Analysis of non-Fourier conduction and volumetric radiation in a concentric spherical shell using...

International Journal of Heat and Mass Transfer 68 (2014) 51–66

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International Journal of Heat and Mass Transfer

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Analysis of non-Fourier conduction and volumetric radiationin a concentric spherical shell using lattice Boltzmann methodand finite volume method

0017-9310/$ - see front matter � 2013 Elsevier Ltd. All rights reserved.http://dx.doi.org/10.1016/j.ijheatmasstransfer.2013.09.014

⇑ Corresponding author. Tel.: +91 361 2582660; fax: +91 361 2690762.E-mail address: [email protected] (S.C. Mishra).

Subhash C. Mishra ⇑, Harsh SahaiDepartment of Mechanical Engineering, Indian Institute of Technology Guwahati, Guwahati 781039, India

a r t i c l e i n f o a b s t r a c t

Article history:Received 11 June 2013Received in revised form 25 August 2013Accepted 7 September 2013Available online 2 October 2013

Keywords:Non-Fourier conductionRadiationConcentric spherical shellLattice Boltzmann methodFinite volume method

Application of the lattice Boltzmann method (LBM) has been extended to formulate and solve the energyequation of a non-Fourier conduction and radiation heat transfer problem in a concentric spherical shell.The enclosed conducting-radiating medium is absorbing, emitting and scattering. The non-Fourier con-duction effect is induced by thermally perturbing one of the boundaries and incorporating the finite prop-agation speed of the thermal wave front in Fourier’s law of heat conduction. The volumetric radiativeinformation needed in the energy equation has been computed using the finite volume method (FVM).To establish the accuracy of the LBM approach, with volumetric radiative information obtained fromthe FVM, the energy equation is also solved using the FVM. Effects of extinction coefficient, scatteringalbedo, conduction–radiation parameter, emissivity, radius ratio and the magnitude of thermal perturba-tions are studied on transient temperature distributions. Effects of the aforesaid parameters on thesteady-state conduction, radiation and total energy flow rates are also studied. Steady-state LBM andFVM results are compared. In all the cases, the LBM results compare exceedingly well with the FVMresults, and LBM has a faster convergence than the FVM.

� 2013 Elsevier Ltd. All rights reserved.

1. Introduction

Analysis of combined mode conduction and radiation heattransfer is important in many thermal systems. Its considerationis paramount in the analysis and design of boilers, furnaces andinsulations [1–4]. Its accounting is equally important for the cor-rect analysis of phase change processes of semi-transparent mate-rials such as glass and silicon [5,6]. Radiation and conduction areimportant modes of heat transfer in burners based on porous med-ium combustion [7,8]. Combined mode conduction and radiationalso finds application in laser based manufacturing process [9,10]and in the area of bioheat transfer pertaining to laser surgeryand ablation of malignant tissues [11–13]. Consideration of ther-mal radiation with and without conduction in a spherical enclosurecontaining an absorbing, emitting, and scattering medium findsapplications in the analysis of nuclear reactors, spherical propul-sion systems, astrophysics, droplet combustion, droplet radiatorsystems for spacecraft thermal control, etc. [14,15].

Combined mode conduction and/or radiation problems in aspherical shell have been analyzed by many [15–21]. In anygeometry, thermal response of the system in transient state isquite different with and without consideration of finite propaga-tion speed of the conduction wave front [21–34]. Conduction heattransfer as per Fourier’s law does not consider any time lag be-tween the cause (thermal perturbation) and the effect (manifesta-tion of energy flow rate). In other words, as per Fourier’s law ofheat conduction, the effect of the thermal perturbation is instanta-neously felt throughout the medium. In reality, however, it is neverso. Thermal wave front does take some finite time to move fromone location to the other. Examples are plenty [21–34]. In any sys-tem, if the next pulse of thermal perturbation is imposed beforethe effects of the previous ones have died out, consideration of fi-nite propagation speed becomes important. Further, even whenany region of the system is thermally perturbed, like suddenlychanging the temperature of any boundaries or changing imposedheat flux, if the temporal change is looked at time scale lower thanthe system time scale L ðmÞ

C ðm s�1Þ where L is the characteristic lengthand C is the propagation speed of the thermal wave front, consid-eration of finite propagation speed becomes essential. The modi-fied form of the heat conduction equation that accounts for thefinite propagation speed of the conduction wave front is knownas non-Fourier conduction.

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Nomenclature

ai weight in LBM formulationbi weight in LBM formulationC speed of thermal wave, (m s�1)Dm directional weight in direction mei non-dimensional propagation velocity in the direction i

in the lattice~er unit radial vectorfi non-dimensional particle distribution function in the i

directionf ð0Þi non-dimensional equilibrium particle distribution func-

tion in the i directionG incident radiation (W m�2)k thermal conductivity (W m�1 K�1)N conduction–radiation parameterNh number of divisions of the polar spaceqC non-dimensional heat fluxqT non-dimensional total heat fluxr1 non-dimensional radius of the inner spherer2 non-dimensional radius of the outer sphereri non-dimensional radial position of the ith node

Greek symbolsa thermal diffusivity (m2 s�1)b extinction coefficient (m-1)e emissivityh polar angle

/ azimuthal anglej absorption coefficient (m�1)m expansion parameterH normalized temperature, T�T2

T1�T2

ja absorption coefficient (m�1)C thermal relaxation time (s)x scattering albedors scattering coefficient (m�1)g normalized radial distance, r�r1

r2�r1

l direction cosinesf non-dimensional timeX solid angle (sr)DX elemental solid angle (sr)W non-dimensional energy flow rate

Superscriptm indices for discrete polar angles

Subscripts1, 2 inner and outer spherical wallsC conductivew wallR radiativeref referenceT total

52 S.C. Mishra, H. Sahai / International Journal of Heat and Mass Transfer 68 (2014) 51–66

Realizing its importance and applications, many researchershave studied combined conduction and/or radiation heat transferwith non-Fourier effect. In the presence of volumetric radiation,owing to the angular dependence of radiation that leads to inte-gro-differential radiative transfer equation, the formulation andsolution become very much involved. Though many researchershave investigated non-Fourier heat conduction alone[21,26,28,31,33], some have done the analyses in the presence ofvolumetric radiation [22–25,27,29,32]. Different numerical radia-tive transfer methods like the P-N approximation [22], discreteordinate method (DOM) [12,34], the finite volume method (FVM)[24,25,27,29,32], etc., have been used to compute the radiativeinformation needed in the energy equation which too has beensolved using different methods like the Green’s function method[21] the finite difference method (FDM) [22] and the FVM [27,29].

During the past two decades, a wide range of problems in sci-ence and engineering have been analyzed using the lattice Boltz-mann method (LBM) [35–37]. LBM has found extensive usage inthe study of fluid flow and thermal problems [35–37]. In the recentpast, Mishra and co-workers and others have used LBM to analyzemany heat transfer problems involving thermal radiation [38–45].Recently Chaabane et al. [43–45] have used the LBM to formulateand solve the energy equations of combined mode conductionand radiation heat transfer problems in 2-D cylindrical enclosure[43] and 2-D rectangular enclosure [44,45]. Heat transfer by con-duction was assumed to follow Fourier’s law. In their work, theyused control volume finite element method to compute the diver-gence of radiative heat flux needed in the LBM formulation. Theusage of the LBM to formulate and solve the energy equations ofdifferent kinds of combined mode problems in various geometriesby all authors was successful, and the experience was encouraging.

Mishra and co-workers have also used LBM to solve non-Fourierconduction and radiation heat transfer in a planar [24,25] andcylindrical geometry [32]. Without radiation, Mishra and Sahai[31] have recently extended application of the LBM to non-Fourier

heat conduction in a cylindrical and spherical geometry. However,as far as application of the LBM to analyze heat transfer with non-Fourier conduction and radiation in a spherical shell is concerned,no work has been reported so far. The present work, therefore,aims at extending the usage of the LBM to formulate and solvethe energy equation of a combined mode non-Fourier conductionand radiation in a concentric spherical shell.

In the present work, the energy equation of a combined modenon-Fourier conduction and radiation in a concentric sphericalshell containing radiating and conducting medium is formulatedand solved using the LBM. The volumetric radiative informationneeded in the solution of the energy equation is computed usingthe FVM. Effects of operating and geometric parameters like theextinction coefficient, the scattering albedo, the conduction–radia-tion parameter, the boundary emissivity, the radius ratio and themagnitudes of the thermal perturbations of the boundaries on nor-malized radial temperature distributions at different instantsincluding the steady-state (SS) are analyzed. At the SS, for all theparameters, radial distributions of conductive, radiative and totalenergy flow rates are also studied. For the same number of controlvolumes and rays, in all cases, distributions of SS temperature andenergy flow rate obtained using the LBM are compared againstthose obtained using the FVM. The number of iterations for theSS results in LBM and FVM are provided.

In the following sections, first the energy equation of problem isformulated in the LBM approach. LBM approach being termed as amesoscopic, concurrently, the governing equation in the macro-scopic (continuum) approach is provided. Using Chapman–Enskogmulti-scale expansion, the consistency of the LBM equation isproved. Expression for calculation of volumetric radiative informa-tion needed in the energy equation is provided and its solutionusing the FVM is briefly discussed. In the next section on resultsand discussion, effects of various parameters on temperature andenergy flow rate distributions are analyzed. Conclusions are madeat the end.

Fig. 1. (a) Schematic of the concentric spherical enclosure and (b) control volumeused in FVM for calculation of radiative information and lattices in the LBM andcontrol volumes in FVM for solving energy equation.

S.C. Mishra, H. Sahai / International Journal of Heat and Mass Transfer 68 (2014) 51–66 53

2. Formulation, validation, grid and ray independency tests

2.1. Formulation

In the LBM approach, following Bhatanagar–Gross–Krook

approximation [46], with collision term given as Xi ¼ðf ð0Þ

iðr;tÞ�fiðr;tÞÞ

sfor a spherical geometry (Fig. 1(a)), in non-dimensional form, con-duction heat transfer by Fourier’s law is proposed as

fiðr þ eiDt; t þ DtÞ � fiðr; tÞ ¼ �Dts½fiðr; tÞ � f ð0Þi ðr; tÞ� �

ð2aiqCDtÞri

;

i ¼ 1; . . . ;M ð1Þ

where the non-dimensional terms fi; r; ei; t; s; f ð0Þi ; ai and qC representparticle distribution functions, position, velocity with which infor-mation propagates along the ith lattice link, time, relaxation time,equilibrium particle distribution functions, lattice specific weightand conduction heat flux, respectively. In a 1-D problem, in anygeometry, the lattice used is D1Q2 (M = 2) (Fig. 1(b)), and for thislattice, with ei ¼ Dr

Dt as the speed with which particle distributionfunction moves to the neighboring lattice, the relaxation time s isgiven by

s ¼ 1e2

i

þ Dt2

ð2Þ

In Eq. (1), for the D1Q2 lattice, the weights a1 ¼ a2 ¼ 12, and the

velocities e1 ¼ �e2 ¼ DrDt. The equilibrium particle distribution func-

tion f ð0Þi in Eq. (1) is given by

f ð0Þi ¼ aiT ð3Þ

With fi known from the solution of Eq. (1), temperature T and con-duction heat flux qC distributions are the zeroeth and the first mo-ments of the particle distribution functions fi.

T ¼X

i

f ð0Þi ¼X

i

fi ð4aÞ

qC ¼X

i

fiei ¼X

i

f ð0Þi ei ð4bÞ

When the finite propagation speed of the conduction wave front isaccounted in non-Fourier conduction, Eq. (1) gets modified to

fiðr þ eiDt; t þ DtÞ � fiðr; tÞ ¼ �Dts½fiðr; tÞ � f ð0Þi ðr; tÞ�

� Dtð2aiqCÞri

� Dtð2bieiqCÞ ð5Þ

where biðb1 ¼ b2Þ ¼ 12 is the weight in the D1Q2 lattice and the equi-

librium distribution function f ð0Þi gets modified to

f ð0Þi ¼ aiT þ bieiqC ð6Þ

In the presence of volumetric radiation, with b as the extinctioncoefficient, k as the thermal conductivity, r is the Stefan–Boltzmannconstant, N ¼ kb

4rT3ref

as the conduction–radiation parameter and qR as

the radiative heat flux, Eq. (5) becomes

fiðr þ eiDt; t þ DtÞ � fiðr; tÞ ¼ �Dts½fiðr; tÞ � f ð0Þi ðr; tÞ�

� Dtð2aiqCÞri

� Dtð2bieiqCÞ

� brref aiDt8N

1r2

@ðr2qRÞ@r

� �ð7Þ

In the macroscopic approach, with conduction and radiation heattransfer through a spherical shell, the energy balance yields

@T@t¼ � 1

r2

@ðr2qCÞ@r

� brref

8N1r2

@ðr2qRÞ@r

ð8Þ

where rref = r2 � r1. When the finite propagation speed of the con-duction wave front is considered, the relationship between the in-duced conduction heat flux qC and the imposed temperaturegradient @T

@r is represented by

@qC

@tþ 2qC ¼ �

@T@r

ð9Þ

Substitution of qC from Eq. (9) in Eq. (8) results in

@2T@t2 þ 2

@T@t¼ 1

r2

@

@rr2 @T@r

� �

� brref

4N1r2

@ðr2qRÞ@r

� �þ 1

2@

@t1r2

@ðr2qRÞ@r

� �� ð10Þ

In Eqs. (7) and (8), the volumetric radiation appearing as 1r2

@ðr2qRÞ@r is

given by

1r2

@ðr2qRÞ@r

¼ brrefð1�xÞp

ð4pT4 � GÞ ð11Þ

wherex is the scattering albedo and G is the incident radiation. Its com-putation using the FVM will be briefly presented after we show theconsistency of the LBM formulation (Eq. (7)) with Eq. (10). This consis-tency is proved through the Chapman–Enskog multi-scale expansion.


If the left hand side of Eq. (7) is expanded in both time andspace dimensions, we get

fiðr þ eiDt; t þ DtÞ ¼ fiðr; tÞ þ Dt@fi

@tþ Dt

@ðeifiÞ@r

þ OðDt2Þ þ OðDr2Þ

ð12Þ

With t as the expansion parameter, with respect to the equilibriumparticle distribution function f ð0Þi , fi can be written as

fiðr; tÞ ¼ fi ¼ f ð0Þi þ tf ð1Þi þ Oðt2Þ; jtj << 1 ð13Þ

Substituting Eq. (13) in the RHS of Eq. (7), we get

@f ð0Þi

@tþ t

@f ð1Þi

@tþ @ðeif

ð0Þi Þ

@rþ t

@ðeifð1Þi Þ

@r

¼ �1s tf ð1Þi � brref ai

8N1r2

@ðr2qRÞ@r

� �� 2aiqC

ri� 4biei

qC

2

þ OðDtÞ þ Oðt2Þ ð14Þ

Summing Eq. (14) over all states, i.e., i ¼ 1; . . . ;M, and noting thatfrom Eqs. (15a)–(15c)

jtj << 1; ð15aÞ

Xi

f ð1Þi ¼ 0 ð15bÞ

Xi

Xi

eifð1Þi

!¼ 0 ð15cÞ

Eq. (14) becomes

Xi

@f ð0Þi

@t|fflfflfflfflffl{zfflfflfflfflffl}I

þX

it@f ð1Þi

@t|fflfflfflfflfflfflffl{zfflfflfflfflfflfflffl}II

þX

it@ðeif

ð1Þi Þ

@r|fflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflffl}III

¼ �X

i

1stf ð1Þi|fflfflfflfflfflfflffl{zfflfflfflfflfflfflffl}

IV

�X

i

brref ai

8N1r2

@ðr2qRÞ@r

� �|fflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl}

V

�X

i

2aiqC

riþ @ðeif

ð0Þi Þ

@r

!|fflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl}

VI

�X

i4biei

qC

2|fflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflffl}VII

þ OðDtÞ þ Oðt2Þ ð16Þ

In Eq. (16), terms II, III, IV and VII = 0.0, and with the remainingterms, the resulting equation (Eq. (17)) is the following

@T@t|{z}

I

¼ � 1r2

@ðr2qCÞ@r

� �|fflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflffl}

VI

� brref

8N1r2

@ðr2qRÞ@r|fflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflffl}

V

þOðDtÞ þ Oðe2Þ ð17Þ

Equation (17) is the same as Eq. (8). This proves recovery of Eq. (8).Multiplying Eq. (14) by the propagation speed ei, and summing

it over all states, we get

@P

iðeifð0Þi Þ

� �@t

þ t@P

iðeifð1Þi Þ

� �@t

þ@P

iðe2i f ð0Þi Þ

� �@r

þ t@P

iðe2i f ð1Þi Þ

� �@r

¼ �1stX

i

eifð1Þi � qC

2

Xi

4aiei

riþ 4bie2

i

� �

� brref aiei

8N1r2

@ðr2qRÞ@r

� �ð18Þ

Noting thatP

ibi ¼ 1,P

iei ¼ 0, and using Eqs. (15b) and (15c), Eq.(18) reduces to

@P

iðeifð0Þi Þ

� �@t|fflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflffl}¼@q@t

þt@P

iðeifð1Þi Þ

� �@t|fflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflffl}¼0

þ@P

iðe2i f ð0Þi Þ

� �@r|fflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflffl}¼@T@r

þt@P

iðe2i f ð1Þi Þ

� �@r|fflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflffl}¼0

¼�1stX

ieifð1Þi|fflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflffl}

¼0

�qC

2

Xi

4aiei

ri

� �|fflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflffl}

¼0

þX

i4bie2

i|fflfflfflfflfflffl{zfflfflfflfflfflffl}¼4

0BBB@

1CCCA

�brref aiei

8N1r2

@ðr2qRÞ@r

� �|fflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl}

¼0

ð19Þ

It is seen that the first order deviation is zero, and with this Eq. (19)leads to the following equation, which is Eq. (9)

@q@tþ @T@r¼ �2qC þ OðDtÞ þ Oðt2Þ ð20Þ

With the recovery of the governing Eqs. (8) and (9), the consistencyof the LBM scheme, (Eq. (7)) is thus established. It is to be noted thatEq. (10) results from Eqs. (8) and (9).

In a spherical geometry, radiation is azimuthally symmetric, i.e.,with hð0 6 h 6 pÞ as the polar angle and /ð0 6 / 6 2pÞ as the azi-muthal angle, the intensity of radiation Iðh;/Þ ¼ IðhÞ. Therefore,intensities need to be traced over the polar space only and, for a gi-ven h, to account for its uniform variation over the azimuthal space/ð0 6 / 6 2pÞ, the expressions of the incident radiation G and theheat flux qR are multiplied by 2p. Therefore, in Eq. (11), the inci-dent radiation G and the radiative heat flux are given by and com-puted from the following

G ¼ 2pZ p

0IðhÞ sin hdh � 2

XNh

m¼1

ImDXm ð21Þ

qR ¼ 2pZ p

0I cos h sin hdh � 2

XNh

m¼1

ImDmn ð22Þ

where in Eqs. (21) and (22), Nh is the total number of discrete polarangles, Im is the intensity in the discrete polar direction having in-dex m, and the weights DXm and Dm

n are given by [42]

DXm ¼Z

DXmdX ¼

Z 2p

/¼0

Z hmþDh2

hm�Dh2

sin hdhd/

¼ 2p sin hm sinDh2

� �ð23Þ

Dm ¼Z

DXmcos hdX ¼ 2p

Z hmþDh2

hm�Dh2

cos h sin hdh

¼ p sin hm cos hm sinðDhmÞ ð24Þ

In Eqs. (21) and (22), intensity I is obtained from the solution ofradiative transfer equation. With an absorbing, emitting andisotropically scattering medium, with l ¼ cos h, it is given by[17,19]

lr2

� � @½r2I�@rþ 1

r

� �@½ð1� l2ÞI�

@l ¼ �bI þ jaIb þrs

4pG ð25Þ

For a diffuse-gray surface having emissivity ew and temperatureTw, Eq. (25) is solved with the following radiative boundarycondition

Iw ¼ewrT4

w

pþ 1� ew

pXNh=2

m¼1

I�;mDm ð26Þ

where in Eq. (26), on the right-hand-side, the first and the secondterms represent emitted and reflected components of intensity Iw,respectively. In the reflected component, I�,m represents intensities


in the half-spherical space that are incident on the surface, and theirweights are accordingly chosen.

In the present work, the solution of radiative transfer equation(Eq. (25)) follows the FVM approach proposed by Kim et al. [18,19].To avoid repetition, specific details have not been included in thissection.

With the volumetric radiative information 1r2

@ðr2qRÞ@r needed in the

energy equation known from the FVM [18,19], the solution of Eq.(7) of the LBM approach yields particle distribution function fi,and then from Eqs. (4a) and (4b), temperature T and conductiveheat flux qC, respectively are calculated. The LBM approach in-volves marching in time. From the knowledge of the initial andboundary conditions, the volumetric radiative information usingthe FVM is calculated first. The LBM equation (Eq. (7)) is solvednext. Implementations of the boundary conditions, and collisionand propagation steps in the LBM for evaluation of fi at a new timelevel, follow that given in Mishra and Roy [42]. At a new time step,with temperature field known from the previous time step, the vol-

umetric radiative information 1r2

@ðr2qRÞ@r is updated, and substituted

back to the energy equation that gives the new temperature field.The process is repeated until the absolute temperature differencebetween two consecutive time levels at each node is6 1:0� 10�6. At this state, the steady-state (SS) was assumed tohave reached.

To ensure the correctness of the LBM formulation for the com-bined mode non-Fourier conduction and radiation problem, for allsets of parameters, the energy equation (Eq. (10)) of the problemwas solved using the FVM, and like the LBM, the needed radiativeinformation was also computed using the FVM. Time and spacediscretization of Eq. (10) followed the steps given in Refs. [27,29].

In the present work, all LBM results for non-Fourier conduc-tion–radiation are compared against the FVM results. Before wepresent results for non-Fourier conduction–radiation, resultsfrom the present formulation with Fourier conduction and radia-tion are validated against those available in the literature.Next, results on grid- and ray-independency are provided for

Table 1Comparisons of non-dimensional conductive heat flux qC and total heat flux qT at the inns2 = 2.0 and e1 = e2 = e = 1.0.

N x Source Conductive heat flux

Inner sphere qC(g1) Ou

1.0 0.9 Present 2.0120 0.5Ref. [16] 2.0062 0.5Ref. [17a] 2.0062 0.5Ref. [17b] 2.0064 0.5

0.5 Present 2.0312 0.5Ref. [16] 2.0303 0.5Ref. [17a] 2.0606 0.5Ref. [17b] 2.0312 0.5


0.1 0.9 Present 2.0612 0.5Ref. [16] 2.0628 0.5Ref. [17a] 2.0629 0.5Ref. [17b] 2.0647 0.5



non-Fourier conduction and radiation. Following that results anddiscussion are provided.

2.2. Validation

With non-Fourier conduction, before we analyze effects of var-ious parameters on temperature and energy flow rate distribu-tions in a concentric spherical shell using the LBM and compareits SS results against the FVM, we first validate the LBM formula-tion for the energy equation with Fourier conduction and calcula-tion of the radiative information using the FVM. This validation inTable 1 is shown against the results of Li et al. [16] and Dlala et al.[17]. It is to be noted that Chebyshev collocation spectral DOMwas used to compute the radiative information in [16], whilethe methods used in [17] were the DOM and the finite Chebyshevtransform. In Table 1, Refs. [17a] and [17b] represent the resultsfrom DOM and the finite Chebyshev transform taken fromRef. [17].

With normalized radial distance defined as g ¼ r�r1r2�r1

, for innerradius g1 = 0.5 and outer radius g2 = 1.0, extinction coefficientb = 2.0, inner surface temperature T1

T2¼ 0:5 and T2 = 1.0, non-

dimensional conductive heat flux qC and the total heat flux qT atthe inner g1 and the outer g2 surfaces of the concentric sphereshell obtained from LBM are compared with that provided in[16,17]. For conduction–radiation parameter N = 1.0 and 0.1,these comparisons are shown for the scattering albedox = 0.9, 0.5 and 0.1. Surfaces are black. For all combinations of(N, x), LBM results are in excellent comparison with those of Liet al. [16] and Dlala et al. [17].

2.3. Grid and ray-independency tests

After validation of the LBM results for Fourier conduction andradiation, we next fix numbers of lattices in the LBM, control vol-umes (CV) in the FVM, and discrete rays in the FVM for calculation

er (g1 = 0.5) and outer (g1 = 1.0) spheres for different combinations of N and x with

Total heat flux

ter sphere qC(g2) Inner sphere qT(g1) Outer sphere qT(g2)

099 2.4102 0.5938042 2.3735 0.5934040 2.3892 0.5973041 2.3851 0.5963162 2.3852 0.5955165 2.3877 0.5969161 2.4032 0.6008160 2.3994 0.5998250 2.3920 0.5995248 2.3994 0.5999243 2.4149 0.6037243 2.4112 0.6028

409 5.7187 1.4658406 5.7350 1.4338395 5.8924 1.4731393 5.8507 1.4627495 5.8522 1.4375493 5.8706 1.4676462 6.0258 1.5064457 5.9862 1.4966172 5.9558 1.5066178 5.9727 1.4923148 6.1279 1.5319135 6.0889 1.5222


of the divergence of radiative heat flux needed in the energyequation. This is done for non-Fourier conduction and radiation.For grid independence, with 12 rays, normalized temperature

H ¼ T�T1T2�T1

� �distributions at time f ¼ C2t

2a

� �¼ 0:2;0:8 and at SS are

shown in Fig. 2(a)–(c), respectively. For 300 CV, Fig. 2(d)–(f) showresults for ray independency. Grid and ray independency results inFig. 2(a)–(f) are shown for b ¼ 1:0, x ¼ 0:5, N ¼ 0:05, e ¼ 1:0 and

(a)

(b)

(c)

Distance, η

Temperature,Θ

0 0.2 0.4 0.6 0.8 10

0.2

0.4

0.6

0.8

1

50

150

300

450

ζ = 0.2

CV

Distance, η

Temperature,Θ

0 0.2 0.4 0.6 0.8 10

0.2

0.4

0.6

0.8

1

50

150

300

450

ζ = 0.8

CV

Distance, η

Temperature,Θ

0 0.2 0.4 0.6 0.8 10

0.2

0.4

0.6

0.8

1

50

150

300

450

SS

CV

Fig. 2. Effect of number of CVs on radial g temperature H distributions at time (a) ndistributions at (d) n = 0.2, (e) n = 0.8 and (f) SS; b ¼ 1:0, x ¼ 0:5, N ¼ 0:05, e ¼ 1:0, RR ¼

RR ¼ r2r1

� �¼ 2:0. It is observed from H distributions shown in

Fig. 2(a)–(c) that no noticeable difference is found beyond300 CVs. Similarly, with 300 CVs, 12 and 16 rays provide almostthe same H distributions. For the same set of parameters, gridand ray independency tests were also performed with the FVMapproach. Findings were the same. Thus, in the following, in bothLBM and FVM all results were obtained with 300 CVs and 12 rays.

(d)

(e)

(f)

Distance, η

Temperature,Θ

0 0.2 0.4 0.6 0.8 10

0.2

0.4

0.6

0.8

1

4

8

12

16

ζ = 0.2

Rays

Distance, η

Temperature,Θ

0 0.2 0.4 0.6 0.8 10

0.2

0.4

0.6

0.8

1

4

8

12

16

ζ = 0.8

Rays

Distance, η

Temperature,Θ

0 0.2 0.4 0.6 0.8 10

0.2

0.4

0.6

0.8

1

4

8

12

16

SS

Rays

= 0.2, (b) n = 0.8 and (c) SS; effect of number of rays on radial g temperature H2:0.


3. Results and discussion

In the following section, LBM results of normalized tempera-

ture H ¼ T�T1T2�T1

� �vs. normalized radial distance g ¼ r�r1

r2�r1

� �are pro-

vided at different instants f ¼ C2t2a

� �including the SS. At the SS,

radial g distributions of conductive QCð¼ 4pr2 � qCÞ, radiative

(a)

(b)

(c)

Distance, η

Temperature,Θ

0 0.2 0.4 0.6 0.8 1

0

0.2

0.4

0.6

0.8

1

SS: LBM (31063)SS: FVM (38121)Cond

ζ = 1.0

ζ = 0.2 ζ = 0.5

ζ = 0.8

β = 0.1

ζ = 2.0

Distance, η

Temperature,Θ

0 0.2 0.4 0.6 0.8 1

0

0.2

0.4

0.6

0.8

1


ζ = 1.0

ζ = 0.2ζ = 0.5

ζ = 0.8

β = 1.0

ζ = 2.0

Distance, η

Temperature,Θ

0 0.2 0.4 0.6 0.8 1

0

0.2

0.4

0.6

0.8

1


ζ = 1.0

ζ = 0.2

ζ = 0.5

ζ = 0.8

β = 2.0ζ = 2.0

Fig. 3. Radial g temperature H at different time n levels including the SS for extinctionradiative WR and total WT energy flow rates for b (d) 0.1, (e) 1.0 and (f) 2.0; x ¼ 0:5, N

QRð¼ 4pr2 � qRÞ and total energy flow rate Q Tð¼ Q C þ Q RÞ areprovided. In all cases, radial distributions of the SS temperatureH and the energy flow rates are compared against those obtainedfrom the FVM. In both LBM and FVM, the volumetric radiativeinformation is obtained using the FVM [18,19]. Withr2 = 2r1 = 1.0, radius ratio RR ¼ r2

r1¼ 2:0, results are analyzed for

the effects of (a) the extinction coefficient b, (b) the scattering

(d)

(e)

(f)

Distance, η

Energyflow

rate,Ψ

0 0.2 0.4 0.6 0.8 10

2

4

6

8

LBMFVM

ΨT

ΨR

ΨC

β = 0.1

Distance, η

Energyflow

rate, Ψ

0 0.2 0.4 0.6 0.8 14

6

8

10

12

14

LBMFVM

ΨT

ΨR

ΨC

β = 1.0

Distance, η

Energyflow

rate,Ψ

0 0.2 0.4 0.6 0.8 14

6

8

10

12

14

16

18

20

LBMFVM

ΨT

ΨR

ΨC

β = 2.0

coefficient b (a) 0.1, (b) 1.0 and (c) 2.0; at SS, radial distributions of conductive WC,¼ 0:1, e ¼ 1:0, RR ¼ 2:0.


albedo x, (c) conduction–radiation parameter N, (d) boundaryemissivity eð¼ e1 ¼ e2Þ, and (e) different magnitudes of thermalperturbation. Results are also analyzed for different values ofthe radius ratio RR. Further, to see the effect of radiation, whilestudying the effects of b, x, N and e on temperature distributions,SS temperature distributions for pure conduction are alsoprovided.

(a)

(b)

(c)

Distance, η

Temperature,Θ

0 0.2 0.4 0.6 0.8 1

0

0.2

0.4

0.6

0.8

1


ζ = 1.0ζ = 0.2ζ = 0.5

ζ = 0.8

ω = 0.1

ζ = 2.0

Distance, η

Temperature,Θ

0 0.2 0.4 0.6 0.8 1

0

0.2

0.4

0.6

0.8

1


ζ = 1.0ζ = 0.2

ζ = 0.5

ζ = 0.8

ω = 0.5

ζ = 2.0

Distance, η

Temperature,Θ

0 0.2 0.4 0.6 0.8 1

0

0.2

0.4

0.6

0.8

1


ζ = 1.0

ζ = 0.2ζ = 0.5

ζ = 0.8

ω = 0.9

ζ = 2.0

Fig. 4. Radial g variations of temperature H at different time n levels including the SS foradiative WR and total WT energy flow rates for x (c) 0.1, (d) 0.5 and (e) 0.9; b = 1.0, N =

3.1. Effect of extinction coefficient on temperature and energy flowrate distributions

In a heat transfer problem involving volumetric radiation,extinction coefficient bð¼ ja þ rsÞ is an important parameter. Ahigher value of b implies smaller mean-free-path of the thermalradiation and hence increased opacity which makes medium

(d)

(e)

(f)

Distance, η

Energyflow

rate,Ψ

0 0.2 0.4 0.6 0.8 14

6

8

10

12

14

LBMFVM

ΨT

ΨR

ΨC

ω = 0.1

Distance, η

Energyflow

rate,Ψ

0 0.2 0.4 0.6 0.8 14

6

8

10

12

14

LBMFVM

ΨT

ΨR

ΨC

ω = 0.5

Distance, η

Energyflow

rate, Ψ

0 0.2 0.4 0.6 0.8 14

6

8

10

12

14

LBMFVM

ΨT

ΨR

ΨC

ω = 0.9

r scattering albedo x (a) 0.0, (b) 0.5 and (c) 0.9; at SS, variations of conductive WC,0.1, e = 1.0, RR = 2.0.


radiatively more participating. For b = 0.1, 1.0 and 2.0, radial

g ¼ r�r1r2�r1

� �temperature distributions H ¼ T�T1

T2�T1at different instants

f including the SS are shown in Fig. 3, respectively. For these re-sults, x = 0.5, N = 0.1, e ¼ 1:0 and RR ¼ 2:0. In all cases, the SSHdistributions are compared against that obtained from the FVM.In all these figures, wave nature of heat transfer by conduction isevident. The wave front starts from the hot (inner) surface which

(a)

(b)

(c)

Distance, η

Temperature,Θ

0 0.2 0.4 0.6 0.8 1

0

0.2

0.4

0.6

0.8

1


ζ = 0.2ζ = 0.8

N = 0.01

ζ = 0.05

Distance, η

Temperature,Θ

0 0.2 0.4 0.6 0.8 1

0

0.2

0.4

0.6

0.8

1


ζ = 1.0

ζ = 0.2ζ = 0.5

ζ = 0.8

N = 0.1

ζ = 2.0

Distance, η

Temperature, Θ

0 0.2 0.4 0.6 0.8 1

0

0.2

0.4

0.6

0.8

1


ζ = 1.0

ζ = 0.2 ζ = 0.5

ζ = 0.8

N = 1.0

ζ = 2.0

Fig. 5. Radial g temperature H at different time n levels including the SS for conduction–WC, radiative WR and total WT energy flow rates for N (d) 0.01, (e) 0.1 and (f) 1.0; b ¼ 1

is the location of the thermal perturbation. As it moves towardsthe cold (outer) surface, it undergoes damping, and the magnitudeof the discontinuity in H profile decreases. When it reaches theouter surface at time f = 1.0, it is not fully damped, and as observedin the figures, it reflects back to the inner surface. With successivereflections, discontinuity in H profile dies out completely, and atthis instant, it is said to have reached the SS.

(d)

(e)

(f)

Distance, η

Energyflow

rate, Ψ

0 0.2 0.4 0.6 0.8 10

10

20

30

40

50

60

70

80

LBMFVM

ΨT

ΨR

ΨC

N = 0.01

Distance, η

Energyflow

rate,Ψ

0 0.2 0.4 0.6 0.8 1

6

8

10

12

14

LBMFVM

ΨT

ΨR

ΨC

N = 0.1

Distance, η

Energyflow

rate,Ψ

0 0.2 0.4 0.6 0.8 10

2

4

6

8

LBMFVM

ΨT

ΨR

ΨC

N = 1.0

radiation parameter N (a) 0.01, (b) 0.1 and (c) 1.0; at SS, distributions of conductive:0, x ¼ 0:5, e ¼ 1:0, RR ¼ 2:0.


With increase in b, at any instant, decrease in the magnitudeof the discontinuity in H profile is attributed to the increasedcontribution of volumetric radiation. At any instant, as can beseen from Fig. 3(a), for b = 0.1, discontinuity in H profile is morethan that for b = 1 (Fig. 3(b)) and b = 2 (Fig. 3(c)). Further,radiation being a fast process, its increased contribution leadsto faster attainment of the SS. For all values of b, SSH profiles

(a)

(b)

(c)

Distance, η

Temperature,Θ

0 0.2 0.4 0.6 0.8 1

0

0.2

0.4

0.6

0.8

1


ζ = 1.0

ζ = 0.2

ζ = 0.5

ζ = 0.8

ε = 0.1

ζ = 2.0

Distance, η

Temperature,Θ

0 0.2 0.4 0.6 0.8 1

0

0.2

0.4

0.6

0.8

1


ζ = 1.0ζ = 0.2ζ = 0.5

ζ = 0.8

ε = 0.5

ζ = 2.0

Distance, η

Temperature,Θ

0 0.2 0.4 0.6 0.8 1

0

0.2

0.4

0.6

0.8

1


ζ = 1.0ζ = 0.2

ζ = 0.5

ζ = 0.8

ε = 1.0

ζ = 2.0

Fig. 6. Radial g variations of temperature H at different time n levels including the Sdistributions of conductive WC, radiative WR and total WT energy flow rates for e1 = e2 =

of LBM and FVM match exceedingly well. In conduction domi-nated case, for lower value of b, both LBM and FVM take longertime to reach the SS, and the difference between the numbersof iterations to reach the SS increases. For b = 0.1, (Fig. 3(a))number of iterations in LBM and FVM are 31,063 and 38,121,respectively, while the same for b = 2.0, are 1394 and 1399,respectively.

(d)

(e)

(f)

Distance, η

Energyflow

rate,Ψ

0 0.2 0.4 0.6 0.8 10

2

4

6

8

10

LBMFVM

ΨT

ΨR

ΨC

ε = 0.1

Distance, η

Energyflow

rate, Ψ

0 0.2 0.4 0.6 0.8 14

6

8

10

12

14

LBMFVM

ΨT

ΨR

ΨC

ε = 0.5

Distance, η

Energyflow

rate,Ψ

0 0.2 0.4 0.6 0.8 14

6

8

10

12

14

16

18

20

LBMFVM

ΨT

ΨR

ΨC

ε = 1.0

S for emissivities of the cylinder walls e1 = e2 = e (a) 0.1, (b) 0.5 and (c) 1.0; at SS,e (d) 0.1, (e) 0.5 and (f) 1.0; b ¼ 1:0, x ¼ 0:5, N ¼ 0:05, RR ¼ 2:0.


In a combined mode problem, temperature is the cumulative ef-fect of all the modes of heat transfer, and contributions of differentmodes cannot be segregated. However, this is no so with heat fluxor energy flow rate. Contributions of respective modes of heattransfer to the total flux or energy flow rates are quantifiable. Ina combined mode conduction–radiation problem, total energy flowrate is the summation of energy flow rates due to conduction andradiation. At the SS, in a sphere, it is not the total heat flux, but the

(a)

(b)

(c)

Distance, η

Temperature,Θ

0 0.2 0.4 0.6 0.8 1

0

0.2

0.4

0.6

0.8

1SS: LBM (29502)SS: FVM (32054)

ζ = 1.0

ζ = 0.2

ζ = 0.5

ζ = 0.8

RR = 1.1

ζ = 2.0

Distance, η

Temperature,Θ

0 0.2 0.4 0.6 0.8 1

0

0.2

0.4

0.6

0.8

1

SS: LBM (4002)SS: FVM (4055)

ζ = 1.0

ζ = 0.2 ζ = 0.5

ζ = 0.8

RR = 2.0

ζ = 2.0

Distance, η

Temperature,Θ

0 0.2 0.4 0.6 0.8 1

0

0.2

0.4

0.6

0.8

1

SS: LBM (1502)SS: FVM (1488)

ζ = 1.0ζ = 0.2

ζ = 0.5

ζ = 0.8

RR = 4.0

ζ = 2.0

Fig. 7. Radial g variation of temperature H at different time n levels including the SS fradiative WR and total WT energy flow rates for RR (a) 1.1, (b) 2.0 and (c) 4.0; b ¼ 1:0, x

total energy flow rate that remains constant along the radial dis-tance. The conductive and radiative energy flow rates vary in sucha way that the total energy flow rate remains constant. Forb = 0.1, 1.0 and 2.0, at the SS, these variations are shown inFig. 3(d)–(f), respectively. Owing to increase in radiative heattransfer, with increase in b, the total energy flow rate increases,and while for b = 0.1, the major contribution to the total energyflow rate comes from conduction (Fig. 3(d)). Opposite is the case

(d)

(e)

(f)

Distance, η

Energyflow

rate,Ψ

0 0.2 0.4 0.6 0.8 10

10

20

30

40

LBMFVM

ΨT

ΨR

ΨC

RR = 1.1

Distance, η

Energyflow

rate,Ψ

0 0.2 0.4 0.6 0.8 14

5

6

7

8

9

10

11

12

13

14

LBMFVM

ΨT

ΨR

ΨC

RR = 2.0

Distance, η

Energyflow

rate,Ψ

0 0.2 0.4 0.6 0.8 13

4

5

6

7

8

9

10

11

12

LBMFVM

ΨT

ΨR

ΨC

RR = 4.0

or radius ratios RR (a) 1.1, (b) 2.0 and (c) 10.0; at SS, variations of conductive WC,¼ 0:5, N ¼ 0:1, e ¼ 1:0.


with b = 2.0. For all values of b, radial distributions of energy flowrates of LBM and FVM match exceedingly well.

3.2. Effect of scattering albedo on temperature and energy flow ratedistributions

Scattering albedo x rsjaþrs

� �provides quantitative information

about the extent of scattering. For x ¼ 0, there is no loss of radiationdue to scattering, hence radiative contribution will be the maximum.On the other hand, all energy is scattered when x ¼ 1:0, and radia-tion does not appear in the governing energy equation. The situationis like that of a pure conduction. With b ¼ 1:0, N ¼ 0:1, e ¼ 1:0 andRR ¼ 2:0, for x ¼ 0:1;0:5 and 0.9, radial temperature H distribu-tions at time f = 0.2, 0.5, 0.8, 1.0, 2.0 and at SS are shown inFig. 4(a)–(c). It is observed from Fig. 4(a)–(c) that with increase inthe value of the scattering albedo x, at any instant f, the magnitudeof discontinuity inHprofile increases with increase inx. A pure con-duction like situation is observed forx = 0.9 (Fig. 4(c)). Since with in-crease in x, radiative contribution decreases, more iterations areneeded to reach the SS. For x = 0.1, in the LBM, the SS is reached at2995 iterations (Fig. 4(a)), while the same for x = 0.9, is 7502. Forall values of x, the SSH profiles from the LBM are in excellent agree-ment with those obtained from the FVM approach.

Effects of the scattering albedo x on radial distributions of con-ductive, radiative and total energy flow rates are shown inFigs. 4(d)–(f). Unlike the effect of the extinction coefficient b(Fig. 3(d)–(f)), effect of x on radial distributions of energy flowrates is not much pronounced. For the chosen values of parametersb ¼ 1:0, N ¼ 0:1, e ¼ 1:0 and RR ¼ 2:0, throughout the medium, thedifference between the radiative and the conductive energy flowrates decrease with increase in x. In all cases, energy flow ratesin LBM and FVM approaches have a good match.

3.3. Effect of conduction–radiation parameter on temperature andenergy flow rate distributions

In a combined mode conduction–radiation problem, relativecontributions of conduction and radiation are dictated by parame-

ter N ¼ kb4rT3

ref

� �. Radiation dominates for a lower value of N. For

N ? 0, the situation tends to that of the radiative equilibrium.For a higher value of N, conduction has dominance over radiation.With b ¼ 1:0, x ¼ 0:5, e ¼ 1:0 and RR ¼ 2:0, effects of N = 0.01,0.1and 1.0 on radial distributions of temperature at different instantsincluding the SS are shown in Fig. 5(a)–(c). It is observed fromFig. 5(a) that owing to the dominance of radiation (N = 0.01), com-pared to higher values of N = 0.1 (Fig. 5(b)) and N = 1.0 (Fig. 5(c)),the non-Fourier effect on temporal radial temperature distribu-tions are less pronounced. For the same reason, radiation being ahighly non-linear process, non-linearity in the SSH profile is morefor a lower value of N. Smaller number of iterations is needed forthe attainment of the SS. For N = 0.01, SS temperature is reachedwith 1342 iterations (Fig. 5(a)), while 9502 iterations are neededfor N = 1.0 (Fig. 5(c)). For all values of N, SS temperature H profilesobtained from the LBM approach match exceedingly well withthose obtained from the FVM approach. LBM is found to convergesooner than the FVM.

For N = 0.01, 0.1 and 1.0, SS radial distributions of conductive,radiative and total energy flow rates are shown in Fig. 5(d)–(f),respectively. It is evident from Fig. 5(d)–(f) that compared to theeffects of the extinction coefficient b (Fig. 3(d)–(f)) and the scatter-ing albedo x (Fig. 4(d)–(f)), the effect of N on distributions of en-ergy flow rates is significant. For N = 0.01, the major contributionto the total energy flow rate comes from radiation (Fig. 5(a)). ForN = 0.1 (Fig. 5(e)), the total energy flow rate decreases drastically,

and except near the boundaries, radiative contribution is slightlymore than that from conduction. For N = 1.0 (Fig. 5(f)), the contri-butions of conduction and radiation should be comparable, butowing to other radiative parameters, i.e., b ¼ 1:0, x ¼ 0:5,e ¼ 1:0, conduction has dominance over radiation. However, for re-duced contribution from radiation, total energy flow rate is lower.For all values of N, profiles of the energy flow rates in the LBM andFVM approaches match very well.

3.4. Effect of emissivity on temperature and energy flow ratedistributions

Emissivity e too has a significant effect on distributions of tem-perature and energy flow rates. If the emissivity of a surface is 1.0,it absorbs all radiations incident on it, and corresponding to itstemperature, emissions from it is the maximum. On the contrary,if a surface is perfectly reflecting (e = 0.0), it does not retain anyincident radiation. It gives back all to the medium. Thus, emissivityof surfaces in a thermal system becomes an important parameter.For b ¼ 1:0, x ¼ 0:5, N ¼ 0:05, RR ¼ 2:0 in Fig. 6(a)–(c), radial tem-perature distributions of temperature are shown fore ðe1 ¼ e2Þ ¼ 0:1;0:5 and 1.0, respectively. It is observed from thesefigures that when the boundaries are reflecting more (Fig. 6(a)), forloss of radiation, conduction has a dominance, and at any instant,discontinuities in temperature profiles are more, and the SS is at-tained late. For all values of e at the SS, LBM and FVM results matchwell, and LBM approach has faster convergence than the FVM.

For eðe1 ¼ e2Þ ¼ 0:1;0:5 and 1.0, SS radial distributions of con-ductive, radiative and total energy flow rates are compared inFig. 6(d)–(f). It is observed from Fig. 6(d)–(f) that with increase inemissivity of the boundaries, the total energy flow rate increases,and this increase is mainly due to enhanced contribution fromradiation. For e = 0.1, conduction dominates over radiation(Fig. 6(a)), while for e = 1.0, major contribution to the total energyflow rate comes from radiation. With increase in e, increase in con-ductive energy flow rate is attributed to the fact that with in-creased radiation owing to increase in e, temperature in themedium is more, and this thus leads to higher conductive energyflow rate. An interesting fact is observed from the profiles of radi-ative energy flow rate in Fig. 6(d)–(f). If the boundaries reflectmore, radiative energy at it will go down. Thus, in Fig. 6(d), fore = 0.1, the radiative energy flow rates at inner and outer bound-aries is lower than that for e = 0.5 and 1.0 as shown in Fig. 6(e)and (f), respectively. In all cases, the distributions of energy flowrates in the two approaches match exceedingly well.

3.5. Effect of radius ratio on temperature and energy flow ratedistributions

In the preceding cases, all the results were provided for radius

ratio RR ¼ r2r1

� �¼ 2:0. For a given value of the extinction coefficient

b, the optical depth b � (r2 � r1) increases with increase in RR andthis will have significant effect on distributions of temperature aswell as energy flow rates. Effect of RR on distributions of tempera-ture and energy flow rates are shown in Fig. 7. With b ¼ 1:0,x ¼ 0:5, N ¼ 0:1, e ¼ 1:0, radial temperature distributions areshown in Fig. 7(a)–(c) for RR = 1.1,2.0 and 4.0, respectively. It is ob-served from Fig. 7(a) that with RR = 1.1, the non-Fourier effect isvery prominent, while for when RR = 4.0 (Fig. 7(c)), the effect is re-duced drastically. With b = 1.0, for RR = 1.1, the optical thicknessb � (r2 � r1) is low, thus the effect of radiation is minimal. At anyinstant, discontinuity in temperature profile is much more as thethermal wave takes less time to travel from one surface to theother. A very high number of iterations (29,502) for the attainmentof the SS in this case further substantiate this fact.


At the SS, distributions of conductive, radiative and total energyflow rates are shown in Fig. 7(d)–(f). With increase in RR, the en-ergy flow rates decrease. With extinction coefficient b fixed, thisdecrease is attributed to increased optical thickness of the mediumthat causes reduction in radiative energy per unit volume. ForRR = 1.1, the major contribution to the total energy flow rate comesfrom conduction, and this is for the reason that for a physicallythinner medium, the temperature gradient is higher and this thusleads to higher energy flow rate by conduction. When the RR has

(a)

(b)

(c)

Distance, η

Temperature,Θ

0 0.2 0.4 0.6 0.8 1

0

0.2

0.4

0.6

0.8

1

SS: LBM (6502)SS: FVM (6887)

ζ = 1.0

ζ = 0.2 ζ = 0.5

ζ = 0.8

θ1 = 1.0θ2 = 0.1

ζ = 2.0

Distance, η

Temperature,Θ

0 0.2 0.4 0.6 0.8 1

0

0.2

0.4

0.6

0.8

1

SS: LBM (4002)SS: FVM (4055)

ζ = 1.0

ζ = 0.2 ζ = 0.5

ζ = 0.8

θ1 = 1.0θ2 = 0.5

ζ = 2.0

Distance, η

Temperature,Θ

0 0.2 0.4 0.6 0.8 1

0

0.2

0.4

0.6

0.8

1

SS: LBM (2002)SS: FVM (2344)

ζ = 1.0ζ = 0.2 ζ = 0.5ζ = 0.8

θ2 = 1.0θ2 = 0.9ζ = 2.0

Fig. 8. Radial g variation of temperature H at different time n levels including the SS for oWC, radiative WR and total WT energy flow rates for outer sphere temperature h2 (a) 0.1

increased to 4.0 (Fig. 7(f)), the temperature gradient is much lowerand hence, conductive energy flow is lower. For all cases, LBM andFVM results have a good match.

3.6. Effect of varying outer surface temperature on temperature andenergy flow rate distributions

While studying the effects of the extinction coefficient b, scat-tering albedo x, conduction–radiation parameter N, boundary

(d)

(e)

(f)

Distance, η

Energyflow

rate, Ψ

0 0.2 0.4 0.6 0.8 1

5

10

15

20

LBMFVM

ΨT

ΨR

ΨC

θ2 = 0.1

Distance, η

Energyflow

rate,Ψ

0 0.2 0.4 0.6 0.8 14

6

8

10

12

14

LBMFVM

ΨT

ΨR

ΨC

θ2 = 0.5

Distance, η

Energyflow

rate,Ψ

0 0.2 0.4 0.6 0.8 10.5

1

1.5

2

2.5

3

3.5

4

LBMFVM

ΨT

ΨR

ΨC

θ2 = 0.9

uter sphere temperature h2 (a) 0.1, (b) 0.5 and (c) 0.9; at SS, variations of conductive, (b) 0.5 and (c) 0.9; b ¼ 1:0, x ¼ 0:5, N ¼ 0:1, e ¼ 1:0.


emissivity e and RR on temperature and energy flow rate distribu-tions, the outer surface was maintained at the initial temperatureH2 = 0.1, and the system was thermally perturbed by raising tem-perature of the inner surface to H1 = 1.0. With thermophysical andoptical properties fixed for a given RR, it will be interesting to studythe effects of magnitudes of perturbations at the boundaries. Weconsider two situations. (a) with inner surface thermally perturbedto H1 = 1.0, the outer surface is at different initial temperaturesðH2 ¼ 0:1;0:5 and 0:9Þ and (b) the inner surface at different initial

)a(

)b(

)c(

Distance, η

Temperature,Θ

0 0.2 0.4 0.6 0.8 1

0

0.2

0.4

0.6

0.8

1

SS: LBMSS: FVM

ζ = 1.0ζ = 0.2

ζ = 0.5ζ = 0.8

θ1 = 0.1θ2 = 1.0

ζ = 2.0

Distance, η

Temperature,Θ

0 0.2 0.4 0.6 0.8 1

0

0.2

0.4

0.6

0.8

1

SS: LBMSS: FVM

ζ = 1.0ζ = 0.2

ζ = 0.5

ζ = 0.8

θ1 = 0.5θ2 = 1.0

ζ = 2.0

Distance, η

Temperature,Θ

0 0.2 0.4 0.6 0.8 1

0

0.2

0.4

0.6

0.8

1

SS: LBMSS: FVM

ζ = 1.0ζ = 0.2

ζ = 0.5

ζ = 0.8

θ1 = 0.9θ2 = 1.0

ζ = 2.0

Fig. 9. Radial g variation of temperature H at different time n levels including the SS for oWC, radiative WR and total WT energy flow rates for outer sphere temperature h2 (a) 0.1

temperatures ðH1 ¼ 0:1;0:5 and 0:9Þ when outer surface is ther-mally perturbed to H2 = 1.0. Temperature and energy flow rate dis-tributions for the aforementioned two cases are shown in Figs. 8and 9. With RR = 2.0, results in Figs. 8 and 9 are shown forb ¼ 1:0, x ¼ 0:5, N ¼ 0:1, e ¼ 1:0.

It is observed from Fig. 8(a)–(c) that with RR fixed, the rise ininitial temperature leads to reduced temperature gradient. Thusthe non-Fourier effect on temperature distribution is most pro-nounced for H2 = 0.1 (Fig. 8(a)), and it is least for H2 = 0.9

)d(

)e(

)f(

Distance, η

Energyflow

rate,Ψ

0 0.2 0.4 0.6 0.8 1-20

-18

-16

-14

-12

-10

-8

-6

-4

LBMFVM

ΨT

ΨR

ΨC

θ1 = 0.1θ2 = 1.0

Distance, η

Energyflow

rate,Ψ

0 0.2 0.4 0.6 0.8 1-16

-14

-12

-10

-8

-6

-4

LBMFVM

ΨT

ΨR

ΨC

θ1 = 0.5θ2 = 1.0

Distance, η

Energyflow

rate,Ψ

0 0.2 0.4 0.6 0.8 1-4

-3.5

-3

-2.5

-2

-1.5

-1

LBMFVM

ΨT

ΨR

ΨC

θ1 = 0.9θ2 = 1.0

uter sphere temperature h2 (a) 0.1, (b) 0.5 and (c) 0.9; at SS, variations of conductive, (b) 0.5 and (c) 0.9; b ¼ 1:0, x ¼ 0:5, N ¼ 0:1, e ¼ 1:0.


(Fig. 8(c)). Since the temperature gradient decreases with increasein H2, the SS is attained faster when H2 is more. For(H1, H2) = (1.0, 0.1) (Fig. 8(a)), in the LBM, 6502 iterations areneeded to reach the SS, while for (H1, H2) = (1.0, 0.9) (Fig. 8(c)),the same takes 2002 iterations. For all values of H2, SSH profilesof LBM and FVM match exceedingly well.

Distributions of conductive, radiative and total energy flowrates are shown in Fig. 8(d)–(f) for H2 ¼ 0:1;0:5 and 0.9, respec-tively. It is observed from these figures that conductive, radiativeand total energy flow rates all decrease with decrease in(H1 �H2). Conductive energy flow rate is less for the obvious rea-son that the temperature gradient decreases, and since the differ-ence in temperatures of the two surfaces decreases, the netradiative heat transfer will be lower. For a higher value of(H1 �H2) (Fig. 8(d)), major contribution to the total energy flowrate comes from conduction, whereas opposite is the case with(H1 �H2) = 0.1 (Fig. 8(f)). In all cases, LBM and FVM results havea very good match.

Fig. 9(a)–(f) show results for the case when the outer surface isperturbed to H2 = 1.0, and the inner surface is kept at different ini-tial temperatures. Temperature H distributions for H1 = 0.1, 0.5and 0.9 are shown in Fig. 9(a)–(c), respectively. In this case sinceouter surface is thermally perturbed, as is observed in Fig. 9(a)–(c), in all cases, wave fronts starts from the outer surface. With de-crease in (H2 �H1), at any instant f, discontinuities in H profile isfound to decrease. Decrease in (H2 �H1), reduces the temperaturegradient which causes faster attainment of the SS.

Distributions of conductive, radiative and total energy flowrates for H1 = 0.1, 0.5 and 0.9 are shown in Fig. 9(d)–(f), respec-tively. The magnitude of the total energy flow rate decreases withdecrease in (H2 �H1). With outer surface at a higher temperature,the net energy flow rate is towards the inner surface. Thus, bothconductive and radiative energy flow rates are negative. In allcases, energy flow rates of LBM and FVM match exceedingly well.

4. Conclusions

Application of the LBM was extended to analyze combinedmode conduction and radiation heat transfer in a concentric spher-ical shell. Fourier’s law of heat conduction was modified by incor-porating the finite propagation speed of the conduction wave front.Energy equation of the non-Fourier conduction and radiation wasformulated using the LBM. The volumetric radiative informationneeded in the energy equation was computed using the FVM. Tojudge the accuracy of the LBM results, the same problem was alsosolved using the FVM in which the radiative information was ob-tained from the FVM. With Fourier conduction and radiation, theLBM formulation was first benchmarked against the results avail-able in the literature. For non-Fourier conduction, normalized ra-dial temperature distributions at different instants including theSS were analyzed for the effects of the extinction coefficient, thescattering albedo, the conduction–radiation parameter, the surfaceemissivities, the radius ratios, and different magnitudes of thermalperturbations of the boundaries. Radial distributions of non-dimensional conductive, radiative and total energy flow rates atSS were also analyzed for all parameters. In all cases, SS tempera-ture and energy flow rate distributions were compared with thoseobtained from the FVM. Excellent comparison was found.

References

[1] J.G. Marakis, C. Papapavlou, E. Kakaras, A parametric study of radiative heattransfer in pulverised coal furnaces, Int. J. Heat Mass Transfer 43 (2000) 2961–2971.

[2] S. Maruyama, R. Viskanta, Active thermal protection system against intenseirradiation, J. Thermophys. Heat Transfer 3 (1989) 389–394.

[3] D. Bai, X.-J. Fan, On the combined heat transfer in the multilayer non-grayporous fibrous insulation, J. Quant. Spectrosc. Radiant. Transfer 104 (2007)326–341.

[4] T. Klason, X.S. Bai, M. Bahador, T.K. Nilsson, B. Sundén, Investigation of radiativeheat transfer in fixed bed biomass furnaces, Fuel 87 (2008) 2141–2153.

[5] P.R. Parida, R. Raj, A. Prasad, S.C. Mishra, Solidification of a semitransparentplanar layer subjected to radiative and convective cooling, J. Quant. Spectrosc.Radiant. Transfer 107 (2007) 226–235.

[6] A. Maroufi, C. Aghanajafi, Analysis of conduction–radiation heat transferduring phase change process of semitransparent materials using latticeBoltzmann method, J. Quant. Spectrosc. Radiant. Transfer 116 (2013) 145–155.

[7] P. Talukdar, S.C. Mishra, D. Trimis, F. Durst, Heat transfer characteristics of aporous radiant burner under the influence of a 2-D radiation field, J. Quant.Spectrosc. Radiant. Transfer 84 (2004) 527–537.

[8] M.M. Keshtkar, S.A.G. Nassab, Theoretical analysis of porous radiant burnersunder 2-D radiation field using discrete ordinates method, J. Quant. Spectrosc.Radiant. Transfer 110 (2009) 1894–1907.

[9] R.M. Colombo, G. Guerra, H. Herty, F. Marcellini, A hyperbolic model for thelaser cutting process, Appl. Math. Modell. (2013). <http://dx.doi.org/10.1016/j.apm.2013.02.031>

[10] D. Zhang, L. Li, Z. Li, L. Guan, X. Tan, Non-Fourier conduction model withthermal source term of ultra-short high power pulsed laser ablation andtemperature evolvement before melting, Phys. B: Condens. Matter 364 (2005)285–293.

[11] K. Kim, Z. Guo, Ultrafast radiation heat transfer in laser tissue welding andsoldering, Numer. Heat Transfer A 46 (2004) 23–40.

[12] M. Jaunich, S. Raje, K. Kim, K. Mitra, Z. Guo, Bio-heat transfer analysis duringshort pulse laser irradiation of tissues, Int. J. Heat Mass Transfer 51 (2008)5511–5521.

[13] A. Narasimhan, S. Sadasivam, Non-Fourier bio heat transfer modelling ofthermal damage during retinal laser irradiation, Int. J. Heat Mass Transfer 60(2013) 591–597.

[14] A. Rozanov, V. Rozanov, J.P. Burrows, A numerical radiative transfer model fora spherical planetary atmosphere: combined differential–integral approachinvolving the Picard iterative approximation, J. Quant. Spectrosc. Radiant.Transfer 69 (2001) 491–512.

[15] T. Sghaier, B. Cherif, M.S. Sifaoui, Theoretical study of combined radiativeconductive and convective heat transfer in semi-transparent porous medium ina spherical enclosure, J. Quant. Spectrosc. Radiant. Transfer 75 (2002) 257–271.

[16] B.-W. Li, Y.-S. Sun, D.-W. Zhang, Chebyshev collocation spectral methods forcoupled radiation and conduction in a concentric spherical participatingmedium, ASME J. Heat Transfer 131 (2009) 062701–062707.

[17] A. Dlala, T. Sghaier, E. Seddiki, Numerical solution of radiative and conductiveheat transfer in concentric spherical and cylindrical media, J. Quant. Spectrosc.Radiant. Transfer 107 (2007) 443–457.

[18] M.Y. Kim, J.H. Cho, S.W. Baek, Radiative heat transfer between two concentricspheres separated by a two-phase mixture of non-gray gas and particles usingthe modified discrete-ordinates method, J. Quant. Spectrosc. Radiant. Transfer109 (2008) 1607–1621.

[19] M.Y. Kim, S.W. Baek, C.Y. Lee, Prediction of radiative transfer between twoconcentric spherical enclosures with the finite volume method, Int. J. HeatMass Transfer 51 (2008) 4820–4828.

[20] K.-Y. Zhu, H. Yong, W. Jun, Combination of spherical harmonics and spectralmethod for radiative heat transfer in one-dimensional anisotropic scatteringmedium with graded index, Int. J. Heat Mass Transfer 62 (2013) 200–204.

[21] T.-M. Chen, C.-C. Chen, Numerical solution for the hyperbolic heat conductionproblems in the radial–spherical coordinate system using a hybrid Green’sfunction method, Int. J. Therm. Sci. 49 (2010) 1193–1196.

[22] H.-S. Chu, S. Lin, C.-H. Lin, Non-Fourier heat conduction with radiation in anabsorbing, emitting, and isotropically scattering medium, J. Quant. Spectrosc.Radiant. Transfer 73 (2002) 571–582.

[23] C.-C. Yen, C.-Y. Wu, Modelling hyperbolic heat conduction in a finite mediumwith periodic thermal disturbance and surface radiation, Appl. Math. Modell.27 (2003) 397–408.

[24] S.C. Mishra, T.B.P. Kumar, B. Mondal, Lattice Boltzmann method applied to thesolution of energy equation of a radiation and non-Fourier heat conductionproblem, Numer. Heat Transfer A 54 (2008) 798–818.

[25] S.C. Mishra, T.B.P. Kumar, Analysis of a hyperbolic heat conduction–radiationproblem with temperature dependent thermal conductivity, ASME J. HeatTransfer 131 (2009) 1–7.

[26] T.-M. Chen, A hybrid Green’s function method for the hyperbolic heatconduction problems, Int. J. Heat Mass Transfer 52 (2009) 4273–4278.

[27] S.C. Mishra, A. Stephen, M.Y. Kim, Analysis of non-Fourier conduction–radiation heat transfer in a cylindrical enclosure, Numer. Heat Transfer A 58(2012) 943–962.

[28] T.-M. Chen, Numerical solution of hyperbolic heat conduction problems in thecylindrical coordinate system by the hybrid Green’s function method, Int. J.Heat Mass Transfer 53 (2010) 1319–1325.

[29] S.C. Mishra, A. Stephen, Combined mode conduction and radiation heattransfer in a spherical geometry with non-Fourier Effect, Int. J. Heat MassTransfer 54 (2011) 2975–2989.

[30] B. Kundu, K.-S. Lee, Fourier and non-Fourier heat conduction analysis in theabsorber plates of a flat-plate solar collector, Sol. Energy 86 (2012) 3030–3039.

[31] S.C. Mishra, H. Sahai, Analyses of non-Fourier heat conduction in 1-Dcylindrical and spherical geometry – an application of the lattice Boltzmannmethod, Int. J. Heat Mass Transfer 55 (2012) 7015–7023.

http://refhub.elsevier.com/S0017-9310(13)00788-6/h0005






















http://dx.doi.org/10.1016/j.apm.2013.02.031

http://dx.doi.org/10.1016/j.apm.2013.02.031




































































[32] S.C. Mishra, H. Sahai, Analyses of non-Fourier conduction and radiation in acylindrical medium using lattice Boltzmann method and finite volumemethod, Int. J. Heat Mass Transfer 61 (2013) 41–45.

[33] R. Shirmohammadi, A. Moosaie, Non-Fourier heat conduction in a hollowsphere with periodic surface heat flux, Int. Commun. Heat Mass Transfer 36(2009) 827–833.

[34] L.H. Liu, H.P. Tan, T.W. Tong, Non-Fourier effects on transient temperatureresponse in semitransparent medium caused by laser pulse, Int. J. Heat MassTransfer 44 (2001) 3335–3344.

[35] S. Succi, The Lattice Boltzmann Method for Fluid Dynamics and Beyond, OxfordUniversity Press, 2001.

[36] H.N. Dixit, V. Babu, Simulation of high Rayleigh number natural-convection ina square cavity using the lattice Boltzmann method, Int. J. Heat Mass Transfer49 (2006) 727–739.

[37] V. Babu, A. Narasimhan, Investigation of vortex shedding behind a poroussquare cylinder using lattice Boltzmann method, Phys. Fluids 22 (2010)053605–053612.

[38] S.C. Mishra, A. Lankadasu, Analysis of transient conduction and radiation heattransfer using the lattice Boltzmann method and the discrete transfer method,Numer. Heat Transfer A 47 (2005) 935–954.

[39] B. Mondal, S.C. Mishra, Application of the lattice Boltzmann method anddiscrete ordinate method for solving transient conduction and radiation heattransfer problems, Numer. Heat Transfer A 52 (2007) 757–777.

[40] B. Mondal, S.C. Mishra, Simulation of natural convection in the presence ofvolumetric radiation using the lattice Boltzmann method, Numer. HeatTransfer A 55 (2009) 18–41.

[41] B. Mondal, X. Li, Effect of volumetric radiation on natural convection in asquare cavity using lattice Boltzmann method with non-uniform lattices, Int. J.Heat Mass Transfer 53 (2010) 4935–4948.

[42] S.C. Mishra, H.K. Roy, Solving transient conduction and radiation heat transferproblems using the lattice Boltzmann method and the finite volume method, J.Comput. Phys. 223 (2007) 89–107.

[43] R. Chaabane, F. Askri, S.B. Nasrallah, Application of the lattice Boltzmannmethod to transient conduction and radiation heat transfer in cylindricalmedia, J. Quant. Spectrosc. Radiant. Transfer 112 (2011) 2013–2027.

[44] R. Chaabane, F. Askri, S.B. Nasrallah, Analysis of two-dimensional transientconduction–radiation problems in an anisotropically scattering participatingenclosure using the lattice Boltzmann method and the control volume finiteelement method, Comput. Phys. Commun. 182 (2011) 1402–1413.

[45] R. Chaabane, F. Askri, S.B. Nasrallah, Parametric study of simultaneoustransient conduction and radiation in a two-dimensional participatingmedium, Commun. Nonlinear Sci. Numer. Simul. 10 (2011) 4006–4020.

[46] P. Bhatnagar, E.P. Gross, M.K. Krook, A model for collision processes in gases I.Small amplitude processes in charged and neutral one-component systems,Phys. Rev. 94 (1954) 511–525.















































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