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The dense medium cyclone – past, present and future Tim Napier-Munn JKMRC, The University of Queensland, Australia
A R T I C L E I N F O
A B S T R A C T
Keywords: Dense medium separation Dense medium cyclone Process analysis Process control Cyclone head Cyclone size History Separation principles
Since the dense medium cyclone (DMC) was first patented in the 1940s it has become the process of choice in coal preparation, and is also widely used for upgrading iron ore and in the pre-concentration of diamonds and metalliferous and industrial minerals. It is in every sense a mature technology. This paper summarises the history of the process, considers its current status in mineral and coal processing, and suggests ways in which the process might evolve. Aspects reviewed include the process principles, process models, the importance of medium behaviour, particle size limitations, process analysis, instrumentation and control, and increases in scale. Particular attention is given to the potential for the use of larger cyclones at lower heads in mineral separations, as practiced in the coal industry. Using operating examples and published modelling results, the paper makes the case that these conditions will work in minerals too, and should be adopted. This will significantly improve the economics of DMC mineral separations at a time when pre-concentration is becoming more important for upgrading lower grade ores.
1. Introduction The dense medium separation process (DMS) is a mature technology, and is widely used in mineral and coal processing. The dense medium cyclone (DMC) is the most ubiquitous of the DMS vessels in use, and deservedly so. It is efficient (when run properly), can process both coarse and fine sizes, and has a relatively small footprint. Unlike other forms of gravity concentration, it also makes a positive separation at a desired density cut-point, due to the presence of the medium whose density is easily controlled. This paper briefly reviews the status of the DMC process, highlights some particular aspects of interest, and suggests some improvements to current design practice in (high density) mineral applications. The paper is not intended as a comprehensive review of the DMC or its literature, ancient or modern. 2. A Brief history The invention of the DMC and its subsequent history is related in Napier-Munn et al. (2013). The accepted mythology is that the DMC was ‘discovered’ in about 1939 by Dutch State Mines (DSM) in Holland when a hydrocyclone processing loess (a clay medium) for a dense medium bath in coal cleaning blocked. While being cleaned out it was noticed that the vortex finder was full of clean coal, suggesting that it was being concentrated in the cyclone overflow. Investigation and development of the principle followed, and after considerable testwork
during the German occupation of Holland in the Second World War the DSM DMC was patented in 1942, together with ancillary equipment such as the sieve bend. DSM formed a company, Stamicarbon, to licence the technology to engineering companies and to provide technical support. Stamicarbon produced a design manual for its licencees which over the decades probably became the most widely photocopied confidential document in mineral processing history. Decades after the expiry of the Stamicarbon patents, the Australian Coal Preparation Society has recently produced a modern version of the manual with additional updated material added (Mathewson and Ryan, 2013). The first applications of the DMC were in coal, reflecting its origins, and the first large-scale use in minerals was probably at Williamson’s Diamonds in Tanzania in 1955 (Chaston and Napier-Munn, 1974). Since its invention, there has been relatively little change in DMC technology other than increases in scale, advances in wear materials (eg polyurethane, and ceramic tiles) and some changes in the geometry, particularly of the inlet (Bosman, 2003; Honaker et al., 2010). There have also been various pretenders to the cyclonic separator throne including the Dyna Whirlpool, Tri-Flo, Vorsyl and Larcodems separators (Wills and Napier-Munn, 2006), all of which have found some application in industry, but the original DSM cyclone principle (Mathewson and Ryan, 2013) is still the most commonly employed. DMCs are widely used in coal preparation, and are also used as the primary concentration step in the recovery of diamonds, in iron ore concentration, in pre-concentration of base metals, and in some industrial minerals. The operating medium density will depend on the
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[email protected]. http://dx.doi.org/10.1016/j.mineng.2017.10.002 Received 12 July 2017; Received in revised form 25 September 2017; Accepted 2 October 2017 0892-6875/ © 2017 Elsevier Ltd. All rights reserved.
Please cite this article as: Napier-Munn, T., Minerals Engineering (2017), http://dx.doi.org/10.1016/j.mineng.2017.10.002
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application. In coal it is generally less than 1.65 RD and in minerals it is generally more than 2.5 RD. Accordingly coal applications use magnetite medium (a natural material) with a density of about 4.5–5.2 RD, and mineral applications use the more expensive ferrosilicon (FeSi, a manufactured product) with a density of about 6.7 RD. These differences are significant in understanding the performance of DMCs, especially in terms of the rheology and behaviour of the medium which is process-determining (Section 3.1). 3. DMCs - What We Know Now 3.1. The principle of separation in DMCs – Medium behaviour is processdetermining The scientific literature on hydrocyclones for classification, dewatering and thickening is very large. This is not true of the DMC, probably because of the difficulty of characterising density separations (Section 3.4) and the complexity of the system, involving as it does a dense medium suspension whose segregation in the cyclone is itself process-determining. There has been an increase in recent years in the application of computational fluid dynamics (CFD) to understanding the principles of separation in DMCs (Section 3.2), and these studies are making good progress. However the understanding is still incomplete. Dimensional analysis gives the following relationship for the cutdensity of the DMC, δ50 (Napier-Munn, 1984):
⎛⎜ δ50−ρf ⎟⎞ = k Re−α d−β i ⎝ ρf ⎠
Fig. 1. Model prediction of partition curves for 400 mm DMC in iron ore concentration (from Dunglison, 1999).
Ep =
(1)
3
(2)
where D is the cyclone diameter, η is the medium viscosity, Q is the flowrate, d is particle size, and K is a constant. Eqs. (1) and (2) imply that the cut-density in a DMC is always greater than the medium density and this is indeed what is observed in normal practice (there can be exceptions in cases of unusual medium conditions). They also capture correctly, at least in sign, the effect of cyclone diameter, flowrate and particle size. However Eqs. (1) and (2) suggest that the cut-density increases with medium viscosity, whereas the consensus of the literature is that in real systems the reverse is true; cut-density decreases with increase in viscosity. This is because these models implicitly assume a stable medium (like a liquid) which does not itself segregate in the cyclone. In practice of course this is not the case; real media are composed of dense solids suspended in water, which partition in the cyclone, leading to a density differential between the underflow and overflow media products, the underflow usually being of higher density than the overflow. It turns out that ore partitioning in the DMC depends to a significant extent on how the medium itself segregates. This leads to some very simple but useful empirical expressions for the cut-density, δ50, in terms of this medium behaviour, such as:
δ50 = a 0 + a1ρf + a2ρu
(4)
where δ75 and δ25 are the densities corresponding to mass recoveries of 75% and 25% respectively. So Ep effectively measures the width of the central section of the partition curve (Wills and Napier-Munn, 2006), a wider curve indicating more misplaced material and therefore a lower efficiency. Ep is generally correlated positively with cut-point, so a relationship of the form of Eq. (3) will work for Ep as well (though of course with different coefficients). Fig. 1 shows this correlation, as well as the increasing cut-point with decrease in particle size, the fact that cut-point indeed exceeds the feed medium density, and the ‘pivot point’ phenomenon which is apparent in many such separations and was in one case used to model the DMC (Scott and Napier-Munn, 1992). In theory the partition number for the pivot point should be equal to the medium split to the underflow, and this has been shown to be true for stable media which do not segregate in the cyclone (Napier-Munn, 1980). However for unstable media (ie the media used in practice) this equality depends on the medium solids concentration and the underflow-overflow density differential (Scott and Napier-Munn, 1992). Feed density ρf in Eq. (3) is an operating variable which is known for a particular system, but ρu has been found to be a function of operating conditions, including medium viscosity. Certainly all the published evidence is that the properties and behaviour of the medium, including its viscosity (Napier-Munn, 1990), are process-determining. Any useful model of the DMC must therefore incorporate the behaviour of the medium. The two most comprehensive process models in the published literature, those of Wood and Dunglison (Section 3.2), both incorporate medium properties and behaviour into their process predictions. Medium behaviour depends in turn on medium properties, including solids density, size distribution and shape. For example, Fig.2 shows how the selection of FeSi medium grade (shape and size distribution) leads to different process outcomes in the processing of a particular iron ore in a 610 mm DMC, based on predictions of the Dunglison model. The five FeSi grades to the left are milled (irregular shape) with fineness (and thus cost) increasing from left to right. As fineness increases (higher viscosity) Fe recovery increases, yield increases, and product Fe grade decreases, at the same cut-density (3.6 RD in this case). The two grades to the right are atomised (rounded shape, lower viscosity), C60 being finer than Special Fine. Atomised media, being of lower viscosity than milled, are generally used where the cut-density and thus medium density needs to be high. They show similar trends with fineness as the milled media but quite different overall behaviour, product grade being much higher than that achievable with the milled media, and recovery and yield being lower. Fig. 2 demonstrates the importance of selecting the correct medium grade for the application.
where ρf is the feed medium density, Rei is the inlet Reynolds Number, d is particle size, and k is a constant. This conforms in general terms to a re-arrangement of the well-known equilibrium orbit hypothesis of hydrocyclone classification to the DMC (Napier-Munn, 1984):
Dη δ50 = ρf + K ⎡ 2 ⎤ ⎢ ⎣ Qd ⎥ ⎦
δ75−δ25 2
(3)
where ρf is the feed medium density, ρu is the underflow medium density, and a0, a1, a2 are coefficients determined from data and specific to a given system. This correlation was found to work both with magnetite media (Davis and Napier-Munn, 1987) and with FeSi media (Napier-Munn, 1984), using density tracers to determine the cut-density. A common (though not very complete) measure of DMC inefficiency is the Ep, defined as: 2
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design and optimisation. 3.3. Particle size limitations As in all fluid-based mineral separation processes the DMC process is sensitive to the size of particle being treated. The maximum particle size that can be processed is governed by the inlet diameter, one third of this diameter being a commonly applied particle size maximum to avoid blockages. In a standard DSM design the inlet diameter is 0.2 times cyclone diameter, so the maximum particle size treatable in the DSM unit is about 0.067 times the cyclone diameter, or 40 mm for a 610 mm cyclone. Material coarser than this limiting value must be treated in other DMS vessels such as DM drums (Wills and NapierMunn, 2006). DMCs are efficient over a remarkably wide size range. However finer particles cut at a higher density (Eqs. (1) and (2), and Fig. 1) and therefore less efficiently than coarser particles, and there is believed to be a size below which cut-point and inefficiency increase sharply. This is called the ‘breakaway’ size (Bosman and Engelbrecht, 1998) and is effectively the finest size that can be efficiently processed in a particular cyclone under given conditions (Fig.3). The breakaway size varies with design and operating conditions, especially the cyclone diameter, the breakaway size increasing with diameter. Process economics will also of course play a role in determining the finest size to be treated. One should note however that not everyone believes in the breakaway size, due to the difficulties of measuring inefficiency (Ep) particularly in the finer sizes (Mathewson and Ryan, 2013; Section 11.8), and model structures that suggest a smooth relationship between inefficiency and size (eg Mathewson and Ryan, 2013) rather than a discontinuity. Nevertheless recent CFD modelling has provided some support for the concept of breakaway size, the limit in that study being found to be around 1 mm for a 1000 mm cyclone in coal preparation (Wang et al., 2014). Larger particle sizes have been quoted by others, eg 2 mm for a 610 mm DMC (Bosman, 1998). Most DMCs in most circumstances will process particles efficiently above about 4-5mm. In fact the Wood coal model assumes the cut-density of +4 mm particles to be constant and provides a separate equation to predict the cut-density of -4mm particles, of the form Ep = constant/d.
Fig. 2. Predicted iron ore separation as a function of FeSi grade; cut-density 3.6 RD.
3.2. Process models of DMCs Process models are useful if they can predict the separation performance, plus ancillary operating and design information such as throughput and (preferably) medium partitioning. The first process model of the DMC was developed by DSM for coal preparation on the basis of regression analysis of data from their original pilot plant work and the first installations made by them and Stamicarbon’s licensees (DSM, 1972). This confidential model became so effective that Stamicarbon was able to offer its licensees process guarantees for the coal plants which they designed and installed. Published models which can be used by anyone with a spreadsheet for routine process prediction over a wide range of operating conditions and cyclone sizes are still rare and always empirical, though progress has been made since they were reviewed at this conference in 1990 (Napier-Munn, 1991). They include a model of a 610 mm DMC in diamond applications (Napier-Munn, 1991), a widely-used model of DMCs in coal preparation (Wood, 1990; Mathewson and Ryan, 2013), and a model which was developed for use in both coal and mineral applications (Dunglison, 1999; Dunglison and Napier-Munn, 1997). Both the Wood and Dunglison models (both developed at the JKMRC at the University of Queensland) predict the partitioning performance of the DMC, capacity, flow split, and the product medium densities, using mostly empirical equations fitted to operating data from pilot and full scale plants. Dunglison’s model is perhaps a little more robust than Wood’s as it uses a published semi-theoretical partitioning model and a published particle terminal velocity correlation to predict medium and ore partitioning, which may be why it is successful in describing both coal (low density) and mineral (high density) DMC performance. However Wood’s model is the generally accepted standard for coal (Mathewson and Ryan, 2013). An important and valuable recent development has been the successful application of computational fluid dynamics code to simulating three phase flow1 in the DMC by application of the fundamental fluid flow equations (eg Narasimha et al., 2007; Wang et al., 2014). The key challenges have been to choose the correct description of turbulence and to support a computational grid fine enough to capture the subtle but process-determining features of the process. These now seem to have largely been solved, and CFD modelling has led to new insights into the principles of the process, including explanations for some of the phenomena discussed in this paper, justification for the form of empirical models (which are much easier to use for the engineer), and suggestions for improved geometry, particularly the inlet. However we are still some way from the routine use of CFD models for process
3.4. Process analysis and control The problem with monitoring and analysing DMC performance is that knowing the true separation performance implies knowledge of the partition (Tromp) curve (Fig. 1), which requires the size-by-size density distribution of the feed and products to be determined. Traditionally this has been done by fractionating the material in heavy liquids which is relatively easy in coal but difficult in minerals (though there are safety concerns in both cases). An alternative is to use particulate density tracers (Napier-Munn, 1985; Davis et al., 1985; Napier-Munn,
Fig. 3. The ‘breakaway size’ for a 1m cyclone in coal preparation (from Bosman and Engelbrecht, 1998).
1 Four phases if one includes the fine media solids and coarse ore solids as separate ‘phases’.
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2014) which are available commercially, including in recent years radio tracers (Partition Enterprises, 2017). Tracers have the advantage of statistical rigour in calculating confidence intervals on the curves and estimating the number of tracers required for the required precision. A disadvantage is that the lower practical size with feed on (generally about 10 mm) is too coarse to monitor performance in the fine sizes, particularly around the breakaway size. De Beers’ company DebTech has now developed a continuous particle density measurement device which promises to overcome the problems of fractionating in heavy liquids (DebTech, 2017). The machine is capable of sorting sized particles on the basis of their true density, and particles collected in chosen density intervals can then be assayed, allowing the full size-density-assay spectrum to be measured, and thus the partition curve characterised and ore/coal washability determined. Another recent development which offers the potential to predict partitioning performance from the analysis of an ore is the use of the standard data produced by an automated mineral analyser such as an MLA which can digitally sort particles into size/density bins whose mean grade can then be determined from the mineralogical composition or direct elemental assay (Wightman, 2015). Fig. 4 shows the theoretical washability of an ore to produce a pyrite pre-concentrate, using only MLA data. The vertical line shows that a perfect separation would reject nearly 60% of the feed with a +500 μm pyrite recovery of over 90% at a separation density of 3.0 RD. A key variable in any DMS process is the density of the medium, which controls the density of separation. From the very early days the medium density has been controlled by over-densifying the circulating medium, measuring the feed medium density with a magnetic pipe coil or nucleonic gauge, and then using PI or PID control to adjust the amount of dilution water added to the pump inlet (or some other part of the process). This approach has proved robust, reasonably precise and easy to implement, and is one of the advantages of DMS over other gravity concentration processes. In recent years good work has been done on developing novel instrumentation and control approaches to other features of the process including optimising control using soft sensors based on empirical models of the kind discussed in Section 3.2. For example Firth et al. (2013) describe how motion analysers on coal product screens were used to continuously infer feed, product and reject mass flow rates (and thus yield), and electrical impedance spectroscopy and Hall effect probes were used to measure the overflow and underflow medium densities and non-magnetic contamination (linked to viscosity). This novel information permits partition performance to be predicted online using soft sensors based on (for example) the Wood or Dunglison models, particularly in terms of medium behaviour (eg Eq. (2)). It also opens up the potential of improved dynamic simulation and control of DMC plants (O’Brien et al., 2016; Meyer and Craig, 2014). Most of this work has been undertaken in coal applications and it would be interesting to apply these new approaches to mineral separations.
Table 1 Economies of scale in the coal industry, for a 1000 t/h DMC plant (from Osborne, 2010). Equipment
No. modules No. deslime screens No. & size DM cyclones No. drain and rinse screens Total no. items Capex
Year 1977
1987
1997
2007
6 6
4 4
2 2
1 1
12 × 500 mm
8 × 660 mm
2 × 1200 mm
1 × 1500 mm
24
16
4
2
164 $26 m
110 $23 m
90 $20 m
82 $18 m
3.5. Increases in scale The main change in DMC practice since its invention has been an increase in scale, especially in coal applications. Cyclones 1500 mm in diameter are now not unusual in coal. However the largest cyclones in mineral applications remain only in the range 700–800 mm (Denysschen and Wagner, 2009; Napier-Munn et al., 2009), and 400 mm is still not uncommon. Cyclone throughput increases approximately as the square of diameter:
Q≈ k H0.5D 2
(5)
where Q = throughput, H = head, D = cyclone diameter and k = constant (measurements suggest that in practice the exponents in Eq. (4) are slightly less than the theoretical values, because of the way energy is dissipated in the cyclone). So increases in cyclone size have a large impact on capacity without much change in footprint. Osborne (2010) showed rather dramatically how increases in cyclone diameter have simplified flowsheets and reduced capital costs in coal preparation over a 20-year period (Table 1). The benefits are clear. The flowsheet complexity has decreased significantly (less modules and equipment items), and capex reduced accordingly. These advantages were also apparent in a trial of a single 800 mm DMC replacing a module of four 400 mm DMCs in the preconcentration plant in the then-Xstrata (now Glencore) zinc-lead concentrator at Mount Isa in Queensland, Australia (Napier-Munn et al., 2009). The benefits noted during the trial, which involved by-passing the distributor feeding the four 400 mm cyclones, included increased cyclone life and reduced wear of other components (and thus changeout downtime), reduced inlet blockages, and improved operability, with no apparent reduction in metallurgical performance. As a consequence the DMC plant recently installed at Glencore’s McArthur River mine in Australia’s Northern Territory, also in a metalliferous preconcentration role, has a similar head and has reported no decline in metal recovery as a consequence (Wallace et al., 2014). 4. A suggestion for improving DMC designs in mineral processing The design and operation of DMCs in mineral processing have been slow to follow the advances achieved in coal preparation, in particular in running vary large cyclones at relatively low heads. The argument for large cyclones was made in Section 3.5 above. The argument for low heads is simply that gravity-fed DMCs (as many mineral DMCs are, but few coal ones) require less tall and therefore cheaper buildings, and all low-head installations require less pumping power and generally lead to lower component wear. The ‘standard’ DMC head in coal preparation is 9D where D is cyclone diameter. (An explanation for why heads are generally expressed in numbers of cyclone diameters is given in the appendix). This arose out of the early DSM work, and the DSM Handbook nominated 9D as the minimum head for coal. It remains the commonest arrangement in
Fig. 4. Reject rate and recovery of +500 μm pyrite using MLA data (data from Wightman, 2015).
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• Coarser FeSi media required, which are cheaper and easier to re-
coal (heads up to 13-15D are sometimes used with large diameter cyclones supposedly to improve the separation efficiency of finer particles, although the evidence for the benefit is weak – see for example Clarkson et al., 2002). Mysteriously, however, the DSM Handbook recommends 15-40D ‘for minerals and ores’, and indeed mineral operations generally do have much higher heads than coal as a consequence. Argyle Diamonds and the two big Pilbara iron ore concentrators (all in Western Australia) have 350–400 mm cyclones with heads greater than 20D, and the ‘standard’ De Beers 610 mm DMC uses a head of 13D. The Mount Isa trial mentioned above reduced the head from 21.6D with the 400 mm cyclones to 9.5D for the 800 mm unit, very close to the coal preparation ‘standard’. (This was driven by plant design and installation requirements rather than a deliberate attempt to evaluate the lower head). So why the higher heads for minerals? The answer may have something to do with the fact that when the first DMCs were installed in mineral applications DSM had no experience or data from such high density separations using mostly higher viscosity FeSi media, and the higher head recommended therefore may represent a safety factor over the existing experience. However there is no evidence in the literature (sparse though it is) for the higher heads actually being necessary in mineral applications, in fact quite the reverse. The following case studies all suggest that there is no advantage to mineral operations of heads much higher than 9D. Each one includes the lower particle size treated (LPS) to demonstrate that these conditions are effective even at the lower end of the particle size range commonly treated in DMCs (around 1 mm).
cycle.
There is no evidence for reduced metallurgical performance under such conditions; in fact there is some weak evidence for an improvement in performance. And in cases where distributors can be eliminated there is a proven benefit of improved plant performance due to the bias which most distributors introduce, leading to different cut-points in individual cyclones and thus decreased efficiency (Mathewson and Ryan, 2013; Section 11.5). 5. Conclusions - does the DMC have a future? Some mineral DMC plants are regarded mainly as a maintenance problem by their owners, who therefore appoint mechanical engineers to run them rather than process engineers, and make no effort to acquire or preserve DMC operating knowledge and experience. The consequence is unsurprising – non-optimal metallurgical performance, and little progress in design innovation. This loss of expertise, in both operations and engineering companies, is troubling. The knowledge to correct this deficiency exists, but it will require an effort of will on behalf of companies and trainers to build a new generation of expertise. The benefits in terms of process performance would be very great, as has been demonstrated by the coal preparation community over the years. However, despite its Cinderella status in some applications, it is difficult to see a serious competitor for the DMC in the foreseeable future in the processing of bulk commodities such as coal and iron ore. It is also clearly an increasingly important option in the pre-concentration role as metalliferous head grades decline. Its greatest competitor in this role is likely to be the growing capability of high tonnage electronic sorting machines, which are in principle capable of multisensing mineral discrimination at relatively coarse sizes (Von Ketelhodt et al., 2009), and even simple screening where the ore is amenable to ‘grade engineering’ (Carrasco et al., 2016). Jigs with optimising control may in due course match the partitioning of DMCs, though many such comparisons in the past have been largely inconclusive. The ability of the DMC to make a positive separation at a chosen density with relatively tight control, combined with its capacity to treat high tonnages at a wide size range and its maturity as a process, would suggest that it will remain an important and possibly even increasing player in the future of mineral processing. Its attraction in mineral separations will be enhanced if some of the developments discussed in this paper are introduced, particularly the use of larger cyclones at lower heads, the exploitation of novel instrumentation and model-based soft sensors for improved control, dynamic simulation, and the better training of practitioners. More speculative developments worthy of attention include novel geometries suggested by CFD simulations (particularly high angle cyclones), and alternative media with improved rheology.
• The 800 mm DMC trial at Mount Isa, discussed in Section 3.5 (head 9.5D, LPS 2.5 mm). • The McArthur River plant discussed in Section 3.5 (head 7.0–9.5D, LPS 1.6 mm). • A model of a 610 mm DMC in diamond recovery based on extensive • •
testwork (LPS 0.5 mm) suggests that, if anything, the Ep is constant in the range 9.7–11.6D and actually rises above 11.6D (NapierMunn, 1991; see also the graph in Napier-Munn et al., 2009). Data collected in seven coal preparation plants with large DMCs by Clarkson et al. (2002) and modelled by Napier-Munn (2014) shows no effect of head on cut-density, over the range 7.5–13.8D (LPS 1 mm). In an iron ore case study (Dunglison et al., 2000) the Dunglison model was used to show that there would be negligible change in metallurgical performance (iron recovery and product grade) when changing from a module of three 355 mm DMCs at a head of 15.5D to a single 710 mm DMC at a head of 10.4D (LPS 1 mm). Tests in the plant confirmed this result.
It is therefore suggested that future DMC plants in mineral processing should be designed with large cyclones at low heads (around 9D), as is common practice in coal preparation. The benefits would include:
• Reduced capital cost. • Reduced wear of cyclones and other components, and thus reduced downtime. • Reduced operating costs for pumping and component replacement. • Simpler flowsheets and greater operability.
Acknowledgement The author thanks the Reviewers for a number of useful comments which have been incorporated into the paper.
Appendix. The Significance of H/D DMC heads (or pressures) are often expressed as numbers of cyclone diameters, ie H/D. The reason for this is not well understood by practitioners, and the author can find no explicit explanation in the published literature. Although some DSM literature uses this approach, it does not explain why. One possible explanation is as follows. Consider a DMC fed by a steady-head tank with a static head of H m, as shown in the diagram. Assume that medium exits the tank and enters the tangential inlet of the DMC at a mean velocity, Vi m/s. Simple fluid mechanics (Bernoulli’s equation) shows that:
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Vi = CD 2gH
(A1)
where g is the acceleration due to gravity, and CD is the discharge coefficient (a function of the inlet characteristics and the inlet velocity loss factor (Napier-Munn, 1986)). The separation of ore particles in the medium is driven largely by the centrifugal forces which they and the medium experience. The centrifugal force, Fc, on an ore particle entering with the medium at the periphery of the cyclone is given by:
Fc =
mV2i r
(A2)
where m is the mass of the particle and r is the radius of rotation, here assumed equivalent to the cyclone radius. Writing D for the diameter of the cyclone gives:
Fc =
mV2i D/2
(A3)
Substituting Eq. (A1) into Eq. (A3) gives
H Fc = K m ⎛ ⎞ ⎝D⎠
(A4)
4gCD2 .
where K= Eq. (A4) shows that the separating force on a given particle in the DMC is proportional to H/D, and therefore that two DMCs with the same H/D ratio will provide approximately the same separating force. For example, a 1500 mm DMC with a head of 13.5 m (9D) will have the same separating force as a 400 mm DMC with a head of 3.6 m (9D). This may be the explanation for why head is often expressed as number of cyclone diameters.
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