Numerical predictions of two-dimensional conduction ... · des problèmes de rayonnement thermique...

Int. J. Therm. Sci. (2000) 39, 332–353 2000 Éditions scientifiques et médicales Elsevier SAS. All rights reservedS1290-0729(00)00222-2/FLA

Numerical predictions of two-dimensional conduction,convection, and radiation heat transfer.

II. Validation

Daniel R. Rousse a*, Guillaume Gautier a, Jean-Francois Sacadura b

a Département de génie mécanique, Université Laval, Cité Universitaire, Québec, Canada G1K 7P4b Centre de Thermique de Lyon (CETHIL)–INSA de Lyon, Bâtiment 404, 69621 Villeurbanne cedex, France

(Received 4 March 1999, accepted 26 August 1999)

Abstract —This paper presents several test problems that were used to validate the formulation and implementation of a CVFEM forcombined-mode heat transfer in participating media. The objective here is to demonstrate that the proposed CVFEM can be used tosolve combined modes of heat transfer in media that emit, absorb, and scatter radiant energy in regularly- and irregularly-shapedgeometries. The paper first assesses briefly the CVFEM for the solution of convection–diffusion problems, then attention is focused onradiative heat transfer problems. Finally, several combined mode problems are investigated. Results show that the proposed methodadequately solves the governing equations (energy and RTE): as the solutions compare favorably with those obtained with otheracknowledged methods or with analytical solutions, when available. The proposed CVFEM could, however, be improved to broadenits scope of application and enhance its numerical efficiency. In a near future, the method will be combined with a code alreadyestablished for fluid flow calculations. 2000 Éditions scientifiques et médicales Elsevier SAS

control volume finite elements method / discrete ordinates / combined modes heat transfer / two-dimensional / validation

Résumé —Prédictions numériques du transfert par conduction, convection, et rayonnement thermiques. II. Validation. Cetarticle présente quelques cas tests ayant été utilisés afin de valider la formulation et l’implantation d’une Méthode d’ÉlémentsFinis basée sur des Volumes de Contrôle (MEFVC) pour la résolution du transfert thermique combiné dans des milieux semi-transparents. L’objectif consiste ici à démontrer que la méthode proposée peut être utilisée pour résoudre des problèmes dethermique couplés dans des milieux qui émettent, absorbent et diffusent le rayonnement en géométries irrégulières ou non. L’articleconfirme tout d’abord rapidement que les problèmes de convection–diffusion peuvent être résolus adéquatement. Par la suite,des problèmes de rayonnement thermique pur sont étudiés. Enfin, quelques problèmes où interviennent convection, diffusion etrayonnement sont présentés. Les résultats présentés permettent d’affirmer que la méthode proposée fournit des solutions adéquatesau modèle mathématique (équations d’énergie et de transfert radiatif) employé pour décrire le phénomène d’intérêt. Toutefois desprolongements et améliorations peuvent être envisagés pour élargir les domaines d’application de la méthode et accroître sonefficacité numérique. Dans un avenir rapproché cette méthode sera combinée à un code établi permettant les prédictions desécoulements fluides. 2000 Éditions scientifiques et médicales Elsevier SAS

méthode éléments finis volumes de contrôle / ordonnées discrètes / modes couplés de transfert thermique /bi-dimensionnel / validation

Nomenclature

A coefficient of the discretization equation,surface area

Bo Boltzmann numbercp medium specific heat . . . . . . . . . . . J·kg−1·K−1

e emissive power . . . . . . . . . . . . . . W·m−2

g incident radiant energy . . . . . . . . . . W·m−2

* Correspondence and [email protected]

i radiative intensity . . . . . . . . . . . . W·m−2·sr−1

EJ dimensionless flux (conductive,convective or radiative)

k medium conductivity . . . . . . . . . . W·m−1·K−1

L characteristic length . . . . . . . . . . . mN Stark number, number of panelsEn unit vector normal to a surfacep control volume surface (panel)Pe Peclet numberEq heat flux . . . . . . . . . . . . . . . . . W·m−2

q′′′ internal heat generation . . . . . . . . . W·m−3

Qml

impinging dimensionless radiative flux

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Numerical predictions of two-dimensional conduction, convection, and radiation heat transfer. II

QinB total incoming dimensionless radiative flux

r reference point for RTE integrations spatial position . . . . . . . . . . . . . . mSφ rate of volumetric generation ofφSI source of radiative intensityT temperature . . . . . . . . . . . . . . . KEu velocity . . . . . . . . . . . . . . . . . . m·s−1

V volume . . . . . . . . . . . . . . . . . . m3

Greek symbols

β extinction coefficient . . . . . . . . . . m−1

ε normalized nodal percentage errorε normalized residualε emissivity of a surfaceγ albedo for single scatteringκ absorption coefficient . . . . . . . . . . m−1

ω solid angle . . . . . . . . . . . . . . . . srEΩ direction of propagationφ dependent variableΦ phase function for scatteringµ,η, ξ direction cosines ofEΩρ medium density . . . . . . . . . . . . . kg·m−3

σ scattering coefficient . . . . . . . . . . m−1

σ Stefan–Boltzmann constant,5.67051E−8 . . . . . . . . . . . . . . . W·m−2·K−4

τ optical depthτ0 optical thicknessEω angular velocity

Subscripts

∗ reference quantityb black bodyB boundaryl control volume surface (panel)m discrete directionP node of referencer, r radiation, or reference point

Superscripts

∗ evaluated at a preceeding step of aniterative procedure

E vectorial quantity′ incoming direction– bulk value∼ a modified quantityC convection termD diffusion term

1. INTRODUCTION

In a companion paper [1], the formulation of a Con-trol-Volume Finite Element Method (CVFEM) for the

solution of two-dimensional, combined-mode heat trans-fer problems in gray participating media is presented.Some of the capabilities of this CVFEM are demon-strated in this paper by its application to several differ-ent test problems involving steady-state combined con-duction, convection, and radiation in emitting, absorb-ing and isotropically scattering media. The assesment ofthe method is carried out in three steps: (1) convection–diffusion problems; (2) radiation heat transfer prob-lems; and (3) combined-modes problems. The followingproblems were considered in this first phase of valida-tion:

• Convection–diffusion problems: (1) radial conductionin a rotating hollow cylinder; (2) heat transfer in a radialflow between concentric cylinders; and (3) heat transferin a recirculating flow within a square enclosure.

• Radiation heat-transfer problems: (1) gray scatteringmedia in a rectangular enclosure; (2) gray absorbingmedia in a rectangular enclosure; (3) gray absorbingmedia in a trapezoidal enclosure; (4) gray absorbing,emitting, and scattering media in an L-shaped enclosure;and (5) gray absorbing media in a curved enclosure.

• Combined-modes problems: (1) conduction–radiationin a rectangular enclosure; (2) conduction–convection–radiation in a rectangular enclosure; and (3) conduction–convection–radiation in a divergent channel.

These test problems were chosen as they: (1) coverseveral commonly encountered features of two-dimen-sional, combined-mode heat transfer in gray gases; (2) al-low for an evaluation of the key ideas put forth inthe development of the CVFEM; (3) provide a meansof determining whether, or not, the proposed CVFEMand its computer implementation are capable of ac-curately solving the mathematical model used in theformulation [1]; (4) demonstrate the possibility to ap-ply the proposed CVFEM in irregularly-shaped geome-tries.

For each problem, steady-state conditions, homoge-neous, constant thermophysical and radiative proper-ties, and radiatively participating gray media are consid-ered [1]. No relaxation was used in the computations. Theiterations were terminated when the absolute value of theresidual on the incident radiationG, from one iteration tothe next, met a prescribed conditionε. In test problemsconsidered hereε = 10−6 was found to be satisfactory: ityielded solutions that were essentially insensitive to fur-ther reductions inε.

Structured-grids, with nodes arranged along straightor curved lines, were employed for all investigations pre-sented here. Various ways of forming triangular elements

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from quadrilaterals were investigated. With regard to an-gular discretization, the angular discretization was al-ways customized so as to match the physical boundariesof the domain withφ− θ azimuthal discretization, exceptin regular geometries for which standard equal weightSndiscretization were used. Results are presented in termsof dimensionless variables and a nodal percentage er-ror. This error is denoted byε, and calculated as fol-lows:

ε = |φref− φnum|φscale

· 100 (1)

where φnum is the quantity of interest as computedwith the proposed CVFEMs,φref is the correspondingquantity as obtained from an analytical solution, an ac-cepted approximate numerical solution, or a nodallyex-act numerical solution; andφscale is a suitable scalefor φ.

The formulations were implemented on a Pentium II266 MHz with 256 MB RAM, using the MSDev FOR-TRAN 90 compiler.

The complete formulation, objectives, computationaldetails, and results of each of the example problems aredescribed in the remainder of this paper. The last sectionpresents the general conclusions of the study.

2. CONVECTION–DIFFUSION PROBLEMS

For all convection–diffusion test cases investigated,the values of the dimensionless temperature were spec-ified using either nodally “exact” numerical solutions [2]or analytical solutions, when available. The internal val-ues of temperature were set to zero at the beginning ofthe iterative process. The problems were solved for sev-eral values of the Peclet number in the range 0.01< Pe<10 000. The low values represent conduction-dominatedproblems, while high values ofPe indicate convection-dominated processes. Conditions forPe= 10 000 arenot physically possible as the flow would be turbulentfor Pr = 1. The convection–diffusion problems permita comparison of the Skew Positive Coefficient Upwind(SPCU) scheme and FLow-Oriented (FLO) scheme, andprovide confidence in the formulation and implementa-tion of the proposed procedure to solve the energy equa-tion.

Figure 1. Heat conduction in a rotating hollow cylinder:problem schematic.

2.1. Heat conduction in a rotatinghollow cylinder

2.1.1. Problem statement

This problem [3] is schematically depicted infigure 1.A hollow cylinder filled with fluid with its axis coinci-dent with theZ axis of a Cartesian coordinate system(X,Y,Z), is considered to rotate as a solid body abouttheZ axis with a constant angular velocity,Eω= ωEk. Theinner surface is maintained at a constant temperatureT1,while the outer surface is maintained at a constant tem-peratureT2.

Considering steady-state conditions, constant thermo-physical properties of the material, and heat generation,q ′′′, this problem is one of pure radial conduction in theradial direction,Er. But, with reference to thefixed two-dimensional calculation domain depicted by dashed linesin figure 1, the problem appears two-dimensional whenit is observed from the fixed Cartesian system(x, y), lo-cated atr1. As a result, the material within the fixed do-main has a steady velocity given by

EV = uEı + v E = Eω× Er with u=−ωy andv = ωx (2)

2.1.2. Numerical details and results

For Peclet numbers in the range 0.01≤ Pe≤ 10 000,and for several values of the constant source term,Q′′′,the exact one-dimensional solution was used to spec-ify the reference values of temperature over the two-dimensional calculation domain. Results for the maxi-mum value ofε (see equation (1)), produced by FLO

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TABLE IHeat conduction in a rotating hollow cylinder. Variation of εmax with Peclet number for a

11× 11 grid and two different discretizations.

Constant source term,Q′′′

0 1.0 10.0

Pe FLO SPCU FLO SPCU FLO SPCU0.01 1.51E−2 1.51E−2 2.64E−2 2.48E−2 6.88E−2 5.96E−2

1.51E−2 1.60E−2 2.64E−2 2.73E−2 6.88E−2 6.88E−2

1.00 1.51E−2 1.93E−2 2.64E−2 1.78E−1 6.88E−2 1.04E+01.49E−2 1.16E−1 2.64E−2 1.13E−1 6.77E−2 8.57E−2

100.00 1.89E+0 3.53E+0 1.35E−1 3.29E+0 2.77E+1 1.22E+13.18E−1 4.18E+0 2.609E−1 3.99E+0 1.65E+0 1.08E+1

10 000.00 5.76E+0 4.28E+0 6.94E+0 4.72E+0 2.77E+1 1.45E+15.13E−1 5.15E+0 3.53E−1 5.19E+0 2.66E+0 1.16E+1

and SPCU schemes, were obtained and solution accuracywith grid refinement was investigated. Results are pre-sented intable I for two types of discretizations (top andbottom values) and for a coarse 11× 11 grid to show themaximum discrepancies between the schemes. The solu-tion with such a grid is relatively sensitive to the orienta-tion and shape of the elements. Here, results are presentedfor the worst case to support the following discussion. Er-rors reported are high as the reference temperature on theexternal cylinder was taken as the scale value,φscale, inequation (1).

In conduction-dominated situations,Pe< 1, both theSPCU and the FLO schemes produce results of similaraccuracy. For small Peclet numbers, the influence of thedifferent approximations of convection embedded in theinterpolation schemes is minimal, as the contributions tothe coefficients of the algebraic discretized equations byconvection transport terms are small compared to thosedue to diffusion. However, the FLO scheme, even forsuch a low Peclet number, was found to produce smallnegative coefficients in the discretized equations with thetwo distorted meshes employed.

For Pe> 1, important negative coefficients were pro-duced with the FLO scheme with distorted grids. Theshape of the elements, with such grids, influences therelative magnitude of the negative coefficients.Table Ishows that the errors differ by an order of magnitude forthe two discretizations employed (that is with quadrilat-erals divided from top-left to bottom-right or from top-right to bottom-left, respectively). Nevertheless, it waspossible to obtain convergent solutions with the FLOscheme for the proposed three values of constant source

term, Q′′′. In table I the error with the FLO schemereaches the order of 10 %. In other test cases (results arenot shown here), involving a regular grid and a square do-main, the relative magnitude of the negative coefficientswas significantly less, and the error,εmax, was always be-low 1 %.

For Pe> 1 andQ′′′ = 10, the SPCU scheme is foundto require a finer spatial discretization than the relativelycoarse 11× 11: the error for both discretizations isgreater than 10 %. One of the difficulties with thisproblem, whenQ′′′ 6= 0, is the presence of importantcross-flow temperature gradients close to radiusr =(r1 + r2)/2, where the dimensionless temperature issubstantially larger than one. Since the fluid flows in thecounter-clockwise direction, the error introduced in thecalculation of the dimensionless temperature at a locationdownstream of pointA in figure 1, is propagated andamplified until the fluid crosses the opposite boundary:the maximum numerical error,εmax, is found in theimmediate vicinity of pointB. More grid points in thecross-flow direction close tor = (r1 + r2)/2 in figure 1should be used to reduce smearing or false diffusion [4].

2.2. Heat transfer in a radial flowbetween concentric cylinders


One of the characteristics of the previous test problemis that the exact solution is independent of the Pecletnumber. Thus, following the discussions of LeDain andBaliga [5], it may not be a very strict and true test of

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the proposed method’s response toPe. To overcome thisdrawback, the following test problem was employed.

A schematic representation of this test problem isshown infigure 2. The temperature boundary conditionson the cylinders are the following:T = T1 at r = r1, andT = T2 at r = r2. In this problem, the cylinders are atrest and convection–diffusion transport of energy is in theradial direction from the inner to the outer cylinder.

Considering steady-state conditions, constant thermo-physical properties of the fluid, and heat generation,q ′′′,this problem is governed by the one-dimensional radialconvection–diffusionequation. Here, to satisfy continuity

Figure 2. Heat transfer in a radial flow: problem schematic.

requirements, the fluid velocity,uR, has to be inverselyproportional to the radius (uR ∝ 1/r). With respect tothefixedtwo-dimensional regular calculation domain de-noted by dashed lines infigure 2, the one-dimensionalproblem in the radial direction becomes two-dimensionalwhen it is observed from the fixed Cartesian system(x, y). The material flowing in the radial direction crossesthe fixed two-dimensional domain boundaries, and thefluid velocity is given by

EV = uEı + v E = 1

rEeR with

r EeR = xEı + y E ; u= x

r2; v = y

r2(3)


The exact analytical solution (and a one-dimensionalnumerical method) were used as a means to validate theresults produced by the CVFEM.Table II shows resultsfor the same grid and source terms as previously. In thecomputation ofεmax, the dimensionless temperature onthe external cylinder was also taken as the scale value,φscale, in equation (1).

For no source term,Q′′′ = 0, it was found thatfor values of the Peclet number above one hundred,Pe> 100, the conditions atr = r1 essentially prevailthroughout the domain, until a very sharp variation takesplace close tor = r2 in order to match the imposedcondition at this boundary.

TABLE IIHeat transfer in a radial flow between concentric cylinders. Variation of εmax with Peclet number

for an 11× 11 grid and two different discretizations.

Source term,Q′′′

0.0 1.0 10.0


5.63E−3 5.66E−3 5.67E−3 5.66E−3 6.05E−3 5.90E−3

1.00 5.71E−4 3.01E−2 1.68E−4 1.46E−1 2.59E−3 6.62E−13.73E−3 4.38E−2 7.98E−3 1.37E−2 2.57E−2 2.81E−1

10.00 1.13E−2 2.69E+0 4.80E−2 1.57E+0 7.49E−2 4.97E+03.57E−2 1.72E+0 2.53E−2 5.72E−1 3.48E−2 2.49E+0

100.00 7.51E−1 2.11E+0 8.51E−1 1.18E+0 1.40E+0 1.37E+05.64E−1 1.44E+0 1.09E+0 2.04E+0 1.14E+0 2.09E+0

10 000.00 – – 2.15E+0 1.77E+0 2.15E+0 1.77E+0– – 1.19E+1 2.89E+0 1.19E+1 2.89E+0

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As long as the convection term is smaller or equalto the diffusion term,Pe≤ 1, for the source terms in-vestigated here, both schemes provide solutions that arewithin 1 % of the reference solution even for a rela-tively coarse regular 11× 11 grid. In simulations where1 < Pe≤ 10, the maximum error occurs in the regionnext to the outer cylinder, and the SPCU scheme pre-dictions of the temperature distribution are significantlyless accurate than those obtained with the FLO scheme.In convection-dominated situations,Pe> 10 000, for allsource terms, the maximum error is found in the imme-diate vicinity of r = r2, and the negative coefficients inthe discretized equations produced by the FLO schemegained in importance: the SPCU scheme produced betterpredictions than the FLO scheme.

Here again the results selected for presentation inta-ble II are those which show the maximum discrepanciesbetween the two schemes and reference solution. For the11× 11 grid, the type (shape) of elements greatly influ-enced the results. However, as the grid was refined, thediscrepancies between the results obtained with differentelements diminished, and the solution accuracy improvedfor both schemes.

2.3. Heat transfer in a recirculatingflow within a square enclosure


The problem, for which an analytical solution can beconstructed [6], is illustrated schematically infigure 3. In

Figure 3. Heat transfer in a recirculating flow within a squareenclosure: problem schematic.

this figure, a 2× 2 square enclosure is depicted, with theorigin of a Cartesian coordinate system(x, y) located atits center.

The recirculating velocity field is specified via:

u= (2x2− x4− 1)[y − y3]

v =−(2y2− y4− 1)[x − x3] (4)

and the following temperature field is proposed as asolution:

T = (1− x2)(1− y2) (5)

When the expressions for velocity and temperature areincorporated in the energy equation, the following ex-pression for the source term is required to satisfy energyconservation:

Q′′′ = 2(

1− x2)+ (1− y2) (6)

In this problem, the temperature, the velocity, and thesource term vary across the domain. As a consequence,it tests the combined effects of velocity, temperature, andsource term distributions on the accuracy of the solutionmethod.


Table III indicates that for both schemes, grid refine-ment leads to accuracy improvement. The FLO schemeaccurately solves the equations at highPe, even on gridsas coarse as the 11× 11 grid: the error was 1.53 %and 2.44 % for the two discretizations used, respectively.However, for the same grid, the SPCU scheme was asfar as 14 % off the expected solution for the tempera-ture distribution. This error was nevertheless cut in half(6.93 %) when the other discretization was used. The er-ror was reduced to 2.52 % when the number of grid nodesper direction was nearly quadrupled (41×41). Moreover,this maximum error obtained forPe= 10 000 further de-creased, by a factor 2 (1.34%), for a 81× 81 grid (notpresented intable III).

The results showed that in conduction-dominatedsituations, both schemes provide excellent accuracy. Atintermediate and high Peclet number, the FLO schemepredictions are significantly more accurate than thoseobtained with the SPCU scheme. This test case doesnot present the sharp temperature gradients found in theprevious two problems. As a result the FLO schemeproduced positive coefficients only, and the first-orderSPCU scheme always provided adequate accuracy.

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TABLE IIIHeat transfer in a recirculating flow within a square enclosure. Variation of εmax with Peclet

number for several regular grids and two different discretizations.

Discretization

11× 11 21× 21 41× 41


3.65E−5 4.76E−4 9.61E−6 2.41E−4 2.53E−6 1.21E−4

1.00 2.98E−5 7.83E−2 7.58E−4 3.95E−2 1.91E−4 1.97E−23.70E−5 4.71E−2 9.52E−4 2.41E−2 2.41E−4 1.21E−2

10.00 3.05E−2 7.48E−1 7.92E−3 3.79E−1 2.00E−3 1.92E−13.97E−2 4.12E−1 1.00E−2 2.15E−1 2.56E−3 1.11E−1

100.00 2.20E−1 4.27E+0 5.81E−2 2.29E+0 1.47E−2 1.20E+02.90E−1 1.36E+0 7.12E−2 7.30E−1 1.78E−2 3.80E−1

10 000.00 1.53E+0 1.40E+1 6.38E−1 8.55E+0 1.76E−1 4.85E+02.44E+0 6.93E+0 5.64E−1 4.73E+0 1.47E−1 2.52E+0

2.4. Conclusion regardingconvection–diffusion

2.4.1. SPCU scheme

The SPCU scheme always gave better results whenthe longest side of the triangular element was nearlyaligned with the average local flow field. For such anelement, the interpolation distances in the direction of thelargest velocity variations are shorter, and this results in abetter approximation of the velocity distribution. Also, inthe evaluation of the interpolated value of the dependentvariable at an integration point, a panel and a nodal valueare involved with such an element, in contrast to a singlenodal value with a short-sided element.

In the implementation of the SPCU scheme, whenthe analytical velocity distribution was available, themass flow rate across the control-volume surfaces (pan-els) was calculated using: (1) linear interpolation of ve-locity; (2) Simpson’s rule or Gaussian quadrature inte-gration of velocity on a panel, with integration pointvalues specified via the analytical profile; and (3) ex-act integration of the velocity profile. The exact inte-gration respects mass conservation precisely—but thiscondition is not guaranteed by the other types of inte-gration. However, in the test cases considered here, itshould be mentioned that no significant gains in accu-racy were observed using exact integration. This is aninteresting result because the analytical velocity field isnot available in general, and because Euler integration

of φ combined with the linear velocity interpolation, issimple and inexpensive. The linear interpolation of ve-locity within each element is thus recommended for theapproximation of mass flow rates involved in the SPCUscheme.

2.4.2. FLO scheme

Generally, whenever the FLO scheme produced con-vergent solutions, its results were more accurate thanthe results of the SPCU scheme. The errors in the FLOscheme predictions for problems with irregular geome-tries were higher than those for problems with regular do-mains. Similar results were also obtained by Saabas [2].When negative coefficients in the discretized equationsare produced by the FLO scheme, but the magnitudeof these negative coefficients is small, the predictionsare generally very accurate for fine grids over the entirerange of Peclet number tested. However, important neg-ative coefficients can lead to solutions that are less ac-curate. Finally, convergence problems were encounteredwith the FLO scheme when the negative coefficients be-came significant at high values of Peclet number and/orwith highly distorted grids. This was observed in prelimi-nary tests, when very skewed elements were employed inthe discretization of the highly-irregular calculation do-mains. Extrapolation, rather than interpolation, occurredin obtuse-angle triangular elements for high Peclet num-bers, leading to negative coefficients in the algebraic dis-cretization equations [7].

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2.4.3. SPCU and FLO schemes

One of the reason for the differences between FLO-and SPCU-based solutions could be that the surface inte-gration of the dependent variable in the FLO scheme iscarried out using Simpson’s rule [5], which is more ac-curate than the prevailing assumption used in Euler in-tegrations involved with the SPCU scheme [1]. More-over, the mass-weighted averages ofφ involved in theSPCU scheme are intrinsically cruder approximations ofthe influence of the direction and the strength of the con-vective transport within an element, than that providedby the flow-oriented Peclet-number-based function usedwith the FLO scheme.

2.4.4. Concluding remarks

The present investigation suggests that: (1) flow-oriented Pe-based interpolation functions or similarshould be used whenever converged solutions—with rel-atively small negative coefficients in the discretized equa-tions, if any—can be obtained; (2) a combination of FLOand SPCU schemes is desirable to avoid computation ofnegative coefficients and still obtain high accuracy; or(3) modifications should be incorporated in the SPCUscheme to enhance its capabilities to approximate the in-fluences of the direction and strength of the flow withinan element. The effect of Peclet number and volumetricsource terms could be explicitly included in the SPCUscheme used to provide algebraic approximations to theintegrated convected flux terms in the governing energyequation. This enhanced scheme would still avoid theappearance of negative coefficients in the algebraic dis-cretization equations.

The results obtained for the three test problems pre-sented here demonstrated that the formulation and imple-mentation of the proposed CVFEM have been carried outcorrectly. The interested reader should contact the corre-sponding author for more details.

3. RADIATION HEAT TRANSFERPROBLEMS

In this section, the issue of validation with respect toselected radiation heat transfer problems is presented.

In the notation for the dimensionless net radiant heatflux, EQr, the vector and the subscript r are dropped forease of presentation, whenever this does not lead to anymisunderstanding. The termnet is also omitted, as thedefinition of EQr accounts for radiation coming from all

directions. Whenever a particular definition of radiantheat flux with respect to a surface is employed, the termincomingor outgoingwill be specified.

For purely absorbing media theexact (reference)solution is calculated with

IP = Ir e−τ + Ib(1− e−τ

)(7)

whereIr is the intensity at an upstream location along adiscrete directionEΩ , τ is the optical depth betweenr andP measured alongEΩ , andIb is the blackbody intensityof the media. To obtain the net dimensionless flux, anumerical quadrature is used. Practically, a very fineangular discretization (ndd directions) is employed tocompute the flux in order to minimize the error due tothe quadrature approximation. The refinement is stoppedwhen further increases in the number of directions doesnot yield a significant change in the solution.

The exponential and upwind schemes were used withthe CVFEM.

3.1. Gray scattering media in arectangular enclosure


In the enclosure depicted infigure 4, surface 4 is main-tained at a constant temperature,T4, and all other sur-faces are assumed to be at a constant zero temperature,T1 = T2 = T3 = 0. The medium is assumed to scatterradiation isotropically and does not absorb radiant en-ergy (γ = 1). Results of the following investigators who

Figure 4. Heat transfer in a rectangular enclosure: problemschematic.

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studied this problem were considered: Fiveland [8], Tru-elove [9], Modest [10], and Crosbie and Schrenker [11].Their numerical solutions are used as references to checkthose obtained with the proposed CVFEM using the ex-ponential and upwind schemes.T4 is used as a referencetemperature to obtain suitable nondimensional quantities.Here,Qr = qr/σT

44 permits a direct comparison with the

results presented in the above-listed references.


The effects of wall emissivities, aspect ratio, andoptical thickness on the dimensionless radiant heat flux,Qr, at the lower surface (wall 4), were investigated.For the results presented in this section, the spatial gridconsisted of 400 uniform rectangular quadrilaterals, eachdivided into two triangular elements. An equal weight(EW) S4 angular discretization was used.

In figure 5(a), results are presented for square enclo-sures,Lx/Ly = 1, with unit optical thickness,τ0 = 1,having three different surface emissivities,ε, namely 1.0,0.5, and 0.1.Figure 5(a) shows that the CVFEM so-lutions, for both schemes, are in good agreement withthe reference solutions, and predictions for a black en-closure are somewhat more accurate than that obtainedby Modest (diamonds) with a differential approximation(DA) [10]. As the emissivity decreases, the intensity leav-ing a boundary becomes increasingly dependent on theincident intensity on this boundary: as a result, the radi-ant heat flux at the hot wall becomes increasingly uniformand small with decreasingε.

The effects of the variation of the enclosure aspectratio,Lx/Ly , are presented infigure 5(b)for τ0= 1 andε = 1. Here again, the results indicate that the CVFEMpredictions are in good agreement with those of the zonalmethod (black diamonds) and Crosbie’s method (∗).Predictions for high aspect ratios are more accurate thanthose obtained by Modest [10].

The effect of the variation of the optical thickness,τ , is shown infigure 5(c) for ε = 1 andLx/Ly = 1:the results indicate once more that the proposed CVFEMcan successfully predict radiant heat transfer in scatteringmedia having medium (τ = 1) to large (τ = 10) opticalthickness, as the predictions are in good agreement withthe reference solutions.

CPU time and convergence rates were also investi-gated. For each scheme, it was found that the numberof iterations required to obtain a converged solution in-creased withτ . It took approximately the same numberof iterations with both schemes, with a slight advantagefor the upwind scheme. However, the CPU time required

(a)

(b)

(c)

Figure 5. Distribution of dimensionless bottom surface radiantheat flux, Qr, for scattering nonabsorbing media, γ = 1.(a) Effect of wall emissivity, ε, with Lx/Ly = 1 and τ0 = 1;(b) effect of aspect ratio, Lx/Ly , with ε = 1 and τ0 = 1; and(c) effect of optical thickness, τ0, with Lx/Ly = 1 and ε = 1.

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TABLE IVGray scattering media in a rectangular enclosure.

Dimensionless radiant heat flux, Qr, at the lower surfaceγ = 1, τ = 1, ε = 1, and Lx/Ly = 1.

Position Exponential scheme

x 11× 11 21× 21 41× 41 81× 81 [11]0.0 0.8781 0.8824 0.8854 0.8872 0.89320.1 0.8262 0.8255 0.8244 0.8236 0.82730.2 0.7938 0.7920 0.7896 0.7888 0.79570.3 0.7741 0.7715 0.7688 0.7676 0.77730.4 0.7636 0.7604 0.7576 0.7564 0.76710.5 0.7608 0.7570 0.7540 0.7532 0.7636

Position Upwind scheme

x 11× 11 21× 21 41× 41 81× 81 [11]0.0 0.8890 0.8886 0.8887 0.8890 0.89320.1 0.8266 0.8259 0.8252 0.8246 0.82730.2 0.7944 0.7934 0.7924 0.7920 0.79570.3 0.7759 0.7756 0.7748 0.7746 0.77730.4 0.7664 0.7655 0.7648 0.7644 0.76710.5 0.7626 0.7604 0.7590 0.7584 0.7636

per iteration for the upwind scheme was about 50 % ofthat of the exponential scheme. This is significant as theaccuracy of both schemes is comparable.

Table IV indicates the numerical values of the di-mensionless radiant heat flux on the lower surface fora unit optical thickness, unit aspect ratio, and blackboundaries. The exact solution obtained by Crosbie andSchrenker [11] is also indicated for comparison. The spa-tial discretization specifications indicate the number ofnodes along thex andy axes. When the incident radi-ant energy is considered, both schemes eventually yieldthe same solution with grid refinement. They both giveabout 0.5 % error, which is due to the EWS4 angular dis-cretization. For a given level of accuracy, the exponentialscheme may require less nodes but about twice as muchCPU time.

3.2. Gray absorbing media in arectangular enclosure


In this test case, the proposed CVFEM is applied toa cold (T1 = T2 = T3 = T4 = Tw = 0) square enclosurefilled with an absorbing medium (γ = 0) at constanttemperatureTm. Here, the dimensionless radiant heatflux,Qr, at any surface (because of symmetry) infigure 4is set equal toqr/(σT

4m).

Figure 6. Distribution of dimensionless surface radiant heatflux, Qr, for nonscattering absorbing media, γ = 0. Effect ofoptical thickness, τ , with ε = 1, and Lx/Ly = 1.


Figure 6 illustrates the effect of the medium opticalthickness,τ , on Qr, for a square (Lx/Ly = 1) blackwalled (ε = 1) enclosure. Results are presented forthree different optical thicknesses, namely, 10.0, 1.0, and0.1. Other tests were carried out but are not presented.Figure 6shows that the CVFEM solutions, with a EWS4discretization, are in good agreement with the exactnumerical solutions, equation (7), for the upwind andexponential schemes. Asτ increases, the intensity atany particular point becomes increasingly dependent onits immediate surroundings. For a boundary pointP ,the incoming intensity is then related to the intensity ofa point in the medium very nearP . As the boundarytemperature is zero (zero outgoing radiant heat flux)and the medium temperature isTm, for τ = 10, Qris approximately unity. To capture the discontinuity oftemperature at the boundary, mesh enrichment is neededin the vicinity of the walls. Otherwise, the CVFEMcannot accurately predict radiant heat fluxes at the walls,showing a 20 % error forτ = 10.

3.3. Gray absorbing media in atrapezoidal enclosure


This test case involves the irregular quadrilateralenclosure depicted infigure 7. The enclosure is assumedto be filled with an absorbing but nonscattering medium(γ = 0) at constant temperatureTm, and the wallsare assumed to be black and held at a constant zerotemperature (T1 = T2 = T3 = T4 = Tw = 0). A FVM

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Figure 7. Gray media in a trapezoidal enclosure: problemschematic.

solution for this problem has been proposed by Chui [12].Since then, the problem has also been used by Chui andRaithby [13].


Figure 8 illustrates the effect of the optical thicknessof the medium,τ , on the values of the dimensionlessradiant heat flux,Qr = qr/(4σT 4

m), at the top wall. Here,a positiveQr indicates a flux that leaves the enclosure.At the implementation level, the (x, y) coordinates ofthe four corners in the enclosure were: (0.00,0.90),(0.36,1.99), (2.36,1.99), and (3.03,0.00). A 41× 41spatial grid was required to match the exact solutionfaithfully. The azimuthal angular grid (16× 4) wasadjusted to make the projections of the edges of the solidangles match the orientations of the walls. That is tosay the value of the azimuthal angle,φ, is first selected,then the values ofθ are defined accordingly. In order tomatch the edges of the solid angles with the irregularquadrilateral domain boundaries, the azimuthal angulargrid had to have a minimum of eight (8) solid angles intheφ (azimuthal) direction to span 2π in the (x, y) plane.For this type of geometry, the advantage of the analyticalintegration of EΩ · En over solid angles is clear. There is noneed to compute the half range first moment and to selectdiscrete directions associated with the solid angles.

The strict respect of the half range first moment isnot critical for this particular problem because as longas all boundaries are black, the discrete representationof the boundary condition for intensity is independent ofthe angular discretization (Ib = IBb). However, the half-range first moment condition needs to be imposed on a

Figure 8. Dimensionless radiant heat flux on the top surfacefor absorbing media bounded by a black trapezoidal enclosure:Effect of optical thickness, τ .

boundary upon which the distribution of the radiant heatflux is desired. Indeed, theincoming radiant heat fluxneeds to be evaluated appropriately on this boundary.

CVFEM solutions for the radiant heat flux leavingthe enclosure via the top wall, obtained with an angu-lar discretization involving 8 directions, are presented forthree different optical thicknesses, namely,τ = 10.0, 1.0,and 0.1.Figure 8 shows that the surface radiant heatfluxes predicted by the proposed CVFEM, with the up-wind scheme, are in good agreement with the exact so-lutions. For thick media, the radiation from distant loca-tions is mostly eliminated before it reaches a particularpoint: the radiative behavior at any point depends onlyon its immediate surroundings. This provides the incen-tive to employ a fine spatial mesh in the corners, as thereare important changes in the radiant heat flux in these ar-eas, especially for a thick medium. For the limit of a thinmedium,τ = 0.1, the surface radiant heat flux dependsessentially on surface temperatures. As a result, the radi-ant heat flux at the top wall decreases with decreasing op-tical thickness because all surface temperatures are zero.

3.4. Gray media in an L-shapedenclosure


This test case involves the irregular-shaped enclosureshown infigure 9. The enclosure is assumed to be filledwith an emitting, absorbing, and isotropically scatteringmedium at constant temperatureTm, and the walls are as-sumed to be black and held at a constant zero temperature(Tw = 0), except for the top wall whereT = T ∗. This isequivalent to assuming an outgoing radiant heat flux,Qin,

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Figure 9. Gray media in an L-shaped two-dimensional enclo-sure: problem schematic.

on the top wall between pointA and pointE in figure 9.Here, the solutions proposed by Chui [12] using a FVMare used as references. Chai et al. [14, 15] also used thistest case to assess their methods.Qin = 1 for purposes ofcomparison with the solutions proposed in [12].


In a preliminary set of simulations, two situationswere investigated: (1) the absorption coefficient,κ =0.01, and the scattering coefficient,σ = 1.00, yieldingγ = 1/1.01; and (2) the case of no scattering,σ = 0,with κ = 0.01. Grid refinement as well as optical thick-ness effects were investigated. The above-described fur-nace involves a particularly interesting feature becausethe non-orthogonal boundaries make a 45 angle withthe (x, y) coordinate system shown infigure 9. There-fore, it is easy to employ aboundary matchingangulardiscretization with at least 2 solid angles per quadrant ofthe (x, y) plane.

Figure 10 illustrates the dimensionless radiant heatflux, Qr, at the left-southwall, identified by pointsA,B, C, andD in figure 9. In figure 10(a)the medium bothscattersσ = 1.00 and absorbsκ = 0.01 radiant energywhile in figure 10(b)the medium absorbsκ = 0.01 radi-ant energy, only. In [12], the investigator employed a rela-tively coarse 11×4 spatial grid and an angular discretiza-tion that involved 2 latitudes and 8 longitudes within anoctant. For this test case, the first spatial computationaldomain employed with the proposed method consisted of52 irregular quadrilaterals, each divided into two triangu-lar elements, and the angular grid was constructed using16 discrete directions, that is one latitude in the positivez direction and 4 discrete solid angles in each quadrantof the (x, y) plane. A finer grid involving 33× 15 nodesand 64 directions, distributed on two latitudes, was also

(a)

(b)

Figure 10. Distribution of dimensionless radiant heat flux,Qr, on the left-south wall A-B-C-D for: (a) κ = 0.01, σ = 1.0;(b) κ = 0.01, σ = 0.0.

employed for direct comparison with Rousse [7]. In thenotation indicated in the legend of the figures, the dis-cretization is identified with 4 numbers: the first numberrefers to the number of grid nodes along the left-southwall (A-B-C-I -D) and right-north wall (E-F -G-H ) ofthe enclosure depicted infigure 9; the second number isthe number of grid nodes used on the top wall (A-E) andright wall (H -D); the third number refers to the num-ber of latitudes employed with the azimuthal angular dis-cretization; and the last number is the number of solid-angle projections in each quadrant of the (x, y) plane.

In figure 10(a), the surface radiant heat flux,Qr,along the left-south wall (A-B-C-I -D) decreases withthe distance away from pointA, measured betweenAandD, because radiation is attenuated by absorption andscattering as it travels away from the top (hot) wall. Whenthere is no scattering,γ = 0 in figure 10(b), the radiantheat flux impinging on a surface is very sensitive to theorientation of this surface with respect to the sourceQin(located on the top wall).

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In figure 10(b)it can be observed that the FVM 11×4fails to predict the two bumps, that correspond to loca-tions B and C, in the surface radiant heat flux distri-bution. Chui’s solution [12] indicates a plateau, whilethe 13× 4 CVFEM solution shows the two peaks atpoint B andC. The CVFEM solution for a coarse gridis shown to be somewhat better than its correspondingFVM counterpart. This may be caused by the treatmentof the boundary conditions and by the fact that in theCVFEM each quadrilateral is divided into two trianglu-lar elements. This yields a finer discretization and hencea better approximation of the radiative fluxes across con-trol volume surfaces.

The orientation of the wall changes atB and C,and this makes the left-south wall lie in a more directline of sight from the source of radiation on the topwall (A-E). There should be discontinuities in theQrdistribution atB andC for σ = 0. The tendency is wellpredicted by the relatively fine grid (33× 15 nodes)combined with fine angular discretization involving 64directions (2 latitudes and 8 projections per quadrant).To capture the flux discontinuities, grid points shouldbe inserted close to pointsB andC. Beyond pointC,alongC-D, after the stepB-C, the radiant heat flux isseen to decrease to zero, as some points (beyond pointI )do not “see” the radiation source on the top wallA-E,and because there is no reflection and no emission fromthe boundaries. When scattering is present,figure 10(a),the changes in the distribution of the radiant heat fluxat pointB andC are less pronounced because a directline of sight toA-E is less important than in the caseof no scattering infigure 10(b). Chui’s solution [12]and the coarse CVFEM solution indicate monotonicdecrease of the radiant heat flux along the left-south wall,whereas the CVFEM solution obtained with a fine grid,which reproduces the physics more faithfully, shows theexpected bumps at pointB andC.

The radiant heat flux at theright-north wall (E-F -G-H in figure 9) was also computed (but not shownhere) with more accuracy with the CVFEM than withthe FVM of Chui [12]. The effect of angular and spatialdiscretization refinement and that of optical thicknesswere also investigated. Interested readers should consultthe corresponding author for details.

3.5. Gray absorbing media in a curvedenclosure


In this last test case for pure radiative heat transfer, aquarter of circle—with a rectangular region added on its

Figure 11. Gray media in a curved computational enclosure:problem schematic.

top—was used as a test geometry. Infigure 11, the curvedwall is hot,T4 6= 0, and black, while all other walls arecold,T1 = T2 = T3 = 0. The medium is cold and purelyabsorbing withκ = 1. The radius of the quarter of circleis equal to the length along thex-axis,Lx = 1.0, andLy = 1.5. Chai et al. [14, 15] and Liu et al. [16] usedthis problem.


A 37× 37 spatial grid and an azimuthal 20× 4 grid intheφ andθ directions were employed with a progressivedeployment of the spatial grid: there are more nodes inthe immediate vicinity of the walls.

The predictions of the radiant heat flux on the rightwall (wall 2) are illustrated infigure 12. It is shown thatthe CVFEM solution, using the upwind scheme, is ingood agreement with that proposed by Chai et al. [15].

As the hot wall is curved, it is impossible on thissurface to customise the angular discretization to havesolid angle edges match the boundaries exactly for eachboundary element. If the value of the incoming flux atthis surface was the quantity of interest or if the curvedwall was reflecting, a special treatment for the boundaryconditions would be required. However, as this curvedsurface is black (no reflection), such a treatment is notnecessary here: only the outgoing flux is needed on thecurved boundary.

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Figure 12. Distribution of dimensionless radiant heat flux, Qr,on the right surface for absorbing media bounded by a blackcurved enclosure, β = 1.

3.6. Conclusion regarding radiation

The first two test cases permitted to compare the solu-tion obtained with the CVFEM formulation with those ofseveral investigators. The results indicate that the methodwas accurately formulated and implemented and that itcan handle emission, absorption, and isotropic scatter-ing in rectangular geometries. The third and fourth prob-lems were introduced to show that irregular geometriescan be handled with the proposed method. Finally, thefifth test case was proposed to demonstrate that the pro-posed CVFEM [1] is not restricted to applications withgeometries where the angular discretization can be ad-justed to make the solid angle edges coincide with thedomain boundaries. In many practical situations, such anapproach could lead to a high number of solid angles or,in other words, a high number of discrete dependent vari-ables.

As boundaries were black for problem five and be-cause the boundary upon which the flux is provided isstraight, there was no need to implement a special treat-ment for the boundary conditions. However, in a moregeneral case, provision should be made to account forsolid angles that straddle boundaries. This particular ap-proach requires care to ensure that the radiant energyis conserved at the gray-diffuse surface boundaries. Thestrict respect of energy conservation may not be that crit-ical: for a sufficiently refined angular discretization, thevery small solid angles straddling boundaries have the dotproduct of their associated direction with the normal tothe boundary very small. This leads to a negligible overalleffect. This, and other considerations that pertain to an-gular discretization in arbitrarily-shaped geometries, willbe addressed in a subsequent study.

The upwind scheme is a low-order scheme that al-lows for a numerical redistribution of radiant energy indirections normal to the direction of propagation. Indeed,the upwind scheme always gave better results when thelongest side of the triangular element was nearly alignedwith the direction of propagation. For such situations, theprevailing assumption of a nodal intensity at a controlvolume face (panel) located downstream is more likelyto describe the reality than when the panel is not in a di-rect line of sight from the node. The result is that whenthe flux and/or the incident radiant energy is calculated,there will be directions for which the mesh alignment isfavorable and others not. Efforts will now be devoted toaccount for the direction of propagation at the spatial in-terpolation level.

Despite its drawbacks, the upwind (also called step)scheme was found to yield accuracy levels that wereessentially similar, from an engineering point of view, tothose obtained with the exponential scheme. The CPUtime ratio for a given problem is also always in favor ofthe upwind scheme.

Isotropic scattering was considered in the test prob-lems involved in this paper. However, provision has beenmade to account for anisotropic scattering and the pre-scription of a suitable phase function is at the discretionof the analyst.

The proposed CVFEM formulation [1] could also beimproved by employing an implicit solution procedurebuilding on ideas developed by Chui [12] in the contextof pure radiative heat transfer with radiative equilibrium.

4. COMBINED MODES HEAT TRANSFERPROBLEMS

In this section, the solutions are obtained with theSPCU scheme when convection is taken into accountand with the upwind and exponential schemes for thecalculation of radiation heat transfer [1]. However, asthe solutions obtained with the upwind and exponentialschemes are very close to each other for all problemsinvolved here, only those produced by using the upwindscheme are presented.

4.1. Conduction–radiation in arectangular enclosure


In the enclosure depicted infigure 4, T4 = Tw andall other surfaces are assumed to be at a constant tem-

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perature,T = 0.5Tw. The lower corners of the enclosureare considered to be atT = 0.75Tw. The medium is as-sumed to be continuous, and only conductive and radia-tive transfer occur in the enclosure. The medium is fur-ther assumed to absorb and emit radiation, but not scat-ter radiant energy,γ = 0. The optical thickness is as-sumed to beτ0= 1.0, andLx/Ly = 1. In the nondimen-sional formulation, the side of this square enclosure is ofunit length. This problem has been investigated by Raz-zaque et al. [17]. In that paper, the authors employed theGalerkin finite element method, with isoparametric, bi-quadratic, quadrilateral elements, and an iterative solu-tion technique.


The effects of the conduction to radiation parameter,the Stark number,N = βk/(4σT 3

w), on the verticalcenterline (lineA-B-C in figure 4) temperature,T/Tw,and on the dimensionlesstotal heat flux,Q = q/(σT 4

w),at the lower surface, were investigated. The results arepresented infigure 13. The CVFEM solutions wereobtained with a EWS4 angular discretization, combinedwith a 41× 41 uniform spatial grid with both the upwindand exponenetial schemes for the calculation of radiation.

In figure 13(a), the distribution of dimensionless tem-perature,T/Tw, along the centerline for three differentvalues ofN is presented. A CVFEM solution is alsogiven for the case of no radiation,N = ∞. This fig-ure shows that the CVFEM solutions (with the upwindscheme) for the centerline temperature profile are in verygood agreement with those presented by Razzaque etal. [17]. When the Stark number is unity, the tempera-ture profile is not greatly affected by radiation when com-pared to the case of pure conduction. This suggests thatthe magnitude of the ratio of the radiant heat transfer tothe conduction heat transfer should be at least unity to re-quire a radiation heat transfer calculation (N ≤ 1). It canbe seen that with decreasingN , the source term in the en-ergy equation becomes more and more important in theupper region of the domain. Infigure 13(b), results arepresented for 0≤ x ≤ 0.5, as the total heat flux distribu-tion is symmetric with respect to pointC on the bottomwall. The CVFEM predictions for the total heat flux,Q,at the bottom wall are also in good agreement with thoseproposed in [17]. In this figure, the peak near the corner(low x position) forN = 1 is due to the fact that lowercorner nodes are assigned a lower temperature than therest of the bottom wall nodes. For this case, the solutionis grid sensitive and the comparison remains meaning-ful as long as general trends are discussed. The sharpercurve close to corner forN = 1 may be explained by the

(a)

(b)

Figure 13. Conduction–radiation in a square black (ε = 1)enclosure with absorbing media, γ = 0, and different values ofthe Stark number, N : (a) centerline temperature distribution;(b) total heat flux, Q, on the bottom surface.

coarse discretization and the interpolation function used.Discrete variations imply strong influences on the fisrt-order upwind scheme.

4.2. Conduction–convection–radiationin a rectangular enclosure


Combined convection–diffusion–radiation is consid-ered in a fully-developed laminar flow within the geom-etry depicted infigure 4. This geometry is consideredto be a straight horizontal gray channel (Lx > Ly ) inwhich the media participate in the radiative heat trans-fer. The temperature is set toT1 = T4 = Tw on the topand bottom surfaces, and these surfaces are considered

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to be opaque, parallel to each other, gray-diffuse, and tohave constant emissivities,ε. The inlet and outlet sec-tions of the channel are assumed to be imaginary porousblack surfaces, through which the media flow without re-strictions. The inlet temperature profile is first assumedto be zero,T3 = 0, and the temperature gradient at theoutlet surface is set to zero. Fully-developed Poiseuilleflow is considered with a velocity given byu(x, y) =6u[(y/Ly)−(y/Ly)2]whereLy is the channel width andu the average velocity. For purpose of comparison, atten-tion was focused on a location of the fully-developed heattransfer region where the centerline dimensionless tem-perature isT = 0.5Tw. This location is located at a dis-tanceLy/2 in figure 4, but not necessarily located at thecenter of the channel,Lx/2. The results obtained usingthe proposed CVFEM [1] are compared with those ob-tained by Viskanta [18], who considered fully-developedheat transfer in emitting and absorbing media,γ = 0.In [18], the axial temperature gradient,∂T /∂x was re-placed by (

Tw − TTw − T

)dT

dx

and conduction as well as radiation in the axial directionwere neglected.


For this test case, many grid nodes were used alongthe axial direction to capture the location, in the fully-developed heat transfer region, where the centerline tem-perature is 0.5. The number of grid nodes employed inthe cross-flow direction was adjusted to obtain triangleswith aspect ratios near one. An EWS4 angular discretiza-tion was employed. The inlet boundary condition forT3had to be adjusted, after preliminary tests, so as to obtainfully-developed heat transfer within a reasonable chan-nel length. The results are presented for a relative opticaldepth measured along segmentA-B in figure 4.

The dimensionless temperature profile along a verticalsegmentA-B, in figure 4, is presented for several valuesof the conduction–radiationparameter,N , in figure 14(a).In this figure, results are presented for a unit opticalthickness,τ = βLy = 1, in the cross flow direction.The effect of the optical thickness variations on thelocal radiant heat fluxEQr along A-B is presented infigure 14(b)for N = 0.01. In this figure, radiant heat fluxprofiles are shown forτ = 1.0 and forτ = 0.1.

The value of the Nusselt number (7.54 based onDh=2Ly ) for pure convection–diffusion was first calculated.Then, radiation heat transfer was combined with con-vection and conduction. AsN decreases, the temper-

(a)

(b)

Figure 14. Conduction–convection–radiation in a gray channelwith absorbing media: (a) effect of Stark number, N , ontemperature; and (b) effect of optical thickness, τ , on the localradiant heat flux.

ature field departs more and more from that of pureconvection–diffusion,N = ∞. In [18], the investigatorfound that the temperature gradient at the wall decreasedwith decreasingN . But Viskanta [18] stated that thiswas an error introduced by the approximation embeddedwithin his numerical method. In [18], the author uses athree terms Taylor series expansion (Barbier’s method)to evaluate the integral terms of the integro-differentialgoverning equation. The temperature gradient at the wallis then in error. Nevertheless, the shape of the curve isgenerally adequate [18].

The CVFEM proposes a temperature gradient that isrelatively unaffected by the Stark number,N . An ex-amination of the results suggests that the proposed two-dimensional CVFEM [1] adequately predicts temperature

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Figure 15. Gray media in a divergent channel: problemschematic.

and heat fluxes, when it is applied to this convection–diffusion–radiation problem in a regular geometry.

4.3. Conduction–convection–radiationin a divergent channel


This last two-dimensional test case involves an irregu-larly-shaped quadrilateral enclosure having the shape of adivergent channel. This geometry is depicted infigure 15.The top and bottom walls of this channel are maintainedat uniform temperatureT1 and T2, respectively. Thesetwo infinitely wide black plates, with an angleδ = 15,are separated by a distanceD0 at the channel exit. Itis assumed that the velocity profile down the channel isfully developed, laminar flow, given by

u(x, y)= 3

2ux

[1− 4

(y

Dx

)2]whereux = (D0/Dx)uL, uL is the average outlet veloc-ity, andDx is the channel width atx. The inlet and out-let sections of the divergent channel are assumed to beimaginary porous black surfaces, through which the me-dia flow without restrictions. There is no heat generation,Q′′′ = 0, in the calculation domain, and the normal tem-perature gradients at the entrance and exit boundaries areset equal to zero. The media flowing along the channelare assumed to emit, absorb, and isotropically scatter ra-diant energy.

This problem was suggested by Chung and Kim [19]in a paper in which the authors used the Galerkin Fi-nite Element Method (FEM) with a total of 72 lineartwo-dimensional isoparametric elements with 91 nodes.Although the overall mass conservation is preserved us-ing the proposedu-velocity profile, continuity require-ments are not satisfied everywhere in the domain. In [19],

the authors did not specify any velocity profile in they-direction, while in this work it is assumed that the cross-section velocity profile is given by

v(x, y)= 3

2ux

[1− 4

(y

Dx

)2]y

Dx

dDxdx

where

dDxdx= 4

5

DL

L

The oulet mean velocity,uL, and the outlet channelwidth, DL, are used as reference velocity and length,respectively, in the specification of the velocity profiles.T4 is assumed to be the reference temperature andT1 =T4/5.


72 quadrilaterals each divided into two triangularelements were first used, that is 13 grid nodes alongthe axial direction and 7 grid nodes in the cross-flowdirection. This relatively coarse grid was employed topermit a more interesting comparison with the resultspresented in [19]. It was also found to yield sufficientaccuracy except in the entrance region, when comparedwith a finer grid (results not shown). The SPCU schemewas used in the approximation of the convective flux, andinflow and outflow boundary conditions were imposed atthe inlet and outlet, respectively. As the velocity profilesare expressed as functions ofx andy, the evaluation ofthe mass flux at the control-volume faces (panels) wascarried out with a second order accurate Simpson’s ruleintegration. An azimuthal angular discretization aboutthe z-axis was used with a singleξ latitude, and 12solid angles, having 7.5 projections in the (x, y) plane,per octant. With reference tofigure 15, the results arepresented in terms of the dimensionless temperaturealong the centerline axial section,A-B, and the middlecross-section,C-D. Investigations were carried out forselected values of the Stark number,N , optical thickness,τ , Peclet number,Pe, and scattering albedo,γ . ThePrandtl number,Pr, was one for all simulations. Resultsfor the upwind scheme used in the interpolation for theradiative flux are reported.

In figures 16and 17, results are presented for sev-eral Peclet numbers,Pe= ρcpuLL/k, a unit opticalthickness,τ = βL = 1, and for a Stark numberN =kβ/(4σT 3

2 )= 0.001.

Figure 16presents results for a purely scattering me-dium, γ = 1. In this case, as the divergence of the ra-diative flux vector is zero, the source term in the en-ergy equation is zero and the solution is that for pure

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(a)

(b)

Figure 16. Conduction–convection–radiation in a divergentchannel. Effect of Peclet number, Pe, on: (a) centerline tem-perature; (b) cross-flow temperature at x = Lx/2. γ = 1, τ = 1,N = 0.001.

convection–diffusion. Hence, the temperature profile atthe axial section,A-B, is not affected by the variationsof the optical thickness,τ , the Stark number,N , and thePeclet number,Pe: a straight line is shown infigure 16(a).However, infigure 16(b), it will be noticed that as con-vection increases with respect to conduction, the heattransfer at the boundaries also increases: the CVFEM so-lution for the cross-sectionC-D indicates strong tem-perature gradients in the vicinity of the walls for highPe. It is surprising to see that Chung and Kim [19] ob-served the inverse phenomenon for a purely scatteringmedium.

Figure 17shows results for a purely absorbing medium,γ = 0. In this case, the divergence of the radiative fluxat a point, E∇ · EQr, is negative except at the hot bound-ary. As a result, a positive source term is present inthe energy equation and the temperature profile is raised

(a)

(b)

Figure 17. Conduction–convection–radiation in a divergentchannel. Effect of Peclet number, Pe, on: (a) centerline tem-perature; (b) cross-flow temperature at x = L/2. γ = 0, τ = 1,N = 0.001.

above that obtained for a purely scattering medium. TheCVFEM solutions for the axial temperature profile atA-B are not very dependent on the Peclet number un-til Pe= 1 000 (seefigure 17(a)). For Pe< 1 000, the di-mensionless temperatureT/T4 reaches about 0.8 veryrapidly for τ = 1. WhenPe= 1 000,N = 0.001, andτ = 1; the Boltzmann number,Bo, is unity: thus, con-vection becomes as important as radiation and the effectof the source term in the energy equation becomes rel-atively less important than forPe< 1 000. As a result,the axial temperature profiles obtained forPe≥ 1 000tend to approach that of a purely scattering medium be-cause convection dominates radiation for such valuesof Pe. For Pe= 1 000, Chung and Kim [19], obtainedtemperatures—in the vicinity of the channel inlet—thatare below 0.6, as if there were a heat sink due to radiationin this area.

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(a)

(b)

Figure 18. Conduction–convection–radiation in a divergentchannel. Effect of scattering albedo, γ , on: (a) centerlinetemperature; (b) cross-flow temperature at x = L/2. Pe= 100,τ = 1, N = 0.001.

If attention is directed towards the results presentedfor the middle cross-sectionC-D in figure 17(b), itcan be observed that the source term due to absorptionraises the temperature profile above what was obtainedfor a scattering medium for the whole range of Pecletnumbers investigated. Since radiation is 1 000 times moreimportant than conduction,N = 0.001, the effect of thesource term in the energy equation is more important forsmallPe.

The effect of the scattering albedo variations is pre-sented infigure 18for τ = 1, Pe= 100, andN = 0.001.As the albedo increases for a constantτ , there is moreand more scattering and proportionally less absorption.This causes the curves presented infigure 18(a)to ap-proach the straight lineT/T4= 0.6 asγ augments. SinceBo= PeN/τ = 0.1, the influence of radiation with re-spect to that of convection is still 10 times more impor-

(a)

(b)

Figure 19. Conduction–convection–radiation in a divergentchannel. Effect of Stark number, N , on: (a) centerline tem-perature; (b) cross-flow temperature at x = L/2. γ = 0, τ = 1,Pe= 100.

tant. This is one of the reasons why the curve forγ = 0.5is closer to that ofγ = 0 than that ofγ = 1. In fig-ure 18(b), it is shown that as the albedo increases, thecurves naturally approach that ofγ = 1.

In figure 19, the dimensionless centerline temperatureand dimensionless cross-flow temperature atx = Lx/2are presented for several values ofN . Results are pre-sented for an absorbing medium,γ = 0, with τ = 1 andPe= 100. It is shown that, as the importance of con-duction over radiation increases for a fixed Peclet num-ber, the centerline temperature profile alongA-B ap-proaches the purely scattering medium solution: that isT/T2 = 0.6. When conduction dominates,N > 1, thesource term in the energy equation has little effect be-cause the Boltzmann number,Bo, becomes large. The so-lution for the midsection temperature profile clearly in-

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TABLE VCombined-modes of transfer in a divergent channel.

Number of iterations and CPU time.

Pe Cond. Conv. Cond. Conv. Rad.

Iter. Time (s) Iter. Time (s)1 2 0.20 30 11.32

100 2 0.33 26 10.851 000 2 0.58 15 8.99

10 000 2 1.66 7 5.83

dicates that whenN →∞, the curve is nearly the onethat is obtained in the case of pure convection–diffusion.WhenN is smaller, the energy absorbed by the mediain the vicinity of the cold wall causes the temperature toincrease along the axis.

When combined modes of heat transfer are simul-taneously treated, CPU time and convergence rates be-come preponderant concerns.Table Vindicates the num-ber of iterations for the energy equation and CPU timeneeded to obtain a convergent solution for the case ofconduction and convection only, and combined conduc-tion, convection, and radiation. Similar discretizations fora purely absorbing medium of unit optical thickness andN = 0.001 were used. WithPe= 1, there is 1 000 timesmore heat transfer due to radiation than to convectionand conduction. This explains why 30 iterations wereneeded for convergence of the algorithm: the source termin the energy equation was very large. AsPe augments,the relative importance of the divergence of the radiativeflux (the source term) in the energy equation diminishesas well as the number of iterations needed for conver-gence. The CPU time ratio with and without radiationchanges from 56 to 8 forPe= 1 to Pe= 10 000, respec-tively.

4.4. Conclusions regarding combinedmodes of heat transfer

The first problem permitted us to verify that thealgorithms developed for conduction heat transfer andradiation heat transfer were compatible and that the samesolver could indeed be used to solve for both phenomena.

The second problem was used to include the effects ofconvection in conjunction with radiation and conduction.Results provided confidence in both the formulationand implementation of the proposed method. The majorconclusion that can be drawn from this second testproblem is that radiation heat transfer has to be about 100times more important than conduction heat transfer,N =

0.01, to introduce significant changes in the temperaturedistribution for a unit optical thickness,τ0= 1.

The conclusions that can be drawn from the last testproblem are that: (1) radiation heat transfer calculationscan be carried out in a geometry that has boundaries un-aligned with the axis of an orthogonal coordinate system;(2) radiation heat transfer has to be much more importantthan conduction and convection heat transfer (smallBo)to introduce significant changes in the temperature distri-bution for a unit optical thickness,τ0 = 1; (3) care mustbe provided to prescribe a flow field that satisfies conti-nuity requirements everywhere in the calculation domainwhen the flow field is not calculated; (4) the convergencerate is slow for small values of the Stark number, as asmallN implies a relatively important source term in thediscretized energy equation.

It should be mentioned that whenE∇ · EQr is incorpo-rated as a constant volumetric source term,SC, in theA∗Pcoefficient of the algebraic discretized energy equation, itcan lead to very slow convergence of the iterative solutionprocedure, especially in radiation dominated situations.The solution may even diverge ifτ0> 1 andN < 0.1. Toensure convergence, it was found necessary to linearizeappropriately the source termE∇ · EQr, and identify suit-ableSC andSP terms [4].

5. CONCLUSION

5.1. Contributions

In this paper, the capabilities of the CVFEMs pro-posed in [1] have been demonstrated by applying themto eleven test problems. Analytical closed-form solu-tions or independent numerical investigations availablein the literature have been used to check the resultsproduced by the CVFEM. These comparisons indicatethat the CVFEM proposed by Rousse [1] can suc-cessfully solve combined-mode heat transfer problemsin irregularly-shaped two-dimensional geometries filledwith gray emitting, absorbing, and isotropically scatter-ing media. Although this paper presents results limited totwo-dimensional enclosures, the method used can be ap-plied to three-dimensional geometries using tetrahedralelements.

5.2. Key features

Globally: (1) the solutions for combined-modes prob-lems were found to be in good agreement with the few

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reference solutions found in the literature. However, sev-eral combined-modes test problems should be proposedto become “benchmarks” problems; (2) the proposedmethod can solve combined-modes problems using thesame assembly procedure for the coefficients of the al-gebraic discretization equations and the same solver forthe three modes of transfer; (3) the computational timebecomes very important with the increasing proportionof radiation heat transfer with respect to convection andconduction heat transfer.

In the context of convection–diffusion: (1) the skewpositive coefficient upwind (SPCU) scheme producespositive coefficients in the algebraic discretization equa-tions; (2) the SPCU scheme provides a reasonable levelof accuracy but is prone to false diffusion. It gives bet-ter results when the longest side of a triangular elementis nearly aligned with the average local flow field; (3) anEuler integration ofφ combined with a linear velocity in-terpolation is adequate when the SPCU scheme is used;(4) the second-order flow-oriented (FLO) scheme pro-duces solutions that are more accurate that those pro-duced with the SPCU scheme, whenever the coefficientsremain positive.

Concerning radiative transfer: (1) Since the CVFEMis a numerical solution of the exact governing equation, itshould provide the exact solution within the limits of nu-merical accuracy and interpolation functions. Discretiza-tion refinement, both angular and spatial, is the key to ob-tain more accuracy. In most problems, coarse grids werefound to yield accuracy levels that were satisfactory froman engineering point of view. (2) The low-order upwindscheme provides accurate solutions even for high opti-cal thicknesses. This scheme was found to be the bestcompromise between accuracy of results and computertime requirements. (3) The computational time becomesimportant with increasing optical thickness and with de-creasing wall emissivities.

5.3. Recommendations

During the testing of the proposed CVFEMs, it wasnoted that several improvements and extensions could beincorporated to increase the efficiency and scope for thesolution of engineering problems. Among them: (1) theSPCU scheme for convection should be modified to in-clude the physical additive correction (PAC) schemessuch as that proposed by Galpin et al. [20]; with thismodification, the effect of Peclet number and volumet-ric source terms could be explicitly included in the in-terpolation scheme; (2) adaptive grid techniques, based

on grid redeployment and mesh enrichment strategies,should be explored and implemented; (3) fluid flow cal-culations should be incorporated; (4) capabilities to ac-count for nongray media and boundaries should be in-vestigated and a suitable model should be implemented;(5) problems involving anisotropically scattering mediashould be investigated; (6) nondiffuse or anisotropic re-flection should be permitted; the concept of a reflec-tion phase function should be investigated and tested;(7) strategies should be developed that allow for efficientassembly rules, and solution procedures on unstructuredgrids; (8) the approximations involved at the interpolationfunction level for the CVFEM for radiation heat transfershould be investigated and tested to determine the influ-ence of such approximations on the accuracy and con-vergence rate of the method; (9) an implicit solution pro-cedure such as that proposed by Chui [12], in the con-text of radiative equilibrium, should be elaborated andtested.

Acknowledgement

The authors gratefully acknowledge EDF for its sup-port and the first author acknowledges the Natural Sci-ences and Engineering Research Council for an operatingresearch grant.

REFERENCES

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[9] Truelove J.S., Discrete-ordinate solutions of theradiation transport equation, ASME J. Heat Tran. 109 (4)(1987) 1048–1052.

[10] Modest M.F., Radiative equilibrium in a rectangularenclosure bounded by gray walls, J. Quant. Spectrosc.Radiat. Transfer 15 (1975) 445–461.

[11] Crosbie A.L., Schrenker P., Radiative transfer in atwo-dimensional rectangular medium exposed to diffuseradiation, J. Quant. Spectrosc. Radiat. Transfer 31 (1984)339–372.

[12] Chui E.H., Modelling of radiative heat transfer inparticipating media by the finite volume method, PhDThesis, University of Waterloo, Canada, 1990.

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[15] Chai J.C., Parthasarathy G., Lee H.S., Patankar S.V.,Finite-volume radiative heat transfer procedure for irregu-

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[17] Razzaque M.M., Howell J.R., Klein D.E., Finite ele-ment solution of heat transfer in a two-dimensional rectan-gular enclosure with gray participating media, ASME J. HeatTran. 105 (4) (1983) 933–939.

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[20] Galpin P.F., Huget R.G., Raithby G.D., Fluid flowsimulations in complex geometries, in: CNS/ANS SecondInt. Conf. on Simulation Methods in Nuclear Engineering,Montreal, 1986.

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