Analysis of non-Fourier conduction and radiation in a cylindrical medium using lattice Boltzmann...

International Journal of Heat and Mass Transfer 61 (2013) 41–55

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International Journal of Heat and Mass Transfer

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Analysis of non-Fourier conduction and radiation in a cylindrical mediumusing lattice Boltzmann method and finite volume method

Subhash C. Mishra ⇑, Harsh SahaiDepartment of Mechanical Engineering, IIT Guwahati, Guwahati 781 039, India

a r t i c l e i n f o

Article history:Received 2 November 2012Received in revised form 5 January 2013Accepted 28 January 2013Available online 26 February 2013

Keywords:Non-Fourier heat conductionVolumetric radiationParticipating mediumConcentric cylinderLattice Boltzmann methodFinite volume method

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

a b s t r a c t

Combined mode heat transfer in a conducting–radiating participating medium bounded by a concentriccylindrical enclosure is studied. The finite propagation speed of heat transfer by conduction is accountedby modifying the Fourier’s law of heat conduction. The energy equation is formulated and solved usingthe lattice Boltzmann method. The finite volume method is used to compute the volumetric radiativeinformation needed in the energy equation. Radial distributions of temporal temperature and thesteady-state conductive, radiative and total energy flow rates are analyzed for a wide range of parame-ters, such as the extinction coefficient, the scattering albedo, the conduction–radiation parameter, theemissivity and the radius ratio. With volumetric radiative information computed using the finite volumemethod, in all cases, the steady-state temperature distributions from the lattice Boltzmann method arecompared with those obtained by solving the energy equation using the finite difference method. Anexcellent comparison is obtained.

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1. Introduction

Accurate analysis of volumetric thermal radiation is importantin the design and analysis of many manmade thermal systemsand in the understanding of natural systems. Owing to the finitetemperature, all objects remain the source of radiation, and theystay in radiative exchange with any object that they can see. Withincrease in temperature, the radiative exchange becomessignificant.

If a surface is diffuse, emission from it is in the hemisphericalspace, but while traveling through a participating medium, be-cause of the absorption, emission and scattering, radiation mani-fests in the spherical space. The radiative transfer processbecomes volumetric. If the geometry is a 3-D one, apart from itsdependence on one temporal and three spatial dimensions, thedependence of intensity on polar and azimuthal angles, transformsthe governing radiative transfer equation (RTE) to an integro-differential one. Radiation is also a spectral quantity. At anytemperature, emission of radiation is over a wide spectrum ofwavelength. Compared to conduction and convection, dependenceof radiation on three extra dimensions, viz., polar angle, azimuthalangle and wavelength, makes its analysis difficult, and its solution,computationally intensive.

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In many systems and process like the fibrous and foam insula-tions [1,2], solidification and melting of semi-transparent materialslike glass and silicon [3–6], combustion in porous media [7–10],heat transfer in tissues [11–12], etc., radiation is coupled with con-duction. In a combined mode transient conduction and radiationproblem, propagation speed of energy transport by radiation isorders of magnitude higher. Thus compared to conduction, owing

to a very large speed of propagation c ¼ 3:0� 108 ms

� �, time

dependant term in radiation transfer equation is ignored untiland unless, transience is looked at time O(1.0 � 10�9 s) or lower[13,14]. In a conduction–radiation problem, if the temperatureevolution is investigated even at time scales of O(1.0 � 10�6 s),only conduction is considered time dependant.

In a combined mode conduction–radiation problem, radiationand conduction are coupled in the energy equation. In the solutionof this equation, at any time level, the volumetric radiative infor-mation needed is taken from the previous time level. With a newtemperature field obtained from the solution of the energy equa-tion, radiative information is updated from the solution of theRTE, which is then fed to the energy equation to calculate thenew temperature field. The process continues until convergenceis reached. Only solving a thermal problem involving radiation,the numerical solution procedure is an iterative one. However, ina combined mode conduction–radiation problem, iteration in theradiation module is bypassed [7,15]. Without any loss of accuracy,this is done to save the computational time. Radiation field evolves

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Nomenclature

ai weight in LBM formulationbi weight in LBM formulationamn

I coefficient of the discretization equation in direction mand n at nodal point I

C speed of thermal wave, m s�1

Dmni directional weight in direction m and n at surface i

Dmn�1=2i angular edge directional weight in direction m and

n ± 1/2 at surface iE, W, N, S east, west, north and south neighboring control vol-

umes of P, respectively~ei propagation velocity in the direction i in the lattice,

m s�1

~er;~ed0 ;~ez cylindrical r- , d0- and z-direction base vectors, respec-tively

fi particle distribution function in the i direction, Kf ð0Þi equilibrium particle distribution function in the i direc-

tion, KG incident radiation, W m�2

N number control volumesni outward unit normal vector at face inw unit normal vector at the wall towards the mediumN conduction–radiation parameterNc number of divisions of the polar spaceNd number of divisions of the angular azimuthal spaceNu number of divisions of the geometric azimuthal spaceP present control volumeq heat flux, W m�2

r1 radius of the inner cylinder, mr2 radius of the outer cylinder, m~r any position along the radial direction, mS source term, W m-2

s direction

Greek symbolsDA, DV surface area and volume of the control volume,

respectivelya thermal diffusivity, m2 s�1

b extinction coefficient, m�1

v expansion parametere emissivityc polar angled, dX angular azimuthal angles measured from the x0- and

r-axis, respectivelyd0 spatial azimuthal angle measured from the x-axis

(Fig. 1(b))H normalized temperature, T�T2

T1�T2

h dimensionless temperature, TT1

ja absorption coefficient, m�1

C thermal relaxation time, sx scattering albedors scattering coefficient, m�1

g normalized radial distance, r�r1r2�r1

ld, gd direction cosines in the r and u directions, respectivelyf non-dimensional timeX solid angle, srDX elemental solid angle, sr

Superscript⁄ dimensionless variablem, n indices for polar and azimuthal angles±1/2 control angle faces

Subscripts1, 2 inner and outer cylinder wallsb blackC conductiveE, W, N, S node points where intensities are locatede, w, n, s control volume faceswall wallP value at the cell centreR radiativeref referenceT total

42 S.C. Mishra, H. Sahai / International Journal of Heat and Mass Transfer 61 (2013) 41–55

with the temporal marching of the energy equation which is thetransience of the conduction.

Transport of heat by conduction is governed by Fourier’s law,which is based on the assumption that the effect of thermalperturbation is established instantaneously in the system, i.e., assoon as the temperature of one boundary of a medium is changed,this effect will be felt instantaneously throughout the medium.This assumption is correct if the change is looked after the systemtime, t ¼ l

C ; where l is the characteristic length and C is the propa-gation speed of heat by conduction.

With a as the thermal diffusivity, propagation speed of con-duction C ¼ a

l ; is material specific. In a thermally conductor, heatflows fast, while through an insulator, it is very slow. Even in aconductor, if the temperature evolution is looked at time scaleslower than t ¼ l

C ; the change will not be felt. Because of the fi-nite propagation speed of conduction, the heat wave front hasnot reached the location of interest. And, Fourier’s law of heatconduction has no answer to this. To account for this, the finitepropagation speed of the energy transport by conduction mustbe accounted [16,17]. This consideration leads to conduction bynon-Fourier effect [18–19]. This consideration becomes impor-tant in situations with periodic boundary conditions in whichthe time period of the boundary condition is lower than the timescale of conduction, cases in which transience is looked at lowertime scales, a material under intense thermal load, and at a very

low temperature [20–25]. Correct analysis of transient heattransfer in a biological tissue too needs this consideration [11].Many researchers have investigated heat transfer problems withnon-Fourier effect [18–29]. Their analyses involve differentgeometry, boundary conditions and applications of differentmethods.

Radiation is a highly non-linear process. If in conduction andconvection, heat transfer is proportional to temperature difference(T1–T2), it is proportional to the difference of the 4th power of theabsolute temperature ðT4

1 � T42Þ in radiation. Thus, in the presence

of radiation, temperature field is much different than that withconduction alone. Further, in a transient process, radiation aug-ments the attainment of the steady-state. In case of combinedmode conduction–radiation heat transfer with non-Fourier effect,radiation too has a significant effect. This has been extensivelystudied by the group of Mishra and co-workers [26–29].

Analysis of combined mode conduction–radiation in a cylindri-cal enclosure finds many applications, and a good number ofresearchers have investigated this problem [28,30–33]. With volu-metric radiative information computed using different variants ofthe discrete ordinate method [34,35], the finite volume method(FVM) [36], etc., the energy equation too has been solved using dif-ferent numerical methods [28,30–33]. Compared to combinedmode conduction–radiation with Fourier effect, study with non-Fourier effect is scarce.

S.C. Mishra, H. Sahai / International Journal of Heat and Mass Transfer 61 (2013) 41–55 43

In the recent times, application of the lattice Boltzmann method(LBM) [37] in the analyses of a wide range of problems has seen asurge. Owing to its mesoscopic nature, it works well for analyses ofproblems at microscopic and macroscopic levels. Compared to thefinite difference method and the FVM, its advantages includeinherent parallelizability, memory overhead, computationaladvantage for complex problems, etc. In fluid mechanics and heattransfer problems, it has been applied to a wide range of problems[37–39]. Its application to the formulation and solution of energyequation of heat transfer problems involving volumetric radiationhas been extended by Mishra and co-workers [3,26,23,35,36,40–42] and others [33,43]. In a combined mode conduction and radia-tion heat transfer problems with non-Fourier conduction, it hasbeen applied to cylindrical geometry by Mishra and Krishna [35].However, as far as its application to combined mode conductionand radiation heat transfer with non-Fourier effect is concerned,no work has been reported so far. The present work, therefore,aims at the application of the LBM to the solution of non-Fourierconduction and radiation heat transfer in a concentric cylindricalenclosure. Consideration is given to a 1-D cylinder. The volumetricradiative information needed in the energy equation is obtainedusing the FVM [44]. Effects of influencing parameters like theextinction coefficient, the scattering albedo, the boundary emissiv-ity, the conduction–radiation parameter and the radius ratio on ra-dial temperature distributions at different time levels including thesteady-state (SS) and the radial distributions of the conductive, theradiative and the total energy flow rates at the SS are analyzed. Toassess the accuracy of the results from the LBM–FVM, at the SS,temperature distributions are compared with those obtained bysolving the same problem using the finite difference method andthe FVM.

2. Formulation

Consideration is given to a 1-D concentric cylindrical enclosure(Fig. 1(a)) containing a homogenous radiating-conducting medium.Radiatively, the medium is absorbing, emitting and scattering. Itsthermophysical and optical properties such as the density q, thespecific heat cp, the thermal conductivity k, the absorption coeffi-cient ja and the scattering coefficient rs are constant. The surfacesof the inner cylinder (radius r1) and the outer cylinder (radius r2)are diffuse gray. Initially at time t = 0, the entire system is attemperature T0 = T2, and for t � 0; the temperature of the inner cyl-inder is maintained at T1 > T2. For the problem under consider-ation, in the macroscopic form, the energy equation can bewritten as

qcp@T@t¼ � 1

r

� �@ðrqTÞ@r

¼ � 1r

� �@ðrqCÞ@r

þ @ðrqRÞ@r

� �ð1Þ

where with qT as the total heat flux, qC and qR are the conductiveand the radiative heat fluxes, respectively. The transport of heatby conduction in a non-Fourier way is given by [16–23]

C@qC

@tþ qC ¼ �k

@T@r

ð2Þ

With a as the thermal diffusivity and C as the propagation speed ofthe conduction wave front, and the thermal time lag given by

C ¼ aC2 ð3Þ

Differentiating Eq. (1) with respect to r and Eq. (2) with respect to

t, and eliminating terms @2qC@t@r and qC, result in the following

equation.

qcp C@2T@t2 þ

@T@t

!¼ k

r

� �@ r @T

@r

� �@r

� 1r

� �@ðrqRÞ@r

� C1r

� � @ @ðrqRÞ@r

� �@t

ð4Þ

When C ? 0, Eqs. (2) and (4) reduce to equations corresponding tothe Fourier’s law of conduction.

Solution of the energy equation (Eq. (4)) requires knowledge ofthe divergence of radiative heat flux 1

r@ðrqRÞ@r ¼ r � qR. It is given by

r � qR ¼1r

� �@ðrqRÞ@r

¼ bð1�xÞ 4prT4

p� G

!ð5Þ

where b = ja + rs is the extinction coefficient, x ¼ rsb is the scatter-

ing albedo and G is the incident radiation. For an isotropically scat-tering medium, with c as the polar angle and d as the angularazimuthal angle, the RTE in any direction s ¼ ðsin c cos dÞiþðsin c sin dÞjþ ðcos cÞk is given by

dIds¼ �bI þ jaIb þ

rs

4pG ð6Þ

where Ib ¼ rT4=p is the blackbody intensity. Resolving Eq. (6) alongthe cylindrical coordinate directions r and d; results in

1r

� �@ðldrIÞ@r

þ 1r

� �@ðgdIÞ@u

� 1r

� �@ðgdIÞ@d

¼ �bI þ jaIb þrs

4pG ð7Þ

where ld ¼ sin c cos d and gd ¼ sin c sin d are direction cosines, andu is the spatial azimuthal angle (Fig. 1(c)). Eq. (7) is an integro-dif-ferential equation. For a diffuse gray boundary having emissivityewall and temperature Twall, the radiative boundary condition forEq. (7) is

Iwallð~rwall; sÞ ¼ ewallrT4

wall

pþ 1� ewall

p

Zs:nwall<0

Ið~rwall; sÞjs0:nwalljdX0;

for nwall � s > 0 ð8Þ

where~rwall is the radial distance to the bounded wall and nwall is theunit normal vector towards the medium at the bounding cylindricalwalls (Fig. 1(b)). In Eq. (8), the first and the second terms on theright hand side represent the emitted and the reflected componentsof the radiosity, i.e., the net intensity emanating from a boundary.

In the present work, Eq. (7) is solved using the FVM approachproposed by Kim and Beak [44]. In this, the RTE is first writtenfor a discrete direction having index mn, where m and n are theindices for discrete polar and azimuthal angles. Next, after inte-grating Eq. (7) over the elemental control volume DV = 2prDrDuand the solid angle DX = sincdcdd, its discretized form becomesXi¼n;s

Imni DAiD

mni þ DAeDmnþ1=2

e Imnþ1=2e þ DAwDmn�1=2

w Imn�1=2w

h iþ bImn

P DVDXmn ¼ SmnP DVDXmn ð9Þ

where with reference to Fig. 1(c), DAe = DAw = (rn � rs) � 1 is thesurface area of the east/west face, DAn = rnDu � 1 is the surfacearea of the north face, DAs = rsDu � 1 is the area of the south face,and with 2Nu as the number of divisions of the geometrical azi-muthal angle uð0 6 u 6 2pÞ; DV ¼ pðr2

n � r2s Þ � 1=2Nu is the vol-

ume of the discrete control volume with center P. The directionalweight Dmn

i represents the inflow and outflow of radiant energyacross the control volume face i(=n,s,e,w) (Fig. 1(c)). It is computedfrom the following [44]

Dmni ¼

ZDXðs � niÞdX ¼

Z dnþ1=2

dn�1=2

Z cmþ1=2

cm�1=2ðs � niÞ sin cdcdd ð10Þ

where the unit direction vector s and the outward normal vector ni

are based on cylindrical coordinates (Fig. 1(c)). Following the


definition given in Eq. (10), Dmnn ; Dmn

s ; Dmnþ1=2e ; Dmn�1=2

w Dmnþ1=2e and

DXmn are given by [44]

Dmnn ¼ �Dmn

s

¼ 12

cmþ1=2 � cm�1=2� �� 1

4sin 2cmþ1=2 � sin 2cm�1=2� ��

� sin dnþ1=2 � sin dn�1=2� �

ð11aÞ

Dmnþ1=2e ¼ � 1

2cmþ1=2 � cm�1=2� �

� 14

sin 2cmþ1=2 � sin 2cm�1=2� �� cos dn � cos dnþ1� �

ð11bÞ

Dmn�1=2w ¼ � 1

2cmþ1=2 � cm�1=2� �

� 14

sin 2cmþ1=2 � sin 2cm�1=2� �� cos dn�1 � cos dn� �

ð11cÞ

DXmn ¼ cos cm�1=2 � cos cmþ1=2� �

dnþ1=2 � dn�1=2� �

ð11dÞ

To reduce the number of unknowns in Eq. (9), the facial intensityImni ; the angular edge intensities Imn�1=2

i and the nodal intensitiesImnI are related to the cell centre intensity Imn

P through a simple stepscheme. This results in

Imni Dmn

i ¼ ImnP maxðDmn

i ;0Þ � ImnI maxð�Dmn

i ;0Þ ð12aÞ

Imnþ1

2e ¼ Imn

P ð12bÞ

Imn��1=2w ¼ Imn�1

W ¼ Imn�1P ð12cÞ

where in Eq. (12a), subscript i represents n and s, while I representscorresponding N and S (Fig. 1(c)). Using the step scheme of Eq. (12),Eq. (9) can be written as

amnp Imn

P ¼XI¼N;S

amnI Imn

I þ bmnP ð13Þ

where

amnP ¼

Xi¼n;s

maxðDAiDmni ;0Þ þ bDVDXmn þ DAeDmnþ1=2

e ð14aÞ

amnI ¼maxð�DAiD

mni ;0Þ ð14bÞ

bmnP ¼ Smn

P DVDXmn � DAwDmn�1=2w Imn�1

P ð14cÞ

SmnP ¼ jaIb þ

rs

4pG ð14dÞ

The radiative boundary condition given by Eq. (8) is written as

Iwallmn ¼ �wall

aT4wall

pþ 1� �wall

pX

m0n0 ;Dm0n0wall <0

Im0n0

wall jDm0n0

wall j

for Dmn

wall > 0

ð15Þ

With intensity distributions Imn known, incident radiation G andradial radiative heat flux qR are numerically calculated from thefollowing:

G ¼ 2XNc

m¼1

XNd

n¼1

ImnDXmn ¼ 2XNc

m¼1

XNu

n¼1

ImnDXmn ð16Þ

qR ¼ 2XNc

m¼1

XNd

n¼1

ImnDmnn ¼ 2

XNc

m¼1

XNu

n¼1

ImnDmnn ð17Þ

where Nc, Nd and Nu are the number of discrete divisions of the po-lar cð0 � c � pÞ; the angular azimuthal dð0 � d � pÞ and geometricazimuthal uð0 � u � pÞ spaces. It is to be noted that the concentriccylindrical geometry (Fig. 1(a)) considered in the present work isazimuthally u symmetric. For a long cylinder, thermally it a 1-Dproblem, i.e., T = T(r). However, while computing the radiativecomponent, unlike a planar geometry, in this case, radiation is azi-muthally not symmetric, i.e., I = I(c,d). At every radial location r,intensities need to be traced over the full spherical space,ð0 � c � p; 0 � d � 2pÞ. For a given polar angle c, to account forthe variation of intensity over the angular azimuthal angle d, owingto symmetry, intensity variation over the geometrical azimuthalspace u yields the same results. This equivalence has been shownby Kim and Beak [44]. Thus, for a 1-D problem, for ease in calcula-tion of radiative information, we resort to 2-D control volumes asshown in Fig. 1(c). However, it is to be noted that the energy equa-tion (Eq. (9)) and its LBM equivalent are solved on a 1-D radial grid(Fig. 1(d)).

For generality, the governing energy equation (Eq. (4)) is writ-ten in the dimensionless form, For this, with various parameterslike time f, distance g, temperature h, conductive heat flux W�C ;radiative heat flux W�R, conduction–radiation parameter N and inci-dent radiation G⁄ defined in the following way,

f ¼ C2t2a

g ¼ Cr2a

h ¼ TTref

W�C ¼qC

rT4ref

W�R ¼qR

rT4ref

N ¼ kb

4rT3ref

G� ¼ G

rT4ref=p

ð18Þ

in non-dimensional form, Eq. (4) is written as

@2h

@f2 þ 2@h@f¼ 1

g@

@gg@h@g

� �� brref

4N1g@ðgW�RÞ@g

� �� brref

8N

� @

@f1g@ðgW�RÞ@g

� �ð19Þ

where rref ¼ 2aC . In non-dimensional form, Eq. (5) for the divergence

of radiative heat flux is written as

r� �W�R ¼1g@ðgW�RÞ@g

¼ brrefð1�xÞp

� �ð4ph4 � G�Þ ð20Þ

Very recently Mishra and Sahai [45] provided a LBM formulation forthe analysis of non-Fourier conduction in cylindrical and sphericalgeometry. In the present work, the same is extended for the com-bined mode conduction and radiation heat transfer. In the LBM for-mulation, with volumetric radiation, the governing equationcorresponding to Eq. (19) takes the form

fiðgþ eiDf; fþ DfÞ � fiðg; fÞ ¼ �Dfs

fiðg; fÞ � f ð0Þi ðg; fÞh i

� brref

4NaiDf

2r� �W�R

Df2

� ðaiW�CÞ

2gþ 4bieiW

�C

� ;

i ¼ 1 and 2 ð21Þ

where fi is the particle distribution function, f ð0Þi is the equilibriumparticle distribution function, ei is the velocity with which fi propa-gates to its nearest neighbor, s is the relaxation time, ai and bi areweights specific to the lattices used. Analysis of the proposed LBMformulation (Eq. (21)) using the Chapman–Enskog multi-scale andTaylor series expansions given in Appendix A establishes the consis-tency of the scheme. Eq. (19) is recovered when Df ? 0 and theexpansion parameter v ? 0.

For the geometry under consideration, viz., 1-D cylindrical(Fig. 1(a)), the lattice used is D1Q2 (Fig. 1(d)). In the D1Q2 lattice,

Fig. 1. (a) Schematic of the 1-D concentric cylindrical enclosure, (b) cylindrical coordinate system used, (c) control volume used in FVM for calculation of radiativeinformation and (d) lattices in the LBM and control volumes in FVM for solving energy equation.


the velocity ei, the corresponding weights ai, bi and the relaxationtime s are given as

e1 ¼ �e2 ¼DgDf

ð22aÞ

a1 ¼ a2 ¼12

; b1 ¼ b2 ¼12

ð22bÞ

s ¼ ae2

i

þ Df2

ð22cÞ

where Df is the time step. It is to be noted for the proposed formu-lation of non-Fourier conduction, with Df = Dg, the relaxation times can it can take any value without affecting the accuracy or stabil-ity of the formulation. However, the expression given in Eq. (22c) ispertinent for the D1Q2 lattice of the most commonly used LBMformulation of Fourier conduction, and the same is used in the pres-ent work.

The solution of Eq. (21) in terms of fi yields temperature distri-bution h(g,f), and this is given by

h ¼X

i

f ð0Þi ¼X

i

fi ð23Þ

With temperature distribution h(g,f), known from the above, heatflux W�C in the LBM methodology is obtained from

W�C ¼X

i

fiei ¼X

i

f ð0Þi ei ð24Þ

It is to be noted that with temperature field obtained from Eq. (23),W�C can also be calculated using a finite difference scheme. The solu-tion of Eq. (21) needs equilibrium particle distribution function f ð0Þi .

This is given by

f ð0Þi ¼ aiT þ bieiW�C ð25Þ

If in Eq. (24), with weight bi = 0, the equation turns to be that obey-ing Fourier’s law of heat conduction.

The solution procedure for computation of the divergence ofradiative heat flux in cylindrical geometry follows references[44,46]. With volumetric radiative information known, theprocedure for solving LBM Eq. (21) follows the approach of Mishraand Roy [36] for conduction–radiation problem with Fourier

0.8

1Nr = 100Nr = 500Nr = 900

β = 1.0, ω = 0.5, Ν = 0.03


conduction. Solution of Eq. (19) using the FDM is according toreference [28]. In the following, we first validate the LBM–FVM for-mulation presented above, and then analyze the effects of variousparameters on radial distributions of temperature and energy flowrates.

(a)

(b)

Distance, η

Tem

pera

ture

, Θ

0 0.2 0.4 0.6 0.8 10

0.2

0.4

0.6

ζ = 0.1 ζ = 0.3

SS

Distance, η

Tem

pera

ture

, Θ

0 0.2 0.4 0.6 0.8 10

0.2

0.4

0.6

0.8

1

4 x 88 x 1212 x 16

ζ = 0.1 ζ = 0.3

β = 1.0, ω = 0.5, Ν = 0.03

SS

Nθ x Nφ =

Fig. 2. Radial g temperature H distributions at three time n levels including the SS,for different numbers of (a) control volumes/lattices and (b) discrete directions.

3. Results and discussion

In a combined mode conduction–radiation problem, parametersof interest are: extinction coefficient b, scattering albedo x, emis-sivity e of the enclosure boundaries and the conduction–radiationparameter N. For a given set of initial and boundary conditions,these parameters influence the temperature field, and hence thedistributions of radiative, conductive and total heat fluxes. In aconcentric cylindrical enclosure, the volume of the enclosed partic-ipating medium, viz., the radius ratio RR also becomes an influenc-ing parameter. In the following, we analyze effects of these

parameters on normalized temperature H ¼ T�T2T1�T2

¼ h�h2h1�h2

� �distri-

butions at different instants f including the SS. In every case, theSS LBM–FVM H distributions are compared with that of theFDM–FVM. Further to know how much SS H distribution is influ-enced by volumetric radiation, in every case, H distribution withpure conduction (x = 1.0) is also shown.

In a combined mode problem, with a change in a given set ofparameters, relative contributions of conduction and radiation tothe total heat flux change. At the SS, thus, it is interesting to knowhow these heat fluxes change. Unlike a planar geometry, in case ofa cylindrical geometry, at SS, it is not the total heat flux qT, but thetotal energy flow rate (qT � 2pr � 1) that remains constant. Tostudy the changes in energy flow rates, in every case, at the SS,radial distributions of conductive, radiative and total energy flowrates are also provided and analyzed.

Accuracy of results from any computational method depends onthe sizes of discrete temporal Df and spatial Dg steps. In case ofvolumetric radiation, the accuracy depends on the number of dis-crete elemental solid angle, i.e., the divisions of the polar space Ncand the geometric azimuthal space Nu(=Nd). Before we proceed toanalyze the effects of various parameters on distributions of tem-perature and energy flow rates, we determine suitable sizes oftime, physical space and angular space discretizations. Since theproblem under consideration is a transient one, and uses an itera-tive procedure, the SS condition was assumed to have beenreached when the change in temperature at all locations at twoconsecutive iterations did not exceed 1.0 � 10�6. In the LBM for-mulation used in the present work, Df

Dg ¼ 1:0: Thus, the size of Df

is determined by Dg. With radius of the inner cylinder g1 = 1.0and the outer cylinder g2 = 2.0, for different number Nr of lattices,plots of normalized temperature H vs normalized radial distanceg ¼ r�r1

r2�r1are shown in Fig. 2(a). With Nc � Nu = 8 � 12 intensities,

for black boundaries, results in Fig. 2(a) are shown for b = 1.0,x = 0.5 and N = 0.03. No significant change in H is observed be-yond 200 lattices. With 200 lattices, for the same set of parameters,effect of number of rays on H distribution is shown in Fig. 2(b).Nc � Nu = 8 � 12 directions are found to suffice.

To ascertain the accuracy of the LBM–FVM formulation for thecombined mode problem with non-Fourier effect, for Fourier con-duction and radiation, SS radial temperature H distributions arecompared with that of Fernandes and Francis [30]. For differentvalues of x, Fig. 3(a) shows comparison for b = 1.0, N = 0.03 ande1 = e2 = 0.5, while with b = 1.0, N = 0.03 and x = 0.5, in Fig. 3(b) re-sults are compared for different values of e. Results of the presentwork are found to compare exceedingly well with those of Fernan-des and Francis [30] obtained using the FDM and Galerkin finiteelement technique. Next we present and analyze results for the

effects of various parameters. In all cases, temperatures of the in-ner and outer cylinders were set to: h1 = 1.0(H1 = 1.0) andh2 = 0.1(H2 = 0.0).

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

Radial g variations of normalized temperature H at 5 time level,viz., f = 0.1, 0.3, 0.6 and 1.2, and at the SS for extinction coefficientb = 0.1, 1.0 and 2.0 are shown in Fig. 4(a)–(c), respectively. For thecorresponding values of b, at the SS, radial distributions of non-dimensional conductive WC, radiative WR and total energy WT

(=WC + WR) flow rates are shown in Fig. 4(d)–(f), respectively. Re-sults in Fig. 4(a)–(f) are shown for x = 0.5, N = 0.1, e = 1.0 andRR ¼ r2

r1¼ 2:0:

It is observed from Fig. 4(a)–(c) that at any instant f, withincrease in extinction coefficient b, the non-Fourier effect, i.e., dis-continuity in the temperature profile becomes less pronounced.After the inner cylinder is thermally perturbed, unlike the Fourierconduction, effect is not felt instantaneously at all points. Temper-ature front starts moving towards the outer cylinder, and afterreaching there, it reflects back towards the inner cylinder. This

(a)

(b)

Distance, η

Tem

pera

ture

, Θ

0 0.2 0.4 0.6 0.8 10

0.2

0.4

0.6

0.8

1

ω = 0.0ω = 0.5ω = 1.0

Markers: Fernandes and Francis [30]

Distance, η

Tem

pera

ture

, Θ

0 0.2 0.4 0.6 0.8 10

0.2

0.4

0.6

0.8

1

ε = 0.1ε = 0.5ε = 1.0

Markers: Fernandes and Francis [30]

Fig. 3. Validation of the steady-state radial temperature H distributions fordifferent values of (a) scattering albedo x and (b) emissivity e.


movement of the thermal wave front continues till the time, tem-perature discontinuity is completely damped. At this stage, it issaid to have reached the SS. At a particular instant f, the disconti-nuity in H is less for higher b, and at any location, H increase withincrease in b; and the SS too is reached sooner. The SS times (fSS =Df � number of iteration store ACH SS) for b = 0.1, 1.0 and 2.0 are6.67, 6.0 and 2.86, respectively. A comparison of the SS H fordifferent b with that for pure conduction (x = 1.0) shows that theradiative effect on temperature is more pronounced for a highervalue of b. The above trend is for the fact that radiative contribu-tion increases with increase in b (Eq. (20)). When b = 0.1(Fig. 4(a)), the situation is like that of a pure conduction. Almostno difference in the SS H distributions between pure conductionand conduction–radiation justifies this. In every case, SS FDM–FVM and LBM–FVM results compare exceedingly well.

Above observation of variations in temperature H is further jus-tified with the SS radial distributions of energy flow rates shown inFig. 4(d)–(f). For all values of b, the total energy flow rate WT

remains independent of radial distance, and its value increaseswith increase in b. For b = 0.1, the total energy flow rate WT ismainly due to conduction (Fig. 4(d)), and WC and WR also remainalmost constant with the radial location g. With increase in b from

0.1 to 1.0, contribution from radiation increases substantially, andthe total energy flow rate also increases. Further increase in energyflow rates is observed for b = 2.0. An observation of Fig. 4(e) and (f)shows that near the boundaries of the cylinder, trends of WC andWR are different. At the hot boundary (inner cylinder), WR is more,and at the cold boundary (outer cylinder), WC is more than WR.Near the hot boundary, higher temperature makes radiation thedominant mode. While when temperature is low near the cold cyl-inder, conduction dominates over radiation.

3.2. Effect of scattering albedo x on temperature and energy flow rate

With extinction coefficient b = 1.0, emissivities of the twoboundaries e1 = e2 = e = 0.5, conduction–radiation parameterN = 0.03 for RR = 2.0, radial variations of H at different instantsincluding the SS for scattering albedo x = 0.0, 0.5 and 0.9 areshown in Fig. 5(a)–(c), respectively. At the SS, energy flow ratedistributions for x = 0.0, 0.5 and 0.9 are shown in Fig. 5(d)–(f),respectively. A higher value of xð¼ rs

b Þ implies increased scatteringof volumetric radiation, which means loss of radiative energy. Thisis also evident from Eq. (20) that with increase in x, the radiativecontribution to the energy equation goes down. Thus, as observed,the non-Fourier effect on H distribution is more noticeable for ahigher value of x (Fig. 5(c)). With reduced radiation, the SS isattained late. At the SS, difference between H from conduction–radiation and pure conduction (x = 1.0) reduces with increase inx. For all values of x, SS H results of LBM–FVM compare exceed-ingly well with that of the FDM–FVM.

It is observed from Fig. 5(d)–(f) that with increase in x, the totalenergy flow rate WT decreases. Unlike the effect of b, the radiativeenergy flow rate WR at the hot wall (inner cylinder) remains almostconstant, and at any location, except near the cold wall, it de-creases with increase in x. Throughout the medium, the conduc-tive energy flow rate WC decreases with increase in x. Near thecold wall (outer cylinder), WC is more than WR. Another importantobservation is that with increase in x, the radial location at whichWC and WR intersect shift towards the outer cylinder. At the innercylinder, the difference between WC and WR increases withincrease in x. Opposite is the trend at the outer cylinder.

3.3. Effect of emissivity e on temperature and energy flow rate

Effect of emissivity e1 = e2 = e of the inner and the outer cylinderson H and W distributions are shown in Fig. 6. These results areshown for b = 1.0, x = 0.0, N = 0.03 and RR = 2.0. At different in-stants f, including the SS, temperature H distributions are shownfor emissivity e = 0.1, 0.5 and 1.0 in Fig. 6(a)–(c), respectively. Atthe SS, variations of energy flow rates W for e = 0.1, 0.5 and 1.0are shown in Fig. 6(d)–(f), respectively.

It is observed from Fig. 6(a)–(c) that the non-Fourier effect on Hdistributions is less pronounced when the emissivity e increases.For e = 0.1 (Fig. 6(a)) discontinuity in temperature H is very muchdistinct at two time levels, viz., f = 0.1 and 0.3. For e = 0.5 and 0.9,there is a sharp decrease in discontinuity in H at f = 0.1 and 0.3.For e = 0.9, this discontinuity is not even visible at f = 0.3. Further,the SS is reached faster when e increases. For e = 0.1 and e = 0.9, theSS times are 3.45 (Fig. 6(a)) and 2.23 (Fig. 6(c)), respectively. This isowing to the fact that with increase in e, radiation becomes morepronounced. Radiation being a much faster process than conduc-tion, when it dominates over conduction, the combined effect isthe faster attainment of the SS. At the SS, H profiles with or with-out non-Fourier effects are the same. Thus, non-Fourier effect isless evident for a higher value of emissivity e. At any location, atany time, a higher H for a higher value of e further substantiatethe observed trend. At the SS, LBM–FVM and FDM–FVM resultscompare exceedingly well.

(d)(a)

(e)(b)

(f)(c)

Distance, η

Tem

pera

ture

, Θ

0 0.2 0.4 0.6 0.8 1

0

0.2

0.4

0.6

0.8

1

SS: LBM (ζ = 6.67)SS: FDMSS: ω = 1.0ζ = 2.0

ζ = 1.2

ζ = 0.1 ζ = 0.3 ζ = 0.6

β = 0.1

Distance, η

Ene

rgy

flow

rate

,Ψ

0 0.2 0.4 0.6 0.8 10

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

ΨT

ΨR

ΨC

β = 0.1

Distance, η

Tem

pera

ture

, Θ

0 0.2 0.4 0.6 0.8 1

0

0.2

0.4

0.6

0.8

1SS: LBM (ζ = 6.00)SS: FDMSS: ω = 1.0ζ = 2.0

ζ = 1.2ζ = 0.1 ζ = 0.3

ζ = 0.6 β = 1.0

Distance, η

Ene

rgy

flow

rate

,Ψ

0 0.2 0.4 0.6 0.8 10.5

1

1.5

2

2.5

3

3.5

ΨT

ΨR

ΨC

β = 1.0

Distance, η

Tem

pera

ture

, Θ

0 0.2 0.4 0.6 0.8 1

0

0.2

0.4

0.6

0.8


ζ = 1.2

ζ = 0.1

ζ = 0.3

ζ = 0.6

β = 2.0

Distance, η

Ene

rgy

flow

rate

,Ψ

0 0.2 0.4 0.6 0.8 10.5

1

1.5

2

2.5

3

3.5

4

4.5

ΨT

ΨR

ΨC

β = 2.0

Fig. 4. Radial g temperature H at different time n levels including the SS for extinction coefficient b (a) =0.1, (b) =1.0 and (c) =2.0; at SS, distributions of conductive WC,radiative WR and total WT energy flow rates for b (d) =0.1, (e) =1.0 and (f) =2.0; x = 0.5, N = 0.1, e = 1.0, RR = 2.0.


The above observation is further substantiated through theplots of SS radial g distributions of conductive WC, radiative WR

and total energy WT flow rates shown in Fig. 6(d)–(f). At all loca-tions, the radiative energy WR flow rate is found to increase with

(d)(a)

(e)(b)

(f)(c)

Distance, η

Tem

pera

ture

, Θ

0 0.2 0.4 0.6 0.8 1

0

0.2

0.4

0.6

0.8


ζ = 0.6ζ = 0.1ζ = 0.3

ζ = 1.2

ω = 0.0

Distance, η

Ene

rgy

flow

rate

, Ψ

0 0.2 0.4 0.6 0.8 1

1

2

3

4

5ΨT

ΨR

ΨC

ω = 0.0

Distance, η

Tem

pera

ture

, Θ

0 0.2 0.4 0.6 0.8 1

0

0.2

0.4

0.6

0.8


ζ = 0.6

ζ = 0.1

ζ = 0.3

ζ = 1.2

ω = 0.5

Distance, η

Ene

rgy

flow

rate

, Ψ

0 0.2 0.4 0.6 0.8 1

1

2

3

4

5 ΨT

ΨR

ΨC

ω = 0.5

Distance, η

Tem

pera

ture

, Θ

0 0.2 0.4 0.6 0.8 1

0

0.2

0.4

0.6

0.8


ζ = 0.6

ζ = 0.1 ζ = 0.3

ζ = 1.2

ω = 0.9

Distance, η

Ene

rgy

flow

rate

,Ψ

0 0.2 0.4 0.6 0.8 1

1

2

3

4

5

ΨT

ΨR

ΨC

ω = 0.9

Fig. 5. Radial g variations of temperature H at different time n levels including the SS for scattering albedo x (a) =0.0, (b) =0.5 and (c) =0.9; at SS, variations of conductive WC,radiative WR and total WT energy flow rates for x (a) =0.0, (b) =0.5 and (c) =0.9; b = 1.0, N = 0.03, e = 0.5, RR = 2.


(d)(a)

(e)(b)

(f)(c)

Distance, η

Tem

pera

ture

, Θ

0 0.2 0.4 0.6 0.8 1

0

0.2

0.4

0.6

0.8


ζ = 1.2

ζ = 0.1

ζ = 0.3 ζ = 0.6

ε = 0.1

Distance, η

Ene

rgy

flow

rate

, Ψ

0 0.2 0.4 0.6 0.8 10

0.5

1

1.5

2

2.5

3

3.5

ΨT

ΨR

ΨC

ε = 0.1

Distance, η

Tem

pera

ture

, Θ

0 0.2 0.4 0.6 0.8 1

0

0.2

0.4

0.6

0.8


ζ = 0.6ζ = 0.1ζ = 0.3

ζ = 1.2

ε = 0.5

Distance, η

Ene

rgy

flow

rate

, Ψ

0 0.2 0.4 0.6 0.8 10

1

2

3

4

5 ΨT

ΨR

ΨC

ε = 0.5

Distance, η

Tem

pera

ture

, Θ

0 0.2 0.4 0.6 0.8 1

0

0.2

0.4

0.6

0.8


ζ = 0.6

ζ = 0.1

ζ = 0.3ζ = 1.2

ε = 0.9

Distance, η

Ene

rgy

flow

rate

, Ψ

0 0.2 0.4 0.6 0.8 10

1

2

3

4

5

6

7ΨT

ΨR

ΨC

ε = 0.9

Fig. 6. Radial g variations of temperature H at different time n levels including the SS for emissivities of the cylinder walls e1 = e2 = e (a) =0.1, (b) =0.5 and (c) =0.9; at SS,distributions of conductive WC, radiative WR and total WT energy flow rates for N (a) =0.05, (b) =0.1 and (c) =0.9; b = 1.0, x = 0.0, N = 0.03, RR = 2.0.


the emissivity, while the conductive energy flow rate WC de-creases. Increase in WR is more significant than the decrease in

WC, as a result, the total energy flow rate increases with increasein emissivity.

Distance, η

Tem

pera

ture

, Θ

0 0.2 0.4 0.6 0.8 1

0

0.2

0.4

0.6

0.8


ζ = 1.2

ζ = 0.1ζ = 0.3

ζ = 0.6

N = 0.05

Distance, η

Ene

rgy

flow

rate

, Ψ

0 0.2 0.4 0.6 0.8 10.5

1

1.5

2

2.5

3

3.5

4

4.5

5

ΨT

ΨR

ΨC

N = 0.05

Distance, η

Tem

pera

ture

, Θ

0 0.2 0.4 0.6 0.8 1

0

0.2

0.4

0.6

0.8


ζ = 1.2ζ = 0.1 ζ = 0.3

ζ = 0.6

N = 0.1

Distance, η

Ene

rgy

flow

rate

, Ψ

0 0.2 0.4 0.6 0.8 11

1.5

2

2.5

3 ΨT

ΨR

ΨC

N = 0.1

Distance, η

Tem

pera

ture

, Θ

0 0.2 0.4 0.6 0.8 1

0

0.2

0.4

0.6

0.8

1

SS: LBM (ζ = 6.67)SS: FDMSS: ω = 1.0ζ = 2.0

ζ = 1.2

ζ = 0.1 ζ = 0.3 ζ = 0.6

N = 1.0

Distance, η

Ene

rgy

flow

rate

, Ψ

0 0.2 0.4 0.6 0.8 1

0

0.5

1

1.5ΨT

ΨR

ΨCN = 1.0

(a) (d)

(b) (e)

(c) (f)

Fig. 7. Radial g temperature H at different time n levels including the SS for conduction–radiation parameter N (a) =0.05, (b) =0.1 and (c) =1.0; at SS, distributions ofconductive WC, radiative WR and total WT energy flow rates for N (a) =0.05, (b) =0.1 and (c) =1.0; b = 1.0, x = 0.5, e = 1.0, RR = 2.0.


Distance, η

Tem

pera

ture

, Θ

0 0.2 0.4 0.6 0.8 1

0

0.2

0.4

0.6

0.8

1

1.2SS: LBM (ζ = 5.00)SS: FDMSS: ω = 1.0

ζ = 0.6ζ = 0.1

ζ = 0.3

ζ = 1.2

RR = 1.1

ζ = 2.0

Distance, η

Ene

rgy

flow

rate

, Ψ

0 0.2 0.4 0.6 0.8 1

2

4

6

8

10

12ΨT

ΨR

ΨC

RR = 1.1

Distance, η

Tem

pera

ture

, Θ

0 0.2 0.4 0.6 0.8 1

0

0.2

0.4

0.6

0.8


ζ = 0.6

ζ = 0.1

ζ = 0.3

ζ = 1.2

RR = 2.0

Distance, η

Ene

rgy

flow

rate

, Ψ

0 0.2 0.4 0.6 0.8 10.5

1

1.5

2

2.5

3

3.5

ΨT

ΨR

ΨC

RR = 2.0

Distance, η

Tem

pera

ture

, Θ

0 0.2 0.4 0.6 0.8 1

0

0.2

0.4

0.6

0.8


ζ = 0.6

ζ = 0.1

ζ = 0.3

ζ = 1.2

RR = 10

Distance, η

Ene

rgy

flow

rate

, Ψ

0 0.2 0.4 0.6 0.8 10

0.2

0.4

0.6

0.8

1

1.2

1.4ΨT

ΨR

ΨC

RR = 10

(a) (d)

(b) (e)

(c) (f)

Fig. 8. Radial g variation of temperature H at different time n levels including the SS for radius ratios RR (a) =1.1, (b) =2.0 and (c) =10.0; at SS, variations of conductive WC,radiative WR and total WT energy flow rates for RR (a) =1.1, (b) =2.0 and (c) =10.0; b = 1.0, x = 0.5, N = 0.1, e = 1.0.



3.4. Effect of conduction–radiation parameter N on temperature andenergy flow rate

In a combined mode conduction–radiation problem, with otherparameters fixed, the conduction–radiation parameter N signifi-cantly affects the distributions of temperature and energy flowrate. For a lower value of N, radiation dominates over conduction.With b = 1.0, x = 0.5, e = 1.0 and RR = 2.0, including the SS, at differ-ent instant f, temperature H distributions are shown in Fig. 7(a)–(c) for N = 0.05, 0.1 and 1.0, respectively. Energy flow rates areshown in Fig. 7(d)–(f) for N = 0.05, 0.1 and 1.0, respectively. It isobserved from Fig. 7(a)–(c) that non-Fourier effect on temperatureH is more significant for higher value of N. When the influence ofradiation is more, i.e., N = 0.05 (Fig. 7(a)), the discontinuity in Halmost dampens after the thermal wave have reached the outercylinder. However, this is not so for a higher value of N = 0.1 and1.0. It is observed from Fig. 7(c) that for N = 1.0, the discontinuityin H has not even died out at f = 2.0. At f = 2.0, the wave hasreached the inner cylinder after reflecting from the outer cylinder.In this case, because of the reduced contribution of radiation, tem-perature evolves slowly, and thus the non-Fourier effect is felt for alonger duration, and the SS is reached late. The SS times forN = 0.05, 0.1 and 1.0 are 3.82, 6.00 and 6.67, respectively. The re-duced effect of radiation with higher value of N is also evident fromthe reduced difference between the SS H distributions for conduc-tion–radiation and pure conduction (x = 1) case. For all values of N,at the SS, LBM–FVM and FDM–FVM results match exceedinglywell.

Plots of conductive WC, radiative WR and total energy WT flowrates for N = 0.05, 0.1 and 1.0 shown in Fig. 7(d)–(f), respectivelyfurther confirm the observed trends of the temperature H distribu-tions. When radiation is dominant (N = 0.05), the total energy flowrate is high and its major contribution comes from radiation(Fig. 7(d)). When N = 0.1, the effect of radiation is relatively lessthan that for N = 0.05, and the total energy flow rate is reduced.Radiation dominates over conduction in most of the regions exceptnear the outer cylinder (Fig. 7(e)). For a higher value of N(=1.0),conduction dominates over radiation, and as observed in Fig. 7(f),throughout the medium, major contribution to the total energyflow rate comes from conduction. And, because of the reducedradiation, the total energy flow rate is low. It is to be noted thatfor N = 1.0, unlike results shown in Fig. 7(f), both conductive andradiative energy flow rates should be equal. However, this is notso, because of the enhanced radiative heat transfer caused by theoptical parameters, b = 1.0, x = 0.5 and e = 1.0.

3.5. Effect of radius ratio RR on temperature and energy flow rate

With all other parameters fixed, the optical thickness of the en-closed medium, which in the present case is b(r2 � r1), and the ra-dial distance (r2 � r1) between the two concentric cylinders have astrong bearing on distributions of temperature and energy flowrate. With radius r2 of the outer cylinder fixed, the above twoparameters can be linked to the radius ratio RR ¼ r2

r1: For RR = 1.1,

2.0 and 10.0, temperature H distributions at different instants fincluding the SS are shown in Fig. 8(a)–(c), respectively. For thecorresponding RR, distributions of energy flow rates at the SS areshown in Fig. 8(d)–(f), respectively. For results in Fig. 8(a)–(f),the parameters used are: b = 1.0, x = 0.5, N = 0.1, e = 1.0.

Fig. 8(a)–(c) shows radial temperature H distributions forRR = 1.1, 2.0 and 10.0, respectively. With decrease in RR, non-Fou-rier effect on temperature becomes significant. For RR ¼ r2

r1¼ 1:1

(Fig. 8(a)), discontinuity in H profile is very significant even afterthe wave front has reached the inner cylinder at time f = 2.0. Inthe present case, with r2 = 1.1r1, the total optical thickness s = (r2 -� r1)b = 0.1 is small, and the radiation effect is low. This is also evi-

dent from radial distributions of WC, WR and WT shown in Fig. 8(d).Conduction contributes much more to WT than radiation. A furtherobservation of Fig. 8(a) reveals that the conductive as well as radi-ative energy flow rates are almost independent of the radial loca-tion. This is again attributed to the fact that for low RR, thegeometry is like a planar medium. With increase in RR, at any in-stant, discontinuity in the temperature profile decreases(Fig. 8(b) and (c)). For RR = 10, the radial distance (r2 � r1) betweenthe inner and the outer cylinder is quite high. This in one handkeeps the temperature source (the inner cylinder) quite far fromthe outer cylinder and on the other, increases the opacity(r2 � r1)b of the medium. Though the total energy flow rate WT islower than that for lower value of RR, because of the increasedopacity, relative contribution of radiation to the total energy flowrate is much more in most of the region except near the outer(cold) cylinder (Fig. 8(f)). Near the outer (cold) cylinder, the tem-perature is low, and thus conduction dominates over radiation.Non-Fourier effect, i.e., discontinuity in H is not observed at anytime level. For all values of RR, SS H distributions from LBM–FVM match exactly with that obtained from the FDM–FVM.

It is to be noted as observed from Fig. 8(a)–(c), that the SS timeincreases with increase in RR. This is for the reason that withRR = 1.1, radial distance between the inner and outer cylinder islower than that for RR = 2.0, and RR = 10.0. Though the non-Fouriereffect is more significant for lower RR for the reason explainedabove, in this case, the wave front takes less time to reach fromthe inner cylinder to the outer cylinder, and vice versa.

4. Concluding remarks

A combined mode transient conduction–radiation problem in aconcentric cylindrical enclosure was solved. Effect of the finitepropagation speed of the thermal perturbation was accounted bynon-Fourier conduction. The energy equation was formulatedand solved using the LBM. The volumetric radiative informationneeded in the energy equation was computed using the FVM. Forthe combined mode problem with Fourier conduction, LBM–FVMresults were validated against those available in the literature.With non-Fourier effect, distributions of temperature, and conduc-tive, radiative and total energy flow rates were analyzed for the ef-fects of the extinction coefficient, the scattering albedo, theemissivity, the conduction–radiation parameter and the radius ra-tio. In all cases, the SS temperature distributions were comparedagainst the FDM–FVM, in which with radiative information com-puted using the FVM, the energy equation was solved using theFDM. Excellent comparison was found in all cases. To study theeffect of radiation, in every case, SS temperature profile for pureconduction was also provided. For all parameters, relative contri-butions of conduction and radiation to the total energy flow ratewere analyzed. When radiation dominated over conduction, tem-perature near the hot cylinder was more, non-Fourier effect waslow, steady-state was attained fast and the total energy flow ratewas higher. Spatially, the problem under consideration was 1-D,and in both FVM and LBM formulations of the energy equations,volumetric radiative information was computed using the FVM.Thus, the computational performances of both the methods wereobserved to be similar. However, for multi-dimensional geometry,as observed in its applications in fluid mechanics and heat transfer,the LBM will have computational advantage over the FDM and theFVM.

Appendix A

To show the consistency of the LBM formulation of the non-Fou-rier conduction with volumetric radiation for the cylindrical geom-etry, we take recourse of the Chapman–Enskog multi-scale


expansion and Taylor series expansion. If the left hand side (LHS) ofEq. (21) is expanded in both time f and space g dimensions, we get

fiðgþ eiDf; fþ DfÞ ¼ fiðg; fÞ þ Df@fi

@fþ Df

@ðeifiÞ@g

þ OðDf2Þ þ OðDg2Þ

ðA1Þ

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

fiðg; fÞ ¼ fi ¼ f ð0Þi þ vf ð1Þi þ Oðv2Þ; jvj < 1 ðA2Þ

Substituting Eq. (A2) in the right hand side (RHS) of Eq. (21), we get

@f ð0Þi

@fþ v

@f ð1Þi

@fþ @ðeif

ð0Þi Þ

@gþ v

@ðeifð1Þi Þ

@g

¼ �1s vf ð1Þi � brref

4NaiDf

2r� �W�R �

W�C2

nai

2gþ 4biei

� þ OðDfÞ þ Oðv2Þ ðA3Þ

Summing Eq. (A3) over all states, i.e., i = 1, . . . ,M, and noting that|v| 1 and from Eqs. (22)–(25) noting thatX

i

f ð1Þi ¼ 0 ðA4Þ

Xi

Xi

eifð1Þi

!¼ 0 ðA5Þ

We get

@h@f¼ �1

g@ðgW�CÞ@g

� �� brref

4NaiDf

2r� �W�R þ OðDfÞ þ Oðv2Þ ðA6Þ

which is the non-dimensional form of Eq. (1). Multiplying Eq. (A3)by the propagation speed ei, we get

@ðX

i

ðeifð0Þi ÞÞ

@fþ e

@ðX

i

ðeifð1Þi ÞÞ

@fþ@ðX

i

ðe2i f ð0Þi ÞÞ

@g

þ e@ðX

i

ðe2i f ð1Þi ÞÞ

@g1seX

i

eifð1Þi � brref

4Naiei

2r� �W�R

�W�C2

Xi

aiei

2gþ 4bie2

i

� ðA7Þ

Noting thatP

ibi ¼ 1;P

iei ¼ 0; and from Eqs. (23), (24), (A4) and(A5) we can write Eq. (A7) as

@P

iðeifð0Þi Þ

� �@f|fflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflffl}@W�

C@f

þ v@P

iðeifð1Þi Þ

� �@f|fflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflffl}¼0

þ@P

iðe2i f ð0Þi Þ

� �@g|fflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflffl}@h@g

þ v@P

iðe2i f ð1Þi Þ

� �@g|fflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflffl}¼0

¼ �1svX

ieifð1Þi|fflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflffl}

¼0

þ brref

4Naiei

2r� �U�R|fflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflffl}

¼0

�U�C2

X aiei

2g|fflfflfflffl{zfflfflfflffl}¼0

þX

4bie2i|fflfflfflfflfflffl{zfflfflfflfflfflffl}

¼4

0BB@

1CCAðA8Þ

It is seen that the first order deviation is zero, and with this Eq. (A8)leads to the following

@W�C@fþ @h@g¼ �2W�C þ OðDfÞ þ Oðv2Þ ðA9Þ

Eq. (A9) is the non-dimensional form of Eq. (4). Thus, like the finitedifference method and FVM approaches, when Df ? 0, and v ? 0,the mesoscopic form of the governing equation (Eq. (21)) convergesto its macroscopic counterpart (Eq. (19)).

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