TECHNICAL NOTES 4 VIBRATING SCREENS

Transcription

1 TECHNICAL NOTES 4 VIBRATING SCREENS Copyrght R P Kng 2000

2 SIZE CLASSIFICATION It s always necessary to control the sze characterstcs of partculate materal that s fed to process equpment that separates the mneralogcal components. It s not possble wthn the producton envronment to exercse precse control over the sze of all partcles n a populaton and, n most cases, sze classfcaton equpment s desgned to splt a feed of partculate materal nto a coarse and a fne product. Occasonally one or two product streams of ntermedate sze may be produced such as by a double or trple deck screen. Probably the most common applcaton of sze classfcaton s ts use to prevent oversze materal leavng a commnuton crcut. Oversze materal s recycled to a commnuton stage for further sze reducton before passng to subsequent stages of processng. The most sgnfcant consderaton when evaluatng classfcaton equpment s the lack of a clean cut at a partcular sze. Even the most effcent ndustral sze classfers wll pass a proporton of oversze materal and wll retan a porton of undersze materal. Sze classfcaton equpment s also subject to capacty lmtatons and these must be consdered when evaluatng the performance of new or exstng classfcaton equpment. In fact older methods of performance evaluaton concentrate entrely on the capacty lmtatons. However, more modern procedures recognze that the effcency of separaton must also be consdered f the performance of a sze classfer n a plant crcut s to be accurately evaluated. The procedures that follow take ths nto account. There are essentally two types of classfer avalable for process operaton: screens and classfers that rely on the varaton of termnal settlng veloctes of partcles of varyng sze when these are mmersed n a vscous flud. In general the former are used for the classfcaton of drer materal at coarser szes. 4.1 Classfcaton Based on Sevng - Vbratng screens The basc method of operaton of a screen s very smple. The screen presents a barrer to the passage of oversze materal whle t readly passes undersze materal. It s only necessary to ensure that each partcle has an opportunty to reach the screen. In practce each partcle s gven several opportuntes to pass through the screen. Screens can be statonary or the screen can vbrate whch ncreases the rate of presentaton of each partcle and asssts n movng oversze materal over and away from the screenng surface Models based on screen capacty The tradtonal method of evaluaton of screen performance s the use of a capacty measure. Ths represents the ablty of the screen to accept and handle the feed tonnage of materal. The most mportant assumpton n ths approach s that the capacty of a screen s drectly proportonal to ts 4-1

3 surface area so that the basc capacty s specfed at tons of feed per hour per square meter of screen. Ths quantty s represented by I u. The basc capacty of any screen s determned under standard operatng condtons usng a predefned standard feed materal. As the nature of the feed materal changes and as the operatng condtons change so the actual capacty of the screen changes - t wll ncrease for condtons less arduous than the standard and decrease for condtons more arduous than the standard. These modfcatons are represented by capacty factors whch multply the standard unt screen capacty to get the actual screen capacty under condtons that the screen wll actually meet n ts poston n the operatng plant Rated screen feed capacty I u K 1 K 2 áá I u K tons/hr m 2 (4-1) where the separate K are the capacty factors for devatons from the standard condtons wth respect to a number of ndvdual condtons. The basc unt capacty vares prmarly wth the sze of the screen openng - screens wth larger openngs beng able to handle large quanttes of feed materal. A typcal relatonshp between I u and mesh sze s I u 0.783h 37 for h25mm (4-2) 20.0h for h<25 mm (4-3) where I u s n tons/hr m 2 and h s the mesh sze n mms. Each screen manufacturer has ts own basc capacty-sze relatonshps for ts range of screens. The above expressons are only meant to defne a typcal trend. The ndvdual capacty factors are provded by screen manufacturers n tabular or graphcal form although the avalablty of computers s encouragng the presentaton of these factors n algebrac form. The followng factors are typcal for woven wre mesh screens. The open area factor K 1 The standard condton s usually 50% open area and the capacty s proportonal to the open area avalable. K 1 % open area 50 (4-4) For materal havng a bulk densty less than 800 kg/m 3 the standard open area s 60% rather than 50% and equaton (4.4) should be modfed accordngly. The half-sze factor K 2 Feed that contans a large proporton of materal that s consderably smaller than the screen mesh sze wll be handled more easly by a screen. The standard condton s defned as feed materal havng 40% smaller than one half of the mesh sze. If the feed has more than 40% smaller than one 4-2

4 half of the screen mesh sze, the half-sze factor wll exceed unty and vce versa. K 2 2P F (0.5h) 0.2 (4-5) The oversze factor K 3 A screen can handle a greater tonnage of feed materal that contans large quanttes of oversze materal because ths materal passes drectly over the screen and need not be transmtted through the mesh. Ths s accounted for by the oversze factor K 3 whch has a value of unty for a standard feed contanng 25% oversze materal. Ths factor ncreases very quckly as the fracton of oversze ncreases and s gven by K exp exp(4.22 P F (h) 3.50) (4-6) In equaton (4.6) P F (h) s the fracton of materal n the feed that has sze greater than the screen mesh sze h. It s related to the cumulatve sze dstrbuton functon by P F (h) 1 P F (h) (4-8) The bulk densty factor K 4 Denser materals wll be transmtted more easly than lghter materals. A factor K 4 accounts for ths effect when bulk densty dffers from the standard of 1600 kg/m 3 K 4! B 1600 (4-9) The deck poston factor K 5 Screens that are lower down n the deck receve undersze from the screen above and can handle less materal than a screen that takes fresh feed. The capacty decreases wth poston accordng to capacty factor K 5 K S (4-10) where S represents the deck poston; 1 for top deck, 2 for 2nd deck and so on. The screen angle factor K 6 The standard nclned screen has an angle of nclnaton of 15 o. Lower angles of nclnaton ncrease the projected area of the screen aperture n the horzontal plane and the screen can handle a greater load. Ths s accounted for by capacty factor K 6 where. s the angle of nclnaton n degrees. K (. 15) (4-11) The wet screen factor K 7 Screenng at fner mesh szes can be mproved by sprayng the screen load wth water. The factor K 7 accounts for ths effect 4-3

5 K (25 h) 2.5 for h 25mm 1.0 for h >25mm (4-12) The aperture shape factor K 8 The standard screen has square openngs and other shapes nfluence the capacty as shown n the table. Table 2.1 Screen capacty factor for dfferent apertures Shape of screen openng K 8 Partcle shape factor K 9 Slabby and elongated partcles are more dffcult to screen than partcles that are essentally sometrc. If the feed contans about 15% of slabby or elongated partcles K 9 should be set at 0.9. Larger amounts of ths type of materal would gve sgnfcant problems and would need to be nvestgated specally. Round Square 2 to 1 rectangular slot 3 to 1 rectangular slot 4 to 1 rectangular slot The surface mosture factor K 10 Surface mosture tends to make the partcles adhere and screen capacty s reduced. Factor K 10 accounts for ths and can be evaluated from the table below Screen transmsson effcency Ideally the screen should be able to transmt all of the undersze materal n the feed. In practce, however, not all of the undersze materal passes through the screen and the fracton of the feed undersze that does pass through s referred to as the screen effcency. The effcency s determned prmarly by the actual feed loadng on the screen relatve to the rated feed capacty as calculated by equaton (4.1). The effcency of transmsson Table 2.2 Surface mosture capacty factor for screens. Condton of feed K 10 Wet, muddy or stcky materal. Wet surface quarred and materal from surface stockples wth up to 14% mosture by volume. Dry crushed materal. Naturally or artfcally dred materal. decreases f the screen must handle feed n excess of 80% of the rated tonnage because the access of ndvdual partcles to the screen surface s hndered to a greater or lesser degree. The effcency also decreases as the actual feed tonnage falls below 80% of rated capacty because partcles tend to bounce on the lghtly loaded screen and make fewer contacts wth the screen surface. If W F represents the actual feed tonnage, then the ratng rato s gven by

6 W F RR I u K screen area (4-13) and the effcency of transmsson s gven by e (RR 0.8) 0.05(RR 0.8) 2 for RR (0.8 RR) 2 for RR < 0.8 The actual tonnage passed to the undersze stream s (4-14) W U ep F (h)w F (4-15) Each sze class smaller than the mesh sze s subject to the same effcency factor e so that the partcle sze dstrbuton n the underflow stream s calculated n dscrete form as p U 0 ep F W F p ep F (h)w F F P F (h) for d p <h for d p h (4-16) where p U s the fracton of the underflow stream n the sze class and p F that fracton n the feed stream. The actual tonnage passed to the overflow stream s W O (1 P F (h))w F (1 e)p F (h)w F W F (1 ep F (h)) and the dscrete sze dstrbuton n the overflow stream s gven by p O (1 e)p F (1 ep F (h)) p F 1 ep F (h) (4-17) for d p <h (4-18) for d p h (4-19) The smple capacty model permts the calculaton of the total screen area that s requred for a specfc duty. The aspect rato of the screen (length/wdth rato) must be determned by consderng the depth of the bed of partcles on the screen. An approxmate gude s the restrcton of the bed depth at the dscharge end to no more than 4 tmes the mesh sze. The thckness of the bed s determned by the total flowrate over the screen surface, the wdth of the screen, b, and the velocty of travel of the materal along the screen. 4-5

7 t b W O bu! b (4-20) where t b s the bed thckness, W d the mass flowrate across the dscharge end, b the screen wdth, u the velocty of travel across the screen surface and! b the bulk densty. The velocty of travel depends prmarly on the angle of nclnaton and the ampltude and mode of vbraton. 4.2 The Classfcaton Functon A more realstc descrpton of the performance of a classfcaton devce s provded by the classfcaton functon. Ths functon defnes the probablty that an ndvdual partcle wll enter the oversze stream that leaves the classfer. Ths functon s also known as the partton functon and when shown graphcally as the partton curve. It s a complcated functon of the partcle propertes and the classfyng acton of the partcular devce under consderaton. The classfcaton functon c(d p ) s defned as the mass fracton of materal n sze nterval n the feed whch fnally leaves n the oversze stream. Once the partton functon s known, the sze dstrbuton n both overflow and underflow can be calculated by a smple mass balance over the solds n sze class. W U p U (1 c(d p ))W F p F (4-21) W O p O c(d p )W F p F (4-22) where the superscrpts O, U and F refer to oversze, undersze and feed respectvely. The total flowrates of sold n the oversze and the undersze are gven by W U M W U p U M (1 c(d p ))W F p F (4-23) W O M W O p O M c(d p )W F p F (4-24) p U [1 c(d p )] W F F W Up M [1 c(d p )]p F [1 c(d p )]p F (4-25) p O c(d p ) W F F W Op M c(d p )p F c(d p )p F (4-26) These formulas are often wrtten n terms of the total yeld of solds to the overflow 4-6

8 Y s W O W F (4-27) p U (1 c(d p ))p F (4-28) 1 Y s p O c(d p )p F (4-29) Y s Y s M c(d p )p F (4-30) The Karra model The approach descrbed n the prevous secton s the tradtonal method used for szng screens. Its chef lmtaton s that the screen must be szed accordng to the amount of feed that s presented to the screen. A more logcal approach s based on the amount of materal that must be actually transmtted by the screen to the underflow stream. Ths approach has been developed by V.K. Karra (CIM Bulletn, Aprl 1979 pp ) nto an effectve descrpton of how a screen may be expected to perform durng plant operaton. The approach s smlar to the tradtonal approach but s based on the capacty of the screen to transmt undersze materal proportonal to the screen area. As n the tradtonal method ths basc capacty s modfed by a number of factors that allow for varatons of the feed materal and the screen from the standard test condtons. Let A represent the basc capacty whch s defned as the tonnage of undersze that a partcular screen can transmt per unt of screen surface area. The basc capacty s ncreased or decreased dependng on the nature of the feed and condtons on the screen. A number of capacty factors allow for the amount of oversze n the feed (factor B), the amount of half sze n the feed (factor C), the locaton of the deck (factor D), factor for wet screenng (factor E) and for materal bulk densty (factor F). These factors all have a value of unty at the nomnal standard operatng condton and move down or up as the screenng duty becomes more or less arduous. Karra has also recognzed that the amount of near-sze materal has a sgnfcant effect on the ablty of the screen to transmt undersze materal and he has ntroduced a near-sze capacty factor G c. The theoretcal amount of undersze that can be transmtted by the screen s gven by Th A.B.C.D.E.F.G c screen area (4-31) 4-7

9 A screen wll be well desgned to handle ts duty n the crcut f Th s approxmately equal to the quantty of undersze n the feed. Each of the capacty factors s related to the qualty of the feed and to the type of screen. Karra bases the screen performance on the effectve throughfall aperture of the screen defned by h T (hd w )cos d w (4-32) where d W s the dameter of the wre and s the angle of nclnaton of the deck. Equaton (4.32) gves the effectve aperture area projected onto the horzontal plane whch s approprate for partcles that must pass through the screen under gravty. The basc capacty A. The basc capacty s prmarly determned by the mesh sze of the screen wth the effectve throughfall aperture beng used as the approprate measure of screen mesh sze. wth h T n mm and A s n metrc tons/hr m 2. A 12.13h 0.32 T 10.3 for h T <51 mm (4-33) A 0.34 h T for h T 51 mm (4-34) The basc capacty wll also depend on the open area of the screen used. The basc capacty calculated from equaton (4.33) and (4.34) are applcable to standard ndustral lght-medum woven wre mesh. For other screen cloths and surfaces, A must be adjusted n proporton to the open area. The percent open area for lght-medum wre mesh s related to the mesh sze h by OA 21.5 log 10 h 37 (4-35) wth h n mllmeters. Thus capacty A must be adjusted to The oversze factor B A actual % open area OA B P F (h T ) for P F (h T )0.87 B P F (h T ) for P F (h T )>0.87 (4-36) (4-37) The half-sze factor C 4-8

10 C P F (0.5h T )forp F (0.5h T )0.3 C P F (0.5h T ) for 0.3<P F (0.5h T )0.55 C 3.35 P F (0.5h T ) 1.37 for 0.55<P F (0.5h T ) 0.8 (4-38) C 5.0 P F (0.5 h T ) 1.5 for P F (0.5h T )>0.8 The deck locaton factor D where S = 1 for top deck, S = 2 for 2nd deck and so on The wet screenng factor E Let T = 1.26 h T (h T n mm) E = 1.0 for T < 1 E = T for 1 T < 2 E = T for 2 T < 4 E = 2.5 for 4 T < 6 E = T for 6 T < 10 E = T for 10 T < 12 E = T for 12 T < 16 E = T for 16 T < 24 E = T for 24 T < 32 E = 1.15 for T > 32 whenever the materal s sprayed wth water. The bulk densty factor F D S (4-39) F! B 1600 (4-40) The near-sze capacty factor G c The capacty of the screen s also affected by the presence of near-sze materal n the feed. The nearsze materal n the feed s n the sze range from 0.75h T to 1.25h T. Consderable quanttes of nearsze materal wll nhbt the passage of undersze materal through the screen. The near-sze capacty factor can be evaluated from G c 0.975(1 near sze fracton n feed) G c P F (1.25h T ) P F (0.75h T ) (4-41) 4-9

11 4.2.2 The screen classfcaton functon In practce, not all of the undersze s transmtted because of varous physcal factors that mpar the effcency of the screen. Ths effect s descrbed by the screen partton functon. Several standard functonal forms are avalable to descrbe ths effect, and Karra uses the functon c(d p ) 1 exp[ 0.693(d p /d 50 ) 5.9 ] (4-42) whch gves the effcency of transfer of partcles of sze d p to oversze. The parameter that wll determne the screenng effcency s d 50. Values of d 50 greater than the mesh sze gve hgh effcences and vce-versa. The actual d 50 acheved wll depend prmarly on the effectve throughfall aperture of the wre mesh used on the screen, on a loadng coeffcent K defned by and on the near-sze factor G c. tons of undersze n the feed / unt of screen area K ABCDEF W F P(h T )/screen area ABCDEF (4-43) Karra has found that expermental screenng data are well represented by d 50 G c (4-44) h T K d 50 can be substtuted n equatons (4.42) and ( ) to calculate the sze dstrbutons n the two product streams. p U M [1 c(d p )]p F [1 c(d p )]p F (4-45) p O M c(d p )p F c(d p )p F (4-46) The model provdes a smulaton of the actual performance of the screen n the crcut. Ths performance can be compared wth the desgn capacty of the screen and the screen performance evaluated. In partcular, the actual operatng effcency can be calculated from smulated effcency tonnage transmtted to undersze tonnage of undersze n feed (4-47) 4-10

12 M [1 c(d p )]p F P F (h) The effectve utlzaton of the screen area can be calculated from AUF area utlzaton factor tonnage transmtted to undersze theoretcal ablty of the screen to pass undersze W F P F (h) smulated effcency A.B.C.D.E.F.G c screen area (4-48) (4-49) An AUF equal to unty ndcates that the screen capacty s exactly balanced to the requred duty. AUF > 1 ndcates that the screen s overloaded whle AUF < 1 ndcates that the screen s underloaded A partcular advantage of the Karra screen model s that no free parameters are requred to be estmated from operatng data. 4.3 A Smple Knetc Model for Screenng Screenng can be consdered to be a knetc process because the rate at whch sold partculate materal s transmtted through a screen s dependent on the nature of the screen, on the load that the screen s carryng and on the nature and sze of partcles. Obvously larger partcles are transmtted more slowly than small partcles and so on. The rate of transmsson of partcles through the screen wll vary from pont to pont on an actual screen because the load that the screen carres vares n the drecton of the materal flow. In general the rate of transmsson depends prmarly on the partcle sze relatve to the screen mesh sze. Observaton of operatng screens shows clearly that the screen acton vares accordng to the load on the screen. Near the feed end, ndustral screens are almost nvarably heavly loaded so that the partculate materal forms a multlayer of partcles on the surface. As the partcles move down the screen surface, the smaller materal falls through and the load on the screen surface decreases steadly untl only a monolayer of partcles reman. These two condtons are referred to as the crowded and separated regmes respectvely. In the crowded condton, the undersze materal must percolate through the upper layers of coarse materal before t has a chance to contact the screen and thus fall through. Thus the rate of transmsson of materal s a functon of the sze dstrbuton n the partcle layer mmedately above the screen surface. On the crowded feed end of the screen, the screen, ths layer s replenshed by downward percolaton of undersze materal through the upper layers of the bed. The net rate at whch partcles of a partcular sze are transmtted through the screen s a functon of the percolaton 4-11

13 process as well as the screen transmsson process Ths model of the screenng operaton s llustrated n Fgure 4.1 Let w (l) = mass flowrate per unt wdth of screen of materal n sze nterval down the screen at dstance l from the feed end (kg/ms). w (l) s related to the dscrete sze dstrbuton densty functon Feed Crowded regon Separated regon L c Oversze L Undersze Fgure 4.1 Varyng rate of transmsson of partcles through a wre mesh screen. by w (l) = W(l)p (l) where W(l) s the total flow of materal down the screen per unt wdth at poston l and the dependence of these functons on poston s shown explctly by the argument l. If the rate of transmsson of partcles n sze nterval s represented by r(d p ) kg/m 2 s, a dfferental mass balance gves dw (l) dl r(d p ) (4-50) Ths equaton can be ntegrated f a model for the rate of transmsson can be developed. The rate of transmsson wll be modeled dfferently under the crowded and separated condtons. Accurate models for the crowded condton are complex snce they must take nto account both the stratfcaton process and the transmsson process. A more realstc model s developed n the next secton. We assume here that the rate of transmsson does not vary along the screen under crowded condtons and that r(d p ) k(d p )M o p F (4-51) where M o s the load on the screen n kg/m 2 at the feed end. There s some expermental evdence that confrms the constant rate of transmsson under crowded condtons. Soldnger M., Interrelaton 4-12

14 of Stratfcaton and passage n the screenng process. Mnerals Engneerng 12, pp , The smplest model for the varaton of k(d p ) wth partcle sze s k(d p ) u k 0 u 1 d p u for d p < h 0 for d p h where k 0 /u s the operatng parameter. k 0 /u wll depend on the mesh sze and the fracton open area and on other operatng parameters such as vbraton ampltude and frequency. Equaton (4.50) can be ntegrated mmedately to gve w (l) w (0) r(d p )l (4-53) At the transton between the crowded regme and the separated regme l = L c w (L c ) w (0) r(d p )L c (4-54) W F b p F k(d p )M o p F L c (4-55) Let M(l) = total load per unt area of screen surface at a dstance l from the feed end n kg/m 2. The flowrate down the screen W(l) s related to the load on the screen by W(l) M(l) u (4-56) where u s the velocty at whch materal travels down the screen. w (L c ) M o up F k(d p )M o p F M o p F u k(d p )L c W F p F 1 k(d p ) u L c L c (4-57) M o p F u k(d p )L c (4-58) L c can be defned as the poston on the screen at whch partcles n the smallest sze class just become depleted u L c k(d pn ) (4-59) Under separated condtons, the rate of transmsson of partcles f sze d p s assumed to be proportonal to the amount of that materal per unt area of screen surface 4-13

15 dw (l) s(d dl p )M(l)p (l) (4-60) where p (l) = the dscrete partcle sze densty functon of the materal on the screen at dstance l from the feed end. s(d p ) = the specfc rate of transmsson for partcles of sze d p (sec -1 ) w (l) s related to M(l) by w (l) p (l)m(l)u (4-61) where u s the velocty at whch the materal travels down the screen surface dw (l) dl s(d p ) u w (l) (4-62) Ths s easy to ntegrate and match the ndvdual mass flowrates for each sze nterval at the boundary between the crowded and separated regons w (l) w (L c )exp s(d p ) u (l L c ) (4-63) w (0) r(d p )L c exp s(d p ) u (l L c ) (4-64) The partton factor can be evaluated from equaton (4.64) c(d p ) w (L) w (0) 1 r(d p )L c w (0) exp s(d p ) u (L L c ) (4-65) Ferrara et. al. (G. Ferrara, U. Pret and G.D. Schena. Internatonal Journal of Mneral Processng 22 (1988) ) have determned the specfc rate constant s(d p )/u n terms of the probablty of passage of a partcle durng a sngle contact wth the screen. Ths s usually computed as the rato of the area of the effectve aperture to the total area of the screen. In order to pass cleanly through a square mesh openng a sphercal partcle must pass wth ts center wthn a square of sde h-d p n the center of the mesh openng. Thus the probablty of passage s gven by 4-14

16 P r (h d p )2 (hd w ) 2 h 2 (hd w ) 2 f 0 1 d p h 1 d p h where f 0 s the fracton open area of the screen. Ths s llustrated n Fgure (4-66) Ferrara et. al. suggest an emprcal functonal form s(d p ) u nf 1/2 0 1 d p h 1 (4-67) where 1 s a constant that should be close to 2. n s the number of presentatons per unt length of bed. Equaton (4.67) can be wrtten s(d p ) s u d p h where s 50 s the value of s(d p )/u when the partcle sze s one half the mesh sze. 1 (4-68) h d w Open area = h 2 (h+d w) 2 d p Area for clear passage (h-d ) p 2 = (h+d w) 2 Fgure 4.2 Geometrcal restrctons on the passage of a partcle through square-mesh screen Unfortunately there s very lttle data avalable on the value of the transmsson rate for crowded condtons, r(d p ), and ts varaton wth partcle sze and wth condtons on the screen. It s also dependent on the sze dstrbuton n the feed. 4-15