the Lewis number, which characterises the ratio of the thermal di€usion and the molecular di€usivity, enhances consistently the solutal exchange (Shm), while the Nusselt

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Fig. 15. Variation of extremum of the stream function for: Da ˆ 10ÿ3 , Rk ˆ 60 and Rd ˆ 1: Ra ˆ 105 : (a) Le ˆ 0:6; (b) Le ˆ 4 ÿ Ra ˆ 106 : (c) Le ˆ 0:6; (d) Le ˆ 4:

number exhibits a smaller decrease. However, when thermal and solutal buoyancy forces are opposed to each other …N < 0), the increase of Le produces an important reduction of the heat exchange …Num ). The buoyancy force ratio, N, is the principal governing parameter of the thermosolutal convection by its magnitude and sign. Particularly, when N ˆ ÿ1 and Le ˆ Rk ˆ Rd ˆ 1, the

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Fig. 16. Variation of extremum of the stream function for: Da ˆ 10ÿ5 , Rk ˆ 60 and Rd ˆ 1: Ra ˆ 105 : (a) Le ˆ 0:6; (b) Le ˆ 4 ÿ Ra ˆ 106 : (c) Le ˆ 0:6; (d) Le ˆ 4:

convective ¯ow in the cavity is completely suppressed. Consequently, in this particular case, the average Nusselt number (or Sherwood) versus N is a V-shaped curve symmetrical about the value N ˆ ÿ1 …Num …jNj† ˆ Num …ÿjNj† ˆ Shm …jNj† ˆ Shm …ÿjNj†). In addition, for Rar104 , when ÿ3RNR ÿ 0:5 and if at least one of the three parameters, Rk, Rd and Le, is di€erent from unity, some multicellular convection ¯uid ¯ow can been observed in the cavity.

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Fig. 17. Variation of Nu( y ) and Sh( y ) for: Da ˆ 10ÿ3 and Rd ˆ 1: (a) Ra ˆ 106 ; Rk ˆ 60; Le ˆ 5 and N ˆ ÿ1:5 …cmin ˆ ÿ1:05; cmax ˆ 67:26). (b) Ra ˆ 106 ; Rk ˆ 1; Le ˆ 0:2 and N ˆ ÿ1:5 …cmin ˆ ÿ2:81; cmax ˆ 6:75).

Fig. 18. E€ect of Rk: streamlines, isotherms and isoconcentrations lines for Ra ˆ 107 ; Da ˆ 10ÿ3 ; Rd ˆ 1; N ˆ ÿ0:5 and Le ˆ 0:1: (a) Rk ˆ 1; (b) Rk ˆ 8; (c) Rk ˆ 60; (d) Rk ˆ 100:

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Furthermore, the structure and the intensity of this multicellular convection ¯uid ¯ow depend greatly on the values of the other governing parameters. The presence of this model of convection is characterised by the existence of two non-zero extrema of the stream function …cmin , cmax † and, some time, by the changes in the slope of the local Nusselt and Sherwood numbers curves.

References [1] Tong TW, Subramanian E. Natural convection in rectangular enclosures partially ®lled with a porous medium. Int J Heat and Fluid Flow 1988;7:3±10. [2] Campo A, Lacoa U, Morales JC, Campos H. Heat transport suppression in a slender annular enclosure containing a ¯uid and a porous medium. Num Simulation of Convection in Electronic Equipment Cooling 1990;121:61±6. [3] Sathe SB, Lin WQ, Tong TW. Natural convection in enclosures containing an insulation with a permeable ¯uid-porous interface. Int J Heat and Fluid Flow 1988;9:389±95. [4] Mbaye M, Vasseur P, Bilgen E. Natural convection heat transfer in cavity partially ®lled with porous medium. In: Proceeding de l'ITEC Fac. Sc. Semlalia Marrakech. 1993. p. 221±4. [5] Beckermann C, Viskanta R, Ramadhyani S. Natural convection ¯ow and heat transfer between a ¯uid layer and a porous layer inside a rectangular enclosure. J of Heat Transfer 1987;109:363±70. [6] Beckermann C, Viskanta R, Ramadhyani S. Natural convection in vertical enclosures containing simultaneously ¯uid and porous layers. J Fluid Mech 1988;186:257±84. [7] Lin TF, Huang CC, Chang TS. Transient binary mixture natural convection in a square enclosures. Int J Heat Mass Transfer 1993;33:1315±31. [8] Chang J, Lin TF, Chien CH. Unsteady thermosolutal opposing convection of a liquid±water mixture in a square cavity. Part I: Flow formation and heat and mass transfer characteristics. Int J Heat Mass Transfer 1990;36:287±99. [9] Trevisan OV, Bejan A. Combined heat and mass transfer by natural convection in a vertical enclosure. J of Heat Transfer 1987;109:104±12. [10] Bghein C, Haghighat F, Allard F. Numerical study of double-di€usive natural convection in a square cavity. Int J Heat Mass Transfer 1992;35:833±46. [11] Raithby GD, Wong HH. Heat transfer by natural convection across vertical air layers. Numerical Heat Transfer 1981;4:447±57. [12] Min Hyun, Wook Lee J. Double-di€usion convection in a rectangle with cooperating horizontal gradients of temperature and concentration. Int J Heat Mass Transfer 1990;33:1605±17. [13] Ghorayeb K. Etude des eÂcoulements de convection thermosolutale en cavite rectangulaire. TheÁse de Doctorat, Univ. Paul Sabatier, Toulouse, 1997. [14] Trevisan OV, Bejan A. Natural convection with combined heat and mass transfer buoyancy e€ects in a porous medium. Int J Heat Mass Transfer 1985;28:1597±611. [15] Trevisan OV, Bejan A. Mass and heat transfer by natural convection in a vertical slot ®lled with porous medium. Int J Heat Mass Transfer 1986;29:403±15. [16] Alavyoon F. On natural convection in vertical porous enclosures due to prescribed ¯uxes of heat and mass at the vertical boundaries. Int J Heat Mass Transfer 1993;36:2479±98. [17] Wang CY, Beckermann C. A two-phase mixture model of liquid±gas ¯ow and heat transfer in capillary porous media. Part I: Formulation. Int J Heat Mass Transfer 1993;36:2747±58. [18] Vasseur P, Robillard L. The Brinkman model for boundary layer regime in a rectangular cavity with uniform heat ¯ux from the side. Int J Heat Mass Transfer 1987;30:717±27. [19] Rosenberg D, Spera FJ. Thermohaline convection in a porous medium heated from below. J of Heat Transfer 1992;35:1261±73.

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M. Mharzi et al. / Energy Conversion & Management 41 (2000) 1065±1090

[20] Goyeau B, Songbe JP, Gobin D. Numerical study of double-di€usive natural convection in a porous cavity using the Darcy±Brinkman formulation. Int J Heat Mass Transfer 1996;39:1363±78. [21] Chen J, Chen CF. Onset of ®nger convection in a horizontal porous layer underlying a ¯uid layer. J Heat Transfer 1988;110:403±9. [22] Chen J, Chen CF. Experimental investigation of convective stability in a superposed ¯uid and porous layer when heated from below. J Fluid Mech 1989;207:311±21. [23] Goyeau B, Mergui S, Songbe JP, Gobin D. Convection thermosolutale en cavite partiellement occupeÂe par une couche poreuse faiblement permeÂable. C.R. Acad. Sc. 1996. T. 323, SeÂrie II b, pp. 447±454. [24] Gobin D, Goyeau B, Songbe JP. Double di€usive natural convection in composite ¯uid-porous layer. J of Heat Transfer 1998;120:234±42. [25] Arquis E, Caltagirone JP. Sur les conditions hydrodynamiques au voisinage d'une interface milieu ¯uide-milieu poreux: application aÁ la convection naturelle. C.R. Acad. Sc. 1984. T. 299, SeÂrie II, No 1, pp. 1±3. [26] Neale G, Nader W. Practical signi®cance of Brinkman's extension of Darcy's law: coupled parallel ¯ows within a channel and a bounding porous medium. Can J Chem Engng 1974;52:475±8. [27] Patankar SV. Numerical heat transfer and ¯uid ¯ow. New York: Hemisphere, McGraw-Hill, 1980. [28] Turki S, Lauriat G. An examination of two numerical procedures for natural convection in enclosures. Num Simulation of Convection in Electronic Equipment Cooling 1990;121:107±13.