176
4 Numerical Methods and Simulation for Pebble Flows
flow features are dominated by kinematic variables, e.g., velocities. It can also be
characterized by a significant correlation between mean force and velocity. However,
in the kinetic flow regime, it is a slow flow or quasi-static flow, which is considerably
intermittent due to the internal sudden “bulk” motion of falling accompanied by a
sudden change in structures. This implies that the transition from slow to fast regime
can be characterized by the transition of the existence of characteristic frequency
or period of variation of the kinetic variable (e.g., contact force) to the kinematic
variable (e.g., velocity).
4.2.3 Energy Span Versus Standard Deviation
4.2.3.1 Numerical Setup and Conditions
A numerical setup is established based on the real pebble-bed reactor HTR-PM [18].
The real pebble bed is composed of a cylinder as the body and a tapered hopper as
the bottom, where a discharging hole is located at the center of the hopper bottom.
In simulations, both dimensions of cylinder-hopper silo and fuel pebbles follow the
1:5 scale of the real reactor HTR-PM and 1:1 scale of the experimental test facility
built by INET at Tsinghua University [40]. The pebbles are identical spheres of the
diameter of 12 mm. The pebble bed is 800 × 1200 × 30 mm in width (x), height (y),
and depth (z), respectively (Fig. 4.8), where the width and height follow 1:5 scale of
the real pebble bed. The computational domain is a layer taken from the 3D bed. This
layer-like domain is a middle longitudinal section of the bed with a depth of 2.5 pebble
diameters. Hence, this domain is still 3D and is able to contain >2 layers of pebbles
in depth. With cyclic boundaries adopted for the depth direction, the computational
domain can be treated as a thin slice crossing the axis of the real three-dimensional
cylinder pebble bed. This means that once a pebble gets out from the front face, it
will enter into the layer-like domain again from the back face at the corresponding
location with the identical motion state. This technique adopted by researchers before
[41–43], will speed up the simulation by reducing the number of simulated pebbles
without disrupting interactions between pebbles in depth (because front and back
pebbles still maintain contacts with other pebbles rather than the wall). Hence, the
slice domain can be assumed as the simplified model of a 3D bed. Diameters of the
bed and discharging hole of the whole geometry in Table (4.3), are equal to their
widths as illustrated in Fig. 4.8.
At first, around 13,090 pebbles are gradually loaded into the pebble bed. These
pebbles fall under gravity and pile up randomly to create the initial stack state with
arbitrary voids and packing structures. When the packing process is done, and the
pebbles become stationary, the reactor is ready to be run. The setup should be in a
circulating condition with pebbles being loaded from the top and discharged from
the bottom. The discharging rate will be set to match the loading rate in order to keep
the total number of pebbles in the bed constant. The circulating rate is controlled as
a constant for each case based on the operating conditions.
4 Numerical Methods and Simulation for Pebble Flows
flow features are dominated by kinematic variables, e.g., velocities. It can also be
characterized by a significant correlation between mean force and velocity. However,
in the kinetic flow regime, it is a slow flow or quasi-static flow, which is considerably
intermittent due to the internal sudden “bulk” motion of falling accompanied by a
sudden change in structures. This implies that the transition from slow to fast regime
can be characterized by the transition of the existence of characteristic frequency
or period of variation of the kinetic variable (e.g., contact force) to the kinematic
variable (e.g., velocity).
4.2.3 Energy Span Versus Standard Deviation
4.2.3.1 Numerical Setup and Conditions
A numerical setup is established based on the real pebble-bed reactor HTR-PM [18].
The real pebble bed is composed of a cylinder as the body and a tapered hopper as
the bottom, where a discharging hole is located at the center of the hopper bottom.
In simulations, both dimensions of cylinder-hopper silo and fuel pebbles follow the
1:5 scale of the real reactor HTR-PM and 1:1 scale of the experimental test facility
built by INET at Tsinghua University [40]. The pebbles are identical spheres of the
diameter of 12 mm. The pebble bed is 800 × 1200 × 30 mm in width (x), height (y),
and depth (z), respectively (Fig. 4.8), where the width and height follow 1:5 scale of
the real pebble bed. The computational domain is a layer taken from the 3D bed. This
layer-like domain is a middle longitudinal section of the bed with a depth of 2.5 pebble
diameters. Hence, this domain is still 3D and is able to contain >2 layers of pebbles
in depth. With cyclic boundaries adopted for the depth direction, the computational
domain can be treated as a thin slice crossing the axis of the real three-dimensional
cylinder pebble bed. This means that once a pebble gets out from the front face, it
will enter into the layer-like domain again from the back face at the corresponding
location with the identical motion state. This technique adopted by researchers before
[41–43], will speed up the simulation by reducing the number of simulated pebbles
without disrupting interactions between pebbles in depth (because front and back
pebbles still maintain contacts with other pebbles rather than the wall). Hence, the
slice domain can be assumed as the simplified model of a 3D bed. Diameters of the
bed and discharging hole of the whole geometry in Table (4.3), are equal to their
widths as illustrated in Fig. 4.8.
At first, around 13,090 pebbles are gradually loaded into the pebble bed. These
pebbles fall under gravity and pile up randomly to create the initial stack state with
arbitrary voids and packing structures. When the packing process is done, and the
pebbles become stationary, the reactor is ready to be run. The setup should be in a
circulating condition with pebbles being loaded from the top and discharged from
the bottom. The discharging rate will be set to match the loading rate in order to keep
the total number of pebbles in the bed constant. The circulating rate is controlled as
a constant for each case based on the operating conditions.