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Combined Radiation And Conduction Heat Transfer
Calculations Using Unstructured Grid
Liu Jing
1, 2
,Zhang Min
1
,John C. Chai
2
,Xu Bin
1
1
School of Power Eng.,Nanjing University of Science & Technology,Nanjing (210094)
2
School of Mechanical and Aero spacing Eng.,Nanyang Tech. University,Singapore (639798)
Abstract
Coupled conduction and radiation heat transfer problems were solved using the based-cell and the finite
volume method (FVM) in the regions with the mediums. The combined heat transfer equations were
discretized in the unstructured meshes and the calculating programs were written. The radiative heat source
of the combined equation was treated specially. Meanwhile, from the relations of the governing equations
and the several of medium characters, the dimensionless numbers were obtained and computed using
unstructured quad grids in various parameters, such as, optical thickness τ, conduction-radiation
parameter N, and scattering albedo ω.
Keywords:unstructured grid/mesh,radiative heat transfer,conduction heat transfer.
In this paper, the problems about radiation heat transfer combined conduction are presented. Some
auxiliary comments are also listed for completeness. This problem was studied by Modest
[1]
, Chai and
Patankar
[2]
and Sparrow and Cass
[3]
, just to name a few.
1. Governing Equations and Boundary Condation
1.1 Radiative Transfer Equation
The Radiative Transfer Equation (RTE) that governs the transfer of radiant energy across a gray
medium can be written as
)
ˆ
,()
ˆ
,()(
)
ˆ
,(
srSsrIr
ds
srdI
R
rrr
r
+−=
β
(1.1)
where the extinction coefficient
β
and the source function
S are
)()()( rrr
s
σκβ
+=
(1.2a)
Ω
′′
+=
∫
dsrI
r
rIrsrS
s
bR
π
π
σ
κ
4
)
ˆ
,(
4
)(
)()()
ˆ
,(
r
rrr
(1.2b)
In equation (1.1) and equation (1.2),
v
and s
ˆ
are the position vector and the unit vector describing
the radiation direction, respectively. Isotropic scattering is assumed in equation (1.2). Equation (1.1)
indicates that the radiant intensity I changes as it travels over distance ds. This is due to the attenuation
of intensity through absorption and outscattering, and the augmentation of intensity through gas
emission and inscattering of radiation from all directions.
Physically, the first term on the right-hand side of equation (1.1) represents the attenuation of
radiant energy due to absorption and outscattering as a radiation beam travels from point a to point b.
The second term describes the augmentation due to emission and inscattering of radiant energy. The
total attenuation or augmentation of radiant energy depends on two factors. These are (1) the relative
importance of absorption and scattering and (2) the “strength” of absorption and scattering. The first
factor is described by the scattering albedo
ω
. This factor is defined as
β
σ
σκ
σ
ω
s
s
s
=
+
≡
(1.3a)