Modelling and simulating dense medium separation processes — A progress report

Minerals Engineering, Vol. 4, No. 3/4, pp. 329-346, 1991 Printed in Great Britain

0892-6875/91 $3.00 + 0.00 © 1991 Pergamon Press pie

MODELLING AND SIMULATING DENSE MEDIUM SEPARATION PROCESSES - A PROGRESS REPORT T.J. N A P I E R - M U N N

Julius Kruttschnitt Mineral Research Centre, Isles Road, Indooroopilly, Qld. 4068, Australia

ABSTRACT

There is an extensive literature on the performance o f dense medium separation (DMS) processes in both coal (low density) and mineral (high density) applications. However, very f e w workers have grappled successfully with the problem of developing effective mathematical models of DMS processes for simulation, other than the trivial option of using partition curves with arbitrary parameter selection. This paper reviews the literature on the modelling of both D M S bath and cyclone separators, and identifies the strengths and weaknesses in the present simulation capabilities. Functions for partition curves are considered in the context of process modelling. Recent work at the JKMRC is summarised, including the development o f models o f DM drums and cyclones, and a new DMS computer simulation module, JKSimDM, is reported. It is concluded that effective simulation models are now available or within reach, and the paper concludes with a statement o f objectives for future research in this area. Keywords Dense medium separation; dense medium baths; dense m e d i u m cyclones; viscosity; modelling; simulation; partition curves INTRODUCTION

The use of computer simulation for mineral process design and optimisation is now a reality. For many years the technique was really accessible only to those involved in the development of the process models themselves. Most practising engineers had neither the time nor the inclination to acquire the skills necessary to interpret and apply the models which had been developed in academia. The experience base, so important in the evolution of such technologies, was therefore small and likely to remain so. This situation changed dramatically with the advent of two important recent innovations: User-friendly simulators : graphics-based computer programs into which the models can be inserted for easy access by professionals who are not necessarily skilled programmers. •

Cheap and powerful desk-top computers to run the software.

Several commercial simulators are now available, and the user base, though still small, is growing rapidly. This has returned the spotlight to the essential ingredient of the simulator - the process models themselves. 329

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T.J. NAPlER-MUNN

McKee and Napier-Munn [1 ] have described the features of the J K M R C mineral processing simulator, JKSimMet, and presented some case study summaries to demonstrate the capability and application of such packages. JKSimMet has until recently incorporated only comminution, screening and classification models, and the simulator stream structure (the protocol for conveying stream information from one unit operation to the next) was originally designed specifically for such models. The reality is that the great majority of mineral processing simulation studies conducted in the industrial environment (ie by operators, designers or consultants) has involved only comminution circuits. This reflects the advanced status and utility of these models compared with those of concentration and separation processes. Even flotation, which has been extensively modelled, cannot yet be reliably simulated in a general sense because the models are not sufficiently comprehensive, and the simulations therefore tend to be very site-specific. Until recently, the same could also be said of gravity concentration and dense medium separation (DMS) processes, which had enjoyed very little attention in the modelling literature. This paper discusses the historical d e v e l o p m e n t of a quantitative understanding of DMS processes, leading to recent advances in the modelling of DM drums and cyclones at the JKMRC. These new models are phenomenological descriptions of the processes based on extensive studies of both pilot scale and industrial operations. They provide the ability to predict process metallurgy and product characteristics from a knowledge of the operating conditions of the process and the feed characteristics. The advent of these models has prompted the development of a derivative of the JKSimMet simulator, "JKSimDM", which incorporates the extended stream structure necessary to carry the densimetric and assay information required by the models. In discussing quantitative DMS process descriptions, it is important to distinguish true predictive process models from trivial representations. In other words, in order to design or optimise a DMS process we need to know not only what cut-point and Ep to target to achieve a given metallurgy, but also how to obtain them. This is a difficult objective for any process model, but its attainment is essential to be able to unlock the full power of simulation. THE PARTITION CURVE DMS separations are conventionally represented by their partition curve, which is shown in Figure 1. It defines the proportion of material of any given density in the feed which reports to either the sink (conventional for mineral separations) or float product (often used for coal separations) - the two options realise curves which are mirror images of each other. This proportion, usually expressed as a percentage, is termed the partition number. The location of the curve is described by the separating density or cut-point, P50, which is the density of material dividing equally between sink and float products. The efficiency of the process is determined by the departure of the curve from a perfect partition represented by a vertical line at the cut-point. An empirical measure of inefficiency is the Ep Cecart probable moyen," or probable error). Although it only describes half the curve (the central portion) it is widely used and has therefore become a de facto standard measure. The error area represents the material misplaced in the separation over the whole density range, but it does not describe the curve itself. Since particles of different size partition in different ways, a separate partition curve must be drawn for each particle size interval. The methods of deriving the experimental partition curve from heavy liquids fractionation of the separation products are well known [2] and will not be repeated here. Suffice it to say that a knowledge of the size-by-size partition curves of a given separation, together with the assays associated with each particle size/density interval, permit the complete metallurgy of the separation to be determined, including the yield and assay of the products. The purpose of any DMS model is therefore to predict the partition curve for a given set of operating conditions.

Dense medium separation processes

331

Error Area

100

A

75 E Z c 0 m

50

t I I I I t I

25

I I

p25 pS0 p75

Particle Density (mean of density interval) Fig.l Partition curve for yield to sink In order to do this, the partition curve itself must be represented by a mathematical function with parameters whose values can be predicted by the DMS model. The curve has the S-shaped form of a cumulative probability distribution, and the process is seen by some workers as stochastic in nature, the experimental partition number (Figure 1) reflecting the probability of a particle of that density reporting to one or other product. Indeed Tromp, who first developed the partition curve [3], used the cumulative normal distribution function to represent the curve. The function must be capable of representing the wide range of curve shapes encountered in practice whilst utilising as few parameters as possible. Stratford and Napier-Munn [4] enumerated the desirable features of such a function: •

It should have natural asymptotes, preferably defined by separate parameters. It should be capable of exhibiting independent asymmetry about the cut-point (there is no a priori reason to assume that real partition curves are symmetrical). It should be mathematically continuous, and amenable to a standard parameter estimation procedure (non-linear if necessary).

In practice, relaxation of some of these criteria is perfectly acceptable if a particular system is shown to produce partition curves of consistent and reproducible character. Several functions for the representation of the partition curve have been suggested in the literature, and some have been evaluated against statistical and other criteria [eg 4, 16]. Some of these are based on the well established two-parameter forms of the reduced efficiency curve for hydrocyclone classification, and have been reviewed by Tarjan [5] and Napier-Munn [6]. One is the function proposed by Whiten and utilised by Lynch and others [7] for classification. In the form suitable for describing the reduced partition curve, this is: f(x) = Y = [exp (ax) - 1]/[exp (ax) + exp (a) - 2]

(1)

where Y = partition number (expressed as a proportion) X = p/p50 where p = particle density p50 = cut-point a = efficiency parameter Equation 1 has a fixed symmetry (defined by a) and cannot reflect independently varying asymptotes, which arise as a consequence of short-circuiting and bypass flow, particularly

332

T.J. NApw_.s-Mtml,,r

with fines, evidenced by a failure of the observed curve to meet the 0% or 100% abscissas. King and Juckes, in a study of fine coal beneficiation in a cyclone [8, 9], introduced two additional parameters to control the tails of the curve, giving (for yield to float, the mirror image of Figure 1): Yc = B + (1 - a - B)f(x)

(2)

where Y_ is the corrected partition number and c~ and 13 are the fractions of feed short. circuiting to the sink and float respectively (which can be estimated directly from the experimental curve). Equation 2 therefore has four parameters (o~, 13, a and p50), which can be fitted simultaneously b y non-linear procedures. The Ep can then be calculated directly from the parameters of the function. King and Juckes [9] used the Imperfection I, defined .

.

U

as

I = (x25 - xTs)/2

(3)

and showed that it could be expressed as: I = (1/2a) In [,-, - 0.75)(0.75 - 13)/{(o~ - 0.25)(0.25 - 13)]

(4)

Further, the imperfection of the corrected partition function is given by: I c = (l/2a) In [(0.75)(0.75)/{(0.25)(0.25)}] = 1.099/a

(5)

The modelling of dense m e d i u m processes at the J K M R C has also been based on the use of Equation 1 as a partition function [10, 11, 12, 13]. For separations in which the tails of the observed curve are asymptotic to Y = 0 and 1, a simplified 2 - p a r a m e t e r (uncorrected) function has been used, assuming that the value of a is large, which it is in most dense medium separations [10, 13]: Y = 1/(I + exp [a (p50 -p) /

Pso])

(6)

In a manner analogous to Equation 5, the Ep can be expressed as Ep = 1.099 pso/a and substituting Equation 7 into Equation 6 gives Y = 1/(1 + exp [1.099 (P50 - P ) / E p ] )

(8)

This has the advantage of incorporating as the two parameters the descriptors most familiar to practitioners. Scott [13] introduced a 3-parameter modification o f Equation 8 based on his modelling of dense medium cyclones, which will be discussed below. A number of other functions have been proposed to describe the partition curve [eg 15]. For example, Erasmus [14] proposed a 4-parameter function based on a conceptual model of coal washing DM cyclones, which assumed that the basic separation was perfect (a vertical partition line), but varied cyclically with time to yield the imperfect performance observed in reality. The function is Y = It 2 - Arctan (Kp - K C ) / ( t 2 - t 1)

(9)

where C --- principal location parameter K = principal scale parameter t 1, t 2 describe the upper and lower tails of the curve respectively (t 2 is negative for the yield-to-sink curve of Figure I). C -- p50 for a symmetrical curve (Itll = It21). The curve is not asymptotic to Y = 0 and 1, and t 1 and t 2 must therefore be constrained to ensure that the curve does not exceed these

Dense medium separation processes

333

bounds. Descriptors such as the Ep and error area can be calculated analytically from Equation 9. A typical partition curve function, then, may incorporate four parameters - one for location (related to Ps0), one for scale (or inefficiency), and two describing the tails of the curve, either in terms of short-circuit or bypass of fines, or to locate the end points of the curve at Y = 0 and 1. The selection of a function will depend on its robustness, its effectiveness in fitting particular datasets, statistical parameter estimation criteria, and no doubt personal prejudice and comfort [4, 16]. However, once the parameters are known, perhaps from published values or from surveys of the process in question, the characteristics of the product from the process can be predicted by numerically applying the partition curve to the feed washability. The process can be "simulated" by exploring the effect on product characteristics of selecting different parameters (or even using look-up tables of partition numbers). The US Department of Energy Coal Preparation Plant Simulator utilises this approach [17]. This can be helpful for example in considering flowsheet options or providing guidance as to the performance (PS0 and Ep) required to achieve a given yield and product specification. It can also help m broad equipment selection decisions (eg cyclones vs baths), if the necessary data are available; the US DOE in particular has acquired large volumes of such data in coal preparation applications. However, it provides no guide as to how to achieve the required performance through the selection of operating conditions (eg medium density or feedrate) or equipment size and other details. This requires a process model. Progress in the development of such models for DM baths and cyclones is now discussed. MODELLING DENSE MEDIUM BATHS The early dense medium processes employed bath-type separators. Most authorities identify Bessemer's 1858 patent for a conical vessel using solutions of metal chloride salts as the first reported DM bath process. However the first commercial plants were only built in the 1920s and these employed the Chance process, which was also the first to use the now conventional unstable aqueous suspension of a finely divided solid (in this case, sand). The modern DMS process involving higher density magnetic media had to await the development of efficient magnetic separators for medium cleaning and regeneration, and plants using magnetite in coal cleaning were operating in the 1940s [12]. The development of 14-16 % corrosion-resistant ferrosilicon powders of specific gravity 6.6 - 7.0 in the late 1950s allowed the principles of DMS to be extended to high density separations, first in diamond and iron ore beneficiation, and later with other minerals. It is significant that medium properties have to some extent driven the evolution of the technology. A particularly important property is the viscosity of the medium. The influence of viscosity on the process, in a hydrodynamic sense, has been recognised from the earliest studies, but very few models have been proposed in which viscosity is explicitly incorporated [18]. A specific objective of the recent JKMRC work on the modelling of DMS processes has therefore been to incorporate quantitatively the effects of medium viscosity, and to decouple this property from the medium density, with which it is strongly correlated. For deep baths (such as cones) and coarse particles, a model is almost superfluous because such separations tend to exhibit perfect partitioning with a cut-point only slightly displaced from the medium density. For other conditions (finer particles and/or shallow baths which are more susceptible to turbulence) this is not the case, and a predictive model would have considerable value in design and optimisation.

There have been several studies of baths (mostly in coal preparation) which have established important process trends and operating guidelines. Some have been reviewed by Napier-

334

T.J. N^pmR-Mtn~

Munn [18] and Baguley [19]. However there exist no published quantitative models of DM baths which can be used to predict a partition curve for a particular separator and set of operating conditions. An intuitive approach to modelling baths is to accept a correlation between particle settling velocity (negative or positive, depending on density), separation characteristics and residence time. For a given separator, particles which rise or fall only slowly, by virtue of their fine size or a density close to that of the medium, may not have time to report to the correct product before they leave the separator. The cut-point and Ep will be defined principally by this mechanism, and the cut-point will also be offset from the medium density, being above the medium density for top-fed units and below the m e d i u m density for bottom-fed units. In deep baths, such as cones, the density gradients set up in the bath due to sedimentation of the medium will also promote misplacement of "middlings" particles of intermediate density. In shallow baths, turbulence is the principal misplacement mechanism. Scott and Lyman [20] illustrated some of these ideas in a simple diagram reproduced as Figure 2, which assumes a bottom-fed bath such as a drum in which float particles must rise in order to be separated. They used simple sedimentation theory to derive a general expression for the cut-point: (P50 -

Pf)/Pf

= K r/a/(d b pfa)

where K is a p50 = pf = r/ = d =

(10)

machine constant (incorporating several effects), cut-point feed medium density medium viscosity particle size

a and b are exponents which depend upon the prevailing particle flow regime. For laminar flow For turbulent flow

a = 1 and b = 2 a = 0 and b = 1.

• ReJectIfloatl V t l > VSO

1 .~,

II

(sPrl ~ t

< v5o

Fig.2 Conceptual model of DM drum separator (after Scott and Lyman [20]) (Terminal velocities should be read as rising positive) Based on theoretical calculations for tracers in the size range 15-60mm in an iron ore application, Scott and Lyman concluded that most of the particles moved in the intermediate regime and suggested values of 0.6 and 1.6 for parameters a and b respectively, though this was not tested experimentally.

Dense medium separation processes

335

Equation 10 only considers the mechanism of sedimentation. Clearly, as has been stated, other mechanisms also play a role and must be incorporated (empirically, if all else fails) in any usable model. Napier-Munn [18] has considered the problem in more detail, in the context of the literature. Some important process trends include: Inefficiency (defined by Ep) increases with viscosity and particle fineness. Whitmore [21] actually developed a model for Ep based on surveys of several production baths in coal preparation. Ep

=

0.0085[r/p/d] 1/2

where r/p = d =

(ll)

plastic viscosity of medium (cP) particle size (inches)

Results on small separators do not necessarily scale to large units because of changes in bath depth, degree of turbulence and other factors; machine variables are therefore important. The yield stress of the medium (characterised as a Bingham plastic) can play a role in partitioning performance, under certain circumstances. In 1986 the JKMRC set out to develop a realistic model of industrial DM drum separators, based on a consideration of the hydrodynamics involved. Details of the model (which is now available in JKSimDM) are given by Baguley [19], and were reported to a closed industry conference by Baguley and Napier-Munn [22]; they will be published in the open literature in due course. The model assumes (as implied above) that there is a correlation between the partitioning performance of particles of a given size and density, as observed in practice, and their terminal velocity in the bath. The model therefore consists of two parts: Computation of the terminal velocity for each particle size/density interval, as a function of medium density and viscosity, based on hydrodynamic theory. Empirical correlation of the calculated terminal velocity with the observed partition number, permitting the prediction of the full partitioning performance. This correlation is then specific to a particular unit. The terminal velocity is calculated from the correlation published by Concha and Almendra [23], with an adjustment for shape factor. The calculation is performed for the mean size and mean density of each size/density interval describing the feed washability. Input data includes medium density and viscosity. The viscosity is at present defined as an apparent equivalent Newtonian viscosity as measured by a Debex on-line viscometer [24, 25]. The model structure available in JKSimDM includes an empirical correlation for viscosity in terms of medium solids concentration, C v (which can' be calculated from medium composition and density), and temperature, t: In (r/) -- C o + C I C v Z - C 2 t

(12)

where C o, C1, C 2 are empirical constants. The value of the constants are specific to a particular medium, and will depend upon the medium composition, size distribution, contamination by slimes, and extent of magnetisation [26]. The relationship must be determined by experiment; Figure 3 shows the relationship obtained from a particular plant using the on-line viscometer. Surveys were made of Wemco drums in two plants, one treating iron ore and one treating manganese ore, in order to generate experimental size-by-size partition curves. Plotting the computed terminal velocity for each size/density interval against the observed partition ME 4/3-.4"--1

336

T.J. NAplr.g-Muss

curves produced S-shaped curves whose characteristics varied with particle size, as shown in Figure 4. These curves were modelled using the empirical expression: Yi = 100[1 - (Vl00 - Vti)2] x where Yi = Vti = VlO 0 = x -A,B =

(13)

partition n u m b e r for size/density interval, i calculated terminal velocity terminal velocity of particles recovered 100% to sinks (a model parameter) A/d 2 + B parameters estimated from data.

120

°I 100

mll

8O

6O

4O 2O 0 2200

2400

2600

2800

3000

3200

3400

3600

Medium Density (kg/m3) Fig.3 M e d i u m viscosity vs density for a particular in-plant medium

'°° t 60

Particle Size

2

-0.3

-0.2

-0.1

-0.0

0.1

Calc. Terminal Velocity (m/s) Fig.4 Partition number vs terminal velocity for the D M drum model The model parameters A,B and V10 0 were either found to be constant for a particular unit, or were empirically correlated with operating conditions such as medium density and viscosity, and feed tonnage. Simulating a drum then consists of the following steps:

Dense medium separation proecsscs

. 2.

337

Calculate an array of terminal velocities based on the size and density intervals for which feed washability information is available. Select the operating conditions (medium density and viscosity, and feed tonnage).

.

Select values of A,B and Vl00 to insert in Equation 13 (these will be specific to a particular installation, and can be used in green field situations if the drum is assumed to have the same characteristics, including feed conditions).

.

Calculate a vector of Yi values from Equation 13. These constitute the predicted partition curve, and can be used to determine the product characteristics, given the feed washability.

As noted earlier, two sets of model parameters have so far been obtained from industrial plants. The model has been used in a number of simulation applications, including: Demonstration in an iron ore application that there is a maximum attainable sink product grade, due to the dominating effects of viscosity at the higher medium densities; this maximum reduces as the proportion of magnetite in magnetite/ferrosilicon mixtures increases (due to the higher viscosity produced by higher proportions of magnetite). Prediction of the performance of an existing drum on new ore sources. Prediction of the performance of a drum in a green field design situation. The model has given plausible predictions in all these applications, although some have yet to be tested. The next phase of development should include the acquisition of circuitspecific parameters from more installations, and the validation of the model in coal applications. In addition, it should be pointed out that the model structure is appropriate for any DM bath separator. Units other than drums should therefore now be modelled. The model structure should also be modified to limit the empirical nature of Equation 13. Terms which should be specifically incorporated include the hole size of the drum lifters, and a Reynolds Number term to represent turbulence in the drum. MODELLING DENSE MEDIUM CYCLONES The DM cyclone has been rather better served by the research community than the DM bath. The process was developed from a fortuitous observation by the Dutch State Mines in 1939 that fine coal concentrated preferentially in the overflow of a thickening cyclone in a DM bath circuit. Patents were granted in 1942, and the DSM licensed this and associated processes (such as the DSM screen, or sieve bend) through a subsidiary, Stamicarbon. Stamicarbon licensees around the world promoted the process and a large installed base developed, in both coal and minerals. As a consequence Stamicarbon accumulated an extensive database of operating data, which was used to develop predictive capabilities sufficiently accurate to enable Stamicarbon and its licensees to offer performance guarantees to its clients. By any definition, this constituted a model, and a powerful one, but it remained proprietary to Stamicarbon and was never placed in the public domain.

The early published work on modelling DM cyclones often approached the unit as a classifier in which the sizing function is suppressed and the density separation enhanced. Lilge, for example, developed his Cone Force Equation [27] from a consideration of the hydrodynamics of the hydrocyclone, which produced an expression for the classification size, d50, in terms of cyclone geometry, fluid density and viscosity, particle density and flowrate. Lilge and Plitt [28] then used the Cone Force Equation to develop a design procedure for DM cyclones based on selecting a finer d50 for the dense component of the ore than the finest size worth recovering in the underflow. The d50 was also constrained to

338

T.J. NxpmR-Mtmr~

be coarse enough to prevent excessive medium segregation. It is fair to say that some of Lilge's ideas were contentious, and certainly the method has not enjoyed widespread application. However it did emphasise the importance of considering the behaviour of the medium separately from that of the density separation. Later research showed the medium behaviour to be process determining, and modelling it is therefore essential. Tar jan had also addressed the behaviour of the m e d i u m in classification terms [29] and used an equilibrium orbit approach to define the conditions under which different radial medium density profiles would occur in the cyclone; the quality of separation would then depend on these profiles and the resulting differential between underflow and overflow medium density. Several workers have equated performance criteria with medium behaviour, expressed as product medium densities. Davies et al. [30] equated the cut-point, p50, with the underflow medium density, #., and also presented an expression for the Ep for a given size in terms of the Ep prevailin"g for coarse (10 m m ) particles. Collins et al. [31] also reported an equivalence of p50 and Pu, until at lower feed densities and a correspondingly high differential ( Pu - Po), p50 dropped below Pu" They interpreted the cyclone performance in terms of the medium behaviour, and in particular the stability of the medium as reflected in the differential. They correlated their data for the 254mm DSM cyclone used, with atomised ferrosilicon and magnetite mixtures as medium, as follows: PS0 - 7s : 0.127 d "1-306

(14)

where % is the cut-point for large particles, which they equated to Pu at differentials below about 500kg m "3. The data of Upadrashta and Venkateswarlu also suggested that Ps0 :#u (although they did not draw attention to it) [32]. However their model for the separation was based on the bulk partitioning of the ore: Ps0 = -ao log gu + al

(15)

gu

(16)

:

a2 (Qu/Qo) a3

Qu/Qo : f(Cv)Qi'°'44 (Du/Do)Z.32

(17)

Ep :(ajd) + a s

(18)

where gu = mass fraction of ore reporting to underflow Qu/Qo = ratio of pulp flowrates to underflow and overflow constants ~0v'"as == material-specific volumetric fraction of solids (FeSi + ore) in feed pulp f(Cv) : 1.44( 1 - Cv) ° ' l a for spray discharge = feed volumetric flowrate Qi (A regression equation for pressure drop factor was also presented). This model can presumably only be used in a design situation if the material-specific constants are known, which must be obtained from testwork. Also, no function was proposed for the partition curve, so one would have to be assumed and the predicted Pso and Ep values inserted. Napier-Munn [6], in studies of a constant geometry 610mm cyclone treating diamond iferons ores, found that in nearly all cases P5o > Pu, which is at variance with the work reported above. He developed an arbitrary regression model for the parameters of the Erasmus function (Equation 9) and the product medium densities, in terms of medium and ore characteristics, feed head and feedrate. The model has good predictive capability within the conditions under which it was developed, but like any such model should be extrapolated only with extreme care. In particular, it is valid only for the particular cyclone configuration tested. Details of the model and the validity range are given in the Appendix.

Dense medium separation proczsscs

339

Later work by Napier-Munn [33] with a 100mm cyclone using density tracers also found that Ps0 > Pu. The data were well correlated by the simple empirical expression PSO = a + b p f

(19)

+ Cpu

where a, b and c were constants which were found to be dependent on tracer size. Pu was predicted from a bulk sedimentation model. Equation 19 was also used by Davis [12] and Davis and Napier-Munn [34] to model both Ps0 and Ep for tracer separations in coal washing, using Equation 1 to model the partition curve. The studies of DM cyclones in coal washing by JKMRC workers in recent years were encapsulated in the form of a regression model which was reported by Wood et al in 1987 [35]. Based on both pilot and full scale data, the model consists of a number of sub-models describing all the features necessary to simulate the process for design or optimisation. These include correlations for feed and product flowrates, product medium densities, cutpoint and Ep. Correlations for predicting the onset of particle retention (which decreases separation efficiency) were also presented. The model (which is currently undergoing revision with the advent of additional data) incorporates some of the concepts outlined earlier, notably the view that the density separation is strongly dependent on medium behaviour, and that the behaviour of coarse particles is a characteristic from which fine particle behaviour can be deduced. One expression given for the cut-point is not dissimilar to Equation 19: P50 (4x2) = a + b Po + c Pu w h e r e Pso(4x2)

Po Pu

(2O)

= cut-point for - 4 + 2mm particles = overflow medium density = underflow medium density

An alternative expression was also given, as a function of vortex finder and apex diameters, feed medium density, head and medium particle size. The cut-point for coarser particles was found to be the same as that for -4 + 2mm. For finer particles of size d, the correlation P50(d) = P50(4x2) (1.05 - 0.25 loglo d)

(21)

was proposed. The experience of most workers is that Ep is correlated with Ps0 and Wood et al. quantified this correlation as follows: EP(d ) = [0.I 25 (Ps0(4x2) - 1000]/d

(22)

If these sub-models can be fully validated on independent data, then this approach will provide a comprehensive simulation facility for coal preparation applications. It has the particular advantage of incorporating cyclone size as a variable (implicit in the definitions of exit orifice dimensions) and can therefore be used for cyclone selection. King and Juckes [8, 9] used Equations 1, 2 and 5 to model partition curves for fine coal beneficiation in a 150mm x 15 ° cyclone, and presented empirical correlations for the model parameters: o= = 0.4 exp (-0.00543 d)

(23)

B = a 1 - a2 d 13 = 0

(24)

for d _< 600 #m for d > 600 ~m

where al, a2 are constants whose value depends on the presence or absence of" slimes. (Pso -

Pf)/Pf

-- 0 . 4 0 2 d "0"32

(25)

340

T.J. NAPlER-MuNN

I: = 0.013 + (3.8/d)

(26)

( d in #m ) It should be remembered that this model was developed for fine coal (- 500 #m ). For coarse particles., a = 13 = 0, and the partition curve function reduces to Equation I. This model form provides a method of predicting the effect of the short-circuiting of fines (leading to partition curves which are not asymptotic to Y = 0 and 1), and further emphasises the strong dependency of cut-point and efficiency on particle size. However the values of the constants are specific to the cyclone and medium utilised in the experiments. Clarkson [36] derived a simple expression for the partition curve: Y/100 = [exp (=a Q ) -1]/[exp (-a) -1] where Q a K Ap Dc

= slurry split K d 2 Ap/(D e )7) = constant = density difference between particle and medium = eddy diffusion coefficient =

(27) (28)

The model was derived from a force balance (centrifugal and radial fluid drag) with a turbulence mechanism imposed, c~ can in practice be fitted directly to experimental partition curves, and Equation 28 then used to simulate performance through plausible selection of the values of the operating variables. Clarkson's model is significant in its consideration of turbulence as a mechanism, one of the very few examples in the DM cyclone literature. Of the cyclone models discussed above, only the JKMRC empirical model developed by Wood et al. [35] for coal preparation makes any attempt to accommodate all the design and operating variables necessary to undertake useful simulation exercises. This model, and the refinements currently being implemented, is therefore being made available within the JKSimDM structure. However, it is not appropriate for high density (mineral) separations. In particular, it does not incorporate medium viscosity as a separate term, which becomes more important at the high operating densities [18]. The objective of the study by Scott [13], therefore, was to develop a model which could decouple viscosity from the other variables, and could also provide a comprehensive description of the process, particularly at the higher operating densities. The work has been described in a closed industry conference [37] and will be published in the open literature in due course. Scott proceeded from the observation that size-by-size partition curves for DM cyclones "pivot" about a point which is controlled by the behaviour of the m e d i u m in the cyclone. In the case of pure liquids or stable (neutrally buoyant) suspensions, such a phenomenon would be expected [38]; the 'pivot point' would be defined by the density of the medium and the yield of medium to underflow, since particles of the same density as the medium would experience no separating force and would therefore partition in the same proportion as the medium. The effect is illustrated in Figure 5. A more surprising result inferred from the literature [e.g.9] and Scott's own work was that the pivot phenomenon prevailed even for cyclones operating with conventional unstable media, despite the segregation of the medium which occurs. He therefore incorporated the phenomenon into the partition curve function and developed empirical models to predict the resulting function parameters. A consequence of the pivot phenomenon is that the P50 and Ep must be positively correlated (for size-by=size curves), The position of the pivot point is defined by the pivot parameters Yp, pp (Figure 5). Substituting these values into Equation 8 gives for the Ep Ep = 1.099 (Pso - pp)/{ln [1 - Yp)/Yp]}

(29)

Dense medium separation prcexsscs

341

where p,, = pivot density ep = pivot partition number (proportion)

Rearranging Equation 29 to provide a definition of Ps0 in terms of Ep, pp and Yp and substituting this value into Equation 8 gives:

(30)

Yij = l/{1 -exp [ In (yp-1 _ 1) + 1.099 (pp - pij)/EPi]) where Yij = partition number for size (i) and density (j).

100

Siz 80'

40'

0

"

;oo

I

,;oo I Pp

'

'

'

'

3000 ;oo Density

3,00

Fig.5 The pivot phenomenon in size-by-size partition curves for DM cyclones It is well known that Ep is an inverse function of particle size (see Equations 18 and 26), and Scott's data confirmed this. He therefore proposed the relation Ep i = kdi "n

(31)

where k is a model parameter, and n is a hydrodynamic constant (1.3 for the pilot plant work). Equations 30 and 31 therefore constitute a 4-parameter model of the process, the parameters being Y,, p,, Ep and k. Scott carried out pilot and industrial experiments on 100, 200 and 400mrh USM cyclones with ferrosilicon media over the density range 1456 3025 kg m "3 to establish predictive correlations for these parameters. In the pilot plant work, the medium viscosity was varied directly by adding molasses and in all eases the viscosity was measured using an on-line viscometer. The general form of the resulting linear regression equations for the 200ram pilot plant unit was (in the order in which they must be computed for simulation): In k = In Rm = In ( Y p / R m ) = I n ( V F R p - 1)=

f [In r/, In (Du/D¢) ] f [In (DJD¢), In (#/Q.pf)] f [In (1 - Vf), (Ap)] f [ l n V f , lnr/]

(32)

342

T.J. NAPmR-MtmN

where D e -- cyclone diameter Q -- volumetric feed flowrate Rm = recovery of pulp to underflow Vf = volume fraction of solids in feed medium VFRp = (pp - Pt)/(Pf -Pt) where Pt = liquid density Ap = (Pu'Po) For the industrial testwork, a modified set of correlations was used. modified by the addition of a crowding factor, z: Ep

=

Equation 31 was (33)

z + k d "n

n was set at unity, and z was a linear function of Ap and volumetric medium/ore ratio. Other relevant correlations were: In k -f (In r/, In De ) Y-o -0.61 Rm In~[(Vp/Vf) - 1] -f [In Vf,ln (o/Q)] (34) In both cases, an estimate is required of Ap, which is a dependent variable. Scott developed two approaches to this problem. The first consisted of empirical correlations for Pu and Po in terms of operating conditions (including viscosity). The second modelled the classification of the medium in terms of the efficiency curve (Equation 1) with a constant value of a, and cut-size and water recovery empirically correlated with feed medium density and viscosity. Scott also reported a procedure for considering the influence of cyclone diameter in terms of a prevailing bulk flow Reynolds number, which modified the correlations for the model parameters. In summary, the JKMRC has developed two models of DM cyclones, one specific to coal and the other a more general model. Both have strong empirical elements, but they do incorporate all the design and operating variables which would normally be considered in a simulation exercise. An important innovation in Scott's model is the decoupling of medium viscosity and medium density which allows the viscosity to be separately manipulated in the model, although this does introduce the requirement that a value for viscosity be known. Also, both models have some scale-up capability through the incorporation of cyclone geometry in some form. Neither model has yet been extensively tested in simulation, and both will undoubtedly benefit from further refinement. However, they do provide an effective and useful structure within which this refinement, which is now underway, can take place. CONCLUSION AND FUTURE WORK This paper has reviewed the modelling of dense medium separation processes in terms of the needs of practical, effective and useful process simulation methodologies. In this context, the modelling of dense medium processes is not yet as advanced as that of comminution and classification processes, although significant progress has been made in recent years. Historical studies of both baths and cyclones have resulted in the eharacterisation of important process effects which must be taken into account in any comprehensive model. These include The influence of medium viscosity on the separation. The dependency of partitioning performance on particle size. The process=determining nature of the medium behaviour particularly in cyclones. Very little DM process simulation has been reported in the literature. This is because the quantitative correlations derived by most workers have tended to be directed to exploring

Dense medium separation processes

343

one particular aspect of the process, or have excluded industrially important variables such as feedrate. Consequently, no comprehensive predictions model of either baths or cyclones has been reported. King and Juckes [8, 9] provided a useful structure for modelling the performance of fine coal separations in cyclones, and indeed any DM cyclone separation in which significant by-pass occurs, and Napier-Munn [6] presented a quantitative cyclone model for one particular industrial application. Apart from these, however, the only models which can claim both industrial utility and some general application is Baguley's model of DM drums [19, 22], the cyclone model of Wood, Davis and Lyman [35] for coal applications, and the general cyclone model of Scott [13, 37]. All these are, or shortly will be, available in the JKMRC dense medium process simulator, JKSimDM. Baguley's drum model has a phenomenological form which is probably suitable for any bath-type separator, although this has not yet been tested for units other than drums. Future work on the model will address the empirical nature of the size-dependency of the correlation between the prevailing partition number and the calculated terminal velocity (Equation 13). The utility of the model will be considerably enhanced by expressing this relationship in terms of design and operating variables, such as a bulk flow Reynolds number (describing turbulent mixing effects) and lifter dimensions (describing the tendency to misplace fine heavies though the lifter drainage holes). Expansion of the operating database is also necessary to develop a parameter library for particular machines, and to calibrate the model for coal applications. Further operating DM cyclone data are also required, for the same reasons. The objective of the next phase of modelling work must be to integrate the various approaches, to synthesise a phenomenological description of the process, possibly in terms of Scott's pivot partition function with elements of Clarkson's model [36] to account for turbulence effects. New surveys of industrial plants have just been conducted with this in mind, and a new phase of model development is underway. The general objective is to continue to refine the capability of JKSimDM in order to provide users with a simulation facility which can be applied with confidence to designing new plants, optimising existing circuits, and specifying the process conditions for new ore and coal sources. Attention is also being given to incorporating these capabilities in a dynamic simulator which will include provision for simulating the medium regeneration circuits; a project to model drain and rinse screens for this purpose is well advanced at the JKMRC. Much work remains to be done, but the ability to simulate industrial dense medium plants effectively and with confidence is for the first time firmly within our grasp. REFERENCES ls

2. 3. 4. ,

6. 7. 8.

McKee D.J., & Napier-Munn T.J., The status of comminution simulation in Australia. Minerals Eng., Vol. 3, No. 1/2, 7 - 21, (1990). Wills B.A., Mineral Processing Technology. Pergamon Press, 4th Ed., (1988). Tromp K.F., New methods of computing the washability of coals. Colliery Guardian, CLIV, 3986, (May 21, 1937). Stratford K.J., & Napier-Munn T.J., Functions for the mathematical representation of the partition curve for dense medium cyclones. Proc. 19th APCOM Syrup., Pittsburgh, (1986). Tar jan G., Application of distribution functions to partition curves. Int. J. Miner. Proc., 1,261-265, (1974). Napier-Munn T.J., Dense medium cyclones in diamond recovery. M.Sc. Thesis, University of the Witwatersrand, (1977). Lynch A.J., Mineral crushing and grinding circuits - their simulation, optimisation, design and control. Elsevier, (1977). King R.P., & Juekes A.H., Cleaning o f f i n e coals by dense-medium hydrocyclone. Powder Tech., 40, 147-160, (1984).

344

.

10.

T.J. NAP~-Mtr~N King R.P. and Juckes A.H. Performance of a dense=medium cyclone when beneficiating fine coal. Coal Prepn., 5, 185 - 210, 1988. Rong R.X. and Lyman G.J. Computational techniques for coal washery optimization parallel gravity and flotation separation. Coal Prepn., 2, 51 - 67, 1985. Scott I.A., Davis J.J. and Manlapig E.V. A methodology for modelling dense medium cyclones. 13th CIMM Cong., Singapore, May 1986 (Aus. IMM). Davis J.J. A study of coal washing dense medium cyclones. PhD Thesis, University of Queensland, 1987. Scott I.A. A dense medium cyclone model based on the pivot phenomenon. PhD Thesis, University of Queensland. Erasmus T.C. Plotting a smooth curve to the experimentally determined co-ordinates of a Tromp curve. Coal, Gold and Base Minerals of S.A., 63 - 67, June 1973. Gottfried B.S. A generalization of distribution data for characterizing the performance of float-sink coal cleaning devices, bzt. J. Miner. Proc., 5, 1 - 20, 1978. Klima M.S. and Luckie P.T. Using model discrimination to select a mathematical function for generating separation curves. Coal Prepn., 3, 33 - 47, 1986. Gottfried B.S. and Tierney J.W. Computer simulation of coal preparation plants. US Dept. of Energy, Dec. 1985. Napier-Munn T.J. The effect of dense medium viscosity on separation efficiency. Coal Preparation, 8 (3-4) 145-165, 1990. Baguley P.J. Modelling and simulation of dense medium drum separators. M.Eng.Sc. Thesis, University of Queensland, 1988. Scott I.A. and Lyman G.J. Metallurgical evaluation of iron ore drum separators using density tracers. Proc. Aus. b~st. Min. Metall., 292, No. 1, 49=56, Feb. 1987. Whitmore R.L. Coal preparation : the separation efficiency of dense medium baths. J.Inst. Fuel, 31,422 - 428, 1958. Baguley P.J. and Napier-Munn T.J. A dense medium drum model for simulation. Proc. 4th Samancor Syrup. on Dense Media Sepn., Cairns, Feb. 1990 (Samancor Ltd.) Concha F. and Almendra E.R. Settling velocities of particulate systems of individual spherical particles. Int. J. Miner. Proc., 5, 349 - 367, 1979. Reeves T.J. On-line viscometer for mineral slurries. Trans. Inst. Min. Met., 94, C201 - C208, 1985. Napier-Munn T.J., Reeves T.J. and Hansen J.O. The monitoring of medium rheology in dense medium cyclone plants. Proc. Austr. Inst. Min. Met., 294, No. 3, 83 - 93, May 1989. Napier-Munn T.J. and Scott I.A. The effect of demagnetisation and ore contamination on the viscosity of the medium in a dense medium cyclone plant. Minerals Eng., 3 (6), 607-613, 1990. Lilge E.O. Hydrocyclone fundamentals, bzst. Min. Met. Bull., 285 = 337, March 1962. Lilge E.O. and Plitt L.R. The cone force equation and hydrocyclone design. Proc. Interamerican Conf. on Marls. Tech., San Antonio, 108 - 118 (ASME), May 1968. Tarjan G. On the heavy suspension developing in the hydrocyclone. Acta Technica (Hungary), 21,387 - 399, 1958. Davies D.S., Dreissen H.H. and Oliver R.H. Advances in hydrocyclone technology for fine ores. Proc. 6th Int. Miner. Proc. Cong. Camzes, 303 - 321, Pergamon, 1963. Collins D.N., Turnbull T., Wright R. and Ngan W. Separation efficiency in dense media cyclones. Trans. Inst. Min. Met., 92, C38 - C51, March 1983. Upadrashta K.R. and Venkateswarlu D. Study of hydrocyclone as heavy media separator. J. Powder and Bulk Solids Tech., 6, No. 3, 22 - 32, 1982. Napier-Munn T.J. The mechanism of separation in dense medium cyclones. 2rid Int. Conf. on Hydrocyclones, Bath, 253 - 280 (BHRA), 1984. Davis J.J. and Napier-Munn T.J. The influence of medium viscosity on the performance of dense medium cyclones in coal preparation. 3rd Int. Conf. on Hydrocyclones, Oxford, 155 - 165, (BHRA), Sept./Oct. 1987. Wood, C.J., Davis J.J. and Lyman G.J. Towards a m e d i u m behaviour based model for coal-washing dense medium cyclones. Dense Medium Operators Conf., Brisbane, 247 = 255 (Aus. Inst. Min. Met.), 1987. -

11. 12. 13. 14. 15. 16. 17. 18. 19. 20. 21. 22. 23. 24. 25. 26. 27. 28. 29. 30. 31. 32. 33. 34. 35.

Dense medium separation processes

345

Clarkson C.J. A model of dense medium cyclones. Dense Medium Operators Conf., Brisbane, 235 - 245 (Aus. Inst. Min. Met.), 1987. Scott I.A. and Napier-Munn T.J. A dense m e d i u m cyclone model for simulation. 4th Samancor Symp. on Dense Media Separation, Cairns (Samancor Ltd), Feb. 1990. Napier-Munn T.J. The influence of medium viscosity on the density separation o f minerals in cyclones. 1st Int. Hydrocyclone Conf., Cambridge, 63 - 82 ( B H R A ) , 1980.

36. 37. 38.

APPENDIX MODEL OF A 610mm DSM DENSE MEDIUM CYCLONE T R E A T I N G D I A M O N D I F E R O U S O R E S [6] The model consists of four regression equations to predict the parameters of the Erasmus partition curve function (Equation 9), and two equations to predict the product medium densities. The independent variables are functions of ore feed characteristics, ferrosilicon medium size distribution, medium density, head and feed rate. The Erasmus parameters exhibit some degree of correlation, and the K and C parameters therefore themselves also appear as independent variables. The equations were developed b y standard stepwise multiple linear regression analysis procedures. exp

C

=

24.376 + (0.1156F/H) + 0.5877M A - (0.4472MA/GA) - (3.9549GA/Gs0) 0.2473[MA1/MB/{In(p_¢- 1)}] + 0.4659(GA1/%) - (3.5531H/pf) + 28.9653[(Gso-Pf)e/pf] (A 1.1)

In (Arctan K) =

In t 1 =

ln(-t2)

:

PM P0

=

-0.00227 + 0.0034M A- 0.7099 Gso - 1.1801 (C-Gso) 2 - (0.00025MA/GA) - 0.0575(C-pf) 2 0.1644 Arctan [I/(C-Gso)] + 0.8582 C (AI.2)

-2.2799 - 0.0361H + 0.0981pf + 0.0006M A + 0.8158G5o - 0.0495 In (MA/10)-0.1083 Arctan [K(G5o-C)] - 0.0036(GA)I/GB - 0.7232C + 1.4039 Arctan K

(AI.3)

-1.9091 - 0.0125 In [MA/10]MB- 0.1027 Arctan [K(Gso2C)] + 0.0051 [HF/100pf] + 0.3208(C-G50) e + 0.0318 (C-pf) ~ + 0.1831 Arctan [I/(C-G50)] + 1.2369 Arctan K

(AI.4)

-0.3496 + 0.9919pf + 0.000675 F + (0.2223 H/p() - 0.0078(MA)I/MB

(AI.5)

0.0144 + 0.9943 pf + (0.0357 H/pf) - 0.1084 M a - (0.0013 MA/MB)

(AI.6)

It should b e remembered that the Erasmus function is not asymptotic to Y = 0 and 100%, and for simulation purposes the constraints Y

=

Y

:

0

100

f o r ! p r e d -< 0

for Ypred -> 100

should be applied, where Ypred is the partition number predicted by Equation 9 using parameter values predicted by Equations A l.l. - A l.4. The medium and ore size distributions are expressed in terms of the parameters of the R o s i n - R a m m l e r function, which were estimated for each dataset: W r = 100 exp [ - ( d / A ) B] where W r = weight % retained.

(AI.7)

346

T . J . NAPmR-MUNN

The cyclone used was a standard g r a v i t y - f e d DSM 610m m d i a m e t e r 20 ° - angle Ni-hard unit, with a 279mm vortex f i n d e r , 178mm spigot and 108ram rectangular inlet. Ferrosilicon medium was used. T he range of conditions was as follows: Ore type Ore medium Feedrate Ferrosilicon Head Medium Observed cu t- p o i nt

12 + 0.4mm kimberlite and - 25 + 2m m alluvial gravels SG 2.6 - 2.95 51 - 147 dry t / h 65D - 270D Samancor grades 5.89 - 8.09m (9.7 - 13.3D) SG 2.40 - 2.98 2.78 - 3.62 SG

NOMENCLATURE A B

C d F GA GB Gs0 H K MA MB P t1

-------=

= ---= = =

t2

=

pf

=

Po #u

= =

location p a r a m e t e r of R o s i n - R a m m l e r distribution. scale p a r a m e t e r o f R o s i n - R a m m l e r distribution. location p a r a m e t e r of Erasmus funct i on. particle size (#m f or m e di um, m m f o r ore) feedrate ( t / hr ) . A - parameter o f R o s i n - R a m m l e r fit to ore f e e d size distribution. B - par a m et er of R o s i n - R a m m l e r fit to ore f e e d size distrtibution. Mean ore SG (determined as 50% point on cum ul at i ve distribution curve) static head (m), or equivalent head = P x 0 . 1 0 1 9 7 / p f . scale p a r a m e t e r of Erasmus function. A - parameter of R o s i n - R a m m l e r fit to f e e d m e d i u m size distribution. B - p a r a m e t e r of R o s i n - R a m m l e r fit to f e e d m e d i u m size distribution. pressure drop (k Pa) scale p a r a m e t e r (upper tail) o f Erasmus f u n c t i o n . scale p a r a m e t e r (lower tail) o f Erasmus f u n c t i o n . feed m e d i u m SG overflow m e d i u m SG u n d e r f l o w m e di um SG