186
4 Numerical Methods and Simulation for Pebble Flows
Table 4.4 Independence test on simulation time
Circulating
rate
(pebbles/s)
Characteristic
value
Simulation time (s)
20
40
80
160
500
1000
1
E span
/
/
1.321
1.358
1.370
1.370
σ
/
/
0.024
0.025
0.025
0.025
10
E span
/
/
1.565
1.573
1.580
1.581
σ
/
/
0.054
0.057
0.056
0.057
80
E span
/
1.667
1.692
1.695
1.696
/
σ
/
0.182
0.180
0.182
0.182
/
500
E span
/
0.980
0.981
0.986
0.987
/
σ
/
0.127
0.127
0.129
0.129
/
1000
E span
0.644
0.652
0.653
0.653
/
/
σ
0.075
0.082
0.080
0.080
/
/
2500
E span
0.368
0.370
0.371
0.371
/
/
σ
0.048
0.049
0.049
0.049
/
/
etc.) and dimensions (pebble bed dimension, particle diameter, etc.). Besides, when
the number of particles and experimental time is large enough to reflect the statistical
rules, E span and σ will not be affected by the particle number and recording time.
Therefore, the E span − σ system provides a visual and simple method to characterize
dense particle flows that is different from old approaches of flow behavior characterization focusing on local physical mechanisms. Necessary data like the instantaneous
velocity field can be acquired through recording a period of fully developed flow and
processing PIV/PTV (Particle Image Velocimetry) analysis.
Values of E span and σ directly depend on the comprehensiveness of the data, which
is related to the recording time. The balance between accuracy and computational
cost should be optimized. For securing the correctness of characteristic values, an
independence test on the simulation time should be conducted before obtaining the
final values. Still, differently, the simulation time rather than mesh density is tested
here. Several typical cases with different flow features are selected to show the test
results.
Table 4.4 shows the procedure of the independence test to determine the final
values. The test principle is explained as follows: when the simulation time is doubled,
if the values of E span and σ remain constant or their variation is tiny enough, the
values can be assumed to be the final ones. Otherwise, the simulation time should be
extended. Independence test has been conducted for each case in Table 4.5, to ensure
accuracy.
Data of E span − σ have been collected for a wide range of gravity-driven dense
particle flows with different mean velocities (refer to Table 4.5 for details). It is
4 Numerical Methods and Simulation for Pebble Flows
Table 4.4 Independence test on simulation time
Circulating
rate
(pebbles/s)
Characteristic
value
Simulation time (s)
20
40
80
160
500
1000
1
E span
/
/
1.321
1.358
1.370
1.370
σ
/
/
0.024
0.025
0.025
0.025
10
E span
/
/
1.565
1.573
1.580
1.581
σ
/
/
0.054
0.057
0.056
0.057
80
E span
/
1.667
1.692
1.695
1.696
/
σ
/
0.182
0.180
0.182
0.182
/
500
E span
/
0.980
0.981
0.986
0.987
/
σ
/
0.127
0.127
0.129
0.129
/
1000
E span
0.644
0.652
0.653
0.653
/
/
σ
0.075
0.082
0.080
0.080
/
/
2500
E span
0.368
0.370
0.371
0.371
/
/
σ
0.048
0.049
0.049
0.049
/
/
etc.) and dimensions (pebble bed dimension, particle diameter, etc.). Besides, when
the number of particles and experimental time is large enough to reflect the statistical
rules, E span and σ will not be affected by the particle number and recording time.
Therefore, the E span − σ system provides a visual and simple method to characterize
dense particle flows that is different from old approaches of flow behavior characterization focusing on local physical mechanisms. Necessary data like the instantaneous
velocity field can be acquired through recording a period of fully developed flow and
processing PIV/PTV (Particle Image Velocimetry) analysis.
Values of E span and σ directly depend on the comprehensiveness of the data, which
is related to the recording time. The balance between accuracy and computational
cost should be optimized. For securing the correctness of characteristic values, an
independence test on the simulation time should be conducted before obtaining the
final values. Still, differently, the simulation time rather than mesh density is tested
here. Several typical cases with different flow features are selected to show the test
results.
Table 4.4 shows the procedure of the independence test to determine the final
values. The test principle is explained as follows: when the simulation time is doubled,
if the values of E span and σ remain constant or their variation is tiny enough, the
values can be assumed to be the final ones. Otherwise, the simulation time should be
extended. Independence test has been conducted for each case in Table 4.5, to ensure
accuracy.
Data of E span − σ have been collected for a wide range of gravity-driven dense
particle flows with different mean velocities (refer to Table 4.5 for details). It is