Effects of graded refractive index on steady and transient heat transfer inside a scattering semitransparent slab

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Journal of Quantitative Spectroscopy & Radiative Transfer 96 (2005) 363–381 www.elsevier.com/locate/jqsrt

Effects of graded refractive index on steady and transient heat transfer inside a scattering semitransparent slab Hong-Liang Yia, He-Ping Tana,, Jian-Feng Luob, Shi-Kui Donga a

School of Energy Science and Engineering, Harbin Institute of Technology, Harbin, 150001, P.R. China b National University of Defense Technology, 410073, P.R. China Received 15 November 2004; accepted 23 December 2004

Abstract Coupled radiative–conductive heat transfer inside an absorbing–emitting–scattering semitransparent slab is solved. The refractive index of the media is distributed spatially in a linear relationship. The two boundary surfaces are diffuse and opaque. In this paper, the media with graded refractive index is simulated by using multilayer composite model, and in each sub-layer the refractive index is supposed to be constant and the rays of thermal radiation travel in a straight line. The multilayer model is developed by ray-tracing method combined with node analysis. A comparison of the present results with previous results shows that the multilayer simulation of media with graded refractive index is rational and correct. Considering isotropic scattering of thermal radiation, with the changes in the extinction coefficient, surface emissivities and the scattering albedo, the influences of refractive index distribution on the temperature and the radiative heat flux fields are investigated. The results show that the gradient distributing of refractive indexes can cause very different thermal behavior concerned with radiative transfer in semitransparent media compared with constant refractive indexes. r 2005 Elsevier Ltd. All rights reserved. Keywords: Coupled radiative–conductive heat transfer; Graded refractive index; Multilayer radiative transfer model; Ray-tracing method; Radiative transfer coefficients

Corresponding author. Tel.: +86 451 86412674; fax: +86 451 86221048.

E-mail addresses: [email protected] (H.-L. Yi), [email protected] (H.-P. Tan). 0022-4073/$ - see front matter r 2005 Elsevier Ltd. All rights reserved. doi:10.1016/j.jqsrt.2004.12.034

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Nomenclature RTC Ak;T i

coefficient Rradiative transferR 1 I ðT Þ dl= i 0 I b;l ðT i Þ dl; fractional spectral emissive power of spectral band Dlk b;l Dlk at nodal temperature T i a1; a2 surface, interface or control volume, used to define one-layer radiant energy quotient transfer functions b1; b2 surface or interface, used to define multilayer radiant energy quotient transfer functions specific heat capacity of bth layer, J kg1 K1 cb F radiant energy quotient transfer function of one-layer model H radiant energy quotient transfer function of multilayer model h1 ; h2 convective heat transfer coefficient at surfaces of S 1 and S 2 ; respectively, W m2 K1 Ib Ith node in bth layer thermal conductivity of bth layer of medium, W m1 K1 kb kie ; kiw harmonic mean thermal conductivity at interface ie and iw, W m1 K1 Lb thickness of bth layer, m total thickness of composite, L1 þ L2 þ    þ Ln ; m Lt number of control volumes of bth layer Mb total number of control volumes of composite, M 1 þ M 2 þ    þ M n Mt n total number of layers of multilayer composite spectral refractive index of bth layer nb;k refractive index of ith control volume; when ipM 1 ; n0i;k ¼ n1;k ; when M 1 þ M 2 þ n0i;k    þ M b1 oipM 1 þ M 2 þ    þ M b ; n0i;k ¼ nb;k n0 ; nnþ1 refractive indexes of the surroundings (equal to the refractive index of air ng ; see Fig. 1) NB total number of spectral bands interface between bth layer and ðb þ 1Þth layer; side of interface Pb facing towards bth Pb layer 0 side of interface Pb facing towards ðb þ 1Þth layer Pb radiative heat flux, W m2 qr q~ dimensionless heat flux, q=ðsT 4r Þ S1 ; S þ1 left and right black surfaces representing the surroundings (Fig. 1) S1 ; S 2 boundary surfaces (Fig. 1) Su ; S v surfaces, u; v ¼ 1; or 2 T absolute temperature, K uniform initial temperature, K T0 reference temperature, K Tr T g1 ; T g2 gas temperature for convection at x ¼ 0 and Lt ; respectively, K T S1 ; T S2 temperature of boundary surfaces S 1 and S 2 ; respectively, K T 1 ; T þ1 temperatures of the black surface S1 and S þ1 respectively, K (Fig. 1)

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t physical time, s the ith control volume, i ¼ 1 to M t Vi V Ib the Ith control volume of the bth layer, I ¼ 1 to M b ðV i V j Þk ; ½V i V j k parts of radiant energy emitted by V i at the kth spectral band ðDlk Þ and arriving at V j for non-scattering and scattering media, respectively ðS u S v Þk ; ½S u S v k parts of radiant energy emitted by S u at the kth spectral band ðDlk Þ and arriving at Sv for non-scattering and scattering media, respectively ðS u V j Þk ; ½S u V j k parts of radiant energy emitted by S u at the kth spectral band ðDlk Þ and arriving at V j for non-scattering and scattering media, respectively ðV i S v Þk ; ½V i S v k parts of radiant energy emitted by V i at the kth spectral band ðDlk Þ and arriving at Sv for non-scattering and scattering media, respectively x coordinate in direction across layer, m xi ; yi geometrical progressions used in tracing radiant energy’s transferring distance between surface a and b, m xba X dimensionless coordinate in direction across layer, X ¼ x=Lt ab;k spectral absorbing coefficient of bth layer, m1 gbo transmissivity of radiant energy propagating from layer ‘‘b’’ to layer ‘‘o’’, 1  rbo Dxb control volume thickness of bth layer, m Dt time step, s ðdxÞie ; ðdxÞiw distance between nodes i and i þ 1; and that between nodes i and i  1; respectively, m (Fig. 1) 0;k ; 1;k emissivity of the outside and inside of surface S 1 ; respectively 2;k ; 3;k emissivity of the inside and outside of surface S 2 ; respectively 1  ob Zb 0 Zi 1  o0i Y dimensionless temperature, T=T r kb;k extinction coefficients of bth layer, ab;k þ ss;b;k ; m1 rb density of bth layer, kg m3 rbo reflectivity of intensity going from layer ‘‘b’’ to layer ‘‘o’’ ss;b;k spectral scattering coefficient of bth layer, m1 tb;k spectral optical thickness of bth layer, kb;k Lb Fri radiative heat source of control volume i spectral scattering albedo of bth layer, ss;b;k =kb;k ob;k 0 oi;k spectral scattering albedo of ith control volume; when ipM 1 ; o0i;k ¼ o1;k ; when M 1 þ M 2 þ    þ M b1 oipM 1 þ M 2 þ    þ M b ; o0i;k ¼ ob;k Superscripts m d 

time step diffuse reflection normalized values

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Subscripts a; b layer index, a; b ¼ 1 to n bo energy propagating from layer ‘‘b’’ to layer ‘‘o’’ c the cth layer, either b or o layer g gas (air) i, j relative to nodes; index of geometrical progression term k relative to spectral band k o the oth layer, either the b  1 or b þ 1 layer o–o refers to a media with opaque surfaces S1 ; S 2 relative to S 1 and S 2 u; v 1 or 2 1; þ1 relative to S1 and S þ1

1. Introduction The refractive index can have large effects on the thermal behavior in semitransparent media. Spuckler and Siegel examined the effects of refractive index on the radiative transfer within a single layer [1,2], two-layer [3], and multilayer [4] translucent media. Tan and coworkers investigated the refractive index effects on the coupled radiative–conductive heat transfer in a semitransparent composite of three layers [5] and multilayers [6]. In the above-mentioned references, the case of constant refractive index in each layer was considered. In recent years, the effects of graded refractive index (GRIN) on the radiative transfer inside semitransparent media have been of interest. Ben Abdallah and Le Dez [7–10] proposed a curved ray-tracing technique for the solution of the radiative transfer problem in GRIN non-scattering media and investigated the radiative transfer [7–9] and the coupled radiative–conductive heat transfer [10] within a slab [7,8,10] and a sphere [9]. Huang et al. examined the radiative heat transfer [11–14] and simultaneous radiation–conduction heat transfer [15] using a curved ray-tracing technique combined with a pseudo-source adding method inside a non-scattering slab with a distribution of sinusoidal refractive index [11], linear refractive index [12,13,15] and arbitrary refractive index [14]. Liu et al. [16] developed a discrete curved ray-tracing method to analyze the radiative transfer in one-dimensional participating non-scattering media with variable spatial refractive index and obtained results with a good accuracy. In the aforementioned papers on the radiative transfer in GRIN media, no scattering of the radiative energy has been included. Tan and Luo [6,17,18] set up a multilayer composite model using ray-tracing method in combination with node analysis and spectral band model and solved the problem of radiative transfer within the participating multilayer composite with specular interfaces. In the multilayered medium, the rays of thermal radiation are straight lines due to the constant refractive index in each sub-layer; while in a single layer medium with a continuous variation of the refractive index (or GRIN), the rays are not straight lines but curved continuous trajectories governed by the Fermat optical principle. In this paper, the multilayer composite model is used to simulate a single

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layer media with the GRIN. The problem of the radiative transfer within a single layer of GRIN media is regarded as that inside a composite of multilayered media, each sub-layer with respective constant refractive index. Boundary surfaces of the single layer of medium with GRIN are opaque and diffuse and the composite slab is considered to be of opaque and diffuse boundary surfaces and semitransparent diffuse interfaces. In each sub-layer of the composite, the refractive index is supposed to be constant and the rays of thermal radiation travel in a straight line instead of a curve. The errors caused by the transfer in a straight line instead of a curve and the diffuse reflection at every interface between two adjacent sub-layers will be reduced with increase of layer number of the composite; if the layer number is big enough, the refractive index variation at each interface of each sub-layer may be considered to be continuous and except in the case of possible total reflection at one interface, the reflected part of the radiative energy can be neglectable compared to the refractive one, which would approach greatly to the physical truth of the radiative transfer inside a medium with GRIN. By the same method used in Refs. [6,17,18], the radiative transfer coefficients (RTCs) are deduced for the multilayer slab with opaque diffuse boundary surfaces and semitransparent diffuse interfaces, which helps to calculate the radiative heat source term, and the transient energy equation for the radiative–conductive coupled heat transfer is solved by the fully implicit control-volume method. In this paper, considering the isotropic scattering of thermal radiation, with the changes in the extinction coefficient, surface emissivities and the scattering albedo, the influences of refractive index distribution on the temperature and the radiative flux fields within a slab are investigated. 2. Physical model and discrete governing equation As shown in Fig. 1, an absorbing, emitting, and isotropically scattering semitransparent slab with GRIN is put between two plane-parallel black surfaces S 1 and S þ1 : The surfaces S1 and S−∞

S1

P1

P2

∆ x1

Tg1

S2

Pn −1

∆ x2

∆ xn

S+∞ Tg 2

h1

n0

T−∞

h2 01

11

21

I1

0

1

2

i


0


1

n1

ρ (θ )g1 ρ (θ )1g

I1 = M1 12 M1

I2

13

In-1 = Mn-1

...

n2

ρ (θ )23 ρ (θ )32

L1

P x( I1+1) 1

Mt +1

Mt nn


2


3

nn+1

T+ ∞

P

2

S x( I1+1) 1

In = Mn

xI n −1

xI 1

1

In

ρ (θ )(n-1)m ρ (θ )n(n-1)m ρ (θ )ng ρ (θ )gn

P

xIS1

1n

M1 + M2 M1 + M2+1M1+ ... + Mn-1

M1 + 1

ρ (θ )12 ρ (θ )21

I2 = M2

n

P x( I1+1) 2

L2

P x( I2+1) 2

P

x( In+−1)1 L 3 + L 4 + ... +Ln-1

n

x(SI2+1)

n

Ln

Fig. 1. Physical model of a plane-parallel slab with graded refractive index divided into n sub-layers.

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S 2 are opaque. The GRIN slab is divided into n layers, each with respective constant refractive index and M 1 ; M 2 ; . . . ; and M n control volumes (inner nodes) respectively along the thickness. The imaginary interfaces, P1 ; P2 ; . . . ; and Pn1 ; are semitransparent and diffuse. I b is used to denote the Ith nodes in the bth layer, and for convenience, all nodes are also denoted by i increasingly along the whole thickness. If b ¼ 1; then i ¼ I b ; else i ¼ M 1 þ M 2 þ    þ M b1 þ I b : The variation of the medium spectral properties with wavelength, such as kb ; ab ; and nb ; etc., can be approximately expressed by a series of rectangular spectral bands. The fully implicit discrete energy equation at control volume i in the bth layer is [6,17,18] rb cb Dxb

mþ1 kmþ1 ðT mþ1 Þ kmþ1 ðT mþ1  T mþ1 T mþ1  Tm iþ1  T i i i i i1 Þ  iw þ Fr;mþ1 , ¼ ie i ðdxÞiw ðdxÞie Dt

(1)

where Fri ; the radiative heat source term, is expressed as [18] ( Mt NB X X d d r 4 02 4 fn02 Fi ¼ s j;k ½V j V i k;oo Ak;T j T j  ni;k ½V i V j k;oo Ak;T i T i g k¼1

þ

j¼1

d 4 fn02 M t ;k ½S 2 V i k;oo Ak;T S2 T S2

d 4  n02 i;k ½V i S 2 k;oo Ak;T i T i g )

d 4 02 d 4 þfn02 1;k ½S 1 V i k;oo Ak;T S1 T S 1  ni;k ½V i S 1 k;oo Ak;T i T i g ;

1pipM i ,

ð2Þ

when i ¼ 0; the radiative heat flux at surface S 1 is ( Mt NB X X d d r 4 02 4 fn02 qS1 ¼ s j; k ½V j S 1 k;oo Ak;T j T j  n1;k ½S 1 V j k;oo Ak;T S1 T S 1 g k¼1

j¼1

)

d 4 02 d 4 þn02 M t ;k ½S 2 S 1 k;oo Ak;T S2 T S 2  n1;k ½S 1 S 2 k;oo Ak;T S1 T S 1 ,

ð3Þ

when i ¼ M t þ 1; the expression of the radiative heat flux at surface S 2 is similar to Eq. (3). The discrete boundary condition at surface S1 is written as qrS1 þ 2k1 ðT 1  T S1 Þ=Dx1 ¼ h1 ðT S1  T g1 Þ þ s

NB X

ð0;k Ak;T S1 T 4S1  Ak;T 1 T 41 Þ.

(4)

k¼1

The discrete boundary condition at surface S 2 is similar to above equation. The major difficulty in solving the coupled radiative–conductive energy equation is the solution to the radiative heat source term Fri ; and from the expression of Fri ; the calculation of RTCs, for example, ½V i V j dk;oo ; ½S2 V i dk;oo ; ½V i S1 dk;oo and so on, is the key to the problem.

3. RTCs for an n-layer slab with opaque and diffuse surfaces RTC of element (surface or control-volume) i to element j is defined as the quotient of the radiant energy that is received by element j in the transfer process of the radiant energy emitted by element i and determined by the geometrical and optical characteristics of media considered in the

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paper. The radiative transfer process in scattering semitransparent medium can be divided into two sub-processes according to its physical mechanism. In the following deduction, RTCs ðV I a V J b Þdoo;k ; etc. correspond to the emitting–attenuating–reflecting sub-process, in which scattering of the radiant energy is not considered, and RTCs ½V I a V J b doo;k ; etc. to absorbing–scattering sub-process, in which the radiant energy scattering and its redistribution are included. 3.1. Multilayer radiant energy transfer model When a bunch of lights is projected on a diffuse surface in some direction, the lights will be reflected back to hemispherical space in all directions. So for the slab with diffuse surfaces we can only examine the radiative transfer process by tracing radiant energy from the whole hemispherical space instead of radiant intensity from some direction. Multilayer radiant energy quotient transfer functions are used to trace radiant energy transferring in the n-layer slab. In Fig. 2, let Pm to denote the interface between the mth and the ðm þ 1Þth layers, then the side of the interface facing the mth layer is specified as Pm ; and that facing the ðm þ 1Þth layer as Pm0 ; P0 denotes surface S1 and Pn denotes surface S2 : b The one-layer radiant energy quotient transfer functions, denoted by symbol Fa2 a1b ;k ; are shown in Appendix A. The multilayer radiant energy quotient transfer functions are denoted by symbol Hb2 b1;mþ1mþDm;k ; which means the total quotient of the radiant energy arrived at superscript b2 (represents Pm0 ; Pmþ10 ; PmþDm1 or PmþDm ) in the process of the radiant energy emitted by subscript b1 (represents Pm0 or PmþDm ) at kth spectral band after ‘‘transferring once’’ within the multilayer model, where subscript m þ 1  m þ Dm denotes the multilayer model consisting of Dm layers (from the ðm þ 1Þth to the ðm þ DmÞth layer). The ‘‘transferring once’’ means the process that the radiant energy is attenuated and reflected again and again until it becomes zero within the layers considered in the model. P 0 P ; HPmþ1 and HPmm00 ;mþ1mþDm are taken as the According to Fig.2, HPPmþDm m0 ;mþ1mþDm m0 ;mþ1mþDm examples to illustrate the deduction of multilayer radiant energy quotient transfer functions, and for convenience, subscript k is omitted.

ρm,m+1

Pm

ρm+1,m ρm+1,m+2

Pm +1

y1

y2

y3

ρm+2,m+1

∆ m -1

Pm+∆m

L m+1

Lm+2 + ... +Lm+∆m

x1

x2

x2

ρm+∆m−1,m+∆m ρm+∆m,m+∆m−1 ρm+∆m,m+∆m+1 ρm+∆m+1,m+∆m

Fig. 2. Multilayer radiative transfer model.

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(1) After ‘‘transferring once’’ inside the ðm þ 1Þth layer, one part quotient, FPPmþ1 gmþ1;mþ2 ; of m0 radiant energy which is emitted by Pm0 penetrating through Pmþ1 enters the following Dm  1 layers, and after ‘‘transferring once’’ in the Dm  1 layers, the quotient arriving at PmþDm for the first time is x1 ¼ FPPmþ1 gmþ1;mþ2 HPPmþDm ; and that arriving at Pmþ10 for the first time is m0 mþ10 ;mþ2mþDm P

y1 ¼ FPPmþ1 gmþ1;mþ2 HPmþ1 : m0 mþ10 ;mþ2mþDm (2) A fraction, gmþ2;mþ1 ; of the above quotient that arrives at Pmþ10 enters the ðm þ 1Þth layer, g ; enters the following Dm  1 and after ‘‘transferring once’’ in the layer, the quotient, FPPmþ1 mþ1 mþ1;mþ2 layers, and after ‘‘transferring once’’ in the Dm  1 layers, the quotient arriving at PmþDm for the gmþ2;mþ1 FPPmþ1 g HPPmþDm ; and that arriving at Pmþ10 for the second second time is x2 ¼ y1 FPPmþ1 mþ1 mþ1;mþ2 m0 mþ10 0

P

time is y2 ¼ y1 gmþ2;mþ1 FPPmþ1 g HPmþ1 : mþ1 mþ1;mþ2 mþ10 ;mþ2mþDm (3) The above quotient y2 repeats step (2), then the quotient arriving at PmþDm for the third time is x3 ¼ y2 FPPmþ1 gmþ2;mþ1 FPPmþ1 g HPPmþDm and that arriving at Pmþ10 for the third time is y3 ¼ mþ1 mþ1;mþ2 m0 mþ10 P

0

g HPmþ1 : y2 gmþ2;mþ1 FPPmþ1 mþ1 mþ1;mþ2 mþ10 ;mþ2mþDm The radiant energy leaving from the interface Pm0 is traced by this way until it finally attenuates to zero. Obviously, xi ðiX2Þ and yi ðiX1Þ are infinite geometric series with a common ratio of P 0 b ¼ gmþ2;mþ1 FPPmþ1 g HPmþ1 ðbo1Þ: From the above analysis, the total quotient of mþ1 mþ1;mþ2 mþ10 ;mþ2mþDm the radiant energy emitted from Pm0 finally arriving at PmþDm can be calculated as follows: ¼ HPPmþDm m0 ;mþ1mþDm

1 X

0

xi ¼ x1 þ

i¼1

þ

x2 gmþ1;mþ2 HPPmþDm ¼ FPPmþ1 m0 mþ10 ;mþ2mþDm 1b

y1 FPPmþ1 gmþ2;mþ1 FPPmþ1 g HPPmþDm mþ1 mþ1;mþ2 m0 mþ10 P

1  gmþ2;mþ1 FPPmþ1 g HPmþ1 mþ1 mþ1;mþ2 mþ10 ;mþ2mþDm 0

,

ð5Þ

and that finally arriving at Pmþ10 and Pm0 can be, respectively, written as follows: P

0

¼ HPmþ1 m0 ;mþ1mþDm

1 X

yi ¼

i¼1

P

P

y1 y1 ¼ P mþ1 1  b 1  gmþ2;mþ1 FP gmþ1;mþ2 HPmþ10 mþ1

P

0

,

(6)

Pmþ10 ;mþ2mþDm

HPmm00 ;mþ1mþDm ¼ HPmþ1 g F m0 . m0 ;mþ1mþDm mþ2;mþ1 Pmþ1

(7)

Besides the above three multilayer radiant energy quotient transfer functions, there are other three P 0 functions: HPPmþDm ; HPPmþDm1 ; HPmmþDm ;mþ1mþDm;k ; and they can also be deduced mþDm ;mþ1mþDm;k mþDm ;mþ1mþDm;k in the same way presented above. Using recursive technique and combining Eqs. (5)–(7) with the one-layer radiant energy ; quotient transfer functions F of the ðm þ 1Þth layer, the quotient functions, HPPmþDm m0 ;mþ1mþDm P

0

P

and HPmm00 ;mþ1mþDm can be evaluated. Similarly, the quotient functions, HPmþ1 m0 ;mþ1mþDm P

0 ; HPPmþDm1 ; HPmmþDm HPPmþDm ;mþ1mþDm;k can be also evaluated. mþDm ;mþ1mþDm;k mþDm ;mþ1mþDm;k

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3.2. RTCs of an n-Layer slab for emitting–attenuating–reflecting sub-process With the help of the multilayer radiant energy quotient transfer function H; RTCs of an n-layer absorbing–emitting slab ðkk ¼ ak Þ can be deduced successfully. For example, the total quotient of the radiant energy emitted by S 1 at the kth spectral band and finally absorbed by S2 after ‘‘transferring once’’ within the n layers can be expressed as ðS 1 S 2 Þdk;oo ¼ HSS21 ;1n;k .

(8)

All of the RTCs, such as ðV i V j Þdk;oo ; ðSu V j Þdk;oo and ðV i Sv Þdk;oo ; can be also obtained using the multilayer radiative energy quotient transfer function H to trace the radiative energy transferring in the n-layer slab, and you can see Refs. [6,17,18] for more detailed information on the deductions of RTCs. 3.3. RTCs of an n-layer slab for absorbing–scattering sub-process Inside a scattering slab, one part of the radiant energy transferred to some control volume is absorbed and the other is scattered; the scattered radiant energy is transferred to another control volume and similarly is partly absorbed and partly scattered. By scattering, the redistribution of the radiant energy is accomplished at each control volume. Using RTCs ðS u S v Þdk;oo ; ðS u V i Þdk;oo ; ðV i S u Þdk;oo ; and ðV i V j Þdk;oo deduced above, RTCs ½Su Sv dk;oo ; ½Su V i dk;oo ; ½V i Su dk;oo ; and ½V i V j dk;oo can be calculated, where the subscripts u; v ¼ 1 or 2. Before the further deduction, the RTCs for emitting–attenuating–reflecting sub-process should be normalized first: ðV i V j Þ ¼ ðV i V j Þ=ð4kb Dxb Þ;

V i 2 bth layer,

(9a)

ðV i S u Þ ¼ ðV i S u Þ=ð4kb Dxb Þ;

V i 2 bth layer,

(9b)

ðS u V j Þ ¼ ðS u V j Þ=u ,

(9c)

ðS u S v Þ ¼ ðSu Sv Þ=u .

(9d)

After the nth-order scattering event, there are ½V i S u nth ¼ ½V i S u ðn1Þth þ a a

Mt X

ðV i V l 2 Þ o0l 2

l 2 ¼1

(

Mt X

 

( Mt X

" 

ðV l n2 V l n1 Þ

o0l n1

¼

½V i V j ðn1Þth a (  

l n1 ¼1

l 4 ¼1 

#)))

o0l n ðV l n S u Þ

,

ð10Þ

l n ¼1

þ

Mt X

ðV i V l 2 Þ



o0l 2

l 2 ¼1 Mt X

l 3 ¼1 Mt X

ðV l n1 V l n Þ

l n1 ¼1

½V i V j nth a

( Mt X ðV l 2 V l 3 Þ o0l 3 ðV l 3 V l 4 Þ o0l 4

ðV l n2 V l n1 Þ o0l n1

( Mt X

"

l 3 ¼1 Mt X l n ¼1



ðV l 2 V l 3 Þ

o0l 3

( Mt X

ðV l 3 V l 4 Þ o0l 4

l 4 ¼1

ðV l n1 V l n Þ o0l n ðV l n V j Þ Z0j

#))) ,

ð11Þ

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½S u S v nth ¼ ½Su Sv ðn1Þth þ a

Mt X

ðS u V l 2 Þ o0l 2

l 2 ¼1

(

Mt X

 

( Mt X l 3 ¼1

ðV l n2 V l n1 Þ



o0l n1

" Mt X

l n1 ¼1

 

Mt X

ðS u V l 2 Þ o0l 2

l 2 ¼1 Mt X

ðV l 3 V l 4 Þ o0l 4

l 4 ¼1 

ðV l n1 V l n Þ

#)))

o0l n ðV l n S v Þ

,

ð12Þ

l n ¼1

½S u V j nth ¼ ½S u V j ðn1Þth þ a a (

ðV l 2 V l 3 Þ o0l 3

( Mt X

( Mt X

ðV l 2 V l 3 Þ o0l 3

l 3 ¼1



ðV l n2 V l n1 Þ

o0l n1

l n1 ¼1

" Mt X

ðV l n1 V l n Þ

( Mt X

ðV l 3 V l 4 Þ o0l 4

l 4 ¼1 

o0l n ðV l n V j Þ Z0j

#))) .

ð13Þ

l n ¼1

After inverse operation and considering the emissive power of control volumes and surfaces, the RTCs for isotropic scattering are obtained: ½V i S u ¼ 4kb Zi Dxb ½V i S u anth ; ½S u S v ¼ u ½S u S v nth a ;

½V i V j ¼ 4kb Zi Dxb ½V i V j nth a ,

½S u V j ¼ u ½Su V j nth a .

(14)

The subscript ‘‘a’’ in Eqs. (10)–(14) denotes absorption quotient of the radiative energy. More detailed information on the deductions of RTCs for absorbing–scattering sub-process can be seen in Ref. [19].

4. Numerical method and verification of RTCs From Eq. (2), the radiative heat source term Fri is a nonlinear function of the temperatures of all nodes; thereby it must be linearized first by Patankar’s method [20]. The temperature fields in the slab considered can be finally obtained by solving the linearized equations using the tri-diagonal matrix algorithm, and then, the radiative flux caused by the coupled radiative–conductive heat transfer can be determined from Eq. (2). The correctness of the RTCs deduced above can be verified using the relativity and integrality relations as follows: ðS1 S2 Þdk;oo ðS2 S1 Þdk;oo ¼ , g12 g23    gn1;n gn;n1 gn1;n2    g21

(15a)

ðS1 V I b Þdk;oo ðV I b S 1 Þdk;oo ¼ , g12 g23    gb1;b gb;b1 gb1;b2    g21

(15b)

ðS 2 V I b Þdk;oo ðV I b S 2 Þdk;oo ¼ , gn;n1 gn1;n2    gbþ1;b gb;bþ1 gbþ1;bþ2    gn1;n

(15c)

ðV I a V J a Þdk;oo ¼ ðV J a V I a Þdk;oo ,

(15d)

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ðV I a V J b Þdk;oo ðV J b V I a Þdk;oo ¼ , ga;aþ1 gaþ1;aþ2    gb1;b gb;b1 gb1;b2    gaþa;a

373

(15e)

where subscripts ‘‘a’’ and ‘‘b’’ denote the ath and the bth layer, respectively, and a; b ¼ 1 to n, b4a: Mt X

½V i V j dk;oo þ ½V i S 1 dk;oo þ ½V i S 2 dk;oo ¼ 4kb;kZb Dxb ;

V i 2 bth layer,

(16a)

j¼1

½S u S v dk;oo þ

Mt X

½S u V j dk;oo þ ½S u Su dk;oo ¼ u;k ;

u; v ¼ 1 or 2.

(16b)

j¼1

5. Results and analyses Using the multilayer model of diffuse reflection developed above to simulate the radiative transfer inside a GRIN slab, the steady and transient coupled radiative–conductive heat transfer in the scattering GRIN slab is investigated. The thickness of the slab with two opaque surfaces keeps a constant value of 0.01 m, and the two boundary temperatures are specified as T S1 ¼ 1000 K and T S2 ¼ 1500 K; respectively. Along the thickness direction, the slab is divided into n ¼ 100 sub-layers. In each sub-layer, the refractive index is constant, and M 1 ¼ M 2 ¼    ¼ M n ¼ 1 inner node is included respectively. Let k ¼ k1 ¼ k2 ¼    ¼ kn ; o ¼ o1 ¼ o2 ¼    on ; k ¼ k1 ¼ k2 ¼    kn 0 ¼ 1 ; 2 ¼ 3 and cr ¼ c1 r1 ¼ c2 r2 ¼    ¼ cn rn : The slab is characterized by a refractive index nðxÞ spatially varying in a linear relationship as bellow: nðxÞ ¼ n1 þ ðn2  n1 Þx=L.

(17)

According to Eq. (17), we can obtain the discrete value of the refractive index of the bth sub-layer as follows: n2  n1 n2  n1 þ  b, (18) nðbÞ ¼ n1  n1 n1 where n1 and n2 are refractive index values of the region adjacent to the two boundary surfaces, respectively. The comparison of the results in this paper with those presented in Ref. [15] for the steady temperature distribution in the non-scattering GRIN media can be seen in Fig. 3. Fig. 3 shows that the results accords well with each other. This proves that the multilayer diffuse reflection radiative transfer model simulating the radiative transfer inside GRIN media is rational and correct. Figs. 4 and 5 show the effects of GRIN on the steady thermal behavior in slab with the changes in surface emissivities. From Fig. 4, we see that increasing the emissivity of the high-temperature surface can increase the temperature level and increasing the emissivity of low-temperature surface can decrease the temperature level; for the case of n1 ¼ 1:2 and n2 ¼ 1:8; high-temperature surface radiation causes the temperature fields close to the high-temperature surface to be higher and, for n1 ¼ 1:8 and n2 ¼ 1:2; low-temperature surface radiation causes the temperature fields

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374

1500

1400

T(K)

1300

1200

n1=1.8, n2=1.2, ref. [15] n1=1.2, n2=1.8, ref. [15] n1=1.8, n2=1.2, this paper

1100

n1=1.2, n2=1.8, this paper

1000 0.0

0.2

0.4

0.6

0.8

1.0

X

Fig. 3. Steady temperature field in a non-scattering media with a linear distribution of refractive index for L ¼ 0:01 m, kk ¼ 1000 m1 ; k ¼ 1:0 W m1 K1 ; 0 ¼ 1 ¼ 0:2 and 2 ¼ 3 ¼ 1:0:

1.5 1.4

1.4

1.4

1.3

1.3

1.3

Θ 1.2

Θ

1.5

Θ

1.5

1.2 n1=1.2, n2=1.8 n1=1.8, n2=1.2

1.1

1.2 n1=1.2, n2=1.8 n1=1.8, n2=1.2

1.1

0.2

0.4

(a)

0.6

n=1.5

n=1.5

n=1.5

1.0 0.0

n1=1.2, n2=1.8 n1=1.8, n2=1.2

1.1

0.8

1.0 0.0

1.0

0.2

0.4

0.6

0.8

1.0

X

(b)

X

1.0 0.0

0.2

0.4

0.6

0.8

1.0

X

(c)

Fig. 4. Effects of GRIN on steady temperature fields with the changes in surface emissivities under k ¼ 0:1 W m1 K1 ; k ¼ 100 m1 and o ¼ 0:9: (a) 1 ¼ 0:1; 2 ¼ 1:0; (b) 1 ¼ 2 ¼ 0:1; (c) 1 ¼ 2 ¼ 1:0:

1.0

q~ r

qr

1.2

0.70

4.8

0.65

4.6

0.60

4.4 4.2

0.55

q~ r

1.4

0.50 0.8

n1=1.2, n2=1.8 n1=1.8, n2=1.2 n=1.5

0.6 0.0

(a)

0.2

0.4

0.6 X

0.8

1.0

0.45 0.35 0.0

(b)

3.8 3.6

n1=1.2, n2=1.8 n1=1.8, n2=1.2 n=1.5

0.40 0.2

0.4

0.6 X

0.8

n1=1.2, n2=1.8 n1=1.8, n2=1.2 n=1.5

4.0

3.4 3.2 0.0

1.0

(c)

0.2

0.4

0.6

0.8

1.0

X

Fig. 5. Effects of GRIN on steady radiative flux fields with the changes in surface emissivities under k ¼ 0:1 W m1 K1 ; k ¼ 100 m1 and o ¼ 0:9: (a) 1 ¼ 0:1; 2 ¼ 1:0; (b) 1 ¼ 2 ¼ 0:1; (c) 1 ¼ 2 ¼ 1:0:

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375

close to the low-temperature surface to be lower as a result of total reflection in GRIN media, which is more obvious especially for the media with two surfaces having equal emissivities. From Fig. 5, it is seen that the radiative flux level can be heightened by increasing the emissivity of the high-temperature surface as well as low-temperature surface; for the media with surfaces of two equal emissivities, compared with the case of constant refractive index, GRIN can weaken the radiative heat transfer; especially for the case of big emissivities, as seen in Fig. 5c, the difference in the radiative flux fields between the media with n1 ¼ 1:2 and n2 ¼ 1:8 and that with n1 ¼ 1:8 and n2 ¼ 1:2 is slight, while the radiative flux field for the media with constant refractive index of n ¼ 1:5 is much bigger than that with GRIN. Figs. 6 and 7 are plotted to display the influences of GRIN on steady temperature fields with the changes in extinction coefficient. Under the strong scattering, for small extinction coefficient, as seen in Figs. 6a and b, GRIN has little effect on the temperature field and consequently, the temperature differences between media with constant refractive index and GRIN are slight; for big extinction coefficient, as seen in Fig. 6c, the effect of GRIN is great, which causes big temperature differences. From Fig. 7, we can see that the less the extinction coefficient is, the bigger the effect of GRIN on the radiative flux is; for the case of large extinction coefficient, compared to the case of constant refractive index, GRIN can weaken the radiative flux in slab.

1.4

1.4

1.4

1.3

1.3

1.3

1.2

Θ

1.5

Θ

1.5

Θ

1.5

1.2 n1=1.2, n2=1.8 n1=1.8, n2=1.2 n=1.5

1.1 1.0 0.0

0.2

0.4

(a)

0.6

0.8

1.2 n1=1.2, n2=1.8 n1=1.8, n2=1.2 n=1.5

1.1

1.0

1.0 0.0

0.2

0.4

(b)

X

0.6

0.8

n1=1.2, n2=1.8 n1=1.8, n2=1.2 n=1.5

1.1 1.0 0.0

1.0

0.2

0.4

(c)

X

0.6

0.8

1.0

X

Fig. 6. Effects of GRIN on steady temperature fields with the changes in extinction coefficient under k ¼ 0:1 W m1 K1 ; 1 ¼ 0:2; 2 ¼ 1:0 and o ¼ 0:9: (a) k ¼ 1 m1 ; (b) k ¼ 10 m1 ; (c) k ¼ 1000 m1 :

1.95

2.0

1.80

1.8 n1=1.2, n2=1.8 n1=1.8, n2=1.2 n=1.5

1.50 1.35

1.4

1.20

(a)

n1=1.2, n2=1.8 n1=1.8, n2=1.2 n=1.5

1.6

qr

1.65

0.0

0.8 0.7

qr

qr

2.10

0.5

0.4

0.6 X

0.8

0.0

1.0

(b)

n1=1.2, n2=1.8 n1=1.8, n2=1.2 n=1.5

0.4

1.2 0.2

0.6

0.3 0.2

0.4

0.6 X

0.8

1.0

0.0

(c)

0.2

0.4

0.6

0.8

1.0

X

Fig. 7. Effects of GRIN on the steady radiative flux fields with the changes in extinction coefficient under k ¼ 0:1 W m1 K1 ; 1 ¼ 0:2; 2 ¼ 1:0 and o ¼ 0:9: (a) k ¼ 1 m1 ; (b) k ¼ 10 m1 ; (c) k ¼ 1000 m1 :

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The dependency of the steady heat transfer on GRIN coupled with the scattering albedo can be seen from Figs. 8 and 9. For the case of moderate extinction coefficient, the temperature fields are significantly influenced by GRIN, except for the case of very strong scattering; for the case of small scattering albedo, see Fig. 9a, the curve of the radiative flux in media with constant refractive index is higher than that with GRIN. Transient effects of thermal behavior in media with constant and GRIN are reported in Figs. 10 and 11. As the heat transfer process evolving, see Fig. 10, temperature curves ascend, and the difference in temperature between media with different distribution of linear refractive index but with the same mean value of n1 and n2 (n1 pn2 ) diminishes gradually. The temperature curves for the medium with bigger gradient of the refractive index are above those with smaller grades of the refractive index in the area close to high-temperature boundary S 2 ; and in the area close to low-temperature boundary S 1 ; the situation is just the opposite, as shown in Fig. 10. It can be explained as the total reflection occurring inside the slab as a result of the gradient of the refractive index in existence. Compared with the case of smaller refractive index gradient media, for the bigger refractive index gradient media, the location where total reflection of the incident radiation from some direction occurs is closer to the higher temperature

1.4

1.4

1.4

1.3

1.3

1.3

1.2

1.2 n1=1.2, n2=1.8 n1=1.8, n2=1.2 n=1.5

1.1 1.0 0.0

Θ

1.5

Θ

1.5

Θ

1.5

0.2

0.4

(a)

0.6

0.8

1.2 n1=1.2, n2=1.8 n1=1.8, n2=1.2 n=1.5

1.1 1.0 0.0

1.0

0.2

0.4

(b)

X

0.6

0.8

n1=1.2, n2=1.8 n1=1.8, n2=1.2 n=1.5

1.1 1.0 0.0

1.0

0.2

0.4

(c)

X

0.6

0.8

1.0

X

Fig. 8. Effects of GRIN on steady temperature fields with the changes in scattering albedo under k ¼ 0:1 W m1 K1 ; 1 ¼ 0:2; 2 ¼ 1:0 and k ¼ 100 m1 : (a) o ¼ 0:1; (b) o ¼ 0:9; (c) o ¼ 0:99:

2.50 2.25

2.0

1.7

1.8

1.6

1.50 1.25

n1=1.2, n2=1.8 n1=1.8, n2=1.2 n=1.5

1.00 0.75 0.0

(a)

1.5

0.2

0.4

0.6 X

0.8

qr

1.6

1.75

qr

qr

2.00 1.4

1.3 n1=1.2, n2=1.8 n1=1.8, n2=1.2 n=1.5

1.2 1.0

1.0

0.0

(b)

n1=1.2, n2=1.8 n1=1.8, n2=1.2 n=1.5

1.4

0.2

0.4

0.6 X

0.8

1.2 1.1 1.0

1.0 0.0

(c)

0.2

0.4

0.6

0.8

1.0

X

Fig. 9. Effects of GRIN on the steady radiative flux fields with the changes in scattering albedo under k ¼ 0:1 W m1 K1 ; 1 ¼ 0:2; 2 ¼ 1:0 and k ¼ 100 m1 : (a) o ¼ 0:1; (b) o ¼ 0:9; (c) o ¼ 0:99:

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377

steady state

1.4 1.3 3s

Φ

1.2 1.1

2s

1.0

n1=1.2,n2=3.0

0.9

n1=1.6,n2=2.6 n1=2.0,n2=2.2

1s

0.8

n1=n2=2.1 0.7 0.0

0.2

0.4

0.6

0.8

1.0

X Fig. 10. Effects of GRIN on transient temperature distribution under cr ¼ 105 J K1 m3 ; k ¼ 0:1 W m1 K1 ; k ¼ 100 m1 ; 1 ¼ 0:2; 2 ¼ 1:0 and o ¼ 0:9: 7

6

6

5 4

4

qr

qr

5

3 3

n1=1.2,n2=3.0

n1=1.2,n2=3.0

n1=1.6,n2=2.6 2 1 0.0

0.2

0.4

0.6

2

n1=1.6,n2=2.6

n1=2.0,n2=2.2

n1=2.0,n2=2.2

n1=n2=2.1

1

n1=n2=2.1

0.8

1.0

X

(a)

0.0

0.2

0.4

0.6

4.0

n1=1.2,n2=3.0

3.0

n1=1.6,n2=2.6

3.5

n1=2.0,n2=2.2 n1=n2=2.1

qr

qr

2.5

3.0

2.0

2.5

n1=1.2,n2=3.0

2.0

1.5

n1=1.6,n2=2.6 1.5

n1=2.0,n2=2.2

1.0

n1=n2=2.1

0.0

1.0

3.5

4.5

(c)

0.8

X

(b)

0.2

0.4

0.6

X

0.8

1.0

1.0

0.0

(d)

0.2

0.4

0.6

0.8

1.0

X

Fig. 11. Effects of GRIN on transient radiative flux distribution under cr ¼ 105 J K1 m3 ; k ¼ 0:1 W m1 K1 ; k ¼ 100 m1 ; 1 ¼ 0:2; 2 ¼ 1:0 and o ¼ 0:9; (a) 1s: (b) at 2s; (c) at 3s; (d) at steady state.

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boundary, and accordingly the influence of the high-temperature boundary on the radiative heat transfer in the region closer to the higher temperature boundary is bigger and that closer to low-temperature boundary is smaller, which results in the higher temperature fields closer to the higher temperature boundary and the lower temperature fields closer to the lower temperature boundary. The radiative heat flux fields evolving with time is presented in Fig. 11. As time goes on, the radiative flux in the region of the medium close to high-temperature boundary gets small in magnitude, and consequently the curves of the radiative flux become flat, resulting in the drop of the radiative flux level. The bigger the gradient of radiative index is, the easier it is to cause total reflection to happen in media, and the lower the corresponding radiative flux curve is.

6. Conclusions By using the ray-tracing technique in combination with spectral band model and node analysis, a multilayer radiative transfer model of diffuse reflection is established to simulate the radiative transfer in GRIN media. Coupled radiative–conductive heat transfer inside a scattering and a linear refractive index distribution media with diffuse and opaque boundary surfaces is solved. From the results and discussion, main conclusions can be drawn as follows: (a) A linear refractive index can either heighten or lower the temperature in medium compared with the case of a constant refractive index, and the effect is great especially for the media with same surface emissivities; the effect that a linear refractive index can weaken the radiative heat transfer compared with the case of a constant refractive index is obvious especially for the case of big emissivities. (b) Under a small thermal conductivity and a big scattering albedo, the effect of a linear refractive index on temperature fields relates to the extinction coefficient and the effect is slight for small extinction coefficient, but great for big extinction; while the effect on the radiative flux decreases with the increase of extinction coefficient and, for the case of large extinction coefficient, GRIN can weaken the radiative flux in slab. (c) Under a small thermal conductivity and a moderate extinction coefficient, the temperature fields are significantly influenced by GRIN, except for the case of very strong scattering; for the case of small scattering albedo, GRIN can weaken the radiative flux in slab. (d) Under a small thermal conductivity, a moderate extinction coefficient and a big scattering albedo, for the case of the refractive index value of the medium adjacent to high-temperature boundary surface bigger than that adjacent to low-temperature boundary surface and keeping the mean value of the two refractive indexes invariable for different linear refractive index media, the total reflection results in the higher transient temperature fields close to hightemperature boundary surface and the lower transient temperature fields close to lowtemperature for media with the bigger gradient of the refractive index compared with media with the smaller gradient of the refractive index; the transient radiative flux decreases with increasing refractive index gradient because of the total reflection.

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Acknowledgements This research is financially supported by the key program of the National Natural Science Foundation of China (Grant. No. 50336010), the China National Key Basic Research Special Funds (Grant. No. 2003CB214500), and the National Natural Science Foundation of China (Grant. No. 50306004).

Appendix A. Single-layer radiant energy quotient transfer functions Different from specular surface, diffuse surface reflects the radiant energy from some direction back to all the hemispherical space. Based on this analysis, the incident radiant energy from all the hemispherical space can be only traced to deduce the single-layer radiant energy quotient transfer functions. According to Fig. 12, we examine the transfer of the radiant energy emitted by surfaces. According to the Bouguer’s law, the direct radiative transfer coefficients are first given as follows: Z 1 expðkb Lb =mÞm dm, (19a) ðsA sB Þk ¼ ðsB sA Þk ¼ 2 0

Z

1

ðsA vI b Þk ¼ 2 0

Z

expðkb xA I b =mÞ½1  expðkb Dxb =mÞ m dm,

(19b)

expðkb xIþ1b =mÞ½1  expðkb Dxb =mÞ m dm.

(19c)

1

ðsB vI b Þk ¼ 2 0

Using these direct radiative transfer coefficients, by tracing radiant energy emitted by surfaces from beginning to end in the radiative transfer process, single-layer radiant energy quotient transfer functions are obtained: 2 d FA A ¼ rb;bþ1 ðsA sB Þ =ð1  b1 Þ,

(20a)

FBA ¼ ðsA sB Þ=ð1  bd1 Þ,

(20b)

VI

FA b ¼ ½ðsA vI b Þ þ ðsA sB Þrb;bþ1 ðsB vI b Þ =ð1  bd1 Þ,

(20c)

FBB ¼ rb;b1 ðsA sB Þ2 =ð1  bd1 Þ,

(20d) ρb −1,b

A VIb

y1

y3

…

ρb,b −1

Ib

xIAb ∆ xb

I + 1b

x1 B

y2

x2

x3

x4

… ρb,b +1 ρb +1,b

Fig. 12. One-layer radiative transfer model.

xIBb

Lb

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d B FA B ¼ FA ¼ ðsA sB Þ=ð1  b1 Þ, V Ib

FB

¼ ½ðsB vI b Þ þ ðsA sB Þrb;b1 ðsA vI b Þ =ð1  bd1 Þ,

(20e) (20f)

where bd1 ¼ rb;b1 rb;bþ1 ðsA sB Þ2 : Using the same ray-tracing method, other radiant energy quotient transfer functions, such as VJ

VI

B b b FA V I ; FV I ; FV I and FV I can also be deduced. b

b

b

b

Appendix B. Determination of reflectivity and transmissivity If the radiant energy from a layer with a bigger refractive index ðnj Þ projects into that with a smaller refractive index ðni Þ; the reflectivity of the suppositional diffuse semitransparent interface between the two layers is given as [21,3]  ) Z p=2 ( tanðy  jÞ 2 sinðy  jÞ 2 sinðyÞ cosðyÞ dy, (21a) þ rij ¼ tanðy þ jÞ sinðy þ jÞ 0 where j is the angle of refraction and j ¼ arc sinðyni =nj Þ; if the radiant energy projects from the inverse direction, the reflectivity is written as [3] rji ¼ rij n2i =n2j þ ð1  n2i =n2j Þ.

(21b)

The effect of the total reflection at interfaces is considered in the Eq. (21b). The corresponding transmissivity is written as follows: gij ¼ 1  rij .

(22)

References [1] Siegel R. Refractive index effects on transient cooling of a semitransparent radiation layer. J Thermophys Heat Transfer 1995;9(1):55–62. [2] Spuckler CM, Siegel R. Refractive index effects on radiative behavior of a heated absorbing–emitting layer. J Thermophys Heat Transfer 1992;6(4):596–604. [3] Siegel R, Spuckler CM. Refractive index effects on radiation in an absorbing, emitting, and scattering laminated layer. J Heat Transfer 1993;115(1):194–200. [4] Siegel R, Spuckler CM. Variable refractive index effects on radiation in semitransparent scattering multilayered regions. J Thermophys Heat Transfer 1993;7(4):624–30. [5] Tan HP, Luo JF, Xia XL. Transient coupled radiation and conduction in a three-layer composite with semitransparent specular. J Heat Transfer 2002;124(3):470–81. [6] Luo JF, Tan HP, Ruan LM. Refractive index effects on heat transfer in multilayer scattering composite. J Thermophys Heat Transfer 2003;17(3):407–19. [7] Ben Abdallah P, Le Dez V. Temperature field inside an absorbing–emitting semitransparent slab at radiative equilibrium with variable spatial refractive index. JQSRT 2000;65(4):595–608. [8] Ben Abdallah P, Le Dez V. Radiative transfer in a plane parallel slab of anisotropic uniaxial semi-transparent medium. JQSRT 2000;67(1):21–42.

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[9] Ben Abdallah P, Le Dez V, Charette A. Influence of a spatial variation of the thermo-optical constants on the radiative transfer inside an absorbing–emitting semi-transparent sphere. JQSRT 2001;70(3):341–65. [10] Ben Abdallah P, Le Dez V. Radiative flux field inside an absorbing–emitting semi-transparent slab with variable spatial refractive index at radiative conductive coupling. JQSRT 2000;67(2):125–37. [11] Huang Y, Xia XL, Tan HP. Radiative intensity solution and thermal emission analysis of a semitransparent medium layer with a sinusoidal refractive index. JQSRT 2002;74(2):217–33. [12] Huang Y, Xia XL, Tan HP. Temperature field of radiative equilibrium in a semitransparent slab with a linear refractive index and gray walls. JQSRT 2002;74(2):249–61. [13] Xia XL, Huang Y, Tan HP. Thermal emission and volumetric absorption of a graded index semitransparent medium layer. JQSRT 2002;74(2):235–48. [14] Tan HP, Huang Y, Xia XL. Solution of radiative heat transfer in a semitransparent slab with an arbitrary refractive index distribution and diffuse gray boundaries. Int J Heat Mass Transfer 2003;46(11):2005–14. [15] Xia XL, Huang Y, Tan HP, Zhang XB. Simultaneous radiation and conduction heat transfer in a graded index semitransparent slab with gray boundaries. Int J Heat Mass Transfer 2002;45(13):2673–88. [16] Liu LH. Discrete curved ray-tracing method for radiative transfer in an absorbing–emitting semitransparent slab with variable spatial refractive index. JQSRT 2004;83(2):223–8. [17] Tan HP, Luo JF, Xia XL, Yu QZ. Transient coupled heat transfer in multilayer composite with one specular boundary coated. Int J Heat Mass Transfer 2003;46(4):731–47. [18] Tan HP, Luo JF, Ruan LM, Yu QZ. Transient coupled heat transfer in a multilayer composite with opaque specular surfaces and semitransparent specular interfaces. Int J Thermal Sciences 2003;42(2):209–22. [19] Tan HP, Luo JF, Xia XL. Transient coupled radiation and conduction in a three-layer composite with semitransparent specular interfaces and surfaces. J Heat Transfer 2002;124(3):470–81. [20] Patankar SV. Numerical heat transfer and fluid flow. New York: McGraw-Hill Book Company; 1980. [21] Siegel R, Howell JR. Thermal radiation heat transfer. 4th ed. New York, London: Taylor & Francis; 2002. p. 77–84.