5.3 Discrete Modeling of Pebble Radiation
277
T
4
j − T
4
i = 4T
3
(T j − T i ) when T j tends to T i [10, 54], k r of SC and FCC packing
can be derived as
k r,SC = 4σ d p T
3 πγ SC
2(1−εr )
εr
+6
6
π
(1 − α f )
1
3
(5.99)
k r,FCC = 4σ d p T
3
2
√
2πγ FCC
2(1−εr )
εr
+12
6
√
2π
(1 − α f )
1
3
(5.100)
For common cases or the cells of random packing, the view factor can be calculated by an analytical approach [5, 37]. For the truncated octahedron of the BCC
packing, the view factor was V i, j = 0.09841929 for hexagons and V i, j = 0.0354409
for squares. The effective thermal conductivity of particle radiation can be written as
k r,HEX = 4σ d p T
3
4π
γ BCC
3
2(1−εr )
εr
+10.1606
8
√
3π
(1 − α f )
1
3
(5.101)
k r,SQU = 4σ d p T
3
4
√
3πγ BCC
2(1−εr )
εr
+28.21595
8
√
3π
(1 − α f )
1
3
(5.102)
For the whole cell, k r is the weighted average of the hexagons and squares. It is
k r,BCC =
8S Hexagon k r,HEX + 6S Square k r,SQU
8S Hexagon + 6S Square
=
12
√
3k r,HEX + 6k r,SQU
12
√
3 + 6
. (5.103)
For the pebble-bed experiments and nuclear reactors [21, 35, 38], the basic pebbles
were graphite spheres of 60 mm in diameter. The surface emissivity was 0.8 and the
average void fraction was 0.39 according to the parameters used in the pebble-bed
study [7]. Using Eq. (5.100) for FCC and Eqs. (5.101) and (5.102) for BCC, together
with the modification factor and view factor for FCC and BCC in Sect. 5.3.8.1, it is
easy to compute the effective thermal conductivity of the pebble bed at the particle
diameter (d p = 60 mm), surface emissivity (ε r = 0.8), and average void fraction
(α f = 0.39). As shown in Fig. 5.31a, the Short-range Radiation Model (SRM) of
BCC packing for the pebble bed is in good agreement with the numerical results
of random packing with the same material properties in [7]. Compared with the
empirical correlations, which are found in [14, 15], it can be seen in Fig. 5.31b that
the SRM results of BCC and FCC are in good agreement with the correlations for
the pebble bed, and it is only slightly higher when the temperature exceeds 1,200
◦ C.
5.3.8.3 Sub-Cell Model (SCM) Solution
Even though the Short-range Radiation Model (SRM) is applicable to the core of
HTGR of dense random packing, a general theoretical approach is essential for
modeling ETC of thermal radiation. Like the fact that the results of BCC and FCC
277
T
4
j − T
4
i = 4T
3
(T j − T i ) when T j tends to T i [10, 54], k r of SC and FCC packing
can be derived as
k r,SC = 4σ d p T
3 πγ SC
2(1−εr )
εr
+6
6
π
(1 − α f )
1
3
(5.99)
k r,FCC = 4σ d p T
3
2
√
2πγ FCC
2(1−εr )
εr
+12
6
√
2π
(1 − α f )
1
3
(5.100)
For common cases or the cells of random packing, the view factor can be calculated by an analytical approach [5, 37]. For the truncated octahedron of the BCC
packing, the view factor was V i, j = 0.09841929 for hexagons and V i, j = 0.0354409
for squares. The effective thermal conductivity of particle radiation can be written as
k r,HEX = 4σ d p T
3
4π
γ BCC
3
2(1−εr )
εr
+10.1606
8
√
3π
(1 − α f )
1
3
(5.101)
k r,SQU = 4σ d p T
3
4
√
3πγ BCC
2(1−εr )
εr
+28.21595
8
√
3π
(1 − α f )
1
3
(5.102)
For the whole cell, k r is the weighted average of the hexagons and squares. It is
k r,BCC =
8S Hexagon k r,HEX + 6S Square k r,SQU
8S Hexagon + 6S Square
=
12
√
3k r,HEX + 6k r,SQU
12
√
3 + 6
. (5.103)
For the pebble-bed experiments and nuclear reactors [21, 35, 38], the basic pebbles
were graphite spheres of 60 mm in diameter. The surface emissivity was 0.8 and the
average void fraction was 0.39 according to the parameters used in the pebble-bed
study [7]. Using Eq. (5.100) for FCC and Eqs. (5.101) and (5.102) for BCC, together
with the modification factor and view factor for FCC and BCC in Sect. 5.3.8.1, it is
easy to compute the effective thermal conductivity of the pebble bed at the particle
diameter (d p = 60 mm), surface emissivity (ε r = 0.8), and average void fraction
(α f = 0.39). As shown in Fig. 5.31a, the Short-range Radiation Model (SRM) of
BCC packing for the pebble bed is in good agreement with the numerical results
of random packing with the same material properties in [7]. Compared with the
empirical correlations, which are found in [14, 15], it can be seen in Fig. 5.31b that
the SRM results of BCC and FCC are in good agreement with the correlations for
the pebble bed, and it is only slightly higher when the temperature exceeds 1,200
◦ C.
5.3.8.3 Sub-Cell Model (SCM) Solution
Even though the Short-range Radiation Model (SRM) is applicable to the core of
HTGR of dense random packing, a general theoretical approach is essential for
modeling ETC of thermal radiation. Like the fact that the results of BCC and FCC