5.3 Discrete Modeling of Pebble Radiation
275
(α f ), are used to study spatial characteristics of the structured packing. For Simple
Cubic (SC) packing, A 0 = a
2 and L 0 = a, in which a is the edge length of the
Voronoï cell. The diameter of the inscribed sphere of the cell is d i = a. A particle
inside the cell is not larger than the inscribed sphere and d p = β 1 d i (0 < β 1 ≤ 1).
The void fraction of the packing is α f = 1 −
πβ
3
1
6
≥ 0.4764.
For Body-Centered Cubic (BCC) packing, the surfaces of the cell are 8 identical
hexagons and 6 squares. The diameter of the inscribed sphere is d i =
√
6a and a is the
edge length of the Voronoï cell. The particle diameter is d p = β 2 d i and 0 ≤ β ≤ 1.
The void fraction is α f = 1 −
√
3πβ
3
2
8
≥ 0.3198. For hexagons, the surface area and
the distance are A 0 =
3
√
3a
2
2
and L 0 =
√
6a, respectively. For squares, it will be
A 0 = a
2 , L 0 = 2
√
2a.
For Face-Centered Cubic (FCC) packing, the surfaces of the cell are 12 identical
rhombic planes. The surface area and the distance are A 0 =
2
√
2a
2
3
and L 0 =
2
√
6a
3
,
respectively, in which a is the cell edge length. The diameter of the inscribed sphere
is d i =
2
√
6a
3
. The particle diameter is d p = β 3 d i and 0 < β 3 ≤ 1. Also, the void
fraction of FCC is α f = 1 −
√
2πβ
3
3
6
≥ 0.25952.
A non-dimensional parameter, namely cell-particle area ratio η, is defined by the
ratio between the surface areas of the cell and the particle. For Simple Cubic (SC)
packing, it can be calculated by
η SC =
A cell
A p
=
6a
2
πβ
2
1 a 2 =
6
π
1
3 (1 − α f )
−
2
3
(5.92)
For the BCC and FCC packing, it can be written as
η BCC =
A cell
A p
=
12
√
3a
2 +6a
2
6πβ
2
2 a 2
=
2
√
3+1
π
8
√
3π
−
2
3 (1 − α f )
−
2
3
(5.93)
η FCC =
A cell
A p
=
8
√
2a
2
8
3 πβ
2
2 a 2 =
3
√
2
π
6
√
2π
−
2
3 (1 − α f )
−
2
3
(5.94)
From the analytical results of Voronoï tessellations [50–52], the cell-particle area
ratio of random packing can be expressed by
η T =
4
3
= Γ
5
3
(1 − α f )
−
2
3
(5.95)
where Γ (·) is the gamma function.
By the comparison shown in Fig. 5.30, it can be seen that the present effective heat
cells are in good agreement with the numerical results [53] and the analytical solution
of random packing. However, for an accurate evaluation, a random modification
factor γ was considered, and the surface area in Eq. (5.91) was defined as A
=
A 0
γ
.
275
(α f ), are used to study spatial characteristics of the structured packing. For Simple
Cubic (SC) packing, A 0 = a
2 and L 0 = a, in which a is the edge length of the
Voronoï cell. The diameter of the inscribed sphere of the cell is d i = a. A particle
inside the cell is not larger than the inscribed sphere and d p = β 1 d i (0 < β 1 ≤ 1).
The void fraction of the packing is α f = 1 −
πβ
3
1
6
≥ 0.4764.
For Body-Centered Cubic (BCC) packing, the surfaces of the cell are 8 identical
hexagons and 6 squares. The diameter of the inscribed sphere is d i =
√
6a and a is the
edge length of the Voronoï cell. The particle diameter is d p = β 2 d i and 0 ≤ β ≤ 1.
The void fraction is α f = 1 −
√
3πβ
3
2
8
≥ 0.3198. For hexagons, the surface area and
the distance are A 0 =
3
√
3a
2
2
and L 0 =
√
6a, respectively. For squares, it will be
A 0 = a
2 , L 0 = 2
√
2a.
For Face-Centered Cubic (FCC) packing, the surfaces of the cell are 12 identical
rhombic planes. The surface area and the distance are A 0 =
2
√
2a
2
3
and L 0 =
2
√
6a
3
,
respectively, in which a is the cell edge length. The diameter of the inscribed sphere
is d i =
2
√
6a
3
. The particle diameter is d p = β 3 d i and 0 < β 3 ≤ 1. Also, the void
fraction of FCC is α f = 1 −
√
2πβ
3
3
6
≥ 0.25952.
A non-dimensional parameter, namely cell-particle area ratio η, is defined by the
ratio between the surface areas of the cell and the particle. For Simple Cubic (SC)
packing, it can be calculated by
η SC =
A cell
A p
=
6a
2
πβ
2
1 a 2 =
6
π
1
3 (1 − α f )
−
2
3
(5.92)
For the BCC and FCC packing, it can be written as
η BCC =
A cell
A p
=
12
√
3a
2 +6a
2
6πβ
2
2 a 2
=
2
√
3+1
π
8
√
3π
−
2
3 (1 − α f )
−
2
3
(5.93)
η FCC =
A cell
A p
=
8
√
2a
2
8
3 πβ
2
2 a 2 =
3
√
2
π
6
√
2π
−
2
3 (1 − α f )
−
2
3
(5.94)
From the analytical results of Voronoï tessellations [50–52], the cell-particle area
ratio of random packing can be expressed by
η T =
4
3
= Γ
5
3
(1 − α f )
−
2
3
(5.95)
where Γ (·) is the gamma function.
By the comparison shown in Fig. 5.30, it can be seen that the present effective heat
cells are in good agreement with the numerical results [53] and the analytical solution
of random packing. However, for an accurate evaluation, a random modification
factor γ was considered, and the surface area in Eq. (5.91) was defined as A
=
A 0
γ
.