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4 Numerical Methods and Simulation for Pebble Flows
Gravity-driven dense particle flows are divided into three sub-regimes and named
based on their primary features. The extremely slow flow describes the flow that
remains stationary most of the time. This type of flow behaves more like static
solids, and it is difficult to notice the motion unless unusual avalanche happens.
The intermittent flow covers the intermediate flow regime with frequent switches
between static and kinetic states. It is the most unstable flow with a large fluctuation
of the system energy. The consistent flow is the fastest flow among gravity-driven
dense particle flows. This type of flow can flow steadily and smoothly like common
liquids. The consistent flow is close to inertial particle flow, but it is difficult for
consistent flow to touch the real inertial flow in practical applications under gravity
at such high solid concentration. In engineering, the intermittent flow and consistent
flow are common, and the extremely slow flow has drawn public attention since its
application in pebble-bed high-temperature gas-cooled reactors.
The gravity-driven dense particle flows present distinguished flow behaviors under
different mean velocities. Especially, the intermittence and avalanche phenomena of
slow flows are highlighted and interpreted. By studying a wide range of the gravitydriven dense particle flows, the transformation of flow behavior is illustrated by E − t
profiles, and a general statistical rule on the time-energy distribution is suggested.
The general rule is that the order of the magnitude of instantaneous kinetic energy
(log 10 E(t)) should follow a “single-peak” distribution for a long enough and fully
developed gravity-driven dense particle flow. Based on this principle, the E span −
σ criteria are developed to characterize gravity-driven dense particle flows from
macroscopic time-energy perspective further. According to the values of the span
of energy magnitude (E span ) and the standard deviation σ ), the gravity-driven dense
particle flows are subdivided into three sub-regimes: an extremely slow flow, an
intermittent flow, and a consistent flow. They, respectively, represent quasi-static,
intermediate and liquid-like states of dense particle materials under gravity
4.2.4 Recirculation Rates and Times
In this section, the effects of recirculation flow rates on the Dense Pebble Flow (DPF)
in a drained pebble bed are investigated through DEM simulation. After validation
by experimental data, the velocity characteristics are analyzed and obvious intermittency characteristics are observed. The profile features and the shapes of the velocity
distributions are estimated considering the flatness and symmetry for all flow rates.
Fluctuation is estimated by the statistical parameter coefficient of variation and the
newly proposed relative fluctuating kinetic energy (RFKE). Furthermore, the multifractal analysis of the velocity signals is carried out by the Multiplicative Cascade
Method (MCM). The pebble flow follows the fractal phenomenon, and the intermittency of pebble flow has been measured quantitatively by the multi-fractal analysis.
The increased flow rates always indicate smaller intermittency and higher frequency
fluctuation, and vice versa. A new intermittency index C is proposed to describe
the intermittency of the DPF quantitatively. The categorization of flow regimes of
4 Numerical Methods and Simulation for Pebble Flows
Gravity-driven dense particle flows are divided into three sub-regimes and named
based on their primary features. The extremely slow flow describes the flow that
remains stationary most of the time. This type of flow behaves more like static
solids, and it is difficult to notice the motion unless unusual avalanche happens.
The intermittent flow covers the intermediate flow regime with frequent switches
between static and kinetic states. It is the most unstable flow with a large fluctuation
of the system energy. The consistent flow is the fastest flow among gravity-driven
dense particle flows. This type of flow can flow steadily and smoothly like common
liquids. The consistent flow is close to inertial particle flow, but it is difficult for
consistent flow to touch the real inertial flow in practical applications under gravity
at such high solid concentration. In engineering, the intermittent flow and consistent
flow are common, and the extremely slow flow has drawn public attention since its
application in pebble-bed high-temperature gas-cooled reactors.
The gravity-driven dense particle flows present distinguished flow behaviors under
different mean velocities. Especially, the intermittence and avalanche phenomena of
slow flows are highlighted and interpreted. By studying a wide range of the gravitydriven dense particle flows, the transformation of flow behavior is illustrated by E − t
profiles, and a general statistical rule on the time-energy distribution is suggested.
The general rule is that the order of the magnitude of instantaneous kinetic energy
(log 10 E(t)) should follow a “single-peak” distribution for a long enough and fully
developed gravity-driven dense particle flow. Based on this principle, the E span −
σ criteria are developed to characterize gravity-driven dense particle flows from
macroscopic time-energy perspective further. According to the values of the span
of energy magnitude (E span ) and the standard deviation σ ), the gravity-driven dense
particle flows are subdivided into three sub-regimes: an extremely slow flow, an
intermittent flow, and a consistent flow. They, respectively, represent quasi-static,
intermediate and liquid-like states of dense particle materials under gravity
4.2.4 Recirculation Rates and Times
In this section, the effects of recirculation flow rates on the Dense Pebble Flow (DPF)
in a drained pebble bed are investigated through DEM simulation. After validation
by experimental data, the velocity characteristics are analyzed and obvious intermittency characteristics are observed. The profile features and the shapes of the velocity
distributions are estimated considering the flatness and symmetry for all flow rates.
Fluctuation is estimated by the statistical parameter coefficient of variation and the
newly proposed relative fluctuating kinetic energy (RFKE). Furthermore, the multifractal analysis of the velocity signals is carried out by the Multiplicative Cascade
Method (MCM). The pebble flow follows the fractal phenomenon, and the intermittency of pebble flow has been measured quantitatively by the multi-fractal analysis.
The increased flow rates always indicate smaller intermittency and higher frequency
fluctuation, and vice versa. A new intermittency index C is proposed to describe
the intermittency of the DPF quantitatively. The categorization of flow regimes of