214
4 Numerical Methods and Simulation for Pebble Flows
body be the origin point (Fig. 4.25), the combined distribution of void fraction in the
r − z plane follows the form of
c (r, z) = c (z) = α c + β c z,
(4.29)
where the coefficients α c and β c can be determined by the mean value of void fraction,
as well as the boundary conditions at the top and the bottom of the cylindrical
volume. The combined distribution is independent of r. In the conical base, the
radial distribution on any height is assumed to follow the Gaussian form
b (r, z) = c b (z) + A b (z)exp
−
r
2
2w
2
b (z)
,
(4.30)
where A b (z), w b (z), c b (z) are the coefficients varied in z-direction. By integrating Eq.
(4.30), in the radial and circumferential directions, the cross-area averaged variation
of the void fraction can be obtained
b (z) =
1
π R 2 (z)
2π
0
d θ
R(z)
0
b (r, z)rdr = c b (z) +
2A b (z)
R 2 (z)
R(z)
0
exp
−
r 2
2w 2
b (z)
rdr
= c b (z) +
2A b (z)w 2
b (z)
R 2 (z)
1 − exp
−
R 2 (z)
2w 2
b (z)
≈ c b (z) +
2A b (z)w 2
b (z)
R 2 (z)
, when
R 2 (z)
w 2
b (z)
1
= α b + β b z,
(4.31)
where R(z) is the radius of the bed at z, and R = max (R(z)) is also the radius of the
cylindrical volume. The last equal sign represents the linear distribution in the axial
direction inside the conical base.
To linearize the right-hand terms of Eq. 4.31, it is assumed that w b (z) = w b =
Const., which means the width of the Gaussian function is independent of the height.
Moreover, as R(z) ∝ z, it is appropriate to assume A b (z) = A b,0 + A b,1 z + A b,2 z
2
∝
R(z)
2 , or
A b (z)
R ( z)
=
A b (0)
R ( 0)
=
A b,0
R 2 = γ , c b (z) = c b0 + c b1 z, where A b,i , c b,i and γ are
constant coefficients. With R(z) = R +
√
3z, we have
A b,0 = γ R
2
, A b,1 = 2
√
3γ R, A b,2 = 3γ.
(4.32)
Then,
2A b (z)w
2
b (z)
R ( z)
= 2γ w
2
b . For the last equal sign of Eq. 4.31, we get
α b = c b,0 + 2γ w
2
b , and β b = c b1 .
(4.33)
The last equal sign of Eq. (4.31), can be written by taking the fitted radial and
axial distributions of the void fraction (Table 4.12), it can depict the steady joint
4 Numerical Methods and Simulation for Pebble Flows
body be the origin point (Fig. 4.25), the combined distribution of void fraction in the
r − z plane follows the form of
c (r, z) = c (z) = α c + β c z,
(4.29)
where the coefficients α c and β c can be determined by the mean value of void fraction,
as well as the boundary conditions at the top and the bottom of the cylindrical
volume. The combined distribution is independent of r. In the conical base, the
radial distribution on any height is assumed to follow the Gaussian form
b (r, z) = c b (z) + A b (z)exp
−
r
2
2w
2
b (z)
,
(4.30)
where A b (z), w b (z), c b (z) are the coefficients varied in z-direction. By integrating Eq.
(4.30), in the radial and circumferential directions, the cross-area averaged variation
of the void fraction can be obtained
b (z) =
1
π R 2 (z)
2π
0
d θ
R(z)
0
b (r, z)rdr = c b (z) +
2A b (z)
R 2 (z)
R(z)
0
exp
−
r 2
2w 2
b (z)
rdr
= c b (z) +
2A b (z)w 2
b (z)
R 2 (z)
1 − exp
−
R 2 (z)
2w 2
b (z)
≈ c b (z) +
2A b (z)w 2
b (z)
R 2 (z)
, when
R 2 (z)
w 2
b (z)
1
= α b + β b z,
(4.31)
where R(z) is the radius of the bed at z, and R = max (R(z)) is also the radius of the
cylindrical volume. The last equal sign represents the linear distribution in the axial
direction inside the conical base.
To linearize the right-hand terms of Eq. 4.31, it is assumed that w b (z) = w b =
Const., which means the width of the Gaussian function is independent of the height.
Moreover, as R(z) ∝ z, it is appropriate to assume A b (z) = A b,0 + A b,1 z + A b,2 z
2
∝
R(z)
2 , or
A b (z)
R ( z)
=
A b (0)
R ( 0)
=
A b,0
R 2 = γ , c b (z) = c b0 + c b1 z, where A b,i , c b,i and γ are
constant coefficients. With R(z) = R +
√
3z, we have
A b,0 = γ R
2
, A b,1 = 2
√
3γ R, A b,2 = 3γ.
(4.32)
Then,
2A b (z)w
2
b (z)
R ( z)
= 2γ w
2
b . For the last equal sign of Eq. 4.31, we get
α b = c b,0 + 2γ w
2
b , and β b = c b1 .
(4.33)
The last equal sign of Eq. (4.31), can be written by taking the fitted radial and
axial distributions of the void fraction (Table 4.12), it can depict the steady joint