4.3 Three-Dimensional Pebble Flow
215
distribution of void fraction throughout the bed. For example, in Table 4.12), we can
find β c = 1.16 × 10
−2 , and
α c = 9.70 × 10
−2
(4.34)
where the analytical counterpart of α c is 0.093, which is regarded as the close hexagonal packing state. Moreover, Table 4.12 shows β b = c b1 = −2.90 × 10
−2 , and
b (r) =
1
h
h
0
b (r, z)dz =
1
h
h
0
(c b,0 + c b,1 z) + (A b,0 + A b,1 z + A b,2 z
2 )exp
−
r 2
2w 2
b
dz
= c b,0 +
h
2
c b,1 + (A b,0 + A b,1
h
2
+ A b,2
h 2
3
)exp
−
r 2
2w 2
b
.
(4.35)
Comparing Eq. 4.35, to the fitted equations of Table 4.12, we can find c b,0 +
h
2
c b,1 =
1.80 × 10
−1 and A b,0 + A b,1
h
2
+ A b,2
h
2
3
= 2.74 × 10
−1 . By substituting Eq. 4.32 and
h = -6.255, R = 15 into above equations, the coefficients can be solved as follows:
⎧
⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨
⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩
γ = 2.08 × 10
−3
,
A b,0 = 4.68 × 10
−1
,
A b,1 = 1.08 × 10
−1
,
A b,2 = 6.24 × 10
−3
,
c b,0 = 8.93 × 10
−2
,
c b,1 = −2.90 × 10
−2
(4.36)
These coefficients in the junction region between the cylindrical volume and the
conical base can be checked. Equations 4.29 and 4.31 are connected in the junction
region at z = 0. The following condition should be satisfied
c (0) ≈ b (0), or α c ≈ α b .
(4.37)
In Table 4.12, the approximation of α c = 9.70 × 10
−2
≈ α b = 1.00 × 10
−1 is reasonable. α b = c b,0 + 2c 2 w
2
b = 1.25 × 10
−1 is obtained by Eq. (4.33). The agreement
between these values seems to be satisfactory, and the small discrepancy between
them is mainly caused by the truncation error in Eq. 4.31.
Thus, the joint radial and axial distribution of void fraction is fully described by
the following equation:
b (r, z) = c b,0 + c b,1 z + (A b,0 + A b,1 z + A b,2 z
2
)exp
−
r
2
2w
2
b
.
(4.38)
It is a combined linear and normal distribution, where the linear distribution is caused
by the effect of gravity and the normal distribution is caused by the void introduction
and transfer in the axial and radial directions. It is still noticed that Eq. (4.38),
is a steady distribution, i.e., time-independent. Moreover, the amplitude of normal
215
distribution of void fraction throughout the bed. For example, in Table 4.12), we can
find β c = 1.16 × 10
−2 , and
α c = 9.70 × 10
−2
(4.34)
where the analytical counterpart of α c is 0.093, which is regarded as the close hexagonal packing state. Moreover, Table 4.12 shows β b = c b1 = −2.90 × 10
−2 , and
b (r) =
1
h
h
0
b (r, z)dz =
1
h
h
0
(c b,0 + c b,1 z) + (A b,0 + A b,1 z + A b,2 z
2 )exp
−
r 2
2w 2
b
dz
= c b,0 +
h
2
c b,1 + (A b,0 + A b,1
h
2
+ A b,2
h 2
3
)exp
−
r 2
2w 2
b
.
(4.35)
Comparing Eq. 4.35, to the fitted equations of Table 4.12, we can find c b,0 +
h
2
c b,1 =
1.80 × 10
−1 and A b,0 + A b,1
h
2
+ A b,2
h
2
3
= 2.74 × 10
−1 . By substituting Eq. 4.32 and
h = -6.255, R = 15 into above equations, the coefficients can be solved as follows:
⎧
⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨
⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩
γ = 2.08 × 10
−3
,
A b,0 = 4.68 × 10
−1
,
A b,1 = 1.08 × 10
−1
,
A b,2 = 6.24 × 10
−3
,
c b,0 = 8.93 × 10
−2
,
c b,1 = −2.90 × 10
−2
(4.36)
These coefficients in the junction region between the cylindrical volume and the
conical base can be checked. Equations 4.29 and 4.31 are connected in the junction
region at z = 0. The following condition should be satisfied
c (0) ≈ b (0), or α c ≈ α b .
(4.37)
In Table 4.12, the approximation of α c = 9.70 × 10
−2
≈ α b = 1.00 × 10
−1 is reasonable. α b = c b,0 + 2c 2 w
2
b = 1.25 × 10
−1 is obtained by Eq. (4.33). The agreement
between these values seems to be satisfactory, and the small discrepancy between
them is mainly caused by the truncation error in Eq. 4.31.
Thus, the joint radial and axial distribution of void fraction is fully described by
the following equation:
b (r, z) = c b,0 + c b,1 z + (A b,0 + A b,1 z + A b,2 z
2
)exp
−
r
2
2w
2
b
.
(4.38)
It is a combined linear and normal distribution, where the linear distribution is caused
by the effect of gravity and the normal distribution is caused by the void introduction
and transfer in the axial and radial directions. It is still noticed that Eq. (4.38),
is a steady distribution, i.e., time-independent. Moreover, the amplitude of normal