Inversion of Multimodal Particle Size Distribution Based on the Artificial Bee Colony Algorithm

Fig. 2. Flowchart of ABC algorithm

Fig. 3. Fitness value variation diagram

Fig. 4. The inversion results of unimodal distribution

Fig. 5. The inversion results of bimodal distribution

Fig. 6. The inversion results of trimodal distribution

Fig. 7. Experimental platform

Fig. 8. Experimental image of unimodal particle system

Fig. 9. Experimental image of bimodal particle system

Fig. 10. Distribution of light energy values

Table 1. The inversion results of unimodal distribution

Table 1. The inversion results of unimodal distribution

Distribution function

Mean value

(σ, M)

Standard deviation

(σ, M)

RRMSE/%

Normal distribution

10.00, 45.00

6.63×10^-6, 6.28×10^-6

2.26×10^-6

RR distribution

10.00, 45.00

9.39×10^-8, 6.61×10^-8

3.53×10^-8

JS_B distribution

10.00, 45.00

0.002, 0.001

0.02

Table 2. The inversion results of bimodal distribution

Table 2. The inversion results of bimodal distribution

Distribution function

Mean value

(σ₁, σ₂, M₁, M₂₎

Standard deviation

（σ₁, σ₂, M₁, M₂)

RRMSE/%

Normal distribution

6.00, 6.04

30.00, 70.03

0.002, 0.001

0.001, 0.002

0.48

RR distribution

5.99, 6.02

30.01, 70.10

0.034, 0.019

0.025, 0.197

0.33

JS_B distribution

5.99, 6.16

30.01, 70.03

0.038, 0.312

0.024, 0.076

1.58

Table 3. The inversion results of trimodal distribution

Table 3. The inversion results of trimodal distribution

Distribution function

Mean value

(σ₁, σ₂, σ₃, M₁, M₂, M₃)

Standard deviation

(σ₁, σ₂, σ₃, M₁, M₂, M₃)

RRMSE/%

Normal distribution

4.98, 4.60, 5.43

14.98, 49.57, 89.34

0.08, 0.94, 3.06

0.05, 0.70, 2.23

8.60

RR distribution

5.00, 5.39, 5.61

15.07, 51.43, 89.60

0.08, 0.33, 3.03

0.13, 4.62, 8.05

5.54

JS_B distribution

5.35, 7.81, 4.55

15.02, 49.93, 89.63

0.61, 4.51, 3.04

0.06, 0.78, 4.90

19.62

Table 4. The inversion results of RR distribution function with random noise

Table 4. The inversion results of RR distribution function with random noise

Distribution function	Random noise	Mean value	Standard deviation	RRMSE/%
Unimodal (σ, M)=(10, 45 μm)	1%	9.97, 45.00	0.002, 0.001	0.26
	3%	10.14, 45.15	0.021, 0.013	2.40
	5%	9.89, 44.67	0.006, 0.004	5.03
	10%	9.05, 45.14	0.013, 0.014	8.73
Bimodal (σ₁, σ₂)=(6, 6)， (M₁, M₂)=(30 μm, 70 μm)	1%	6.04, 6.26 29.90, 70.02	0.005, 0.023 0.012, 0.073	2.41
	3%	6.10, 5.85 29.84, 70.68	0.008, 0.043 0.007, 0.056	3.34
	5%	5.71, 4.99 30.80, 72.11	0.014, 0.099 0.038, 0.341	14.27
	10%	6.33, 8.39 29.53, 69.58	0.012, 0.050, 0.014, 0.047	18.55
Trimodal (σ₁, σ₂, σ₃)=(5, 5, 5)，(M₁, M₂, M₃)=(15 μm, 50 μm, 90 μm)	1%	4.95, 5.47, 7.48 15.12, 50.61, 93.79	0.14, 0.45, 4.90 0.12, 2.99, 3.14	12.00
	3%	4.99, 5.17, 5.88 15.27, 54.62, 89.37	0.11, 3.52, 4.85 0.20, 1.77, 3.37	14.45
	5%	4.57, 5.30, 4.88 15.50, 55.72, 89.68	0.13, 1.25, 6.47 0.17, 2.07, 11.81	18.87
	10%	5.58, 5.71, 6.18 15.11, 57.02, 100.02	0.11, 0.70, 2.39 0.08, 2.10, 1.93	20.14

Table 5. The simulation results of ABC algorithm, NNPT algorithm and Chahine algorithm

Table 5. The simulation results of ABC algorithm, NNPT algorithm and Chahine algorithm

Theoretical value	Random noise	NNPT RRMSE/%	Chahine RRMSE/%	ABC RRMSE/%
(σ, M)=(2, 30 μm)	0%	0.87	3.49	0.02
(σ, M)=(2, 30 μm)	5%	29.88	14.13	1.48
(σ, M)=(4, 30 μm)	0%	21.82	1.16	2.27×10^-8
(σ, M)=(4, 30 μm)	5%	40.27	14.31	3.65
(σ, M)=(6, 30 μm)	0%	45.98	0.20	1.72×10^-5
(σ, M)=(6, 30 μm)	5%	52.93	12.13	2.41
(σ₁, σ₂)=(2, 2)，(M₁, M₂)=(30 μm, 60 μm)	0%	3.38	2.70	1.53
(σ₁, σ₂)=(2, 2)，(M₁, M₂)=(30 μm, 60 μm)	5%	23.11	25.56	10.34
(σ₁, σ₂)=(4, 4)，(M₁, M₂)=(30 μm, 60 μm)	0%	20.94	8.06	0.07
(σ₁, σ₂)=(4, 4)，(M₁, M₂)=(30 μm, 60 μm)	5%	26.86	25.17	5.75
(σ₁, σ₂)=(6, 6)，(M₁, M₂)=(30 μm, 60 μm)	0%	41.85	6.36	1.23
(σ₁, σ₂)=(6, 6)，(M₁, M₂)=(30 μm, 60 μm)	5%	44.99	24.36	18.22

Table 6. The inversion results of unimodal particle system
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Table 6. The inversion results of unimodal particle system
Theoretical value (M) Initial value range Inversion algorithm Inversion value（M） Relative error (M) /%
30 µm M∈[3~100] ABC 30.97 µm 3.23
NNPT 32.27 µm 7.57

Table 7. The inversion results of bimodal particle system