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THE METHOD OF LINES SOLUTION OFTHE DISCRETE ORDINATES METHODFOR RADIATIVE HEAT TRANSFER INENCLOSURES CONTAINING SCATTERINGMEDIANevin Selçuk a & Işıl Ayrancı aa Department of Chemical Engineering, Middle East TechnicalUniversity, Ankara, TurkeyPublished online: 02 Feb 2011.

To cite this article: Nevin Selçuk & Işıl Ayrancı (2003) THE METHOD OF LINES SOLUTION OFTHE DISCRETE ORDINATES METHOD FOR RADIATIVE HEAT TRANSFER IN ENCLOSURES CONTAININGSCATTERING MEDIA, Numerical Heat Transfer, Part B: Fundamentals: An International Journal ofComputation and Methodology, 43:2, 179-201, DOI: 10.1080/713836169

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THE METHOD OF LINES SOLUTION OF THE DISCRETEORDINATES METHOD FOR RADIATIVE HEAT TRANSFERIN ENCLOSURES CONTAINING SCATTERING MEDIA

Nevin Selcuk and Isıl AyrancıDepartment of Chemical Engineering, Middle East Technical University,Ankara, Turkey

A radiation code based on the method of lines (MOL) solution of the discrete ordinates

method (DOM) for three-dimensional radiative heat transfer in rectangular enclosures

containing gray, absorbing, emitting, isotropically and anisotropically scattering media was

developed. Predictive accuracy of the code was evaluated by applying the code to 1-D and

3-D problems containing scattering media and benchmarking its predictions against exact

solutions, the zone method, and Monte Carlo solutions. Favorable comparisons reveal that

MOL solution of the DOM provides accurate solutions for modeling radiative heat transfer

in 3-D rectangular enclosures containing purely scattering or absorbing, emitting, and

scattering media with isotropic or anisotropic scattering properties.

1. INTRODUCTION

Accurate and computationally efficient prediction of transient behavior ofturbulent reacting and radiating flows typical of industrial combustion chambersnecessitates solution of the radiative transfer equation (RTE) in conjunction with thetime-dependent conservation equations for mass, momentum, energy, and chemicalspecies. Method of lines (MOL) solution of the discrete ordinates method (DOM) isa promising solution method for multidimensional radiative heat transfer in parti-cipating media. It is based on implementation of a false-transients approach to thediscrete ordinates representation of the RTE and discretizing the spatial derivativesin the resulting equations and hence transforming the original boundary-valueproblem to an initial-value problem which can be solved by using a powerfulordinary differential equation (ODE) solver. As time goes to infinity, solution to theoriginal boundary-value problem is obtained. Solution of the RTE by using thisapproach not only makes its coupling with computational fluid dynamics (CFD)codes easier, it also alleviates the slow-convergence problem encountered in theimplementation of the classical DOM to steady-state problems involving stronglyscattering media [1].

Received 16 November 2001; accepted 6 July 2002.

Address correspondence to Prof. Dr. Nevin Selcuk, Department of Chemical Engineering, Middle

East Technical University, Ankara, 06531, Turkey. E-mail: selcuk@metu.edu.tr

Numerical Heat Transfer, Part B, 43: 179–201, 2003

Copyright # 2003 Taylor & Francis

1040-7790/03 $12.00 + .00

DOI: 10.1080/10407790390122041

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This method was first applied to the determination of heat fluxes in a 2-D blackenclosure containing purely isotropically scattering medium and tested for accuracyby comparing its predictions with exact solutions [2]. Predictive accuracy of theMOL solution of the DOM was then investigated by comparing its steady-statepredictions with the analytical solution of the DOM and exact solution of the RTEon a 1-D slab containing a gray, absorbing, emitting medium with uniform tem-perature profile [3, 4]. The method was also extended to a 3-D problem, a box-shaped enclosure with black walls bounding a gray, absorbing, emitting mediumwith steep temperature gradients typical of operating furnaces and combustors, andvalidated against exact solutions for the same problem [4, 5]. Recently, the accuracyand computational efficiency of the method were assessed by applying it to theprediction of incident wall fluxes on the freeboard of a pilot-scale atmosphericbubbling fluidized bed combustor, treated as a 3-D rectangular enclosure containingabsorbing, emitting, isotropically scattering medium and benchmarking its predic-tions against those of the zone method and measurements [6, 7]. Favorable com-parisons obtained to date have led to the assessment of the method on morechallenging test problems with prespecified temperatures or source terms involvinganisotropically scattering media.

NOMENCLATURE

ei unit vector in the coordinate direc-

tion i

G total intensity of incident radiation,

W=m2

H height of the enclosure, m

I radiative intensity, W=m2 sr

kt time constant, (m=s)�1

‘m0,i direction cosine of Om0 with respect

to eiL length, m

m discrete direction

M total number of ordinates

n unit normal vector

N order of approximation

NEQN number of ODEs

NX,

NY, NZ total number of interior grid points

in x; y; z directions

q heat flux vector, W=m2

qþ incoming wall heat flux, W=m2

q� leaving wall heat flux, W=m2

q00 net radiative heat flux, W=m2

q000 nonradiative source term, W=m3

Q� dimensionless heat flux

r position vector, m

s geometric path length, m

t pseudo-time variable, s

T temperature, K or +C

TF final time of integration, s

TP print interval, s

w quadrature weight

W width of the enclosure, m

x; y; z Cartesian coordinates, m

b extinction coefficient, m�1

e emissivity

Z direction cosine

y polar angle, rad

Y scattering angle, deg or rad

k absorption coefficient, m�1

m direction cosine

x direction cosine

s Stefan-Boltzmann constant

ð¼ 5:6 � 10�8 W=m2 K4Þss scattering coefficient, m�1

t optical thickness

f azimuthal angle, rad

F scattering phase function, sr�1

o single scattering albedo

V direction of radiation intensity

dO solid angle, sr

Subscripts and Superscripts

b blackbody

i index for spatial coordinate of

interest

inc incident

m outgoing ordinate direction

m0 incoming ordinate direction

r radiative

w wall

� dimensionless

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In the present article, a three-dimensional radiation code based on the MOLsolution of the DOM is developed for multidimensional radiative heat transfer inscattering media and tested for predictive accuracy by applying it to (1) a 1-D testproblem containing purely anisotropically scattering medium and comparing itssteady-state solutions with exact solutions and DOM solutions, (2) a 3-D idealizedfurnace problem containing isotropically scattering medium and benchmarking itspredictions against DOM and zone method solutions, and (3) cubical enclosureproblems containing purely scattering and absorbing, emitting, scattering media withisotropic and anisotropic scattering properties and validating the solutions againstMonte Carlo solutions available in the literature.

2. DESCRIPTION OF THE METHOD

The physical situation to be considered is that of a radiatively gray, absorbing,emitting, scattering medium surrounded by gray, diffuse walls. Radiative propertiesof the medium are assumed to be uniform throughout the enclosure.

2.1. Radiative Transfer Equation

The radiative transfer equation takes the following form for the problem underconsideration:

dI

ds¼ ðV HÞIðr;VÞ ¼ �k Iðr;VÞ þ k IbðrÞ � ss Iðr;VÞ

þ ss

4pZ

4pFðV0;VÞ Iðr;V0Þ dO0 ð1Þ

where Iðr;OÞ is the radiation intensity at position r in the direction V; k and ss

represent the absorption and scattering coefficients of the medium, respectively,IbðrÞ½� sT 4ðrÞ=p� stands for the black-body radiation intensity, FðV0;VÞ is thephase function for scattering, and dV0 is the differential solid angle for incomingradiation. The expression on the left-hand side represents the change of the intensityin the specified direction V. The terms on the right-hand side stand for absorption,emission, out-scattering, and in-scattering, respectively.

The phase function, which describes the fraction of energy scattered fromincoming direction V0 to outgoing direction O, can be approximated by the Legendreseries as

FðV0;VÞ ¼XNj¼0

Cj PjðcosYÞ ð2Þ

where Y is the scattering angle between incoming and outgoing directions; Pj’s andCj’s are the Legendre polynomials of order j and the expansion coefficients,respectively. For linear anisotropic scattering the phase function takes a linear formðN ¼ 1Þ, and for isotropic scattering the phase function equals unity.

As the surface bounding the medium is a diffuse, gray wall at specified tem-perature, Eq. (1) is subject to the boundary condition

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Iðrw;VÞ ¼ ew Ib;w þ ð1 � ewÞp

Zn:V0<0

n V0�� �� Iðrw;V0Þ dO0 n V > 0 ð3Þ

where Iðrw;VÞ represents the radiative intensity leaving the surface at a boundarylocation, ew denotes the surface emissivity, Ib;wð� s Tw4=pÞ denotes the black-bodyradiation intensity at the surface temperature, n is the local outward surface normal,and n V0 stands for the cosine of the angle between incoming direction V0 and thesurface normal.

The RTE, Eq. (1), and its boundary conditions, Eq. (3), are closed if thetemperature field throughout the medium is specified. Otherwise, coupling betweenthe overall energy conservation equation and the RTE is required in order to solvefor the temperature field from a prespecified source term field. The steady-stateenergy conservation equation in which total heat generation due to convection,conduction, and other external sources such as chemical reaction is represented bya volumetric heat generation term and takes the following form:

H qr ¼ q000 ð4Þ

where qr represents the radiative heat flux vector and q000 is the volumetric non-radiative heat generation term. From conservation of radiative energy, the diver-gence of radiative heat flux is obtained by integration of the RTE over all Vdirections, leading to

H qr ¼ k ½4p IbðrÞ � GðrÞ� ð5Þ

where G is the incident radiation, defined by

GðrÞ ¼Z

4pIðr;VÞ dO ð6Þ

Finally, coupling between the RTE and the energy equation is achieved bymaking the right-hand sides of Eq. (4) and Eq. (5) equal and substituting theresulting expression for IbðrÞ into Eq. (1):

ðV HÞIðr;VÞ ¼ �ðkþ ssÞ Iðr;VÞ þ q000

4pþ k

4p GðrÞ

þ ss

4pZ

4pFðV0;VÞ Iðr;V0Þ dO0 ð7Þ

Equation (7) and the boundary conditions are closed when the nonradiative heatgeneration term q000 is specified or when the medium is assumed to be in radiativeequilibrium ðH qr ¼ q000 ¼ 0Þ.

Once the radiation intensities are calculated by solving either Eq. (1) or Eq. (7)together with the boundary conditions, Eq. (3), quantities of interest such asradiative flux can be readily evaluated. The net radiative flux on a surface element isdefined as

q00 ¼ qþ � q� ð8Þ

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where qþ and q� are incident and leaving wall heat fluxes, respectively. For a diffusegray wall, qþ and q� are obtained from

qþ ¼ZnV<0

n Vj j I dO ð9Þ

q� ¼ZnV>0

n Vj j I dO ð10Þ

The coordinate component of radiative heat flux vector inside the medium isgiven by

qi ¼Z

4pðei VÞ I dO ð11Þ

where ei is the unit normal vector in the coordinate direction i.

2.2. Discrete Ordinates Method (DOM)

The discrete ordinates method is based on representation of the continuousangular domain by a discrete set of ordinates with appropriate angular weights,spanning the total solid angle of 4p steradians. The RTE is replaced by a discrete setof equations for a finite number of directions and each integral is replaced by aquadrature summed over the ordinate directions [8]. The discrete ordinates repre-sentation of RTE for a 3-D enclosure containing a uniform, gray, absorbing,emitting and scattering medium takes the following form for a rectangular coordi-nate system (see Fig. 1):

mm@Im

@xþ Zm

@Im

@yþ xm

@Im

@z¼ �k Im þ k Ib � ss Im

þ ss

4pXMm0¼1

FðVm0 ;VmÞ wm0 Im0 ð12Þ

where Im½� Iðr;mm;Zm; xmÞ� is the radiation intensity at position rðx; y; zÞ in thediscrete ordinate direction Vm, m denotes the discrete ordinate ðm ¼ 1; 2; . . . ;MÞ,M is the total number of ordinates used in the approximation, mm;Zm, and xmare the direction cosines of Vm with x; y, and z axes, respectively (m ¼cos y;Z ¼ sin y sinf and x ¼ sin y cosf), and wm0 is the angular quadrature weightassociated with the incoming direction Vm0 . The discrete ordinates representation ofcosine of the scattering angle, which is the argument of the phase function given inEq. (2), is

cosY ¼ mm mm0 þ Zm Zm0 þ xm xm0 ð13Þ

where subscripts m and m0 denote outgoing and incoming ordinates, respectively.

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If nonradiative source term q000 is specified instead of the temperature field,discrete ordinates representation of Eq. (7) takes the following form:

mm@Im

@xþ Zm

@Im

@yþ xm

@Im

@z¼ �ðkþ ssÞ Im þ q000

4pþ k

4p GðrÞ

þ ss

4pXMm0¼1

FðVm0 ;VmÞ wm0 Im0 ð14Þ

where incident radiation is evaluated from

GðrÞ ¼XMm0¼1

wm0 Im0 ð15Þ

The boundary conditions at the two opposite, diffuse, gray surfaces withnormal vectors parallel to the x axis can be written as

at x ¼ 0; Im ¼ ew Ib;w þ ð1 � ewÞp

Xmm0<0

wm0 mm0j j Im0mm > 0 ð16Þ

at x ¼ L; Im ¼ ew Ib;w þ ð1 � ewÞp

Xmm0>0

wm0 mm0j j Im0mm < 0 ð17Þ

Figure 1. Coordinate system.

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where Im is the intensity of radiation leaving the surface. Similar expressions hold forthe boundaries in the other coordinate directions.

Using the DOM, the RTE is transformed into a set of simultaneous partialdifferential equations containing only space coordinates as independent variables.Spatial discretization may be accomplished by using a variety of methods includingfinite-volume, finite-element, or finite-difference techniques. In the classical DOMapplications [8–13], spatial differencing is carried out by using standard cell-centered,finite-volume technique. In this approach, the discrete ordinates equations areintegrated over a typical control volume and interpolation schemes are defined torelate face intensities to cell-centered intensities. An iterative, ordinate sweepingtechnique described in [10] is applied to solve for the intensities at each ordinate andat each control volume.

Once the radiation intensities are solved from Eq. (12) or Eq. (14) together withboundary conditions, the net radiative heat flux along a direction i can be obtainedfrom

q00i ¼XMm0¼1

wm0 ‘m0;i Im0 ð18Þ

where ‘m0; i is the direction cosine of ordinate Vm0 with respect to the unit vector ei.The accuracy of the discrete ordinates method is affected by the accuracy of the

angular quadrature scheme, order of approximation, and spatial differencing schemeadopted for the solution.

2.3. Method of Lines (MOL) Solution of the DOM

The solution of discrete ordinates equations by the MOL is carried out byadoption of the false-transients approach, which involves incorporation of a pseudo-time derivative of intensity into the discrete ordinates equations [2]. Application ofthe false-transients approach to Eq. (12) yields

kt @Im

@t¼ �mm

@Im

@x� Zm

@Im

@y� xm

@Im

@z� k Im þ k Ib � ss Im

þ ss

4pXMm0¼1

FðVm0 ;VmÞ wm0 Im0 ð19Þ

where t is the pseudo-time variable and kt is a time constant with dimension ðm=sÞ�1

which is introduced to make the equation dimensionally consistent and is taken asunity.

The system of partial differential equations (PDEs) with initial and boundary-value independent variables is then transformed into an ODE initial-value problemby using the method of lines approach [14]. The transformation is carried out byrepresentation of the spatial derivatives with algebraic finite-difference approxima-tions. Starting from an initial condition for radiation intensities in all discretedirections, the resulting ODE system is integrated until steady state by using apowerful ODE solver. The ODE solver takes the burden of time discretization andchooses the time steps in a way that maintains the accuracy and stability of the

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evolving solution. Any initial condition can be chosen to start the integration, as itseffect on the steady-state solution decays to insignificance. As a result, evolution ofradiative intensity with time at each node and ordinate is obtained. The steady-stateintensity values obtained from Eq. (19) are also the solutions to Eq. (12), as theartificial time derivative vanishes at steady-state.

Once the steady-state intensities at all grid points are available, the netradiative heat fluxes on enclosure boundaries and incident radiation at interior gridpoints can be evaluated by using Eqs. (18) and (15), respectively.

3. PERFORMANCE OF THE MOL SOLUTION OF THE DOM

The predictive accuracy of the MOL solution of the DOM was investigated onthree test cases by comparing its predictions with exact solutions or benchmarksolutions available in the literature. The first test case is a 1-D parallel-plate problemwith purely anisotropically scattering medium in radiative equilibrium, the second isa 3-D idealized furnace problem containing absorbing, emitting, and isotropicallyscattering medium with uniform nonradiative source term, and the third test case is a3-D cubical enclosure containing participating medium with isotropic and aniso-tropic scattering properties.

The MOL solution of the DOM was implemented to these test cases by usingthe finite-differencing schemes, DSS012 and DSS014, of Schiesser [14], which aretwo- and three-point upwind differencing schemes, respectively. The ODE solverutilized in the present study is the RKF45 (Runge-Kutta-Fehlberg integration)subroutine [14, 15], previously found to be as accurate but less CPU intensive thananother quality ODE solver, LSODE (Livermore Solver for Ordinary DifferentialEquations) [4, 5].

3.1. One-Dimensional Slab with Purely Scattering Mediumin Radiative Equilibrium

The physical system chosen for investigation is a uniform, plane-parallel, gray,purely scattering medium in radiative equilibrium, confined within gray, diffusewalls. Scattering is assumed to be linearly anisotropic and the phase function inEq. (2) is utilized by assigning N ¼ 1;C0 ¼ 1, and C1 ¼ �0:7.

For this problem, the RTE coupled with the energy equation, Eq. (7), and itsdiscrete ordinates representation, Eq. (14), were reduced to one-dimensional forms.Since radiative equilibrium prevails within the medium, the volumetric heat gen-eration term, q000, equals zero. The resulting equations were solved by the MOLapproach. The spatial discretization scheme adopted for solution is the three-pointupwind differencing scheme (DSS014), which was found to be more accurate thanthe two-point upwind scheme in a previous study [3, 4]. For implementation of theDOM in this study, the SN angular quadrature scheme proposed by Carlson andLathrop [16] was selected. The choice was based on an assessment study carried outby Selcuk and Kayakol [17].

The predictive accuracy of the method for various optical thicknesses (t ¼ b xÞbetween 0.1 and 3.0 was examined by comparing its steady-state dimensionless heat

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fluxes with exact solutions provided by Dayan and Tien [18]. The definition ofdimensionless heat flux is given by

Q� ¼ q00

s ðT 41 � T 4

2 Þ � q00 ð1=e1 � 1=e2 � 2Þ ð20Þ

where q00 is the net radiative heat flux, T and e stand for wall temperature andemissivity, and the subscripts 1 and 2 denote the two walls at x ¼ 0 and x ¼ L0,respectively. Comparisons are presented in terms of percentage errors defined as

%Rel. Err. ¼Q�

predicted �Q�exact

Q�exact

� 100 ð21Þ

The effect of grid number on the accuracy of the MOL solution of the DOMwas investigated by analyzing the percentage errors for different grid numbers.Figure 2a shows variation of percent relative error in dimensionless heat fluxes withnumber of grids using an S8 scheme for various optical thicknesses. It can be seenfrom the figure that the predictions are in good agreement with the exact solutions.Although the errors increase with increasing optical thickness, errors beyond 150grids are all less than 0.5% for all cases. The variation of required computationaltime with number of grids is presented in Figure 2b. The CPU times reported are foran IBM F80. From the analysis of the effect of spatial discretization on accuracy andcomputational efficiency, 150 grids was chosen to be the optimum grid number fromthe viewpoint of computational time and accuracy.

Figure 3a illustrates variation of percentage errors in dimensionless heat fluxeswith the order of approximation using 150 grids and DSS014 for various opticalthicknesses. As can be seen from the figure, the effect of order of approximation onthe accuracy of results is significant, especially for cases with higher optical thick-nesses. For t ¼ 0:1 the S2 scheme produces reasonable accuracy, whereas for opticalthicknesses greater than that, it is required to employ at least an S6 scheme to obtainerrors less than 0.5% Figure 3b shows the effect of order of approximation on CPUtimes on an IBM F80. It can be seen from the figure that application of higher orderof approximation is computationally more expensive for cases with lower opticalthickness.

Comparative testing of the predictive accuracies of the MOL solution of theDOM and the DOM was carried out by comparing the solutions obtained in thisstudy and those obtained by DOM [9], for various optical thicknesses. Table 1 dis-plays exact solutions, predictions of the DOM with S4, predictions of the MOLsolution of the DOM with S4 and three-point upwind differencing, and the absolutepercentage relative errors of the two approximations. As can be seen from the table,the MOL solution of the DOM yields slightly more accurate solutions than the DOM.

3.2. Three-Dimensional Idealized Furnace Problem with SpecifiedSource Term

This test case is a rectangular enclosure bounded by gray, diffuse walls,containing a uniform, gray, absorbing, emitting, and isotropically scattering mediumwith specified nonradiative source term. This test case is the idealized furnace

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Figure 2. Variation of percent relative error in dimensionless heat flux and CPU time with number of grids

for various optical thicknesses (S8, DSS014).

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Figure 3. Variation of percent relative error in dimensionless heat flux and CPU time with order of

approximation for various optical thicknesses (150 grids, DSS014).

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problem of Menguc and Viskanta [19]. The dimensions of the enclosure, emissivities,and temperatures of the walls and properties of the medium are given in Table 2.

Specification of the nonradiative source term instead of the temperature fieldinside the medium requires coupling between the RTE and the energy equation. Forthis reason, in this problem Eq. (7) is solved together with the boundary conditionsgiven in Eq. (3).

The MOL solution of the DOM was applied to the prediction of net radiativeheat flux distributions on the firing and exit ends of the idealized furnace and thetemperature profiles inside the enclosure. The accuracy of the method was assessedby comparing its predictions with those of the zone method with 565610 sub-divisions and DOM solutions with 868615 grids and S4, provided by Truelove [13].The MOL solution of the DOM was applied by using 767615 grids in the spatialdomain with respect to the x; y; z axes, respectively, and S4 order of approximationso that the angular and spatial discretizations of MOL solutions are consistent withthose of DOM solutions. This spatial discretization in the MOL solution of theDOM provides solutions at the centerlines, where Truelove’s solutions are reported.

Table 1. Comparative testing between dimensionless heat flux predictions of DOM (S4) and MOL

solution of DOM (S4, 150 grids, DSS014)

DOM MOL solution of DOM

Optical

thickness

Exact solution

Q� Q�Absolute

% errora Q�Absolute

% errora

0.1 0.901 0.897 0.44 0.900 0.11

0.5 0.663 0.651 1.81 0.664 0.15

1.0 0.505 0.495 1.98 0.506 0.20

3.0 0.260 0.258 0.77 0.261 0.38

aAbsolute percent relative error ¼ ðjQ�predicted �Q�

exactj=Q�exact�100.

Table 2. Input data for the idealized furnace problem

Dimensions:

Length, L 2 m

Width, W 2 m

Height, H 4 m

Medium:

Extinction coefficient, b ¼ kþ ss 0.5 m7 1

Scattering albedo, o ¼ ss=b 0.7

Volumetric heat generation, q000 5.0 kW=m3

Boundaries:

Temperature Emissivity

Firing end 1,200 K 0.85

Exit end 400 K 0.70

Others 900 K 0.70

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Figure 4. Comparison between predictions of MOL solution of DOM, DOM, and the zone method for

temperature distributions along the x axis at three axial locations and net heat flux profiles along the x axis

on firing and exit ends ðy ¼ 1mÞ.

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Figure 4a shows the predictions for the temperature distributions in themedium at three axial locations obtained from the zone method, the DOM, and theMOL solution of the DOM. The net heat flux predictions obtained from the threemethods are compared in Figure 4b. As can be seen from the two figures, predictionsof the MOL solution of the DOM are in good agreement with zone method solu-tions, and they are as accurate as DOM solutions. The CPU time requirement for theMOL solution of the DOM is 200 s on an IBM F80 computer.

3.3 Three-Dimensional Cubical Enclosure Containing ScatteringMedium

The physical system under consideration is a 3-D cubical enclosure containinguniform, gray, isotropically and anisotropically scattering media confined withindiffuse, gray walls. In this physical system, two different problems were analyzed.The first one is characterized by a purely scattering medium and nonsymmetricboundary conditions, and the second problem contains an isothermal, absorbing,emitting, and scattering medium with symmetric boundary conditions.

The scattering phase functions used to examine the effects of anisotropy inboth problems are those studied by Kim and Lee [11]. Phase functions described bythe expansion coefficients listed in Table 3 and illustrated in Figure 5 were utilized inthis test case. Two of the phase functions, F1 and F2, are for forward scattering, theother two, B1 and B2, are for backward scattering.

This test case has been defined by Kim and Huh [20], who tested accuracies ofvarious radiation models against dimensionless Monte Carlo solutions defined as

Q�i ¼

q00is T 4

0

ð22Þ

G� ¼ G

4s T 40

ð23Þ

Table 3. Expansion coefficients for phase functions [11]

Cj

j F1 F2 B1 B2

0 1.00000 1.00000 1.00000 1.00000

1 2.53602 2.00917 �0.56524 �1.20000

2 3.56549 1.56339 0.29783 0.50000

3 3.97976 0.67407 0.08571

4 4.00292 0.22215 0.01003

5 3.66401 0.04725 0.00063

6 3.01601 0.00671

7 2.23304 0.00068

8 1.30251 0.00005

9 0.53463

10 0.20136

11 0.05480

12 0.01099

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T0 is the reference temperature. Accuracy of the MOL solution of the DOM wastested by applying it to the two problems and validating predicted radiation variablesagainst Monte Carlo solutions [20].

3.3.1. Purely scattering medium with nonsymmetric boundaryconditions. In this case, the cubical enclosure with dimension L0 contains gray,purely scattering medium with dimensionless scattering coefficient, s�

s ð¼ ss L0Þ,equal to unity. Temperature of the wall at z ¼ 0 is T0 and the other surfaces arecold. The enclosure boundaries are black.

Performance of the MOL solution of the DOM for this problem was assessedby comparing its predictions for radiative heat flux Q�

z along the centerline of theenclosure (L0=2;L0=2; z) with those of Monte Carlo solutions with 25625 sub-surfaces and 106 energy bundles emitted from each subsurface.

Figure 5. Polar plots of the phase functions.

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As the accuracy of the MOL solution of the DOM depends on grid spacing,spatial discretization scheme, and angular approximation, a parametric study wascarried out for this problem.

The comparisons between the predictions of heat fluxes along the centerlineobtained with finer subdivisions and Monte Carlo solutions are shown in Figure 6.The spatial discretization scheme and order of approximation used in this analysisare DSS012 scheme and S8, respectively. As can be seen from the figure, satisfactoryagreement is achieved with 25625625 subdivision.

The effect of spatial discretization schemes on the predictions was analyzed byrunning the program with two-point upwind (DSS012) and three-point upwind(DSS014) schemes for S8 and 25625625 control volumes. Comparisons are shownin Figure 7. As can be seen from the figure, DSS012 produces good agreement,whereas DSS014 results in oscillatory solutions. Similar behavior was also observedwhen comparisons are carried out on the same problem with anisotropically scat-tering media. These findings are also in agreement with those of Yucel [2] for purelyscattering media contained in a 2-D enclosure with one hot wall surrounded by coldwalls and with an optical thickness of unity. The similarity between the two test casesand the failure of DSS014 in both problems are noteworthy. The resulting oscilla-

Figure 6. Effect of spatial discretization on the dimensionless heat flux predictions of MOL solution of

DOM (DSS012 scheme, S8).

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tions due to the use of higher-order discretization scheme may be considered to bedue to the use of the most severe condition, i.e., strong temperature gradient in apurely scattering medium, as no such oscillations were encountered in absorbing,emitting, scattering media bounded by hot walls (Section 3.2 of this study, [6]).Therefore for the trials that follow, DSS012 was selected.

Effect of order of approximation on the predictions was studied by running theprogram for S4;S6;S8 with LSO (Level Symmetric Odd) [16] and S10 with LSH(Level Symmetric Hybrid) [21] quadratures for 25625625 subdivision and DSS012spatial discretization scheme. Figure 8 shows comparison between heat fluxes pre-dicted by the MOL solution of the DOM and the Monte Carlo solutions. As can beseen from the figure, satisfactory agreement is obtained except for the lowest order ofapproximation employed (S4). For the applications that follow, S10 order ofapproximation was selected in order to minimize ray effects in anisotropicallyscattering media.

Having selected the parameters for spatial and angular discretizations, theMOL solution of the DOM was applied to the prediction of dimensionless heat fluxdistributions along the centerline of the enclosure for anisotropically scatteringmedia with forward and backward scattering phase functions. Figure 9 displays the

Figure 7. Effect of order of spatial discretization on the dimensionless heat flux predictions of MOL

solution of DOM (S8 with 25625625 grids).

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comparisons between the predictions and Monte Carlo solutions for the isotropicallyscattering case and anisotropically scattering cases with phase functions F1, F2, B1,and B2, respectively. As can be seen from the figure, discrepancies between the heatflux predictions of the MOL solution of the DOM and the Monte Carlo methodincrease from the cold surface toward the hot surface. The maximum and averagepercentage errors for different phase functions are given in Table 4. As can be seenfrom the table, errors in the predictions (< 5%) are of the same order of magnitudefor isotropic and anisotropic phase functions.

3.3.2. Absorbing, emitting, scattering medium with symmetricboundary conditions. The physical situation under consideration is a cubicalenclosure with side length L0 containing isothermal, gray, absorbing, emitting,and scattering medium at temperature T0. The dimensionless extinctioncoefficient, b�ð¼b L0Þ, and scattering albedo, o ð¼s�

s=b�Þ, are 10 and 0.5,

respectively. The boundaries are cold and black.Performance of the MOL solution of the DOM for this problem was investi-

gated by comparing its predictions for dimensionless radiative heat flux Q�z along the

centerline of a wall (x;L0=2;L0) and dimensionless incident radiation G� along the

Figure 8. Effect of order of approximation on the dimensionless heat flux predictions of MOL solution of

DOM (DSS012 scheme with 25625625 grids).

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centerline of the enclosure (x;L0=2;L0=2) with those of Monte Carlo solutions for15615615 subvolumes and 36106 energy bundles emitted from each subvolume.The angular and spatial discretization parameters selected in Section 3.3.1 were alsoutilized for this problem. Figure 10 illustrates the heat flux and incident radiation

Figure 9. Comparison between predictions of MOL solution of DOM and Monte Carlo method for

dimensionless heat flux profiles along the certerline for isotropically and anisotropically scattering media

(S10 and DSS012 with 25625625 grids).

Table 4. Comparative testing between dimensionless heat flux predictions of Monte Carlo method

and MOL solution of DOM for various phase functions for the cubical enclosure problem with purely

scattering medium

Phase function

Avg. absolute

% error aMax. absolute

% error a

Isotropic 2.45 4.43

F1 2.31 4.38

F2 2.47 4.82

B1 2.36 4.31

B2 2.14 4.23

aAbsolute percentage error¼ðjQ�MOL �Q�

MCj=Q�MC� 100.

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Figure 10. Comparison between predictions of MOL solution of DOM and Monte Carlo method for

dimensionless heat flux and incident radiation profiles for isotropically and anisotropically scattering

media (S10 and DSS012 with 25625625 grids).

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predictions of the MOL solution of the DOM and Monte Carlo method for variousphase functions.

The discrepancies between the solutions of the two methods are given incondensed form in Table 5. As can be seen from the figures and the table, the MOLsolution of the DOM predictions are in reasonable agreement with those of theMonte Carlo method, and the average errors are of the same order of magnitude forisotropic and anisotropic cases.

4. CONCLUSION

The predictive accuracy of a radiation code based on the MOL solution of theDOM for 3-D radiative heat transfer in rectangular enclosures containing gray,absorbing, emitting, and scattering media was developed and tested by applying it to(1) a 1-D problem with purely anisotropically scattering medium in radiative equi-librium, (2) a 3-D problem containing isotropically scattering medium with a uni-form, nonradiative source term, (3) a cubical enclosure problem with purelyisotropically and anisotropically scattering medium and nonsymmetric boundaryconditions, (4) a cubical enclosure problem with absorbing, emitting, isotropically,and anisotropically scattering medium and symmetric boundary conditions andvalidating its predictions against exact solutions, the zone method, or Monte Carlosolutions, reported in the literature.

Effects of order of approximation (S4;S6;S8;S10Þ and spatial discretizationscheme (two- and three-point upwind schemes) on the predictive accuracy of gridindependent solutions were investigated.

Comparisons show that

� The MOL solution of the DOM is a promising technique for prediction ofradiative heat transfer in 3-D rectangular enclosures containing purelyscattering or absorbing, emitting, scattering media with isotropic scatteringproperties.

� The same order of magnitude of accuracy is achieved in both isotropicallyand anisotropically scattering media.

Table 5. Comparative testing between dimensionless heat flux and incident radiation predictions of Monte

Carlo method and MOL solution of DOM for various phase functions for the cubical enclosure problem

with absorbing, emitting, scattering medium

Heat flux (Q�zÞ Incident radiation (G�)

Avg. abs. Max. abs. Avg. abs. Max.abs.

Phase function % errora % errora % errora % errora

Isotropic 2.86 3.63 2.52 3.83

F1 2.01 3.28 2.46 5.23

F2 2.54 3.10 2.51 4.64

B1 2.86 3.79 2.32 3.50

B2 2.89 3.68 1.95 2.43

aAbsolute percentage error¼ðjQ�MOL �Q�

MCj=Q�MC� 100.

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� In problems involving pure scattering with strong temperature gradients,oscillations may be encountered when a higher-order upwind spatial dis-cretization scheme is utilized.

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