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A cascade wire separator
Int. J. Miner. Process. 78 (2005) 40 – 48 www.elsevier.com/locate/ijminpro
A cascade wire separator Lenka Muchova´ a,*, Marieke Mooij a, Laurens van Kooy b, Peter Berkhout a a
Delft University of Technology, Mijnbouwstraat 120, 2628 RX Delft, The Netherlands b AEB, Australie¨havenweg 21, 1045 BA Amsterdam, The Netherlands
Received 15 February 2005; received in revised form 25 July 2005; accepted 26 July 2005 Available online 13 September 2005
Abstract Separation of wires from recycling streams is often desired for material recovery, quality improvement or for operational reasons. It increases the recovery of non-ferrous metals while congestion of material flows can be avoided. Tests with a new type of separator give good results for removing wires and other long particles from a granular stream of material. High recovery is achieved by repeating a simple separation step many times within one unit process. The resulting cascade system was tested with 700 kg of bottom ash, showing good agreement with theory and resulting in a complete recovery of wires exceeding 50 mm in length. D 2005 Elsevier B.V. All rights reserved. Keywords: wire separation; shape sorting; recycling
1. Introduction In most recycling operations the presence of wires complicates the separation process. Such processes are often designed for granulate particles. The curved shape of wires and their large ratio of length and diameter create different transport behaviour than for granulate particles. For this reason, a major problem caused by wires is blocking of the material flow. Secondary benefits of removing wires are enhanced metal recovery or the reduction of the metal content of basically nonmetal material streams such as plastics or wood. The present research aims to design a separation process with a total recovery of strongly elongated particles. * Corresponding author. E-mail address: [email protected] (L. Muchova´). 0301-7516/$ - see front matter D 2005 Elsevier B.V. All rights reserved. doi:10.1016/j.minpro.2005.07.005
Various solutions have been proposed to remove wires for different kinds of applications. Fabrizi et al. (2003) propose a nail roll separator for separating automotive shredder residue. Their results show a maximum recovery of 70% while the grade of the wire fraction varies from 35% to 87%. Another device, the Hamos CUX L brush belt separator, used for pre-treated electronic scrap, has a recovery between 75% and 80% with a product grade of 70%. Other solutions are known from shape sortation (e.g., Beunder and Rem, 1999; Beunder et al., 2002; Feruuchi and Gotoh, 1992; Shinohara, 1986). An example is the inclined belt, which can be used to concentrate wires based on their curvature and friction properties. The poor selectivity of shape separators usually prevents the production of relatively pure fractions: the efficiency depends strongly on the prop-
L. Muchova´ et al. / Int. J. Miner. Process. 78 (2005) 40–48
41
Fig. 1. Single step wire separator.
erties of the processed mixture. Known differences as textile content and wire shape have their influences on the result. The nail roll separator, for example, performs well on partly bent or hooked wires. Straight wires find their way to the non-wire fraction. All of the mentioned devices remove wires partially, leaving some of the wires in the bulk stream. The recovery of these separators could be improved by repeating the process many times in a cascade system. A cascade of separation steps results in an increase of grade and recovery. However, this would generally require a lot of space and energy for most separators. An important advantage of the design described here is that it is particularly suitable to repeat itself many times in one unit operation.
2. The cascade separator The basic concept of each step of the cascade separator is the ability of elongated particles to pass
over a gap or slot while granular particles of the same diameter will not (see Fig. 1). By placing the construction on top of a conventional vibrating feeder, a single step separator is created. Wires move along the top level of the separator while the granulate particles fall down through the dfingersT in the gap to a lower level. The dfingersT are positioned over the gap with a parallel orientation, to prevent wires that have a different orientation from falling through the gap. The separator works for dry and wet applications, although wet materials show some negative side effects due to sticky granulate particles, which result in a lower throughput and a lower grade of the wire fraction. A single step separation can be sufficient for applications with wires that are long compared to the size of the granular particles and for not too high capacities. At higher loads on the separator, the granular particles form a multi-layer and may cross the gap, while short wires occasionally fall through the gap. Therefore, positioning multiple gaps in a cascade is the solution
Fig. 2. Cascade wire separator with 5 5 steps.
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L. Muchova´ et al. / Int. J. Miner. Process. 78 (2005) 40–48
for applications with a need for high capacity and sharp separation on wire length. Such a design is shown in Fig. 2. The dimensions and the number of separation steps can be changed according to the application. A high throughput of granulate with few wires will demand many gaps in the horizontal direction, while for a high recovery of short wires more separating steps in the vertical direction are required. A 5 5 cascade wire separator was built for testing the concept. The total height of this cascade separator is 360 mm and the length is 600 mm. The gap size is 17 mm by 20 mm (width times height). The distance between the fingers is 20 mm.
3. Theory In order to develop a predictive theory for the performance of the separator (single step and cascade system), it is helpful to take a closer look at the various elements of the process. The mechanics of the single step separation is a combination of transport and two mechanisms that prevent wires from falling through the gap. Going from the single step separation to a cascade separator can be described by statistics.
. Single step separation
.
o Transport velocity of particles o Mechanisms that prevent wires from falling through the gap n Mechanism A (fingers) n Mechanism B (step) Cascade separator
Fig. 4. Motion of the particles (dashed) and the plate (bold): vertical position (bottom) and horizontal velocity (top).
Note that the coordinate system is oriented so that the x-axis is parallel to the plate. This coordinate system is used for all displacements, velocities and accelerations. The plate vibration is defined by its amplitude a 0, the frequency 1/(2s) and the angle of vibration a. For simplicity, we assume a triangular motion. Then the plate equations of motion are, vxplate ¼ vðt Þcosa vyplate ¼ vðt Þsina vð t Þ ¼ vo ¼
ao s
0 bt b s ao s
3.1. Transport velocity of particles
vð t Þ ¼ vo ¼
An important parameter for the single step process, next to the length of the wire, is the transport velocity of the particles as a result of the vibratory feeder (Lim, 1997; Soldinger, 2002). The situation is explained in Fig. 3.
A particle moving on the vibrating plate can have three possible interactions with the surface. These cases are shown in Fig. 4 (1, 2, and 3). Case 1: for small vibration amplitudes or frequencies, the particle will catch up with the plate during the upward motion and it will leave the plate, hop forward and fall down at some point A during the down stroke. If we assume that the friction l of the particle and the plate is sufficiently high, i.e.,
Fig. 3. Particle transported by a vibratory feeder.
2ao ðcosa þ lsinaÞblg s2
sb t b2s
ð1Þ
ð2Þ
L. Muchova´ et al. / Int. J. Miner. Process. 78 (2005) 40–48
43
0,16
Velocity (m/s)
0,14 0,12
3,5 mm wire (wet)
0,10
5 mm wire (wet)
0,08
12,5 mm wire (wet)
0,06 Theoretical curve (Case 2)
0,04 0,02 0,00 3,2
3,4
3,6
3,8
Amplitude (mm) Fig. 5. Measured transport velocities versus theory for a frequency of 19 Hz.
the average particle transport velocity is vx ¼
4a2o sina gs3
ð3Þ
cosa
Otherwise, we get: ao 1 2ao sina 2ao sina vx ¼ cosa lgs 1 2 gs2 gs2 s ð4Þ Case 2: for larger values of the vibration amplitude or frequency, the particle reaches a higher velocity than for Case 1 and jumps high enough to catch up with the plate during the upward stroke. This mode of operation is optimal. It is required that: g ao sina b bg: 4 s2
ð5Þ
In this case, the transport velocity is given by: ao ð6Þ vx ¼ cosa: s Case 3: for still larger values of the amplitude or frequency, the feeding becomes suboptimal (see Fig. 4). This mode will never be used in practice. To check the theoretical equations, the transport velocity of the particles was measured for different settings of the amplitude and compared to Eq. (6) (see Fig. 5). The transport velocity for the 2nd (optimal) case was applied to the wire separation model. 3.2. Wire recovery by fingers (Mechanism A) The first mechanism to keep wires from falling to a lower level is by being carried over by the fingers: this
Mechanism A 1,2
Probability
1 0,8 0,6 0,4 0,2 0 0
10
20 = W
30
40 = 2W
Length of wires (lsinϕ) (mm) Fig. 6. Recovery curve for Mechanism A.
50
60
44
L. Muchova´ et al. / Int. J. Miner. Process. 78 (2005) 40–48
mechanism is shown in Fig. 1 (right). If a wire of length l is transported with an orientation u (the angle between the axis of the wire and the parallel direction), it is clear that the probability of recovery to the wire fraction depends on l sinu and the distance w between the two fingers. If lsinu b w, the wire will always be lost to the lower level, whereas for lsinu b 2w, the wire will always be recovered. In between these values, the wire has a linearly increasing probability of recovery. The probability curve for the test device is shown in Fig. 6.
Fig. 8. Motion of the tipping wire.
The critical wire length for the test device is about l c = 52 mm (for very low speed). The recovery for Mechanism B is plotted in Fig. 9.
3.3. Motion of particles falling down a step 3.4. Single step separation Wires that pass through the fingers will remain on the top level if their projection in the parallel direction, lcosu, is sufficiently large to reach the next plate on the top level (Figs. 7 and 8). The problem is to estimate the critical wire length in the parallel direction. The time that elapses between the moment of tipping and the touch down on the next plate is approximately: sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 2 3 hl cos u t¼ 2vx gd
ð7Þ
The critical length of the wire is therefore approximately given by: sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 2 pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 3 hl v cos u x lcosu=2 þ vx tlcosu=2 þ N h 2 þ d2 : 2gd ð8Þ
The single step separation combines mechanisms A and B. The recovery of a single step can now be computed for a given wire length by assuming that each orientation 0 b u b k/2 of the wire has the same probability. The recovery is found by averaging the probability PA(u) + (1 PA(u))P B(u) over all orientations, where PA is the recovery of Mechanism A and P B is the recovery of Mechanism B. The theoretical graph of Fig. 10 is split into 3 main sections. The first part P 1, for wires with w b l b 2w b l c, is given by:
P 1 ðl Þ ¼
2 p
2 ¼ p
Z
p 2
u¼arcsin wl
lsinu w du w
ffi pffiffiffiffiffiffiffiffiffiffiffiffiffiffi l 2 w2 w þ arcsin 1: l w
ð9Þ
The second part of the graph P 2, for 2w b l b l c , is given by:
2 P 2 ðl Þ ¼ p
Z
2 þ p
Fig. 7. Wire movement seen from the top.
arcsin wl u¼0
Z
p 2
2 0du þ p
Z
2 1du¼ 2w p u¼arcsin l
arcsin 2w l u¼arcsin wl
lsinu w du w
ffi pffiffiffiffiffiffiffiffiffiffiffiffiffiffi l 2 w2 w þ arcsin l w
pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi l 2 4w2 2w 2arcsin þ1 l w
ð10Þ
L. Muchova´ et al. / Int. J. Miner. Process. 78 (2005) 40–48
45
Mechanism B 1,2 1
Probability
0,8 0,6 0,4 0,2 0 0,00
20,00
40,00
lc
60,00
80,00
100,00
120,00
Length of wires (lcosϕ)(mm) Fig. 9. Recovery curve for Mechanism B.
The third part P 3, for lc blb 2 P3 ðl Þ ¼ p
Z
arccos lcl u¼0
Z
2 1du þ p
pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi lc2 þ w2 is:
Z
2 p
pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi lc2 þ w2 b l b lc2 þ 4w2 is:
P3 ðl Þ ¼
2 p
arcsin wl
0du u¼arccos lcl
Z
arcsin 2w l
Z
arccos lcl
1du þ
u¼0
2 p
Z
arcsin 2w l u¼arccos lcl
lsinu w du w
Z p 2 2 2 lc lc þ 1du ¼ 2arccos þ p u¼arcsin 2wl p l w pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2w l 2 4w2 2arcsin þ 1: ð12Þ l w
p 2
lsinu w 2 du þ 1du w p u¼arcsin 2wl u¼arcsin wl pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi l 2 w2 2 lc w ¼ þ arcsin arccos þ p l l w pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi l 2 4w2 2w 2arcsin þ1 ð11Þ l w þ
and for
The theoretical curves for the critical length of l c = 42 mm (Fig. 10) corresponds with a feeding speed of 50 mm/s.
1,0
Probability of recovery
0,9
Sigmoid fit
0,8 0,7 0,6
Theoretical curve, lc=52 mm
0,5 0,4 0,3
Theoretical curve, lc=42 mm
0,2 0,1 0,0 0
10
20
30
40
50
60
70
80
90
Length (mm) Fig. 10. Single step operation (theoretical recovery curve for wires versus a sigmoid fit); the results are shown for two values of l c, corresponding to different values of the vibrating amplitude.
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L. Muchova´ et al. / Int. J. Miner. Process. 78 (2005) 40–48
Probability of recovery
1,2 1,0 Probability curve single step
0,8
Probability curve 3 by 3 cascade
0,6 0,4
Probability curve 5 by 5 cascade
0,2 0,0 0
10
20
30
40
50
60
70
80
90
Length (mm) Fig. 11. Theoretical wire recovery curves for a 3 by 3 and a 5 by 5 cascade operation.
3.5. Cascade system
Table 1 Recovery of wires for different gap dimensions
The sigmoid fit in Fig. 10 shows the standard recovery probability curve:
Gap size (mm)
Wires b 50 mm (%)
50 mm b Wires b 80 mm (%)
Wires N 80 mm (%)
Width: 17.0, Height: 20.0 Width: 17.0, Height: 15.0 Width: 16.0, Height: 15.0
10.0
66.0
87.5
40.0
76.0
93.8
46.7
89.8
93.8
1 n l50 1þ l
ð13Þ
In this formula, l 50 is the length at which 50% of the wires goes to the wire concentrate and 50% of the wires goes to the tailings. The parameter n is related to the steepness of the curve. The largest disadvantage of the single step operation is that it does not show a sharp separation at a specific wire length. A way to improve the recovery curve is by using a cascade operation. Fig. 11 shows the probability curves for a 3 by 3 and a 5 by 5 cascade operation based on the sigmoid approximation. The formulas below for the 3 by 3 and the 5 by 5 cascade operations were calculated from Eq. (13). The curves show that the more steps in the cascade, the steeper the slope of the curve. 1 þ 6ð150 =1Þn þ15ð150 =1Þ2n þ10ð150 =1Þ3n ð1þ 3ð150 =1Þn þ 3ð150 =1Þ2n þð150 =1Þ3n Þ2
ð33Þ ð14Þ
4. Experimental results During the development of the separator, a set of experiments has been done to improve and optimise the design. The material mix used in these experiments is a 2–6 mm bottom ash fraction (slag product of waste incineration) sieved on a bar screen. This material consists of mainly granular and flat shaped particles, but also wires of all lengths and shapes. At first a series of experiments were conducted with the single step separator mounted on a standard vibrating feeder with wires of three length categories. The first goal was to optimise the width and height of the gap. Table 1 shows the wire recovery results for three different gap dimensions. The recovery of wires increases with decreasing width and height, although
1 þ 10ð150 =1Þn þ 45ð150 =1Þ2n þ 120ð150 =1Þ3n þ 210ð150 =1Þ4n þ 126ð150 =1Þ5n ð1 þ 5ð150 =1Þn þ 10ð150 =1Þ2n þ 10ð150 =1Þ3n þ 5ð150 =1Þ4n þ ð150 =1Þ5n Þ2
ð5 5Þ:
ð15Þ
L. Muchova´ et al. / Int. J. Miner. Process. 78 (2005) 40–48
47
100% Theoretical curve, lc=42 mm
Probability of recovery
90% 80% 70%
Theoretical curve, lc=52 mm
60% 50%
Single step (experiment al curve)
40% 30%
Sigmoid fit (single step)
20% 10% 0% 0
10
20
30
40
50
60
70
80
90
Length (mm) Fig. 12. Experimental recoveries of a single step operation and a sigmoid fit, compared to the theoretical recovery for the feeder setting corresponding to l c = 42 mm and l c = 52 mm.
there is a minimum dimension of the gap to ensure a limited loss of granulates to the wire fraction. After optimisation of the gap dimensions, the influence of the frequency and the amplitude of the feeder was tested. In a second experiment, wires from four different categories (35–39 mm, 40–44 mm, 45–50 mm and 50–54 mm) were fed with a gap of 17 mm by 20 mm (width by height) for two feeder settings (hereafter called 1 and 2). The purpose of the experiment was to find the relation between the feeder settings and l 50. Data in black in Fig. 12 show the experimental result. The experimental results were fitted to a sigmoid curve (Eq. (13)). The n in the formula was determined as 4.
A final experiment was executed to verify the theory using a 5 by 5 cascade, processing 700 kg of 2–6 mm bottom ash containing 632 wires of at least 35 mm long. The results of this experiment are shown in Table 2. The recovery of the larger wires shows that the separation of larger wires is outstanding (100% recovery). A comparison of theoretical and experimental data for the 5 5 test separator is shown in Fig. 13. The experimental curve follows the shape of the calculated recovery curves although the slope is less steep. It is believed that the decrease in steepness is caused by changes of the operation conditions during the separation. The first one is the feeder amplitude. It
Table 2 Experimental results on the wire recovery Length categories of wires (mm)
Wires in wire product (number)
Wires in tailings (number)
Total amount of wires (number)
Average wire length of category
Recovery wires in product (%)
35–39 40–44 45–49 50–54 55–59 60–64 65–69 70–74 75–79 80–84 85–89 N90
147 145 85 55 30 29 15 8 7 3 4 8
64 26 3 0 1 1 1 0 0 0 0 0
211 171 88 55 31 30 16 8 7 3 4 8
37.0 41.8 47.0 52.2 57.2 61.9 67.6 72.5 76.7 82.7 87.5 117.4
69.7 84.8 96.6 100.0 96.8 96.7 93.8 100.0 100.0 100.0 100.0 100.0
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L. Muchova´ et al. / Int. J. Miner. Process. 78 (2005) 40–48 1,2
Probability
1,0 Theoretical curve, for lc=52
0,8 0,6
Experimental curve
0,4 0,2 0,0 0
10
20
30
40
50
60
70
80
90
100 110 120 130
Length (mm) Fig. 13. Cascade system (experimental data versus theoretical calculation).
was difficult to keep the amplitude constant due to irregular pressure of the compressor. The second factor of influence is the moisture content of the bottom ash. The 700 kg sample was collected in a box containing a lot of water concentrated at the bottom of the box. The top material was therefore dryer than the material at the bottom of the box.
5. Conclusion A cascade wire separator has been successfully applied to a granular material from bottom ash. The experimental recovery for wires longer than 50 mm was essentially complete. With increasing the speed of the wires (for larger values of the vibration amplitude or frequency) the wire has a larger probability to pass the gap. The shape of the recovery curves depends of two factors: the critical length of the wires, which is a function of the feeding speed and the cascade system. These parameters define the position and the steepness of the curve. Notation ao Track vibration amplitude (m) d Diameter of the gap (mm) F x , F y Effective force in the x- and y-directions, respectively (N) g Gravitational constant h Height of the gap (mm) l Length of the wires (mm) lc Critical length of the wires (mm)
m P t vx, vy w x, y a l s u
Body mass (kg) Probability (%) Time (s) Velocities in the horizontal and vertical directions, respectively (ms- 1) Diameter between two fingers (mm) Displacements along the x- and y-directions (m) Angle of vibration (o) Coefficient of friction between body and the track Half period of vibration (s) Orientation of the wire (o)
References Beunder, E.M., Rem, P.C., 1999. Screening kinetics of cylindrical particles. International Journal of Mineral Processing 57, 73 – 81. Beunder, E.M., van Olst, K.A., Rem, P.C., 2002. Shape separation on a rotating cone. Minerals Processing 67, 145 – 160. Fabrizi, L., de Jong, T.P.R., Bevilacqua, P., 2003. Wire separation from automotive shredder residue. Physical Separation in Science and Engineering 12 (3), 145 – 165. Feruuchi, M., Gotoh, K., 1992. Shape separation of particles. Powder Technology 73, 1 – 9. Lim, G.H., 1997. On the conveying velocity of a vibratory feeder. Computers & Structures 62 (1), 197 – 203. Shinohara, K., 1986. Fundamental analysis on gravitational separation of differently shaped particles on inclined plates. Powder Technology 48, 151 – 159. Soldinger, M., 2002. Transport velocity of a crushed rock material bed on a screen. Minerals Engineering 15, 7 – 17.
A cascade wire separator Lenka Muchova´ a,*, Marieke Mooij a, Laurens van Kooy b, Peter Berkhout a a
Delft University of Technology, Mijnbouwstraat 120, 2628 RX Delft, The Netherlands b AEB, Australie¨havenweg 21, 1045 BA Amsterdam, The Netherlands
Received 15 February 2005; received in revised form 25 July 2005; accepted 26 July 2005 Available online 13 September 2005
Abstract Separation of wires from recycling streams is often desired for material recovery, quality improvement or for operational reasons. It increases the recovery of non-ferrous metals while congestion of material flows can be avoided. Tests with a new type of separator give good results for removing wires and other long particles from a granular stream of material. High recovery is achieved by repeating a simple separation step many times within one unit process. The resulting cascade system was tested with 700 kg of bottom ash, showing good agreement with theory and resulting in a complete recovery of wires exceeding 50 mm in length. D 2005 Elsevier B.V. All rights reserved. Keywords: wire separation; shape sorting; recycling
1. Introduction In most recycling operations the presence of wires complicates the separation process. Such processes are often designed for granulate particles. The curved shape of wires and their large ratio of length and diameter create different transport behaviour than for granulate particles. For this reason, a major problem caused by wires is blocking of the material flow. Secondary benefits of removing wires are enhanced metal recovery or the reduction of the metal content of basically nonmetal material streams such as plastics or wood. The present research aims to design a separation process with a total recovery of strongly elongated particles. * Corresponding author. E-mail address: [email protected] (L. Muchova´). 0301-7516/$ - see front matter D 2005 Elsevier B.V. All rights reserved. doi:10.1016/j.minpro.2005.07.005
Various solutions have been proposed to remove wires for different kinds of applications. Fabrizi et al. (2003) propose a nail roll separator for separating automotive shredder residue. Their results show a maximum recovery of 70% while the grade of the wire fraction varies from 35% to 87%. Another device, the Hamos CUX L brush belt separator, used for pre-treated electronic scrap, has a recovery between 75% and 80% with a product grade of 70%. Other solutions are known from shape sortation (e.g., Beunder and Rem, 1999; Beunder et al., 2002; Feruuchi and Gotoh, 1992; Shinohara, 1986). An example is the inclined belt, which can be used to concentrate wires based on their curvature and friction properties. The poor selectivity of shape separators usually prevents the production of relatively pure fractions: the efficiency depends strongly on the prop-
L. Muchova´ et al. / Int. J. Miner. Process. 78 (2005) 40–48
41
Fig. 1. Single step wire separator.
erties of the processed mixture. Known differences as textile content and wire shape have their influences on the result. The nail roll separator, for example, performs well on partly bent or hooked wires. Straight wires find their way to the non-wire fraction. All of the mentioned devices remove wires partially, leaving some of the wires in the bulk stream. The recovery of these separators could be improved by repeating the process many times in a cascade system. A cascade of separation steps results in an increase of grade and recovery. However, this would generally require a lot of space and energy for most separators. An important advantage of the design described here is that it is particularly suitable to repeat itself many times in one unit operation.
2. The cascade separator The basic concept of each step of the cascade separator is the ability of elongated particles to pass
over a gap or slot while granular particles of the same diameter will not (see Fig. 1). By placing the construction on top of a conventional vibrating feeder, a single step separator is created. Wires move along the top level of the separator while the granulate particles fall down through the dfingersT in the gap to a lower level. The dfingersT are positioned over the gap with a parallel orientation, to prevent wires that have a different orientation from falling through the gap. The separator works for dry and wet applications, although wet materials show some negative side effects due to sticky granulate particles, which result in a lower throughput and a lower grade of the wire fraction. A single step separation can be sufficient for applications with wires that are long compared to the size of the granular particles and for not too high capacities. At higher loads on the separator, the granular particles form a multi-layer and may cross the gap, while short wires occasionally fall through the gap. Therefore, positioning multiple gaps in a cascade is the solution
Fig. 2. Cascade wire separator with 5 5 steps.
42
L. Muchova´ et al. / Int. J. Miner. Process. 78 (2005) 40–48
for applications with a need for high capacity and sharp separation on wire length. Such a design is shown in Fig. 2. The dimensions and the number of separation steps can be changed according to the application. A high throughput of granulate with few wires will demand many gaps in the horizontal direction, while for a high recovery of short wires more separating steps in the vertical direction are required. A 5 5 cascade wire separator was built for testing the concept. The total height of this cascade separator is 360 mm and the length is 600 mm. The gap size is 17 mm by 20 mm (width times height). The distance between the fingers is 20 mm.
3. Theory In order to develop a predictive theory for the performance of the separator (single step and cascade system), it is helpful to take a closer look at the various elements of the process. The mechanics of the single step separation is a combination of transport and two mechanisms that prevent wires from falling through the gap. Going from the single step separation to a cascade separator can be described by statistics.
. Single step separation
.
o Transport velocity of particles o Mechanisms that prevent wires from falling through the gap n Mechanism A (fingers) n Mechanism B (step) Cascade separator
Fig. 4. Motion of the particles (dashed) and the plate (bold): vertical position (bottom) and horizontal velocity (top).
Note that the coordinate system is oriented so that the x-axis is parallel to the plate. This coordinate system is used for all displacements, velocities and accelerations. The plate vibration is defined by its amplitude a 0, the frequency 1/(2s) and the angle of vibration a. For simplicity, we assume a triangular motion. Then the plate equations of motion are, vxplate ¼ vðt Þcosa vyplate ¼ vðt Þsina vð t Þ ¼ vo ¼
ao s
0 bt b s ao s
3.1. Transport velocity of particles
vð t Þ ¼ vo ¼
An important parameter for the single step process, next to the length of the wire, is the transport velocity of the particles as a result of the vibratory feeder (Lim, 1997; Soldinger, 2002). The situation is explained in Fig. 3.
A particle moving on the vibrating plate can have three possible interactions with the surface. These cases are shown in Fig. 4 (1, 2, and 3). Case 1: for small vibration amplitudes or frequencies, the particle will catch up with the plate during the upward motion and it will leave the plate, hop forward and fall down at some point A during the down stroke. If we assume that the friction l of the particle and the plate is sufficiently high, i.e.,
Fig. 3. Particle transported by a vibratory feeder.
2ao ðcosa þ lsinaÞblg s2
sb t b2s
ð1Þ
ð2Þ
L. Muchova´ et al. / Int. J. Miner. Process. 78 (2005) 40–48
43
0,16
Velocity (m/s)
0,14 0,12
3,5 mm wire (wet)
0,10
5 mm wire (wet)
0,08
12,5 mm wire (wet)
0,06 Theoretical curve (Case 2)
0,04 0,02 0,00 3,2
3,4
3,6
3,8
Amplitude (mm) Fig. 5. Measured transport velocities versus theory for a frequency of 19 Hz.
the average particle transport velocity is vx ¼
4a2o sina gs3
ð3Þ
cosa
Otherwise, we get: ao 1 2ao sina 2ao sina vx ¼ cosa lgs 1 2 gs2 gs2 s ð4Þ Case 2: for larger values of the vibration amplitude or frequency, the particle reaches a higher velocity than for Case 1 and jumps high enough to catch up with the plate during the upward stroke. This mode of operation is optimal. It is required that: g ao sina b bg: 4 s2
ð5Þ
In this case, the transport velocity is given by: ao ð6Þ vx ¼ cosa: s Case 3: for still larger values of the amplitude or frequency, the feeding becomes suboptimal (see Fig. 4). This mode will never be used in practice. To check the theoretical equations, the transport velocity of the particles was measured for different settings of the amplitude and compared to Eq. (6) (see Fig. 5). The transport velocity for the 2nd (optimal) case was applied to the wire separation model. 3.2. Wire recovery by fingers (Mechanism A) The first mechanism to keep wires from falling to a lower level is by being carried over by the fingers: this
Mechanism A 1,2
Probability
1 0,8 0,6 0,4 0,2 0 0
10
20 = W
30
40 = 2W
Length of wires (lsinϕ) (mm) Fig. 6. Recovery curve for Mechanism A.
50
60
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L. Muchova´ et al. / Int. J. Miner. Process. 78 (2005) 40–48
mechanism is shown in Fig. 1 (right). If a wire of length l is transported with an orientation u (the angle between the axis of the wire and the parallel direction), it is clear that the probability of recovery to the wire fraction depends on l sinu and the distance w between the two fingers. If lsinu b w, the wire will always be lost to the lower level, whereas for lsinu b 2w, the wire will always be recovered. In between these values, the wire has a linearly increasing probability of recovery. The probability curve for the test device is shown in Fig. 6.
Fig. 8. Motion of the tipping wire.
The critical wire length for the test device is about l c = 52 mm (for very low speed). The recovery for Mechanism B is plotted in Fig. 9.
3.3. Motion of particles falling down a step 3.4. Single step separation Wires that pass through the fingers will remain on the top level if their projection in the parallel direction, lcosu, is sufficiently large to reach the next plate on the top level (Figs. 7 and 8). The problem is to estimate the critical wire length in the parallel direction. The time that elapses between the moment of tipping and the touch down on the next plate is approximately: sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 2 3 hl cos u t¼ 2vx gd
ð7Þ
The critical length of the wire is therefore approximately given by: sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 2 pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 3 hl v cos u x lcosu=2 þ vx tlcosu=2 þ N h 2 þ d2 : 2gd ð8Þ
The single step separation combines mechanisms A and B. The recovery of a single step can now be computed for a given wire length by assuming that each orientation 0 b u b k/2 of the wire has the same probability. The recovery is found by averaging the probability PA(u) + (1 PA(u))P B(u) over all orientations, where PA is the recovery of Mechanism A and P B is the recovery of Mechanism B. The theoretical graph of Fig. 10 is split into 3 main sections. The first part P 1, for wires with w b l b 2w b l c, is given by:
P 1 ðl Þ ¼
2 p
2 ¼ p
Z
p 2
u¼arcsin wl
lsinu w du w
ffi pffiffiffiffiffiffiffiffiffiffiffiffiffiffi l 2 w2 w þ arcsin 1: l w
ð9Þ
The second part of the graph P 2, for 2w b l b l c , is given by:
2 P 2 ðl Þ ¼ p
Z
2 þ p
Fig. 7. Wire movement seen from the top.
arcsin wl u¼0
Z
p 2
2 0du þ p
Z
2 1du¼ 2w p u¼arcsin l
arcsin 2w l u¼arcsin wl
lsinu w du w
ffi pffiffiffiffiffiffiffiffiffiffiffiffiffiffi l 2 w2 w þ arcsin l w
pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi l 2 4w2 2w 2arcsin þ1 l w
ð10Þ
L. Muchova´ et al. / Int. J. Miner. Process. 78 (2005) 40–48
45
Mechanism B 1,2 1
Probability
0,8 0,6 0,4 0,2 0 0,00
20,00
40,00
lc
60,00
80,00
100,00
120,00
Length of wires (lcosϕ)(mm) Fig. 9. Recovery curve for Mechanism B.
The third part P 3, for lc blb 2 P3 ðl Þ ¼ p
Z
arccos lcl u¼0
Z
2 1du þ p
pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi lc2 þ w2 is:
Z
2 p
pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi lc2 þ w2 b l b lc2 þ 4w2 is:
P3 ðl Þ ¼
2 p
arcsin wl
0du u¼arccos lcl
Z
arcsin 2w l
Z
arccos lcl
1du þ
u¼0
2 p
Z
arcsin 2w l u¼arccos lcl
lsinu w du w
Z p 2 2 2 lc lc þ 1du ¼ 2arccos þ p u¼arcsin 2wl p l w pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2w l 2 4w2 2arcsin þ 1: ð12Þ l w
p 2
lsinu w 2 du þ 1du w p u¼arcsin 2wl u¼arcsin wl pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi l 2 w2 2 lc w ¼ þ arcsin arccos þ p l l w pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi l 2 4w2 2w 2arcsin þ1 ð11Þ l w þ
and for
The theoretical curves for the critical length of l c = 42 mm (Fig. 10) corresponds with a feeding speed of 50 mm/s.
1,0
Probability of recovery
0,9
Sigmoid fit
0,8 0,7 0,6
Theoretical curve, lc=52 mm
0,5 0,4 0,3
Theoretical curve, lc=42 mm
0,2 0,1 0,0 0
10
20
30
40
50
60
70
80
90
Length (mm) Fig. 10. Single step operation (theoretical recovery curve for wires versus a sigmoid fit); the results are shown for two values of l c, corresponding to different values of the vibrating amplitude.
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L. Muchova´ et al. / Int. J. Miner. Process. 78 (2005) 40–48
Probability of recovery
1,2 1,0 Probability curve single step
0,8
Probability curve 3 by 3 cascade
0,6 0,4
Probability curve 5 by 5 cascade
0,2 0,0 0
10
20
30
40
50
60
70
80
90
Length (mm) Fig. 11. Theoretical wire recovery curves for a 3 by 3 and a 5 by 5 cascade operation.
3.5. Cascade system
Table 1 Recovery of wires for different gap dimensions
The sigmoid fit in Fig. 10 shows the standard recovery probability curve:
Gap size (mm)
Wires b 50 mm (%)
50 mm b Wires b 80 mm (%)
Wires N 80 mm (%)
Width: 17.0, Height: 20.0 Width: 17.0, Height: 15.0 Width: 16.0, Height: 15.0
10.0
66.0
87.5
40.0
76.0
93.8
46.7
89.8
93.8
1 n l50 1þ l
ð13Þ
In this formula, l 50 is the length at which 50% of the wires goes to the wire concentrate and 50% of the wires goes to the tailings. The parameter n is related to the steepness of the curve. The largest disadvantage of the single step operation is that it does not show a sharp separation at a specific wire length. A way to improve the recovery curve is by using a cascade operation. Fig. 11 shows the probability curves for a 3 by 3 and a 5 by 5 cascade operation based on the sigmoid approximation. The formulas below for the 3 by 3 and the 5 by 5 cascade operations were calculated from Eq. (13). The curves show that the more steps in the cascade, the steeper the slope of the curve. 1 þ 6ð150 =1Þn þ15ð150 =1Þ2n þ10ð150 =1Þ3n ð1þ 3ð150 =1Þn þ 3ð150 =1Þ2n þð150 =1Þ3n Þ2
ð33Þ ð14Þ
4. Experimental results During the development of the separator, a set of experiments has been done to improve and optimise the design. The material mix used in these experiments is a 2–6 mm bottom ash fraction (slag product of waste incineration) sieved on a bar screen. This material consists of mainly granular and flat shaped particles, but also wires of all lengths and shapes. At first a series of experiments were conducted with the single step separator mounted on a standard vibrating feeder with wires of three length categories. The first goal was to optimise the width and height of the gap. Table 1 shows the wire recovery results for three different gap dimensions. The recovery of wires increases with decreasing width and height, although
1 þ 10ð150 =1Þn þ 45ð150 =1Þ2n þ 120ð150 =1Þ3n þ 210ð150 =1Þ4n þ 126ð150 =1Þ5n ð1 þ 5ð150 =1Þn þ 10ð150 =1Þ2n þ 10ð150 =1Þ3n þ 5ð150 =1Þ4n þ ð150 =1Þ5n Þ2
ð5 5Þ:
ð15Þ
L. Muchova´ et al. / Int. J. Miner. Process. 78 (2005) 40–48
47
100% Theoretical curve, lc=42 mm
Probability of recovery
90% 80% 70%
Theoretical curve, lc=52 mm
60% 50%
Single step (experiment al curve)
40% 30%
Sigmoid fit (single step)
20% 10% 0% 0
10
20
30
40
50
60
70
80
90
Length (mm) Fig. 12. Experimental recoveries of a single step operation and a sigmoid fit, compared to the theoretical recovery for the feeder setting corresponding to l c = 42 mm and l c = 52 mm.
there is a minimum dimension of the gap to ensure a limited loss of granulates to the wire fraction. After optimisation of the gap dimensions, the influence of the frequency and the amplitude of the feeder was tested. In a second experiment, wires from four different categories (35–39 mm, 40–44 mm, 45–50 mm and 50–54 mm) were fed with a gap of 17 mm by 20 mm (width by height) for two feeder settings (hereafter called 1 and 2). The purpose of the experiment was to find the relation between the feeder settings and l 50. Data in black in Fig. 12 show the experimental result. The experimental results were fitted to a sigmoid curve (Eq. (13)). The n in the formula was determined as 4.
A final experiment was executed to verify the theory using a 5 by 5 cascade, processing 700 kg of 2–6 mm bottom ash containing 632 wires of at least 35 mm long. The results of this experiment are shown in Table 2. The recovery of the larger wires shows that the separation of larger wires is outstanding (100% recovery). A comparison of theoretical and experimental data for the 5 5 test separator is shown in Fig. 13. The experimental curve follows the shape of the calculated recovery curves although the slope is less steep. It is believed that the decrease in steepness is caused by changes of the operation conditions during the separation. The first one is the feeder amplitude. It
Table 2 Experimental results on the wire recovery Length categories of wires (mm)
Wires in wire product (number)
Wires in tailings (number)
Total amount of wires (number)
Average wire length of category
Recovery wires in product (%)
35–39 40–44 45–49 50–54 55–59 60–64 65–69 70–74 75–79 80–84 85–89 N90
147 145 85 55 30 29 15 8 7 3 4 8
64 26 3 0 1 1 1 0 0 0 0 0
211 171 88 55 31 30 16 8 7 3 4 8
37.0 41.8 47.0 52.2 57.2 61.9 67.6 72.5 76.7 82.7 87.5 117.4
69.7 84.8 96.6 100.0 96.8 96.7 93.8 100.0 100.0 100.0 100.0 100.0
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L. Muchova´ et al. / Int. J. Miner. Process. 78 (2005) 40–48 1,2
Probability
1,0 Theoretical curve, for lc=52
0,8 0,6
Experimental curve
0,4 0,2 0,0 0
10
20
30
40
50
60
70
80
90
100 110 120 130
Length (mm) Fig. 13. Cascade system (experimental data versus theoretical calculation).
was difficult to keep the amplitude constant due to irregular pressure of the compressor. The second factor of influence is the moisture content of the bottom ash. The 700 kg sample was collected in a box containing a lot of water concentrated at the bottom of the box. The top material was therefore dryer than the material at the bottom of the box.
5. Conclusion A cascade wire separator has been successfully applied to a granular material from bottom ash. The experimental recovery for wires longer than 50 mm was essentially complete. With increasing the speed of the wires (for larger values of the vibration amplitude or frequency) the wire has a larger probability to pass the gap. The shape of the recovery curves depends of two factors: the critical length of the wires, which is a function of the feeding speed and the cascade system. These parameters define the position and the steepness of the curve. Notation ao Track vibration amplitude (m) d Diameter of the gap (mm) F x , F y Effective force in the x- and y-directions, respectively (N) g Gravitational constant h Height of the gap (mm) l Length of the wires (mm) lc Critical length of the wires (mm)
m P t vx, vy w x, y a l s u
Body mass (kg) Probability (%) Time (s) Velocities in the horizontal and vertical directions, respectively (ms- 1) Diameter between two fingers (mm) Displacements along the x- and y-directions (m) Angle of vibration (o) Coefficient of friction between body and the track Half period of vibration (s) Orientation of the wire (o)
References Beunder, E.M., Rem, P.C., 1999. Screening kinetics of cylindrical particles. International Journal of Mineral Processing 57, 73 – 81. Beunder, E.M., van Olst, K.A., Rem, P.C., 2002. Shape separation on a rotating cone. Minerals Processing 67, 145 – 160. Fabrizi, L., de Jong, T.P.R., Bevilacqua, P., 2003. Wire separation from automotive shredder residue. Physical Separation in Science and Engineering 12 (3), 145 – 165. Feruuchi, M., Gotoh, K., 1992. Shape separation of particles. Powder Technology 73, 1 – 9. Lim, G.H., 1997. On the conveying velocity of a vibratory feeder. Computers & Structures 62 (1), 197 – 203. Shinohara, K., 1986. Fundamental analysis on gravitational separation of differently shaped particles on inclined plates. Powder Technology 48, 151 – 159. Soldinger, M., 2002. Transport velocity of a crushed rock material bed on a screen. Minerals Engineering 15, 7 – 17.