4.2 Gravity-Driven Flow Regime Characterization
163
industry and the natural world, they only represent a part of flow regimes of particle
materials. According to Campbell’s studies [13–15], particle flows can be divided
into two global regimes, elastic and inertial. The elastic regime is dominated by
force chains. It is divided into the elastic–quasi-static regime and the elastic– inertial
regime depending on whether there is a noticeable dependence of the stresses on the
shear rate. The inertial regime, which is free of force chains and has stresses that
scale with the square of the shear rate, can be divided into the inertial-non-collisional
regime and the inertial-collisional (or rapid flow) regime depending on whether the
dominant particle interaction is binary collisions. Moreover, Campbell also introduced a dimensionless parameter that governs the ratio of elastic to inertial effects
that is defined by k
=
k
ρd 3 γ 2 ; where k, ρ, d , and γ represent the inter-particle stiffness, solid density, particle diameter, and shear rate, respectively. By setting solid
concentration v and dimensionless stiffness k
as y-coordinate and x-coordinate,
Campbell drew a complete flow map for flow of particle materials (refer to Fig. 4.1).
An important conclusion can be drawn from the flow map. At solid concentrations
v larger than 0.6, one cannot leave the elastic regimes by changing k
(usually by
changing shear rate γ ), and can only have a transition between elastic–quasi-static
and elastic–inertial regimes. Furthermore, out of a laboratory environment, it is difficult to reach the elastic–inertial regime at high solid concentration, because it mostly
requires considerable shear rate. Therefore, at high solid concentrations, the elastic–
quasi-static regime which covers the common useful gravity-driven dense particle
flow needs to be investigated.
However, the current flow regime division is not detailed enough and does not
consider some of the distinct flow behaviors within the elastic–quasi-static regime.
For instance, dry and powder-like materials in an hourglass can smoothly flow similar
to liquids. However, the gravity-driven dense pebble flow in High-Temperature Gascooled Reactor (HTGR) [16–18], in nuclear engineering behaves differently with
apparent flow discontinuity [19]. In the pebble-bed HTGR, when a pebble-bed reactor
runs in a circulating way, new pebbles are loaded from the top of the core, and used
pebbles are discharged from the bottom at the same rate. Hundreds of thousands of
fuel pebbles move downward under gravity with a very small velocity around 10
−4
−
10
−3 m/h. This forms the special very slow particle flow in pebble-bed reactors. Due
to the very small average velocity, the overall flow shows a significant intermittence.
Most of the time, the system is static, but the static state is disrupted by different
sizes of pebble avalanches at random times and locations.
One of the applications of the slow dense particle flow is the pebble flow in HTGR
which shows the importance of a more detailed investigation on such flow. Therefore,
this chapter aims to further describe gravity-driven dense particle flow by studying
its major flow regime characteristics through numerical simulations. The phenomena
of pebble avalanche and flow intermittence are illustrated in detail. A new general
statistical criterion on the time-energy distribution of gravity-driven dense particle
flow is proposed to subdivide the elastic–quasi-static regime, which may help to
characterize dense particle flows under gravity (Fig. 4.1).
163
industry and the natural world, they only represent a part of flow regimes of particle
materials. According to Campbell’s studies [13–15], particle flows can be divided
into two global regimes, elastic and inertial. The elastic regime is dominated by
force chains. It is divided into the elastic–quasi-static regime and the elastic– inertial
regime depending on whether there is a noticeable dependence of the stresses on the
shear rate. The inertial regime, which is free of force chains and has stresses that
scale with the square of the shear rate, can be divided into the inertial-non-collisional
regime and the inertial-collisional (or rapid flow) regime depending on whether the
dominant particle interaction is binary collisions. Moreover, Campbell also introduced a dimensionless parameter that governs the ratio of elastic to inertial effects
that is defined by k
=
k
ρd 3 γ 2 ; where k, ρ, d , and γ represent the inter-particle stiffness, solid density, particle diameter, and shear rate, respectively. By setting solid
concentration v and dimensionless stiffness k
as y-coordinate and x-coordinate,
Campbell drew a complete flow map for flow of particle materials (refer to Fig. 4.1).
An important conclusion can be drawn from the flow map. At solid concentrations
v larger than 0.6, one cannot leave the elastic regimes by changing k
(usually by
changing shear rate γ ), and can only have a transition between elastic–quasi-static
and elastic–inertial regimes. Furthermore, out of a laboratory environment, it is difficult to reach the elastic–inertial regime at high solid concentration, because it mostly
requires considerable shear rate. Therefore, at high solid concentrations, the elastic–
quasi-static regime which covers the common useful gravity-driven dense particle
flow needs to be investigated.
However, the current flow regime division is not detailed enough and does not
consider some of the distinct flow behaviors within the elastic–quasi-static regime.
For instance, dry and powder-like materials in an hourglass can smoothly flow similar
to liquids. However, the gravity-driven dense pebble flow in High-Temperature Gascooled Reactor (HTGR) [16–18], in nuclear engineering behaves differently with
apparent flow discontinuity [19]. In the pebble-bed HTGR, when a pebble-bed reactor
runs in a circulating way, new pebbles are loaded from the top of the core, and used
pebbles are discharged from the bottom at the same rate. Hundreds of thousands of
fuel pebbles move downward under gravity with a very small velocity around 10
−4
−
10
−3 m/h. This forms the special very slow particle flow in pebble-bed reactors. Due
to the very small average velocity, the overall flow shows a significant intermittence.
Most of the time, the system is static, but the static state is disrupted by different
sizes of pebble avalanches at random times and locations.
One of the applications of the slow dense particle flow is the pebble flow in HTGR
which shows the importance of a more detailed investigation on such flow. Therefore,
this chapter aims to further describe gravity-driven dense particle flow by studying
its major flow regime characteristics through numerical simulations. The phenomena
of pebble avalanche and flow intermittence are illustrated in detail. A new general
statistical criterion on the time-energy distribution of gravity-driven dense particle
flow is proposed to subdivide the elastic–quasi-static regime, which may help to
characterize dense particle flows under gravity (Fig. 4.1).