conditioning in flotation of gold, uranium, pyrite

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  • 8/8/2019 Conditioning in Flotation of Gold, Uranium, Pyrite

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    Sam ple A Sam ple B

    7, 6 :t 0,46 7, 3 :t 0,8817 2 :t 2 14 1 :t I1,74 :t 0,01 1,77 :t Om

    MAY 1 99 1 16 9

    J. S. Afr. Inst. M in. Metal/., vo!. 91, no. 5.M ay 199 1. pp. 169-174

    Conditioning in the flotation of gold, uraniumoxide, and pyrite by F.J.N . STASSEN *

    SYNOPSISThe effect of conditioning energy on the flotation of gold, UaOe, and pyrite was investigated in the range 0,1 to100 kW h per tonne of dry ore for various com binations of conditioning tim e and impeller speed in a cylindrical con-ditioning tank. It was found that, when the conditioning energy was increased to between 5 and 10 kW h per tonneof dry ore, the total recovery and flotation rate of the valuable m inerals (expressed as Klim pel param eters) increas-ed substantia lly. The Klim pel param eters are dependent on conditioning energy, but are independent of condition-ing time or impeller speed (at constant conditioning energy). The Klim pel param eters of the gangue are indepen-dent of conditioning energy.S AM EV A T TIN GDie uitwerking van die kondisioneringsenergie op die flottasie van goud, UaOe, en piriet is met verskillendekom binasies van kondisioneertyd en stuwerspoed in 'n silindriese kondisioneertenk oor die strek 0,1 tot 100 kW hper ton droe erts ondersoek. Daar isgevind dat indien die kondisioneringsenergie tot tussen 5 en 10 kW h per tondroe erts verhoog word, die herwinning en flottasietempo van die waardevolle m inerale (uitgedruk asKlim pelparam eters) aansienlik toeneem . D ie Klimpelparameters is afhanklik van die kondisioneringsenergie eno nafh an klik van d ie kon disio ne ertyd o f stuw erspo ed (b y 'n k on sta nte k on dis io ne rin gs en er gie ). D ie K lim pelp aramete rsva n d ie uitsko t is on afh an klik va n d ie ko ndision ere ing se nerg ie .

    IntroductionT he effect o f conditioning has b een w idely studied foragglom eration flotation and is discussed elsew here'.C orresponding data for froth flotation are scarce andsketchy. H ow ever, it has been show n that conditioninge ne rg y, o fte n ex pre ss ed as c on ditio nin g tim e a t c on sta ntim pe lle r sp ee d o r a s impelle r sp ee d a t c on stan t c on ditio n-ing tim e, has a pronounced effect on the recovery andflotation rate of valuable m inerals and on the grade ofthe concentrate2-1O. The m ost important results ares ummariz ed b elow.F lo ta ti on Rate . D uring the froth flotation of gold,030S , pyrite, and arsenide and sulphide m inerals3-9, theflotation rate can be increased by an increase in condi-tioning energy until a lim it is reached. T he flotation rateof the gangue (norm ally silica or silicate m inerals) cand ec re as e3 , o r s ta y con sta nt6 , o r i nc re as e ma rg in al ly S, w ithin cr ea sin g condit io nin g ene rg y.Recovery. The total recovery of nickel sulphidem in era ls 2 a nd c op pe r m in era ls (m ain ly m ala ch ite )1O c anbe increased by an increase in conditioning en ergy untila lim it is reached. T he equilibrium recovery of gold caneither stay constanf.7,9 or increase m arginally until alim it is reacheds,6,s. T he equilibrium recov ery o f 30Scan stay constanf,6,S , or decreases, or decrease after aninitial increase7. T he equilibrium recovery of pyrite cane ith er s ta y con sta nf, 6,7 o r in cre as e ma rg in ally S,S w ith in -cre as in g co nd itio nin g e ne rg y. T he e qu ilib rium re co ve ryof arsen ide and sulphide m inerals is independent of con-ditioning energ~. The equilibrium recovery of gangued ecre as es w ith in cre asin g co nd itio nin g e ne rg y3 's .. Form erly of Anglo Vaal Lim ited, Johannesburg; now P.O. Box

    3 51 10 , M en lo P ark , 0 10 2 T ra ns va al.@ The South African Institute of M ining and M etallurgy, 1991. SAISSN oo38-223X/ 3.00 + 0.00. Paper received 12th June, 1990;m odified pap er receiv ed 19 th D ecem ber, 1 99 0.

    Grade. Norm ally the grade of the flotation concen -trate in the froth flotation of copper m inerals (m ainlymalachite)10,gold3,S,6,s,30S3,S,S,and pyrite3,s increasesw ith in cre asin g co nd itio nin g e nerg y u ntil a lim it is re ac h-ed. H ow ever, it can also stay constant in the froth flota-tio n o f g old 9, 3 0S6, p yri teS,6 , an d a rs en id e and sulp hid eminerals9. The increase in grade can be ascribed to areduced recovery of gangue, rather than to an increasedre co ve ry o f v alu ab le m in eral.B ecause conditioning form s an integral part of flota-tion and has a pronounced effect on both the recoveryand the flotation rate of valuable m inerals, and becauseo f th e fi na nc ia l imp li ca tio ns con sequ en tly in vo lv ed , it h asbecom e im perative to m ake a thorough investigation ofthe effect of conditioning on the flotation of W itw aters-rand ore and to obtain an understanding of the m ore im -portant param eters and m echanism s involved.

    ExperimentalT wo sam ples of W itw atersrand ore (identified here asore A and ore B ) w ere used in this investigation. The par-ticle size w as m inus 10 mm and the analysis w as as show nin Table I. The samples were individually crushed in ajaw crusher to minus 10 mesh (minus 1,68 mm), werestored in drum s, and w ere thoroughly m ixed before use.Ore samples of 2 kg were milled with 1 litre of water(R and W ater B oard) in a stainless-steel rod m ill t065 percent minus 75 /Lm.

    TABLE IANALYSIS OF THE ORE SAMPLES

    Constituent

    Gold, gltU 3O 8, gltSulphur, %

    JO URNAL O F TH E SOU TH A FR IC AN IN ST ITU TE O F M IN IN G AND META LLU RGY

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    The experimental conditioning tank consisted of acylindrical PVC container with an inside diameter of300 mm, a flat bottom , a height of 600 mm, and four ver-tical baffles that were one-twelfth the tank diameter(25 mm) at 90 to one another and that ended 10 mmfrom the b ottom . T his size is th e m inim um for experim en -tal workll if scale-up problems are to be avoided. Theimpe lle r sh aft w as mou nte d ce ntra lly in th e c on ditio nin gtank, and w as driven by a variable-speed electric m otor.The m otor, driving gear, and gearbox form ed a unit thatwas moun ted ver ti ca lly on a ver ti ca lly adjus tabl e p la tf orm .The im peller speed was m easured by a photo-electrictachometer.The condi tio ns d ur in g conditio ni ng (Tab le I I) we re k ep tcon stant, w hile the item s sh ow n in T ab le III w ere varied.The conditioning tim e w as m easured from the m om entof reagent addition, and included only the tim e the pulpw as conditioned in the conditioning tank.

    TABLE IIEXP ER IM EN TA L CONDIT IONS DUR ING CONDIT IONING

    F lo ta tio n c on ditio n ValueM ass of ore, kgHeight of pulp, mmD en sity of pu lp , kg /m 3T em pe ra tu re , .CpHReage nts , g ltCuSO.S od ium n -p ro py l x an th ateA ero pr om ote r 3 47 7

    1030 013002511,5

    10 06015

    TABLE IIIEXPERIMENTAL VARIABLESVariable RangeP itc he d- bla de im pe lle rN, r/mint" minD,mmDID,Number o f te sts: O re AOre BL IGHTNIN A31O impel le rN, rlminl e, m inD,mmHID,Number o f te sts, O re B

    300 to 11004,5 to 7275 to 1500,25 to 0,507244

    500 to 11004,5 to 7296 to 1600,32 and 0,5317

    T he conditioning occurred in the turbulent region, i.e.Re>10000, with Re d efin ed ll a sR e = pD2IN lp..(A ll the symbols used are defined at the end of thepaper.) The power dissipated during conditioning isg iv en ll b yP = NppN3D',

    with Np as a function of the system geometry and Reil.17 0 MAY1991 JOURNAL OF THE SOUTH AFRICAN INSTITUTE OF MINING AND METALLURGY

    In the turbulent region, N p is independent of Re, and isa function only of im peller type and system geom etry.F or th e s usp en sio n o f s olid s, it is n ec es sary to u se an ax ial-flow im pellerll such as the pitched-blade im peller or theh ydrofoil or aerofoil im peller (e.g. the L IGHTNIN A31Oimpeller). A t standard geometry, Np is a constant foreach im peller type, e.g. 1,37 for the standard four-bladepitched-blade im pellerll and 0,30 for the LIG HTN INA 31O im peller!2. By different com binations of thesev aria ble s, th e co nd itio nin g en erg y in th e tes ts w as v arie dover three orders of m agnitude (0,1 to 10 0 kWh per tonneof dry ore).A fter the conditio ning, the contents of the condition-ing tank were split in a sample splitter, and 40 per centw as used for flotation. C are w as taken so that heavy par-ticles could not stay b ehind and consequently cause largelosses of gold and pyrite.Flotation w as conducted in a laboratory-size 9-litreD en ver 1 2 flo ta tio n c ell a t a n impeller sp ee d o f 1 50 0 r /m inand at constant air-feed rate. Approxim ately 13 g /t oftri-ethoxy butane w as added to the condition ed pu lp, andthe conditioning lasted for roughly 1 m inute. E xactly 2m inutes after the conditioning had been com pleted, theair inlet w as opened and flotation started, the flotationtim e being m easured from that m om ent. C are w as takento ensure a constant procedure and rate of froth rem ovalbecause O f the sensitivity of flotation rate to both the rateand the method of froth rem oval.F ive sam ples of concentrate w ere taken for the follow -ing periods: 0-1 m in, 1-2 min, 2-4 min, 4-8 min, and8 -1 2 m in . T he fiv e s ample s a nd th e ta ilin gs w ere p re pare d,and analysed for gold, U3O8 ' and sulphur (and thuspyrite).A curve w as fitted to the cum ulative recoveries of thevaluable m inerals at the different flotation tim es so th atthe K lim pel param eters, R and k, used in the K lim pelf lo ta ti on -r at e equat ion !3 ,

    r = R[1 - (Ilk/rH1 - e-klf)],c ou ld b e d ete rm in ed .

    ResultsT he K lim pel param eters for ore A are depicted in F igs.1 and 2 as a function of the natural logarithm of the con-d itio nin g e ne rg y. S im ila r re su lts w ere o bta in ed fo r o re B .F rom the results it is clear that the co nditioning energyhas a pronounced effect o n both the equilibrium recoveryand the flotation rate of the valuable minerals (gold,U3O8' a nd p yrite ). If th e c on ditio nin g e ne rg y is in crea se dto more than 5 to 10 kWh per tonne of dry ore, theequilibrium recovery of the valuable m inerals can be im -proved as shown in Table IV .If the conditioning energy is increased by the sam eam ount, the K lim pel flotation rate constant can also beincreased substantially, as show n in Table V .F rom Fig. 1 it is clear that the relationship betw een thenatural logarithm of the conditioning energy and bothK lim pel param eters follow s an S-shaped curve for thethree valuable m inerals, w hich can be represented b y them odified G ompertz equation !4. F or the system in w hichthe K lim pel param eters are a function of the logarithmo f th e c on ditio nin g e ne rg y, th is e qu atio n c an b e w ritte n a sIn[(ct>max - ct/(ct>max - ct>mio)] = -k:W.

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    RAu k.Au99 13* 0 0c9 129a 0 11 1J~* 0 *7 10 + 0 0X * 0* 09 0 096 X * a 0 0+ ** 07 P 0 * 05 * X * ** 0094 X+ 'It x 0/01-0,25 5 *~ X 0/01-0,25*~ DD + 0/01-0.33 cl - + 0/01-0,334 X93 * + * 0/01-0.40 X * 0/01-0,40X0 0/01-0,50 3 * 0 0/01-0.50+92 20,1 10 100 0,1 10 10 0E (kWh/t) E (kWh/t)R.S k.S99 11 * 09a 0 0 CD 0*097 0996 0ax 05 X+ 0x 794 * 1* 693 *+** 592 t1-X* 0 +~ 491 + x 0/01-0.25 3 0 X 0/01-0,2590 0 0 + 0/01-0.33 + 0/01-0,33a9 * 2 * 0/01-0,40/01-0.40aa 0 0/01-0.50 0 0/01-0,50087 00,1 10 10 0 0,1 10 10 0E (kWh/t) E (kWh/t)F ig . 1 -K llmpe l p arame te rs fo r g old a nd su lp hu r a s a fu nc tio n o f c on ditio nin g e ne rg y (p ltc he d-b la de Impe lle r a t sta nd ardgeometr y, o re A )

    Without Withconditioning co nditio nin g Im prov em en t

    Ore M ineral 07 0 07 0 07 0OreA Gold 93,8 97,5 3, 7

    U3Os 40,1 47,9 7, 8Pyrite 92,7 97,6 4,9Ore B Gold 95,8 97,7 1,9

    U3Os 37,6 42,4 4,8Pyrite 95,7 97,9 2, 2

    Without! WithConstit- conditioning conditioning ImprovementOre uent m in-I m in-t (factor)

    Ore A Gold 3,8 10,0 2,6U3Os 1, 4 3,0 2,1Pyrite 1,5 9,1 6,1

    Ore B Gold 4, 7 11,1 2,4U3Os 0,95 2, 7 2,8Pyrite 1, 7 11,3 6,6

    TABLE IVEQUIL IBRIUM RECOVERY OF VALUABLEMINERALS

    This c urv e is c ha ra cte riz ed b y a v ery small in itia l slo pe ,Le. nearly constant minimum values of the Klimpelparameters (Rmin an d kmin), followed by a period ofra pid ly in cre asin g slo pe (0 ,5 to 1 kWh/t), w hic h g iv es w ayto an interval of nearly constant slope, succeeded by ap erio d o f ra pid ly d ec re asin g slo pe a pp ro ac hin g z ero slo pe(at above 5 to 10 kW h/t). The Klimpel param eters thenlevel off at a constant m axim um value (Rmax a nd kmax).

    TABLE VKLIMPEL FLOTATION RATE CONSTANT OF THE VALUABLE

    MINERALS

    Rmin , Rmax , kmin, an d kmax are functions of both the typeof mineral and the type of ore.. Some scattering of the data occurs as a result of thenorm al variab ility en countered in flo tatio n testw ork .The m inim um conditioning energy required for op-tim um flotation results lies in the range 5 to 10 kW h pertonne of dry ore. This is in accord with the results ofp revious inves tiga to rs . Wes tonlS,16sugges ted the fol low-

    JOURNA L O F THE SOU TH AFRIC AN IN STIT UTE O F M IN ING AND MET ALL URGY MAY 1991 17 1

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    k.U3084

    3,53

    2,52 :; :

    1,5+X '"

    +0,5 -'------10 0 0,1

    RU30850 r -4948474645 ~44 ~43 ~42 ~41 ~

    *

    x

    Cb

    *0 ~

    II X 0/01 -0.25I + 0/01-0.33* 0/01-0.400 0/01-0.50;~ [ *;:1- '" 038 ~

    ~ ~ L.c.t -L .L -. L L Lj~ .L L - - -- '- -- l- -0,1 10E (kW h/t)

    R.Gangue109 0

    0a X 0 0 [JIJXX bJ dd ~O~*o '" *,t;' ~ *** *0 ~!O 0 C f : J 0

    * Oi-% *iV* Jj ~ ~*T'~ '"t;

    43 X 0/01-0,25

    -i- 0 /0 1-0 ,3 3101

    * 0/01-0,40I 0 0/01-0,50

    """" """'!, , , " " ,0, 1 10E (kW h/t)

    00IfJ 0** 0

    X 0/01-0.25+ 0/01-0,33'" 0/01-0.400 0/01-0.5010 10 0

    E (kW h/t)k.Gangue0,4

    0,3 *+ * #+ 00 +# +*0 0 *. . 0 *8 nO ~O []J * I2J *~ * '* J J* * O~O*X 0 0**00 0 0

    00

    0,2'" x+*

    x x 0+ x+ X 0 /01 -0 .2 5+ 0/01-0.33* 0/01-0.400 0/01-0.50

    0,1

    10 0 00,1 10 00E (kW h/t)

    Fig. 2-K llm pel param eters for U 30. and gangue as a function of conditioning energy (pltched-blade Im peller at standardgeom etry, ore A )ing for froth flotation: 0,3 to 2,0 kWh/t for coppersulphide m inerals, 0,15 to 0,6 kW h/t for m olybdenumsulphide minerals, and 0,6 to 4,0 kWh/t for nickelsulphide minerals, but alleged that the energy mayam ount to as m uch as 5 kW h/t. For agglom eration flota-tion, conditioning energies of the sam e order have beendetermined! .N o correlation betw een the am ount of conditioningand the Klimpel parameters of the gangue could bedetected.

    P ossib le R ea ctio n Mecha nismNo attempt has thus far been made to propose am echanism to explain the effect of conditioning uponf ro th f lo ta tion .It has been found!7 that the repulsive energy barriersbetween collector m olecules and m ineral surfaces inagglom eration flotation can be overcom e by the hydro-dy nam ic fo rce s e ncou ntered d urin g cond ition ing .T he K lim pel param eters o f the valuable m in era ls showan increase w ith in crea sing cond ition ing energy, bu t theKlimpel parameters for the gangue show no corre-spon ding decrease, w hich w ou ld have be en exp ected w itha m echan ism of rea gent tra nsfer from particles of gan gue172 MAY 1991 JOURNA L O F TH E SOU TH A FR IC AN IN ST ITU TE O F M IN ING AND MET ALLURGY

    to particles of valuable m inerals w ith increasing condi-tioning energy!8. It is therefore concluded that there isno transfer of reagents. This is in accordance with theresu lts o f K arjalah ti1 9 for the ag glo meration flo tation o fapatite.There is some indication of support for an attritionmechanism, by which the collector is desorbed from thesurface of m ineral particles at very high conditioningin te nsity o r e ne rg ylO ,2 1.This wou ld re su lt in d ec re asin gK lim pel param eters at very high conditioning energy asdisplayed by kAu and ku O' These decreases are verysm all but m ay be caused by attrition. T his very high con-d itio nin g e ne rg y a t w hic h a ttritio n ta ke s p la ce is p ro ba blythe result of the large energy needed to break the verystrong chem ical bond between the xanthate and the sur-faces o f th e valuab le m ine rals1 7.B ased o n th e find ing s of th is inv estig atio n, the fo llo w-ing m echanism is postulated to explain the effect of con-ditioning on the flotation of W itwatersrand ore:(1) dispersion of the reagents into the pulp(2 ) sp ecific adso rp tio n of th e co llector o nto the surfacesof all the m ineral particles according to the relativere pu lsiv e e ne rg y b arrie rs a nd re la tiv e a ffin itie s o f th ediffe ren t m in erals for th e collector

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    Coef fi ci en t o fk, determinationO re cp cpmin cpmax (W/m3)-'min-1 'Y R2

    areA RA" 93,8 97,5 0,0167 1,0 0,7061kA " 3, 8 10,0 0,0244 1,4 0,4556RU30S 40,1 47,9 0,0150 1,0 0,7646kU30S 1,4 3,0 0,0440 1,4 0,7068Rs 92,7 97,6 0,0100 1,25 0,6103ks 1, 5 9, 1 0,0161 1,1 0,8420

    Ore B RA " 95,8 97,7 0,0288 1, 0 0,5045kA " 4, 7 11,1 0,0229 1, 4 0,6367RU30S 37,6 42,4 0,0172 1,05 0,7885kU30S 0,95 2, 7 0,0273 1,4 0,5939Rs 95,7 97,9 0,0207 1,0 0,5888ks 1, 7 11,3 0,0133 1,15 0,8959

    Coef fi ci en t o fk; determinationOre cp cpnUn cpmax (kWh/t)-' 'Y R2

    areA RA" 93,8 97,5 0,24 0,90 0,7077kA " 3, 8 10,0 0,43 0,75 0,4238RU30S 40,3 47,7 0,21 0,90 0,7668kU30S 1,4 3,0 0,57 1,2 0,6530Rs 92,7 97,6 0,093 1,3 0,6137ks 1,5 9,1 0,18 1,3 0,8542

    OreB RA" 95,7 97,7 0,275 1,3 0,5126kA " 4, 6 11,1 0,26 1,05 0,5258RU30S 37,6 42,4 0,16 1,25 0,7973kU30S 0,95 2,7 0,41 0,85 0,4545Rs 95,7 97,9 0,17 1,3 0,5979ks 1,7 11,3 0,215 0,90 0,8924

    (3) saturation of all the mineral surfaces w ith thecollector(4) desorption of the collector from the surfaces of thev al uable -m in era l p ar tic le s a nd o f th e g angu e p ar tic le sa t v er y h ig h le ve ls o f c onditi on in g ene rg y by a ttr itio n.

    The follow ing equation (regression m odel) for bothK lim pel param eters h as been developed!:In[(cf>max- cf/(cf>max-

    cf>min)]= -kc(PIV)Jtcwith cf>equivalent to any K lim pel parameter (R or k) .

    Application of R eg ression ModelThe cons tan ts cf>min,f>max,c, and'Y in the above equa-tion w ere determ ined by m inim izing the expression forthe residual sum of squares for the regression m odel, asshown in Table VI.TABLE V I

    COEFFICIENTS FOR REGRESSION M ODEL (FU NCTION OFCO ND ITION ING PO WER AN D TIM E)

    R2 (th e c oe ffic ie nt o f d eterm in atio n2 2) is th e fra ctio nof the total variation that is accounted for by the regres-sion model. A very high percentage (up to 90 per centin the case of ks for ore B) of the variability of the datais exp lained by the equation.In the case of RAUan d Rs, the im prov em ent as a resultof conditioning is only a few percentage points, and then orm al v aria bility as a re su lts o f th e te stw ork is th ere fo rere la tiv ely la rg e c ompa re d w ith th e to ta l v aria bility . In th ecase of kA u and ku 0' a large fraction of the variance isn ot explained by theSm odel equation. T his is believed tobe the result of attrition at high conditioning energy.T he c lo se ne ss o f'Y to th e e xp on en t o f th e c on ditio nin gtim ~ suggests that the regression m odel m ay be approx-imated as a function of conditioning energy, E' ex(P I V)"ItJ , giving

    In[(cf>max- cf/(cf>max- cf>min)]= -k:E"I.T he coefficients listed in T able V II w ere obtained bym in im izin g th e re sid ua l sum o f s qu are s fo r th is e qu atio n.There is a close correlation between the Klimpelparam eters and the conditioning energy, and this equa-

    tion can be used for the accurate prediction of bothK lim pel param eters for all three valuable m inerals as afunction of the conditioning energy. The Klimpel

    param eters predicted by the regression m odel using thecoefficients given in Table V II are given as solid blacklines in Figs. I and 2.TABLE VII

    CO EFFICIENTS FOR REGRESSION M ODEL (FU NCTIO N OFC ON DIT IO NIN G E NE RG Y)

    A com parison of the coefficients of determ inationgiven in T ables V I and V II reveals that both regressionequations (Le. as a function of E"I and p "I t respectively)account for the variability in the data with the samedegree of adequacy. It is therefore not clear w hether theK lim pel param eters are functions of the conditioningenergy or of the intensity of conditioning. This result isin ac co rd w ith th at o f p re vio us in ve stig ato rs . Run no lin nae t 0 1.2 3alleged that the flotatio n result o f con dition ingdepends only on the conditioning energy, and that it isindependent of the method and rate of conditioning.However, Lapidot and M ellgren20 and Karjalahti!9found that the intensity of conditioning, rather than thetotal energy input, w as the im portant variable for thattype of ore, but that this difference is rather sm alll9.C oefficients w ere obtained for the regression m odelw ith v ar yin g impell er h eig ht (Z/ D), v ary in g pulp h eig ht(HIDt), and the LIG HTN IN A 31O im pellers, and it w asfound! that this equation can be used for the accurateprediction o f the K lim pel param eters for non-standardgeom etry or other types of im pellers.Lim ited published data on the use of conditioning inthe flotation of other m inerals are available. W hen suchc oe fficie nts w ere u se d in th e re gre ss io n mod el fo r c op pe rm in era ls (m ain ly m ala ch ite ), g old , a rs en id e an d s ulp hid em in era ls, m an gan es e o xid e m in erals, a nd ilm en ite , it w asfoundl that this equation also accurately predicts theeffect of conditioning energy on the flotation of otherminerals.

    RecommendationsThere is enough reason to believe that an equation ofthe type of the m odified G om pertz equation presentedhere, or a more general equationl, for the Klimpelparam eters accurately predicts the influence of othervariables during flotation. It is well-known that therecovery and flotation rate of m any m inerals follow anS-shaped curve w ith the logarithm of variables such asc olle cto r c on ce ntra tio n, h yd ro ge n-io n c on ce ntra tio n (o r

    JOURNAL OF THE SOUTH AFRICAN INSTITUTE OF MINING AND METALLURGY MAY1991 17 3

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