Double-Diffusive Convection from a Discrete Heat and Solute Source in a Vertical Porous Annulus

Transcription

1 Transp Porous Med (2012) 91: DOI /s Double-Diffusive Convection from a Discrete Heat and Solute Source in a Vertical Porous Annulus M. Sankar Youngyong Park J. M. Lopez Younghae Do Received: 10 March 2011 / Accepted: 2 September 2011 / Published online: 16 September 2011 Springer Science+Business Media B.V Abstract This article reports a numerical study of double-diffusive convection in a fluidsaturated vertical porous annulus subjected to discrete heat and mass fluxes from a portion of the inner wall. The outer wall is maintained at uniform temperature and concentration, while the top and bottom walls are adiabatic and impermeable to mass transfer. The physical model for the momentum equation is formulated using the Darcy law, and the resulting governing equations are solved using an implicit finite difference technique. The influence of physical and geometrical parameters on the streamlines, isotherms, isoconcentrations, average Nusselt and Sherwood numbers has been numerically investigated in detail. The location of heat and solute source has a profound influence on the flow pattern, heat and mass transfer rates in the porous annulus. For the segment located at the bottom portion of inner wall, the flow rate is found to be higher, whereas the heat and mass transfer rates are higher when the source is placed near the middle of the inner wall. Further, the average Sherwood number increases with Lewis number, while for the average Nusselt number the effect is opposite. The average Nusselt number increases with radius ratio (λ); however, the average Sherwood number increases with radius ratio only up to λ = 5, and for λ>5, the average Sherwood number does not increase significantly. Keywords Double-diffusive convection Porous annulus Discrete heating and salting Radius ratio Darcy model M. Sankar Y. Park Y. Do (B) Department of Mathematics, Kyungpook National University, 1370 Sangyeok-Dong, Buk-Gu, Daegu , Republic of Korea yhdo@knu.ac.kr M. Sankar manisankarir@yahoo.com Y. Park dhdkwnfk@naver.com J. M. Lopez School of Mathematical and Statistical Sciences, Arizona State University, Tempe, AZ 85287, USA lopez@math.asu.edu

2 754 M. Sankar et al. List of symbols 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 the 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 heat and solute source Le Lewis number N Buoyancy ratio Nu Average Nusselt number Sh Average Sherwood number p Fluid pressure (Pa) q h Heat flux (W/m 2 ) j h Mass flux (kg/m 2 s) Ra T Thermal Darcy Rayleigh number S Dimensional concentration T Dimensionless temperature t Dimensional time (s) (r i, r o ) Radius of inner and outer cylinders (m) (r, x) Dimensional radial and axial co-ordinates (m) (R, X) Dimensionless co-ordinates in radial and axial directions (u,w) Dimensional velocity components in (r, x) direction (m/s) (U, W ) Dimensionless velocity components in (R, X) direction Greek letters α T Thermal diffusivity (m 2 /s) α C Mass diffusivity of the solute in the fluid (m 2 /s) β T Thermal expansion coefficient (1/K) β C Solutal expansion coefficient (1/K) σ Heat capacity ratio ε Dimensionless length of the heat and solute source ζ Dimensionless vorticity θ Dimensional temperature (K) λ Radius ratio υ e Effective kinematic viscosity of the porous medium (m 2 /s) υ f Fluid kinematic viscosity (m 2 /s) ρ Fluid density (kg/m 3 ) τ Dimensionless time φ Dimensional porosity φ Normalized porosity ψ Dimensionless stream function

3 Double-Diffusive Convection from a Discrete Heat Introduction Double-diffusive convection driven in finite porous enclosures by the combined buoyancy effect due to temperature and concentration variations has been extensively investigated in recent years. Interest in this phenomenon has been motivated by diverse engineering problems such as drying processes, migration of moisture contained in fibrous insulation, grain storage installations, food processing, crystal growth applied to semiconductors, contamination transport in saturated soil, and the underground disposal of nuclear wastes. Natural convection flow in porous media, due to thermal buoyancy alone, has been widely studied and well-documented in the literature (Bear 1988; Inghamand Pop2005; Vafai 2005; Nield and Bejan 2006; Vadasz 2008). Among the finite porous enclosures, free convective heat transfer in a differentially heated vertical porous annulus has received considerable attention owing to its importance in high performance insulation for buildings and porous heat exchangers (Hickox and Gartling 1985; Char and Lee 1998; Al-Zahrani and Kiwan 2009; Shivakumara et al. 2003; Prasad 1986). Natural convection in square and rectangular porous enclosures subject to discrete heating has drawn much attention in recent years. Using Darcy model, Saeid andpop (2005) numerically investigated natural convection in a porous square cavity with an isoflux and isothermal discrete heater. 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 (2006) numerically studied natural convection induced by two isothermal heat sources on a vertical plate channel filled with a porous medium. Other notable studies involving the discrete heating of a rectangular cavity are due to Saeid and Pop (2004) and Sivasankaran et al. (2011). Recently, Sankar et al. (2011a) reported on the effects of size and location of a discrete heater on the natural convective heat transfer in a vertical annulus. Natural convection resulting from thermal and solutal buoyancy forces in rectangular and annular cavities has also been investigated (Trevisan and Bejan 1986; Goyeau et al. 1996; Sivasankaran et al. 2008; Shipp et al. 1993a,b; Chen et al. 2010). Interest in double-diffusive natural convection in a fluid saturated porous annulus has mainly been motivated by diverse applications such as melting and solidification processes in binary mixtures and storage of liquefied gases. Nithiarasu et al. (1997) applied the finite-element method to investigate double-diffusive natural convection in a vertical porous annulus using a generalized porous medium model. They studied both Darcy and non-darcy flow regimes for a wide range of Darcy and Rayleigh numbers, and radius ratios. Using the Darcy model, Marcoux et al. (1999) reported a 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. For high aspect ratios, their numerical and analytical solutions are in good agreement. Beji et al. (1999) performed the numerical simulation of double-diffusive convection in a vertical porous annulus, whose vertical walls are maintained at uniform temperatures and concentrations for a wide range of physical and geometrical parameters. Later, Bennacer et al. (2000) carried out a numerical study on thermosolutal convection in a vertical porous annulus using the Brinkman extended Darcy model. More recently, the research community has shifted their attention to understanding the mechanism of heat and mass transfer in square or rectangular enclosures with discrete energy and solute sources placed at either of the side walls. This is due to the fact that many engineering systems may be characterized by double-diffusive convective flow with a discrete heat and solute source on one of the vertical walls, such as zone melting, alloy solidification, hazardous thermo-chemical spreading and liquid fuel storage tank. Using the Darcy model, Zhao et al. (2007) conducted a numerical study of double-diffusive convection from a single thermal and solute source in a square enclosure. Their numerical simulations for a wide

4 756 M. Sankar et al. parameter range reveals that the source location plays a vital role in altering the heat and mass transfer rates in the cavity. Later, Liu et al. (2008) analyzed the effect of partial heating and salting on the thermosolutal convection in a square enclosure. The heating and salting segments were subjected to constant heat and mass fluxes, and the Darcy model was used to predict the heat and mass transfer rates in the porous cavity. An in-depth literature survey has revealed that all published works on double-diffusive convection in the annular cavity have been restricted to uniform heat and mass fluxes (Marcoux et al. 1999) or constant temperature and concentration (Nithiarasu et al. 1997; Beji et al. 1999; Bennacer et al. 2000) at one of the vertical walls of the annulus. The existing studies on double-diffusive natural convection subject to partial heating and salting have been restricted to square enclosures (Zhao et al. 2007; Liu et al. 2008). Although the influence of discrete heating and salting on the thermosolutal convection in a rectangular porous cavity have been addressed, they do not always adequately represent the important practical situations in which the flow domain is a porous layer bounded by two vertical concentric cylinders since curvature effects can be important. The lack of information on the thermosolutal natural convective heat and mass transfer in an annular enclosure with a discrete heat and solute source motivates the present investigation. The main objective of the present investigation is to examine the effects of the location of a discrete heating and salting segment on double-diffusive natural convection in a vertical porous annulus. In the following, the physical model and mathematical formulation of the problem is first given. Subsequently, the numerical solution of the governing equations is carried out for a wide range of parameters. Finally, the numerical results are discussed in detail followed by some important conclusions. 2 Mathematical Formulation Consider a vertical annulus of height H, inner radius r i, and outer radius r o as shown in Fig. 1. The top and bottom portions are closed by two insulated disks, which are impermeable to mass transfer. At the inner wall, a heating and salting element of length h is subjected to constant heat and mass fluxes q h and j h, while the remaining portion is insulated and impermeable. The distance between the centre of the heating and salting segment and the bottom wall is l. The outer wall is maintained at a lower temperature θ 0 and lower concentration S 0. Also, the fluid is assumed to be Newtonian with negligible viscous dissipation and gravity acts in the negative x-direction. The flow is assumed to be axisymmetric. The annular region is filled with a rigid, fluid-saturated porous medium, and the fluid is in local thermodynamic equilibrium with the solid matrix. In the porous medium, Darcy s law is assumed to hold, and hence the viscous drag and inertial terms of the momentum equations are neglected. The heat flux produced by the concentration gradient (Dufour effect) and the mass flux produced by the temperature gradient (Soret effect) is neglected. Further, the fluid is assumed to be a Boussinesq fluid, i.e., both the porous matrix and the saturating fluid are incompressible, and all thermo-physical properties of the medium are constant, except the density of the mixture which depends linearly on the temperature and concentration and is given by ρ(θ, S) = ρ 0 [1 β T (θ θ 0 ) β C (S S 0 )]. By employing the above approximations, the equations governing the conservation of mass, momentum in the Darcy regime, energy and solute concentration in an isotropic and homogeneous porous medium can be written in a cylindrical co-ordinate system with radial and axial directions (r and x), and corresponding velocity components (u and w) as (Beji et al. 1999; Marcoux et al. 1999)

5 Double-Diffusive Convection from a Discrete Heat 757 x, w r i r o D H h q h θ 0 j h S 0 l r, u Fig. 1 Geometry of the porous annulus subject to discrete heat (q h ) and mass ( j h ) fluxes at the inner wall, and coordinate system u r + w x + u r = 0, (1) p r, (2) u = K μ p x + ρg [β T(θ θ 0 ) + β C (S S 0 )], (3) w = K μ σ θ t + u θ r + w θ x = α T 1 2 θ, (4) φ S t + u S r + w S x = α c 1 2 S, (5) where 1 2 = r 1 r ( ) r r + 2,φ is the porosity of the porous medium, and σ is the heat x 2 capacity defined as σ = φ (ρc) f +(1 φ )(ρc) p (ρc) f (Beji et al. 1999; Nield and Bejan 2006). Here (ρc) f and (ρc) p are the heat capacity of the fluid and the saturated porous medium, respectively. Since the flow depends only on two spatial coordinates, the stream function formulation is used. Hence, by eliminating the pressure terms from the Eqs. 2 and 3, and using the following non-dimensionless variables, U = ud α T, W = wd, T = θ θ 0 α T θ, C = S S 0 S, R = r D, X = x D, τ = tα T σ D 2, P = pk, D = r o r i, θ = q h D μα T k, S = j h D, α C the governing Eqs. 1 5 may be written in dimensionless form as:

6 758 M. Sankar et al. T τ + U T R + W T X = 2 T, (6) φ C τ + U C R + W C X = 1 Le 2 C, (7) 2 R 2 1 R R + 2 X 2 = R Ra T U = 1 R X, W = 1 R [ T R + N C ], (8) R R, (9) where 2 = R 1 ( ) R R R + 2. X 2 Now, the dimensionless parameters governing double-diffusive natural convection are the thermal Darcy Rayleigh number,ra T, the Lewis number, Le, the buoyancy ratio, N, and the normalized porosity, φ, defined by: Ra T = gkβ T q h D 2 υkα T, Le = α T α C, N = β C S φ β T θ,φ = σ. In addition to the above dimensionless parameters, this study also involves the following geometrical parameters: λ = r o ri, the radius ratio, L = H l, non-dimensional location of the heater, A = H D, the aspect ratio, and ε = H h, non-dimensional length of the heating and salting segment. The dimensionless initial and boundary conditions are: 1 τ = 0 : U = W = T = 0, = C = 0; λ 1 R λ λ 1, 0 X A τ>0 : = 0, T R = C R = 0; R = 1 λ 1 and 0 X < L ε 2 = 0, T R = C R = 1; R = 1 λ 1 and L ε 2 X L+ ε 2 = 0, T R = C R = 0; R = 1 λ 1 and L + ε 2 < X A = 0, T = C = 0; R = λ λ 1 and 0 X A = 0, T X = C R = 0; X = 0andX = A The average Nusselt (Nu) and Sherwood (Sh) numbers on the surface of the energy and solute sources at the inner wall of the annulus is defined as Nu = 1 ε L+ 2 ε NudX. (10) L ε 2 Sh = 1 ε L+ 2 ε ShdX. (11) L ε 2 where Nu and Shin Eqs. 10and 11are, respectively, the local Nusselt and Sherwood numbers along the energy and solute sources, which can be written as Nu = 1 T (R, X) R= 1 λ 1, (12)

7 Double-Diffusive Convection from a Discrete Heat 759 and 1 Sh =, (13) C (R, X) R= 1 λ 1 where T (R, X)andC(R, X) are the dimensionless temperature and concentration along the heat and solute sources at the inner wall of the annulus. 3 Numerical Technique and Code Validation The nonlinear system of governing partial differential equations, namely the species equation, the energy equation and the stream function equation has been numerically solved using an implicit finite difference method. The discretized equations are iterated until steady state, using the alternating direction implicit (ADI) method and 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 (Sankar et al. 2011b; Sankar and Do 2010; Venkatachalappa et al. 2011). 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 numerical study, a grid independence test has been conducted. Grid size dependency is studied by verifying the variation of the predicted results from a coarse grid to a refined grid. The average Nusselt and Sherwood numbers are used as sensitivity measures of the accuracy of the solution. Based on these tests, all the computations are performed with a grid, which gives a good compromise between accuracy and CPU time. The thermal boundary conditions between the segment and the rest of the regions are discontinuous. Such discontinuous thermal boundary conditions have been addressed in the literature (Ameziani et al. 2009). With local methods, such as finite differences, if the grid spacing is small compared to the boundary layer thickness near the discontinuity in the boundary condition, there is no need to regularize the discontinuity. This is in sharp contrast to global methods such as spectral methods where a discontinuous boundary condition leads to Gibbs phenomenon and a regularization of the boundary condition is required (Lopez and Shen 1998). The steady state solution to the problem has been obtained as an asymptotic limit to the transient solutions. A FORTRAN code has been developed for the present model and has been successfully validated against the available benchmark solutions in the literature before obtaining the simulations. 3.1 Validation The numerical technique implemented in this study has been successfully employed in our recent papers to investigate the effects of magnetic field on the double-diffusive convection (Venkatachalappa et al. 2011) and thermocapillary convection (Sankar et al. 2011b) in a vertical non-porous annulus, and also to understand the effect of discrete heating on the natural convection in a vertical porous annulus (Sankar et al. 2011a). Further, 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 pure thermal convection, and thermosolutal convection in a cylindrical porous annulus. First, the numerical results for different Darcy Rayleigh numbers and radius ratios are obtained for natural convection, driven by thermal buoyancy alone, in a vertical porous annulus. The inner and outer walls of the annulus are, respectively, maintained at uniform heat flux and constant

8 760 M. Sankar et al. Table 1 Comparison of average Nusselt number with the results of Prasad (1986) for pure thermal convection (N = 0) in a uniformly heated porous annulus at A = 1 Radius Thermal Darcy Rayleigh Prasad (1986) Present study Relative ratio (λ) number (Ra T ) difference (%) temperature, and the horizontal walls are assumed to be adiabatic. Table 1 shows the comparison of average Nusselt numbers between the present study and that of Prasad (1986) in an annular enclosure for different thermal Darcy Rayleigh numbers and radius ratios. From the table, an overall good degree of agreement can be observed between the present results and the correlation data with the maximum difference being 0.5% at higher values of thermal Darcy Rayleigh number and radius ratio. Next, a comparison is made with double-diffusive convection in a vertical porous annulus. For this, the average Nusselt and Sherwood numbers are determined for isothermal and isoconcentrations at the inner and outer walls, insulated and impermeable horizontal walls of the annulus. These quantitative results are compared with the Darcy flow model results of Nithiarasu et al. (1997) forle = 2, N = 1, and λ = 5, and are given in Table 2. The comparison with their finite element method using non-uniform grids is quite good. Also, for uniform temperature and concentration at the inner wall, the flow pattern, temperature, and concentration fields are obtained to compare with the corresponding results of Beji et al. (1999). Figure 2 exhibits the good agreement between the present streamlines, isotherms and isoconcentrations and that of Beji et al. (1999) in a uniformly heated and salted porous annulus. In addition to the above validation, we also compare our results with Goyeau et al. (1996) andbennacer et al. (2001) in a rectangular porous cavity (λ=1). In Table 2 Comparison of average Nusselt and Sherwood numbers with Nithiarasu et al. (1997) for doublediffusive convection in a porous annulus at Le = 2, N = 1, A = 1, and λ = 5 Thermal Darcy Rayleigh Nithiarasu et al. (1997) Present study Relative number (Ra T ) difference (%) 100 Nu Sh Nu Sh

9 Double-Diffusive Convection from a Discrete Heat 761 Fig. 2 Comparison of streamlines (top), isotherms (middle) and isoconcentrations (bottom) between the present results and that of Beji et al. (1999)forRa T = 500, Le = 10, N = 0, and λ = 5 theory, the case of infinite curvature characterized by λ = 1 represents a rectangular cavity. The comparison shown in Table 3 reveals that the detected maximum difference with the results of Goyeau et al. (1996) andbennacer et al. (2001) is less than 2.3%. From Fig. 2 and Tables 1, 2, and 3, the agreement between the present results and benchmark solutions is quite acceptable.

10 762 M. Sankar et al. Table 3 Comparison of average Nusselt and Sherwood numbers with Goyeau et al. (1996)and Bennacer et al. (2001)for double-diffusive convection in a rectangular porous cavity at Le = 10, N = 0, A = 1, and λ = 1 Thermal Darcy Rayleigh number (Ra T ) Goyeau et al. (1996) Nu Sh Bennacer et al. (2001) Nu Sh Present study Nu Sh Results and Discussion In this section, the results of numerical simulations are presented with an objective to understand the influence of the location of a heat and solute source on the double-diffusive convective flows, and to evaluate the corresponding heat and mass transfer in a vertical porous annulus. Although this tudy involves eight parameters, for the sake of brevity, only a selected number of them are varied. In the present study, the aspect ratio (A) of the annulus and normalized porosity (φ) are kept at unity. Also, the size of the heating and salting element (ε) is fixed at 0.25; however, its location (L) is varied from to The thermal Darcy Rayleigh number ( Ra T ) and Lewis number (Le) are, respectively, varied in the range 10 Ra T 500 and 1 Le 10. In order to investigate the effects of curvature, the radius ratio (λ) of the annular cavity is examined for a vast range (1 λ 10), with r i kept constant and r o varied. The dynamic parameters representing the driving forces are varied through a wide range of buoyancy ratio (N) 10 N +10, covering the concentration-dominated opposing flow (N = 10), pure thermal-convection dominated flow (N = 0), and concentration-dominated aiding flow (N = 10). The flow fields, temperature, and concentration distributions in the porous annulus are illustrated through streamlines, isotherms, and isoconcentrations. In all contour figures, the left and right vertical sides correspond to the inner and outer cylinders, respectively. In addition, the variation of heat and mass transfer rates are presented in terms of the average Nusselt and Sherwood numbers for different thermal Darcy Rayleigh numbers, Lewis numbers, segment locations, buoyancy ratios, and radius ratios. 4.1 Effect of Buoyancy Ratio and Location of Heat and Solute Source First, the influence of buoyancy ratio on the flow pattern, thermal and solute distributions is analyzed in the porous annulus for three different locations of the heat and solute source (henceforth heat and solute source is referred to as the segment), namely bottom, middle and top portions of the inner wall. The streamlines, isotherms and isoconcentrations are illustrated in Figs. 3, 4,and5 for different combinations of buoyancy ratio and segment location by fixing the values of Ra T, Le,andλ, respectively, at 500, 10, and 2. Figures 3a c report the influence of negative buoyancy ratio (N = 5) on the flow pattern, temperature, and concentration fields. Negative values of N represent the opposing nature of two buoyancy forces, due to the negative coefficient of concentration expansion. For N = 5, at all three

11 Double-Diffusive Convection from a Discrete Heat 763 Fig. 3 Streamlines (top), isotherms (middle), and isoconcentrations (bottom) for opposing flow (N = 5) with Ra T = 500, Le = 10, λ = 2 at three different locations of the heat and solute source. (a) L = 0.125, (b) L = 0.5, (c) L = locations of the segment, we observe counter-rotating cells in the cavity. The clockwise rotating cell at the central upper zone is driven by thermal buoyancy, and in the bottom part of the annulus, where the fluid is denser, the solute buoyancy-driven counterclockwise cell can be observed. At this value of N, although the driving forces are still opposed to each other, the thermal buoyant force takes over the solutal force which is further supported by the magnitude of maximum stream function contour for thermal buoyancy. This is expected due to the high thermal Darcy Rayleigh number considered in this case. Since the thermal diffusivity is higher than the mass diffusivity, the isotherms are less affected by the flow than the isoconcentrations. When the segment is moved to the middle portion of the inner wall, the thermal buoyancy-driven cell increases in size and magnitude, but decreases as the segment is placed at the top. Interestingly, as the segment move upwards, the size of the solute buoyancy-driven cell increases, and it occupies the major portion of the annulus when the segment is placed at the top. The plots of isotherms manifest an upward thermal flow, while

12 764 M. Sankar et al. Fig. 4 Streamlines (top), isotherms (middle), and isoconcentrations (bottom) for the heat-driven flow (N = 0) with Ra T = 500, Le = 10, λ = 2 at three different locations of the heat and solute source. (a) L = 0.125, (b) L = 0.5, (c) L = flow due to solute buoyancy is directed downwards. As the segment location is shifted toward the top, a distortion is present in the temperature and concentration contours at the interface between the solute and thermal buoyancy-driven cells. The distortion is more pronounced in the concentration contours. A careful observation of the streamline pattern reveals the distinct effect of segment location on the thermal and solute flow circulation. Higher thermal flow circulation is observed when the segment is placed at the middle, whereas the solute flow circulation is found to be higher as the segment is placed at the top. The influence of segment location on the flow pattern, thermal and solutal fields are presented in Fig. 4a c for the heat-driven flow limit (N = 0). In this limit, the flow is mainly driven by thermal buoyancy forces, and the solute buoyancy does not influence the flow field and heat transfer rate. Comparing the streamlines in Figs. 3 (opposing flow) and 4 (heat-driven flow), it can be observed that the solute driven-cell in the cavity has completely disappeared in Fig. 4 due to the absence of solute buoyancy force for N = 0. In this case, a unicellular

13 Double-Diffusive Convection from a Discrete Heat 765 Fig. 5 Streamlines (top), isotherms (middle), and isoconcentrations (bottom) for aiding flow (N = 5) with Ra T = 500, Le = 10, λ = 2 at three different locations of the heat and solute source. (a) L = 0.125, (b) L = 0.5, (c) L = flow driven by thermal buoyancy is observed, a situation similar to thermal convection in the annulus. Also, unlike in the opposing buoyancy forces (N = 5) case, the thermal and solutal boundary layers are developing in the same direction. As the segment is placed at the bottom, the flow originated from the heating source travel diagonally between the bottom and top corners of the inner and outer walls, respectively, (Fig. 4a). The temperature variation around the heaters is found to be large, while the variation of temperature in the core region is linear in the vertical direction and is nearly constant in the horizontal direction. The isoconcentrations reveal that the solutal boundary layer is sharper than the thermal boundary layer, and this can be attributed to the high value of Le (Trevisan and Bejan 1986; Beji et al. 1999; Liu et al. 2008). As the segment is moved to the middle, shown in Fig. 4b, the flow and main eddy have changed to the horizontal direction. The isoconcentrations reveal that the mass flow near the sink (outer wall) travels faster than the thermal flow. When the segment is placed at the top of the inner wall, the flow intensity is high near the upper portion of the annulus,

14 766 M. Sankar et al. with weak flow in the lower part. This is further reflected in isotherms and isoconcentrations (Fig. 4c). The main eddy is moved toward the outer wall, and the formation of hydrodynamic boundary layers are visible around the segment, top and outer walls. A similar observation was predicted by Zhao et al. (2007)andLiu et al. (2008) for discrete heating and salting on the right wall of a rectangular porous cavity. A close observation of the streamlines, isotherms, and isoconcentrations reveals that the location of the segment has a distinct effect on the flow pattern, temperature, and concentrations fields. The flow circulation is found to be highest when the segment is placed at the bottom. This is due 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 (sink). In fact, the closer the segment is to the bottom wall, the higher the magnitude for the stream function that is achieved. This observation is qualitatively in good agreement with the predictions of Zhao et al. (2007) and Saeid and Pop (Saeid and Pop 2005) in a rectangular porous cavity. However, when the segment is positioned in the middle, a larger displacement of isotherms and isoconcentrations from the segment can be observed, which indicates higher rates of heat and mass transfer for that location compared to other locations. Figures5a c exemplifies results for aiding double-diffusive flow in the annulus. When the buoyancy ratio is increased above zero, the thermal and solutal buoyancy forces are acting in the same direction, and hence the flow is accelerated by the combined buoyancies. As the buoyancy ratio increases, it may lead to a concentration-dominated flow, and such a configuration is identified as mass transfer driven flow (Trevisan and Bejan 1986; Beji et al. 1999). In the aiding flow, the temperature field is advected by the flow field caused by solute buoyancy force. However, due to the higher Lewis number (Le = 10), the isotherms are less affected by the flow than the isoconcentrations. Therefore, the isotherms for the case of heat-driven flow (Fig. 4) and solutal dominated aiding flow (Fig. 5) do not show any significant variation. At all three locations of the segment, the streamline pattern for the aiding flow case is akin to the heat-driven flow limit (N = 0). However, the hydrodynamic boundary layer is more pronounced for the aiding flow compared to heat-driven flow. The isoconcentrations, at all three locations, reveal a strengthened stratification and strong horizontal intrusion layers in the annulus than their thermal counterparts. The blocking (stabilizing) effect of the vertical stratification of the combined density field in the core of the annulus can be clearly observed from the magnitude of the maximum stream function for the aiding flow (N = 5). Although thermal and solutal buoyancy effects augment each other for aiding flow, the magnitude of maximum stream function is lower for the N = 5 compared to N = 0. However, compared to the uniform heating and salting conditions at the inner wall of the annulus (Beji et al. 1999), 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 effect of segment location on the average Nusselt and Sherwood numbers for the buoyancy ratio range of 10 N +10 is presented in Fig. 6. In general, the average Nusselt and Sherwood numbers are less in the opposing flow region (N < 0) than for the corresponding N in the aiding flow region (N > 0), which is consistent with the earlier findings in a uniformly heated vertical annulus (Bennacer etal. 2000; Shippet al. 1993a,b). Interestingly, the magnitude of average Sherwood number for both aiding and opposing buoyancy ratios is higher compared to the average Nusselt number. This is due to the fact that the magnitude of solutal buoyancy increases and overpowers the thermal buoyancy as the magnitude of N increases in opposing and aiding flow ranges. An overview of Fig. 6 reveals that the rates of heat and mass transfer strongly depend on the segment location and buoyancy ratios. For opposing flows, it is interesting to observe that the minimum values of average Nusselt and Sherwood numbers is detected at a same segment location (L = 0.125), but the maximum

15 Double-Diffusive Convection from a Discrete Heat 767 Fig. 6 Variation of average Nusselt and Sherwood numbers with buoyancy ratios and segment locations for Le = 10, λ = 2, and Ra T = 200 values are found at different locations, say at L = (Nu)andL = (Sh)(Fig.6). The different locations for maximum heat and mass transfer rates can be expected due to the opposing nature N < 0(β C < 0andβ T > 0). In contrast, for aiding flows the average Nusselt and Sherwood numbers attain their minimum value at L = and maximum value at the location L = (Fig. 6). For aiding flows, the segment location (L = 0.875) for minimum heat and mass transfer rates are due to the severe restrictions imposed by the top wall of the annulus. For any location of the segment, the average Nusselt and Sherwood

16 768 M. Sankar et al. numbers move toward a minimum value in the transitional range of flow reversal. However, this minimum value depends on the location of the heat and solute source. Further, for aiding flow, the heat and mass transfer rates are higher for the segment location L = 0.375, whereas the corresponding location for higher heat and mass transfer rates in a rectangular porous cavity (Zhao et al. 2007; Liu et al. 2008)isL = The difference in the segment location for maximum heat and mass transfer rates are due to the curvature of the annulus, boundary conditions and placement of segment in the vertical wall. 4.2 Effect of Lewis Number and Radius Ratio The influence of Lewis number on the streamlines, isotherms and isoconcentrations is reported in Fig. 7. The values of thermal Darcy Rayleigh number, segment location, radius Fig. 7 Streamlines (top), isotherms (middle), and isoconcentrations (bottom) for three different Lewis numbers with Ra T = 200, L = 0.5, λ = 2, and N = 2. (a) Le = 1, (b) Le = 5, (c) Le = 10

17 Double-Diffusive Convection from a Discrete Heat 769 ratio, and buoyancy ratio are, respectively, fixed at Ra T = 200, L = 0.5, λ = 2, and N = 2, while three different Lewis numbers (Le = 1, 5, and 10) are considered. The Lewis number characterizes solute transport relative to thermal diffusion. At Le = 1, a solute-dominated unicellular flow in the counterclockwise direction is observed, due to negative buoyancy ratio. For unit Lewis number, the heat and solute diffuse in equal proportions, leading to identical temperature and concentration profiles in the annulus (Fig. 7a). As the Lewis number is increased (Le = 5), the effect of solute buoyancy force is reduced, and a thermal buoyancy-driven cell intrudes from the top, moving the solutal cell to the bottom wall. Further increasing the Lewis number (Le = 10) results in a significant change in the flow structure, and the onset of transitional flow can be observed from Fig. 7b and c. For higher Lewis number, the diffusivity of concentration decreases, which results in the reduction of the solute boundary layer thickness. As a result, at Le = 10, the cells of solutal origin are confined near the bottom corners of the inner cylinder, and the thermal buoyancy-driven cell occupies major portion of the annulus. Figure 8 depicts the combined effects of Lewis number and radius ratio on the heat and mass transfer rates for fixed values of thermal Darcy Rayleigh number, segment location, and buoyancy ratio, respectively, at Ra T = 200, L = 0.5, and N = 1. The Lewis number, which measures the relative importance of thermal to mass diffusion, has a direct influence on the heat and mass transfer coefficients. For all radius ratios, it can be seen in Fig. 8 that when the Lewis number increases, Nu begins to decrease and then becomes constant; however, Sh increases with Le. ForLe > 3, mass transfer occurs by convection whereas heat is transferred by diffusion. Under these circumstances, the driving force is produced by solutal buoyancy. In addition to the Lewis number effect, Fig. 8 also illustrates the effect of radius ratio on the heat and mass transfer rates in the porous annulus. From Fig. 8, itis observed that, for a given Le, introduction of curvature effects (λ >1) considerably increases the average Nusselt and Sherwood numbers. These results are similar to those reported by Marcoux et al. (1999) for double-diffusive convection in a uniformly heated vertical porous annulus. 4.3 Effect of Darcy Rayleigh Number and Radius Ratio The heat and mass transfer rates for different segment locations and thermal Darcy Rayleigh numbers are important quantitative measures of the problem. These are investigated in Fig. 9, where the Lewis number Le = 10, radius ratio λ = 2 and buoyancy ratio N = 1. Since the thermal Darcy Rayleigh number characterizes the influence of external forces on the convective motion driven by the combined buoyancies, it can be seen from Fig. 9 that the heat and mass transfer rates from the segment increase with Ra T. The streamlines in Figs. 4 and 5 show a strong flow circulation in the annulus when the segment is placed at the bottom portion of the inner wall. However, a careful observation of Fig. 9 reveals that the heat and mass transfer rates are higher when the segment is placed around the middle rather than placing it near the bottom or top portion of the inner wall. It can be expected that the rising binary fluid cannot wipe the entire surface of the segment when it is placed very near to the bottom or top wall of the annulus. Therefore, 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 reveals an important fact that the segment location of higher flow circulation may not lead to higher rates of heat and mass transfer. For a rectangular porous cavity, this prediction was pointed out by Zhao et al. (2007) that the relationship between the heat and mass transfer rates and rate of flow circulation is surprisingly complex. They found that a high or low rate of circulation may be possible for

18 770 M. Sankar et al. Fig. 8 Variation of average Nusselt and Sherwood numbers with Lewis number and radius ratios for Ra T = 200, L = 0.5, and N = 1 the same rates of heat and mass transfer by choice of the segment location, and conversely, the same rate of circulation can be obtained for two different heat and mass transfer rates. In the study of natural convection heat and mass transfer in a vertical annulus, the knowledge of radius ratio effect on the heat and mass transfer rates is important in designing many engineering applications. Figure 10 exemplify the effects of radii ratio on the average Nusselt

19 Double-Diffusive Convection from a Discrete Heat 771 Fig. 9 Variation of average Nusselt and Sherwood numbers with Darcy Rayleigh number and segment locations for Le = 10, λ = 2, and N = 1 and Sherwood numbers for different values of Ra T and fixed values of Le, L and N. The results obtained for the case of a rectangular cavity (λ = 1), are similar to those reported in the literature for a double-diffusive convection in a rectangular porous cavity (Zhao et al. 2007). An increase in λ above unity produces a thinner thermal boundary layer around the heat and mass source on the inner wall and a thicker thermal boundary layer on the outer

20 772 M. Sankar et al. Fig. 10 Variation of average Nusselt and Sherwood numbers with Darcy Rayleigh number and radius ratios for Le = 10, L = 0.5, and N = 1 wall. This results in an increase in the average Nusselt number as the radius ratio increases. Choukairy et al. (2004) presented an analysis of the variation of average Nusselt number with radius ratio for thermal convection in a non-porous annulus. Our results in a porous annulus, for a fixed Lewis number, show variations with radius ratio which are consistent with that theory.

21 Double-Diffusive Convection from a Discrete Heat 773 In contrast, the radius ratio affects the mass transfer rate for λ 5, but for λ>5the variation of average Sherwood number is minimal, as shown in Fig. 10. The explanation for this is due to the solute boundary layer is thinner than the thermal boundary layer due to the lower diffusivity of concentration (Le > 1,see Fig.7). As a result, the concentration remains constant in the core of the annulus, and this constant concentration decreases sharply as λ increases. Thus, an increase in the radius ratio beyond λ>5 does not produce significant changes in the average Sherwood number. 5 Conclusions In this work, double-diffusive convection in a vertical porous annulus induced by the combined buoyancy effects of thermal and mass diffusion has been investigated. A discrete heat and solute source is placed on the inner wall, while the outer wall is maintained at uniform temperature and concentration. The flow pattern, temperature, and concentration fields, and rates of heat and mass transfer in the porous annulus has been examined for several segment locations and wide range of buoyancy ratios, reflecting the entire range of flow configurations. Also, the effect of segment location on the heat and mass transfer is determined for various radius ratios, Lewis and thermal Darcy Rayleigh numbers. What follows is a brief summary of the major results obtained from the present investigations. A counter-rotating flow is observed in the annulus for opposing buoyancy forces, while a strong unicellular flow exists when the buoyancy forces are augmenting each other. The segment location influences the flow pattern and rates of heat and mass transfer in a complex fashion. For example, a stronger flow circulation in the annulus does not produce higher heat and mass transfer rates. The rate of flow circulation is found to be higher as the segment is positioned at the bottom portion of inner wall, while placing the segment near the middle of the inner wall, higher heat and mass transfer rates are achieved. This result is consistent with the predictions for a rectangular porous cavity with a discrete heat and solute source. The present results are compared with the double-diffusion convection in uniformly heated cylindrical annular and rectangular enclosures, and good agreement is found. The segment location for minimum and maximum values of average Nusselt and Sherwood numbers greatly depends on the magnitude of the buoyancy ratio. For any segment location, the average Nusselt and Sherwood numbers tend to approach a minimum value in the transitional region at which the flow reversal takes place. Further, the heat and mass transfer rates can be effectively controlled by the segment location. Higher heat transfer rates can be achieved for an annulus with a larger radius ratio due to the larger combined buoyancy effects, while an increase in the radius ratio beyond a particular value (λ >5) does not increase the rate of mass transfer. Also, curvature affects the symmetric structure of the flow, thermal, and solutal fields. As Lewis number increases, the rate of mass transfer increases, whereas the heat transfer rate decreases with Le. Further, heat and mass transfer rates in the vertical porous annulus behave in a slightly different fashion from that of a rectangular porous cavity with different segment locations due to the effects of curvature. Acknowledgments 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 ). The author M. Sankar would like to acknowledge the support and encouragement of Chairman and Principal of East Point College of Engineering and Technology, Bangalore, and to Visvesvaraya Technological University (VTU), Belgaum, India. The authors would like to extend their appreciation to the referees for their helpful comments to improve the quality of this article.

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