240
5 Numerical Models for Pebble-Bed Heat Transfer
k c =
N c
6
ρ 0 H 1 d
2
,
(5.10)
where H 1 = H (d) =
1
R i j
and R i j is the average contact resistance.
On the other hand, the radiative heat transfer in a packed bed is of long-range
interaction. The ETC of the thermal radiation in Eq. (5.8) is then
k r =
8π
3
ε r σ T
3 A i ρ
2
0
+∞
0
X (r i j )g(r i j )r
4
i j dr i j .
(5.11)
For black radiation (ε r = 1), i.e., T j = 0 for all surrounding ones, it is Q i = E b,i in
Eq. (5.5). Additionally, the radial distribution function is subject to the normalization
condition for the radiation:
ρ 0
R 3
X (r i j )g(r i j )dx j dy j dz j = 1
(5.12)
5.2.1.1 Validation
In general, the average number density in a packed bed can be written as
ρ 0 =
1 − α f
V i
(5.13)
where α f and V i are porosity of the bed, and the particle volume, respectively. The
ETC of the conduction in Eq. (5.10) is given as
k c =
N c
π R i j d
(1 − α f ).
(5.14)
For the structured Simple Cubic (SC) packing, the coordination number N c is 6
and α f = 1 −
π
6
. For the Face-Centered Cubic (FCC) packing, it is N c = 12 and
α f = 1 −
√
2π
6
. The ETC becomes
k c,SC =
1
R i j d
k c,FCC =
2
√
2
R i j d
(5.15)
Equation (5.15) is the same as in [11]. For the random packing, the correlation [13]
to calculate the coordination number is given as
N c = 2.02
1 + 87.38(1 − α f )
4
1 + 25.81(1 − α f ) 4
(5.16)