Experimental Validation of Extended Stratification Model - Springer Link

Trans Indian Inst Met (2017) 70(2):359–373 DOI 10.1007/s12666-016-0980-y

TECHNICAL PAPER

Experimental Validation of Extended Stratification Model: Part A—Ore with Tracer Particle Studies in a Batch Jig Operation B. Venkoba Rao1 • Rishi Ramesh1 • N. M. Hemanth Kumar1 • S. J. Gopalkrishna2

Received: 18 June 2016 / Accepted: 17 October 2016 / Published online: 27 October 2016 Ó The Indian Institute of Metals - IIM 2016

Abstract Particle stratification in gravity concentrators is influenced by both particle size and particle density. King’s stratification model defines very well the segregation of particles in bed forming separators but is based on monosized feed particles that are distributed in density. The model has been extended to incorporate particle size effects into it and to define segregation patterns of the bed (Venkoba Rao in Int J Miner Process 85:50, 2007). The extended stratification model shows that the segregation patterns evaluated from it at various cut heights conform to size-density partition surfaces. The model is more theoretical when proposed and lack experimental evidence. In Part A of this series of papers, the model is validated with segregation patterns measured from a batch jig operation by introducing small amount of tracer particles (of various sizes and densities) along with feed particles of fluorite ore. Such patterns formed by tracers are measured by slicing the segregated bed at various cut heights. The patterns of partition surfaces by tracers are in support of the extended stratification model. This exercise also shows that small amount of size-density tracers along with feed particles can quickly help to characterize the segregation patterns of a jig bed. Also, by inferring segregation patterns from Part B of this paper series, a single go approach is proposed to correct the errors of partition surfaces of the entire bed, arising from sparse measured tracer data at various bed heights. & B. Venkoba Rao [email protected] 1

Tenova India Private Limited, Tenova Delkor, 108/D, 6th Main Road, III Phase, Peenya, Bangalore 560058, India

2

P. G. Center, Vijayanagara Sriksrishnadevaraya University, Nandihalli, Sandur 5831019, India

Keywords Extended stratification model Tracer particles Batch jig Partition surface Particle segregation

1 Introduction Segregation of particles in various unit operations is a topic of interest in many fields including mineral beneficiation. It has been well established that both particle size and particle density of individual particles influence their probabilistic separation in an unit operation. This has been well established in literature with various four parameter models that define partitioning of particles in a separator with respect to size and density attributes [1–4]. A partition surface defines the segregation of particles from feed into one of the streams under a given set of operating conditions. In mathematical terms, it defines the probabilistic recovery of feed particles of given size and given density into the sink stream, from which recovery into float stream can be easily calculated. Thus, this surface along with the bivariate feed distribution defines the bivariate product distributions in terms of particle size and particle density [5]. This partition surface is found common to all gravity concentrators, hydrocyclone/aircyclone classifiers, fluidized beds and dense-media separators, despite differences in their mode of operation and equipment design profiles. Thus it unifies all of them into one category of separators [3] where size and density of particles influence their separation under hindered conditions. King [6] and Tavares and King [7] proposed an elegant model of particle segregation in bed-forming gravity-concentrators such as jigs, Richert cones and sluices. The model defined the partition curve in terms of particle

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Trans Indian Inst Met (2017) 70(2):359–373

Fig. 2 Tracers of various sizes and densities used in the jig experiments

Fig. 1 Tenova Lab Jig

density at any given slice height of the segregated bed and the bed formed out of mono-sized particles. Despite the constraint of mono-sized particles, the model was validated for many industrial jig operations for partition curves in terms of particle density [7]. Recently, Woollacott et al. [8] revisited the King stratification model with many monosized tracer particles jigged separately in a batch jig and confirmed that the model holds good for their data. It is to be noted that any separator feed is always distributed in terms of particle size and density and evaluating their performance with one attribute will be a marginal approach. Therefore King’s stratification model was extended to incorporate particle size effects [9], in order to show the partition surfaces at various cut heights of the segregated bed that were formed from a bivariate distribution of coal sample. Hence, the model was considered more theoretical/empirical till date, as it was based on simulation, and lacked any experimental validation [10]. The lacuna in its validation was due to non availability any suitable data in literature at the time of publication. In this paper, we have validated the extended stratification model for a batch jig operation that conforms to partition surfaces at various cut heights using small amount of tracer particles along with a fluorite ore. Seven experiments with two pulse wave forms and various jig times have been utilized for its

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validation in terms of partition surfaces arising from particle segregation trends. The logic of jig control currently practiced for industrial operation are based on bed density measurements alone, by using a float that reveals the position of desired density bed height in the operating jig and thereby helps to adjust the pulsation characteristics and material discharge withdrawal [11, 12] to suit the incoming feed characteristics. However, the particle size effect is not essentially measured/utilized in current control practice. The jig feed is generally maintained as truncated distribution of relatively coarse particles, with maximum to minimum particle size ratio at 2.5–5. It is essential to have a better model incorporating the size effects to understand separation characteristics, as the ore deposits in future get leaner and complexity of liberation characteristics increase. Tracers of known size and known density can be used to assess the segregation of particles in the jig bed. Such studies also have been done in the past on industrial scale jigs and heavy media cyclones [2, 13]. Naude et al. [14] and Woollacott et al. [8] used complete tracer feed particles to assess the batch jig performance. This tracer separation concepts have been used here to assess jig segregation patterns.

2 Brief Description of Partition Surface A density partition curve, defined in terms of particle density, when extended to explain both particle size and density effects, forms partition surface. It defines the recovery of feed stream particles of given size and density into the heavy/sink stream. There are a number of probabilistic equations that explain this phenomenon mathematically [1–4].


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Table 1 Tracer count details used in the tests Density, g/cc

Tracer particle size 3 mm

4 mm

1.2

10

1.6

11

1.7

9

5 mm

8 mm

10

10

9

10

2.3

10

2.5 3

10

3.2 3.4 3.53

9

10

10

10

10

10 10

11

Table 2 Experimental details of tracer batch jig tests Exp no.

Duration of jigging, s

Frequency Wave form

Stroke length, mm

T2, s

T3, s

1

15

60

0.5

0.5

120

2

30

60

0.5

0.5

120

3

60

60

0.5

0.5

120

4

90

60

0.5

0.5

120

5

15

60

0.7

0.3

110

6

30

60

0.7

0.3

110

7

45

60

0.7

0.3

110

parameters can be estimated from experimental data of size density separation. ð1Þ Y ¼ 50 1 erf Ad c q qp B where A, B, c and qp are partition surface parameters respectively representing the acceleration force on particles, liquid drift force on particles, turbulence level of separation and density of separation. These parameters are dependent on the operational settings and location of separation within the separator. The key performance indices (KPIs) of the partition surface, namely, Ecart Probable, Ep , and cut density, q50 , can be obtained from the partition surface parameters at any given slice height as given below. Conversely, if these indices are known for a given separator performance for at least three size classes of particles, then they can also redefine partition surface inversely [15]. 0:476936 Ad c B ¼ qp þ c Ad

Ep ¼

ð2Þ

q50

ð3Þ

The by-pass fraction Yp can be calculated from Eq. (1) when particle size tends to zero size to get Yp ¼ 50½1 þ erf ðBÞ

ð4Þ

The cut size, d50 , where particle settling velocity equals fluid drift velocity can be calculated as !ð1cÞ B d50 ¼ ð5Þ Aðq qp Þ Jigs follow partition surface at given slice point, as reported earlier in a few publications [3, 11, 14, 16]. Although it is shown in literature that jigs follow partition surface at a given slice height, it is not clear how these partition surfaces vary across the bed height when sliced. Extended stratification model [9] brings out on how these partition surfaces vary across the bed when sliced at different height of the bed. This has been verified and explained in this paper.

3 Experimental Details Fig. 3 A typical view of jig segregated bed with ore and tracer particles

Stochastic model [1] defines partition surface by imposing a random walk on settling particles in a separator that are drifted by rising fluid. According to this model, the partition number of specified size, d, and density, q, is given by Eq. (1) that has four parameters. These

All experiments were done using Tenova Lab Jig shown in Fig. 1. The feed consisted of 2.8 kg of fluorite ore particles of 3–12.5 mm size and 39 grams of tracers of various sizes and densities as shown in Fig. 2. Approximately #10 tracers of each size and density were used in the tests conducted. Table 1 gives the details of the tracers used in the tests. The jig had a PLC control system to vary the wave form with adjustable times, and to define the wave frequency. A

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Table 3 Partition parameters for Experiment-5 evaluated individually for each layer Parameters

Experiment-5 Layer 1

Layer 2

Layer 3

Layer 4

Layer 5

Layer 6

Layer 7 2.4322

qp 9 103

2.3807

2.4570

2.9950

3.1342

2.7353

2.6317

c

2.0138

1.0644

0.6106

0.4127

1.8705

2.7532

2.5010

-A 9 10-3

0.0399

0.1548

0.3464

0.4500

0.0336

0.0067

0.0103

-B

-0.5841

-0.2988

-0.5716

-0.4047

0.2301

0.5574

1.1085

SSE

0.1043

0.1570

0.1387

0.3663

0.4205

0.2165

0.0717

hydraulically operated piston with its adjustable stroke length pushed water into & out of the jig chamber. Basically it produced a trapezoidal wave. A round hole sieve at the bottom of the jig chamber (which could be changed to suit particle size requirement of the test) served to hold the material in the chamber. The jig chamber was made up of attachable and detachable rings. Each ring was about a centimetre high, and allowed the stratified bed to be sliced layer wise after the completion of the test. Hutch water could be removed from the bottom of the jig chamber after the test. It could be noted that there was no hutch material in the current tests. Two sets of experiments were done with differing ramp up and ramp down of the pulse wave form. The first set consisted of 4 experiments and second set consisted of 3 experiments. All experiments were done at 60 cycles/min of wave frequency and stroke lengths of 120 mm and 110 mm in first and second test sets respectively. Each test corresponded to different jigging time, varying between 15 and 90 s of jigging time. Table 2 gives the experimental details of test conditions of these 7 tests. Each test was conducted by thoroughly mixing tracer particles with feed particles which were poured into the jig chamber during first up pulse. After the completion of the jig test for the specified time period of jigging, the layers were sliced into different trays. The tracers in each layer were recovered using a magnet, as tracers were magnetic in nature and also were colour coded for easy identification. Their count of recovery in each layer was noted down against size and density they belonged to. Based on the count of tracers recovered below the particular layer slice and by knowing total amount of tracers introduced, it was possible to measure approximate partition number for the given size and density of tracers. Equation (6) gives the calculation of partition number from tracer count values. Partition Coefficient; Y ¼

# Tracers recovered below the slice layer # Tracers introduced for the test

ð6Þ The partition coefficients of tracer for the given size and density could now be fitted using Eq. (1) to get partition surface and its parameters. This partition surface would be same for fluorite feed particle segregation as well.

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Figure 3 shows a view of segregated jig bed along with tracer particles before slicing the bed.

4 Results and Discussion In the current tests, feed particle distribution cannot be same as tracer bivariate distribution. Both these are independent distributions and their combined distribution will be distribution of the jig feed. It is assumed that we are not going to alter the feed particle distribution majorly by adding small amount of tracers. But eventually, the segregation of tracer particles in the jig bed will be similar to that of feed particles that segregate as per their size and density during jig operation. Hence, small amount of tracers give the segregation pattern of the bed without affecting feed distribution majorly. Also, the limitation of absence of true measurement of feed bivariate distribution will not allow for complete mathematical treatment of extended stratification model [9], with partly added tracers to ore particles. This will be the topic of Part B of this paper [17]. However, the measured tracer data in the current experiments is sparse and how to make use of the this data in a meaningful way in the presence of experimental measurement errors under small quantity of tracers and also to understand the segregation patterns formed thereby, are the question to be answered. Table 3 gives the estimated parameters of the individual layer fits carried out for Experiment-5 using Levenberg– Marquardt least-square minimization technique, along with the sum of squared errors (SSE) of the fits. Figure 4 gives the individual partition surface fits [refer to Eq. (1)] of the sliced tracer data for Experiment-5 (in Table 2). The solid dots represent the partition number from experimental tracer count and lined surface represents the fitted partition surface. The partition number here represents the recovery of tracers of given size and density from the bottom layers of the bed at that cut position. Layer 1 is the topmost layer in the stratified bed. Tracers recovered below layers 1, 2, 3 etc. in the jig bed, respectively, represent partition surfaces below the layers layer 1, layer


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Fig. 4 Tracer partition coefficients and individual partition surface fits at various cut slice positions for Experiment-5. Tracer data is represented as dots and surface fit is given by lines

2, layer 3 etc. This convention is followed in all subsequent figures. Thus for layer-1, partition numbers represent tracers of given size and density reporting to all bottom layers below layer 1.

The fits in Fig. 4 show that the extended stratification model [9] very well describes the trend of segregation patterns of a stratified jig bed in terms of partition surfaces, though the model was published much earlier to this verification.

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364


Table 4 Best fit parameters with new concept for all the experiments Layers

Parameters

Experiments Exp 1

All Layers Layer 1

Exp 3

Exp 4

Exp 5

Exp 6

Exp 7

qp 9 103

2.4829

2.3847

2.1743

2.4710

2.5219

2.4189

2.3696

c

0.9867

0.6381

0.7278

0.7579

1.4560

0.9395

1.4336

-A 9 10-3

0.2285

0.0316

0.3818

0.5221

0.1005

0.2823

0.2738

-1.3294

-0.4723

-0.3892

-0.1533

-0.7653

-0.6288

-0.2190

0.3155

0.7190

0.5468

0.5407

0.0835

0.3985

0.1938

-0.3406

-0.0369

0.2232

0.3178

-0.3622

-0.1692

0.2862

0.2134 0.3279

0.7214 0.5553

0.6588 0.8679

0.6515 0.9849

0.0932 -0.1290

0.3011 0.2308

0.1941 0.4550

-B Layer 2

Exp 2

-A 9 10-3 -B

Layer 3

-A 9 10-3 -B

Layer 4

-A 9 10-3

0.4677

0.5731

0.5322

0.8893

0.0753

0.2959

0.1420

-B

0.6571

0.7878

0.9900

1.5254

0.1265

0.5066

0.7450

Layer 5 Layer 6 Layer 7 Layer 8

-A 9 10-3

0.6241

0.5512

0.6057

1.1881

0.0646

0.2770

0.1370

-B

1.0831

0.9231

1.5410

2.4296

0.3731

0.6981

0.8787

-A 9 10-3

0.4315

0.7264

0.9153

0.8303

0.0768

0.3067

0.1217

-B

0.9417

1.3648

2.4070

2.0177

0.6970

1.1254

0.9451

-A 9 10-3

0.4148

0.5830

0.7650

0.3818

0.1267

0.4574

0.0951

-B

1.1966

1.1672

2.5495

1.2414

1.5724

1.9339

1.0058

-A 9 10-3

0.3751

0.4873

0.6394

-B

1.2680

1.3096

2.3545

Except layers 3 and 4, which show slightly higher bypass fraction to the subsequent layers, there is a continuous decrease in bypass fraction from top to the bottom of the jig bed. This aberration in the values is due to some erratic fit of partition surface with relatively sparse data of tracers, which have very less partition numbers to fit the four parameter surface represented by Eq. (1). In order to maintain the correct trend of partition surface fits across the bed height in a more meaningful way, we have imposed the parameter trend from Part B of this paper series [17], where the experiments have been conducted with more detailed partition numbers for many sizes and densities. Hence we consider the idea that partition parameters c and qp more or less remain unchanged for a given jig operation (for all layers). Recently it has been shown that parameters A and B are more sensitive to partition surface representation than parameters c and qp [15]. Using this concept, Table 4 represents the parameters simultaneously estimated for all layers in each individual jig experiment, so as to minimize the error between measured tracer partition numbers and calculated partition numbers. The best fitted parameters have been found using Levenberg–Marquardt least-square minimization technique with consideration of parameterized Eq. (1) to represent each layer partition surface for each experiment. Figures 5, 6, 7, 8, 9, 10 and 11 depict these fits along with measured tracer partition numbers. These figures clearly show that partition surfaces at various cut

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heights of stratified jig bed are similar to what is defined by extended stratification model for a batch jig operation [9]. Analyses of the test conditions from Table 2 and corresponding Figs. 5, 6, 7, 8, 9, 10 and 11 depict clearly on how the increase in jigging time period influences the change in partition surface profiles across the jig bed height. Undoubtedly, as jig time increases, more finer particles report to top layer and hence by-pass fraction at that slice position (representing the fines recovery from the lower bed layers of sink stream) decreases. Also it can be seen that with progressing jigging time, more change in partition surface profiles happen in upper part of the bed than the bottom layers. So it can be thought that with progressive time, the particles are arranged from bottom to top of the bed. Ultimately the jig bed attains the dynamic equilibrium [7] and no further changes happen thereafter in the bed segregation patterns (except the local rearrangement of particles). Also the jig bed attains dynamic equilibrium quite fast in a lab jig [18]. This is against the conventional notion of 20–30 min of lab jigging [8], which says that more the jig time, the better is the stratification. This is the reason why Table 2 shows less jig times (in seconds) for the experiments conducted. Figure 12 shows the by-pass fraction using Eq. (4) at various layers, from parameter B values given in Table 4. It should be noted that the by-pass fraction (i.e., the amount of fines reporting to sink/underflow) is more notional concept in the present case, as the jig feed is a truncated distribution of coarse particles and lacks presence


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Fig. 5 Tracer partition coefficients and partition surface fits at various cut slice positions for Experiment-1. Tracer data is represented as dots and surface fit is given by lines

of fines in the feed. So we can consider this number as the amount of fines that would have reported to the sink stream, if present in the feed stream. It can be noticed that by-pass fraction in the bottom layers is quite small, as indicated by flat curve for bottom layers in Fig. 12. It can

also be noticed that with progress in jigging time, more fines report to top layers and hence the by-pass fraction at top layer slice position decreases. From Figs. 4, 5, 6, 7, 8, 9, 10 and 11, it can be noticed that at any given slice position, the separation of finer

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particles shows flat portion on the partition surface. This is the reason why the jig feed will be truncated to a coarse distribution for normal operation. Usually jig feed is

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scrubbed and washed ahead of jig operation (like the fines removed ahead of a spiral concentrator with a hydrocyclone operation), which removes the fines entering the jig


367


chamber to improve the efficiency of particle separation during jig operation, and thus making the jig feed, a truncated coarse distribution.

Figure 13 shows the Ecart Probable, Ep, values for Experiment-5 at slice positions below the #layers mentioned therein. The values lie within a narrow band, with

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368



relatively inefficient separation in bottom layers compared to top layers for a given particle size. It should be noted that, lower the Ep value, the sharper is the separation.

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Figure 14a, b show respectively for Experiment-5, the cut density for a given particle size & the cut size for a given particle density, for the various layer slice


369


positions mentioned therein. These graphs seem to be complementary to each other, as by definition, the cut size is the particle size where particle density

corresponds to 50% partition number. Similarly, the cut density is the particle density that has 50% partition number on the partition surface. The dotted line corre-

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370



sponds to particle density equal to parameter qp , which divides the particle separation into positive and reverse separation zones [1]. In these plots, the top 3 layer slices are indicated on one side of qp and bottom 4 layer

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lie on the other side of qp . This is because the by-pass fractions for top layers is greater than 50%, while that of bottom 4 layers is less than 50%. These curves are asymptotic to qp value.


371


5 Conclusions Small amount of tracer particles of various sizes and densities were used along with feed particles in batch jig operation to assess partition surfaces of the segregated bed.

The tests were conducted with two wave forms, for various times of jigging. The segregation patterns at various cut height, for each of the seven experiments showed that the partition surfaces at various cut heights followed the same patterns defined by extended stratification model. Also, a

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372


(a) 10000

1 0.9 0.8 0.7 0.6

90 Sec

0.5

60 Sec

0.4

30 Sec

0.3

Layer 2 Slice Layer 3 Slice Layer 4 Slice Layer 5 Slice Layer 6 Slice Layer 7 Slice

0.2

1000 0

2

4

0.1 0

8

10

(b) 1

2 3 4 5 6 7 Layer Number from jig top to boom

8

40 35

1 0.9 0.8 0.7 0.6

30

Layer 1 Slice

25

Layer 2 Slice Layer 3 Slice

20

Layer 4 Slice

15

Layer 5 Slice

10

Layer 6 Slice

5

0.5

15 Sec

0.4

30 Sec

0.3

45 Sec

0 1200

Layer 7 Slice 1600

2000

2400

2800

3200

3600

Parcle Density, kg/ cu.m

Fig. 14 a Cut density as function of particle size, b cut size as function of particle density at various slice positions for Jig Experiment-5

0.2 0.1 0

6

Parcle Size, mm

Cut Size, d50 (in mm)

Bypass fracon at Layer Slice Posion, Yp

Layer 1 Slice

15 Sec

(b)

1

2 3 4 5 6 Layer Number from jig top to boom

7

qp . The evolving partition surfaces with time showed that the layers rearranged from bottom to top in a jig bed.

Fig. 12 By-pass fractions at various slice positions across the bed, from top to bottom of the bed: a for first set of experiments, b for second set of experiments 10000

Ecart Probable, Ep, kg/cu.m

Cut Density, ρ 50 , kg/cu.m

Bypass Fracon at Layer Slice Posion, Yp

(a)

Layer 1 Slice

Acknowledgements BVR acknowledges Mr. Ramesh Mahadevan, Regional Managing Director, Tenova Delkor, Bangalore and Dr. Pradip, TRDDC, Pune for their individual encouragement in this work.

References

Layer 2 Slice Layer 3 Slice

1000

Layer 4 Slice Layer 5 Slice Layer 6 Slice Layer 7 Slice

100 0

2

4

6

8

10

Parcle Size, mm

Fig. 13 Ecart probable values as function of particle size for various slice positions of the segregated jig bed for Experiment-5 in Table 2

method was proposed to estimate the partition surface parameters effectively from sparse tracer data for the whole bed slices using the concept of constant parameters c and

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