Chapter 4
Numerical Methods and Simulation
for Pebble Flows
4.1 Discrete Element Methods
The Discrete Element Method (DEM) has become a significant approach to understand granular materials over the past two decades better, and hence has been
employed widely to study various applications of granular systems in the industry,
especially for hopper and silo facilities. The three-dimensional DEM has been widely
used for granular flow simulation, which was originally proposed by [1]. Numerical
results show good quantitative agreement with the experiment results. Moreover,
DEM modeling on many complicated issues of granular materials has been carried
out, like particle cohesion [2], dense packing of the random irregular particle [3],
granular mixing [4], etc. These studies help improve the design and efficiency of
granular systems. For instance, the stagnant region appearing in a pebble bed should
be reduced to maintain flow uniformity [5]. Mechanism study of particle arches helps
to predict hopper-discharge flow rate [6], and explain flow-rate fluctuation [7].
In DEM [8], the particles are discretized into a collection of many unique “discrete elements”. Each particle is traced deterministically by Newton’s law of motion
and the interaction between particles is governed by some kinds of contact models.
The Hertz-Mindlin granular contact model [9], is selected for particle motion. The
directional constant torque model [10], is adopted for computing the rolling friction
resistance between particles. In the most widely adopted soft-sphere contact model
(spring-slider model), spring and slider are used to model the normal and tangential
forces between any pair of particles, which is a simplified approach to solve the contacts between particles. In general, the governing equations of each particle i with
mass m i and radius R i at time t can be depicted as follows:
m i
d V i
dt
=
n
j=1
(F n,ij + F t,ij ) + m i g,
(4.1)
I i
d ω i
dt
=
n
j=1
R i × F t,ij −
ω ij
|ωij| μ r R i F n,ij
,
(4.2)
© Tsinghua University Press 2021
S. Jiang et al., Multiphase Flow and Heat Transfer in Pebble Bed Reactor Core,
https://doi.org/10.1007/978-981-15-9565-3_4
161
Numerical Methods and Simulation
for Pebble Flows
4.1 Discrete Element Methods
The Discrete Element Method (DEM) has become a significant approach to understand granular materials over the past two decades better, and hence has been
employed widely to study various applications of granular systems in the industry,
especially for hopper and silo facilities. The three-dimensional DEM has been widely
used for granular flow simulation, which was originally proposed by [1]. Numerical
results show good quantitative agreement with the experiment results. Moreover,
DEM modeling on many complicated issues of granular materials has been carried
out, like particle cohesion [2], dense packing of the random irregular particle [3],
granular mixing [4], etc. These studies help improve the design and efficiency of
granular systems. For instance, the stagnant region appearing in a pebble bed should
be reduced to maintain flow uniformity [5]. Mechanism study of particle arches helps
to predict hopper-discharge flow rate [6], and explain flow-rate fluctuation [7].
In DEM [8], the particles are discretized into a collection of many unique “discrete elements”. Each particle is traced deterministically by Newton’s law of motion
and the interaction between particles is governed by some kinds of contact models.
The Hertz-Mindlin granular contact model [9], is selected for particle motion. The
directional constant torque model [10], is adopted for computing the rolling friction
resistance between particles. In the most widely adopted soft-sphere contact model
(spring-slider model), spring and slider are used to model the normal and tangential
forces between any pair of particles, which is a simplified approach to solve the contacts between particles. In general, the governing equations of each particle i with
mass m i and radius R i at time t can be depicted as follows:
m i
d V i
dt
=
n
j=1
(F n,ij + F t,ij ) + m i g,
(4.1)
I i
d ω i
dt
=
n
j=1
R i × F t,ij −
ω ij
|ωij| μ r R i F n,ij
,
(4.2)
© Tsinghua University Press 2021
S. Jiang et al., Multiphase Flow and Heat Transfer in Pebble Bed Reactor Core,
https://doi.org/10.1007/978-981-15-9565-3_4
161