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5 Numerical Models for Pebble-Bed Heat Transfer
5.3.10.1 Analytical Description
The Radiation Transfer Equation (RTE) can be simplified as a type of heat diffusion
[83, 84]. The radiation flux is formulated as
q r = −k r ∇T,
(5.132)
where k r is the radiative Effective Thermal Conductivity (ETC). This equation is
commonly called as Rosseland approximation. Generally, for the conductive and
radiative heat transfer, the total heat flux in the pebble beds can be written as
q t = −k e ∇T,
(5.133)
where k e is the total Effective Thermal Conductivity (ETC) of the bed, which includes
the radiation part (k r ) and the conduction part (k c ), i.e., k e = k r + k c . T is the absolute particle temperature in Kelvin, and it is directly measured by thermocouples
at different positions in the experimental facility [15, 21, 24]. Moreover, the radiation flux significantly increases with the temperature and dominates the heat transfer
processes at high-temperature ranges. When the effect of thermal conductivity of
particle (k s ) on the radiation is considered, the radiant effective thermal conductivity
for the packed beds of spherical particles can be obtained by the widely accepted
Schotte equation [17], which is given by Eq. (5.108). In Eq. (5.108),
k ∞ = F · 4σ T
3 d p
(5.134)
is the ETC when k s is infinity, and F is the radiation exchange factor, given by (Eq.
(5.112)), i.e.,
F =
3
8Γ
5
3
√
2π
6
1
3 (1 − α f )
1
3
α f
1
1−ε r
ε r
+
1
1+V 12
(5.135)
In SCM, it indicates that when the solid conductivity is far larger than the radiative
effective thermal conductivity, i.e., k s k r , k r is directly proportional to the cube
of the particle absolute temperature. In Eq. (5.135), V 12 is the view factor, and is
0.3955 at α f = 0.39 and 0.3561 at α f = 0.44. Table 5.1 lists the models developed
by scientists for radiation exchange factor computation.
5.3.10.2 Comparison with Experiment
Comparisons between the Sub-Cell radiation Model (SCM) (Eqs. (5.108), (5.134),
and (5.135)) and experimental data under various conditions are shown in Figs. 5.48
and 5.49. In Fig.5.48, the experimental data of the pebble bed are taken from the
HTTU experiment [21] and TF-PBEC [24]. Both beds are filled with many graphite
5 Numerical Models for Pebble-Bed Heat Transfer
5.3.10.1 Analytical Description
The Radiation Transfer Equation (RTE) can be simplified as a type of heat diffusion
[83, 84]. The radiation flux is formulated as
q r = −k r ∇T,
(5.132)
where k r is the radiative Effective Thermal Conductivity (ETC). This equation is
commonly called as Rosseland approximation. Generally, for the conductive and
radiative heat transfer, the total heat flux in the pebble beds can be written as
q t = −k e ∇T,
(5.133)
where k e is the total Effective Thermal Conductivity (ETC) of the bed, which includes
the radiation part (k r ) and the conduction part (k c ), i.e., k e = k r + k c . T is the absolute particle temperature in Kelvin, and it is directly measured by thermocouples
at different positions in the experimental facility [15, 21, 24]. Moreover, the radiation flux significantly increases with the temperature and dominates the heat transfer
processes at high-temperature ranges. When the effect of thermal conductivity of
particle (k s ) on the radiation is considered, the radiant effective thermal conductivity
for the packed beds of spherical particles can be obtained by the widely accepted
Schotte equation [17], which is given by Eq. (5.108). In Eq. (5.108),
k ∞ = F · 4σ T
3 d p
(5.134)
is the ETC when k s is infinity, and F is the radiation exchange factor, given by (Eq.
(5.112)), i.e.,
F =
3
8Γ
5
3
√
2π
6
1
3 (1 − α f )
1
3
α f
1
1−ε r
ε r
+
1
1+V 12
(5.135)
In SCM, it indicates that when the solid conductivity is far larger than the radiative
effective thermal conductivity, i.e., k s k r , k r is directly proportional to the cube
of the particle absolute temperature. In Eq. (5.135), V 12 is the view factor, and is
0.3955 at α f = 0.39 and 0.3561 at α f = 0.44. Table 5.1 lists the models developed
by scientists for radiation exchange factor computation.
5.3.10.2 Comparison with Experiment
Comparisons between the Sub-Cell radiation Model (SCM) (Eqs. (5.108), (5.134),
and (5.135)) and experimental data under various conditions are shown in Figs. 5.48
and 5.49. In Fig.5.48, the experimental data of the pebble bed are taken from the
HTTU experiment [21] and TF-PBEC [24]. Both beds are filled with many graphite