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Numerical Heat Transfer, Part A: Applications

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Radiation Element Method Coupled with the Lattice Boltzmann Method Applied to the Analysis of Transient Conduction and Radiation Heat Transfer Problem with Heat Generation in a Participating Medium

Atsushi Sakuraia; Subhash C. Mishrab; Shigenao Maruyamac a Department of Mechanical and Production Engineering, Niigata University, Niigata-city, Japan b Department of Mechanical Engineering, Indian Institute of Technology Guwahati, Guwahati, India c Institute of Fluid Science, Tohoku University, Sendai, Japan Online publication date: 08 March 2010

To cite this Article Sakurai, Atsushi , Mishra, Subhash C. and Maruyama, Shigenao(2010) 'Radiation Element Method

Coupled with the Lattice Boltzmann Method Applied to the Analysis of Transient Conduction and Radiation Heat Transfer Problem with Heat Generation in a Participating Medium', Numerical Heat Transfer, Part A: Applications, 57: 5, 346 — 368 To link to this Article: DOI: 10.1080/10407780903583008 URL: http://dx.doi.org/10.1080/10407780903583008

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Numerical Heat Transfer, Part A, 57: 346–368, 2010 Copyright # Taylor & Francis Group, LLC ISSN: 1040-7782 print=1521-0634 online DOI: 10.1080/10407780903583008

RADIATION ELEMENT METHOD COUPLED WITH THE LATTICE BOLTZMANN METHOD APPLIED TO THE ANALYSIS OF TRANSIENT CONDUCTION AND RADIATION HEAT TRANSFER PROBLEM WITH HEAT GENERATION IN A PARTICIPATING MEDIUM

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Atsushi Sakurai1, Subhash C. Mishra2, and Shigenao Maruyama3

Department of Mechanical and Production Engineering, Niigata University, Niigata-city, Japan 2 Department of Mechanical Engineering, Indian Institute of Technology Guwahati, Guwahati, India 3 Institute of Fluid Science, Tohoku University, Sendai, Japan

This article deals with the implementation of the radiation element method (REM) with the lattice Boltzmann method (LBM) to solve a combined mode transient conduction-radiation problem. Radiative information computed using the REM is provided to the LBM solver. The planar conducting-radiating participating medium is contained between diffuse gray boundaries, and the system may contain a volumetric heat generation source. Temperature and heat ?ux distributions in the medium are studied for different values of parameters such as the extinction coef?cient, the scattering albedo, the conduction-radiation parameter, the emissivity of the boundaries, and the heat generation rate. To check the accuracy of the results, the problem is also solved using the ?nite-volume method (FVM) in conjunction with the LBM. In this case, the data for radiation ?eld are calculated using the FVM. The REM has been found to be compatible with the LBM, and in all the cases, results of the LBM-REM and the LBM-FVM have been found to provide an excellent comparison.

1

1. INTRODUCTION Consideration of thermal radiation combined with conduction and=or convection ?nds applications in the design of many thermal systems such as insulations, gas turbine engines, boilers, furnaces, and ?re protection devices [1–7]. It also ?nds applications in materials processing [8–10], weather forecasting [11], and in biomedical science [12]. Radiative heat transfer in a participating medium is a volumetric phenomenon. Unlike conduction and convection, it depends on the contributions received from and given to the entire spherical space surrounding the area under consideration.

Received 11 March 2009; accepted 12 December 2009. Under the Invitation Fellowship of the Japan Society for Promotion of Science (JSPS), the second co-author (SCM) contributed to the present work during his stay at the Institute of Fluid Science, Tohoku University, Sendai, Japan. SCM gratefully acknowledges the support of the JSPS. Address correspondence to Dr. Subhash C. Mishra, Professor, Department of Mechanical Engineering, IIT Guwahati, Guwahati – 781039, India. E-mail: scm_iitg@yahoo.com

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NOMENCLATURE

AR i cp e Eb Fm

eq fm A Fi;j Fi;j E Fi;j G h I K k L M

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q QR,j t T V w X x

effective radiation area speci?c heat at constant pressure ? ? propagation speed Dx Dt black-body emissive power particle distribution function in the m direction equilibrium particle distribution function in the m direction absorption view factor diffuse scattering view factor extinction view factor incident radiation fraction of energy intensity number of radiation elements thermal conductivity number of discrete directions number of propagation directions in a lattice heat ?ux radiant energy for element j dimensional time dimensional temperature volume weight in the LBM length of the geometry space variable

a b e d m q r rs s x h n X

thermal diffusivity extinction coef?cient emissivity polar angle direction cosine density Stefan-Boltzmann constant, 5.67 ? 10?8 W=m2 ? K4 scattering coef?cient relaxation time in the LBM scattering albedo dimensionless temperature dimensionless time weight factor in the REM

Superscript ? dimensionless variable A absorption E extinction Subscripts b boundary C conductive E,W east, west l lth quadrature m mth ray R radiative T total

Its correct analysis is, therefore, important for the accurate design of any thermal system or prediction of thermal ?eld in any medium which interacts with radiation. Except for very simple cases, that too for a planar geometry, analytic solutions to radiative transfer problems are dif?cult. Further, even for the planar geometry, in combined mode radiation, conduction, and=or convection problems radiative information required for the energy equation computed using analytic solution is not preferred [13, 14]. Unlike conduction and convection, dependence of radiation on three extra dimensions, viz., polar angle, azimuthal angle and wavelength, causes dif?culties in obtaining analytic solutions. Thus, for the calculation of radiative information, whether it is a pure radiation problem or a combined mode radiation, conduction, and=or convection problem numerical methods are always preferred. Since the beginning of engineering applications of radiative heat transfer in the 1950s, over a dozen numerical methods have been proposed. Each method has strong and weak points. With the development of more versatile methods, some of the methods such as the zonal method and the diffusion approximation [13, 14] do not ?nd many applications. Among the available methods, the Monte Carlo method (MCM) [15, 16], the discrete ordinates method [17, 18], the discrete transfer method [19, 20], and the ?nite-volume method [21–27] ?nd extensive applications. For pure radiative transport problems, for the lack of experimental results, the MCM results are normally considered a benchmark [15]. However, owing to its

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appetite for computational time and statistical error, MCM has not found applications in the combined mode heat transfer problems. The DTM, the DOM, and the FVM have been applied to a wide range of problems. Since the FVM of radiative heat transfer follows the same approach as that of the FVM of the computational ?uid dynamics, the FVM of radiative heat transfer is very much amenable to the combined mode radiation, conduction, and=or convection problems. The spatial discretization of the DOM [17, 18] is the same as that of the FVM [21–27]; thus, the DOM has also found a wide usage. Since in the FVM, unlike the DOM, the radiative transfer equation is integrated over the elemental solid angle, in cases of highly scattering medium, the FVM becomes less prone to ray effect. The DTM provides accurate results, but in complex geometries, this is only at the expense of computational time. If the time dependent term is not important, thermal radiation is a function of three space and two angular dimensions. In a multidimensional problem, the number of variables to store intensity is thus equal to the number of total dimensions involved in the problem. For example, in a 3-D geometry, at any point, storage of intensity will require ?ve variables—three for space and two for angles. For numerical radiative transfer methods like the DTM, the DOM, and the FVM, for a multidimensional problem this requirement becomes very critical and demands a high memory. To overcome the problem, Maruyama and Aihara [28, 29] proposed the radiation element method by ray emission model (REM2), which is a generalized numerical method in participating media in multidimensional geometry. The REM2 can be easily applied to complex systems having specular surfaces, diffuse surfaces, nongray participating gases, and anisotropic scattering media. Since the development of the REM2 [28– 35], it has been successfully applied to a wide range of simple to complex geometries [28–35]. In the following pages, REM2 is referred to as REM. Recently, the lattice Boltzmann method [36, 37] has emerged as an ef?cient computational tool to analyze problems in science and engineering. Proponents of the LBM consider this method to be a potential versatile CFD platform. In applications to ?uid mechanics problems, the usage of the LBM has seen a great surge [36, 37]. Interest in using the LBM for heat transfer problems has also gained momentum. Very recently, usage of the LBM has been extended to formulate and solve heat transfer problems involving thermal radiation [38–46], and its application to some other cases can be found [47, 48]. Compatibility of different numerical radiative transfer methods such as the collapsed dimension method [39], the DTM [38], the DOM [40], and the FVM [42, 44–46] with the LBM has been tested for a wide range of problems. However, as far as the REM [28–35] is concerned, its compatibility with the LBM has not been explored yet in solving any combined mode radiation and conduction heat transfer problems. The present work, therefore, is aimed at extending the application of the REM to solve a combined mode heat transfer problem in which the energy equation is formulated using the LBM. To test the compatibility of the REM with the LBM for a combined mode problem, we consider transient conduction and radiation heat transfer in a planar geometry. The energy equation is formulated in the LBM and the radiative information required for the energy equation is computed using the REM. To validate the LBM–REM results, the same problem is also solved using the LBM–FVM in which radiative information is obtained from the FVM and the energy equation is solved using the LBM. Dimensionless temperature and heat ?ux distributions are

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本文的绪论可用

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studied and compared for a wide range of parameters such as the extinction coef?cient, the scattering albedo, the conduction-radiation parameter, the hot boundary emissivity, and the heat generation rate. As mentioned above, the REM as such has been applied to a wide range of problems [28–35], and for the conduction-radiation problems, the application of the LBM has been extensively studied by Mishra and colleagues [38–46]. Thus, to avoid repetition with the literature, in the following pages, we provide only brief formulations of the LBM and the REM. The FVM for radiation is well established and its application to various conduction-radiation problems and its application with the LBM can be found in the work of Mishra and colleagues [42, 44–46]. Therefore, in this work, we are not providing any formulation of the FVM. The benchmarked FVM code of the second co-author (SCM) was coupled with the LBM solver to generate the results for the LBM–FVM.

2. FORMULATION The planar participating medium (Figure 1) is initially at temperature Tref. At time t ? 0.0, its west and the east boundaries are suddenly raised to temperatures TW (> Tref) and TE (? Tref), respectively, and then, these temperatures are maintained constant for all the time t > 0. The medium has a uniform volumetric heat generation source g. Thermophysical and optical properties of the medium are constant. For the problem under consideration, the governing energy equation is given by qcp qT qqC qqR ?? ? ?g qt qx qx ?1?

where q is the density, cp is the speci?c heat, qC is the conduction heat ?ux, and qR is the radiation heat ?ux. If heat transfer by conduction is assumed to follow the Fourier’s law, Eq. (1) can be written as qcp qT q2 T qqR ?k 2 ? ?g qt qx qx ?2?

where k is the thermal conductivity. Radiative contribution in the governing energy equation (Eq. (2)) appears in the form of the divergence of radiative heat ?ux qqR . It is given by [13, 14] qx qqR rT 4 ? b?1 ? x??4pIb ? G? ? b?1 ? x? 4p ?G qx p

4

?3?

where b is the extinction coef?cient, x is the scattering albedo, Ib ? rT is the blackp body intensity, and G is the incident radiation. For any elemental control volume, Eq. (3) represents the difference between volumetric emission and absorption. In the present work, for a combined mode transient conduction-radiation problem, we test the ability of the REM [28–35] in computing the divergence of

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radiative heat ?ux qqR information and its compatibility with the LBM solver. Below qx we provide brief formulations of the LBM and the REM to analyze a conductionradiation problem. In the LBM formulation, the equivalent of the energy equation (2) is given by [38, 41, 42] i Dt h ?0? fm ?~; t? ? fm ?~; t? fm ?~ ?~m Dt; t ? Dt? ? fm ?~; t? ? x e x x x s Dt qqR ? ?wm Dt?g m ? 1; 2; . . . M ? wm qcp qx

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?4?

where fm is the particle distribution function, em ? Dx is the velocity, s is the relaxDt ?0? ation time, fm is the equilibrium distribution function, wm is the weight corresponding to the direction m, and M is the number of particle distribution functions. For the 1-D planar medium problem under consideration, with D1Q2 lattice, M ? 2. For the D1Q2 lattice, the relaxation time s is given by [37] a Dt s ? ? ?2 ? Dx 2

Dt

?5?

where a is the thermal diffusivity. Solution of Eq. (4) will yield fm, and once the particle distribution functions fm are known, in a conduction-radiation problem temperature is calculated from the following relation [38, 39]. T?~; t? ? x

M X m?1

fm ?~; t? x

?6?

Solution of Eq. (4) requires knowledge about the evolution of the equilibrium ?0? particle distribution function fm . For the problem under consideration, this is given by

?0? fm ?~; t? ? wm T?~; t? x x

?7?

P For any type of lattice, M wm ? 1. Thus, from Eqs. (6) and (7), we have the followm?1 ing relation which is used to compute unknown particle distribution function at the boundary.

M X m?1 ?0? fm ?~; t? ? x M X m?1

wm T?~; t? ? T?~; t? ? x x

M X m?1

fm ?~; t? x

?8?

Below we provide a brief formulation of the REM. Its details for solving problems dealing with only radiative heat transfer can be found in references [28–35]. Further, below we present only those details of the REM which are pertinent for the problem considered in the present study. In the REM formulation, the computational domain is divided into K radiation elements consisting of surface and volume elements. Volume elements result

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from the discretization of the medium, and discretization of the boundary yields surface elements. For any given radiation element i, the divergence of radiative heat ?ux qqR given by Eq. (3) is computed from the following equation [28, 29]. qx ! K X qqR;i 1 R A ?9? ? Ai ei Eb;i ? Fj;i QR;j Vi qx j?1 where for the element i, Vi is volume, AR is the effective radiation area, ei is emissivity, i A Eb,i ? rT4 is blackbody emissive power, Fj;i is the absorption view factor, and QR,j is the radiant energy. It is to be noted that for the surface elements, 0.0 ei 1.0, but since emission from a gas volume is always black, for volume elements, ei ? 1.0. A The absorption view factor Fj;i appearing in Eq. (9) is the fraction of the radiative energy leaving the radiation element i which is absorbed by the radiation E element j. It is related with the extinction view factor Fi;j as [28, 29]

A E Fi;j ? ej Fi;j E where the extinction view factor Fi;j is computed from [28,29] E Fi;j ? L ? ? ??? ? ?? ? 1 X? hi;j ?ml ?ml 1 ? exp ?bDxi ml 1 ? exp ?bDxj ml Xl R pAi l?1

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?10?

?11?

where for a discrete direction having index l, m ? cosd and X are the direction cosine and the weight, respectively, and L is the total number of discrete directions over the polar space 0 d p. In the present study, the discrete directions have been chosen according to the S8 quadrature set proposed by Fiveland [17], and accordingly, values of ordinates m and weights X have been taken from reference [17]. In Eq. (11), hi,j(ml) is the fraction of the energy emitted from element i in the direction having direction cosine ml which reaches element j. Considering contributions from all surface and volume elements in Eq. (11), for any radiation element i, the radiative energy QR,i is given by QR;i ? AR ei Eb;i ? i

K X j?1

4

Fj;i QR;j ? AR ei Eb;i ? i

K X j?1

? ? Fj;i ?1 ? xj ?Ib;j ? xj Ij AR j

?11?

where x is the scattering albedo, Ib ? rT is the blackbody emission and for a given p direction, and I is the average intensity in a radiation element. The effective radiation area AR is given by i AR ? i

L X l?1 E where Fj;i appearing in Eq. (11) is related with Fj;i as follows. E Fj;i ? xi Fj;i

?1 ? exp??bDxi ml ??ml Xl

?12?

?13?

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Implementation of the initial and boundary conditions in the LBM was applied according to the procedure described in references [38, 39]. The solution methodology is described below. 1. From the known initial temperature, in the ?rst iteration from Eq. (7), calculate ?0? the equilibrium particle distribution function fm ?~; 0?. For the ?rst iteration, x ?0? also set fm ?~; 0? ? fm ?~; 0?. Calculate the relaxation time s from Eq. (5). x x 2. With initial temperature and boundary conditions known, calculate qqR using qx Eq. (9) in the REM. 3. Now calculate the particle distribution functions fm ?x ?~i Dt; t ? Dt? using e Eq. (4). 4. Propagate the particle distributions to the neighboring lattice centers. 5. Calculate the new temperature ?eld T?~; t? using Eq. (6). x 6. Check for convergence and terminate the process, if appropriate. 7. Modify the particle distribution functions locally to satisfy the boundary conditions. ?0? 8. Compute the equilibrium particle distribution functions fm ?~; t? from the new x temperature ?eld using Eq. (7) for every lattice. 9. Go to step 2. In the iteration loop of the LBM, at any given time level, in the REM the solution procedure to calculate qqR is as follows. qx 1. Divide the solution domain into K number radiation elements which is the same as the number of lattices in the LBM. Also, choose the number of divisions L of the angular space and accordingly select ordinates ml and weights Xl from Fiveland [17]. 2. Calculate the effective radiation area AR using Eq. (12). i E 3. Calculate the extinction view factor Fi;j using Eq. (11). 4. Using Eq. (12), for every element, calculate radiant energy QR,i . 5. Use Eq. (9) to calculate qqR . qx In the following pages, we provide results of the present study.

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3. RESULTS AND DISCUSSION We present results in the dimensionless form. In calculating the results, in dimensionless form we have de?ned distance x? , time n, temperature h, conductive heat ?ux WC, radiative heat ?ux WR, conduction-radiation parameter N, and heat generation g? in the following way. x? ? x X n? at X2 N? h? T Tref WC ? g? ? qC 4 rTref

WR ?

qR 4 rTref

k 3 4X rTref

X 2g kTref

?14?

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The dimensional thickness X? of the planar medium is unity. Initially, the entire system is at a ?nite temperature Tref. Thus, the nondimensional initial temperature h(x? , 0) ? 1.0. For time n > 0.0, the west and the east boundaries are at temperatures hW and hE, respectively. We assume, TE ? Tref and TW ? 2TE. Thus, dimensionless temperatures of the west and the east boundaries are hW ? 1.0 and hE ? 0.5, respectively. Lattices in the LBM and the radiation elements in the REM are arranged as shown in Figure 1. Lattices in the LBM and volume elements in the REM are of equal sizes and they overlap, as shown in Figure 1. In the REM, the divergence of radiative heat ?ux qqR is calculated at the center of the volume element. In the qx LBM, qqR is required at the lattice centre. As seen from Figure 1, these two locations qx are offset by a distance of Dx. Thus, with qqR known from the REM, for its use in the 2 qx LBM, values of qqR at the lattice centers were calculated by an averaging procedure. qx For grid independent solution, 100 equal size lattices in the LBM and volume elements in the REM were used. In calculation of qqR , discretization of the angular qx space was considered as per S8 approximation of the DOM as proposed by Fiveland [17]. For time marching, the time step Dn ? 0.0001 was considered. For every time step, a separate iteration loop in the REM module for calculating qqR was found qx unnecessary. Thus, the LBM and the REM were subjected to the same iteration loop. With an updated temperature ?eld obtained from the LBM solution, for the next iteration qqR was calculated, which for the following iteration became the qx known radiative information. The steady-state condition was assumed to have been achieved when at all locations the difference of temperature h between two consecutive iterations did not change beyond 1.0 ? 10?6. The approach of the REM for calculation of the radiative information is completely different from the FVM, and the FVM is considered an accurate method whose results can be assumed benchmark. Thus, to validate the LBM–REM results, the same problem was also solved using LBM in which qqR was calculated using the FVM. qx Also in the FVM, the same numbers of control volumes and discrete directions as used in LBM–REM combination were used. In the following pages, for different parameters such as the extinction coef?cient b, the scattering albedo x, the

Figure 1. Schematic of the 1-D planar medium with the arrangements of the lattices in the LBM and the radiation elements in the REM.

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conduction-radiation parameter N, the boundary emissivity e, and the heat generation rate g? at different time levels n including the steady-state (SS) condition, we compare the LBM–REM and LBM–FVM results of dimensionless temperature h and heat ?ux W distributions in the medium. Comparison for the Effect of the Extinction Coef?cient With x ? 0.0, N ? 0.1, eW ? eE ? 1.0, and g? ? 0.0 at 5 different time n levels temperature h distributions in the medium have been compared for four values of the extinction coef?cient, viz., b ? 0.1, 1.0, 3.0, and 5.0, respectively, in Figures 2a–2d. For a given value of b, h distributions have been plotted at n ? 0.002, 0.01, 0.02, 0.05, and at the SS. When extinction coef?cient b increases, medium becomes radiatively more participating, and as can be seen from Eq. (3), the divergence of radiative heat ?ux qqR increases. qx With an increase in the effect of radiation in a conduction-radiation problem, nonlinearity in the temperature pro?le will increase. This effect on the h pro?le is evident from Figures 2a–2d. Increased nonlinearity is more visible in the SS h pro?le, although for b > 1.0, the effect is there and is not that signi?cant. At all time levels, for all values of b, LBM–REM results are in excellent comparison with the LBM–FVM results. For x, N, and g? values the same as that for h distributions given in Figures 2 for b ? 1.0, 3.0, and 5.0, distributions of conduction heat ?ux WC, radiation heat ?ux WR, and total heat ?ux WT ? WC ? WR are given in Figure 3. For a given value of b, these distributions are given at time n ? 0.05 and at the SS condition. It is seen from Figures 3a–3f that with increase in b, both in the transient as well as in the SS, the total heat ?ux WT at any location decreases. In the SS, in the absence of heat generation, the total heat ?ux WT has to remain constant. This fact is observed in Figures 3b, 3d, and 3f. In all cases, the LBM–REM results are found to have a good comparison with that of the LBM–FVM results. Comparison for the Effects of the Scattering Albedo In Figures 4 and 5, temperature h and heat ?ux W distributions, respectively, are presented for the effects of the scattering albedo x. For these results, b ? 1.0, N ? 0.1, eW ? eE ? 1.0 and g? ? 0.0. In Figures 4a–4c, h distributions have been presented for x ? 0.0, 0.5, and 0.9. For a given value of x, these results are plotted at time n ? 0.002, 0.01, 0.02, 0.05, and at the SS. With increase in x, medium scatters more. Therefore, with increase in x, qqR will decrease (see Eq. (3)) and this will result in an qx increased conduction effect. This fact is observed from Figures 4a–4c. For x ? 0.9, it is seen from Figure 4c that the SS h pro?le is almost linear. For all values of x, LBM–REM results are in excellent agreement with those of the LBM–FVM results. For the effects of x, distributions of conduction heat ?ux WC, radiation heat ?ux WR, and total heat ?ux WT have been plotted in Figure 5. These distributions are given at time n ? 0.05 and at the SS. Values of others parameters are the same as that considered for the h distributions (Figure 4) for the effect of x. In the SS, in the absence of heat generation (g? ? 0.0), for a given set of parameters, the total heat ?ux WT will always remain constant in the medium. This trend is observed from Figures 5b, 5d, and 5f. With an increase in x, because of the strong scattering, variation of WR in the medium decreases. At a higher value of x (x ? 0.9), both in the

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Figure 2. Comparison of distributions of dimensionless temperature h at different instants for extinction coef?cient. (a) b ? 0.1, (b) b ? 1.0, (c) b ? 3.0, and (d) b ? 5.0; x ? 0.0, N ? 0.1, g? ? 0.0, and eW ? eE ? 1.0.

transient as well as in the SS, radiation heat ?ux WR in the medium remains almost constant. For x ? 0.9, in the SS (Figure 5f), even the conduction heat ?ux WC does not change much in the medium. This is because the total heat ?ux WT has to be a constant and because of the high level of scattering, WR will not vary much; therefore, WC will also not vary much. In this case too, for all values of x, LBM–REM results compare very well with the LBM–FVM results. Comparison for the Effects of the Conduction-Radiation Parameter In a combined mode conduction-radiation problem, conduction radiation parameter N signi?es relative importance of one mode over the other. A problem is said to be radiation-dominated when N is small. With an increase in the value of N, the effect of radiation decreases and that of conduction increases. When N ! 0, the problem approaches the case of a radiative equilibrium.

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Figure 3. Comparison of distributions of dimensionless conduction heat ?ux WC, radiation heat ?ux WR, and total heat ?ux WT at n ? 0.05 and SS for extinction coef?cient. (a) b ? 1.0, (b) b ? 1.0, (c) b ? 3.0, (d) b ? 3.0, (e) b ? 5.0, and (f) b ? 5.0; x ? 0.0, N ? 0.1, g? ? 0.0, and eW ? eE ? 1.0.

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Figure 4. Comparison of distributions of dimensionless temperature h at different instants for scattering albedo. (a) x ? 0.0, (b) x ? 0.5, and (c) x ? 0.9; b ? 1.0, N ? 0.1, g? ? 0.0, and eW ? eE ? 1.0.

In Figures 6 and 7, results of h and W distributions have been presented, respectively. These results are given for b ? 1.0, x ? 0.5, eW ? eE ? 1.0, and g? ? 0.0. Figures 6a–6d show h distributions for N ? 0.001, 0.01, 0.1, and 1.0, respectively. When N is small, because of the dominance of radiation, SS is achieved fast and temperature gradient near the boundaries will be very large. In the limiting case of N ? 0.0, both the boundaries will notice a temperature slip. This phenomenon is characteristic of the radiative equilibrium situation. In Figure 6a, results for N ? 0.001 represent a high dominance of radiation, and in this case, SS was found to reach very fast. In this case, near the boundaries, a very sharp gradient in the h pro?le which is characteristic of a radiation-dominated situation, is observed. For N ? 1.0, the effect of conduction is signi?cant. Because of this reason, as observed in Figure 6d, in this case, in the SS, almost a linear h pro?le is obtained. For all values of N, results of the two sets of methods show a good comparison.

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Figure 5. Comparison of distributions of dimensionless conduction heat ?ux WC, radiation heat ?ux WR, and total heat ?ux WT at n ? 0.05 and SS for scattering albedo. (a) x ? 0.0, (b) x ? 0.0, (c) x ? 0.5, (d) x ? 0.5, (e) x ? 0.9, (f) x ? 0.9; b ? 1.0, N ? 0.1, g? ? 0.0, and eW ? eE ? 1.0.

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For three different values of the conduction-radiation parameter, viz., N ? 0.01, 0.1, and 1.0 at n ? 0.05 and at the SS, distributions of conduction heat ?ux WC, radiation heat ?ux WR and total heat ?ux WT have been shown in Figures 7a–7f. Values of different parameters for this case are b ? 1.0, x ? 0.5, eW ? eE ? 1.0, and g? ? 0.0. Both in the transient as well as in the SS, the relative importance of radiation and conduction modes are very much evident from Figures 7a–7f. When the value of N decreases, the effect of radiation decreases, and, therefore, the contribution of radiation to the total heat ?ux also decreases. For N ? 0.01 (Figures 7a and 7b), the major contribution to the total heat ?ux WT comes from the radiation heat ?ux WR. The case is almost reversed for N ? 1.0 (Figures 7e and 7f) in which case, conduction is the dominant mode. Like in the previous cases, in this case too, the LBM–REM and the LBM–FVM results show an excellent comparison. Having shown that the LBM–REM results show excellent comparison with the LBM–FVM results for the effects of the extinction coef?cient b, the scattering

Figure 6. Comparison of distributions of dimensionless temperature h at different instants, for conduction-radiation parameter. (a) N ? 0.001, (b) b ? 0.01, (c) b ? 0.1, and (d) b ? 1.0; b ? 1.0, x ? 0.5, g? ? 0.0, and eW ? eE ? 1.0.

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Figure 7. Comparison of distributions of dimensionless conduction heat ?ux WC, radiation heat ?ux WR, and total heat ?ux WT at n ? 0.05 and SS for conduction-radiation parameter. (a) N ? 0.01, (b) N ? 0.01, (c) N ? 0.1, (d) N ? 0.1, (e) N ? 1.0, (f) N ? 1.0; b ? 1.0, x ? 0.5, g? ? 0.0, and eW ? eE ? 1.0.

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albedo x, and the conduction-radiation parameter N, next we show comparisons for the effects of the emissivity of the hot boundary and the heat generation rate. Comparison for the Effects of the Emissivity of the Hot (West) Boundary With b ? 1.0, x ? 0.5, N ? 0.1, g? ? 0.0, and eE ? 1.0, Figures 8 and 9, respectively, show the effects of the emissivity of the hot (west) boundary on temperature h and heat ?ux W distributions in the medium. Since cases of black boundaries have already been discussed, in Figures 8a–8c, we have given temperature h distributions for eW ? 0.3, 0.5, and 0.7, respectively. For the three values of the emissivity of the hot boundary, heat ?ux distributions have been plotted in Figures 9a–9f. Compared to other parameters, temperature h pro?le is less affected with change in eW. When the hot boundary becomes more re?ecting, the boundary retains less radiative energy

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Figure 8. Comparison of distributions of dimensionless temperature h at different instants for emissivity of the hot (west) boundary. (a) eW ? 0.3, (b) eW ? 0.5, and (c) eW ? 0.7; b ? 1.0, x ? 0.5, N ? 0.1, and g? ? 0.0.

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Figure 9. Comparison of distributions of dimensionless conduction heat ?ux WC, radiation heat ?ux WR, and total heat ?ux WT at n ? 0.05 and SS for emissivity of the hot (west) boundary. (a) eW ? 0.3, (b) eW ? 0.3, (c) eW ? 0.5, (d) eW ? 0.5, (e) eW ? 0.7, and (f) eW ? 0.7; b ? 1.0, x ? 0.5, N ? 0.1, and g? ? 0.0.

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Figure 10. Comparison of distributions of dimensionless temperature h at different instants for heat generation rate. (a) g? ? 0.0, (b) g? ? 1.0, and (c) g? ? 3.0; b ? 1.0, x ? 0.5, N ? 0.1, and eW ? eE ? 1.0.

and this causes a conduction-like situation. In this case, the SS h pro?le tends to be a linear one (Figure 8a). The effect of the emissivity eW of the hot boundary can be better appreciated from the heat ?ux distributions given in Figures 9a–9f. With change in the value of eW, variation of the conduction heat ?ux in the medium is not much changed. However, as the hot boundary becomes more re?ecting, contribution of radiative heat ?ux WR decreases and this causes a decrease in the value of the total heat ?ux WT. From Figures 8 and 9, it is observed that in all cases, results of the LBM–REM and LBM–FVM are in excellent agreement with each other.

Comparison for the Effects of the Heat Generation Rate Comparison of the LBM–REM and the LBM–FVM results for the effects of heat generation are shown in Figures 10 and 11. Temperature h distributions for

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Figure 11. Comparison of distributions of dimensionless conduction heat ?ux WC, radiation heat ?ux WR, and total heat ?ux WT at n ? 0.05 and SS for emissivity of the hot (west) boundary. (a) g? ? 0.0, (b) g? ? 0.0, (c) g? ? 1.0, (d) g? ? 1.0, (e) g? ? 3.0, and (f) g? ? 3.0; b ? 1.0, x ? 0.5, N ? 0.1, and eW ? eE ? 1.0.

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three values of dimensionless heat generation rate g? ? 0.0, 1.0, and 3.0 are shown in Figures 10a–10c, respectively. For these three values, heat ?ux distributions are shown in Figures 11a–11f. Results in Figures 10 and 11 have been plotted for b ? 1.0, x ? 0.5, eW ? eE ? 1.0, and N ? 0.1. Heat generation will augment nonlinearity in the temperature h pro?le. This fact is evident from Figures 10a–10c. When g? increases, nonlinearity in the SS h pro?le will increase, and at any location, in the SS, temperature h increases when g? increases. For the results in Figure 10c, since heat generation is very intense near the hot boundary dimensionless temperature is found to go beyond the unity value. For all values of g, at all time levels, results of the LBM–REM and the LBM–FVM are found to be in excellent agreement. At time n ? 0.05 and at the SS, for three values of g? distributions of conduction heat ?ux WC, radiation heat ?ux WR, and total heat ?ux WT have been shown in Figures 11a–11f. For g? 6? 0.0 in the SS, unlike the case of g? ? 0.0, the total heat ?ux WT will be a linear function of distance. The slope of the WT pro?le will increase with an increase in the value of g? . This trend is observed from Figures 11b, 11d, and 11f. It is seen from Figure 11f that in the SS near the hot boundary WC 0.0 and WR > WT. For a high value of the heat generation rate g? ? 3.0, this trend can be explained with the SS temperature h pro?le in Figure 10f in which it is observed that near the hot boundary, slope of the h pro?le is negative, and thus the region in which the slope of the h pro?le is negative, WC 0.0. In transient as well as in the SS, g? is found to have more of an effect on WC distributions than on WR distributions. In all cases, results of the LBM–REM and LBM–FVM are observed to be in excellent agreement.

4. CONCLUSION Compatibility of the REM for radiative heat transfer with the LBM for solving a combined mode transient conduction and radiation heat transfer problem was investigated. The radiative information computed using the REM was provided to the LBM formulation of the energy equation. To validate the LBM–REM results, in all cases, the problem was also solved using the LBM–FVM combination in which the FVM was used to compute the radiative information. For the 1-D transient conduction and radiation heat transfer problem considered in the present work, at various time levels including the steady-state, dimensionless temperature, conduction heat ?ux, radiation heat ?ux, and total heat ?ux distributions in the medium were compared for different values of the extinction coef?cient, the scattering albedo, the conduction-radiation parameter, the hot boundary emissivity, and the heat generation rate. For the solution of a conduction-radiation problem, the REM was found to be compatible with the LBM, and in all cases, results of the LBM–REM and the LBM–FVM were found in excellent agreement.

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