5.5 Further Issues
373
F =
k e
4σ d p T 3 = ε r (1 − ϕ)
1
N
tr
X h
T
,
(5.250)
where d p is the particle diameter. Herein, the dimensionless distance matrix is defined
as h =
h
2
i j
N ×N
and h i j =
r i j
d p
.
5.5.3.2 Obstructed View Factor
It can be seen in the above section that the view factor matrix is the critical issue
in predicting the full ranges of radiative flux and the effective thermal conductivity
in large-scale packed beds. Thus, the important task is to accurately compute the
obstructed view factor to surrounding ones for every particle in the bed. For the
three-dimensional bed of mono-sized spheres, the view factor function between two
spheres (i and j), which may be obstructed by particles, like 1, 2, …, n, should
satisfy the restrictions of (1). X i j ≥ 0; (2). X i j = 0 for i = j; (3). X i j = X ji . It is a
complex continuous function of particle positions, i.e.,
X i j = f
P i , P j , S n
,
(5.251)
where P i and P j are the sphere centers of particle i and j, respectively. S n is the set
of all surrounding spheres and S n = {P 1 , P 2 , . . . , P n }. It is defined that the distance
from P k+1 to the line of P i P j is larger than that from the point P k to P i P j , and
S 1 = {P 1 }, for S k = S k−1 ∪ {P k }. The view factor function fulfills
f
P i , P j , S n
≤ f
P i , P j , S n−1
≤ f
P i , P j , S n−2
≤ · · · ≤ f
P i , P j
,
(5.252)
where f
P i , P j
is the view factor of two spheres without obstruction. Equation
(5.252) means that if any particle P k is located between P i and P j , the P k plays the
role of obstruction, which may cause the view factor f
P i , P j
between P i and P j
reduced. More P k leads to more severely reduced view factor f
P i , P j
. If P k is
far from particle P i and particle P j , it can be deleted from the set S n . This means that
if f
P i , P j , P n
= f
P i , P j
, it may have f
P i , P j , S n
= f
P i , P j , S n−1
.
The Monte Carlo method by the ray tracing technique [158] is applied here to
building the dataset of the obstructed view factor. For the case with only two spheres, it
becomes f
P i , P j
= f
| P i , P j |
. Figure 5.111 shows that the numerical results
having 1.2×10
8 rays from each sphere are in good agreement with the analytical
equation of [5]. There are no general analytical solutions for the complex cases of
f
P i , P j , P n
and f
P i , P j , S n
. Thus, in the Monte Carlo method, 1.2 × 10
8
rays are required for every sphere in the following simulations.
For example, Fig. 5.112a illustrates the case with three spheres, the view factor
f ( P 1 , P 2 , P 3 ) from sphere P 1 to sphere P 2 may be affected by the position of
sphere P 3 . The center of sphere P 3 may be located in the regions of A, B, C, or D
(Fig. 5.112b). Then