Radiation Element Method Coupled with the Lattice Boltzmann Method Applied to the Analysis of Transi

PLEASE SCROLL DOWN FOR ARTICLEThis article was downloaded by: [EBSCOHost EJS Content Distribution - Current]On: 14 December 2010Access details: Access Details: [subscription number 925797205]Publisher Taylor & FrancisInforma Ltd Registered in England and Wales Registered Number: 1072954 Registered office: Mortimer House, 37-41 Mortimer Street, London W1T 3JH, UKNumerical Heat Transfer, Part A: ApplicationsPublication details, including instructions for authors and subscription information:/smpp/title~content=t713657973Radiation 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 MediumAtsushi Sakurai a ; Subhash C. Mishra b ; Shigenao Maruyama c aDepartment 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 2010To cite this Article Sakurai, Atsushi , Mishra, Subhash C. and Maruyama, Shigenao(2010) 'Radiation Element MethodCoupled 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 — 368To link to this Article: DOI: 10.1080/10407780903583008URL: /10.1080/10407780903583008Full terms and conditions of use: /terms-and-conditions-of-access.pdf This article may be used for research, teaching and private study purposes. 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The publisher shall not be liable for any loss,actions, claims, proceedings, demand or costs or damages whatsoever or howsoever caused arising directly or indirectly in connection with or arising out of the use of this material.RADIATION ELEMENT METHOD COUPLED WITH THE LATTICE BOLTZMANN METHOD APPLIED TO THE ANALYSIS OF TRANSIENT CONDUCTION ANDRADIATION HEAT TRANSFER PROBLEM WITH HEAT GENERATION IN A PARTICIPATING MEDIUMAtsushi Sakurai 1,Subhash C.Mishra 2,and Shigenao Maruyama 31Department of Mechanical and Production Engineering,Niigata University,Niigata-city,Japan 2Department of Mechanical Engineering,Indian Institute of Technology Guwahati,Guwahati,India 3Institute of Fluid Science,Tohoku University,Sendai,JapanThis 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 flux distributions in the medium are studied for different values of parameters such as the extinction coefficient,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 finite-volume method (FVM)in conjunction with the LBM.In this case,the data for radiation field 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.INTRODUCTIONConsideration of thermal radiation combined with conduction and =or convection finds applications in the design of many thermal systems such as insulations,gas turbine engines,boilers,furnaces,and fire protection devices [1–7].It also finds 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 11March 2009;accepted 12December 2009.Under the Invitation Fellowship of the Japan Society for Promotion of Science (JSPS),the secondco-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 MechanicalEngineering,IIT Guwahati,Guwahati –781039,India.E-mail:scm_iitg@Numerical Heat Transfer,Part A ,57:346–368,2010Copyright #Taylor &Francis Group,LLC ISSN:1040-7782print =1521-0634online DOI:10.1080/10407780903583008346D o w n l o a d e d B y : [E B S C O H o s t E J S C o n t e n t D i s t r i b u t i o n - C u r r e n t ] A t : 01:03 14 D e c e m b e r 2010Its correct analysis is,therefore,important for the accurate design of any thermalsystem or prediction of thermal field 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 difficult.Further,even for the planar geometry,in combined mode radiation,conduction,and =or convection problems radiative infor-mation required for the energy equation computed using analytic solution is not pre-ferred [13,14].Unlike conduction and convection,dependence of radiation on three extra dimensions,viz.,polar angle,azimuthal angle and wavelength,causes difficult-ies 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 find 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 finite-volume method [21–27]find 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 itsNOMENCLATUREA R i effective radiation area c p specific heat at constant pressure e propagation speed D x D tÀÁE b black-body emissive powerF mparticle distribution function in the m directionf eq mequilibrium particle distribution function in the m direction F A i ;j absorption view factorF i ;j diffuse scattering view factor F E i ;j extinction view factorG incident radiation h fraction of energy I intensityK number of radiation elements k thermal conductivityL number of discrete directionsM number of propagation directions in a lattice q heat fluxQ R,j radiant energy for element j t dimensional timeT dimensional temperature V volumew weight in the LBM X length of the geometry xspace variablea thermal diffusivityb extinction coefficient e emissivity d polar angle m direction cosine q densityr Stefan-Boltzmann constant,5.67Â10À8W =m 2ÁK 4r s scattering coefficients relaxation time in the LBM x scattering albedoh dimensionless temperature n dimensionless timeX weight factor in the REMSuperscript Ãdimensionless variableA absorption E extinction Subscripts b boundary C conductive E,W east,west l l th quadrature m m th ray R radiative T totalCONDUCTION AND RADIATION WITH HEAT GENERATION 347D o w n l o a d e d B y : [E B S C O H o s t E J S C o n t e n t D i s t r i b u t i o n - C u r r e n t ] A t : 01:03 14 D e c e m b e r 2010appetite 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 radi-ative heat transfer follows the same approach as that of the FVM of the computational fluid dynamics,the FVM of radiative heat transfer is very much amenable to the com-bined mode radiation,conduction,and =or convection problems.The spatial discreti-zation 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 five 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 over-come the problem,Maruyama and Aihara [28,29]proposed the radiation element method by ray emission model (REM 2),which is a generalized numerical method in participating media in multidimensional geometry.The REM 2can be easily applied to complex systems having specular surfaces,diffuse surfaces,nongray participating gases,and anisotropic scattering media.Since the development of the REM 2[28–35],it has been successfully applied to a wide range of simple to complex geometries [28–35].In the following pages,REM 2is referred to as REM.Recently,the lattice Boltzmann method [36,37]has emerged as an efficient computational tool to analyze problems in science and engineering.Proponents of the LBM consider this method to be a potential versatile CFD platform.In applica-tions to fluid 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 momen-tum.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 extend-ing 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 infor-mation 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 flux distributions are348 A.SAKURAI ET AL.D o w n l o a d e d B y : [E B S C O H o s t E J S C o n t e n t D i s t r i b u t i o n - C u r r e n t ] A t : 01:03 14 D e c e m b e r 2010parameters such as the extinction coef-ficient,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 for-mulations 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.FORMULATIONThe planar participating medium (Figure 1)is initially at temperature T ref .At time t ¼0.0,its west and the east boundaries are suddenly raised to temperatures T W (>T ref )and T E (¼T ref ),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 byq c pq T q t ¼Àq q C q x Àq q Rq xþg ð1Þwhere q is the density,c p is the specific heat,q C is the conduction heat flux,and q R is theradiation heat flux.If heat transfer by conduction is assumed to follow the Fourier’s law,Eq.(1)can be written asq c p q T ¼k q 2T Àq q Rþg ð2Þwhere k is the thermal conductivity.Radiative contribution in the governing energy equation (Eq.(2))appears inthe form of the divergence of radiative heat flux q q Rq x .It is given by [13,14]q q R q x ¼b ð1Àx Þð4p I b ÀG Þ¼b ð1Àx Þ4p r T 4pÀGð3Þwhere b is the extinction coefficient,x is the scattering albedo,I b ¼r T4is the black-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 ofWITH HEAT GENERATION349D o w n l o a d e d B y : [E B S C O H o s t E J S C o n t e n t D i s t r i b u t i o n - C u r r e n t ] A t : 01:03 14 D e c e m b e r 2010radiative heat flux q q Rq x information and its compatibility with the LBM solver.Below we provide brief formulations of the LBM and the REM to analyze a conduction-radiation problem.In the LBM formulation,the equivalent of the energy equation (2)is given by [38,41,42]f m ~x þ~e m D t ;t þD t ðÞ¼f m ~x ;t ðÞÀD t s f m ~x ;t ðÞÀf ð0Þm ~x ;t ðÞh i ÀD t q c p w mq q Rq xþðw m D t Þg m ¼1;2;...M ð4Þwhere f m is the particle distribution function,e m ¼D xD t is the velocity,s is the relax-ation time,f ð0Þm is the equilibrium distribution function,w m is the weight correspond-ing to the direction m ,and M is the number of particle distribution functions.For the 1-D planar medium problem under consideration,with D1Q2lattice,M ¼2.For the D1Q2lattice,the relaxation time s is given by [37]s ¼a D x D tÀÁþD t 2ð5Þwhere a is the thermal diffusivity.Solution of Eq.(4)will yield f m ,and once the particle distribution functions f mare known,in a conduction-radiation problem temperature is calculated from the following relation [38,39].T ð~x ;t Þ¼X M m ¼1f m ~x ;t ðÞð6ÞSolution of Eq.(4)requires knowledge about the evolution of the equilibrium particle distribution function f ð0Þm .For the problem under consideration,this is given byf ð0Þm ð~x ;t Þ¼w m T ð~x ;t Þð7ÞFor any type of lattice,PM m ¼1w m ¼1.Thus,from Eqs.(6)and (7),we have the follow-ing relation which is used to compute unknown particle distribution function at the boundary.X M m ¼1f ð0Þm ~x ;t ðÞ¼X M m ¼1w m T ð~x ;t Þ¼T ð~x ;t Þ¼X M m ¼1f m ~x ;t ðÞð8ÞBelow we provide a brief formulation of the REM.Its details for solving problemsdealing 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 radi-ation elements consisting of surface and volume elements.Volume elements result350 A.SAKURAI ET AL.D o w n l o a d e d B y : [E B S C O H o s t E J S C o n t e n t D i s t r i b u t i o n - C u r r e n t ] A t : 01:03 14 D e c e m b e r 2010from the discretization of the medium,and discretization of the boundary yields surface elements.For any given radiation element i ,the divergence of radiative heatflux q q Rq x given by Eq.(3)is computed from the following equation [28,29].q q R ;i q x ¼1V i A R i e i E b ;i ÀX K j ¼1F A j ;i Q R ;j!ð9Þwhere for the element i ,V i is volume,A R i is the effective radiation area,e i is emissivity,E b ,i ¼r T 4is blackbody emissive power,F A j ;i is the absorption view factor,and Q R ,j is the radiant energy.It is to be noted that for the surface elements,0.0 e i 1.0,but since emission from a gas volume is always black,for volume elements,e i ¼1.0.The absorption view factor F A j ;i appearing in Eq.(9)is the fraction of the radi-ative energy leaving the radiation element i which is absorbed by the radiation element j .It is related with the extinction view factor F E i ;j as [28,29]F A i ;j ¼e j F Ei ;jð10Þwhere the extinction view factor F E i ;j is computed from [28,29]F E i ;j¼1p A R i XL l ¼1h i ;j ðm l Þm l 1Àexp ÀbD x i m lÀÁÂÃ1Àexp ÀbD x j m l ÀÁÂÃX l ÈÉð11Þwhere for a discrete direction having index l ,m ¼cos d and X are the direction cosineand 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 S 8quadrature set proposed by Fiveland [17],and accordingly,values of ordinates m and weights X have been taken from reference [17].In Eq.(11),h i ,j (m l )is the fraction of the energy emitted from element i in the direction having direction cosine m l which reaches element j .Considering contributions from all surface and volume elements in Eq.(11),for any radiation element i ,the radiative energy Q R ,i is given byQ R ;i ¼A R i e i E b ;iþX K j ¼1F j ;i Q R ;j ¼A R i e i E b ;iþX K j ¼1F j ;i ð1Àx j ÞI b ;j þx j I j ÂÃA R jð11Þwhere x is the scattering albedo,I b ¼r T4pis the blackbody emission and for a given direction,and I is the average intensity in a radiation element.The effective radiation area A R i is given byA R i¼X L l ¼1½1Àexp ðÀbD x im l Þ m l X lð12Þwhere F j ;i appearing in Eq.(11)is related with F E j ;i as follows.F j ;i ¼x i F E j ;ið13ÞCONDUCTION AND RADIATION WITH HEAT GENERATION 351D o w n l o a d e d B y : [E B S C O H o s t E J S C o n t e n t D i s t r i b u t i o n - C u r r e n t ] A t : 01:03 14 D e c e m b e r 2010Implementation of the initial and boundary conditions in the LBM was applied according to the procedure described in references [38,39].The solution method-ology is described below.1.From the known initial temperature,in the first iteration from Eq.(7),calculatethe equilibrium particle distribution function f ð0Þm ð~x ;0Þ.For the first iteration,also set f m ð~x ;0Þ¼f ð0Þm ð~x ;0Þ.Calculate the relaxation time s from Eq.(5).2.With initial temperature and boundary conditions known,calculate q q Rq x using Eq.(9)in the REM.3.Now calculate the particle distribution functions f m ðx þ~e i D t ;t þD t Þusing Eq.(4).4.Propagate the particle distributions to the neighboring lattice centers.5.Calculate the new temperature field T ð~x ;t Þusing Eq.(6).6.Check for convergence and terminate the process,if appropriate.7.Modify the particle distribution functions locally to satisfy the boundary conditions.pute the equilibrium particle distribution functions f ð0Þm ð~x ;t Þfrom the new temperature field 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 thesolution procedure to calculate q q Rq x is as follows.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 m l and weights X l from Fiveland [17].2.Calculate the effective radiation area A R i using Eq.(12).3.Calculate the extinction view factor F Ei ;j using Eq.(11).ing Eq.(12),for every element,calculate radiant energy Q R ,i .e Eq.(9)to calculate q q Rq x .In the following pages,we provide results of the present study.3.RESULTS AND DISCUSSIONWe present results in the dimensionless form.In calculating the results,in dimensionless form we have defined distance x Ã,time n ,temperature h ,conductive heat flux W C ,radiative heat flux W R ,conduction-radiation parameter N ,and heat generation g Ãin the following way.x üx Xn ¼a t X h ¼T T refW C ¼q C r T refW R ¼q Rr T ref N ¼k4X r T refg üX 2g kT refð14Þ352 A.SAKURAI ET AL.D o w n l o a d e d B y : [E B S C O H o s t E J S C o n t e n t D i s t r i b u t i o n - C u r r e n t ] A t : 01:03 14 D e c e m b e r 2010The dimensional thickness X Ãof the planar medium is unity.Initially,the entire system is at a finite temperature T ref .Thus,the nondimensional initial temperature h (x Ã,0)¼1.0.For time n >0.0,the west and the east boundaries are at temperatures h W and h E ,respectively.We assume,T E ¼T ref and T W ¼2T E .Thus,dimensionless tem-peratures of the west and the east boundaries are h W ¼1.0and h E ¼0.5,respectively.Lattices in the LBM and the radiation elements in the REM are arranged as shown in Figure ttices 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 ofradiative heat flux q q Rq x is calculated at the center of the volume element.In theLBM,q q Ris required at the lattice centre.As seen from Figure 1,these two locationsare offset by a distance of D x.Thus,with q q R known from the REM,for its use in theLBM,values of q q Rq x at the lattice centers were calculated by an averaging procedure.For grid independent solution,100equal size lattices in the LBM and volumeelements in the REM were used.In calculation of q q Rq x ,discretization of the angular space was considered as per S 8approximation of the DOM as proposed by Fiveland [17].For time marching,the time step Dn ¼0.0001was considered.For every timestep,a separate iteration loop in the REM module for calculating q q Rq x was found unnecessary.Thus,the LBM and the REM were subjected to the same iteration loop.With an updated temperature field obtained from the LBM solution,forthe next iteration q q Rq x was calculated,which for the following iteration became the known radiative information.The steady-state condition was assumed to have been achieved when at all locations the difference of temperature h between two consecu-tive iterations did not change beyond 1.0Â10À6.The approach of the REM for calculation of the radiative information is com-pletely 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 sameproblem was also solved using LBM in which q q Rq x was calculated using the FVM.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 coefficient b ,the scattering albedo x ,theFigure 1.Schematic of the 1-D planar medium with the arrangements of the lattices in the LBM and the radiation elements in the REM.CONDUCTION AND RADIATION WITH HEAT GENERATION 353D o w n l o a d e d B y : [E B S C O H o s t E J S C o n t e n t D i s t r i b u t i o n - C u r r e n t ] A t : 01:03 14 D e c e m b e r 2010conduction-radiation parameter N ,the boundary emissivity e ,and the heat gener-ation rate g Ãat 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 flux W distributions in the medium.Comparison for the Effect of the Extinction CoefficientWith x ¼0.0,N ¼0.1,e W ¼e E ¼1.0,and g ü0.0at 5different time n levels tem-perature h distributions in the medium have been compared for four values of the extinc-tion coefficient,viz.,b ¼0.1,1.0,3.0,and 5.0,respectively,in Figures 2a –2d .For a given value of b ,h distributions have been plotted at n ¼0.002,0.01,0.02,0.05,and at the SS.When extinction coefficient b increases,medium becomes radiatively more participat-ing,and as can be seen from Eq.(3),the divergence of radiative heat flux q q Rincreases.With an increase in the effect of radiation in a conduction-radiation problem,nonlinear-ity in the temperature profile will increase.This effect on the h profile is evident from Figures 2a –2d .Increased nonlinearity is more visible in the SS h profile,although for b >1.0,the effect is there and is not that significant.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 Ãvalues the same as that for h distributions given in Figures 2for b ¼1.0,3.0,and 5.0,distributions of conduction heat flux W C ,radiation heat flux W R ,and total heat flux W T ¼W C þW R are given in Figure 3.For a given value of b ,these distributions are given at time n ¼0.05and 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 flux W T at any location decreases.In the SS,in the absence of heat generation,the total heat flux W T 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 AlbedoIn Figures 4and 5,temperature h and heat flux W distributions,respectively,are presented for the effects of the scattering albedo x .For these results,b ¼1.0,N ¼0.1,e W ¼e E ¼1.0and g ü0.0.In Figures 4a–4c ,h distributions have been presented for x ¼0.0,0.5,and 0.9.For a given value of x ,these results are plotted at time n ¼0.002,0.01,0.02,0.05,and at the SS.With increase in x ,medium scatters more.Therefore,with increase in x ,q q Rq x will decrease (see Eq.(3))and this will result in an increased conduction effect.This fact is observed from Figures 4a–4c .For x ¼0.9,it is seen from Figure 4c that the SS h profile 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 flux W C ,radiation heat flux W R ,and total heat flux W T have been plotted in Figure 5.These distributions are given at time n ¼0.05and 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 ü0.0),for a given set of parameters,the total heat flux W T 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,vari-ation of W R in the medium decreases.At a higher value of x (x ¼0.9),both in the354 A.SAKURAI ET AL.D o w n l o a d e d B y : [E B S C O H o s t E J S C o n t e n t D i s t r i b u t i o n - C u r r e n t ] A t : 01:03 14 D e c e m b e r 2010。