5.3 Discrete Modeling of Pebble Radiation
299
Fig. 5.51 Voronoï tessellations of two bonded particles of three sphere elements
l
i j
k
m
=
l
i
k
i
+
l
j
k
l
(5.139)
where l
i and l
j are the distances from the sphere center to the Voronoï face shown
in Fig. 5.51. k
i of the sphere “i” is
k
i = k c + k r,SCM (T i , α
f,i , ε
r,i , d
p , k s ),
(5.140)
where k r,SCM (T i , α
f,i , ε
r,i , d
p , k s ) is the effective thermal conductivity of the SubCell radiation Model (SCM) (Eqs. (5.108), (5.134), and (5.135)). The local porosity
α
f,i is defined as
α
f,i = 1 −
V sphere
V
cell
(5.141)
where V sphere and V
cell are the volume of the element sphere and the volume of its
Voronoï cell, respectively. Specifically, for the sphere-packed bed of Simple Cubic
(SC) packing (α
f,i = 0.476), the flux of Eq. (5.138) based on its Voronoï cell [6] is
re-written as
Q e,i j =
T j − T i
R t
=
T j − T i
1
k
i d p
(5.142)
299
Fig. 5.51 Voronoï tessellations of two bonded particles of three sphere elements
l
i j
k
m
=
l
i
k
i
+
l
j
k
l
(5.139)
where l
i and l
j are the distances from the sphere center to the Voronoï face shown
in Fig. 5.51. k
i of the sphere “i” is
k
i = k c + k r,SCM (T i , α
f,i , ε
r,i , d
p , k s ),
(5.140)
where k r,SCM (T i , α
f,i , ε
r,i , d
p , k s ) is the effective thermal conductivity of the SubCell radiation Model (SCM) (Eqs. (5.108), (5.134), and (5.135)). The local porosity
α
f,i is defined as
α
f,i = 1 −
V sphere
V
cell
(5.141)
where V sphere and V
cell are the volume of the element sphere and the volume of its
Voronoï cell, respectively. Specifically, for the sphere-packed bed of Simple Cubic
(SC) packing (α
f,i = 0.476), the flux of Eq. (5.138) based on its Voronoï cell [6] is
re-written as
Q e,i j =
T j − T i
R t
=
T j − T i
1
k
i d p
(5.142)