5.3 Discrete Modeling of Pebble Radiation
287
F = 1 −
k r
4σ d p T 3
(5.122)
From the mechanism of the thermal radiation, the radiation exchange factor should
fulfill the following conditions
lim
ε r →0
F(ε r ) = 0,
d F(ε r )
dε r
> 0.
(5.123)
Based on the experimental measurement, Nasr model [15] is given as
F = ε r .
(5.124)
From Argo model [27], the radiation exchange factor is written as
F =
1
2
ε r
− 1
(5.125)
And in Wakao model [73], F is formulated as
F =
2
2
ε r
− 0.264
(5.126)
The results of SCM and the models mentioned above in nuclear-packed pebble beds
of dense packing (α f = 0.39) are shown in Fig. 5.41. For all considered models, the
radiation exchange factor increases with the surface emissivity, and F approaches 0
when ε r goes to 0. The SCM is in good agreement with the Nasr model and Wakao
model, and it is slightly larger than the Argo model.
5.3.9.3 Particle-Scale Heat Transfer Model
Combined with the Discrete Element Method (DEM), the particle motion of elastic
Hertz–Mindlin model [74, 75] and the heat transfer equations at time t are written
as
m i
du i
dt
=
n
j=1
(F n,i j + F t,i j ) + m i g + F f,i ,
I i
dω i
dt
= M r + R i ×
n
j=1
(F n,i j + F t,i j ),
m i C p,i
dT i
dt
= −k i j
A i j
L i j
(T i − T j ) + Q f + Q s ,
(5.127)