Comparison of the Accuracy and Performance of Different Numbers of Classes in Discretised Solution Method for Population Balance Model

Table 1

Calibration parameters and error of simulation using different numbers of classes in discretised solution method for PBM at different velocity gradients.

Number of classes

Ratio of breakage rate coefficient and collision efficiency, (m^{−1}⋅s^{−1})

Average error of volume

Computational time (s)

Dimensionless number , D

Note

s^{−1}

24 (k = 2.0)

1032

0.0247

13.8

0.0900

The total simulation time at which the steady-state PSD can be obtained was 5400 s. The possible values of the parameter were chosen in the range of , and a constant step-length 1 was applied in the exhaust algorithm for parameter calibration.

28 (k = 1.8)

1045

0.0235

18.7

0.0980

35 (k = 1.6)

1025

0.0181

30.4

0.0760

48 (k = 1.4)

1052

0.0163

61.5

0.0977

88 (k = 1.2)

1050

0.0128

213.3

0.0900

s^{−1}

24 (k = 2.0)

2602

0.0243

7.6

0.0900

The total simulation time at which the steady-state PSD can be obtained was 3000 s. The possible values of the parameter were chosen in the range of , and a constant step-length 1 was applied in the exhaust algorithm for parameter calibration.

28 (k = 1.8)

2598

0.0231

10.1

0.1020

35 (k = 1.6)

2617

0.0165

14.6

0.0640

48 (k = 1.4)

2611

0.0150

25.5

0.0770

88 (k = 1.2)

2626

0.0123

89.4

0.0900

s^{−1}

24 (k = 2.0)

4267

0.0287

5.5

0.0900

The total simulation time at which the steady-state PSD can be obtained was 2500 s. The possible values of the parameter were chosen in the range of , and a constant step-length 1 was applied in the exhaust algorithm for parameter calibration.

28 (k = 1.8)

4185

0.0268

8.2

0.1130

35 (k = 1.6)

4228

0.0214

10.3

0.0842

48 (k = 1.4)

4226

0.0185

18.7

0.0959

88 (k = 1.2)

4199

0.0149

55.6

0.0900

: the cumulative volume percentage distribution was first calculated from the simulated PSD using different numbers of classes and then interpolated at the pivots of the measured PSD grid. Finally, the average error of volume percentage was recalculated by comparing with the measured PSD. : a dimensionless number; : relating error and computational time; see (10).