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A mathematical model of the duplex concentrator
Minerals Engineering, Vol. 4, No. 3/4, pp. 347-354, 1991 Printed in Great Britain
Pergamon Press plc
A MATHEMATICAL MODEL OF THE DUPLEX CONCENTRATOR*
M. PEARL, K.A. LEWIS and P. TUCKER Warren Spring Laboratory, Gunnels Wood Road, Stevenage, Herts. SGI 2BX, England
ABSTRACT This paper describes the development of a mathematical model of the Duplex concentrator. The model is formulated in terms of the spatial distribution of material (ie. banding) of each size/ SG component down the length of the Duplex deck. In the model, analytical equations are derived for these distribution profiles for the point in time at which the concentrate is rinsed from the deck. Auxiliary equations are presented to describe how the distribution profiles vary with operating conditions. The physical significance of these relationships is briefly discussed. The model has been based on experimental data collected at South Crofty and at Beralt Tin and Wolfram S.A. (BTW) where a unit has been in production since late 1988. Metallurgical performance data for this unit are summarised. Model validation results are reported and, for completeness, a brief description of the Duplex is also given.
Keyword$ Duplex concentrator; mathematical model 1. INTRODUCTION The Duplex separator is a gravity concentration device for slimes separation. The device has only been introduced, fairly recently, into the industry. Beralt Tin and Wolfram S.A. (BTW) have been one of the first commercial operations to make use of it in their beneficiation process. Separation on the Duplex makes use of the flowing film separation mechanism (cf. fines shaking tables)though, unlike the shaking table, the Duplex is run in a semi-batch mode with concentrate accumulating on the deck surface rather than being continuously discharged. The machine consists of two decks, side by side, one accumulating material whilst the other is washed and the batch of concentrate discharged. The decks have an adjustable downwards tilt in the direction of feed flow. An even (sinusoidal) vibratory motion is applied to the deck perpendicular to the tilt. The feed, wash and rinse periods are controlled by pre-set programs. The basic program cycle is as follows: (1)
Feed slurry is introduced on to the deck. Whilst the denser material settles onto the deck, the less dense material flows down the deck and is discharged to tailing ('tails during feed').
(2)
The feed is switched off and wash water applied. This cleaning process generates a second tailings discharge ('tails during wash').
*UK Crown CopyriRht 347
348
(3)
M. PEARL et al.
The whole deck tips sideways and the remaining settled material is rinsed off into separate discharge launders. Two products are generally collected at this stage, a concentrate from material which settled out at the top of the deck and a middlings from that which settled lower on the deck.
The work reported in this paper sets out to quantify Duplex performance in production and to investigate the relative effects, on performance, of changing operating variables. The results of the study are formulated in terms of a mathematical model of the Duplex separation. The experimental studies, carried out to underpin the development of the mathematical model, were undertaken at two sites using three different feedstocks. (i) Tests at South Crofty (using tin fines) quantified the effects of wash water flow rates and vibration frequency on separation performance. Cycle times were kept constant. (ii) Tests at BTW (July 1989) investigated the effects of stroke length, wash water flow rates and cycle times. The Duplex was set up in these tests to process stockpiled Bartles (or slimes circuit) feed. The BTW slimes circuit configuration, at that time is shown in figure 1. A preliminary test (March 1989) had also been undertaken to characterise the production Duplex operation, at BTW, in its then normal duty of concentrate cleaning.
BARTLES FEED (MOZLEY U/F}
TAIL
L BARTLES (4 OFF')
TAIL
FLOAT CELL
PLAT"O" 4 OFF I
F
-t
l
MIDDS f
TAIL
TAIL
CONC
MIDDS
TAIL
Fig.l BTW Slimes Section 2. EXPERIMENTAL RESULTS The shape of the partition curves for the concentrate, and jointly, for the concentrate and middlings were broadly similar over all tests. Characteristic curves (BTW; July 1989) are shown in figures 2 and 3, though it should be noted that these curves were obtained under non-optimum conditions. For all size fractions, the curves are sigmoidal in shape with the steepest increase over the density range 3.0 - 3.7. The curves become progressively less steep with decreasing particle size. For +45 micron material, the separation density (i.e.DS0) increases with particle size. The recovery of high SG material decreases with increasing particle size; this drop being more pronounced for the BTW March 1989 sample and the South Crofty tests than for the BTW July 1989 samples. The effect can be best visualised in figures 4 and 5, which show the size transfer (recovery) curves for discrete density fractions. The observed behaviourial difference at coarse sizes is thought to be due (primarily) to the relatively low quantity of coarse material in the BTW July 1989 feedstock compared with the other two sets of samples. This coarse material is deposited on the deck
349
A mathematical model of the Duplex concentrator
in a band in the concentrate but close to the middlings cutter position. If the amount of coarse material were larger, this band would broaden, spanning the cutter position, letting some of the material carry over into the middlings product. Separation efficiency decreases below 45 microns and recoveries also decrease below this size.
100
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70
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E 50 40
3O
2O
10
0
'
2.5
'
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:.o
3.5
:.5
,'.o
DENSITY, 9/ml
Fig.2 Partition to concentrate- July 1989, Test 2
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. ~
- ~__ -_._--~___" ~-~-__-- __--~____- ---__.~
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70
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20 x
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/
JxJ
20
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3'°
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,o'
,'5
.Io
DENSITY. g/ml
Fig.3 Partition to concentrate + middlings- July 1989, Test 2 Overall performance data for the production operation (at BTW;March 1989) are additionally given in tables I and 2. The feed size distribution, at that time, was 84% passing 180 microns with 21% below 45 microns. Over 98% of the wolframite was recovered to the concentrate at an upgrade ratio of approximately 2.5. The Duplex was operated to cut midway through the sulphide band; the bulk of the arsenopyrite being recovered to the concentrate, whilst the chalcopyrite was almost totally rejected. The rejected material also contained significant amounts of the siderite present in the feed. Rejection of this material
M . PEARL et al.
350
is particularly important at BTW as it poses a significant problem downstream in the final magnetic cleaning of the wolfram concentrate.
l!
> 3.3 "~ EXPERIMENTAL <3,3J
TEST 1
1O0
PREDICTED
8
80
=~
~- 60 Z
o
80
rl
eo
z
o
40
40
20
20
,
•
40
~
, =,
80
120
=
J
160
0
¥
0
200
I
I
|
I
40
80
120
160
J
200
SIZE, MICRONS
SIZE, MICRONS
TRANSFER TO CONCENTRATE AND MIDDLINGS
TRANSFER TO CONCENTRATE
Fig.4 Beralt, July 1989 campaign 0 >3.3 1 / A 2.8-3.3 t EXPERIMENTAL
Ix < 2 . 8 J
TEST R6
100
1O0
80
80 8 60 uJ
80
z
~ 4o
Z < 40
20
20
0 0
I 40
I 80
I 120
I 160
J 200
I
I
I
I
I
40
80
120
160
200
SIZE, MICRONS
SIZE, MICRONS
TRANSFER TO CONCENTRATE AND MIDDLINGS
TRANSFER TO CONCENTRATE
Fig.5 South Crofty campaign TABLE I: BTW Duplex: Balanced Data %NO 3
Feed Concentrate Middllngs Tails/feed Tails/wash
15.06 38.99 0.58 0.81 0.21
%As
%Cu
T/hr
0.24 0.46 0.02 0.30 0.09
0.01 0.00 0.02 0.02 0.01
0.2086 0.0792 0.0368 0.0228 0.0698
351
A mathematical model of the Duplex concentrator
TABLE 2: BTW Duplex: Recoveries WO 3
To To To To
Concentrate Middlings Tails/feed Tails/wash
98.28 0.67 0.59 0.46
AS
72.45 1.46 13.46 12.63
CU
1.2 23.4 29.7 45.7
3. MODEL DEVELOPMENT The trends seen in the experimental results are not too dissimilar to those commonly observed for conventional shaking tables. On this basis, a prototype Duplex model could be built on the same concepts as those used for modelling the shaking table. The WSL Duplex model thus closely parallels the WSL shaking table model described in Reference 1. The model is formulated in terms of the spatial distribution of material (of each s i z e / S G component) on the separating surface, in this case down the length of the Duplex table. The frame of reference is taken as x=0 (feed end) to x=l (tailings end) with an intermediate value of x=0.5 representing the usual cutter position separating the concentrate from the middling. The prototype model, described here, does not attempt to describe the time dependant build up of the material distributions (or their modification during the wash cycle), rather it simply models the distribution profiles at the instant in time immediately following the wash cycle and prior to the start of the rinse cycle. In the model, a separate distribution profile is formulated for each SG fraction (i) of each size fraction (j). These distribution profiles (D) are described in a cumulative form and modelled by a summation of two functions: Dl(i,j) = tanh (0.05. [x/d5(i,j)] zl ) D2(i,j) = tanh (0.54932. [(x - d5(i,j))/d50(i,j)] zz ) where D = DI and D = 0.05 + 0.95 . D2
for x < d5 for x > d5
d5(i,j) and d50(i,j) are the x positions corresponding to 5% and 50% of each component distribution respectively and the exponents ZI and Z2 are model constants which control the shapes of the two curve segments. In the formulation, the d5 is taken as the position of the 'top' of the individual component band down the table and the d50 represents the halfwidth of the band. (Note that in the solution space, x, d5 and d50 are not constrained to lie below unity. In physical terms, values greater than unity simply mean 'off the end of the table - in the tailings product'). The observed 'fuzziness' of the band edge is modelled by allocating the first 5% of the material to a transition zone whilst D2 describes the main band.The probability of transfer, or recovery, T(Kij) of material of size (j) and SG (i) to the concentrate (K= 1) or the middlings (K,.2) is computed from the component distribution by introducing a cutter fraction: x = CP(K) MI: 4/3-4--J
; CP(I)=0.5 CP(2)=I.0 (normally)
352
M. PEARL et al.
The size/ SG dependencies of the d5 and d50 are found to be built up out of two component factors. Firstly there is an analytical dependence on the numerical values of the sizes and SGs themselves. Secondly there is a dependence on the amount of material within the component fractions. This second dependence is a consequence of particle crowding or competition. The two dependencies can be best visualised as follows: When there is substantial material of one component present, its own band will be wide and will also force all succeeding bands further down the table (particle crowding). The positioning of the band however, in the absence of any preceding bands, is not fixed in position (as it would be if the first effect was absent) but moves progressively down the table with decreasing SG (and increasing size). The two dependencies appear comparable in effect with the Duplex, in contrast to the shaking table where the particle crowding effect dominates (Ref. 1). For a given feed rate FW, the weight of material in any component (i,j) which is fed onto the table, in time t,is given by FW(i,j).t. If this were deposited as a band n particles thick, the volume of the band would be proportional to [n.size(j).W(i,j).b] where W(i,j) is the width of the band and b is the breadth of the table. W could then be computed from the following expression: W (i,j) ct FW (i,j) / ( SG ( i ) . size ( j ) ) For the real flow conditions, and with overlapping bands, this expression is too simplistic and was found not to give a good enough fit to data. It was found, however, that by introducing a simple empirical correction factor (e), the goodness of fit could be substantially improved. The model relationship thus became: W (i,j) ~ [FW (i,j) / ( SG (i). size (j))]e The half width of the band (d50) can then be modelled by: dSO (i,j) = P2. W (i,j). SS where P2 is a model parameter. SS is another empirical correction factor which serves to increase d50 for sizes significantly greater or less than a reference size, sO. This factor accounts for the poorer separation efficiencies found with unclassified feed material. The band start positions (ie the d5s) are modelled to take account of the two prime dependencies discussed above: d5 (i,j) ffi P3. function [ SG(i), size (j)] + PI . E W(i,j) rank
where PI, P3 are model coefficients. The second term, which is a measure of the cumulative width of all preceding bands, explicitly models the particle crowding. The rank order can be fixed by a simple empirical function (as for the shaking table; Ref. I). 4. MODEL DEPENDENCE ON OPERATING VARIABLES The model is calibrated for a given ore and a given set of operating conditions through deriving the appropriate parameter set [PI, P2 ,P3]. The exact form and positions of these parameters in the model equations were carefully chosen such that they could fulfil another role, enabling the model to be extrapolated to different operating conditions. In this respect, machine performance variations with operational regime were modelled using a suite of relationships, which linked each parameter (Pn) to each of the operating variables. These relationships were based on a combination of empirical and conceptual premises. The identified relationships are described below:
A mathematical model of the Duplex concentrator
353
Taliings Wash Water Addition Tailings wash water variations have a small but significant effect on the value of the parameters for low levels of wash water addition, with a very much larger effect at the higher wash water levels (fig. 6a). The concept of a threshold value (in this case around 400 kg/hr water addition) might be defined. The overall variations can be described by equations of the form: Pn = a(n). (wash water addition)q(n)+ b(n) where a, b and q are separate constants for parameters n=l to 3 respectively. The 400 kg/hr, threshold appears to define the activation energy at which the dense material starts to be swept off the deck with band widths (related to P2) and band start positions (related to PI) both increasing dramatically. Below 400 kg/hr, the main effect of increasing wash water is to move the lower SG material bodily down the deck (as portrayed by the increase in P3).
Amplitude of Stroke For P1, P2 and P3, the general form of the variations with stroke amplitude (Fig. 6c) can also be given by the function: Pn = a(n) . (amplitude)q(n) + b(n) with a, b and q representing the new set of coefficients. As with wash water level, a threshold stroke length of about 8 cm. can be defined, beyond which the dense material starts to be lost from the table. WASHWATER
¢
1 e
WASHWATER,~ h r
T
I
t
10
TAIL8 WASH DURATION bl FREQUENCYOF STROKE
~1
STROKE AMPLrrUDF-© m s
0
1,0~
,.
0.8~
O
t 0.2 0,0
....
FREQUENCY.&'TROKF~,'mln
:. . . .
,
J,3
TALL8 WASH DURATION.mln4
Fig.6 Model parameter values against operating variables
Frequency of Vibration The variations in PI, P2 and P3 (Fig. 6b) can be modelled adequately by the Rosin-Ramler relationship: Pn = g(n). (l - exp
(-0.693.(frequency/h(n))r(n))
where g, h and r are separate constants for parameters n=l to 3 respectively.
354
M. PEARL et al.
Here again a threshold value (ffi78 Strokes per minute) appears to be evident. This threshold appears to determine a 'step' from closely spaced to more widely spaced bands (related to PI and P3) with little further change to band spacing as frequencies move away from this value. A less pronounced step in band widths also occurs at the threshold frequency though there is also a residual progressive increase in band width with frequency over the whole frequency range. Tailings Wash Duration The effects of tailings wash duration were studied over a limited number of tests at high and low wash water additions. The variations of PI, P2 and P3 (Fig. 6d) have the following relationships: Pn ffi a(n). (wash duration) + b(n) where a and b are the coefficient sets for this relationship. Variations in Pl, P2 and P3 are relatively small over the wash duration values of the study, a higher duration moving the bands slightly further down the deck (slight increases in Pl and P3) and slightly increasing the widths of each band (small increase in P2). This agrees with observation, where it has been seen that the majority of the cleaning action takes place early in the wash cycle, with extended wash periods not adding much to this. In physical terms, the wash water level provides the energy to mobilise the particles. Each class of particles will have a spectrum of mobilisation energies. Those particles with mobilisation energies below the supplied energy (in the wash) will be free to move 'instantaneously'. Those particles with mobilisation energies just above the supplied energy will not move, but may subsequently be mobilised as 'random' perturbations in the energies occur. The scale of these random perturbations is likely to be small relative to the spectrum of mobilisation energies present and, as such, may only affect a small proportion of the particles in each class. 5. VALIDATION AND CONCLUSIONS The main model function has been validated against measured performance data and has been found to provide acceptable fits to both the Beralt and the South Crofty data (eg. Figs. 4 and 5). Although generally successful, the model still has some difficulty in accounting for performance at the coarsest sizes. The largest errors in the calculated transfer curves generally being for these sizes, underestimating the recovery of the high SG fractions in the BTW July 1989 tests and overestimating the recoveries for the South Crofty and BTW March 1989 data sets. (Note this is where the measured data show their most pronounced differences). Although some improvement could be made through further empirical tuning, it is believed that more experimental data is first needed to substantiate the observed effect and its proposed explanation before such corrections could be applied with any degree of confidence. ACKNOWLEDGEMENT The work was co-funded by the Department of Trade and Industry and by the Commission of the European Communities 3rd Raw Materials Programme. The contributions of Beralt Tin and Wolfram SA and Carnon Consolidated Ltd to the model development are gratefully acknowledged. REFERENCE .
Tucker P., Lewis K.A. & Wood P., Computer Optimisation of a Shaking Table, Minerals Engineering, 4 (3/4) (1991).
Pergamon Press plc
A MATHEMATICAL MODEL OF THE DUPLEX CONCENTRATOR*
M. PEARL, K.A. LEWIS and P. TUCKER Warren Spring Laboratory, Gunnels Wood Road, Stevenage, Herts. SGI 2BX, England
ABSTRACT This paper describes the development of a mathematical model of the Duplex concentrator. The model is formulated in terms of the spatial distribution of material (ie. banding) of each size/ SG component down the length of the Duplex deck. In the model, analytical equations are derived for these distribution profiles for the point in time at which the concentrate is rinsed from the deck. Auxiliary equations are presented to describe how the distribution profiles vary with operating conditions. The physical significance of these relationships is briefly discussed. The model has been based on experimental data collected at South Crofty and at Beralt Tin and Wolfram S.A. (BTW) where a unit has been in production since late 1988. Metallurgical performance data for this unit are summarised. Model validation results are reported and, for completeness, a brief description of the Duplex is also given.
Keyword$ Duplex concentrator; mathematical model 1. INTRODUCTION The Duplex separator is a gravity concentration device for slimes separation. The device has only been introduced, fairly recently, into the industry. Beralt Tin and Wolfram S.A. (BTW) have been one of the first commercial operations to make use of it in their beneficiation process. Separation on the Duplex makes use of the flowing film separation mechanism (cf. fines shaking tables)though, unlike the shaking table, the Duplex is run in a semi-batch mode with concentrate accumulating on the deck surface rather than being continuously discharged. The machine consists of two decks, side by side, one accumulating material whilst the other is washed and the batch of concentrate discharged. The decks have an adjustable downwards tilt in the direction of feed flow. An even (sinusoidal) vibratory motion is applied to the deck perpendicular to the tilt. The feed, wash and rinse periods are controlled by pre-set programs. The basic program cycle is as follows: (1)
Feed slurry is introduced on to the deck. Whilst the denser material settles onto the deck, the less dense material flows down the deck and is discharged to tailing ('tails during feed').
(2)
The feed is switched off and wash water applied. This cleaning process generates a second tailings discharge ('tails during wash').
*UK Crown CopyriRht 347
348
(3)
M. PEARL et al.
The whole deck tips sideways and the remaining settled material is rinsed off into separate discharge launders. Two products are generally collected at this stage, a concentrate from material which settled out at the top of the deck and a middlings from that which settled lower on the deck.
The work reported in this paper sets out to quantify Duplex performance in production and to investigate the relative effects, on performance, of changing operating variables. The results of the study are formulated in terms of a mathematical model of the Duplex separation. The experimental studies, carried out to underpin the development of the mathematical model, were undertaken at two sites using three different feedstocks. (i) Tests at South Crofty (using tin fines) quantified the effects of wash water flow rates and vibration frequency on separation performance. Cycle times were kept constant. (ii) Tests at BTW (July 1989) investigated the effects of stroke length, wash water flow rates and cycle times. The Duplex was set up in these tests to process stockpiled Bartles (or slimes circuit) feed. The BTW slimes circuit configuration, at that time is shown in figure 1. A preliminary test (March 1989) had also been undertaken to characterise the production Duplex operation, at BTW, in its then normal duty of concentrate cleaning.
BARTLES FEED (MOZLEY U/F}
TAIL
L BARTLES (4 OFF')
TAIL
FLOAT CELL
PLAT"O" 4 OFF I
F
-t
l
MIDDS f
TAIL
TAIL
CONC
MIDDS
TAIL
Fig.l BTW Slimes Section 2. EXPERIMENTAL RESULTS The shape of the partition curves for the concentrate, and jointly, for the concentrate and middlings were broadly similar over all tests. Characteristic curves (BTW; July 1989) are shown in figures 2 and 3, though it should be noted that these curves were obtained under non-optimum conditions. For all size fractions, the curves are sigmoidal in shape with the steepest increase over the density range 3.0 - 3.7. The curves become progressively less steep with decreasing particle size. For +45 micron material, the separation density (i.e.DS0) increases with particle size. The recovery of high SG material decreases with increasing particle size; this drop being more pronounced for the BTW March 1989 sample and the South Crofty tests than for the BTW July 1989 samples. The effect can be best visualised in figures 4 and 5, which show the size transfer (recovery) curves for discrete density fractions. The observed behaviourial difference at coarse sizes is thought to be due (primarily) to the relatively low quantity of coarse material in the BTW July 1989 feedstock compared with the other two sets of samples. This coarse material is deposited on the deck
349
A mathematical model of the Duplex concentrator
in a band in the concentrate but close to the middlings cutter position. If the amount of coarse material were larger, this band would broaden, spanning the cutter position, letting some of the material carry over into the middlings product. Separation efficiency decreases below 45 microns and recoveries also decrease below this size.
100
9O
8O
70
6O S
E 50 40
3O
2O
10
0
'
2.5
'
3.O
:.o
3.5
:.5
,'.o
DENSITY, 9/ml
Fig.2 Partition to concentrate- July 1989, Test 2
tO0
°
. ~
- ~__ -_._--~___" ~-~-__-- __--~____- ---__.~
9O
8O
70
60
_g
50 qa
ii
40
/
20 x
/
]
"45"10 MOCRONS
~
"O, ~ -90+63 MICRONS --3t~--- - -180+90 MICRONS ~ - ~80 MICRONS
/
JxJ
20
°2~
3'°
~'~
,o'
,'5
.Io
DENSITY. g/ml
Fig.3 Partition to concentrate + middlings- July 1989, Test 2 Overall performance data for the production operation (at BTW;March 1989) are additionally given in tables I and 2. The feed size distribution, at that time, was 84% passing 180 microns with 21% below 45 microns. Over 98% of the wolframite was recovered to the concentrate at an upgrade ratio of approximately 2.5. The Duplex was operated to cut midway through the sulphide band; the bulk of the arsenopyrite being recovered to the concentrate, whilst the chalcopyrite was almost totally rejected. The rejected material also contained significant amounts of the siderite present in the feed. Rejection of this material
M . PEARL et al.
350
is particularly important at BTW as it poses a significant problem downstream in the final magnetic cleaning of the wolfram concentrate.
l!
> 3.3 "~ EXPERIMENTAL <3,3J
TEST 1
1O0
PREDICTED
8
80
=~
~- 60 Z
o
80
rl
eo
z
o
40
40
20
20
,
•
40
~
, =,
80
120
=
J
160
0
¥
0
200
I
I
|
I
40
80
120
160
J
200
SIZE, MICRONS
SIZE, MICRONS
TRANSFER TO CONCENTRATE AND MIDDLINGS
TRANSFER TO CONCENTRATE
Fig.4 Beralt, July 1989 campaign 0 >3.3 1 / A 2.8-3.3 t EXPERIMENTAL
Ix < 2 . 8 J
TEST R6
100
1O0
80
80 8 60 uJ
80
z
~ 4o
Z < 40
20
20
0 0
I 40
I 80
I 120
I 160
J 200
I
I
I
I
I
40
80
120
160
200
SIZE, MICRONS
SIZE, MICRONS
TRANSFER TO CONCENTRATE AND MIDDLINGS
TRANSFER TO CONCENTRATE
Fig.5 South Crofty campaign TABLE I: BTW Duplex: Balanced Data %NO 3
Feed Concentrate Middllngs Tails/feed Tails/wash
15.06 38.99 0.58 0.81 0.21
%As
%Cu
T/hr
0.24 0.46 0.02 0.30 0.09
0.01 0.00 0.02 0.02 0.01
0.2086 0.0792 0.0368 0.0228 0.0698
351
A mathematical model of the Duplex concentrator
TABLE 2: BTW Duplex: Recoveries WO 3
To To To To
Concentrate Middlings Tails/feed Tails/wash
98.28 0.67 0.59 0.46
AS
72.45 1.46 13.46 12.63
CU
1.2 23.4 29.7 45.7
3. MODEL DEVELOPMENT The trends seen in the experimental results are not too dissimilar to those commonly observed for conventional shaking tables. On this basis, a prototype Duplex model could be built on the same concepts as those used for modelling the shaking table. The WSL Duplex model thus closely parallels the WSL shaking table model described in Reference 1. The model is formulated in terms of the spatial distribution of material (of each s i z e / S G component) on the separating surface, in this case down the length of the Duplex table. The frame of reference is taken as x=0 (feed end) to x=l (tailings end) with an intermediate value of x=0.5 representing the usual cutter position separating the concentrate from the middling. The prototype model, described here, does not attempt to describe the time dependant build up of the material distributions (or their modification during the wash cycle), rather it simply models the distribution profiles at the instant in time immediately following the wash cycle and prior to the start of the rinse cycle. In the model, a separate distribution profile is formulated for each SG fraction (i) of each size fraction (j). These distribution profiles (D) are described in a cumulative form and modelled by a summation of two functions: Dl(i,j) = tanh (0.05. [x/d5(i,j)] zl ) D2(i,j) = tanh (0.54932. [(x - d5(i,j))/d50(i,j)] zz ) where D = DI and D = 0.05 + 0.95 . D2
for x < d5 for x > d5
d5(i,j) and d50(i,j) are the x positions corresponding to 5% and 50% of each component distribution respectively and the exponents ZI and Z2 are model constants which control the shapes of the two curve segments. In the formulation, the d5 is taken as the position of the 'top' of the individual component band down the table and the d50 represents the halfwidth of the band. (Note that in the solution space, x, d5 and d50 are not constrained to lie below unity. In physical terms, values greater than unity simply mean 'off the end of the table - in the tailings product'). The observed 'fuzziness' of the band edge is modelled by allocating the first 5% of the material to a transition zone whilst D2 describes the main band.The probability of transfer, or recovery, T(Kij) of material of size (j) and SG (i) to the concentrate (K= 1) or the middlings (K,.2) is computed from the component distribution by introducing a cutter fraction: x = CP(K) MI: 4/3-4--J
; CP(I)=0.5 CP(2)=I.0 (normally)
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The size/ SG dependencies of the d5 and d50 are found to be built up out of two component factors. Firstly there is an analytical dependence on the numerical values of the sizes and SGs themselves. Secondly there is a dependence on the amount of material within the component fractions. This second dependence is a consequence of particle crowding or competition. The two dependencies can be best visualised as follows: When there is substantial material of one component present, its own band will be wide and will also force all succeeding bands further down the table (particle crowding). The positioning of the band however, in the absence of any preceding bands, is not fixed in position (as it would be if the first effect was absent) but moves progressively down the table with decreasing SG (and increasing size). The two dependencies appear comparable in effect with the Duplex, in contrast to the shaking table where the particle crowding effect dominates (Ref. 1). For a given feed rate FW, the weight of material in any component (i,j) which is fed onto the table, in time t,is given by FW(i,j).t. If this were deposited as a band n particles thick, the volume of the band would be proportional to [n.size(j).W(i,j).b] where W(i,j) is the width of the band and b is the breadth of the table. W could then be computed from the following expression: W (i,j) ct FW (i,j) / ( SG ( i ) . size ( j ) ) For the real flow conditions, and with overlapping bands, this expression is too simplistic and was found not to give a good enough fit to data. It was found, however, that by introducing a simple empirical correction factor (e), the goodness of fit could be substantially improved. The model relationship thus became: W (i,j) ~ [FW (i,j) / ( SG (i). size (j))]e The half width of the band (d50) can then be modelled by: dSO (i,j) = P2. W (i,j). SS where P2 is a model parameter. SS is another empirical correction factor which serves to increase d50 for sizes significantly greater or less than a reference size, sO. This factor accounts for the poorer separation efficiencies found with unclassified feed material. The band start positions (ie the d5s) are modelled to take account of the two prime dependencies discussed above: d5 (i,j) ffi P3. function [ SG(i), size (j)] + PI . E W(i,j) rank
where PI, P3 are model coefficients. The second term, which is a measure of the cumulative width of all preceding bands, explicitly models the particle crowding. The rank order can be fixed by a simple empirical function (as for the shaking table; Ref. I). 4. MODEL DEPENDENCE ON OPERATING VARIABLES The model is calibrated for a given ore and a given set of operating conditions through deriving the appropriate parameter set [PI, P2 ,P3]. The exact form and positions of these parameters in the model equations were carefully chosen such that they could fulfil another role, enabling the model to be extrapolated to different operating conditions. In this respect, machine performance variations with operational regime were modelled using a suite of relationships, which linked each parameter (Pn) to each of the operating variables. These relationships were based on a combination of empirical and conceptual premises. The identified relationships are described below:
A mathematical model of the Duplex concentrator
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Taliings Wash Water Addition Tailings wash water variations have a small but significant effect on the value of the parameters for low levels of wash water addition, with a very much larger effect at the higher wash water levels (fig. 6a). The concept of a threshold value (in this case around 400 kg/hr water addition) might be defined. The overall variations can be described by equations of the form: Pn = a(n). (wash water addition)q(n)+ b(n) where a, b and q are separate constants for parameters n=l to 3 respectively. The 400 kg/hr, threshold appears to define the activation energy at which the dense material starts to be swept off the deck with band widths (related to P2) and band start positions (related to PI) both increasing dramatically. Below 400 kg/hr, the main effect of increasing wash water is to move the lower SG material bodily down the deck (as portrayed by the increase in P3).
Amplitude of Stroke For P1, P2 and P3, the general form of the variations with stroke amplitude (Fig. 6c) can also be given by the function: Pn = a(n) . (amplitude)q(n) + b(n) with a, b and q representing the new set of coefficients. As with wash water level, a threshold stroke length of about 8 cm. can be defined, beyond which the dense material starts to be lost from the table. WASHWATER
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1 e
WASHWATER,~ h r
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Frequency of Vibration The variations in PI, P2 and P3 (Fig. 6b) can be modelled adequately by the Rosin-Ramler relationship: Pn = g(n). (l - exp
(-0.693.(frequency/h(n))r(n))
where g, h and r are separate constants for parameters n=l to 3 respectively.
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Here again a threshold value (ffi78 Strokes per minute) appears to be evident. This threshold appears to determine a 'step' from closely spaced to more widely spaced bands (related to PI and P3) with little further change to band spacing as frequencies move away from this value. A less pronounced step in band widths also occurs at the threshold frequency though there is also a residual progressive increase in band width with frequency over the whole frequency range. Tailings Wash Duration The effects of tailings wash duration were studied over a limited number of tests at high and low wash water additions. The variations of PI, P2 and P3 (Fig. 6d) have the following relationships: Pn ffi a(n). (wash duration) + b(n) where a and b are the coefficient sets for this relationship. Variations in Pl, P2 and P3 are relatively small over the wash duration values of the study, a higher duration moving the bands slightly further down the deck (slight increases in Pl and P3) and slightly increasing the widths of each band (small increase in P2). This agrees with observation, where it has been seen that the majority of the cleaning action takes place early in the wash cycle, with extended wash periods not adding much to this. In physical terms, the wash water level provides the energy to mobilise the particles. Each class of particles will have a spectrum of mobilisation energies. Those particles with mobilisation energies below the supplied energy (in the wash) will be free to move 'instantaneously'. Those particles with mobilisation energies just above the supplied energy will not move, but may subsequently be mobilised as 'random' perturbations in the energies occur. The scale of these random perturbations is likely to be small relative to the spectrum of mobilisation energies present and, as such, may only affect a small proportion of the particles in each class. 5. VALIDATION AND CONCLUSIONS The main model function has been validated against measured performance data and has been found to provide acceptable fits to both the Beralt and the South Crofty data (eg. Figs. 4 and 5). Although generally successful, the model still has some difficulty in accounting for performance at the coarsest sizes. The largest errors in the calculated transfer curves generally being for these sizes, underestimating the recovery of the high SG fractions in the BTW July 1989 tests and overestimating the recoveries for the South Crofty and BTW March 1989 data sets. (Note this is where the measured data show their most pronounced differences). Although some improvement could be made through further empirical tuning, it is believed that more experimental data is first needed to substantiate the observed effect and its proposed explanation before such corrections could be applied with any degree of confidence. ACKNOWLEDGEMENT The work was co-funded by the Department of Trade and Industry and by the Commission of the European Communities 3rd Raw Materials Programme. The contributions of Beralt Tin and Wolfram SA and Carnon Consolidated Ltd to the model development are gratefully acknowledged. REFERENCE .
Tucker P., Lewis K.A. & Wood P., Computer Optimisation of a Shaking Table, Minerals Engineering, 4 (3/4) (1991).