coupled radiation–conduction heat transfer in an anisotropically scattering slab with mixed...

Journal of Quantitative Spectroscopy &Radiative Transfer 83 (2004) 667–698

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Coupled radiation–conduction heat transfer in ananisotropically scattering slab with mixed boundaries

He-Ping Tana, Hong-Liang Yia ;∗, Hao-Chun Zhanga, Ping-Yang Wangb,Timothy W. Tongc

aSchool of Energy Science and Engineering, Harbin Institute of Technology, 92, West Dazhi Street,Harbin 150001, People’s Republic of China

bShanghai Jiao Tong University, Shanghai, People’s Republic of ChinacSchool of Engineering and Applied Science, the George Washington University, Washington, DC 20052, USA

Received 1 November 2002; accepted 27 February 2003

Abstract

Ray tracing method combined with Hottel’s zonal method is used to establish radiation transfer model inanisotropic scattering media. The transmission progress of radiation intensity in the absorbing, emitting andanisotropically scattering medium is divided into two sub-progresses: emitting–attenuating–re<ecting progressand absorbing–scattering progress. For the medium with both surfaces being semitransparent and mirror-like,radiative transfer coe>cients are developed. Energy equation of transient coupled radiation and conductionheat transfer is solved by the fully implicit control-volume method. In this paper, it needs only to dispersespatial location while spatial solid angle is integrated directly. On this basis, eBects of conduction–radiationparameter, spectral refractive index and scattering phase functions etc., on the temperature Celd and radiativeheat <ux Celd within an anisotropic scattering medium are examined. The results show that: the linear part inlinear or nonlinear scattering phase function can produce great diBerence of the temperature Celd and radiativeheat <ux Celd between anisotropic scattering medium and isotropic scattering medium.? 2003 Elsevier Ltd. All rights reserved.

Keywords: Coupled radiation and conduction heat transfer; Semitransparent; Ray tracing; Radiative transfer coe>cients;Anisotropic scattering; Phase functions

1. Introduction

The semitransparent media containing particles has wide range applications in the engineering,for example, the space droplet radiator [1]; the thermal insulting layer in the surface of the turbine

∗ Corresponding author.E-mail addresses: hongliang [email protected] (H.-L. Yi), king [email protected] (Hao-Chun Zhang).

0022-4073/04/$ - see front matter ? 2003 Elsevier Ltd. All rights reserved.doi:10.1016/S0022-4073(03)00112-2

mailto:[email protected]

mailto:[email protected]

668 H.-P. Tan et al. / Journal of Quantitative Spectroscopy & Radiative Transfer 83 (2004) 667–698

Nomenclature

RTC radiative transfer coe>cientRTF radiative transfer functionAk;Ti

∫I�k

Ib;�(Ti) d�=∫∞0 Ib;�(Ti) d�, fractional spectral emissive power of spectral

band I�k at nodal temperature TiC unit heat capacity, J m−3 K−1

FSz(z), radiative transfer functions without regard to anisotropic scattering (Eqs. (7))FT z(z)FDp(z), radiative transfer functions considering anisotropic scattering (Eqs. (11)–(13))FRp(z),FRRp(z)H1; H2 convection–radiation parameters, H1 = h1=�T 3

rf and H2 = h2=�T 3rf , respectively

h1; h2 convective heat transfer coe>cients at surfaces of S1 and S2, respectively,W m−2 K−1

k phonic thermal conductivity, W m−1 K−1

L thickness of slab, mN k=(4�T 3

rfL), conduction–radiation parameterNB total number of spectral bandsNM total number of control volumes of slabnm;k spectral refractive index of slabqr heat <ux of radiation, W m−2

qr dimensionless heat <ux of radiation, qr=(4�T 4rf )

Su; Sv black surfaces representing the surroundings, S−∞ or S+∞, respectively(SuSv)k ; radiative transfer coe>cients of black surface vs. black surface, black surface vs.(SuVj)k ; volume, and volume vs. volume without considering scattering relative to the(ViVj)k spectral band I�k

[SuSv]k ; radiative transfer coe>cients of black surface vs. black surface, black surface[SuVj]k ; vs. volume, and volume vs. volume considering multiple scattering in[ViVj]k anisotropic media relative to the spectral band I�k

(ViVj)fk ; radiation transfer coe>cients of volume vs. volume and surface vs. volume

(ViVj)bk ; considering incident direction of Vj

(SuVj)fk ;

(SuVj)bk

(ViVj);k ; radiation transfer coe>cients of volume vs. volume and volume vs. surface

(ViVj)fbk ; considering scattering direction

(ViVj)bfk ,

(ViVj)bbk ;

(ViSv)ftk ;

(ViSv)btk

S1; S2 boundary surfaces (Fig. 1)S−∞; S+∞ black surfaces representing the surroundings (Fig. 1)

H.-P. Tan et al. / Journal of Quantitative Spectroscopy & Radiative Transfer 83 (2004) 667–698 669

T absolute temperature, KT dimensionless absolute temperature, T=TrfTg1; Tg2 gas temperatures for convection at X = 0 and 1, KTrf reference temperature, KT0 uniform initial temperature, Kt physical time, st∗ dimensionless time, (4�T 3

rf =CL)t

t∗c dimensionless time when diBerence of dimensional temperatures gets the largestvalue

t∗s steady-state dimensionless timeVi control-volume iX dimensionless coordinate in direction across layer, X = x=Lx1; i normal distance of ray transfer between both subscripts, mz the normal distance of ray transfer, m$k spectral absorption coe>cient of slab, m−1

It, It time interval and dimensionless time interval, respectivelyIx spacing interval between two nodes, mIX dimensionless spacing interval between two nodes, Ix=L% amount of the control-volume per optical thickness, % = NM=&o' 1− !) dimensionless temperature, (T − T0)=(Trf − T0)*c critical angle, arcsin(1=nm)*i; * incident, scattering included angle between the ray and x-axis+k spectral extinction coe>cient of slab, m−1

,(*) re<ectivity of surface at angle *� Stefan–Boltzmann constant, =5:6696× 10−8 W m−2 K−4

�s; k spectral scattering coe>cient of slab, m−1

&o +L, optical thickness of slab. scattering phase function.r

i radiative heat source of Vi

.ri dimensionless radiative heat source of Vi; .r

i =(4�T 4rf )

!k �s; k =(�s; k + $k), spectral single-scattering albedo of slab

Subscripts

a absorbed quotient in the overall radiative heat transfer coe>cientie; iw right and left interface of Vi (Fig. 1)k relative to spectral band I�k

s scattered quotient in the overall radiative heat transfer coe>cient1; 2 relative to boundary surfaces S1 and S2, respectively−∞;+∞ relative to black surfaces S−∞ and S+∞, respectivelyL; L component for parallel and perpendicular polarization, respectively


Superscripts

1st, 2nd the Crst-order scattering, the second-order scattering, respectivelyb, f, t incidence radiation from negative, positive and both direction relative to the

x-axis, respectively, for RTC onlyh; q backward scattering and forward scattering relative to the incident direction,

respectivelym; m + 1 time stepp q; hr radiations specular re<ection

blades of the aircraft engine, zirconium oxide and ceramic [2–4]; gases containing ash particlesin the furnace of coal burning boiler [5] and etc. Considering anisotropic scattering in the mediacontaining particles, the distribution of radiant intensity is related to the incidence direction as wellas the scattering direction, as a result, the radiation source term is more di>cult to solve thanthat for an isotropic one, and there is fewer research on the anisotropic scattering [6–12] than thaton the isotropic scattering. Busbridge [6] who adopted analyzing method and numerical method andMaruyama [12] who adopted the radiation element method combined with ray emission model (REM)analyzed the one-dimension radiation heat transfer in an anisotropically scattering gray media withblack surfaces and temperature boundary conditions presented. Siewert [10] combined PN methodwith Hermite cubic spline and gained an iterative solution to study one-dimension steady coupledheat exchange in the absorbing, emitting and anisotropically scattering media with the isothermalgray walls under the condition of mirror and diBuse re<ection respectively. Machali et al. [11]studied the radiation heat transfer in one-dimensional anisotropically scattering media under the Crsttype of boundary condition by projection method (PM) and presented the forward, backward and thetotal heat <ux under the condition of linear anisotropically scattering and Rayleigh mode scatteringrespectively.

Since the complexity of the anisotropic scattering, the isotropic scattering assumption has com-monly been adopted in the engineering computation. As we know, semitransparent materials withsome particles behave anisotropically in nature. So, the isotropic scattering treatment would no doubtcause errors. In order to make results more accurate, anisotropic scattering must be considered inradiative heat transfer. In this paper, ray tracing method combined with Hottel’s zonal method [13]has been employed to set up radiation transfer model in the absorbing, emitting and anisotropicallyscattering media under the condition of bilaterally semitransparent boundaries. The radiative transfercoe>cients for the one-dimensional anisotropially scattering media with both surfaces being semi-transparent and mirror-like is developed, which helps to calculate the radiative heat source term,and transient energy equation for the coupled radiative and conductive heat transfer is solved bythe fully implicit control-volume method under the mixed boundary conditions. In<uences of theoptical thickness, scattering albedo, and scattering phase functions etc., on coupled heat transfer areinvestigated in the paper.


Fig. 1. Discrete model of space zone.

2. Discrete model of space zone

As Fig. 1 shows, the tabular media is located between two black surfaces S−∞ and S+∞ whichare representative of the surroundings with their temperatures being T−∞ and T+∞, respectively.The interfaces of the slab are both semitransparent. There are NM + 2 nodes along the thicknessdirection, indicated by i. Here, i = 1 represents surface S1 and i = NM + 2 represents surface S2.The variation of the extinction coe>cient +k , scattering albedo !k and index of refraction nm;k withthe wavelength can be expressed by a group of rectangular spectral bands. The total number of thespectral bands is NB and the subscript ‘k’ denotes the kth zone of the spectral bands.Within the time interval of t and t + It, the fully implicit discrete energy equation of control

volume i in dimensionless form for the transient coupled radiation–conduction heat transfer is writtenas

IXT m+1

i − T mi

It=

N m+1ie (T m+1

i+1 − T m+1i )− N m+1

iw (T m+1i − T m+1

i−1 )IX

+ .r;m+1i : (1)

By the nodal analysis method [14], the dimensionless radiation source term of the control-volume ican be expressed as

.ri = qrie(T )− qriw(T ) = qrie(T )− qr(i−1)e(T ): (2)

When the surfaces of the two sides S1 and S2 are semitransparent, radiation heat <ux qrie throughinterface “ie” (between node i and node i + 1) can be expressed as [15]

qrie =14

NB∑k=1

i∑j=2

([S+∞Vj]kAk;T+∞ T 4+∞ − n2

m;k[VjS+∞]kAk;Tj T4j )

+NM+1∑j=i+1

i∑l=2

(n2m;k[VjVl]kAk;Tj T

4j − n2

m;k[VlVj]kAk;Tl T4l )


+NM+1∑j=i+1

(n2m;k[VjS−∞]kAk;Tj T

4j − [S−∞Vj]kAk;T−∞ T 4

−∞)

+ ([S+∞S−∞]kAk;T+∞ T 4+∞ − [S−∞S+∞]kAk;T−∞ T 4

−∞)

}; (3)

where, i = 2 ∼ NM + 1.When i = 1 and i = NM + 2, the dimensionless radiative heat <uxes qrS1 and qrS2 of the interfaces

can be written as

qrS1 = qr1e =14

NB∑k=1

([S+∞S−∞]kAk;T+∞ T 4

+∞ − [S−∞S+∞]kAk;T−∞ T 4−∞)

+NM+1∑

j=2

(n2m;k[VjS−∞]kAk;Tj T

4j − [S−∞Vj]kAk;T−∞ T 4

−∞)

; (4a)

qrS2 = qr(Mt+1)e =14

NB∑k=1

[S+∞S−∞]kAk;T+∞ T 4

+∞ − [S−∞S+∞]kAk;T−∞ T 4−∞

+NM+1∑

j=2

([S+∞Vj]kAk;T+∞ T 4+∞ − n2

m;k[VjS+∞]kAk;Tj T4j )

: (4b)

The discrete boundary conditions for two semitransparent surfaces can be written as

8N1e(T 2 − T S1) = H1(T S1 − T g1)IX; (5a)

8N(Mt+1)e(T S2 − T Mt+1) = H2(T g2 − T S2)IX: (5b)

3. Derivations of RTCs

The RTC of element (surface or control-volume) i to element j is deCned as the quotient ofthe radiative energy that is received by element j in the transfer process of the radiative energyemitted by element i. The radiative transfer process in scattering medium can be divided into twosub-processes [16]. The Crst sub-process is emitting–attenuating–re<ecting sub-process and RTCs aredenoted by (SiSj)sk ; (SiVj)sk ; (ViVj)sk . The second sub-process is multiple absorbing–multiple scatteringsub-process and RTCs are denoted by [SiSj]sk ; [SiVj]sk ; [ViVj]sk . For convenience, in the deduction ofRTCs, subscript ‘k’ of variables denoting spectral properties and superscript ‘s’ of variables denotingspecular re<ection are omitted. (S−∞Vj) and [S−∞Vj] are used as the examples to deduce all RTCsin the paper.


In the derivation of RTCs, the re<ection of a specular surface must be considered. Fresnel’sre<ection law is employed to calculate re<ectivities. When a ray enters a medium with its refrac-tive index being n2 at an incident angle * from a medium with its refractive index being n1, thedirectional–hemispherical re<ectivities of the interface are given by [17]

,‖(*) ={(n2=n1)2 cos * − [(n2=n1)2 − sin2 *]1=2

(n2=n1)2 cos * + [(n2=n1)2 − sin2 *]1=2

}2

; (6a)

,⊥(*) ={[(n2=n1)2 − sin2 *]1=2 − cos *[(n2=n1)2 − sin2 *]1=2 + cos *

}2

; (6b)

where, ,‖(*) is the parallel polarization component of specular re<ection and ,⊥(*) is the perpen-dicular polarization component of specular re<ection. n1 is refractive index of media on the incidentside, and n2 is refractive index or media on the refractive side; * is incident angle. When a rayenters a medium from gas or vacuum, total re<ection does not occur. So the re<ectivity in all angularincident directions can be calculated according to Eqs. (6), When a ray enters gas or vacuum from amedium with its refractive index bigger than one: if incident angle is within 06 * ¡ *c, (*c meanscrititcal angle for the total re<ection, arcsin(1=nm)) total re<ection will not occur, and the re<ectivitycan be still calculated according to Eqs. (6); if incident angle is within *c6 *6 4=2, total re<ectionwill occur and ,‖(*) = ,⊥(*) = 1.In the paper, two specular re<ection components of radiant intensity are traced respectively. By

this method, RTCs for the parallel component and for the perpendicular component are deduced.Then, total RTCs can be also gained.

3.1. RTCs for emitting–attenuating–re<ecting sub-process

Without regard to the eBect of scattering, the emitting–attenuating–re<ecting sub-process onlyconsiders the whole process in which radiant energy emitted or transmitted from some control-volumeis multiply re<ected by surfaces and is multiply absorbed by control-volumes in a medium until iteventually attenuates to zero. In the process, the deduction of RTCs can be divided into three steps:(1) considering surface transmission and multiple re<ection of radiant intensity; (2) radiative transferfunctions without regard to anisotropic scattering, and (3) RTCs considering the incident directionof radiant intensity.

3.1.1. Considering surface transmission and multiple re<ection of radiant intensityA ray emitted from environment S−∞ penetrates the surface S1 and reaches the control-volume j

by two paths as shown in Fig. 2:

(a) A ray penetrates the surface S1 and directly reaches the control-volume j, and its geometricprogression as shown in Fig. 2a: the ray penetrates through the control-volume j and reachesthe surface S2, and after being re<ected, it reaches the surface S1, and after being re<ected again,it reaches j again, and again and again, until it attenuates to zero.

(b) A ray penetrates the surface S1 and reaches the surface S2, and after being re<ected, it reachesthe control-volume j, and its geometric progression as shown in Fig. 2b: the ray penetratesthrough the control-volume j and reaches the surface S1, and after being re<ected, it reaches


Fig. 2. Sketch of two transfer paths from S−∞ to Vj .

the surface S2, and after being re<ected again, it reaches j again, and again and again, until itattenuates to zero.

3.1.2. Radiative transfer functions without regard to anisotropic scatteringAccording to Fig. 2, regardless of anisotropic scattering, two radiative transfer functions FT z(z)

and FSz(z) are deCned as

FT z(z) = 2∫ 4=2

0

(1− ,1(*i)) exp(−+z=cos *r)sin *i cos *i1− ,1(*i),2(*i) exp(−2&0=cos *r)

d*i; (7a)

FSz(z) = 2∫ 4=2

0

(1− ,1(*i)),2(*i) exp(−+z=cos *r)sin *i cos *i1− ,1(*i),2(*i) exp(−2&0=cos *r)

d*i: (7b)

FT z(z) is used to describe the radiative transfer as shown in Fig. 2a; FSz(z) is used to describe theradiative transfer as shown in Fig. 2b. In Eqs. (7), &0=+L is optical thickness; the variable z is normaltransfer distance; *i and *r are incident angle and refractive angle, respectively. The relation betweenthem can be described by the Snell’s refractive law: sin *i=sin *r = nr=ni, where ni is the refractiveindex of the medium on the incident side and nr is that on the refractive side. For parallel componentsof radiant intensity, FT z(z) = FT z

‖(z), FSz(z) = FSz‖(z), ,1(*i) = ,2(*i) = ,‖(*i); for perpendicular

components of radiant intensity, FT z(z) = FT z⊥(z), FSz(z) = FSz

⊥(z), ,1(*i) = ,2(*i) = ,⊥(*i).

3.1.3. RTCs considering incidence direction of radiant intensityThe energy from environment S−∞ which is radiated into control-volume j is divided into two

parts: one comes from the positive direction of x-axis (forward direction), which is denoted by thesymbol (S−∞Vj)f and it is deCned as the forward incidence RTC; the other comes from the negativedirection of x-axis (backward direction), which is denoted by the symbol (S−∞Vj)b and it is deCnedas the backward incidence RTC. According to Fig. 2 and using the radiative transfer functions givenabove, (S−∞Vj)f , (S−∞Vj)b and total incidence RTC (S−∞Vj) can be obtained as below:

(S−∞Vj)f‖ = FT z‖(x1; j)− FT z

‖(x1; j+1); (8a)

(S−∞Vj)f⊥ = FT z⊥(x1; j)− FT z

⊥(x1; j+1); (8b)

(S−∞Vj)f = 12 [(S−∞Vj)f‖ + (S−∞Vj)f⊥]; (8c)


Fig. 3. Sketch of four transfer paths from Vl to Vj .

(S−∞Vj)b‖ = FSz‖(L + x2; j+1)− FSz

‖(L + x2; j); (9a)

(S−∞Vj)b⊥ = FSz⊥(L + xz; j+1)− FSz

⊥(L + xz; j); (9b)

(S−∞Vj)b = 12 [(S−∞Vj)b‖ + (S−∞Vj)b⊥]; (9c)

(S−∞Vj) = (S−∞Vj)f + (S−∞Vj)b: (10)

3.2. RTCs for multiple absorbing–multiple scattering sub-process

The multiple absorbing–multiple scattering sub-process considers the redistributing of the radiantenergy quotient in an anisotropic scattering medium. The deduction of RTC [S−∞Vj] includes threesteps: (1) radiative transfer functions considering anisotropic scattering; (2) RTCs considering scat-tering direction, and (3) RTCs considering the multiple-scattering eBect and the redistribution ofradiant energy in a medium.

3.2.1. Radiative transfer functions considering anisotropic scatteringUnit radiant energy emitted from the environment S−∞ penetrates the surface S1 and enters a

semitransparent medium. According to transfer mechanism of the radiant intensity, the radiant inten-sity gets to Vj in two diBerent ways. One is shown in Fig. 2: it directly reaches Vj without beingscattered. The other is shown in Fig. 4: it Crstly reaches Vl and after being scattered by Vl, (itreaches Vl2 and after being scattered by Vl2 (it reaches Vl3 and after being scattered by Vl3 ...)) itreaches Vj. For an anisotropic scattering medium, the distribution of radiation intensity depends onnot only its transfer paths, but also its transfer directions. So, in order to calculate the RTC [S−∞Vj],the RTC (VlVj) considering scattering direction must be Crstly derived.

Radiant energy emitted from l is transferred to j by four paths as showed in Fig. 3:

(a) The radiative energy emitted from the control-volume l directly reaches the control-volume jand its geometric progression: energy travels from j to surface S2, and after being re<ected, itreaches surface S1, and after being re<ected again, it reaches j again, and again and again, untilit attenuates to zero (see Fig. 3a).


(b) The radiative energy emitted from the control-volume l is transferred to surface S2, and afterbeing re<ected, it reaches the control-volume j and its geometric progression: energy travelsfrom j to surface S1, after being re<ected, it reaches surface S2, and after being re<ected again,it reaches j again, and again and again, until it attenuates to zero (see Fig. 3b).

(c) The radiative energy emitted from the control-volume l is transferred to surface S1, and afterbeing re<ected, it reaches the control-volume j and its geometric progression: energy travelsfrom j to surface S2, after being re<ected, it reaches surface S1, and after being re<ected again,it reaches j again, and again and again, until it attenuates to zero (see Fig. 3c).

(d) The radiative energy emitted from the control-volume l is transferred to surface S1, and af-ter being re<ected, it reaches the surface S2, and after being re<ected again, it reaches thecontrol-volume j and its geometric progression: energy travels from j to surface S1, after beingre<ected, it reaches surface S2, and after being re<ected again, it reaches j again, and again andagain, until it attenuates to zero (see Fig. 3d).

For one-dimensional problem, scattering of the radiative intensity is divided into forward scat-tering and backward scattering with regard to incident direction. Anisotropic scattering character ofa medium is described by phase function .(*; *i), where * is scattering angle and *i is incidentangle. If * falls in the range (06 *6 4=2), .(*; *i) is denoted by .q(*; *i) deCned as the forwardscattering phase; if * falls in the range (46 *6− 4=2), .(*; *i) is denoted by .h(*; *i) deCned asthe backward scattering phase. According to Fig. 3, the radiative transfer functions (RTFs) FD(z),FR(z) and FRR(z) considering anisotropic scattering within a medium are given as follows:

FDp(z) = 2∫ *c

0

exp(−+z=cos *i) sin *i cos *i (1=4=2)∫

5 .p(*; *i) d*1− ,1(*i),2(*i) exp(−2&0=cos *i)

d*i

+ 2∫ 4=2

*c


5 .p(*; *i) d*1− exp(−2&0=cos *i)

d*i; (11)

FRp(z) = 2∫ *c

0

,(*i) exp(−+z=cos *i) sin *i cos *i (1=4=2)∫

5 .p(*; *i) d*1− ,1(*i),2(*i) exp(−2&0=cos *i)

d*i

+ 2∫ 4=2

*c


5 .p(*; *i) d*1− exp(−2&0=cos *i)

d*i; (12)

FRRp(z) = 2∫ *c

0

,1(*i),2(*i) exp(−+z=cos *i) sin *i cos *i (1=4=2)∫

5 .p(*; *i) d*1− ,1(*i),2(*i) exp(−2&0=cos *i)

d*i

+ 2∫ 4=2

*c


5 .p(*; *i) d*1− exp(−2&0=cos *i)

d*i; (13)

where the superscript p denotes q, forward scattering, or h, backward scattering, relative to incidencedirection. When p denotes q, the integral limit 5 is within (06 *6 4=2) and when p denotes h,5 is in (46 *6−4=2). FD(z) is used to describe the radiative transfer paths as shown in Fig. 3a:


not being re<ected by any surface, radiant intensity reaches Vj for the Crst time; FR(z) is used todescribe the radiative transfer paths as shown in Fig. 3b or c: radiant intensity reaches Vj for theCrst time, after being re<ected by S2 or S1; FRR(z) is used to describe the radiative transfer pathsas shown in Fig. 3d: radiant intensity reaches Vj for the Crst time, after being re<ected by S2 andS1. Corresponding to the two specular re<ection components, FDp(z), FRp(z) and FRRp(z) all havetwo forms.

3.2.2. RTCs considering scattering directionUsing radiative transfer functions given above, RTC (VlVj) considering scattering direction can

be derived. For the convenience of deductions, according to radiative transfer paths shown in Fig.3, and considering the energy conservation relation, such expressions are Crstly given as follows:

GDp = FDp(xl+1; j)− FDp(xl; j)− FDp(xl+1; j+1) + FDp(xl; j+1); (14)

GR2p = FRp(xl+1;2 + x2; j+1)− FRp(xl;2 + x2; j+1)− FRp(xl+1;2 + x2; j) + FRp(xl;2 + x2; j); (15)

GR1p = FRp(xl;1 + x1; j)− FRp(xl+1;1 + x1; j)− FRp(xl;1 + x1; j+1) + FRp(xl+1;1 + x1; j+1); (16)

When l6 j,

GR1R2p =FRRp(xl;1 + L + x2; j+1)− FRRp(xl+1;1 + L + x2; j+1)

−FRRp(xl;1 + L + x2; j) + FRRp(xl+1;1 + L + x2; j); (17)

When l ¿ j,

GR2R1p =FRRp(xl+1;2 + L + x1; j)− FRRp(xl;2 + L + x1; j)

−FRRp(xl+1;2 + L + x1; j+1) + FRRp(xl;2 + L + x1; j+1) (18)

In the following derivation, two superscripts are used in (VlVj) to describe incident directions ofVl and Vj relative to x-axis. The Crst superscript means the direction of incident radiation into Vl

and the second one means the direction of incident radiation into Vj. Superscript f denotes positiveincidence and superscript b denotes negative incidence. If the second superscript is ‘t’, it denotesthe total radiative energy quota absorbed by the second control-volume j:

(VlVj)ft = (VlVj)B + (VlVj)fb; (19a)

(VlVj)bt = (VlVj)bf + (VlVj)bb: (19b)

(1) l ¡ j, for the positive incidence (see Fig. 4a)

(VlVj)B‖ = GDq‖ + GR1h

‖;

(VlVj)B⊥ = GDq⊥ + GR1h

⊥;

(VlVj)B = 12 [(VlVj)B‖ + (VlVj)B⊥]; (20a)


Fig. 4. Sketch of transfer direction from Vl scattered to Vj: (a) l ¡ j, positive incidence, (b) l ¡ j, negative incidence,(c) l= j, positive incidence, (d) l= j, negative incidence, (e) l ¿ j, positive incidence, and (f) l ¿ j, negative incidence.

(VlVj)fb‖ = GR2q‖ + GR1R2h

‖;

(VlVj)fb⊥ = GR2q⊥ + GR1R2h

⊥;

(VlVj)fb = 12 [(VlVj)fb‖ + (VlVj)fb⊥]: (20b)


For the negative incidence (see Fig. 4b)

(VlVj)bf‖ = GDh‖ + GR1q

‖;

(VlVj)bf⊥ = GDh⊥ + GR1q

⊥;

(VlVj)bf = 12 [(VlVj)bf‖ + (VlVj)bf⊥ ]; (21a)

(VlVj)bb‖ = GR2h‖ + GR1R2q

‖;

(VlVj)bb⊥ = GR2h⊥ + GR1R2q

⊥;

(VlVj)bb = 12 [(VlVj)bb‖ + (VlVj)bb⊥ ]: (21b)

(2) l = j, for the positive incidence (see Fig. 4c)

(VlVj)B‖ = GR1h‖ + GR2R1q

‖;

(VlVj)B⊥ = GR1h⊥ + GR2R1q

⊥;

(VlVj)B = 0:25FV + 12 [(VlVj)B‖ + (VlVj)B⊥]; (22a)

(VlVj)fb‖ = GR2q‖ + GR1R2h

‖;

(VlVj)fb⊥ = GR2q⊥ + GR1R2h

⊥;

(VlVj)fb = 0:25FV + 12 [(VlVj)fb‖ + (VlVj)fb⊥]: (22b)

For the negative incidence (see Fig. 4d)

(VlVj)bf‖ = GR1q‖ + GR2R1h

‖;

(VlVj)bf⊥ = GR1q⊥ + GR2R1h

⊥;

(VlVj)bf = 0:25FV + 12 [(VlVj)bf‖ + (VlVj)bf⊥ ]; (23a)

(VlVj)bb‖ = GR2h‖ + GR1R2q

‖;

(VlVj)bb⊥ = GR2h⊥ + GR1R2q

⊥;

(VlVj)bb = 0:25FV + 12 [(VlVj)bb‖ + (VlVj)bb⊥ ]: (23b)

where, FV = 4+Ix − 2[1 − 2E3(+Ix)], is the direct exchange area of volume l vs. volumel [14].


(3) l ¿ j, for the positive incidence (see Fig. 4e)

(VlVj)B‖ = GR1h‖ + GR2R1q

‖;

(VlVj)B⊥ = GR1h⊥ + GR2R1q

⊥;

(VlVj)B = 12 [(VlVj)B‖ + (VlVj)B⊥]; (24a)

(VlVj)fb‖ = GDh‖ + GR2q

‖;

(VlVj)fb⊥ = GDh⊥ + GR2q

⊥;

(VlVj)fb = 12 [(VlVj)fb‖ + (VlVj)fb⊥]: (24b)

For the negative incidence (see Fig. 4f)

(VlVj)bf‖ = GR1q‖ + GR2R1h

‖;

(VlVj)bf⊥ = GR1q⊥ + GR2R1h

⊥;

(VlVj)bf = 12 [(VlVj)bf‖ + (VlVj)bf⊥ ]; (25a)

(VlVj)bb‖ = GDq‖ + GR2h

‖;

(VlVj)bb⊥ = GDq⊥ + GR2h

⊥;

(VlVj)bb = 12 [(VlVj)bb‖ + (VlVj)bb⊥ ]: (25b)

From Eqs. (20)–(25), each (VlVj) considering scattering direction includes two parts: forwardscattering denoted by the superscript q and backward scattering denoted by the superscript h.

3.2.3. RTCs considering multi-scattering e;ect and redistribution of radiative energy in a mediumThe transfer process of radiative energy, which can be absorbed and scattered many times until

it attenuated to zero, is further traced. Before that, RTCs must be normalized as follows:

(ViVj)∗ftk = (ViVj)ftk =(4+kIx); (ViVj)∗btk = (ViVj)btk =(4+kIx);

(ViSu)∗fk = (ViSu)fk =(4+kIx); (ViSu)∗bk = (ViSu)bk =(4+kIx);

(SuVi)∗k = (SuVi)k =9u;k ; (SuSv)∗k = (SuSv)k =9u;k ;

where, Su; Sv = S−∞, or S+∞; i; j = 2 ∼ NM + 1 and the superscript “∗” indicates the normalizedparameter.


The detailed derivation of [S−∞Vj] is given here. For simplicity, when n¿ 3, two functions aredeCned as follows:

H (Vln+1Vln)∗ft =

NM+1∑ln=2

[(Vln+1Vln)∗BH (VlnVln−1)

∗ft + (Vln+1Vln)∗fbH (VlnVln−1)

∗bt]; (26a)

H (Vln+1Vln)∗bt =

NM+1∑ln=2

[(Vln+1Vln)∗bfH (VlnVln−1)

∗ft + (Vln+1Vln)∗bbH (VlnVln−1)

∗bt]: (26b)

If n=2, and the RTC is for the control-volume to the control-volume, i.e. (ViVj)∗nth, or the surfaceto the control-volume, i.e. (SuVj)∗nth, then:

H (Vl3Vl2)∗ft =

NM+1∑l2=2

[(Vl3Vl2)∗B (Vl2Vj)∗ft + (Vl3Vl2)

∗fb(Vl2Vj)∗bt]; (27a)

H (Vl3Vl2)∗bt =

NM+1∑l2=2

[(Vl3Vl2)∗bf (Vl2Vj)∗ft + (Vl3Vl2)

∗bb(Vl2Vj)∗bt]: (27b)

If n = 2, and the RTC is for the control-volume to the surface, i.e. (ViSv)∗nth, or the surface to thesurface, i.e. (SuSv)∗nth, then

H (Vl3Vl2)∗ft =

NM+1∑l2=2

[(Vl3Vl2)∗B (Vl2Sv)∗f + (Vl3Vl2)

∗fb(Vl2Sv)∗b]; (28a)

H (Vl3Vl2)∗bt =

NM+1∑l2=2

[(Vl3Vl2)∗bf (Vl2Sv)∗ft + (Vl3Vl2)

∗bb(Vl2Sv)∗bt]: (28b)

(1) After the Crst-order scattering

[S−∞Vj]∗1sta = (S−∞Vj)∗'; [S−∞Vj]∗1sts = (S−∞Vj)∗!:

(2) After the second-order scattering

[S−∞Vj]∗2nda = [S−∞Vj]∗1sta +NM+1∑l2=2

[(S−∞Vl2)∗f (Vl2Vj)∗ft + (S−∞Vl2)

∗b(Vl2Vj)∗bt]!';

[S−∞Vj]∗2nds =NM+1∑l2=2

[(S−∞Vl2)∗f (Vl2Vj)∗ft + (S−∞Vl2)

∗b(Vl2Vj)∗bt]!2:

(3) After the third-order scattering

[S−∞Vj]∗3rda = [S−∞Vj]∗2nda

+NM+1∑l3=2

{(S−∞Vl3)

∗fNM+1∑l2=2

{(Vl3Vl2)∗B (Vl2Vj)∗ft + (Vl3Vl2)

∗fb(Vl2Vj)∗bt}


+ (S−∞Vl3)∗b

NM+1∑l2=2

{(Vl3Vl2)∗bf (Vl2Vj)∗ft + (Vl3Vl2)

∗bb(Vl2Vj)∗bt}}

!2'

= [S−∞Vj]∗2nda +NM+1∑l3=2

{(S−∞Vl3)∗fH (Vl3Vl2)

∗ft + (S−∞Vl3)∗bH (Vl3Vl2)

∗bt}!2';

[S−∞Vj]∗3rds =NM+1∑l3=2



∗bt}!3:

(4) After the fourth-order scattering

[S−∞Vj]∗4tha = [S−∞Vj]∗3rda +NM+1∑l4=2

{(S−∞Vl4)

∗fNM+1∑l3=2

{(Vl4Vl3)

∗B

×NM+1∑l2=2


∗fb(Vl2Vj)∗bt}

+ (Vl4Vl3)∗fb

NM+1∑l2=2


∗bb(Vl2Vj)∗bt}}

+ (S−∞Vl4)∗b

×NM+1∑l3=2

{(Vl4Vl3)

∗bfNM+1∑l2=2


∗fb(Vl2Vj)∗bt}

+ (Vl4Vl3)∗bb

NM+1∑l2=2


∗bb(Vl2Vj)∗bt}}}

!2'

= [S−∞Vj]∗3rda +NM+1∑l4=2



∗bt}!3';

[S−∞Vj]4ths =NM+1∑l4=2



∗bt}!4:

(5) The rest are deduced by analogy. After the (n + 1)th-order scattering

[S−∞Vj]∗(n+1)tha = [S−∞Vj]∗nth

a +NM+1∑ln+1=2

{(S−∞Vln+1)∗fH (Vln+1Vln)

∗ft

+(S−∞Vln+1)∗bH (Vln+1Vln)

∗bt}!n'; (29)

[S−∞Vj]∗(n+1)ths =

NM+1∑ln+1=2

{(S−∞Vln+1)∗fH (Vln+1Vln)

∗ft + (S−∞Vln+1)∗bH (Vln+1Vln)

∗bt}!n+1: (30)


Other RTCs are given by analogy

[S+∞Vj]∗(n+1)tha = [S+∞Vj]∗nth

a +NM+1∑ln+1=2

{(S+∞Vln+1)∗fH (Vln+1Vln)

∗ft

+(S+∞Vln+1)∗bH (Vln+1Vln)

∗bt}!n'; (31a)

[S+∞Vj]∗(n+1)ths =

NM+1∑ln+1=2

{(S+∞Vln+1)∗fH (Vln+1Vln)

∗ft + (S+∞Vln+1)∗bH (Vln+1Vln)

∗bt}!n+1; (31b)

[ViVj]∗(n+1)tha = [ViVj]∗nth

a +NM+1∑ln+1=2

{(ViVln+1)∗fH (Vln+1Vln)

∗ft

+ (ViVln+1)∗bH (Vln+1Vln)

∗bt}!n'; (32a)

[ViVj]∗(n+1)ths =

NM+1∑ln+1=2


∗ft + (ViVln+1)∗bH (Vln+1Vln)

∗bt}!n+1; (32b)

[ViSv]∗(n+1)tha = [ViSv]∗nth

a +NM+1∑ln+1=2


∗ft + (ViVln+1)∗bH (Vln+1Vln)

∗bt}!n;

(33a)

[SuSv]∗(n+1)tha =[SuSv]∗nth

a +NM+1∑ln+1=2

{(SuVln+1)∗fH (Vln+1Vln)

∗ft + (SuVln+1)∗bH (Vln+1Vln)

∗bt}!n: (34)

After inverse operation and considering E=4+k'kIx, the RTCs for anisotropic scattering is obtained:

[ViVj]k = 4+k'kIx[ViVj]∗(n+1)tha;k ; [SuVj]k = 9u;k[SuVj]

∗(n+1)tha;k ;

[ViSv]k = 4+k'kIx[ViSv]∗(n+1)tha;k ; [SuSv]k = 9u;k[SuSv]

∗(n+1)tha;k : (35)

4. Numerical method and its veri cation

4.1. Relativity of the RTCs and integrality of the RTCs

Theoretically, radiative transfer coe>cients of one-dimensional <at medium must satisfy the rela-tivity and integrality of the RTCs as followed:

(S−∞Vj)k = nm(VjS−∞)k ; (36a)

(S+∞Vj)k = nm(VjS+∞)k ; (36b)


NM+1∑j=2

[ViVj]k + [ViS−∞]k + [ViS+∞]k = 4+k'k Ix; (37a)

[S−∞S−∞]k +NM+1∑

j=2

[S−∞Vj]k + [S−∞S+∞]k = 1; (37b)

[S+∞S−∞]k +NM+1∑

j=2

[S+∞Vj]k + [S+∞S+∞]k = 1; (37c)

where, j = 2 ∼ NM + 1, and i = 2 ∼ NM + 1. In computation, radiative transfer coe>cientsconsidering anisotropically scattering can satisfy well above Eqs. (36) and (37) with very highprecision. Since other published papers on coupled radiative and conductive heat transfer withinsingle layer semitransparent, anisotropic scattering media with semitransparent boundaries cannot befound, the results of this paper cannot be compared with other ones and the correctness of the modelput forward in the paper can only be mainly validated by Eqs. (36) and (37).

4.2. Numerical method

The dimensionless radiative heat source term .ri of the control-volume i, is the nonlinear function

of T i, the dimensionless temperature of the control-volume i, so it must be Crstly treated withlinearization as in the following [18]:

.r;m;n+1i = .r;m;n

i + (d.ri =dT i)m;n(T m;n+1

i − T m;ni )

= Scm;n+1i + Spm;n+1

i T m;n+1i ; (38)

where, the superscript ‘m’ denotes the mth time step, ‘n’ and ‘n+1’ denote the nth and the (n+1)thiterative calculations in the mth time step; Sci represents the constant part of .

ri ; Spi is the modulus of

Ti, and Spi6 0 [19]. After linearization treatment of the nonlinear term in Eq. (1), linear equationscan be solved by TDMA (tri-diagonal matrix algorithm), and also the temperatures of all nodescan be evaluated. The control precision is chosen as: MAX|T m;n+1

i − T m;ni |6EPS1 = 10−5. In the

process of calculating, suppose that temperature Celd reaches the steady state if MAX|T m+1; nfi −

T m;nfi |6EPS2 = 10−5. Here, the superscript nf designates the Cnal iterative step.Radiative transfer functions Eqs. (11)–(13), are numerically calculated by a 30-point improved

Gaussian quadrature scheme, and the integral precision is EPS3 = 10−9. In the process of calculat-ing the redistribution of the radiative energy, suppose that all the radiative energy is absorbed bycontrol-volumes and the anisotropic scattering within a medium terminates and the redistribution ofthe radiative energy is Cnished, if the sum of radiative energy scattered by all control-volumes isnot bigger than EPS0 that is chosen as 10−9.

5. Results and analyses

In this paper, coupled radiation and conduction heat transfer is studied in a one-dimensional <atwith mixed boundary conditions. EBects of conduction–radiation parameter, scattering albedo, optical


thickness, spectral refractive index of slab and scattering phase function on temperature Celds andradiative heat <ux Celds in a medium are explored. Results for isotropic scattering and anisotropicscattering are analyzed and compared. Then, when a medium can be treated as isotropic scatteringand when a medium must be treated as anisotropic scattering is investigated. The mathematicalmodel considering anisotropic scattering established in this paper is suitable for non-gray mediumas well as gray medium. Coupled radiation and conduction heat transfer only for a gray medium isstudied in the next sections.

In the calculation, environment temperatures are taken as T−∞=T+∞=1500 K. A uniform initialtemperature T0 = 293 K is adopted and gas temperatures are chosen as Tg1 = Tg2 = 350 K; thereference temperature is Trf = 1500 K. At the left boundary, H1 = 0:5, which is the third type ofboundary condition and at the right boundary, H2 = ∞, which is the Crst boundary condition: TS2keeps unchanged and is equal to the right gas temperature.

5.1. E;ect of conduction–radiation parameter

EBects of conduction–radiation parameter denoted by N on the transient and steady-state tempera-ture Celd and radiative heat <ux Celd in a medium is investigated. Calculating parameters are givenas: +k =4000:0, nm;k =2:0, !k =0:8, L=0:01 m, C=106 J m−3 K−1, %=1:25, N =0:001; 0:01; 0:1; 1:0.The following isotropic scattering and linear anisotropic scattering phase functions are considered inthe section:

.0(*; *i) = 1; (39)

.2(*; *i) = 1− cos * cos *i: (40)

DeCne D) as the diBerence of dimensionless temperatures and Dqr as the diBerence of dimensionlessradiative heat <uxes, and

D)20 = )(.2)− )(.0) =T (.2)− T (.0)

Trf − T0; (41)

Dqr20 = qr(.2)− qr(.0): (42)

EBects of the conduction–radiation parameter on the coupled radiation–conduction heat transfer ina medium are shown in Figs. 5 and 6. In Fig. 5, the ordinate denotes the diBerence of dimensionlesstemperatures D)20 between the temperature for anisotropic scattering and that for isotropic scattering,and in Fig. 6, it denotes the diBerence of dimensionless radiative heat <uxes Dqr20; the symbol t∗cdenotes the dimensionless time when D)20 gets the largest value, and t∗s denotes the dimensionlesstime when the temperature Celd is in steady state.

Extrema of diBerences in temperature for the various values of N are given in Table 1. Accordingto Fig. 5 and Table 1, it can be seen that, the smaller the value of N is, the more signiCcant thediBerence of temperature Celds within two mediums with diBerent scattering characters is, and thebigger the value of N is, the more slight the diBerence of temperature Celds is. When N is chosenas 0.001, the extremum of the diBerence in temperature between the isotropic scattering medium andthe anisotropic scattering medium, i.e., the absolute extremum of D)20, is equal to 206:81 K. Withthe value of N increasing, the extremum decreases rapidly, and when N is chosen as 1.0, it decreasesto −5:35 K. The reason is that: scattering in a medium mainly in<uences radiation heat transfer. A


0.0 0.2 0.4 0.6 0.8 1.0

-0.16

-0.12

-0.08

-0.04

0.0020

DΘ

X

*

*

* *

* *

0.076541.1481

c

s

ttt tt t

====

X0.0 0.2 0.4 0.6 0.8 1.0

-0.08

-0.06

-0.04

-0.02

0.00

20D

Θ

*

*

* *

* *

0.076541.1481

c

s

ttt tt t

====

0.0 0.2 0.4 0.6 0.8 1.0

-0.015

-0.012

-0.009

-0.006

-0.003

0.000

X

*

*

* *

* *

0.076541.1481c

s

ttt tt t

====20

DΘ

0.0 0.2 0.4 0.6 0.8 1.0

-0.005

-0.004

-0.003

-0.002

-0.001

0.000

X

*

*

* *

* *

0.076541.1481c

s

ttt tt t

====

20D

Θ

(a)

(c) (d)

(b)

Fig. 5. Sketch of the eBect of conduction–radiation parameter on temperature Celds: (a) N = 0:001, (b) N = 0:01, (c)N = 0:1, and (d) N = 1:0.

small value of N means that radiation plays a dominant role in the coupled conduction and radiationheat transfer process, while a big value of N means that conduction plays a dominant role in theprocess. As a result, the mathematical model must be treated with the anisotropic scattering when Nis small, or a big error will occur. When N is big, the anisotropic scattering model can be simpliCedwith the isotropic scattering one, which causes a little error. But when the temperature Celd is insteady state, D)20 is little independent of the value of N , which means that the eBect of anisotropicscattering on the temperature Celd in the steady state is slight.

The eBect of N on radiation heat <ux is shown in Fig. 6. In the case of steady state, when N issmall, Dqr20 is close to zero, and when N is big, the diBerence of the radiation heat <ux Celds withintwo mediums with diBerent scattering characters is large. But for the transient state, the diBerencewhich is independent of the value of N is signiCcant.

5.2. E;ect of scattering albedo

The in<uence of scattering albedo !k on temperature Celd and radiation heat <ux Celd is investi-gated. Here, N = 0:001, !k = 0:10; 0:40; 0:90; 0:99, and other calculating parameters are the sameas those in the section above.


0.0 0.2 0.4 0.6 0.8 1.0

-0.04

-0.02

0.00

0.02

0.0420r

~D

q

X

*

*

*

* *

0.076541.14813.5208s

tttt t

====

20r~

Dq

X0.0 0.2 0.4 0.6 0.8 1.0

-0.04

-0.02

0.00

0.02

0.04*

*

*

* *

0.076541.14813.5208s

tttt t

====

0.0 0.2 0.4 0.6 0.8 1.0

-0.04

-0.02

0.00

0.02

0.04

20r~

Dq

X

*

*

*

* *

0.076541.14813.5208s

tttt t

====

0.0 0.2 0.4 0.6 0.8 1.0

-0.04

-0.02

0.00

0.02

0.04

20r~

Dq

X

*

*

*

* *

0.076541.14813.5208

s

tttt t

====

(a) (b)

(d)(c)

Fig. 6. Sketch of eBect of conduction–radiation parameter on radiative <ux Celds: (a) N =0:001, (b) N =0:01, (c) N =0:1,and (d) N = 1.

Table 1Extrema of temperature diBerences for the various values of N

N 0.001 0.01 0.1 1.0

D)20 −0:17134 −0:07041 −0:01525 −0:00443T (.2)− T (.0) −206:81 K −84:98 K −18:41 K −5:35 K

Extrema of diBerences in temperature for the various values of !k are presented in Table 2.The eBect of scattering albedo on the temperature Celd is shown in Fig. 7. From the Fig. 7 andTable 2, !k has a critical value. When !k is chosen as the smaller one than the critical value,the smaller !k is, the less the diBerence of the transient temperature Celds within two mediumswith diBerent scattering characters is. When !k is chosen as 0.40, the extremum of diBerence intemperature is 111:12 K, and when !k is 0.10, the extremum is only −28:29 K. However, when!k is bigger than the critical value, the smaller !k is, the more signiCcant the diBerence becomes.In the case of !k = 0:99, the extremum is 160:47 K, and in the case of !k = 0:90, it increases to218:50 K. The transient diBerence becomes largest when !k gets the critical value. When attenuation


Table 2Extrema of temperature diBerences for the various values of !k

!k 0.10 0.40 0.90 0.99

D)20 −0:02344 −0:09206 −0:18103 −0:13295T (.2)− T (.0) −28:29 K −111:12 K −218:50 K −160:47 K

X0.0 0.2 0.4 0.6 0.8 1.0

-0.024

-0.018

-0.012

-0.006

0.000

20D

Θ

*

*

* *

* *

0.076541.1481

c

s

ttt tt t

====

X0.0 0.2 0.4 0.6 0.8 1.0

-0.10

-0.08

-0.06

-0.04

-0.02

0.00

*

*

* *

* *

0.076541.1481c

s

ttt tt t

====

20D

Θ

0.0 0.2 0.4 0.6 0.8 1.0-0.20

-0.16

-0.12

-0.08

-0.04

0.00

20D

Θ

X

*

*

* *

* *

0.076541.1481

c

s

ttt tt t

====

0.0 0.2 0.4 0.6 0.8 1.0

-0.12

-0.09

-0.06

-0.03

0.00

20D

Θ

X

*

*

* *

* *

0.076541.1481

c

s

ttt tt t

====

(a) (b)

(c) (d)

Fig. 7. Sketch of eBect of spectral single-scattering albedo of slab on temperature Celds: (a) !k = 0:1, (b) !k = 0:4, (c)!k = 0:9, and (d) !k = 0:99.

coe>cient +k is kept as constant, the bigger !k is, the less the absorption coe>cient is, and theweaker the absorption of the radiative energy in the medium becomes. Accordingly, the secondsub-process, absorbing–scattering sub-process gets longer in the process of radiative energy transfer,and the eBect of scattering on the temperature Celd becomes more signiCcant, which causes greaterdiBerence between the temperature Celds. On the other hand, when the albedo gets the big value,the absorption coe>cient becomes small, and the absorption of radiative energy in a medium getslow. So the contribution of radiation to the temperature Celd becomes slight under such condition,which results in the slight diBerence of temperature Celds. That is the reason for the existenceof the critical value of the scattering albedo. From Fig. 7, in the steady state the diBerence of the


0.0 0.2 0.4 0.6 0.8 1.0

-0.006

-0.003

0.000

0.003

0.00620r

~

X

*

*

*

* *

0.076541.14812.7554

s

tttt t

====

20r~

Dq

Dq

X0.0 0.2 0.4 0.6 0.8 1.0

-0.03

-0.02

-0.01

0.00

0.01

0.02 *

*

*

* *

0.076541.14812.7554

s

tttt t

====

0.0 0.2 0.4 0.6 0.8 1.0

-0.06

-0.03

0.00

0.03

0.06

20r~

Dq

X

*

*

*

* *

0.076541.14812.7554

s

tttt t

====

0.0 0.2 0.4 0.6 0.8 1.0

-0.04

-0.02

0.00

0.02

0.04

20r~

Dq

X

*

*

*

* *

0.076541.14812.7554

s

tttt t

====

(a) (b)

(d)(c)

Fig. 8. Sketch of eBect of spectral single-scattering albedo of slab on radiative <ux Celds: (a) !k =0:1, (b) !k =0:4, (c)!k = 0:9, and (d) !k = 0:99.

temperature Celds independence of !k is always slight. From Fig. 8, the eBect of !k on the radiationheat <ux is almost the same as that on the temperature Celd in an anisotropic scattering medium.According to the analysis above, in the case of N = 0:001, when !k gets the big value (near tothe critical value, including the critical value), the isotropic scattering model does not suit and theanisotropic scattering model must be considered in the coupled conduction–radiation heat transferprocess.

5.3. E;ect of optical thickness

The eBect of optical thickness &0 on the coupled conduction–radiation heat transfer is investigated.In the section, !k = 0:80, +k = 40:0; 400:0; 4000:0; 40000:0, i.e., &0 = +kL = 0:4; 4:0; 40:0; 400:0,and other parameters is the same as those presented in Section 5.2.

Table 3 presents the extrema of diBerences in temperature for the various values of &0. The eBectof &0 on the temperature Celd can be seen from the Fig. 9. There is also a critical value of &0 inexistence from Fig. 9 and Table 3. When &0 is smaller than the critical value, the diBerence of thetemperature Celds decreases with &0 decreasing. Especially for the optically thin medium, it getsvery slight. From Table 3, when &0 is chosen as 40.0, the extremum of the diBerence in temperature


Table 3Extrema of temperature diBerences for the various values of &0

&0 0.40 4.0 40.0 400.0

D)20 −5:03563E-4 −0:01877 −0:17134 −0:04095T (.2)− T (.0) −0:61 K −22:66 K −206:81 K −49:43 K

0.0 0.2 0.4 0.6 0.8 1.0

-0.0005

-0.0004

-0.0003

-0.0002

-0.0001

0.0000

0.0001

20D

Θ

X

*

*

* *

* *

0.076541.1481

c

s

ttt tt t

====

0.0 0.2 0.4 0.6 0.8 1.0

-0.021

-0.014

-0.007

0.000

0.007

0.014

20D

Θ

X

*

*

* *

* *

0.076541.1481

c

s

ttt t

t t

====

0.0 0.2 0.4 0.6 0.8 1.0

-0.16

-0.12

-0.08

-0.04

0.00

20D

Θ

X

*

*

* *

* *

0.076541.1481

c

s

ttt tt t

====

0.0 0.2 0.4 0.6 0.8 1.0

-0.04

-0.03

-0.02

-0.01

0.00

20D

Θ

X

*

*

* *

* *

0.076541.1481

c

s

ttt tt t

====

(a) (b)

(d)(c)

Fig. 9. Sketch of eBect of optical thickness of slab on temperature Celds: (a) &0 = 0:4, (b) &0 = 4:0, (c) &0 = 40:0, and (d)&0 = 400:0.

is 206:81 K, and it decreases to 22:66 K while &0 = 4:0, and it is only 0:61 K under the conditionof &0 = 0:4. But when &0 is above the critical value, with &0 increasing, the diBerence decreases:while &0 increases from 40.0 to 400.0, the extremum of the diBerence in temperature decreases from206.81 to 49:43 K.

The reason for the existence of the critical value of &0 is given as follows. Keep the media thicknessand !k unchanged. While &0 decreases, the absorption of coe>cient decreases (i.e., the absorptionof the radiative energy gets low) and the contribution of radiation to the temperature Celd becomeslittle, which causes the diBerence to becomes small gradually; while &0 increases, the absorption ofcoe>cient increases, and the scattering eBect lowers relatively in the second sub-process, which also


-0.00015

-0.00010

-0.00005

0.00000

0.00005

0.00010

0.0001520r~

Dq

*

*

*

* *

0.076541.14813.827

s

tttt t

====

-0.02

-0.01

0.00

0.01

0.02

20r~

Dq

*

*

*

* *

0.076541.14813.827

s

tttt t

====

0.0 0.2 0.4 0.6 0.8 1.0

-0.04

-0.02

0.00

0.02

0.04

20r~

Dq

X

*

*

*

* *

0.076541.14813.827

s

tttt t

====

0.0 0.2 0.4 0.6 0.8 1.0

-0.008

-0.004

0.000

0.004

0.008

20r~

Dq

X

*

*

*

* *

0.076541.14813.827

s

tttt t

====

0.0 0.2 0.4 0.6 0.8 1.0X

0.0 0.2 0.4 0.6 0.8 1.0X(a) (b)

(d)(c)

Fig. 10. Sketch of eBect of optical thickness of slab on radiative <ux Celds: (a) &0 = 0:4, (b) &0 = 4:0, (c) &0 = 40:0, and(d) &0 = 400:0.

causes the diBerence to becomes small. From the Fig. 9, in the steady state, the diBerence of thetemperature Celds in two diBerent scattering mediums is very slight. The eBect of &0 on the radiationheat <ux is almost same as that on the temperature Celd, as shown in Fig. 10.

From the above analyse, if &0 is chosen as a very small or very big value, the diBerence of transienttemperature Celds and that of transient radiation heat <uxes are all not signiCcant. Especially for theoptically thin medium, the anisotropic scattering can be simpliCed with the isotropic scattering andthe error introduced is very small. Except for the cases mentioned above, the eBect of the anisotropicscattering must be considered.

5.4. E;ect of refractive index

In this section, the case of &0 = 40:0, nm;k = 1:1; 2:0; 3:0; 6:0 (other parameters are the same asthat mentioned above) is taken as working condition for the calculation, and the eBect of spectralrefractive index of a medium nm;k on the coupled conduction–radiation heat transfer is investigated.

The eBect of the refractive index on the temperature Celd is shown in Fig. 11. For the variousvalue of nm;k , Table 4 presents the extrema of diBerences in temperature. For the semitransparent,participating medium, as shown in Fig. 11, with the refractive index increasing, the re<ection of theexternal radiative energy becomes strong and the part of energy entering the medium gets small,


0.0 0.2 0.4 0.6 0.8 1.0

-0.20

-0.15

-0.10

-0.05

0.0020

DΘ

X

*

*

* *

* *

0.076540.7654

c

s

ttt tt t

====

0.0 0.2 0.4 0.6 0.8 1.0

-0.16

-0.12

-0.08

-0.04

0.00

20D

Θ

X

*

*

* *

* *

0.076540.7654

c

s

ttt tt t

====

0.0 0.2 0.4 0.6 0.8 1.0-0.12

-0.09

-0.06

-0.03

0.00

0.03

20D

Θ

X

*

*

* *

* *

0.076540.7654

c

s

ttt tt t

====

0.0 0.2 0.4 0.6 0.8 1.0-0.06

-0.04

-0.02

0.00

0.02

20D

Θ

X

*

*

* *

* *

0.076540.7654

c

s

ttt tt t

====

(a) (b)

(d)(c)

Fig. 11. Sketch of eBect of spectral refractive index of slab on temperature Celds: (a) nm;k = 1:1, (b) nm;k = 2:0, (c)nm;k = 3:0, and (d) nm;k = 6:0.

Table 4Extrema of temperature diBerence for the various values of nm;k

nm;k 1.1 2.0 3.0 6.0

D)20 −0:20489 −0:17072 −0:11409 −0:05192T (.2)− T (.0) 247:30 K 206:06 K 137:71 K 62:67 K

which causes the eBect of scattering in the medium to become weak in the process of the radiativeenergy transfer, so the diBerence of temperature Celds within two mediums with diBerent scatteringcharacters decreases. The trend can be seen from the Table 4, while nm;k = 1:1, the extremum is247:30 K; in the case of nm;k =2:0; 6:0, it is equal to 206:06 K and 62:67 K, respectively. In steadystate, from the Cgure, the temperature Celd for the anisotropic scattering medium accords well withthat for the isotropic scattering medium. From the Fig. 12, the eBect of nm;k on the radiation <ux isalmost identical with that on the temperature Celds. According to the analysis above, for the mediumwith a small refractive index, the anisotropic scattering must be considered.


0.0 0.2 0.4 0.6 0.8 1.0

-0.04

-0.02

0.00

0.02

0.0420r~

Dq

X

*

*

*

* *

0.076540.76543.827

s

tttt t

====

0.0 0.2 0.4 0.6 0.8 1.0

-0.04

-0.02

0.00

0.02

0.04

20r~

Dq

X

*

*

*

* *

0.076540.76543.827

s

tttt t

====

0.0 0.2 0.4 0.6 0.8 1.0

-0.04

-0.02

0.00

0.02

0.04

20r~

Dq

X

*

*

*

* *

0.076540.76543.827s

tttt t

====

0.0 0.2 0.4 0.6 0.8 1.0

-0.03

-0.02

-0.01

0.00

0.01

0.02

0.03

20r~

Dq

X

*

*

*

* *

0.076540.76543.827

s

tttt t

====

(a) (b)

(d)(c)

Fig. 12. Sketch of eBect of spectral refractive index of slab on radiative <ux Celds: (a) nm;k = 1:1, (b) nm;k = 2:0, (c)nm;k = 3:0, and (d) nm;k = 6:0.

5.5. E;ect of scattering phase function

EBects of the linear scattering and non-linear scattering on the coupled conduction–radiation heattransfer are investigated in this section. Calculating parameters are identical with those in the Section5.4 except for nm;k = 2:0. Considering the phase functions as follows:

.0(*; *i) = 1:0; (39)

.2(*; *i) = 1:0− cos * cos *i; (40)

.1(*; *i) = 1:0 + cos * cos *i; (43)

.3(*; *i) = 1:0 + 0:5(1:5 cos2 * − 0:5)(1:5 cos2 *i − 0:5); (44)

.4(*; *i) = 1:0 + cos * cos *i + 0:5(1:5 cos2 * − 0:5)(1:5 cos2 *i − 0:5); (45)

.5(*; *i) = 1:0 + 1:5 cos * cos *i + 0:5(1:5 cos2 * − 0:5)(1:5 cos2 *i − 0:5); (46)


0.0 0.2 0.4 0.6 0.8 1.0

-0.16

-0.12

-0.08

-0.04

0.0020

DΘ

X

*

*

* *

* *

0.076541.1481

c

s

ttt tt t

====

0.0 0.2 0.4 0.6 0.8 1.0

0.00

0.04

0.08

0.12

0.16

10D

Θ

X

*

*

* *

* *

0.076541.1481

c

s

ttt tt t

====

(a) (b)

Fig. 13. Sketch of eBect of linear scattering functions on temperature Celds: (a) linear backward scattering function .2,(b) linear forward scattering function .1.

0.0 0.2 0.4 0.6 0.8 1.0

-0.04

-0.02

0.00

0.02

0.04

20r~

Dq

X

*

*

*

* *

0.076541.14812.6789

s

tttt t

====

0.0 0.2 0.4 0.6 0.8 1.0

-0.04

-0.02

0.00

0.02

0.04

10r~

Dq

X

*

*

*

* *

0.076541.14812.6789

s

tttt t

====

(b)(a)

Fig. 14. Sketch of eBect of linear scattering functions on radiative <ux Celds: (a) linear backward scattering function .2,(b) linear forward scattering function .1.

where, Eq. (39) is for the isotropic scattering, .1 is the linear forward scattering phase function, .2

is the linear backward scattering phase function, and others are nonlinear scattering phase functions.DeCnitions of D) and Dqr are the same as those in Eqs. (41) and (42). The eBect of linearscattering phase functions on the coupled conduction–radiation heat transfer is shown in Figs. 13and 14. EBect of nonlinear scattering phase functions is shown in Figs. 15 and 16. Table 5 givesextrema of diBerence in temperature for diBerent phase functions.

From Figs. 13 and 14, we can know that, compared with linear forward scattering phase function,eBects of linear backward scattering phase function on temperature Celd and radiation heat <ux Celdin the anisotropic scattering medium is coincidentally opposite. From Fig. 13, can we know that in alinear backward scattering medium, temperature is lower than that in the isotropic scattering mediumat the same instant of the same point except positions near the boundary; in a linear forward scatteringmedium, temperature is higher than that in the isotropic scattering medium at the same instant of thesame point except positions near the boundary. It can be also seen that, the temperature Celd in the


0.0 0.2 0.4 0.6 0.8 1.0

0.000

0.003

0.006

0.009

0.012

30D

Θ

X

*

*

* *

* *

0.076541.1481

c

s

ttt tt t

====

0.0 0.2 0.4 0.6 0.8 1.0

0.00

0.04

0.08

0.12

0.16

40D

Θ

X

*

*

* *

* *

0.076541.1481

c

s

ttt tt t

====

0.0 0.2 0.4 0.6 0.8 1.0

0.00

0.05

0.10

0.15

0.20

0.25

50D

Θ

X

*

*

* *

* *

0.076541.1481

c

s

ttt tt t

====

(a)

(b) (c)

Fig. 15. Sketch of eBect of nonlinear scattering functions on temperature Celds: (a) nonlinear scattering function .3, (b)nonlinear scattering function .4, and (c) nonlinear scattering function .5.

linear backward scattering medium is symmetrical with that in the linear forward scattering mediumwith regard to x-axis. From Fig. 14, we can know that, in linear backward scattering medium, theradiation heat <ux of the former half part points are lower than that of isotropic scattering medium,the radiation heat <ux of the latter half parts is higher than that of isotropic scattering medium; heat<ux distribution in linear forward scattering media is opposite to that in linear backward scatteringmedia. At the former part, radiation heat <ux is higher than that in isotropic scattering medium,while at the latter part, radiation heat <ux is lower than that in isotropic scattering medium.

EBects of nonlinear scattering functions on media temperature and distribution of radiation heat<ux can be seen from Figs. 15 and 16. From Fig. 15a and Table 5, we can know that, when nonlinearscattering function is .3, the diBerence of media temperature Celd and radiative heat <ux Celd isvery little, compared with isotropic scattering medium. Its biggest diBerence in temperature is only16.49. When nonlinear scattering functions are .4 and .5, the diBerence of medium temperatureCeld and radiation heat <ux Celd is great, compared with isotropic scattering medium. Its biggesttemperature diBerence is 217.66 and 311.88 respectively. From Eqs. (44)–(46), it can be discoveredthat nonlinear scattering function consists of three parts: isotropic part, linear part and nonlinear part;while .3 consists of two parts: isotropic part and nonlinear part. From analysis above, we can knowthat, the linear part in linear or nonlinear scattering function can produce greater diBerence of the


0.0 0.2 0.4 0.6 0.8 1.0

-0.004

-0.002

0.000

0.002

0.004

X

*

*

*

* *

0.076541.14812.6789

s

tttt t

====

0.0 0.2 0.4 0.6 0.8 1.0-0.06

-0.03

0.00

0.03

0.06

40r~

Dq

X0.0 0.2 0.4 0.6 0.8 1.0

X

*

*

*

* *

0.076541.14812.6789

s

tttt t

====

-0.08

-0.04

0.00

0.04

0.08

50r~D

q

*

*

*

* *

0.076541.14812.6789

s

tttt t

====

30rD

q~

(a)

(b) (c)

Fig. 16. Sketch of eBect of nonlinear scattering functions on radiative <ux Celds: (a) nonlinear scattering function .3, (b)nonlinear scattering function .4, and (c) nonlinear scattering function .5.

Table 5Extrema of temperature diBerence for various phase functions

. .1 .2 .3 .4 .5

D) 0.16854 −0:17134 0.01366 0.18033 0.25839T (.)− T (.0) 203:43 K −206:81 K 16:49 K 217:66 K 311:88 K

temperature Celd and radiation heat <ux Celd between anisotropic scattering medium and isotropicscattering medium, while the nonlinear part in the nonlinear scattering phase function produce lessdiBerence of the temperature Celd and radiation heat <ux Celd between isotropic scattering mediumand anisotropic scattering medium. For nonlinear scattering medium, the diBerence of its temperatureCeld and radiation heat Celd in the nonlinear scattering medium is the combined result of the linearpart and nonlinear part in phase function. Further investigation shows that, the linear part in nonlinearscattering phase function .5 is 1.5 times as much as that in .4, while D)50 of all points in Fig.15b can be approximately 1.5 times as much as D)40 of all points in Fig. 15a, at the same timeand at the same point.


6. Conclusions

This paper utilizes ray tracing method combined with Hottel’s zonal method and has studiedone-dimensional coupled heat transfer of radiation–conduction in a bilaterally semitransparent, partic-ipating gray slab medium considering its anisotropic scattering under the mixed boundary condition.It mainly calculates and analyses eBects of conduction–radiation parameter, scattering albedo, opticalthickness, spectral refractive index and phase functions on the temperature Celd and radiation heat<ux Celd within an anisotropic scattering medium and some conclusions are made below as

(1) The less conduction–radiation parameter is, the greater are the diBerences of temperature Celdand radiation heat <ux Celd between anisotropic scattering medium and isotropic scatteringmedium, and there is more need to consider the anisotropic scattering property of medium.

(2) A greater critical albedo value exists. The closer sampling albedo is to the value, the more needthere is to consider the anisotropic scattering within a medium.

(3) A greater critical optical thickness value exists. When a medium is optically thin or its opticalthickness is large, the isotropic scattering eBect of medium can be only considered; when opticalthickness of medium is greater, i.e. close to the critical value, there is more need to consideranisotropic scattering eBect in the medium.

(4) The less spectrum index of refraction is, the greater is the diBerence of the temperature Celdand radiation heat <ux between the anisotropic and the isotropic scattering medium, and thereis more need to consider the anisotropic scattering property of the medium.

(5) For an anisotropic scattering medium, the linear part in linear or nonlinear scattering phasefunction can produce greater diBerence of the temperature Celd and radiation heat <ux Celdbetween anisotropic scattering medium and isotropic scattering medium; the nonlinear part inthe nonlinear scattering phase function produces less diBerence of the temperature Celd andradiation heat <ux Celd between the isotropic and the anisotropic scattering medium.

Acknowledgements

This research is supported by the Chinese National Science Fund for Distinguished Young Scholars(No. 59725617), the Science & Technology Foundation of Heilongjiang, and the National NaturalScience Foundation of China (No. 59806003).

References

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[5] Liu LH, Yu QZ, Tan HP. Irradiation characteristics of domestic boiler coal particulates. Power Eng 1998;18(3):65–9 (in Chinese).

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[10] Siewert CE. An improved iterative method for solving a class of coupled conductive–radiative heat transfer problems.JQSRT 1995;54(4):599–605.

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[12] Maruyama S. Radiative heat transfer in anisotropic scattering media with specular boundary subjected to collimatedirradiation. Int J Heat Mass Transfer 1998;41(18):2847–56.

[13] Hottel HC, SaroCm AF. Radiative transfer. New York: McGraw-Hill Book Company, 1967. p. 265, 6.[14] Saulnier JB. La modXelisation thermique et ses applications aux transferts couplXes et au controle actif. ThZese de

Doctorat d’Etat, UniversitXe de Poitiers, France, 1980.[15] Tan HP. Transfert couplXe rayonnement-conduction instationnaire dans les milieux semi-transparents Za frontiZeres

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coupled radiation–conduction heat transfer in an anisotropically scattering slab with mixed...

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