Efficient Methodologies for Processing of Fluids by Thermal Convection within Porous Square Cavities

Efficient Methodologies for Processing of Fluids by Thermal Convection withinPorous Square Cavities

Ram Satish Kaluri and Tanmay Basak*

Department of Chemical Engineering, Indian Institute of Technology Madras, Chennai-600036, India

Effective use of thermal energy is the key for the energy-efficient processing of materials. A properunderstanding of heat flow would be very useful in designing the systems of high energy efficiency with aminimal waste of precious energy resources. In the current study, a distributed heating methodology is proposedfor the efficient thermal processing of materials. A detailed investigation on the processing of various fluidsof industrial importance (with a Prandtl number of Pr ) 0.015, 0.7, 10, and 1000) in differentially and discretelyheated porous square cavities is presented. Analysis of laminar convective heat flow within a range of Darcynumber, Da ) 10-6-10-3 and Rayleigh number, Ra ) 103-106 has been carried out, based on a heatlinevisualization approach. The effect of Da and the role of distributed heating in enhancing the convection inthe cavities is illustrated via heatline distributions, which represent the paths of the heat flow, the magnitudeof heat flow, and zones of high heat transfer. It is observed that distributed heating plays an important rolein enhancement of thermal mixing and temperature uniformity. Furthermore, the effect of Da for various Prvalues on the variation of local Nusselt number (Nu) is analyzed, based on heatline distributions.

1. Introduction

Porous materials continue to play an important role in variousfields, such as chemical processing,1-7 food,8 and biomedical,9

environmental,4,10,11reservoir,12andelectrokineticapplications,13,14

as well as thermal management in electronics,15-18 fuel cells,19

electrochemical applications,20 combustion,21,22 building insula-tion,23 heat exchangers,24 etc. The subject of natural convectionin porous media has significant importance, because manyprocesses are governed by this phenomena. For example, Brookset al.25 discussed the problem of spontaneous combustion ofcoal stockpiles and concluded that natural convection is theprimary flow mechanism. Ejlali et al.26 investigated the keyparameters such as approaching wind speed, porosity, andpermeability of the porous medium and presented a criterionfor the design of reactive coal stockpiles. Bejan27 presented acomplete theory of the melting that occurs in a confined porousmedium saturated with phase-change material and obtained thematched boundary layer solution for natural convection-dominated melting in the quasi-steady region. Gatica et al.28

investigated the interaction between chemical reaction andnatural convection in porous media to determine the conditionsunder which secondary flows would develop, and showed thatthe presence of buoyancy-driven flows will drastically modifythe topology of heterogeneous reaction system in an adiabaticpacked bed. Benneker et al.29 experimentally illustrated theinfluence of natural convection on fluid mixing in packed bedsat elevated pressure by varying flow rates, pressures, particlesizes, tube diameters, and flow directions, and they concludedthat the usual correlations for mass and heat transfer may notbe valid in high-pressure equipment.

In light of numerous applications mentioned above, manystudies have been conducted to gain a fundamental understand-ing of natural convection in porous media. A comprehensiveexperimental and numerical study on natural convection in aporous medium was carried out by Prasad et al.,30 and a modelfor effective thermal conductivity was proposed to account forthe enhanced effect of fluid thermal conductivity during

convective flow. Chamkha31 numerically studied the transienthydromagnetic three-dimensional natural convection from aninclined porous stretching surface for metallurgical applications.Baytas and Pop32 numerically investigated the effects of naturalconvection on the heat transfer and temperature distributionwithin a tilted porous trapezoidal enclosure with cylindrical topand bottom walls at different temperatures and adiabaticsidewalls. Kathare et al.33 experimentally studied buoyantconvection in superposed metal foam and water layers andconcluded that the enhancement of heat transfer by naturalconvection could be obtained via varying the sublayer thicknessof foam and its placement.

In recent years, the approach of discrete/distributed heatingis being considered as the potential methodology for enhancedthermal processing of materials. Distributed heating has advan-tages such as improved thermal mixing, greater temperatureuniformity, greater control over flow field, and, importantly, highenergy efficiency. A few interesting studies have been reportedfor enhanced processing of materials based on the distributedheating. Plumat34 reported the enhanced melting of glass byemploying heated strips. Sarris et al.35,36 showed that the flowcurrents, temperature distribution, and thermal penetration couldbe controlled by heated strips for a homogeneous melting ofglass which results in better quality of the final product.Enhancement of oxygen transport in liquid metal by naturalconvection during oxidation of liquid lead and lead-bismutheutectic in a discretely heated enclosure was investigated byMa et al.37

Studies on natural convection in porous media due todiscrete heat sources are mainly focused on electronic coolingapplications.38-41 However, thermal treatment of variousfluids in porous media have several other important applica-tions such as molten metal infiltration in porous media,42-45

drying and transport of gases in porous media,46-48 enhancedoil recovery by hot-water flooding in porous beds,49 combus-tion of heavy oils in porous reservoirs,50,51 etc. Thus, anunderstanding of the role of discrete heating for various fluidsin porous media is important and such studies are yet toappear in the literature. The current work aims to address

* To whom correspondence should be addressed. E-mail: [email protected].

Ind. Eng. Chem. Res. 2010, 49, 9771–9788 9771

10.1021/ie100569w 2010 American Chemical SocietyPublished on Web 09/28/2010

issues such as thermal mixing and temperature uniformityby distributed heating during natural convection in porousmedia, in the context of material processing applications.

Numerical results of natural convection are mostly pre-sented and analyzed by streamlines and isotherms. Althoughvisual representation of fluid flow can be adequately il-lustrated by streamlines, isotherms are inadequate for convec-tive heat flow visualization. The information of heat flowmay be very useful, because that may be helpful with regardto devising strategies to control the temperature distributionin the cavity. A numerical tool to visualize heat flow wasfirst proposed by Kimura and Bejan.52 They formulated a“heatfunction”, which is analogous to a streamfunction, byaccounting for conductive and convective fluxes. The iso-lines of the heatfunction are called “heatlines”, and thetrajectories of these heatlines represent the paths of heat flow.In addition, they indicate the magnitude of heat flow andzones of high heat transfer. The concept of heatlines has beenemployed and extended by several researchers to describevarious physical phenomena.53-57 Few studies on the analysisof natural convection in porous media based on heatlineconcept have also been reported.58-60 In current work, theapproach of heatlines is employed to study the heat flowdistribution and thermal mixing within the square cavity inthe presence of distributed heating of various walls.

The main objective of the present study is to analyze the roleof distributed heating and permeability of porous media (in termsof the Darcy number, Da) in enhancing the thermal mixing andtemperature uniformity during natural convection in squarecavities filled with fluid-saturated porous media, based on theheatline approach. Three different cases are considered: (1) hot-isothermal bottom wall with cold-isothermal side walls, (2)discretely heated cavity with isothermal heat sources at thecentral regions of bottom and side walls, and (3) cavity withmultiple isothermal heat sources, located at central as well as

lower corner regions. The porous medium is modeled based ona generalized non-Darcy model, neglecting the Forchheimerinertia term. This generalized model, based on volume averagingprinciples, was developed by Vafai and Tien.61 Cases 1-3 arestudied for a range of parameters (Rayleigh number (Ra) of103 e Ra e 106, Darcy number (Da) of 10-6 e Da e 10-3).Fluids of industrial importance, viz, molten metals (Pr ) 0.015),gases (Pr ) 0.7), aqueous chemical solutions (Pr ) 10), andolive/engine oils (Pr ) 1000) are used as working fluids. Adetailed analysis on fluid and heat flow is presented, based onstreamlines, heatlines, and isotherms. Thermal mixing andtemperature uniformity in various cases is further analyzed usingthe cup-mixing temperature and the root-mean-square deviation(RMSD). Finally, the effect of Da on the heat transfercharacteristics are studied using the local Nusselt number (Nu),and various qualitative and quantitative features of Nu areadequately explained, based on heatline distributions.

2. Mathematical Modeling and Simulation

2.1. Governing Equations: Temperature and Velocity.The physical domains for various cases are shown in Figures1a-c. The top wall is maintained adiabatic in all of the cases.Case 1 involves an isothermal hot bottom wall with side wallsbeing maintained isothermally cold (Figure 1a). Case 2represents discrete heating of the cavity with heat sourcesapplied along the walls of the cavity (Figure 1b), whereasCase 3 involves multiple heat sources, which are placed atthe central portions of the walls and at the lower corners ofthe cavity. The location and the length of the heat sourcesmay be noted from Figure 1c. The dimensionless length ofthe hot isothermal zone of the bottom wall is 0.5, while thatalong the side walls is 0.25. Note that the total dimensionlesslength of hot zones in Cases 2 and 3 is equal to that in case1. All the physical properties are assumed to be constant,

Figure 1. Schematic diagrams of the cavities filled with a fluid-saturated porous medium for different cases. The top wall is adiabatic. Thick lines representsthe uniformly heated sections, while the remaining sections are maintained cold.

9772 Ind. Eng. Chem. Res., Vol. 49, No. 20, 2010

except for the density in buoyancy term, and the change indensity due to temperature variation is obtained usingBoussinesq’s approximation. Also, it is assumed that the localthermal equilibrium (LTE) is valid, i.e., the temperature ofthe fluid phase is equal to the temperature of the solid phasewithin the porous medium. The momentum transfer in aporous medium is based on the generalized non-Darcy modelof Vafai and Tien.61 However, the velocity squared term (orForchheimer term), which models the inertia effect, isneglected in the present case, because this work involves onlynatural convection flow in a porous medium in an enclosedcavity. This inertia effect is more important for non-Darcyeffects on the convective boundary-layer flow over the surfaceof the body embedded in a high-porosity medium. Underthese assumptions, the governing equations for steady two-dimensional natural convection flow in a porous squarecavity, using the conservation of mass, momentum, andenergy, can be written with the following dimensionlessvariables or numbers:

as

The boundary conditions for the velocities are

and the boundary conditions for the temperature in Cases1-3 are

Note that, in eqs 1-7, X and Y are dimensionless coordinatesvarying along the horizontal and vertical directions, respec-tively; U and V are dimensionless velocity components alongthe X- and Y-directions, respectively; θ is the dimensionlesstemperature; P is the dimensionless pressure; Ra is theRayleigh number; Pr is the Prandl number; and Da is theDarcy number.

The momentum and energy balance equations (eqs 3-5)are solved using the Galerkin finite element method. The

continuity equation (eq 2) has been used as a constraint,because of mass conservation, and this constraint may beused to obtain the pressure distribution. To solve eqs 3 and4, we use the penalty finite-element method, where thepressure P is eliminated by a penalty parameter γ and theincompressibility criteria that is given by eq 2, which resultsin

The continuity equation (eq 2) is automatically satisfied forlarge values of γ. Typical value of γ that yield consistentsolutions is 107. Using eq 8, the momentum balance equations(eqs 3 and 4) reduce to

and

The system of equations (eqs 5, 9, and 10) with boundaryconditions is solved using the Galerkin finite elementmethod.62 Expanding the velocity components (U, V) andtemperature (θ), using basis set Φkk)1

N , as

for

The Galerkin finite element method yields the followingnonlinear residual equations for eqs 9, 10, and 5, respectively,at nodes of the internal domain Ω:

X ) xL

, Y ) yL

, U ) uLR

, V ) υLR

, θ )T - Tc

Th - Tc

P ) pL2

FR2, Pr ) ν

R, Da ) K

L2,

K )e3dp

2

150(1 - e)2, Ra )

g(Th - Tc)L3Pr

ν2(1)

∂U∂X

+ ∂V∂Y

) 0 (2)

U∂U∂X

+ V∂U∂Y

) -∂P∂X

+ Pr(∂2U

∂X2+ ∂

2U

∂Y2) - PrDa

U (3)

U∂V∂X

+ V∂V∂Y

) -∂P∂Y

+ Pr(∂2V

∂X2+ ∂

2V

∂Y2) - PrDa

V + RaPrθ

(4)

U∂θ∂X

+ V∂θ∂Y

) ∂2θ

∂X2+ ∂

2θ∂Y2

(5)

U(X, 0) ) U(X, 1) ) U(0, Y) ) U(1, Y) ) 0V(X, 0) ) V(X, 1) ) V(0, Y) ) V(1, Y) ) 0

(6)

θ ) 1 (for the hot region)θ ) 0 (for the cold region)∂θ∂Y

) 0 (for adiabatic topwall)(7)

P ) -γ(∂U∂X

+ ∂V∂Y ) (8)

U∂U∂X

+ V∂U∂Y

) γ ∂

∂X(∂U∂X

+ ∂V∂Y ) + Pr(∂2U

∂X2+ ∂

2U

∂Y2) - PrDa

U

(9)

U∂V∂X

+ V∂V∂Y

) γ ∂

∂Y(∂U∂X

+ ∂V∂Y ) + Pr(∂

2V

∂X2+ ∂

2V

∂Y2) -

PrDa

V + RaPrθ (10)

U ≈ ∑k)1

N

UkΦk(X, Y), V ≈ ∑k)1

N

VkΦk(X, Y), and

θ ≈ ∑k)1

N

θkΦk(X, Y) (11)

0 e X, Y e 1

Ri(1) ) ∑

k)1

N

Uk∫Ω [(∑k)1

N

UkΦk)∂Φk

∂X+

(∑k)1

N

VkΦk)∂Φk

∂Y ]Φi dX dY + γ[∑k)1

N

Uk∫Ω

∂Φi

∂X

∂Φk

∂XdX dY +

∑k)1

N

Vk∫Ω

∂Φi

∂X

∂Φk

∂YdX dY] + Pr ∑

k)1

N

Uk∫Ω [∂Φi

∂X

∂Φk

∂X+

∂Φi

∂Y

∂Φk

∂Y ] dX dY + PrDa ∫Ω (∑

k)1

N

UkΦk)Φi dX dY (12)

Ind. Eng. Chem. Res., Vol. 49, No. 20, 2010 9773

and

Biquadratic basis functions with three-point Gaussian quadra-ture are used to evaluate the integrals in the residual equationsexcept the second term in eqs 12 and 13. In eqs 12 and 13,the second term, which contains the penalty parameter (γ),are evaluated with two-point Gaussian quadrature (reducedintegration penalty formulation62). The nonlinear residualequations (eqs 12-14) are solved using the Newton-Raphsonmethod to determine the coefficients of the expansions in eq11. The detailed solution procedure is given in an earlierwork.41

2.2. Nusselt Number and the Overall Heat Balance. Theheat-transfer coefficient, in terms of the local Nusselt number(Nu) is defined by

where n denotes the normal direction on a plane. The localNusselt numbers at the bottom wall (Nub), at the left wall (Nul),and at the right wall (Nur) are respectively defined as

and

Note that i is the local node number in an biquadratic element,which consists of nine local nodes and hence, the summationis written within the limits 1 to 9. In absence of any distributedheat sources, the average Nusselt number for the isothermalbottom and side walls (Nub and Nus, respectively) is given by

In Cases 2 and 3, the cold and hot sections are present onthe same wall, because of distributed heating. Therefore, theaverage Nusselt numbers for individual sections on the bottomwall in Case 2 are calculated as follows:

and

In a similar manner, Nul, 1, Nul, 2, and Nul, 3 on the left verticalwall and Nur, 1, Nur, 2, and Nur, 3 on the right vertical wall arecalculated. Note that subscripts “1” and “3” refer to coldsections, whereas the subscript “2” refers to the hot section oneach wall. Following a similar manner, the average Nusseltnumber on hot and cold sections are evaluated in Case 3. Theintegrals are numerically evaluated using the adaptive trapezoi-dal rule. Furthermore, the convergence of integration results ischecked with a heat balance, in terms of percentage error(discussed below). The heat balance was verified to ensure thatthe energy balance across the cavity is satisfied, i.e., the totalheat lost by the hot sections is equal to the total heat gained bythe cold sections. The heat balance equation for each case maybe written as

(i) Case 1:

(ii) Case 2:

(iii) Case 3:

The error in heat balance is calculated as

where l′ refers to the length of cold or hot sections and thesubscript j refers to the left (l), bottom (b), or right (r) walls.Subscripts m ) 1, 3 (cold sections) and 2 (hot section) on each

Ri(2) ) ∑

k)1

N

Vk∫Ω [(∑k)1

N

UkΦk)∂Φk

∂X+

(∑k)1

N

VkΦk)∂Φk

∂Y ]Φi dX dY + γ[∑k)1

N

Uk∫Ω

∂Φi

∂Y

∂Φk

∂XdX dY +

∑k)1

N

Vk∫Ω

∂Φi

∂Y

∂Φk

∂YdX dY] + Pr ∑

k)1

N

Vk∫Ω [∂Φi

∂X

∂Φk

∂X+

∂Φi

∂Y

∂Φk

∂Y ] dX dY + PrDa ∫Ω (∑

k)1

N

VkΦk)Φi dX dY -

RaPr∫Ω (∑k)1

N

θkΦk)Φi dX dY (13)

Ri(3) ) ∑

k)1

N

θk∫Ω [(∑k)1

N

UkΦk)∂Φk

∂X+ (∑

k)1

N

VkΦk)∂Φk

∂Y ]Φi dX dY +

∑k)1

N

θk∫Ω [∂Φi

∂X

∂Φk

∂X+

∂Φi

∂Y

∂Φk

∂Y ] dX dY (14)

Nu ) -∂θ∂n

(15)

Nub ) ∑i)1

9

θi

∂Φi

∂Y(16)

Nul ) ∑i)1

9

θi

∂Φi

∂X(17)

Nur ) -∑i)1

9

θi

∂Φi

∂X(18)

Nub ) ∫0

1Nub dX (19)

Nus ) ∫0

1Nus dY (20)

Nub,1 )∫0

0.25Nub dX

0.25(for the left cold section)

(21)

Nub,2 )∫0.25

0.75Nub dX

0.5(for the middle hot section)

(22)

Nub,3 )∫0.75

1Nub dX

0.25(for the right cold section)

(23)

Nub ) Nul + Nur (24)

lb,2′ Nub,2 + ll,2′ Nul,2 + lr,2′ Nur,2 ) lb,1′ Nub,1 +

lb,3′ Nub,3 + ll,1′ Nul,1 + ll,3′ Nul,3

+ lr,1′ Nur,1 + lr,3′ Nur,3

(25)

lb,1′ Nub,1 + lb,3′ Nub,3 + lb,5′ Nub,5 + ll,1′ Nul,1 + ll,3′ Nul,3 +lr,1′ Nur,1 + lr,3′ Nur,3 ) lb,2′ Nub,2 + lb,4′ Nub,4 + ll,2′ Nul,2 +

ll,4′ Nul,4 + lr,2′ Nur,2 + lr,4′ Nur,4 (26)

ε )| ∑

m)1

Nh

lj,m′ Nuj,m|hot

- | ∑m)1

Nc

lj,m′ Nuj,m|cold

min(| ∑m)1

Nh

lj,m′ Nuj,m|hot

, | ∑m)1

Nc

lj,m′ Nuj,m|cold

)× 100

(27)


wall in case 2 whereas in case 3, m ) 1, 3, 5 (hot sections)and 2, 4 (cold sections) and Nh and Nc refer to the total numberof hot and cold sections in the cavity for given case. For eachcase, heat balance (eqs 24-26) is verified at all Ra values, andthe error (ε), based on eq 27, is determined to be <3%.

2.3. Visualization. 2.3.1. Streamfunction. The fluid motionis displayed using the streamfunction (ψ) obtained from velocitycomponents U and V. The relationships between streamfunctionand velocity components for two-dimensional flows are

which yield a single equation,

Using the above definition of the streamfunction, the positivesign of ψ denotes anticlockwise circulation and the clockwisecirculation is represented by the negative sign of ψ. Expandingthe streamfunction (ψ) using the basis set Φkk)1

N as ψ )∑k)1

N ψkΦk(X, Y) and the relationships for U and V from eq 11,the Galerkin finite element method yields the following linearresidual equations for eq 29:

The no-slip condition is valid at all boundaries, because thereis no cross-flow, hence, ψ ) 0 is used as the residual equationsat the nodes for the boundaries. The biquadratic basis functionis used to evaluate the integrals in eq 30 and ψ values areobtained by solving the N linear residual equations (eq 30).

2.3.2. Heatfunction. The heat flow within the enclosure isdisplayed using the heatfunction Π obtained from conductiveheat fluxes (-∂θ/∂X, -∂θ/∂Y), as well as convective heat fluxes(Uθ, Vθ). The heatfunction satisfies the steady energy balanceequation (eq 5),52 such that

which yield a single equation,

Using the above definition of the heatfunction, the positivesign of Π denotes anticlockwise heat flow and the clockwiseheat flow is represented by the negative sign of Π. Expandingthe heatfunction Π, using the basis set Φkk)1

N as Π )∑k)1

N ΠkΦk(X, Y) and the relationship for U, V and θ from eq11, the Galerkin finite element method yields the followinglinear residual equations for eq 32.

The residual equation (eq 33) is further supplemented withvarious Dirichlet and Neumann boundary conditions, to obtainan unique solution of eq 32. The Neumann boundary conditionsof Π are obtained for the isothermal (hot or cold) walls, asderived from eq 31, and are specified as follows:

The Dirichlet boundary condition for the top insulated wall isobtained from eq 31, which is simplified to ∂Π/∂X ) 0 for anadiabatic wall. A reference value of Π is assumed to be 0 at X) 0, Y ) 1, and, hence, Π ) 0 is valid for Y ) 1,∀X. Note thatcurrent work is based on situations of differential and distributedheating that correspond to various nonhomogeneous Dirichletconditions for Π at the hot-cold junctions. The unique solutionof eq 32 is strongly dependent on the nonhomogeneous Dirichletconditions, and the boundary conditions at these junctions areobtained by integrating eq 31 along boundaries from thereference point until the end point of each junction. Representa-tive calculations for Cases 1 and 2 are given as follows:

(i) At (X ) 0, Y ) 1),

(ii) At (X ) 0, Y ) 0),

The boundary conditions for the heatfunction in Case 2(Figure 1b) are obtained as follows:

(i) At (X ) 0, Y ) 1),

(ii) At (X ) 0, Y ) Ya),

(iii) At (X ) 0, Y ) Yb),

In a similar manner, boundary conditions are derived at all ofthe hot-cold junctions. All boundary conditions for the heat-function Π, for Cases 1-3, are listed in the Appendix.

2.4. Thermal Mixing: Average Temperature versusRoot-Mean-Square Deviation (RMSD). To compare thethermal mixing in various cases, the bulk average temperature

U ) ∂ψ∂Y

and V ) -∂ψ∂X

(28)

∂2ψ

∂X2+ ∂

2ψ∂Y2

) ∂U∂Y

- ∂V∂X

(29)

Ris ) ∑

k)1

N

ψk∫Ω [∂Φi

∂X

∂Φk

∂X+

∂Φi

∂Y

∂Φk

∂Y ] dX dY -

∫ΓΦin ·∇ψ dΓ + ∑

k)1

N

Uk∫ΩΦi

∂Φk

∂YdX dY -

∑k)1

N

Vk∫ΩΦi

∂Φk

∂XdX dY (30)

∂Π∂Y

) Uθ - ∂θ∂X

-∂Π∂X

) Vθ - ∂θ∂Y

(31)

∂2Π

∂X2+ ∂

2Π∂Y2

) ∂

∂Y(Uθ) - ∂

∂X(Vθ) (32)

Rih ) ∑

k)1

N

Πk∫Ω [∂Φi

∂X

∂Φk

∂X+

∂Φi

∂Y

∂Φk

∂Y ] dX dY -

∫ΓΦin ·∇Π dΓ + ∑

k)1

N

Uk∫Ω (∑k)1

N

θkΦk)Φi

∂Φk

∂YdX dY +

∑k)1

N

θk∫Ω (∑k)1

N

UkΦk)Φi

∂Φk

∂YdX dY -

∑k)1

N

Vk∫Ω (∑k)1

N

θkΦk)Φi

∂Φk

∂XdX dY -

∑k)1

N

θk∫Ω (∑k)1

N

VkΦk)Φi

∂Φk

∂XdX dY (33)

n ·∇Π ) 0 (isothermal hot or cold wall) (34)

Π(0, 1) ) ∂Π∂X

) ∂θ∂Y

) 0 (35)

Π(0, 0) ) Π(0, 1) + ∫Y)1

Y)0 ∂Π∂Y

) Nul (36)

Π(0, 1) ) ∂Π∂X

) ∂θ∂Y

) 0 (37)

Π(0, Ya) ) Π(0, 1) + ∫Y)1

Y)Ya (∂Π∂Y ) dY ) ll,3′ Nul,3 (38)

Π(0, Yb) ) Π(0, Ya) + ∫Y)Ya

Y)Yb (∂Π∂Y ) dY ) Π(0, Ya) + ll,2′ Nul,2

(39)


across the cavity, i.e., the “cup-mixing temperature”, is defined.The cup-mixing temperature is the velocity-weighted averagetemperature, which is most suitable when convective flow exists.The cup-mixing temperature (Θcup) and spatial average or areaaverage temperature (Θavg) are given as

where

and

Furthermore, the “root-mean-square deviation” (RMSD) isdefined to quantify the degree of temperature uniformity in eachcase. Various forms of RMSDs are defined based on the cup-mixing temperature (Θcup) and spatial average temperature (Θavg)as follows:

and

Note that the lower RMSD values indicate higher temperatureuniformity in the cavity and vice versa. In addition, the RMSD

value cannot exceed 1, because the dimensionless temperaturevaries only between 0 and 1.

3. Results and Discussion

3.1. Numerical Procedure and Validation. The computa-tional domain consisting of 20 × 20, 24 × 24, and 28 × 28biquadratic elements, which correspond to 41 × 41, 49 × 49,and 57 × 57 grid points, respectively, are considered for thestudy. A uniform grid is used for 20 × 20 elements, whereasan adaptive grid with local refinement near the distributed heatsources has been used for the 24 × 24 and 28 × 28 sizes.Figures 2 and 3 show the influence of various grid sizes for thevariation of the local Nusselt number (Nu) and cup-mixingtemperature (Θcup) for Pr ) 0.015 and 1000 at Ra ) 105. Notethat the qualitative trends are identical for 20 × 20 and 28 ×28 elements. However, 28 × 28 elements are used for the com-putations, to preserve numerical accuracy. It is worth mentioningthat the biquadratic elements smoothly capture the nonlinearvariations of the field variables with relatively lesser numberof nodes, in contrast with finite difference/finite volume solu-tions. The iterative process is terminated based on the followingtwo-norm criteria:

where Rij corresponds to residual vectors, given by eqs 12, 13,

14, 30, and 33.For discrete heating situations, jump discontinuities exist at

the edges of the discrete heat sources, which correspond to

Figure 2. Variation of the local Nusselt number (Nu) with distance in Case 3 for (a) Pr ) 0.015 and (b) Pr ) 1000 at Da ) 10-3 and Ra ) 106 for differentmesh sizes ((O) 20 × 20, (]) 24 × 24 and (s) 28 × 28).

Θcup )∫ ∫Vk(X, Y)θ(X, Y) dX dY

∫ ∫Vk(X, Y) dX dY(40)

Vk(X, Y) ) √U2 + V2

Θavg )∫ ∫ θ(X, Y) dX dY

∫ ∫ dX dY(41)

RMSDΘcup) ∑

i)1

N

(θi - Θcup)2

N(42)

RMSDΘavg) ∑

i)1

N

(θi - Θavg)2

N(43)

Figure 3. Cup-mixing temperature (Θcup) in Case 3 for Pr ) 0.015 and1000 at Da ) 10-3 and Ra ) 106 with various mesh sizes ((O) 20 × 20,(]) 24 × 24, and (3) 28 × 28).

[ ∑ (Rij)2]0.5

e 10-5


mathematical singularities. The problem is resolved by specify-ing the average temperature of the two walls at the hot-coldjunction points and keeping the adjacent grid nodes at therespective wall temperatures. The unique solution for such typesof situations involves implementation of exact boundary condi-tions at those singular points. The Gaussian quadrature-basedfinite element method has been used in the current investigation,and this method provides smooth solutions in the computationaldomain; including the singular points as an evaluation ofresiduals is dependent on the interior Gauss points. The currentsolution scheme produces grid-invariant results, as discussedin an earlier article.63 Generally, Nusselt numbers for finitedifference/finite volume-based methods are calculated at anysurface using some interpolation functions, which are nowavoided in current work. The present Galerkin finite elementapproach offers special advantages in the evaluation of Nu atthe various cold or hot portions of the wall are used as elementbasis functions to evaluate the heat flux.

To validate the present numerical approach, benchmarkstudies have been carried out for the case where a porous squarecavity, with air (Pr ) 0.71) as the fluid, is heated isothermallyon the left wall, while the right wall is maintained coldisothermal and the horizontal walls are maintained adiabatic.Table 1 shows a comparison of the average Nusselt number(Nu) on the left hot wall, based on the current numericalprocedure, with the result of the same case obtained by Bejan,64

Manole and Lage,65 Saied and Pop,66 and Baytas and Pop67 ata modified Rayleigh number (Ram ) 1000, where Ram ) Ra ×Da. Note that the results of the Darcy model (employed byearlier works64-67) may be compared with a non-Darcy model(used in the present case) only at low Da values.

Benchmark studies to validate heatfunction (Π) contours arecarried out for the pure fluid case, and the results agree wellwith the work of Kimura and Bejan52 (the results are not shown,for the sake of brevity). In addition, simulations are also carriedout for the present cases (Cases 1-3) with a porous mediumfor a range of Rayleigh numbers (Ra ) 0, 10, 100, 103) andDarcy numbers (Da ) 10-6-10-3). At low Ra and Da values,fluid is virtually stagnant without any flow and heat transfer isprimarily due to conduction. Under these conditions, heatlinesessentially represent “heatflux” lines, which are commonly usedfor conductive heat transport.68 In addition, heatflux lines, bydefinition, are perpendicular to isothermal surfaces and parallelto adiabatic surfaces. As shown in Figure 4, results are inagreement with theoretical definitions. It is observed that theheatlines emanate from a hot surface and end on a cold surfaceand they are perpendicular to isothermal surfaces, similar toheatflux lines, during the conduction-dominant regime. As theyapproach the adiabatic wall, they slowly bend and becomeparallel to that surface. Also, the heatlines and isotherms arefound to be smooth curves, without any distortion in thepresence of a dominant conductive transport of heat.

The sign of the heatfunction Π requires special mention. Thesolution of the heatfunction, which is a Poisson equation, isstrongly dependent on nonhomogeneous Dirichlet boundaryconditions and the sign of heatfunction is governed by the signof the “nonhomogeneous” Dirichlet condition, which is specifiedat the edges of the discrete heat sources. For the current situation,a negative sign for the heatlines represents a clockwise flow ofheat, whereas a positive sign refers to anticlockwise flow. Thisassumption is in agreement with the sign convention for thestreamfunction (ψ). The streamfunction ψ and the heatfunctionΠ have identical signs for convective transport. The detaileddiscussion on heat transport based on heatlines for various casesis presented in the following sections.

3.2. Case 1. Figure 1a depicts the schematic diagram of Case1. The bottom wall is isothermally heated while the side wallsare isothermally cold. The top wall is adiabatic. At low Davalues (Da ) 10-6) and low Ra values (Ra ) 103) with Pr )0.015, fluid circulation is weak, because of low permeabilityof the porous medium and, hence, the buoyancy force is low.Therefore, the heat transfer is primarily due to conduction (figurenot shown). To show the effect of Da on convection, simulationshave been carried out at Ra ) 106, which corresponds to aconvection for a nonporous case or a pure fluid medium.63

Figure 4 illustrates streamlines, isotherms, and heatlines for Pr) 0.015 at Da ) 10-6 and Ra ) 106. At low Da, the hydraulicresistance of porous medium is high and, as a result, fluid flowis very weak, which is indicated by low magnitudes of thestreamfunction ψ. Corresponding heatlines start from the hotbottom wall and end at the cold side walls, illustrating thetrajectories or paths of heat flow. It is interesting to observethat the heatlines are perpendicular to the walls, signifying theconduction-dominant heat transfer at Da ) 10-6, even at ahigher Ra value (Ra ) 106). The magnitude of heatlinesindicates the amount of heat flux drawn from the hot surface. Itis observed that, at the bottom portion of the side wall, theheatlines are denser and they have high magnitude, whichindicates large heat flow to the lower portion, compared to thetop portions of the side walls. As a result of less heat flow tothe upper portion of the cavity, the thermal boundary layerthickness is found to be large at those regions and thetemperature (θ) is maintained at 0.1-0.2 at the top centralregion.

Figure 5 illustrates the effect of higher Da values on the fluidflow and heat flow patterns in the cavity. At Da ) 10-3 and Ra) 106 with Pr ) 0.015, convection is enhanced in the cavity asthe permeability of the porous medium increases or the hydraulicresistance decreases with increase in Da. The magnitude of thestreamfunctions increases and small secondary vortices tend toform at lower corners of the cavity. Heat flow during enhancedconvection may be visualized by heatlines. A large amount ofheat is drawn from the hot bottom region, as indicated by denseheatlines with large magnitudes of Π occurring along thecenterline. Therefore, the isotherms are compressed toward thebottom wall, depicting a high thermal gradient. The large amountof heat rises along the vertical centerline of the cavity and,hence, the top central portion is maintained at a highertemperature (θ ) 0.3-0.4), in contrast to that at low Da values.The boundary layer thickness is found to be small near the upperportion of the side walls as large heat flux from bottom wall isdelivered at those portions. On the other hand, a large boundarylayer thickness (i.e., a low thermal gradient) occurs at Y ) 0.2,as heatlines are observed to be dispersed at that region. Notethat the heatlines occur as end-to-end lines as well as closed-loop cells. The end-to-end heatlines indicate the transfer of heat

Table 1. Comparison of the Present Results with the BenchmarkSolutions of Earlier Works for Natural Convection in a PorousSquare Cavity with Air (Pr ) 0.71) as the Fluid Medium at Ram )Ra × Da ) 1000a

reference Nu

Bejan64 15.800Manole and Lage65 13.637Saeid and Pop66 13.726Baytas and Pop67 14.060present work 14.165

a The present study employs 28 × 28 bi-quadratic elements (57 × 57grid points).


from a hot region to a cold region, whereas the closed-loopheatline cells illustrate the recirculation of heat energy, whichcorresponds to the thermal mixing in the cavity. It may be seenthat, at high Da values, the thermal mixing is significantlyenhanced, as seen from |Π|max ) 2.87. It is observed that the

secondary circulations at the lower corners also recirculate asmall amount of heat. The results for Pr ) 1000 are similar tothat for the Pr ) 0.015 case at identical Da and Ra values,where the fluid flows in two symmetrical rolls in each verticalhalf of the cavity. However, the streamlines and heatlines are

Figure 4. Streamfunction (ψ), heatfunction (Π), and temperature (θ) contours for Da ) 10-6, Ra ) 106, and Pr ) 0.015 (molten metal) in Case 1. Clockwiseand anticlockwise flows are shown via negative and positive signs of the streamfunctions and heatfunctions, respectively.



observed to be diagonally skewed and their magnitudes arefound to be higher for higher Pr (figure not shown). It may beobserved that, as a result of uniform heating (Figure 5),temperature distribution is nonuniform, where the high tem-perature occurs at the central portions while a low temperatureis found along the side walls. The nonuniformity in temperaturedistribution may be effectively circumvented by distributing theheat source, which is discussed in the next section.

3.3. Case 2. In this case (see Figure 1b), the heat source isdivided into three parts, which are placed at the central locationsof the bottom and side walls of the cavity. The length of theheat source on the bottom wall is 0.5 and that along the sidewalls is 0.25. Note that the total length of the heat sources isthe same as that in Case 1. Distributions for fluid and heat flowfor Da ) 10-6 and Ra ) 106 with Pr ) 0.015 are shown inFigure 6. As a result of discrete heat sources, multiple circulationsare induced in the cavity. Because the bottom heat source is large,the primary circulation is stronger near the lower portion of thecavity and the secondary and tertiary circulations are relativelyweaker at central and upper portions, respectively. The conductiveheat flow from various heat sources at low Da values may bevisualized from heatlines distribution. The bottom heat sourcetransfers heat to the lower cold portions as well as to the top coldportions of the side walls. High heat flux with |Π|g 0.4 is observednear the edges of bottom heat source, as depicted by denseheatlines. In contrast, heatlines from the central region of the heatsource are weak, illustrating the lesser amount of heat flux. Theheat sources of the side walls transfer heat to the cold regions aboveand below them. It is observed that, as a consequence of distributedheating, the central region of the cavity is maintained at θ )0.3-0.5, even with the low permeability of the porous medium(low Da), in contrast to that in Case 1 at identical parameters.

As Da increases to 10-3 (see Figure 7), the resistance to fluidflow due to the porous medium is reduced, resulting in enhancedconvection with intense fluid flow and heat transfer. Multiplecirculations are observed at higher Da values for Pr ) 0.015

and the eye of the primary vortex occurs in the upper portionof the cavity. Visualization of the heat flow based on heatlinesfor this case depicts that a large amount of heat, correspondingto dense heatlines with |Π| e 1.5, from the bottom heat sourceis transferred to the upper cold portions of the side walls within0.78 e Y e 1. The thickness of the thermal boundary layer isobserved to be small along the cold upper portion of the sidewall at high Da values. The heat sources on the side wallsrecirculate a large amount of heat in the upper region of thecavity, resulting in enhanced thermal mixing with |Π|max ) 4.3.Thus, the temperature is observed to be uniform in the upperregion of the cavity, with θ ) 0.5-0.6. Note that the heatsources of the side walls also transfer heat to the cold portionsbelow them (0 e Y e 0.625), as well as to the cold portions ofthe bottom wall. However, it may be observed that the heatlinescorresponding to 1e |Π|e 2 occur sparsely in the lower cornersand thus, the temperature at those zones is found to be <0.2.Heat drawn from the bottom heat source is also found to berecirculated in the lower central region of the cavity with |Π|max

) 3 and, hence, isotherms are found to be dispersed. A similarexplanation follows for the heat flow distribution of the Pr )1000 fluid and, hence, the result is not shown for the sake ofbrevity.

From the above results, it is observed that distributed heatingeffectively improves thermal mixing in the cavity with enhanceduniformity in temperature distribution. Heatline distributionsillustrate that temperature distribution within cavities aregoverned by heat flow from various heat sources. However, notethat the lower corner portions remain colder, even at higher Davalues, because of inadequate heat flow, which is indicated bysparse heatlines at those regions. As a result, a bulk amount offluid is maintained cold as seen from θ ) 0.1-0.4 at lowercorners. To avoid these undesirable thermal dead zones atcorners regions, heat sources may be further divided and placedat corner portions. A detailed analysis of this situation isaddressed in the next section.



3.4. Case 3. In this case, the heat sources are divided intoseven parts and placed at lower corners and central portions ofthe bottom and side walls. The schematic diagram (Figure 1c)of this case illustrates the location of heat sources on the wallsof the cavity. The streamlines, isotherms, and heatlines for Pr) 0.015 at Da ) 10-6 and Ra ) 106 are shown in Figure 8.Tiny cells are formed near hot sources at the bottom corners

apart from circulations at the lower, middle, and upper portionsof the cavity. The magnitudes of the streamfunctions are low,because of a higher hydraulic resistance to flow at Da ) 10-6.Because the heat source distributions are different from that inCase 2, multiple flow circulations occur near the heat sourcesand, consequently, the heat flow patterns change accordingly.It is observed that the central bottom heat source transfers very




little heat to the top cold portions of the side walls, in contrastto that in Case 2 at low Da values. On the other hand, the centralheat sources of the side walls transfer heat to the top coldportions. These heat sources also transfer heat to the cold regionsbelow them. The central bottom heat source transfers heat tothe adjacent cold regions and a small amount of heat to thelower cold sections of side walls. The corner hot regions alsotransfer heat to their adjacent cold portions. As a result ofdistributed heating with multiple heat sources, the central coreregion is maintained at θ ) 0.3-0.4 at low Da values.

At Da ) 10-3 (Figure 9), the resistance offered by the porousmedium is reduced and, consequently, the convection in thecavity is enhanced. As Da increases, the primary circulation inthe upper half of the cavity is enhanced (|ψ|max ) 5) and theintensity of the primary circulation is stronger than those ofthe secondary and tertiary circulations at the bottom and thecentral portion of side walls, respectively. The quaternarycirculations at the lower corners are now reduced at higher Davalues. Visualization of the heat flow via heatlines depicts thata large amount of heat is drawn from the heat sources at thecentral portion of the side walls and that is recirculated at theupper portion of the cavity (see Figure 9). These heat sourcestransfer heat to the cold portions above them with 0.65 e Y e0.8 and, thus, the thermal boundary layer is compressed towardthe side walls at those portions. Enhanced convection at Da )10-3 leads to enhanced thermal mixing as seen from thecirculation cells of the heatlines. As a result, the temperatureof upper portion of cavity is uniformly maintained within θ )0.3-0.4. At higher Da, a large amount of heat is drawn fromthe central heat source of the bottom wall and that is depictedby dense heatlines along the vertical central line (see Figure9). It is interesting to note that, at higher Da values, the centralbottom heat sources transfer heat to the top portion of the sidewalls within a larger region (0.9 e Y e 1). Furthermore, thecentral bottom heat sources transfer heat to the lower coldportions of side walls and to the cold portions adjacent to the

heat source. The thermal mixing at the lower portion of thecavity is only due to the recirculation of heat from the centralbottom heat source. The heat sources at the corner portions ofthe bottom wall and side walls transfer heat to the cold portionsof side walls. It is observed that the heat sources at the cornerregions transfer a much smaller amount of heat to the coldportions above them. The cold regions at the lower portion ofside walls receive heat primarily from the heat sources at thecentral region of the side walls. It may be noted that the fluidin corner regions is adequately heated, in contrast to that inCase 2. Almost 80% of the cavity is maintained within θ )0.3-0.5 with increased distribution of heat sources. It isobserved that the temperature at the core region of the cavity isless, compared to θ ) 0.5-0.6 in Case 2 at the same Da value.It is interesting to note that the temperature near the bottom cornersand the cold portion of the bottom wall is higher in Case 3,compared to that in Case 2. This enhancement of temperature isdue to the larger distribution of heat, as seen from heatlines.

As the value of Pr increases to 0.7 at Da ) 10-3 and Ra )105 (Figure 10), the primary circulations occur at the lowerportion of the cavity. The strengths of all circulations are foundto be increased, including the quaternary circulations at the lowercorners of the cavity. Heatlines indicate that the heat flow patternwith larger Pr values significantly differs from that with thelower Pr case. It is seen that the central heat source of the bottomwall plays a major role in heat distribution and thermal mixingin the lower portion of the cavity. Enhanced convection at Da) 10-3 results in efficient thermal mixing in the lower portionof the cavity where |Π|max ) 2.8 is observed. The central heatsource of the bottom wall transfers heat to the lower coldportions of the side walls. Also, a small amount of heat istransferred to the cold regions at the top portion of the sidewalls within 0.88 e Y e 1. However, it is interesting to observethat the central heat source of the bottom wall transfers only avery small amount of heat to the adjacent cold regions of thesame wall. However, much of the heat to those cold portions is

Figure 9. Streamfunction (ψ), heatfunction (Π), and temperature (θ) contours for Da ) 10-3, Ra ) 106, and Pr ) 0.015 (molten metal) in Case 3. Clockwiseand anticlockwise flows are shown via negative and positive signs of streamfunctions and heatfunctions, respectively.


transferred by heat sources at the lower corner regions. Notethat the thermal mixing is further enhanced by the corner hotsources, as seen from small heatline cells.

The central heat sources of the side walls also transfer heatto the cold portions above and below them. The heat sources atthe central portion of the side walls play an active role inenhancing the thermal mixing at the central region of the cavity,compared to that with Pr ) 0.015, as seen from stronger heatlinecells with |Π|max ) 1.7 (see Figure 10). The heat sources inducethermal mixing in the top portion of the cavity at higher Da.However, a large amount of heat is recirculated near the heatsource at the central region of the cavity, as seen from the denseheatlines. Because of enhanced heat flow from the heat sourcesat central regions of the walls of the cavity, the central core ofthe cavity is maintained at θ ) 0.4-0.5 at higher Da values. Itmay be noted that the heat transfer to the upper portion of thecavity is relatively low, and, thus, the top central portion of thecavity is maintained only at θ ) 0.3-0.4. Also, a smallerthermal gradient is observed near the cold portion above thecentral heat sources of side walls, because of less heat transfer,as seen from heatlines. As Pr increases to 1000 (Figure 11),the qualitative features of fluid and heat flow are similar to thatof Pr ) 0.7 at Da ) 10-3 and Ra ) 105, but their intensitiesare higher, because of higher momentum diffusivity of higherPr fluid. The boundary layer thickness with high thermalgradient is observed for Pr ) 1000 at the upper portion of theside walls, in contrast to that for the Pr ) 0.7 case. It isinteresting to note that, at higher Da values, the heat flow fromthe central bottom heat source is enhanced significantly, as seenfrom the increased magnitude of dense heatlines. Therefore,relatively a large area in the cavity is maintained at θ ) 0.4-0.5at Da ) 10-3 near the top portion. Also, as a result of distributedheating with multiple heat sources in this case, the lower cornerregions are adequately hot and temperature is maintained withinθ g 0.3, in contrast to that in Case 2, where θ e 0.1 wasobserved.

3.5. Local Nusselt Number and Heatlines. Figure 12illustrates the local Nusselt number versus the distance overthe bottom and side walls for Pr ) 0.015 and 1000 at Da )10-6 (upper panels) and Da ) 10-3 (lower panels) and Ra )106. The hot sections on each wall are represented by the thicklines. The positive values of the Nusselt number (Nu) indicateheat “taken away” from the (hot) surface, while negative valuesindicate that heat is “received” by the (cold) surface. It may benoted from eq 31 that the magnitude of the gradient of theheatfunction, ∂Π/∂Y or ∂Π/∂X, is equal to the temperaturegradient (∂θ/∂X or ∂θ/∂Y) at the wall and the gradient oftemperature is equal to Nu (see eq 15). In this way, theheatfunction (Π) is inherently related to Nu and, therefore, thequalitative and quantitative features of heatlines may also beused for analysis of the variation of Nu on the walls of the cavity,apart from studying the path of heat flow.

At low Da () 10-6), the variation of each of Nub and Nul isidentical for Pr ) 0.015 and 1000 in all cases (see the upperpanels in Figures 12a-f) as the heat transfer is conductiondominant, because of the high hydraulic resistance of the porousmedium and the rate of heat transfer is similar for both Pr. Alongthe bottom wall in Case 1, the rate of heat transfer is maximumat the lower corners, where the hot bottom wall is in directcontact with the cold side walls. However, Nub remains almostinvariant and is ∼1.8-2.4 within 0.3 e X e 0.7 of the bottomwall (see Figure 12a). The small values of Nub are due to thefact that heatlines are almost parallel along that portion and |Π|varies within a range of 0.01-0.9 in a large region (see Figure4). On the other hand, the local Nusselt number on the left sidewall, Nul (see upper panel of Figure 12b), is high near the lowerportion as the large amount of heat from the bottom wall isdelivered within the section Y ) 0-0.1, as depicted by heatlineswith |Π| > 1 within that section (see Figure 4). A plateau in Nul

is observed within a large section (0.2 e Y e 1) of the side wall.As Da increases to 10-3, the magnitude of Nub for Pr ) 1000

is higher than that for Pr ) 0.015 for Case 1 (see the lower

Figure 10. Streamfunction (ψ), heatfunction (Π), and temperature (θ) contours for Da ) 10-3, Ra ) 106, and Pr ) 0.7 (air) in Case 3. Clockwise andanticlockwise flows are shown via negative and positive signs of streamfunctions and heatfunctions, respectively.


panel of Figure 12a). A sharp decrease in Nub is observed forPr ) 0.015 and remains almost invariant within 0.15 e X e0.85. In contrast, Nub decreases at a slower rate from the endportions of bottom wall to the center for Pr ) 1000. Thevariation of Nub may be adequately explained based on heatlines.The heatlines for Pr ) 0.015 (see Figure 5) and Pr ) 1000(figure not shown) illustrate that small corner regions correspondto large magnitudes of heatfunctions, indicating high heat flow,and, thus, maxima in Nub are observed near X ) 0 and 1. Similarto the low Da case, heatlines are found to be dispersed alongthe central region and, hence, a plateau of Nub occurs along80% of the bottom wall. The local Nusselt number on the leftside wall (Nul) is found to be large for Pr ) 0.015 along thetop portion (see the lower panel of Figure 12b). This larger valuefor Pr ) 0.015 is due to the fact that a large amount of heat istransported to the top portions, as seen from dense heatlines(recall Figure 5).

Variations of Nub and Nul for the bottom and left walls inCase 2 for Da ) 10-6 and 10-3 at Ra ) 106 and Pr ) 0.015and 1000 are shown in Figures 12c and 12d. Note that, in thiscase, the heat sources are located at central portions on therespective walls. The maxima in Nub and Nul occur at the hot-cold junctions of the discrete heat sources. At low Da, themaxima of Nub on the bottom heat source is confined only within∆X ) 0.05 near the edges. The maxima of Nub at the junctionsare due to dense heatlines, as seen from Figure 6. The centralregion of the bottom wall corresponds to lesser Nub, because asmall amount of heat is drawn from the bottom heat source,which is depicted by small magnitudes of heatfunctions withina range of |Π| ) 0.01-0.1 (Figure 6). It is also interesting toobserve that Nub along the cold portion of the bottom wall isalmost zero, because of dispersed heatlines along those portions,as seen in Figure 6. Similar to Case 1, Nub is found to beidentical at Pr ) 0.015 and Pr ) 1000. As the value of Daincreases to 10-3, Nub is found to be high on the heat source ofthe bottom wall for Pr ) 1000, compared to that for Pr ) 0.015.

However, Nub is almost invariant with Pr on the cold sectionsof the bottom wall, except near hot-cold junctions. The largevalues of Nub along the hot-cold junctions are due to a largeheat flux, which is transferred to the top portions of side walls,as seen in Figure 7. On the other hand, Nub is higher on theadjacent cold regions at low Da values (see the upper panel ofFigure 12c). This is due to high conductive heat transport tothe adjacent cold sections at low Da values, which is depictedby short dense heatlines (|Π|g 0.4) near the edges of the bottomheat source (see Figure 6).

The local Nusselt number for the left wall (Nul) is identicalfor both Pr and that varies symmetrically about Y ) 0.5 at lowDa values in Case 2 (see the upper panel of Figure 12d). A fewinteresting features are observed for Nul at Da ) 10-3 and Ra) 106. The maxima in Nul for Pr ) 0.015 occurs at the topedge of the heat source, whereas the maxima for Pr ) 1000occurs at the lower edge of the heat source. This may beexplained via heatlines. In the case of low Pr (Figure 7), it isseen that dense heatlines emanate from the top edge of the heatsource, indicating that a large amount of heat is being drawnfrom the top portion and is recirculated throughout almost theentire cavity. In contrast, it was found that dense heatlinesoccurred near the lower edge of the central heat sources of theside walls for the high Pr fluid (figure not shown), indicating alarge amount of heat transfer. Similar to that in the previouscase (Case 1), Nul is higher for Pr ) 1000 near the top coldportions of the side walls within 0.85 e Y e 1.

The variations of Nub and Nul for Pr ) 0.015 and 1000 forCase 3 at Da ) 10-6 and 10-3 are shown in Figures 12e and12f. Similar to that in previous cases, high heat flux is observednear the edges of the hot regions, because dense heatlines occurat the junctions (see Figures 8-11). At low Da values, thevariations in Nub and Nul for both Pr values are identical andthey are almost invariant along the cold regions, except nearthe junctions. At high Da (Da ) 10-3), the magnitude of Nub

is higher for Pr ) 1000 on the hot sections of the bottom wall,

Figure 11. Streamfunction (ψ), heatfunction (Π), and temperature (θ) contours for Da ) 10-3, Ra ) 106, and Pr ) 1000 (olive/engine oil) in Case 3.Clockwise and anticlockwise flows are shown via negative and positive signs of the streamfunctions and heatfunctions, respectively.


whereas slightly higher Nub values are observed for Pr ) 0.015on the cold sections of the bottom wall. This may be explainedbased on heatlines that are depicted in Figures 9 and 11. Theheatlines emanating from the bottom heat source in the Pr )0.015 case transfer a large amount of heat to the adjacent coldsection, while a significant amount of heat in the Pr ) 1000case is recirculated in the lower central region with a smallamount of heat transfer to the adjacent cold section. Similar tothat in Case 2, interesting features on variations in Nul areobserved on the side walls for Case 3. It is found that, in contrastto Nub, the magnitude of Nul is slightly higher for Pr ) 0.015than that for Pr ) 1000 at the hot section along the bottom

corner of the side wall (within 0 e Y e 0.1). This may beexplained based on the distributions of heatlines as illustratedin Figure 9. It is observed that, for Pr ) 0.015 case, the heatlinescorresponding to |Π| e 0.37 emanate from the heat source atthe lower portion of the side walls and heat is transferred to thecold sections of the bottom wall. On the other hand, the gradientof heatfunctions is small along that heat source for the Pr )1000 case (see Figure 11), indicating that only a small amountof heat is drawn. Hence, Nul is observed to be low for Pr )1000, compared to that for the Pr ) 0.015 case on the lowerhot region of the side walls. It is also interesting to note thatNul for Pr ) 1000 is higher than that for Pr ) 0.015 near the

Figure 12. Variations of the local Nusselt number (Nub and Nul) with distance on the bottom and left walls in various cases for (- - -) Pr ) 0.015 and(s) Pr ) 1000, Da ) 10-6 (upper panels) and Da ) 10-3 (lower panels) at Ra ) 106. Variations of Nub and Nul for both Pr values are identical for Da )10-6. Thick lines along the “distance” axis represent hot sections, and the other portions indicate cold sections on respective walls. The vertical dotted linesdenote the boundary between cold and hot regions.


lower portions of central heat sources of the side walls, whereasa larger Nul is observed near the top portion of those heat sources(see the lower panel of Figure 12d). This may be clearly seenfrom the heatline distribution. Dense heatlines occur near bothedges of the heat source of the side wall for the Pr ) 0.015case, indicating large heat transport to the adjacent upper andlower cold portions (see Figure 9). In contrast, the heatlinesare found to be clustered only near the bottom edge of the heatsource of the side wall for the Pr ) 1000 case (see Figure 11)and much of the heat drawn is transferred to the top cold portionof the side walls. The variation in Nul for both Pr values isqualitatively similar to that in Case 2 along the top cold sectionof side walls within 0.65 e Y e 1.

3.6. Degree of Thermal Mixing and Temperature Uni-formity. The bulk mean temperature in the cavity is estimatedby evaluating the cup-mixing temperature (Θcup), which isdefined by eq 40. The bulk mean temperature in the cavity isadequately represented by the velocity-weighted temperaturewhen convective flow exists. In addition, the spatial or area-averaged temperature (Θavg; see eq 41) is also presented, forthe sake of comparison. The upper panels of Figures 13a-dillustrate Θcup and Θavg at Ra ) 106 for Da ) 10-6 and Da )10-3. To quantify the degree of temperature uniformity in eachcase, the root-mean-square deviation based on the cup-mixingtemperature (RMSDΘcup

) and average temperature (RMSDΘavg)

are evaluated according to eqs 42 and 43, and they are illustratedin the lower panels of Figures 13a-d. As mentioned earlier, alow RMSD value indicates a high degree of temperatureuniformity across the cavity.

Note that, in the convection-dominant regime (high Da and Ravalues), a high value of Θcup indirectly indicates a high degree ofthermal mixing in the cavity due to convective flow. However,that may not be true for a conduction-dominant regime, because ahigh Θcup value may also be possible for low Da and Ra values.Such types of situations occur when there is a high degree oftemperature stratification in the cavity, where θ values woulddominate over Vk values in the evaluation of Θcup (see eq 40). Undersuch situations, RMSDΘcup

values can provide insight on thermalmixing. For identical Θcup values at different Da, the degree ofthermal mixing may be compared, based on RMSDΘcup

. Thethermal mixing for the corresponding Da value is low for a highRMSDΘcup

value and vice versa. This type of situation will beillustrated in the following discussion, to compare various casesfor enhanced thermal mixing and temperature uniformity.

Comparison of the cup-mixing temperatures for various casesat low Da values (see the “0” symbols in upper panels ofFigures 13a-d) indicates that Θcup is high in Case 2 for all Prvalues. The corresponding RMSDΘcup

in the lower panelsindicate that temperature uniformity is poor in Case 1 and highin Case 3. The smaller values of RMSDΘcup

with large Θcup in

Figure 13. Upper panels: Cup-mixing temperature (Θcup, denoted by open squares (0) and circles (O)) and average temperature (Θavg, denoted by crosssymbols (+) and times symbols (×)). Lower panels show (0, O) RMSDΘcup

and (+, ×) RMSDΘavgvalues. Note that, for all the plots, Da ) 10-6 (denoted

by 0 and + symbols), Da ) 10-3 (denoted by O and × symbols), and Ra ) 106.


Cases 2 and 3 are attributed to the distributed heating of thecavity, which results in relatively higher thermal mixing andtemperature uniformity (See Figures 6 and 8), compared to thatin Case 1 (see Figure 4), even at low Da values. Note that Θcup

is high in Case 1 for Pr ) 1000, compared to that in Case 3(see Figure 13d). This is due to the high thermal diffusivity ofthe high Pr fluid. However, the RMSDΘcup

value is found to besignificantly low in Case 3.

At Da ) 10-3, it is found that Θcup ≈ Θavg and RMSDΘcup≈ RMSDΘavgfor each of Cases 1-3. Interesting features of

Θcup are observed in the convection-dominant region at highDa values. Similar to that in the low Da case, Θcup is observedto be high in Case 2 for all Pr values (see the “O” symbols in theupper panels of Figures 13a-d). The distributed heating in Case 3results in low Θcup values (except for the Pr ) 0.015 case).However, the corresponding RMSDΘcup

is low in Case 3 for all Prvalues, indicating a high degree of temperature uniformity in thecavity (see Figures 5, 7, 9, and 11). It is interesting to note that theΘcup values in Cases 1 and 3 for Pr ) 0.015 and Cases 1 and 2for Pr ) 1000 at Da ) 10-3 are almost equal to those at Da )10-6. As mentioned earlier, this is due to high-temperaturestratification in those cases at low Da values, and that may beconfirmed from high RMSDΘcup

values at low Da values. Also,temperature stratification may be seen from the isotherms of Figure4.

Overall, a high Θcup value is observed in Case 2 for all Pr valuesat Da ) 10-3 and Ra ) 106, and that is followed by Cases 1 and3, respectively (except at Pr ) 0.015, where Case 3 is followedby Case 1). However, the corresponding RMSDΘcup

value indicatesthat, although the thermal mixing is enhanced in Case 2, as a resultof distributed heating, the uniformity in temperature distributionis poor. On the other hand, in Case 3, the RMSDΘcup

value isobserved to be low, indicating a greater degree of temperatureuniformity in the cavity, despite a relatively lower degree of thermalmixing. This is a result of efficient heat distribution by discreteheat sources. The results suggest that a high degree of thermalmixing is not always needed to ensure high uniformity intemperature and a distributed heating methodology with moderatethermal mixing may be adequate in attaining a desired temperaturedistribution.

4. Conclusion

The prime objective of the current investigation is toanalyze the role of distributed heating on thermal mixing andtemperature uniformity in square cavities filled with fluid-saturated porous media. Three different configurations of thecavities are considered. A detailed analysis of heat flow basedon a heatline approach is presented. At low Darcy number(Da ) 10-6), heatlines are observed to be smooth lines withlow magnitudes of the heat source (Π), indicating conduction-dominant heat transfer in the cavity. At higher Da values(Da ) 10-3), the hydraulic resistance of the porous mediumdecreases, resulting in enhanced convection. In convection-dominant mode, the magnitude of the heatlines increases andthey are found to be distorted. Enhanced heat flow from thediscrete heat sources in a convection-dominant regime isdepicted by dense heatlines. Also, circular heatline cells areobserved during convection mode in addition to end-to-endheatlines, which illustrate enhanced thermal mixing in thecavity at higher Da values.

Heatline analysis of Case 1 indicates that the nonuniformtemperature distribution occurs within the cavity with hightemperature at the central regions and low temperature alongthe side walls at Da ) 10-3 and Ra ) 106. The heat distribution

and thermal mixing for Case 2 is greatly enhanced, comparedto that in Case 1, because of distributed heating at high Davalues. However, the lower corner portions were found to remaincolder, even at higher Da values, because of inadequate heatflow from the bottom and side wall heat sources. Furthermore,in Case 3, heat sources are placed at corner regions apart fromthe central regions on the bottom and side walls for effectiveheat distribution throughout the cavity. Consequently, a smallreduction in thermal mixing is observed, but the temperaturedistribution is enhanced significantly.

Thermal mixing in the three cases is quantitativelycompared, using cup-mixing and spatial average temperatures,and temperature uniformity is compared by root-mean squaredeviations (RMSDΘcup

/RMSDΘavg). It is found that high

thermal mixing is achieved in Case 2 for all Pr values butwith nonuniform temperature distribution in the cavity. Onthe other hand, only moderate thermal mixing is achieved inCase 3, but a large uniformity in temperature distribution isobserved. Furthermore, the effect of Da on the variation oflocal Nusselt number (Nu) is studied. It is found that Nusseltnumbers at the bottom and left side walls (Nub and Nul,respectively) for lower and higher Pr limits vary identicallyat low Da values. However, at high Da values, the heattransfer rate is higher for high Pr fluids. Distinctive featuresof the local Nusselt number, such as that in Case 2, wherethe maxima in Nul occur at opposite edges of the heat sourcefor lower and higher Pr values, and these features areadequately explained based on the heat flow paths representedby the heatlines. Overall, it has been established that thedistributed heating plays an important role in enhancementof the thermal mixing and temperature uniformity in thecavity, and it may be further explored for efficient thermalprocessing of the materials and proper utilization of thethermal energy.

Acknowledgment

Authors would like to thank anonymous reviewers for theircritical comments and suggestions, which improved the qualityof the manuscript.

Nomenclature

Da ) Darcy numberdp ) particle diametere ) porosityg ) acceleration due to gravity (m s-2)k ) thermal conductivity (W m-1 K-1)K ) permeability of porous medium (m2)l′ ) length of hot/cold section (m)L ) side of the square cavity (m)N ) total number of nodesNc ) total number of cold sectionsNh ) total number of hot sectionsNu ) average Nusselt numberNu ) local Nusselt numberp ) pressure (Pa)P ) dimensionless pressurePr ) Prandtl numberR ) residual of weak form


Ra ) Rayleigh numberRam ) modified Rayleigh numberRMSD ) root-mean-square deviationT ) temperature of the fluid (K)Tc ) temperature of cold section (K)Th ) temperature of hot section (K)u ) x component of velocityU ) x component of dimensionless velocityV ) y component of velocityV ) y component of dimensionless velocityVk ) dimensionless velocityX ) dimensionless distance along the x-coordinatex ) distance along the x-coordinateY ) dimensionless distance along the y-coordinatey ) distance along the y-coordinate

Greek Symbols

R ) thermal diffusivity (m2 s-1) ) volume expansion coefficient (K-1)γ ) penalty parameterε ) percentage errorθ, Θ ) dimensionless temperatureν ) kinematic viscosity (m2 s-1)F ) density (kg m-3)Φ ) basis functionsψ ) streamfunctionΠ ) heatfunctionΩ ) domain

Subscripts

b ) bottom walli ) residual numberj ) wallk ) node numberl ) left wallm ) hot or cold sectionr ) right wall1, 3 ) cold sections2 ) hot sectioncup ) cup mixingavg ) spatial average

Superscript

j ) equation number

Appendix: Boundary Conditions for Heatfunction Π

Case 1:

Case 2:

Case 3:

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Π ) 0 at (X ) 0, Y ) 1)) Nul at (X ) 0, Y ) Ya)

) 0 at (X ) 1, Y ) 1)) - Nur at (X ) 1, Y ) 0)

(44)

Π ) 0 at (X ) 0, Y ) 1)) ll,3′ Nul,3 at (X ) 0, Y ) Ya)

) Π(0, Ya) + ll,2′ Nul,2 at (X ) 0, Y ) Yb)

) Π(0, Yb) + ll,1′ Nul,1 + lb,1′ Nub,1 at (X ) Xc, Y ) 0)

) 0 at (X ) 1, Y ) 1)) - lr,3′ Nur,3 at (X ) 1, Y ) Yf)

) Π(1, Yf) - lr,2′ Nur,2 at (X ) 1, Y ) Ye)

) Π(1, Y)e - lr,1′ Nur,1 - lb,3′ Nub,3 at (X ) Xd, Y ) 0)(45)

Π ) 0 at (X ) 0, Y ) 1)) ll,4′ Nul,4 at (X ) 0, Y ) Ya)

) Π(0, Ya) + ll,3′ Nul,3 at (X ) 0, Y ) Yb)

) Π(0, Y)b + ll,2′ Nul,2 at (X ) 0, Y ) Yc)

) Π(0, Yc) + ll,1′ Nul,1 + lb,1′ Nub,1 at (X ) Xd, Y ) 0)

) Π(Xd, 0) + lb,2′ Nub,2 at (X ) Xe, Y ) 0)

) 0 at (X ) 1, Y ) 1)) - lr,4′ Nur,4 at (X ) 0, Y ) Yj)

) Π(0, Yj) - lr,3′ Nur,3 at (X ) 0, Y ) Yi)

) Π(0, Y)i - lr,2′ Nur,2 at (X ) 0, Y ) Yh)

) Π(0, Yh) - lr,1′ Nur,1 - lb,5′ Nub,5 at (X ) Xg, Y ) 0)

) Π(Xg, 0) - lb,4′ Nub,4 at (X ) Xf, Y ) 0)(46)


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ReceiVed for reView March 10, 2010ReVised manuscript receiVed August 11, 2010

Accepted August 23, 2010

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Efficient Methodologies for Processing of Fluids by Thermal Convection within Porous Square Cavities

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