TECHNICAL NOTES 3. Hydraulic Classifiers

Transcription

1 TECHNICAL NOTES 3 Hydraulc Classfers

2 3.1 Classfcaton Based on Dfferental Settlng - The Hydrocyclone General prncples of the operaton of the hydrocyclone The prncple of operaton of the hydrocyclone s based on the concept of the termnal settlng velocty of a sold partcle n a centrfugal feld. The condtons n an operatng hydrocyclone can be descrbed by reference to Fgures 3.1 and 3.2. The feed enters tangentally nto the cylndrcal secton of the hydrocyclone and follows a crculatng path wth a net nward flow of flud from the outsde to the vortex fnder on the axs. The crculatng veloctes are very hgh and these generate large centrfugal felds nsde the hydrocyclone. The centrfugal feld s usually hgh enough to create an ar core on the axs that often extends from the spgot openng at the bottom of the concal secton through the vortex fnder to the overflow at the top. In order for ths to occur the centrfugal force feld must be many tmes larger than the gravtatonal feld. Vortex fnder Overflow Inlet Typcal trajectory of a small lght partcle Ar core Typcal trajectory of a larger heaver partcle Spgot or apex Underflow Fgure 3.1 Typcal partcle trajectores n a hydrocyclone. Partcles that experence ths centrfugal feld wll tend to move outwards relatve to the carrer flud because of ther relatvely greater densty. The larger, heaver partcles wll mgrate rapdly to the outsde walls of the cylndrcal secton and wll then be forced to move downward on the nsde of the concal wall. Small, lght partcles, on the other hand wll be dragged nwards by the flud as t moves toward the vortex fnder. The drag force experenced by any partcle wll be a complex functon of the hydrodynamc condtons nsde the hydrocyclone and the shape and sze of the partcle.

3 3.1.2 The Equlbrum Orbt Hypothess The classfcaton acton of the hydrocyclone s determned by the net effect of the two competng forces that act on every partcle - the outward centrfugal force and the nward drag force. A rough gude to the effect of varous operatng varables on the performance of the devce can be establshed usng the so-called equlbrum orbt hypothess. Any partcle that experences an equlbrum between these two forces nsde the hydrocyclone wll have an equal chance to ext through ether the underflow or the overflow because they wll tend to crculate on a crcular orbt n the hydrocyclone and wll be moved Outward movement of larger partcles Smaller and lghter partcles Net nward flow of carrer flud Larger and heaver partcles Fgure 3.2 Schematc representaton of the net flow of water and the counter flow of larger partcles n the hydrocyclone. toward one or other outlet by random mpacts wth other partcles and the random eddy moton n the hghly turbulent flow feld nsde. An orbt on whch a partcle experences a balance between the centrfugal and drag forces s called an equlbrum orbt. The condtons that defne an equlbrum orbt can be defned as follows: If v r s the radal velocty of the flud at a pont n the hydrocyclone and u r the radal velocty of the partcle, the drag force s gve by equaton (3.9. Drag force 0.5C D (v r u r 2 A c (3.1 where s the flud densty, A c the cross-sectonal area of the partcle and C D s the drag coeffcent. It s not dffcult to show that, wthn the tme taken for a partcle to make a sngle orbt n the cyclone, the drag force s balanced by the centrfugal force due to the crculatng moton and the partcles move at ther termnal settlng veloctes relatve to the nward movng flud. The centrfugal force s gven by Centrfugal force v 2 # r p (3.2 where v s the tangental component of the partcle velocty vector, and r the radus of the tangental moton. # p the volume of the partcle and! s the densty of the sold. Balancng these forces 0.5C D (v r u r 2 A c v 2 # r p (3.3

4 It s usually assumed that partcles that have a 50% chance of passng to overflow wll establsh an equlbrum orbt somewhere wthn the cyclone. The poston of ths equlbrum orbt s not precsely defned although some authors clam that t s at the pont where the locus of zero vertcal velocty meets the spgot openng. The uncertanty regardng the actual poston of the equlbrum orbt of the 50% partcle s not mportant snce we are nterested only n establshng a functonal form for the relatonshp among the partcle propertes that gve t a 50% chance of leavng n ether the overflow or underflow. All partcles that have combnatons of densty and sze that produce a 50% splt n the cyclone are assumed to have equlbrum orbts at the same locaton n the cyclone and ths assumpton allows a useful correlaton to be developed for the cutpont as shown n the followng analyss. On an equlbrum orbt u r = 0 and Eq. (3.3 can be wrtten for a partcle havng a 50% chance of passng to overflow. # p A c 0.5C D. rv 2 r v 2 Equaton (3.4 defnes the so-called cutpont for the hydrocyclone. Ths s the sze of the partcle that has a 50% chance of leavng n ether the underflow or overflow. The cutpont s normally represented by the symbol d 50 and for sphercal partcles so that (3.4 # p A c 2 3 d p (3.5 d 50c 0.75 C 2 D rvr! s v 2 (3.6 For a partcular cyclone the partcles that have a 50% chance of passng to overflow wll satsfy Eq. (3.6 at a partcular value of rv 2 r. These partcles defne the mdpont of the partton curve as v 2 shown n Fgure 3.3 and Eq. (3.6 provdes a correlaton for the cut pont as a functon of partcle propertes. The key to the equlbrum orbt theory s the drag force experenced by a partcle due to the relatve moton between t and the flud. It s known that these relatve veloctes are small and the partcle Reynolds number s usually so low that t s commonly assumed that the drag force can be calculated from the formula for the slow relatve moton between a sphere and a Newtonan flud. Ths s the well-known Stokes formula. Ths approach neglects two mportant phenomena: the hghly turbulent nature of the flud nsde the hydrocyclone and the relatvely hgh concentraton of partcles. Thus the use of Stokes flow theores whch assumes that only a sngle solated partcle s present s not very realstc.

5 Four lmtng cases can be consdered: solated partcles n a lamnar flow feld (Stokes regme, solated partcles n a turbulent flow feld (Newton regme, nteractng partcles n a lamnar flow feld (Blake-Kozeny regme and nteractng partcles n a turbulent flow feld (Burke-Plummer regme. The drag coeffcent n Eq. (3.6 s a functon of the partcle sze and shape and the envronment that the partcle fnds on ts equlbrum orbt. Each of the four envronments defned above gve dfferent expressons for C D. Stokes Regme In ths regme the partcles do not nteract wth each other and are surrounded by flud n lamnar moton. The drag coeffcent s gven by equaton 3.25 C D 23 Re p 23µ f d p (v r u r (3.7 23µ f d p v r on an equlbrum orbt Substtutng n Eq. (3.6 d 50c 17.25µ f rv r d 50 v 2 d 2 50c K 1 rv r v 2 (3.8 Newtonan Regme In ths regme solated partcles are surrounded by flud n turbulent moton and from the Abraham equaton Substtutng n equaton (3.7 C D 0.28 (3.9 d 50c 0.22 rv! s d 50c K 2 rv 2 r v 2 2 r v 2 (3.10

6 The two lmtng cases for the equlbrum orbt hypothess (equatons 3.8 and 3.10 ndcate that the varaton of d 50c wth the densty dfference and wth the volumetrc feed rate should take the form d 50c KD k c (3.11 m Q n c where m and k are constants havng a values between 0.5 and 1 and n s a constant between 0 and 0.5. The lower lmt of 0.5 for m and k represents lamnar flow condtons and turbulence nsde the hydrocyclone wll gve rse to hgher values of m. and k and lower values of n. Expermental observatons have shown that the rato v r /v s ndependent of total flowrate at every pont nsde the hydrocyclone and the assumpton of turbulent flow (equaton 3.10 ndcates that d 50c should be at most a weak functon of the total flowrate whle the assumpton of lamnar flow condtons (equaton 3.8 ndcates that d 50c should be nversely proportonal to the square root of the volumetrc flowrate. Under turbulent condtons, d 50c may be expected to ncrease lnearly wth the cyclone dameter. No one set of assumptons s lkely to descrbe the operatng behavor of the hydrocyclone under all condtons. Consequently only comparatvely crude emprcal correlatons are currently avalable for the predcton of the varaton of d 50c wth hydrocyclone geometry and operatng condtons Emprcal Performance odels for Hydrocyclones The technology of commnuton s ntmately connected wth classfcaton devces through the concept of closed-crcut mllng. All commnuton operatons are not selectve n that they can potentally reduce the sze of all partcles n the unt. It s undesrable to reduce the sze of any partcle beyond the desred product sze for that unt snce that consumes addtonal energy and further down-stream processng can be adversely effected. A classfer placed at an approprate pont n the crcut can selectvely remove all of these partcles that meet the product sze crtera for the crcut and return coarse partcles back to the commnuton unt. Recovery to underflow Actual partton curve c(d p Ideal partton curve e(d p Unfortunately no classfer operates perfectly and classfers wll not dvde a populaton of partcles nto two groups separated at a defnte and partcular sze. All classfers are charactersed by a dstrbuton functon whch gves the effcency of separaton at any sze and the dstrbuton functon s more or less d50 d25 d75 d50c Partcle sze Fgure 3.3 Typcal partton curves for a hydrocyclone.

7 sharp dependng on the effcency of separaton. A typcal classfcaton curve s shown n Fgure 3.3. Ths s called the partton curve (sometmes called the Tromp curve and t shows the fracton of partcles at a partcular sze that wll be parttoned to the coarser fracton. The S shaped curve s typcal of all practcal classfers and a varety of quanttatve expressons have been used to descrbe the shape of the curve. A characterstc of vrtually all practcal classfers s the phenomenon of short crcutng. All classfers explot some physcal process to separate partcles on the bass of sze. Ths wll be the dfferental settlng velocty n a vscous flud n spral, rake and hydrocyclone classfers or the physcal sevng acton n a screenng operaton. Some partcles pass through the equpment wthout beng subjected to the physcal separaton acton. In practce t s only the short crcutng of the fne partcles to the coarser product stream that s sgnfcant. Ths shows up as a non-zero ntersecton on the partton axs at zero sze. In the hydrocyclone ths s due to the water carryng fne partcles nto the boundary layer on the outer wall of the concal secton and dschargng them wth the underflow. In other classfers such as spral, rake and the varous screenng operatons, fne partcles are physcally carred wth the large partcles nto the coarse product stream. Ths short crcut effect can easly be accounted for by reference to Fgure 3.4. Fne product If the deal classfcaton acton of the unt s descrbed by classfcaton functon e(d p and a fracton. of the feed short crcuts drectly to the coarser product then a smple mass balance gves the actual classfcaton curve as Feed Ideal classfer c(d p. (1.e(d p (3.12 e(d p s called the corrected classfcaton functon and c(d p the actual classfcaton functon. The mportant parameter that characterzes the operaton of the classfer s the sze at whch e(d p s 0.5. Ths sze s usually called the corrected d 50c (corrected snce the effect of the short crcut flow has been elmnated as shown n Fgure 3.3. d 50c and d 50 are defned by the equatons and Shortcrcut flow Actual classfer Coarse product Fgure 3.4 A hydrocyclone wth short-ccut to the underflow. e(d 50c 0.5 (3.13 c(d (3.14 d 50 s clearly an ndcator of the sze at whch the classfer cuts the partcle populaton. Although phenomenologcal models have been developed for the man classfer types, no completely satsfactory procedure exsts for the calculaton of the classfcaton functon n any of

8 the man types of classfer. In the hydrocyclone the functon s determned prmarly by the turbulent dsperson wthn the centrfugal velocty feld n the concal secton and ths has been a fertle feld of research for many years. On the varous types of screenng devces, the classfcaton functon s determned by the knetcs of the transmsson process of partcles through the apertures of the screen. These processes have been comprehensvely studed but t s not yet possble to make accurate calculatons of the classfcaton functon n terms of the dmensons of and operatng load on the classfer. However, a number of useful emprcal functonal forms are avalable to descrbe the deal classfcaton functon. The most commonly used are: 1. Rosn-Rammler e(d p 1 exp( 0.693x ( Exponental Sum e(d p exp(x 1 exp(x exp( 2 ( Logstc e(d p 1/(1 x (3.17 In these equatons x = d p /d 50c and s a parameter that quantfes the sharpness of the classfcaton. It s relatvely easy to apprecate the sharpness of classfcaton n terms of the sharpness ndex defned by SI d 25 /d 75 (3.18 wth e(d 25 = 0.25 and e(d 75 = SI has a value between 0 and 1wth low values ndcatng poor and neffcent separaton whle a value of 1.0 ndcates perfect classfcaton at the cut sze d 50c. The parameter s related to the sharpness ndex for each case gven above as follows: 1. Rosn-Rammler 2. Exponental Sum (SI SI exp( 1.572/ ( ln(si (3.21 ln[(exp 2/3] SI (3.22 ln[3exp 2] s usually qute large and equaton (3.22 can be solved approxmately for n terms of SI as

9 follows 3. Logstc (1SI (1 SI (SI 9 (3.23 SI exp( / ( (3.26 ln(si These functons are useful for the quanttatve descrpton of the behavor of the varous classfcaton unts but t s necessary to evaluate the three parameters., d 50 and n terms of the physcal dmensons and the actual operatng condtons relevant to the unt n queston The Pltt odel for the Hydrocyclone The classfcaton acton of the hydrocyclone depends on a balance between the hydrodynamc drag forces that tend to convect the partcle toward the axs and the centrfugal force that tends to move the partcle outward towards the wall of the cone. The d 50 sze wll be determned for the partcle that fnds these two forces n balance and ths gves rse to the equlbrum orbt hypothess that s dscussed n secton Ths hypothess allows some general concluson to be drawn for predctng d 50 as a functon of hydrocyclone geometry and operatng condtons. The performance of the hydrocyclone s strongly nfluenced by the short crcut to underflow and ths s determned by the volumetrc flow splt between over- and underflow. The volumetrc flow splt, S, s a functon prmarly of the rato of spgot to vortex fnder dameters but s also a strong functon of the total feedrate. As the flowrate through the hydrocyclone ncreases, the dameter of the ar core ncreases chokng off the underflow. Ths effect has not been comprehensvely studed and we use a correlaton developed by Pltt (CI Bulletn, Dec. 1976, pp based on expermental data. S a(d u /D o b (D 2 u D 2 o c h d exp(0.543 D f c H g (3.27 where S = volumetrc flowrate n underflow / volumetrc flowrate n overflow D u = spgot dameter D o = vortex fnder dameter D c = cyclone dameter 3 = volume fracton solds n the feed h = vortex fnder to spgot dstance H = slurry feed head

10 Pltt recommends the followng values for hydrocyclone operatng wth free dscharge a = 3.79 b = 3.31 c = 0.36 d = 0.54 f = 1.11 g = 0.24 The nverse dependence on the feed head H should be noted. As the head ncreases the flow through the hydrocyclone ncreases, the centrfugal feld ncreases and the ar core expands chokng off the dscharge from the spgot The followng recoveres are defned R v = volumetrc flowrate n the underflow / volumetrc flowrate n the feed R f = recovery of flud phase to the underflow R s = recovery of sold phase to the underflow. The volumetrc recovery to underflow s related to S by S R v (3.28 S1 The bypass fracton. s assumed to be equal to the fracton of water that reports to the underflow R f, and ths s related to the recovery of solds R s by R v R s 3 R f (1 3 (3.29. R f R v R s (3.30 R s s determned by the actual classfcaton functon whch n turn s tself a functon of R f R s c(d p p F (3.31 where p F represents the partcle sze dstrbuton n the feed. Usng equaton (3.12 R s [R f (1 R f e(d p ]p F R f (1 R f e(d p p F (3.32 Substtuton nto equaton (3.30 and smplfyng gves R f R v 3 e(d p p F 1 3 e(d p p F (3.33 Thus the hydrocyclone performance s completely determned f the corrected classfcaton functon e(d p s known together wth the volumetrc flow splt S. Pltt uses the Rosn-Rammler model (3.15 to descrbe the partton functon whch requres two parameters d 50 and. Pltt has correlated these parameters n terms of the cyclone geometry and the operatng varables as follows

11 d 50c ad b c D c D d o exp[6.33] D f u h (3.34 g Q 0.5 D s the nlet dameter and Q the volumetrc flowrate to the cyclone. Recommended values for the constants are a = and b = 0.46 c = 0.6 d = 1.21 f = 0.71 g = 0.38 = Wth ths value of a, d 50c from equ ton (3.34 wll be n mcrons. Comparson of equaton (3.34 wth equaton (3.11 ndcates that Pltt's model for d 50c s consstent wth the man conclusons drawn from the equlbrum orbt hypothess n that d 50c vares roughly n proporton to the cyclone dameter (cyclone sze to the power b+c+d-f-g = 1.18 and nversely wth feed rate to a power less than 1. The 0.5 power dependence on! s - n dctates that the nteracton between partcles and flud s governed by Stokes' Law but hgher values can be used. The parameter n equaton (3.15 s correlated by b D 2 c a h exp( 1.58R Q v (3.35 Recommended values for the constants are a = 2.96 and b = Ths equaton reveals that the effcency of separaton depends on the volumetrc recovery R v but s a comparatvely weak functon of the cyclone sze partcularly when the feed rate Q s properly matched to the cyclone sze. It s useful to convert equaton (3.35 to a consstent bass for all classfcaton functons usng equaton (3.21 as follows. ln(si 1.24 Q D 2 c h 0.15 exp(1.58r v (3.36 The value of the selectvty ndex calculated from equaton (3.36 can be used n equatons (3.23 and (3.26 to evaluate for exponental sum and logstc partton functons. The values of the parameters a n formulas (3.27, (3.34 and (3.35 are often estmated from expermental data obtaned from an exstng cyclone nstallaton n order to make the model correspond to the actual operatng performance and these parameters can each be multpled by a separate calbraton factor.. In spte of the emprcal nature of the Pltt hydrocyclone model t has proved to be robust for practcal work. The chef source of uncertanty s n the predcton of the flow splt S. The partcle sze dstrbuton n the overflow stream s gven by

12 p O (1 c(d p p F (1 c(d p p F (3.37 p U c(d p p F c(d p p F (3.38 It often happens that the dfferent mnerals present n the sold have dfferent denstes. Then the classfcaton functon wll be a functon of the partcle sze and the partcle composton and the sze dstrbuton densty n the overflow wll be gven by p O j p O j (d p,g (3.39 j (1 c(d p,g p F j (d p,g j (1 c(d p,g p F j (d p,g (3.40 The recovery of solds s gven by R s R f (1 R f! F e(d p,g j s j! s (g j p F j (d p,g j (3.41 where! F s s the average densty of the solds n the feed and! s (g s the densty of a partcle of composton g. Equaton (3.33 becomes. R f R v 3! F s j e(d p,g j! s (g j 1 3! F s j e(d p,g j! s (g j p F j (d p,g j p F j (d p,g j ( Capacty Lmtatons of the Hydrocyclone The capacty of the hydrocyclone s essentally lmted by the ablty of the spgot openng to dscharge solds. If the solds load s too hgh the solds content of the underflow ncreases and the vscosty of the dscharged pulp becomes hgh. The tangental velocty of the slurry at the spgot decreases and the usual umbrella-shaped dscharge becomes a rope-lke stream of dense pulp. Under

13 these condtons the cyclone s sad to be ropng and the classfer performance s sgnfcantly degraded wth the dscharge of sgnfcant quanttes of oversze materal through the overflow. Hydrocyclones should not normally be operated n the ropng condton and they are often nstalled n clusters to spread the feed over several unts to ensure that no one cyclone s overloaded. When a cyclone s operatng properly the dscharge should flare out n an umbrella shape.. Prelmnary selecton of hydrocyclones s usually done usng manufacturers charts. These show regons of operaton of each cyclone n the manufacturers range as a functon of the nlet pressure and the volumetrc flowrate that must be processed. Typcally a range of cut szes s assocated wth each hydrocyclone. A typcal chart s shown n the fgure. It s generally not possble to use these selecton charts effectvely untl an estmate of the crculatng load s known. It s however a smple matter to make a tentatve selecton of an approxmate cyclone sze and then determne the number of cyclones n the cluster to handle the requred load. Effcent smulaton s the most effcent way to acheve a satsfactory desgn.