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International Journal of Heat and Mass Transfer 55 (2012) 4116–4128

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International Journal of Heat and Mass Transfer

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Thermosolutal convection from a discrete heat and solute sourcein a vertical porous annulus

M. Sankar a, Beomseok Kim b, J.M. Lopez b,c, Younghae Do b,⇑a Department of Mathematics, East Point College of Engineering and Technology, Bangalore 560049, Indiab Department of Mathematics, Kyungpook National University, 1370 Sangyeok-Dong, Buk-Gu, Daegu 702-701, Republic of Koreac School of Mathematical and Statistical Sciences, Arizona State University, Tempe, AZ 85287, USA

a r t i c l e i n f o a b s t r a c t

Article history:Received 7 June 2011Received in revised form 3 March 2012Accepted 3 March 2012Available online 18 April 2012

Keywords:Double-diffusive convectionPorous annulusHeat and solute sourceRadius ratio

0017-9310/$ - see front matter � 2012 Elsevier Ltd. Ahttp://dx.doi.org/10.1016/j.ijheatmasstransfer.2012.03

⇑ Corresponding author. Tel.: +82 53 950 7951; faxE-mail addresses: [email protected] (M

(B. Kim), [email protected] (J.M. Lopez), yhdo@knu

Double-diffusive convection in a vertical annulus filled with a fluid-saturated porous medium is numer-ically investigated with the aim to understand the effects of a discrete source of heat and solute on thefluid flow and heat and mass transfer rates. The porous annulus is subject to heat and mass fluxes from aportion of the inner wall, while the outer wall is maintained at uniform temperature and concentration.In the formulation of the problem, the Darcy–Brinkman model is adopted to the fluid flow in the porousannulus. The influence of the main controlling parameters, such as thermal Rayleigh number, Darcy num-ber, Lewis number, buoyancy ratio and radius ratio are investigated on the flow patterns, and heat andmass transfer rates for different locations of the heat and solute source. The numerical results show thatthe flow structure and the rates of heat and mass transfer strongly depend on the location of the heat andsolute source. Further, the buoyancy ratio at which flow transition and flow reversal occur is significantlyinfluenced by the thermal Rayleigh number, Darcy number, Lewis number and the segment location. Theaverage Nusselt and Sherwood numbers increase with an increase in radius ratio, Darcy and thermalRayleigh numbers. It is found that the location for stronger flow circulation is not associated with higherheat and mass transfer rates in the porous annular cavity.

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1. Introduction

Double-diffusive convective flow caused by the combinedinfluence of thermal and solutal buoyancy forces through porousmedium has numerous industrial, natural and geophysical applica-tions. Some of the prominent applications are petrochemical pro-cesses, the food industry, grain storage installations, fuel cells,crystal growth applied to semiconductors, solar ponds, migrationof moisture through air contained in fibrous insulation, contamina-tion transport in saturated soil, and the underground disposal of nu-clear wastes. A comprehensive overview of double-diffusiveconvection in saturated porous media, its relevance in the under-standing of many natural systems and its wide variety of engineer-ing applications are well documented in the literature [1–5].

The combined effects of thermal and solutal buoyancy forceslead to complex flow structures in a vertical porous annulus, andthe understanding of their interaction with heat and mass trans-port is relevant to the above mentioned applications. Using theDarcy model, Beji et al. [6] numerically investigated double-diffu-sive convection in a vertical porous annulus, with a uniform

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temperature and concentration difference applied across the verti-cal walls. They found that the buoyancy ratio at which flow transi-tion and flow reversal occur depends strongly on the physical andgeometrical parameters. A combined numerical and analyticalstudy of double-diffusive convection in a fluid saturated porousannulus subjected to uniform heat and mass fluxes from the sidewalls was reported by Marcoux et al. [7]. For large aspect ratios,their analytical results are in good agreement with numericalsolutions, and they found that the flow, thermal and solutal fieldsare significantly influenced by curvature effects. In modeling theflow in porous media, Darcy’s law is one of the most popularmodels. However, it is generally recognized that Darcy’s modelmay over predict the convective flows for large values of Darcynumber. Using a finite-element method, Nithiarasu et al. [8] pro-posed a generalized model to study both Darcy and non-Darcy flowregimes of double-diffusive convection in a vertical porous annu-lus. They found that the generalized model predicts lower heatand mass transfer rates compared to other porous medium models,such as Darcy, Brinkman and Forchheimer models. Benzeghibaet al. [9] applied the Brinkman-Forchheimer–Darcy model to ana-lyze the thermosolutal convection in a partly filled porous annulus.Recently, the Darcy–Brinkman formulation has been used to studythe influence of the Darcy number on the double-diffusive naturalconvection in a vertical porous annulus [10,11]. Bahloul et al. [12]

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Nomenclature

A aspect ratioC dimensionless concentrationcp specific heat at constant pressureD width of the annulus (m)Da Darcy numberg acceleration due to gravity (m/s2)H height of the annulus (m)h dimensional length of heat and solute source (m)K permeability of the porous medium (m2)k thermal conductivity (W/(m K))l distance between the bottom wall and centre of the

source (m)L dimensionless location of the sourceLe Lewis numberN buoyancy ratioNu average Nusselt numberSh average Sherwood numberp fluid pressure (Pa)Pr Prandtl numberqh heat flux (W/m2)jh mass flux (kg/m2s)Ra thermal Rayleigh numberRaT thermal Darcy–Rayleigh number RaT ¼ gKbT qhD2

tkaT

� �S dimensional concentrationT dimensionless temperature

t⁄ dimensional time (s)t dimensionless time(ri, ro) radius of inner and outer cylinders (m)(r, x) dimensional radial and axial co-ordinates (m)(R, X) dimensionless radial and axial co-ordinates(u, w) dimensional velocity components in (r, x) direction

(m/s)(U, W) dimensionless velocity components in (R, X) direction

Greek lettersaT thermal diffusivity (m2/s)aC mass diffusivity of the solute in the fluid (m2/s)bT thermal expansion coefficient (1/K)bC solutal expansion coefficient (1/K)r heat capacity ratioe dimensionless length of the sourcef dimensionless vorticityh dimensional temperature (K)k radii ratio[ kinematic viscosity (m2/s)q fluid density (kg/m3)/ porosity of the porous mediumw dimensionless stream function

M. Sankar et al. / International Journal of Heat and Mass Transfer 55 (2012) 4116–4128 4117

investigated analytically and numerically the behaviour of a binarymixture saturating a vertical annular porous medium with Soret-induced convection. Approximate expressions for Nusselt andSherwood numbers are obtained for the heat-driven and solute-driven flow regimes.

Double-diffusive convective flows in a differentially heated ver-tical annulus have been intensively studied in relation to applica-tions such as oxidation of surface materials, cleaning and dyingoperations, fluid storage components and energy storage in solarponds. Shipp et al. [13,14] performed a detailed analysis on dou-ble-diffusive convection in a vertical annulus. The effects of ther-mal Rayleigh number, Lewis number and buoyancy ratio areinvestigated on flow transition and flow reversal for fixed valuesof radius ratio, aspect ratio and Prandtl number. Lee et al. [15]studied numerically double-diffusive convection of a salt-watersolution in a uniformly rotating annulus with particular attentionpaid to a multilayered flow regime. They found that the azimuthalvelocity induced by the rotation of the system suppressed the gen-eration of rolls at the hot wall, and the merging of adjacent layers.The effects of film evaporation and condensation on the heat andmass transfer rates were examined by Yan and Lin [16] in anopen-ended vertical annular duct. Retiel et al. [17] investigatedthe influence of curvature on double-diffusive convection in a ver-tical annulus subjected to cooperating gradients of temperatureand solute concentration at the vertical walls. Notable among therecent studies on double-diffusive convection in a vertical annulusare due to Chen et al. [18] and Venkatachalappa et al. [19].

Natural convection flow in a vertical porous annulus, due tothermal buoyancy alone, has been widely studied and well-docu-mented in the literature, owing to its importance in building insu-lation, porous heat exchangers and many others applications.Prasad and co-workers [20,21] numerically investigated naturalconvection in a vertical porous annulus with constant temperatureand constant heat flux conditions at the inner wall for a wide rangeof parameters. A combined analytical and numerical study of

natural convection in a vertical annular porous layer with the innerwall maintained at a constant heat flux and insulated outer wallhas been carried out by Hasnaoui et al. [22]. Using Darcy–Brink-man model, Shivakumara et al. [23] numerically investigatednatural convection in a vertical porous annulus. More recently,Sankar et al. [24] reported on the effects of size and location of adiscrete heater on the natural convective heat transfer in a verticalporous annulus.

Natural convection resulting from thermal and solutal buoy-ancy forces in rectangular porous enclosures have also been thesubject of a vast number of investigations. Trevisan and Bejan[25] investigated in detail the flow characteristics, heat and masstransfer rates in a rectangular enclosure subjected to uniform heatand mass fluxes from the vertical walls. They obtained the analyt-ical solution in the boundary layer regime, and numerical solutionsvalid for the entire flow regime. Later, Alavyoon [26] extended thework of Trevisan and Bejan [25] for a much wider range of param-eters. For thermosolutal flows in a square porous cavity, Goyeauet al. [27] found that the influence of Darcy number on the heattransfer is more complex than in thermal convection, and the ther-mosolutal flow behavior in porous media is much different fromthat of pure convection. The influence of other factors such asanisotropy [28], heat generation or absorption [29], and Soret ef-fect [30] on the double-diffusive flow, heat and mass transfer ratesin rectangular porous enclosures have also been investigated. Morerecently, Rahli et al. [31] numerically investigated three-dimen-sional double-diffusive mixed convection in a horizontal rectangu-lar duct.

In numerous applications involving finite porous enclosures,the heating is only imposed over a portion of the wall rather thanover the entire wall. Natural convection in square and rectangularporous enclosures subject to discrete heating has drawn muchattention in recent years. Natural convection in a porous squarecavity with an isoflux and isothermal discrete heater has beennumerically studied by Saeid and Pop [32] using the Darcy model.

4118 M. Sankar et al. / International Journal of Heat and Mass Transfer 55 (2012) 4116–4128

They found that maximal heat transfer can be achieved when theheater is placed near the bottom of one of the vertical walls. Later,Saeid and Pop [33] numerically studied mixed convection inducedby two isothermal heat sources on a vertical plate channel filledwith a porous medium. Recently, using Bejan’s heatlines method,Kaluri et al. [34] analyzed the optimal heat transfer by consideringthree different heating conditions in a square porous cavity.

The existing studies on double-diffusive convection subject to adiscrete heat and solute source have been restricted only to squareenclosures [35–37], in which the effects of curvature are missing.Also, the available studies related to double-diffusive convectionin a vertical porous annulus are restricted only to uniform heatingand salting of inner wall by either uniform heat/mass fluxes or con-stant temperature or concentration [6–11]. In spite of the impor-tant applications of the annular cavity, the influence of discreteheat and solute source on double-diffusive convection in a verticalporous annulus remains poorly understood in the literature, whichmotivates the present investigation. The purpose of the present pa-per is to make up for the lack of understanding on double-diffusiveconvection in a vertical porous annulus filled with a two-compo-nent mixture, and subjected to partial heating and salting at the in-ner wall. Here, we numerically examine the effects of a discreteheat and solute source on the double-diffusive natural convectionin a porous annular cavity, and in particular explore the effects ofsource location and curvature effects. The mathematical formula-tion and methods of solution are respectively presented in Sections2 and 3. Section 4 is devoted to the results of the numerical simu-lations, and the conclusions are given in Section 5.

2. Mathematical formulation

Consider a vertical annulus filled with a homogeneous isotropicporous medium, closed at the top and bottom ends by two insu-lated disks, which are impermeable to mass transfer. The heightof the annulus is denoted by H, and the inner and outer radiusby ri and ro respectively, as shown in Fig. 1. A heat and solutesource of length h (=H/4), placed on the inner wall, is subjectedto constant heat and mass fluxes of strengths qh and jh, while therest of the inner wall is insulated and impermeable. The outer wallis maintained at a lower temperature h0 and lower concentrationS0. The distance between the centre of the heat and solute sourceand the bottom wall is l. Also, the fluid is assumed to be Newtonianwith negligible viscous dissipation, and the flow is assumed to be

Fig. 1. Flow configuration and coordinate system.

axisymmetric. Gravity acts in the negative x-direction. The porousmatrix is assumed to be rigid, and in local thermodynamic equilib-rium with the fluid. In the porous medium, the Darcy–Brinkmanformulation is assumed to hold, and hence the Forchheimer qua-dratic drag term of the momentum equation is neglected[10,11,23]. The heat flux produced by the concentration gradient(Dufour effect) and the mass flux produced by the temperaturegradient (Soret effect) is neglected. The following non-dimension-less variables are used:

ðR;XÞ ¼ ðr; xÞ=D; ðU;WÞ ¼ ðu;wÞD=aT ; t ¼ t�aT=D2;

T ¼ ðh� h0Þ=Dh; C ¼ ðS� S0Þ=DS; P ¼ pD2=q0a2;

f ¼ f�D2=aT ; w ¼ w�=DaT ; where D ¼ ro � ri;

Dh ¼ qhD=k; DS ¼ jhD=aC :

Further, we assume that all thermophysical properties are constant,except for the effect of density variations in the buoyancy term. Accordingto the Boussinesq approximation, the density of the mixture is related tothe temperature and solute concentration through the following linearequation of state: qðh; SÞ ¼ q0½1� bTðh� h0Þ � bCðS� S0Þ�; where bT

and bC are respectively the coefficients for thermal and concentrationexpansions. Based on the above assumptions, the dimensionless govern-ing equations for the conservation of mass, momentum, heat and soluteconcentration in a porous annulus are written in the vorticity-streamfunction formulation as [8,11,13,14]

r oTotþ U

oToRþW

oToX¼ r2T; ð1Þ

/oCotþ U

oCoRþW

oCoX¼ 1

Ler2C; ð2Þ

1/

ofotþ C1

1/2 U

ofoRþW

ofoX� Uf

R

� �

¼ C2Pr/r2f� f

R2

� �� Pr

Daf� PrRa

oToRþ N

oCoR

� �; ð3Þ

f ¼ 1R

o2w

oR2 �1R

owoRþ o2w

oX2

" #; ð4Þ

U ¼ 1R

owoX

; W ¼ �1R

owoR

; ð5Þ

where

r2 ¼ o2

oR2 þ1R

o

oRþ o2

oX2 :

In Eq. (3), the coefficients C1 and C2 can be set equal to 0 or 1 in or-der to obtain the Darcy or Darcy–Brinkman models. In the presentstudy, the values of fluid kinematic viscosity (tf) and effective kine-matic viscosity (te) are assumed to be equal (tf = te = t). The dimen-sionless parameters governing double-diffusive natural convectionin the porous annulus are the thermal Rayleigh number, Ra, theDarcy number, Da, the Lewis number, Le, the Prandtl number, Pr,the buoyancy ratio, N, and the heat capacity ratio, r, defined by:

Ra ¼ gbTDhD3

taT; Da ¼ K

D2 ; Le ¼ aT

aC; Pr ¼ t

a; N ¼ bCDS

bTDh;

r ¼/ðqcpÞf þ ð1� /ÞðqcpÞs

ðqcpÞf;

where ðqcpÞf and ðqcpÞs are respectively the heat capacity of thefluid and the saturated porous medium. In addition to the abovedimensionless parameters, the present study also involves the geo-metrical parameters, such as the radius ratio, k, the aspect ratio, A,


the location of heat and solute source, L, and the length of heat andsolute source, e, which are defined as:

k ¼ ro

ri; A ¼ H

D; L ¼ l

H; and e ¼ h

H:

The initial and boundary conditions in dimensionless form are:

t¼0 : T ¼w¼ C¼ 0; 1k�16R6 k

k�1 ; 06X6A

t>0 : w¼ owoR¼ 0; oT

oR¼ oCoR¼0; R¼ 1

k�1 and 06X< L� e2 ; Lþ e

2<X6A

w¼ owoR¼ 0; oT

oR¼ oCoR¼�1; R¼ 1

k�1 and L� e26X6 Lþ e

2

w¼ owoR¼ 0; T ¼C¼0; R¼ k

k�1 and 06X6A

w¼ owoX¼ 0; oT

oX¼ oCoR¼0; X¼ 0 and X¼A:

The average Nusselt (Nu) and Sherwood (Sh) numbers on thesurface of the heat and solute source at the inner wall of the annu-lus are defined as

Nu ¼ 1e

Z Lþe2

L�e2

NudX and Sh ¼ 1e

Z Lþe2

L�e2

ShdX; ð6Þ

where Nu and Sh in Eq. (6) are respectively the local Nusselt andSherwood numbers, which can be written as

Nu ¼ 1TðR;XÞjR¼ 1

k�1

and Sh ¼ 1CðR;XÞjR¼ 1

k�1

; ð7Þ

where T(R, X) and C(R, X) is the dimensionless temperature and con-centration along the heat and solute source at the inner wall of theannulus.

3. Numerical technique and code validation

The non-dimensional governing equations along with the initialand boundary conditions were discretized using an implicit finitedifference method. The heat, the species concentration and thevorticity transport equations are iterated until steady state usingthe Alternating Direction Implicit (ADI) method, and the streamfunction equation is solved by Successive Line Over Relaxation(SLOR) method. This technique is well described in the literatureand has been widely used for natural convection in rectangularand annular cavities. For brevity, the details of the numerical meth-od are not repeated here, and can be found in our recent works[24,38,39]. A uniform grid is used in the R–X plane of the annulus,and in order to determine a proper grid size for the present study, agrid independence test has been conducted with different gridsizes. The effect of grid resolution has been examined in order toselect the appropriate grid size. Grid size independency is validatedby obtaining the predicted results from a coarse grid of 81�81 torefined one of 201 � 201. The average Nusselt and Sherwood num-bers are used as sensitivity measures of the accuracy of the solu-tion. Based on these tests, a 161 � 161 uniform grid is found tomeet the requirements of both the accuracy of the solution and areasonable computational time. The steady state solution to theproblem has been obtained as an asymptotic limit to the transientsolutions. The numerical method was implemented by developingan in-house FORTRAN program for the present model and it hasbeen successfully validated against the available benchmark solu-tions in the literature before obtaining the simulations.

3.1. Validation

The numerical technique implemented in the present study hasbeen successfully employed in our recent papers to investigate theeffects of a magnetic field on double-diffusive convection [19] andthermocapillary convection [39] in a vertical nonporous annulus,and also to investigate the effects of discrete heating on naturalconvection in a vertical porous and nonporous annulus

[24,38,39]. However, in order to verify the accuracy of the currentnumerical results, simulations of the present model are tested andcompared with different reference solutions available in the litera-ture for thermosolutal convection in the cylindrical annular andrectangular enclosures. First, a comparison is made with double-diffusive convection in a vertical porous annulus for the Darcymodel by setting the coefficients C1 = C2 = 0 in Eq. (3). For this,the flow pattern, temperature and concentration fields are ob-tained for the uniform temperature and concentration at the innerwall, and are compared with the corresponding results of Beji et al.[6]. Fig. 2 exhibits a good agreement between the present stream-lines, isotherms and isoconcentrations and that of Beji et al. [6] inthe porous annulus.

The accuracy of the numerical results are further checked bycomputing the average Nusselt and Sherwood numbers for uni-form temperature and concentration at the inner wall of the por-ous annulus. These quantitative results are compared with theDarcy flow model results of Nithiarasu et al. [8] for Le = 2,N = 1 and k = 5, and are given in Table 1. The comparison withtheir finite element method using non-uniform grids is quitegood. In addition to the above validations, we also compareour results with Goyeau et al. [27] and Bennacer et al. [28] ina rectangular porous cavity (k = 1). In theory, the case of infinitecurvature characterized by k = 1 represents a rectangular cavity.The comparison shown in Table 2 reveals that the detected max-imum difference with the results of Goyeau et al. [27] and Benn-acer et al. [28] is less than 2.3%. From Fig. 2, and Tables 1 and 2,the agreement between the present results and benchmark solu-tions is quite acceptable.

4. Results and discussion

The numerical results are presented in this section with themain objective of investigating the effect of a heat and solutesource (henceforth heat and solute source is referred to as the seg-ment) location and buoyancy ratio on the double-diffusive convec-tive flow, and the corresponding heat and mass transfer rates in avertical porous annulus. Since the present study involve a largenumber of non-dimensional parameters (Ra, Da, Le, Pr, N, e, r, , A,k, L), only the main controlling parameters are varied. In the pres-ent study, the aspect ratio (A) of the annulus, Prandtl number (Pr),heat capacity ratio (r) and porosity of the porous medium () arekept at unity. Also, the size (e) of the segment is fixed at 0.25; how-ever, its location (L) is varied from 0.125 to 0.875. The thermal Ray-leigh number (Ra), Darcy number (Da) and Lewis number (Le) arerespectively varied in the ranges 5� 104

6 Ra 6 107; 10�56 Da 6

100 and 1 6 Le 6 10: Curvature effect can be important for theannular cavity flows, and so the effect of radius ratio (k) on the heatand mass transfer rates is also examined for a wide rangeð1 6 k 6 10Þ, with ri kept constant and ro being varied. The relativeimportance of thermal and solutal buoyancy forces is denoted bythe buoyancy ratio (N), and is defined as the ratio of the solutalbuoyancy force to thermal buoyancy force. This parameter is var-ied through a wide range �10 6 N 6 þ10; covering the concentra-tion-dominated opposing flow (N = -10), pure thermal convection-dominated flow (N = 0), and concentration-dominated aiding flow(N = 10). In the following sections, the flow fields, temperatureand concentration distributions in the porous annulus are illus-trated through streamlines, isotherms and isoconcentrations. Inall contour graphs, the left and right vertical sides correspond tothe inner and outer cylinders, respectively. Further, the effects ofthermal Rayleigh number, Darcy number, Lewis number and radiusratio on the heat and mass transfer rates are evaluated in terms ofthe average Nusselt and Sherwood numbers at different segmentlocations and buoyancy ratios.

Fig. 2. Comparison of streamlines (left), isotherms (middle) and isoconcentrations (right) between the present results and that of Beji et al. [6] for RaT = 500, Le = 10,N = 0 and k = 5.

Table 1Comparison of average Nusselt and Sherwood numbers with Nithiarasu et al. [8] fordouble-diffusive convection in a porous annulus at Le = 2, N = 1, A = 1 and k = 5.

Thermal Darcy–Rayleighnumber (RaT)

Nithiarasuet al. [8]

Presentstudy

Relativedifference (%)

100 8.5 8.45 0.59 Nu14.27 14.22 0.35 Sh

500 21.42 20.95 2.24 Nu34.66 35.42 2.19 Sh

Table 2Comparison of average Nusselt and Sherwood numbers with Goyeau et al. [27] andBennacer et al. [28] for double-diffusive convection in a rectangular porous cavity atLe = 10, N = 0, A = 1 and k = 1.

Thermal Darcy–Rayleigh number (RaT)

100 200 400

Goyeau et al. [27] 3.11 4.96 7.77 Nu13.25 19.86 28.41 Sh

Bennacer et al. [28] 3.11 4.96 7.77 Nu13.24 19.83 29.36 Sh

Present study 3.11 4.91 7.72 Nu13.24 19.92 28.71 Sh


4.1. Effect of buoyancy ratio and segment location

As a first step towards studying the influence of segment loca-tion, the effect of buoyancy ratio on the flow pattern, thermal

and solute concentration distributions is examined for three differ-ent segment locations, namely at the bottom, middle and top por-tions of the inner wall of the porous annulus. Figs. 3–5 illustratesthe streamlines, isotherms and isoconcentrations for three differ-ent combinations of buoyancy ratio and segment location withthe values of Ra, Da, Le and k are respectively fixed at 107, 10�3,10 and 2. The intervals of streamlines, isotherms and isoconcentra-tions are Dn ¼ ðnmax � nminÞ=15; where stands for , T or C. Since theheat and solute source is placed on the inner wall, the direction ofthermal flow is always clockwise, whereas the direction of solutalflow strongly depends on the sign of N or bC. The solutal flow isclockwise for N (or bC) > 0 and counterclockwise for N (or bC) < 0.Fig. 3 provides exemplary results on the flow pattern, temperatureand concentration fields for the opposing buoyancy forces(N = -10). In the case of opposing flow, regardless of the segmentlocation, two or three counter-rotating cells are observed in theannulus. When the segment is located at the bottom portion ofthe inner wall, two weak counter rotating cells with equal magni-tude are observed in the annulus. The streamlines divide the cavityinto a thermal buoyancy-driven cell at the upper zone, and a solutebuoyancy-driven cell in the bottom part of the annulus (Fig. 3). Asthe segment shifts to the middle portion, the strength of the ther-mally driven cell is significantly promoted over the solute buoy-ancy-driven cell. Interestingly, as the segment move upwards,the flow is characterized by a solute-driven counterclockwiserotating cell in the core region and two thermal-driven clockwisecirculations located at the top and the bottom of the annulus.The isotherms are more skewed towards the segment, but on theother hand, the concentration lines in the interior core are ob-served to be diagonal. A careful observation of the streamline pat-tern reveals the distinct location effect of the segment on thethermal and solute flow circulation. It is found that the rate of

Fig. 3. Effect of segment location on the streamlines (top), isotherms (middle), and isoconcentrations (bottom) for opposing flow (N = �10) with Ra = 107, Da = 10�3, Le = 10,k = 2: (a) L = 0.125, (b) L = 0.5, (c) L = 0.875.


thermal flow circulation is higher for the middle location of thesegment, whereas higher solutal flow circulation rate is observedwhen the segment is placed at top of the inner wall.

The results of the heat-driven flow (N = 0) are shown in Fig. 4 forthree different segment locations. For N = 0, the flow is driven bythe thermal buoyancy force, and the effect of concentration doesnot exist. The streamlines and isotherms obtained in the presentstudy are similar to those reported previously for the thermal con-vection in a vertical porous annulus [24]. The flow structure con-sists of one main circulation occupying the entire enclosure andthe solute driven-cell in the cavity has been annihilated. Also, astriking difference can be seen in the direction of isotherms andisoconcentrations between the opposing and heat-driven flows. Ahighly stratified medium with almost parallel and horizontal flowresults in the core region when the segment is placed at the bottomportion of the inner wall (Fig. 4a). As the segment location shiftstowards the top portion, the main vortex reduces in size and movesnear the top portion of the outer wall. Further, the symmetric

structure of the flow pattern, thermal and solutal fields has beendisturbed when the segment location is shifted to the top portionof the inner wall. The relative strength of the flow as indicatedby the maximum absolute stream function reduces as the segmentmoves upwards. That is, the rate of fluid circulation is found to behigher, when the segment is placed at the bottom portion of the in-ner wall. This may be attributed to the distance that the fluid needsto travel in the circulating cell to exchange the heat and solute con-centration between the segment and outer wall. These predictionsare consistent with those reported by Zhao et al. [37] for double-diffusive convection with a heat and solute source mounted onthe right wall of a square porous enclosure. As the segment isplaced at the top, a strong flow exists in the upper zone of theannulus with a main vortex near the outer wall, and a weak flowat the bottom portion of the annular enclosure, which is vividly re-flected in the corresponding isotherms and isoconcentrations(Fig. 4c). This may be expected to be due to the presence of animpermeable and insulated top wall, which results in a severe

Fig. 4. Effect of segment location on the streamlines (top), isotherms (middle), and isoconcentrations (bottom) for heat-driven flow (N = 0) with Ra = 107, Da = 10�3, Le = 10,k = 2: (a) L = 0.125, (b) L = 0.5, (c) L = 0.875.


restriction to the flow emerging from the segment. The formationof hydrodynamic, thermal and solutal boundary layers is visiblearound the segment and on the upper portion of the outer wall.

The influence of segment locations on the mutually augmentingthermal and solutal buoyancies, termed as aiding double-diffusiveflow (N > 0), is illustrated in Fig. 5(a)–(c). With the buoyancy ratioincreased to N = 10, the solutal and thermal buoyancy forces areaugmenting each other, resulting in an accelerated clockwise flowin the annulus. The location of the segment significantly alters theflow pattern as indicated by the streamlines. When the segment ispositioned at the bottom portion of the inner wall, the flow patterninduced by the combined thermal and solutal buoyancy forces con-sists of a single cell with its maximum near the inner wall of theannulus. The cell elongated in the horizontal direction when thesegment is shifted to the middle. For top location of the segment,the flow strength is stronger in the upper part of the annulus,and the main eddy moves towards the outer wall. In the bottomportion of the annulus, the flow is weak and a secondary eddy isgenerated. It is worth mentioning that similar flow structures,

but for a square porous cavity (k = 1) have been reported by Zhaoet al. [37]. The isoconcentrations, at all three locations, reveal astrengthened stratification and stronger horizontal intrusion layersin the annulus than their thermal counterparts. Although the ther-mal and solutal buoyancy effects augment each other, a carefulobservation of the streamlines reveals that the magnitude of max-imum stream function is lower for N = 10 compared to N = 0. Thismay be attributed to the stabilizing or blocking effect of the verti-cal stratification of the combined density field in the core of theannulus. However, compared to the uniform heating and saltingconditions at the inner wall of the annulus (Bennacer et al. [10]),the blocking effect in the present study is reduced to a great extentdue to the discrete heating and salting of the inner wall. The tem-perature and concentration fields reveal a symmetric structurewhen the segment is at the bottom. However, as the segmentmoves upwards, the symmetric structure of the thermal and solu-tal fields is completely lost. The thermal and solutal boundary layeralong the heater and outer wall reveals that solutal boundary layeris thinner than the thermal boundary layer due to lower mass

Fig. 5. Effect of segment location on the streamlines (top), isotherms (middle), and isoconcentrations (bottom) for aiding flow (N = 10) with Ra = 107, Da = 10�3, Le = 10, k = 2:(a) L = 0.125 (b) L = 0.5, (c) L = 0.875.


species diffusivity (Le = 10). Further, the solutal field is well strati-fied, while the thermal field in the interior core is observed to bediagonal.

The variation of average Nusselt and Sherwood numbers underthe combined effects of segment location and buoyancy ratio canbe examined in Fig. 6. The buoyancy ratio (N) and segment location(L) are varied in the range of �10 6 N 6 þ10 and 0:125 6 L 60:875; while the parameters Le, Ra, Da and k are respectively fixedat 5, 107, 10�5 and 2. An overview of the figure reveals that in theopposing flow region (N < 0), the heat and mass transfer rates arelower compared to the corresponding N in the aiding flow region(N > 0). A similar observation was predicted in earlier studies in auniformly heated vertical porous and nonporous annulus[9,10,13,14]. This can be expected due to the lower flow rate nearthe annular walls for the opposing flow than that of the aidingflow. Although the value of N decreases in the opposing flow re-gion, the heat and mass transfer rates are constantly increasing.This may be attributed to the fact that the heat and mass transferrates are enhanced due to the magnitude of velocity and not

because of the direction of flow. The figure clearly shows that theheat and mass transfer rates are strongly depends on the segmentlocation. Moreover, the segment location for maximum heat andmass transfer rates in the opposing flow region is not same forthe aiding flow region. For any value of N in the opposing flow re-gion, the average Nusselt (Nu) and Sherwood (Sh) numbers in-creases as the value of L increases from 0.125 to 0.625, butdecreases for L > 0.625. Similarly, in the aiding flow region, Nuand Sh increases as the value of L decreases from 0.875 to 0.375,but decrease when L < 0.375. This result reveals the complex rela-tionship between the segment location and the rates of heat andmass transfer.

From Fig. 6, it can be seen that for aiding flows (N > 0) the heatand mass transfer rates are highest for segment location L = 0.625,while highest heat and mass transfer rates for the opposing flows(N < 0) are achieved by placing the segment at L = 0.375. Irrespec-tive of the segment location, as |N| decreases in the opposing flowregion, the heat and mass transfer rates decreases to the point offlow reversal. The minimum value of the average Nusselt number

Fig. 6. Influence of buoyancy ratio on the average Nusselt (Nu) and Sherwood (Sh)numbers for different segment locations with Le = 5, Ra = 107, Da = 10�5 and k = 2.

Fig. 7. Influence of buoyancy ratio on the average Nusselt (Nu) and Sherwood (Sh)numbers for different Darcy numbers with Le = 5, Ra = 107, L = 0.5 and k = 2.


is noticeably different when the segment is placed either near tothe bottom (L = 0.125 or 0.25) or to the top (L = 0.75 or 0.875) por-tion of the inner wall, but the minimum value for other locations ofthe segment is almost the same. However, the average Sherwoodnumber has different minimum values only for segment locationsnear the bottom portion (L = 0.125 or 0.25) of the inner wall, andfor the remaining segment locations the minimum values are al-most the same. The influence of segment location on the flow pat-tern reveals a higher flow circulation when the segment is placedat the bottom portion of the inner wall. However, the variation ofaverage Nusselt and Sherwood numbers with segment locationsexhibits that the heat and mass transfer rates are higher, whenthe segment is placed around middle portion (L = 0.375) ratherthan placing it near the bottom or top portion of the inner wall.This reveal an important fact that the optimal location for maxi-mum heat and mass transfer not only depends on the circulationintensity, but also depends on the shape of the thermal and solutalbuoyancy-driven flow. This observation is in agreement with therecent works of by Zhao et al. [37] for double-diffusive convectionin a square porous cavity.

4.2. Effect of Darcy number

In this section, we discuss the effects of Darcy number togetherwith buoyancy ratio and segment location on the heat and mass

transfer rates. First, the combined influence buoyancy ratio andDarcy number on the average Nusselt and Sherwood numbers isillustrated in Fig. 7 for fixed values of Le = 5, Ra = 107, L = 0.5 andk = 2. The non-dimensional parameter N characterizes the ratio be-tween solutal and thermal buoyancy forces is varied in the range�10 6 N 6 þ10: For all Darcy numbers, the average Nusselt andSherwood numbers have a tendency to reach a minimum valueof N, Nmin, in the transitional range, at which the flow reversal fromtransitional to thermal dominated flow occurs. However, this min-imum value, Nmin, strongly depends on the Darcy number. The fig-ure vividly reveals that the value of Nmin decreases as thepermeability of porous matrix increases. Further, the presence ofthe porous medium strongly influences the heat and mass transferrate, particularly in the transitional range. For aiding flows (N > 0),as N increases, the solutal buoyancy force increases relative to thethermal buoyancy force. As a result, the net upward buoyancyforce near the inner wall increases, and so does the intensity ofthe circulation. Hence, the average Nusselt and Sherwood numbersincrease steadily with the magnitude of the buoyancy ratio N.However, for opposing case (N < 0), the variation of average Nusseltand Sherwood numbers are not monotonic, particularly at largeDarcy numbers. As |N| decreases, initially Nu and Sh decrease untilthey reach Nmin, and then they increase with N. This is in agree-ment with the uniform heating and salting results in a vertical por-ous annulus [9,10]. Also, it should be pointed out that the main

Fig. 8. Influence of Darcy number on the average Nusselt (Nu) and Sherwood (Sh)numbers for different segment locations with Le = 10, Ra = 107, N = 8 and k = 2.

Fig. 9. Influence of radius ratio on the average Nusselt (Nu) and Sherwood (Sh)numbers for different segment locations with Le = 10, Ra = 107, N = 8 and L = 0.5.


contribution of the presence of a porous medium is to substantiallyreduce the rate of heat transfer compared to mass transfer rate.

Next, the combined effects of Darcy number and segment loca-tion on the average Nusselt and Sherwood numbers is reported inFig. 8. The values of Lewis number, thermal Rayleigh number,buoyancy ratio and radius ratio are respectively fixed at Le = 10,Ra = 107, N = 8 and k = 2, while the Darcy number and segmentlocation are varied. At low Darcy numbers, the porous medium ex-erts a resistance to the double-diffusive convective flow, and hencethe rate of heat and mass transfer is less compared to higher valuesof Da. The slopes of the average Nusselt and Sherwood numbercurves decrease with an increase in the value of Da, and finally ap-proach to zero for all segment locations. This indicate that thereexists an asymptotic double-diffusive convective regime wherethe heat and mass transfer rates are independent of the Darcynumber, but depends strongly on the segment location. This trendhas also been demonstrated in the numerical results of Zhao et al.[37] for double-diffusive convection in a square porous enclosure.On comparing the variations of average Nusselt and Sherwoodnumbers with the Darcy number, it is apparent that the permeabil-ity of the porous medium significantly affects the heat transfer ratemore than the rate of mass transfer. The combined influence ofDarcy number and segment location on the average Nusselt num-ber reveals that the permeability of the porous medium has a

distinct effect on the heat transfer rate at different segment loca-tions. At lower values of Darcy number (Da < 10�3), the variationof the average Sherwood number with segment locations is notice-ably small compared to that of the heat transfer whereas, at higherDarcy numbers, a larger variation is observed with segment loca-tions. Further, at all Darcy numbers, the heat and mass transferrates are lower when the segment is placed at the top of the innerwall. A higher heat transfer rate is observed for the segment loca-tion L = 0.125 at all values of Da, while the higher mass transferrate for the segment location strongly depend on the Darcynumber.

4.3. Effect of radius ratio

In the study of natural convection heat and mass transfer in avertical annulus, knowledge of radius ratio effect (curvature) onthe heat and mass transfer rates is important in designing manyengineering applications. Fig. 9 exemplify the effects of radius ra-tio on the average Nusselt and Sherwood numbers for differentvalues of Da and for fixed values of Le, Ra, N and L. The resultsobtained for the case of a rectangular cavity (k = 1), are similarto those reported in the literature for double-diffusive convectionin a rectangular porous cavity (Zhao et al. [37]). In general, theheat and mass transfer rates increases as the Darcy numberand radius ratio increases. Also, when the radius ratio (k)


increases, the annular gap between the inner and outer cylindersincreases, and the surface area of the outer cylinder becomesincreasingly larger than that of the inner cylinder. Therefore,for a given Darcy number, an increase in k above unity producesa thinner thermal boundary layer around the heat and solutesource on the inner wall and a thicker thermal boundary layeron the outer wall. This results in an increase in the average Nus-selt number as the radius ratio increases. However, the radius ra-tio affects the mass transfer rate for k 6 5, whereas for k > 5 thevariation of average Sherwood number is minimum at smallDarcy numbers (Da < 10�3). Due to the lower diffusivity of soluteconcentration (Le = 10), the solute boundary layer is thinner thanthe thermal boundary layer. As a result, the concentration re-mains stagnant in the core of the annulus, and thus an increasein the radius ratio beyond k > 5 does not produce significantchanges in the average Sherwood number.

4.4. Effect of Lewis number and buoyancy ratio

Fig. 10 depicts the combined effects of Lewis number and buoy-ancy ratio on the heat and mass transfer rates for fixed values ofDarcy number, thermal Rayleigh number, segment location and ra-dius ratio respectively at Da = 10�5, Ra = 107, L = 0.5 and k = 2. The

Fig. 10. Influence of buoyancy ratio on the average Nusselt (Nu) and Sherwood (Sh)numbers for different Lewis numbers with Da = 10�5, Ra = 107, k = 2 and L = 0.5.

Lewis number, which measures the relative importance of thermalto mass diffusion, has a stronger influence on the heat and masstransfer rates. For unit Lewis number, the diffusion of heat and sol-ute concentration in the mixture is in equal proportions, and hencethe average Nusselt and Sherwood number curves for Le = 1 areidentical (Fig. 10). For any buoyancy ratio, an increase in the Lewisnumber tends to decrease the heat transfer rate, except in the ther-mally dominated opposing flow region between N = 0 and thepoint of flow reversal (Nmin). For a fixed value of Le, an increasein the magnitude of N enhances the flow strength, which in turnincreases the heat transfer rate. However, the mass transfer rateconstantly increases with the Lewis number for all ranges of buoy-ancy ratio. This may be attributed to the fact that, for Le > 1, the dif-fusivity of concentration decreases, while the thermal diffusivityincreases. As a result, mass transfer occurs by convection, whereasheat is transferred by diffusion. The combined influence of Lewisnumber and buoyancy ratio on the average Nusselt number revealsan important observation that the rate of heat transfer is the same(Nu ¼ 4:8312) at all values of Le when N = 0. This observation hasbeen previously confirmed by Shipp et al. [14] while investigatingdouble-diffusive convection in a vertical nonporous annulus. Final-ly, the critical buoyancy ratio, (Nmin), strongly depends on the Le-wis number. The flow reversal from transitional to thermaldominated flow (Fig. 10a) and the onset of transitional flow

Fig. 11. Influence of buoyancy ratio on the average Nusselt (Nu) and Sherwood (Sh)numbers for different values of Ra with Le = 10, Da = 10�3, k = 2 and L = 0.5.


(Fig. 10b) has also been shifted to lower buoyancy ratios as the Le-wis number is increased beyond unity.

4.5. Effect of thermal Rayleigh number

The heat and mass transfer rates for different buoyancy ratiosand thermal Rayleigh numbers are important quantitative mea-sures of the problem. These are investigated in Fig. 11, where theLewis number Le = 10, the Darcy number Da = 10�3, radius ratiok = 2 and segment location L = 0.5. To analyze the influence of thetransition zone, the buoyancy ratio was taken in the range -106N610, and the thermal Rayleigh number was considered inthe range 5�104

6Ra6107. For lower values of Ra (Ra6105), theheat and mass transfer rates are modest at all ranges of buoyancyratio, and the onset of transition flow occurs at the same buoyancyratio. However, the heat and mass transfer rates are significantlyenhanced with an increase in Ra beyond 105, since the thermalRayleigh number stimulates the convective motion driven by thecombined buoyancies. An increase in the thermal Rayleigh number(Ra > 105) shifts the onset of flow transition to a lower buoyancyratio. The average Sherwood number significantly increases withthe thermal Rayleigh number, although the porous medium actsas an barrier to the mass transfer. However, as mentioned before,the effect of increasing Ra enhances the thermally induced flowand hence the heat transfer rate, as can be seen from the variationof average Nusselt number. This thermally induced flow within theporous matrix is the main source of mass transfer in the annulus.

5. Conclusions

In this paper, we have numerically investigated the double-dif-fusive convection in a vertical porous annulus with a discretelyheated and salted segment at the inner wall. The effects of themain controlling parameters, such as segment location, buoyancyratio, Darcy number, thermal Rayleigh number, Lewis numberand radius ratio were investigated in detail to gain new insightsinto the flow patterns, thermal and solutal fields, and the rates ofheat and mass transfer. Many of the observations of the presentstudy are in good agreement with the similar studies in the litera-ture. The main findings of the present investigation are summa-rized as follows:

1. The flow structure, thermal and solutal fields, rates of heat andmass transfer are profoundly affected by the relative magni-tudes of buoyancy ratio and segment location.

2. The critical buoyancy ratio at which the transitional flow occursstrongly depends on the segment location, Lewis, Darcy andthermal Rayleigh numbers.

3. The segment location of higher flow circulation does not producehigher heat and mass transfer rates. The heat and mass transferrates can be effectively controlled by the segment location.

4. For concentration-dominated opposing case (N = -10), the flowin the annulus is characterized by two separate flow circula-tions in opposite direction, while for heat-driven (N = 0) andconcentration-dominated aiding (N = 10) flows, a strong unicel-lular flow structure is observed in the annulus.

5. The Darcy number significantly affects the heat transfer ratemore than the rate of mass transfer. Furthermore, the influenceof porous medium on the heat and mass transfer rates stronglydepends on the segment location.

6. For all Lewis numbers, the average Nusselt number attains thesame value for the heat driven flow (N = 0). An increase in Lewisnumber will increase the mass transfer rate, while it decreasesthe heat transfer rate, except for the regions between N = 0 andNmin, where the heat transfer rate increases.

7. The segment location influences the heat and mass transferrates in a different fashion with respect to buoyancy ratio andDarcy number.

8. An increase in Ra beyond 105 tends to increase the mass trans-fer rate noticeably through the thermally induced flow circula-tions in the porous annulus.

Acknowledgements

This work was supported by WCU (World Class University) pro-gram through the Korea Science and Engineering Foundationfunded by the Ministry of Education, Science and Technology(Grant No. R32-2009-000-20021-0).

References

[1] B. Gebhart, L. Pera, The nature of vertical natural convection flows resultingfrom the combined buoyancy effect of thermal and mass diffusion, Int. J. HeatMass Transfer 14 (1971) 2025–2050.

[2] J.S. Turner, Double diffusive phenomena, Annu. Rev. Fluid Mech. 6 (1974) 37–56.

[3] D.B. Ingham, I. Pop (Eds.), Transport Phenomena in Porous Media, Elsevier,Oxford, 2005.

[4] D.A. Nield, A. Bejan, Convection in Porous Media, third ed., Springer, New York,2006.

[5] H.J. Sung, W.K. Cho, J.A. Hyun, Double-diffusive convection in a rotatingannulus with horizontal temperature and vertical solutal gradients, Int. J. HeatMass Transfer 36 (1993) 3773–3782.

[6] H. Beji, R. Bennacer, R. Duval, P. Vasseur, Double diffusive natural convection ina vertical porous annulus, Numer. Heat Transfer, Part A: Appl. 36 (1999) 153–170.

[7] M. Marcoux, M.C. Charrier-Mojtabi, M. Azaiez, Double-diffusive convection inan annular vertical porous layer, Int. J. Heat Mass Transfer 42 (1999) 2313–2325.

[8] P. Nithiarasu, K.N. Seetharamu, T. Sundararajan, Non-Darcy double-diffusivenatural convection in axisymmetric fluid saturated porous cavities, Heat MassTransfer 32 (1997) 427–433.

[9] M. Benzeghiba, S. Chikh, A. Campo, Thermosolutal convection in a partlyporous vertical annular cavity, ASME J. Heat Transfer 125 (2003) 703–715.

[10] R. Bennacer, H. Beji, R. Duval, P. Vasseur, The Brinkman model forthermosolutal convection in a vertical annular porous layer, Int. Commun.Heat Mass Transfer 27 (1) (2000) 69–80.

[11] R. Bennacer, A.A. Mohamad, M. El Ganaoui, Thermodiffusion in porous media:multi-domain constitutant separation, Int. J. Heat Mass Transfer 52 (2009)1725–1733.

[12] A. Bahloul, M.A. Yahiaoui, P. Vasseur, R. Bennacer, H. Beji, Natural convectionof a two-component fluid in porous media bounded by tall concentric verticalcylinders, J. Appl. Mech. 73 (2006) 26–33.

[13] P.W. Shipp, M. Shoukri, M.B. Carver, Double diffusive natural convection in aclosed annulus, Numer. Heat Transfer, Part A: Appl. 24 (1993) 339–356.

[14] P.W. Shipp, M. Shoukri, M.B. Carver, Effect of thermal Rayleigh and Lewisnumbers on double diffusive natural convection in closed annulus, Numer.Heat Transfer, Part A: Appl. 24 (1993) 451–465.

[15] J. Lee, S.H. Kang, Y.S. Son, Numerical study of multilayered flow regime indouble-diffusive convection in a rotating annulus with lateral heating, Numer.Heat Transfer, Part A: Appl. 38 (2000) 467–489.

[16] W.M. Yan, D. Lin, Natural convection heat and mass transfer in vertical annuliwith film evaporation, Int. J. Heat Mass Transfer 44 (2001) 1143–1151.

[17] N. Retiel, E. Bouguerra, M. Aichouni, Effect of curvature ratio on cooperatingdouble-diffusive convection in vertical annular cavities, J. Appl. Sci. 6 (2006)2541–2548.

[18] S. Chen, J. Tolke, M. Krafczyk, Numerical investigation of double-diffusive(natural) convection in vertical annuluses with opposing temperature andconcentration gradients, Int. J. Heat Fluid Flow 31 (2010) 217–226.

[19] M. Venkatachalappa, Y. Do, M. Sankar, Effect of magnetic field on the heat andmass transfer in a vertical annulus, Int. J. Eng. Sci. 49 (2011) 262–278.

[20] V. Prasad, F.A. Kulacki, Natural convection in a vertical porous annulus, Int. J.Heat Mass Transfer 27 (1984) 207–219.

[21] V. Prasad, Numerical study of natural convection in a vertical, porous annuluswith constant heat flux on the inner wall, Int. J. Heat Mass Transfer 29 (1986)841–853.

[22] M. Hasnaoui, P. Vasseur, E. Bilgen, L. Robillard, Analytical and numerical studyof natural convection heat transfer in a vertical porous annulus, Chem. Eng.Commun. 131 (1995) 141–159.

[23] I.S. Shivakumara, B.M.R. Prasanna, N. Rudraiah, M. Venkatachalappa,Numerical study of natural convection in a vertical cylindrical annulus usinga non-Darcy equation, J. Porous Med. 5 (2) (2003) 87–102.

[24] M. Sankar, Y. Park, J.M. Lopez, Y. Do, Numerical study of natural convection in avertical porous annulus with discrete heating, Int. J. Heat Mass Transfer 54(2011) 1493–1505.


[25] O.V. Trevisan, A. Bejan, Mass and heat transfer by natural convection in avertical slot filled with porous medium, Int. J. Heat Mass Transfer 29 (1986)403–415.

[26] F. Alavyoon, On natural convection in vertical porous enclosures due toprescribed fluxes of heat and mass at the vertical boundaries, Int. J. Heat MassTransfer 36 (1993) 2479–2498.

[27] B. Goyeau, J.P. Songbe, D. Gobin, Numerical study of double-diffusive naturalconvection in porous cavity using the Darcy–Brinkman formulation, Int. J. HeatMass Transfer 39 (1996) 1363–1378.

[28] R. Bennacer, A. Tobbal, H. Beji, P. Vasseur, Double diffusive convection in avertical enclosure filled with anisotropic porous media, Int. J. Therm. Sci. 40(2001) 30–41.

[29] A.J. Chamkha, Double-diffusive convection in a porous enclosure withcooperating temperature and concentration gradients and heat generation orabsorption effects, Numer. Heat Transfer, Part A: Appl. 41 (2002) 65–87.

[30] A. Khadiri, A. Amahmid, M. Hasnaoui, A. Rtibi, Soret effect on double-diffusiveconvection in a square porous cavity heated and salted from below, Numeri.Heat Transfer, Part A: Appl. 57 (2010) 848–868.

[31] O. Rahli, R. Bennacer, K. Bouhadef, D.E. Ameziani, Three-dimensional mixedconvection heat and mass transfer in a rectangular duct: case of longitudinalrolls, Numer. Heat Transfer, Part A: Appl. 59 (2011) 349–371.

[32] N.H. Saeid, I. Pop, Natural convection from discrete heater in a square cavityfilled with a porous medium, J. Porous Med. 8 (1) (2005) 55–63.

[33] N.H. Saeid, I. Pop, Mixed convection from two thermal sources in a verticalporous layer, Int. J. Heat Mass Transfer 48 (2005) 4150–4160.

[34] R.S. Kaluri, T. Basak, S. Roy, Bejan’s heatlines and numerical visualization ofheat flow and thermal mixing in various differentially heated porous squarecavities, Numer. Heat Transfer, Part A: Appl. 55 (5) (2009) 487–516.

[35] F.Y. Zhao, D. Liu, G.F. Tang, Free convection from one thermal and solute sourcein a confined porous medium, Transport Porous Med. 70 (2007) 407–425.

[36] D. Liu, F.Y. Zhao, G.F. Tang, Thermosolutal convection in saturated porousenclosure with concentrated energy and solute sources, Energy Convers.Manage. 49 (2008) 16–31.

[37] F.Y. Zhao, D. Liu, G.F. Tang, Natural convection in a porous enclosure withpartial heating and salting element, Int. J. Therm. Sci. 47 (2008) 569–583.

[38] M. Sankar, M. Venkatachalappa, Y. Do, Effect of magnetic field on the buoyancyand thermocapillary driven convection of an electrically conducting fluid in anannular enclosure, Int. J. Heat Fluid Flow 32 (2011) 402–412.

[39] M. Sankar, J. Park, Y. Do, Natural convection in a vertical annuli with discreteheat sources, Numer. Heat Transfer, Part A: Appl. 59 (2011) 594–615.

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