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A Model for Performance Prediction of Hydro Cyclones

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Chemical Engineering Journal 84 (2001) 7–14 A model for performance prediction of hydrocyclones M.A.Z. Coelho a , R.A. Medronho b,∗ a Biochemical Engineering Department, School of Chemistry, Universidade Federal do Rio de Janeiro, Centro de Tecnologia, Bloco E, Ilha do Fundão, 21949-900 Rio de Janeiro-RJ, Brazil b Chemical Engineering Department, School of Chemistry, Universidade Federal do Rio de Janeiro, Centro de Tecnologia, Bloco E, Ilha do Fundão, 21949-900 Rio de Janeiro-RJ, Brazil Rece
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  Chemical Engineering Journal 84 (2001) 7–14 A model for performance prediction of hydrocyclones M.A.Z. Coelho a , R.A. Medronho b , ∗ a  Biochemical Engineering Department, School of Chemistry, Universidade Federal do Rio de Janeiro,Centro de Tecnologia, Bloco E, Ilha do Fundão, 21949-900 Rio de Janeiro-RJ, Brazil b Chemical Engineering Department, School of Chemistry, Universidade Federal do Rio de Janeiro,Centro de Tecnologia, Bloco E, Ilha do Fundão, 21949-900 Rio de Janeiro-RJ, Brazil Received 8 August 1999; received in revised form 4 September 2000; accepted 13 September 2000 Abstract The equilibrium orbit theory and the residence time theory predict that the product between the Stokes number and the Euler number( Stk  50  Eu ) should be constant for a family of geometrically similar hydrocyclones. It has been already shown that Stk  50  Eu is, in reality,a function of feed concentration and water flow ratio. For different families, this product is also a function of the cyclone geometricalproportions. In this work, data obtained with seven hydrocyclones was used to generate a model based on dimensionless groups capable topredict performance of hydrocyclones. Unlike other models that can be found in the literature, where the parameters have to be adjustedfor each data set, the proposed model was able to reproduce data from the classical works of Rietema, Bradley and Kelsall. The assumptionof invariance of the reduced grade efficiency curve adopted in this work seems to be a good approximation for performance prediction of hydrocyclones. The fish-hook effect could not be found in any of the 160 experiments, probably due to the apparatus used to determinesize distributions, which measure a dynamically equivalent diameter known as Stokes diameter. © 2001 Elsevier Science B.V. All rightsreserved. Keywords: Hydrocyclones; Model; Performance; Geometrical proportions; Efficiency 1. Introduction Two well-known theories for particle separation in hydro-cyclones are the equilibrium orbit theory [1] and the resi-dence time theory [2]. The equilibrium orbit theory assumesthat particles of a given size will reach an equilibrium radialorbit position inside the hydrocyclone where their outwardterminal settling velocity is equal to the inward radial veloc-ity of the liquid. Accordingly to this theory, larger particleswill attain a radial orbit position near the wall, where the ax-ial fluid velocity has a downward direction. These particleswill, therefore, leave the cyclone through the underflow. Theradial orbit position of smaller particles will be located nearthe centre, inside the region where the axial fluid velocity isupward. These particles will, therefore, escape through theoverflow. The cut size is defined as the particle size whoseequilibrium orbit is coincident with the locus of zero vertical ∗ Corresponding author. Present address: Gessellschaft fur Biotechno-logische Forschung (GBF), Biochemical Engineering Division BVT,Mascheroder Weg 1, 38124 Braunschweig, Germany.Tel.: + 49-531-618-1106; fax: + 49-531-618-1111.  E-mail addresses: medronho@gbf.de, medronho@ufrj.br(R.A. Medronho). velocity of the fluid. Such a particle will have equal chanceto escape the hydrocyclone either through the underflowor through the overflow. According to the residence timetheory, a particle will be separated as a function of both theposition it enters the cyclone and the available residencetime. The cut size will be the size of the particle which en-tering the equipment exactly in the centre of the inlet pipewill just reach the wall in the residence time available.Although the first patent of a hydrocyclone is about 110years old, research works are still in progress aiming atdeveloping new applications [3–7] or at understanding thecomplex flow inside this equipment. In the recent years, theadvance of computer power has allowed numerical solutionsfor the differential equations that constitute the equation of motion. The use of computational fluid dynamics (CFD)is beginning to give a better understanding of the stronglyswirling turbulent flow inside hydrocyclones. These highswirl effects induce anisotropic turbulence. As the conven-tional k – ε model assumes isotropic turbulence, it usuallygives incorrect predictions of the flow patterns. This prob-lem can be overcome either by modifying the srcinal k – ε model [8,9] or by using the Reynolds stress model [10–12].The problem still to be properly solved is the coupling be-tween the particulate phase and the liquid. Turbulent eddies, 1385-8947/01/$ – see front matter © 2001 Elsevier Science B.V. All rights reserved.PII: S1385-8947(00)00265-5  8 M.A.Z. Coelho, R.A. Medronho/Chemical Engineering Journal 84 (2001) 7–14 Nomenclature C  v feed volumetric concentration C  vu underflow volumetric concentration d  particle diameter (m) d   50 reduced cut size (m)  D c hydrocyclone diameter (m)  D i feed inlet diameter (m)  D o overflow diameter (m)  D u underflow diameter (m)  E  T total efficiency E  T reduced total efficiency  Eu Euler number G  reduced grade efficiency k  parameter in Eqs. (1)–(3)  vortex finder length (m)  L hydrocyclone length (m)  L 1 height of hydrocyclone cylindrical part (m) m parameter in Eq. (9) n parameter in Eqs. (1)–(3)  P pressure drop (Pa) Q feed volumetric flow rate (m 3 s − 1 ) Q u underflow volumetric flow rate (m 3 s − 1 )  R w water flow ratio  Re Reynolds number Stk  50 Stokes number  y cumulative particle size distribution(undersize) of feed suspension Greek symbols α parameter in Eq. (8) µ liquid viscosity (Pas) θ angle of the hydrocyclone cone ρ liquid density (kgm − 3 ) ρ s solid density (kgm − 3 )which are random in nature, and hindered settling make atheoretical solution for performance very complex. There-fore, there is still a need for empirical and semi-empiricalmodels to describe hydrocyclone performance.Several authors have used either the equilibrium orbittheory or the residence time theory to derive different equa-tions for the cut size. It has been shown that most of theseequations lead to the conclusion that the product betweenStokes number and Euler number ( Stk  50  Eu ) is constant forgeometrically similar hydrocyclones [13,14]. It has alsobeen shown that this product depends on the cyclone design,but is not affected by the relative size of the inlet orifice(  D i  /   D c ) [15]. Both theories were developed under the as-sumption that hindered settling does not occur, i.e., that thefeed suspension is diluted. As concentration reduces the ter-minal settling velocities of the particles, it can be expectedthat the product Stk  50  Eu will vary with feed concentra-tion. Some hydrocyclones have a set of different underfloworifice sizes (  D u ) in order to permit adjustments in theirperformance. It is also expected that the product Stk  50  Eu will be a function of  D u or of an operational variable greatlyaffected by D u , like the water flow ratio.Experimental work has shown that the product Stk  50  Eu is a function of water flow ratio R w and volumetric feedconcentration C  v [14,16]. Based on this work, a modelcomposed of Eqs. (1)–(3), which describes the operation of geometrically similar hydrocyclones, was proposed. Stk  50  Eu = k 1  ln  1 R w  n 1 exp (n 2 C v ) (1)  Eu = k 2 Re n 3 exp (n 4 C v ) (2) R w = k 3  D u D c  n 5  Eu n 6 (3)where k  and n are the parameters of the equations and theproduct Stk  50  Eu , the Euler number Eu , the Reynolds number  Re , and the water flow ratio R w are given by Eqs. (4)–(7),respectively. Stk  50  Eu = π(ρ s − ρ) PD c (d   50 ) 2 36 µρQ (4)  Eu = π 2  PD 4c 8 ρQ 2 (5)  Re = 4 ρQπµD c (6) R w = Q u ( 1 − C vu )Q( 1 − C v ) (7)Table 1 gives the geometrical proportions of the Rietema[2], Bradley [17], and Demco 4H hydrocyclones. The firsttwo are families of geometrically similar hydrocyclones andthe last is a commercial one produced by Demco. Table 2gives the values for the constants in Eqs. (1)–(3) for thesethree types of cyclones [14,16,18].If the feed size distribution y and the reduced grade effi-ciency curve G  are known, Eqs. (1)–(3) can fully describethe performance of a hydrocyclone [19]. G  can be given bythe Lynch and Rao [20] model or by a modification of theclassical Rosin and Rammler equation [21,44] as proposedby Plitt [22]. These two models are given by Eqs. (8) and(9), respectively. G  = exp (αd/d   50 ) − 1exp (αd/d   50 ) + exp (α) − 2(8) G  = 1 − exp  − 0 . 693  d d   50  m  (9) Table 1Geometrical proportions of the hydrocyclones used in this work Hydrocyclone D i  /   D c D o  /   D c L  /   D c L 1  /   D c   /   D c θ Rietema 0.28 0.34 5.0 – 0.40 20 ◦ Bradley 1/7 1/5 – 1/2 1/3 9 ◦ Demco 4H 0.26 0.33 3.3 0.55 0.55 18 ◦   M.A.Z. Coelho, R.A. Medronho/Chemical Engineering Journal 84 (2001) 7–14 9Table 2Parameters of Eqs. (1)–(3) for Rietema, Bradley, and Demco 4H hydro-cyclones [14,16,18]Constant HydrocycloneRietema Bradley Demco 4H k  1 0.0474 0.0550 0.0088 k  2 371.5 258 3300 k  3 1218 1 . 21 × 10 6 0.127 n 1 0.74 0.66 2.31 n 2 9.0 12.0 15.5 n 3 0.12 0.37 0.00 n 4 − 2.12 0.00 0.00 n 5 4.75 2.63 0.78 n 6 − 0.30 − 1.12 0.00 where α and m are the parameters of the equations. Table 3gives the values of these parameters for Rietema, Bradley,and Demco 4H hydrocyclones [14,16,18].The reduced total efficiency E  T and the total efficiency  E  T can be calculated based on Eqs. (10) and (11). E  T =   10 G  d y (10) E  T = E T − R w 1 − R w (11)The model described by Eqs. (1)–(3) can be used eitherin design or in performance prediction of hydrocyclonesof a given geometry. The problem posed here is that hy-drocyclones with geometrical proportions other than thosementioned in Table 1 need to be tested in order to find theconstants of the aforementioned equations.Some authors [20,22–27] have proposed empirical mod-els to describe the performance of hydrocyclones of anygeometrical proportions. Unfortunately, these empiricalequations are only capable to fit well with their srcinaldata, i.e., the equation constants must be recalculated foreach new data set [25–28].A model that could be applicable to a cyclone of anydesign would have more chances to succeed if it has a the-oretical background. In the present work, a semi-empiricalmodel based on dimensionless groups like the model givenby Eqs. (1)–(3) is proposed. In order to validate it, the valuescalculated with this model are compared with data sets thatcan be found in the classical works of Rietema [2], Bradleyand Pulling [29,45], and Kelsall [30,46]. Table 3Parameters of Eqs. (8) and (9) for the reduced grade efficiency curve[14,16,18]Hydrocyclone α m Rietema 4.23 2.45Bradley 5.1 3.12Demco 4H 5.4 3.30Table 4Size of the hydrocyclones and underflow orifices used in this work Hydrocyclone D c (mm) D u (mm)Rietema 1 22 2.0–4.6–6.0Rietema 2 44 4.0–8.2–11.5Rietema 3 88 8.5–16.0–24.7Bradley 1 15 1.0–1.5–2.0Bradley 2 30 2.0–3.0–4.0Bradley 3 60 4.0–6.0–8.0Demco 4H 122 4.5–6.0–11.0–19.0 2. Materials and methods Data obtained with seven hydrocyclones were used inthis work. Three of the cyclones obey the geometry rec-ommended by Rietema [2], three obey Bradley’s geometry[17], and the last was a Demco 4H. Two Bradley and allRietema hydrocyclones were manufactured from brass, thelargest Bradley cyclone was manufactured from PVC, andthe Demco 4H is made of steel with an internal protectionof polyurethane. Table 4 gives the diameters of these hydro-cyclones and also their underflow orifice sizes. The rangeof geometrical proportions covered in this work is given inTable 5. This range was thought to be wide enough aimingto include most of the well-known hydrocyclone designs.The test rigs obeyed all recommendations given bySvarovsky [13] and Heiskanen [31] and details can be foundelsewhere [14–16,18]. Both the underflow and the overflowwere discharged into atmospheric pressure. The pressuredrop used ranged from 70 to 280kPa, and the pressuregauge was placed close to the hydrocyclone inlet in orderto exclude the pressure losses in the inlet pipe.The suspending liquid both in the hydrocyclone exper-iments and in the particle size analysis was water with1.0gl − 1 of Calgon as a dispersing agent. The test mate-rials used were CaCO 3 ( ρ s = 2450kgm − 3 ), chalk ( ρ s = 2780kgm − 3 ), alumina hydrate ( ρ s = 2420kgm − 3 ), andbarite ( ρ s = 3750kgm − 3 ). The particle size distributionsof the feed solids are shown in Fig. 1. The feed volumetricconcentration varied from 0 to 10%. The dispersion beforeeach particle size analysis was aided by using an ultrasonicbath at 80W for 1min [32]. The particle size analyses werecarried out using the Andreasen pipette method [33] or aLadeq equipment [16]. Table 5Range of geometrical proportions of the hydrocyclones used in this work Geometrical proportion Range  D i  /   D c 0.14–0.28  D o  /   D c 0.20–0.34  D u  /   D c 0.04–0.28   /   D c 0.33–0.55  L  /   D c 3.30–6.93 θ 9 ◦ –20 ◦  10 M.A.Z. Coelho, R.A. Medronho/Chemical Engineering Journal 84 (2001) 7–14 Fig. 1. Particle size distributions of the feed materials used in this work. 3. Results and discussion An amount of 160 experiments were considered and mul-tiple linear regression was used to derive Eqs. (12)–(14). Stk  50  Eu = 0 . 12  D c D o  0 . 95  D c L −   1 . 33  ln  1 R w  0 . 79 × exp ( 12 . 0 C v ) (12)  Eu = 43 . 5 D 0 . 57c  D c D i  2 . 61  D c D 2o + D 2u  0 . 42 ×  D c L −   0 . 98  Re 0 . 12 exp ( − 0 . 51 C v ) (13) R w = 1 . 18  D c D o  5 . 97  D u D c  3 . 10  Eu − 0 . 54 (14)Eqs. (12) and (14) are totally based on dimensionlessgroups, therefore, they can be used with any coherent systemof units as, for instance, the international system of units(SI). This is not case for Eq. (13), where only the SI unitsmust be employed.Eqs. (12)–(14) can predict the performance of hydrocy-clones with geometrical proportions within the limits usedin this work and given in Table 5. For instance, the reducedcut size, the flow rate, and the underflow orifice size can becalculated based on Eqs. (4) and (12); (5), (6) and (13); (5)and (14), respectively, leading to Eqs. (15), (16) and (17). d   50 = 1 . 173 D 0 . 64c D 0 . 475o (L − ) 0 . 665  µρQ(ρ s − ρ)P   0 . 5 ×  ln  1 R w  0 . 395 exp ( 6 . 0 C v ) (15) Q = 0 . 184 D − 0 . 217c D 1 . 231i (D 2o + D 2u ) 0 . 198 × (L − ) 0 . 462 µ 0 . 0566 ρ − 0 . 528 P  0 . 472 exp ( 0 . 241 C v ) (16) Fig. 2. Comparison between the experimental values of the reduced cutsizes obtained in this work and the values predicted by Eq. (15). D u = 0 . 983 D 1 . 926o D 0 . 229o  P ρQ 2  0 . 174 R 0 . 323w (17)Eqs.(15)and(17)canbeusedwithanycoherentsystemof units as, for instance, the SI units. Eq. (16), however, must beused only in SI units. Figs. 2–4 show a comparison betweenthe experimental results of reduced cut size, flow rate andunderflow orifice diameter and the values calculated withEqs. (15)–(17), respectively. The good agreement betweenthe experimental and the calculated values shown in thesefigures means that Eqs. (15)–(17) are capable to adjust wellthe experimental data that has srcinated these equations.In order to test the capability of the model to fit differentdata sets, the experimental values reported in classical works[2,29,30,45,46] were compared with the values predicted bythe model.Rietema [2] presented the results obtained with a hydro-cyclone of 75mm diameter. In order to establish his best Fig. 3. Comparison between the experimental values of the flow ratesobtained in this work for pressure drops between 70 and 280kPa and thevalues predicted by Eq. (16).
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