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In presenting this thesis in partialfulfilment of the requirements for an advanced degree at theUniversity of Bri t i sh Columbia, I agree that the Library shall make it freely available forreference and study. I further agree thatpermission for extensive copying of this thesis forscholarly purposes may be granted by the head of my department or by his or herrepresentatives. It is understood thatcopying or publication of this thesis for financial gainshallnot beallowedwithout my written permission.

I a nM. F l i n t

Departmentof M i n i n gandM i n e r a l P r o c e s s i n g E n g i n e e r i n gThe U n i v e r s i t y of B r i t i s h C o l umb iaV a n c o u v e r , Canada

D a t e : T h u r s d a y ,November 22, 2 01

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Abstract

Thisresearchpresents threemodels of froth flotationthatrecognize flotation as an interfacialphenomenon wherein the rateof solids-surface-area removal is related to the bubble-surface-area flux. These models use two "streams", air and l iquid , within the vessel. The solidswithin the column are associated withone, or both, ofthesestreams. The procedures andbenefits of batchtestsfor flotation columns are also outlined. The first modelusesthis batchdata and the simplex method of non-linear regression to determine four model parameters: akinetic rate constant, maximum recovery, entrainment and carrying capacity. Theseparameters are then used in the second model in order to estimate continuous columnperformance. W i t h i n the continuous model, carrying capacity is determined by overflowbubble size; recoverywithin the froth zone is determined by loss of bubble surface area; andsolids transfer from l iquid- to gas-phase is estimated by kinetic relationships and the axialdispersion model. The prediction performance of the model is verified using both batchmechanical cel l and column flotation cell batch data. The final model "framework"characterizes the effect of bubble residence time and bubble loadingrateswithin a flotationcel lusing mechanistic bubble - particleco l l i s ionandattachmentrelationships.

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TableofContentsA B S T R A C T iiT A B LE O F CONTENTS iiiLIST O F F I G U R E S viiiLIST O F T A B L E S x iACKNOWLEDGEMENTS xiiiP R E F A C E xivNE W KNOWLEDGE xv i i i1 I N T R O D U C T I O N 11.1 PURPOSE 11 2 SCOPE 1

1.2.1 Lim itation s 11.2.2 Eq uipment G oals 21.2.3 M od elin g G oals 21.3 B A T C HFLOTATION 41.3.1 M ec hanic al C ell 41.3.2 C olum n C ells 41.3.3 H ydrod yna mic Zones 71.3.4 C olum n Flow s 91.3.5 M od e of Operation 11

2 B A C K G R O U N D 132.1 INTROD UCT ION 132.2 MODELINGTHEORY 13

2.2.1 Introduction 132.2.2 Ph ys ical M odels 152.2.3 Sy mbol ic M odels 152.2.4 M athematical M ode ls 152.2.5 D evelopme nt 162.2.6 Ap plic abil i ty 172.2.7 De termination of M od el Parameters 18

2.3 CURRENTFLOTATIONMODELS 192.3.1 Introductionto K inet ic M odels 192.3 .2 Batch M echanica l -C el l K inet ic M odels 192.3.3 C olumn K inet ic M odels 23

2.4 CARRYING CAPACITY 312.4.1 Introduction 312.4.2 Cross-Sect ional Area C arry ing C apaci ty 312.4.3 Li p Lo adin g 332.4.4 Sum mary 332.4.5 Bu bble Load ing 34

2.5 ENTR AINME NT 352.5.1 Introduction 352.5.2 Entrainment in Flotation Co lumns 35

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2.5 .3 Feed W ater Re covery Entrainment M ode ls 3 72.5 .4 M oys and F inch( 1 9 9 1 )- Feed W ater R eco very 4 0

2 .6 G A S P H A S E 4 12.6.1 Introduction 412.6 .2 Bub ble Flow Regime , 412.6 .3 Bubble Rise-veloc i ty 4 22.6 .4 M ax imum Gas Rate 4 52.6 .5 Superf ic ial Bubble-S urface-Area Rate 4 6

2 . 7 S E T T L I N G V E L O C I T Y 4 82.7 .1 Introduction 4 82.7 .2 Sett ling V eloc i ty of aSphere in W ater 4 82.7 .3 S ett ling V eloc ity of Spheres in Suspensions 4 82.7 .4 Non-S pher ical Part ic le V eloc i ty 5 0

2. 8 M I X I N G 5 22.8.1 Introduction 5 22.8 .2 Tanks-in-Series 5 32.8 .3 Ax ia l D ispers ion M odel 5 32.8 .4 Sol idsDistribution 5 92.8 .5 Conc lus ions 61

2.9 M A C R O - P H E N O M E N O L O G I C A L K I N E T I C S 6 22.9.1 Introduction 6 22.9 .2 Re action Order 6 32.9 .3 Non -Floating Partic les 6 42.9 .4 Flotation Phases 6 52.9 .5 M ode l Stages 6 62 . 9 . 6 Recovery M odels 6 72.9 .7 Determination of M acro-P henom enologica l Rates 7 12.9 .8 InterfaceM ass Transfer 7 3

2 . 1 0 M I C R O - P HE N O M E N O L O G I C A L M O D E L S 7 42.10.1 Introduction 7 42 . 1 0 . 2 Col lec t ion Ef f i c iency 7 52.10.3 C ol l ision Ef f i c iency 7 62 . 1 0 . 4 Attachment E ff ic ienc y 8 5

2 . 11 S U M M A R Y '. 913 A P P A R A T U S '. 9 2

3 .1 I N T R O D U C T I O N 9 23 . 2 T E S T C O L U M N 9 3

3.2.1 Introduction 9 33 .2 .2 C o lum n Body 9 43.2.3 C ontrol System 9 43 .2 .4 Wash-WaterDistributor 9 43.2.5 Launder 9 73 .2 .6 Sparger 9 8

3.3 B A T C H M E C H A N I C A L - C E L L T E S T S 9 93 . 4 C O N T I N U O U S C O L U M N S 9 9

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4 P R O C E D U R E S 1014 .1 L A B O R A T O R Y B A T C H M E C H A N I C A L - C E L L 101

4 . 1 . 1 Introduction 1014 . 1 . 2 Feed M aterial ; 1 0 24 . 1 . 3 M ech anical-C el l Base Line Tests 1 0 9

4 . 2 L A B O R A T O R Y - S C A L E B A T C H C O L U M N 1 1 24 . 2 . 1 Equipment 1 1 24 . 2 . 2 Procedures 1 1 24 . 2 . 3 Surface Are a D etermination 1 1 4

5 D I S C U S S I O N 1 1 95. 1 M O D E L I N T R O D U C T I O N 1 1 95 .2 V A R I A B L E S 1 2 05 .3 B A T C H K I N E T I CM O D E L 121

5 . 3 . 1 Introduction 1215 . 3 . 2 M od el Structure 1 2 25 . 3 . 3 M od el Descr ipt ion 1 2 25 . 3 . 4 Liberation 1 2 55 . 3 . 5 Ves s el M ix ing 1 2 65 . 3 . 6 C olumn Dead Zone 1 2 75 . 3 . 7 C olum n Residence-time 1 2 85 . 3 . 8 Column M ax imum Recyc le Rate 1 3 05 . 3 . 9 Entrainment 1 3 15.3.10 Sp ecif ic Surface 1 3 65.3.11 Flotation 1 3 75.3.12 Carry ing Capaci ty 1415.3.13 Appl icat ion of the Batch M ode l 1 4 95.3.14 M odel Val idation 1 5 25.3.15 INC O Batch M echanica l -Cel l Tests 1 5 35.3.16 Example - INC O Batch C olumn Appl icat ion 1 5 95.3.17 Quinto Batch M echanica l -Cel l 1635.3.18 Quinto C olumn 1 6 65.3.19 Batch M od el Sensit iv ity 1 6 95.3.20 Batch M od el Summary 1 6 9

5 . 4 C O N T I N U O U S C O L U M N K I N E T I CM O D E L 1 7 05 . 4 . 1 Introduction 1 7 05 . 4 . 2 M od el Structure 1 7 25 . 4 . 3 Froth Zone 1 7 45 . 4 . 4 Pulp Zone 1 7 85 . 4 . 5 App lication of the Continuous M od el 1 9 25 . 4 . 6 M odel Val idation 1 9 5

5 .5 C O M P A R I S O N O F A P P L I C A T I O N S 1 965 . 5 . 1 IN C O Industr ial-Scale C olumn 1 9 65 . 5 . 2 Q uinto G raphite Continuous C olumn Com parison 2 0 15 . 5 . 3 M ode l Sen sitiv ity 2 0 2

5 .6 M E C H A N I S T I C B U B B L E L O A D I N G " M O D E L F R A M E W O R K " 2 1 1

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5.6.1 Introduction 2115.6 .2 M od el Assumptions 2135.6.3 Ba tc h C o l lec t ion M od e l 2155.6 .4 M o d e l useforFlotat ion Characterization 2255.6.5 Future D evelopment 2295.6.6 Testsfor M od e l Va l ida tion 2295.6 .7 M echanis tic Bubb le Loading M od el S teps 230

6 C O N C L U S I O N S 2336.1 B A T C H C O L U M N T E S T S 2336.2 B A T C H K I N E T I C M O D E L 2346.3 C O N T I N U O U S K I N E T I C M O D E L 2376.4 M E C H A N I S T I C B U B B L E L O A D I NG M O D E L : 240

6.4.1 M i x i n g 2406.4 .2 Parameter C omparisonwith K inet ic M odels 2406.4 .3 DeterminationofInductionT imes 2416.4 .4 DeterminationofP ack ing Factor 2416.4.5 R elationship Between P ack ing Factor and B ubble Loa d 2426.4 .6 Estimated C rit ical Ang le 2426.4 .7 Bub ble Residence TimeatCr i t ical Angle 242

7 R E C O M M E N D A T IO N S 2437.1 T E S T I N G 2437.2 B A T C H M O D E L 2447.3 C O N T I N U O U S M O D E L 2447.4 B U B B L E L O A D I N G O R M E C H A N I S T I C M O D E L 2447.5 B A T C H C O L U M N 245

8 N O M E N C L A T U R E 2468.1 V A R I A B L E S , P A R A M E T E R S A N D C O N S T A N T S 2468.2 S U B S C R I P T S 250

9 R E F E R E N C E S 25210 D A T A A P P E N D I X 271

10.2 I N T R O D U C T I O N 27110.2 I N C O D A T A 271

10.2.1 Ba tc h M ec han ic a l -C e l l 27110.2.2 Batch M ec hanic al-C el l Performance 27810.2.3 NoEntrainmentAc count ing 28610.2 .4 Batch Column 289

10.3 Q U I N T OD A T A 30510.3.1 Ba tc h M ec han ic a l -C e l l 30510.3.2 Batch M ech anica l-Ce l l Performance 31010.3 .3 Batch Column 31310.3 .4 Batch C olumn Performance 317

11 C O N T I N U O U S C O L U M N T E ST S 320

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11.1 I N C O S E C O N D C O P P E R C L E A N E R 32011.2 Q U I N T O P I L O T C O L U M N 324

12 P A R T I C L E S I Z EA N A L Y S I S ; 32 612.1 P A R T I C L E D I S T R I BU T I O N 326

12.1.1 Introduction 32612.1.2 Q uinto Overflow 32612.1.3 IN C O Feed Stream 33612.1.4 IN C O Overflow Stream 340

12.2 A V E R A G E P A R T I C L E S I ZE 34312.2.1 Introduction 34312.2.2 Q uinto Ove rflow Stream 34512.2.3 IN C O Average Part ic le Size 346

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List ofFiguresFigure1 C olum n hydrodynamic zones 7Figure 2 Froth-zone water flow diagram 9Figure 3 Bias schematic 10Figure 4 Co ntinuous co lumn -ce ll operation 12Figure 5 B atch colum n-ce ll operation 12Figure 6 Ty pes of models andtheir components from Sastry(1990) 14Figure 7 Two-stage flotation model of Harris and Rimmer(1966) 20Figure 8 Phaseso fflotation fromSz atkowski 1987 65Figure 9 C oll is i on model and experimental comparison fromNguyen et. al.(1998) 78Figure 10 R ec ove ry response to particle sizefromTrahar and Warren (1976) 79Figure 11 D epic tion of particle - bubble c ollisio n fromNguyen and Km et (1992) 80Figu re 12 P artic le behavior in water streamlines fromReayand R atc l i f f (1973) 81Figure 13 P article - bubble co llisionfromSc hulze(1992)or S chulze et. al.(1989) 81Figure 14 S lidin g action between bubble and particle fromNgu ye n 1993 87Figure 15 B atch colum n schematic drawing 93Figure 16 Internalflowpipe wash-waterdistributor 95Figure 17 Ex ternalflowrod w ash-waterdistributor 95Figure 18 W ash-water rodpattern 96Figure 19 Launde r system 97Figure 20 Sparger Sy stem 98Figure 21 Stages of IN C O M atte Separation Flotation fromW ilso n 1990 102Figure 22 IN C O #2 copper c leaner colum n feed partic le sizedistribution 104Figure 23 IN C O #2 copper cleaner overflow particle sizedistribution 105Figure 24 S E M image of the Q uinto ore including graphite and mic a 106Figure 25 Q uinto graphite pilot column overflow partic le sizedistribution 107Figure 26 Q uintopilotplant graphite circuit andtestco lum n sample loca tion 108Figure 27 D ead-zone volume design 127Figure 28 Theratio of active residence-time (t [s]) to elapsedtest time (ti [s]) 129Figure 29 E xam ple entrainment withoutkinetic flotation 135Figure 30 Illustrationof theeffectof delays inflotation time 140Figure 31 P article major and minor axisillustration 142Figure 32 Response of bubble loading to bubble diameter 145Figure 33 No rma lize d solids surfaces areafluxwithtime 146Figure 34 E xamp le of aloadlimited flotationresponse 147Figure 35 Illustrationof how error in bubble measurement affects loa ding 148Figure 36 Load ingwithtime in batch mec hanical-cell 155Figure 37 M echan cialcel l , semi-log -4 4 urn cha lcoc ite mod el performanc e) 156Figure 38 Figure 36 showingbothpredicted entrainment andflotationcom ponents 157Figure 39 M echan icalcel l , cumulative recoveryof-44 um chalcoc ite 158Figure 40 C olumn cel l , semi log, -44 cha lcocite mod el performance 161Figure 41 C olumn cel l , cumulative recoveryof-44 um chalcoc ite 162Figure 42 M od el performance for the Q uinto graphite fraction 164

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Figure 43 M echa nical batch model ,without entrainment, Quinto other fraction 165Figure 44 C olum n ce ll batch mod el, Q uinto graphite fraction 167Figure 45 C olum n batch mode l,withoutentrainment for the Qu into other fraction.... 168Figure 46 M od el stage locations 172Figure 47 Total colum n solidsflow 173Figure 48 Froth-zone superficial surface-area flows 174Figure 49 Froth-zone liquid balance 176Figure 50 P ulp-zo ne superficial surface-arearatebalance 178Figure 51 C olum n l iquid flow schema tic 182Figure 52 Co mparison - model andplantperformance, IN C O 85 % chalcoc ite 197Figure 53 Co mparison - model andplantperformance, IN C O 83% chalcoc ite 198Figure 54 Co mparison - model andplantperformance, IN C O 80% chalcoc ite 199Figure 55 Co mparison - model andplantperformance, Q uinto 1.4% graphite 201Figure 56 S ensitivity of mode l response to particle sizevariations 202Figure 57 S ensitivity of mode l response to particle sizevariations 205Figure 58 G rade - recovery response, chang ing bubble size under loaded conditions....206Figure 59 M od el sensitivity to changes inrateconstant; kf [s 1] 207Figure 60 M od el sensitiv ity to changes in maxim um recovery; R o o 208Figure 61 The generalizedeffect of negative bias on overflo w grade 209Figure 62 M od el sensitiv ity to changes in maxim um bubble load; ( J ) S B 210Figure 63 Lo ad ing cond itions for induc tion time determination: 0i > 0 m 211Figure 64 Lo ad ing cond itions for solids pac king determination: 0] 0 m 212Figure 65 An gle measurements basedfromthe forwardstagnationpoint t 214Figure 66 Ac tive Are a of Attachment 215Figure 67 M ax imu m angle of co ll is ionthat leads to attachment (9a) 218Figure 68 Massrateversustime for batch mechanical-celltest#1, totalmasses 272Figure 69 Figure 68, log normal scale ..273Figure 70 B ubb le diameter versustime for batch mechanical-celltest#1 273Figure 71 44 um cha lcoc ite response on a linearplot 278Figure 72 -74/+44um chalcocite response on a semi-logplot 279Figure 73 +74 um cha lcoc ite response on a sem i-logplot 280Figure 74 -4 4 um heazelwoodite response on a sem i-logplot 281Figure 75 -74/+44um heazelwoodite response on a sem i-logplot 282Figure 76 +74 um heazelwoodite response showing on a sem i-logplot 283Figure 77 -4 4 urn other response o n a se mi-logplot 284Figure 78 -74/+44 um other response on a semi-logplot 285Figure 79 -44 urn cha lcoc ite response,withoutentrainment 287Figure 80 Figure 79plottedon a linear scale.. 288Figure 81 -44 um , recoverywithtimewithoutentrainment 289Figure 82 -74/+44 um cha lcocite response, batch co lumn 293Figure 83 -74/+44um chalcocite, batch column, recoverywithtime 294Figure 84 +74 um chalcoc ite, on a semi-logplot,batch colum n 295Figure 85 +74 jam cha lcoc ite, batch co lum n, recoverywithtime. 296Figure 86 -44 um heazelwoodite, batch co lumn , on a sem i-logplot) 297Figure 87 -44 um heazelwoodite, batch co lumn , recoverywithtime 298

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Figure 88 -74/44 um heazelwoodite, batch co lumn , on a sem i-logplot 299Figure 89 74/+44 urn heazelwoodite, batch column, recoverywithtime 300Figure 90 +74 um heazelwoodite, batch co lumn , on a sem i-logplot 301Figure 91 +74 um heazelwoodite, batch co lumn , recoverywithtime 302Figure 92 -44 um other ,batch colum n, on a semi-logplot 303Figure 93 74/+44um other ,batch c olumn on a semi-logplot 304Figure 94 Massrateversustimefor batch mec hanic al-ce ll 305Figure 95 Massrateversustimefor batch mec hanic al-ce ll on a log-norm al scale 306Figure 96 B ubb le diameter versustimefor batch mec hanical-c ell 307Figure 97 W ater volume transfer rateacross the mec han ical-c ell overflow 308Figure 98 G raphitic carbon versustimefor the Q uinto batch mec hanic al-c ell tests 309Figure 99 G raphite model response on a linearplot 311Figure 100 G raphite model response, mec hanic al-cell tests, semi-logplot 312Figure 101 Massrateversustime for batch column cel l :totalmasses 314Figure 102 Massrateversustime for batch column,totalmasses, log-n orm al scale 315Figure 103 G raphitic carbon (cG ) versustime for the Q uinto batch co lumn tests 315Figure 104 Ove rflow water rateversustimefor batch co lumn 316Figure 105 B ubb le size versustimefor the Q uinto batch co lumn tests 317Figure 106 G raphite co lulmn model response on a log-linear plot 318Figure 107 Other minera l, colum n parameters, model performance o n a sem i-logplot)..319Figure 108 P artial construction drawings of the Q uinto pilot-sca le co lumn c ell 325Figure 109 Q uinto W eibu lldistribution predicted sizedistribution 328Figure 110 Q uinto overflow size class data W ieb ulldistributionun-weighted residuals..329Figure 111 Q uinto overflow size class data W ieb ull distributionweigh ted residuals 329Figure 112 Linea rizedplotof Q uinto data acc ording to E qua tion 225 332Figure 113 R osin-R amm lerplot of Quinto overflow particle sizedistribution 333Figure 114 Q uinto G ate-Gaudin-Sc huhmann predicted sizedistribution 335Figure 115 Q uinto G ates-G audin-S chuh mann un-weighted residuals 336Figure 116 IN C O feed W eibul ldistributionpredicted sizedistribution 337Figure 117 IN C O feed W iebul ldistributionun-we ighted residuals 338Figure 118 R osin-R amm lerplotof IN C O feed partic le sizedistribution 339Figure 119 IN C O overflow W eibu ll distrubition predicted sizedistribution 341Figure 120 IN C O feed size class data W ieb ulldistributionun-we ighted residuals 341Figure 121 R osin-R amm lerplot of INCO overflow partic le sizedistribution 342

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List o f TablesTable1 Casti l lo(1988)version of the Finc h and D obby M od el 27Table 2 T ypic al f lows : .; 28Tab le 3 R ec overy equations 29Table 4 Transit ion G as Ratesfromvarious sources (Fin ch and D ob by ; 1991) 42Tab le 5 R ey no lds' number general equation functions 44Table 6 S ym bol D efinit ions for Equ ation 49 and Eq uation 50 57Ta ble 7 S olids dispersion datafromvarious sources 61Ta ble 8 Constants of SutherlandtypeE c equations 77Table 9 V ortic ity correlation for20

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Table 44 Batc h columntest#2, nic ke l assays 291Table 45 IN C O batch columntestwash-water and bias rates (Test #1) 292Table 46 IN C O batch column testwash-water and bias rates (Test #2) 292Table 47 IN C O batch column testwash-water and bias rates (Test #3) 292Table 48 Ba tch mecha nical-celltest#1, totalmasses 305Table 49 B atch mec hanic al-ce ll bubble diameter 307Table 50 B atch mec hanical-cell overflow waterrate 308Tab le 51 B atch mec hanic al-ce ll graphitic carbon assays (cG ) 309Tab le 52 Q uinto mech ancial c ell tests - overflow graphite-surface-area flux 310Table 53 B atch columntotalmasses and graphite assays 314Table 54 B atch co lumn - overflow water volume and bubble diameter 316Table 55 Q uinto superficial graphite-surface-area flux to the overflow, column tests...317Table 56 Feed Ch aracteristics, 2 n d C opper Cleane r, 1.8 m co lumn (W ilso n 1990) 321Table 57 Ove rflow Results of the 2 n d copper cleaner 1.8 m co lumn (W ilso n 1990) 322Table 58 Und erflow Re sults of the 2 n d copper cleaner 1.8 m co lumn (W ilso n 1990)... .323Table 59 Q uinto Continuous P ilot C olumn Performance - S olids Balanc e 324Table 60 Q uinto Continuous P ilot C olum n Performance - Liq uid Balanc e 324Table 61 Q uinto Overflow Size ClassData and R osin-R amm ler Predicted Va lues 327Table 62 Linearization of Q uinto Data 331Table 63 Q uii i to Overflow Size ClassData and G ates-Gaud in-Schuhm ann Va lues 334Table 64 IN C O feed stream particle sizedistribution 336Table 65 IN C O overflow stream particle sizedistribution 340Tab le 66 C ompa rison of average particle size calc ulation methods using Q uinto data. .345Tab le 67 C ompa rison of average particle size calc ulation methods using IN C O data.. .346

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Acknowledgements

Appreciation isgiven to theprinc iple sponsorsof this project forboth funding and expertadvice: Com inco L td ., Teck C orp . , INC O Ltd . , and C A N M E T operating through M I T E C .Thanks also to A n d y M u l a r , Dr.Bern K le in and thefaculty and staffof the Universi tyofBri t ish Columbia andto my colleages and studentsatD alhousie Universi ty . For the yearof1994-1995 thanks isalso extendedtothe Br i t ish C olumb ia Science C oun ci lforfunding.For1999 thanks is extended to the Universi ty of Br i t i sh C o lumbia , and the Federal HumanResources M inist ry . Other companies have generously provide d ore, advice , assays,andfunding fortesting expenses. T his l ist inc ludes Process R esearc h Associates,K lohn-Cr ippenConsultants, A L C A N , theQuinto M ining C orp. , and Com inco Engineering Services Ltd. no w CanadianP rocess Technology.

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PrefaceThis research presents three models for frothflotation. M os t of the variables usedwithinthesemodels are similar to those used by other researchers. H owe ver, this research recognizesthatflotation is an interfacial phenomenon whe rein a solids-surface-area is attached to, and thus,removed by a bubble-surface-area. The rate o f solids-surface-area rem oval is related to thebubble-surface-area flux. Th e one-dimen sional equivalent to bubb le surface areaflux is thesuperfic ial bubble surface-areaflux; & B [s 1] wh ile the equivalent to solids-surfa ce-a rea rate,attached to the bubble, is SSB[s '] A lso , the l iquid stream is characterized by the superficiall iquid flow rate (Ji [cm s*1]) and an associated sup erfic ial suspend ed-particle surface-areaflux(9SL [s1]). The concentration of solids in the l iqu id stream is charac terized by the spec ificsolids surface or the concentration of solids-surface-area( C p[cm 1]). T hus, these models usedtwo streams within the vessel: bubble (air) and liquid . Th e solidswithin the column areassociatedwithone orbothof these streams.

Industrial-scale columns are currently designed using data gathered from either continuouscolumn or ba tch mec hanical-cel l testing. A laboratory-scale colum n operating in ba tchoffers m any advantages over other testing methods. W hen comp ared to continuous c olum ntests, ba tc h colum ns use only a fraction o f the feed mate rial, have an equal residence-time ofall particle sizes and densities, and have a slightly p ositive bias. Testwork done to predictindustrial-scale column performance using a laboratory-scale column, when compared to testsusing ba tc h me cha nical-c ells, eliminates differences in flotation due to impeller agitationand turbulence, reduces the effect of entrainment and allows for an estimate of carryingcapacity.

B a t c h column testing is not yet practised due to a lack of procedures to run the test andmodelsthat relate the test results to continuous co lum n perform anc e. T his thesis defines thenecessary test procedures and introduces models that allow ba tc h test data to be used topredict continuous colum n performance. T he ba tc h kinetic mo del is used to determineflotation parameters. The continuous, kinetic m odel uses the parameters determined in the

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first model to predict continuous column performance. The final model is a mechanisticframework.

The empirical batch kinetic model determines four flotation related parameters: flotationrate constants (kf(m>n) [s 1]),entrainment proportionality constants (ke(m,n))> maximum recovery(Roo(m,n)) and the maximum solids-surface-area to bubble-surface-area ratio (

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T he adjusted parameters determined in the batch kinetic model (Equation 1)are used in acontinuous, column, kinetic model to predict performance. The continuous model useshydrodynamic principles, mixing theory and mass balances. This model is an alternative toexisting laboratory-scale, batch , mechanical-cell and continuous, column, kinetic models.

Thismodel uses the total superficial solids-surface-areaflux (&s(o,i) [s 1])that is fundamentalto flotation, rather than mass or volume. This value is determined by dividing the solids-

2 1 2surface-arearate [cm s ] by the cross-sectional area of the vessel [cm ].

T he continuous kinetic model uses superficial surface-area fluxes. A distinction is madebetween solids suspended in water and those attached to bubbles. Three hydrodynamicregions are used: the collection zone (between the feedport and spargers), the recollectionzone (between the feed port and froth interface), and the froth zone. The collection andrecollection zones are modeled using axial dispersion and carrying capacity, or loading,considerations. Thefrothzone accounts for loss of bubble surface-area. Solids bias is usedthroughout the model to predict the bulk transport of solids across zone boundaries(entrainment).

T he mechanistic model framework has been developed in response to validity questionsconcerningthatchemical rate analogy of flotation kinetics and conventional assumptionthatbubble residence time can be ignored. A bubble that has no residence-time cannot carryparticles since no collisionswillhave occurred. Bubble load increaseswithbubble residence-timeas more particles collide and attach. As such, the short residence-time of laboratory testunits meanthatmaximum loading in test- and plant-scales may bedifferent.

T he mechanistic model uses descriptions of bubble-particle attachment, induction time, asolidspacking factor, solids- and bubble-surface-area fluxes and specific surface (surface-areaconcentration), to predict solids-surface-area removal rates (Equation 2) and bubble loading(Equation3).

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sinA3SB =3BCptb(ub+up)Ec

0. 2t;VaP b J

Equa tion 2

s in 2 9..

Yc / a\i Equation 3

Equation 2 calculates the change in superficial attached-solids surface-area flux ( A 9 S B [s 1])passing through a control volume. W ithin that control volum e, 9 B [S * 1] is the averagesuperficial bubble-surface-area flux; C p is the solids specific surface [cm 1]; ' V [s] is thebubble residence-time; ub [cm s 1] is the bubb le rise-v eloc ity (assumes an upward flow );

u p [cm s 1] is the particle s ettling-veloc ity (assumes a dow nward flo w) ; E c is the coll isionefficiency; 9 e [radians] is the m axim um angle of particle contact on the bub ble; ti [s] is theparticle induction time;, ve [cm s 1] is the particle velocity along the bubble surface while incontact; d p [cm] is the particle diameter, and db [cm] is the bub ble diameter.

Equation 3 gives the bubble loading ( S B [dimensionless]); the ratio between particle-surface-area and bubble-surface-area, in terms of the solids packing ( r [dimensionless]) and themaximum angle of the solids-surface-area cap (0j [radians]).

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New KnowledgeT his thesis develops three new models. The first two: the ba tc h kinetic and continuouskinetic models, are designed to work together in order to use b at c h test data to predictcontinuous industrial-scale co lumn performance. Th ethird is a mechanistic modelthat couldbe adapted for either ba tc h - or continuous co lum n m ode ling. Th e follo win g newinformation is presented in one or more of thesem odels:

1. D evelopmen t of test methods to obtain b atc h c olum n data.2. Development of the ba tch column mod el, as a whole, wh ich is used to determine

the flotation parameters for the continuous model.3. C harac terization of the solids as beingpart of, or being transferred by, either the

liquid or gasphases.4. The use of surface-area conc entration, or spec ific surface area, to characterize

solids concentrations in flotation.5. A mo del o f entrainment based upon the bu lk transport o f solids-surface-areawithin

the liquid phase.6. Incorporation of bubble loading and solids pac king , expressed in relation to bubble

surface,intoboth batc h - and continuous kinetic models.7. A froth-zone recovery mod el based on the bulk transport o f water and the loss of

bubble-surface-area.8. Th e distinct treatment of the recollection zone as a unique mixing and collection

environment.9. The determination ofsolids residence-time in ba tc h co lum n tests.10. M od elin g of bubble loading in terms of bubble residence-time.11. Determination of the maxim um bat ch recycle rate.12. De velopment of a wash-water system designed to m inim ize radial differences in

wash-water rates,13. D esign of a test columnwithan adjustable feedport to enable the characterization

of the recollection zone.

X V l l l

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1 Introduction1.1 PurposeThepurpose of this work is the development of equipment, procedures and models to allowthe batch testing of columns to be used in the prediction of plant-scale continuous columnperformance.

1. 2 Scope1.2.1 LimitationsTheemphasis of this thesis is column flotation. Dissolved air-, cavitation/nucleation-, andelectro-flotationdevices are not discussed. Mechanical-cellsare reviewedonlyin the contexto f "batch" testing when they are used to determine column-modeling parameters. Thechemistryof flotation isonlymentioned as regards to induction time.

Theprimary emphasis is onrateconstants, recovery equations, and the effect ofparticlesize.Thefroth zone is treated in terms of bubble loading and the loss of bubble-surface-area. It isassumed that the "batch" columnacts in aplug-flow manner. The continuous models usethree stages: the froth, recollection and collection zones. Work by previous researchers isused, whenever possible, to formulate models, to predict the effect of variables, and forcomparative data.

Themechanistic flotation model has been developed solely for "batch" operation. The modelcanbe directly applied to continuous operation by means ofsubdividingthe column intosmallzones along the axial dimension. M i x i n g , however, has not been accounted for. The basicstrength of this description of flotation probably lieswithin its' future applicationwithin amodelthataccounts for l iquidand bubble movement in bothaxialand radial dimensions.

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1.2.2 Equipment GoalsThe following objectives were considered to minimize error when designing the testequipment:

M i n i m i z a t i o nof dead zoneswithinthe column M i n i m i z a t i o nofrecyclevolumes M i n i m i z a t i o nof required sample size Production of an even wash-water distribution in the froth zone Production ofavariable feed port elevation

The equipment was designed to test small amounts of feed material. This allows multipleteststo be run orteststo be performed on core or small grab samples.

1.2.3 Modeling GoalsThree flotation models have been constructed each with a different purpose. The first is a"batch"-model that may be used with either "batch" mechanical-cell or column generateddata. The second is a continuous-column modelthat uses the flotationparameters generatedby the first model. The third, is highly mechanistic modelthathas been developed, but notyet tested, in order to predict solids removal as a function of bubble residence-time andinductiontime and to predict loading as a function ofparticlepacking on the bubble.

1.2.3.1 BatchModelCurrently, both laboratory-scale "batch" mechanical-cells and test- or pilot-scale continuouscolumns are used to predict performance of industrial-scale columns, or to provide sizinginformation. Mechanical-celltestsare both simple and inexpensive but error may be high dueto entrainment or turbulent conditions. In contrast, continuous pilot- or laboratory-scalecolumn tests offer the potential of more accurate sizing parameter determination but at

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substantially higher costs. The"batch" modelwasdeveloped inorder toprovideanalternative tothesetwo types oftestsand was designedwiththefollowingobjectives:

Togenerate flotation parameters, including kinetic rate constants, entrainmentconstants, maximum recovery and bubble loading,

Toreflect thesurface transfer natureofflotationbyusing specific surfaceandsurface-are;: fluxeswithinthe model,

Tosimplifydatacollectionand model usage, and To estimate column flotation parameters from either "batch" columnor"batch"

mechanical-celldata

1.2.3.2 ContinuousColumnModelColumn sizingand performance models, already in existence, have been extensivelymodifiedto receive data fromthelaboratory-scale "batch" kinetic models developed in this thesis. Inaddition, theentrainment modelsofother researchers have been incorporated inordertosimulate negative bias conditions. The continuous model also uses specific surfaceandsurface-area fluxes.

1.2.3.3 Mechanistic BubbleL oad ModelA mechanistic particle load (carrying capacity) model frameworkhas been developedas analternativetoearlier models. These relationships remain untested,butpresent amethod ofpredictingsolids-surface-area removal dependent upon bubble surface flux. The mechanisticapproach also characterizes thebubble load as a function ofbubble residence-timeandincorporates bothinductiontime and particle packing on the bubble surface.

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1.3 BatchFlotation

1.3.1 Mechanical CellMechanicalflotation cells form the bulk of flotation separation equipment on both industrial-and laboratory-scales. "Batch" mechanical-cells are the standard for laboratory test-work. Asmall flotation cel l , typically between two (2) and five (5) liters in volume, is filled withslurry. Reagents are added and conditioned prior to the aerationthat causesflotation. No newsolids are placed into the system while the test is underway; air, however, is appliedcontinuously, and water may be added to make-up the volume lost through flotation.Technicallythesetestsare "batch" onlywithrespect to the solids content.

1.3.2 ColumnCells

1.3.2.1 IntroductionA column flotation cel l is a vertical device wherein slurry containing the minerals to beseparated is passed through a column, from upper feed to lower underflow against a risingswarmof bubbles. Materialthatis collected by the bubbles, or entrainedwithoverflow water,rises into a froth zone where a descending flowof water washes the entrained material backinto the collection volume. This system takes advantageof the concentration gradient thatexists as the newly formed bubbles are exposed to the lowest concentration of floatableparticles at thebaseof the column (Rice et. al.; 1974).

Test columns have been used in solids batch for many years to either fill columns, or whilethe system is on stand-by to preserve feed. This mode of operation, however, has not beenused to performtestwork.

2 2Industrial-scale columnsrangein area from 0.1 m to 20 m ("Ac")and between 3.0 and 18 min total elevation (h(V)). Application ofthese vessels include the flotation of such diverseminerals as: clay ("dp" < 10 um), potash ("dp" > 4000 um, "p s" 1.7) and galena ("ps" 6.5),and also the removal of micrometer sized organics in solvent extraction plants, newspaper de-

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inking (Petri and Dobby; 1993), ion concentration (Walkowiak; 1991 and Mezhov et. al.;1992), cel l separation, biologicalwaste material(Martiet. al.; 1994) and other environmentalapplications.

1.3.2.2 SizingDesign of the earliest columns was based on "rules-of-thumb" residence-time and volumeconsiderations. These columns were typically retrofit applications and were made withvolumes equivalent to the mechanical-cells they replaced. Dobby's 1984 thesis, the firstattempt to model ilotation columns used in mineral processing, introduced mixing theory,bubble size and particle settling to the field. Two empirical sizing-parameters have beenadded since: from M c G i l lUniversity;area carrying capacity (Espinosa-Gomez, Yianatos, andFi n c h ; 1988), and from C E S L ; the lip loading capacity as presented by Amelunxen (1990).To-date, models have assumed a fractional recovery through the froth zone. Additionalworkhas been done on column froths in recent years;thesestudies, however, are beyond the scopeo f this thesis. The following are some of theparametersand variablesthat are important inthe kinetic models presented:

Rate constants, "kf" [s"1], Entrainment constant, "ke" [dimensionless], Maximumrecovery; "R>"and Maximumand actual bubble load;

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Currentmethods, as practiced by industry to collect flotation column scale-up data, are bothtime-consuming and expensive. Pilot-column testing is performed for accurate modelparameter determinations. Bothcarrying capacities andrateconstants are inferred from thisdata by adjusting the current continuous model parameters sothatthe model predicts thetestresults.

Abench-scale batchmechanical-cellis often used, rather than a continuous column, to reducecosts. Prediction error, however, is probably increased due to different bulk transport,col l i s ion and attachment environments. The use of batch mechanical-cell generated datameans thatan increased design safety factor must be used. Some columncel lsuppliers designcolumn cells directlyfrom batch mechanical-celltest results when carrying capacities areknown and the particle size; "dp",rangesbetween thirty (30) and seventy (70) micrometers.This type of estimation, however, cannot predict the behavior of middlings, locked, orentrained particles and ignores differences in the col l i s ion and attachment environmentsthatexistbetweenthesevessels.

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1.3.3 HydrodynamicZones

1.3.3.1 IntroductionColumnsoperated in either continuous- or batch have two distinct zones: the pulp and frothzones. The pulp zoneisnecessary for operation; the froth zone, however, may be omitted incertain applications suchasscavenging operations. These zones areshowninFigure1.Figure 1also showsthesub-zones ofthe pulp-zone:therecollection, feed, collection andsparger zones.

RecollectionzoneFeedzone

Collectionzone

Sparger zone

FrothZone

PulpZone(s)

Froth Interface

Feed

SpargerElevation

Figure 1: Columnhydrodynamiczones

1.3.3.2 PulpZone(s)The pulp zone is thevolume between the froth-pulp interface and thelowest descentofbubbles producedby thespargers. The bubble-surface-area rate introducedat thespargerelevation,or thesuperficial bubble-surface-area flux (&B(sPa)[s"1]),doesnotcarry any solidssurface (SsB(spa) [s"1]=0). The sub-zones withinthepulp "zone"arecharacterized, under

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normaloperation, by a continuous swarm of independentlyrisingbubbles in quiescent slurry.Hydrophobic particles may collide and attach to thesebubbles. Those solids that becomeattached are carried upwards (& S B X [ S 1 ]>0)and eventually reach the interface between thepulp and the froth zone. The solids areathat attaches to the bubbles is removed from l iquidsuspension (&SL [s1]). Hydrophilic particles that do not attach to the bubble phase fallthrough thecollectionzone and flowout the column underflow.

The pulp zone may be further divided into four sub-zones (Figure1). The volume below thefroth interface, and above the upper turbulence of the feed, is often called the "re-collectionzone". The volume of moderately intensemixingwhere the feed is introduced may betreatedseparately and is termed the "feed zone". Asimilarzone may be defined around thespargerscalled the "sparger zone" and the remainder is the collection zone. Neither the "feed", northe "sparger" zones are used in the models presented in this thesis.

1.3.3.3 Froth ZonesThe froth zone is primarily gas, with solids and l iquid in the lamella between the bubbles.Wash-water, whichmay be added at the top of the column, filters down through this bed toremove entrained particles from the froth. Particles attached to a bubble w i l l overflow thecolumn unless bubble coalescence occurswhichmay force a fraction of the floated particlesto return to the waterphasein the froth.

The froth zone may be further subdivided intothreesub-zones. The volume above an in-tankwash waster system is characterized by a dry, draining froth and may be termed the "dry frothzone", or "drainage zone". The volume under the wash-water addition system is a washedfroth and can be termed the "wet froth zone" or "washing zone". The interface between pulpand froth zones may also be considered aseparatezone. The models presented in this thesisconsider the froth to be a single zone.

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+Washwater

Feed Water+ ++ + +Figure3:Biasschematicwherein arrowsshowdirectionofwater flowandwidthshowsvolumeflow. Theleft sideof the diagramshowspositive bias conditionswhereas therightshowsnegativebiasconditions.

A n average positive bias across the entire column cross-section may contain within itselfareasof substantial negativebias ifwash-wateraddition is not uniform. A l s o ,if channeling ofthe water occurs, large volumes ofwash-water w i l l descend through a small area therebyloweringtheaveragebiasthroughoutthe remaining cross-sectional area.

1.3.4.2 AirThere are limits to the airratesthat can be applied to a column. The maximum air rateisdependentupon many factors including slurry viscosity, density, and downward flow ratesasw el l as the bubble-size distribution. In general, air cannot be added at a superficial rategreaterthan the bubble rise-velocity at any column elevation. The minimumamountof airthat can be added is limited by the stability of the froth zone; lowerratesresult in a slowermovingfroth thatmaydecreasethe froth stability. The optimum airrateis the minimumratethat provides sufficient bubble surface area to remove the solids surface. Excess rate wi l lincrease the gas fraction contained within the vessel, andthus, reducesolids residence times.Typical ly, columns are designed with a maximum superficial air rate (Jg), at local pressureandtemperature,ofabout3[cm s"1] for slurry flotation. The column minimum superficial air

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ratevaries with column cel ldesign, wash-water rate, reagentsand froth height, howeverareusuallyin therangeof 0.5 cms"1.

1.3.4.3 FeedFlowColumns usually receive slurry feed at a rate; " J s " ,ofbetween 0.3 and 1.5 [cm s"1](volumetric feed rate divided bythecolumn cross-sectional area). Specific designs canbemadetoshift this range. For example:al iquid flow increase reduces the maximum airrate.The practicalfeed-ratelower-limitmust be high enough to prevent solids from settling out ofsuspension withinthe columnorsupport piping. Particlessettlewithin the slurry,thussolidsresidence-time w i l lbe less than slurry residence-time.

1.3.5 Mode of Operation

1.3.5.1 IntroductionThere are two general modes inwhichaflotation cel lmaywork:batch and continuous. Batchmeans batch with respect to the solids only. Flotation,on a plant- or pilot-scale,iscontinuous. Onalaboratory-scale, batch column flotation has been possible but, in thepast,testshave been performed using continuous operation. Another operational mode variantisthe "semi-batch" column wherein a positive bias is maintained by water addition andasurplusvolume accumulates.

1.3.5.2 Continuous-ModeTo-date, only continuous columns have been used for both test-work andindustrialapplications. Feedentersthe column continuously and is met byacontinual stream ofrisingbubbles in the column. Wash-water may or may not be used. Bothoverflow and underflowproducts arealways being produced. Figure4 is a schematic diagram ofthis modeofoperation. Some solvent-liquid extraction columns collect the organicphasecontinuously onthe top of the column but only dump itperiodically.

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Feed

A i r

Washwater

'Overflow

-Underflow

Figure4: Continuous column-cell operation

1.3.5.3 Batch-ModeA batch column receives an ini t ial solids feed. Thereafter, thefeed isrecycled fromtheunderflowto thefeed. Wash-water is addedtomake upthevolume lostto theoverflow. Thisoperation isactually "pseudo-batch" since both water and air areadded to thesystem. Tomaintainapositive biastheslurry-froth interface levelrises duringatest.

Washwater

Overflow

Underflow

Figure5: Batch column-cell operation

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2 Background2.1 IntroductionThis section reviews information relevant to this research found in literature. The modelingtheorypresented: types, development and application of models, is used as a framework forthe batch surface-area models. A selection of published flotation models are reviewedthatpertain to column or batch flotation models and many of the relationships used in thesemodels, such as carrying capacity, entrainment, rise or settling velocities, the axial dispersionmodel,kinetics and collection efficiencies are reviewed.

2.2 Modeling Theory

2.2.1 IntroductionSimulation, or modeling, of flotation vessels is complex because of the large number ofprocess variables involved (Lynch et. al.; 1981 and Frew and Davey; 1988). Processconditions and mineral kinetics do play an important role (Schuhmann; 1942), and should beconsidered in a working model. The parameters of many models, such as most macro-phenomenological kinetic models, deal with many process variables as groups. Therefore,little information about the process itself can be inferred (Reuter and van Deventer; 1992).There are three aspects of flotation thatmust be taken into consideration when models areconstructed and used: equipment, operational and chemical factors.

E q u i p m e n t Factors C e l ldesign, type Configuration Control Froth removal and

depth

C hem ic a l Reagents Dissolvable species Ionic concentration

Operat ions Mass flowrates Minerals, liberation Particle size Pulp density Solids densities A i r flow

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There must be an understanding of the advantages and disadvantages of columns, incomparison to other flotation vessels, in order to identify proper applications. A goodmathematical model is an important tool for systematic and consistent process analysis(Sastry; 1990). A good model can assist in each oftheseaspects. Modeling and simulationcangive the fol lowing:

A structured understanding anddefinitionof a process, A basis forplanning,evaluation,optimization,and process control, A trainingtoolfor operational staff and Aminimizationof experimental costs.

Inorder to achievethesegoals, the model must fit the experimental data independently of theprocess and also be accurate.

Thepurpose of a model is torepresenta "real" process, or an assembly of elements linkedtogether. These elements may be as small as a unit operation or encompass an entireprocessing plant. A model is an approximation of the "real" systemthat is neither completenor exact. There are two types of models: physical and symbolic (Jacoby and K o w a l i k ; 1980).These models and their inter-relationships are shown in Figure 6 (Sastry; 1990)

M o d e lPhysical Symbolic

Bench Pi lot Verbal Mathematical

Experimental Computer

Parameter EstimationM o d e lValidationorApplication

Figure6: Types of models and their components fromSastry (1990).

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2.2.2 Physical ModelsPhysical models, scale models, drawings and diagrams are concrete representations of the"real" system. Scale models are used for experimentation and the gathering of informationthat concerns the "real" system. Common scale models include bench- and pilot-scales.Szekelyet. al. (1987)statethatfor meaningful information on the "real" system scale modelsmust be constructed and used appropriately, including considerations of geometric, dynamic,kinematicand process similarity. The batch column is a scale model.

2.2.3 SymbolicModelsSymbolic,or process models, may be either verbal or mathematical. A verbal model may beeither knowledge or fuzzy logic based such as Reuter and van Deventer (1992). Thesemodels are "symbolicstatementsof [the] structure and behavioralaspectsof the real process"(Sastry; 1990). There is a l inkbetween physical and symbolic modelsbecauseinvestigationo f the physical leads to knowledge used to construct and verify the symbolic. Simulationsusingsymbolic models are used to plan laboratory- and pilot-scale testing and are helpful toplanor modify "real" systems.

2.2.4 Mathematical ModelsMathematical models may be stochastic or deterministic in nature, or may be comprised ofelements of both. Stochastic models use probability, or chance, and are rarely used in mineralprocessingwith the exception of the modeling of specific phenomena such as particle-bubbleattachment. Deterministic models, using either empirical or mechanistic methods, assumeaconsistent, repeatablebehavior. E mpi r i c a lmodels are based on statistical correlation betweencausal elements and results, while phenomenological (mechanistic) models relate themechanics of the process mathematically to the result. In mineral processing,phenomenologicalmodels are not usuallylinkeddirectly to the basic chemistry or physics ofthe system, but are empirical models of sub-processes; thus most ofthese models must beconsidered to be semi-empirical innatureor empirical on a basic l e v e l . Mechanistic models

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have the potential to be more accurate when extrapolating from known performancerelationships (Box et. al.; 1978).

2.2.5 DevelopmentM o d e l development is an iterative procedure used to maximize utility and accuracy whileminimizing research costs. One possible development flowchart involves the following fivestepiterative process (based on Sastry; 1990). ,

1. Determine the scope and purpose of the model. Possible objectives are tounderstand the fundamentals of the process, to determine scale-up criteria andto optimize existing circuits. These models may also be used in the design andexecution oftestwork and the design of full-scale processes.

2. Collect information: existing knowledge should be surveyed including adescription of mechanistic information, experimental data of others, and priormodels. A determination should be made whether more experimentation isneeded based on the collected information.

3. Experimentation: A test work program should be designed and carried out todetermine any information missing from the collectionstage.

4. M o d e lDevelopment: The model is composed of an axiomatic and mechanisticdescription of the process. Thefollowingelementsare used:

Statevariables andparametersare defined. Laws of conservation are identified. Mathematical model equations along with inferential, analytical or

numerical solutions are developed. A l s o , all sub-processes that causechanges within the vessel are characterized either mechanistically orempirically.

A l l necessary equations including a clear statement of interdependence ofvariables andparametersare introduced.

A l l ini t ialand boundary conditions are defined.

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1. Those models that are used to predict industrial-scale equipment performancetypically are concerned with component recovery and are often used duringoperation to perform material balances, minimize circulating loads, anddetermine thebestoperating conditions of a vessel or circuit

2. Those modelsthatare used to size and configure equipment.

2.2.7 De ter min ati on of M o d e l ParametersThe traditional method of flotation kinetics and performance evaluation has been throughbench-scale tests(physical models), performed over a sufficient time period to determine finalrecovery values. This type oftestgives a single point on a time-recovery curvethatcan beused to estimate the kinetic rate constant if both entrainment and maximum recovery areassumed. According to Bushell (1962), and K l i m p e l (1980), this assumption is unjustifiedwhen relating bench-scale tests to full-scale mechanical-cells. In order to minimize time-dependency error, afull time-recovery curve is needed (Dowlinget. a l ; 1986).

The confidence in this method ofsizingparameter determination may be further eroded whencomparing different types of machines as is the case when batch mechanical-cell rougherresults are used to predict full-scale column performance. In order to use this information anassumption must be made of time-recovery equivalency between these tests and the "real"flotation systems.

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2 .3 CurrentFlotationModels2.3.1 IntroductiontoKinetic ModelsThe mechanisms governing flotation ineither mechanical-cellsorcolumns are not fullyknown. Asaresult, mathematical models cannot be formulated directly from theory. Thus,tosome extent, all current models are empirical. These models may or may not be based onthekineticflotationrate.

The kinetic models currently availabletypicallyconsider the frothas asimple zone wherein anexperimentallydetermined fraction of the solids particles are rejected and returnedtothe pulpzones. While there areinsights into the actions occurring, thesemay not bemodeled.Performance maybesignificantly different between laboratory-, pilot-,andindustrial-scaleunits depending on froth stability,mixingand residence-times.

Batch mechanical cel lmodels are consideredbecausethey form the basis of the batch columncel lmodel developed in this research.

2.3.2 BatchMechanical-Cell Kinetic Models

2.3.2.1 IntroductionThe mechanical flotation cel lcan betreatedas asingle entity (singlestage),orasamultistagesystem wherein twoormore entities (Harris; 1978) are joinedtoformacomplete, coherentsystem. In twostagemodels the pulp and frothstagesaretreatedseparately. In modelswithmore than twostages,the froth and pulp are subdivided into additionalstages.2.3.2.2 SingleStage .The U S E V l P A C simulators ofmechanical-cellsuse either (1) sub-populations of each mineralincluding non-floating, slowandfast floating components (FicheraandChudacek; 1991,MehrotraandPodmanabhan; =1990,andK e l l y andCarlson; 1991),or (2) adistributionof

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kineticrateconstantswith particle size. The flotation rateconstantof singlestagemodelsis acomposite ofmany mechanisms including flotation, drop-back, and operator technique.Themechanical-cell isassumed to be aperfect mixer. Thetotal recovery predicted bytheseequationsmay be modified by anentrainmentfactor suchasthato fKirjavainen (1992).

2.3.2.3 TwoStage2.3.2.3.1 Introduction

Arbiter andHarris (1961) propose a two-stage mechanical-cell kinetic model. HarrisandRimmer's (1966)version ofthismodel is shown in Figure7.

V(froth),m(froth), C(froth)

am(Coi)

Q(F),C(F)

Q(0),C(o)

brri(froth)

V(Coi),m(Coi),C(Coi)

Q(U),C(u)

Figure7:Two-stageflotationmodelof Harris andRimmer(1966)

In Figure7,"Q" [cm3 s"1] is thevolumetric flow rate;"C " is thecomponent concentration(wheretheunits dependonthetypeof concentration); "a" [s"1] is thetransportratefrom pulpto froth; "b" [s"1]is thetransportratefrom frothtopulp; "m" [g]isthe componentmassand" V " [cm3] is the stage volume. Thesubscripts used inFigure 7 are "F " (feed)," U "(underflow), "O" (overflow), " C o l " (collectionzone)and "froth" (froth zone).

Neither theArbiterandHarris (1961)orHarrisandRimmer (1966) models describe interstagewaterorgas flow. The Harris and Rimmer(1966)modelassumesthateachpulpstageisperfectly mixed. This assumption is valid for smaller sized particles considering thatmechanical-celltestunitsoperatewith Reynolds' numberson theorderof10,000 (Arbiter and

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Harris; 1961). In the Harris and Rimrner (1966) modelmassis assumed to flow into, and outof, eachstagefrom theother.

2.3.2.3.2 PulpStage

The pulp stageismodeled using continuously stirred tank reactor ( C S T R ) kinetics whereinmass recovered is a function ofthe mineral/particle-size rate constant and residence-timeunder perfectly mixed conditions. Input to the system is fromafeed stream and "drop-back"isfrom the frothstage. The pulpstagebalance is found in Equation4(Harris; 1978).

d m (co. ) n m(co,) , Equation4r =QFf(F) ~ +bm{frolh) - amCol)(Col)

In Equation4thesymbols usedare thesameasin Figure7. In thepulp stage,the massbalance consists ofmass entering via new feed (assuming continuous operation), minusthemassremoved to the froth, plus anymassreturned to the pulpstagefrom the froth.

Sastry and Fuerstenau (1970) performed an analysis of the flotation column. They assumedaxiallydispersed plug-flow of both l iquidand gasstages.2.3.2.3.3 FrothStage

The frothmass flow ratebalance ofHarris(1978) is shown as Equation 5.

dm{frolh)_ W) E

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There are many frothstagemodelsincludingthose byM o y s(1989), Cutting et. al. (1986), andRoss(1991b). Bisshop and White (1976)claimthat the most importantraterecovery factor isthe froth residence-time.

2.3.2.3.4 Stage Interactions

Sadler (1973) used a single-stage, gamma distributed, first-orderratecoefficient model and abubble-surface-area model to describe flotation. The transfer of floatable mineralfrom thefrothback to the pulp is related to bubble-surface-area and loading density.

I fa constant volume is maintained then Q(F)=Q(0)+Q(U) - At steady stateboth Smooth/dt"and"

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2.3.2.4 Mult iStageHarris(1978) mathematically modeledathree-stage system and showed that,atsteady state,the model reducesto thetwo-stage equations, althoughtherateconstantsare acomposite ofthethree-stagerates.An example of a three-stagesystemisthatof Hanumans and Wil l iams(1992) who modeled the frothstageas twoseparatezones.

2.3.3 Column Kinetic Models

2.3.3.1 IntroductionC o l u m n kinetic models of other researchers formthebasis of the continuous kinetic modeldeveloped in this thesis. Therearethreecommon kinetic models of column flotation.Thefirstis awidelyused model by Dobby (1984) and F i n c hand Dobby (1990). The secondis avariationofthe firstbyY o o n and Lut t re l lat V i r g in i a Polytechnical Institute ( Y o o n ; 1993,Y o o n and Lut t re l l ; 1989 and Y o o net.al.; 1991). The third, alsoavariant of the first,is byAlfo rd (1992). A l l threeofthesekinetic modelsuse theaxialdispersion theory inthepulpzone. Bo t h the F i n c hand Dobby model and the Y o o n and Lut t re l lmodel assignarecovery tothe froth zoneorincorporate froth-zone affects intothepulp stage model. Al fo rd usesasingle-stage model. The m i x i n g parameter is thePeclet number "Pe"orits' inverse "Np".WhenthePeclet number approaches infinity,aplug flow condition exists.APeclet numberthatapproaches zerorepresentsperfect mi x i ng .

2.3.3.2 SingleStageThe J K M R C model (Alford; 1992)is asingle stage equivalentofthe Dobbyand Finch's(1990) two-stage model. Thismodel assumes that froth depth and wash-water ratehaveaninsignificant effect on the flotation rate parameter. In addition, Al fo rd (1992) usesanempiricalmodification of therateconstant to compensate for airratesandviscosity(Equation7).

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/ y Equation7kf(m,n)~C\{m,n){jg)

1.5J E, Equation8kf=-g k

Equation 7 is an empirical equation that is similar toEquation 8(Jameson et. al.;1977).Equation7useswaterviscosity (pi[g cm'V1])and slurry viscosity (usi)but isonlyvalid whenthe Peclet numberislow andthereis ahigh froth recovery. A l s o inEquation7" C i (m , n ) " is adimensionless characteristic of themineral andparticle size as w e l l as afunction ofspargercharacteristicsand frotherconcentration. "Rf" istherecovery throughthefroth zone.

In Equation8,"k"[s"1] is thekineticrateconstant;" Jg "is thesuperficialgasvelocity [cms"1];" d b " [cm]is thebubble diameterand " E k "[dimensionless] is thecollection efficiency.

The adjusted gasvelocityofEquation 7 isfound inEquation 9. This equation attemptstocompensatefor the gasvelocity term ofEquation 7 byusing loading andcolumn diameter(Alford; 1992).

bxAJQg15 Equation9m0

In Equation9,"bi"[gcm"2] isindependentof mineral and particle size. A l s o , in Equation9,2

" m 0 " [g] is themassoffloatable particles thatare in theoverflow stream; " A c "[cm] is thecolumn cross-sectional areaand " J g "[cms"1]isthesuperficialgasvelocity.ReferringtoEquation 7, A l f or d (1992)usesEquation 10topredict the" C i (m , n ) "value.

C = f d 2e~ dp Equation10\(m,n) \(m) p

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The model parameters of Equation 10 are determined by curve-fitting techniques and areapplied for "scale-up" with the axialdispersion model. Alford's (1992) curve fitting wasdone to minimize the chi-squared statistic using the Levenberg-Marquardt method ( Koj ov i c ;1988,as referred to by Al fo rd ; 1992). Redundancy is a requirement of this method and Boxand Draper (1987) recommendthatthe number of observations befiveto ten times the numbero festimated parameters.

The A l f o r d (1992) J K M R C model is an empirical simplification of the Dobby model thatassumes the froth and pulp zone cannot be, or need not be, separated. This model alsoassumes thatbaffles do not reduce mixing andthatthe effect of variations in froth depth andbias,withintherangeused by the model, haveminimalimpact upon performance.

2.3.3.3 Two Stage -Finchand DobbyM odelThe development of the F i n c h and Dobby type model beganwithDobby's 1984 P h . D . thesis.Since that time it has been refined by many others. The purpose of the F i n c h and Dobbymodelis to determine modelingparametersfrom continuous pilot-scale columntestsand thento use theseparameters to predict the performance of continuous industrial-scale columns.This model was not designed to use parameters determined in batch mechanical-celltests,althoughwithmodifications it has been used forthatpurpose.

Dobby and Finch's two-stage model is based on axial dispersion model and kinetics. Nocomprehensive froth model is given and recovery through thisstage is set as an input value.A n assumption is made thatperformance is dictated within the collection zone. The modeltakesinto consideration solids loading of bubbles through a carrying capacity term at the topof the collection zone, solids overloading in the froth, misalignment of the column andbaffling andusesmineral and size-classrateconstants(Castillo; 1988).

The Dobby and F i n c hmodel is similar to thatof Harris and Rimmer (1966) inthat both aretwo-stage. The firststageis the collection,or pulp zone. Feed to this section is composed ofvessel feed and drop-back from the froth zone and includes water flow terms. Recovery iscalculated in the collection zone by using a dispersion model with a cap on recovery as

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dictated by carrying capacity constraints. Entrainment of non-attached particles, eitherwithinthe bubble wake orwithin the boundary layer, is neglected under a positive bias assumption.The collection rate is assumed to be first-order with respect to concentration. Material notfloated in thecollection-zoneflowswiththe underflow stream.

Recovery in the Dobby and F i n c h (1986a) collection-zone model is a function of rateconstants, particle retention time andmixing. In this model, the solids are assumed to have thesame axial dispersion as the l iquid. The equations in Table 1 are used to simulate columnperformance.

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Table1:Castillo(1988)versionoftheFinchandDobbyModelDescription Units Definition , '

A c Ar ea c m 2 AAC =nd2c Adee C o l u m ndiameter cm A=dCeNb B

h(Coi) Collect ion height cm h(Coi)- h(V)-h(f)-h(spa) CQ s i Volumetricrate c m 3s"1 MSL(F) | mSL(f) | MLL(F) + MLL f)

PSL(F) PsL(f) PL

D

Ul Interstitir.lliquidflowrate

cms"1Ac

E

tl L i q u i dretentiontime sF

Up Particlesettling-velocity

cms"1gdl{pp-p){\-e,)1J1 8 ( l+0.15Re/8 ' )

Masliyah(1979)

G

tp Particleretentiontime

s u,u,+up

H

D i DispersionDobby(1984)

c m 2s"1 Di =0.0547dCeJg 3[l+(l100adCe)]2 I

N p Dispersion Np = 7 Th{Co,)\J, +Up)

J

N p Dispersionnumber

1 0 . 0 5 4 7 W c . / 3Pe~ ( j \ +u h,r ,x1 P i' (c

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In Table 1, n i s L [g] is thefeed solidsmass suspendedin the l iquid , " mL L ( S ) [g] is the l iquidmassinthel iquidphaseinthefeed; " P S L "[gcm"3] is thesuspendedsolids density; " P L "[gcm"3] is the l iquid density; "Np"is thedispersion number (inverse Peclet number); "Nb"is thenumber ofbaffles; " J " [cm s"1] is the superficial l iquid flow rate and "a" is thecolumnmisalignment in radians. The subscripts usedare"F"(feed)and "f ' (drop-back).The particle Reynolds'number(Rep Table 1-N)andparticle settling-velocity(up[cms"1]Table 1-G)calculations use the solids fractional holdup within the l iquid phase (es). Thissolids holdup canbecalculated using Equation11.

V Equation11

s= K+v,

In Equation 11," V s "[cm3] is solids volume; " V "[cm3] is the l iquid volume and "ss"[dimensionless] is thevolumetric solids holdup. Table 2 shows typical velocity rangesasstatedby Dobby aridF i n c h(1986a). Columns mayberunundernegativebias conditions.

Table2:Typical flowsSymbol Description Units Minimum Maximumh Superficialgas velocity cms" 1.0 3.0Jsl Superficialslurryvelocity cms~' 0.5 2.0Jb Superficialbias velocity cms"' 0.1 0.5

Vesselrecovery is calculated using Table3limitedbythecarrying capacity.

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Table 3:RecoveryequationsSymbol Description Units D e f i n i t i o n SourceR-s(v) Vessel recovery R Vo'V*) Do bb y (1986a)S(V) D D 4-1RRs (Coi) Colle ction-zone recovery Wehner -W i l h e l m Lev enspi el (1972)Rs(froth) Froth-zone recovery 0.5 C a s t i l l o (1988)

0.4 -> 0.8 Falutsu (1989a)rrio/t Overflow solids f l o w rate gs-'C a ( m a x ) M a x im u m carrying capacity g c m " 2 s"' Ca, ,=h n ,xd , N Espinosa, Yianatos

and F i n c h (1988)C a Carrying capacity g c m " 2 s"1 Ca= ^

The total vessel recovery (RS(V)) is calculated when determining overflow recovery. Then thesolids flow ratein the overflow (mo/t" [g s"1]) is calculated using the feed dataand recovery.The actual carrying capacity (Ca [gcmV])is then calculated and compared to the maximumcarrying capacity (Ca(max)). If" C a " isgreaterthen "Ca(max)", recovery (RS(V)) is decreased andthesecalculations arerepeated. Material rejected from the froth zone is then calculated, addedto the feed (for collection zonepurposes only), and the entire model is iterated. Note: "dso"size is quoted in cm.

The secondstage,or froth zone,assumesa certain recovery of materialpresentedto it from thecollectionstage. The material recovered then becomes overflow product while everything elseisreturned to the collection zone through drop-back (Table 3).

Carrying capacity is determined using therelationship ofEspinosa-Gomez, F i n c handYianatos (1988) as shown in Equation 12.

C a M =bn P s { o )dm { o ) Equation 12

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2 1 ' * 3In Equation 12, "Ca( ma x)" [g cm" sec"]is the maximumareacarrying capacity; "p S(0)" [g cm"]istheoverflow solids density and "dgo(O)" [cm] is theoverflow particle sizeatwhich 80% ofthe particle massissmaller than. The value of "bn" [s"1] dependsupon theunits used intheequation. Use of this model involvestheminimization of errorbetween actual and predictedperformance by theadjustmentof modelparameters.2.3.3.4 TwoStage -V P I ModelY o o n et. al.(1991, 1993) also model columns using axial dispersion and theWehnerandW i l he l m (1956) equation. Intheir calculations, thePeclet number (Pe[dimensionless]) iscalculated using the empirical relationship found in Equation 13.

] 0 5 Equation13Pe =0.6\\col)

0.63

J

Equation 13usescollection-zone height (h(C oi) [cm]), columndiameter(dc[cm]), superficialslurry velocity (Jsi [cms"1]),superficial gas velocity (Jg [cms"1]),and fractional gas-holdup (sg[dimensionless]). Y o o n et.al. (1991, 1993) usesJameson et. al's(1977) relationship, foundin Equation 8, to further relate the flotation rate parameter (kf[s"1]) to the mechanisticcollectionprocess using aprobability factor thatisequivalent to Dobby's (1984) collectionefficiency "Ek". Yoon's carrying capacity limitation is themaximum solids-surface-arearemoved as abubble-surface-area percentage. Furthermore, amaximum gasrateisused thatis determined bycoalescing andslugging factors. Similar equations can bederived fromDobby(1984) and the bubble size and floodingpatternsfrom F l i n t (1989).

2.3.3.5 SummaryThe simplifications andempirical natureofthe single-stage J K M R C model (Al fo rd ;1992)make this model difficult toconvert for usewith bubble loading andparticle surface-area.The VPI model follows theform ofthe F i n c h andDobby model with modifications to themixingterms. However,moredataisavailable on the F i n c hand Dobby model andit ismorewidelyused. Thus,theF i n c h and Dobby continuous kinetic column model isused as abasiso fthe continuous kineticsurface-areamodel in thisresearch.

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found this equation lacking. Xu (1987) estimated the gas carrying capacity (Ca) as therelationship shown in Equation 15. This equation is the same as Equation 14 when theproportionality constant, " b i 4 " [s"1], is replaced by 7rJg/(2db)"and "dgo" [cm] is replacedwith" d p "[cm].

C a = JsdPPs(o) Equation 152 d

InEquation 15, "J g " [cm s"1] is the superficial gasrate;"db" and "dp " [cm] are the bubble andparticle diameters respectively, and "pS (0)" [g cm"3] is the average overflow solids density.Thisequation should only be used when mono-dispersed particles are assumed.

X u (1987) calculates the bubble density as shown in Equation 16 by assuming that themaximum loading on the bubble is 50% ["

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Ityokumbul and Trubelja (1998) propose the relationship shown in Equation 18 for area"carryingcapacity" (Ca) when a mono-dispersed solidis assumed.

Ca = b.240J.

Ps{0)dPEquation18

18

Equation 18 issimilar to Equation 14 when "big" is equivalent to b i4[ 2 4 0 J g / d b ]and "dso" isequivalent to "d p". Flotation occurs in slurries of poly-dispersed solids, thus, a value must bedetermined for particle size such as the "d 8 0 " term ofEspinosa-Gomez,Yianatos and F i n c h(1988). Ityokumbul and Trubelja (1998) experimentally determined the value of ( J > S B at 0.5,which issimilartothatassumed byX u (1987).

2.4.3 Lip LoadingAmelunxen(1990) presented a second empirical "carrying capacity" relationship after findingthat " C a " [gcnfV1]decreasesfor column diameters greaterthan one meter. This relationshipassumes that the total froth-zone lip-length is a constraining factor. Amelunxen's (1990) liploading; C L [g cm''s'1],usesslurrymass ratherthan only solidsmass(Equation 19) forzinc.

Equation 19 is an empirical equation where the constant "big" and the exponent a i 9(determined to be 0.3 for sphalerite) are determined strictly by regression analysis. Both theproportionality constant b 2 and the exponent "0.3" vary depending on the operation inquestion.

2.4.4 SummaryNeither the simplification of carrying capacity found in Equation 14 nor the empirical liploading relationship (Equation 19) include all variables necessary to describe maximumloading of bubbles. The expanded area carrying capacity relationship described by Equation15 is more comprehensive. The area carrying capacity relationship shown in Equation 17uses

Equation19

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the bubble density as calculated in Equation 16which assumesthat the bubble size is muchlarger than the particle size.

2 . 4 . 5 Bubble LoadingFlotation is an interfacial phenomenon wherein a solids-surface-area is attached to, and thus,removed by a bubble-surface-area. The rate of removal of solids-surface-area is related to thebubble-surface-area flux as shown, through regression, by a modified form of Ityokumbul andTrubelja's ( 1998)relationship (Equation20) . Bradshaw and O'Connor( 1 9 9 6 )reportsimilarresultsusingpyrite.

Rs - b20 c \ o Equation 20

*B(o)

KS*B(0)J

InEquation 20," R s " [dimensionless] is the solids recovery; " S B ( o ) "[cm ] is the total surface-areasof bubble, " S S B ( O ) " [cm2] is the total surface of particles attached to the bubbles in theoverflow, b2o is determined to be 0 . 4 4 4 5and "a2o is determined to be 1.05. This equation,however, should work just as w e l lwith surface-area per time or surface-area flux ( 3B[s"1] and& S B [s"1]). The superficial bubble-surface-area flux (Xu; 1987) is the bubble-surface-areapassing through the column per time and cross-sectional area ( & B [S 1])as shown in Equation2 1(modified from X u ; 1 9 8 7 ) . & S B [S " 1]attached superficial total-solids surface-area rate.

J Equation 21d

InEquation2 1, " x "indicates any elevation.

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2.5 Entrainment

2.5.1 IntroductionEntrainment is the non-selective recovery, to the overflow, of both hydrophobic andhydrophilic solid particles andoccurs inmost flotation vessels. A nadequate modelofentrainment must be incorporated inorder to model flotation properly. The entrainmentrecoveryisinfluencedbymany factors, butwater flow between flotationstagesisgenerallyrecognized as themedium forgangue transport (Harriset. al.;1963, Jowett; 1966,Sadler;1973, Fl in t ; 1974, BisshopandWhite; 1976, and M o y s; 1978). Some variablesthataffecttherate of solids entrainment are: pulp density (Kirjavainen; 1989), particle size (TraharandWarren; 1976; BisshopandWhite; 1976), particleshape(KirjavainenandLaapas; 1988),andfroth properties suchasstability, drainage (Subrahmanyam and Forssberg; 1988, Ross; 1990band Ross; 1991a). removal rate (Flynn andWoodburn; 1987), and froth residence-time(BisshopandWhite;1976,M o y s; 1984 and Ross; 1990c).

2.5.2 EntrainmentinFlotation ColumnsUndernormal industrial-scale column operatinganddesign conditions, overflow entrainmentis minimaland isusually ignored. However, entrainment can become important i f any of thefollowingoperation conditions exist:

Closetoneutralornegative bias, H i g hconcentration ratios,or Variableorcycl ical flows.

A s in themechanical-cell, flotation column entrainment rateisassumedto beproportionaltothe overflow feed water recoveryrate. Thisrateisdetermined by twofactors: therelativewater magnitudes requiredtotransport thefroth comparedto thequantity of wash-water used,andthemixingthatoccurs inthefroth zone.

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Columnwash-water additionratesare usuallyknown. The volume of water, however, used totransport the froth to the overflow is often unknown but can be estimated using empiricallyderivedrelationships. The flowmagnitude isdependenton the quantity, hydrophobicity, sizeand shapeof floating particles, on the reagent suite used in the process, and on the bubblequantity and size distribution. Operating variables such as interface level andvariabilityalsohave an impact.

A t best, wash-water introduced into the froth would replace an equivalent volume of feedwater. Industrial-scale column froths, however, do not occur in a plug flow environment andbias may not be consistent throughout the cross-section. Some areas of negative bias mayoccureven when the average bias is positive. Factorsthataffect mixingin the froth zone are:radial pulp-zone bubble distribution, placement of internal baffles, wash-water distributordesign,wash-waterrate,and operational practices.

The results of both Maachar (1992), and Pal and Masliyah (1990) show thatgas rate has asignificant impact on feed-water recovery. Water content in the froth increases withincreasinggas ratesince water enters the froth zonewith the rising bubbles. As more froth-zone feed-water is recovered biasdecreasesand entrainment increases.

Bias increases in a linear fashion with wash-water rate within the range testedby Maachar(1992) and Pal and Masliyah (1990). An increase in bias, when all else is held constant,results in diminished feed water in the overflow (Maachar; 1992). A higher average biasrateismore l ikely toensure that there is a positive bias throughout the froth zone. At very highwash-water rates,the wash-water may channel or froth destruction may occur.

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2.5.3 FeedWaterRecoveryEntrainment Models

2.5.3.1 IntroductionMany entrainment models have been published. Three have been presented hereinordertoillustratethedifferent methods used. Theseare themodels of Ross (1990b), Trahar (1981)andKirjavainen (1992) and Warren (1985).

2.5.3.2 Ross (1990b)Ross (1990b) uses transfer functions to characterize the entrainment andtotal ratesofoverflow solids production--shown inmodified form inEquation22 andEquation 23,respectively. Ross assumes thatonly entrainment occursatthe end of a batch flotationtestofsufficient residence-time (no flotation). Thisassumption means thatthe transfer function " X "approaches " Y " withsufficient residence-time.

mSL Q) = m ^ )mSL z) mLL{z)

mSB Q) + mSL Q) _ y ^ m L L Q )mSL z) mLL z)

Thetransfer function, " X ( t ) " [dimensionless] characterizesthesolidsin theoverflow carriedby suspensionin thel iquid phase, and " Y ( t ) " [dimensionless] characterizesthesolids intheoverflowcarried both in suspension and attached to bubbles. In Equation 22 and Equation23m is themassofsolidsto theoverflow (O)orwithinazone (z). That mass (m) maybe

eitherthatof the solids (S)orl iquid(L) associatedwitheither the l iquidphase(L) or attachedtothebubblesphase(B). The pulp density inaflotation columnis notconstant; thus,theproper values of solids and water concentrations may notbeknown. The ratio of "mn(o)"to"nin(Z)"is the recovery of feed water to the overflow inacontinuous system.

2.5.3.3 Trahar (1981)Trahar determines therelationship between solids and water recovery,onmineraland size-class base, by measuring solids andwater recovery with andwithout collectors - the

Equation22

Equation23

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difference between thesetwotests is assumed to be the trueflotationrate. Recovery withoutcollector is assumed to be due only to entrainment. This test assumes that tests with, orwithout collectors, have thesamewater recoverywhichmay not be the case.

2.5.3.4 Warren (1985) andKirjavainen(1992)Warrenrelatessolids to water recovery by a series ofteststhatvary the ratio of water to solidsrecovery by altering froth height and removalrates. The procedure assumesthatthe functionslope is the rateof entrainment and the intercept, at zero water recovery, is the rateoftrueflotation (Equation 24: Warren; 1985, andVilleneuveet. al.; 1995) for each counter: mineral(m) and size-class (n).

R = R + B Equation 24xv S(m,n)(i) IVSf(m,n)(i) lJ(m,n)(i)1Kl(i)

In Equation 24, "Rs " is the total vessel solids recovery; "Rsf" is the solids recovered byflotation; " R " is the water recovery and "P" is a mineral, size, andshape-dependentconstant.A l l variables in Equation 24 are dimensionless. A proper time interval must be chosenbecause the relationship between the degree of entrainment and the water recovery maydeterioratewith time (Warren; 1985).

Villenueveet. al. (1995), based on work by Kirjavainen (1992), use a simple proportionalitywith water recovery to estimate recovery by entrainment. The mineral recovery relationshipused is Equation 24 when the recovery of solids due to flotation (Rsf(o) is assumed zero. Theslope of this relationship, " pm,n ( t ) " ,is calculated byKirjavainen (1992) using Equation 25.

n lo.7 Equation 25a I ^ W J

M- 7 +*c> ms{0 ) m,n)

InEquation 25, " R " [dimensionless] is the water recovery to the overflow; "m" [g] is themasso fsolids;" us ] " [g cmV] is the slurry viscosity and "b" [s"'] is a mineralconstant.

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Kirjavainen (1992) obtains the slurry viscosity using the relationship found in Equation 26where "b" is a mineral specific constant (Kirjavainen; 1992). The value for quartz, forexample, is 1.83.

InEquation 26,"usi"isthe slurryviscosity;"p" is the waterviscosity;"ss" is the solids holdupand the proportionality constant and the constant "b" has the units ofviscosity[g cm"'s"1].

2 .5 .3 .5 SummaryThe model ofRoss(1990b)assumes thatthe material recovered after a long flotation residencetime is due only to entrainment. This is not necessarily true. The entrainment predicted byTrahar (1981) may not be v a l i d since solids may stabilize the froth. Subrahmanyam andForssberg (1988) use Trahar's method but Ross (1991)statesthatthis method underestimatesentrainment. The number oftestsneeded to determine entrainment using Warren's (1985)method is adeterrentto this method's use. A l s o ,the varying removalratesand frothdepthsofWarren's method could change the entrainment rate (Flynn and Woodburn; 1987).Kirjavainen's(1992) model for(3 usesa mineral constant thatmay not be known.

The method of Ross (1990b) was chosen to determine an ini t ia l , temporary value. O n l y anapproximation is needed of the entrainment in order to use a minimization routine. Thismethod eliminated the need for special mineralconstantsand extensivetestwork.

Equation26

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2.5.4 Moys and Finch(1991) - Feed WaterRecoveryWater recoveryisrecognizedas akey variable intheentrainment of solidsinflotation.Theamount of feed water recoveredtothe overflow can be calculated usingatemperaturebalanceon thefeed, wash-water andoverflowasshowninEquation27(Moys andF i n c h ; 1991).Uribe-Salaset. al. (1990) usesimilartestswithconductivity.

QKF..O) = T{o)-T{w) E q u a t i 0 n 2 7Ql{w.O) T{F)~TW)

InEquation 27, " Q I ( F O > " [cm3s"1] is thevolumetric, overflow, water flow originatingin thefeed and"Qi(wo)"[cm3s"1]is the total volumetric, overflow water. A l s o ,"T " istemperatureofthe overflow (O), feed (F) and wash-water (W). Inthis equation,theratio of feed waterthatgoesto theoverflowto thetotal feed water equalstheratio of overflowtofeed temperatureswhenbothtemperatureshave the wash-water temperaturesubtracted from them. Maacharet.al. (1992) proves thatthetemperaturemethodofdetermining overflow feed water recoveryhasavery close correlation to the results of tracertests.

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2 .6 GasPhase

2.6.1 IntroductionThere are four gasphase"characteristics"thatare important in flotation: bubble flow regime,bubble risevelocity,maximum gasrateand superficial bubble surface area flux.

2.6.2 BubbleFlow RegimeIn the gas phase, the bubble flow regime, surface-area-flux and size are important becauseflotation is a separation based on the removal of solid-surface-area on a bubble-surface-area.A column collection-zonetypically operates in the "bubbly f l ow" regime. Kumar et. al.(1976) further divide the "bubbly regime" into the "dispersed regime"where bubbles flowfreely (gas holdup,eg< 0.10), and the "fluidized regime" where bubbles rise as swarms(0.10

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Table4: Transition Gas Ratesfromvarious sources (Finchand Dobby; 1991)d b (mm) Kasireday Mankosa Dobby Fi n c h Xu(1991)

(1989) (1990) (1986a) (1990)Jsi= 0.4 J g cm/s J g cm/s J g cm/s J g cm/s J g cm/s1.0 1.9 >2 1.5 2.9 2.50.6 1.1 1.5 1.0 1.3 1.40.2

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InEquation 28, " J g " , " J " , and " J s "are the gas, l iquidand solids superficial flow rates [cm s"1]respectively and "eg" is the gas holdup [dimensionless]. Yianatos, F i n c h , Dobby and Xu(1988) use a modification of the Masliyah (1979) hindered settling relationship shown inEquation29 (from F i n c hand Dobby; 1990) to calculate bubble "swarm" rise-velocity.

K2(P-A)(I-^) ' E Q U A I O N 2 918ft(l +0.15Rer7)

InEquation 29, "g" [cm s"2] is gravitational acceleration; "dt," [cm] is bubble diameter; " ps i [gcm"3] is slurry density; "pb"[g cm"3] is bubble density; "p" [g cm"'^1] is l iquidviscosity and"Ret," is the bubble Reynolds' number [dimensionless].

Equation 29 assumes that bubbles act as solid spheres (Flint and Howarth; 1971). In theprevious equation "m", a dimensionless number found as a function of Reynolds' number,takesthe form illustrated in either Equation 30 (1< Reb< 200) or Equation 31 (200< Reb

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a15 cm diametercolumn,however, assuminga0.1 cm bubble, the difference between the twoequations w i l l be on theorderof10%. Undertheseconditions,itmaybepossibletoscaleeach equation sothatno incongruence occurs.

Reynolds' numberis anindexthatcharacterizes thesystem turbulence. The general equationdefining Reynolds' number is found inEquation32. This dimensionless number can bedescribedas"a ratio ofinertialtoviscous forces" (Roberson andCrowe; 1965.)

VDp Equation 32Re=

M

InEquation32,"D" [cm]is theparticle (droplet) diameter; " V " [cms"1] is thevelocityofthatparticle;"p"[gcm"3]is theparticle-to-liquiddensity difference and u.[gs^cm"1] is thel iquidviscosity.

Bubbles,inflotation columns,aretypicallybetween0.8and 2.0 mm. The Reynolds' numberforthis bubble sizerangeis between approximately 120 and 750. For particles upto1 mmtheReynolds' numberisexpectedto bebetween 20 and 650. These numbers shouldonlybe usedasanapproximate guide since they have been generated without regardtothree-phasedensity.There aremany variationson thegeneral Reynolds' number equation. Some ofthesearepresented in Table5.

Table5:Reynolds' number general equation functionsAuthor Symbol V D P H DescriptionF i n c hand Dobby 1990 Re p Up dp (pP-psi)(l-eg) Psi particleSmithand Reuther 1986 Re p Jg dp Pi particleF i n c hand Dobby 1990 Re b Ub d b Pl-Pb Psi 2phasebubbleF i n c hand Dobby 1990 Re b Ub d b (Psl-Pb)(l-Sg) Psi 3phasebubbleSmithand Reuther 1986 Re g Jg d c P i column

The Reynolds' number of amechanical-cellisdescribed bySchubert andBischofberger(1978).

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Many functions have been described that relate the terminal bubble rise-velocity of a singlebubble ( U T [cm s"1]) tothatin a "swarm" (Lockett andKirkpatrick; 1975, Shah et. al.; 1982).One such relationship, byWal l i s(1962), is shown as Equation 33.

U b=UT( l -S gr ' Equation 33In Equation 33, "ub " [cm s"1] is the bubble-swarm rise-velocity; U T [cm s"1] is the bubblerise-velocityand "sg" is the gas holdup [dimensionless]. Equation 29 is based on Equation 33and the bubblerise-velocityofSchillerand Naumann (1933), as shown in Equation 34.

gd2b(p,-pg) Equation 3418^(1+0.15Re"68 7)

Equation 34 is a two-phase relationship that does not account for solids. As such, waterviscosity(ui [gcm'V1])and l iquiddensity (pi [g cm"3]) are used alongwithgas density (pg [gcm"3], assumed zero) and the bubble Reynolds' number (Ret,). Terminal bubble rise-velocity( U T [cm s"1]) is about twenty-one (21) for bubble sizes between 0.15 and 0.10 cm (Clift et. al.;1978). This bubble velocity relationship is v a l idfor bubble sizes (db[cm]), smaller than 0.20.There arethreeunknowns in the drift-flux calculation used in flotation vessel models: bubblesize (db [cm]), gas holdup (sg) and rise-velocity (ub [cm s"1]). Three equations thatmay beused to determine these unknowns are Equation 28, Equation 29 and the Reynolds' numbercalculatedusing th