nodal synthetic kernel (n-skn) method for solving radiative heat transfer problems in one- and...

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Journal of Quantitative Spectroscopy &Radiative Transfer

Journal of Quantitative Spectroscopy & Radiative Transfer 129 (2013) 214–235

0022-40http://d

n CorrE-m

tmesut@

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Nodal synthetic kernel (N-SKN) method for solving radiativeheat transfer problems in one- and two-dimensionalparticipating medium with isotropic scattering

Zekeriya Altaç a,n, Mesut Tekkalmaz b

a Department of Mechanical Engineering, Eskisehir Osmangazi University, Batı-Meşelik 26480, Eskisehir, Turkeyb Department of Materials and Metallurgy Engineering, Eskisehir Osmangazi University, Eskisehir, Turkey

a r t i c l e i n f o

Article history:Received 6 April 2013Received in revised form26 May 2013Accepted 15 June 2013Available online 22 June 2013

Keywords:Radiative transferParticipating mediumNodal SKN methodTwo-dimensionalRectangular geometry

73/$ - see front matter & 2013 Elsevier Ltd.x.doi.org/10.1016/j.jqsrt.2013.06.017

esponding author. Tel.: +90 222 2393750; faail addresses: [email protected] (Z. Altaç),ogu.edu.tr, [email protected] (M. Tekkalma

a b s t r a c t

In this study, a nodal method based on the synthetic kernel (SKN) approximation isdeveloped for solving the radiative transfer equation (RTE) in one- and two-dimensionalcartesian geometries. The RTE for a two-dimensional node is transformed to one-dimensional RTE, based on face-averaged radiation intensity. At the node interfaces,double P1 expansion is employed to the surface angular intensities with the isotropictransverse leakage assumption. The one-dimensional radiative integral transfer equation(RITE) is obtained in terms of the node-face-averaged incoming/outgoing incident energyand partial heat fluxes. The synthetic kernel approximation is employed to the transferkernels and nodal-face contributions. The resulting SKN equations are solved analytically.One-dimensional interface-coupling nodal SK1 and SK2 equations (incoming/outgoingincident energy and net partial heat flux) are derived for the small nodal-mesh limit.These equations have simple algebraic and recursive forms which impose burden onneither the memory nor the computational time. The method was applied to one- andtwo-dimensional benchmark problems including hot/cold medium with transparent/emitting walls. The 2D results are free of ray effect and the results, for geometries of afew mean-free-paths or more, are in excellent agreement with the exact solutions.

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1. Introduction

In the design of high-temperature equipment involvingparticipating medium such as combustion chambers, fur-naces etc., an accurate estimation of the radiative intensity,incident energy and heat flux is essential. In these pro-blems, the radiative transfer equation (RTE) is the basicequation coupled with the energy equation to be solved tocompute the radiative energy distribution and the heat flux.Unfortunately, the RTE contains not only spatial but alsoangular variables. This inherent nature of the RTE makes it amulti-variable problem even for the simplest cases.

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z).

In the last two decades, with rapidly developing computertechnology, the most widely used method to solve the RTEemerged to be the Discrete Ordinates Method (DOM). On theother hand, in multi-dimensional geometries, the DOM andits derivative methods are plagued with the so-called theray-effect caused due to angular discretization schemes [1,2].The ray-effect which is significant especially near the enclos-ing boundaries is observed as non-physical oscillations in thesolutions. The diffusion approximation (or P1 approximation)is relatively simple and compatible with the finite-differenceand finite-volume methods, but it is only accurate in opticallythick medium [3–6].

The RTE could be converted into an integral form byintegrating the RTE over the solid angle in which theangular dependency is completely eliminated. The exactsolutions are generally obtained from the grid independentnumerical solutions of the radiative integral transfer

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Nomenclature

a, b geometry dimensions (m)CG,i,Cq,i coefficients of the interface nodal SKN

equationsDn artificial diffusion coefficient (¼ μ2n=β)En(x) nth order exponential integral functionsFG, Fq node surface contribution functions for inci-

dent energy and heat fluxG incident energyGn SKN equation for the nth componentI radiation intensityLx,Ly node transverse leakage termsN order of the SKN approximationq radiative heat fluxQ internal heat sourceR reflectivityS isotropic medium sourceT transmissivitywn,μn Gauss quadraure absicas and weightsX,Y coordinate variables (m)x,y node-based dimensionless coordinatesx0,y0 optical dimensions (¼aβ, ¼bβ)β extinction coefficient (m−1)φ azimuth

κ absorption coefficient (m−1)μ direction cosiness scattering coefficient (m−1)τ0 optical thickness (¼βa)ω0 scattering albedo (¼ss=β)Γ nodal-face functions defined by Eqs. (17), (18),

(22) and (23)

Subscripts

E East faceN North faceS South facex,y direction/dimensionW West face

Superscripts

+ forward direction (West to East or South toNorth)

− backward direction (East to West or Northto South)

k node indices

Z. Altaç, M. Tekkalmaz / Journal of Quantitative Spectroscopy & Radiative Transfer 129 (2013) 214–235 215

equations (exact RITEs). However, exact solutions generallyinvolve idealized problems of simple geometries [7–21].The main shortcomings of the RITE methods are: (i) themethod is restricted to isotropic and linearly isotropicscattering; (ii) the radiative transfer kernels and the inci-dent energy from the boundary contributions possessmathematical singularities, and they must be derivedand/or treated analytically specifically for each geometry;(iii) the resulting system of linear equations from thenumerical solutions are dense matrices which imposerestrictions on the computer memory and/or the computa-tion time. Therefore, the exact RITE methods are notpractical to be used in actual design problems; they arerather sought and used for benchmarking purposes.

The SKN method which is also an approximation wasdeveloped directly from the RITE as an alternative to solvethe radiative transfer problems. The method was success-fully employed to one-dimensional plane parallel, solidand hollow cylindrical and spherical, and two-dimensionalcartesian and cylindrical geometries [22–34]. In thismethod, the radiative integral transfer kernels are approxi-mated by a sum of exponentials—synthetic kernels. Thenthe RITE is reducible to a set of coupled ODEs in 1D or PDEsin 2D or 3D geometries which are referred to as the SKN

equations. The SKN equations do not involve angularvariables and have the form of the general diffusionequation, subjected to mixed type boundary conditions.Since one has to deal with only ODEs or PDEs, the SKN

equations could be easily solved by the conventionalnumerical methods involving finite-element, finite-difference or control-volume approaches yielding signifi-cant reduction in computation time over the numerical

solution of the RITEs. Even though the ray effect problem iscompletely eliminated, just as formulating the RITEs, theSKN method also requires analytical derivations for thesurface contributions of the incident energy and heat fluxfor a specific geometry. These boundary integrals are alsosingular in nature, and these singularities require for mostpart analytical and/or numerical treatments. This issueremains as the major handicap of the SKN method whichmakes it unpractical to employ to complex irregulargeometries.

The neutron transport equation (the Boltzmann's equa-tion) is also used in nuclear reactor design to estimate theneutron distribution and leakage. The methods such as PN,SN and Monte Carlo used widely in reactor physics topredict the neutron distribution also found application inthe thermal radiation analysis. In the last three-decades,advances in computer hardware and improvements insolution algorithms have made it possible to solve increas-ingly complicated problems. However, in reactor physicsbesides these advances, practical limitations on computerstorage and execution time generally prohibit the explicitmodeling of each fuel pin in a nuclear reactor. Instead ofcomputing the neutron distribution pin-by-pin, the fuelassemblies are homogenized as regions consisting of entirefuel assemblies. Thus computationally more efficient solu-tion techniques are oriented towards the determination ofthe neutron flux averaged over each homogeneous regionor node. These methods have become known as the nodalmethods. The basic idea in the nodal methods is to solve theone-dimensional transport equation for each node withsome high-order methods (SN, PN etc.) while node bound-ary fluxes are constrained to a simple form.

Fig. 1. Schematic of a nodal element with illustration of nodal interfacevariables.

Z. Altaç, M. Tekkalmaz / Journal of Quantitative Spectroscopy & Radiative Transfer 129 (2013) 214–235216

Lawrence [35] was first to report on a nodal methodbased on the integral transport equation. In his work, theinterface coupling nodal equations were derived using theintegral transport theory along with the isotropic leakageapproximation. The node face-averaged incoming/out-going fluxes were used in connection with the node BCs.The neutron flux was approximated with up to seconddegree polynomials in which the coefficients were com-puted using a moment method [35,36]. His results exhib-ited significant improvement over the diffusion theory.However, for mesh widths of the order of a mean-free-path, 1D nodal solution was generally much more accuratethan its small-mesh limit formulations. Gelbard et al. [37]reported that in the small nodal-mesh limit, the nodaltransport equations, the double P1 approximation used forthe boundaries yielded worse results than double P0approximation. They proposed to replace the exponentialintegral functions that appear in the node-face contribu-tions with the Gauss quadratures equivalents (known alsoas double P1 quadratures) which in our case correspond tothe SK2 quadratures. They also reported that using theapproximate exponential integral functions yielded moreaccurate results than that standard DIF3D code results forall mesh widths [37].

The nodal method presented here is primarily based onLawrence's nodal integral equation formulation [35]. How-ever, instead of using a moment type approach to predictthe incident energy and the heat flux within a specifiednode, we have adapted and extended the SKN methodol-ogy to not only the integral equation kernels but also tothe node-face contributions of the nodal integral equa-tions. In the same manner with the standard SKN metho-dology, the integral equation for the transverse-averagedincident energy is reducible to a set of coupled ODEs forwhich mathematical boundary conditions and analyticalsolutions could be found. Then 1D interface-couplingnodal SKN equations are obtained in a simple recursivealgebraic form. In this paper, the computational accuracyand efficiency of the nodal SKN method using benchmarkproblems in one- and two-dimensional homogeneousmedia are investigated and reported.

2. Mathematical formulation

Two dimensional radiative transfer equation with iso-tropic scattering and uniform properties within a two-dimensional node (X,Y)∈Vk is defined by

μ∂I∂X

þffiffiffiffiffiffiffiffiffiffiffi1−μ2

pcos φ

∂I∂Y

þ β IðX;Y ; μ;φÞ ¼ 14π

SðX;YÞ;

ðVk : Xk ≤X≤Xkþ1; Yk ≤Y ≤Ykþ1Þ ð1ÞWe transform the RTE to a node-based coordinate

system:

μ

ΔXk

∂Ik

∂xþ

ffiffiffiffiffiffiffiffiffiffiffi1−μ2

pcos φ

ΔYk

∂Ik

∂yþ β Ikðx; y; μ;φÞ ¼ 1

4πSkðx; yÞ;

Vk : −12≤x≤

12;−

12≤y≤

12

� �ð2Þ

where ΔXk ¼ Xkþ1−Xk and ΔYk ¼ Ykþ1−Yk and x and y arethe dimensionless coordinates in terms of the node mesh

spacing; x¼ −1=2þ ðX−XkÞ=ΔXk, y¼−1=2þ ðY−YkÞ=ΔYk.Integrating Eq. (2) over y∈(−½,½) yields one-dimensionalequation for kth node

μ

ΔXk

∂Ik

∂xðx; μ;φÞ þ βkIkðx; μ;φÞ ¼ 1

4πSkðxÞ−Lkyðx; μ;φÞ;

x∈− �12;12

� �ð3Þ

where

Ikðx; μ;φÞ ¼Z 1=2

y ¼ −1=2Ikðx; y; μ;φÞdy ð4Þ

SkðxÞ ¼Z 1=2

y ¼ −1=2Skðx; yÞdy¼QkðxÞ þ sksG

kðxÞ ð5Þ

Lkyðx; μ;φÞ ¼1

ΔYk

Z 1=2

y ¼ −1=2

∂∂y


pcos φ Ikðx; y; μ;φÞ

� �dy

ð6Þ

Here Sk(x) is the isotropic source term with contributionsdue to internal sources and Lkyðx; μ;φÞ is the transverseleakage term. The incoming and outgoing face-averageradiation intensity, the incident energy and the heat fluxinto and from kth node are illustrated in Fig. 1 where thesubscripts denote (E)ast, (W)est, (N)orth and (S)outh face.

Although the analysis is based on uniform propertiesand isotropic scattering assumption, anisotropic scatteringcould also be easily incorporated into the source term S(X,Y) in Eq. (1). On the other hand, Eq. (3) resulting from theforegoing analysis also allow inhomogeneous medium tobe easily incorporated since medium properties areexpressed as average values (βk; sks etc.) of the kth node.

For simplicity, k subscripts/superscripts used for thenode identification will be dropped. Hereafter for theincoming and outgoing quantities into and from a node,we adapt the notations presented in Fig. 1. Next we assumethe incoming angular incidence at nodal (East or West)faces can be approximated by double P1 approximation;


that is, the radiation intensity is assumed to behave as

Iðμ;φÞ ¼a1 þ a2μ; μ40b1 þ b2μ; μo0

(ð7Þ

where a1≠b1, a2≠b2. Introducing the nodal surface variablesasR 1

μ ¼ 0

R 2πφ ¼ 0 I

þW ðμ;φÞdφ dμ¼ Gþ

W ;R 0μ ¼ −1

R 2πφ ¼ 0 I

−E ðμ;φÞdφ dμ¼ G−

E ;R 1μ ¼ 0

R 2πφ ¼ 0 I

þW ðμ;φÞdφμ dμ¼ qþx;W ;R 0

μ ¼ −1

R 2πφ ¼ 0 I

−E ðμ;φÞdφμ dμ¼−q−x;E;

ð8Þ

and using Eq. (7) in Eq. (8) to solve for a's and b's for Eastand West faces separately, the nodal face intensities yieldfor the west wall (0≤μ≤1; 0≤φ≤2π)

IþW ðμ;φÞ ¼ 12π

ð4GþW−6qþx;W Þ þ 1

2πð−6Gþ

W þ 12qþx;W Þμ; ð9Þ

and for the east wall (−1≤μ≤0; 0≤φ≤2π)

I−E ðμ;φÞ ¼12π

ð4G−E−6q

−x;EÞ−

12π

ð−6G−E þ 12q−x;EÞμ; ð10Þ

Thus two equations for the outgoing and incomingintensity are obtained in terms of the face-average incom-ing and outgoing incident energy and the heat flux at thenode faces. It should be noted that Eqs. (9) and (10) do notaccount for the azimuthal dependence.

Now solving Eq. (3) for Iðx; μ;φÞ and μ40 yields

Iðx; μ;φÞ ¼ IþW ðμ;φÞe−βðð1=2ÞþxÞ=μ

þΔXμ

Z x

−1=2

14π

Sðx′Þ−Lyðx′; μ40;φÞ� �

e−βðx−x′Þ=μdx′

ð11Þwhereβ¼ΔX β and IþW ðμ;φÞ ¼ Ið−1

2; μ;φÞ is the incomingface-average radiation intensity of the west face. Similarlysolving Eq. (3) for Iðx; μ;φÞ and replacing μ with −μ, weobtain

Iðx;−μ;φÞ ¼ I−E ð−μ;φÞe−βðð1=2Þ−xÞ=μ

þΔXμ

Z 1=2

x

14π

Sðx′Þ−Lyðx′; μ40;φÞ� �

e−βðx′−xÞ=μdx′

ð12Þwhere I−E ð−μ;φÞ ¼ Ið12;−μ;φÞ is the face-average incomingradiation intensity of the east face.

The incident energy, at a location x inside the node, canthen be obtained from

GðxÞ ¼ GþðxÞ þ G−ðxÞ

¼Z 2π

φ ¼ 0dφ

Z 1

μ ¼ 0Iðx; μ40;φÞdμþ

Z 0

μ ¼ −1Iðx; μo0;φÞdμ

!ð13Þ

which yields

GðxÞ ¼Z 2π

φ ¼ 0

Z 1

μ ¼ 0IþW ðμ;φÞe−βðð1=2ÞþxÞ=μdμ

þZ 1

μ ¼ 0I−E ð−μ;φÞe−βðð1=2Þ−xÞ=μdμ

!dφ

þΔX2

Z 1=2

x′ ¼ −1=2ðSðx′Þ−Lyðx′ÞÞE1ðβjx−x′jÞdx′ ð14Þ

where E1(x) is the first order exponential integral function.Substituting Eqs. (11) and (12) in Eq. (14) and usingR 1

0 μn−2expð−x=μÞdμ¼ EnðxÞ, we obtain the one-dimensionalintegral transfer equation

GðxÞ ¼ FGðxÞ þΔX2

Z 1=2

x′ ¼ −1=2ðSðx′Þ−Lyðx′ÞÞE1ðβjx−x′jÞdx′ ð15Þ

where G(x) is the node transverse-average incident energyfunction,

FGðxÞ ¼ Γ1ðxÞGþW þ Γ1ð−xÞG−

E þ Γ2ðxÞqþx;W þ Γ2ð−xÞq−x;E ð16Þ

Γ1ðxÞ ¼ 4E2 βð12 þ xÞh i

−6E3 βð12 þ xÞh i

; ð17Þ

Γ2ðxÞ ¼−6E2 βð12 þ xÞh i

þ 12E3 βð12 þ xÞh i

; ð18Þ

and En(x) is the nth order exponential integral function.Similarly the node transverse-average net radiative

heat flux along the x-direction is obtained from

qxðxÞ ¼ qþx ðxÞ−q−x ðxÞ

¼Z 2π

φ ¼ 0dφ

Z 1

μ ¼ 0Iðx; μ40;φÞμ dμ

þZ 0

μ ¼ −1Iðx; μo0;φÞμ dμ

!ð19Þ

Using Eqs. (11) and (12) in Eq. (19) yields

qxðxÞ ¼ FqðxÞ þ ΔX2

Z 1=2

x′ ¼ −1=2ðSðx′Þ−Lyðx′ÞÞsgnðx−x′ÞE2ðβjx−x′jÞdx′

ð20Þwhere sgn is the sign function and

FqðxÞ ¼ Γ3ðxÞGþW þ Γ3ð−xÞG−

E þ Γ4ðxÞqþx;W þ Γ4ð−xÞq−x;E ð21Þ

Γ3ðxÞ ¼ 4E3 βð12 þ xÞh i

−6E4 βð12 þ xÞh i

; ð22Þ

Γ4ðxÞ ¼−6E3 βð12 þ xÞh i

þ 12E4 βð12 þ xÞh i

ð23Þ

Evaluating the outgoing incident energy and the partialheat flux at the East face of the node by setting x¼1/2 atEqs. (15) and (19) and noting that Gð1=2Þ ¼ Gþ

E þ G−E ,

Γ1ð−1=2Þ ¼ Γ4ð−1=2Þ ¼ 1 and Γ2ð−1=2Þ ¼ Γ3ð−1=2Þ ¼ 0,after algebraic manipulations one obtains

GþE ¼ ΔX

2

Z 1=2

x′ ¼ −1=2ðSðx′Þ−Lyðx′ÞÞE1ðβ 1

2−x′� �Þdx′

þCG;1ðβÞGþW þ CG;2ðβÞqþx;W ð24Þ

and

qþx;E ¼ΔX2

Z 1=2

x′ ¼ −1=2ðSðx′Þ−Lyðx′ÞÞE2ðβ 1

2−x′� �Þdx′

þCq;1ðβÞGþW þ Cq;2ðβÞqþx;W ð25Þ

where

CG;1ðβÞ ¼ 4E2ðβÞ−6E3ðβÞ; CG;2ðβÞ ¼ −6E2ðβÞ þ 12E3ðβÞ ð26Þ

Cq;1ðβÞ ¼ 4E3ðβÞ−6E4ðβÞ; Cq;2ðβÞ ¼−6E3ðβÞ þ 12E4ðβÞ ð27Þ


The accompanying equations for the outgoing quanti-ties from the west face (G−

W ; q−x;W ) are obtained in a similarfashion. These equations have basically the same formwith Eqs. (24) and (25); however, subscript “E” is replacedwith “W”, and “+” sign is interchanged with “−”. All fourequations are referred to as x-direction interface couplingnodal equations.

The incident energy and the transverse leakage whichare present under the integral sign of the interface nodalequations are unknown. Information on the incident energyfunction, G(x), and the transverse leakage, Ly(x), for the kthnode are needed. Thus, some approximations to thesequantities are required. Lawrence [35] uses up to secondorder polynomial approximations for the unknown quanti-ties. The coefficients of the polynomials are computed bymatching the weighted moments of Eq. (15). Lawrence wasable to obtain very good results for coarse nodes; however,as the nodal-mesh was further refined, the solutions wan-dered in the wrong direction. Gelbard et al. [37] studied thisphenomenon analytically and numerically in detail. Forsmall arguments, the exponential integral functions yieldE1(x)≅x−(γ+ln x), E2(x)≅1+x(γ−1+ln x) and E3(x)≅1/2−x+x2/4� (3γ−2−2ln x) in which singularity is present as thelogarithm function. They concluded that the problem withthe small node-mesh limit originated from this singularity ofthe exponential integral functions, and as a remedy theyrecommended the use of double P1 quadratures for theexponential integral functions in Eqs. (16) and (21). Gel-bard's this particular finding was also taken into account inour analysis which suited perfectly to the spirit of thesynthetic kernel method.

3. The synthetic kernel (SKN) method

In this study, the SKN method is employed to Eqs. (15)and (20) to obtain the incident energy and the radiativeheat flux (or the transverse leakage) distribution, insteadof using polynomial approximations. The SKN methodrequires the placement of the RITE kernels with thesynthetic kernels (sum of exponentials). The simplestway of obtaining the synthetic kernels is to replace theexponential integral functions with their Gaussian sum asfollows [23]:

EkðxÞ ¼Z 1

μ ¼ 0μk−2 e−x=μdμ≅ ∑

N

n ¼ 1wnμ

k−2n e−x=μn for k≥1;

ð28Þwhere wn; μn are the Gauss quadratures in the prescribed(0,1) interval. Upon substituting these approximations intoEqs. (15) and (20), we define

GnðxÞ ¼ΔX2μn

Z 1=2

x′ ¼ −1=2ðSðx′Þ−Lyðx′ÞÞe−βjx−x′j=μndx′ ð29Þ

and

qx;nðxÞ ¼ΔX2

Z 1=2

x′ ¼ −1=2ðSðx′Þ−Lyðx′ÞÞsgnðx−x′Þe−βjx−x′j=μndx′

ð30ÞAt this point, when evaluating nodal-face contributions

given by Eqs. (16) and (21), however, we will choose a

different route from the standard SKN practice in whichthe boundary terms were kept in the exact forms. Here, wealso extend the synthetic approximations for FG(x) andFq(x) functions by replacing the interface functions (Γ' s)with the approximated expressions. Then, Eqs. (15) and(20) for x∈ð−1=2;1=2Þ become

GðxÞ ¼ eFGðxÞ þ ∑N

n ¼ 1wnGnðxÞ ð31Þ

qxðxÞ ¼ eF qðxÞ þ ∑N

n ¼ 1wnqx;nðxÞ ð32Þ

To obtain the approximate interface functions, theexponential integral functions are replaced by Eq. (28)using the second order Gauss quadratures (μ1;2 ¼ ð37

ffiffiffi3

pÞ=

6, w1;2 ¼ 1=2) which yield

eΓ1ðxÞ ¼ e−3βðð1=2ÞþxÞðcoshðffiffiffi3

pβð12 þ xÞÞ−

ffiffiffi3

psinhð

ffiffiffi3

pβð12 þ xÞÞÞ;

eΓ2ðxÞ ¼ 2ffiffiffi3

pe−3βðð1=2ÞþxÞsinhð

ffiffiffi3

pβð12 þ xÞÞ;eΓ3ðxÞ ¼− 1ffiffi

3p e−3βðð1=2ÞþxÞsinhð

ffiffiffi3

pβð12 þ xÞÞ;eΓ4ðxÞ ¼ e−3βðð1=2ÞþxÞðcoshð

ffiffiffi3

pβð12 þ xÞÞ þ

ffiffiffi3

psinhð

ffiffiffi3

pβð12 þ xÞÞÞ;

ð33ÞThen, the coefficients defined by Eqs. (26) and (27)

becomeeCG;1ðβÞ ¼ e−3βðcoshðffiffiffi3

pβÞ−

ffiffiffi3

psinhð

ffiffiffi3

pβÞÞ;eCG;2ðβÞ ¼ 2

ffiffiffi3

pe−3βsinhð

ffiffiffi3

pβÞ ð34Þ

eCq;1ðβÞ ¼− 1ffiffi3

p e−3βsinhðffiffiffi3

pβÞ;eCq;2ðβÞ ¼ e−3βðcoshð

ffiffiffi3

pβÞ þ

ffiffiffi3

psinhð

ffiffiffi3

pβÞÞ ð35Þ

Now to develop the SKN equations, we first take thederivative of Eq. (29) with respect to x and inspecting Eq.(30), we find

qx;nðxÞ ¼−DndGn

dxð36Þ

where Dn ¼ μ2n=β. Taking the derivative of Eq. (36) againwith respect to x yields

−Dnd2Gn

dx2þ β GnðxÞ ¼ΔXðSðxÞ−LyðxÞÞ ð37Þ

Combining Eqs. (5) and (31), the SKN equations for thekth node are obtained as

−Dnd2Gn

dx2þ β GnðxÞ−ss ∑

N

m ¼ 1wmGmðxÞ

¼ΔXðQ ðxÞ−LyðxÞÞ þ sseFGðxÞ; x∈ð−12 ;

12Þ ð38Þ

where ss ¼ΔX ss.Evaluating Eqs. (29) and (30) at the east and west faces

(for x¼71/2), the boundary conditions for the SKN

equations are obtained as

7qx;nð712 Þ−μnGnð71

2Þ ¼ 0 ð39ÞOnce the SKN solutions are obtained and substituted

into Eqs. (24) and (25), we obtain

GþE ¼ eCG;1ðβÞGþ

W þ eCG;2ðβÞqþx;W þ ∑N

n ¼ 1wnGnð12Þ ð40Þ


and

qþx;E ¼ eCq;1ðβÞGþW þ eCq;2ðβÞqþx;W þ ∑

N

n ¼ 1wnqx;nð12Þ ð41Þ

Under prescribed boundary conditions—Eq. (39)—ana-lytical solution of the SKN equations—Eq. (38)—for the kthnode can be obtained in a straightforward manner. Theanalytical solutions of the SKN equations involve lengthyexpressions; to save space, these expressions are notpresented here. Besides, computationally it would not beappropriate to incorporate the analytical solutions into theinterface coupling nodal SKN equations due to their lengthand excessive computational efforts. Instead, in order toobtain equations valid in the small nodal-mesh limit, wechoose to expand the analytical solutions and the coeffi-cients of Eqs. (40) and (41) into Taylor series near ΔX ¼ 0and truncate them to the order of accuracy desired (seeAppendix A).

For one-dimensional plane geometry, the transverseleakage, Ly, is zero; however, in two dimensional rectan-gular geometries, the transverse leakage must also betaken into account. So, next, we turn our attention to thetransverse leakage term. Using Eq. (6), the transverseleakage can be integrated out as

Lyðx; μ;φÞ ¼1ΔY

ðffiffiffiffiffiffiffiffiffiffiffi1−μ2

pcos φ Iðx; 12 ; μ;φÞ−


pcos φ Iðx;−1

2; μ;φÞÞ

ð42ÞNoting that Ωy ¼


pcos φ and

RΩIðx; y;ΩÞ Ωy dΩ¼

qyðx; yÞ, the simplest approximation to the angular depen-dence of the transverse leakage is to assume that it isisotropic; that is,Z 2π

φ ¼ 0Lyðx; μ;φÞdφ≡Lyðx; μÞ ¼

12LyðxÞ ¼

12qyðx; 12 Þ−qyðx;−1

2ÞΔY

ð43ÞHowever, Eq. (43) is not yet suitable to be used in

connection with the interface-coupling nodal equationssince the net heat flux qy is not face-average quantity forthe North and South faces. Thus, Eq. (43) can be related tothe face-averaged partial heat fluxes by considering theaverage over x as

Ly≡Z 1=2

x ¼ −1=2LyðxÞdx¼

qy;N−qy;SΔY

¼ 1ΔY

ðqþy;N−q−y;N−ðq−y;S−qþy;SÞÞ

ð44ÞAs presented in Appendix A, the interface coupling

nodal SK2 equations of the third order truncation error, forforward (West to East sweep) direction, are obtained as

GþE ¼ 3ΔX

2 1−2β þ 3ss2

� �ðQ−LyÞ

þ 1−6β þ 15β2 þ 3ss

2 1−5β þ 3ss2

� �h iGþW

þ 6β−18β2 þ 9

2βssh i

qþx;W þ 3ss2 1−5β þ 3ss

2

� �G−E

þ92βss q−x;E þ O ðΔXÞ3

h ið45Þ

and

qþx;E ¼ ΔX2 1−3κ

2

ðQ−LyÞ þ −β þ 3β2 þ ss

2 1−3ss−9κ2

h iGþW

þ 1−3β2 þ 3ss β

2

h iqþx;W þ ss

2 1−3ss−9κ2

G−E

þ3ss β2 q−x;E þ O ðΔXÞ3

h ið46Þ

where κ¼ΔX κ. Noting that the quantities likeβ2 ¼ ðΔXÞ2β2, ssβ¼ ðΔXÞ2ssβ and so on are second order,

the interface coupling nodal SK2 equations with secondorder truncation errors can be written as

GþE ¼ 3ΔX

2 ðQ−LyÞ þ 1−6β þ 3ss2

� �GþW þ 6β qþx;W

þ3ss2 G−

E þ O ðΔXÞ2h i

ð47Þ

and

qþx;E ¼ ΔX2 ðQ−LyÞ þ −β þ ss

2

� �GþW þ qþx;W

þss2G

−E þ O ðΔXÞ2

h ið48Þ

The interface coupling nodal SK2 equations for theincident energy and the partial heat flux for the backwarddirection (from East to West) are obtained in a similarfashion. These equations have the same form as those ofthe forward direction; we could simply obtain the equa-tions for the backward direction by interchanging “E” with“W” and “+” signs with “−” sign.

It should be pointed out that the interface-nodal equationsin the small nodal-mesh limit preserve the conservation ofenergy. To illustrate this, we use Eq. (48) and the accompany-ing equation for outgoing heat flux for the West face:

q−x;W ¼ ΔX2 ðQ−LyÞ þ −β þ ss

2

� �G−E

þq−x;E þ ss2G

þW þ O ðΔXÞ2

h ið49Þ

The internal source and transverse leakage present inthese equations are evaluated at the node center. Addingup Eqs. (48) and (49) and rearranging yields

qþx;E−q−x;E−ðqþx;W−q−x;W Þ

¼ΔXðQ−LyÞ−κ ΔXðG−E þ Gþ

W Þ þ O ðΔXÞ2h i

ð50Þ

Noting that for sufficiently small ΔX, we may writeG−E þ Gþ

W ¼ GðX;YÞ, Q ¼Q ðX;YÞ and we note thatLy ¼ ðqyðX;Y þ ðΔY=2ÞÞ−qyðX;Y−ðΔY=2ÞÞÞ=ΔY . On the otherhand, the left hand side of Eq. (50) can be written in termsof net radiative heat fluxes as qx;E ¼ qþx;E−q

−x;E ¼ qxðX þ

ðΔX=2Þ;YÞ and qx;W ¼ qþx;W−q−x;W ¼ qxðX−ðΔX=2Þ;YÞ. ThenEq. (50) may be expressed as

qxðX þ ðΔX=2Þ;YÞ−qxðX−ðΔX=2Þ;YÞΔX

¼Q ðX;YÞ− qyðX;Y þ ðΔY=2ÞÞ−qyðX;Y−ðΔY=2ÞÞΔY

−κ GðX;YÞ þ OðΔXÞ ð51ÞIn the small node-mesh limit of Eq. (51) ðΔX-0;

ΔY-0Þ, one arrives at

∂qx∂X

þ ∂qy∂Y

¼ div q¼Q ðX;YÞ−κ GðX;YÞ ð52Þ

which is basically the conservation of energy. In the aboveproof, it is sufficient to use the second order truncatedequations since O[(ΔX)2] and higher order terms will alsogo to zero.

Solution algorithm. In the plane-parallel geometry, thetransverse leakage term is zero; Ly¼0. At the boundingEast and West walls, either the incoming or the outgoing


radiation intensity is specified. However, for the symmetryboundary condition, an initial guess is required. Thus, forthe nodes adjacent to the walls, in addition to specifiedintensity, the incident energy and the heat flux can becomputed from Eq. (8). The numerical solution algorithmis similar to that of the DOM. Once initial guesses areassigned to unknown incoming/outgoing nodal interfacevariables, one can start with the known quantities, say, atthe West wall and sweep in the East direction (forward) tocompute the forward-directed incident energy and theheat flux using Eqs. (45) and (46). Then starting with theknown quantities at the East wall and using backwardinterface coupling SKN equations, the nodes are sweptfrom the East to West direction to compute backward-directed nodal variables. This iterative scheme is repeatedback-and-forth until the convergence of both the incidentenergy and the net heat flux for each node-face is achievedwithin the prescribed tolerance. The convergence could befurther speeded up if old estimates for nodal variables arereplaced as soon as the new ones become available.

In two-dimensional rectangular geometries, we alsoneed additional interface coupling equations for the y-direction. These equations are also analogous to those forthe x-direction; however, the transverse leakage Ly isreplaced with Lx, and the radiative heat fluxes qx arereplaced with qy. The subscripts W and E are replacedwith S and N, respectively while the quantities β, ss and κare evaluated with respect to ΔY nodal mesh spacing. Thealgorithm for 2D cases is as follows: the transverse leakage(Ly) is also computed for each node during the West–Eastand East–West sweeps using x-direction interface nodalSKN equations. This procedure is accompanied by South–North and North–South sweeps using the y-interfacecoupling SKN equations in red–black checkerboard itera-tion manner. It should also be noted that the node-face-averaged net qx is numerically available on the mid pointsof the East and West node faces while the node-face-averaged net qy is obtained on the mid points of the Southand North faces (see Fig. 1); that is, all the heat fluxcomponents are not available on all faces of each node.To obtain these quantities, one has to resort to interpolation.

Fig. 2. (a) BMP-3 (b) BMP-4 (c) the paths where solutions are compared.

4. Results and discussion

4.1. Benchmark problems

Four benchmark problems (BMPs) have been set fortesting the numerical performance of the nodal SKN

method.Benchmark Problem 1 (BMP-1). This problem involves

radiative transfer in a plane-parallel slab with constantproperties (homogeneous medium, β¼1 m−1) and a trans-parent boundary. The medium is cold (Q¼0). The onlysource in the medium is due to the externally isotropicunit incidence of radiation at the west wall, I+(0)¼1, andthe east wall is transparent, I−(τ0)¼0. The hemisphericalreflectivity, R¼q−(0)/q+(0), and the transmissivity, T¼q+

(τ0)/q+(0), are computed for geometries of optical thick-nesses ranging from τ0¼0.1 to τ0¼5 and for scatteringalbedos from ω0¼0.2 to ω0¼0.995. The thickness of the

slab, a, is adjusted in accordance with the opticalthickness.

Benchmark Problem 2 (BMP-2). This benchmark pro-blem is basically the same as the BMP-1; however, themedium additionally has a uniform internal source; Q¼1.Similarly the hemispherical reflectivity and the transmis-sivity are computed for plane medium having opticalthickness ranging from τ0¼0.1 to τ0¼5 and scatteringalbedos from ω0¼0.2 to ω0¼0.995. The exact RITE solu-tions reported here for comparisons were also produced inthe course of this study. This benchmark problem wasprimarily set up in order to investigate the effect of aninternal source (hot medium) on the nodal SKN solutions.


Benchmark Problem 3 (BMP-3). This problem involvesradiative transfer in a square homogeneous cold medium(Q¼0) whose enclosing walls are subjected to the exter-nally isotropic unit incidence of radiation (Fig. 2a). Theincident energy and partial heat flux for square geometrieswith the dimensions of a¼b¼1, 2.5 and 5 m are computedfor ω0¼0.2, 0.5 and 0.8. The extinction coefficient is alsotaken unity so that the enclosure dimensions correspondone-on-one to the optical dimensions, x0 and y0. Forcomparisons, the reference exact and the DOM S8 solu-tions were also produced for this benchmark problem. Thesource term of the interface coupling nodal SKN equationsconsists of internal source and the transverse leakageterm. This benchmark problem, without the internalsource, is setup to observe the effect of the transverseleakage approximation on the numerical solutions of theN-SKN method.

Benchmark Problem 4 (BMP-4). This problem is the sameas the BMP-3, however, the medium has an internalisotropic unit source (Q¼1) and the walls are transparent(Fig. 2b). The incident energy and partial heat flux forsquare geometries with optical dimensions of x0¼y0¼1,2.5 and 5 mfp are computed for various scattering albedos.This problem serves to explore the effect of internal sourceon the numerical solutions with the N-SKN method withsignificant transverse leakage. The incident energy and thenet partial heat flux solutions of BMP-3 and BMP-4 arecompared with those of exact and the DOM S8 along ABand BC lines illustrated in Fig. 2c. Due to symmetry, thepartial heat fluxes qx and qy along BC are identical to qy andqx along CD line. The reference RITE and DOM solutionswere also produced for this study.

Table 1Convergence of Lawrence's method with nodal-mesh refinement for τ0 ¼ 1 of B

ω0 ¼ 0:2 ω0 ¼ 0:5

Exacta 0.0439 0.2463 0.1342 0.3067Nodes R T R T

10 0.0480 0.2553 0.1429 0.319550 0.0431 0.2545 0.1315 0.3165

100 0.0423 0.2542 0.1297 0.3163200 0.0418 0.2540 0.1286 0.3162400 0.0415 0.2538 0.1279 0.3161

a Ref. [23].

Table 2Convergence of the N-SK2 with nodal-mesh refinement for τ0 ¼ 1 of BMP-1 us

ω0 ¼ 0:2 ω0 ¼ 0:5

Exacta 0.0439 0.2463 0.1342 0.306Nodes R T R T

10 0.0432 0.2511 0.1328 0.31050 0.0446 0.2498 0.1355 0.308

100 0.0446 0.2497 0.1356 0.308200 0.0446 0.2497 0.1356 0.308

a Ref. [23].

4.2. On the solution of 1D benchmark problems

One-dimensional BMPs suit well to analyze the con-vergence and accuracy issues of the N-SKN method in theabsence of the transverse leakage. We first compare theaccuracy and the convergence of Lawrence's nodal integralmethod. The solutions presented here assume flat incidentenergy (zeroth order polynomial) approximation in Eqs.(14) and (19); that is, GðxÞ≅Gþ

W þ G−E . In Table 1, the effect of

nodal-mesh refinement on the computed reflectivity (R)and transmissivity (T) for τ0¼1 and scattering albedos ofω0¼0.2, 0.5, 0.8 and 0.995 of BMP-1 are tabulated. Theresults depict maximum absolute error of 1.4% for ω0¼0.5and 0.8 for 10 nodes, and just below 1% for 50 nodal-mesh.However, as the nodal mesh is further refined, the absoluteerrors increase for all scattering albedo cases, and thesolutions converge to values distant from those of theexact. This phenomenon is not surprising as Gelbard et al.reported that the nodal integral solutions in small nodal-mesh limit wandered in the wrong direction [37]. Thisproblem, as explained earlier, is caused by the singular ln(x) term present in the series forms of the exponentialintegral functions for small x. The quadratic polynomialapproximation also suffers from the same problem.

In Table 2, the computed reflectivity and the transmis-sivity with the N-SK2 using the third order interface-nodalSK2 equations—Eqs. (45) and (46)—are tabulated for var-ious nodal meshes for BMP-1, τ0¼1 and the scatteringalbedos of ω0¼0.2, 0.5, 0.8 and 0.995. The maximumabsolute errors for N¼10 are below 0.5% with the largesterrors being in the transmissivity. As the number of nodesis increased, the absolute errors for both quantities are

MP-1.

ω0 ¼ 0:8 ω0 ¼ 0:995

0.2802 0.4163 0.4412 0.5488R T R T

0.2880 0.4296 0.4375 0.55360.2745 0.4270 0.4316 0.55860.2721 0.4272 0.4301 0.56010.2707 0.4275 0.4290 0.56120.2697 0.4279 0.4281 0.5620

ing the third order interface nodal SK2 equations.

ω0 ¼ 0:8 ω0 ¼ 0:995

7 0.2802 0.4163 0.4412 0.5488R T R T

0 0.2788 0.4180 0.4402 0.54983 0.2820 0.4155 0.4435 0.54653 0.2820 0.4154 0.4436 0.54643 0.2821 0.4154 0.4437 0.5464

Table 3Convergence of theN-SK2 with grid refinement for τ0 ¼ 1 of BMP-1 using the second order interface nodal SK2 equations.

ω0 ¼ 0:2 ω0 ¼ 0:5 ω0 ¼ 0:8 ω0 ¼ 0:995

Exacta 0.0439 0.2463 0.1342 0.3067 0.2802 0.4163 0.4412 0.5488Nodes R T R T R T R T

10 0.0496 0.2295 0.1478 0.2879 0.3008 0.3946 0.4654 0.524650 0.0455 0.2457 0.1379 0.3043 0.2856 0.4114 0.4478 0.5422

100 0.0451 0.2477 0.1367 0.3063 0.2838 0.4134 0.4457 0.5443200 0.0448 0.2487 0.1362 0.3073 0.2830 0.4144 0.4447 0.5454400 0.0447 0.2492 0.1359 0.3078 0.2825 0.4149 0.4442 0.5459800 0.0447 0.2494 0.1358 0.3079 0.2823 0.4151 0.4439 0.5462

1600 0.0446 0.2495 0.1357 0.3081 0.2822 0.4152 0.4438 0.5463

a Ref. [23].

Fig. 3. (a) Incident energy (b) radiative heat flux distributions for BMP-1and τ0 ¼ 1;ω0 ¼ 0:5 using the exact, P7, nodal SK1 and nodal SK2 methods.

Fig. 4. (a) Incident energy (b) radiative heat flux distributions for BMP-1and τ0 ¼ 5;ω0 ¼ 0:8 using the exact, P7, nodal SK1 and nodal SK2 methods.


reduced below 0.3%. The interface coupling nodal SK2

equations convergence very fast, and the grid-independent solutions are obtained with relatively smallnumber of nodes (N¼100) in comparison to Lawrence'smethod (N≈500). This benchmark problem was solved forall scattering albedo values, and no convergence problemswere encountered for all scattering albedos includingω0¼1.

In Table 3, the computed reflectivity and transmissivitywith the N-SK2 using the second order interface nodal SK2

equations—Eqs. (47) and (48)—are tabulated for variousnodal meshes for BMP-1,τ0¼1 and the scattering albedosof ω0¼0.2, 0.5, 0.8 and 0.995. The maximum absoluteerrors for N¼10 are in the order of 2.5%, and the errorsincrease with increasing optical thickness. As the numberof nodes are increased, the solutions also converge to


those of given in Table 2; however, the number of nodesrequired to obtain grid-independent solutions are higheras expected.

At this point, a word is in order as to explain as to whyEqs. (34) and (35) were also expanded to Taylor series.When the node-face contributions are kept in the presentform, the solutions yield worse results than those obtainedusing the second or third order nodal coupling equations.We understand that this is a result of expanding theanalytical solution of the SKN equations to a truncatedseries whereas keeping the exact forms of the coefficientsof the nodal surface variables which yields an imbalance inconservation of energy. However, when all terms areexpanded to truncated series of the same order, theconservation of energy is satisfied in the same order ofmagnitude which yields better solutions in the small-mesh limit.

In solving BMP-1 and BMP-2, the grid-independentsolutions were obtained between 100 and 200 nodes forall geometries. A high order spherical harmonics, P7,solutions were also obtained to depict comparative spatialagreement in Figs. 3 and 4. The same grid intervals wereused to obtain the exact, P7 and N-SKN solutions to keepthe discretization error at the same magnitude, and theincident energy and heat flux distributions were obtainedfor the solution domain. The exact RITE solutions for thereflectivity and transmissivity from Ref. [23] were trun-cated to four significant decimal places. The convergencecriterion was 10−6 for all methods. The convergence of theinterface nodal SKN equations is very fast; in 1D casesconsidered here, the convergence was achieved between 7and 15 iterations.

In Table 4, the reflectivity and the transmissivity forBMP-1 using the exact RITE, N-SK1 and N-SK2 solutions aretabulated. The absolute errors of the N-SK1 solutions areless than 0.5% for τ0¼0.1; however, the absolute errors in Rand/or T increase with increasing optical thickness. Themaximum absolute errors are about 1.2%, 2.55%, 3.4% and3.9% for τ0¼0.5, 1, 2 and 5, respectively. As the highesterrors are observed for ω0¼0.8 case, for both R and T, theerrors of ω0¼0.995 are less than 1.27%. On the other hand,the absolute errors with the N-SK2 are less than 0.5% for all

Table 4The transmissivity and reflectivity values of BMP-1 with the exact, N-SK1 and N

ω0 Method τ0 ¼ 0:1 τ0 ¼ 0:5

R T R T

0.2 Exacta 0.0146 0.8470 0.0371 0.4743N-SK1 0.0161 0.8424 0.0431 0.4753N-SK2 0.0156 0.8418 0.0383 0.4693

0.5 Exact 0.0384 0.8704 0.1077 0.5350N-SK1 0.0415 0.8676 0.1199 0.5429N-SK2 0.0406 0.8665 0.1106 0.5300

0.8 Exact 0.0649 0.8966 0.2056 0.6220N-SK1 0.0684 0.8942 0.2167 0.6299N-SK2 0.0678 0.8934 0.2102 0.6170

0.995 Exact 0.0838 0.9152 0.2932 0.7018N-SK1 0.0865 0.9125 0.2946 0.7012N-SK2 0.0868 0.9122 0.2985 0.6965

a Ref. [23].

plane geometries considered regardless of the scatteringalbedo of the medium. These errors are reduced below0.22% for τ0¼2, and below 0.12% for τ0¼5. The solutionssignificantly improve with increasing optical thicknessconverging to the exact solutions.

The accuracy assessment solely based on R and T maybe misleading as far as the accuracy of the overalldistribution of the incident energy and the heat flux isconcerned. Since these quantities, by definition, involvethe ratios of incoming and outgoing partial heat fluxes atthe walls, their numerical values are sensitive to slightchanges in these quantities, yet very good agreement inthe distributions may be achieved. Thus, in order to assessthe overall agreement of the distributions over the entiresolution domain, the radiative heat flux and the incidentenergy are comparatively depicted for two cases in Figs.3and 4.

In Fig. 3(a), the incident energy distribution obtainedusing 100 nodes/grids for τ0¼1 and ω0¼0.5 of BMP-1using the exact, P7, N-SK1 and N-SK2 are depicted. Theincident energy distribution with the N-SK2 is in perfectagreement with that of the exact solution. P7 slightly over-estimates the incident energy near the west wall while theN-SK1 underestimates the incident energy throughout thesolution domain. In Fig. 3(b), τ0¼1 and ω0¼0.5 case ofBMP-1, the distributions of net radiative heat flux usingthe exact RITE, P7, N-SK1 and N-SK2 are depicted. Theagreement between the exact and the N-SK2 solution isperfect while the N-SK1 yields slight deviations from theexact solution. On the other hand, P7 approximation over-estimates the heat flux by as much as 50% near the source(west) wall.

In Fig. 4, the incident energy and the heat flux dis-tributions obtained using 200 nodes/grids for τ0¼5 andω0¼0.8 of BMP-1 are depicted. In Fig. 4(a), the incidentenergy distributions with the N-SK2 and P7 are in perfectagreement with that of the exact RITE while the N-SK1

slightly under-estimates the incident energy near the westwall. On the other hand, as seen in Fig. 4(b), as the opticalthickness is increased to τ0¼5 (a medium of larger opticalthickness), the net radiative heat flux distribution of P7improves; however, the agreement of net heat flux

-SK2 methods.

τ0 ¼ 1 τ0 ¼ 2 τ0 ¼ 5

R T R T R T

0.0439 0.2463 0.0461 0.0727 0.0463 0.00240.0520 0.2579 0.0553 0.0804 0.0556 0.00250.0446 0.2497 0.0467 0.0750 0.0469 0.00210.1342 0.3067 0.1451 0.1071 0.1466 0.00530.1524 0.3291 0.1686 0.1268 0.1713 0.00740.1356 0.3083 0.1464 0.1091 0.1478 0.00500.2801 0.4162 0.3280 0.1973 0.3418 0.02290.2978 0.4417 0.3582 0.2311 0.3809 0.03560.2821 0.4154 0.3290 0.1983 0.3428 0.02280.4412 0.5488 0.5988 0.3815 0.7637 0.18920.4359 0.5562 0.5909 0.3943 0.7623 0.20290.4437 0.5464 0.5993 0.3811 0.7636 0.1892

Table 5The transmissivity and reflectivity values of BMP-2 with the exact, N-SK1 and N-SK2 methods.

ω0 Method τ0 ¼ 0:1 τ0 ¼ 0:5 τ0 ¼ 1 τ0 ¼ 2 τ0 ¼ 5

R T R T R T R T R T

0.2 Exact 0.1011 0.9335 0.3425 0.7796 0.4876 0.6899 0.5968 0.6235 0.6409 0.5971N-SK1 0.1087 0.9350 0.3994 0.8316 0.5981 0.8041 0.7677 0.7928 0.8428 0.7896N-SK2 0.1047 0.9309 0.3460 0.7770 0.4856 0.6908 0.5956 0.6239 0.6413 0.5964

0.5 Exact 0.1296 0.9616 0.4650 0.8923 0.6933 0.8658 0.8930 0.8549 0.9948 0.8535N-SK1 0.1367 0.9628 0.5192 0.9421 0.8099 0.9865 1.1019 1.0600 1.2786 1.1146N-SK2 0.1335 0.9594 0.4700 0.8894 0.6917 0.8644 0.8909 0.8536 0.9950 0.8521

0.8 Exact 0.1612 0.9928 0.6366 1.0530 1.0392 1.1752 1.5149 1.3842 1.9303 1.6115N-SK1 0.1664 0.9922 0.6708 1.0840 1.1249 1.2687 1.7248 1.5978 2.3700 2.0245N-SK2 0.1649 0.9904 0.6423 1.0492 1.0385 1.1718 1.5109 1.3801 1.9289 1.6087

0.995 Exact 0.1837 1.0151 0.7912 1.1998 1.4333 1.5409 2.5645 2.3473 5.4812 4.9068N-SK1 0.1866 1.0123 0.7933 1.1999 1.4307 1.5510 2.5677 2.3712 5.5667 5.0074N-SK2 0.1867 1.0121 0.7965 1.1946 1.4356 1.5384 2.5645 2.3464 5.4804 4.9060


distributions of the N-SK1 or N-SK2 with the exact solutionis clearly superior to that of P7.

Using the exact RITE, N-SK1 and N-SK2, the computedreflectivity and the transmissivity for BMP-2 are tabulatedin Table 5. The absolute errors with the N-SK1 are high incomparison to those of the BMP-1. The maximum absoluteerrors encountered with N-SK1 are about 0.8%, 5.7%, 12.1%,21.4% and 43.7% for τ0¼0.1, 0.5, 1, 2 and 5, respectively. Themaximum absolute errors with the N-SK2 in either thereflectivity or the transmissivity are about 0.4%, 0.6%, 0.4%,0.45% and 0.4% for τ0¼0.1, 0.5, 1, 2 and 5, respectively.Similar to BMP-1, the overall net radiative heat flux andthe incident energy distributions with the N-SK2 agreevery well with those of the exact solution.

The foregoing analysis on 1D benchmark problemsclearly indicate that in the absence of the transverseleakage for hot and/or cold participating medium, theincident energy and the net radiative heat flux computedwith the N-SK2 method approaches to the exact solutionsand exhibits better performance than Lawrence's nodalintegral method. It is also observed that in optically thingeometries even the N-SK1 approximation produces muchbetter heat flux solutions than a high order PN approxima-tion; however, the N-SK2 approximation yields improvedsolutions for 1D medium with and/or without an internalsource.

4.3. On the solution of 2D Benchmark problems

The reference exact RITE solutions for BMP-3 and BMP-4 were obtained using the subtraction of singularitytechnique [17,19]. For the numerical integrations, fourthorder Simpson's rule was used with 100�100 grid con-figurations. Also the DOM S8 solutions were obtained withthe same grid configurations along with the level-symmetric quadratures. For the N-SK2, the grid-indepen-dent solutions were obtained with 100�100 nodes. Theconvergence in 2D benchmarks was generally achievedbetween 50 and 75 iterations.

In Fig. 5, for BMP-3, the exact RITE, N-SK2 and DOM S8solutions for the incident energy and the net radiative heatflux for x0¼y0¼1 and ω0¼0.2 case are comparativelydepicted along AB and CD lines. The net qy along the AB

line is zero due to symmetry; thus, it is not depicted in thefigures. Both the DOM S8 and N-SK2 solutions of theincident energy and the net qx along the AB line (interiorregion) are in very good agreement with those of the exactRITE solution (Fig. 5a). In Fig. 5b, along BC line, while theincident energy is under-estimated by 2–3% in comparisonto the exact solution, the net qx distributionwith the N-SK2

approximation is better in agreement with those of theexact and the DOM S8. The net heat flux qx distribution isover-estimated about 2% while the distribution improvestowards the point C. However, qx is under-estimated nearpoint C. In Fig. 5b, the net qy distribution along the BC lineis depicted. The agreement of the N-SK2 solutions with theexact and S8 solutions are consistent, qy with the N-SK2

deviates towards point C. Although the DOM S8 solutionsfor the incident energy are better than those of the N-SK2,the net partial heat flux distributions with the N-SK2,overall, exhibit slightly better performance. It should alsobe noted that, due to symmetry, the partial heat fluxes qxand qy along BC are identical to qy and qx along CD line,respectively.

In Fig. 6, the exact RITE, N-SK2 and DOM S8 solutions forthe incident energy and the net radiative heat flux forx0¼y0¼1 and ω0¼0.8 case are comparatively depictedalong AB and BC lines of BMP-3. As the scattering albedo ofthe medium increases, the incident energy distributionsimprove significantly. The net qx distribution also improvesover the entire boundary, the N-SK2 solutions are in betteragreement than that of the DOM S8; however, the net qxsolution near the corner point C is under-estimated byabout 3.5% (Fig. 6b). The net qx distribution with the DOMS8 depicts smooth oscillation around the exact solution(Fig. 6b) which is caused by the ray-effect. The deviation inthe qx distribution of the N-SK2 solutions is also observed;however, these deviations occurring as a result of theisotropic-leakage assumption which are smaller in magni-tude. Similarly, while the incident energy is consistentlyunder-estimated by 1–2%, the net heat flux qy is inagreement with the exact solution for the most part ofBC line; however, again it under-estimates the heat fluxnear the corner point C.

In Fig. 7, the exact RITE, N-SK2 and DOM S8 solutions forthe incident energy and the net radiative heat flux for

Fig. 5. The incident energy and the partial heat flux distributions for x0 ¼ y0 ¼ 1;ω0 ¼ 0:2 of BMP-3 along (a) AB line (b) BC line with exact, N-SK2 andDOM S8.


x0¼y0¼2.5 and ω0¼0.5 case are comparatively depictedalong AB and BC lines of BMP-3. The incident energy andthe net qx heat flux distributions with the N-SK2 approx-imation at interior region (AB line) are in perfect agree-ment with those of the DOM S8 and the exact RITE(Fig. 7a). Along BC line (Fig. 7b), the agreement in boththe incident energy and the net radiative heat fluxes

improve. The incident energy is off from the exact solutionby less than 1%; however, the net qx partial heat fluxdistributions with the N-SK2 along BC line is in betteragreement than that of the DOM S8. The net qy heat fluxwith the N-SK2 along the BC line slightly over-estimatesthe solution in general, but the distribution under-estimates the heat flux near point C (Fig. 7b). This behavior

G

0.5 0.6 0.7 0.8 0.9 1x

-0.2

-0.16

-0.12

-0.08

-0.04

0

0.04

ABExactN-SK2S8-DOM

G

0.5 0.6 0.7 0.8 0.9 1yy

-0.2

-0.18

-0.16

-0.14

-0.12

-0.1BCExactN-SK2S8-DOM

0.5 0.6 0.7 0.8 0.9 1y

-0.16

-0.12

-0.08

-0.04

0

BCExactN-SK2S8-DOM



indicates that as the optical thickness of the mediumincreases, both the incident energy and the partial netheat flux solutions for all scattering albedo cases improvesignificantly.

In Fig. 8, the exact RITE, N-SK2 and DOM S8 solutions forthe incident energy and the net radiative heat flux forx0¼y0¼5 and ω0¼0.5 case are comparatively depicted

along AB and BC lines of BMP-3. Both the incident energyand the partial net heat flux solutions are in perfectagreement with those of the exact RITE throughout themedium (Fig. 8a,b). Moreover, the net heat flux solutionsare better than the DOM S8, and the N-SK2 producessolutions identical with the exact solution for both theincident energy and the heat flux for all scattering albedo

Fig. 7. The incident energy and the partial heat flux distributions for x0 ¼ y0 ¼ 2:5;ω0 ¼ 0:5 of BMP-3 along (a) AB line (b) BC line with exact, N-SK2 andDOM S8.


cases; however, the N-SK2 qy heat flux solutions along withthe BC line is slightly over-estimated. The magnitude ofthe deviations decreases with increasing optical dimen-sions. It should be pointed out that the DOM S8 solutionsno longer exhibit oscillations in this geometry. As theoptical dimensions increase, the incident energy and theheat flux solutions with the N-SK2 significantly improve on

the enclosure boundaries and yields solutions equivalentand/or better than the DOM S8.

In Fig. 9, the exact RITE, N-SK2 and DOM S8 solutions forthe incident energy and the net radiative heat flux forx0¼y0¼1 and ω0¼0.5 case are comparatively depicted alongAB and BC lines of BMP-4. Similar to BMP-3, both the DOM S8and N-SK2 distributions of the incident energy and the net qx



at the interior region are in very good agreement with thoseof the exact RITE solutions within a few percent (Fig. 9a).While the incident energy along BC line is over-estimated by5–6%, the net qx heat flux distribution along AB line with theN-SK2 is slightly under-estimated yet better than the DOM S8and in good agreement with that of the exact solution(Fig. 9b). The ray-effect in the qx heat flux solution of the

DOM S8 along this line is apparent. On the other hand, withthe N-SK2, the net qy along AB line is overall in betteragreement than that of qx distribution; however, the partialheat flux slightly deviates near the corner point C.

In Fig. 10, the exact RITE, N-SK2 and DOM S8 solutions forthe incident energy and the net radiative heat flux forx0¼y0¼1 and ω0¼0.8 case are comparatively depicted along

Fig. 9. The incident energy and the partial heat flux distributions for x0 ¼ y0 ¼ 1;ω0 ¼ 0:5 of BMP-4 along (a) AB line (b) BC line with exact, N-SK2 and DOM S8.


AB and BC lines of BMP-4. The solutions with N-SK2 exhibitsimilar behavior as in ω0¼0.5 case. As the incident energy isover-predicted along the boundaries (Fig. 10a,b), the N-SK2

net heat flux distributions (Fig. 10a,b) are in better agreementwith the exact than the DOM S8 where slight deviations in qxattributed to ray effect is observed (Fig. 10b).

In Fig. 11, the exact RITE, N-SK2 and DOM S8 solutions forthe incident energy and the net radiative heat flux forx0¼y0¼2.5 and ω0¼0.5 case are comparatively depictedalong AB and BC lines of BMP-4. As the optical thickness ofthe square enclosure is increased, the incident energy and theheat flux solutions improve significantly as in the BMP-3 case



(Fig. 11a). Although the absolute errors along the boundariesare reduced within a few percent, the incident energydistribution is still over-estimated (Fig. 11b). On the otherhand, the net qx distribution with the N-SK2 along BC linedepict much better agreement than that of the DOM S8(Fig. 11b) whereas the net qy distribution show slightdeviation.

In Fig. 12, the exact RITE, N-SK2 and DOM S8 solutions forthe incident energy and the net radiative heat flux forx0¼y0¼5 and ω0¼0.5 case are comparatively depicted alongAB and BC lines of BMP-4. For this case, a square mediumwithlarger optical dimensions, both the incident energy and thenet partial heat flux distributions improve significantly at theinterior region and along the enclosure walls (Fig. 12a,b).

Fig. 11. The incident energy and the partial heat flux distributions for x0 ¼ y0 ¼ 2:5;ω0 ¼ 0:5 of BMP-4 along (a) AB line (b) BC line with exact, N-SK2 andDOM S8.


4.4. On the isotropic leakage assumption

The foregoing analysis of 2D cases indicate that for acold square medium (1 mfp�1 mfp) with emitting enclo-sure walls, BMP-3, the N-SK2 method yield 2–3% error inthe distribution of the incident energy and the net partialheat flux. The agreement of the incident energy and thepartial heat flux distributions at interior region generallyagree well with the exact solution. Similarly, for medium

with an internal source, BMP-4, for enclosures with opticaldimensions of a few mean-free-path or more, the N-SK2

solutions also yields excellent results for a wide range ofscattering albedos. However, due to the presence of aninternal source, the incident energy is over-predicted. Alsofor BMP-3 and BMP-4, the largest deviations in N-SK2 areobserved especially along the walls of the enclosureand especially near the corner point C. We attributethis behavior, on the major part, to the isotropic leakage

b

a



approximation. It should be pointed out that thedeviations for enclosures of smaller (o1 mfp) opticaldimensions are much larger indicating the failure of theisotropic leakage assumption for optically very thin media.

As the medium becomes optically very thin, the radiantenergy streaming into neighboring nodes is inadequatelydistributed especially at the enclosing boundary nodeswhere the radiative intensity is highest and angular-


dependent. However, for mediumwith optical thickness ofa few mean-free-paths or more, the isotropic leakageassumption yields much better performance.

The slight deviations observed in N-SK2 solutions maybe mistaken as a result of the ray-effect. As it has beenwidely studied and documented, the ray effect is a result ofangular discretization of the solid angle. Though its effectscould be reduced, it is inevitably observed in the DOM andits derivative methods. In BMP-3 and BMP-4 of DOM S8solutions, the ray-effect does not manifest itself withstrong oscillations along the walls. On the other hand,the derivation of the SKN equations and/or the nodal SKN

equations is based on the exact integral transfer equations.The studies involving homogeneous and non-homogeneousproblems using the SKN method [25,27,28,33,34] confirmthat the SKN method is free-off ray effect since angulardependence of the RTE is completely eliminated. In thenodal SKN method, however, the nature of the problem isdifferent. The interface-coupling nodal equations are derivedbased on the double P1 approximation used to describe theradiation intensity at the nodal interfaces in terms of theface-average nodal variables (G, q) and cosine of zenith angle(μ). This angular dependence is also eliminated by integrat-ing over incoming/outgoing directions—Eqs. (14) and (19).However, the azimuthal (φ) dependence of the radiationintensity at the nodal interfaces is not represented whichreduces the angular dependence of the intensity (Eq. 7) tosimple one-dimensional double P1 approximation. In theirstudies, Lawrence and Gelbard do not also make reference tothe ray effect and acknowledge the major source of error asthe isotropic transverse-leakage assumption. This phenom-enon is only observed for square medium with opticaldimensions less than 2 mfp. As the optical dimensions areincreased, the N-SK2 solutions approach the exact solutiondemonstrating the validity of the isotropic leakage assump-tion for optically fairly small to large medium.

The benchmark problems tackled in this paper are forthe homogeneous medium. On the other hand, the nodalneutron transport was incorporated into DIF3D code as theversion DIF3D-NTT (Nodal Transport Theory) [35–37]. Thenuclear reactor design problems involve stepwise varyingmedium properties (strong discontinuities present due tofuel, moderator, control rod regions, etc.) and multi-group(non-monochromatic) calculations. Thus, the nuclear reac-tor applications pose severe test for the nodal integralmethod of Lawrence and Gelbard. With the use of nodaltransport method, the authors reported that the methodimproved solutions significantly over the diffusion method[35–37]. Based on the 1D analysis presented in Section 4.2,the N-SKN method was shown to exhibit better perfor-mance than the original nodal transport method. Thus wepredict that the N-SKN should similarly depict betterperformance in inhomogeneous media as well. Investiga-tion of this aspect is currently underway.

4.5. Memory requirements and computation time

Memory requirement and computation time are twoimportant aspects of an efficient numerical method and/oralgorithm. In this respect, when using the DOM in 2D x−ygeometry, the number of discrete directions using the

Level Symmetric quadratures is N(N+2)/2; thus the radia-tion intensities for each mesh point must be stored. Forinstance, in DOM S8, the intensity for 40 directions needsto be stored for every mesh point. On the other hand, thenodal SKN method requires two equations for the forwarddirected incident energy (Gþ) and the partial heat flux(qþx ), and two backward directed quantities (G−; q−x ) forx-direction sweep. Likewise, four equations for y-directionforward and backward sweeps are needed at north orsouth faces. Thus, the method requires total of 8 arrays formemory allocation of interface coupling nodal N-SKN

equations. This requirement is also independent of theorder of the method whereas the memory requirement ofradiation intensity in the DOM increases quadraticallywith the order of the approximation which in turn restrictsthe use of high order approximations.

As mentioned earlier, the numerical solution algorithmof N-SKN method resembles the DOM. The interface-coupling N-SKN equations could be thought of as faceintensity equations for the x- or y-direction. Since theinterface-coupling nodal equations consist of 8 equations,it is evident that computation time will be shorter thanthat of the DOM S8. To illustrate this aspect, the exact RITE,DOM S8 and N-SK2 solutions with 200�200 grids wereobtained on a personal computer equipped with Intel i3-2100 cpu @3.10 GHz. For two-dimensional medium, BMP-4and 1�1 mfp geometry, typical cpu-time for ω0¼0.5 is0.56 and 1.22 s for N-SK2 and DOM S8, respectively. As thescattering albedo increases, the number of source itera-tions also increase. The same problem for ω0¼0.8 yields1.06 and 1.74 s for N-SK2 and DOM S8, respectively. On theother hand, the cpu-time for the exact RITE solution isabout 55 and 74 min for ω0¼0.5 and 0.8, respectively. It isevident that the N-SKN method in terms of memoryrequirement and computation time is more efficient thanthe DOM.

The interface coupling nodal SKN equations have verysimple recursive forms in which the coefficients consist ofalgebraic expressions of the medium properties. Forhomogeneous medium, the coefficients of the interfacecoupling N-SKN equations can be pre-calculated before theiterations to further speed up the computations. In thisstudy, we have not looked into the ways to speed up theconvergence of the nodal SKN equations. However, thismatter may also worth investigating.

5. Conclusions

The N-SKN method presented relies on one-dimen-sional RITE equation based on face-averaged incidentenergy and the partial heat fluxes. The incident energyand partial heat fluxes at the nodal interfaces are assumedto behave as double P1 in small nodal-mesh limit. Thesecond and/or third dimensions can be easily incorporatedinto 1D N-SKN equations as transverse leakage. Thus, themethod can be easily extended to 3D-cartesian geometry.The synthetic kernel approximation is introduced to thesource integrals as well as the nodal face contributions(Γs), and 1D interface coupling nodal equations areobtained. Analytical solutions for the SKN equations areobtained and expanded to series for the small-mesh limit


case. The N-SK2 approximation, in the absence of trans-verse leakage, yields excellent results in one-dimensionalplane-parallel geometries for all scattering cases and widerange of (0.1≤τ0≤5) optical thickness. In two-dimensionalsquare medium, hot/cold medium and/or various bound-ary conditions (transparent/emitting walls), the N-SK2

yields excellent and comparable solutions to the DOM S8for medium of a few mean-free-paths or more. Foroptically very small geometries (o2 mfp), the isotropicleakage approximation seems to inadequately representthe radiative intensity distribution especially along theenclosure boundaries where the source intensity is thehighest. The method is easy to implement, and it does notrequire excessive memory and/or cpu-time. Furthermore,the N-SKN method eliminates the tedious derivations ofthe boundary conditions and excessive analytical treat-ments to remove the singularities as observed in thestandard SKN method. This nodal approach can be easilyextended to include all types of boundary conditions(reflecting, gray wall etc.) that could be imposed to aboundary in practical applications.

Appendix A

In this analysis, throughout the kth node, the mediumproperties, internal source and the transverse leakage areassumed to be constant at an average value. The analyticalsolutions of the SK1 and SK2 equations—a set of ODEequations in BVP form—are obtained. The solution ODEsobtained in this study is straightforward. The analyticalexpressions which consist of lengthy expressions involvingexponentials are not given here to save space, and also itwould not be appropriate to use the analytical solutions inthe interface coupling equations from the computationalstand point. On the other hand, since small nodal-meshlimit solution is sought, Gn(1/2) and qn(1/2) terms requiredin Eqs. (40) and (41), to obtain the interface couplingequations; thus we chose to expand the analytical solu-tions to Taylor series near ΔX¼0.

The series forms for the SK1 approximation (w1¼1,μ1¼1/2) with third order truncation error are

w1G1ð12Þ ¼ΔXð1−κÞðQ−LyÞ þ ssð1−4β þ ssÞðGþW þ G−

E Þþ3ssβðqþx;W þ q−x;EÞ þ O ðΔXÞ3

h iðA:1Þ

w1qx;1ð12 Þ ¼ ΔX2 ð1−κÞðQ−LyÞ þ ss

2 ð1−4β þ ssÞðGþW þ G−

E Þþ3ss β

2 ðqþx;W þ q−x;EÞ þ O ðΔXÞ3h i

ðA:2Þ

and for SK2 approximation (μ1;2 ¼ ð37ffiffiffi3

pÞ=6, w1;2 ¼ 1=2)

we obtain

∑2

n ¼ 1wnGnð12Þ ¼ 3ΔX

2 1−2β þ 3ss2

� �ðQ−LyÞ

þ3ss2 1−5β þ 3ss

2

� �ðGþ

W þ G−E Þ

þ9βss4 ðqþx;W þ q−x;EÞ þ O ðΔXÞ3

h iðA:3Þ

∑2

n ¼ 1wnqx;nð12Þ ¼ ΔX

2 1−3κ2

ðQ−LyÞ þ ss2 1−9β

2 þ 3ss2

� �ðGþ

W þ G−E Þ

þ3βss2 ðqþx;W þ q−x;EÞ þ O ðΔXÞ3

h iðA:4Þ

where κ¼ΔX κ.Alternatively the coefficients defined by Eqs. (34) and

(35) are also expanded to Taylor series for small nodal-mesh limit:

eCG;1ðβÞ≅1−6β þ 15β2 þ O ðΔXÞ3

h i; eCG;2ðβÞ≅6β−18β

2 þ O ðΔXÞ3h i

ðA:5Þ

eCq;1ðβÞ≅−β þ 3β2 þ O ðΔXÞ3

h i; eCq;2ðβÞ≅1−3β

2 þ O ðΔXÞ3h i

ðA:6ÞSubstituting Eqs. (A.1)–(A.4) into Eqs. (40) and (41) and

collecting the terms under the same interface nodalvariables, the SK1 solution for outgoing incident energyand the heat flux at the East face of the node yields

GþE ¼ΔXð1−κÞðQ−LyÞ þ eCG;1ðβÞ þ ssð1−κ−3βÞ

h iGþW

þ eCG;2ðβÞ þ 3ssβh i

qþx;W

þssð1−κ−3βÞG−E þ 3ssβ q−x;E þ O ðΔXÞ3

h iðA:7Þ

qþx;E ¼ ΔX2 ð1−κÞðQ−LyÞ þ eCq;1ðβÞ þ ss

2 ð1−κ−3βÞh i

GþW

þ eCq;2ðβÞ þ 3ss β2

h iqþx;W þ ss

2 ð1−κ−3βÞG−E


h iðA:8Þ

On the other hand, for the SK2 approximation, weobtain

GþE ¼ 3ΔX

2 1−2β þ 3ss2

� �ðQ−LyÞ

þ eCG;1ðβÞ þ 3ss2 1−5β þ 3ss

2

� �h iGþW þ eCG;2ðβÞ þ 9

2βssh i

qþx;W

þ3ss2 1−5β þ 3ss

2

� �G−E þ 9

2βss q−x;E þ O ðΔXÞ3h i

ðA:9Þ

qþx;E ¼ ΔX2 1−3κ

2

ðQ−LyÞ þ eCq;1ðβÞ þ ss2 1−3ss−9κ

2

h iGþW

þ eCq;2ðβÞ þ 3ss β2

h iqþx;W þ ss

2 1−3ss−9κ2

G−E


h iðA:10Þ

These interface equations are used to compute theoutgoing incident energy and the heat flux at the Eastface of the node progressively from in forward (West toEast) direction. A set of equations covering backward (Eastto West) direction are obtained in the same manner. Theequations for the incoming incident energy and the heatflux (G−

W ; q−W ) are also analogous to equations of theforward direction. These interface equations can beobtained by simply replacing the subscripts W with S,E with N and exchanging “+” and “−” signs.

It has been observed that expanding the coefficientseCG; eCq also into Taylor series consistent with the trunca-tion error of interface-coupling equations produces bettersolutions.


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