international journal of heat and mass transferlopez/pdf/ijhmt_spld11.pdftion heat transfer in a...

International Journal of Heat and Mass Transfer 54 (2011) 1493–1505

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International Journal of Heat and Mass Transfer

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Numerical study of natural convection in a vertical porous annuluswith discrete heating

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

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

Article history:Received 13 September 2010Received in revised form 11 November 2010Accepted 11 November 2010Available online 20 December 2010

Keywords:Natural convectionAnnulusDiscrete heatingPorous mediumRadii ratioBrinkman-extended Darcy model

0017-9310/$ - see front matter � 2010 Elsevier Ltd. Adoi:10.1016/j.ijheatmasstransfer.2010.11.043

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

(Y. Park), [email protected] (J.M. Lopez), yhdo@knu

In this paper natural convection flows in a vertical annulus filled with a fluid-saturated porous mediumhas been investigated when the inner wall is subject to discrete heating. The outer wall is maintained iso-thermally at a lower temperature, while the top and bottom walls, and the unheated portions of the innerwall are kept adiabatic. Through the Brinkman-extended Darcy equation, the relative importance of dis-crete heating on natural convection in the porous annulus is examined. An implicit finite differencemethod has been used to solve the governing equations of the flow system. The analysis is carried outfor a wide range of modified Rayleigh and Darcy numbers for different heat source lengths and locations.It is observed that placing of the heater in lower half of the inner wall rather than placing the heater nearthe top and bottom portions of the inner wall produces maximum heat transfer. The numerical resultsreveal that an increase in the radius ratio, modified Rayleigh number and Darcy number increases theheat transfer, while the heat transfer decreases with an increase in the length of the heater. The maxi-mum temperature at the heater surface increases with an increase in the heater length, while it decreaseswhen the modified Rayleigh number and Darcy number increases. Further, we find that the size and loca-tion of the heater effects the flow intensity and heat transfer rate in the annular cavity.

� 2010 Elsevier Ltd. All rights reserved.

1. Introduction

The analysis of natural convection heat transfer in fluid satu-rated porous media plays an important role in many practicalapplications. These include geothermal engineering, thermalinsulation systems, packed bed chemical reactors, porous heatexchangers, oil separation from sand by steam, underground dis-posal of nuclear waste materials, food storage, electronic devicecooling, to name a few applications. The literature concerning theexperimental and numerical studies on convective flow in porousmedia is abundant and a comprehensive bibliography concernedwith this topic can be found in the monographs and books by Vafaiand Hadim [1], Ingham and Pop [2], Vafai [3], Nield and Bejan [4]and Vadasz [5]. Natural convection in finite porous enclosureshas received considerable attention over the last several yearsand, in particular, non-Darcy effects on natural convection in por-ous media have received a great deal of attention recently [6–10].

Among the finite enclosures, free convective heat transfer in adifferentially heated vertical porous annulus has been investigatedwidely in the literature, owing to its importance in high perfor-

ll rights reserved.

: +82 53 950 7954.nkar), [email protected] (Y. Do).

mance insulation for building, porous heat exchangers and manyothers applications. Havstad and Burns [11] used a perturbationmethod and a finite difference technique to analyze the heat trans-fer characteristics in a vertical annulus filled with a porous med-ium, and presented correlations for the heat transfer in theannulus. Using a finite element technique, Hickox and Gartling[12] studied natural convection flow in a vertical annular enclosurefor a wide range of radius and aspect ratios, and also used anapproximate analysis to obtain a closed form solution for theNusselt number when the aspect ratio of the annulus is high.Natural convection in a vertical porous annulus has been carriedout for isothermal heating [13] as well as by applying a constantheat flux [14] at the inner wall for a much wider range of Rayleighnumbers, aspect ratios and radius ratios than those considered in[11,12]. A combined analytical and numerical study of natural con-vection in a vertical annular porous layer with the inner wall main-tained at a constant heat flux and insulated outer wall has beencarried out by Hasnaoui et al. [15]. It is worth mentioning thatthe above cited works in the porous annulus [11–15] mainly dealtwith the Darcy formulation.

Natural convection in a vertical porous annulus has beenexperimentally investigated by many researchers. Notable amongthem are Reda [16], Prasad et al. [17], Prasad et al. [18]. Usingthe Brinkman-extended Darcy–Forchheimer model, Marpu [19]

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Nomenclature

A aspect ratioD width of the annulus (m)Da Darcy numberg acceleration due to gravity (m/s2)H height of the annulus (m)h dimensional length of heater (m)K permeability of the porous medium (m2)k thermal conductivity (W/(m K))l distance between the bottom wall and centre of the

heater (m)L dimensionless location of the heaterNu average Nusselt numberp fluid pressure (Pa)Pr Prandtl numberqh heat flux (W/m2)Ra Rayleigh number for isothermal heating

Ra ¼ gbðTh�TcÞD3

tj

� �Ra* modified Rayleigh number for isoflux heatingT dimensionless temperatureTmax maximum temperature of the heaterTh & Tc temperature at inner and outer walls

t 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 direc-

tions(u, w) dimensional velocity components in (r, x) direction (m/s)(U, W) dimensionless velocity components in (R, X) direction

Greek lettersb coefficient of thermal expansion (1/K)e dimensionless length of the heaterf dimensionless vorticityh dimensional temperature (K)j thermal diffusivity (m2/s)k radii ratiote effective kinematic viscosity of the porous medium (m2/s)tf fluid kinematic viscosity (m2/s)q fluid density (kg/m3)s dimensionless timeu porosityW dimensionless stream functionWmax maximum value of the dimensionless stream function

D

M N

P O

x

r

ri

ro

H

qhh

θc

x

r

u

w

l

Fig. 1. Physical configuration and co-ordinate system

1494 M. Sankar et al. / International Journal of Heat and Mass Transfer 54 (2011) 1493–1505

numerically investigated the natural convection heat transfer in avertical cylindrical porous annulus. Char and Lee [20] applied afinite difference method to examine the natural convection of coldwater in a vertical porous annulus under density inversion. Theyfound that both the Forchheimer inertia parameter and the Darcynumber have the same influence on the heat transfer characteris-tics. Shivakumara et al. [21] made a numerical study of naturalconvection in a vertical annulus using the Brinkman-extendedDarcy equation. The effect of porous inserts on the natural convec-tion heat transfer in a vertical open-ended annulus has beennumerically investigated by Kiwan and Al-Zahrani [22]. Threeporous inserts are attached at the inner cylinder of the annulusin the form of porous rings and the flow equations are solved byconsidering two different set of equations for fluid and porousdomains. Later, Al-Zahrani and Kiwan [23] numerically analyzedthe mixed convection in an open-ended vertical annulus with aporous layer attached to the outer surface of the inner cylinder.They considered the Darcy–Brinkman–Forchheimer model for theporous region and the Navier–Stokes equation for the clear fluidregion, and found that the heat transfer can be enhanced by usingporous layers of high thermal conductivity ratios. Recently, Reddyand Narasimhan [24] have numerically examined the effect ofinternal heat generation on the natural convection heat transferin a vertical porous annulus.

On the other hand, natural convection in a vertical annuluswithout porous media have been extensively investigated in theliterature for uniform or discrete heating (Khan and Kumar [25],Sankar and Do [26]), and stationary or rotating side walls (Reeveet al. [27], Abu-Sitta et al. [28]). Among the finite porous enclo-sures, a special interest has also been devoted to buoyancy drivenconvection in a vertical porous cylinder with closed-ends (Changand Hsiao [29], Barletta et al. [30]) as well as with a open topand bottom ends (Amara et al. [31]). Natural convection in rectan-gular enclosures subject to discrete heating has been extensivelyinvestigated in recent years because of the increasing interest inunderstanding the flow and heat transfer in MEMS applications[32] and the cooling of electronic devices [33–35]. Natural convec-tion in a porous square cavity with an isoflux and isothermal dis-crete heater placed at the left wall has been numerically studied

by Saeid and Pop [36] using the Darcy model. They found thatthe maximum heat transfer can be achieved when the heater isplaced near the bottom of the left wall. Later, Saeid [37] numeri-cally studied the natural convective flow induced by two isother-mal heat sources on a vertical plate channel filled with a porouslayer. Natural convection heat transfer in a square porous enclo-sure due to non-uniformly heated walls has been investigated inthe literature by Basak et al. [38] and Sathiyamoorthy et al. [39].Using Bejan’s heatlines method, Kaluri et al. [40] analyzed the opti-mal heating in a square cavity filled with a fluid saturated porousmedium for three different thermal conditions. Recently, mixedconvection heat transfer of a laminar slot-jet impinging on a metal-lic porous block mounted along the bottom surface of the channelhas been numerically studied by Marafie et al. [41].

Relative to a large volume of investigations in the porous annu-lus, we noticed that the preceding works on natural convection in avertical porous annulus are limited to uniform heating of the innerwall by either isothermal or isoflux wall-heating conditions [11–24]. However, in many practical applications, heating takes place

Table 1Grid-independence study for Ra* = 107, Pr = 0.7, e = 0.4, L = 0.5, Da = 10�4, u = 0.9 andk = 2.

Grid Nu Tmax

51 � 51 9.0172 0.145981 � 81 8.9062 0.1469101 � 101 9.2743 0.1473121 � 121 9.2786 0.1473

Table 2Comparison of present results with a uniformly heated rectangular porous cavity(A = 1, k = 1, u = 1 and Pr = 0.71).

Rayleigh number (Ra) Darcy number (Da) Waheed [42] Present study

104 10�3 1.0301 1.032110�2 1.5849 1.586210�1 2.1526 2.1552

105 10�3 2.0940 2.130410�2 4.0634 4.081210�1 4.4915 4.5216

106 10�3 6.6452 6.752110�2 8.5284 8.634010�1 8.7564 8.8721

M. Sankar et al. / International Journal of Heat and Mass Transfer 54 (2011) 1493–1505 1495

over a portion of one of the vertical walls, where the size and loca-tion of the heating segment may significantly affect the heat trans-port process in the annular enclosure filled with a fluid-saturatedporous media. The earlier works on natural convection in porousenclosures with discrete heating have mainly focused on rectangu-lar or square configurations [36,37,40]. Although the annular por-ous enclosure is employed in many practical applications, it hasnot been well investigated as compared to the rectangular enclo-sures. To the best of our knowledge, no studies have been foundin the literature on natural convection in a vertical porous annularenclosure with one of its vertical walls heated discretely. Thismotivates the present study, where the main objective is to exam-ine the effects of the size and location of an isoflux discrete heateron the natural convective flows in a porous annular cavity formedby two vertical coaxial cylinders. In the following, the physicalmodel and mathematical formulation of the problem is first given.Subsequently, the numerical solution of the governing equations iscarried out for a wide range of parameters of the problem. Finally,the numerical results are discussed in detail.

2. Mathematical formulation

The physical domain under investigation is a two-dimensional,cylindrical annular enclosure filled with a fluid-saturated porousmedium as shown in Fig. 1. The important geometrical parametersand the co-ordinate system with the corresponding velocity com-ponents are also indicated in Fig. 1. The width and height of theannular enclosure are D and H respectively. An isoflux heat sourceof length h and strength qh is placed on the inner wall of the annu-lus. The distance between the centre of the heater and the bottomwall is l. The outer wall is kept at a constant temperature hc, whilethe top and bottom walls as well as the unheated portions of theinner wall are maintained at adiabatic condition. Also, the fluid isassumed to be Newtonian with negligible viscous dissipation and

Fig. 2. Comparison between the present numerical results and the correlationequation of Khan and Kumar [25] for a cylindrical annular cavity with the inner wallmaintained at uniform heat flux.

gravity acts in the negative x-direction. In addition, the flow is as-sumed to be axisymmetric, laminar and the thermophysical prop-erties of the fluid are assumed as constant, except for the density inthe buoyancy term of the momentum equations, which is treatedaccording to the Boussinesq approximation. Since axisymmetry isassumed, a vertical r–x plane, marked as MNOP in Fig. 1, in theannular region is considered for the analysis.

Further, it is assumed that the temperature of the fluid phase isequal to the temperature of the solid phase everywhere in the por-ous region, and the Local Thermal Equilibrium (LTE) model is appli-cable in the present investigation. The widely used models in theliterature to study the flow problems in porous media are the Darcymodel, the Darcy–Brinkman model, and the Darcy–Brinkman–Forchheimer models. Apart from these models, the Brinkman-extended Darcy model with the convective terms has also beenextensively used in modeling the flow and heat transfer in finiteporous enclosures. In the present study, the Brinkman-extendedDarcy model with the inclusion of convective and transient termshas been adopted in the governing equations of the problem. TheForchheimer inertia term in the momentum equations is neglectedand a brief discussion of the exclusion of this term can be found inSathiyamoorthy et al. [39] and Kaluri et al. [40]. The Brinkman-extended Darcy model, adopted in the present study, has beenused in a large number of investigations for natural convection inannular and rectangular porous enclosures [21,38–40]. By employ-ing the aforementioned approximations, the time dependentgoverning equations for the conservation of mass, momentumand energy in an isotropic and homogeneous porous medium canbe written as

@u@rþ @w@xþ u

r¼ 0; ð1Þ

1/@u@tþ 1

/2 u@u@rþw

@u@x

� �¼ � 1

q0

@p@rþ te r2

1u� ur2

h i� tf

Ku; ð2Þ

1/@w@tþ 1

/2 u@w@rþw

@w@x

� �¼ � 1

q0

@p@xþ ter2

1w� gbðh� hcÞ �tf

Kw;

ð3Þ

Table 3Comparison of present results with the discretely heated rectangular cavity (A = 1,k = 1, Da =1, u = 1 and Pr = 0.7). The isothermal heater, whose non-dimensionallength is 0.5, is placed at the inner wall of the annulus between X = 0.25 and X = 0.75.

Rayleigh number (Ra) Corcione and Habib [35] Present study

103 0.985 0.9731104 1.879 1.8934105 3.630 3.6452106 6.737 6.7486


@h@tþ u

@h@rþw

@h@x¼ jr2

1h; ð4Þ

where r21 ¼ @2

@r2 þ 1r@@r þ @2

@x2.In the present study, the values of the fluid kinematic viscosity

(tf) and effective kinematic viscosity of the porous medium (te) areassumed to be equal. This approximation provides good agreementwith the experimental data available in the literature. Also, sincethe flow depends only on two spatial co-ordinates, a vorticity–stream function approach is chosen for the present numericalstudy. Hence, by eliminating the pressure terms from the Eqs. (2)and (3), and using the following non-dimensionless variables,

U ¼ uDj

A; W ¼ wHjA

; T ¼ kðh� hcÞðqhDÞ ; R ¼ r � ri

D;

X ¼ xH; s ¼ tj

D2 ; f ¼ f�D2

tf; W ¼ w

rij; D ¼ ro � ri;

the governing Eqs. (1)–(4) reduce to the following dimensionlessvorticity–stream function formulation:

@T@sþ U

A@T@RþW

A@T@X¼ r2T; ð5Þ

1/@f@sþ 1

/2

UA@f@RþW

A@f@X� U

AD

RDþ ri

� �f

� �

¼ Pr r2f� DRDþ ri

� �2

f

" #þ Ra�

@T@R� Pr

Daf; ð6Þ

f ¼ 1Pr

ri

DRþ ri

� �@2W

@R2 �D

DRþ ri

� �@W@Rþ 1

A2

@2W

@X2

" #; ð7Þ

U ¼ ri

DRþ ri

� �@W@X

; W ¼ � ri

DRþ ri

� �@W@R

; ð8Þ

where f ¼ 1Pr

1A2

@U@X � @W

@R

h iand r2 ¼ @2

@R2 þ DDRþri

� �@@Rþ 1

A2@2

@X2.

In the above equations Ra� ¼ gbqhD4

ktf j; Pr ¼ tf

j ; Da ¼ KD2 ; A ¼ H

D are

the modified Rayleigh number, the Prandtl number, the Darcynumber and the aspect ratio. In addition to the above parameters,the present study also involves the parameters, k ¼ ro

rithe radii

Fig. 3. Plots of streamlines (top) and isotherms (bottom) for Ra* = 107, u = 0.9, e = 0.4 an|Wmax| = 16.09.

ratio, L ¼ lH, non-dimensional location of the heater, and e ¼ h

H,non-dimensional length of the heater. It may be noted that, regard-less of the notation used in the cylindrical co-ordinates, thegoverning Eqs. (5)–(8) reduces to that of the Cartesian rectangularco-ordinates when D = 0 (or k = 1) [14,15]. Hence, for the validationof present study, simulations are obtained from the present code tocompare with the existing results of the rectangular cavity.

The dimensionless initial and boundary conditions of the prob-lems under consideration are:

s ¼ 0 : U ¼W ¼ T ¼ 0; W ¼ f ¼ 0; 0 6 R 6 1; 0 6 X 6 1;

s > 0 : W ¼ @W@R¼ 0;

@T@R¼ 0; R ¼ 0 and 0 6 X < L� e

2;

W ¼ @W@R¼ 0;

@T@R¼ �1; R ¼ 0 and L� e

26 X 6 Lþ e

2;

W ¼ @W@R¼ 0;

@T@R¼ 0; R ¼ 0 and Lþ e

2< X 6 1;

W ¼ @W@R¼ 0; T ¼ 0; R ¼ 1 and 0 6 X 6 1;

W ¼ @W@X¼ 0;

@T@X¼ 0; X ¼ 0 and X ¼ 1:

The boundary condition for the vorticity is deduced from Taylor’sseries expansion of the stream function W near the walls and itcan be computed from the following expressions:

f ¼ ri

PrðRDþ riÞ

� �@2W

@R2 ; R ¼ 0;R ¼ 1 and 0 6 X 6 1

f ¼ ri

A2PrðRDþ riÞ

!@2W

@X2 ; X ¼ 0;X ¼ 1 and 0 6 R 6 1

The local Nusselt number along the heat source is defined by

Nu ¼ hDk¼ qhD

kðhh � hcÞ; ð9Þ

d L = 0.5. (a) Da = 10�6, |Wmax| = 0.28, (b) Da = 10�4, |Wmax| = 5.44 and (c) Da = 10�2,

Fig. 4. Effect of Darcy number on the average Nusselt number for L = 0.5 and twodifferent heater lengths. (a) e = 0.4, u = 0.4, (b) e = 0.4, u = 0.9 and (c) e = 0.8,u = 0.9.


where, h is the local heat transfer coefficient at some point of theheater and is given by h ¼ qh=ðhh � hcÞ; where hh is the local temper-ature of the heat source. Using the relation T ¼ kðh�hcÞ

ðqhDÞ , the local Nus-selt number expression (9) can be written as

Nu ¼ 1TðXÞ ; ð10Þ

where T(X) is the dimensionless temperature along the heat source.The average Nusselt number is determined by integrating Nu along

the heat source and is defined as Nu ¼ 1e

R Lþe2

L�e2

NudX.

3. Numerical technique and code validation

The system of coupled partial differential equations (5)–(8)along with the boundary conditions are numerically solved usingan implicit finite difference method. The vorticity transport andenergy equations are solved using the ADI (Alternating DirectionImplicit) method and the stream function equation is solved bySLOR (Successive Line Over Relaxation) method. This technique iswell described in the literature and has been widely used for nat-ural convection in porous enclosures [6,21,25,26]. The SLOR meth-od converges in less iteration than the point iteration methods andimmediately transmits the boundary condition information to theinterior domain. The over relaxation parameter for the streamfunction equation is chosen as 1.7, after several trial runs. Centraldifferencing is used for the buoyancy and diffusion terms, whilethe second upwind difference is preferred for the non-linear con-vection terms for the sake of numerical stability. The velocity com-ponents at every grid point are evaluated using the centraldifference approximations to velocity–stream function relation.Finally, the average Nusselt number is obtained by using theSimpson’s rule.

A uniform grid is used in the R–X plane of the annulus and thenumerical results are checked for the grid independence. In orderto determine a proper grid size for the present numerical study,a grid independence test has been conducted for Ra* = 107,Pr = 0.7, e = 0.4, L = 0.5, u = 0.9, Da = 10�4 and k = 2. Four differentgrids 51 � 51, 81 � 81, 101 � 101 and 121 � 121 were used. Theaverage Nusselt number and maximum temperature were usedas sensitivity measures of the accuracy of the solution. Table 1shows that the two grids 101 � 101 and 121 � 121 give nearlyidentical results. Hence, considering both the accuracy and thecomputational time, all the computations were performed with a101 � 101 grid. The steady state solution to the problem has beenobtained as an asymptotic limit to the transient solutions. That is,the steady state solution is obtained when the following conver-gence criterion is satisfied:P

i

Pj Unþ1

i;j �Uni;j

�� Pi

Pj Unþ1

i;j

�� 6 C:

Here U is any variable W, f, T, and C is a pre-specified constant, usu-ally set to 10�7. Also, in the above expression, (i, j) refers to spaceco-ordinates and n refers to time. A FORTRAN code has been devel-oped for the present numerical method and it has been successfullyvalidated against the available benchmark solutions in the litera-ture before obtaining the simulations.

3.1. Validation

To verify the numerical code, simulations of the present modelare tested and compared with different benchmark solutions avail-able in the literature for the cylindrical and rectangular cavities,filled with a porous media or a clear fluid, and with uniform as wellas discrete heating of the inner wall. First, the numerical results for

different Rayleigh numbers and radius ratios are obtained for nat-ural convection in a vertical annulus without porous medium(Da =1). The inner and outer walls of the annulus are respectivelymaintained at uniform heat flux and constant temperature, and thehorizontal walls are kept adiabatic. Fig. 2 illustrates the compari-son of average Nusselt numbers between the present study andthe correlation data of Khan and Kumar [25] for a vertical annulusat different radius ratios. From the figure, an overall good degree of


agreement can be observed between the present results and thecorrelation data. Further, the present numerical technique hasbeen successfully used to study the effect of discrete heating in avertical non-porous annulus and more validation of the presentmethod can be found in Sankar and Do [26].

To further validate the present numerical results, the averageNusselt numbers are obtained by putting D = 0 in the governingequations of present study to compare with the rectangular porouscavity. The quantitative results are compared with the correspond-ing solutions of Waheed [42] for a rectangular porous cavity andare given in Table 2. As can be seen from Table 2, the results ofthe present simulation agree well with the results of Waheed[42] over the entire range of Rayleigh and Darcy numbers. Finally,due to lack of suitable theoretical or experimental results for a dis-cretely heated cylindrical annular cavity, the present model hasbeen validated against the discretely heated rectangular cavityinvestigated by Corcione and Habib [35] in the absence of porousmedium. To perform this validation, the average Nusselt numbersare measured along the inner wall of the annular cavity by puttingD = 0 in Eqs. (5)–(8) and by considering an ‘‘isothermal’’ heatsource. The non-dimensional length of the heat source is takenas 0.5 and is placed between X = 0.25 and X = 0.75. The comparison,shown in Table 3, reveals a good agreement between our resultsand that of Corcione and Habib [35]. From Fig. 2, and Tables 2and 3, the correspondence between the present results and litera-ture data is widely satisfactory. Through these validation tests, theaccuracy of the present numerical computation is assured.

4. Results and discussion

The effect of discrete heating caused by an isoflux discrete hea-ter on the buoyancy driven convection in an annular enclosure,

Fig. 5. Effect of heater length on the streamlines (left) and isotherms (right) for Da = 10�3

which is filled with a fluid-saturated porous media is numericallyinvestigated. One of the fundamental problems in cooling of elec-tronic devices is the optimal size and positioning of a discrete heatsource in finite enclosures. In many situations, a heat transfer de-signer prefers to avoid the use of mechanical fans or other activedevices for fluid circulation, due to power consumption, excessiveoperating noise or reliability concerns. Hence, the placement ofthese heaters within the enclosure requires to be optimized so thatthe heat transfer from the isoflux heater to the adjacent fluid ismaximum. Accordingly, five different locations (L = 0.2, 0.4, 0.5,0.6 and 0.8) of the heater are considered in the present study. Also,since the electronic devices involve heater strips of different sizes,the isoflux heater embedded on the inner wall of the annulus hasbeen considered of four different lengths (e = 0.2, 0.4, 0.6 and0.8). For each value of e and L, the physical parameters are variedover a wide range of values (the modified Rayleigh number1036 Ra* 6 107, the Darcy number 10�6

6 Da 6 10�1 and theradius ratio 1 6 k 6 5). Thus, the computations are carried out fora total of more than 250 combinations of the above parameters.Since the present study involves eight parameters, the numericalsimulation for all combinations of parameters is quite large. Hence,the computations are limited to the annular enclosure with unitaspect ratio and the value of Prandtl number is respectively fixedat Pr = 0.7. The porosity (u) values are taken as 0.4 and 0.9. Theflow and temperature fields in the porous annulus are presentedin terms of streamlines and isotherms to illustrate the influenceof the isoflux heater. Also, the variation of heat transfer rate fromthe heater are presented in terms of the average Nusselt numberfor different values of Darcy number and modified Rayleigh num-bers. In addition, the variation of maximum temperature at theheater is discussed in detail. The above parametric simulationsare performed to develop basic information on understanding the

, Ra* = 107, u = 0.9 and L = 0.5. (a) e = 0.2, |Wmax| = 9.29 and (b) e = 0.8, |Wmax| = 14.44.

Fig. 6. Effect of heater length on the velocity and temperature profiles along X = 0.5for Da = 10�3, Ra* = 107, u = 0.9 and L = 0.5.


size and location effects of an isoflux discrete heater on the fluidflow and corresponding heat transfer characteristics in an annularenclosure filled with a fluid-saturated porous media, a situationthat has not been attempted in the existing literature.

4.1. Effect of Darcy number

Fig. 3 exhibits the streamlines and isotherms to reveal thehydrodynamic flow and thermal fields inside the porous cavityfor three different Darcy numbers, as the representative cases.The Darcy numbers are chosen as 10�6, 10�4 and 10�2, to simulatethe limiting situations of Darcy and viscous flows. An overview ofthe figure reveals that the flow strongly depends on the Darcynumber. This is expected, since the Darcy number is directly asso-ciated to the permeability of the porous medium, and as a result, itsignificantly affects the flow and heat transfer in the annulus. Asimple circulating flow pattern with the centre of rotation at themiddle of the cavity is observed for a low value of Darcy number(Da = 10�6). At low Darcy number, Da = 10�6, the convective mo-tion in the annulus is found to be weak due to the resistance gen-erated by the boundary friction and also the bulk frictional draginduced by the solid matrix. This is quite apparent from the lowDarcy number result in Fig. 3a. It reveals that, although the modi-fied Rayleigh number is relatively high, the flow is unable to pen-etrate deeper into the porous medium due to the high resistanceproduced by the porous medium. From the parallel isotherms, itcan be seen that the transfer of heat from the discrete heater ismainly controlled by the conduction-dominated mechanism dueto the porous drag.

However, as the Darcy number is increased from10�6 to 10�4,viscous effects become more important. As pointed out by Lauriatand Prasad [6] for a rectangular porous cavity, the value ofDa = 10�4 represents the limit where the viscous effects are impor-tant in a porous medium and the resulting viscous force increasesthe velocity as Darcy number increases. At this Darcy number, con-vection prevails and thus the magnitude of maximum stream func-tion increases. The nearly-parallel isotherms, at Da = 10�6, revealsa significant variation due to the presence of convection. As theDarcy number further increases to 10�2, the permeability of theporous medium increases and hence the resistance from theboundary friction has been gradually reduced, and the flow is akinto pure buoyancy induced flow. At this Darcy number, the effects ofviscous forces will be dominant and hence the flow velocitybecomes significant. As a result, the streamlines exhibit a strongflow pattern with the main vortex moved towards the cold wall.On comparing Fig. 3a and c, it can be noticed that the strength ofconvective flow becomes stronger as the value of Darcy numberis increased. Further, as the Darcy number increases, the flowpenetrates deeper into the porous medium as can be seen fromthe streamlines and isothermal fields. The isotherms show thepresence of relatively stronger gradients at the middle of the annu-lus. The general conclusion based on these plots is that increasingthe Darcy number helps the flow to penetrate deeper into the por-ous layer.

Fig. 4 depicts the influence of Darcy number on the averageNusselt number at different values of modified Rayleigh numbers.Two different heater lengths (e = 0.4 and 0.8) and porosities(u = 0.4 and 0.9) are considered for a fixed heater location atL = 0.5. An overview of the figure reveals that the average Nusseltnumber increases with Darcy number and porosity, due to thehigher permeability of the medium which results in larger flowvelocity. However, an increase in Da beyond 10�2 has little effecton the average Nusselt number for all values of Ra*. Also, the heattransfer rate decreases with an increase in the heater length for allvalues of Ra*. At low value of Darcy number, the fluid flow experi-ences more resistance, and hence the average Nusselt number is

almost flat at all values of the modified Rayleigh number. It is alsoobserved that the heat transfer increases sharply for Da > 10�5,while the variation is minute when Da < 10�5. Further, the steepincrease in the Nusselt number curve is progressively delayed asthe Darcy number decreases. This is quite evident from the curvesof Da = 10�5 and 10�6. This can be attributed due to the additionalresistance to the flow caused by the porosity of the medium at lowDarcy numbers. These results are consistent with the fact that forhigh values of Da, the Darcy term becomes small, while theBrinkman term in the momentum equation becomes small forlow values of Da. Another important feature of Fig. 4 is the fact thata smaller size heater (e = 0.4) may transport a larger amount ofheat compared to the heater with larger length (e = 0.8). The heattransfer at a low value of Darcy number (Da = 10�6) representsthe physical limit of an almost impervious porous medium.

4.2. Effect of heater length

Fig. 5 demonstrate the effects of heat source length on thestreamlines and isotherms for two different values of e (e = 0.2and 0.8) with the heat source located in the middle of the innerwall (L = 0.5). It is worth noticing that the size of the heater has adirect influence on the intensity of the flow. Since the discrete heatsource remains in the middle of the inner wall, the flow structure isnot altered in spite of changing the length of the heat source.


Although the shape of the circulating cells does not change appre-ciably with heat source length, their intensity increases as the heatsource length increases. This can be attributed to the fact thathigher heat transfer rates are generated as the heat source lengthincreases. A careful observation of Fig. 5 reveals that the magnitudeof |Wmax|, which determines the extent of flow circulation, is rela-tively higher for larger heater length (e = 0.8) compared to smallerheater length (e = 0.2). An overview of the figures reveals that theconvective cells in the annulus are intensified and higher temper-ature patterns can be observed from the isotherms, as the heatsource length increases from e = 0.2 to e = 0.8. In order to have abetter understanding of the flow and thermal behaviour with theheat source length, the axial velocity along the mid-section ofthe annulus and the temperature at the heater wall are presentedin Fig. 6. It is clear that the absolute magnitude of axial velocity andwall temperature increases by increasing the heat source length,due to the greater heat inducing strong buoyant flow in theannulus.

As regards to the influence of heater size on the heat transfer,the average Nusselt numbers are displayed for four heater lengthsat various values of Ra* and two values of Da. To demonstrateproperly the true effects of discrete heating in the porous annulus,the average Nusselt numbers are also obtained for various Darcynumbers and heater lengths by fixing the modified Rayleigh num-ber at 107. The porosity values are chosen as u = 0.4 and 0.9. Anoverview of Fig. 7a and b reveals two different kinds of variationof average Nusselt number that greatly depends on the value of

Fig. 7. Effect of heater length on the average Nusselt number for different values of RRa* = 107, u = 0.4 and L = 0.5 and (d) Ra* = 107, u = 0.9 and L = 0.5.

Darcy number. Since an increase in Ra* characterizes the enhance-ment in buoyancy forces, the average Nusselt number shouldincrease with modified Rayleigh number. However, whenDa = 10�5, the average Nusselt number curve is almost flat untilRa* � 106 and then increases sharply due to high buoyancy forcesat Ra* = 107. This can be expected, since at low values of Da, theDarcy resistance is greater, and therefore, the flow intensity insidethe annulus decreases, which results in a reduction of convectiveheat transfer for Ra* < 106 (Fig. 7a). On the contrary, when Darcynumber increases, the heat transfer rate monotonically increaseswith the modified Rayleigh number (Fig. 7b).

The effects of heat source length on the cooling performance ofthe discrete heater in the annulus can be more clearly understoodfrom Fig. 7, where the average Nusselt number is plotted for arange of modified Rayleigh and Darcy numbers. In general, increas-ing the heat source length decreases the average Nusselt number.That is, the average Nusselt number is found to be higher for asmaller heater length (e = 0.2) rather than the heater with largerlength (e = 0.8) at all modified Rayleigh numbers (Fig. 7a and b).On the other hand, when considering the velocity and temperatureprofiles in Fig. 6, the observed variation in temperature and veloc-ity was opposite. That is, the temperature profile along the heaterwall is found to be higher when the size of the discrete heater islarge (Fig. 6b). This is due to the fact that the temperature at theheater wall is not uniform for a constant heat flux condition, andhence one would expect that the wall temperature has a maximumvalue where the temperature difference between the heater wall

a* and Da. (a) Da = 10�5, u = 0.9 and L = 0.5, (b) Da = 10�1, u = 0.9 and L = 0.5, (c)

Fig. 8. Effect of heater position on the streamlines (left) and isotherms (right) forDa = 10�3, Ra* = 107, u = 0.9 and e = 0.4. (a) L = 0.2, |Wmax| = 13.94, (b) L = 0.4,|Wmax| = 12.51, (c) L = 0.6, |Wmax| = 10.89 and (d) L = 0.8, |Wmax| = 9.57.


and the adjacent fluid is minimum. Also, the local Nusselt number(Eq. (10)) is the reciprocal of the dimensionless temperature for theconstant heat flux condition, and hence at the point where the walltemperature is maximum, the corresponding Nusselt number as-sumes the minimum value. These predictions are in good agree-ment with Sharif and Mohammad [33] for natural convection inan inclined rectangular enclosure with a constant flux heating ele-ment at the bottom wall. On the contrary, for the case of isother-mal discrete heating, the average Nusselt number increases withincreasing the size of the heat source [35]. The variation of averageNusselt number for different values of Darcy number and heaterlength is shown in Fig. 7(c) and (d) for fixed values of Ra*, L andu = 0.4, 0.9. As stated above, at low Darcy numbers, the fluid flowis resisted by the porous medium and the resulting heat transfer isless compared to higher values of Da. Also, the slope of the Nusseltnumber curve decreases with an increase in the value of Da, andfinally approaches zero. This reveals an important fact that thereexists an asymptotic convection regime where the heat transferrate is independent of the Darcy number, and depends only onthe modified Rayleigh number. This has been clearly demonstratedin the numerical results of Lauriat and Prasad [6] for a rectangularporous enclosure. On comparing the Fig. 7(c) and (d), it is observedthat the average Nusselt number increases, at high values of Da, asthe porosity (u) increases from 0.4 to 0.9.

4.3. Effect of heater location

In this part of the study, the effect of heater location on the flowand temperature distributions, and the corresponding heat transferis discussed by fixing the values of Ra* = 107, Da = 10�3, k = 2,u = 0.9 and e = 0.4. Fig. 8 depicts the effect of heater location onthe development of the streamlines (left) and isotherms (right)for four different locations of the heater. A highly stratified med-ium with almost parallel, horizontal flow in the core results whenthe heat source is located at the bottom portion of the inner wall.As the heat source moves towards the top wall, the main vortex re-duces in size and shift towards the cold wall, and also the symmet-ric structure of the streamlines is destroyed. Further, the relativestrength of the flow as indicated by the maximum absolute streamfunction reduces as the heat source move upwards. That is, the rateof fluid circulation is found to be higher, when the isoflux heater isplaced near the bottom wall. This can be explained due to the dis-tance that the fluid needs to travel in the circulating cell to ex-change the heat between the heat source and outer cold wall. Infact, the closer the heat source is to the bottom wall, the higherthe magnitude for the stream function that is achieved. These pre-dictions are consistent with those reported by Saeid and Pop [36]for natural convection in a square porous cavity with a single iso-thermal or isoflux heat source mounted on the left wall. As theheater moves upwards, the flow strength is reduced which resultsin a portion of the fluid remaining stagnant at the bottom of theenclosure, and is vividly reflected in the corresponding isotherms(Fig. 8d).

Fig. 9 depicts the effects of the heat source location on the aver-age Nusselt number at different modified Rayleigh and Darcy num-bers. In Fig. 9(a) and (b), the Darcy number is fixed respectively at10�5 and 10�1 and the values of Ra* and L are varied, whereas inFig. 9(c) and (d), the value of Ra* is fixed at 107 and the Darcy num-ber and heater locations are varied for two different porosities(u = 0.4 and 0.9). The effect of low Darcy number or low perme-ability on the heat transfer is very much apparent from Fig. 9(a).When the modified Rayleigh number is in the range of 103–105,the magnitude of average Nusselt number at all five different loca-tions is same. However, this trend changes when the Darcy numberis increased to 10�1 due to the dominance of convection. Anotherimportant observation that can be made from Fig. 9(a) and (b) is

the location of maximum average Nusselt number for different val-ues of Ra*. It is observed that, at low values of modified Rayleighnumber, the average Nusselt number attains the maximum valuewhen the heat source is placed at L = 0.5. But, the location of max-imum average Nusselt number shifts towards the lower half of theinner wall as the modified Rayleigh number increases. This indi-cates that the location of the heat source plays a crucial role indetermining the removal of heat from the heater to the surround-ing fluid at different Rayleigh numbers. Since conduction is the ma-jor mode of heat transfer at low values of Ra*, higher value of the


average Nusselt number is found near the middle portion of the in-ner wall, and for convection dominant cases, the location shifts to-wards the bottom wall of the annulus. A similar observation wasreported by Saeid and Pop [36] for a square porous enclosure withdiscrete heating.

The effect of heater location on the streamline pattern reveals astrong flow circulation in the annulus when the heat source isplaced close to the bottom wall (Fig. 8). But, a careful observationof Fig. 9 reveals that the heat transfer is maximum when the iso-flux heater is placed around the mid-height of the enclosure ratherthan placing the heater near the bottom portion of the inner wall.This may be explained due to the fact that the rising fluid cannotwipe the entire surface of the heater, when the heat source isplaced very near to the bottom or top wall of the enclosure. There-fore, the optimal heat source location for maximum heat transfernot only depends on the circulation intensity, but also dependson the shape of the buoyancy driven flow. This trend has been ob-served in the previous investigation for natural convection heattransfer in a tilted rectangular cavity with an isothermal heatsource [35]. The variation of heat transfer with the different loca-tions of the heater at various Darcy numbers is shown in Fig. 9(c)and (d) for Ra* = 107, e = 0.4 and two values of u. For all Darcynumbers, the average Nusselt number increases up to the middleportion of the inner wall and then decreases. As stated before,Fig. 9(c) and (d) vividly illustrates the influence of heat source loca-tion on the maximum average Nusselt number for different Darcynumbers. For low Darcy numbers, the average Nusselt number ismaximum near L = 0.5, and as the value of Da increases, the loca-tion for maximum heat transfer moves towards the lower half ofthe inner wall. Further, it is observed that the effect of porosity

Fig. 9. Variation of average Nusselt number with different heater positions. (a) Da = 10�5,and (d) Ra* = 107, u = 0.9 and e = 0.4.

on the average Nusselt number is mild for Da = 10�6 to 10�4, whileits effect is noticeable for Da > 10�4 at all locations of theheater.

4.4. Effect of radius ratio

The influence of radius ratio on the flow patterns and tempera-ture fields for Ra* = 107, L = 0.5, e = 0.4, u = 0.9 and Da = 10�3 arepresented in Fig. 10. It is observed from the flow pattern that withan increase in radius ratio, the main vortex of the stream functionshifts towards the top right corner of the annulus with enhancedstrength. Also, the isotherms accumulate near the bottom of theheat source, and this can be attributed to flow acceleration towardsthe inner wall at higher value of radius ratio (Fig. 10b). A similarshift in isotherms has also been previously reported by Havstadand Burns [11], Hickox and Gartling [12] and Prasad and Kulacki[13]. Furthermore, the temperature gradient near the hot wall in-creases rapidly as the radius ratio increases, and the crowded iso-therms indicate a thin thermal boundary layer around the heater,which further influences the heat transfer. This result follows fromthe fact that with increasing k, the annulus width increases, and thefluid volume associated with the strong temperature and velocitygradients near the inner boundary increases. The packed stream-lines near the top right corner of the cavity also indicate thin veloc-ity boundary layers in that region. This trend is in full accordancewith what was previously observed by Prasad and Kulacki [13]and Prasad [14] for the natural convection heat transfer in a verticalporous annulus for isothermal as well as isoflux heating conditions.Further, it can be observed that the curvature of the annulusdestroys the centrosymmetric properties of the temperature and

u = 0.9 and e = 0.4, (b) Da = 10�1, u = 0.9 and e = 0.4, (c) Ra* = 107, u = 0.4 and e = 0.4

Fig. 10. Plots of streamlines (top) and isotherms (bottom) for Ra* = 107, u = 0.9, L = 0.5, e = 0.4, Da = 10�3 at (a) k = 1 and (b) k = 5. The values of |Wmax| are (a) 8.33 and (b)18.45.

Fig. 11. Effect of radii ratio on the average Nusselt number for e = 0.4, L = 0.5,u = 0.9 at two different Darcy numbers.


flow fields observed in the case of unit radius ratio (rectangularcavity).

In order to have a better understanding of curvature effects onthe convective heat transfer in the porous annulus, the averageNusselt number is depicted in Fig. 11 for three different values ofradius ratios and two values of Da. An overview of the figure re-veals that the average Nusselt number increases with radius ratio,since increasing k decreases the effective sink temperature for theboundary layer on the inner wall. At low Darcy numbers, the aver-age Nusselt number curve remains almost flat until Ra* = 106, andthen increases slightly. This feature can be attributed to the exis-tence of a thin thermal boundary layer adjacent to the inner wallat low Darcy numbers. However, the magnitude of the averageNusselt number is high for the combination of higher values ofRa* and Da, since the buoyancy driven flow is predominant at thiscombination of Ra* and Da. In general, the heat transfer curve isnon-monotonic for low value of Darcy number, whereas it ismonotonic for high value of Darcy number.

4.5. Maximum temperature

One of the important problems in the cooling of electronic de-vices is the maximum temperature or hot spots appearing along

the chips (discrete heaters). The hot spots, if they exist, may havean adverse effect on the circuitry system if it is not paid due atten-tion and hence this quantity is carefully examined for dependence


on different lengths and locations of the heat source. The variationof maximum surface temperature with the modified Rayleighnumber is shown in Fig. 12 for four heater lengths and four heaterlocations. In the figure, two different Darcy numbers are chosen tounderstand the effect of low and high permeability of the porousmedium on the maximum temperature. It can be seen that in theconvection dominated flow regime, the maximum temperature de-creases as the modified Rayleigh number increases due to strongbuoyancy forces. As the length of the heat source increases, themaximum temperature continuously increases due to the higherheat flux generated by the heat source. Fig. 12 shows that at lowDarcy numbers (Da = 10�5), regardless of the size and location ofthe heat source, the maximum temperature remains constant forRa* 6 106. This can be attributed to the conduction dominated heattransfer mechanism due to porous resistance. However, the maxi-mum temperature decreases rapidly with modified Rayleigh num-ber at high values of Da. Also, the maximum dimensionlesstemperature is found be high when the heater length is larger forboth low and high Darcy numbers. As regards to the variation ofmaximum temperature with the heater location, we found thatthe maximum temperature is lower when the heater is placed nearthe lower half of the inner wall, whereas it is higher when the hea-ter is placed near the top adiabatic wall. This result is consistentwith the earlier predictions that the hot spots or maximum tem-perature always appear when the heater is placed near the top hor-izontal wall for convection dominated heat transfer. At low Darcynumber, the magnitude of maximum temperature reveals only

Fig. 12. Variation of maximum temperature with (a) heater lengths and (b) heaterpositions. In (a) L = 0.5, u = 0.9 and (b) e = 0.4, u = 0.9.

two types of variations at four different locations for Ra* 6 105,and then decreases for Ra* > 105. That is, the maximum tempera-ture is the same for the cases L = 0.2 and 0.8, and L = 0.4 and 0.6.In general, the figure reveals that the maximum surface tempera-ture increases with heater length, but decreases with the Darcynumber.

5. Conclusions

The present numerical investigation exhibits many interestingresults concerning the effect of discrete heating on the naturalconvective heat transfer in a vertical porous annulus using theBrinkman-extended Darcy equation. An isoflux discrete heater isplaced at the inner wall of the annular cavity, while the outer wallis kept at a lower temperature. Our efforts have been focused onthe size and location effects of the heater on the fluid flow and heattransfer characteristics for a wide range of parameters of theproblem.

The size and location of the heat source have different effects onthe fluid motion intensity and the rate of heat transfer. That is, theannular enclosure is significantly affected by the buoyancy drivenflow when the discrete heater is either larger or occupies a lowerposition in the cavity. However, the rate of heat transfer is foundto be higher when the heater is small or placed at mid-height ofthe cavity. For a fixed heater length, the maximum heat transferrate changes with the Darcy number and location of the heater.For low Darcy number (10�6), the rate of heat transfer is higherwhen the heater is located at the middle of the inner wall, and asthe value of Da increases, this location shifts towards the lowerend of the inner wall. At low Darcy number (10�5 and 10�6), forall combination of parameters, the fluid flow is weak and the heattransfer in the annulus is conduction-dominant due to the hydrau-lic resistance of the porous medium.

An increase in the radius ratio is seen to shift the fluid towardsthe cold wall and the average Nusselt number increases as the ra-dius ratio increases for both low and high Darcy numbers. Themaximum temperature increases with an increase in the heatsource length, while it decreases with an increase in the modifiedRayleigh number and Darcy number. As regards to the location ofthe heater, the magnitude of maximum temperature is lower, forboth the low and high Darcy numbers, when the heater is placedat lower half of the inner wall. At low Darcy number, the maximumtemperature remains in variant for Ra* < 106. For different size andlocations of the heater, the effects of porosity on the overall heattransfer rate is small at low Darcy numbers, while its effect be-comes significant at higher values of Darcy number. Also, the aver-age Nusselt number increases with an increase in porosity of themedium.

Acknowledgements

This work was supported by WCU (World Class University) pro-gram through the Korea Science and Engineering Foundationfunded by the Ministry of Education, Science and Technology(Grant No. R32-2009-000-20021-0). The author Sankar would liketo acknowledge the support and encouragement of the Chairmanand Principal of East Point College of Engineering and Technology,Bangalore, India.

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