www. Combined heat transfer within an anisotropic scattering slab
with diffu surfaces
Hong-Liang Yi, He-Ping Tan*
School of Energy Science and Engineering, Harbin Institute of Technology, Harbin, 150001, P.R.China
150080, P.R.China
Abstract
Under various interface reflecting modes, different transient thermal respon will occur in media. Combined radiative-conductive heat transfer is investigated within a participating, anisotropic scattering gray planar slab. The two interfaces of the slab are considered to be diffu and mitransparent. Using the ray tracing method, an anisotropic scattering radiative transfer model for diffu reflection at boundaries is t up, and with the help of direct radiative transfer coefficients, corresponding radiative transfer coefficients (RTCs) are deduced. RTCs are ud to calculate the radiative source term in energy equation. Transient energy equation is solved by the full implicit control-volume method under the external radiative-convective boundary conditions. The influences of two reflecting modes including
both specular reflection and diffu reflection are examined on transient temperature fields and steady heat flux. According to numerical results obtained in this paper, it is found that there exits great difference in thermal behavior between slabs with diffu interfaces and that with speccular interfaces for slabs with big refractive index.
Keywords: Combined radiative-conductive heat transfer / ray tracing method / Radiative transfer coefficients / Anisotropic scattering / Diffu reflection
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*Corresponding author. Tel.: 86-451-86412028, Fax: 86-451-86221048. Email:
Nomenclature
Abbreviation RTC
Radiative Transfer Coefficient
,i k T A
,,()()k
b i b i I T d I T d λλλλ
λ∞
∆∫∫, fractional spectral emissive power of spectral band k λ∆
at nodal temperature i T
C unit heat capacity of media, 11JKg K −− ()n E x
exponential integral function of the th n rank
12,h h
convective heat transfer coefficient at surfaces of 1S and 2S , respectively , 21Wm K −− k
phonic thermal conductivity, 11Wm K −− L thickness of slab, m NB total number of spectral bands NM total number of control volumes of slab ,k m n
spectral refractive index of slab r q
heat flux of radiation, 2Wm −
r q % dimensionless heat flux of radiation, ()
4
/4r rf
q T σ c q
heat flux of conduction, 2Wm −
c q % dimensionless heat flux of conduction, ()
4/4c rf
q T σ t q
total heat flux, r c q q +
t q % dimensionless total heat flux, r c q q +%% ,u v S S
S −∞or S +∞
()()(),
,u v k u j k i j k
S S S V V V
radiation transfer coefficients of surface vs surface, surface vs volume, and volume vs
volume in non-scattering media relative to the spectral band k λ∆
[],
,u v k u j k i j k
S S S V V V ⎡⎤⎣⎦⎡⎤⎣⎦ radiation transfer coefficient of surface vs surface, surface vs volume, and volume vs volume in isotropic or anisotropic scattering media relative to the spectral band k λ∆
12,S S
boundary surfaces
,S S −∞+∞
black surfaces reprenting the surroundings T
absolute temperature, K
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12,g g T T
gas temperatures for convection at 0X = and 1, K rf T
reference temperature, K 0T
uniform initial temperature, K t
physical time, s
X dimensionless coordinate in direction across layer, X x L = ,i j x normal distance of ray transfer between both subscripts, m z
normal distance of ray transfer, m t ∆, time interval
x ∆
spacing interval between two nodes, m
β
the common ratio of the infinite geometric ries,()2
1,in 2,in 12s s ρρ 1,2,, k k εε emissivity of surface 1S , 2S , respectively η 1ω−
,,i s r θθθ, incident, scattering, refractive angle between the ray and x -axis k κ spectral extinction coefficient of slab, 1m − ρ density of media, 3kg/m
12,ρρ
reflectivity of surface 1S , 2S , respectively
σ
Stefan −Boltzmann constant, =8106696.5−×24Wm K −− Φ
scattering pha function
r i Φ
radiative heat source of the control-volume i
o τ
L κ, optical thickness of slab ω scattering albedo of slab
ζ control volumes in units of optical thickness, 0NM τ Subscripts
, d s diffu reflection, specular reflection iw ie ,
right and left interface of control volume i , in out
refer to inner, outer surface, respectively k
relative to the th k region of spectral bands
t t −, -o o
refer to media with mitransparent , opaque interfaces, respectively 1, 2 refer to the boundary surfaces 1S and 2S , respectively ,−∞+∞
refer to the black surfaces S −∞ and ∞+S , respectively Superscripts
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, d s diffu reflection, specular reflection
t f b ,, incidence radiation from negative, positive and both direction relative to the x axis, respectively
q h , backward scattering and forward scattering relative to the incident direction, respectively r
refer to radiation
1. Introduction
Semitransparent participating materials such as ceramic, glass, fiber, zirconia, silica gel and so on, have many engineering applications in which radiation transport and coupled radiative-conductive heat transfer play a dominant role to control their transient thermal behavior. The applications include heat insulation and thermal protection [1,2], ignition of mitransparent solid fuels [3,4], measurement of thermophysical properties of translucent materials [5,6], infrared heating and infrare
d drying, design of furnaces, prediction of the effect of dust, CO 2 and other participating gas on the global environment, and so forth.
In the past micentury, much work has been done on the radiative transfer and coupled radiative-conductive heat transfer within participating media. Relative to isotropic scattering, much less literatures on thermal radiation involving in anisotropic scattering have been found. Using discrete ordinates method, Krishnaprakas et al. [7] examined conduction-radiation heat transfer through a gray planar nonlinearly anisotropic scattering medium with two plane-parallel surfaces reflecting both diffuly and specularly. M. Lazard et al. [8] developed a mi-analytical model bad on the exponential-kernel method for the transient combined conduction-radiation heat transfer for an anisotropically scattering participating slab excited by heat pul stimulation on the front face. Prabal Talukdar et al. [9] solved combined conduction-radiation problem using the collapd dimension method for gray planar absorbing, emitting and anisotropically scattering medium. Zekeriya Altaç [10] propod the Synthetic Kernel approximation to solve radiative transfer problems in linearly anisotropically scattering homogeneous and inhomogeneous participating plane-parallel medium.
There are generally two reflecting modes at interfaces: specular reflection and diffu reflection. If the interface is optically smooth, it can be assumed specular; if the interface is optically rough, it can
be assumed diffu. As discusd by Siegel [11], the diffu reflectivity of mitransparent interface can be obtained from the F resnel interface relations by integrating the reflected energy over all incident directions if assuming that each bit of roughness acts as a smooth facet. For the mitransparent specular interface, the reflectivity can be determined by
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Fresnel’s reflective law and Snell’s refractive law [12]. In the prent investigation, we adopt the methods propod in Refs [11, 12] for the treatment of specular and diffu interface conditions.
In the last five years, He-Ping Tan, Ping-Yang Wang and Jian-F eng Luo et al. have solved transient coupled radiative-conductive problem by ray tracing method through a participating media with one-layer [13], two-layer [14], three-layer [15] and multi-layer [16], respectively. In their studies, anisotropic scattering was not considered. While for some materials, such as ceramics and zirconia, which are porous mitransparent media with highly anisotropic scattering [17], the assumption of isotropic scattering can cau big error. In this paper, bad on the radiative transfer model developed for anisotropic scattering media with specular and mitransparent boundaries in Ref. [18] and using ray-tracing method, we t up an anisotropic scattering radiative transfer model for di
ffu reflection at interfaces and with the help of direct radiative transfer coefficients, corresponding radiative transfer coefficients (RTCs) are deduced in combination with nodal analysis bad on the zone method. A gray slab with two mitransparent surfaces is considered here. The radiative heat source term is calculated by the RTCs and is linearized using the method prent in Ref. [19]. The transient energy equation is solved by the control-volume method under the external radiative-convective boundary conditions. The influences of two reflecting modes on the transient temperature fields and steady heat flux are investigated under the various values of refractive indexes, albedo, and extinction coefficients.
2. Governing equations
A
mitransparent slab with thickness of L is considered as shown in Fig.1 and is confined within two black surfaces
T +∞
S +∞
2
S 2
ρ
22g T h
S −∞
T −∞ 1
1
g T h 1
1
12
i i j j S S S S S S ++
Fig.1 Discrete model of space zone
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S −∞ and S +∞, denoting the environment of temperatures ∞−T and ∞+T , respectively. Both of boundary surfaces are
mitransparent and diffu. The slab is divided into 2+NM nodes along its thickness and denoted by i . Here, 1=i and 2i NM =+ reprent surfaces 1S and 2S , respectively.
Between the time interval t and t t +∆, the fully implicit discrete energy equation of control volume i for the transient coupled radiation and conduction is obtained as
()()(
)11
11111
11,1
m m
m m m m m m i
i ie
i i iw i i
r m i
T T k T
T k T T C x
t
x
++++++++−+−−+−∆=+Φ
∆∆ (1) where ie k , iw k are harmonic mean media thermal conductivities at the interfaces ‘ie ’ and ‘iw ’, respectively. The spectral parameters, such as extinction coefficient k κ, emissivity k ε, and refractive index ,k m n etc., vary with the wavelength, and they can be approximately expresd in a ries of rectangular spectral bands. The radiative source term at the control volume i can be expresd as
(
)
(
)
(
)1
244,,,,,1
2
244
,,,,,424,,,,,j i S i S i
NB
NM d d
r
i k m j i k T j i j k T i k t t k t t k j d
d
S k m i k T i k T i k t t k t t d d
S i k T k m i k T i
k t t k t t n V V A T V V A T n S V A T V S A T S V A T n V S A T σ+∞+∞−∞−∞+−−==+∞+∞−−−−∞−∞−−⎧⎪
⎡⎤⎡⎤Φ=−⎨⎣⎦⎣⎦⎪⎩
+−⎡⎤⎡⎤⎣⎦⎣⎦⎫⎪ +⎡⎤⎡⎤⎬
⎣⎦⎣⎦⎪⎭
∑
∑
21i NM ≤≤+ (2)
where, symbol NB indicates the total spectral band number and k the th k region of spectral bands. Discrete boundary conditions at 1S and 2S are now given as follows: ()
()
1
112212S S g k T T x h T T −∆=− (3a)
()
()
2
221122NM NM S S g k T T x h T T ++−∆=− (3b)
In the Eq. (2), for the local radiative heat source term, besides temperatures at all nodes, the symbols, such as
,d i j k t t V V −⎡⎤⎣⎦, ,d
k t t S S −∞+∞−⎡⎤⎣⎦ and ,d
j k t t S V −∞−⎡⎤⎣⎦, where the superscript d denotes diffu reflection and the subscript
t t −denotes two mitransparent interfaces, etc., defined as radiative transfer coefficients (RTCs), are unknowns to
be calculated. Therefore, the radiative transfer coefficients must be firstly evaluated before solving the local radiative heat source. In the next ction, RTCs are deduced for anisotropic scattering media with diffu boundaries
3. Deduction of RTCs
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