Energy Conversion and Management 67 (2013) 287–296

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Energy Conversion and Management

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Heatline based thermal management for natural convection in porous rhombicenclosures with isothermal hot side or bottom wall

R. Anandalakshmi, Tanmay Basak ⇑Department of Chemical Engineering, Indian Institute of Technology Madras, Chennai 600 036, India

a r t i c l e i n f o

Article history:Received 4 July 2012Received in revised form 23 November 2012Accepted 29 November 2012Available online 8 January 2013

Keywords:Porous rhombic enclosureDifferential heatingRayleigh–Benard convectionHeatlineDarcy numberConvection based dynamic solution

0196-8904/$ - see front matter � 2012 Elsevier Ltd. Ahttp://dx.doi.org/10.1016/j.enconman.2012.11.022

⇑ Corresponding author. Tel.: +91 44 2257 4173; faE-mail addresses: anandalakshmir@gmail.com (R. A

m.ac.in (T. Basak).

a b s t r a c t

An accurate prediction of the flow structure and heat distribution in rhombic configurations are of greaterimportance due to its significant engineering i.e. cooling of electronics devices as well as natural appli-cations i.e. geothermal extraction. Heatline method is used to analyze natural convection in porous rhom-bic enclosures with various inclination angles, u for differential (case 1) and Rayleigh–Benard heatingsituations (case 2). Increase in u(u = 90�) results in pure conduction dominant heat transfer with stag-nant fluid condition for u = 90� at Da = 10�5 and a slight perturbation of u at higher Da (Da P 4 � 10�5)leads to convection based dynamic solution for u = 90� in case 2 irrespective of Pr. At Da = 10�3, strengthof fluid and heat flow increase with u due to enhanced convection effect and u = 90� shows maximummagnitude of streamfunction (wmax) and heatfunction (Pmax) values in both cases except convectionbased solution at u = 90� for Pr = 7.2. Both cases are compared based on local (Nu) and average Nusseltnumbers ðNuÞ and those are adequately explained based on heatlines. Also, Nu increases with Da in bothcases except convection based solution at u = 90� for Pr = 7.2. Overall, Nu is higher for case 2 at u 6 45�whereas case 1 shows larger Nu for u P 45� irrespective of Pr at Da = 10�3. Hence, u = 45� is the criticalrhombic angle which demarcates the heating strategies of case 1 and case 2 to achieve higher heat trans-fer rates ðNuÞ in various applications.

� 2012 Elsevier Ltd. All rights reserved.

1. Introduction

The problem of natural convection in porous media have severaldifferent physical phenomena over convective flows in non-porousmedia and they have been widely studied by the researchers due toits vivid practical applications which can be modeled as transportphenomena in porous media. These types of flows extensively ap-pear in industrial applications such as cooling of electronic compo-nents, packed bed reactors, food processing, thermal insulationengineering, casting and welding of manufacturing processes, alloycasting, turbomachinery, and natural applications such as geother-mal extraction, storage of nuclear waste material, ground waterflows, agricultural water distribution, oil recovery processes, pollu-tant dispersion, dispersion of chemical contaminants, and soil pol-lution. A comprehensive review concerning those processes maybe found in the recent books [1,2].

A significant amount of literatures is available on the convec-tion patterns in enclosures for various applications which analyzevarious aspects of problems such as type of convective flow, hori-zontal/vertical thermal gradient, strength of flow, stability of flow,and net heat transfer, [3–11]. Although there are several compli-

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x: 91 44 2257 0509.nandalakshmi), tanmay@iit-

cated enclosure models, which are being used for investigatingand describing natural convection processes in energy relatedapplications [12–18], considerable efforts are still needed to obtaina unique porous enclosure design for efficient heat transfer pro-cesses since a fundamental understanding of transport phenomenain porous media is very much essential in various applications.

Most of the recent studies analyzes several aspects of the prob-lem of convection flows in fluid saturated porous media [19–21].Analysis on natural convection in square porous enclosures, wavyporous enclosures and thermosolutal convection in square porousenclosures were also considered by several authors [22–24] toinvestigate the effect of porous media for various applications.Though the problems in fluid saturated porous media are wellunderstood, almost all existing theoretical and experimental stud-ies consider the case which involves simple geometries.

In view of various applications of energy efficient processes, acomprehensive understanding of heat transfer and flow circula-tions within non-rectangular cavities are very much essential forindustrial development. Rhombic enclosures have received atten-tion for current work as various shapes including squares can beformed based on inclination angles. Also, rhombic cavities are sim-ple geometrical shapes which can be used in many areas such asbuilding structures, solar collectors, electronics thermal control,and geothermal applications. Although a few number of numericalinvestigations have been carried out in rhombic enclosures

Nomenclature

cf drag coefficientd diameter of the packing materials in the porous bedDe equivalent diameter of the porous bedDa Darcy numberg acceleration due to gravity, m s�2

K permeability, m2

k thermal conductivity, W m�1 K�1

L side of the rhombic cavity, mN total number of nodesn normal direction to a planeNu local Nusselt numberNu average Nusselt numberp pressure, PaP dimensionless pressurePr Prandtl numberRa Rayleigh numberRam Modified Rayleigh numberS dimensionless distance along inclined wallst tangential direction along a planeT temperature of the fluid, KTh temperature of hot wall, KTc temperature of cold wall, Ku x component of velocity, m s�1

U x component of dimensionless velocityv y component of velocity, m s�1

V y component of dimensionless velocityX dimensionless distance along x coordinate

x distance along x coordinate, mY dimensionless distance along y coordinatey distance along y coordinate, m

Greek symbolsa thermal diffusivity, m2 s�1

b volume expansion coefficient, K�1

c penalty parameterh dimensionless temperaturem kinematic viscosity, m2 s�1

q density, kg m�3

u inclination angle with the positive direction of X axisU basis functionsw dimensionless streamfunctionP dimensionless heatfunctionX two dimensional domain

Subscriptsb bottom walleff effective properties of fluid saturated porous mediaf fluidk local node numberl left wallr right walls side wall

288 R. Anandalakshmi, T. Basak / Energy Conversion and Management 67 (2013) 287–296

[25–27], the detailed analysis of heat flow was poorly understooddue to improper visualization tool.

Design of efficient thermal management system involves posi-tioning the components to facilitate convection cooling andarranging several such components to dissipate significant amountof heat [28]. Also, if the flow of air is blocked by the componentswithin enclosures during convective air cooling, the performanceof cooling reduces significantly. Hence, it is necessary to considerthe effect of permeability and heterogeneity on flow velocities inaddition to clear fluid flow. In sum, geometrical orientation of por-ous domains plays an important role in the ability to design opti-mal thermal management design to save possible device failures.The motivation for this work arises to address the effects of geo-metrical orientation, thermal boundary conditions involving differ-ential heating (case 1) and Rayleigh Benard convection (case 2) onenhanced heat transfer rate during natural convection in rhombiccavities filled with porous media, in the context of various energyrelated applications.

The present article attempts to analyze natural convectionwithin porous rhombic cavities with various inclination angles ofside walls based on heatlines approach. The heat flow visualizationin case of convection heat transfer through a two-dimensional do-main is non-trivial as heat flux lines are non-orthogonal to the iso-therms to analyze the direction and intensity of the convectivetransport processes. In order to analyze the convection heat trans-fer, there is a need for proper tool by which one can visualize theheat flow distributions and heatlines have been used to illustratethe flow of heat within cavities.

The ‘heatline’ concept was first developed by Kimura and Bejan[29] to visualize convective heat transfer. Bejan [30] also analyzedthe heatline approach for various physical situations. Heatfunc-tions are energy analog of streamfunctions in such a way thatthe former intrinsically satisfies the thermal energy equation for

energy flows while the later plays the same role in mass continuityequation to explain fluid dynamics. Heatlines are mathematicallyrepresented by heatfunctions which are in turn related to Nusseltnumber based on proper dimensionless form. A few number ofarticles were presented using this heatline concept for variousphysical situations [31–35].

The objective of this article is to analyze the thermal energy andfluid flow through the porous rhombic enclosure of insulated hor-izontal walls with isothermally heated left wall and isothermallycooled right wall (case 1: differential heating) and insulated verti-cal walls with isothermally heated bottom wall and isothermallycooled top wall (case 2: Rayleigh Benard convection) for variouspractical applications. Generalized non-Darcy model without theForchheimer inertia term is used to predict the flow in porousmedium. This generalized model based on volume averaging prin-ciples was developed by Vafai and Tien [36]. Numerical investiga-tions on heating characteristics within porous rhombic cavitieshave been carried out based on coupled partial differential equa-tions of momentum and energy which are solved using Galerkin fi-nite element method with penalty parameter to obtain thenumerical simulations in terms of streamfunctions, heatfunctionsand isotherms. The streamfunctions and heatfunctions of Poissonequation are also solved using Galerkin finite element method.Nonlinear dynamics related aspects of natural convection andmixed convection problems are considered by various researchersin the recent past and many of them were solved by well-tested ro-bust finite element methods (FEM) [37–42]. A number of modelfluid for almost all industrial application ranges (0.015 6 Pr 6 7.2)have been considered for current analysis. The heat transfer char-acteristics are studied with the help of local and average Nusseltnumbers for all us in both heating situations. Qualitative trendsof local and average Nusselt numbers with Darcy number are ex-plained based on heatfunction gradients.

R. Anandalakshmi, T. Basak / Energy Conversion and Management 67 (2013) 287–296 289

2. Mathematical modeling and simulations

2.1. Governing equations and boundary conditions

The physical domain is shown in Fig. 1a and b. Thermophysicalproperties of the fluid in the flow field are assumed to be constantexcept the density in buoyancy term and change in density due totemperature variation is calculated using Boussinesq’s approxima-tion. It may be noted that the local thermal equilibrium (LTE) is va-lid i.e., the temperature of the fluid phase (Tf) is equal to thetemperature of the solid phase (Ts) within the porous mediumand similar approximations were also used by earlier researchers[1,43]. The momentum transport in porous medium is based ongeneralized non-Darcy model. However, the velocity square termor Forchheimer term which models the inertia effect, is neglectedhere in the present case as this work deals with natural convectionflow within porous enclosed cavity. This inertia effect is moreimportant for non-Darcy effects on the convective boundary-layerflow over the surface of the body embedded in a high-porositymedium. Also, inertia effects are significant for high velocity flowsespecially in forced convection [44,45].

The quadratic form drag is known as Forchheimer term and thisterm depends on drag coefficient, Cf = 0.55[1 � (5.5d/De)], whered = the diameter of the packing materials in the porous bed andDe = equivalent diameter of the porous bed [1]. The drag coeffi-cient, Cf, is function of porous bed geometry. The coefficient willtend to zero for d/De = 0.1 � 0.2 and the Forchheimer quadraticdrag term will be negligible. The term may be important at lowPr(Pr 6 0.1) for a narrow porous bed of 1 cm thickness [46], butfor a particular geometry and dimensions with proper d/De ratio,the Forchheimer quadratic drag will be negligible even at muchlower Pr. Needless to say that, the drag term may be safely ne-glected for Pr� 0.1 in the range of Ra considered in this studybased on the previous works [46]. Current work is based on thegeometry and dimension of the porous bed (specific packing mate-rials) such that the Forchheimer inertial term may be neglected,which was also adopted from the previous studies [36,43,47,48].However current model involves advection term as well as Brink-man terms to incorporate non-Darcy effects [36]. Under theseassumptions, the governing equations for steady two-dimensionalnatural convection flow in a porous rhombic cavity for conserva-tion of mass, momentum and energy may be written with follow-ing dimensionless variables or numbers:

X ¼ xL; Y ¼ y

L; U ¼ uL

a; V ¼ vL

a; h ¼ T � Tc

Th � Tc

P ¼ pL2

qa2 ; Pr ¼ ma; Da ¼ K

L2 ; Ra ¼ gbðTh � TcÞL3Prm2 : ð1Þ

X,UX,U

ϕϕ

CD

Y,V

ADIABATIC

ADIABATIC A B

g

Porousθ=1

θ=0

(a)

Fig. 1. Schematic diagram of a porous rhombic cavity for various inclination angles u

The governing equations in dimensionless forms are:

@U@Xþ @V@Y¼ 0; ð2Þ

U@U@Xþ V

@U@Y¼ � @P

@Xþ Pr

@2U

@X2 þ@2U

@Y2

!� Pr

DaU; ð3Þ

U@V@Xþ V

@V@Y¼ � @P

@Yþ Pr

@2V

@X2 þ@2V

@Y2

!� Pr

DaV þ Ra Prh; ð4Þ

U@h@Xþ V

@h@Y¼ @2h

@X2 þ@2h

@Y2 : ð5Þ

and the governing equations [Eqs. (3)–(5)] are subject to followingboundary conditions for case 1 and case 2:

Case 1: Differential heating

Y,V

in (a)

U ¼ 0; V ¼ 0;@h@Y¼ 0; Y ¼ 0; 0 6 X 6 1 on AB;

U ¼ 0; V ¼ 0; h ¼ 1; X sinðuÞ � Y cosðuÞ ¼ 0; 0 6 Y 6 sinðuÞ on AD;U ¼ 0; V ¼ 0; h ¼ 0; X sinðuÞ � Y cosðuÞ ¼ sinðuÞ; 0 6 Y 6 sinðuÞ on BC;

U ¼ 0; V ¼ 0;@h@Y¼ 0; Y ¼ sinðuÞ; cosðuÞ 6 X 6 1þ cosðuÞ on DC: ð6Þ

Case 2: Rayleigh Benard convection

U ¼ 0; V ¼ 0; h ¼ 1; Y ¼ 0; 0 6 X 6 1 on AB;

U ¼ 0; V ¼ 0;@h@n¼ 0; X sinðuÞ � Y cosðuÞ ¼ 0; 0 6 Y 6 sinðuÞ on AD;

U ¼ 0; V ¼ 0;@h@n¼ 0; X sinðuÞ � Y cosðuÞ ¼ sinðuÞ; 0 6 Y 6 sinðuÞ on BC;

U ¼ 0; V ¼ 0; h ¼ 0; Y ¼ sinðuÞ; cosðuÞ 6 X 6 1þ cosðuÞ on DC: ð7Þ

Note that, in Eqs. (1)–(7), x and y are the distances measuredalong the horizontal and vertical directions, respectively; u and vare the velocity components in the x and y directions, respectively;T denotes the temperature; m and a are kinematic viscosity andthermal diffusivity, respectively; K is the medium permeability; pis the pressure and q is the density; Thand Tcare the temperaturesat hot and cold walls, respectively; g is the acceleration due togravity; b is the volume expansion coefficient; L is the each sideof the rhombic cavity; Note that Ra, Pr and Da are Rayleigh, Prandtland Darcy numbers, respectively. X and Y are the dimensionlessdistances measured along horizontal and vertical directions,respectively; U and V are dimensionless velocity components inthe X and Y directions, respectively; P is the dimensionless pres-sure; h is the dimensionless temperature; n is the distance mea-sured along the outward perpendicular direction of left or rightwall; u is the inclination angle with the positive direction of X axis.

θ=1

θ=0

X,U

ϕ

A B

CD

ADIABATIC

ADIABATIC

g

Porous

(b)

Case 1: differential heating and (b) Case 2: Rayleigh Benard convection.

290 R. Anandalakshmi, T. Basak / Energy Conversion and Management 67 (2013) 287–296

2.2. Solution procedure

The momentum and energy balance equations [Eqs. (3)–(5)] aresolved using Galerkin finite element method. The continuity equa-tion [Eq. (2)] has been used as a constraint due to mass conserva-tion and this constraint may be used to obtain the pressuredistribution. In order to solve Eqs. (3) and (4), we use the penaltyfinite element method where the pressure, P is eliminated by apenalty parameter c and the incompressibility criteria given byEq. (2) which results in

P ¼ �c@U@Xþ @V@Y

� �: ð8Þ

The continuity equation [Eq. (2)] is automatically satisfied for largevalues of c. Typical values of c that yield consistent solutions are107. Using Eq. (8), the momentum balance equations [Eqs. (3) and(4)] reduce to

U@U@Xþ V

@U@Y¼ c

@

@X@U@Xþ @V@Y

� �þ Pr

@2U

@X2 þ@2U

@Y2

!� Pr

DaU ð9Þ

and

U@V@Xþ V

@V@Y¼ c

@

@Y@U@Xþ @V@Y

� �þ Pr

@2V

@X2 þ@2V

@Y2

!� Pr

DaV

þ RaPr h: ð10Þ

The system of equations [Eqs. 5, 9 and 10] with boundary conditionsis solved by using Galerkin finite element method. Expanding thevelocity components (U,V) and temperature (h) using basis setfUkgN

k¼1 as,

U �XN

k¼1

UkUkðX;YÞ; V �XN

k¼1

VkUkðX;YÞ and h

�XN

k¼1

hkUkðX;YÞ: ð11Þ

The Galerkin finite element method yields the nonlinear residualequations for Eqs. 9, 10 and 5, respectively, at nodes of internal do-main X. The detailed solution procedure is given in an earlier work[49].

2.3. Post processing: streamfunction, Nusselt number and heatfunction

2.3.1. StreamfunctionThe fluid motion is displayed using the streamfunction (w) ob-

tained from velocity components (U and V). The relationships be-tween streamfunction and velocity components for twodimensional flows are

U ¼ @w@Y

and V ¼ � @w@X

; ð12Þ

which yield a single equation

@2w

@X2 þ@2w

@Y2 ¼@U@Y� @V@X

: ð13Þ

The no slip condition is valid at all boundaries as there is no crossflow. Hence, w = 0 is used in residual equations at the boundarynodes. Using the above definition of the streamfunction, the posi-tive sign of w denotes anticlockwise circulation and the clockwisecirculation is represented by the negative sign of w. Expandingthe streamfunction (w) using the basis set fUkgN

k¼1 asw ¼

PNk¼1wkUkðX;YÞ and the relationships for U, V from Eq. (11),

the Galerkin finite element method yields the linear residual equa-tions for Eq. (13) and the detailed solution procedure to obtain ws ateach node points are given in an earlier work [49].

2.3.2. Nusselt numberThe heat transfer coefficient in terms of the local Nusselt num-

ber (Nu) is defined as

Nu ¼ � @h@n

; ð14Þ

where n denotes the normal direction on a plane. The normal deriv-ative is evaluated by the bi-quadratic basis set in n � g domain. Thelocal Nusselt numbers at the bottom wall (Nub), top wall (Nut), leftwall (Nul) and right wall (Nur) are defined based on basis functions[50]. The average Nusselt number at the bottom, top and inclinedside walls are given by

Nub ¼R 1

0 NubdX

Xj10¼Z 1

0NubdX;

Nut ¼Z 1þcos u

cos uNutdX;

Nul ¼Z 1

0NuldS1;

and

Nur ¼Z 1

0NurdS2: ð15Þ

Here dS1 and dS2 are the small elemental lengths along the left andright walls, respectively. Average Nusselt numbers are also useful tobenchmark overall heat balance within the cavity. Note thatNul ¼ Nur for case 1 and Nub ¼ Nut for case 2.

2.3.3. HeatfunctionThe heat flow within the enclosure is displayed using the heat-

function (P) obtained from conductive heat fluxes � @h@X ;� @h

@Y

� �as

well as convective heat fluxes (Uh,Vh). The heatfunction satisfiesthe steady energy balance equation [Eq. (5)] [29] such that

@P@Y¼ Uh� @h

@X;

� @P@X¼ Vh� @h

@Yð16Þ

which yield a single equation

@2P

@X2 þ@2P

@Y2 ¼@

@YðUhÞ � @

@XðVhÞ ð17Þ

Using the above definition of the heatfunction, the positive sign of Pdenotes anti-clockwise heat flow and the clockwise heat flow is rep-resented by the negative sign of P. Expanding the heatfunction (P)using the basis set fUkgN

k¼1 as P ¼PN

k¼1PkUkðX;YÞ and the relation-ship for U, V and h from Eq. (11), the Galerkin finite element methodyields the linear residual equations for Eq. (17) and the detailedsolution procedure to obtain Ps at each node points is given in anearlier work [49].

The residual equation [Eq. (17)] is further supplemented withvarious Dirichlet and Neumann boundary conditions in order toobtain an unique solution of Eq. (17). Neumann boundary condi-tions of P, derived from Eq. (16) is specified as,

n � rP ¼ 0 ðfor isothermal hot=cold wall in both casesÞ ð18Þ

The top and bottom insulated walls may be represented by Dirichletboundary condition for case 1 as obtained from Eq. (16) which issimplified into @P

@X ¼ @h@Y ¼ 0 for an adiabatic wall and a reference va-

lue of P = 0 may be assumed at Y = 0 which leads to P = 0"X at Y = 0and P ¼ Nu8X at Y = sinu for case 1. Similarly, the left and rightinsulated walls may be represented by Dirichlet boundary conditionfor case 2 as obtained from Eq. (16) which is simplified into@P@t ¼ @h

@n ¼ 0 for an adiabatic wall and a reference value of P = 0

R. Anandalakshmi, T. Basak / Energy Conversion and Management 67 (2013) 287–296 291

may be assumed at Xsin(u) � Ycos(u) = sinu which leads toP = 0"Y at Xsin(u) � Ycos(u) = sinu and P ¼ Nu8Y at Xsin(u) �Ycos(u) = 0 for case 2. It may also be noted that, most of earlierworks [29,31,35] are also limited within two adiabatic walls whereDirichlet boundary condition is either 0 or Nu at the adiabatic wallsfor fluids confined within square cavities.

3. Results and discussion

3.1. Numerical procedure and validation

The computational grid within the rhombus is generated viamapping the rhombus into square domain in n � g coordinate sys-tem as mentioned in our earlier work [50]. In order to assess theaccuracy of our numerical procedure, we have tested our algorithmbased on the grid size (57 � 57) for a square enclosure as carriedout in an earlier work [51] and results are in good agreement. Com-parison of results are not shown for the brevity of the manuscript.To validate our present numerical approach, benchmark studieshave also been carried out for porous square cavity filled with air(Pr = 0.71), in presence of isothermally heated left wall, while theright wall is maintained isothermally cold and the horizontal wallsare maintained adiabatic (case 1). Table 1 shows a comparison ofthe average Nusselt number ðNuÞ based on the current numericalprocedure, with the result of the same case (case 1) obtained byBaytas and Pop [52] and Saeid and Pop [53,54] at a modified Ray-leigh number (Ram = 1000) where Ram = Ra � Da. Note that the re-sults of the Darcy model (employed by earlier works [52–54]) maybe compared with a non-Darcy model (used in the present case)only at low Da values.

Detailed computations were carried out for various values of Pr(Pr = 0.015 and 7.2), inclination angles (u = 30�, 45�, 75� and 90�)and Da (Da = 10�5 � 10�3). Galerkin finite element approach asused in current work also offers special advantage on evaluationof local Nusselt number at side walls and horizontal walls as ele-ment basis functions have been used to obtain heat flux whereasfinite difference/finite volume based methods involve interpola-tion functions to calculate Nusselt numbers at the surface [55].

In order to validate heatfunction contours or heatlines, we havecarried out simulations for a range of Darcy numbers (Da = 10�5 -� 10�3) and Prandtl numbers (Pr = 0.015 and 7.2). At low Da, fluidis almost stagnant and heat transfer is conduction dominant. Inaddition, heatflux lines, by definition, are perpendicular to isother-mal surfaces and parallel to adiabatic surfaces. It is observed thatthe heatlines emanate from hot surface and end on cold surfacewhich are perpendicular to isothermal surfaces, similar to heatfluxlines, during conduction dominant regime. As they approach theadiabatic wall, they slowly bend and become parallel to that sur-face. Also, the heatlines and isotherms are found to be smoothcurves, without any distortion in presence of dominant conductiveheat transport. These features are discussed in Sections 3.2 and 3.3.

The sign of heatfunction needs special mention. The sign ofheatfunction is governed by the sign of ‘non-homogeneous’

Table 1Comparison of the present results with the benchmark resolutions of earlier works[52–54] for natural convection in a air (Pr = 0.71) filled porous square cavity in thepresence of isothermally heated left wall and isothermally cooled right wall withadiabatic horizontal walls for 28 � 28 bi-quadratic elements (57 � 57 grid points) atRam = Ra � Da = 1000.

S. No. References Nu

1. Baytas and Pop [52] 14.0602. Saeid and Pop [53] 13.7263. Saeid and Pop [54] 13.4424. Present work 14.182

Dirichlet condition. In the current situation, negative sign of heat-lines represents clockwise flow of heat while positive sign refers toanti-clockwise flow. This assumption is in accordance with the signconvention for streamfunction. The streamfunction and heatfunc-tion have identical signs for convective transport. The detailed dis-cussion on heat transport based on heatlines for various cases ispresented in following sections.

3.2. Case 1: differential heating

The distribution of streamfunction (w), heatfunction (P) andtemperature (h) contours for u = 30� � 90� at Da = 10�5 andPr = 0.015 have been studied (figures not shown). Due to buoyancyand imposed temperature gradient between hot isothermal leftwall and cold isothermal right wall, hot fluid rises from the leftwall and cold fluid flows down along right walls with clockwiseunicellular flow pattern within the cavity. The fluid circulation cellis not circular and elongated diagonally due to geometrical asym-metry with respect to central vertical line for u = 30� and 45�. As uincreases, fluid circulation shape gradually approaches to the cir-cular shape and that is almost circular near the core for 90�. Thestrength of the fluid circulation cell tends to become stronger withu. However, the flow strength within the cavity is very weak dueto low permeability within the porous media and that leads to con-duction dominant heat transfer at Da = 10�5 (figures not shown).Heatlines illustrate that the heat flow occurs from hot to cold walland the negative heatfunction corresponds to clockwise circulatoryheat flow. The closed loop heatlines denote convection heat circu-lation due to onset of convection for all us and the strength of theheat circulation increases with u at Pr = 0.015 and Da = 10�5. It isalso found that, the characteristic features of heatlines in all config-urations signify convection heat transfer despite its low magni-tudes as the heatlines are distorted enough even at Da = 10�5 forPr = 0.015 (figures not shown). The boundary layer thickness ishigh near lower left portion (h P 0.9) and top right portion(h 6 0.1) of the cavity as indicated by sparse heatlines at u = 30�.As u increases, the boundary layer thickness near lower left por-tion of the cavity and near top right portion of the cavity decreasessignificantly as indicated by dense heatlines near those regions dueto significant heat flow by the presence of closed loop heatlines(figures not shown).

Dominant convective transport is clearly illustrated by stream-lines and heatlines at Da = 10�3 irrespective of u. Fig. 2a–d showsthe effect of enhanced convection at Da = 10�3 due to high perme-ability through porous media. The circulation cells near top rightcorner and bottom left corner of the cavity form neck type second-ary circulation cells for u = 30� and 45� (see Fig. 2a and b). As u in-creases, the central primary circulation cell grows bigger and fillsthe entire part of the cavity at Da = 10�3 and Pr = 0.015 (Fig. 2cand d). It is also interesting to note that, the circulation cells nearcore of the cavity split into two cells of equal magnitude foru = 75� and 90� (Fig. 2c and d). It is interesting to observe thattrend on heat flow patterns via heatline cells and heat flow distri-bution within the cavity are greatly enhanced from Da = 10�5 toDa = 10�3 corresponding to each specific angle. It is also interestingto observe that heatline circulation cells which are indicative ofconvective transport, are much stronger for Da = 10�3 than thosefor Da = 10�5. Also, as u increases, the convective heat transportin the central portion of the cavity gradually becomes strongerfor Da = 10�3. Localized heat circulation occurs near the bottom leftcorner of the cavity due to the presence of tiny heat circulation cellin that regime for u = 30� and 45� (see Fig. 2a and b). Secondaryheat circulation cells corresponding to secondary fluid circulationcells are observed near lower left portion of the cavity and theyare absent near top right portion of the cavity despite the presenceof secondary fluid circulation cells at u = 30� and 45�. This is due to

Fig. 2. Streamfunction (w), heatfunction (P) and temperature contours (h) for case 1 with Pr = 0.015 and Da = 10�3 for (a) u = 30�, (b) u = 45� (c) u = 75� and (d) u = 90�.Clockwise and anti-clockwise flows are shown via negative and positive signs of streamfunction and heatfunction, respectively. White and black shades in h represent hot andcold fluid inside the cavity, respectively.

292 R. Anandalakshmi, T. Basak / Energy Conversion and Management 67 (2013) 287–296

the fact that the thermal gradient is not high enough for convectiveheat circulation of fluid (see Fig. 2a and b). As a result of high heattransport to right cold wall, boundary layer thickness along theside walls are reduced for all us (see Fig. 2a–d).

It is also interesting to note that, closed loop heatlines convectlarge amount of heat from hot wall to cold wall. Hence, the amountof heat transfer from hot left wall to cold right wall is high near theregion where the convective closed loop heatline cells are present.This is illustrated by highly dense heatlines near middle portion ofthe hot and cold walls for u = 30� (see Fig. 2a). This effect is morepronounced as u increases since the size of the closed loop heat-line cells increase with u and therefore the zone of dense heatlinesmoves towards the lower portion of the hot wall and upper portionof the cold wall. As a result of highly compressed isotherms,boundary layer thickness is small near middle portions of the sidewalls for all us (see Fig. 2a–d). Due to strong convective closedloop heatline cells, isotherms are highly distorted in the middleportion of the cavity for all us (see Fig. 2a–d). It is also interestingto note that the size of the closed loop heatline cells near core ofthe cavity shrinks in size and that result in the formation of strat-ification zone in the circulatory heat flow for 75� and 90�. Hence,isotherms near core of the cavity also form the stratification zoneof temperature for 75� and 90� (see Fig. 2c and d).

The qualitative trends of the streamlines and heatlines are sim-ilar for Pr = 0.7 and Pr = 7.2 for all us. Hence, the distributions areshown only at Pr = 7.2 for the brevity of the manuscript. As Pr in-creases further, the intensity of flow circulations increases due tohigh momentum diffusivity compared to the case of fluids withPr = 0.015 and 7.2 as seen from the maximum values of stream-

functions and heatfunctions (jwjmax and jPjmax) for all us (Figs. 3a–c and 2a–d). The neck type fluid circulation cells near lower leftand top right corners of the cavity as observed for Pr = 0.015 atu = 30� and 45� is absent for Pr = 7.2 (Fig. 3a–c). Fig. 3a–c illustratethat, larger portion of the hot wall delivers heat to the cold wallbased on dense heatlines at higher Pr (Pr = 7.2). Also, heat transferto the top right portions of the cavity is enhanced as seen from largethermal gradient based on highly dense heatlines. In contrast toPr = 0.015, as u increases, heat circulation cell near the core of theclosed loop heatline cells move towards the left hot wall illustratingthat more convective heat transfer takes place from the hot left wallto cold right wall for Pr = 7.2 and Da = 10�3 due to convection in-duced by flow circulation as shown in Fig. 3a–c. The amount of heattransfer is high near the region, where the convective closed loopheatline cells are present as illustrated by largely compressed iso-therms. In contrast to previous case with Pr = 0.015, isotherms arecompressed along the side walls illustrating high heat transfer inthose regions for all us (Fig. 3a–c). Thus, the boundary layer thick-ness is small near those regions compared to the previous case withPr = 0.015 (Figs. 3a–c and 2a–d).

3.3. Case 2: Rayleigh–Benard convection

Fig. 4a and d illustrate the streamlines, heatlines and isothermsfor Rayleigh Benard convection. Qualitative convective transportcharacteristics are almost similar for Pr = 0.015 and Pr = 7.2 atDa = 10�3 except multiple flow and heat circulation cells atPr = 0.015. Hence, the characteristics of isotherms, streamlinesand heatlines are discussed only for Pr = 7.2 and Da = 10�3 in case

−1−0.5 1

−0.5

Π

−6−4 −2

−4−2

ψ(a) θ

−2−1.5

0.14

−1−0.5

Π

−8.8−8

−4

−7−5.75

−2

ψ(b) θ

−2.5 −2.3−2

−113 5 6

−1−0.1

Π

−11.5−10−8−5

−1

−10 −8−3

ψ(c) θ

Fig. 3. Streamfunction (w), heatfunction (P) and temperature contours (h) for case 1 with Pr = 7.2 and Da = 10�3 for (a) u = 30�, (b) u = 45� and (c) u = 75�. Clockwise andanti-clockwise flows are shown via negative and positive signs of streamfunction and heatfunction, respectively. White and black shades in h represent hot and cold fluidinside the cavity, respectively.

R. Anandalakshmi, T. Basak / Energy Conversion and Management 67 (2013) 287–296 293

2 (see Fig. 4a–d). Intensity of fluid flow near the core is higher forthe present case at h 6 75� (see Fig. 4a–d) compared to that of case1 whereas that is lower for h P 75� at Pr = 7.2. As u increases(u = 75�), heatlines are highly distorted and the closed loop heat-lines almost fill the entire cavity. It may also be noted that thepresence of closed loop heatlines further compress the wall-to-wall heatlines and hence, heat flow is enhanced near top and bot-tom portions of the cavity as illustrated by highly compressed iso-therms (see Fig. 4a–c). This further results comparatively lessboundary layer thickness near isothermal walls (see Fig. 4a–c). Atu = 90�, the heatlines perfectly satisfy the conduction situation,where heatlines are perpendicular to isothermal surface and paral-lel to adiabatic surface. There is no fluid motion ðV!� 0Þ even atDa = 10�3 for Pr = 7.2 (see Fig. 4d(i)). The conduction dominant sta-tic solution is the characteristics of square domain in Rayleigh Be-nard convection heating situation for Pr = 0.015 (figures notshown) and 7.2 at Da = 10�3 (high permeability through porousmedia) (see Fig. 4d(i)). It is found that, convection dominant heattransfer at u = 90� (see Fig. 4d(ii)) may be achieved by perturbingu and that further leads to secondary fluid circulation cells(jwjmax = 1) near top left and bottom right corners with strong pri-mary fluid circulation cells (jwjmax = 13.59) near the core of the cav-ity. Secondary fluid circulation cells further reduce the strength ofconvective heat flow, which is lower for the present case (u = 90�)compared to u = 75� (see Figs. 4d(ii) and 4c). But, the thermal mix-ing is increased further (h = 0.4 � 0.6) as seen from the isothermcontours of Fig. 4d(ii) compared to u = 75� (Fig. 4c).

Rhombic cavity may be an alternative choice for efficient heattransfer in Rayleigh Benard convection at high Darcy regime(Da = 10�3) for all types of fluids (Pr) considered here for the anal-ysis as a slight perturbation of u at higher Da (Da P 4 � 10�5)leads to convection based dynamic solution irrespective of Pr(see Fig. 4a–d). The multiple solutions involving conduction basedstatic fluid solutions were also reported by Venturi et al. [8]. Thepresent analysis is restricted to only conduction based static solu-tion and convection based dynamic solution at u = 90�. Our futureanalysis will include results of all possible steady states at variousRa involving multiple circulations in Rayleigh Benard convection.Overall, the heat distribution and thermal mixing in Rayleigh Be-nard convection (case 2) is higher than that of differential heating

case (case 1) and that is clearly indicated by large magnitudes ofstreamfunctions, heatfunctions and shades of isotherm contoursin gray scale plots.

3.4. Heat transfer rates: average Nusselt numbers

The overall effects on heat transfer rates via average Nusseltnumber for left and bottom walls (Nul and Nub) vs logarithmicDarcy number for all the cases are displayed in panel plots ofFig. 5a and b. Panel plots of Fig. 5a and b illustrate that averageNusselt number increases with Darcy number irrespective of uand Pr except u = 90� in case 2 at Pr = 7.2. It may also be noted thatthe average heat transfer rate is high at Da = 10�3 (see panel plotsof Fig. 5a–d) as high permeability enhances the convective heatflow within the cavity except u = 90� in case 2 at Pr = 7.2. Averageheat transfer rates at the left wall ðNulÞ are maintained constantover a range of Da (10�5

6 Da 6 4 � 10�5) for u = 30� and 45� incase 1 due to conduction dominant heat transfer in that region.However, Nul starts increasing from Da P 10�5 for u = 75� and90� due to convection dominant heat transfer at even low Da incase 1 (see upper panel of Fig. 5a). This is further illustrated bythe high dense and closed loop heatlines at Da = 10�3 andPr = 0.015 in Fig. 2c and d. Due to higher inclination of left hot wall(u = 30� and 45�), there is no significant heat flow from the lowerportion of the hot left wall for u = 30� and 45� compared to u = 75�and 90�. This is also illustrated by the sparse heatlines of low heat-function gradient at u = 30� and 45� in case 1 for Pr = 0.015 andDa = 10�3 (see upper panel of Figs. 5a and 2a–d). It is found that,maximum heat flow occurs from the hot left wall to the cold rightwall due to dense heatlines of high heatfunction gradient foru = 90� in case 1 at Da = 10�3 and Pr = 0.015 (see upper panel ofFig. 5a). The qualitative trends of Nu for case 2 ðNubÞ are similarto that of case 1 ðNulÞ except constant Nu occurs over certain rangeof Da due to conduction dominant heat transfer for all us in case 2at Pr = 0.015 (see lower panel of Fig. 5a). Similar to case 1, the max-imum heat flow occurs from hot bottom wall to cold top wall foru = 75� and 90� in case 2 at Da = 10�3 and Pr = 0.015 (see lower pa-nel of Fig. 5a). Overall, heat transfer rates (Nul and Nub) increaseswith u in both cases at Pr = 0.015 (see Fig. 5a). It may be noted that,Nub is constant throughout the range of Da(10�5

6 Da 6 10�3) for

−3−1 0.5

−2−1

1.55

Π

−11−8 −4

−11−8−6

ψ θ

−4.5−3

−0.50.13.5

−4−1.82

Π

−14−12

−6

−14−12−8−4

ψ θ

−6−4

−113.5

−5−4−22.5

Π

−17.5−15−8

−1

−16−15−9−5

ψ θ

1.5−0.5−2−4

−5−2

−0.13

Π

1−3−9−13

−11−9−41

ψ(ii) θ

0.1

0.4

0.6

0.8

Π

0.0005 −0.0005 −0.001

5e−05

ψ

(a)

(b)

(c)

(i) θ(d)

Fig. 4. Streamfunction (w), heatfunction (P) and temperature contours (h) for case 2 with Pr = 7.2 and Da = 10�3 for (a) u = 30�, (b) u = 45� (c) u = 75�, (d) (i) conduction atu = 90� and (ii) convection at u = 90�. Clockwise and anti-clockwise flows are shown via negative and positive signs of streamfunction and heatfunction, respectively. Whiteand black shades in h represent hot and cold fluid inside the cavity, respectively.

294 R. Anandalakshmi, T. Basak / Energy Conversion and Management 67 (2013) 287–296

conduction based static fluid solution at u = 90� and Pr = 0.015 incase 2.

It is also interesting to note that, u = 30� shows lower heat trans-fer rate in case 1 compared to case 2 at Pr = 0.015 and Da = 10�3 dueto high intense convective heat transfer in case 2 compared to case 1atu = 30� at Pr = 0.015 and Da = 10�3 (see Fig. 5a). On the other hand,u = 90� shows higher heat transfer rate in case 1 compared to case 2at Pr = 0.015 and Da = 10�3. This is due to the presence of high in-tense convective heat circulation cells, which further enhance thelocalized heat flow and decrease the overall heat flow for u = 90�in case 2 at Pr = 0.015 and Da = 10�3 (see Fig. 5a). The variations ofNul and Nub for Pr = 7.2 are qualitatively similar with those ofPr = 0.015 except decrease in average heat transfer rates foru = 90� at Pr = 7.2 and Da = 10�3 due to sparse heatlines as there isno significant heat transfer from the right corner of the hot bottomwall. This is due to the fact that the presence of low strength second-ary heatline circulation cells near bottom wall of the cavity affect theheat flow inside the cavity as seen from Fig. 4d(ii). Further, increasein Pr increases momentum diffusivity and therefore Nu values forPr = 7.2 are higher compared to Pr = 0.015 (see Fig. 5a–d) for all us.Overall, average heat transfer rate is maximum for case 1 atu P 45� due to high heatfunction gradient despite wmax and Pmax

values are high in case 2 for all Pr at Da = 10�3 (see Fig. 5a and b).

Fig. 5c and d shows the variations of average heat transfer ratewith rhombic angles (u) for case 1 and case 2 at Da = 10�3 forPr = 0.015 and 7.2. As mentioned earlier, conduction based staticfluid solution in addition to convection based dynamic fluid solu-tion exists for u = 90� in case 2. It is interesting to note that, Nu in-creases with u for convection based dynamic solution in case 1 forall Pr. It is observed that, u = 45� is the critical angle as case 2shows higher heat transfer rate at u 6 45� whereas case 1 showshigher heat transfer rate at u P 45� for all Pr (see Fig. 5c and d).It is interesting to note that Nu for convection based dynamic fluidsolution in case 2 decreases with u at u P 80� for Pr = 0.015whereas that decreases with u at u P 55� for Pr = 7.2 due to theformation of multiple circulations near bottom wall which furtheraffects the heat flow from bottom wall (see Figs. 5c and d, 3d, 4cand d(ii)).

3.5. Conclusion

The role of inclination angle (u) for improvement of heat distri-bution and thermal mixing inside porous rhombic cavities duringnatural convection is analyzed using heatlines and streamlines.The direction of heat flow inside the porous rhombic cavities is

0

2

4

6

Nu l

Case 1

(a) Pr=0.015

10-5 10-4 10-3

Darcy number, Da

0

2

4

6 Case 2N

u b0

2

4

6

8

Nu l

Case 1

(b) Pr=7.2

10-5 10-4 10-3

Darcy number, Da

0

2

4

6 Case 2

Nu b

20 30 40 50 60 70 80 90

Inclination angle

0

1.5

3

4.5

6

Ave

rage

Nus

selt

Num

ber,

Nu l

or

Nu b

Dua

l sol

utio

n

convection in case 1convection in case 2conduction in case 2

(c) Pr=0.015

20 30 40 50 60 70 80 90

Inclination angle

0

2

4

6

8

10

Ave

rage

Nus

selt

Num

ber,

Nu l

or

Nu b

solu

tion

Dua

l

convection in case 1convection in case 2conduction in case 2

(d) Pr=7.2

Fig. 5. Variations of average Nusselt number with Darcy number for case 1 (differential heating) ½Nul � and case 2 (Rayleigh–Bernard convection) ½Nub� at (a) Pr = 0.015 and (b)Pr = 7.2 for various inclination angles u = 30� (����), u = 45� (- - -), u = 75� (– –) and u = 90� (—) [for case 2: convection at u = 90� (-–) and conduction at u = 90� (-�–)]. Plotsfor cases 1 and 2 are shown in upper and lower panels, respectively. Variations of average Nusselt numbers (Nul or Nub) with rhombic angles in conduction and convectionregime for (c) Pr = 0.015 and (d) Pr = 7.2 at Da = 10�3 for cases 1 and 2.

R. Anandalakshmi, T. Basak / Energy Conversion and Management 67 (2013) 287–296 295

explained based on heatlines in differential heating and RayleighBenard convection cases.

� At Da = 10�5, heat transfer is conduction dominant for u = 30�and less intense closed loop heatlines with slightly distortedisotherms are observed for u P 30� irrespective of Pr(Pr = 0.015 � 7.2) even at Da = 10�5 in case 1. But, heat transferis primarily conduction dominant for all us in case 2.� At Da = 10�3, multiple flow circulations are observed at

Pr = 0.015 for all us. Multiple circulation cells are suppressedand single flow circulation cells are found to occupy the entirecavity at a higher Pr (Pr = 7.2) for all us in both cases.� Local distribution of heat from hot wall to cold wall mainly

depends on closed loop convective heatline cells for all us espe-cially at u P 45�, where enlarged closed loop convective heatlinecells are observed in both cases irrespective of Pr at Da = 10�3.� Conduction dominant static fluid solution is the unique charac-

teristics of square cavity (u = 90�) in case 2 for all Pr atDa 6 10�3. It is observed that convection based dynamic solu-tion is also possible with a slight perturbation of u at higherDa (Da P 4 � 10�5) irrespective of Pr.� Rhombic cavity may be an alternative choice for efficient heat

transfer in Rayleigh Benard convection at high Darcy regime(Da = 10�3) as a slight perturbation of u at higher Da(Da P 4 � 10�5) leads to convection based dynamic solutionirrespective of Pr.

� Maximum average heat transfer rate from the hot wall to coldwall (maximum Nul) occurs for higher u (u = 90�) irrespectiveof Pr in case 1 at Da = 10�3. Due to the formation of multiple cir-culations near bottom wall, Nub decreases with u at u P 55� incase 2 for Pr = 7.2.� Overall, case 2 shows larger Nu compared to case 1 at u 6 45�

whereas case 1 shows larger Nu compared to case 2 atu P 45� irrespective of Pr at Da = 10�3. Hence, heating patterns(case 1 and 2) may be changed based on critical rhombic angle(u = 45�) to achieve higher heat transfer rates in variousapplications.

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