Technology, Chonbuk National University, Jeonju, Chonbuk, Republic of Korea 3 Institute of Fluid Science, Tohoku University, Sendai, Japan

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1 Numerical Heat Transfer, Part A, 60: , 2011 Copyright # Taylor & Francis Group, LLC ISSN: print= online DOI: / ANALYSIS OF CONDUCTION AND RADIATION HEAT TRANSFER IN A 2-D CYLINDRICAL MEDIUM USING THE MODIFIED DISCRETE ORDINATE METHOD AND THE LATTICE BOLTZMANN METHOD Subhash C. Mishra 1, Ch. Hari Krishna 2, and Man Young Kim 3 1 Department of Mechanical Engineering, Indian Institute of Technology Guwahati, Guwahati, India 2 Department of Aerospace Engineering, Research Center of Industrial Technology, Chonbuk National University, Jeonju, Chonbuk, Republic of Korea 3 Institute of Fluid Science, Tohoku University, Sendai, Japan This article deals with the analysis of radiative transport with and without conduction in a finite concentric cylindrical enclosure containing absorbing, emitting, and scattering medium. Isothermal medium as the radiation source confined between the cold cylinders and a nonisothermal medium with the inner cylinder as the radiation source are the two nonradiative and radiative equilibrium problems. They involve only calculation of radiative information. In the third problem, a conducting-radiating medium is thermally perturbed by raising the temperature of the inner cylinder. In all problems, radiative information is computed using the modified discrete ordinate method (MDOM), and in the third problem, the lattice Boltzmann method (LBM) is used to formulate and solve the energy equation. Depending on the problems, effects of various parameters such as the extinction coefficient, the scattering albedo, the boundary emissivity, the conduction-radiation parameter, and the radius ratio are studied on temperature and heat flux distributions. The MDOM and the LBM-MDOM results are compared with those available in the literature. To further establish the accuracy of the MDOM and the LBM-MDOM results, in all problems, comparisons are made with the results obtained from the finite volume method (FVM) and the finite difference method-fvm approach, in which FVM provides the radiative information. The selection of the FDM-FVM for the third problem is also with the objective that for this problem, not much work is reported in which the FVM is used to calculate the radiative information. MDOM and LBM-MDOM results are found to compare well with those available in the literature, and in all cases they are in excellent agreement with FVM and FDM-FVM approaches. 1. INTRODUCTION As temperature cannot go to absolute zero, so is the radiation. Whatever magnitude it may have, it is always there. Every entity must have a finite temperature, and thermal radiation is the one which is caused by the finite temperature of an Received 19 November 2010; accepted 18 April Ch. Hari krishna contributed to this work when he was an MTech student at IIT Guwahati. Address correspondence to Subhash C. Mishra, Department of Mechanical Engineering, IIT Guwahati, Guwahati , India. scm_iitg@yahoo.com 254

2 CONDUCTION AND RADIATION IN A 2-D CYLINDRICAL MEDIUM 255 NOMENCLATURE DA, DV surface area and volume of the control volume, respectively E, W, N, S, T, B east, west, north, south, top, and bottom neighbor control volumes of P, respectively e, w, n, s, t, b east, west, north, south, top, and bottom face of the control volumes of P, respectively. c p specific heat ~e i propagation velocity in the direction i in the lattice ~e r ;~e u ;~e z cylindrical r, u, and z directions base vectors, respectively f i particle distribution function in the i direction f ð0þ i G I k m N N h equilibrium particle distribution function in the i direction incident radiation intensity thermal conductivity index for direction conduction-radiation parameter, jb=ð4rtref 3 Þ number of discrete h directions N / number of discrete / directions N r number of control volumes in radial direction N z number of control volumes in axial direction ^n outer normal P cell center r radius of the cylinder q R radiative heat flux S source term T temperature t time w weight in the LBM Z height of the cylinder a thermal diffusivity b extinction coefficient e emissivity h polar angle H nondimensional temperature /, / X angular azimuthal angles measured from the x 0 - and r- axes, respectively (Figure 2b) u spatial azimuthal angle measured from the x-axis (Figure 2b) j a absorption coefficient n nondimensional time, at=l 2 q density r Stefan-Boltzmann constant (¼ W=m 2 K 4 ) r s scattering coefficient m, g, f direction cosines in the r, u, and z directions, respectively. s relaxation time U nondimensional emissive power / azimuthal angle x scattering albedo X direction in the MDOM and rate of change of the particle distribution function f i in the LBM DX elemental solid angle W dimensionless heat flux Subscripts 1, 2, 3, 4 inner, outer, top, and bottom walls of the cylinder b black=boundary C Conductive E, W, N, S, T, B node points where intensities are located n, s control volume faces P value at the cell center R radiative T Total ref reference Superscripts m r z dimensionless variable discrete direction radial direction axial direction entity. Because it is of the order of O(rT 4 ), where r ¼ Wm 2 K 4 is the Stephen-Boltzmann constant and T(K) is the absolute temperature, compared to conduction and convection which are of the order of O(DT), consideration of thermal radiation becomes important in high temperature applications. In science and

3 256 S. C. MISHRA ET AL. engineering, we deal with many high temperature applications. These are not limited to boilers, furnaces, IC engines, and insulations [1 4]. Thus, for the correct analysis of these systems, radiation needs to be properly accounted. Further, analysis of thermal radiation finds applications in atmospheric sciences [5] for weather forecasting, remote sensing [6], thermal characterization during the phase change of semitransparent materials like glass and semiconductors [7], diagnosis of tumors in the biological systems [8], and treatment of cancers=tumors by irradiating the nanoparticles [9], etc. Almost all of the applications mentioned above deal with situations in which radiation penetrates the medium, and absorption, emission, and scattering influence the process of radiative transport. In contrast to the radiation transport in a vacuum, which is characterized by surface phenomenon, radiative transfer in a participating medium becomes a volumetric phenomenon and the governing radiative transfer equation turns out to be an integro-differential one. Even though in many situations radiation can be considered gray (wavelength independent) and an instantaneous process and one can thus ignore spectral and temporal dimensions, its angular dependence introduces two variable, viz., polar and azimuthal angles. Thus, apart from the spatial dimensions, transport of thermal radiation involves two more independent variables, and the much talked about difficulties in the analysis of thermal radiation is mainly because of these two variables. The analytic solutions for radiative transport problems are rare and for the available cases they are with limitations that in a combined mode problems, they may not be compatible with the solvers of the other modes of heat transfer. Results from the Monte Carlo method (MCM) [10] in thermal radiation is considered exact, but because of its appetite for computational time and its inability to work efficiently with solvers of other modes of heat transfer, its application to provide radiative information in combined mode radiation, conduction, and=or convection problem is rarely seen. The discrete ordinate method (DOM) [11], first proposed by Chandrasekhar for stellar application, is even older than the MCM, but because of its elegance and although it has gone through several modifications over the last 50 years it has survived the test of time. It is being widely used by scientists and engineers [12 19]. For complex situations, like a multidimensional geometry, the DOM was found to have some weaknesses such as the ray effect and false scattering, and motivated researchers have come up with the discrete transfer method (DTM) [20] and the finite volume method (FVM) [21]. In computational fluid dynamics, the FVM has become very handy and it has wide application. Since the FVM for radiation works on the same grid as the FVM for conduction and convection, this method has found good acceptance. Apart from the MCM, DOM, DTM and FVM, there are many other methods like the P N approximations [22], zonal method [23], YIX [24], ray emission method [25], collapsed dimension method [20], etc., each having some strong and weak points. However, the DOM is still a preferred method by many as, like the FVM, it is well adaptable to the grids of other modes of heat transfer. In engineering, the first application of the DOM was reported by Fiveland in 1984 [26]. He used it to solve the radiative transport problem in a 2-D square enclosure with cold black boundaries containing isothermal absorbing-emitting medium. Its ordinates and corresponding weights were found to have problems with

4 CONDUCTION AND RADIATION IN A 2-D CYLINDRICAL MEDIUM 257 moments, and subsequently, improvements were suggested by Truelove [27]. Even after Truelove [27], several modifications were proposed but none could free the DOM with the restriction of choosing directions already predefined according to the order of approximations, and in line with the one given by Carlson and Lathrop [28]. Thus in this method, for the sake of moment matching, although different versions of tables of ordinates and weights were suggested, the selection of ray directions were not as per the DTM and FVM. In Beak and Kims [18] and Kim et al. [19] they proposed a new variant of the DOM, but the procedure described there is much akin to the FVM. Therefore, going in line with the DOM in references [12 19], efforts on easing the constraints on selection of ordinates and their corresponding weights from the look up tables continued. In 2005, Mishra et al. [29] proposed a modification in the DOM called the modified DOM (MDOM), that focused on freedom in selecting the directions (ordinates) like any other method including the option to have them similar to the conventional DOM or in any new way, and a simple formula to calculate their corresponding weights. With ray directions decided as per the need or according to the wish of the user [29], direction cosines are calculated and the radiative transfer equation is written in the coordinate directions the same way as the conventional DOM. Each ray is assumed isotropic over the elemental solid angle with which the 4p spherical space is made of, and the corresponding weights are calculated by a simple integration. Other procedures are similar to the conventional DOM. The MDOM has previously been applied to different types of radiative transport problems in Cartesian as well as in cylindrical and spherical geometries. It has been applied to a medium with diffuse as well as collimated loadings [30]. It has been found to work well for the analysis of transport of a short-pulse radiation [31]. In combined mode conduction-radiation problems, it has also worked well. Very recently, its application was, extended to radiative transport problems with and without radiation in a 1-D cylindrical enclosure [32], and a concentric spherical enclosure containing participating media [33]. In all cases, the MDOM has been found to work well. Because of the shape, radiative transfer analysis in a 2-D cylindrical geometry is relatively difficult, and so far the MDOM proposed by Mishra et al. [29] has not been applied to radiative transport with and without radiation in this geometry. The lattice Boltzmann method (LBM) is coming up as a versatile computational tool to analyze a wide range of problems in science and engineering [34 36]. Recently, it has also been applied to formulate and solve the energy equations of heat transfer problems involving volumetric radiation [25, 37 42]. Its compatibility with various numerical radiative transfer methods has also been established [25, 37 42], and in complex problems involving heavy computations it has shown an advantage over its CFD counterparts [39]. However, as far as its application to formulate an energy equation of a combined mode transient conduction-radiation heat transfer problems in a 2-D cylindrical geometry is concerned, no work has been reported so far. Thus, this work, while extending the application of the MDOM [29] to calculate volumetric radiative information in radiation with and without conduction in a 2-D concentric cylindrical enclosure, also aims at application of the LBM and MDOM to analyze the combined mode transient conduction-radiation problem. The FVM for radiation [43, 44] is too robust a method and since it was proposed in 1990s, its application to various problems is increasing. Although for pure

5 258 S. C. MISHRA ET AL. radiative transfer problems it has been applied to various geometries [2, 39, 40 46], as far as its application to a combined mode transient conduction-radiation heat transfer in a 2-D concentric cylindrical geometry is concerned, not much study has been reported so far. Another objective of this article to extend the applicability of the FVM for the combined mode problems in 2-D cylindrical geometry. Since literature on the analysis of this problem even by other methods is scarce, to validate the MDOM for pure radiative transport problems and the LBM-MDOM for the combined mode problems, we solve the pure radiative transports problems using the FVM and the combined mode problems using the finite difference method (FDM)-FVM, in which volumetric radiative information is computed using the FVM and the FDM is used to solved the energy equation. The FVM formulation follows the one given by Kim [2], and for the sake of brevity here we skip its description. As described in the next section, we consider three types of problems, and depending upon the problems we analyze and compare the effects of various parameters such as the extinction coefficient, the scattering albedo, the conduction-radiation parameter, the radius ratio, and the wall emissivity on variations of nondimensional temperature=emissive power and conductive, radiative and total heat fluxes. 2. FORMULATION Towards the application of the MDOM in calculating the radiative information in 2-D finite concentric cylindrical enclosure, and the LBM for formulating and solving the energy equation of a combined mode transient conduction-radiation problem, we consider the following three problems.. Problem 1. The four (inner, outer, bottom, and top) boundaries of the 2-D concentric cylindrical enclosure (Figure 1a) are cold and the contained isothermal homogeneous medium inside it is the radiation source. Radiative energy flow rate=heat flux varies in both radial and axial directions. This case is the representative of a nonradiative equilibrium benchmark problem in which r~q R 6¼ 0:0; where q R is the radiative heat flux.. Problem 2. In this case, temperature of the participating medium inside the concentric cylindrical enclosure (Figure 1a) is unknown. The inner cylinder is the radiation source and the other three boundaries are kept cold. In this case, the divergence of radiative heat flux r~q R ¼ 0:0: This problem represents the case of a benchmark radiative equilibrium problem in which either conduction and=or or convection are absent or because of the very high temperature they are negligible.. Problem 3. The medium is both radiating and conducting, and the presence of conduction cannot be ignored. The entire system, which is initially at temperature T 2, is perturbed by raising the temperature of the inner cylinder to T 1 > T 2, while temperatures of the other three boundaries remain at T 2. In the absence of further perturbations, with the passage of time the system heads towards the steady-state (SS). Distributions of temperature T as well as conductive q C, radiative q R and total heat q T fluxes change with time, and depending upon the values of parameters relative contributions of conductive heat flux q C and radiative heat flux

6 CONDUCTION AND RADIATION IN A 2-D CYLINDRICAL MEDIUM 259 Figure 1. (a) Geometry of the 2-D concentric cylindrical enclosure, and (b) arrangement of lattices in the LBM and control volumes in the MDOM in the solution (r z) plane of the 2-D cylindrical enclosure. q R to the total heat flux q T change. This problem belongs to the class of a nonradiative equilibrium problem in which thermal equilibrium is the combined effect of radiation and conduction. Out of the three problems, problem 3 is the most involved one. In this case, with radiative information r~q R computed using any of the numerical radiative transfer methods, in the present case, such as the MDOM, the energy equation needs to be solved using a suitable numerical scheme such as the FDM, the FVM, the LBM, etc. Since the present work aims at extending the applications of the MDOMB to the 2-D cylindrical geometry, and to the best of the knowledge of the authors the LBM has not been applied to a conduction-radiation problem in this geometry, in the present work we formulate the energy equation using the LBM and use the MDOM to provide the radiative information required in the LBM formulation.

7 260 S. C. MISHRA ET AL. Below, we provide the formulations of the LBM and the MDOM, and outline the steps for solving a combined mode problem in a 2-D cylindrical geometry using the LBM-MDOM. For the case considered in problem 3, the energy equation is given by qc p qt qt ¼ kr2 T r~q R ð1þ where q is the density, c p is the specific heat, k is the thermal conductivity, and ~q R is the radiative heat flux. At any time level, radiative information r~q R appearing in Eq. (1) is given by r~q R ¼ j a ð4pi b GÞ where j a is the absorption coefficient, I b ¼ rt 4 =p is the blackbody intensity, and G is the incident radiation. With r~q R computed using any of the methods, such as the MDOM, the energy equation can be solved using any suitable methods such as the FDM or the FVM, or it can be formulated and solved in the LBM. Since LBM is gaining importance in the CFD, and for a combined mode conduction-radiation problem in a 2-D cylindrical geometry it has not been applied. In the present work, we use the LBM to formulate and solve the problem Lattice Boltzmann Method (LBM) Since in the spatial azimuthal direction u thermally the boundary conditions of the concentric cylinders are constant, mathematically in the spatial coordinates (r, u, z) the problem is a 2-D one, i.e., temperature T ¼ T(r, z) and heat flux q ¼ q(r, z). Thus, owing to the axisymmetric nature of the problem the solution plane is a 2-D one, and in the front view the solution r z plane is a rectangular one (Figure 1b). Therefore, like the 2-D rectangular geometry, in the LBM formulation for the 2-D cylindrical geometry considered in the present work, in the LBM information propagation will be in 2-D, i.e., in the r z plane. Like a 2-D rectangular geometry [39], in the LBM for information propagation in 2-D, we can use the D2Q9 lattice (Figure 2a). Therefore, for the 2-D cylindrical geometry under consideration, by taking into account the variable area effect with the BGK approximation [34], the discrete Boltzmann equation with D2Q9 lattice (Figure 2a) inany direction i is written as ð2þ qf i ð~r; tþ þ~e i rf i ð~r; tþ ¼ 1 qt s ½f ið~r; tþ f ð0þ i ð~r; tþš þ 1 ~r a ~e i Dt qfi qr i ¼ 1; ð3þ where f i is the particle distribution function, ~e i is the propagation velocity, f ð0þ i is the equilibrium particle distribution function, s is the single relaxation time, and a ¼ q=kc p is the thermal diffusivity.

8 CONDUCTION AND RADIATION IN A 2-D CYLINDRICAL MEDIUM 261 Figure 2. (a) D2Q9 lattice used in 2-D geometry, (b) cylindrical coordinate system used in MDOM formulation, (c) a 3-D control volume, (d) 2-D computational domain in front view, (e) schematic of the control volume in a cylindrical enclosure with P located at the control volume center in top view, (f) schematic of 3-D intensity distribution in top view, (g) schematic view of 2-D intensity distribution in top view, and (h) schematic of direction X m in the center of the elemental sub-solid angle DX m.

9 262 S. C. MISHRA ET AL. In the discrete form, Eq. (3) can be written as f i ð~r þ~e i Dt; t þ DtÞ ¼f i ð~r; tþ Dt s ½f ið~r; tþ f ð0þ i ð~r; tþš þ 1 adt ½f i ð~r þ~e i Dt; tþ f i ð~r; tþš ~r ~e i Dt ð4þ To account for the volumetric radiation rq R, Eq. (4) gets modified to f i ð~r þ~e i Dt; t þ DtÞ ¼f i ð~r; tþ Dt s ½f ið~r; tþ f ð0þ i ð~r; tþš þ 1 adt ½f i ð~r þ~e i Dt; tþ f i ð~r; tþš w ð5þ idt r~q ~r ~e i Dt qc R p where the divergence of radiative heat flux r~q R is given by Eq. (2). In Eq. (3), w i is the weight corresponding to the lattice direction. The relaxation time s for the D2Q9 (Figure 2a) lattices is given by [34] s ¼ 3a j~e i j 2 þ Dt 2 The nine velocities ~e i and their corresponding weights w i in the D2Q9 lattice are as follows. e 0 ¼ð0; 0Þ; e 1;3 ¼ð1; 0ÞC; e 2;4 ¼ð0; 1ÞC; e 5;6;7;8 ¼ð1; 1ÞC ð7þ w 0 ¼ 4 9 ; w 1;2;3;4 ¼ 1 9 ; w 5;6;7;8 ¼ 1 36 In the above equations, C ¼ Dr=Dt ¼ Dz=Dt and the weights satisfy the relation P 9 i¼1 w i ¼ 1. In a conduction-radiation problem, equilibrium particle distribution function f eq i is given by ð6þ ð8þ f ð0þ i ð~r; tþ ¼w i Tð~r; tþ ð9þ From the solution of Eq. (5), once the particle distribution functions f i are known temperature is calculated from the following. Tð~r; tþ ¼ X9 i¼1 f i ð~r; tþ ð10þ Implementation of the boundary conditions in the LBM for the conductionradiation problem for the current 2-D cylindrical geometry follows the procedure described in Mishra et al. [39] for a 2-D rectangular geometry, and with radiative information known from the MDOM, the solution procedure is described shortly.

10 CONDUCTION AND RADIATION IN A 2-D CYLINDRICAL MEDIUM 263 Below, we first provide the MDOM formulation to calculate the divergence of radiative heat flux r~q R Modified Discrete Ordinate Method (MDOM) Unlike the classical DOM and several of its variants available in the literature [26, 27], the MDOM proposed by Mishra et al. [29], although it has no restriction, does not select ray directions=ordinates and their corresponding weights in a pre-defined way from the look up tables. The MDOM has a freedom to select ray directions as per the requirements and the user s wish, and once the ray directions= ordinates are fixed, their corresponding weights are calculated using a simple formula. The robustness of the MDOM has earlier been exemplified in 1-D planar [30], 2-D [38] rectangular, 1-D cylindrical [32], and spherical [33] geometries. In this work, we use MDOM for a more relatively complicated situation, 2-D cylindrical geometry, and along with that we also exemplify the application of the LBM to a combined mode conduction-radiation problem in this geometry. Like any other numerical radiative transfer methods, in the MDOM first we select a coordinate system, and in the present case, as shown in Figure 2b, the appropriate coordinate system is the cylindrical one. Next, for any discrete direction ~X ¼ðsin h cos /Þ~e r þðsin h sin /Þ~e u þðcos hþ~e z ; we write the radiative transfer equation (RTE) as qi qs ¼ bi þ j ai b þ r Z s IðX 0 ÞHðX; X 0 ÞdX 0 4p X 0 ¼4p where s is the distance in the direction ~ X, r s is the scattering coefficient, and H is the scattering phase function. For the geometry (Figure 1a) and the coordinate system (Figure 2b) considered in the present work, Eq. (11) can be written as 1 q r qr ½mrIŠþ1 q r qu ½gIŠþ q qz ½fIŠ 1 q r q/ ½gIŠ ¼ bi þ j ai b þ r Z s 4p X 0 ¼4p ð11þ IðX 0 ÞHðX; X 0 ÞdX 0 where with h as the angular polar angle and / as the angular azimuthal angle, m ¼ sinhcos/, g ¼ sinhsin/ and f ¼ cosh are the direction cosines, and u is the spatial azimuthal angle. In the MDOM, the spatial (r, u, z) and the angular (h, /) coordinates are based on the cylindrical base vectors, ~e r ; ~e u and ~e z (Figure 2b). Therefore, angular azimuthal angles / (measured from the cylindrical base vector~e r ) and spatial azimuthal angle u X (measured from the Cartesian x 0 coordinate (Figure 2b)), are related to the spatial azimuthal angle u (measured from the Cartesian x coordinate (Figure 2b)), through the relation / ¼ u X u. For the 2-D cylindrical enclosure (Figure 1a), Eq. (12) can be simplified to ð12þ 1 q r qr ½mrIŠþ q qz ½fIŠ 1 q ½gIŠ ¼ bi þ S ð13þ r q/

11 264 S. C. MISHRA ET AL. where the source term S is given by S ¼ k a I b þ r Z s IðX 0 ÞHðX; X 0 ÞdX 0 ð14þ 4p X 0 ¼4p Since, spatially the problem is a 2-D one and it has no dependence on the spatial azimuthal angle u, because of the symmetry, to be explained shortly, and for simplicity instead of tracing intensity over the angular azimuthal angle /, instead of 2-D control volumes (rings), we construct 3-D control volumes (Figure 2c), each having the volume of DV ¼ pðr 2 n r2 s ÞDZ=2N / where N / is the number of divisions of the angular (also the spatial) azimuthal angle. It is to be noted that to exploit the axisymmetric nature of the cylindrical geometry, as shown in Figures 2d and 2e, we consider only half of the cylinder and to consider the contribution from the second half of the cylinder, the integrated quantities such as the incident radiation G and heat flux q R are multiplied by 2. Next, like the conventional DOM [26, 27], in the MDOM too, Eq. (13) is written for a discrete direction (X m ¼ (h m, / m )) having index m and is then integrated over the elemental control volume (Figure 2c) having center P. Z 1 q DV r qr ½mm ri m ŠdV fflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl} I Z þ q DV qz ½fm I m ŠdV fflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflffl} Z þ DV II Z bi m dv ¼ fflfflfflfflfflfflffl{zfflfflfflfflfflfflffl} IV Z 1 q DV r q/ ðgm I m ÞdV fflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl} III S m dv DV fflfflfflfflfflffl{zfflfflfflfflfflffl} V Integration of Eq. (15) over the 3-D control volume (Fig. 2c) results in m m ðda n In m DA sis m fflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl} Þ þ f m ðda t It m DA b Ib m Þ ðda e g m e fflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl} I e m DA wg m w I w m fflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl} Þ I II III þ bi m P DV fflfflffl{zfflfflffl} IV ¼ S m P DV fflffl{zfflffl} V where in the direction having index m, In m, I s m, I t m,andi b m, are the intensities at the north, south, top, and bottom faces of the control volume (Figure 2d), respectively, and Ie m and Iw m are the intensities through east and west faces (Figure 2e), respectively. It is to be noted that Figures 2d and 2e are the front and top views of the 2-D cylinder (Figure 1a), respectively. In Eq. (16), IP m and Sm P are the volumeaveraged intensities and source term in the direction m, respectively. The direction cosines appearing in Eq. (16) are as follows. m m ¼ sin h m cos / m ; f m ¼ cos h m ; g m e ¼ sin hm sin / m þ D/ ; 2 g m w ¼ sin hm sin / m D/ ð17þ 2 ð15þ ð16þ

12 CONDUCTION AND RADIATION IN A 2-D CYLINDRICAL MEDIUM 265 It is to be noted that the west and east faces of the 3-D control volume (Figure 2c) are azimuthally separated by D/, and hence the expression of g m e and gm w are different (Figure 2e). It is to be further noted that for the fact that the spatial azimuthal angle u is analogous to the angular azimuthal angle / (to be shown shortly) in Eq. (17) above, to show the angles between the east and west faces in the 3-D control volume (Figure 2c) instead of the spatial azimuthal angle Du, we have used the angular azimuthal angle D/. With reference to the control volume shown in Figure 2c, area terms appearing in Eq. (16) above are as follows. DA n ¼ r n D/ DZ; DA e ¼ DA w ¼ðr n r s ÞDZ; DA s ¼ r s D/ DZ; DA t ¼ DA b ¼ pðr2 n r2 s Þ 2N / To cover all directions, the spherical space (X ¼ 4p) is divided into some finite number of elemental solid angles DX, and in each solid angle the intensity is assumed isotropic with its value equal to that of the one at the center of the elemental solid angle. In this process, for a particular direction having index m, the intensities are identified with polar h m and azimuthal / m angles. Since the number of subdivisions of the polar space and the azimuthal space need not be the same, to have this freedom the polar space can have divisions like Dh ¼ p=n h and for the azimuthal space D/ ¼ 2p=N /. Thus, a direction having index m will be identified with the polar and azimuthal angles as ðh m k ; /m l Þ, where k(1 k N h ) and l(1 l N / ) are the indices for polar and azimuthal angles, respectively, and with this 1 m (N h N / ). Above, we have stated that for a given polar angle h m k, in the present work variation of the intensity over the angular azimuthal / m l ð0 / m l 2pÞ is exactly the same as the variation of intensity over spatial azimuthal angle u(0 u 2p). At any radial distance, for the current 2-D problem, this procedure frees us from the complicated procedure of tracing intensities over the complete span of the angular azimuthal angle /, but instead it makes us trace intensities over the 3-D (instead of 2-D) control volumes spanned over the spatial azimuthal space. Further, because of the spatial symmetry we do this tracing over 0 u p, i.e., half cylinder (Figures 2f and 2g). A similar procedure has been adopted by Kim [2] in the FVM for a cylindrical geometry. To explain the above, we take recourse of Figures 2f and 2g. The intensities in the 3-D control volume denoted by I1 1; I 2 1; I 3 1, and I 4 1 shown in Figure 2f all have the same radius r and z. For the four token intensities considered over the 0 / p, in Figure 2f the spatial azimuthal angle u between all consecutive intensities is p=4. Therefore, the angular azimuthal angle / ¼ u X u ¼ 2p u is 9p=8; 11p=8; 13p=8 and15p=8, for the intensities I1 1; I 2 1; I 3 1, and I 4 1, respectively. The intensities at point 1 in Figure 2g have the same radius r, but all are located at the spatial azimuthal angle / ¼ p=2 with the angular azimuthal / angle 15p=8, 13p=8, 11p=8, and 9p=8 for intensities I1 4; I 1 3; I 1 2, and I 1 1 ; respectively. Therefore, the intensities Ij 1 and I j 1 have the same values of r, z, h, and /, leading to I j 1 ¼ I j 1. Thus, a mapping exists between the intensities in the two figures. Having established the mapping of the intensities variation over angular azimuthal angles in 2-D (Figure 2g) to its variation of the spatial azimuthal angle in ð18þ

13 266 S. C. MISHRA ET AL. a 3-D control volume (Figure 2f), we next proceed towards further simplification that will lead to the calculation of intensity distribution, radiative heat flux, and divergence of radiative heat flux (Eq. (2)). To reduce the number of unknowns in Eq. (16), we relate the facial intensities I m n, I m s ; I m t, and I m b, the angular edge intensities I m e and I m w to the nodal intensity I m P (Figures 2d and 2e), and to ensure positive intensity, we use the simple step scheme. In m ¼ I P m ; I s m ¼ IS m ; I t m ¼ IP m ; Ib m ¼ I B m ; I e m ¼ IP m ; I w m ¼ I W m for mm > 0 and f m > 0 ð19aþ In m ¼ I N m ; I s m ¼ IP m ; I t m ¼ IP m ; Ib m ¼ I B m ; I e m ¼ IP m ; I w m ¼ I W m for mm < 0 and f m > 0 ð19bþ In m ¼ I N m ; I s m ¼ IP m ; I t m ¼ IT m ; Ib m ¼ I P m ; I e m ¼ IP m ; I w m ¼ I W m for mm < 0 and f m < 0 ð19cþ In m ¼ I P m ; I s m ¼ IS m ; I t m ¼ IT m ; Ib m ¼ I P m ; I e m ¼ IP m ; I w m ¼ I W m for mm > 0 and f m < 0 ð19dþ where IW m ; I E m; I N m; I S m; I T m,andi B m are the nodal intensities in the six control volumes surrounding the control volume under consideration. By using the above step scheme, to cover the complete span of the polar space, depending on whether we are marching from the inner and bottom of the cylinder to the outer and top of the cylinder (in this case m m > 0 and f m > 0, first quadrant), from the outer and bottom of the cylinder to the inner and top of the cylinder (in this case m m < 0andf m > 0, second quadrant), from the outer and top of the cylinder to the inner and bottom of the cylinder (in this case m m < 0 and f m < 0, third quadrant), or from the inner and top of the cylinder to the outer and bottom of the cylinder (in this case m m > 0 and f m < 0, fourth quadrant) (Figure 2d), by using Eqs. (19a) (19d) and Eq. (16), the nodal intensity IP m can be written as I m P ¼ mm DA s I m S þ fm DA b I m B DA wg m w I m W þ DVSm P m m DA n þ f m DA t DA e g m e þ bdv for m m m > 0 and f m > 0 I m P ¼ mm DA n I m N þ fm DA b I m B DA wg m w I m W þ DVSm P m m DA s þ f m DA t DA e g m e þ bdv for m m > 0 and f m > 0 ð20aþ ð20bþ I m P ¼ mm DA n I m N fm DA t I m T DA wg m w I m W þ DVSm P m m DA s f m DA b DA e g m e þ bdv for m m > 0 and f m > 0 ð20cþ I m P ¼ mm DA s I m S fm DA t I m T DA wg m w I m W þ DVSm P m m DA n f m DA b DA e g m e þ bdv for m m > 0 and f m > 0 ð20dþ

14 CONDUCTION AND RADIATION IN A 2-D CYLINDRICAL MEDIUM 267 It is to be noted that with the above step scheme, for calculation of the facial intensities Ie m and Iw m on u faces, no additional expressions of I P m as given by Eqs. (20a) (20d) are required. For isotropic scattering H(X, X 0 ) ¼ 1.0, the expression of the source term S given in Eq. (14) is given by rt 4 S ¼ k a þ r s p 4p G ð21þ where the incident radiation G is given by and numerically computed from the following. G ¼ Z 4p X¼0 where with reference to Figure 2h, Z DX m ¼ dx ¼ DX m Z / m l þ D/ 2 / m l D/ 2 Iðh; /ÞdX 2 XN / Z h m k þdh 2 h m k Dh 2 X N h l¼1 k¼1 I m DX m sinhdhd/ ¼ 2sinh m k ð22þ Dh sin D/ ð23þ 2 where N h and N / are the discrete points considered over the complete span of the polar angle (0 h p) and the azimuthal angle (0 / p), respectively. Therefore, total solid angle 2p is divided into N h N / directions. While the spatial azimuthal angle is discretized as N¼N / with D ¼ D/. For a diffusely reflecting and emitting wall=boundary having emissivity e b and temperature T b, the boundary intensity is calculated from I m b ¼ e brt 4 b p þ 1 e b p X Ib m0 m0;d m0 b <0 D m0 b for D m b > 0 ð24þ where the first and the second terms on the right-hand side of Eq. (24) represent boundary emission and reflection, respectively. In Eq. (24), D m b is given by D m b ZDX ¼ X ~ ~nb dx m where, as shown in Figure 2b, at the bounding cylindrical wall ~n b is the unit normal vector pointing towards medium. In the present case of a 2-D cylindrical enclosure, although we have to deal with 3-D control volumes, the radiative boundary condition given by Eq. (24) will apply only to the radial and axial faces, i.e., inner, outer, top, and bottom sides of the cylinder. For these four faces, with reference to Figure 2d, the weights D m n and Dm t given by Eq. (25) are calculated from and are given by ð25þ Z D m n ZDX ¼ X ~ ~nw dx ¼ mn Z / m l þ D/ 2 sinhcos/dx ¼ DX m / m l D/ 2 ¼ cos/ m l sin D/ Dh cos2h m k 2 Z h m k þdh 2 h m k Dh 2 sin ð Dh cos/sin 2 hdhd/ Þ ð26þ

15 268 S. C. MISHRA ET AL. D m t Z ¼ DX mn X ~ ~nw Z Z / m l þ D/ Z 2 h m dx ¼ coshdx ¼ DX m / m l D/ 2 ¼ sinh m k coshm l sin Dh k þdh 2 h m k Dh 2 cos h sinhdhd/ The radiative heat flux in the radial direction q r R and axial direction qz R are given by and computed from the following. ð27þ q r R ZDX ¼ I m X ~ ~nn dx ¼ 2 XN / m X N h l¼1 k¼1 I m D m n ð28þ q z R ZDX ¼ I m X ~ ~nt dx ¼ 2 XN / m X N h l¼1 k¼1 I m D m t where D m n and Dm t are given by Eqs. (26) and (27), respectively. For generalization, problems have been solved in dimensionless form, with dimensionless time n, radial distance r, axial distance z, temperature H, particle distribution function fi, conductive heat flux W C, radiative heat flux W R, conduction-radiation parameter N, incident radiation G, and total heat flux W T defined in the following way. n ¼ at L 2 r ¼ r L z ¼ z L W R ¼ q R N ¼ kb rt 4 ref 4rT 3 ref H ¼ T T ref G ¼ G rt 4 ref p f i ¼ f i T ref The governing energy Eq. (1) takes the following form. W C ¼ dh dr W T ¼ dh dr þ bl 4N W R qh qn ¼ q2 H qr 2 þ 1 qh r qr þ q2 H qz 2 b2 L 2 ð1 xþ 4H 4 G 4N p fflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl} r W R where the divergence of radiative heat flux r~q R given in Eq. (2) takes the following form. r W R ¼ b2 L 2 ð1 xþ 4H 4 G ð32þ 4N p ð29þ ð30þ ð31þ where x ¼ r s =b is the scattering albedo. In calculating the radiative information r W R, intensity I and the source term S given in Eq. (21) have been nondimensionalized as I ¼ I rt 4 ref p S ¼ S rt 4 ref p ¼ bð1 xþh 4 þ bx 4p G ð33þ

16 CONDUCTION AND RADIATION IN A 2-D CYLINDRICAL MEDIUM 269 In nondimensional form in the LBM approach, the governing equation (Eq. (5)) becomes f i ~r þ~e i Dn; n þ Dn ¼ f i ð~r ; nþ Dn h i s fi ð~r ; nþ f ð0þ i ð~r ; nþ þ ~r ~e fi ~r þ~e i Dn; n f i ~r ð ; nþ i! b 2 L 2 ð1 xþ ðw i DnÞ 4H 4 G 4N p fflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl} r W R where the relaxation time given in Eq. (6) in nondimensional form is written as s ¼ 3 2 þ Dn 2 ~e i It is to be noted that in Eqs. (31) and (35) for the divergence of radiative heat flux, instead of using the left-hand side (LHS) of Eq. (2) (its nondimensional form is given in Eq. (32)), the rationale behind using its right-hand side is to avoid the discretization error that will get associated with the consideration of the LHS of Eq. (2), as we calculate the divergence of radiative heat flux rw R, not its variation with r and z, i.e., r W R ðr ; z Þ¼ð1=r Þqðr W R Þ=qr þ qw R =qz : Thus, going by the wide experience of the lead co-author (SCM) in solving conduction-radiation problems, it is always prudent to use the right side of Eq. (32). Below, we provide the solution procedure for problem 3. With initial and boundary conditions known, calculate the divergence of radiative heat flux r W R. In doing so, in the MDOM and also in the LBM, first discretize the 2-D solution space, i.e., r z plane as shown in Figure 1b. Itistobe noted that in the LBM, for the boundary lattices, lattice centers lie along the boundaries. Thus as shown, they stretch half distance beyond the respective boundaries. Therefore, if N r N z is the number of control volumes in the MDOM, the LBM will have (N r þ 1) (N z þ 1) lattices, and they overlap as shown in Figure 1b. In the LBM, r W R are required at lattice centers, whereas in the MDOM they are calculated at the centers of the MDOM control volumes. This requirement of the LBM is met for taking the average of the r W R values calculated at four centers of the MDOM control volumes. Having discretized the solution space, the angular space is divided into N h N / elemental solid angles with polar and azimuthal angular spans as 0 h pd/¼ 2p=2N / ; respectively. Although, in the MDOM there is no restriction that the polar space 0 h p should have the same size Dh ¼ p=n h divisions, and in the same way the azimuthal space 0 / 2p be divided equally in the present work, Dh is the same for all divisions of the polar space. In the same way, D/ ¼ 2p=2N / is also the same for all divisions of the azimuthal space. It is to be noted that because of the axisymmetric nature of the geometry, with a mapping between angular azimuthal angles and spatial azimuthal angles shown previously only for the sake of calculation of the radiative information, instead of 2-D control ð35þ ð36þ

17 270 S. C. MISHRA ET AL. volumes, we construct 3-D control volumes (Figure 2c). With N h N / fixed, angular weights given by Eqs. (26) (27) are calculated once and for all, and they are kept outside the iteration=time marching loop. Following the space marching procedures given in Eqs. (19a) (19d), nodal intensities are calculated from Eqs. (20a) (20d). While marching from any boundary, the boundary intensities are calculated from Eq. (24). Now with the divergence of radiative heat flux r W R calculated, in the LBM, in the first iteration we calculate the equilibrium particle distribution function f ð0þ i from Eq. (9). Next, the new particle distribution functions fi ð~r þ~e i Dn; n þ DnÞ are calculated from Eq. (35) and then they are propagated to its neighboring lattices. New temperature H is computed from Eq. (10). If the convergence is not achieved, to satisfy the boundary conditions the particle distribution functions are locally modified and the new particle distribution functions are calculated. With the new temperature field, an updated value of the r W R is computed. The procedure repeats until the convergence is achieved. Results of the present work obtained using the LBM-MDOM for problem 3 and MDOM for problem 1 have been compared with those available in the literature [44]. However, to further ascertain the robustness of the results, problem 3 was solved using the FDM-FVM in which the FVM provided the radiative information, and at any time step, with r W R known, the energy equation (27) was solved using the alternate direction implicit (ADI) finite difference scheme. Problems 1 and 2 were solved using the FVM. While solving the three problems, the FVM approach proposed by Kim [2] was followed, and in the combined mode transient conduction-radiation problem, the number of iterations and CPU times in the LBM- MDOM and FDM-FVM approaches were noted. In both approaches, the steady-state (SS) conditions were assumed to have been reached when the temperature difference between two consecutive time levels at each node did not exceed For a pure radiative transfer process, i.e., problems 1 and 2, unlike problem 3, the RTE is solved in an iterative mode and its process is terminated when either the difference of the source term S or the incident radiation G between two consecutive iterations at all nodes are within the prescribed limit RESULTS AND DISCUSSION 3.1. Grid and Ray Independency Test In the following pages, depending upon the problems, for different sets of parameters we provide variations of emissive power=temperature and heat flux distributions along the radial and axial directions. First, we provide results on grid and ray independency tests. For this, we choose the case of problem 1 in which the absorbing-emitting (x ¼ 0.0) isothermal medium is contained inside a concentric black cold cylindrical enclosure having radius ratio r 1 =r 2 ¼ 0:5 and height Z ¼ 1:0. For N h N / ¼ rays, the effect of the number of control volumes on variation of nondimensional heat flux W r R along the outer cylinder in the axial direction z is given in Figure 3a. No noticeable change in W r R is observed beyond N r N z ¼ control volumes. In Figure 3b, with N r N z ¼ control volumes fixed, the effect of number of rays is shown and N h N / ¼ rays

18 CONDUCTION AND RADIATION IN A 2-D CYLINDRICAL MEDIUM 271 Figure 3. (a) Grid test for r 1 =r 2 ¼ 0:5, Z ¼ 1:0 and N h N / ¼ 16 16, and (b) ray independency test for r 1 =r 2 ¼ 0:5, Z ¼ 1:0 and N r N z ¼ 20 40; medium is in nonradiative equilibrium. are found sufficient. Above, grid and ray independency tests are shown for the MDOM, but the same trend was found for the FVM too Isothermal Medium Inside 2-D Concentric Cylinders Problem 1 For the case of problem 1, variations of nondimensional radial wall heat flux W r R ¼ qr R =rt g 4 where T g is the medium temperature on the outer cylinder along the axial z direction, are shown in Figures 4a d. With the height of the cylinder Z ¼ Z=L ref ¼ 1:0 (Z ¼ L ref ¼ 2.0), radius of the outer cylinder r 2 ¼ r 2=L ref ¼ 0:5, and radius ratio r 1 =r 2 ¼ 0:5, variations of Wr R with z are shown for the effects of the optical thickness s ¼ r 2 b in Figure 4a, the scattering albedo x in Figure 4b and outer cylinder emissivity e 2 in Figure 4c. In Figure 4d, these variations are shown for different radius ratios. In Figure 4a, MDOM and FVM results are compared with those available in Kim and Beak [44]. Results are in good agreement. With dimensions of the enclosure fixed, increase in optical thickness s ¼ r 2 b means an increase in the value of the extinction coefficient b. Since the medium is isothermal and boundaries are cold, with an increase in opacity s ¼ r 2 b, in other sense the extinction coefficient, the radiation strength of the medium becomes stronger and this results in higher values of the wall heat flux. This behavior is seen in Figure 4a. Scattering means a loss of energy. The scattering albedo x ¼ r s b (0.0 x 1.0) is indicative of the extent of scattering. x ¼ 0.0 means no scattering, whereas for x ¼ 1.0 the medium is fully scattering. With other parameters fixed, an increase in scattering albedo will yield a decrease in the heat flux. This trend is seen in Figure 4b. Compared to x ¼ 0.0, for x ¼ 0.9 a considerable decrease in heat flux is observed. For a diffuse gray boundary, reflectivityq ¼ 1 e. With a decrease in emissivity, a wall reflects more to the medium. A black boundary (e ¼ 1, q ¼ 0) does not reflect

19 272 S. C. MISHRA ET AL. Figure 4. Comparison of dimensionless radial heat flux W r R with distance z for different values of (a) optical thicknesses s, (b) scattering albedo x, (c) outer wall emissivity e 2, and (d) radius ratio r 1 r 2. any radiation. Thus, if a boundary is cold ðt b ¼ 0:0Þ, which is the case in the present benchmark problem since the boundary emission ertb 4 =p is zero, with a decrease in emissivity meaning an increase in reflectivity, the wall heat flux has to decrease. This behavior is seen in Figure 4c. Here, the effect of the emissivity e 2 of the outer wall on heat flux variations are shown for b ¼ 5.0 and x ¼ 0.0. With the height of the cylinder Z ¼ 1:0 fixed, effect of the gas volume, meaning the effect of the radius ratio r 1 =r 2 on variations of the outer wall heat flux, is shown in Figure 4d. Heat flux is found to decrease with an increase in r 1 =r 2 : This is because of the fact that with an increase in r 1 =r 2, volume of the radiation source and the optical thickness in the radial direction b(r 2 r 1 ) decrease, and hence, the influence of the isothermal medium will decrease. With all boundaries black, results in Figure 4d are given for b ¼ 5.0 and x ¼ 0.0.

20 CONDUCTION AND RADIATION IN A 2-D CYLINDRICAL MEDIUM Nonisothermal Medium Inside 2-D Concentric Cylinders Problem 2 For the case considered in problem 2, variations of nondimensional radial heat flux W r R ¼ qr R =rt 1 4 along the outer wall in the axial direction z are given in Figure 5a. These results are given for five values of the extinction coefficients, viz., b ¼ 0.1, 1.0, 3.0, 5.0, and Heat flux is found to decrease with an increase in b. With radius ratio fixed at r 1 =r 2 ¼ 0:5 and all boundaries black, in the absence of a medium or in the case of a nonparticipating medium, the inner cylinder being the radiation source (which is hot), the heat flux at the outer cylinder will be the maximum, and since the center of the outer cylinder will get maximum radiation, the heat flux will be maximum at this point. With the presence of the medium, as the extinction nature of the medium, i.e., b increases, radiation with less magnitude will reach the outer cylinder. Hence, the observed trend is that heat flux decreases with an increase in b. For the situation considered in Figure 5a, the nondimensional emissive power U ¼ G =4p distributions in the radial direction r at axial location z ¼ 0:5 are given. Problem 2 is representative of a radiative equilibrium problem, r~q R ¼r W R ¼ 0:0; and this case is characterized by temperature slips at the boundaries. The medium becomes, optically thicker i.e., b increases, because the situation approaches that of the diffusive (conductive) nature, and the slips on the boundaries are less. Thus, as is seen from Figure 5b, on both the cylinder walls for b ¼ 10.0, the slips are the minimum and for b ¼ 0.1 they are the maximum. It is to be noted that since in this case r W R ¼ 0:0, from Eq. (32) we calculate emissive power as H 4 ¼ G =4p: It is seen from Figures 4 and 5 that in all cases, the MDOM and the FVM results compare exceedingly well with each other. Figure 5. Comparison of variations of (a) wall heat flux W r R and (b) emissive power U for different values of extinction coefficient b.

21 274 S. C. MISHRA ET AL Combined Mode Transient Conduction-Radiation Heat Transfer Inside 2-D Concentric Cylinders Problem 3 Having validated MDOM for two benchmark cases, in the following we consider the case of the combined mode conduction and radiation heat transfer described in problem 3, and show the robustness of the LBM formulation in solving the energy equation and that of the MDOM in providing the radiative information. For the purpose of comparison, in all cases the problem is also solved using the FDM-FVM approach, in which with radiative information calculated from the FVM the energy equation is solved using the ADI scheme. In Figures 6a and 6b, we benchmark our LBM-MDOM and FDM-FVM results against those available in the literature, and in Figure 6c, by making the radius ratio r 1 =r 2! 1:0; we simulate the condition of a 1-D long cylinder and check Figure 6. Validation of temperature H with distance r at the midplane ðz ¼ 0:5Þ for different values of (a) inner radius r 1,(b) conduction-radiation parameter, and (c) with 1-D cylinder code.

22 CONDUCTION AND RADIATION IN A 2-D CYLINDRICAL MEDIUM 275 if the results of our 2-D codes for the LBM-MDOM and the FDM-FVM match with that of the 1-D FDM-MDOM code for a long cylinder. For Z ¼ 1:0, extinction coefficient b ¼ 1.0, scattering albedo x ¼ 0.0, and conduction-radiation parameter N ¼ 0.1 for a black concentric cylindrical enclosure having an outer cylinder radius fixed at r 2 ¼ 1:0, for four values of the inner cylinder radius, viz., r 1 ¼ 0:3; 0:5; 0:7, and 0.9, variations of non-dimensional temperature H ¼ T=T ref with distance r ¼ r r 1 =r 2 r 1 at the mid-plane z ¼ 0:5 are given in Figure 6a. For these, the inner cylinder is insulated ðqq R =qr ¼ 0:0Þ, the bottom wall is hot H 4 ¼ T 4 =T ref ð¼ T 4 Þ ¼ 1:0, and the other two walls are cold H 2 ¼ H 3 ¼ 0.0. The LBM-MDOM and FDM-FVM results are compared with that of Li and Ozisik [14] obtained using the FDM and the classical DOM with S-6 approximation. For the boundary conditions that are the same as that of the case considered in Figure 6a, for r 1 =r 2 ¼ 0:5 for two values of the conduction-radiation parameter N ¼ 0.1 and 1.0, temperature H ¼ T=T ref distributions have been compared with that of Li and Ozisik [14]. LBM-MDOM and FDM-FVM results compare well with that available in the literature [14]. With an inner wall temperature H 1 ¼ 1.0 and the rest all boundary temperatures at H 2 ¼ H 3 ¼ H 4 ¼ 0.25, for b ¼ 1:0; x ¼ 0:0; N ¼ 0:001, and Z ¼ 1:0 for r 1 =r 2 ¼ 0:9; i.e., r 2 r 1 ¼ 0:1; the centerline temperature distribution obtained from the 2-D LBM-MDOM and 2-D FDM-FVM codes are compared with our 1-D code for FDM-MDOM. With Z ¼ 1:0 and r 2 r 1 ¼ 0:1 in the centerline z ¼ 0:5 the end effects are negligible, and it is seen from Figure 6c that results the 2-D codes matching well with that of the 1-D code. Because of the relative difficulty in calculating radiative information, compared to other geometries, not much research has been done to analyze the combined mode transient conduction-radiation problems in 2-D cylindrical geometries. To fill this gap (Figures 7 14), we investigate the effects of various parameters such as the extinction coefficient b, the scattering albedo x, the conduction-radiation parameter N, and the emissivity e 1 of the inner cylinder on variations of temperature H, conductive W C, radiative W R, and total W T heat fluxes. We also do this to establish the applicability of the MDOM in providing the volumetric radiative information, and of the LBM to solve the energy equation. To check the robustness of the LBM-MDOM formulation, we solve the same problem using the FDM-FVM approach in which with radiative information calculated from the FVM, the FDM is employed to solve the energy equation. To see which of the two approaches converge fast in the temperature plots, we show the steady-state (SS) times. For some cases, the CPU times are reported in the texts. The heat flux results are given only for the LBM-MDOM. Since temperature distributions in the two approaches match well, in any figure, this has been done to avoid the crowding of the curves. For all results presented in Figures 7 14, the inner wall is at a higher temperature H 1 ¼ T 1 =T ref ¼ 1:0 and temperatures of the rest of the three walls are H 2 ¼ H 3 ¼ H 4 ¼ 0.25.For the concentric cylinder, the inner and outer radii were r 1 ¼ 0:2 and r 2 ¼ 1:0 and the cylinder length Z ¼ 1:0: The initial temperature of the system was Hðr ; z ; 0Þ ¼ 0:25: In Figures 7, 9, 11, and 13 variations of nondimensional temperature H ¼ T=T 1 along the radial direction r ¼ r=r 2 are given at the mid-plane z ¼ 0:5 of the cylinder, and in Figures 8, 10, 12, and 14 variations of the conductive heat flux W r C the radiative heat flux Wr R and the total heat flux