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EVALUATION OF THE FINITE-VOLUME SOLUTIONSOF RADIATIVE HEAT TRANSFER IN A COMPLEXTWO-DIMENSIONAL ENCLOSURE WITHUNSTRUCTURED POLYGONAL MESHES

Man Young Kim1, Seung Wook Baek2, and Il Seouk Park3

1Department of Aerospace Engineering, Chonbuk National University, Jeonju,Chonbuk, Korea2Department of Aerospace Engineering, Korea Advanced Institute of Scienceand Technology, Daejeon, Korea3Technical Research Laboratories, POSCO Co., Ltd., Pohang, Gyeongbuk,Korea

Radiative heat transfer in a complex two-dimensional enclosure with an absorbing, emitting,

and isotropically scattering gray medium is investigated by using the finite-volume method.

In particular, an implementation of the unstructured polygonal meshes, based on an unstruc-

tured triangular system, is introduced, thereby reducing the computation time required for

the analysis of thermal radiation. By connecting each center point of the triangular meshes

rather than joining the centroids of the triangular elements to the midpoints of the corre-

sponding sides to form a polygonal element, it not only prevents concave shapes of adjacent

faces, it also reduces the number of control-volume faces of the polygonal mesh. After a

mathematical formulation and corresponding discretization equation for the radiative trans-

fer equation are derived, the final discretization equation is introduced with the conventional

finite-volume approaches. The present formulation is then validated by comparing results

with those obtained in previous related works. All the results presented in this work show

that the present method is accurate and computationally efficient for the analysis of

radiative heat transfer problems in complex geometries.

INTRODUCTION

During the past few decades an unstructured triangular mesh in two dimensionsand a tetrahedral mesh in three dimensions have been widely accepted in computationof fluid flow and heat transfer due to the ease of mesh adaptation fitted to realisticcomplex geometries. These unstructured methods are now available for solving com-pressible and incompressible industrial flows with turbulence, combustion, and otherphysical complexities [1]. Also, problems of engineering applications such as boilers,furnaces, and high-temperature equipment often require that the effect of radiative

Received 26 February 2008; accepted 7 April 2008.

This work was supported by the Research Center of Industrial Technology at Chonbuk National

University, Korea.

Address correspondence to Man Young Kim, Department of Aerospace Engineering, Chonbuk

National University, 664-14 Duckjin-Dong, Duckjin-Gu, Jeonju, Chonbuk 561-756, Korea. E-mail:

manykim@chonbuk.ac.kr

116

Numerical Heat Transfer, Part B, 54: 116–137, 2008

Copyright # Taylor & Francis Group, LLC

ISSN: 1040-7790 print=1521-0626 online

DOI: 10.1080/10407790802154215

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heat transfer be included. Therefore, it is highly desirable to implement an unstruc-tured radiation solver consistent with that used for fluid flow and heat transfer ones.

In recent years, considerable efforts have been exerted to solve the radiativeheat transfer in complex geometries with unstructured meshes by using variousapproaches, including the finite-volume method (FVM) [1–4], the discrete ordinatesmethod (DOM) [5, 6], the discrete-transfer method (DTM) [7], and the finite-elementmethod (FEM) [8–11]. Most of these unstructured radiation solvers are based on tri-angular or quadrilateral meshes in two-dimensional cases and tetrahedral or hexa-hedral ones in three-dimensional applications. With these developments ofradiation solution methods, commercial software such as FLUENT has adoptedvarious mesh systems to solve the fluid and heat transfer problems. The most recentdevelopment in FLUENT 6.3 [12] includes a new cell type of polyhedral meshsystems for three-dimensional analysis. The polyhedral cells are generated from abaseline tetrahedral or hybrid meshes, and so they possess an arbitrary numberof neighboring cells. Despite numerous studies in this field and high demand intwo-dimensional applications, there are still no works for fully two-dimensionalpolygonal unstructured mesh systems with the application of the FVM.

NOMENCLATURE

amI coefficient of the

discretization equation in

direction m at nodal point I

Dmci directional weight in

direction m at surface i,

Eq. (4b)~eex;~eey;~eez x-, y-, and z-direction base

vectors, respectively

I radiative intensity, W=m2 sr

Ib blackbody radiative intensity

ð¼rT4=pÞ;W=m2 sr

~nni outward unit normal vector

at face i, Eq. (4d)~nnw unit normal vector at the wall

toward the medium

Nh;N/ number of polar and

azimuthal directions,

respectively

P present control volume

qRw radiative wall heat flux,

Eq. (9), W=m2

~rr position vector

Smr radiative source term in

direction m, Eq. (4e)~ss direction vector

ð¼~eex sinh cos/þ~eeysinh sin /þ~eez coshÞ,Eq. (4c)

b0 extinction coefficient

ð¼ja þ rsÞ;m�1

DA;DV surface area and volume of

the control volume,

respectively

DXm discrete control angle,

Eq. (4f), sr

ew wall emissivity

ja;rs absorption and scattering

coefficients, respectively, m�1

r Stefan-Boltzmann constant

ð¼ 5:67� 10�8Þ;W=m2 K4

U scattering phase function

x0 scattering albedo ð¼ rs=b0Þ

Subscripts

i surface integration point

between P and I

I nodal point where intensity is

stored

P present nodal point

w wall

Superscripts

m; n polar and azimuthal

directions, respectively, see

Figure 4

mþ;m�; nþ; n� boundaries of the radiation

directions, m� � m � mþand n� � n � nþ, see

Figure 4

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The polygonal mesh systems have two different features compared with thetriangular meshes, especially in two-dimensional cases. First of all, polygonal meshesmay have such arbitrary shapes as quadrilateral, pentagonal, hexagonal, heptagonal,and so on, depending on the number of neighboring cells which share the vertex of thetriangular meshes. Therefore, they have so many control surfaces and correspondingneighboring nodes, in which scalar quantities such as intensity and temperature arestored, while triangular meshes have only three fixed control faces and neighboringnodes. The second characteristic is related to this feature, i.e., it can be reduced inthe total number of nodes to be visited in order to calculate the radiative intensity,therefore the computation time is dramatically decreased. Murthy and Mathur [1]first introduced this polygonal concept, but they only presented computation resultsusing the triangular and rectangular finite-volume approaches. Rousse [8] and Roesseet al. [9] formulated and validated the control-volume finite-element method (CV-FEM) by using polygonal meshes. Here, polygonal control volumes are generatedaround each triangular node by joining the centroids of each triangular mesh tothe midpoints of corresponding sides. To calculate the intensity at the integrationpoint which lies in the center of each control face, interpolation schemes that guaran-tee positive contributions to the coefficients in the algebraic discretization equationsare used to approximate the convective and radiative fluxes across the control faces.Following Roesse [8], Ben Salah et al. [10] adopted the CV-FEM for the calculation ofthe radiative heat transfer in complex 2-D geometries. They also used polygonal con-trol volumes based on triangular unstructured meshes. Recently, Asllanaj et al. [4]investigated the solution of radiative heat transfer in 2-D geometries by introducinga modified FVM based on a cell vertex scheme using polygonal control volumes basedon unstructured triangular meshes. The polygonal meshes used by Rousse [8], BenSalah et al. [10], and Asllanaj et al. [4], however, may be skewed or concave-shapedin the control faces, because polygonal meshes are generated by connecting the cen-troids of the triangular meshes into midpoints of the corresponding faces. This featuremay cause not only concave shape between adjacent faces but also slow convergencedue to the increase in the number of surface integration points of radiant flux.

In this work, a particular implementation of the FVM with polygonal unstruc-tured meshes is introduced that applies to the problems of radiative heat transfer intwo-dimensional complex enclosures. The polygonal meshes are generated by con-necting the adjacent centroids of triangular meshes, similar to the approach adoptedin FLUENT 6.3 [12]. Once the polygonal meshes are formed, the triangular meshesare no longer important, because conventional finite-volume approaches using thequadrilateral structured FVM [13, 14] are adopted. The medium may be absorbing,emitting, and isotropically scattering. The gray gas assumption is implicitly usedthroughout the present article, although this is not a requirement for the presentmethod. The contributions of this work include a new discretization scheme withpolygonal meshes in the context of the FVM, and demonstration and evaluationof the performance of the proposed method through various engineering problemswith complex geometries. In the following, mathematical formulations and corre-sponding discretization equations for the radiative transfer equation (RTE) arederived with two-dimensional unstructured polygonal cells. In order to validatethe formulations derived here, six benchmark problems, i.e., an equilateral triangularenclosure, a quadrilateral enclosure, an L-shaped enclosure, a rhombic enclosure, a

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hemispherical enclosure with a circular hole, and a furnace with embedded coolingpipes, are examined with respect to the accuracy of the solution and the computa-tional time required. Finally, some concluding remarks are given.

MATHEMATICAL FORMULATION

Radiative Transfer Equation

The radiation intensity for a gray medium at any position ~rr along a path ~ssthrough an absorbing, emitting, and scattering medium is given by [15]

1

b0

dIð~rr;~ssÞds

¼ �Ið~rr;~ssÞ þ ð1� x0ÞIbð~rrÞ þx0

4p

ZX0¼4p

Ið~rr;~ss 0ÞUð~ss 0 !~ssÞdX0 ð1Þ

where b0 ¼ ja þ rs is the extinction coefficient, and x0 ¼ rs=b0 is the scatteringalbedo. Uð~ss 0 !~ssÞ is the scattering phase function for radiation from the incomingdirection~ss 0 to the scattered direction~ss. In the case of isotropic scattering, the scatter-ing phase function becomes unity. This equation, if the temperature of the medium,Ibð~rrÞ, and boundary conditions for intensity are given, provides a distribution ofradiation intensity in a medium. Therby, the radiative heat flux can be obtained.The boundary condition for a diffusely emitting and reflecting wall can be denoted by

Ið~rrw;~ssÞ ¼ ewIbð~rrwÞ þ1� ew

p

Z~ss 0�~nnw< 0

Ið~rrw;~ss0Þ ~ss 0 �~nnwj j dX0 ð2Þ

where ew is the wall emissivity and subscript w denotes the location of the wall, while~nnw is the unit normal vector toward the medium. The above equation illustrates thatthe leaving intensity at the wall is a summation of the emitted and reflected intensities.

Spatial and Angular Discretizations

In order to generate the polygonal mesh systems from a baseline triangularmesh, first the domain of interest is discretized into the desired number of unstruc-tured triangular elements as shown in Figure 1, which is an illustrative example with13 triangular vertices (marked by diamonds in the boundary and rectangles in theinternal vertices, respectively) and 15 corresponding triangular elements, which aredepicted as solid lines. Then, centroids of each internal triangular element (markedby circles) from 19 to 33 are generated to form vertices in a polygonal system. Here,the boundary vertices from 1 to 9 (marked by diamonds) in Figure 1 also becomevertices in the polygonal mesh systems. The boundary vertices, for example, from10 to 18 marked in a circle, are assumed to be located in the midpoints of eachboundary lines. By connecting all the neighboring vertices represented by circles, apolygonal mesh system with 33 vertices and 13 polygonal elements is established,as shown by the dashed lines in Figure 1. The approach adopted in this procedurehas different features compared with the works conducted by Rousse [8], Feldheimand Lybaert [7], and Asllanaj et al. [4]. By connecting each center point of thetriangular meshes rather than joining the centroids of the triangular elements to

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the midpoints of the corresponding sides to form polygonal elements, it can not onlyprevent concave shapes of adjacent faces, but also reduce the number of control vol-ume faces consisting of polygonal meshes. Therefore, it is expected that the compu-tation time will be significantly reduced.

Figure 2 is a schematic representation of the sample polygonal controlvolumes, which comes from elements marked as 12 and 1, which are internal andnear-wall elements, respectively, as shown in Figure 1. Here, it is noted that eachvertex is renumbered in a counterclockwise direction and the volume of the poly-gonal elements is subdivided into triangular ones for convenience. Here, specialemphasis should be placed on the near-wall elements in Figure 2b. The new poly-gonal node, denoted by 1, encapsulated by a rectangle is located at the geometriccenter of this near-wall polygonal element, and becomes a polygonal node wherethe intensities are located. In Figures 2a and 2b, the geometric properties such asDAi;~nni, and DVI denote the surface area, the outward unit normal vector, and thevolume of the triangular subelement making up a polygonal element, respectively.These are easily calculated from the geometric relations, for example:

DAi ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiðxiþ1 � xiÞ2 þ ðyiþ1 � yiÞ2

qð3aÞ

~nni ¼~eexðyiþ1 � yiÞ=DAi �~eeyðxiþ1 � xiÞ=DAi ð3bÞ

and

DVi ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffisiðsi � DAiÞðsi � DAi2Þðsi � DAi3Þ

pð3cÞ

Figure 1. Schematics of the spatial meshes with the unstructured triangular (bold line) and polygonal

(dashed line) control volumes. Numbers in a rectangle and a diamond represent the vertices of the triangu-

lar meshes, while numbers in a circle and a diamond are the vertices of the polygonal mesh systems.

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where DAi2 and DAi3 are the remaining internal surface areas of the triangularsubelements, and si ¼ ðDAi þ DAi2 þ DAi3Þ=2 is the semiperimeter. Finally, thevolume of the polygonal element is obtained by summing up each volume ofthe triangular subelement, i.e., DV ¼

PDVi. Figure 3 shows the evolution of the

polygonal meshes from the triangular meshes adopted in this work. As previouslydiscussed, the polygonal meshes are generated by connecting the centers of the

Figure 2. Typical polygonal control volumes (a) in the interior and (b) near the boundary of the

calculation domain.

Figure 3. Schematics of the typical triangular and polygonal control volumes. The final unstructured

polygonal meshes in (c) are generated based on unstructured triangular meshes in (a) by connecting the

geometric center of each triangular element as shown in (b).

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triangular meshes sharing a triangular vertex. Therefore, it is noted that no controlfaces of the polygonal mesh can see each other.

There exist two types of control angle implementations in the FVM. The first isthe SN DOM-type angular discretization, which uses the direction cosine and corre-sponding angular weights spanning the range of 4p solid angle [16]. Anotherapproach includes the works of Chui and Radithby [17], Chai et al. [13], and Baeket al. [14]. The total solid angle, 4p sterad, is divided into Nh by N/ directions, whereh is the polar angle and / is the azimuthal angle, ranging from 0 to p and from 0 to2p as shown in Figure 4, respectively. For the analysis of a two-dimensional case,consideration of only 2p sterad, i.e., 0 � h � 0:5p and 0 � / � 2p, is necessarybecause of angular symmetry.

Unstructured Finite-Volume Method

To derive the discretization equation, first Eq. (1) is integrated over a controlvolume DV in Figure 2, and a control angle DXm in Figure 4. By assuming that themagnitude of intensity is constant but its direction varies within a given control volumeand control angle, the following finite-volume formulation can be obtained [13, 14]:

Xi

Imi DAi Dm

ci ¼ b0 �Im þ Smr

� �PDV DX ð4aÞ

where

Dmci ¼

ZDXmð~ss �~nniÞ dXm ð4bÞ

Figure 4. Schematics of the control angle. The angular polar angle h in (a) is measured from the z axis,

while azimuthal angle / in (b) is measured from the x axis.

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~ss ¼ sin h cos /~eex þ sin h sin /~eey þ cos h~eez ð4cÞ

~nni ¼ nx;i~eex þ ny;i~eey þ nz;i~eez ð4dÞ

Smr ¼ ð1� x0ÞIb þ

x0

4 p

ZX0¼4 p

Im0Um0mdX0 ð4eÞ

DXm ¼Z mþ

m�dXm ¼

Z /mþ

/m�

Z hmþ

hm�sin h dh d/ ð4f Þ

This equation indicates that a net outgoing radiant energy across control-volume facesmust be balanced by a net generation of radiant energy within the control volume andcontrol angle. In the FVM, the directional weights [13, 14, 17] Dm

ci should be carefullyevaluated, since they determine an inflow or outflow of radiant energy across thecontrol-volume face, depending on their sign. Detailed derivation of the angular direc-tional weights, for example, is given by Baek et al. [17].

Next, required is a scheme which relates the control-volume face intensity to anodal one. Among many others, such as diamond, positive-intensity, variable-weight, and exponential-type schemes [18], the one adopted here is a step schemein which a downstream face intensity is set equal to the upstream nodal value [13,14]. It is not only simple and convenient, it also ensures positive intensity whilenot considering complex geometric and directional information. According to thisscheme, typical relations are as follows:

Imi Dm

ci ¼ ImP maxðDm

ci ; 0Þ � ImI maxð�Dm

ci ; 0Þ ð5Þ

where subscript I represents the neighboring nodal point, while i represents thecorresponding face. This simplicity and convenience, however, may degenerate thesolution accuracy. By using this scheme, Eq. (4a) can be recast into the followinggeneral discretization equation:

amP Im

P ¼X

I

amI Im

I þ bmP ð6aÞ

where

amI ¼ maxð�DAiD

mci ; 0Þ ð6bÞ

amP ¼

Xi

maxðDAiDmci ; 0Þ þ b0;PDVDXm ð6cÞ

bmP ¼ ðb0Sm

r ÞPDVDXm ð6dÞ

Note that~nni; DAi, and DV have to be calculated carefully according to grid skewnessor curvature, as shown in the previous section. The boundary condition in Eq. (2) for

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a diffusely emitting and reflecting wall can be discretized as

Imw ¼ ewIbw þ

1� ew

p

XDm0

cw< 0

Im0

w Dm0

cw

�� �� ð7Þ

The iterative solution is terminated when the following convergence is attained:

max ImP � Im;old

P

������ Im

P

�� �� 10�6 ð8Þ

where Im;oldP is the previous iteration value of Im

P . Once the intensity field is obtained,the wall radiative heat flux can be estimated as

qRw ¼

ZX¼4p

Ið~rrw;~ssÞð~ss �~nnwÞdX ¼XMm¼1

Imw Dm

cw ð9Þ

where Dmcw is the directional weight at the bounding wall composed of~ss and ~nnw.

RESULTS AND DISCUSSION

The solution approach presented in this work is applied to two-dimensional radi-ative heat transfer problems of (1) an equilateral triangular enclosure, (2) a quadrilat-eral enclosure, (3) an L-shaped enclosure, (4) a rhombic enclosure, (5) a hemisphericalenclosure with a circular hole, and (6) a furnace with embedded cooling pipes. For allcases presented below, the total solid angle 2p is divided into Nh by N/ directions withuniform Dhm ¼ hmþz� hm� ¼ 0:5 p=Nh and D/n ¼ /nþ � /n� ¼ 2 p=N/, while thespatial domain is discretized as polygonal unstructured elements based on triangularmeshes depending on the problem dealt with.

An Equilateral Triangular Enclosure

Based on the present formulations, the unstructured polygonal FVM is appliedto an equilateral triangular enclosure with an absorbing and emitting but nonscatter-ing medium, maintained at constant temperature Tg ¼ 1; 000 K as shown inFigure 5a. The length of each side is 1 m, and the walls are held cold (Tw ¼ 0 K)and black (ew ¼ 1). In this case, the intensity, which is exact at any location withinthe enclosure, can be obtained by summing up all the intensities from the enclosurewalls as well as the local emission by the medium, such as

IðsÞ ¼ Ibwe�jas þ Ibð1� e�jasÞ ð10Þ

where Ib and Ibw are the local blackbody intensities of the medium and walls, respec-tively, while s is the path length of the radiant beam. Then, the exact wall heat fluxcan be obtained by numerically integrating IðsÞðs �~nnwÞ over all incident solid angleusing the approach of Gaussian quadrature numerical integration. Figure 5b showsthe triangular mesh system consisting of 400 elements and 231 vertices. As discussedin the previous section, each vertex in a triangular mesh system becomes the node

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points of the polygonal meshes in which the temperatures and intensities are located,so there exist 231 polygonal elements as illustrated in Figure 5c. The angular discri-tization used in this problem consists of Nh ¼ 4 and N/ ¼ 16.

The dimensionless incident radiative heat flux onto the bottom wall is pre-sented in Figure 6 with three different absorption coefficients of 0.1, 1.0, and10.0 m�1. When the absorption coefficient is as large as ja ¼ 10:0 m�1, the radiantenergy arriving on the wall approaches the blackbody intensity of the mediumIb ¼ rT4

g due to the heat blockage effect, i.e., the thick medium absorbs nearly allthe radiation from the neighboring medium, and the intensity impinging on the wallis influenced only by the emission of the hot gas near the enclosing walls. Near thecorner, however, a sharp decrease in radiant heat flux is observed because of theeffect of neighboring cold side walls. When the absorption coefficient is as smallas ja ¼ 0:1 m�1, the emission of the medium is weak, and the radiant heat flux is sig-nificantly reduced. This is due to the far-reaching effect of the other cold walls andnegligible self-extinction of the optically thin gas. In Figure 6, as a comparison, theexact and CV-FEM solutions [10] are also plotted alongside the present solutionsobtained using the triangular and polygonal FVM. Clearly, the present results arein good agreement with the exact solutions regardless of the absorption coefficients.The maximum error compared with the exact solution becomes 1.8% and 2.5% withthe triangular and polygonal meshes, respectively, which occurs with the opticallythick case of ja ¼ 10:0 m�1 as shown in Table 1. The computation times spent in thisexample are 1.02, 1.13, and 1.27 s for ja ¼ 10:0, 1.0, and 0:1 m�1 with the triangularmeshes, 0.57, 0.51, and 0.61 s for ja ¼ 10:0, 1.0, and 0:1 m�1 with the polygonal meshsystems, respectively, on a Pentium 1.7-GHz processor.

A QUADRILATERAL ENCLOSURE

The second benchmark problem deals with a quadrilateral enclosure with anabsorbing and emitting medium as illustrated in Figure 7a. The enclosure wallsare set black, and are assumed to have nonzero temperature. The nonscatteringmedium has constant temperature of Tg ¼ 100 K and three different absorption

Figure 5. An equilateral triangular enclosure: (a) schematic; (b) triangular meshes with 400 elements;

(c) polygonal meshes with 231 elements.

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coefficients of 0.1, 1.0, and 10.0 m�1. In this configuration, the enclosed medium ishomogeneous, absorbing and emitting, and isothermal at Tg, with intensity is givenby Eq. (14), therefore, the exact heat flux can be obtained as previously discussed.The spatial triangular and polygonal mesh systems used are plotted in Figures 7band 7c, respectively, where 2,020 triangular and 1,076 polygonal mesh systems are

Table 1. Comparison of the nondimensional wall heat flux in the case of an equi-

lateral triangular enclosure

jaðm�1Þ s (m) Exact Triangular FVM Polygonal FVM

10.0 0.2 0.954054 0.937106=1.8a 0.929784=2.5

0.4 0.993920 0.990059=0.4 0.986734=0.7

0.5 0.996120 0.993298=0.3 0.990561=0.6

1.0 0.2 0.396492 0.395076=0.4 0.390451=1.5

0.4 0.472184 0.468258=0.8 0.463476=1.8

0.5 0.480884 0.474332=1.4 0.470764=2.1

0.1 0.2 0.054993 0.055340=�0.6 0.054961=0.1

0.4 0.065362 0.065300=0.1 0.064886=0.7

0.5 0.066594 0.065856=1.1 0.065759=1.3

aA=B, where A ¼ nondimensional wall heat flux, B ¼ relative solution error (%).

Figure 6. Comparison of the dimensionless radiative wall heat flux for three different absorption

coefficients, ja ¼ 0:1, 1.0, and 10:0 m�1. The temperatures of the medium and wall are Tg ¼ 1; 000 K

and Tw ¼ 0 K with ew ¼ 1:0, respectively.

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illustrated. The angular mesh system used in this problem consists of Nh ¼ 4 andN/ ¼ 32, respectively.

The triangular and polygonal solutions of the radiative heat flux along thenorth cold wall is plotted in Figure 8 for three degrees of absorption coefficients withexact and structured finite-volume solution given by Chui [19]. When the medium isoptically thin, i.e., ja ¼ 0:1 m�1, the walls which are hotter than the medium canclearly see the north wall, therefore, more radiant energy arrives at the north wall.This can be explained as a far-reaching effect of the wall, as discussed in the previousexample. As the medium becomes optically thick, i.e., as the absorption coefficientincreases to ja ¼ 10:0 m�1, the incident radiant intensity impinging on the north wallapproaches Ib ¼ rT4

g because of the higher extinction effect of the radiant energywithin the medium. This implies that radiative energy emerging from distant hot wallboundaries is mostly eliminated as it travels the optically thick medium, therefore,the radiative heat flux on the north wall is influenced only by the emission of thehot medium adjacent to the wall. Overall, both the proposed triangular and poly-gonal unstructured solutions are seen to match the exact and structured finite-volume solutions very well for various optical thicknesses. The computation timesrequired for the case of ja ¼ 1:0 m�1 are 77.60 and 28.06 s for the triangular andpolygonal meshes, respectively, with the same Pentium 1.7-GHz processor.

An L-Shaped Enclosure

For the third test problem, the L-shaped enclosure is chosen to study the appli-cability of the present solution methods. The schematic of the test problem is shown

Figure 7. A quadrilateral enclosure: (a) schematic; (b) triangular meshes with 2,020 elements; (c) polygonal

meshes with 1,076 elements.

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in Figure 9a. The enclosure contains an absorbing medium with ja ¼ 0:01 m�1,which is maintained at constant temperature Tg ¼ 0 K, while all walls except thetop wall between points A and H are cold and black. The temperature of the topblack wall is set to Tw ¼ 100 K, which simulates an incident wall heat flux denotedby qin. In this configuration, since not only the wall below the dashed line connectingpoints A and I in Figure 9a can see the hot top wall, but there is also no scattering inthe medium, the incident heat flux on the top wall cannot reach the lower walllocated between points I and D. The triangular and polygonal unstructured meshsystems are illustrated in Figures 9b and 9c, respectively. Here, 1,006 triangularand 562 polygonal elements are used, while Nh ¼ 4 and N/ ¼ 24 angular discretiza-tions are adopted.

Figure 10 shows the dimensionless radiative heat flux impinging on the left-south wall, connecting points A, B, C, I, and D in Figure 9a. As the distanceincreases along the wall from point A to point D, the radiative heat flux on the wallsuffers significant variations, according to the orientation of the correspondingwalls, with respect to the heat source located on the top wall as illustrated inFigure 10. Along the wall from A to B, the heat flux decreases because radiant energyoriginated from the hot top wall is absorbed by the cold medium. The orientationof the wall changes at points B and C, therefore, there exists a sharp increase in

Figure 8. Comparison of the dimensionless radiative wall heat flux on north wall for three different

absorption coefficients, ja ¼ 0:1, 1.0, and 10:0 m�1. The temperature of the medium is Tg ¼ 1; 000 K,

while the walls have different temperatures of Tw ¼ 250 K, 450 K, 0 K, and 150 K for the south, east,

north, and west walls, respectively.

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the radiative heat flux because the wall can see a greater portion of the top wall.Beyond the point I, i.e., s ¼ 5:13 m, which is the end point that the top wall AHcan see, the incident heat flux along the surface becomes zero because there is noemission and scattering in the medium and enclosed cold wall.

The dimensionless heat fluxes obtained using the triangular and polygonalmeshes are compared with solutions obtained by the zone [9], FVM [9], DTM [7],and polygonal FVM [4] as shown in Figure 10. The present solutions follow reason-ably the above-mentioned thermal behavior and are seen to be very close to the othersolutions. It takes about 19.11 and 8.78s on a Pentium 1.7-GHz processor for thetriangular and polygonal mesh systems used, respectively.

A Rhombic Enclosure

The schematic description of the fourth test example is illustrated in Figure 11a.The temperature of the medium is set to Tg ¼ 1; 000 K, while all the walls areassumed to be cold and black. The medium has five different scattering albedos,x0 ¼ 0:0; 0:2; 0:5; 0:8, and 0.99, with an extinction coefficient of b0 ¼ 1:0 m�1.Scattering is assumed isotropic. The triangular and polygonal unstructured meshsystems used in this example are illustrated in Figures 11b and 11c, respectively.Here, 516 triangular and 299 polygonal elements are used, while an Nh ¼ 4 andN/ ¼ 16 angular discretization is adopted to capture of the physics of the problem.

Figure 9. An L-shaped enclosure: (a) schematic; (b) triangular meshes with 1,006 elements; (c) polygonal

meshes with 562 elements.

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Figure 12 shows the predicted nondimensional radiative heat flux on the northwall with five different scattering albedos. It is important to note that, for the case ofnonscattering medium, i.e., x0 ¼ 0:0, more heat is transferred to the north cold wallfrom the hot medium than for the case with scattering medium. Also, it can be foundthat the heat flux near the left corner is greater than near the right corner because ofthe orientation of the adjacent cold walls. As the scattering albedo increases with thesame extinction coefficient, less heat is transferred to the north wall since less radiantenergy is emitted from the medium and more radiant energy emitted is scattered tothe surrounding medium rather than to the wall. When the medium is assumed as tobe nearly pure scattering, i.e., x0 ¼ 0:99, radiative heat flux arrived at the wall is verysmall, mainly because of the absolutely small emission of the medium. As expected,when the medium is purely scattering, no radiative heat is emitted and transferred inthe medium. Figure 12 also shows a comparison between the present finite-volumeand the available Monte Carlo solutions [20, 21], and it can be seen that the presentsolutions are close to the Monte Carlo solutions. The calculation time taken in thisexample increases from 1.33 to 2.77s with the increment of scattering albedo fromx0 ¼ 0 to x0 ¼ 0:99 with the polygonal meshes, because more radiant energy isscattered. The corresponding times with the triangular mesh system are 2.83 and5.96s on the same computer.

Figure 10. Comparison of the dimensionless radiative wall heat flux on the west and south walls for

absorption coefficient ja ¼ 0:01 m�1. The temperature of the medium is Tg ¼ 0 K, while the walls are cold

and black, except that the temperature of the north wall is set to 1; 000 K.

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A Hemispherical Enclosure with a Circular Hole

Based on the validation of the present unstructured polygonal FVM solver asabove, a semicircle enclosure with one inner circular hole is now examined as anexample of a more complex configuration with an obstacle in the medium. As pre-sented in Figure 13a, a circular hole 0.2 m in radius is located at y ¼ 0.4 m fromthe bottom wall of the semicircle outer enclosure with R0 ¼ 1 m. The mediumenclosed is absorbing as well as emitting, with uniform gas temperatureTg ¼ 1; 000 K, while all boundaries are cold and black. Three different absorptioncoefficients, ja ¼ 0:1, 1.0, and 10:0 m�1, are considered. The triangular and poly-gonal unstructured mesh systems used in this example are illustrated in Figures13b and 13c, respectively. Here, 1,356 triangular and 737 polygonal elements areused, while the angular discretization adopted in this problem is Nh ¼ 4 andN/ ¼ 16.

Comparisons of the radiative heat flux on the bottom wall of the outer hemi-spherical enclosure are presented in Figure 14 with other solutions of exact inte-gration and multiblock FVM [22]. When ja ¼ 10:0 m�1, the dimensionlessradiative heat flux along the bottom wall is near unity except at the center and theside corners. This is because the intensity impinging on the wall is influenced mainlyby the neighboring hot medium rather than the inner medium away from the wall forthe case of large absorption coefficient. The heat flux near the center is slightlyreduced due to the existence of the cold inner circle, since the wall of the inner circleabsorbs the radiant energy emitted by the participating hot medium. Near the sidecorner, however, the radiative heat flux rapidly decreases, since that is the place

Figure 11. A rhombic enclosure: (a) schematic; (b) triangular meshes with 516 elements; (c) polygonal

meshes with 299 elements.

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Figure 12. Comparison of the dimensionless radiative wall heat flux on the north wall for four scattering albedos

x0 ¼ 0:0, 0.2, 0.5, and 0.8. The temperature of the medium is Tg ¼ 1; 000 K, while all the walls are cold and black.

Figure 13. A hemispherical enclosure with a circular hole: (a) schematic; (b) triangular meshes with 1,356

elements; (c) polygonal meshes with 737 elements.

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where the cold walls meet each other. As the absorption coefficient decreases, theradiative heat flux at the bottom wall is significantly reduced. This is because theemissive power by the medium is reduced since the absorption coefficient is smaller.Results obtained by the triangular and polygonal unstructured FVM are in goodagreement with the exact and multiblock finite-volume solutions [22]. Also, it canbe found that the computational time with the polygonal meshes is almost 40% ofthat of the triangular mesh systems.

A Furnace with Embedded Cooling Pipes

Finally, a more practical model of a furnace is considered, which is similar tothe three-dimensional one of Adams and Smith [23], who analyzed the radiative heattransfer using the discrete-ordinates method. In the present example, however, thefurnace is simplified as a two-dimensional enclosure 2.0 m in width and 6.29 m inlength as presented in Figure 15a. Here, the west, north, and west boundaries arewalls, while the south boundary is considered as a symmetric line. There exist 22cooling pipes along the north wall between 2.22 and 6.29 m. The pipe of 0.03 m indiameter is 0.025 m away from the north wall. The furnace wall and pipe emissivitiesare 0.5 and 0.8, respectively. The furnace wall and gas medium temperatures areobtained from the work by Michelfelder and Lowes [24], while the cooling pipe

Figure 14. Comparison of the dimensionless radiative wall heat flux on the south wall for three different

absorption coefficients ja ¼ 0:1, 1.0, and 10:0 m�1. The temperature of the medium is Tg ¼ 1; 000 K, while

all the walls are cold and black.

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temperature is assumed to be 300 K. At the point where a measured value does notexist, the temperature is calculated by interpolating the neighboring data. The gasabsorption coefficient is set as ja ¼ 0:2 m�1. The triangular and polygonal unstruc-tured meshes used are illustrated in Figures 15b and 15c, respectively. Here, 12,838triangular and 6,774 polygonal elements are used, while Nh ¼ 4 in polar and N/ ¼ 16in azimuthal angular discretization is adopted. Figure 15d shows an enlarged view ofpolygonal mesh systems near the cooling pipes adjacent to the furnace wall. It can befound that there exist various types of meshes such as quadrilateral, pentagonal,hexagonal, heptagonal, and octagonal ones.

Figure 16 shows the incident radiative heat flux distribution along the northwall. It is shown that the incident fluxes are seen to fluctuate dramatically wherethe cooling pipes exist, since they intercept incident radiation coming from the hotmedium toward the wall. The present polygonal finite-volume solutions are seento be as good as the results obtained with the triangular unstructured FVM. A dif-ferent type of control-angle overlap manipulation makes only a minor difference. Inthis example, a more dramatic reduction in computation time is observed, i.e.,858.15s with the triangular and 341.59s with the polygonal meshes on a Pentium1.7-GHz processor, respectively.

Figure 15. A two-dimensional furnace with embedded cooling pipes: (a) schematic; (b) triangular meshes

with 12,838 elements; (c) polygonal meshes with 6,774 elements; (d) enlarged view.

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CONCLUSIONS

The implementation of the finite-volume method with unstructured polygonalmeshes has been described for the analysis of radiative heat transfer in a two-dimensionalcomplex enclosure with an absorbing, emitting, and isotropically scattering graymedium. Based on the unstructured triangular meshes, the unstructured polygonalelements are generated by connecting each node enveloping a vertex of the triangularmeshes, thereby developing convex polygonal meshes. After a mathematical formula-tion and corresponding discretization equation for the RTE are derived, the finaldiscretization equation is introduced by using the directional weight, which is the keyparameter in the FVM since it represents the inflow or outflow of radiant energy acrossthe control-volume faces depending on its sign, as is the conventional FVM. The presentapproach is then validated by comparing the present results with those of previousworks. All the results presented in this work show that the present method is accurateand valuable for the analysis of radiative heat transfer problems in complex geometries.In particular, it can be found that the computation time with the polygonal meshescan be dramatically reduced compared with the triangular mesh systems.

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Figure 16. Comparison of the incident radiative wall heat flux on the north wall of the furnace for absorp-

tion coefficient ja ¼ 0:2 m�1.

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