5.2 Continuum Modeling of Pebble Radiation
251
Additionally, for the packed beds of an isothermal absorbing medium of uniform
temperature, the particle–particle radiative flux is then formulated as follows
ˆ
Q r,i j = f (ε r )A i X i j σ (T
4
i − T
4
j ) exp(−β · r i j ),
(5.47)
where β is an attenuation coefficient. Similarly, in this case, the radiation exchange
factor of the packed bed is
F =
k r
4σ T 3 d
= ε r (1 − α f )
+∞
1
h(η)g(η) exp (−βdη)η
4 dη
+∞
1
h(η)g(η)η 2 dη
(5.48)
It is noted that βd 1 should be met for common gases, such as air and helium. Thus,
the effect of the stagnant gas atmosphere on particle–particle radiation is neglected
in the following sections.
5.2.2.2 Validation by Detailed DEM
The approximation function model can be applied to analyze numerical radiation
methods in a particle scale. For example, in the Short-range Radiation Model (SRM)
[5], only radiation flux between Voronoï neighbors was considered. For a bed packed
by particles arranged in the Face-Centered Cubic (FCC) lattice, the radial distribution
function and porosity for Voronoï neighbors are given by Eqs. (5.17) and (5.18).
The radiative effective thermal conductivity of Eq. (5.41) is Eq. (5.19) at ε r = 1.
It is the same as the results of [22] for the structured packing of black radiation.
Moreover, the Radiation Interaction Function (RIF) in SRM of the random packing
is reduced to
h
0
(η) =
erfc(bη), η ≤ η 0
0, else
(5.49)
where η 0 =
L v
d
and L v is the average distance of two Voronoï neighboring pairs.
From DEM results shown in Fig. 5.7a, it is η 0 = 1.11 at ε r = 0.39, and the radiation
exchange factor F of Eq. (5.44) is 0.6484 for black radiation. In [8, 29], RIF in the
radiation model of CFD-DEM simulation can be equivalently written as
h
0
(η) =
1, η ≤ 1.5
0, else
(5.50)
The result of random packing is F = 0.7968 at α f = 0.39 and ε r = 1.
In the current approximation function model, the view factor for a packed particle
bed is replaced by the Radiation Interaction Function (RIF). For the HTR-10 packed
with spheres, a general agreement can be seen in Fig. 5.9 between the RIF and detailed
251
Additionally, for the packed beds of an isothermal absorbing medium of uniform
temperature, the particle–particle radiative flux is then formulated as follows
ˆ
Q r,i j = f (ε r )A i X i j σ (T
4
i − T
4
j ) exp(−β · r i j ),
(5.47)
where β is an attenuation coefficient. Similarly, in this case, the radiation exchange
factor of the packed bed is
F =
k r
4σ T 3 d
= ε r (1 − α f )
+∞
1
h(η)g(η) exp (−βdη)η
4 dη
+∞
1
h(η)g(η)η 2 dη
(5.48)
It is noted that βd 1 should be met for common gases, such as air and helium. Thus,
the effect of the stagnant gas atmosphere on particle–particle radiation is neglected
in the following sections.
5.2.2.2 Validation by Detailed DEM
The approximation function model can be applied to analyze numerical radiation
methods in a particle scale. For example, in the Short-range Radiation Model (SRM)
[5], only radiation flux between Voronoï neighbors was considered. For a bed packed
by particles arranged in the Face-Centered Cubic (FCC) lattice, the radial distribution
function and porosity for Voronoï neighbors are given by Eqs. (5.17) and (5.18).
The radiative effective thermal conductivity of Eq. (5.41) is Eq. (5.19) at ε r = 1.
It is the same as the results of [22] for the structured packing of black radiation.
Moreover, the Radiation Interaction Function (RIF) in SRM of the random packing
is reduced to
h
0
(η) =
erfc(bη), η ≤ η 0
0, else
(5.49)
where η 0 =
L v
d
and L v is the average distance of two Voronoï neighboring pairs.
From DEM results shown in Fig. 5.7a, it is η 0 = 1.11 at ε r = 0.39, and the radiation
exchange factor F of Eq. (5.44) is 0.6484 for black radiation. In [8, 29], RIF in the
radiation model of CFD-DEM simulation can be equivalently written as
h
0
(η) =
1, η ≤ 1.5
0, else
(5.50)
The result of random packing is F = 0.7968 at α f = 0.39 and ε r = 1.
In the current approximation function model, the view factor for a packed particle
bed is replaced by the Radiation Interaction Function (RIF). For the HTR-10 packed
with spheres, a general agreement can be seen in Fig. 5.9 between the RIF and detailed