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Modeling classification in small-diameter hydrocyclones under variable rheological conditions
Minerals Engineering 15 (2002) 613–622 This article is also available online at: www.elsevier.com/locate/mineng
Modeling classification in small-diameter hydrocyclones under variable rheological conditions L.M. Tavares a
a,*
, L.L.G. Souza b, J.R.B. Lima b, M.V. Possa
c
Department of Metallurgical and Materials Engineering, COPPE, Universidade Federal do Rio de Janeiro, UFRJ, Cx. Postal 68505, CEP 21945-970, Rio de Janeiro, RJ, Brazil b Department of Mining Engineering, Universidade de S~ ao Paulo, S~ ao Paulo, Brazil c Centro de Tecnologia Mineral, CETEM/MCT, Rio de Janeiro, RJ, Brazil Received 12 December 2001; accepted 20 May 2002
Abstract Although several mathematical models describing hydrocyclone performance exist in the literature, only a few suitably describe classification of non-Newtonian slurries in small-diameter hydrocyclones (smaller than 75 mm of diameter). In order to develop a mathematical model accounting for the effects of slurry rheology on classification, experiments have been conducted with phosphate ore slurries of variable solids concentration and chemical environment, covering a wide range of rheological conditions, both Newtonian and non-Newtonian. Semi-empirical models for estimating the hydrocyclone capacity and corrected cut size and an empirical model of water split have been proposed and/or modified from the literature in order to describe classification under such extreme rheological conditions. Of particular interest is the application of a model developed by the authors and based on the residence time theory, that is used to predict the corrected cutpoint. Experimental results demonstrated that, under the conditions studied, hydrocyclone capacity and sharpness of separation were not affected by slurry rheology. It was also found that the plastic viscosity is the most significant parameter for modeling the corrected cut size in 50 and 25 mm diameter hydrocyclones. 2002 Elsevier Science Ltd. All rights reserved. Keywords: Classification; Hydrocyclone; Modeling; Rheology; Viscosity
1. Introduction Size reduction of weathered ores, such as phosphates from Brazil, generate large amounts of slimes that must be removed in order to allow selective flotation. This desliming operation, often carried out in small-diameter hydrocyclones, poses an important technical challenge, given the high viscosity and non-Newtonian rheology often associated with these slurries. Technical information on the classification of nonNewtonian slurries in small-diameter (smaller than 75 mm) hydrocyclones is scarce in the literature. Important advances have been made in the last few decades in the development of phenomenological models of the hydrocyclone under Newtonian flow conditions (Hsieh and Rajamani, 1991; Rajamani and Devulapalli, 1994; Barrientos and Concha, 1992). However, the application of * Corresponding author. Tel.: +55-21-2562-8538; fax: +55-21-22906626. E-mail address: [email protected] (L.M. Tavares).
such approach to classification in non-Newtonian fluids is even a more formidable task, given the limited knowledge yet available of the behavior of particles in non-Newtonian fluids (He et al., 2001) and of the flow of such fluids in the hydrocyclone (Upadrashta et al., 1987). While phenomenological models are not yet widely available for robust process optimization, empirical and semi-empirical models are the best alternative within reach for this task. Although a large number of papers deal with the empirical or semi-empirical modeling of hydrocyclones, most of these models were developed over quite a limited range of hydrocyclone diameters (100–380 mm) (Lynch and Rao, 1975; Plitt, 1976; Nageswararao, 1978), and their applicability to smalldiameter hydrocyclones has not yet been verified (Napier-Munn et al., 1996). Modeling studies of smalldiameter hydrocyclones are fairly scarce (Bradley and Pulling, 1959; Brookes et al., 1984; Vallebuona et al., 1995), and correspond to new equations or to modification of those already used for larger diameter hydrocyclones.
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Nomenclature a
consistency index of the fluid in Ostwald and Waele model (–) b slurry type flow behavior index (¼1 for Newtonian, <1 for pseudoplastic; >1 for dilatant) (–) Cv volume fraction of solids in the hydrocyclone feed (–) d particle size (m) Dc cylindrical section diameter of the hydrocyclone (m) Di inlet diameter of the hydrocyclone (m) Do vortex finder diameter of the hydrocyclone (m) Du apex diameter of the hydrocyclone (m) d50c corrected cutpoint (m) E partition coefficient (–) E1 , E2 , E3 model parameters (–) f1 , f2 dimensionless numbers that describe the effect of geometry in cyclone models (–) g acceleration due to gravity (m/s2 ) K1 , K2 , K3 scale-up parameter in hydrocyclone models (–) Lc vortex finder clearance (m) n flow pattern constant (–) P inlet pressure (Pa) Q slurry flowrate in hydrocyclone feed (m3 /h) r radial distance inside the hydrocyclone (m)
Viscosity, which represents the fluid’s resistance to shear, plays a significant role in the classification using hydrocyclones. A variation in slurry viscosity alters the particle settling velocities and the slurry velocities within the hydrocyclone, directly affecting its performance. Although several mathematical models for cut size developed in the past have viscosity terms, most of them were developed for dilute suspensions (Kawatra et al., 1996). The classical models of Lynch and Rao (1975) and Plitt (1976) do not have an explicit viscosity term, accounting for its effect rather indirectly, usually through a term that depends on solids concentration. However, it is well known that solids concentration is not the only parameter that affects slurry rheology. Temperature, size distribution of the solids, and chemical environment can also alter rheology (Shi and Napier-Munn, 1996). More recent models overcome this limitation by using slurry viscosities at high shear rates, measured using online viscometers (Kawatra et al., 1996; Asomah and Napier-Munn, 1997). For highly non-Newtonian slurries the slurry viscosity (apparent or plastic) varies significantly with shear rate, which in turn varies with the hydrocyclone geometry and flowrate. Recognizing that the major limitation of hydrocyclone models is that they are not generally amenable to
r50 Rf vc vi vt
Greeks a b k DP c q qp qs g gp h s so
dimensionless radial distance corresponding to the envelope of zero axial velocity (–) water split to the underflow (–) mean fluid velocity in the cylindrical part of the hydrocyclone (m/s) inlet velocity of the fluid in the hydrocyclone (m/s) tangential velocity of the fluid in the hydrocyclone (m/s)
sharpness index of the partition curve (–) friction factor (–) velocity reduction parameter (–) pressure drop in the hydrocyclone (Pa) shear rate (s1 ) fluid density (kg/m3 ) slurry density (kg/m3 ) solids density (kg/m3 ) viscosity of a Newtonian fluid (Pa s) plastic viscosity of a non-Newtonian fluid (Pa s) included angle of the conical section of the hydrocyclone (–) shear stress in the fluid (Pa) critical shear stress of the fluid (Pa)
extrapolation, this paper describes work conducted to investigate classification of a feed material containing a large proportion of ultrafines in small-diameter hydrocyclones under widely varying rheological conditions, achieved through independent control of solids concentration and chemical environment.
2. Experimental 2.1. Test material A sample of phosphate ore (with a specific gravity of 3430 kg/m3 ) from the feed to the desliming circuit at Serrana, SA (Araxa, Brazil) was collected for the experiments. The rock is of magmatic origin and is part of an alkaline–carbonatite complex. Particle size distribution of the sample was measured by laser scattering using a Malvern Mastersizer-Microplus and is shown in Fig. 1. The very large proportion of ultrafine material (over 40% <1 lm) demonstrates the severe degree of weathering of the ore. Detailed mineralogical studies (Possa, 2000) indicated that the main mineral constituents (by weight) were goethite (43%), apatite (17%), gorceixite (16%), barite (8%), vermiculite (5%), quartz
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Fig. 1. Size distribution of the feed determined by laser scattering.
(4%) and smaller proportions of anatase, hollandite and pyrochlore. Slurries containing different solids concentrations (15%, 25% and 35% by weight) and chemical environment were prepared. Slurry rheology was controlled either through the addition of a dispersant––polycarboxylic sodium salt (PSA) from BASF––or thickening agents (carboximethylcellulose or glycerin). A total of seven different slurries were prepared (see Table 1), with widely variable rheological characteristics. Their apparent viscosities at low shear rates (in the order of 75 s1 ) have been measured with a Brookfield viscometer. Further, their rheological behavior was determined at 30 C with variable shear rates from 10 to 4000 s1 using a rotational rheometer (Haake Rotovisco model RS 100) equipped with a cylinder cup and bob arrangement, and a summary is given in Table 1. 2.2. Test procedure Experiments were conducted in a closed-circuit test rig fitted with a by-pass line. KREBS hydrocyclones with diameters of 25 and 50 mm and inlet pressures of 170, 240, 310 and 450 kPa were used. Due to pumping limitations, experiments were not conducted at the
615
highest pressure (450 kPa) with the 50 mm hydrocyclone. In addition, experiments were not conducted at the low feed pressure (170 kPa) with the slurries thickened with carboxymethylcellulose (CMC) and glycerin, due to roping. Geometry was maintained constant for each hydrocyclone in the tests, so that ratios of Di =Dc , Do =Dc , Du =Dc were 0.28, 0.25 and 0.19 for the 25 mm hydrocyclone and 0.24, 0.37 and 0.37 for the 50 mm hydrocyclone. The internal conical angle of both hydrocyclones was 9. All test runs were carried out in random order to avoid potential sources of experimental bias. A total of 41 experiments were conducted and no experimental results were discarded. The solids content in the slurries were controlled within 2% of the set values (Table 1) while the temperature was maintained between 25 and 35 C during the tests. Samples at each experimental condition studied (feed pressure, solids concentration, reagent concentration and hydrocyclone diameter) were collected simultaneously from the underflow and overflow discharges. Whenever sampling was not being carried out the products were continuously recombined and returned to the slurry tank. Samples collected were weighed, filtered and dried. In order to guarantee an accurate measurement and prevent particle agglomeration, samples for size analysis in the Malvern Mastersizer were tested prior to filtration and drying, through dilution in water and dispersion with PSA. The chemical composition of samples from the feed and the underflow streams was also determined by X-ray fluorescence. The resultant test data (size analyses and water and solids weights) were reconciled using the method of Lagrange multipliers with the aid of the computer software MATBAL (Laguitton, 1985) and mass-balanced. Reconstituted feeds for different tests were compared among each other and to the measured size distribution (Fig. 1), indicating no systematic errors in the experiments. Partition curves were then prepared from the reconciled size analyses and slurry flowrates. Parameters of the mathematical models used to describe hydrocyclone capacity, water split and cutpoint have been fitted by least-squares to experimental data using a quasi-Newton optimization technique available
Table 1 Summary of the characteristics of the slurries used in the hydrocyclone experiments % Solids (w/w)
% Solids (v/v)
Additive (weight of additive per ton of dry ore)
Brookfield viscosity (mPa s)
Rheological model (Eq. (10))
15 25 35 15 15 25 35 35
5.0 9.0 13.4 5.0 5.0 9.0 13.4 13.4
– – – 27 kg of CMC 4900 kg of glycerin 1.0 kg of PSA 0.95 kg of PSA 2.6 kg of PSA
23–24 108–140 232–240 114–120 100–108 37–40 112–130 20–24
s ¼ 0:32 þ 0:010c0:80 s ¼ 1:40 þ 0:026c0:78 s ¼ 5:80 þ 0:084c0:69 s ¼ 0:48 þ 0:52c0:58 s ¼ 1:45 þ 0:040c0:96 s ¼ 0:044 þ 0:0027c0:96 s ¼ 2:30 þ 0:021c0:80 s ¼ 0:10 þ 0:0060c0:85
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in the software STATISTICA (Statsoft, Inc.). The significance of individual regression coefficients has been verified with Student’s t-tests (Hogg and Ledolter, 1987). If the p-value is less than or equal to the chosen significance level of 0.05, the hypothesis that they are equal to zero was rejected. Therefore, variables whose parameters presented p-values above 0.05 were deleted from the model. 3. Model development, results and discussion 3.1. Capacity A relationship between the pressure drop in the hydrocyclone and the feed slurry flowrate can be derived considering that the friction factor b for the turbulent flow from fluid mechanics (Batchelor, 1973) is given by b¼
DP qv2c =2
ð1Þ
The mean velocity of the fluid in the cylindrical section of the hydrocyclone (vc ) is vc ¼
Q pD2c =4
ð2Þ
Substituting Eq. (2) in (1), replacing the fluid density q by the slurry density qp , and replacing the friction factor by empirical factors related to the material type and hydrocyclone geometry, gives !0:5 DP 2 f1 ðgeometryÞ ð3Þ Q ¼ K1 Dc qp where K1 is the scale-up parameter that must be calibrated with experimental data. This, along with the function of geometry f1 , represent the effect of the friction factor b. The influence of the hydrocyclone geometry can be taken into account with a dimensionless group proposed by Nageswararao (1978) 0:45 0:68 0:20 Di Do Lc f1 ðgeometryÞ ¼ h0:10 ð4Þ Dc Dc Dc Using data from all 41 tests, covering a range of slurry flowrates from 1 to about 5 m3 /h, the scale-up parameter K1 was determined from the least-squares best-fit. A comparison between measured and calculated feed capacities is given in Fig. 2. The good agreement demonstrates that the model can appropriately describe the experimental data. It has been suggested (Bloor et al., 1980) that b––and therefore K1 ––may also be influenced by slurry rheology. An attempt has been made to include a term corresponding to slurry viscosity raised to a constant in Eq. (3). The parameter, however, was not found to be sta-
Fig. 2. Correspondence between measured and calculated flowrates in the hydrocyclones. The least-squares best-fit parameter K1 is 445.85 and the coefficient of correlation is 0.975.
tistically significant. Indeed, other authors (Coelho and Medronho, 2001) deduced from theory that feed flowrate varies only with the 0.05 power of fluid viscosity. Among some of the more widely used models found in the literature (Lynch and Rao, 1975; Plitt, 1976; Nageswararao, 1978; Vallebuona et al., 1995; Asomah and Napier-Munn, 1997) the model given by Eqs. (3) and (4) was the one that best combined quality of fit to the data and simplicity. It requires fitting only one parameter to experimental data. Further, it is based on a well-known correlation from fluid mechanics derived from the conservation of energy and momentum in the equipment. 3.2. Water split Defined as the ratio between the water volumetric flowrates in the underflow and the feed streams, the water split is often the most controversial part of hydrocyclone models (Napier-Munn et al., 1996). It has been described with a variety of expressions in the literature (Lynch and Rao, 1975; Nageswararao, 1978; Asomah and Napier-Munn, 1997). Based on classification data in Krebs hydrocyclones with diameters in the range 102–381 mm, Nageswararao (1978) proposed an expression of the form for the water split, given by !E3 P E2 E1 Rf ¼ K2 Dc gðCv Þ f2 ðgeometryÞ ð5Þ qp gDc where gðCv Þ ¼ 101:82Cv =ð1 Cv Þ2 is essentially a ratio of free to hindered settling terminal velocity suggested by Steinour (1944). Although it was first proposed for spherical particles, fitting of the parameter E2 to exper-
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Table 2 Summary of statistics of the water split model (Eq. (5)) Variable
Estimate
Standard error
p-value
K2 E1 E2 E3
221.5 1.27 0.23 0.049
56.89 0.0631 0.0429 0.0358
0.0198 0.0000 0.0000 0.1805
imental data allows its application to non-spherical particles, such as those found in the present study. f2 describes the effect of geometry and is given by (Nageswararao, 1978) f2 ðgeometryÞ ¼
Di Dc
0:50
Do Dc
1:19
Du Dc
2:40
Lc Dc
0:22
h0:24
ð6Þ Experimental conditions covered in the experimental runs resulted in water splits varying from 0.3 to about 0.8. Model parameters (K2 , E1 , E2 and E3 ) were fitted to these data and a summary is presented in Table 2. Considering that there is over 18% probability that we are in error if we consider that E3 is statistically significant, then the pressure term was omitted from the model and the remaining parameters were estimated again. Therefore, Eq. (5) may be rewritten as Rf ¼ 199:1D1:33 c gðCv Þ
0:22
f2 ðgeometryÞ
ð7Þ
which is valid for volume fractions of solids in the slurry (Cv ) from 0.05 to 0.134. A comparison between measured and calculated results from Eq. (7) is presented in Fig. 3, which shows a very good fit. The relative independence of the water split in respect to pressure, also evident in Fig. 4, has also been found by Vallebuona et al. (1995) for small-diameter (25 and
Fig. 4. Effect of solids concentration in the water split. Experimental results from 0% solids were included for reference only, given that they were not used in fitting the model. The solid lines represent the calculation of the model (Eq. (7)).
50 mm) hydrocyclones. Fig. 4 also shows that, under the conditions studied, the water split decreases with an increase in feed solids concentration. An increase in water split with the solids concentration, as predicted by Plitt (1976) and Nageswararao (1978) for larger diameter hydrocyclones, has only been observed in the present study for low volume concentrations (Fig. 4). Using data from a 102 mm diameter hydrocyclone, Kawatra et al. (1996) showed that the water split increases as the slurry viscosity increases in the range from about 0.5 to 4 mPa s, becoming relatively constant for higher slurry viscosities. An attempt has also been made to incorporate its effect in the model by including a power term of slurry viscosity in Eq. (5). The data, however, showed no direct correlation between water split and slurry viscosity, only indirectly through the solids concentration term gðCv Þ. These results are not necessarily in contradiction with findings from Kawatra et al. (1996) given that in the present work the apparent viscosities of the slurries in the range of shear rates of interest are typically higher than 4 mPa s (Table 1). Reasons are not clear for the apparently different effects of solids concentration and slurry viscosity found in the present study and those observed for larger diameter hydrocyclones, so more in-depth studies are necessary to clarify the matter. 3.3. Partition curve
Fig. 3. Correspondence between measured and calculated water splits in the hydrocyclones.
The performance of the hydrocyclone can be fully defined by its capacity, water split to the underflow and its partition curve. Lynch and Rao (1975) proposed the following empirical model for the partition curve
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Fig. 5. Normalized partition data for the tests. Fig. 6. Vertical and tangential velocity profiles in a hydrocyclone, showing the envelope of zero vertical velocity (after Kellsal (1952)).
E ¼ Rf þ ð1 Rf Þ
expðad=d50c Þ 1 expðad=d50c Þ þ expðaÞ 2
ð8Þ
where d50c is the corrected cutpoint and a is the classification index, which gives the sharpness of separation. Data from the experiments are plotted in Fig. 5 in normalized coordinates. The fact that they appear to conform to a single normalized partition curve, as precluded by Lynch and Rao (1975), suggests that slurry rheology and pressure drop may have limited effect on the sharpness of separation parameter. Indeed, this relative independence of the reduced efficiency curve in respect to slurry viscosity has also been previously observed by Kawatra et al. (1996). The value of the sharpness of separation (a) determined by fitting Eq. (8) to the entire data set (Fig. 5) was found to be 2.4. This value was used in the simulations described later. 3.4. Corrected cutpoint d50c The corrected cutpoint ðd50c Þ is the size of particles, after discounting the effect of short-circuit flow, that have the same chance of reporting to the underflow or the overflow streams (Eq. (8)). A variety of models have been proposed to describe the effect of design and operating variables in the hydrocyclone on the corrected cutpoint. Only a fairly limited number, however, describe explicitly the relationship between the cut size and the viscosity. One phenomenological description of the effect of slurry viscosity on the cutpoint is provided by the equilibrium orbit hypothesis (Rietema, 1961; Bradley, 1965; Plitt, 1976). It states that each particle tends to be in equilibrium between the centrifugal force acting towards the wall of the hydrocyclone and the drag force from the fluid acting towards the axis. For the cut size, the equilibrium orbit overlaps with the envelope of zero vertical velocity (Fig. 6a) and so for this size, the par-
ticles have equal chances of reporting to the underflow and overflow streams. Bradley (1965) showed that, for a given hydrocyclone geometry and laminar flow, models of this kind reduced to the following form 1=2 d50c gDc ¼ K3 ð9Þ Dc Qðqs qÞ where g and q are the viscosity and density of the fluid, respectively. Dc is the hydrocyclone diameter, qs is the density of the solids and K3 is a dimensionless parameter that depends on the geometry of the hydrocyclone. The increase in cutpoint with viscosity from Eq. (9) is due to the fact that with higher viscosities the particles will tend to remain in suspension and be recovered in the overflow. An initial attempt has been made to fit the parameter K3 in Eq. (9) to the experimental data in this work, substituting the fluid viscosity by the average Brookfield viscosity of the slurry (Table 1). However, the resulting correlation obtained turns out to be very poor, with a coefficient of correlation below 0.5. This is not surprising, given that most of the slurries in the present study demonstrated time-independent non-Newtonian behavior, so that the viscosity (either apparent or plastic) depends on the shear rate (Fig. 7). A more rational approach, therefore, consists of replacing the viscosity of the fluid in Eq. (9) by the viscosity (apparent or plastic) of the slurry at the point where the axial velocity in the hydrocyclone is zero (Fig. 6). This viscosity may be calculated using the procedure described as follows. A variety of rheological responses of fluids can be described by Ostwald and Waele model (Bird et al., 1960), which is given by s ¼ so þ acb
ð10Þ
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c¼
8nkQ pD2i Dc rnþ1
619
ð17Þ
where r is the dimensionless radial distance in the hydrocyclone, r ¼ 2r=Dc . Finally, substituting Eq. (17) in (12), gives b1 8knQ ð18Þ gp ðr Þ ¼ ab pD2i Dc rnþ1
Fig. 7. Rheological behavior of mineral slurries used in the experiments (lines represents Ostwald and Waele model, Eq. (10)).
where so is the critical shear stress ð¼ 0 for Newtonian fluids), b is the flow behavior index (¼1 for Newtonian or Bingham plastic fluid, <1 for pseudoplastic fluids, >1 for dilatant fluids) and a is the consistency index. The plastic viscosity gp is given by gp ¼
ds dc
ð11Þ
Substituting Eq. (10) in (11) gives gp ¼ abcb1
ð12Þ
Inside the hydrocyclone, the shear rates vary according to the radial position. The shear rates inside the cyclone can be calculated from the tangential velocity of the fluid vt , given by dvt ð13Þ dr where r is the radial distance in the hydrocyclone from the central axis. The relationship between the tangential velocity of the fluid and the radius of the hydrocyclone may be described by Bradley (1965) (see Fig. 6b)
c¼
vt rn ¼ constant
ð14Þ
where n is a constant. At the hydrocyclone wall the tangential velocity of the fluid is vt ¼ kvi
at r ¼ Dc =2
ð15Þ
The inlet velocity vi can be expressed as a function of the feed flowrate Q and the equivalent inlet diameter Di by vi ¼
4Q pD2i
ð16Þ
Substituting Eq. (14) in (13) and differentiating with the initial condition given by Eq. (15), gives
which allows the estimation of the plastic viscosity of the slurry at any radial distance within the cyclone for which Eq. (14) provides a valid approximation of the tangential velocities. If it is assumed that the plastic viscosity offers a valid description for the effect of viscosity in the equilibrium orbit model, then by replacing Eq. (18) evaluated at the envelope of zero vertical velocity (at a radial distance of r50 ) in Eq. (9), gives " b1 #12 ab 8nk 4b b2 d50c ¼ K3 Dc Q ð19Þ nþ1 qs q pD2i r50 Most of the constants in Eq. (19) can be estimated using data from the literature. The parameter n typically varies between 0.6 and 0.9, with the value 0.8––suggested by Bradley (1965)––being used in this work. The parameter r50 , which represents the locus of null vertical velocity, is suggested to be typically about 0.38 (Bradley, 1965). Dyakowski et al. (1994) observed that it is relatively independent of the rheological behavior of the fluid. The parameter k, which represents a correction in the inlet velocity when the slurry enters the hydrocyclone, has been suggested to depend only on the hydrocyclone geometry (Lilge, 1962). 1:13 Di k ¼ 4:5 ð20Þ Dc The parameters a and b describing the rheological response of the slurry (Eq. (10)) contained exactly in the locus of null vertical velocity of the hydrocyclone must be known in order to apply Eq. (19). In practice, however, these can only be estimated (offline using a rheometer) for the feed slurry. The validity of this approximation is discussed later. The least-squares bestfit parameters of Eq. (10) for the various feed slurries tested in the present study are summarized in Table 1. Fig. 7 also shows that the slurries used in the present work demonstrated widely varying rheological responses, several exhibiting a strong pseudoplastic behavior with yield stress. While the model for the corrected cut size, given by Eqs. (19) and (20), describes explicitly the effects of slurry flowrate, solid and fluid densities, hydrocyclone diameter and pulp rheology, the influence of the hydrocyclone geometry must be determined from calibration of parameter K3 in the model with experimental
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Fig. 8. Correspondence between measured and calculated corrected cut-size for both hydrocyclones.
data. Fitting the data from classification in the 25 and 50 mm hydrocyclones gave values for the parameter K3 of 0.0589 for the 25 mm hydrocyclone (with a coefficient of correlation of 0.972) and 0.0451 for the 50 mm hydrocyclone (with a coefficient of correlation of 0.911). Fig. 8 illustrates the very good correspondence of the model to the data. It is important to observe that, as the rheological behavior of the slurry was taken into consideration, there was no need to include a term accounting for the solids concentration of the slurry. In general, the most commonly used descriptor of fluid viscosity is the apparent viscosity, defined as the ratio of shear stress and shear strain. Therefore, an expression analogous to Eq. (19) has been derived using this definition, instead of that of plastic viscosity. However, fitting the resulting model to the data yields coefficients of correlation of 0.948 and 0.903 for the 25 and 50 mm diameter hydrocyclones, respectively. This, justifies the use of the plastic viscosity in describing the effect of slurry rheology in the classification in hydrocyclones. A comparison between experimental and modeling results is also shown in Fig. 9, where the effects of pressure drop and pulp rheology on the cut size are investigated. It shows the good predictive capabilities of the model over a very wide range of rheological conditions. The significant effect of hydrocyclone diameter (also shown in Fig. 9) is partially explained by the effective shear rates, which were estimated to be in the order of 400–600 s1 for the 50 mm hydrocyclone and between about 1500 and 3000 s1 for the 25 mm hydrocyclone. The predictive capabilities of the model are demonstrated by comparing the measured solids concentration
Fig. 9. Variation of corrected cut size with pressure drop for classification of slurries at 15% and 35% of solids (in weight) in the 25 mm diameter hydrocyclone. Data for 35% of solids slurry with the 50 mm diameter hydrocyclone is also included for comparison.
in the underflow to that calculated from the solids and water balance. The excellent agreement in Fig. 10 demonstrates the validity of the model equations. One key assumption in the model is regarding the rheological response of the fluid in the locus of zero vertical velocity, which is measured using the feed material. At least two situations can be identified when this assumption should be valid. This is the case of very dilute slurries, where the slurry rheology is dominated by the rheological response of the liquid. This approach should also be valid whenever the feed contains a significant proportion of fine particles. This is because
Fig. 10. Correlation between measured and calculated solids concentration in the underflow of the hydrocyclone.
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these particles have a more marked effect on the rheological response of the fluid (if compared to coarser ones that also may be present in the feed) and that they remain in suspension throughout the entire classification process, being separated solely depending on the water split. Although it is difficult to determine a priori what the minimum proportion of slimes required to validate this assumption is, this approach was certainly valid for the material studied in the present investigation. Evidence of this is the good agreement between model and experiments (Fig. 8) obtained by fitting a single parameter. While the limit of validity may not be clearly stated, the analysis of the partition data from all 41 tests demonstrated that at least all material smaller than 2 lm, which corresponded to more than 50% of the material, remained in suspension throughout all the tests. Another difficulty associated with the application of the present model is regarding the fact that it requires measuring the rheological curve of the slurry, which can only be carried out offline. However, this may be partially overcome by existing mathematical models, which allow the prediction of the effect of several variables, including size distribution and solids concentration in the rheological behavior of the slurry (Shi and NapierMunn, 1996). Finally, the assumption that an average value could be used to describe the density of the solids (qs ) in Eq. (19), so that no significant density segregation occurred during the experiments, was also investigated. If density segregation is to occur, the heavier minerals, such as barite (BaSO4 ) and goethite (Fe2 O3 H2 O), would more likely concentrate in the underflow stream, whereas lighter minerals, such as vermiculite (Mg3 (Al,Si)4 O10 (OH)2 4H2 O) and quartz (SiO2 ), would be more prone to report to the overflow stream. This was verified by comparing the Fe2 O3 , BaO and SiO2 contents in the feed and in the underflow streams from selected experiments. Whereas a marginally higher grade of BaO was found in the underflow stream (6.2–8.0%) in comparison to the feed (5.9%), an opposite trend was found for Fe2 O3 , where the underflow concentrations (34.1–40.4%) were either marginally lower or nearly equal to that of the feed (40.3%). No distinct trend has been found for SiO2 , which varied from 5.8% and 7.1% in the underflow stream, comparing to the 6.7% content in the feed. The lack of a distinct overall trend suggests that, in spite of the fact that the density of individual minerals differed by as much as 30% from that of the average feed solids, no significant density segregation occurred in the experiments.
4. Conclusions The following conclusions can be drawn from this investigation:
621
• Semi-empirical models have been successfully used to describe capacity and cut size in classification using small-diameter hydrocyclones, while an empirical model has been satisfactorily used to predict the water split. • Although it has a major effect on cutpoint, no direct influence is found of slurry rheology in capacity, water split and sharpness of separation under the conditions studied. This suggests that the addition of rheology modifiers is an effective way of controlling the cutsize in hydrocyclones, having no side effects on the other performance criteria. • Water split has not been found to be significantly dependent on inlet pressure under the conditions studied. • The effect of solids concentration in hydrocyclone cutpoint can be solely explained by its effect on the slurry rheology. • The rheological response of the feed has been successfully used to describe the rheology in the locus of zero vertical velocity inside the hydrocyclone for modeling the cutpoint for the material studied. However, the validity of such approach is probably limited to the classification of feeds containing large proportions of slimes, and assumptions associated with the existence of a steady state locus of zero velocity in the hydrocyclone.
Acknowledgements The authors would like to thank the financial support from the Centro de Tecnologia Mineral (CETEM/ MCT) and from the Conselho Nacional de Desenvolvimento Cientıfico e Tecnol ogico (CNPq) of Brazil for this investigation. The authors are also gratefulto Serrana, SA for providing the samples for this study.
References Asomah, A.K., Napier-Munn, T.J., 1997. An empirical model of hydrocyclones incorporating angle of cyclone inclination. Minerals Engineering 10 (3), 339–347. Barrientos, A., Concha, F., 1992. Phenomenological model of classification in conventional hydrocyclones. In: Kawatra, S.K. (Ed.), Comminution––Theory and Practice. SME Inc., Littleton, pp. 287– 305. Batchelor, G.K., 1973. An introduction to fluid dynamics. Cambridge University Press, Cambridge. Bird, R.B., Stewart, W.E., Lightfoot, E.N., 1960. Transport Phenomena. John Wiley & Sons. Bloor, M.I.G., Ingham, D.B., Laverack, S.D., 1980. An analysis of boundary layer effects in a hydrocyclone. In: First International Conference on Hydrocyclone, Cambridge, UK, 1980, p. 49. Bradley, D., 1965. The Hydrocyclone. Pergamon Press, New York.
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Bradley, D., Pulling, D.J., 1959. Flow patterns in the hydraulic cyclone and their interpretation in terms of performance. Transaction of the Institution of Chemical Engineers 37, 34–45. Brookes, G.F., Miles, N.J., Clayton, J.S., 1984. Hydrocyclone performance related to velocity parameters. In: Proceedings of the Second International Conference of Hydrocyclones, Bath, 1984, paper C1, pp. 67–82. Coelho, M.A.Z., Medronho, R.A., 2001. A model for performance prediction of hydrocyclones. Chemical Engineering Journal 84, 7–14. Dyakowski, T., Hornung, G., Williams, R.A., 1994. Simulation of non-Newtonian flow in a hydrocyclone. Transaction of the Institution of Chemical Engineers, part A 72, 513–520. He, Y.B., Laskowski, J.S., Klein, B., 2001. Particle movement in nonNewtonian slurries: the effect of yield stress on dense medium separation. Chemical Engineering Science 56, 2991–2998. Hogg, R.V., Ledolter, J., 1987. Engineering Statistics. Macmillan Publishing Company. Hsieh, K.T., Rajamani, R.K., 1991. Mathematical model of the hydrocyclone based on physics of fluid flow. AIChE Journal 37 (5), 735–746. Kawatra, S.K., Bakshi, A.K., Rusesky, M.T., 1996. Effect of viscosity on the cut (d50) size of hydrocyclone classifiers. Minerals Engineering 8 (8), 881–891. Kellsal, D.F., 1952. A study of the motion of solid particles in a hydraulic cyclone. Transaction of the Institution of Chemical Engineers 30, 87–108. Laguitton, D., 1985. MATBAL––computer program for material balance data adjustment. In: The SPOC Manual, 1984, CANMET, Chapter 3.2, 61 p. Lilge, E.O., 1962. Hydrocyclone fundamentals. Bulletin of the Institution of the Mining and Metallurgy 71, 285–337.
Lynch, A.J., Rao, T.C., 1975. Modelling and scale-up of hydrocyclone classifiers. In: Proceedings 11th International Mineral Processing Congress, 1975, Paper 9, Cagliari. Nageswararao, K., 1978. Further developments in the modelling and scale-up of industrial hydrocyclones. Ph.D. thesis. University of Queensland, 1978. Napier-Munn, T.J., Morrel, S., Morrison, R.D., Kojovic, T., 1996. Mineral comminution circuits: their operation and optimization, vol. 2. JKRMC Monograph Series. Plitt, L.R., 1976. A mathematical model of the hydrocyclone classifier. CIM Bulletin 69 (776), 114–123. Possa, M.V., 2000. Efeitos da viscosidade no processo de deslamagem em microciclones em polpa n~ao-Newtoniana de rocha fosfatica. D.Sc. thesis (in Portuguese). Universidade de S~ao Paulo, S~ao Paulo, Brazil, 154 p. Rajamani, R.K., Devulapalli, B., 1994. Hydrodynamic modeling of swirling flow and particle classification in large-scale hydrocyclones. KONA Powder and Particle 12, 95–104. Rietema, K., 1961. Performance and design of hydrocyclones. Chemical Engineering Science 15, 298–325. Shi, F.N., Napier-Munn, T.J., 1996. A model of slurry rheology. International Journal of Mineral Processing 47, 103–123. Steinour, H.H., 1944. Rate of sedimentation (in 3 parts). Industrial and Engineering Chemistry 36, 618–624, 840–847, 901–907. Upadrashta, K.R., Ketcham, V.J., Miller, J.D., 1987. Tangential velocity profile for pseudoplastic power-law fluids in the hydrocyclone––a theoretical derivation. International Journal of Mineral Processing 20, 309–318. Vallebuona, G., Casali, A., Ferrara, G., Leal, O., Bevilacqua, P., 1995. Modelling for small diameter hydrocyclones. Minerals Engineering 7 (3), 321–327.
Modeling classification in small-diameter hydrocyclones under variable rheological conditions L.M. Tavares a
a,*
, L.L.G. Souza b, J.R.B. Lima b, M.V. Possa
c
Department of Metallurgical and Materials Engineering, COPPE, Universidade Federal do Rio de Janeiro, UFRJ, Cx. Postal 68505, CEP 21945-970, Rio de Janeiro, RJ, Brazil b Department of Mining Engineering, Universidade de S~ ao Paulo, S~ ao Paulo, Brazil c Centro de Tecnologia Mineral, CETEM/MCT, Rio de Janeiro, RJ, Brazil Received 12 December 2001; accepted 20 May 2002
Abstract Although several mathematical models describing hydrocyclone performance exist in the literature, only a few suitably describe classification of non-Newtonian slurries in small-diameter hydrocyclones (smaller than 75 mm of diameter). In order to develop a mathematical model accounting for the effects of slurry rheology on classification, experiments have been conducted with phosphate ore slurries of variable solids concentration and chemical environment, covering a wide range of rheological conditions, both Newtonian and non-Newtonian. Semi-empirical models for estimating the hydrocyclone capacity and corrected cut size and an empirical model of water split have been proposed and/or modified from the literature in order to describe classification under such extreme rheological conditions. Of particular interest is the application of a model developed by the authors and based on the residence time theory, that is used to predict the corrected cutpoint. Experimental results demonstrated that, under the conditions studied, hydrocyclone capacity and sharpness of separation were not affected by slurry rheology. It was also found that the plastic viscosity is the most significant parameter for modeling the corrected cut size in 50 and 25 mm diameter hydrocyclones. 2002 Elsevier Science Ltd. All rights reserved. Keywords: Classification; Hydrocyclone; Modeling; Rheology; Viscosity
1. Introduction Size reduction of weathered ores, such as phosphates from Brazil, generate large amounts of slimes that must be removed in order to allow selective flotation. This desliming operation, often carried out in small-diameter hydrocyclones, poses an important technical challenge, given the high viscosity and non-Newtonian rheology often associated with these slurries. Technical information on the classification of nonNewtonian slurries in small-diameter (smaller than 75 mm) hydrocyclones is scarce in the literature. Important advances have been made in the last few decades in the development of phenomenological models of the hydrocyclone under Newtonian flow conditions (Hsieh and Rajamani, 1991; Rajamani and Devulapalli, 1994; Barrientos and Concha, 1992). However, the application of * Corresponding author. Tel.: +55-21-2562-8538; fax: +55-21-22906626. E-mail address: [email protected] (L.M. Tavares).
such approach to classification in non-Newtonian fluids is even a more formidable task, given the limited knowledge yet available of the behavior of particles in non-Newtonian fluids (He et al., 2001) and of the flow of such fluids in the hydrocyclone (Upadrashta et al., 1987). While phenomenological models are not yet widely available for robust process optimization, empirical and semi-empirical models are the best alternative within reach for this task. Although a large number of papers deal with the empirical or semi-empirical modeling of hydrocyclones, most of these models were developed over quite a limited range of hydrocyclone diameters (100–380 mm) (Lynch and Rao, 1975; Plitt, 1976; Nageswararao, 1978), and their applicability to smalldiameter hydrocyclones has not yet been verified (Napier-Munn et al., 1996). Modeling studies of smalldiameter hydrocyclones are fairly scarce (Bradley and Pulling, 1959; Brookes et al., 1984; Vallebuona et al., 1995), and correspond to new equations or to modification of those already used for larger diameter hydrocyclones.
0892-6875/02/$ - see front matter 2002 Elsevier Science Ltd. All rights reserved. PII: S 0 8 9 2 - 6 8 7 5 ( 0 2 ) 0 0 0 8 5 - 7
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Nomenclature a
consistency index of the fluid in Ostwald and Waele model (–) b slurry type flow behavior index (¼1 for Newtonian, <1 for pseudoplastic; >1 for dilatant) (–) Cv volume fraction of solids in the hydrocyclone feed (–) d particle size (m) Dc cylindrical section diameter of the hydrocyclone (m) Di inlet diameter of the hydrocyclone (m) Do vortex finder diameter of the hydrocyclone (m) Du apex diameter of the hydrocyclone (m) d50c corrected cutpoint (m) E partition coefficient (–) E1 , E2 , E3 model parameters (–) f1 , f2 dimensionless numbers that describe the effect of geometry in cyclone models (–) g acceleration due to gravity (m/s2 ) K1 , K2 , K3 scale-up parameter in hydrocyclone models (–) Lc vortex finder clearance (m) n flow pattern constant (–) P inlet pressure (Pa) Q slurry flowrate in hydrocyclone feed (m3 /h) r radial distance inside the hydrocyclone (m)
Viscosity, which represents the fluid’s resistance to shear, plays a significant role in the classification using hydrocyclones. A variation in slurry viscosity alters the particle settling velocities and the slurry velocities within the hydrocyclone, directly affecting its performance. Although several mathematical models for cut size developed in the past have viscosity terms, most of them were developed for dilute suspensions (Kawatra et al., 1996). The classical models of Lynch and Rao (1975) and Plitt (1976) do not have an explicit viscosity term, accounting for its effect rather indirectly, usually through a term that depends on solids concentration. However, it is well known that solids concentration is not the only parameter that affects slurry rheology. Temperature, size distribution of the solids, and chemical environment can also alter rheology (Shi and Napier-Munn, 1996). More recent models overcome this limitation by using slurry viscosities at high shear rates, measured using online viscometers (Kawatra et al., 1996; Asomah and Napier-Munn, 1997). For highly non-Newtonian slurries the slurry viscosity (apparent or plastic) varies significantly with shear rate, which in turn varies with the hydrocyclone geometry and flowrate. Recognizing that the major limitation of hydrocyclone models is that they are not generally amenable to
r50 Rf vc vi vt
Greeks a b k DP c q qp qs g gp h s so
dimensionless radial distance corresponding to the envelope of zero axial velocity (–) water split to the underflow (–) mean fluid velocity in the cylindrical part of the hydrocyclone (m/s) inlet velocity of the fluid in the hydrocyclone (m/s) tangential velocity of the fluid in the hydrocyclone (m/s)
sharpness index of the partition curve (–) friction factor (–) velocity reduction parameter (–) pressure drop in the hydrocyclone (Pa) shear rate (s1 ) fluid density (kg/m3 ) slurry density (kg/m3 ) solids density (kg/m3 ) viscosity of a Newtonian fluid (Pa s) plastic viscosity of a non-Newtonian fluid (Pa s) included angle of the conical section of the hydrocyclone (–) shear stress in the fluid (Pa) critical shear stress of the fluid (Pa)
extrapolation, this paper describes work conducted to investigate classification of a feed material containing a large proportion of ultrafines in small-diameter hydrocyclones under widely varying rheological conditions, achieved through independent control of solids concentration and chemical environment.
2. Experimental 2.1. Test material A sample of phosphate ore (with a specific gravity of 3430 kg/m3 ) from the feed to the desliming circuit at Serrana, SA (Araxa, Brazil) was collected for the experiments. The rock is of magmatic origin and is part of an alkaline–carbonatite complex. Particle size distribution of the sample was measured by laser scattering using a Malvern Mastersizer-Microplus and is shown in Fig. 1. The very large proportion of ultrafine material (over 40% <1 lm) demonstrates the severe degree of weathering of the ore. Detailed mineralogical studies (Possa, 2000) indicated that the main mineral constituents (by weight) were goethite (43%), apatite (17%), gorceixite (16%), barite (8%), vermiculite (5%), quartz
L.M. Tavares et al. / Minerals Engineering 15 (2002) 613–622
Fig. 1. Size distribution of the feed determined by laser scattering.
(4%) and smaller proportions of anatase, hollandite and pyrochlore. Slurries containing different solids concentrations (15%, 25% and 35% by weight) and chemical environment were prepared. Slurry rheology was controlled either through the addition of a dispersant––polycarboxylic sodium salt (PSA) from BASF––or thickening agents (carboximethylcellulose or glycerin). A total of seven different slurries were prepared (see Table 1), with widely variable rheological characteristics. Their apparent viscosities at low shear rates (in the order of 75 s1 ) have been measured with a Brookfield viscometer. Further, their rheological behavior was determined at 30 C with variable shear rates from 10 to 4000 s1 using a rotational rheometer (Haake Rotovisco model RS 100) equipped with a cylinder cup and bob arrangement, and a summary is given in Table 1. 2.2. Test procedure Experiments were conducted in a closed-circuit test rig fitted with a by-pass line. KREBS hydrocyclones with diameters of 25 and 50 mm and inlet pressures of 170, 240, 310 and 450 kPa were used. Due to pumping limitations, experiments were not conducted at the
615
highest pressure (450 kPa) with the 50 mm hydrocyclone. In addition, experiments were not conducted at the low feed pressure (170 kPa) with the slurries thickened with carboxymethylcellulose (CMC) and glycerin, due to roping. Geometry was maintained constant for each hydrocyclone in the tests, so that ratios of Di =Dc , Do =Dc , Du =Dc were 0.28, 0.25 and 0.19 for the 25 mm hydrocyclone and 0.24, 0.37 and 0.37 for the 50 mm hydrocyclone. The internal conical angle of both hydrocyclones was 9. All test runs were carried out in random order to avoid potential sources of experimental bias. A total of 41 experiments were conducted and no experimental results were discarded. The solids content in the slurries were controlled within 2% of the set values (Table 1) while the temperature was maintained between 25 and 35 C during the tests. Samples at each experimental condition studied (feed pressure, solids concentration, reagent concentration and hydrocyclone diameter) were collected simultaneously from the underflow and overflow discharges. Whenever sampling was not being carried out the products were continuously recombined and returned to the slurry tank. Samples collected were weighed, filtered and dried. In order to guarantee an accurate measurement and prevent particle agglomeration, samples for size analysis in the Malvern Mastersizer were tested prior to filtration and drying, through dilution in water and dispersion with PSA. The chemical composition of samples from the feed and the underflow streams was also determined by X-ray fluorescence. The resultant test data (size analyses and water and solids weights) were reconciled using the method of Lagrange multipliers with the aid of the computer software MATBAL (Laguitton, 1985) and mass-balanced. Reconstituted feeds for different tests were compared among each other and to the measured size distribution (Fig. 1), indicating no systematic errors in the experiments. Partition curves were then prepared from the reconciled size analyses and slurry flowrates. Parameters of the mathematical models used to describe hydrocyclone capacity, water split and cutpoint have been fitted by least-squares to experimental data using a quasi-Newton optimization technique available
Table 1 Summary of the characteristics of the slurries used in the hydrocyclone experiments % Solids (w/w)
% Solids (v/v)
Additive (weight of additive per ton of dry ore)
Brookfield viscosity (mPa s)
Rheological model (Eq. (10))
15 25 35 15 15 25 35 35
5.0 9.0 13.4 5.0 5.0 9.0 13.4 13.4
– – – 27 kg of CMC 4900 kg of glycerin 1.0 kg of PSA 0.95 kg of PSA 2.6 kg of PSA
23–24 108–140 232–240 114–120 100–108 37–40 112–130 20–24
s ¼ 0:32 þ 0:010c0:80 s ¼ 1:40 þ 0:026c0:78 s ¼ 5:80 þ 0:084c0:69 s ¼ 0:48 þ 0:52c0:58 s ¼ 1:45 þ 0:040c0:96 s ¼ 0:044 þ 0:0027c0:96 s ¼ 2:30 þ 0:021c0:80 s ¼ 0:10 þ 0:0060c0:85
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in the software STATISTICA (Statsoft, Inc.). The significance of individual regression coefficients has been verified with Student’s t-tests (Hogg and Ledolter, 1987). If the p-value is less than or equal to the chosen significance level of 0.05, the hypothesis that they are equal to zero was rejected. Therefore, variables whose parameters presented p-values above 0.05 were deleted from the model. 3. Model development, results and discussion 3.1. Capacity A relationship between the pressure drop in the hydrocyclone and the feed slurry flowrate can be derived considering that the friction factor b for the turbulent flow from fluid mechanics (Batchelor, 1973) is given by b¼
DP qv2c =2
ð1Þ
The mean velocity of the fluid in the cylindrical section of the hydrocyclone (vc ) is vc ¼
Q pD2c =4
ð2Þ
Substituting Eq. (2) in (1), replacing the fluid density q by the slurry density qp , and replacing the friction factor by empirical factors related to the material type and hydrocyclone geometry, gives !0:5 DP 2 f1 ðgeometryÞ ð3Þ Q ¼ K1 Dc qp where K1 is the scale-up parameter that must be calibrated with experimental data. This, along with the function of geometry f1 , represent the effect of the friction factor b. The influence of the hydrocyclone geometry can be taken into account with a dimensionless group proposed by Nageswararao (1978) 0:45 0:68 0:20 Di Do Lc f1 ðgeometryÞ ¼ h0:10 ð4Þ Dc Dc Dc Using data from all 41 tests, covering a range of slurry flowrates from 1 to about 5 m3 /h, the scale-up parameter K1 was determined from the least-squares best-fit. A comparison between measured and calculated feed capacities is given in Fig. 2. The good agreement demonstrates that the model can appropriately describe the experimental data. It has been suggested (Bloor et al., 1980) that b––and therefore K1 ––may also be influenced by slurry rheology. An attempt has been made to include a term corresponding to slurry viscosity raised to a constant in Eq. (3). The parameter, however, was not found to be sta-
Fig. 2. Correspondence between measured and calculated flowrates in the hydrocyclones. The least-squares best-fit parameter K1 is 445.85 and the coefficient of correlation is 0.975.
tistically significant. Indeed, other authors (Coelho and Medronho, 2001) deduced from theory that feed flowrate varies only with the 0.05 power of fluid viscosity. Among some of the more widely used models found in the literature (Lynch and Rao, 1975; Plitt, 1976; Nageswararao, 1978; Vallebuona et al., 1995; Asomah and Napier-Munn, 1997) the model given by Eqs. (3) and (4) was the one that best combined quality of fit to the data and simplicity. It requires fitting only one parameter to experimental data. Further, it is based on a well-known correlation from fluid mechanics derived from the conservation of energy and momentum in the equipment. 3.2. Water split Defined as the ratio between the water volumetric flowrates in the underflow and the feed streams, the water split is often the most controversial part of hydrocyclone models (Napier-Munn et al., 1996). It has been described with a variety of expressions in the literature (Lynch and Rao, 1975; Nageswararao, 1978; Asomah and Napier-Munn, 1997). Based on classification data in Krebs hydrocyclones with diameters in the range 102–381 mm, Nageswararao (1978) proposed an expression of the form for the water split, given by !E3 P E2 E1 Rf ¼ K2 Dc gðCv Þ f2 ðgeometryÞ ð5Þ qp gDc where gðCv Þ ¼ 101:82Cv =ð1 Cv Þ2 is essentially a ratio of free to hindered settling terminal velocity suggested by Steinour (1944). Although it was first proposed for spherical particles, fitting of the parameter E2 to exper-
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617
Table 2 Summary of statistics of the water split model (Eq. (5)) Variable
Estimate
Standard error
p-value
K2 E1 E2 E3
221.5 1.27 0.23 0.049
56.89 0.0631 0.0429 0.0358
0.0198 0.0000 0.0000 0.1805
imental data allows its application to non-spherical particles, such as those found in the present study. f2 describes the effect of geometry and is given by (Nageswararao, 1978) f2 ðgeometryÞ ¼
Di Dc
0:50
Do Dc
1:19
Du Dc
2:40
Lc Dc
0:22
h0:24
ð6Þ Experimental conditions covered in the experimental runs resulted in water splits varying from 0.3 to about 0.8. Model parameters (K2 , E1 , E2 and E3 ) were fitted to these data and a summary is presented in Table 2. Considering that there is over 18% probability that we are in error if we consider that E3 is statistically significant, then the pressure term was omitted from the model and the remaining parameters were estimated again. Therefore, Eq. (5) may be rewritten as Rf ¼ 199:1D1:33 c gðCv Þ
0:22
f2 ðgeometryÞ
ð7Þ
which is valid for volume fractions of solids in the slurry (Cv ) from 0.05 to 0.134. A comparison between measured and calculated results from Eq. (7) is presented in Fig. 3, which shows a very good fit. The relative independence of the water split in respect to pressure, also evident in Fig. 4, has also been found by Vallebuona et al. (1995) for small-diameter (25 and
Fig. 4. Effect of solids concentration in the water split. Experimental results from 0% solids were included for reference only, given that they were not used in fitting the model. The solid lines represent the calculation of the model (Eq. (7)).
50 mm) hydrocyclones. Fig. 4 also shows that, under the conditions studied, the water split decreases with an increase in feed solids concentration. An increase in water split with the solids concentration, as predicted by Plitt (1976) and Nageswararao (1978) for larger diameter hydrocyclones, has only been observed in the present study for low volume concentrations (Fig. 4). Using data from a 102 mm diameter hydrocyclone, Kawatra et al. (1996) showed that the water split increases as the slurry viscosity increases in the range from about 0.5 to 4 mPa s, becoming relatively constant for higher slurry viscosities. An attempt has also been made to incorporate its effect in the model by including a power term of slurry viscosity in Eq. (5). The data, however, showed no direct correlation between water split and slurry viscosity, only indirectly through the solids concentration term gðCv Þ. These results are not necessarily in contradiction with findings from Kawatra et al. (1996) given that in the present work the apparent viscosities of the slurries in the range of shear rates of interest are typically higher than 4 mPa s (Table 1). Reasons are not clear for the apparently different effects of solids concentration and slurry viscosity found in the present study and those observed for larger diameter hydrocyclones, so more in-depth studies are necessary to clarify the matter. 3.3. Partition curve
Fig. 3. Correspondence between measured and calculated water splits in the hydrocyclones.
The performance of the hydrocyclone can be fully defined by its capacity, water split to the underflow and its partition curve. Lynch and Rao (1975) proposed the following empirical model for the partition curve
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Fig. 5. Normalized partition data for the tests. Fig. 6. Vertical and tangential velocity profiles in a hydrocyclone, showing the envelope of zero vertical velocity (after Kellsal (1952)).
E ¼ Rf þ ð1 Rf Þ
expðad=d50c Þ 1 expðad=d50c Þ þ expðaÞ 2
ð8Þ
where d50c is the corrected cutpoint and a is the classification index, which gives the sharpness of separation. Data from the experiments are plotted in Fig. 5 in normalized coordinates. The fact that they appear to conform to a single normalized partition curve, as precluded by Lynch and Rao (1975), suggests that slurry rheology and pressure drop may have limited effect on the sharpness of separation parameter. Indeed, this relative independence of the reduced efficiency curve in respect to slurry viscosity has also been previously observed by Kawatra et al. (1996). The value of the sharpness of separation (a) determined by fitting Eq. (8) to the entire data set (Fig. 5) was found to be 2.4. This value was used in the simulations described later. 3.4. Corrected cutpoint d50c The corrected cutpoint ðd50c Þ is the size of particles, after discounting the effect of short-circuit flow, that have the same chance of reporting to the underflow or the overflow streams (Eq. (8)). A variety of models have been proposed to describe the effect of design and operating variables in the hydrocyclone on the corrected cutpoint. Only a fairly limited number, however, describe explicitly the relationship between the cut size and the viscosity. One phenomenological description of the effect of slurry viscosity on the cutpoint is provided by the equilibrium orbit hypothesis (Rietema, 1961; Bradley, 1965; Plitt, 1976). It states that each particle tends to be in equilibrium between the centrifugal force acting towards the wall of the hydrocyclone and the drag force from the fluid acting towards the axis. For the cut size, the equilibrium orbit overlaps with the envelope of zero vertical velocity (Fig. 6a) and so for this size, the par-
ticles have equal chances of reporting to the underflow and overflow streams. Bradley (1965) showed that, for a given hydrocyclone geometry and laminar flow, models of this kind reduced to the following form 1=2 d50c gDc ¼ K3 ð9Þ Dc Qðqs qÞ where g and q are the viscosity and density of the fluid, respectively. Dc is the hydrocyclone diameter, qs is the density of the solids and K3 is a dimensionless parameter that depends on the geometry of the hydrocyclone. The increase in cutpoint with viscosity from Eq. (9) is due to the fact that with higher viscosities the particles will tend to remain in suspension and be recovered in the overflow. An initial attempt has been made to fit the parameter K3 in Eq. (9) to the experimental data in this work, substituting the fluid viscosity by the average Brookfield viscosity of the slurry (Table 1). However, the resulting correlation obtained turns out to be very poor, with a coefficient of correlation below 0.5. This is not surprising, given that most of the slurries in the present study demonstrated time-independent non-Newtonian behavior, so that the viscosity (either apparent or plastic) depends on the shear rate (Fig. 7). A more rational approach, therefore, consists of replacing the viscosity of the fluid in Eq. (9) by the viscosity (apparent or plastic) of the slurry at the point where the axial velocity in the hydrocyclone is zero (Fig. 6). This viscosity may be calculated using the procedure described as follows. A variety of rheological responses of fluids can be described by Ostwald and Waele model (Bird et al., 1960), which is given by s ¼ so þ acb
ð10Þ
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c¼
8nkQ pD2i Dc rnþ1
619
ð17Þ
where r is the dimensionless radial distance in the hydrocyclone, r ¼ 2r=Dc . Finally, substituting Eq. (17) in (12), gives b1 8knQ ð18Þ gp ðr Þ ¼ ab pD2i Dc rnþ1
Fig. 7. Rheological behavior of mineral slurries used in the experiments (lines represents Ostwald and Waele model, Eq. (10)).
where so is the critical shear stress ð¼ 0 for Newtonian fluids), b is the flow behavior index (¼1 for Newtonian or Bingham plastic fluid, <1 for pseudoplastic fluids, >1 for dilatant fluids) and a is the consistency index. The plastic viscosity gp is given by gp ¼
ds dc
ð11Þ
Substituting Eq. (10) in (11) gives gp ¼ abcb1
ð12Þ
Inside the hydrocyclone, the shear rates vary according to the radial position. The shear rates inside the cyclone can be calculated from the tangential velocity of the fluid vt , given by dvt ð13Þ dr where r is the radial distance in the hydrocyclone from the central axis. The relationship between the tangential velocity of the fluid and the radius of the hydrocyclone may be described by Bradley (1965) (see Fig. 6b)
c¼
vt rn ¼ constant
ð14Þ
where n is a constant. At the hydrocyclone wall the tangential velocity of the fluid is vt ¼ kvi
at r ¼ Dc =2
ð15Þ
The inlet velocity vi can be expressed as a function of the feed flowrate Q and the equivalent inlet diameter Di by vi ¼
4Q pD2i
ð16Þ
Substituting Eq. (14) in (13) and differentiating with the initial condition given by Eq. (15), gives
which allows the estimation of the plastic viscosity of the slurry at any radial distance within the cyclone for which Eq. (14) provides a valid approximation of the tangential velocities. If it is assumed that the plastic viscosity offers a valid description for the effect of viscosity in the equilibrium orbit model, then by replacing Eq. (18) evaluated at the envelope of zero vertical velocity (at a radial distance of r50 ) in Eq. (9), gives " b1 #12 ab 8nk 4b b2 d50c ¼ K3 Dc Q ð19Þ nþ1 qs q pD2i r50 Most of the constants in Eq. (19) can be estimated using data from the literature. The parameter n typically varies between 0.6 and 0.9, with the value 0.8––suggested by Bradley (1965)––being used in this work. The parameter r50 , which represents the locus of null vertical velocity, is suggested to be typically about 0.38 (Bradley, 1965). Dyakowski et al. (1994) observed that it is relatively independent of the rheological behavior of the fluid. The parameter k, which represents a correction in the inlet velocity when the slurry enters the hydrocyclone, has been suggested to depend only on the hydrocyclone geometry (Lilge, 1962). 1:13 Di k ¼ 4:5 ð20Þ Dc The parameters a and b describing the rheological response of the slurry (Eq. (10)) contained exactly in the locus of null vertical velocity of the hydrocyclone must be known in order to apply Eq. (19). In practice, however, these can only be estimated (offline using a rheometer) for the feed slurry. The validity of this approximation is discussed later. The least-squares bestfit parameters of Eq. (10) for the various feed slurries tested in the present study are summarized in Table 1. Fig. 7 also shows that the slurries used in the present work demonstrated widely varying rheological responses, several exhibiting a strong pseudoplastic behavior with yield stress. While the model for the corrected cut size, given by Eqs. (19) and (20), describes explicitly the effects of slurry flowrate, solid and fluid densities, hydrocyclone diameter and pulp rheology, the influence of the hydrocyclone geometry must be determined from calibration of parameter K3 in the model with experimental
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Fig. 8. Correspondence between measured and calculated corrected cut-size for both hydrocyclones.
data. Fitting the data from classification in the 25 and 50 mm hydrocyclones gave values for the parameter K3 of 0.0589 for the 25 mm hydrocyclone (with a coefficient of correlation of 0.972) and 0.0451 for the 50 mm hydrocyclone (with a coefficient of correlation of 0.911). Fig. 8 illustrates the very good correspondence of the model to the data. It is important to observe that, as the rheological behavior of the slurry was taken into consideration, there was no need to include a term accounting for the solids concentration of the slurry. In general, the most commonly used descriptor of fluid viscosity is the apparent viscosity, defined as the ratio of shear stress and shear strain. Therefore, an expression analogous to Eq. (19) has been derived using this definition, instead of that of plastic viscosity. However, fitting the resulting model to the data yields coefficients of correlation of 0.948 and 0.903 for the 25 and 50 mm diameter hydrocyclones, respectively. This, justifies the use of the plastic viscosity in describing the effect of slurry rheology in the classification in hydrocyclones. A comparison between experimental and modeling results is also shown in Fig. 9, where the effects of pressure drop and pulp rheology on the cut size are investigated. It shows the good predictive capabilities of the model over a very wide range of rheological conditions. The significant effect of hydrocyclone diameter (also shown in Fig. 9) is partially explained by the effective shear rates, which were estimated to be in the order of 400–600 s1 for the 50 mm hydrocyclone and between about 1500 and 3000 s1 for the 25 mm hydrocyclone. The predictive capabilities of the model are demonstrated by comparing the measured solids concentration
Fig. 9. Variation of corrected cut size with pressure drop for classification of slurries at 15% and 35% of solids (in weight) in the 25 mm diameter hydrocyclone. Data for 35% of solids slurry with the 50 mm diameter hydrocyclone is also included for comparison.
in the underflow to that calculated from the solids and water balance. The excellent agreement in Fig. 10 demonstrates the validity of the model equations. One key assumption in the model is regarding the rheological response of the fluid in the locus of zero vertical velocity, which is measured using the feed material. At least two situations can be identified when this assumption should be valid. This is the case of very dilute slurries, where the slurry rheology is dominated by the rheological response of the liquid. This approach should also be valid whenever the feed contains a significant proportion of fine particles. This is because
Fig. 10. Correlation between measured and calculated solids concentration in the underflow of the hydrocyclone.
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these particles have a more marked effect on the rheological response of the fluid (if compared to coarser ones that also may be present in the feed) and that they remain in suspension throughout the entire classification process, being separated solely depending on the water split. Although it is difficult to determine a priori what the minimum proportion of slimes required to validate this assumption is, this approach was certainly valid for the material studied in the present investigation. Evidence of this is the good agreement between model and experiments (Fig. 8) obtained by fitting a single parameter. While the limit of validity may not be clearly stated, the analysis of the partition data from all 41 tests demonstrated that at least all material smaller than 2 lm, which corresponded to more than 50% of the material, remained in suspension throughout all the tests. Another difficulty associated with the application of the present model is regarding the fact that it requires measuring the rheological curve of the slurry, which can only be carried out offline. However, this may be partially overcome by existing mathematical models, which allow the prediction of the effect of several variables, including size distribution and solids concentration in the rheological behavior of the slurry (Shi and NapierMunn, 1996). Finally, the assumption that an average value could be used to describe the density of the solids (qs ) in Eq. (19), so that no significant density segregation occurred during the experiments, was also investigated. If density segregation is to occur, the heavier minerals, such as barite (BaSO4 ) and goethite (Fe2 O3 H2 O), would more likely concentrate in the underflow stream, whereas lighter minerals, such as vermiculite (Mg3 (Al,Si)4 O10 (OH)2 4H2 O) and quartz (SiO2 ), would be more prone to report to the overflow stream. This was verified by comparing the Fe2 O3 , BaO and SiO2 contents in the feed and in the underflow streams from selected experiments. Whereas a marginally higher grade of BaO was found in the underflow stream (6.2–8.0%) in comparison to the feed (5.9%), an opposite trend was found for Fe2 O3 , where the underflow concentrations (34.1–40.4%) were either marginally lower or nearly equal to that of the feed (40.3%). No distinct trend has been found for SiO2 , which varied from 5.8% and 7.1% in the underflow stream, comparing to the 6.7% content in the feed. The lack of a distinct overall trend suggests that, in spite of the fact that the density of individual minerals differed by as much as 30% from that of the average feed solids, no significant density segregation occurred in the experiments.
4. Conclusions The following conclusions can be drawn from this investigation:
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• Semi-empirical models have been successfully used to describe capacity and cut size in classification using small-diameter hydrocyclones, while an empirical model has been satisfactorily used to predict the water split. • Although it has a major effect on cutpoint, no direct influence is found of slurry rheology in capacity, water split and sharpness of separation under the conditions studied. This suggests that the addition of rheology modifiers is an effective way of controlling the cutsize in hydrocyclones, having no side effects on the other performance criteria. • Water split has not been found to be significantly dependent on inlet pressure under the conditions studied. • The effect of solids concentration in hydrocyclone cutpoint can be solely explained by its effect on the slurry rheology. • The rheological response of the feed has been successfully used to describe the rheology in the locus of zero vertical velocity inside the hydrocyclone for modeling the cutpoint for the material studied. However, the validity of such approach is probably limited to the classification of feeds containing large proportions of slimes, and assumptions associated with the existence of a steady state locus of zero velocity in the hydrocyclone.
Acknowledgements The authors would like to thank the financial support from the Centro de Tecnologia Mineral (CETEM/ MCT) and from the Conselho Nacional de Desenvolvimento Cientıfico e Tecnol ogico (CNPq) of Brazil for this investigation. The authors are also gratefulto Serrana, SA for providing the samples for this study.
References Asomah, A.K., Napier-Munn, T.J., 1997. An empirical model of hydrocyclones incorporating angle of cyclone inclination. Minerals Engineering 10 (3), 339–347. Barrientos, A., Concha, F., 1992. Phenomenological model of classification in conventional hydrocyclones. In: Kawatra, S.K. (Ed.), Comminution––Theory and Practice. SME Inc., Littleton, pp. 287– 305. Batchelor, G.K., 1973. An introduction to fluid dynamics. Cambridge University Press, Cambridge. Bird, R.B., Stewart, W.E., Lightfoot, E.N., 1960. Transport Phenomena. John Wiley & Sons. Bloor, M.I.G., Ingham, D.B., Laverack, S.D., 1980. An analysis of boundary layer effects in a hydrocyclone. In: First International Conference on Hydrocyclone, Cambridge, UK, 1980, p. 49. Bradley, D., 1965. The Hydrocyclone. Pergamon Press, New York.
622
L.M. Tavares et al. / Minerals Engineering 15 (2002) 613–622
Bradley, D., Pulling, D.J., 1959. Flow patterns in the hydraulic cyclone and their interpretation in terms of performance. Transaction of the Institution of Chemical Engineers 37, 34–45. Brookes, G.F., Miles, N.J., Clayton, J.S., 1984. Hydrocyclone performance related to velocity parameters. In: Proceedings of the Second International Conference of Hydrocyclones, Bath, 1984, paper C1, pp. 67–82. Coelho, M.A.Z., Medronho, R.A., 2001. A model for performance prediction of hydrocyclones. Chemical Engineering Journal 84, 7–14. Dyakowski, T., Hornung, G., Williams, R.A., 1994. Simulation of non-Newtonian flow in a hydrocyclone. Transaction of the Institution of Chemical Engineers, part A 72, 513–520. He, Y.B., Laskowski, J.S., Klein, B., 2001. Particle movement in nonNewtonian slurries: the effect of yield stress on dense medium separation. Chemical Engineering Science 56, 2991–2998. Hogg, R.V., Ledolter, J., 1987. Engineering Statistics. Macmillan Publishing Company. Hsieh, K.T., Rajamani, R.K., 1991. Mathematical model of the hydrocyclone based on physics of fluid flow. AIChE Journal 37 (5), 735–746. Kawatra, S.K., Bakshi, A.K., Rusesky, M.T., 1996. Effect of viscosity on the cut (d50) size of hydrocyclone classifiers. Minerals Engineering 8 (8), 881–891. Kellsal, D.F., 1952. A study of the motion of solid particles in a hydraulic cyclone. Transaction of the Institution of Chemical Engineers 30, 87–108. Laguitton, D., 1985. MATBAL––computer program for material balance data adjustment. In: The SPOC Manual, 1984, CANMET, Chapter 3.2, 61 p. Lilge, E.O., 1962. Hydrocyclone fundamentals. Bulletin of the Institution of the Mining and Metallurgy 71, 285–337.
Lynch, A.J., Rao, T.C., 1975. Modelling and scale-up of hydrocyclone classifiers. In: Proceedings 11th International Mineral Processing Congress, 1975, Paper 9, Cagliari. Nageswararao, K., 1978. Further developments in the modelling and scale-up of industrial hydrocyclones. Ph.D. thesis. University of Queensland, 1978. Napier-Munn, T.J., Morrel, S., Morrison, R.D., Kojovic, T., 1996. Mineral comminution circuits: their operation and optimization, vol. 2. JKRMC Monograph Series. Plitt, L.R., 1976. A mathematical model of the hydrocyclone classifier. CIM Bulletin 69 (776), 114–123. Possa, M.V., 2000. Efeitos da viscosidade no processo de deslamagem em microciclones em polpa n~ao-Newtoniana de rocha fosfatica. D.Sc. thesis (in Portuguese). Universidade de S~ao Paulo, S~ao Paulo, Brazil, 154 p. Rajamani, R.K., Devulapalli, B., 1994. Hydrodynamic modeling of swirling flow and particle classification in large-scale hydrocyclones. KONA Powder and Particle 12, 95–104. Rietema, K., 1961. Performance and design of hydrocyclones. Chemical Engineering Science 15, 298–325. Shi, F.N., Napier-Munn, T.J., 1996. A model of slurry rheology. International Journal of Mineral Processing 47, 103–123. Steinour, H.H., 1944. Rate of sedimentation (in 3 parts). Industrial and Engineering Chemistry 36, 618–624, 840–847, 901–907. Upadrashta, K.R., Ketcham, V.J., Miller, J.D., 1987. Tangential velocity profile for pseudoplastic power-law fluids in the hydrocyclone––a theoretical derivation. International Journal of Mineral Processing 20, 309–318. Vallebuona, G., Casali, A., Ferrara, G., Leal, O., Bevilacqua, P., 1995. Modelling for small diameter hydrocyclones. Minerals Engineering 7 (3), 321–327.