International Journal of Heat and Mass Transfer

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1 International Journal of Heat and Mass Transfer 55 (2012) Contents lists available at SciVerse ScienceDirect International Journal of Heat and Mass Transfer journal homepage: Thermosolutal convection from a discrete heat and solute source in 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 , India b Department of Mathematics, Kyungpook National University, 1370 Sangyeok-Dong, Buk-Gu, Daegu , Republic of Korea c School of Mathematical and Statistical Sciences, Arizona State University, Tempe, AZ 85287, USA article info abstract Article history: Received 7 June 2011 Received in revised form 3 March 2012 Accepted 3 March 2012 Available online 18 April 2012 Keywords: Double-diffusive convection Porous annulus Heat and solute source Radius ratio Double-diffusive convection in a vertical annulus filled with a fluid-saturated porous medium is numerically investigated with the aim to understand the effects of a discrete source of heat and solute on the fluid flow and heat and mass transfer rates. The porous annulus is subject to heat and mass fluxes from a portion 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 porous annulus. The influence of the main controlling parameters, such as thermal Rayleigh number, Darcy number, Lewis number, buoyancy ratio and radius ratio are investigated on the flow patterns, and heat and mass transfer rates for different locations of the heat and solute source. The numerical results show that the flow structure and the rates of heat and mass transfer strongly depend on the location of the heat and solute source. Further, the buoyancy ratio at which flow transition and flow reversal occur is significantly influenced by the thermal Rayleigh number, Darcy number, Lewis number and the segment location. The average Nusselt and Sherwood numbers increase with an increase in radius ratio, Darcy and thermal Rayleigh numbers. It is found that the location for stronger flow circulation is not associated with higher heat and mass transfer rates in the porous annular cavity. Ó 2012 Elsevier Ltd. All rights reserved. 1. Introduction Double-diffusive convective flow caused by the combined influence of thermal and solutal buoyancy forces through porous medium has numerous industrial, natural and geophysical applications. Some of the prominent applications are petrochemical processes, the food industry, grain storage installations, fuel cells, crystal growth applied to semiconductors, solar ponds, migration of moisture through air contained in fibrous insulation, contamination transport in saturated soil, and the underground disposal of nuclear wastes. A comprehensive overview of double-diffusive convection in saturated porous media, its relevance in the understanding of many natural systems and its wide variety of engineering applications are well documented in the literature [1 5]. The combined effects of thermal and solutal buoyancy forces lead to complex flow structures in a vertical porous annulus, and the understanding of their interaction with heat and mass transport is relevant to the above mentioned applications. Using the Darcy model, Beji et al. [6] numerically investigated double-diffusive convection in a vertical porous annulus, with a uniform Corresponding author. Tel.: ; fax: addresses: manisankarir@yahoo.com (M. Sankar), kbs84@knu.ac.kr (B. Kim), lopez@math.asu.edu (J.M. Lopez), yhdo@knu.ac.kr (Y. Do). temperature and concentration difference applied across the vertical walls. They found that the buoyancy ratio at which flow transition and flow reversal occur depends strongly on the physical and geometrical parameters. A combined numerical and analytical study of double-diffusive convection in a fluid saturated porous annulus subjected to uniform heat and mass fluxes from the side walls was reported by Marcoux et al. [7]. For large aspect ratios, their analytical results are in good agreement with numerical solutions, and they found that the flow, thermal and solutal fields are significantly influenced by curvature effects. In modeling the flow in porous media, Darcy s law is one of the most popular models. However, it is generally recognized that Darcy s model may over predict the convective flows for large values of Darcy number. Using a finite-element method, Nithiarasu et al. [8] proposed a generalized model to study both Darcy and non-darcy flow regimes of double-diffusive convection in a vertical porous annulus. They found that the generalized model predicts lower heat and mass transfer rates compared to other porous medium models, such as Darcy, Brinkman and Forchheimer models. Benzeghiba et al. [9] applied the Brinkman-Forchheimer Darcy model to analyze the thermosolutal convection in a partly filled porous annulus. Recently, the Darcy Brinkman formulation has been used to study the influence of the Darcy number on the double-diffusive natural convection in a vertical porous annulus [10,11]. Bahloul et al. [12] /$ - see front matter Ó 2012 Elsevier Ltd. All rights reserved.

2 M. Sankar et al. / International Journal of Heat and Mass Transfer 55 (2012) Nomenclature A aspect ratio C dimensionless concentration c p specific heat at constant pressure D width of the annulus (m) Da Darcy number g acceleration due to gravity (m/s 2 ) H height of the annulus (m) h dimensional length of heat and solute source (m) K permeability of the porous medium (m 2 ) k thermal conductivity (W/(m K)) l distance between the bottom wall and centre of the source (m) L dimensionless location of the source Le Lewis number N buoyancy ratio Nu average Nusselt number Sh average Sherwood number p fluid pressure (Pa) Pr Prandtl number q h heat flux (W/m 2 ) j h mass flux (kg/m 2 s) Ra thermal Rayleigh number Ra T thermal Darcy Rayleigh number Ra T ¼ gkb T q h D 2 tkat S dimensional concentration T dimensionless temperature t dimensional time (s) t dimensionless time (r i, r o ) 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 letters a T thermal diffusivity (m 2 /s) a C mass diffusivity of the solute in the fluid (m 2 /s) b T thermal expansion coefficient (1/K) b C solutal expansion coefficient (1/K) r heat capacity ratio e dimensionless length of the source f dimensionless vorticity h dimensional temperature (K) k radii ratio [ kinematic viscosity (m 2 /s) q fluid density (kg/m 3 ) / porosity of the porous medium w dimensionless stream function investigated analytically and numerically the behaviour of a binary mixture saturating a vertical annular porous medium with Soretinduced convection. Approximate expressions for Nusselt and Sherwood numbers are obtained for the heat-driven and solutedriven flow regimes. Double-diffusive convective flows in a differentially heated vertical annulus have been intensively studied in relation to applications such as oxidation of surface materials, cleaning and dying operations, fluid storage components and energy storage in solar ponds. Shipp et al. [13,14] performed a detailed analysis on double-diffusive convection in a vertical annulus. The effects of thermal Rayleigh number, Lewis number and buoyancy ratio are investigated on flow transition and flow reversal for fixed values of radius ratio, aspect ratio and Prandtl number. Lee et al. [15] studied numerically double-diffusive convection of a salt-water solution in a uniformly rotating annulus with particular attention paid to a multilayered flow regime. They found that the azimuthal velocity induced by the rotation of the system suppressed the generation of rolls at the hot wall, and the merging of adjacent layers. The effects of film evaporation and condensation on the heat and mass transfer rates were examined by Yan and Lin [16] in an open-ended vertical annular duct. Retiel et al. [17] investigated the influence of curvature on double-diffusive convection in a vertical annulus subjected to cooperating gradients of temperature and solute concentration at the vertical walls. Notable among the recent studies on double-diffusive convection in a vertical annulus are due to Chen et al. [18] and Venkatachalappa et al. [19]. Natural convection flow in a vertical porous annulus, due to thermal buoyancy alone, has been widely studied and well-documented in the literature, owing to its importance in building insulation, porous heat exchangers and many others applications. Prasad and co-workers [20,21] numerically investigated natural convection in a vertical porous annulus with constant temperature and constant heat flux conditions at the inner wall for a wide range of parameters. A combined analytical and numerical study of natural convection in a vertical annular porous layer with the inner wall maintained at a constant heat flux and insulated outer wall has been carried out by Hasnaoui et al. [22]. Using Darcy Brinkman model, Shivakumara et al. [23] numerically investigated natural convection in a vertical porous annulus. More recently, Sankar et al. [24] reported on the effects of size and location of a discrete heater on the natural convective heat transfer in a vertical porous annulus. Natural convection resulting from thermal and solutal buoyancy forces in rectangular porous enclosures have also been the subject of a vast number of investigations. Trevisan and Bejan [25] investigated in detail the flow characteristics, heat and mass transfer rates in a rectangular enclosure subjected to uniform heat and mass fluxes from the vertical walls. They obtained the analytical solution in the boundary layer regime, and numerical solutions valid for the entire flow regime. Later, Alavyoon [26] extended the work of Trevisan and Bejan [25] for a much wider range of parameters. For thermosolutal flows in a square porous cavity, Goyeau et al. [27] found that the influence of Darcy number on the heat transfer is more complex than in thermal convection, and the thermosolutal flow behavior in porous media is much different from that of pure convection. The influence of other factors such as anisotropy [28], heat generation or absorption [29], and Soret effect [30] on the double-diffusive flow, heat and mass transfer rates in rectangular porous enclosures have also been investigated. More recently, Rahli et al. [31] numerically investigated three-dimensional double-diffusive mixed convection in a horizontal rectangular duct. In numerous applications involving finite porous enclosures, the heating is only imposed over a portion of the wall rather than over the entire wall. Natural convection in square and rectangular porous enclosures subject to discrete heating has drawn much attention in recent years. Natural convection in a porous square cavity with an isoflux and isothermal discrete heater has been numerically studied by Saeid and Pop [32] using the Darcy model.

3 4118 M. Sankar et al. / International Journal of Heat and Mass Transfer 55 (2012) They found that maximal heat transfer can be achieved when the heater is placed near the bottom of one of the vertical walls. Later, Saeid and Pop [33] numerically studied mixed convection induced by two isothermal heat sources on a vertical plate channel filled with a porous medium. Recently, using Bejan s heatlines method, Kaluri et al. [34] analyzed the optimal heat transfer by considering three different heating conditions in a square porous cavity. The existing studies on double-diffusive convection subject to a discrete heat and solute source have been restricted only to square enclosures [35 37], in which the effects of curvature are missing. Also, the available studies related to double-diffusive convection in a vertical porous annulus are restricted only to uniform heating and salting of inner wall by either uniform heat/mass fluxes or constant temperature or concentration [6 11]. In spite of the important applications of the annular cavity, the influence of discrete heat and solute source on double-diffusive convection in a vertical porous annulus remains poorly understood in the literature, which motivates the present investigation. The purpose of the present paper is to make up for the lack of understanding on double-diffusive convection in a vertical porous annulus filled with a two-component mixture, and subjected to partial heating and salting at the inner wall. Here, we numerically examine the effects of a discrete heat and solute source on the double-diffusive natural convection in a porous annular cavity, and in particular explore the effects of source location and curvature effects. The mathematical formulation and methods of solution are respectively presented in Sections 2 and 3. Section 4 is devoted to the results of the numerical simulations, and the conclusions are given in Section Mathematical formulation Consider a vertical annulus filled with a homogeneous isotropic porous medium, closed at the top and bottom ends by two insulated disks, which are impermeable to mass transfer. The height of the annulus is denoted by H, and the inner and outer radius by r i and r o respectively, as shown in Fig. 1. A heat and solute source of length h (=H/4), placed on the inner wall, is subjected to constant heat and mass fluxes of strengths q h and j h, while the rest of the inner wall is insulated and impermeable. The outer wall is maintained at a lower temperature h 0 and lower concentration S 0. The distance between the centre of the heat and solute source and the bottom wall is l. Also, the fluid is assumed to be Newtonian with negligible viscous dissipation, and the flow is assumed to be axisymmetric. Gravity acts in the negative x-direction. The porous matrix is assumed to be rigid, and in local thermodynamic equilibrium with the fluid. In the porous medium, the Darcy Brinkman formulation is assumed to hold, and hence the Forchheimer quadratic 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 temperature gradient (Soret effect) is neglected. The following non-dimensionless variables are used: ðr; XÞ ¼ðr; xþ=d; ðu; WÞ ¼ðu; wþd=a T ; t ¼ t a T =D 2 ; T ¼ðh h 0 Þ=Dh; C ¼ðS S 0 Þ=DS; P ¼ pd 2 =q 0 a 2 ; f ¼ f D 2 =a T ; w ¼ w =Da T ; where D ¼ r o r i ; Dh ¼ q h D=k; DS ¼ j h D=a C : Further, we assume that all thermophysical properties are constant, except for the effect of density variations in the buoyancy term. According to the Boussinesq approximation, the density of the mixture is related to the temperature and solute concentration through the following linear equation of state: qðh; SÞ¼q 0 ½1 b T ðh h 0 Þ b C ðs S 0 ÞŠ; where b T and b C are respectively the coefficients for thermal and concentration expansions. Based on the above assumptions, the dimensionless governing equations for the conservation of mass, momentum, heat and solute concentrationinaporousannulusarewritteninthevorticity-stream function formulation as [8,11,13,14] r ot ot þ U ot or þ W ot ox ¼ r2 T; / oc ot þ U oc or þ W oc ox ¼ 1 Le r2 C; 1 of / ot þ C 1 1 / U of 2 or þ W of ox Pr ¼ C 2 / r2 f f R 2 f ¼ 1 R U ¼ 1 R where Uf R ð1þ ð2þ Pr ot f PrRa Da or þ N oc ; ð3þ or " # o 2 w or 1 ow 2 R or þ o2 w ; ð4þ ox 2 ow ox ; W ¼ 1 R ow or ; ð5þ r 2 ¼ o2 or þ 1 o 2 R or þ o2 ox : 2 In Eq. (3), the coefficients C 1 and C 2 can be set equal to 0 or 1 in order to obtain the Darcy or Darcy Brinkman models. In the present study, the values of fluid kinematic viscosity (t f ) and effective kinematic viscosity (t e ) are assumed to be equal (t f = t e = t). The dimensionless parameters governing double-diffusive natural convection in the porous annulus are the thermal Rayleigh number, Ra, the Darcy number, Da, the Lewis number, Le, the Prandtl number, Pr, the buoyancy ratio, N, and the heat capacity ratio, r, defined by: Ra ¼ gb TDhD 3 ; Da ¼ K ta T D ; Le ¼ a T ; Pr ¼ t 2 a C a ; N ¼ b CDS b T Dh ; r ¼ /ðqc pþ f þð1 /Þðqc p Þ s ðqc p Þ f ; Fig. 1. Flow configuration and coordinate system. where ðqc p Þ f and ðqc p Þ s are respectively the heat capacity of the fluid and the saturated porous medium. In addition to the above dimensionless parameters, the present study also involves the geometrical parameters, such as the radius ratio, k, the aspect ratio, A,

4 M. Sankar et al. / International Journal of Heat and Mass Transfer 55 (2012) the location of heat and solute source, L, and the length of heat and solute source, e, which are defined as: k ¼ r o r i ; A ¼ H D ; L ¼ l H ; and e ¼ h H : The initial and boundary conditions in dimensionless form are: t ¼ 0 : T ¼ w ¼ C ¼ 0; 1 k 1 k 1 t > 0 : w ¼ ow ot or or or R ¼ 1 and 0 6 X < L e ; L þ e < X 6 A k w ¼ ow ot or or or k w ¼ ow ¼ 0; T ¼ C ¼ 0; or R ¼ k and 0 6 X 6 A k 1 w ¼ ow ot ¼ 0; ¼ 0; ox ox or X ¼ 0 and X ¼ A: The average Nusselt (Nu) and Sherwood (Sh) numbers on the surface of the heat and solute source at the inner wall of the annulus are defined as Nu ¼ 1 e Z Lþ e 2 L e 2 NudX and Sh ¼ 1 e Z Lþ e 2 L e 2 ShdX; where Nu and Sh in Eq. (6) are respectively the local Nusselt and Sherwood numbers, which can be written as 1 1 Nu ¼ TðR; XÞj R¼ 1 k 1 and Sh ¼ CðR; XÞj R¼ 1 k 1 ð6þ ; ð7þ where T(R, X) and C(R, X) is the dimensionless temperature and concentration along the heat and solute source at the inner wall of the annulus. 3. Numerical technique and code validation The non-dimensional governing equations along with the initial and boundary conditions were discretized using an implicit finite difference method. The heat, the species concentration and the vorticity transport equations are iterated until steady state using the Alternating Direction Implicit (ADI) method, and the stream function equation is solved by Successive Line Over Relaxation (SLOR) method. This technique is well described in the literature and has been widely used for natural convection in rectangular and annular cavities. For brevity, the details of the numerical method 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, a grid independence test has been conducted with different grid sizes. The effect of grid resolution has been examined in order to select the appropriate grid size. Grid size independency is validated by obtaining the predicted results from a coarse grid of 8181 to refined one of The average Nusselt and Sherwood numbers are used as sensitivity measures of the accuracy of the solution. Based on these tests, a uniform grid is found to meet the requirements of both the accuracy of the solution and a reasonable computational time. The steady state solution to the problem has been obtained as an asymptotic limit to the transient solutions. The numerical method was implemented by developing an in-house FORTRAN program for the present model and it has been successfully validated against the available benchmark solutions in the literature before obtaining the simulations Validation The numerical technique implemented in the present study has been successfully employed in our recent papers to investigate the effects of a magnetic field on double-diffusive convection [19] and thermocapillary convection [39] in a vertical nonporous annulus, and also to investigate the effects of discrete heating on natural convection in a vertical porous and nonporous annulus [24,38,39]. However, in order to verify the accuracy of the current numerical results, simulations of the present model are tested and compared with different reference solutions available in the literature for thermosolutal convection in the cylindrical annular and rectangular enclosures. First, a comparison is made with doublediffusive convection in a vertical porous annulus for the Darcy model by setting the coefficients C 1 = C 2 = 0 in Eq. (3). For this, the flow pattern, temperature and concentration fields are obtained for the uniform temperature and concentration at the inner wall, and are compared with the corresponding results of Beji et al. [6]. Fig. 2 exhibits a good agreement between the present streamlines, isotherms and isoconcentrations and that of Beji et al. [6] in the porous annulus. The accuracy of the numerical results are further checked by computing the average Nusselt and Sherwood numbers for uniform temperature and concentration at the inner wall of the porous annulus. These quantitative results are compared with the Darcy flow model results of Nithiarasu et al. [8] for Le =2, N = 1 and k = 5, and are given in Table 1. The comparison with their finite element method using non-uniform grids is quite good. In addition to the above validations, we also compare our results with Goyeau et al. [27] and Bennacer et al. [28] in a rectangular porous cavity (k = 1). In theory, the case of infinite curvature characterized by k = 1 represents a rectangular cavity. The comparison shown in Table 2 reveals that the detected maximum difference with the results of Goyeau et al. [27] and Bennacer et al. [28] is less than 2.3%. From Fig. 2, and Tables 1 and 2, the agreement between the present results and benchmark solutions is quite acceptable. 4. Results and discussion The numerical results are presented in this section with the main objective of investigating the effect of a heat and solute source (henceforth heat and solute source is referred to as the segment) location and buoyancy ratio on the double-diffusive convective flow, and the corresponding heat and mass transfer rates in a vertical porous annulus. Since the present study involve a large number of non-dimensional parameters (Ra, Da, Le, Pr, N, e, r,, A, k, L), only the main controlling parameters are varied. In the present study, the aspect ratio (A) of the annulus, Prandtl number (Pr), heat capacity ratio (r) and porosity of the porous medium () are kept at unity. Also, the size (e) of the segment is fixed at 0.25; however, its location (L) is varied from to The thermal Rayleigh number (Ra), Darcy number (Da) and Lewis number (Le) are respectively varied in the ranges Ra ; Da and 1 6 Le 6 10: Curvature effect can be important for the annular cavity flows, and so the effect of radius ratio (k) on the heat and mass transfer rates is also examined for a wide range ð1 6 k 6 10Þ, with r i kept constant and r o being varied. The relative importance of thermal and solutal buoyancy forces is denoted by the buoyancy ratio (N), and is defined as the ratio of the solutal buoyancy force to thermal buoyancy force. This parameter is varied through a wide range 10 6 N 6 þ10; covering the concentration-dominated opposing flow (N = -10), pure thermal convectiondominated flow (N = 0), and concentration-dominated aiding flow (N = 10). In the following sections, the flow fields, temperature and concentration distributions in the porous annulus are illustrated through streamlines, isotherms and isoconcentrations. In all contour graphs, the left and right vertical sides correspond to the inner and outer cylinders, respectively. Further, the effects of thermal Rayleigh number, Darcy number, Lewis number and radius ratio on the heat and mass transfer rates are evaluated in terms of the average Nusselt and Sherwood numbers at different segment locations and buoyancy ratios.

5 4120 M. Sankar et al. / International Journal of Heat and Mass Transfer 55 (2012) Fig. 2. Comparison of streamlines (left), isotherms (middle) and isoconcentrations (right) between the present results and that of Beji et al. [6] for Ra T = 500, Le = 10, N = 0 and k =5. Table 1 Comparison of average Nusselt and Sherwood numbers with Nithiarasu et al. [8] for double-diffusive convection in a porous annulus at Le = 2,N = 1,A = 1 and k = 5. Thermal Darcy Rayleigh number (Ra T ) Nithiarasu et al. [8] Present study 4.1. Effect of buoyancy ratio and segment location Relative difference (%) Nu Sh Nu Sh Table 2 Comparison of average Nusselt and Sherwood numbers with Goyeau et al. [27] and Bennacer et al. [28] for double-diffusive convection in a rectangular porous cavity at Le = 10, N =0,A = 1 and k =1. Thermal Darcy Rayleigh number (Ra T ) Goyeau et al. [27] Nu Sh Bennacer et al. [28] Nu Sh Present study Nu Sh As a first step towards studying the influence of segment location, the effect of buoyancy ratio on the flow pattern, thermal and solute concentration distributions is examined for three different segment locations, namely at the bottom, middle and top portions of the inner wall of the porous annulus. Figs. 3 5 illustrates the streamlines, isotherms and isoconcentrations for three different combinations of buoyancy ratio and segment location with the values of Ra, Da, Le and k are respectively fixed at 10 7,10 3, 10 and 2. The intervals of streamlines, isotherms and isoconcentrations are Dn ¼ðn max n min Þ=15; where stands for, T or C. Since the heat and solute source is placed on the inner wall, the direction of thermal flow is always clockwise, whereas the direction of solutal flow strongly depends on the sign of N or b C. The solutal flow is clockwise for N (or b C ) > 0 and counterclockwise for N (or b C )<0. Fig. 3 provides exemplary results on the flow pattern, temperature and concentration fields for the opposing buoyancy forces (N=-10). In the case of opposing flow, regardless of the segment location, two or three counter-rotating cells are observed in the annulus. When the segment is located at the bottom portion of the inner wall, two weak counter rotating cells with equal magnitude are observed in the annulus. The streamlines divide the cavity into a thermal buoyancy-driven cell at the upper zone, and a solute buoyancy-driven cell in the bottom part of the annulus (Fig. 3). As the segment shifts to the middle portion, the strength of the thermally driven cell is significantly promoted over the solute buoyancy-driven cell. Interestingly, as the segment move upwards, the flow is characterized by a solute-driven counterclockwise rotating cell in the core region and two thermal-driven clockwise circulations located at the top and the bottom of the annulus. The isotherms are more skewed towards the segment, but on the other hand, the concentration lines in the interior core are observed to be diagonal. A careful observation of the streamline pattern reveals the distinct location effect of the segment on the thermal and solute flow circulation. It is found that the rate of

6 M. Sankar et al. / International Journal of Heat and Mass Transfer 55 (2012) Fig. 3. Effect of segment location on the streamlines (top), isotherms (middle), and isoconcentrations (bottom) for opposing flow (N = 10) with Ra =10 7, Da =10 3, Le = 10, k = 2: (a) L = 0.125, (b) L = 0.5, (c) L = thermal flow circulation is higher for the middle location of the segment, whereas higher solutal flow circulation rate is observed when the segment is placed at top of the inner wall. The results of the heat-driven flow (N = 0) are shown in Fig. 4 for three different segment locations. For N = 0, the flow is driven by the thermal buoyancy force, and the effect of concentration does not exist. The streamlines and isotherms obtained in the present study are similar to those reported previously for the thermal convection in a vertical porous annulus [24]. The flow structure consists of one main circulation occupying the entire enclosure and the solute driven-cell in the cavity has been annihilated. Also, a striking difference can be seen in the direction of isotherms and isoconcentrations between the opposing and heat-driven flows. A highly stratified medium with almost parallel and horizontal flow results in the core region when the segment is placed at the bottom portion of the inner wall (Fig. 4a). As the segment location shifts towards the top portion, the main vortex reduces in size and moves near the top portion of the outer wall. Further, the symmetric structure of the flow pattern, thermal and solutal fields has been disturbed when the segment location is shifted to the top portion of the inner wall. The relative strength of the flow as indicated by the maximum absolute stream function reduces as the segment moves upwards. That is, the rate of fluid circulation is found to be higher, when the segment is placed at the bottom portion of the inner wall. This may be attributed to the distance that the fluid needs to travel in the circulating cell to exchange the heat and solute concentration between the segment and outer wall. These predictions are consistent with those reported by Zhao et al. [37] for doublediffusive convection with a heat and solute source mounted on the right wall of a square porous enclosure. As the segment is placed at the top, a strong flow exists in the upper zone of the annulus with a main vortex near the outer wall, and a weak flow at the bottom portion of the annular enclosure, which is vividly reflected in the corresponding isotherms and isoconcentrations (Fig. 4c). This may be expected to be due to the presence of an impermeable and insulated top wall, which results in a severe

7 4122 M. Sankar et al. / International Journal of Heat and Mass Transfer 55 (2012) Fig. 4. Effect of segment location on the streamlines (top), isotherms (middle), and isoconcentrations (bottom) for heat-driven flow (N = 0) with Ra =10 7, Da =10 3, Le = 10, k = 2: (a) L = 0.125, (b) L = 0.5, (c) L = restriction to the flow emerging from the segment. The formation of hydrodynamic, thermal and solutal boundary layers is visible around the segment and on the upper portion of the outer wall. The influence of segment locations on the mutually augmenting thermal and solutal buoyancies, termed as aiding double-diffusive flow (N > 0), is illustrated in Fig. 5(a) (c). With the buoyancy ratio increased to N = 10, the solutal and thermal buoyancy forces are augmenting each other, resulting in an accelerated clockwise flow in the annulus. The location of the segment significantly alters the flow pattern as indicated by the streamlines. When the segment is positioned at the bottom portion of the inner wall, the flow pattern induced by the combined thermal and solutal buoyancy forces consists of a single cell with its maximum near the inner wall of the annulus. The cell elongated in the horizontal direction when the segment 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 bottom portion of the annulus, the flow is weak and a secondary eddy is generated. It is worth mentioning that similar flow structures, but for a square porous cavity (k = 1) have been reported by Zhao et al. [37]. The isoconcentrations, at all three locations, reveal a strengthened stratification and stronger horizontal intrusion layers in the annulus than their thermal counterparts. Although the thermal and solutal buoyancy effects augment each other, a careful observation of the streamlines reveals that the magnitude of maximum stream function is lower for N = 10 compared to N = 0. This may be attributed to the stabilizing or blocking effect of the vertical stratification of the combined density field in the core of the annulus. However, compared to the uniform heating and salting conditions at the inner wall of the annulus (Bennacer et al. [10]), the blocking effect in the present study is reduced to a great extent due to the discrete heating and salting of the inner wall. The temperature and concentration fields reveal a symmetric structure when the segment is at the bottom. However, as the segment moves upwards, the symmetric structure of the thermal and solutal fields is completely lost. The thermal and solutal boundary layer along the heater and outer wall reveals that solutal boundary layer is thinner than the thermal boundary layer due to lower mass

8 M. Sankar et al. / International Journal of Heat and Mass Transfer 55 (2012) Fig. 5. Effect of segment location on the streamlines (top), isotherms (middle), and isoconcentrations (bottom) for aiding flow (N = 10) with Ra =10 7, Da =10 3, Le = 10, k =2: (a) L = (b) L = 0.5, (c) L = species diffusivity (Le = 10). Further, the solutal field is well stratified, while the thermal field in the interior core is observed to be diagonal. The variation of average Nusselt and Sherwood numbers under the combined effects of segment location and buoyancy ratio can be 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 6 0:875; while the parameters Le, Ra, Da and k are respectively fixed at 5, 10 7,10 5 and 2. An overview of the figure reveals that in the opposing flow region (N < 0), the heat and mass transfer rates are lower compared to the corresponding N in the aiding flow region (N > 0). A similar observation was predicted in earlier studies in a uniformly heated vertical porous and nonporous annulus [9,10,13,14]. This can be expected due to the lower flow rate near the annular walls for the opposing flow than that of the aiding flow. Although the value of N decreases in the opposing flow region, the heat and mass transfer rates are constantly increasing. This may be attributed to the fact that the heat and mass transfer rates are enhanced due to the magnitude of velocity and not because of the direction of flow. The figure clearly shows that the heat and mass transfer rates are strongly depends on the segment location. Moreover, the segment location for maximum heat and mass transfer rates in the opposing flow region is not same for the aiding flow region. For any value of N in the opposing flow region, the average Nusselt (Nu) and Sherwood (Sh) numbers increases as the value of L increases from to 0.625, but decreases for L > Similarly, in the aiding flow region, Nu and Sh increases as the value of L decreases from to 0.375, but decrease when L < This result reveals the complex relationship between the segment location and the rates of heat and mass transfer. From Fig. 6, it can be seen that for aiding flows (N > 0) the heat and 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 = Irrespective of the segment location, as N decreases in the opposing flow region, the heat and mass transfer rates decreases to the point of flow reversal. The minimum value of the average Nusselt number

9 4124 M. Sankar et al. / International Journal of Heat and Mass Transfer 55 (2012) Fig. 6. Influence of buoyancy ratio on the average Nusselt (Nu) and Sherwood (Sh) numbers for different segment locations with Le =5,Ra =10 7, 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 =10 7, L = 0.5 and k =2. is noticeably different when the segment is placed either near to the bottom (L = or 0.25) or to the top (L = 0.75 or 0.875) portion of the inner wall, but the minimum value for other locations of the segment is almost the same. However, the average Sherwood number has different minimum values only for segment locations near the bottom portion (L = or 0.25) of the inner wall, and for the remaining segment locations the minimum values are almost the same. The influence of segment location on the flow pattern reveals a higher flow circulation when the segment is placed at the bottom portion of the inner wall. However, the variation of average Nusselt and Sherwood numbers with segment locations exhibits that the heat and mass transfer rates are higher, when the segment is placed around middle portion (L = 0.375) rather than placing it near the bottom or top portion of the inner wall. This reveal an important fact that the optimal location for maximum heat and mass transfer not only depends on the circulation intensity, but also depends on the shape of the thermal and solutal buoyancy-driven flow. This observation is in agreement with the recent works of by Zhao et al. [37] for double-diffusive convection in a square porous cavity Effect of Darcy number In this section, we discuss the effects of Darcy number together with buoyancy ratio and segment location on the heat and mass transfer rates. First, the combined influence buoyancy ratio and Darcy number on the average Nusselt and Sherwood numbers is illustrated in Fig. 7 for fixed values of Le =5,Ra =10 7, L = 0.5 and k = 2. The non-dimensional parameter N characterizes the ratio between solutal and thermal buoyancy forces is varied in the range 10 6 N 6 þ10: For all Darcy numbers, the average Nusselt and Sherwood numbers have a tendency to reach a minimum value of N, N min, in the transitional range, at which the flow reversal from transitional to thermal dominated flow occurs. However, this minimum value, N min, strongly depends on the Darcy number. The figure vividly reveals that the value of N min decreases as the permeability of porous matrix increases. Further, the presence of the porous medium strongly influences the heat and mass transfer rate, particularly in the transitional range. For aiding flows (N > 0), as N increases, the solutal buoyancy force increases relative to the thermal buoyancy force. As a result, the net upward buoyancy force near the inner wall increases, and so does the intensity of the circulation. Hence, the average Nusselt and Sherwood numbers increase steadily with the magnitude of the buoyancy ratio N. However, for opposing case (N < 0), the variation of average Nusselt and Sherwood numbers are not monotonic, particularly at large Darcy numbers. As N decreases, initially Nu and Sh decrease until they reach N min, and then they increase with N. This is in agreement with the uniform heating and salting results in a vertical porous annulus [9,10]. Also, it should be pointed out that the main

10 M. Sankar et al. / International Journal of Heat and Mass Transfer 55 (2012) Fig. 8. Influence of Darcy number on the average Nusselt (Nu) and Sherwood (Sh) numbers for different segment locations with Le = 10, Ra =10 7, N = 8 and k =2. contribution of the presence of a porous medium is to substantially reduce the rate of heat transfer compared to mass transfer rate. Next, the combined effects of Darcy number and segment location on the average Nusselt and Sherwood numbers is reported in Fig. 8. The values of Lewis number, thermal Rayleigh number, buoyancy ratio and radius ratio are respectively fixed at Le = 10, Ra =10 7, N = 8 and k = 2, while the Darcy number and segment location are varied. At low Darcy numbers, the porous medium exerts a resistance to the double-diffusive convective flow, and hence the rate of heat and mass transfer is less compared to higher values of Da. The slopes of the average Nusselt and Sherwood number curves decrease with an increase in the value of Da, and finally approach to zero for all segment locations. This indicate that there exists an asymptotic double-diffusive convective regime where the heat and mass transfer rates are independent of the Darcy number, but depends strongly on the segment location. This trend has 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 Sherwood numbers with the Darcy number, it is apparent that the permeability of the porous medium significantly affects the heat transfer rate more than the rate of mass transfer. The combined influence of Darcy number and segment location on the average Nusselt number reveals that the permeability of the porous medium has a Fig. 9. Influence of radius ratio on the average Nusselt (Nu) and Sherwood (Sh) numbers for different segment locations with Le = 10, Ra =10 7, N = 8 and L = 0.5. distinct effect on the heat transfer rate at different segment locations. At lower values of Darcy number (Da <10 3 ), the variation of the average Sherwood number with segment locations is noticeably small compared to that of the heat transfer whereas, at higher Darcy numbers, a larger variation is observed with segment locations. Further, at all Darcy numbers, the heat and mass transfer rates are lower when the segment is placed at the top of the inner wall. A higher heat transfer rate is observed for the segment location L = at all values of Da, while the higher mass transfer rate for the segment location strongly depend on the Darcy number Effect of radius ratio In the study of natural convection heat and mass transfer in a vertical annulus, knowledge of radius ratio effect (curvature) on the heat and mass transfer rates is important in designing many engineering applications. Fig. 9 exemplify the effects of radius ratio on the average Nusselt and Sherwood numbers for different values of Da and for fixed values of Le, Ra, N and L. The results obtained for the case of a rectangular cavity (k = 1), are similar to those reported in the literature for double-diffusive convection in a rectangular porous cavity (Zhao et al. [37]). In general, the heat and mass transfer rates increases as the Darcy number and radius ratio increases. Also, when the radius ratio (k)

11 4126 M. Sankar et al. / International Journal of Heat and Mass Transfer 55 (2012) increases, the annular gap between the inner and outer cylinders increases, and the surface area of the outer cylinder becomes increasingly larger than that of the inner cylinder. Therefore, for a given Darcy number, an increase in k above unity produces a thinner thermal boundary layer around the heat and solute source on the inner wall and a thicker thermal boundary layer on the outer wall. This results in an increase in the average Nusselt number as the radius ratio increases. However, the radius ratio affects the mass transfer rate for k 6 5, whereas for k > 5 the variation of average Sherwood number is minimum at small Darcy numbers (Da <10 3 ). Due to the lower diffusivity of solute concentration (Le = 10), the solute boundary layer is thinner than the thermal boundary layer. As a result, the concentration remains stagnant in the core of the annulus, and thus an increase in the radius ratio beyond k > 5 does not produce significant changes in the average Sherwood number Effect of Lewis number and buoyancy ratio Fig. 10 depicts the combined effects of Lewis number and buoyancy ratio on the heat and mass transfer rates for fixed values of Darcy number, thermal Rayleigh number, segment location and radius ratio respectively at Da =10 5, Ra =10 7, L = 0.5 and k = 2. The Lewis number, which measures the relative importance of thermal to mass diffusion, has a stronger influence on the heat and mass transfer rates. For unit Lewis number, the diffusion of heat and solute concentration in the mixture is in equal proportions, and hence the average Nusselt and Sherwood number curves for Le = 1 are identical (Fig. 10). For any buoyancy ratio, an increase in the Lewis number tends to decrease the heat transfer rate, except in the thermally dominated opposing flow region between N = 0 and the point of flow reversal (N min ). For a fixed value of Le, an increase in the magnitude of N enhances the flow strength, which in turn increases the heat transfer rate. However, the mass transfer rate constantly increases with the Lewis number for all ranges of buoyancy ratio. This may be attributed to the fact that, for Le > 1, the diffusivity of concentration decreases, while the thermal diffusivity increases. As a result, mass transfer occurs by convection, whereas heat is transferred by diffusion. The combined influence of Lewis number and buoyancy ratio on the average Nusselt number reveals an 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 has been previously confirmed by Shipp et al. [14] while investigating double-diffusive convection in a vertical nonporous annulus. Finally, the critical buoyancy ratio, (N min ), strongly depends on the Lewis number. The flow reversal from transitional to thermal dominated flow (Fig. 10a) and the onset of transitional flow Fig. 10. Influence of buoyancy ratio on the average Nusselt (Nu) and Sherwood (Sh) numbers for different Lewis numbers with Da =10 5, Ra =10 7, k = 2 and L = 0.5. 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.

12 M. Sankar et al. / International Journal of Heat and Mass Transfer 55 (2012) (Fig. 10b) has also been shifted to lower buoyancy ratios as the Lewis number is increased beyond unity Effect of thermal Rayleigh number The heat and mass transfer rates for different buoyancy ratios and thermal Rayleigh numbers are important quantitative measures of the problem. These are investigated in Fig. 11, where the Lewis number Le = 10, the Darcy number Da =10 3, radius ratio k = 2 and segment location L = 0.5. To analyze the influence of the transition zone, the buoyancy ratio was taken in the range - 106N610, and the thermal Rayleigh number was considered in the range Ra For lower values of Ra (Ra610 5 ), the heat and mass transfer rates are modest at all ranges of buoyancy ratio, and the onset of transition flow occurs at the same buoyancy ratio. However, the heat and mass transfer rates are significantly enhanced with an increase in Ra beyond 10 5, since the thermal Rayleigh number stimulates the convective motion driven by the combined buoyancies. An increase in the thermal Rayleigh number (Ra >10 5 ) shifts the onset of flow transition to a lower buoyancy ratio. The average Sherwood number significantly increases with the thermal Rayleigh number, although the porous medium acts as an barrier to the mass transfer. However, as mentioned before, the effect of increasing Ra enhances the thermally induced flow and hence the heat transfer rate, as can be seen from the variation of average Nusselt number. This thermally induced flow within the porous matrix is the main source of mass transfer in the annulus. 5. Conclusions In this paper, we have numerically investigated the double-diffusive convection in a vertical porous annulus with a discretely heated and salted segment at the inner wall. The effects of the main controlling parameters, such as segment location, buoyancy ratio, Darcy number, thermal Rayleigh number, Lewis number and radius ratio were investigated in detail to gain new insights into the flow patterns, thermal and solutal fields, and the rates of heat and mass transfer. Many of the observations of the present study are in good agreement with the similar studies in the literature. The main findings of the present investigation are summarized as follows: 1. The flow structure, thermal and solutal fields, rates of heat and mass transfer are profoundly affected by the relative magnitudes of buoyancy ratio and segment location. 2. The critical buoyancy ratio at which the transitional flow occurs strongly depends on the segment location, Lewis, Darcy and thermal Rayleigh numbers. 3. The segment location of higher flow circulation does not produce higher heat and mass transfer rates. The heat and mass transfer rates can be effectively controlled by the segment location. 4. For concentration-dominated opposing case (N = -10), the flow in the annulus is characterized by two separate flow circulations in opposite direction, while for heat-driven (N = 0) and concentration-dominated aiding (N = 10) flows, a strong unicellular flow structure is observed in the annulus. 5. The Darcy number significantly affects the heat transfer rate more than the rate of mass transfer. Furthermore, the influence of porous medium on the heat and mass transfer rates strongly depends on the segment location. 6. For all Lewis numbers, the average Nusselt number attains the same value for the heat driven flow (N = 0). An increase in Lewis number will increase the mass transfer rate, while it decreases the heat transfer rate, except for the regions between N = 0 and N min, where the heat transfer rate increases. 7. The segment location influences the heat and mass transfer rates in a different fashion with respect to buoyancy ratio and Darcy number. 8. An increase in Ra beyond 10 5 tends to increase the mass transfer rate noticeably through the thermally induced flow circulations in the porous annulus. Acknowledgements This work was supported by WCU (World Class University) program through the Korea Science and Engineering Foundation funded by the Ministry of Education, Science and Technology (Grant No. R ). References [1] B. Gebhart, L. Pera, The nature of vertical natural convection flows resulting from the combined buoyancy effect of thermal and mass diffusion, Int. J. Heat Mass Transfer 14 (1971) [2] J.S. Turner, Double diffusive phenomena, Annu. Rev. Fluid Mech. 6 (1974) [3] D.B. Ingham, I. Pop (Eds.), Transport Phenomena in Porous Media, Elsevier, Oxford, [4] D.A. Nield, A. Bejan, Convection in Porous Media, third ed., Springer, New York, [5] H.J. Sung, W.K. Cho, J.A. 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