2

3

(PCB)

, FR-4 ,

/

JKSimMet, Metsim, Modsim

PCB

Trial/Error

4

,

Recycling

Programming tool: MATLAB R2015a

PCB

2 (Copper, FR-4)

(Dliberation ≒ 600 )

Zhang and Forssberg, 1997, Wen et al., 2005

5

2. :

(a) FR-4 PCB , (b)

(a) (b)

Modeling

Shredder, Cut crusher

Dmax = 4 mm

, FR-4

: 30%

500

6

3.

graph

D50 Dmax

Andrews-Mika diagram

Beta distribution

modeling

7

4.

Andrews-Mika diagram

𝑝 𝑔 = (1 − 𝐿0 − 𝐿1)𝑔𝛼−1 1 − 𝑔 𝛽−1

Beta(𝛼, 𝛽)

𝑔: 품위

𝑝(𝑔) : 특정입도에서품위𝑔의질량분율

𝐿0: 품위가 0인입자들의질량분율

𝐿1: 품위가 1인입자들의질량분율

/

graph

Andrews-Mika diagram

: 500

: 10-60%

8

5.

/

1. Feed

2.

3. Screen size screen Product

4. Screen size

5. Screen size 3,4

ProductGrinding

MillScreen undersize

oversize

( - )

( + )

Feed

9

6.

𝑓: Feed (𝑛 × 1)

𝑝: Product (𝑛 × 1)

𝐵: Breakage matrix (𝑛 × 𝑛)

𝑆: Selective matrix (𝑛 × 𝑛)

𝐶: Screening matrix (𝑛 × 𝑛)

𝐼: Identity matrix (𝑛 × 𝑛)

Grinding and screening matrix (1st stage)

𝑝 = 𝐵𝑆 + 𝐼 − 𝑆 𝑓 = 𝐷𝑓

𝑝1∘ = 𝐶𝑝 = 𝐶𝐷𝑓 = 𝐶 𝐵𝑆 + 𝐼 − 𝑆 𝑓

𝑝1∗ = 𝐼 − 𝐶 𝑝 = 𝐼 − 𝐶 𝐷𝑓 = 𝐼 − 𝐶 𝐵𝑆 + 𝐼 − 𝑆 𝑓

Circulation (nth stage)

𝑝𝑛∘ = 𝐶𝐷𝑝𝑛−1

∘ = 𝐶𝐷𝐶𝐷𝑝𝑛−2∘ = ⋯ = 𝐶𝐷 𝑛𝑓

𝑝𝑛∗ = 𝐼 − 𝐶 𝐷𝑝𝑛−1

∘

𝑝𝑛 = σ𝑘=1𝑛 𝑝𝑛

∗

Run the circulation until; 𝑝𝑛∘ ≈ 0

oversize

undersize

10

Matrix

Breakage matrix: RR dist’n model (b=0.1, n=1)

Selective matrix: GGS dist’n model (a=0.5, k=1)

Screening matrix: Ideal partition curve

* Both breakage and selective functions are size independent

11

7. Breakage, Selective, Screening function graphical expression

𝐹 𝑥 = 1 − 𝑒−

𝑥𝑏

𝑛𝐹 𝑥 =

𝑥

𝜅

𝛼

Start

Stop

𝑝∘ ≈ 0 ?

𝑝∘ ← 𝐶𝐷𝑓𝑝∗ ← 𝐼 − 𝐶 𝐷𝑓

𝑝 ← 𝑝 + 𝑝∗Enter 𝑓, 𝐵, 𝑆, 𝐶

Print 𝑝𝐷 ← 𝐵𝑆 + 𝐼 − 𝑆

Initialize 𝑝

𝑓 ← 𝑝∘

yes

no

Algorithm

12

8. Algorithm

Knelson concentrator

5 chamber (fluidizing water)

chamber

chamber

13

𝑁

𝑄

PCB 분쇄물FR-4

9. Knelson concentrator

Knelson concentrator ( )

(𝐹𝑑) (𝐹𝑐)

𝐹𝑑 : , ,

𝐹𝑐 : , , chamber ,

Fd

Particle properties (𝑑, 𝜌𝑠)

Operating condition (𝑄,𝑁)

𝑓(𝑑, 𝜌𝑓 , 𝑄)

𝑁

r

𝑄 Fc𝑓(𝑑, 𝜌𝑠, 𝑟, 𝑁)

14

10. Knelson concentrator

1. Feed Knelson concentrator ( KC)

2. KC chamber /

3. 2.

4. 2. 3. Product1, Product2

15

FeedKnelson

Concentrator

Operating Condition

Product1

Product2

11.

Mathematical expression

𝐹𝑑 =1

2𝜌𝑓𝑣

2𝐴𝑠𝐶𝐷 =𝜋

8𝜌𝑓𝐷

2 𝑄

𝐴

2𝐶𝐷

𝐹𝑐 =𝑚𝑉2

𝑟=

4

6𝜋3𝜌𝑠𝐷

3𝑅𝑁2

𝑋 =𝐹𝑑

𝐹𝑐=

241

𝜋2×

1

𝐴2𝑅×

𝜌𝑓

𝜌𝑠×

𝐶𝐷

𝐷×

𝑄

𝑁

2

𝑋 > 1: overflow (tailings)

𝑋 < 1: underflow (concentrate)

시료의변수

𝜌𝑠: 입자, 유체의밀도

𝐷: 입자의직경

공정변수

𝑄: 유동수의유입량

𝑁: chamber의회전수

기타상수

𝐶𝐷: 입자의저항계수 (Drag coefficient)

𝐴: 유동수(fluidizing water)의유입면적

𝑅: 입자의회전반경

1st

chamberFeed2nd

chamber

3rd

chamber

4th

chamber

5th

chamber Tailingso/f o/f o/f o/f

u/f

o/f

u/f u/f u/f u/f

Concentrate

16

12. Knelson concentrator u/f, o/f

Algorithm

17

Start

𝑗 ← 1(grade class)

Enter 𝑄,𝑁, 𝜌𝑓 , 𝑓

𝑖 ← 1(particle size)

Initiate 𝑝1, 𝑝2

𝑝1 ← 𝑝1 + 𝑓𝑖,𝑗

𝑋𝑓𝑖,𝑗 < 1 ?

calc. 𝑋𝑓𝑖,𝑗 in nth chmb.

𝑛 ← 1(chamber no.)

End of 𝑗?

End of 𝑖?

𝑛 ← 𝑛 + 1

𝑗 ← 𝑗 + 1

𝑝2 ← 𝑝2 + 𝑓𝑖,𝑗

𝑛 = 5 ?

𝑖 ← 𝑖 + 1

Print 𝑝1, 𝑝2

Stopno

no

no

no

yes yes

yes

yes13.

algorithm

Simulation

stream

(Particle size distribution)

/ (Particle size / grade distribution)

/

(Grade) vs. (Recovery)

(Newton’s efficiency)

18

Grinding MillKnelson

Concentrator

Product1

Feed

Product2

14.

Screen size

Feed

D80: 2,000 → 110

D50: 1,800 → 100

Feed

Screen size 500

19

15. screen size

simulation

/

20

16. / / simulation

(aperture size: 500 )

Fluidizing water /

Q = 6, 12 L/min ,

/

u/f o/f

Fluidizing water

o/f

Yield, Recovery

21

17. Fluidizing water

u/f, o/f / (N=1,000 rpm)

Q = 6 L/min, overflow Q = 12 L/min, overflow

Q = 6 L/min, underflow Q = 12 L/min, underflow

Fluidizing water

22

18. Fluidizing water

Recovery vs. Grade graph (N = 1,000 rpm)

0

0.2

0.4

0.6

0.8

1

0 0.2 0.4 0.6 0.8 1

Re

co

ve

ry

Grade of concentrate

Q = 3 L/min

Q = 6 L/min

Q = 9 L/min

Q = 12 L/min

27%

63%

11%

5%

0%

10%

20%

30%

40%

50%

60%

70%

Newton's efficiency

3 L/min 6 L/min 9 L/min 12 L/min

19. Fluidizing water

Newton’s efficiency (N = 1,000 rpm)

Chamber /

N=500, 1,000 rpm

/

Chamber

, u/f

23

20. Chamber

u/f, o/f (Q=6 L/min)

N = 500 rpm, overflow N = 1,000 rpm, overflow

N = 500 rpm, underflow N = 1,000 rpm, underflow

Chamber /

24

21. Chamber

Recovery vs. Grade graph (Q = 6 L/min)

0

0.2

0.4

0.6

0.8

1

0 0.2 0.4 0.6 0.8 1

Re

co

ve

ry

Grade of concentrate

N = 1,250 rpm

N = 1,000 rpm

5%

30%

63%59%

0%

10%

20%

30%

40%

50%

60%

70%

Newton's efficiency

500 rpm 750 rpm 1,000 rpm 1,250 rpm

22. Chamber

Newton’s efficiency (Q = 6 L/min)

N = 750 rpm

N = 500 rpm←

N

Q: 3, 6, 9, 12 L/min

N: 500, 750, 1,000 1,250 rpm

Max. Newton effi.: 63%

#1. Q: 3 L/min, N: 500 rpm

#2. Q: 6 L/min, N: 1,000 rpm

25

63%

5%

0% 0%

43%

30%

5%

1%

27%

63%

11%

5%

0%

59%

44%

11%

0%

10%

20%

30%

40%

50%

60%

70%

3 L/min 6 L/min 9 L/min 12 L/min

500 rpm

750 rpm

1,000 rpm

1,250 rpm

23. Fluidizing water

Chamber

Newton’s efficiency

𝑋 =𝐹𝑑𝐹𝑐

=241

𝜋2×

1

𝐴2𝑅×𝜌𝑓

𝜌𝑠×𝐶𝐷𝐷×

𝑄

𝑁

2

: 34.56%

: 89.29 %

: 67.74%

26

Grinding MillKnelson

Concentrator

Product1

Feed

Product2

Feed Ground product Concentrate Tailings

24. /

① ②③

④

① ② ③ ④

,

,

Knelson concentrator (Newton efficiency)

#1. Q: 3 L/min, N: 500 rpm

#2. Q: 6 L/min, N: 1,000 rpm

: 34.56%

: 89.29 %

: 67.74%

27

28

29

30

Particle

(Flowrate) FlowRate 1 x 1

( )

(Components) Componentsi 1 x 2

i ( ), text

ex> {‘Copper’, ‘FR-4’}

(Density) Densityi 1 x 2

i

ex> [2 9]

(Particle size range) PSRi 1 x 13

i (Nominal size)

ex> [45 62.5, 90, 125, … 2,800]

31

Particle ( )

(Particle size distribution) PSDi 1 x 13

i

ex> [0.1, 0.15, … 0.1]

(Grade distribution) GDi,j 13 x 12

i j

ex> [0 0.1 0.12, … 0.1]

(Drag coefficient) C_D 1 x 1

ex> 0.47

32

/

Particle size, Particle size distribution of feed

Breakage, Selective and Screening matrix of grinding mill

Particle size distribution of product

ProductGrinding

MillScreen undersize

oversize

( - )

( + )

Feed

33

6.

Flowrate, Density of solid, Particle size, Particle size distribution,

Drag coefficient of feed

Rotating number, Flowrate of fluidizing water, Density of the fluid

Flowrate, Particle size distribution of concentrate and tailings

34

FeedKnelson

Concentrator

Operating Condition

Product1

Product2

11.