206
4 Numerical Methods and Simulation for Pebble Flows
4.3.1.2 Radial Void Fraction
The concept of the radial void fraction is widely utilized in early studies, which is
defined in terms of a summation of solid volume segments ΔV i of the sphere “i”
within the radial layers ΔV rl , namely [64, 65]
(r) = 1 −
1
ΔV rl
N
i=1
ΔV i ,
(4.24)
where N is the number of spheres intersected by the radial layer. In addition, an
area-based numerical procedure is used by Toit [57], to evaluate the void fraction
(r) = 1 −
1
A r
N
i=1
A i,r
(4.25)
where A is the area of the cylindrical plane at radius r, and A i,r is the intersection
area between sphere i and the cylindrical plane. In this analysis, a three-dimensional
pebble-discharging bed is simulated. Thus, Eq. (4.2), is used based on the volume method for the void fraction calculation as it is suitable for the present threedimensional bed [65].
As the pebble bed is axisymmetrical, the circumferential distribution of void
fraction should be omitted. Thus, the three-dimensional void fraction characteristics
can adequately be described by the radial and axial distributions. Moreover, the
pebble bed can be divided into a cylindrical volume and a conical base (Fig. 4.25).
They are needed to be studied separately.
Radial Distribution of Void Fraction
The Cylindrical Volume
As a first step, this section shows the radial void fraction profile in the cylindrical
volume of the pebble bed. The radial distribution is formulated as follows
(r) = 1 −
1
π(r
2
i − r
2
i−1 )δh
N
j=1
ΔV j | h 1 ≤z(δV j )≤h 2 ,r i−1 ≤r(δV j )
where δh = h 2 − h 1 is the height of the cylindrical volume. r i is the division of bed
radius. In Eq. 4.26, the bed radius is divided into 360 divisions, with division width
about 1/24 of a pebble diameter, i.e. R p δ h .
4 Numerical Methods and Simulation for Pebble Flows
4.3.1.2 Radial Void Fraction
The concept of the radial void fraction is widely utilized in early studies, which is
defined in terms of a summation of solid volume segments ΔV i of the sphere “i”
within the radial layers ΔV rl , namely [64, 65]
(r) = 1 −
1
ΔV rl
N
i=1
ΔV i ,
(4.24)
where N is the number of spheres intersected by the radial layer. In addition, an
area-based numerical procedure is used by Toit [57], to evaluate the void fraction
(r) = 1 −
1
A r
N
i=1
A i,r
(4.25)
where A is the area of the cylindrical plane at radius r, and A i,r is the intersection
area between sphere i and the cylindrical plane. In this analysis, a three-dimensional
pebble-discharging bed is simulated. Thus, Eq. (4.2), is used based on the volume method for the void fraction calculation as it is suitable for the present threedimensional bed [65].
As the pebble bed is axisymmetrical, the circumferential distribution of void
fraction should be omitted. Thus, the three-dimensional void fraction characteristics
can adequately be described by the radial and axial distributions. Moreover, the
pebble bed can be divided into a cylindrical volume and a conical base (Fig. 4.25).
They are needed to be studied separately.
Radial Distribution of Void Fraction
The Cylindrical Volume
As a first step, this section shows the radial void fraction profile in the cylindrical
volume of the pebble bed. The radial distribution is formulated as follows
(r) = 1 −
1
π(r
2
i − r
2
i−1 )δh
N
j=1
ΔV j | h 1 ≤z(δV j )≤h 2 ,r i−1 ≤r(δV j )
where δh = h 2 − h 1 is the height of the cylindrical volume. r i is the division of bed
radius. In Eq. 4.26, the bed radius is divided into 360 divisions, with division width
about 1/24 of a pebble diameter, i.e. R p δ h .