漫反射边界下各向异性散射介质内的耦合换热

Combined heat transfer within an anisotropic scattering slabwith diffuse surfacesHong-Liang Yi, He-Ping Tan*School of Energy Science and Engineering, Harbin Institute of Technology, Harbin, 150001, P.R.China150080, P.R.ChinaAbstractUnder various interface reflecting modes, different transient thermal response will occur in media. Combined radiative-conductive heat transfer is investigated within a participating, anisotropic scattering gray planar slab. The two interfaces of the slab are considered to be diffuse and semitransparent. Using the ray tracing method, an anisotropic scattering radiative transfer model for diffuse reflection at boundaries is set up, and with the help of direct radiative transfer coefficients, corresponding radiative transfer coefficients (RTCs) are deduced. RTCs are used to calculate the radiative source term in energy equation. Transient energy equation is solved by the full implicit control-volume method under the external radiative-convective boundary conditions. The influences of two reflecting modes including both specular reflection and diffuse reflection are examined on transient temperature fields and steady heat flux. According to numerical results obtained in this paper, it is found that there exits great difference in thermal behavior between slabs with diffuse interfaces and that with speccular interfaces for slabs with big refractive index.Keywords: Combined radiative-conductive heat transfer / ray tracing method / Radiative transfer coefficients / Anisotropic scattering / Diffuse reflection___________________________________*Corresponding author. Tel.: 86-451-86412028, Fax: 86-451-86221048. Email: tanheping77@NomenclatureAbbreviation RTCRadiative Transfer Coefficient,i k T A,,()()kb i b i I T d I T d λλλλλ∞∆∫∫, fractional spectral emissive power of spectral band k λ∆at nodal temperature i TC unit heat capacity of media, 11JKg K −− ()n E xexponential integral function of the th n rank12,h hconvective heat transfer coefficient at surfaces of 1S and 2S , respectively , 21Wm K −− kphonic thermal conductivity, 11Wm K −− L thickness of slab, m NB total number of spectral bands NM total number of control volumes of slab ,k m nspectral refractive index of slab r qheat flux of radiation, 2Wm −r q % dimensionless heat flux of radiation, ()4/4r rfq T σ c qheat flux of conduction, 2Wm −c q % dimensionless heat flux of conduction, ()4/4c rfq T σ t qtotal heat flux, r c q q +t q % dimensionless total heat flux, r c q q +%% ,u v S SS −∞or S +∞()()(),,u v k u j k i j kS S S V V Vradiation transfer coefficients of surface vs surface, surface vs volume, and volume vsvolume in non-scattering media relative to the spectral band k λ∆[],,u v k u j k i j kS S S V V V ⎡⎤⎣⎦⎡⎤⎣⎦ radiation transfer coefficient of surface vs surface, surface vs volume, and volume vs volume in isotropic or anisotropic scattering media relative to the spectral band k λ∆12,S Sboundary surfaces,S S −∞+∞black surfaces representing the surroundings Tabsolute temperature, K12,g g T Tgas temperatures for convection at 0X = and 1, K rf Treference temperature, K 0Tuniform initial temperature, K tphysical time, sX dimensionless coordinate in direction across layer, X x L = ,i j x normal distance of ray transfer between both subscripts, m znormal distance of ray transfer, m t ∆, time intervalx ∆spacing interval between two nodes, mβthe common ratio of the infinite geometric series,()21,in 2,in 12s s ρρ 1,2,, k k εε emissivity of surface 1S , 2S , respectively η 1ω−,,i s r θθθ, incident, scattering, refractive angle between the ray and x -axis k κ spectral extinction coefficient of slab, 1m − ρ density of media, 3kg/m12,ρρreflectivity of surface 1S , 2S , respectivelyσStefan −Boltzmann constant, =8106696.5−×24Wm K −− Φscattering phase functionr i Φradiative heat source of the control-volume io τL κ, optical thickness of slab ω scattering albedo of slabζ control volumes in units of optical thickness, 0NM τ Subscripts, d s diffuse reflection, specular reflection iw ie ,right and left interface of control volume i , in outrefer to inner, outer surface, respectively krelative to the th k region of spectral bandst t −, -o orefer to media with semitransparent , opaque interfaces, respectively 1, 2 refer to the boundary surfaces 1S and 2S , respectively ,−∞+∞refer to the black surfaces S −∞ and ∞+S , respectively Superscripts, d s diffuse reflection, specular reflectiont f b ,, incidence radiation from negative, positive and both direction relative to the x axis, respectivelyq h , backward scattering and forward scattering relative to the incident direction, respectively rrefer to radiation1. IntroductionSemitransparent participating materials such as ceramic, glass, fiber, zirconia, silica gel and so on, have many engineering applications in which radiation transport and coupled radiative-conductive heat transfer play a dominant role to control their transient thermal behavior. These applications include heat insulation and thermal protection [1,2], ignition of semitransparent solid fuels [3,4], measurement of thermophysical properties of translucent materials [5,6], infrared heating and infrared drying, design of furnaces, prediction of the effect of dust, CO 2 and other participating gases on the global environment, and so forth.In the past semicentury, much work has been done on the radiative transfer and coupled radiative-conductive heat transfer within participating media. Relative to isotropic scattering, much less literatures on thermal radiation involving in anisotropic scattering have been found. Using discrete ordinates method, Krishnaprakas et al. [7] examined conduction-radiation heat transfer through a gray planar nonlinearly anisotropic scattering medium with two plane-parallel surfaces reflecting both diffusely and specularly. M. Lazard et al. [8] developed a semi-analytical model based on the exponential-kernel method for the transient combined conduction-radiation heat transfer for an anisotropically scattering participating slab excited by heat pulse stimulation on the front face. Prabal Talukdar et al. [9] solved combined conduction-radiation problem using the collapsed dimension method for gray planar absorbing, emitting and anisotropically scattering medium. Zekeriya Altaç [10] proposed the Synthetic Kernel approximation to solve radiative transfer problems in linearly anisotropically scattering homogeneous and inhomogeneous participating plane-parallel medium.There are generally two reflecting modes at interfaces: specular reflection and diffuse reflection. If the interface is optically smooth, it can be assumed specular; if the interface is optically rough, it can be assumed diffuse. As discussed by Siegel [11], the diffuse reflectivity of semitransparent interface can be obtained from the F resnel interface relations by integrating the reflected energy over all incident directions if assuming that each bit of roughness acts as a smooth facet. For the semitransparent specular interface, the reflectivity can be determined byFresnel’s reflective law and Snell’s refractive law [12]. In the present investigation, we adopt the methods proposed in Refs [11, 12] for the treatment of specular and diffuse interface conditions.In the last five years, He-Ping Tan, Ping-Yang Wang and Jian-F eng Luo et al. have solved transient coupled radiative-conductive problem by ray tracing method through a participating media with one-layer [13], two-layer [14], three-layer [15] and multi-layer [16], respectively. In their studies, anisotropic scattering was not considered. While for some materials, such as ceramics and zirconia, which are porous semitransparent media with highly anisotropic scattering [17], the assumption of isotropic scattering can cause big error. In this paper, based on the radiative transfer model developed for anisotropic scattering media with specular and semitransparent boundaries in Ref. [18] and using ray-tracing method, we set up an anisotropic scattering radiative transfer model for diffuse reflection at interfaces and with the help of direct radiative transfer coefficients, corresponding radiative transfer coefficients (RTCs) are deduced in combination with nodal analysis based on the zone method. A gray slab with two semitransparent surfaces is considered here. The radiative heat source term is calculated by the RTCs and is linearized using the method present in Ref. [19]. The transient energy equation is solved by the control-volume method under the external radiative-convective boundary conditions. The influences of two reflecting modes on the transient temperature fields and steady heat flux are investigated under the various values of refractive indexes, albedo, and extinction coefficients.2. Governing equationsAsemitransparent slab with thickness of L is considered as shown in Fig.1 and is confined within two black surfacesT +∞S +∞2S 2ρ22g T hS −∞T −∞ 11g T h 1112i i j j S S S S S S ++Fig.1 Discrete model of space zoneS −∞ and S +∞, denoting the environment of temperatures ∞−T and ∞+T , respectively. Both of boundary surfaces aresemitransparent and diffuse. The slab is divided into 2+NM nodes along its thickness and denoted by i . Here, 1=i and 2i NM =+ represent surfaces 1S and 2S , respectively.Between the time interval t and t t +∆, the fully implicit discrete energy equation of control volume i for the transient coupled radiation and conduction is obtained as()()()111111111,1m mm m m m m m ii iei i iw i ir m iT T k TT k T T C xtx++++++++−+−−+−∆=+Φ∆∆ (1) where ie k , iw k are harmonic mean media thermal conductivities at the interfaces ‘ie ’ and ‘iw ’, respectively. The spectral parameters, such as extinction coefficient k κ, emissivity k ε, and refractive index ,k m n etc., vary with the wavelength, and they can be approximately expressed in a series of rectangular spectral bands. The radiative source term at the control volume i can be expressed as()()()1244,,,,,12244,,,,,424,,,,,j i S i S iNBNM d dri k m j i k T j i j k T i k t t k t t k j ddS k m i k T i k T i k t t k t t d dS i k T k m i k T ik t t k t t n V V A T V V A T n S V A T V S A T S V A T n V S A T σ+∞+∞−∞−∞+−−==+∞+∞−−−−∞−∞−−⎧⎪⎡⎤⎡⎤Φ=−⎨⎣⎦⎣⎦⎪⎩+−⎡⎤⎡⎤⎣⎦⎣⎦⎫⎪ +⎡⎤⎡⎤⎬⎣⎦⎣⎦⎪⎭∑∑21i NM ≤≤+ (2)where, symbol NB indicates the total spectral band number and k the th k region of spectral bands. Discrete boundary conditions at 1S and 2S are now given as follows: ()()1112212S S g k T T x h T T −∆=− (3a)()()2221122NM NM S S g k T T x h T T ++−∆=− (3b)In the Eq. (2), for the local radiative heat source term, besides temperatures at all nodes, the symbols, such as,d i j k t t V V −⎡⎤⎣⎦, ,dk t t S S −∞+∞−⎡⎤⎣⎦ and ,dj k t t S V −∞−⎡⎤⎣⎦, where the superscript d denotes diffuse reflection and the subscriptt t −denotes two semitransparent interfaces, etc., defined as radiative transfer coefficients (RTCs), are unknowns tobe calculated. Therefore, the radiative transfer coefficients must be firstly evaluated before solving the local radiative heat source. In the next section, RTCs are deduced for anisotropic scattering media with diffuse boundaries3. Deduction of RTCsDefine RTC of element (surface or control-volume) i to element j as the quotient of the radiative energy that is received by element j in the transfer of the radiative energy emitted by element i . The radiation transfer process in scattering medium can be divided into two sub-processes [13]: emitting-attenuating-reflecting sub-process andmultiple absorbing-multiple scattering sub-process. Take ,d i j k t tV V −⎡⎤⎣⎦as an example to illustrate the deduction ofRTCs. F or convenience, we omit superscript ‘d ’ and subscript ‘k ’ in the following text. According to twosub-processes of the radiative transfer, the deduction of i j t tV V −⎡⎤⎣⎦can be accomplished by two steps: the first step iswithout considering scattering and the second step is with considering scattering. For scattering media, i j t tV V −⎡⎤⎣⎦includes two parts. One part is the radiative energy emitted by control-volume i V that directly reaches j V and is absorbed partly without being scattered by any other control-volumes; the other part is the radiative energy emitted by i V that reaches j V and is absorbed partly after being scattered many times by other control-volumes.3.1. RTCs without scattering consideredIn the emitting-attenuating-reflecting sub-process, scattering is not considered, and the deduction ofi j t t V V −⎡⎤⎣⎦begins with the first step. Figs.2 shows four paths through which the radiant energy emitted from i V gets toj V for the first time. According to Figs.2, several direct radiative transfer coefficients are defined as follows:()()()1122102exp /t t t ts s s s kL d µµµ−−==−∫ (4a)()()()()1111,02exp /1exp /i i i t t t t s v v s kx k x d µµµµ−−==−−−∆⎡⎤⎣⎦∫ (4b)()()()()1222,102exp /1exp /i i i t t t ts v v s kx k x d µµµµ+−−==−−−∆⎡⎤⎣⎦∫ (4c)()()()()121,02exp /1exp / i j i j t t v v kx k x d i j µµµµ+−=−−−∆≠⎡⎤⎣⎦∫ (4d)()()34212i i t t v v k x E k x −=∆−−∆⎡⎤⎣⎦ (4e)As shown in Figs.2, the radiant energy emitted by i V reaches j V in two directions: positive direction and negative direction relative x -axis. Using the direct radiative transfer coefficients presented above, we can obtain such expressions of the radiative transfer coefficients without considering scattering as follows:1S 2S iV j V(a)1S 2S jV iV(b)1S 2SiV(c)Figs. 2 Transfer paths from control-volume i to control-volume j :(a)i j <, (b)>i j , and (c)i j =.(1) i j < (see Fig.2a)()()()()()11,in 11fi ji j i j t t t t t t t tV V v v v s s v ρβ=−−−−+− (5a)()()()()()()()22,in 211,in 122,in 21bi ji j i j t t t t t t t t t t t t V V v s s v v s s s s v ρρρβ=−−−−−−⎡⎤+−⎢⎥⎣⎦(5b)(2) i j = (see Fig.2b) ()()()()()()()()0.511,in 122,in 211,in 11f i i i i i i i i t t t t t t t t t t t t t t V V v v v s s v v s s s s v ρρρβ=−−−−−−−⎡⎤++−⎢⎥⎣⎦(5c)()()()()()()()()0.522,in 211,in 122,in 21b i i i i i i i i t t t t t t t t t t t t t t V V v v v s s v v s s s s v ρρρβ=−−−−−−−⎡⎤++−⎢⎥⎣⎦(5d) (3) i j > (see Fig.2c)()()()()()()()11,in 122,in 211,in 11fi ji j i j t t t t t t t t t t t t V V v s s v v s s s s v ρρρβ=−−−−−−⎡⎤+−⎢⎥⎣⎦(5e)()()()()()22,in 21bi j i j i j t t t t t t t tV V v v v s s v ρβ=−−−−+− (5f)And the ()i j t tV V −, denoting i j t tV V −⎡⎤⎣⎦when scattering is not considered, equals the sum of ()fi jt tV V −and ()bi jt t V V −:()()()fbi j i j i j t t t t t t V V V V V V =+−−− (6)In above equations, the superscript ‘f ’denotes positive incidence on j V , the superscript ‘b ’denotes negative incidence on j V , the subscript ‘in ’ denotes interior surface, and βis the common ratio, equal to ()21,in 2,in 12s s ρρ, of the infinite geometric series.3.2. RTCs with anisotropic scattering consideredWhen the effect of scattering is considered in the multiple absorbing-multiple scattering sub-process, the quotient of radiant energy represented by RTC ()i jt t V V − will be redistributed. F or anisotropic media, the radiant energyscattered by media is relevant to the scattering direction and is the function of scattering phase function. Scattering directions are relative to incident directions. For one-dimensional problem, if the scattering direction is the same as the incident one, the scattering can be defined as forward scattering; if the scattering direction is the opposite to the incident one, the scattering can be defined as backward scattering. In the multiple absorbing-multiple scattering sub-process, the radiant energy emitted from i V firstly gets to 1l V in two directions (positively incident direction and negatively incident direction), after being scattered, it reaches 2l V in two directions, after being scattered again, it reaches 3l V in two directions …, after being scattered again and again, it finally arrives at j V in two directions. According to above analyse, we must firstly derive such radiative transfer coefficients ()mnff l lt tV V −, ()mnfb l lt tV V −,()mnbf l lt tV V − and ()mnbb l lt tV V −, with anisotropic scattering considered, where ‘f ’ denotes positively incident direction,‘b ’ denotes negatively incident direction, and the first superscript denotes the direction of radiation incidence on ml V ,the second superscript denotes the direction of radiation incidence on nl V .S−∞S S S+∞S S S S+∞(a) (b)S−∞S S S+∞S−∞S S S+∞(c) (d)S S S S+∞S−∞S S S+∞(e) (f)F igs. 3 Sketch of transfer directions fromm lV scattered ton lV: (a)m nl l<, positive incidence,(b)m nl l<, negative incidence, (c)m nl l=, positive incidence, (d)m nl l=, negative incidence,(e)m nl l>, positive incidence, and (f)m nl l>, negative incidence.Figs.3 give four transfer directions fromm lV scattered ton lV. In Figs.3, thick lines with big arrowhead indicatedirection of radiative incidence onm lV, symbol ‘q’ denotes forward scattering and symbol ‘h’ denotes backward scattering relative to incident direction. According to Figs.3, several direct radiative transfer coefficients caused by anisotropic scattering are present as bellows:( vi s1 )tq−tπ 2=2∫0exp ( − kx1,i / cos θ i ) ⎡1 − exp ( − k ∆x / cosθ i ) ⎤ cos θ i sin θ i ⎣ ⎦1π 20π 21∫ Φ (θ s ,θi ) dθ s dθ iΦ (θ s ,θ i ) dθ s dθ i(7a)( vi s1 )th−t( vi s2 )tq−tπ 2=2∫0exp ( − kx1,i / cos θ i ) ⎡1 − exp ( − k ∆x / cosθ i ) ⎤ cos θ i sin θ i ⎣ ⎦π 2 π∫21π(7b)π 2=2∫0exp ( −kx2,i +1 / cos θ i ) ⎡1 − exp ( − k ∆x / cos θ i ) ⎤ cosθ i sin θ i ⎣ ⎦π 20π 21∫ Φ (θ s ,θi ) dθ s dθ iΦ (θ s ,θ i ) dθ s dθ i(7c)( vi s2 )th−tπ 2=2∫0exp ( −kx2,i +1 / cos θ i ) ⎡1 − exp ( − k ∆x / cos θ i ) ⎤ cosθ i sin θ i ⎣ ⎦π 2 π∫2π 20π(7d)q ( vi v j )t −t = 2 ∫ exp ( −kxi +1, j / cosθi ) ⎡1 − exp ( −k ∆x / cosθi )⎤ 2 cosθi sin θi π12 ∫ Φ (θ s ,θi ) dθ s dθi (i ≠ j ) ⎣ ⎦π 20(7e)h ( vi v j )t −t = 2 ∫ exp ( −kxi +1, j / cosθi ) ⎡1 − exp ( −k ∆x / cosθi )⎤ 2 cosθi sin θi π12 ∫ Φ (θ s ,θi ) dθ s dθi (i ≠ j ) ⎣ ⎦π 20π(7f)π 2where, Φ (θ s ,θ i ) is scattering phase function, θ i denotes incident angle and θ s denotes scattering angle; superscript ‘ q ’ denotes forward scattering and superscript ‘ h ’ denotes backward scattering relative to incident direction. Using Eqs. (7a)~ (7f) and Eqs. (4a)~ (4e), RTCs caused by anisotropic scattering can be derived as follows: (1) lm < ln , for the positive incidence on Vlm (see Fig.3a),(V(Vlm Vlnlm Vln)ff t −t=(v v )lm lnq t −t+ vl s1m())h t −tρ1,in s1vln()t −t(1 − β )(8a))fb t −t=⎡ ⎢ vlm s2 ⎣()q t −tρ 2,in s2 vln()t −t+ vl s1m(h t −tρ1,in ( s1s2 )t −t ρ 2,in s2 vln()t −t ⎥ ⎦⎤(1 − β )(8b)for the negative incidence on Vlm (see Fig.3b),(V(Vlm Vlnlm Vln)bf t −t=(v v )lm lnh t −t+ vl s1m())q t −tρ1,in s1vln()t −t(1 − β )(8c))bb t −t=⎡ ⎢ vlm s2 ⎣()h t −tρ 2,in s2 vln()t −t+ vl s1m(q t −tρ1,in ( s1s2 )t −t ρ 2,in s2 vln()t −t ⎥ ⎦⎤(1 − β )(8d)(2) lm = ln , for the positive incidence on Vlm (see Fig.3c),(V (Vlm Vlm) )ff t −t= 0.5(vlm vlm) )t −t⎡ + ⎢ vl s1 ⎣ m ⎡ + ⎢ vl s2 ⎣ m() )h t −tρ1,in s1vl(m t −t)+ vl s2m()q t −tρ 2,in ( s2 s1 ) ρ1,in s1vl(m t −t)⎤ ⎥ ⎦ ⎤ ⎥ ⎦(1 − β ) (1 − β )(8e)fb t −tlm Vlm= 0.5(vlm vlmt −t(qt −tρ 2,in s2 vl(m)t −t+ vl s1m()h t −tρ1,in ( s1s2 ) ρ 2,in s2 vl(m t −t)(8f)− 11 −for the negative incidence on Vlm (see Fig.3d),(V (Vlm Vlm) )bf t −t= 0.5(vlm vlm) )t −t⎡ + ⎢ vl s1 ⎣ m ⎡ + ⎢ vl s2 ⎣ m() )q t −tρ1,in s1vl(m t −t)+ vl s2m()h t −tρ 2,in ( s2 s1 ) ρ1,in s1vl(m t −t)⎤ ⎥ ⎦ ⎤ ⎥ ⎦(1 − β ) (1 − β )(8g)bb t −tlm Vlm= 0.5(vlm vlmt −t(ht −tρ 2,in s2 vl(m)t −t+ vl s1m()q t −tρ1,in ( s1s2 ) ρ 2,in s2 vl(m t −t)(8h)(3) lm > ln , for the positive incidence on Vlm (see Fig.3e),(Vlm Vln)ff t −t=⎡ ⎢ vlm s1 ⎣()ht −tρ1,in s1vln()t −t+ vl s2m()qt −tρ 2,in ( s2 s1 )t −t ρ1,in s1vln()t −t ⎥ ⎦⎤(1 − β )(8i)(V(V ) (lm Vln)fb t −t=(v v ) + (v s )h lm ln t −t lm 2q t −tρ 2,in s2 vln()t −t(1 − β )(8j)for the negative incidence on Vlm (see Fig.3f),bf t −t lm Vln =⎡ ⎢ vlm s1 ⎣)qt −tρ1,in s1vl(n)t −t+ vl s2m()ht −tρ 2,in ( s2 s1 )t −t ρ1,in s1vl(n)t −t ⎥ ⎦⎤(1 − β )(8k)(Vlm Vln)bb t −t=(v v )lm lnq t −t+ vl s2m()h t −tρ 2,in s2 vln()t −t(1 − β )(8l)For the forwardly incident on Vl , the total quotient, absorbed by Vl , of the radiative energy scattered out by Vl , ism n m(Vmlm Vln)ft t −t=(Vlm Vln) (Vff t −t +lm Vln)fb t −t(9a)For the backwardly incident on Vl , the total quotient, absorbed by Vl , of the radiative energy scattered out by Vl ,n mis(Vlm Vln)bt t −t=(Vlm Vln) (Vbf t −t +lm Vln)bb t −t(9b)And then, with the help of RTCs derived above, considering the redistribution of radiant energy, ⎡ViV j ⎤ can be ⎣ ⎦ t −t educed. For simplicity, when n ≥ 3 , four functions are defined as follows:ft H (Vln +1Vln ) a ,t −t = NM +1 ln = 2 NM +1 ln = 2 NM +1 ln = 2∑ ∑ft ⎡ (Vl Vl )tff t ω H (Vl Vl ) a ,t −t + (Vl Vl )tfbt ω H (Vl Vl )bt,t −t ⎤ n n −1 n +1 n − n n −1 a ⎣ n +1 n − ⎦(9a)H (Vln +1Vln )bt,t −t = aft ⎡ (Vl Vl )bf t ω H (Vl Vl ) a ,t −t + (Vl Vl )bbt ω H (Vl Vl )bt,t −t ⎤ n n −1 n +1 n t − n n −1 a ⎣ n +1 n t − ⎦(9b)H (Vln +1Vln ) sftt −t = ,∑⎡(Vl Vl )tff t ω H (Vl Vl ) sftt −t + (Vl Vl )tfbt ω H (Vl Vl )btt −t ⎤ n n −1 , n +1 n − n n −1 s , ⎣ n +1 n − ⎦(9c)− 12 −H (Vln +1Vln )btt −t = s,NM +1 ln = 2∑⎡ (Vl Vl )bf t ω H (Vl Vl ) sftt −t + (Vl Vl )bbo ω H (Vl Vl )btt −t ⎤ n n −1 , n +1 n t − n n −1 s , ⎣ n +1 n t − ⎦(9d)The subscript ‘ a ’ is for absorption and ‘ s ’ for scattering. If n = 2 , thenH Vl3 Vl2( ()a,t −t = ∑ ⎡(Vl Vl )t −t ω (Vl V j )t −t η + (Vl Vl )t −t ω (Vl V j )t −t η ⎤ ⎢ ⎥ ⎦ l =2 ⎣ft ff ft fb bt3 2 2 3 2 2 2NM +1(10a)H Vl3 Vl2)a,t −t = ∑ ⎡(Vl Vl )t −t ω (Vl V j )t −t η + (Vl Vl )t −t ω (Vl V j )t −t η ⎤ ⎢ ⎥ ⎦ l =2 ⎣bt bf ft bb bt3 2 2 3 2 2 2NM +1(10b)H Vl3Vl2( ()s,t −t = ∑ ⎡(Vl Vl )t −t ω (Vl V j )t −t ω + (Vl Vl )t −t ω (Vl V j )t −t ω ⎤ ⎢ ⎥ ⎦ l =2 ⎣ft ff ft fb bt3 2 2 3 2 2 2NM +1(10c)H Vl3Vl2)s,t −t = ∑ ⎡(Vl Vl )t −t ω (Vl V j )t −t ω + (Vl Vl )t −t ω (Vl V j )t −t ω ⎤ ⎢ ⎥ ⎦ l =2 ⎣bt bf ft bb bt3 2 2 3 2 2 2NM +1(10d)(1) After the first scattering,⎡ViV j ⎤ ⎣ ⎦ a ,t −t = ViV j1st()t − t η⎡ViV j ⎤ ⎣ ⎦ s ,t −t = ViV j1st()t −t ω(2) After the second scattering,⎡ViV j ⎤ ⎣ ⎦ a ,t −t = ⎡ViV j ⎤ a ,t −t + ⎣ ⎦2 nd 1st NM +1 l2 = 2∑⎡ ⎢ ViVl2 ⎣()t −t ω (Vl V j )t −t η + (ViVl )t −t ω (Vl V j )t −t ηf ft b bt2 2 2⎤ ⎥ ⎦⎡ViV j ⎤ ⎣ ⎦ s ,t − t =2 ndNM +1 l2 = 2∑⎡ ⎢ ViVl2 ⎣()t −t ω (Vl V j )t −t ω + (ViVl )t −t ω (Vl V j )t −t ωf ft b bt2 2 2⎤ ⎥ ⎦(3) After the third scattering,⎡ViV j ⎤ ⎣ ⎦ a ,t −t = ⎡ViV j ⎤ a ,t −t ⎣ ⎦ +NM +1 l3 = 2 3rd 2 nd∑⎧ ⎪ ⎨ ViVl3 ⎪ ⎩()t − t ω ∑ l =2f2NM +1⎧ ⎨ Vl3 Vl2 ⎩()t −t ω (Vl V j )t −t η + (Vl Vl )t −t ω (Vl V j )t −t ηff ft fb bt2 3 2 2⎫ ⎬ ⎭+ ViVl3()t − t ω ∑ l =2b2NM +1⎧ ⎨ Vl3 Vl2 ⎩()t −t ω (Vl V j )t −t η + (Vl Vl )t −t ω (Vl V j )t −t ηbf ft bb bt2 3 2 2⎫ ⎬ ⎭⎫ ⎪ ⎬ ⎪ ⎭ ⎫ ⎪ ⎬ ⎪ ⎭= ⎡ViV j ⎤ ⎣ ⎦2 nd a ,t − t+NM +1 l3 = 2∑⎧ ⎨ ViVl3 ⎩()t −t ω H (Vl Vl )a,t −t + (ViVl )t −t ω H (Vl Vl )a,t −tf ft b bt3 2 3 3 2⎡ViV j ⎤ ⎣ ⎦ s ,t − t =3rdNM +1 l3 = 2∑⎧ ⎨ ViVl3 ⎩()t −t ω H (Vl Vl )s,t −t + (ViVl )t −t ω H (Vl Vl )s,t −tf ft b bt3 2 3 3 2⎫ ⎪ ⎬ ⎪ ⎭(4) The rest are deduced by analogy and after the (n + 1)th scattering, − 13 −⎡ ⎤ ⎣ViV j ⎦ a ,t −t( n +1)th⎡ ⎤ = ⎣ViV j ⎦( n +1) thnth a ,t − t+⎧ ⎫ f ft b bt ⎨ (ViVln +1 )t −t ω H (Vln +1Vln ) a ,t −t + (ViVln +1 )t −t ω H (Vln +1Vln ) a ,t −t ⎬ ⎭ ln +1 = 2 ⎩NM +1∑(11a)⎡ViV j ⎤ ⎣ ⎦ s ,t − t=⎧ ⎫ f ft b bt ⎨ (ViVln +1 )t −t ω H (Vln +1Vln ) s ,t −t + (ViVln +1 )t −t ω H (Vln +1Vln ) s ,t −t ⎬ ⎭ ln +1 = 2 ⎩NM +1∑(11b)If ⎡ViV j ⎤ ⎣ ⎦ s ,t − t( n +1)this less than the value of 10−10 , the process of redistribution of radiant energy can be supposed to befinished and ⎡ViV j ⎤ can be obtained as follows: ⎣ ⎦ t −t⎡ ⎤ ⎡ ⎤ ⎣ViV j ⎦ t −t = η ⎣ViV j ⎦ a , t −td d d( n +1)th(12)dOther RTCs for instance ⎡ S+∞Vi ⎤ k ,t −t , ⎡ S−∞Vi ⎤ k ,t −t , ⎡Vi S−∞ ⎤ k ,t −t and ⎡Vi S+∞ ⎤ k ,t −t can be also educed by the same ⎣ ⎦ ⎣ ⎦ ⎣ ⎦ ⎣ ⎦ method as that adopted for the deduction of ⎡ViV j ⎤ ⎣ ⎦ k , t −t .d3.3. Determination of diffuse reflectivityIf assuming that each bit of roughness of diffuse surface acts as a smooth facet, as discussed by Siegel [11], the diffuse reflectivity of semitransparent interface can be obtained from the Fresnel interface relations by integrating the reflected energy over all incident directions. If radiant energy projecting into a medium with a refractive index bigger than one from surroundings with refractive index of 1.0 , the diffuse reflectivity of outer surface is [20]ρ1,out = ρ 2,out =π 2⎧∫0⎪ ⎡ tan (θ i − θ r ) ⎤ ⎡ sin (θ i − θ r ) ⎤ ⎥ +⎢ ⎥ ⎨⎢ ⎪ ⎢ tan (θ i + θ r ) ⎥ ⎢ sin (θ i + θ r ) ⎥ ⎦ ⎣ ⎦ ⎩⎣22⎫⎪ ⎬ sin (θ i ) cos (θ i ) dθ i ⎪ ⎭(13)where, θ r is an angle of refraction, equal to arcsin (θ i nm ) . When going in the reverse direction from the medium to the surroundings, the diffuse reflectivity of inner surface is [11]ρ1,in = ρ 2,in =ρ1,out2 nm⎛ 1 ⎞ + ⎜1 − 2 ⎟ ⎜ n ⎟ m ⎠ ⎝(14)It is worth noting that the effect of total reflection occurring at the inner surface under the condition of the incident angle being equal to or bigger than the critical angle, arcsin(1/ nm ) , is considered in Eq. (14).4. Results and Analyses4.1. Dimensionless heat fluxes comparison with Ref. [21]There is no published data for the study of radiative transfer within an anisotropic scattering media with diffuse and semitransparent surfaces to be compared with the results of this paper. We can but compare dimensionless heat − 14 −。