double-diffusive convection from a discrete heat and solute …lopez/pdf/tpm_spld12.pdf · 2012. 2....

Transp Porous Med (2012) 91:753–775DOI 10.1007/s11242-011-9871-1

Double-Diffusive Convection from a Discrete Heatand 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. 2011

Abstract This article reports a numerical study of double-diffusive convection in a fluid-saturated vertical porous annulus subjected to discrete heat and mass fluxes from a portion ofthe inner wall. The outer wall is maintained at uniform temperature and concentration, whilethe top and bottom walls are adiabatic and impermeable to mass transfer. The physical modelfor the momentum equation is formulated using the Darcy law, and the resulting governingequations are solved using an implicit finite difference technique. The influence of phys-ical and geometrical parameters on the streamlines, isotherms, isoconcentrations, averageNusselt and Sherwood numbers has been numerically investigated in detail. The location ofheat and solute source has a profound influence on the flow pattern, heat and mass transferrates in the porous annulus. For the segment located at the bottom portion of inner wall, theflow rate is found to be higher, whereas the heat and mass transfer rates are higher when thesource is placed near the middle of the inner wall. Further, the average Sherwood numberincreases 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 Sherwoodnumber increases with radius ratio only up to λ = 5, and for λ > 5 , the average Sherwoodnumber 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 702-701, Republic of Koreae-mail: [email protected]

M. Sankare-mail: [email protected]

Y. Parke-mail: [email protected]

J. M. LopezSchool of Mathematical and Statistical Sciences, Arizona State University, Tempe, AZ 85287, USAe-mail: [email protected]

123

754 M. Sankar et al.

List of symbolsA Aspect ratioC Dimensionless concentrationcp Specific heat at constant pressureD Width of the annulus (m)Da Darcy numberg Acceleration due to gravity (m/s2)

H Height of the annulus (m)h Dimensional length of the heat and solute source (m)K Permeability of the porous medium (m2)

k Thermal conductivity (W/(m K))l Distance between the bottom wall and centre of the source (m)L Dimensionless location of the heat and solute sourceLe Lewis numberN Buoyancy ratioNu Average Nusselt numberSh Average Sherwood numberp Fluid pressure (Pa)qh Heat flux (W/m2)

jh Mass flux (kg/m2s)RaT Thermal Darcy–Rayleigh numberS Dimensional concentrationT Dimensionless temperaturet Dimensional time (s)(ri, ro) 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 lettersαT Thermal diffusivity (m2/s)αC Mass diffusivity of the solute in the fluid (m2/s)βT Thermal expansion coefficient (1/K)βC Solutal expansion coefficient (1/K)σ Heat capacity ratioε Dimensionless length of the heat and solute sourceζ Dimensionless vorticityθ Dimensional temperature (K)λ Radius ratioυe Effective kinematic viscosity of the porous medium (m2/s)υf Fluid kinematic viscosity (m2/s)ρ Fluid density (kg/m3)

τ Dimensionless timeφ∗ Dimensional porosityφ Normalized porosityψ Dimensionless stream function

123

Double-Diffusive Convection from a Discrete Heat 755

1 Introduction

Double-diffusive convection driven in finite porous enclosures by the combined buoyancyeffect due to temperature and concentration variations has been extensively investigated inrecent years. Interest in this phenomenon has been motivated by diverse engineering problemssuch as drying processes, migration of moisture contained in fibrous insulation, grain stor-age installations, food processing, crystal growth applied to semiconductors, contaminationtransport in saturated soil, and the underground disposal of nuclear wastes.

Natural convection flow in porous media, due to thermal buoyancy alone, has been widelystudied and well-documented in the literature (Bear 1988; Ingham and Pop 2005; Vafai 2005;Nield and Bejan 2006; Vadasz 2008). Among the finite porous enclosures, free convectiveheat transfer in a differentially heated vertical porous annulus has received considerableattention owing to its importance in high performance insulation for buildings and porousheat exchangers (Hickox and Gartling 1985; Char and Lee 1998; Al-Zahrani and Kiwan2009; Shivakumara et al. 2003; Prasad 1986). Natural convection in square and rectangularporous enclosures subject to discrete heating has drawn much attention in recent years. UsingDarcy model, Saeid and Pop (2005) numerically investigated natural convection in a poroussquare cavity with an isoflux and isothermal discrete heater. They found that maximal heattransfer can be achieved when the heater is placed near the bottom of one of the verticalwalls. Later, Saeid (2006) numerically studied natural convection induced by two isothermalheat sources on a vertical plate channel filled with a porous medium. Other notable studiesinvolving the discrete heating of a rectangular cavity are due to Saeid and Pop (2004) andSivasankaran et al. (2011). Recently, Sankar et al. (2011a) reported on the effects of sizeand 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 andannular 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 hasmainly been motivated by diverse applications such as melting and solidification processesin binary mixtures and storage of liquefied gases. Nithiarasu et al. (1997) applied the finite-ele-ment method to investigate double-diffusive natural convection in a vertical porous annulususing a generalized porous medium model. They studied both Darcy and non-Darcy flowregimes for a wide range of Darcy and Rayleigh numbers, and radius ratios. Using the Darcymodel, Marcoux et al. (1999) reported a numerical and analytical study of double-diffusiveconvection in a fluid saturated porous annulus subjected to uniform heat and mass fluxesfrom the side walls. For high aspect ratios, their numerical and analytical solutions are ingood agreement. Beji et al. (1999) performed the numerical simulation of double-diffusiveconvection in a vertical porous annulus, whose vertical walls are maintained at uniform tem-peratures 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 verticalporous annulus using the Brinkman extended Darcy model.

More recently, the research community has shifted their attention to understanding themechanism of heat and mass transfer in square or rectangular enclosures with discrete energyand solute sources placed at either of the side walls. This is due to the fact that many engi-neering systems may be characterized by double-diffusive convective flow with a discreteheat 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 sin-gle thermal and solute source in a square enclosure. Their numerical simulations for a wide

123


parameter range reveals that the source location plays a vital role in altering the heat andmass transfer rates in the cavity. Later, Liu et al. (2008) analyzed the effect of partial heatingand salting on the thermosolutal convection in a square enclosure. The heating and saltingsegments were subjected to constant heat and mass fluxes, and the Darcy model was used topredict the heat and mass transfer rates in the porous cavity.

An in-depth literature survey has revealed that all published works on double-diffusive con-vection in the annular cavity have been restricted to uniform heat and mass fluxes (Marcouxet 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 stud-ies on double-diffusive natural convection subject to partial heating and salting have beenrestricted to square enclosures (Zhao et al. 2007; Liu et al. 2008). Although the influenceof discrete heating and salting on the thermosolutal convection in a rectangular porous cav-ity have been addressed, they do not always adequately represent the important practicalsituations in which the flow domain is a porous layer bounded by two vertical concentriccylinders since curvature effects can be important. The lack of information on the thermo-solutal natural convective heat and mass transfer in an annular enclosure with a discreteheat and solute source motivates the present investigation. The main objective of the pres-ent investigation is to examine the effects of the location of a discrete heating and saltingsegment on double-diffusive natural convection in a vertical porous annulus. In the follow-ing, the physical model and mathematical formulation of the problem is first given. Subse-quently, the numerical solution of the governing equations is carried out for a wide range ofparameters. Finally, the numerical results are discussed in detail followed by some importantconclusions.

2 Mathematical Formulation

Consider a vertical annulus of height H , inner radius ri, and outer radius ro as shown in Fig. 1.The top and bottom portions are closed by two insulated disks, which are impermeable tomass transfer. At the inner wall, a heating and salting element of length h is subjected toconstant heat and mass fluxes qh and jh, while the remaining portion is insulated and imper-meable. The distance between the centre of the heating and salting segment and the bottomwall is l. The outer wall is maintained at a lower temperature θ0 and lower concentration S0.Also, the fluid is assumed to be Newtonian with negligible viscous dissipation and gravityacts in the negative x-direction. The flow is assumed to be axisymmetric. The annular regionis filled with a rigid, fluid-saturated porous medium, and the fluid is in local thermodynamicequilibrium 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. Theheat flux produced by the concentration gradient (Dufour effect) and the mass flux producedby the temperature gradient (Soret effect) is neglected.

Further, the fluid is assumed to be a Boussinesq fluid, i.e., both the porous matrix andthe saturating fluid are incompressible, and all thermo-physical properties of the medium areconstant, except the density of the mixture which depends linearly on the temperature andconcentration and is given by ρ(θ, S) = ρ0 [1 − βT(θ − θ0) − βC(S − S0)]. By employingthe above approximations, the equations governing the conservation of mass, momentum inthe Darcy regime, energy and solute concentration in an isotropic and homogeneous porousmedium 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; Marcouxet al. 1999)

123


ri

ro

qh

h

jh

x, w

r, u

D

θ0

S0

l

H

Fig. 1 Geometry of the porous annulus subject to discrete heat (qh) and mass ( jh) fluxes at the inner wall,and coordinate system

∂u

∂r+ ∂w

∂x+ u

r= 0, (1)

u = − K

μ

∂p

∂r, (2)

w = − K

μ

∂p

∂x+ ρg [βT(θ − θ0) + βC(S − S0)] , (3)

σ∂θ

∂t+ u

∂θ

∂r+ w

∂θ

∂x= αT ∇2

1θ, (4)

φ∗ ∂S

∂t+ u

∂S

∂r+ w

∂S

∂x= αc∇2

1 S, (5)

where ∇21 = 1

r∂∂r

(r ∂

∂r

) + ∂2

∂x2 , φ∗ is the porosity of the porous medium, and σ is the heat

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, respec-tively.

Since the flow depends only on two spatial coordinates, the stream function formulation isused. Hence, by eliminating the pressure terms from the Eqs. 2 and 3, and using the followingnon-dimensionless variables,

U = u D

αT, W = wD

αT, T = θ − θ0

�θ, C = S − S0

�S, R = r

D, X = x

D, τ = tαT

σ D2 ,

P = pk

μαT, D = ro − ri, �θ = qh D

k, �S = jh D

αC,

the governing Eqs. 1–5 may be written in dimensionless form as:

123


∂T

∂τ+ U

∂T

∂ R+ W

∂T

∂ X= ∇2T, (6)

φ∂C

∂τ+ U

∂C

∂ R+ W

∂C

∂ X= 1

Le∇2C, (7)

∂2�

∂ R2 − 1

R

∂�

∂ R+ ∂2�

∂ X2 = −R · RaT

[∂T

∂ R+ N

∂C

∂ R

], (8)

U = 1

R

∂�

∂ X, W = − 1

R

∂�

∂ R, (9)

where ∇2 = 1R

∂∂ R

(R ∂

∂ R

) + ∂2

∂ X2 .

Now, the dimensionless parameters governing double-diffusive natural convection are thethermal Darcy–Rayleigh number,RaT, the Lewis number, Le, the buoyancy ratio, N , and

the normalized porosity, φ, defined by: RaT = gKβT qh D2

υkαT, Le = αT

αC, N = βC�S

βT�θ, φ = φ∗

σ.

In addition to the above dimensionless parameters, this study also involves the followinggeometrical parameters: λ = ro

ri, the radius ratio, L = l

H , non-dimensional location of the

heater, A = HD , the aspect ratio, and ε = h

H , non-dimensional length of the heating andsalting segment.

The dimensionless initial and boundary conditions are:

τ = 0 : U = W = T = 0, � = C = 0; 1

λ − 1≤ R ≤ λ

λ − 1, 0 ≤ X ≤ A

τ > 0 : � = 0,∂T

∂ R= ∂C

∂ R= 0; R = 1

λ − 1and 0 ≤ X < L− ε

2

� = 0,∂T

∂ R= ∂C

∂ R= −1; R = 1

λ − 1and L − ε

2≤ X ≤ L+ ε

2

� = 0,∂T

∂ R= ∂C

∂ R= 0; R = 1

λ − 1and L + ε

2< X ≤ A

� = 0, T = C = 0; R = λ

λ − 1and 0 ≤ X ≤ A

� = 0,∂T

∂ X= ∂C

∂ R= 0; X = 0 and X = A

The average Nusselt (Nu) and Sherwood (Sh) numbers on the surface of the energy andsolute sources at the inner wall of the annulus is defined as

Nu = 1

ε

L+ ε2∫

L− ε2

NudX . (10)

Sh = 1

ε

L+ ε2∫

L− ε2

ShdX . (11)

where Nu and Sh in Eqs. 10 and 11 are, respectively, the local Nusselt and Sherwood numbersalong the energy and solute sources, which can be written as

Nu = 1

T (R, X)|R= 1λ−1

, (12)

123


and

Sh = 1

C (R, X)|R= 1λ−1

, (13)

where T (R, X)andC(R, X) are the dimensionless temperature and concentration along theheat 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 equa-tion, the energy equation and the stream function equation has been numerically solved usingan 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 usedfor natural convection in rectangular and annular cavities. For brevity, the details of thenumerical 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 theR–X plane of the annulus, and in order to determine a proper grid size for the present numer-ical study, a grid independence test has been conducted. Grid size dependency is studiedby verifying the variation of the predicted results from a coarse 81 × 81 grid to a refined201×201 grid. The average Nusselt and Sherwood numbers are used as sensitivity measuresof the accuracy of the solution. Based on these tests, all the computations are performed witha 161 × 161 grid, which gives a good compromise between accuracy and CPU time. Thethermal boundary conditions between the segment and the rest of the regions are discontin-uous. 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 spacingis small compared to the boundary layer thickness near the discontinuity in the boundarycondition, there is no need to regularize the discontinuity. This is in sharp contrast to globalmethods such as spectral methods where a discontinuous boundary condition leads to Gibbsphenomenon and a regularization of the boundary condition is required (Lopez and Shen1998). The steady state solution to the problem has been obtained as an asymptotic limit tothe transient solutions. A FORTRAN code has been developed for the present model and hasbeen successfully validated against the available benchmark solutions in the literature beforeobtaining the simulations.

3.1 Validation

The numerical technique implemented in this study has been successfully employed in ourrecent papers to investigate the effects of magnetic field on the double-diffusive convec-tion (Venkatachalappa et al. 2011) and thermocapillary convection (Sankar et al. 2011b) ina vertical non-porous annulus, and also to understand the effect of discrete heating on thenatural convection in a vertical porous annulus (Sankar et al. 2011a). Further, in order toverify the accuracy of the current numerical results, simulations of the present model aretested and compared with different reference solutions available in the literature for purethermal convection, and thermosolutal convection in a cylindrical porous annulus. First, thenumerical results for different Darcy–Rayleigh numbers and radius ratios are obtained fornatural convection, driven by thermal buoyancy alone, in a vertical porous annulus. The innerand outer walls of the annulus are, respectively, maintained at uniform heat flux and constant

123


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 Relativeratio (λ) number (RaT) difference (%)

2 103 6.4934 6.4815 0.18

104 16.0498 16.0271 0.14

3 103 7.1659 7.1804 0.20

104 17.2691 17.2226 0.27

5 103 8.0036 8.0262 0.28

104 18.8055 18.8631 0.31

10 103 9.3975 9.4452 0.51

104 20.7498 20.8325 0.40

temperature, and the horizontal walls are assumed to be adiabatic. Table 1 shows the com-parison of average Nusselt numbers between the present study and that of Prasad (1986) inan annular enclosure for different thermal Darcy–Rayleigh numbers and radius ratios. Fromthe table, an overall good degree of agreement can be observed between the present resultsand the correlation data with the maximum difference being 0.5% at higher values of thermalDarcy–Rayleigh number and radius ratio.

Next, a comparison is made with double-diffusive convection in a vertical porous annu-lus. For this, the average Nusselt and Sherwood numbers are determined for isothermal andisoconcentrations at the inner and outer walls, insulated and impermeable horizontal wallsof the annulus. These quantitative results are compared with the Darcy flow model resultsof Nithiarasu et al. (1997) for Le = 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, temper-ature, and concentration fields are obtained to compare with the corresponding results ofBeji 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 andsalted porous annulus. In addition to the above validation, we also compare our results withGoyeau et al. (1996) and Bennacer 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 double-diffusive convection in a porous annulus at Le = 2, N = 1, A = 1, and λ = 5

Thermal Darcy–Rayleigh Nithiarasu et al. (1997) Present study Relativenumber (RaT) difference (%)

100

Nu 8.5 8.45 0.59

Sh 14.27 14.22 0.35

500

Nu 21.42 20.95 2.24

Sh 34.66 35.42 2.19

123


Fig. 2 Comparison of streamlines (top), isotherms (middle) and isoconcentrations (bottom) between thepresent results and that of Beji et al. (1999) for RaT = 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 theresults of Goyeau et al. (1996) and Bennacer et al. (2001) is less than 2.3%. From Fig. 2and Tables 1, 2, and 3, the agreement between the present results and benchmark solutionsis quite acceptable.

123


Table 3 Comparison of averageNusselt and Sherwood numberswith Goyeau et al. (1996) andBennacer et al. (2001) fordouble-diffusive convection in arectangular porous cavity atLe = 10, N = 0, A = 1, andλ = 1

Thermal Darcy–Rayleigh number (RaT )

100 200 400

Goyeau et al. (1996)

Nu 3.11 4.96 7.77

Sh 13.25 19.86 28.41

Bennacer et al. (2001)

Nu 3.11 4.96 7.77

Sh 13.24 19.83 29.36

Present study

Nu 3.11 4.91 7.72

Sh 13.24 19.92 28.71

4 Results and Discussion

In this section, the results of numerical simulations are presented with an objective to under-stand the influence of the location of a heat and solute source on the double-diffusive con-vective flows, and to evaluate the corresponding heat and mass transfer in a vertical porousannulus. Although this tudy involves eight parameters, for the sake of brevity, only a selectednumber of them are varied. In the present study, the aspect ratio (A) of the annulus andnormalized 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 0.125 to 0.875. The thermalDarcy–Rayleigh number ( RaT) and Lewis number (Le) are, respectively, varied in the range10 ≤ RaT ≤ 500 and 1 ≤ Le ≤ 10. In order to investigate the effects of curvature, the radiusratio (λ) of the annular cavity is examined for a vast range (1 ≤ λ ≤ 10), with ri kept constantand ro varied. The dynamic parameters representing the driving forces are varied through awide range of buoyancy ratio (N) −10 ≤ N ≤ +10, covering the concentration-dominatedopposing flow (N = −10), pure thermal-convection dominated flow (N = 0), and concen-tration-dominated aiding flow (N = 10). The flow fields, temperature, and concentrationdistributions in the porous annulus are illustrated through streamlines, isotherms, and iso-concentrations. In all contour figures, the left and right vertical sides correspond to the innerand outer cylinders, respectively. In addition, the variation of heat and mass transfer ratesare presented in terms of the average Nusselt and Sherwood numbers for different thermalDarcy–Rayleigh numbers, Lewis numbers, segment locations, buoyancy ratios, and radiusratios.

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 distributionsis 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 andtop portions of the inner wall. The streamlines, isotherms and isoconcentrations are illus-trated in Figs. 3, 4, and 5 for different combinations of buoyancy ratio and segment locationby fixing the values of RaT, Le, and λ, respectively, at 500, 10, and 2. Figures 3a–c reportthe influence of negative buoyancy ratio (N = −5) on the flow pattern, temperature, andconcentration fields. Negative values of N represent the opposing nature of two buoyancyforces, due to the negative coefficient of concentration expansion. For N = −5, at all three

123


Fig. 3 Streamlines (top), isotherms (middle), and isoconcentrations (bottom) for opposing flow (N = −5)with RaT = 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 = 0.875

locations of the segment, we observe counter-rotating cells in the cavity. The clockwiserotating cell at the central upper zone is driven by thermal buoyancy, and in the bottom partof the annulus, where the fluid is denser, the solute buoyancy-driven counterclockwise cellcan be observed. At this value of N , although the driving forces are still opposed to eachother, the thermal buoyant force takes over the solutal force which is further supported bythe magnitude of maximum stream function contour for thermal buoyancy. This is expecteddue to the high thermal Darcy–Rayleigh number considered in this case. Since the thermaldiffusivity is higher than the mass diffusivity, the isotherms are less affected by the flow thanthe 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 seg-ment is placed at the top. Interestingly, as the segment move upwards, the size of the solutebuoyancy-driven cell increases, and it occupies the major portion of the annulus when thesegment is placed at the top. The plots of isotherms manifest an upward thermal flow, while

123


Fig. 4 Streamlines (top), isotherms (middle), and isoconcentrations (bottom) for the heat-driven flow (N = 0)with RaT = 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 = 0.875

flow due to solute buoyancy is directed downwards. As the segment location is shifted towardthe top, a distortion is present in the temperature and concentration contours at the interfacebetween the solute and thermal buoyancy-driven cells. The distortion is more pronouncedin the concentration contours. A careful observation of the streamline pattern reveals thedistinct effect of segment location on the thermal and solute flow circulation. Higher thermalflow circulation is observed when the segment is placed at the middle, whereas the soluteflow 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 pre-sented in Fig. 4a–c for the heat-driven flow limit (N = 0). In this limit, the flow is mainlydriven by thermal buoyancy forces, and the solute buoyancy does not influence the flow fieldand heat transfer rate. Comparing the streamlines in Figs. 3 (opposing flow) and 4 (heat-drivenflow), it can be observed that the solute driven-cell in the cavity has completely disappearedin Fig. 4 due to the absence of solute buoyancy force for N = 0. In this case, a unicellular

123


Fig. 5 Streamlines (top), isotherms (middle), and isoconcentrations (bottom) for aiding flow (N = 5) withRaT = 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 = 0.875

flow driven by thermal buoyancy is observed, a situation similar to thermal convection inthe annulus. Also, unlike in the opposing buoyancy forces (N = −5) case, the thermal andsolutal boundary layers are developing in the same direction. As the segment is placed at thebottom, the flow originated from the heating source travel diagonally between the bottomand top corners of the inner and outer walls, respectively, (Fig. 4a). The temperature variationaround the heaters is found to be large, while the variation of temperature in the core region islinear in the vertical direction and is nearly constant in the horizontal direction. The isocon-centrations 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 maineddy have changed to the horizontal direction. The isoconcentrations reveal that the massflow near the sink (outer wall) travels faster than the thermal flow. When the segment is placedat the top of the inner wall, the flow intensity is high near the upper portion of the annulus,

123


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 hydrodynamicboundary layers are visible around the segment, top and outer walls. A similar observationwas predicted by Zhao et al. (2007) and Liu et al. (2008) for discrete heating and salting on theright 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 flowpattern, temperature, and concentrations fields. The flow circulation is found to be highestwhen the segment is placed at the bottom. This is due to the distance that the fluid needsto travel in the circulating cell to exchange the heat and solute concentration between thesegment and outer wall (sink). In fact, the closer the segment is to the bottom wall, the higherthe magnitude for the stream function that is achieved. This observation is qualitatively ingood agreement with the predictions of Zhao et al. (2007) and Saeid and Pop (Saeid and Pop2005) 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 otherlocations.

Figures 5a–c exemplifies results for aiding double-diffusive flow in the annulus. When thebuoyancy ratio is increased above zero, the thermal and solutal buoyancy forces are actingin the same direction, and hence the flow is accelerated by the combined buoyancies. As thebuoyancy ratio increases, it may lead to a concentration-dominated flow, and such a configu-ration is identified as mass transfer driven flow (Trevisan and Bejan 1986; Beji et al. 1999). Inthe aiding flow, the temperature field is advected by the flow field caused by solute buoyancyforce. However, due to the higher Lewis number (Le = 10), the isotherms are less affectedby the flow than the isoconcentrations. Therefore, the isotherms for the case of heat-drivenflow (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 akinto the heat-driven flow limit (N = 0). However, the hydrodynamic boundary layer is morepronounced for the aiding flow compared to heat-driven flow. The isoconcentrations, at allthree locations, reveal a strengthened stratification and strong horizontal intrusion layers inthe annulus than their thermal counterparts. The blocking (stabilizing) effect of the verticalstratification of the combined density field in the core of the annulus can be clearly observedfrom the magnitude of the maximum stream function for the aiding flow (N = 5). Althoughthermal and solutal buoyancy effects augment each other for aiding flow, the magnitude ofmaximum stream function is lower for the N = 5 compared to N = 0. However, comparedto 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 discreteheating and salting of the inner wall.

The effect of segment location on the average Nusselt and Sherwood numbers for the buoy-ancy ratio range of −10 ≤ N ≤ +10 is presented in Fig. 6. In general, the average Nusseltand Sherwood numbers are less in the opposing flow region (N < 0) than for the correspond-ing N in the aiding flow region (N > 0), which is consistent with the earlier findings in auniformly heated vertical annulus (Bennacer et al. 2000; Shipp et al. 1993a,b). Interestingly,the magnitude of average Sherwood number for both aiding and opposing buoyancy ratiosis higher compared to the average Nusselt number. This is due to the fact that the magnitudeof 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 ratesof heat and mass transfer strongly depend on the segment location and buoyancy ratios. Foropposing flows, it is interesting to observe that the minimum values of average Nusselt andSherwood numbers is detected at a same segment location (L = 0.125), but the maximum

123


Fig. 6 Variation of average Nusselt and Sherwood numbers with buoyancy ratios and segment locations forLe = 10, λ = 2, and RaT = 200

values are found at different locations, say at L = 0.625 (Nu) and L = 0.875 (Sh) (Fig. 6).The different locations for maximum heat and mass transfer rates can be expected due tothe opposing nature N < 0 (βC < 0 and βT > 0). In contrast, for aiding flows the averageNusselt and Sherwood numbers attain their minimum value at L = 0.875 and maximumvalue at the location L = 0.375 (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 thetop wall of the annulus. For any location of the segment, the average Nusselt and Sherwood

123


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 aidingflow, the heat and mass transfer rates are higher for the segment location L = 0.375, whereasthe corresponding location for higher heat and mass transfer rates in a rectangular porouscavity (Zhao et al. 2007; Liu et al. 2008) is L = 0.125. The difference in the segment locationfor maximum heat and mass transfer rates are due to the curvature of the annulus, boundaryconditions 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 isreported 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 num-bers with RaT = 200, L = 0.5, λ = 2, and N = −2. (a) Le = 1, (b) Le = 5, (c) Le = 10

123


ratio, and buoyancy ratio are, respectively, fixed at RaT = 200, L = 0.5, λ = 2, andN = −2, while three different Lewis numbers (Le = 1, 5, and 10) are considered. TheLewis number characterizes solute transport relative to thermal diffusion. At Le = 1, a sol-ute-dominated unicellular flow in the counterclockwise direction is observed, due to negativebuoyancy ratio. For unit Lewis number, the heat and solute diffuse in equal proportions, lead-ing to identical temperature and concentration profiles in the annulus (Fig. 7a). As the Lewisnumber is increased (Le = 5), the effect of solute buoyancy force is reduced, and a ther-mal 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 flowstructure, and the onset of transitional flow can be observed from Fig. 7b and c. For higherLewis number, the diffusivity of concentration decreases, which results in the reduction ofthe solute boundary layer thickness. As a result, at Le = 10, the cells of solutal origin areconfined near the bottom corners of the inner cylinder, and the thermal buoyancy-driven celloccupies major portion of the annulus.

Figure 8 depicts the combined effects of Lewis number and radius ratio on the heat andmass transfer rates for fixed values of thermal Darcy–Rayleigh number, segment location,and buoyancy ratio, respectively, at RaT = 200, L = 0.5, and N = 1. The Lewis number,which measures the relative importance of thermal to mass diffusion, has a direct influenceon the heat and mass transfer coefficients. For all radius ratios, it can be seen in Fig. 8that when the Lewis number increases, Nu begins to decrease and then becomes constant;however, Sh increases with Le. For Le > 3, mass transfer occurs by convection whereasheat is transferred by diffusion. Under these circumstances, the driving force is producedby solutal buoyancy. In addition to the Lewis number effect, Fig. 8 also illustrates the effectof radius ratio on the heat and mass transfer rates in the porous annulus. From Fig. 8, it isobserved that, for a given Le, introduction of curvature effects (λ > 1) considerably increasesthe average Nusselt and Sherwood numbers. These results are similar to those reported byMarcoux et al. (1999) for double-diffusive convection in a uniformly heated vertical porousannulus.

4.3 Effect of Darcy–Rayleigh Number and Radius Ratio

The heat and mass transfer rates for different segment locations and thermal Darcy–Ray-leigh numbers are important quantitative measures of the problem. These are investigated inFig. 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 onthe convective motion driven by the combined buoyancies, it can be seen from Fig. 9 thatthe heat and mass transfer rates from the segment increase with RaT. The streamlines inFigs. 4 and 5 show a strong flow circulation in the annulus when the segment is placed atthe bottom portion of the inner wall. However, a careful observation of Fig. 9 reveals thatthe heat and mass transfer rates are higher when the segment is placed around the middlerather than placing it near the bottom or top portion of the inner wall. It can be expected thatthe rising binary fluid cannot wipe the entire surface of the segment when it is placed verynear to the bottom or top wall of the annulus. Therefore, the optimal location for maximumheat and mass transfer not only depends on the circulation intensity, but also depends on theshape of the thermal and solutal buoyancy-driven flow. This reveals an important fact thatthe segment location of higher flow circulation may not lead to higher rates of heat and masstransfer. 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 issurprisingly complex. They found that a high or low rate of circulation may be possible for

123


Fig. 8 Variation of average Nusselt and Sherwood numbers with Lewis number and radius ratios for RaT =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 knowl-edge of radius ratio effect on the heat and mass transfer rates is important in designing manyengineering applications. Figure 10 exemplify the effects of radii ratio on the average Nusselt

123


Fig. 9 Variation of average Nusselt and Sherwood numbers with Darcy–Rayleigh number and segmentlocations for Le = 10, λ = 2, and N = 1

and Sherwood numbers for different values of RaT and fixed values of Le, L and N . Theresults obtained for the case of a rectangular cavity (λ = 1), are similar to those reportedin 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 theheat and mass source on the inner wall and a thicker thermal boundary layer on the outer

123


Fig. 10 Variation of average Nusselt and Sherwood numbers with Darcy–Rayleigh number and radius ratiosfor 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 withradius 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 thattheory.

123


In contrast, the radius ratio affects the mass transfer rate for λ ≤ 5, but for λ > 5 thevariation of average Sherwood number is minimal, as shown in Fig. 10. The explanation forthis is due to the solute boundary layer is thinner than the thermal boundary layer due to thelower diffusivity of concentration (Le > 1, see Fig. 7). As a result, the concentration remainsconstant 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 significantchanges in the average Sherwood number.

5 Conclusions

In this work, double-diffusive convection in a vertical porous annulus induced by the com-bined buoyancy effects of thermal and mass diffusion has been investigated. A discrete heatand solute source is placed on the inner wall, while the outer wall is maintained at uniformtemperature and concentration. The flow pattern, temperature, and concentration fields, andrates of heat and mass transfer in the porous annulus has been examined for several segmentlocations and wide range of buoyancy ratios, reflecting the entire range of flow configura-tions. Also, the effect of segment location on the heat and mass transfer is determined forvarious radius ratios, Lewis and thermal Darcy–Rayleigh numbers. What follows is a briefsummary of the major results obtained from the present investigations.

A counter-rotating flow is observed in the annulus for opposing buoyancy forces, whilea strong unicellular flow exists when the buoyancy forces are augmenting each other. Thesegment location influences the flow pattern and rates of heat and mass transfer in a complexfashion. For example, a stronger flow circulation in the annulus does not produce higher heatand mass transfer rates. The rate of flow circulation is found to be higher as the segment ispositioned at the bottom portion of inner wall, while placing the segment near the middleof the inner wall, higher heat and mass transfer rates are achieved. This result is consistentwith 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 heatedcylindrical annular and rectangular enclosures, and good agreement is found.

The segment location for minimum and maximum values of average Nusselt and Sher-wood numbers greatly depends on the magnitude of the buoyancy ratio. For any segmentlocation, the average Nusselt and Sherwood numbers tend to approach a minimum valuein the transitional region at which the flow reversal takes place. Further, the heat and masstransfer rates can be effectively controlled by the segment location. Higher heat transferrates can be achieved for an annulus with a larger radius ratio due to the larger combinedbuoyancy 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 struc-ture of the flow, thermal, and solutal fields. As Lewis number increases, the rate of masstransfer increases, whereas the heat transfer rate decreases with Le. Further, heat and masstransfer rates in the vertical porous annulus behave in a slightly different fashion fromthat of a rectangular porous cavity with different segment locations due to the effects ofcurvature.

Acknowledgments This work was supported by WCU (World Class University) program through the KoreaScience and Engineering Foundation funded by the Ministry of Education, Science, and Technology (Grant No.R32-2009-000-20021-0). The author M. Sankar would like to acknowledge the support and encouragement ofChairman and Principal of East Point College of Engineering and Technology, Bangalore, and to VisvesvarayaTechnological University (VTU), Belgaum, India. The authors would like to extend their appreciation to thereferees for their helpful comments to improve the quality of this article.

123


References

Al-Zahrani, M.S., Kiwan, S.: Mixed convection heat transfer in the annulus between two concentric verticalcylinders using porous layers. Transp. Porous Med. 76, 391–405 (2009)

Ameziani, D.E., Bennacer, R., Bouhadef, K., Azzi, A.: Effect of the days scrolling on the natural convectionin an open ended storage silo. Int. J Therm. Sci. 48, 2255–2263 (2009)

Bear, J.: Dynamics of Fluids in Porous Media. Dover Publications, New York (1988)Beji, H., Bennacer, R., Duval, R., Vasseur, P.: Double diffusive natural convection in a vertical porous annu-

lus. Numer. Heat Transf. A 36, 153–170 (1999)Bennacer, R., Beji, H., Duval, R., Vasseur, P.: The Brinkman model for thermosolutal convection in a vertical

annular porous layer. Int. Commun. Heat Mass Transf. 27(1), 69–80 (2000)Bennacer, R., Tobbal, A., Beji, H., Vasseur, P.: Double diffusive convection in a vertical enclosure filled with

anisotropic porous media. Int. J. Therm. Sci. 40, 30–41 (2001)Char, M.-I., Lee, G.-C.: Maximum density effects on natural convection in a vertical annulus filled with a

non-Darcy porous medium. Acta Mech. 128, 217–231 (1998)Chen, S., Tolke, J., Krafczyk, M.: Numerical investigation of double-diffusive (natural) convection in vertical

annuluses with opposing temperature and concentration gradients. Int. J. Heat Fluid Flow 31, 217–226 (2010)

Choukairy, K., Bennacer, R., Vasseur, P.: Natural convection in a vertical annulus boarded by an inner wallof finite thickness. Int. Commun. Heat Mass Transf. 31(4), 501–512 (2004)

Goyeau, B., Songbe, J.P., Gobin, D.: Numerical study of double-diffusive natural convection in porous cavityusing the Darcy–Brinkman formulation. Int. J. Heat Mass Transf. 39, 1363–1378 (1996)

Hickox, C.E., Gartling, D.K.: A numerical study of natural convection in a vertical annular porous layer. Int.J. Heat Mass Transf. 28, 720–723 (1985)

Ingham, D.B., Pop, I. (eds.): Transport Phenomena in Porous Media. Pergamon, Oxford (2005)Liu, D., Zhao, F.Y., Tang, G.F.: Thermosolutal convection in saturated porous enclosure with concentrated

energy and solute sources. Energy Convers. Manage. 49, 16–31 (2008)Lopez, J.M., Shen, J.: An efficient spectral-projection method for the Navier–Stokes equations in cylindrical

geometries I. axisymmetric cases. J. Comput. Phys. 139, 308–326 (1998)Marcoux, M., Mojtabi, M.C.C., Azaiez, M.: Double-diffusive convection in an annular vertical porous

layer. Int. J. Heat Mass Transf. 42, 2313–2325 (1999)Nield, D.A., Bejan, A.: Convection in Porous Media, 3rd edn. Springer, New York (2006)Nithiarasu, P., Seetharamu, K.N., Sundararajan, T.: Non-Darcy double-diffusive natural convection in axisym-

metric fluid saturated porous cavities. Heat Mass Transf. 32, 427–433 (1997)Prasad, V.: Numerical study of natural convection in a vertical, porous annulus with constant heat flux on the

inner wall. Int. J. Heat Mass Transf. 29, 841–853 (1986)Saeid, N.H.: Natural convection from two thermal sources in a vertical porous layer. ASME J. Heat

Transf. 128, 104–109 (2006)Saeid, N.H., Pop, I.: Maximum density effects on natural convection from a discrete heater in a cavity filled

with a porous medium. Acta Mech. 171, 203–212 (2004)Saeid, N.H., Pop, I.: Natural convection from discrete heater in a square cavity filled with a porous medium. J.

Porous Med. 8(1), 55–63 (2005)Sankar, M., Do, Y.: Numerical simulation of free convection heat transfer in a vertical annular cavity with

discrete heating. Int. Comm. Heat Mass Transf. 37, 600–606 (2010)Sankar, M., Park, Y., Lopez, J.M., Do, Y.: Numerical study of natural convection in a vertical porous annulus

with discrete heating. Int. J Heat Mass Transf. 54, 1493–1505 (2011a)Sankar, M., Venkatachalappa, M., Do, Y.: Effect of magnetic field on the buoyancy and thermocapillary driven

convection of an electrically conducting fluid in an annular enclosure. Int. J. Heat Fluid Flow 32, 402–412 (2011b)

Shipp, P.W., Shoukri, M., Carver, M.B.: Double diffusive natural convection in a closed annulus. Numer. HeatTransf. A 24, 339–356 (1993a)

Shipp, P.W., Shoukri, M., Carver, M.B.: Effect of thermal Rayleigh and Lewis numbers on double diffusivenatural convection in closed annulus. Numer. Heat Transf. A 24, 451–465 (1993b)

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

Sivasankaran, S., Kandaswamy, P., Ng, C.O.: Double diffusive convection of anomalous density fluids in aporous cavity. Transp. Porous Med. 71, 133–145 (2008)

Sivasankaran, S., Do, Y., Sankar, M.: Effect of discrete heating on natural convection in a rectangular porousenclosure. Transp. Porous Med. 86, 291–311 (2011)

123


Trevisan, O.V., Bejan, A.: Mass and heat transfer by natural convection in a vertical slot filled with porousmedium. Int. J. Heat Mass Transf. 29, 403–415 (1986)

Vadasz, P. (ed.): Emerging Topics in Heat and Mass Transfer in Porous Media. Springer, New York (2008)Vafai, K. (ed.): Handbook of Porous Media, 2nd edn. Taylor & Francis, New York (2005)Venkatachalappa, M., Do, Y., Sankar, M.: Effect of magnetic field on the heat and mass transfer in a vertical

annulus. Int. J. Eng. Sci 49, 262–278 (2011)Zhao, F.Y., Liu, D., Tang, G.F.: Free convection from one thermal and solute source in a confined porous

medium. Transp. Porous Med. 70, 407–425 (2007)

123

double-diffusive convection from a discrete heat and solute …lopez/pdf/tpm_spld12.pdf · 2012. 2....

Documents