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Vortex finder optimum length in hydrocyclone separation
Available online at www.sciencedirect.com
Chemical Engineering and Processing 47 (2008) 192–199
Vortex finder optimum length in hydrocyclone separation Luc´ıa Fern´andez Mart´ınez ∗ , Antonio Guti´errez Lav´ın ∗ , Manuel Mar´ıa Mahamud, Julio L. Bueno Department of Chemical Engineering and Environmental Technology. University of Oviedo, C/Juli´an Claver´ıa s/n, 33071 Oviedo, Spain Received 4 August 2006; received in revised form 27 October 2006; accepted 7 March 2007 Available online 16 March 2007
Abstract Effectiveness of hydrocyclone separations is highly dependent on their geometrical characteristics such as: chamber dimensions, aperture diameters or feed inlet geometry, for instance. Moreover, slight modifications of any of these features might severely affect separation efficiency. This work highlights the fundamental significance of the position of the vortex finder, showing how small changes in its length have meaningful effects on mass recovery and particle size distribution in overflow and underflow streams. This parameter has been scarcely considered in design studies. In order to establish the importance of the vortex finder length different and complementary methodologies were used such as mass balance, granulometric analysis and efficiency evaluation. Results obtained using theses methodologies were in agreement, showing that the highest efficient length of the vortex finder is 10% of the total length of the cyclone (0.1 Lt ). This result was found for two hydrocyclones of different sizes, giving a more consistent conclusion. © 2007 Elsevier B.V. All rights reserved. Keywords: Hydrocyclones; Particle analysis; Solid–liquid separation; Vortex finder
1. Introduction Cyclones are widely known in practice, mainly due to their use in particle separation from gaseous streams [1,2]. Applying this basic knowledge to the separation of suspended solids from liquid streams the inertial devices known as “hydrocyclones” appear. A hydrocyclone is able to separate or concentrate suspended particles from a fluid stream. Nowadays, hydrocyclones are used to separate solid–fluid streams [3] fluid–fluid streams [4,5] and gas–liquid streams [6]. Cyclones are inertial devices that allow separation or concentration due to the difference between inertial forces that induce the movement of suspended solids in a liquid bulk. Unlike conventional centrifuges, which use a similar separation principle, hydrocyclones present many advantages [7,8], such as the absence of moving parts, low energy consumption and low residence time.
∗
Corresponding authors. Tel.: +34 985103518; fax: +34 985103434. E-mail addresses: [email protected] (L.F. Mart´ınez), [email protected] (A.G. Lav´ın). 0255-2701/$ – see front matter © 2007 Elsevier B.V. All rights reserved. doi:10.1016/j.cep.2007.03.003
Generally, the feed slurry is introduced into the hydrocyclone flowing tangentially to the cylindrical upper zone, allowing a progressive separation of the suspended solids from the feed stream. The separation principle is based on inertial forces, since the circular trajectory induces a radial acceleration. If the density of solid particles is higher than the fluid density, these particles are moved towards the wall and leave the hydrocyclone preferentially through the lower exit. If the particles are lighter than the liquid, they are drawn mainly to the upper exit. 2. Separation mechanism The predicted helicoidal flow determines both the particle separation performance and the solids distribution within a hydrocyclone [9]. There exist two theories on particle separation within a hydrocyclone. The classical Eulerian one, establishes that the flow within a hydrocyclone is a balance between the radial inward drag force and the outward radial centrifugal force. On the other hand, the Lagrangian model or particle tracking theory establishes that the separation mechanism is driven by turbulent radial fluctuations and explicitly enforced force balance. In fact, both theories are complementary. For dilute systems where the volume occupied by particles may be overlooked. The Eulerian–Lagrangian model can be used
L.F. Mart´ınez et al. / Chemical Engineering and Processing 47 (2008) 192–199
193
as proposed by Ma et al. [10]. Other models are proposed by Nowakowski et al. [11], if the concentration of particles exceeds 5% by volume. The particle concentration has an influence on viscosity stresses and, if concentration rises to 10% by volume, the larger particles move towards the less shear strain zone (towards the air-core) and the smaller particles move towards the wall (greater shear strain). In the mentioned study, low solid concentration fields are measured by dual-planar laser-induced fluorescence and the high solids concentration fields by electrical impedance tomography (EIT) [11]. 3. Experimental set-up Two hydrocyclone sizes (5 and 10 cm i.d.) have been used to carry out the experiments described here and to contrast data. They have been designed according to Rietema criterion [12]. Hydrocyclone design parameters following this criterion are calculated based on semi-empirical equations and dimensionless numbers proposed by Svarovsky [13] and Castilho and Medronho [14]. These equations can also help to predict hydrocyclone performance. The geometry of the hydrocyclones used is illustrated in Figs. 1 and 2. Table 1 shows the dimensions of the prototypes used in our study. As it is shown, the inlet pipe is a cylindrical tube, of 9 cm in length for the 5 cm i.d. hydrocyclone and of 18 cm in length for the 10 cm i.d. hydrocyclone that
Fig. 2. Schematic diagram of a conventional hydrocyclone.
enters tangentially to the cylindrical hydrocyclone body which length is defined as . A scheme of the experimental apparatus is shown in Fig. 3. This consists in a tank provided with a stirrer, which maintains the particles in suspension. The closed-circuit mode of operation means that the discharged underflow and overflow are returned to the feed tank maintaining the concentration at a constant value. Water at room temperature (15 ◦ C) is fed using a 3 kW centrifugal pump P050/30T, passing through a by-pass that regulates the flow, which is controlled by a Khrone Aquaflux 090 K/D DN40 PN 40 electromagnetic flow meter. The hydrocyclones are connected in parallel. The existence of two valves allows operating with a single hydrocyclone in all experiments. 4. Materials and methods The feed sample was taken directly from the tank. Overflow and underflow samples were taken from the tank return lines. A granulometric analysis was carried out and the suspended solids (SS) concentration was measured for all samples. The mass of suspended solids is measured after drying a known volume (at 105 ◦ C for a minimum of 6 h) and weighting by difference. The suspended solids tested are composed by CaCO3 with a purity of 81%. According to Perry [15], the value
Fig. 1. Some typical views (ground and side) and a schematic 3D diagram of a hydrocyclone. Table 1 Dimensions (centimetres) of the hydrocyclones used in the experiments DC (internal)
Di
Do
Du
Lt −
θ
5 (m) 10 (cm)
1.4 2.9
1.7 3.4
4 8
1 2
21 42
11.4◦ 11.4◦ Fig. 3. Simplified flow sheet of the experimental plant.
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of density for CaCO3 is within 2 and 2.8 g/mL. The real density of the CaCO3 used in this study (2.75 g/mL) is measured by AccuPyc 1330 V2.04N. Particle size distribution (by volume percentage) is measured by a laser granulometer MALVERN MASTERSIZER QS (small volume sample dispersion unit). 5. Results and discussion 5.1. System overall balance Laser analysis provides the volume of particles of a given diameter. From this analysis D50 is defined [16] as the particle diameter for which the accumulated mass is 50%, that is, in fact, the mass median of the distribution. Only with the determination of D50 is it not possible to obtain the real efficiency of a hydrocyclone, which makes it necessary to carry out other determinations or calculations in the processing of the experimental data. Considering a two-phase system (solid–liquid system) as schematically shown in Fig. 4, being mF the mass of fluid, mS the total mass of solids and mSij the mass of a class of solids whose diameter is between dpi and dpj (dpi < dp < dpj ). Thus, the concentration can be expressed in different ways, depending on its reference basis which may be the suspension, the fluid or the dried solid. mS = yS ; mass fraction of solids (1) mT mS = Y S ; mass relation of solids (2) mL mSij = yij ; mass fraction of class ij (3) mT mSij = Y ij ; mass relation of class ij (4) mL mSij = xˆ ij ; fraction of the mass of solids of class ij (5) mS Thus, xij = mT yij mSij = mS
yij =
mS xij = yS xij mT
Fig. 4. Schematic macroscopic representation of two-phase system.
(6) (7)
Fig. 5. Differential forms. (a) Absolute mass distribution, (b) relative fraction frequency curve f(dp ), (c) cumulative fraction frequency curve F(dp ).
Being yS provided by drying and weighting and xij by particle size analysis. The characteristics of the different streams may be described by their composition, expressed not only as total solid concentration but also in terms of concentration of each class (discrete or differential) of solids. The distribution of solids may thus be expressed by functions which are shown in Fig. 5. The left hand-side image shows mass versus particle diameter (objective function in our case), the right hand-side figure being its normalised form, frequency versus particle diameter. Frequency may be expressed, according to the method of granulometric analysis as frequency in mass or volume, in surface or in number. D50 , is the median, a previously mentioned form of mean diameter. Once the concentration variables have been defined, a global balance may be applied to the separator. Following the nomenclature defined in Fig. 6, let us first define the ratio between the underflow and the feed flow as R1 , and the ratio between concen tration in the underflow and feed as R2 . W is used to define mass flow, the subscript “W” being used for the underflow stream,
Fig. 6. Separator global mass balance.
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the subscript “F” for the feed stream and “O” for the overflow stream. WO WW = 1 − R1 ; = R1 ; 0 ≥ R1 ≤ 1 (8) WF WF y R2 = SW ; ∞ ≥ R2 ≥ 1 (9) ySF A global balance of suspension may be written as: ˆF=W ˆ O+W ˆW W
(10)
and the solids balance as: ˆ SF = W ˆ SO + W ˆ SW W
(11)
ˆ F ySF = W ˆ O ySO + W ˆ W ySW W
(12)
In the same way, the partial solids balance of ij class is: ˆ ijF = W ˆ ijO + W ˆ ijW W
(13)
ˆ F ˆ O ˆ W W yijF = W yijO + W yijW
(14)
ˆ F xˆ ijF = W ˆ O xˆ ijO + W ˆ W xˆ ijW W
(15)
Thus, the total separation efficiency (E) is:
W SW WW y underflow solids = SW = feed solids W SF W F ySF y = R1 SW = R1 R2 ySF
E=
(16)
and the partial separation efficiency (Eij ) is:
WijW (ij)class of solids in the underflow = (ij)class of solids in the feed WijF yijW xijW WW yijW = = R1 = R1 R2 WF yijW yijW xijW
Eij =
(17)
5.2. Calculation of the vortex finder optimum length The insertion of the vortex finder attempts to avoid the re-entrainment of particles in the overflow stream [17]. This element avoids the so-called “short-circuit” generated at the top portion of the hydrocyclone, close to the feed inlet and the overflow upper exit (Fig. 7). Thanks to the vortex finder, the particles are induced to flow down guided by the outside wall. Increasing the vortex finder length, more time is given for particle re-entrainment in the underflow stream and this increases separation efficiency. Nonetheless, if the vortex finder tip reaches the conical zone, some coarse particles might reach the return overflow stream instead of exiting through the apex and this causes a decrease in efficiency. Optimum length depends on feed particle size and distribution and this should be determined preferably by experimentation. There does not exist a complete agreement of vortex finder length due to the fact that this depends on geometry, feed particle size and feed concentration. There are several values recorded in the bibliography that express the ratio of the length of the vortex finder to the hydrocyclone diameter (Lvortexfinder − DC ). This is
Fig. 7. Elimination of short-circuit by means of a vortex finder.
the case of Rietema [12] who gives a value of 0.4; Bradley [17] and Hass et al. [18] propose a figure of 1/3, Wang et Yu [19] accept as valid a value of 0.67; Narasimha et al. [20] use two values (0.67 and 0.5). Kraipech [21] juggle with different vortex finder insert depths but for different hydrocyclone geometries, obtaining ratios ranging from 0.28 to 0.93. In this study, different vortex finder lengths have been tested in order to understand the influence of this parameter on concentration and particle distribution, that is, their influence on efficiency; comparing their value with those proposed by other authors. For the hydrocyclone measuring 5 cm in i.d., working with different feed flows the effect of vortex finder depth on concentration has been evaluated (see Fig. 8). It can be seen that trend is similar for the different flows. However, the higher the flow, the more elevated concentration. According to the maximum efficiency point, when the ratio between the vortex finder length and total length is 0.1, the underflow concentration reaches the highest value, and thus resulting in maximum efficiency. For further understanding of separation behaviour, the above results are compared with those obtained by granulometric analysis. Fig. 9 shows the particle size distribution curves for the most efficient point, obtained when the depth of the vortex finder is 2.5 cm, (vortex finder depth-hydrocyclone length ratio
Fig. 8. Solids concentration in the underflow stream vs. fraction length of vortex finder for the 5 cm i.d. hydrocyclone.
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Fig. 9. Granulometric analysis for the most and least efficient configurations for the 5 cm i.d. hydrocyclone.
of 0.1) and for the least efficient point (vortex finder depthhydrocyclone length ratio of 0.2). It can be seen that for the optimum ratio (0.1) a higher number of coarser particles are recovered in the underflow stream. When the ratio is 0.2 the curve corresponding to the underflow stream is displaced to the right-handside and lower recovery values are achieved. A hydrocyclone is more effective in solid separation, the higher the mass and the coarser the particles recovered by the spigot. Fig. 10 shows a scheme of the most and least efficient vortex finder configurations. The changes in efficiency with the length of the vortex finder might also be explained by the pressure turbulences within a hydrocyclone (as tested by the authors in recent experiments on pressure patterns within a hydrocyclone). This
Fig. 10. Most and least efficient vortex finder lengths (left and right, respectively) in the 5 cm hydrocyclone (dimensions given in centimetres).
study shows a high turbulence zone in the region of transition between the conical and cylindrical part that leads to the least efficient point. This is in accordance with predictions of Dai et al. [22] and Yang et al. [23]. When there is no introduction of the vortex finder, some of the particles go through the upper exit without passing into the hydrocyclone. Introducing the vortex finder, this effect can be avoided. Nevertheless, if there is an excessive depth, some of the particles can reach the overflow stream instead of the underflow stream due to the swirls generated in the conical part. In our experiments, we have tried to find the optimum point. Fig. 10 shows that at the most efficient point (2.5 cm vortex finder length), the vortex finder tip is situated about one centimetre under the lowest point of the feed inlet section. This allows a re-entrainment of particles within the hydrocyclone, avoiding the “short-circuit”. Nonetheless, if the vortex finder reaches the conical section, the existing turbulence could cause coarser particles to exit through the overflow stream instead of through the underflow stream as expected. This has also been demonstrated by other further experiments carried out by the authors regarding pressure fields. By using 10 cm i.d. hydrocyclone, and following the same experimental procedure, it has also been proved that the higher the feed flow, the better the efficiency. For the intermediate flow (7.5 m3 /h), the mass balance also shows an optimum condition (highest solids amount eliminated by the underflow) when the ratio between the vortex finder length and the total hydrocyclone length is again 0.1, 5 cm being the introduced length of the vortex finder. Once the mass balance has been studied, it is necessary to test the reliability of the results obtained by laser granulometric analysis. For the same intermediate flow, Fig. 11 shows the particle size distribution for the most efficient configuration (ratio between the vortex finder length and the total length of 0.1) and for the least efficient (ratio equal to 0.2). If these two plots are compared, it can be seen that the highest percentage in volume of coarser particles in the underflow is obtained when the ratio is 0.1. Hydrocyclones should preferentially separate coarser particles. So, parameter D50 is used in this article to complement information obtained by mass balance. These results are shown in Fig. 12 when the maximum value of D50 is obtained at the vortex finder–total length ratio equal to 0.075 and 0.1 for the hydrocyclones of 10 and 5 cm i.d., respectively. These values are close to optimum ratio (0.1) obtained by mass balance. For further analysis of what occurs within a hydrocyclone, granulometric analysis is carried out when the ratio reaches 0.3 (Fig. 13). It can be seen that overflow and underflow streams overlap, so there is a division but no separation. This again evidences that the introduction of the vortex finder in the conical space does not aid the separation process. Similar results were presented by Kraipech et al. [21]. This study shows how hydrocyclone geometry and type of feed slurry determine the separation process. Due to this fact, only by modifying the vortex finder length, may the entire separation process be modified. This is an essential characteristic of the versatility of hydrocyclones for different industrial purposes.
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Fig. 12. D50 values vs. fractional length of the vortex finder for the 5 and 10 cm i.d. hydrocyclone.
Fig. 11. Granulometric analysis for the most and least efficient configurations for the 10 cm i.d. hydrocyclone.
Fig. 14 shows different configurations of the vortex finder within the 10 cm i.d. hydrocyclone in order to explain the influence of the vortex finder depth on the separation process. When the tip of the vortex finder is still placed in the cylindrical part, this allows a re-entrainment of particles within the hydrocyclone avoiding “short-circuiting”. Nonetheless, if the tip of the vortex finder is placed in the juncture between the conical and cylindrical regions, the turbulences generated within this zone avoid a correct separation and could cause coarser particles to exit through the overflow stream instead of through the underflow stream as expected. Fig. 14 also shows a longer vortex finder (15 cm) its tip being situated in the conical part of the hydrocyclone when the ratio of this length to the total hydrocyclone length is 0.3.
Fig. 13. Granulometric analysis for 0.3 vortex finder length–total length ratio for the 10 cm i.d. hydrocyclone.
For a more in depth approach to the reliability of the results obtained, and taking into account the mathematical model previously explained, complementary analysis of the data obtained is carried out based on hydrocyclone efficiency. Efficiency is calculated according to the Eq. (16); using the parameters R1 and R2 defined in Eqs. (8) and (9), respectively, the underflow concentration and a constant value of feed con-
Fig. 14. Three vortex finder lengths in the 10 cm hydrocyclone (dimensions given in centimetres).
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Table 2 Calculus of efficiency of 10 cm diameter hydrocyclone (R1 = 0.098; CF = 0.8 g/L) Lvortex finder /Ltotal 0 0.07 0.1 0.2 0.3
Underflow concentration (g/L)
R2
3.03 3.13 3.89 1.32 2.32
3.79 3.91 4.86 1.65 2.90
Appendix A. Nomenclature
Efficiency (E) 0.37 0.38 0.48 0.16 0.28
centration (0.8 g/L). These values together with length ratios are shown in Table 2. As proven by the previous analysis, the optimum lengths ratio corresponds to 0.1 with efficiencies close to 50%. Hence, an agreement in data is demonstrated using different methodologies such as mass balance, calculus of efficiency and granulometric analysis. This allows us to carry out an in-depth study of hydrocyclone behaviour, finding a value of optimum vortex finder depth between those mentioned in the bibliography. 6. Conclusions The following conclusions can be drawn from this study: • The depth at which the vortex finder tip is placed greatly influences hydrocyclone efficiency. • This optimum vortex finder length value corresponds to a vortex finder length-hydrocyclone total length ratio of 0.1. This optimum value is obtained by the analysis of both the solid mass separation in the underflow stream. Moreover, this value is in the range of those presented in the literature. • For hydrocyclones designed considering geometrical similarity (proportional dimensions), the optimum ratio of vortex finder length to total hydrocyclone length tends to be the same, regardless of the hydrocyclone size and thus differences in values of concentration and D50 . • In the absence of vortex finder, the short-circuit generated in the upper part of the hydrocyclone avoids a clear separation. In the same way, when the depth of the vortex finder tip is excessive, a substantial decrease in efficiency may be observed due to the swirls generated at the bottom of the hydrocyclone. • Less efficient conditions were found when the vortex finder depth is near the juncture between the cylindrical and the conical part, where a high turbulence may appear due to the synergy of two phenomena: the change of trajectory by entering in the conical part and the turbulence associated to the vortex finder itself. This is in accordance with predictions of Dai et al. [22] and Yang et al. [23]. Acknowledgments The authors would like to acknowledge the financial support from the “Ministerio de Educaci´on y Ciencia” (Spain) for this investigation within the Project (REN2003-09389).
feed concentration (kg/L) particle size (m) particle diameter for which half of accumulated mass achieved (m) Dc hydrocyclone cylindrical section diameter (m) Di hydrocyclone inlet diameter (m) Do hydrocyclone vortex finder diameter (m) Du hydrocyclone apex diameter (m) E total separation efficiency Eij partial separation efficiency cylindrical part length (m) Lt total length (m) Lvortex finder vortex finder clearance (m) mL mass of fluid (kg) mS total mass of solids (kg) mSij fraction of a class of solids whose diameter is dpi < dp < dpj (kg) total mass (kg) mT R1 underflow-feed mass flow ratio R2 underflow-feed concentration ratio fraction of the mass of solids of class ij xˆ ij yS mass fraction of solids yij mass fraction of class ij YS mass relation of solids Y ij mass relation of class ij ˆO W overflow mass flow (kg/s) ˆ ijO W overflow mass flow of class ij (kg/s) ˆ WF feed mass flow (kg/s) ˆ ijF feed mass flow of class ij (kg/s) W ˆW W underflow mass flow (kg/s) ˆ ijW underflow mass flow of class ij (kg/s) W
CF dp D50
References [1] M. Crawford, Air Pollution Control Theory, McGraw-Hill, NY, 1976. [2] D.F. Ciliberti, B.W. Lancaster, An improvement of the simple model for rotatory flow cyclones, AIChE J. 22 (1976) 1150–1152. [3] M.J. Doby, W. Kraipech, A.F. Nowakowski, Numerical prediction of outlet velocity patterns in solid–liquid separators, Chem. Eng. J. 111 (2005) 173–180. [4] C.A. Petty And, S.M. Parks, Flow structures within miniature hydrocyclones, Miner. Eng. 17 (2004) 615–624. [5] R. Delfos, S. Murphy, D. Stanbridge, Z. Olujic, P.J. Jansens, A design tool for optimising axial liquid–liquid hydrocyclones, Miner. Eng. 17 (2004) 721–731. [6] J.R. Parga, D.L. Cocke, Oxidation of cyanide in a hydrocyclone reactor by chlorine dioxide, Desalination 140 (2001) 289–296. [7] K. Dwari, M.N. Biswas, B.C. Meikap, Performance characteristics for particles of sand FCC and fly ash in a novel hydrocyclone, Chem. Eng. Sci. 59 (2004) 671–684. [8] K. Udaya Bhaskar, B. Govindarajan, J.P. Barnwal, K.K. Rao, B.K. Gupta, T.C. Rao, Classification studies of lead–zinc ore fines using water-injection cyclone, Int. J. Miner. Process. 77 (2005) 80–94. [9] L. Fern´andez, A.G. Lav´ın, M.M. Mahamud, J.L. Bueno, Hydrocyclones in water depuration (Spain), Tecnolog´ıa del Agua 275 (2006) 42–56. [10] L. Ma, D.B. Ingham, X. Wen, Numerical modelling of the fluid and particle penetration through small sampling cyclones, J. Aerosol. Sci. 31 (2000) 1097–1119.
L.F. Mart´ınez et al. / Chemical Engineering and Processing 47 (2008) 192–199 [11] A.F. Nowakowski, J.C. Cullivan, R.A. Williams, T. Dyakowski, Application of CFD to modelling of the flow in hydrocyclones. Is this a realizable option or still a research challenge? Miner. Eng. 17 (2004) 661– 669. [12] K. Rietema, Performance and design of hydrocyclones, Parts I–IV, Chem. Eng. Sci. 15 (1961) 298–325. [13] L. Svarovsky, Solid-Liquid Separation, fourth ed., ButterworthHeinemann, Oxford, UK, 2000. [14] I.R. Castilho, R.A. Medronho, A simple procedure for design and performance prediction of Bradley and Rietema Hydrocyclones, Miner. Eng. 13 (2000) 183–191. [15] R.H. Perry, Perry’s Chemical Engineers’ Handbook, seventh ed., McGrawHill, NY, 1999. [16] C. Puprasert, G. Hebrard, L. Lopez, Y. Aurelle, Potential of using hydrocyclone and hydrocyclone equipped with grit pot as a pre-treatment in run-off water treatment, Chem. Eng. Process. 43 (2004) 67–83.
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[17] D. Bradley, The Hydrocyclone, Pergamon Press, Oxford, 1965. [18] P.A. Haas, E.O. Nurmi, M.E. Whatley, J.R. Engel, Midget hydroclones remove micron particles, Chem. Eng. Progr. 53 (1957) 203–207. [19] B. Wang, A.B. Yu, Numerical study of particle-fluid flow in hydrocyclones with different body dimensions, Miner. Eng. 19 (2006) 1022–1033. [20] M. Narasimha, M. Brennan, P.N. Holtham, Large eddy simulation of hydrocyclone-prediction of air-core diameter and shape, Int. J. Miner. Process. 80 (2006) 1–14. [21] W. Kraipech, W. Chen, F.J. Parma, T. Dyakowski, Modelling the fish-hook effect of the flow within hydrocyclones, Int. J. Miner. Process. 66 (2002) 49–65. [22] G.Q. Dai, J.M. Li, W.M. Chen, Numerical prediction of the liquid flow within a hydrocyclone, Chem. Eng. J. 74 (1999) 217–223. [23] I.H. Yang, C.B. Shin, T.-H. Kim, S. Kim, A three-dimensional simulation of a hydrocyclone for the sludge separation in water purifying plants and comparison with experimental data, Miner. Eng. 17 (2004) 637–641.
Chemical Engineering and Processing 47 (2008) 192–199
Vortex finder optimum length in hydrocyclone separation Luc´ıa Fern´andez Mart´ınez ∗ , Antonio Guti´errez Lav´ın ∗ , Manuel Mar´ıa Mahamud, Julio L. Bueno Department of Chemical Engineering and Environmental Technology. University of Oviedo, C/Juli´an Claver´ıa s/n, 33071 Oviedo, Spain Received 4 August 2006; received in revised form 27 October 2006; accepted 7 March 2007 Available online 16 March 2007
Abstract Effectiveness of hydrocyclone separations is highly dependent on their geometrical characteristics such as: chamber dimensions, aperture diameters or feed inlet geometry, for instance. Moreover, slight modifications of any of these features might severely affect separation efficiency. This work highlights the fundamental significance of the position of the vortex finder, showing how small changes in its length have meaningful effects on mass recovery and particle size distribution in overflow and underflow streams. This parameter has been scarcely considered in design studies. In order to establish the importance of the vortex finder length different and complementary methodologies were used such as mass balance, granulometric analysis and efficiency evaluation. Results obtained using theses methodologies were in agreement, showing that the highest efficient length of the vortex finder is 10% of the total length of the cyclone (0.1 Lt ). This result was found for two hydrocyclones of different sizes, giving a more consistent conclusion. © 2007 Elsevier B.V. All rights reserved. Keywords: Hydrocyclones; Particle analysis; Solid–liquid separation; Vortex finder
1. Introduction Cyclones are widely known in practice, mainly due to their use in particle separation from gaseous streams [1,2]. Applying this basic knowledge to the separation of suspended solids from liquid streams the inertial devices known as “hydrocyclones” appear. A hydrocyclone is able to separate or concentrate suspended particles from a fluid stream. Nowadays, hydrocyclones are used to separate solid–fluid streams [3] fluid–fluid streams [4,5] and gas–liquid streams [6]. Cyclones are inertial devices that allow separation or concentration due to the difference between inertial forces that induce the movement of suspended solids in a liquid bulk. Unlike conventional centrifuges, which use a similar separation principle, hydrocyclones present many advantages [7,8], such as the absence of moving parts, low energy consumption and low residence time.
∗
Corresponding authors. Tel.: +34 985103518; fax: +34 985103434. E-mail addresses: [email protected] (L.F. Mart´ınez), [email protected] (A.G. Lav´ın). 0255-2701/$ – see front matter © 2007 Elsevier B.V. All rights reserved. doi:10.1016/j.cep.2007.03.003
Generally, the feed slurry is introduced into the hydrocyclone flowing tangentially to the cylindrical upper zone, allowing a progressive separation of the suspended solids from the feed stream. The separation principle is based on inertial forces, since the circular trajectory induces a radial acceleration. If the density of solid particles is higher than the fluid density, these particles are moved towards the wall and leave the hydrocyclone preferentially through the lower exit. If the particles are lighter than the liquid, they are drawn mainly to the upper exit. 2. Separation mechanism The predicted helicoidal flow determines both the particle separation performance and the solids distribution within a hydrocyclone [9]. There exist two theories on particle separation within a hydrocyclone. The classical Eulerian one, establishes that the flow within a hydrocyclone is a balance between the radial inward drag force and the outward radial centrifugal force. On the other hand, the Lagrangian model or particle tracking theory establishes that the separation mechanism is driven by turbulent radial fluctuations and explicitly enforced force balance. In fact, both theories are complementary. For dilute systems where the volume occupied by particles may be overlooked. The Eulerian–Lagrangian model can be used
L.F. Mart´ınez et al. / Chemical Engineering and Processing 47 (2008) 192–199
193
as proposed by Ma et al. [10]. Other models are proposed by Nowakowski et al. [11], if the concentration of particles exceeds 5% by volume. The particle concentration has an influence on viscosity stresses and, if concentration rises to 10% by volume, the larger particles move towards the less shear strain zone (towards the air-core) and the smaller particles move towards the wall (greater shear strain). In the mentioned study, low solid concentration fields are measured by dual-planar laser-induced fluorescence and the high solids concentration fields by electrical impedance tomography (EIT) [11]. 3. Experimental set-up Two hydrocyclone sizes (5 and 10 cm i.d.) have been used to carry out the experiments described here and to contrast data. They have been designed according to Rietema criterion [12]. Hydrocyclone design parameters following this criterion are calculated based on semi-empirical equations and dimensionless numbers proposed by Svarovsky [13] and Castilho and Medronho [14]. These equations can also help to predict hydrocyclone performance. The geometry of the hydrocyclones used is illustrated in Figs. 1 and 2. Table 1 shows the dimensions of the prototypes used in our study. As it is shown, the inlet pipe is a cylindrical tube, of 9 cm in length for the 5 cm i.d. hydrocyclone and of 18 cm in length for the 10 cm i.d. hydrocyclone that
Fig. 2. Schematic diagram of a conventional hydrocyclone.
enters tangentially to the cylindrical hydrocyclone body which length is defined as . A scheme of the experimental apparatus is shown in Fig. 3. This consists in a tank provided with a stirrer, which maintains the particles in suspension. The closed-circuit mode of operation means that the discharged underflow and overflow are returned to the feed tank maintaining the concentration at a constant value. Water at room temperature (15 ◦ C) is fed using a 3 kW centrifugal pump P050/30T, passing through a by-pass that regulates the flow, which is controlled by a Khrone Aquaflux 090 K/D DN40 PN 40 electromagnetic flow meter. The hydrocyclones are connected in parallel. The existence of two valves allows operating with a single hydrocyclone in all experiments. 4. Materials and methods The feed sample was taken directly from the tank. Overflow and underflow samples were taken from the tank return lines. A granulometric analysis was carried out and the suspended solids (SS) concentration was measured for all samples. The mass of suspended solids is measured after drying a known volume (at 105 ◦ C for a minimum of 6 h) and weighting by difference. The suspended solids tested are composed by CaCO3 with a purity of 81%. According to Perry [15], the value
Fig. 1. Some typical views (ground and side) and a schematic 3D diagram of a hydrocyclone. Table 1 Dimensions (centimetres) of the hydrocyclones used in the experiments DC (internal)
Di
Do
Du
Lt −
θ
5 (m) 10 (cm)
1.4 2.9
1.7 3.4
4 8
1 2
21 42
11.4◦ 11.4◦ Fig. 3. Simplified flow sheet of the experimental plant.
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of density for CaCO3 is within 2 and 2.8 g/mL. The real density of the CaCO3 used in this study (2.75 g/mL) is measured by AccuPyc 1330 V2.04N. Particle size distribution (by volume percentage) is measured by a laser granulometer MALVERN MASTERSIZER QS (small volume sample dispersion unit). 5. Results and discussion 5.1. System overall balance Laser analysis provides the volume of particles of a given diameter. From this analysis D50 is defined [16] as the particle diameter for which the accumulated mass is 50%, that is, in fact, the mass median of the distribution. Only with the determination of D50 is it not possible to obtain the real efficiency of a hydrocyclone, which makes it necessary to carry out other determinations or calculations in the processing of the experimental data. Considering a two-phase system (solid–liquid system) as schematically shown in Fig. 4, being mF the mass of fluid, mS the total mass of solids and mSij the mass of a class of solids whose diameter is between dpi and dpj (dpi < dp < dpj ). Thus, the concentration can be expressed in different ways, depending on its reference basis which may be the suspension, the fluid or the dried solid. mS = yS ; mass fraction of solids (1) mT mS = Y S ; mass relation of solids (2) mL mSij = yij ; mass fraction of class ij (3) mT mSij = Y ij ; mass relation of class ij (4) mL mSij = xˆ ij ; fraction of the mass of solids of class ij (5) mS Thus, xij = mT yij mSij = mS
yij =
mS xij = yS xij mT
Fig. 4. Schematic macroscopic representation of two-phase system.
(6) (7)
Fig. 5. Differential forms. (a) Absolute mass distribution, (b) relative fraction frequency curve f(dp ), (c) cumulative fraction frequency curve F(dp ).
Being yS provided by drying and weighting and xij by particle size analysis. The characteristics of the different streams may be described by their composition, expressed not only as total solid concentration but also in terms of concentration of each class (discrete or differential) of solids. The distribution of solids may thus be expressed by functions which are shown in Fig. 5. The left hand-side image shows mass versus particle diameter (objective function in our case), the right hand-side figure being its normalised form, frequency versus particle diameter. Frequency may be expressed, according to the method of granulometric analysis as frequency in mass or volume, in surface or in number. D50 , is the median, a previously mentioned form of mean diameter. Once the concentration variables have been defined, a global balance may be applied to the separator. Following the nomenclature defined in Fig. 6, let us first define the ratio between the underflow and the feed flow as R1 , and the ratio between concen tration in the underflow and feed as R2 . W is used to define mass flow, the subscript “W” being used for the underflow stream,
Fig. 6. Separator global mass balance.
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195
the subscript “F” for the feed stream and “O” for the overflow stream. WO WW = 1 − R1 ; = R1 ; 0 ≥ R1 ≤ 1 (8) WF WF y R2 = SW ; ∞ ≥ R2 ≥ 1 (9) ySF A global balance of suspension may be written as: ˆF=W ˆ O+W ˆW W
(10)
and the solids balance as: ˆ SF = W ˆ SO + W ˆ SW W
(11)
ˆ F ySF = W ˆ O ySO + W ˆ W ySW W
(12)
In the same way, the partial solids balance of ij class is: ˆ ijF = W ˆ ijO + W ˆ ijW W
(13)
ˆ F ˆ O ˆ W W yijF = W yijO + W yijW
(14)
ˆ F xˆ ijF = W ˆ O xˆ ijO + W ˆ W xˆ ijW W
(15)
Thus, the total separation efficiency (E) is:
W SW WW y underflow solids = SW = feed solids W SF W F ySF y = R1 SW = R1 R2 ySF
E=
(16)
and the partial separation efficiency (Eij ) is:
WijW (ij)class of solids in the underflow = (ij)class of solids in the feed WijF yijW xijW WW yijW = = R1 = R1 R2 WF yijW yijW xijW
Eij =
(17)
5.2. Calculation of the vortex finder optimum length The insertion of the vortex finder attempts to avoid the re-entrainment of particles in the overflow stream [17]. This element avoids the so-called “short-circuit” generated at the top portion of the hydrocyclone, close to the feed inlet and the overflow upper exit (Fig. 7). Thanks to the vortex finder, the particles are induced to flow down guided by the outside wall. Increasing the vortex finder length, more time is given for particle re-entrainment in the underflow stream and this increases separation efficiency. Nonetheless, if the vortex finder tip reaches the conical zone, some coarse particles might reach the return overflow stream instead of exiting through the apex and this causes a decrease in efficiency. Optimum length depends on feed particle size and distribution and this should be determined preferably by experimentation. There does not exist a complete agreement of vortex finder length due to the fact that this depends on geometry, feed particle size and feed concentration. There are several values recorded in the bibliography that express the ratio of the length of the vortex finder to the hydrocyclone diameter (Lvortexfinder − DC ). This is
Fig. 7. Elimination of short-circuit by means of a vortex finder.
the case of Rietema [12] who gives a value of 0.4; Bradley [17] and Hass et al. [18] propose a figure of 1/3, Wang et Yu [19] accept as valid a value of 0.67; Narasimha et al. [20] use two values (0.67 and 0.5). Kraipech [21] juggle with different vortex finder insert depths but for different hydrocyclone geometries, obtaining ratios ranging from 0.28 to 0.93. In this study, different vortex finder lengths have been tested in order to understand the influence of this parameter on concentration and particle distribution, that is, their influence on efficiency; comparing their value with those proposed by other authors. For the hydrocyclone measuring 5 cm in i.d., working with different feed flows the effect of vortex finder depth on concentration has been evaluated (see Fig. 8). It can be seen that trend is similar for the different flows. However, the higher the flow, the more elevated concentration. According to the maximum efficiency point, when the ratio between the vortex finder length and total length is 0.1, the underflow concentration reaches the highest value, and thus resulting in maximum efficiency. For further understanding of separation behaviour, the above results are compared with those obtained by granulometric analysis. Fig. 9 shows the particle size distribution curves for the most efficient point, obtained when the depth of the vortex finder is 2.5 cm, (vortex finder depth-hydrocyclone length ratio
Fig. 8. Solids concentration in the underflow stream vs. fraction length of vortex finder for the 5 cm i.d. hydrocyclone.
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Fig. 9. Granulometric analysis for the most and least efficient configurations for the 5 cm i.d. hydrocyclone.
of 0.1) and for the least efficient point (vortex finder depthhydrocyclone length ratio of 0.2). It can be seen that for the optimum ratio (0.1) a higher number of coarser particles are recovered in the underflow stream. When the ratio is 0.2 the curve corresponding to the underflow stream is displaced to the right-handside and lower recovery values are achieved. A hydrocyclone is more effective in solid separation, the higher the mass and the coarser the particles recovered by the spigot. Fig. 10 shows a scheme of the most and least efficient vortex finder configurations. The changes in efficiency with the length of the vortex finder might also be explained by the pressure turbulences within a hydrocyclone (as tested by the authors in recent experiments on pressure patterns within a hydrocyclone). This
Fig. 10. Most and least efficient vortex finder lengths (left and right, respectively) in the 5 cm hydrocyclone (dimensions given in centimetres).
study shows a high turbulence zone in the region of transition between the conical and cylindrical part that leads to the least efficient point. This is in accordance with predictions of Dai et al. [22] and Yang et al. [23]. When there is no introduction of the vortex finder, some of the particles go through the upper exit without passing into the hydrocyclone. Introducing the vortex finder, this effect can be avoided. Nevertheless, if there is an excessive depth, some of the particles can reach the overflow stream instead of the underflow stream due to the swirls generated in the conical part. In our experiments, we have tried to find the optimum point. Fig. 10 shows that at the most efficient point (2.5 cm vortex finder length), the vortex finder tip is situated about one centimetre under the lowest point of the feed inlet section. This allows a re-entrainment of particles within the hydrocyclone, avoiding the “short-circuit”. Nonetheless, if the vortex finder reaches the conical section, the existing turbulence could cause coarser particles to exit through the overflow stream instead of through the underflow stream as expected. This has also been demonstrated by other further experiments carried out by the authors regarding pressure fields. By using 10 cm i.d. hydrocyclone, and following the same experimental procedure, it has also been proved that the higher the feed flow, the better the efficiency. For the intermediate flow (7.5 m3 /h), the mass balance also shows an optimum condition (highest solids amount eliminated by the underflow) when the ratio between the vortex finder length and the total hydrocyclone length is again 0.1, 5 cm being the introduced length of the vortex finder. Once the mass balance has been studied, it is necessary to test the reliability of the results obtained by laser granulometric analysis. For the same intermediate flow, Fig. 11 shows the particle size distribution for the most efficient configuration (ratio between the vortex finder length and the total length of 0.1) and for the least efficient (ratio equal to 0.2). If these two plots are compared, it can be seen that the highest percentage in volume of coarser particles in the underflow is obtained when the ratio is 0.1. Hydrocyclones should preferentially separate coarser particles. So, parameter D50 is used in this article to complement information obtained by mass balance. These results are shown in Fig. 12 when the maximum value of D50 is obtained at the vortex finder–total length ratio equal to 0.075 and 0.1 for the hydrocyclones of 10 and 5 cm i.d., respectively. These values are close to optimum ratio (0.1) obtained by mass balance. For further analysis of what occurs within a hydrocyclone, granulometric analysis is carried out when the ratio reaches 0.3 (Fig. 13). It can be seen that overflow and underflow streams overlap, so there is a division but no separation. This again evidences that the introduction of the vortex finder in the conical space does not aid the separation process. Similar results were presented by Kraipech et al. [21]. This study shows how hydrocyclone geometry and type of feed slurry determine the separation process. Due to this fact, only by modifying the vortex finder length, may the entire separation process be modified. This is an essential characteristic of the versatility of hydrocyclones for different industrial purposes.
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197
Fig. 12. D50 values vs. fractional length of the vortex finder for the 5 and 10 cm i.d. hydrocyclone.
Fig. 11. Granulometric analysis for the most and least efficient configurations for the 10 cm i.d. hydrocyclone.
Fig. 14 shows different configurations of the vortex finder within the 10 cm i.d. hydrocyclone in order to explain the influence of the vortex finder depth on the separation process. When the tip of the vortex finder is still placed in the cylindrical part, this allows a re-entrainment of particles within the hydrocyclone avoiding “short-circuiting”. Nonetheless, if the tip of the vortex finder is placed in the juncture between the conical and cylindrical regions, the turbulences generated within this zone avoid a correct separation and could cause coarser particles to exit through the overflow stream instead of through the underflow stream as expected. Fig. 14 also shows a longer vortex finder (15 cm) its tip being situated in the conical part of the hydrocyclone when the ratio of this length to the total hydrocyclone length is 0.3.
Fig. 13. Granulometric analysis for 0.3 vortex finder length–total length ratio for the 10 cm i.d. hydrocyclone.
For a more in depth approach to the reliability of the results obtained, and taking into account the mathematical model previously explained, complementary analysis of the data obtained is carried out based on hydrocyclone efficiency. Efficiency is calculated according to the Eq. (16); using the parameters R1 and R2 defined in Eqs. (8) and (9), respectively, the underflow concentration and a constant value of feed con-
Fig. 14. Three vortex finder lengths in the 10 cm hydrocyclone (dimensions given in centimetres).
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Table 2 Calculus of efficiency of 10 cm diameter hydrocyclone (R1 = 0.098; CF = 0.8 g/L) Lvortex finder /Ltotal 0 0.07 0.1 0.2 0.3
Underflow concentration (g/L)
R2
3.03 3.13 3.89 1.32 2.32
3.79 3.91 4.86 1.65 2.90
Appendix A. Nomenclature
Efficiency (E) 0.37 0.38 0.48 0.16 0.28
centration (0.8 g/L). These values together with length ratios are shown in Table 2. As proven by the previous analysis, the optimum lengths ratio corresponds to 0.1 with efficiencies close to 50%. Hence, an agreement in data is demonstrated using different methodologies such as mass balance, calculus of efficiency and granulometric analysis. This allows us to carry out an in-depth study of hydrocyclone behaviour, finding a value of optimum vortex finder depth between those mentioned in the bibliography. 6. Conclusions The following conclusions can be drawn from this study: • The depth at which the vortex finder tip is placed greatly influences hydrocyclone efficiency. • This optimum vortex finder length value corresponds to a vortex finder length-hydrocyclone total length ratio of 0.1. This optimum value is obtained by the analysis of both the solid mass separation in the underflow stream. Moreover, this value is in the range of those presented in the literature. • For hydrocyclones designed considering geometrical similarity (proportional dimensions), the optimum ratio of vortex finder length to total hydrocyclone length tends to be the same, regardless of the hydrocyclone size and thus differences in values of concentration and D50 . • In the absence of vortex finder, the short-circuit generated in the upper part of the hydrocyclone avoids a clear separation. In the same way, when the depth of the vortex finder tip is excessive, a substantial decrease in efficiency may be observed due to the swirls generated at the bottom of the hydrocyclone. • Less efficient conditions were found when the vortex finder depth is near the juncture between the cylindrical and the conical part, where a high turbulence may appear due to the synergy of two phenomena: the change of trajectory by entering in the conical part and the turbulence associated to the vortex finder itself. This is in accordance with predictions of Dai et al. [22] and Yang et al. [23]. Acknowledgments The authors would like to acknowledge the financial support from the “Ministerio de Educaci´on y Ciencia” (Spain) for this investigation within the Project (REN2003-09389).
feed concentration (kg/L) particle size (m) particle diameter for which half of accumulated mass achieved (m) Dc hydrocyclone cylindrical section diameter (m) Di hydrocyclone inlet diameter (m) Do hydrocyclone vortex finder diameter (m) Du hydrocyclone apex diameter (m) E total separation efficiency Eij partial separation efficiency cylindrical part length (m) Lt total length (m) Lvortex finder vortex finder clearance (m) mL mass of fluid (kg) mS total mass of solids (kg) mSij fraction of a class of solids whose diameter is dpi < dp < dpj (kg) total mass (kg) mT R1 underflow-feed mass flow ratio R2 underflow-feed concentration ratio fraction of the mass of solids of class ij xˆ ij yS mass fraction of solids yij mass fraction of class ij YS mass relation of solids Y ij mass relation of class ij ˆO W overflow mass flow (kg/s) ˆ ijO W overflow mass flow of class ij (kg/s) ˆ WF feed mass flow (kg/s) ˆ ijF feed mass flow of class ij (kg/s) W ˆW W underflow mass flow (kg/s) ˆ ijW underflow mass flow of class ij (kg/s) W
CF dp D50
References [1] M. Crawford, Air Pollution Control Theory, McGraw-Hill, NY, 1976. [2] D.F. Ciliberti, B.W. Lancaster, An improvement of the simple model for rotatory flow cyclones, AIChE J. 22 (1976) 1150–1152. [3] M.J. Doby, W. Kraipech, A.F. Nowakowski, Numerical prediction of outlet velocity patterns in solid–liquid separators, Chem. Eng. J. 111 (2005) 173–180. [4] C.A. Petty And, S.M. Parks, Flow structures within miniature hydrocyclones, Miner. Eng. 17 (2004) 615–624. [5] R. Delfos, S. Murphy, D. Stanbridge, Z. Olujic, P.J. Jansens, A design tool for optimising axial liquid–liquid hydrocyclones, Miner. Eng. 17 (2004) 721–731. [6] J.R. Parga, D.L. Cocke, Oxidation of cyanide in a hydrocyclone reactor by chlorine dioxide, Desalination 140 (2001) 289–296. [7] K. Dwari, M.N. Biswas, B.C. Meikap, Performance characteristics for particles of sand FCC and fly ash in a novel hydrocyclone, Chem. Eng. Sci. 59 (2004) 671–684. [8] K. Udaya Bhaskar, B. Govindarajan, J.P. Barnwal, K.K. Rao, B.K. Gupta, T.C. Rao, Classification studies of lead–zinc ore fines using water-injection cyclone, Int. J. Miner. Process. 77 (2005) 80–94. [9] L. Fern´andez, A.G. Lav´ın, M.M. Mahamud, J.L. Bueno, Hydrocyclones in water depuration (Spain), Tecnolog´ıa del Agua 275 (2006) 42–56. [10] L. Ma, D.B. Ingham, X. Wen, Numerical modelling of the fluid and particle penetration through small sampling cyclones, J. Aerosol. Sci. 31 (2000) 1097–1119.
L.F. Mart´ınez et al. / Chemical Engineering and Processing 47 (2008) 192–199 [11] A.F. Nowakowski, J.C. Cullivan, R.A. Williams, T. Dyakowski, Application of CFD to modelling of the flow in hydrocyclones. Is this a realizable option or still a research challenge? Miner. Eng. 17 (2004) 661– 669. [12] K. Rietema, Performance and design of hydrocyclones, Parts I–IV, Chem. Eng. Sci. 15 (1961) 298–325. [13] L. Svarovsky, Solid-Liquid Separation, fourth ed., ButterworthHeinemann, Oxford, UK, 2000. [14] I.R. Castilho, R.A. Medronho, A simple procedure for design and performance prediction of Bradley and Rietema Hydrocyclones, Miner. Eng. 13 (2000) 183–191. [15] R.H. Perry, Perry’s Chemical Engineers’ Handbook, seventh ed., McGrawHill, NY, 1999. [16] C. Puprasert, G. Hebrard, L. Lopez, Y. Aurelle, Potential of using hydrocyclone and hydrocyclone equipped with grit pot as a pre-treatment in run-off water treatment, Chem. Eng. Process. 43 (2004) 67–83.
199
[17] D. Bradley, The Hydrocyclone, Pergamon Press, Oxford, 1965. [18] P.A. Haas, E.O. Nurmi, M.E. Whatley, J.R. Engel, Midget hydroclones remove micron particles, Chem. Eng. Progr. 53 (1957) 203–207. [19] B. Wang, A.B. Yu, Numerical study of particle-fluid flow in hydrocyclones with different body dimensions, Miner. Eng. 19 (2006) 1022–1033. [20] M. Narasimha, M. Brennan, P.N. Holtham, Large eddy simulation of hydrocyclone-prediction of air-core diameter and shape, Int. J. Miner. Process. 80 (2006) 1–14. [21] W. Kraipech, W. Chen, F.J. Parma, T. Dyakowski, Modelling the fish-hook effect of the flow within hydrocyclones, Int. J. Miner. Process. 66 (2002) 49–65. [22] G.Q. Dai, J.M. Li, W.M. Chen, Numerical prediction of the liquid flow within a hydrocyclone, Chem. Eng. J. 74 (1999) 217–223. [23] I.H. Yang, C.B. Shin, T.-H. Kim, S. Kim, A three-dimensional simulation of a hydrocyclone for the sludge separation in water purifying plants and comparison with experimental data, Miner. Eng. 17 (2004) 637–641.