394
K . L. SUTHERLASD
(5) CORBIN,s., ALEXANDER, AI., AXD EGLOFF,G.: J. Phys. Colloid Chem. 51, 528 (1947). (6) CORBIK, S., ALEXANDER, hI., A N D EGLOFF, G.: Ind. Eng. Cheni. 39, 1147 (1947). (7) EGLOFF,G. : Physical C o m t a n t s of Hydrocarbons, Reinhold Publishing Corporation, Yew 1-ork: Vol. I. Paragins, Olejins, Acetylenes, and Other Aliphalic Hydrocarbons (1939) ; T-01. 11. Cyclanes, C'yclcnes, C'yclynes, and Other Alicyclic Hydrocarbons (1940); Vol. 111. Mononuclear Aromatic Hydroca&ons (1946) ; 1701. IV. Polynuclear Aronialic Hydrocarbons (1947). (8) EGLOFF,G., ASD K U D E RR.: , J. Phys. Cheni. 45, 836 (1941); 46, 296 (1942); I n d . Eng. Cherii. 34,372 (1942). (9) EGLOFF,G., ASD K G U E RR.: , J. I'liys. Chem. 46, 28 (1942). (10) EGLOFF,G., SHERXIS, J., ASD DULL,R. B.:J. Phys. Chem. 44, 730 (1940). (11) FRASCIS, A . W.:I n d . Eng. Chern. 33, 554 (1941); 35, 442 (1943); 36, 256 (1944). (12) GLASGOK, A . R.,JR.,STREIFF, h. J . , . ~ S D ROSSINI,F . D.: J. Research S a d . Bur. Standards 35,355 (1945). (13) HUGGISS, 11.L.: J. din. Clieni. Sac. 63, 116 (1941). (14) HUGGISS,AI. L.: J. h i i . Cheni. SOC.63, 916 (1941). (15) IVANOVSKI-, L., ASD B R - ~ X C K EAR. ,T-.: Petroleum 5, KO.10, 169 (1942). (16) KIKNEY, C. R.: I n d . Eng. Cheni. 32, 550 (1940). (17) I~ISSEY, C. R.: Ind. Eng. Chern. 33, 791 (1941). (18) RISSEY,C. R.: J. Ani. Cheni. h e . 60, 3032 (1935). (19) KIXXEY, C . R.: ,J. Org. Chem. 6, 220 (1941). F.: Ber. 76, 7bS (1943). (20) KLAGES, (21) KURTZ, S.S., J R . , ASD LIPKIX,11. R.: Ind. Eng. Chem. 33, 779 (1941). (22) KURTZ, S.S., ,JR,, A N D LIPKIN,AI. R.: J . Am. Chem. Soe. 63, 2158 (1941). (23) MIBASHAN, -4.:Trans. Faraday Soc. 41, 374 (1945). (24) T.LYI.ort, W.J . , I'IGXOCCO, J. 11.,ASD ROSSINI,F. D.: J. Research Natl. Bur. Standards 34, 413 (1945). (25) WIESER, H . : J. Am. (:hem. SOC.69, 17 (1947). (26) WIESER, IT. : Private communication (April 25, 1947). (27) WILLINOHAM, C . B., TAYLOR, W. J., PIGNOCCO, J. M., AND ROSSINI,F. D.: J. Research Natl. Bur. Standards 35, 219 (1945).
PHYSICAL CHEMISTRY OF FLOTATION. XI
KIYETICSOF
THE
FLoTa.rroN
PROCESS
T i . I,. d U T H E R 1 , h S D 1)irzszon of Induotrial Cheinzsiry, Comnionweulth Council for Scientific and Industrial Research, Melbourne, Australia Received A u g u s t 27. 1947 I . ISTRODL7C'l'IOh-
To detrimiiie tlw tliroretical rate of flotation of mineral in a cell, it is necessary t o make 1i.e of ;I Yimplified model of the -!.stem, nhich ignores unimportant variables u-liile retaining e*sentinl teatiire-. Theories of air-mineral adhesion are discussed and tlic directt enc~oiinterhypothesis is chosen for detailed investigation. .lny fiiilure of the subsequent theory t o describe tlie kinetics of the process is certainly due t o oversimplification of the encounter hypothesis and some attempt is made to a the importance of the simplifications.
K I S E T I C S O F THE FLOTATIO?; 11. T H E O R I E S O F I I R - M I X E R A L
PROCESS
395
ADHESIOS
‘I‘aggart (21) observed that gas is deposited on non-polar surfaces from solutions supersaturated m-ith the gas. In the subaeration and agitation machines there are regions of high pressure and lo^ pressure before and after the impeller blade. These pressures produce super- and iinder-sntmittiori, respectively, of the pulp. The air can then precipitate from the supei,satiirated liquid on the particles of mineral? and it is postulated that all mineral ivhich floats has heen buoyed t o the surface by air attached in this way. Taggart also observed that, in one test at least, conditioned particles falling on stationary air 1)uhbles {lo not adhere, and he therefore declares the encoiinter proceps inadequate. Despite Taggart’s assertion that ’This idea (i.e., the encounter hypothesis) (\\-as) conceived in ignorance and horn in litigation, ivas foaered 1)y selfish interest and, unfortunately. n-as copied into some textbooks”, k g d a n o v nnd Filanol obtain opposite results in similar tests. They piwent diiwt evidence I matographic record for the collision theory. l’artirles \\-it11 i t hydropliobic surface fell on a stationary air bubble. They xlheiwl. Iluhblc~rising n-ith a speed of 18 cm. set.-' (about 0.09 cm. radius) collided ivith and adhered t o .mspended mineral particles (about 0.0075 cm. iadiiis). ‘The time required for. adhesion t o the bubble of suitably conditioncd miiirixl TVW h t i v w n 2 :md 10 millisec. Gaudin (Ga) and earlier investigators po,qtulatcd :i diiwt encounter, leading to adhesion between suqiended mineral and the rising h b b l e . Treating this process theoretically Gaudin shon.ed that no encounter \vas possible Iietwen a bubble and a mineral particle in an itleal fluid (comp:ire later). Further, he stated that in some experiments he found that “it is very difficult t o cawe coursing air bubbles to pick up n-ell-prepared mineral particles” ((j, 11. 92). Despite these difficulties Gaudin considers that the theory is more plaiisihle than the gasprecipitation hypothesis, which lie rriticizes on the grounds that ( 1 ) \-;iciiiim and pressure near the impeller blades follon. so mpitlly that buhble gro\vth is not possible; (.2) it is not posihle for one bubhle to grow siniultaneoi particles, yet heavily armored biibbles are c.ommon in s;nitable precipitation theory must lead to equal rates of fiotation’ of small and large particles, n-hich is not so. The first objection i> not important, since supersaturated pulp could be flung from the impeller arid the gron-th of the hubble coiild proceed s l o ~ d yin pulp distant from agitation (see also page -109). The second and third objections are sufficiently important to ivnrrant dismissal of the theory as the major method b - n-hich mineral is attached to b ~ r b b k which s biiog it t,o the surface. This conclusion is strengthened by the observation of lhlozemoff and Kamsey (1317who found that the operating rapacity of a mechanical flotation cell is chiefly dependent upon that volume of the cell through which air bubbles are passing. Design of the cell i d 1 influence the aerated volume but cannot markedly influence the supersaturation of dissolved air produced hg the impeller. In a pneumatic cell supersaturation is still possihle but is smaller than in the T h e term “rate of flotation” is used in this paragraph in the sense of the number of particles floating per unit time rather t h a n in the sense of the weight of mineral floating per unit time.
396
I on rnirieral particles during flotation in a cell, subsequent flotation niay still be ascribed t o collision b e t m e n an air bubble and an aggregate corisisting of a iiiineral particle and a minute air bubble. The aggregate may have a density greater than 01’ even less than that of the pulp. The theoretical approach adopted in this paper nil1 still be valid, but for the smaller sizes of mineral the aggregate will be very much larger than the particle. It then follows from our theory that the chances of an aggregate floating are better than those for the mineral particle 1,y itself, particularly in the smaller size ranges. 111, ISSI‘MPTIOSS IS THEORETICAL TRCATJIEST O F FLOTATIOK RATE, USING THE DIRECT ENCOUNTER HYPOTHESIS
111 1832 Gaudin (Cjc) (*onsideredcolliqion between an air bubble and a mineral particle in a flotation cell, awiming that ( A rmter is non-viscous and incompressible; ( 2 ) \\ ater i- infinitely divisible (i.e., the mineid particles are large compared n itli tlir moleculcs or their meail free path) ; ( 3 ) bubble and particle ( 6) tii~hblea i d p:irticle :ire the only disturbing factors in the cell; (.j)niotion 01 1)uhble and particle is irrotational (i.e.*there is streamline motion uf t l p 11 titer pa\f these s p l i c i d . IIr deduced that collision is possible only if thc cwi1cy of the particle lies on the rentral line of motion of the bubble.
KISETICS O F THE FLOrATIOK PROCESS
397
His malysis, hoirever, T X I ~b a w l on an implicit, hut eiwnwus, a-wmption that two bodies move independently i n a fluid, thnf i4, there i q no intei.nrtion of thc4r velocity field,. The motion of tn-o spheiw t on-aid one another hi not hrcw succe-sfully analyzed except when the spheres move aloiig their line of (’enters ( 1 7). llathematical treatment i b po4hle. honever, if the ma- of one of the bpheres is zero, Le., if the particle is inertia-le-; it uill then travel along a stieamline of the liquid passing around the qecond *phere. Oiir treatment of collision in an ideal fluid rill t>mhodythe awimption that inritial effect< he neglected for the .ni:iller sphere. The question of the forceq of adhesion b e t w e n the air bubble and mineral being sufficient to pievent the aggregate parting is diicutsctl in Section TX, Considering these assumptions in detail. n e may say n i t h reqpect to the first, that the results obtained for a non-viscous, incompressible fluid are an excellent approximation to the behavior in a viscoiis, dightly coniprewible fluid s i i p h as n-ater. The second assumption nill lead to significant errois only if the size of the particle (or air bubble) is le*>than 0.1 micron. Particles as wiall as this nill not be considered. Particles are not spherical, h i t n e can define a hpherical particle which hehaves essentially as the mineral. Calculations based on this assumption for other hydrodynamic problems show a good approximation to actual behavior. Departure from sphericity of the bubble is more serious. The bubbles are deformed (flattened on top) and the motion of bigger bubbles shon-s instability (oscillation). Hence the velocity of the bubble relatire to the particle used in this paper vi11 be determined from an esperiniental cquation clesciibing the motion. There will be an error, honever. in deriving the “colli4on area”. 71-hich error is difficiilt to evaliiate. The moyt serious error t i p p ~ a r ti o be in :wimption -1. It i- implicitly assumed that ( a ) there is no coalewence of bubbles: ( 1 ) ) huhhlei do riot cluster around one particle (19): (c) buhbles do not hinder each other while ri4ng (i). AIF;siimption ( a ) was proved valid by experiments debcrihed in Section IX. lye have riot studied the importance of factor and farlor (c) i i *hoTI-n in &IppendixI to be relatirely unimportant. ;1seuniption 5 is not seriously in error, though for the bubhle ,-ize,; studied the motion is not completely streaniline (irrotational). The theory can be applied into the region of tmbulent fiov-, e g., Stokes’:, lair (’an he extrapolated n-ell into the region of ttiihiilence n-ithoiit errors eKceeding. say, 10 per cent. Seglect of inertial effect? (asslimption 6) slightly lowers the calrulaterl collision rate for the small partirleq common in flotation. IV. SYMBOLS USED IU THE THEORCTICAL TREATMEST
-1
=
o\-erAon- rntc of froth from re11 per iinit of re11 rolurn? ( v c . -
= h l l ( J ~ a n c yforce on the P = constant of integration.
f?
(,tinsen
’
hitbble irhich is rletrrmind h?. the -ti(vtrnline
398
E;. L. SUTHERLASD
maximum distance at which the center of a particle can lie from the line of motion of the bubble (figure 1) for collision t o be possible (cm ) d = d e n d y of mineral particle to be floated ( g . ~ m . - ~ ) d,,, = density of pulp ( g . cm.-') E = distance of particle from center of line of motion of the bubble (cm.) 8' = Stokesian resistance to a sphere L) =
f G G,
= = = = = = =
/
/
U ' f U'O
grade of unfloated pulp grade of concentrate g acceleration due to gravity (980 cm. sec.?) h height of attached bubble k , kl, Ji? constant. 1 distance travelled by bubble in cell (em.) L = capillary attraction of particle and bubble corrected for hydrostatic forces -1-= number of particles removed per unit vohime of pulp after time t see. (crn.-') SO= number oi particles initially present per unit volume of pulp (c n1.-I*'= number of bubbles per unit volume of pulp (em.- ') AS, = num1,er of particles of radius r per unit volume of pulp ( ~ m . - ~ ) A S A = number of particles \lit11 an induction period A initially present per unit volume ( c m . 3 &So = number of particles of radius T initially present per unit volume (em-? AA17k= number of bubbles of radius R per unit volume of pulp (~rn.-~) n = number of collisions per unit time (set.-') nf = number of fruitful collisions per unit time (set.-') rnf = number of fiuitful collisions in the cell between bubbles and particles of radius r crn. lnf = number of iruitiul collisions when bubble is loaded n L = number of particles attached to a bubble P = volume of pulp entering cell per unit time per unit cell volume
(set.-? Q
=
weight of mineral particle in n-ater
p = 37ro sech? ( -~ ~ ) R r ~(sec.1) ~ \
KIXETICS OF THE FLOTATIOS PROCESS
T , TI,
399
ArRm = rate of flotation of particles of radius I' ( g . set.-' ~ m . - ~ ) A R R m= rate of flotation of particles by bubbles of radius R (g. set.-' ~m.-~) s,,,= average rate of flotation (g. set.-' cm.?) ,R, = rate of flotation of sample containing only particles of mean radius f R , = rate of flotation from cell in continuous operation (g. see.-' ernsp3) ,Rf,, = rate of flotation horn cell in continuous operation when conditions are steady (g. see.-1 ern.?) tR, = rate of removal of ore in tailing (g. sec.-1 cm.?) 7-2, rk, r, = radius of particles (cm.) s = distance travelled by particle on bubble surface (cm.) t , = time of contact betn-een bubble and particle (sec.) t = time variable (sec.) T = surface tension (dynes cm.-') V = velocity of bubble relative to the particles (em. set.-') P = velocity of average-sized bubble (cm. set.-') 1'' = velocity of particle around bubble surface, arcial velocity (cm. see.-'> TT- = total weight of gangue in concentrate (g.) ec = concentration of mineral removed from pulp after a time t sec. (g. zco = Concentration of mineral in the feed (g. ~ m . - ~ ) zc: = concentration of gangue in the pulp (g. ~ r n . - ~ ) t c i = concentration of gangue in the overflow liquid (g. ~ m . - ~ ) w C = concentration of mineral removed from unit volume of pulp into the concentrate a t a time f (g. ~ m . - ~ ) tc; = concentration of gangue removed from unit volume of pulp into the concentrate a t a time t (g, ~ m . - ~ ) Ar" = concentration of particles of radius T removed from the pulp after a time t (9. c m . 7 ArwO = concentration of particles of radius I' initially in the pulp (g. ~m.-~) R: = polar coordinate a = contact angle measured from mineral through pulp p = radius of curvature of bubble in the planes perpendicular t o the plane of contact PO = radius of curvature of bubble a t its apes 6' = proportion of particles retained in the froth after fruitful collision 4 = velocity potential of the liquid relative t o the bubble X = induction period necessary for air-mineral adhesion (sec.) 9 = viscosity of pulp (poises) v = kinematic viscosity of the pulp (stokes)
400
{(T,
K . L. SUTHERLBSD
u ='density of air bubble (g. cn?) a,:=jdensity of gas in air bubble (g. mi.?) 4 = polar coordinate # = probability of gangue adhesion R, etc.):= function of T , R , etc. V. COLLISION BETWEEN BUBBLE AND PARTICLE
Ramsey (17) shows that the equation to the streamlines of a fluid moving past a sphere of radius R (the bubble) is sin2 4
C'X = ___
x3 - R3
where (z, 4) is the polar coordinate of the streamline (figure 1). As the inertialess particle will travel along a streamline it will just touch the bubble if it is travelling on the streamline whose closest approach to the bubble is r cm., the
FIG.1. P a t h of particle in streamline
radius of the particle (figurc 1). This streamline v-hich passes through the point z =E
+
T,
4
=
ciiitbl~si i b to determine the constant in equation 1, ciz.:
C'
=
(R
+
T ) ~ R3 R f r
\lThen the particle is at an infinite diitance above the bubble, then 4 = 0 and we require the value of s sin 4 vhich is equal to L ) em., the maximum distance a t which the center of the particle ran lie from the center of the line of motion of the bubble in order to j u i l collide nith it. Subgtituting in equation 1, we have
and taking tlie limit as + + 0 ii.c., R'
----f
.t)
If, as is true in flotation, i' is smnll comparcd u-ith R,t h m terms containing r3 only will he negligible and hence I1 = 4 3 2 (4) r ,
Ihis cIu:intity, I ) , defines t h e "collision radius" of thc htbhlr for i? particle of radius r . Llllparticles lying n-ithin Ihis dietnncc from thc line of motion of the
40 1
KISETICS O F THE FLOTATION PROCESS
bubble will collide with it. The number of collisions per second the bubble and a suspension of particles ( S o per ~ m . is ~ therefore ) n
=
TD~V.I-~ = 37~RrV~l~~
(71)
between (5 )
where V is the velocity of the bubble relative to the particles. If the bubble is an oblate spheroid (about the axis of motion) the number of collisions is n = 3X-nRrVLYo
where tk is a factor which is less than unity, the amount depending upon the eccentricity of the ellipsoid ( R is here the radius of the ellipsoid perpendicular to the axis of motion). If S’bubbles are present per cubic centimeter, the number of collisions is n
=
3aRrVX0S’
(6)
VI. RATE OF FLOTATIOX
The rate of flotation is assumed to be governed by the rate of collision between mineral and air bubbles in the pulp. It is observed that bubbles sometimes drop their load when they reach the froth, ou-ing to coalescence and subsequent decrease in area of surface holding the mineral. T5’e will suppose that a fraction (1 - e) of the particles is returned at any instant to the pulp by coalescence either a t froth-pulp interface or in pulp owing to coalescence of bubbles or to some turbulent condition which will strip the particle from the bubble. Usually e will be close to unity. If the bubble is practically completely covered with particles before it has risen into the froth, the last part of its journey t o the surface is ineffective, for collision cannot lead to adhesion. If the pulp density of floating mineral is large or the grade of feed is high (as it is in cleaning operations) then “crowding” becomes important. Under usual conditions the “collision area” on top of the bubble will be free from particles which slide to the bottom of the bubble (3). I n addition to the above factors, not every collision will be fruitful, since a finite time of contact between bubble and particle is needed t o ensure adhesion (the induction period A ) . This period is of the order of magnitude of 0.005-0.1 see. ( 5 , 20), although in special systems it is considerably longer (25). TYhen a particle is a t a distance d z r cm. from the line of’ motion of the bubble, it is obvious that the bubble will only be touched for an infinitesimally short time before the particle leave> it following the streamline flow of the liquid. As the particle lies nearer to the line of motion of the center of the bubble, it nil1 be in contact for longer and longer periods before it reaches a position where it can leaye the bubble. Let the particle be E cm. distant from the center of line of I?,+) at nhich the particle motion of the bubble. Then the point (s = strikes the bubble surface is derived from equation 1. Since lim (asin Q ) is equal t o E , the constant C‘ is ecliial to E’. Equation 1
+
z-m
becomes
402
K. L. SUTHERLASD
and the polar coordinate of the point n-here the particle strikes the bubble is
x
=
r
+ R, 4
=
arsin E
or neglecting r3
+ = arsin
(E/.\/=T)
The velocity potential of the liquid relative t o the bubble is given by
The arcial velocity-
+
Although the center of the particle travels on a circle of radius (12 r) after touching the bubble, we may consider it as moving on a circle of radius R without being in error by more than 10 per cent for the size of bubble and particle usual in flotation. This simplification is used in equation 15 and subsequently. From equation 10
+
The time to travel from the point (12 I’, (6) where the particle collides to the point ( R r , s - 4 ) n-here it leaves the bubble is given by
+
since dt,
=
dS
- and ds V’
=
( R + I’) d+ is the distance travelled on the bubble surface
in a time dt,. Hence
Substituting for 4 from equation 8
2
A more accurate approximation would be
KIKETICS O F THE FLOTATION PROCESS
403
If the particle is to adhere t o the bubble, then the induction period A must be less than or equal to t,. Equation 15 then defines E for a given A :
Figure 2 shows equation 14 plotted for different values of R arid X and also s h o m the effective area of contact for each bubble after reducing to unit volume of air. T' is calculated from ,Allen's equation (see -Appendix I).
Radius( R cm)
FIG.2. The relationship betn-een ( a ) bubble size arid the effective contact radius E and ( b ) bubble size and the effective area of contact per unit volume of bubble (EZ/R3).
We non- replace equation 6 by
where ?I,+ is the number of collisions per unit time per unit volume which lead to adhesion. The rate of flotation (g. ~ m . set.-') -~ is then
n-here 0 is introduced to allow for particles removed from the bubble after fruitful collision (page 401) and d is the density of the particles. Hence I?,, is pro-
40-1
L. SUTIIEKLAND
K.
portional to the pulp density and the rate of‘aeiation. It is directly proportional to the radius of the mineral h i t is related in a more romples niariner t o the bubble size, z i x : RT- qech’
(y;) __
Suppose that in a batch cell the rate of flotation i:, qtudied uitli time.
w o g. ~ m . -be~ the initial concentration of the niineral. Then if ( v ~ -
I,et is
u‘)
, ~ rate of flotation is giren by the concentration at f s e ~ .the
where p is a constant. Integrating h e t n w n the limits 0 and t
i 20)
u t = ~ ~- (e P1) -3
But a t the concentration (zco - w) g. cam.
we haye from equation 18
where S o and S itre the number of particles per ~ m of. cell ~ pulp which were present originally and which were removed re;spectively, i.e., (So- -Y)is the number remaining. Since
me have from equations 21 and 22
R,,
=
390 seclil
(g)
RrVS’(wo -
(23)
UJ)
Comparison of equations 19 and 23 s h o w that p = 390 sech’
(,,>3 1’X
IZrVS’
and hence equation 20 becomes w =
wojl
-3rOsech?(?rX
- e
4R)RrF.Y’t 1 I
(23)
and substituting from equation 23 in equation 23 for 10, U P c a n tierire the variation of rate with time. il Denver subaeration flotation re11 of depth 25 em. roritains 104r n of~ pulp. ~ The follon ing values are typical:
R
=
0.03 cm.
S’= 20 brtbbles em.-’
I’
=
3
8=1
x
1W’cm.
I’ = 6 cm. ace. ti’il
= 0.01 g.
- I
mi.
*’
3 S o minerd will appear i n the froth uiitil :t time t sec. a f t e r aeration co~iiinencea,i.e., until bubbles have risen from bottom of cell t o the top. t is dependent upon bubble size, Vt being equal t o the depth of the (%ell. The tinw can br considered as starting after this initial period.
405
KISETICS OF THE FLOTlTIOX PROCESS
-1typical time reyuired for a 90 per cent recovery is GO sec. (e.g.,for sized quartz floated with cetyltrimethylammonium bromide). 'Then from equation 20 the d u e of X is 0.009 sec., a value which is plausible. It is generally knoivn that the rate of flotation iq governed approximately by an exponential lan-. Thub Beloglazov ( 2 ) ,Zuniga (28), and Grunder and Iiadur (9) found that the rate of flotation at any time \\-as approximately proportional to t h e amount of floatable material remaining in the cell. VII. VARIBTIOS IX GRADE OF COSCESTRATE KITH THE TIME
WHET\' GASGI-E IS PRESEXT
Let zcg g. cm.-d be the initial concentration of mineral to be floated, and ~ ~ g.' 0 cm.-3 the concentration of gangue. Let $ be the probability of flotation of gangue after collision ($ to include the sech and 0 terms). Since will be small ( is usually large compared with tr0 we can, with good approximation, and u consider L U ~ constant throughout a batch test. Mineral and gangue n-ill be carried into the concentrate by the walls of the bubble cells. The extent of draining (proportional to the height of froth) determines the amount of pulp left in the walls. This pulp will not have the same composition as that in the cell, because gangue and mineral settle from it. I n the mineral concentrate, the overflow of liquid is d sec.-l per ~ m of. cell ~ volume carrying Aw; g. sec.-1 of gangue into the concentrate (IC; = concentration of gangue in the overflow liquid). The value of w;may be considered constant throughout the experiment and, judging from Schuhmann's results (18), may have values between 0.5 Z P ~and 0.1 for sizes of particles of less than 7 microns and 35 microns, respectively. The coarser the mineral the more rapidly does the gangue leave a froth. The grade of the feed is
+
If we,wi are the weights of mineral and gangue, respectively, in the concentrate (produced from 1 cme3of pulp), then the grade is
.
ZL', G, = ___
'lcc
+
i.e.,
VI-
l-G,-wb --
cc
a',
(27)
From equation 23 for the gangue
R,t
=
3n-y5RrVLV'w!$!
= p'$u(t where p' = 37rRrX'T'. I n the same time .iwSt g. is carried over by the froth liquid. The total weight (11') of gangue appearing in the concentrate is
TI' = p'$zcit =
u 4 { p'$
+ AW;t + ,L41
(29)
406
K. L. SCTHERLAXD I
/
where f = i t f ,t~~ is constant for any >ize range and has values less than unity and greater than zero. The amount of mineral floated in the time 1 is and from equations 25, 26, 27, 28, and 29 rve have: u i .
If, in a batch float, the grade of concentrate is studied, then it is apparent from equation 30 that the grade falls steadily as time increases. VIII. RATE OF FLOTATIOK O F I I I S E R A L WHICH EXHIBITS A SIZE RANGE
Hitherto the mineral floated has been considered to be of uniform size. Since the rate of flotation depends upon r. the radius of particle floated, the average rate of flotation does not equal the rate of flotation of the average size particles. I n the analysis which follon-s the presence of gangue is neglected. Let the number of particles A S , with a radius between r and r Ir be given by AL\-, = -\-"{(r)Ar (31)
+
where { ( r ) is a bize-distribution function. Corresponding to equation 17 n e have that the number of' fruitful collisions Ar for particles in the size range r , I'
+
and the weight of mineral floated in time Af is
+
If AriYo particles of radius between r and I' Ar are originally present and AX, are removed after a time f, the rate of' flotation i.;
and
Hence
-1-0’7
I i I S E T I C d OF THE FLOTATIOS PROCESS
and substituting for
&ZL
Q F . CY = 45'. Velocity of ascent from appropriate Reynolds number and coefficient of resistance. Density of irticles = 4 g.cm.-3
+
+
DIAMETER OF PARTICLE
microns
Q f F
cm
cm
d>nes
sec-1
25 50 75 100 200 500 2.5 50 75 100 200
I
0.18
0.40 0.78
0.39 0.75 1.08 1.39 2.35 3.80
18
500
dines
x 3.9 x 1.1 x 2.1 x 8.1
R
1
dynes
10-4
10-3 10-2 10-2 1.3 X 10-1 1.7
9.9 x 10-4 4.6 X 1 . 2 x 10-2 2.3 x 10-2 1 . 3 X 10-1 1.7
~
28.2
1
2.99
+
Table 1 compares the value of F Q, where Q is the weight of the particle in the fluid, with the buoyancy force of bubbles (I? = 4rX3d,g) and the adhesional force L which is given by K a r k (26) as
s>
Id = srT sin a - -
(44)
where a is the contact angle, I' the surface tension of the pulp, r the radius of circle of contact, and p the principal radius of curvature a t the circle of contact in the plane perpendicular t o the solid. T o calculate p it is necessary to know the forces acting on bubble and particlr. Thu.: \\-ark's equation for the bubble contour becomes
where
po
ir the curvature of' the hubblc a t its apex and it is the distance of the
412
K. L. SUTHERLAHD
apes from the plane of contact. X-olkova (23) s h o w that for small areas of adhesion 1%may be put equal to 2 R , the diameter of' the bubble before adhesion, and that po is approsimately equal to X. Thus:
+
The (Q F ) term replaces the usual buoyancy force due to the bubble. The impulsive force is probably so small that it can be neglected. It can only be large if t is small (equation d3d). For the example cited in table 1 the impulsive force does not exceed the adhesional force, L , except if the particle can be detached from the bubble in less than sec. Thib is not feasible and for a period of even so short a time as lo-? see., the force is from 30 to 10' times smaller than L , and is aln-ays smaller than the (Q F ) force. Calculation from equation 46 s h o w that the bubble shape involves no reentrant surface and consequently the aggregate is stable. It is apparent that the adhesional forces are sufficient for flotation however small the particle.
+
X. RATE OF FLOTATIOS OF A SAXIPLE THE PARTICLES OF WHICH DO S O T
HAVE A CXIFOIIXI INDUCTION PERIOD
Let the number of particles AAlvh \vliirh have an induction period between AX sec. be given by:
X and X
+
L1-A
(47)
= S,{(X)AX
Then an analysis corresponding t o that in the preceding section leads to
where p' = 3xRrT;S' and zuc = + ~ r ~ ~ Y ~ d . Corresponding t o equation 25 the weight of mineral floated a t a time t is given by ' B t sechz (3VX/4R)
I
dX (49)
Since the total number of particles originally present in the sample is it follows from equation 4 i that
A-0,
Hence equation 49 becomes (51)
413
KISETIC,S O F THE FLOTATIOS PROCESS
If n-e put z
=
(y;)
p'0 swh" __ "
, then equation 48 becomes
TABLE 2 Weight of mineral (w Q . ) Pouted i n a D e n z o cell originally containing w o g.(= 469 g . ) of spheres a f t e r carious times t sec. wo - w wo
0
30 60 120 180 240 300
182 333 385 408 420 427
I
I I
0.6119 0.2857 0.1791 0.1301 0.1045 0.0896
It is shown in -Appendix I11 that the second integral is negligible for certain numerical values of p'0. If it is neglected, then the inverse Laplace transformation can be applied and the integral i.j (33)
47.
where i = If therefore a batch rest is made in a flotation cell and an analytic function determined empirically for (vi) - zu), z c o from the experimental data then, provided this function fulfils the condition. of the I.aplace transformation, the function {(A) can be evaluated. To evaluate the distribution likely in a flotation pulp, spheres of a calcium glass n-ere prepared. The fraction betn-een 52 mid 7 2 mesh was separated and each particle allon-ed to roll don-n an inclined slope. Only those which n-ere completely spherical rolled straight and w r e collected. -1sample was photographed. and more than 98 per cent of the particles n-ere spheres. The average diameter vas 0.0252 cm. -After cleaning by ignition a t G5OoC.,the spheres were floated in a 2000-g. subaeration Denver cell, using 20 mg. of laurylamine hydrochloride per liter as collector and frother. The pH value as 7.0, the temperature 20°C. The rate of air flon- into the cell n-as constant at 55 per second. The weiqhts of concentrate taken at various times are shown in table 2. The
414
E;.
1,. SUTHERLAND
low rate of air flmv \\as essential t o enable e a s - removal of the concentrate: a t the full rate the float I\-as over curnpletely in less than 1 min. From table 2 an analytic function relating (w0I w0 to the time t with an accuracy of better than 1 per cent and nf the required form is
-w 1.614 X - ______
1 ~ 0 ~
100
(47.46
(47.46
x
+ r ) 3 +--(47.46 + 2.747
- ____22.62
+ f)’
10‘ +
2.39 X 108 (47.46 t)5
+
(jq)
Integrating equation 53 after subhtitution from equation 54 we have:
i1.614 X LOw8 - 11.312
0
4.Q05
+ 4.578 x
0.om
10’~‘ + 9.975 X 106z3} (55)
0.0,s
o.oz0
h
FIG.5 . Distribution in values of the induction period for flotation of spheres
For the sample studied it is assumed that 6 is unity and the following measured values r = 0.0126 em., R = 0.064 cm., T’ = 12 cm. set.-', S’= 5 bubbles ~ m . of -~ pulp enable p’ t o be calculated.
.)Y;
The value of z is found from p’ sech’
~
x
plot showing A4Vh’ X Ois given in figure 5. The curve s h o w a typical distribution of X values. With a sample containing particles with different induction periods the curve relating weight of mineral floated with time will no longer be a simple exponential one. Calculation of the effect of varying induction period on the rate of flotation under continuous operating conditions is difficult, but qualitatively the result resembles that for a distrihution of particle sizes (see hppendix 11). XI. STATE OF COSTIAXOCS OPERATIOS
I n correlating the batch test and a continuously operating cell, factors such as the accumulation of soluble products which may affect adsorption of collector,
415
KISETICS OF THE FLOTATIOS PROCESS
rate of adsorption of collector, etc. are neglected. Let the rate of flow of ore into -1 cell be Pw0 g. sec. , where P ~ m per . C~ M . of ~ cell volume is the volume of pulp entering the cell per second. At a time t , the total weight of ore that has been in the unit volume of the cell n-hich originally contained w0g. of ore is (1
+ Pow0 g.
The weight of ore floated per unit volume of cell when w g. moved is f R t n = p(wo(1 Pt) - U ' ] At the same time mineral is removed in the tailing a t the rate
has been re-
+
=
(P - A ) jU'o(1
(56)
+ Pt) - w 1
(57) is the volume of liquid removed per second per unit volume of cell where A with the concentrate. The total rate (fRn8 iX,n) is tKn,
+
(58)
For the boundary condition t = 0, ut = 0 , (59) where p is as usual. of flotation is f R m
Substituting for u1 from equation 59 in equation 56 the rate
-
(60)
When t is large (steady conditions) equation 60 becomes
This equation is of the form
k2 r and will be of 1 + A3
value in determining the vari-
ation of rate of flotation with size in continuous test. IRA can be calculated from batch cell tests, since p is deducible from these. If we calculate the ratio of equation GO to equation 61 then we obtain curves of the type shown in figure 6. Data presented by Schuhmann (18) for the rate of flotation of a copper mineral with time are also plotted in the same figure. The experimental curve cannot be accurately described by an equation of the type given. This is discussed in A%ppendixI1 of this paper, where Schuhmann'n curve is shon-n t o he compounded probably of a number of esponential Curves. X I . VrlRIATIOS OF FLOTATIOS HrSTE WITH P.IRTICLE SIZE
'l'lie variation in flotation rate for particlei of different size in continuous flow
P - 3 tests is drdiicwJ from eqiiation 61. Figure 7 b h o w c u r v c ~for n-hich ___ P
4l(i
K. L. SUTHERLAND
t0
40
60
/oo
80 T/M€
-
120
no
an
/e0
Coo
t SECQVDS
FIG.6 . Ratio of flotation rate t o rate under steady conditions during a continuous test
PARTICLE SIZE - LOGARITHMC SCALE - r ,n MICRONS FIG.7 . Relationship betffeen rate of flotation and size of particle floated during continuous tests.
is varied and data obtained by Gaudin, Schuhmann, and Schlechten (8). The curve does not fit the smaller values of r and unfortunately there are insufficient data to determine whether the flotation rate increases linearly with radius, as
KIXETICS OF THE FLOTATION PROCESS
417
claimed by the author, or whether it is described by the relation above. Gaudin Schuhmann, and Schlechten hare suggested that the smaller particles aggregate to non-aerated flocs. If this is true it is difficult to see why they were able t o find smaller particles in their method of measurement of particle size. Their method v a s to follow the rate of sedimentation, n-hich rate depends upon the size of particle or aggregate. If special precautions were taken to disperse these aggregates, then their hypothesis may be proved by conducting the sedimentation experiments without dispersing agents. Because the flotation rate is constant, the number of particles to maintain the size of the flocs constant must increase enormously as the mineral size is decreased.
D/A*I€T€R
OF
DROPLETS ( Y / C R O M )
FIG.8. Curve showing‘distribution of particle size in a n emulsion (after Clayton (3))
bliming” is an esample of the flocrulation of small particles on larger ones, but excellent flotation of mineral is still obtainable nhen the conditions in the pulp are such that “sliming,” a n d therefore presumably flocculation of small particles, is absent. I t is poqsible that conditions for *bsliming”were present in the above experiments. .An :malogoiis system is shonn in the distribution of droplet sizes in a n emulsion. One would expect, a priori, a distribution range from large particles to almost infinitely sniall particles. Figure 8 4io~rhan actual frequency curre (4). Dropleti of less than 2 microns are almost completely absent-this being common to many emidiionq iescliiding those in nhich the viscosity is yery high) and it is usually postulated that becau-e Bran-ninn morement iq PO marked for particles of less than 2 micron diameter frequent collisions ensure coalescence, which gives a smaller surface and hence a decrease in free energy for the system. Khilst flocculation presumably leads to only a small decrease in free energy due to ( 1 1
118
K. L. SUTHERLASD
diminution in “free surface,” the study of colloids s h o w that flocculation is frequent and is a stable condition. That the rate of flotation becomes constant when the mineral particles possess a diameter less than 2 microns and that emulsion droplets aggregate to form drops of the same size may be a coincidence, but it provides a useful starting point for discussion. O d h (14) has measured the size of flocs produced by the coagulation of barium sulfate suspensions and found sizes between 1 and 13 microns, according to the concentration of flocculating agent. These flocs contained betn-een 50 and 120,000 particles. The flocs were formed under quiescent conditions, and the larger-sized particles may disintegrate if placed, for example, in a flotation cell. S o data are available to show the variation in size of floc when a standard amount of coagulating agent is added to suspensions containing particles of different sizes. APPENDIX I
The effect of bobble loads o n rate of Jlotntiori Allen’s equation (1) describes the relationship between bubble size and tcrminal velocity for the size of bubble common in flotation:
d, is the density of the pulp, u that of the bubble, and v the kinematic viscosity. For the experimental values used in thi, paper T-
=
229(12 - 0.0034)
(63)
Allen only claims that hi3 equation is accurate to a bubble radius of 0.05 cm.7 but his data do not extend beyond this d u e . -in assumption made in our use of Allen’s equation is that when many bubbles are present the hindered rate of rise will not differ greatly from that of the single bubble. Gaudin ( 7 ) discusses the data of Iiermack, N’Iiendrick, and Ponder (12) for solids and finds corrections to the Stokes equation of the order of 5 per cent for an 0.005 ratio of volume of dispersed solid to volume of liquid. If the same data are typical of a dispersion of gas bubbles in water, the error due to the use of Allen’s equation will be less than 5 per cent, since Allen’q equation is more accurate than Stokes’s equation. Efect of load on bubblr : If the bubble carries a load vhich is sufficiently large, the terminal velocity is considerably derreased. Rewrite -Illen’s equation
I‘
=
k,(d,, - ~)~”(h’ - 0.0034)
(64)
where 1.1 2 = 0.5y2 -113 2 / 3 v d,,
The density of the bubble, \\-hen ? l L particles of radiu. tached, is
I‘
and density d ale at-
KINETICS OF THE FLOTATION PROCESS
419
+ 43 a r 3 n L d
4 - .lrR3ug 3
-
rR3 where u,,
=
density of the gas. Hence T3
21 3
d, - ug - - n L d )
R3
( R - 0.0034) (66)
since uo
= 0.001 g. cmGP3
Kow the number of fruitful collisions in a distance A1 is A 1 n j = 3.lrRrNN‘k
If it is assumed that the bubbles carry the full load the whole distance, then we get an upper limit t o the extent to which loading would increase the flotation rate. Taking the ratio of collision rate for loaded to unloaded bubbles we have from equations 67 and 17:
For d = 3 g. ~ m . - ~r , = 0.005 cm., R = 0.1 cm., d, = 1.2 g. ~ m . - ~and , X = 0.0166 it is calculated that 10, 100, and 1000 particles increase the flotation rate by 0.0, 2.0, and 15 per cent, respectively. The effect is relatively unimportant unless the particles arr large and the bubble small. APPENDIX I1
Flotation rate when sample has a distribiition o j particle sizes Equation 35 gives the flotation rate when the distribution of mineral sizes is described by a function ( ( r ) . I t is more usual to determine the size range of a sample by screening and weighing the mineral lying between two screen sizes. The screen openings are generally related by a geometric factor, e.g., each screen opening is 1/2 to l/d2 the preceding sizes. We have chosen therefore the geometric mean of each screen size as representing the sample.
420
K. L. SUTHERLAND
It is convenient to replace the integral of equation 35 by the sum,
where r1, r2 . . . T , ~are the geometric mean si7e of each screening and z~'~{(Ill.)is the amount of mineral of radius ~ k . TABLE 3 .lliIl f e e d ( a j l e r Gaudin) WEIGHT PER CENT
~
SIZE RASGE (DIALIETER IN IAICRONS)
0.5 3 7 13 17.5 14 10 7 9 6 11 2
GEOMETRIC XEAN RADIUS (MICRONS)
416-295 295-20s 205-147 147-1 04 104-74 74-52 52-37 37-26 26-13 134.5 6.5-0.5 0.5-0 .0
175.2 123.9 87.4 61.8 43.6 31.0 21.93 15.51 0.10 4.60 0,902
TABLE 4 Rate ofjlotation of each size and total rate compared with average rate Time (seconds).
,R,
x
............ ... . . ;
104
10
~
50
'
100
'
150
~
200
~
300
~
560
I
I
1 176.1
47.3
16.5
S.10
5.49
2.43 I
1.11
Gaudin has given an average mill feed which will serve as an example ( 6 , p. 142) (see table 3). The mean radiu\ for the >ample excluding the 0.5-0.0 micron size is 35.9 microns.
Lissuniing a value of 37d sech? f$)RVA-'
rate of flotation of particles of tile nican r3diiLs ,n
of 7.%, the
ia
R,,, = 0.02;(ju)oe-o 0 2 S 2 t
(74
If the rate of flotation for each of the hize distiibutiona is calculated, then on summing the%erates we may compare them n-it11 the rate for the average particle (table 4). For thi, sample the rate of flotation of the sample u-ould be smaller a t first than that of a sample conristing of the average-sized parricles. A sample could a130 be chosen such that the initial rate i:, greater than that consisting of arerage-sized particles.
421
KINETICS OF THE FLOTATIOX PROCESS
The behavior of an ore containing a size range of particles is important when steady flotation conditions are observed in continuous tests. It will be noticed above that the variation of the flotation rate with time is no longer a simple exponential curve but is the sum of a number of exponential terms. If data determined by Schuhmann (18) are examined, the curve showing flotation rate n-ith time is found not to be a simple esponential curve. This may be due to either variation in particle size or induction period or both. The ratio of rates of flotation of initial to steady conditions is given by equations 60 and 61, xhich have the following form for a distribution of mineral sizes:
where the symbols are as previously and pl
=
3 d sech2
31'1 (=) RTW
Consider a sample containing 50 per cent of particles of 1l)O-micron radius and 50 per cent 20-micron radius. Here
S!-(lT71) and let
PI =
=
t(JT2) = 0.5
=
100
5, A = 5 X
x
IO-* em.,
rZ
=
20
and P = 5 X
x io-* cm. Then
For this sample the arithmetic mean is GO microns, the geometric mean 95 microns. Hence the rates of flotation of samples of these sizes are = 1+
5.ge-0.0345'
(72)
and
Figure 9 shows the ratio of rate of flotation a t different times to the final rate calculated from equations 71, 72, and 73. The behavior of the sample possessing particles whose size is the geometric mean of the sample above shon-s an initial rate lower than that given by the simple exponential function. It is then higher and finally is lower again. This is the behavior when a single exponential curve is compared with Schuhmann's data.
422
E;. L. STTHERLAXD
APPEXDIX I11
C'orzditiorzs ;Tor riqlccf o j spcond integral of eyuatio?r 41 From equation 55 we can substitute in equation 52 and deduce the value of (wo - w)/wo,i.e.
where A = 1.614 X p'0 = x , and 1' = 1
B
+ 47.46.
=
-11.31, C'
7/U€
-P
=
4.578 X to3, D
=
9.475 X IO6,
rp.KC0ND.f
FIG.9. Comparison of ratio of initial rates of flotation t o rate under steady conditions during a continuous test for a sample with a size distribution of particles and a sample with particles with the geometric and arithmetic mean.
where
423
KISETICS OF THE FLOTATIOS PROCESS
Thi.;
iq
ZCC
- uq
smaller than the experimental value of -by the term
{l(t)
(see equa-
U70
tion 5 2 ) . Son-rl(t) will have its largest value when t For an accuracy of 1 per cent in the function
U‘O ~
5
0, Le., when
2 =
47.43.
- w it is only necessary that the
WO
ratios of each of the following terms
-4 23
and
il -
23
-
-
22
+
e-zm ( x 2 m 2 22m
+ 2), etc.
shall not exceed 1.01. The dominant term for experiments conducted over periods of less than 1.5 min. is D/z’ and we can consider that the total error will not exceed twice that in the Djs‘ term. Hence require
$/[$
(1 -
For
2 =
Since x
e-zm(s4m4
+ 4 2 3 m ~+ 122’nz’ + 24sm + 211
rg)
50 this requires a value of x greater than 0.25. =
3aerRVLYfsech‘ -1
and assuming that X
=
1
5 1 . 0 0 ~ (77)
0.002 sec., R = 0.1
cm., 1’ = 20 cm. sec. , -I-’= 5 then we estimate that the smallest particle size to which this section of the mathematical treatment will apply is 30 microns provided the constant 47.46 is not altered as particle size is diminished. If is larger than 0.002 sec. and particularly if the rate of aeration is increased, this minimum bize can be reduced. It is essential to calculate the error for each set of psperimental conditions. SUMMARY
The direct-encounter hypothesis for adhesion of mineral to bubble in a flotation cell is investigated. I t is shown theoretically that: (I) the “collision area” of a bubble is proportional to the product of bubble radius ( 8 )and mineral particle radius ( r ); (2) the area over which fniitful collision is possible is given by
(z )
3rR sech’ 3 T’X
(The induction period (A) almost entirely governs the sech term, because the ratio of liubble wlocity (17 to its radius (I?) does not alter appreciably.) induction period? can be calculated and these are probably as accurate as the experimental value,.; when there i. a distribution of particle sizes the rate of flotation i b the sum of a number of exponential functions of time; when there i. a distribution of hubble sizes the mean rate of flotation i- equal to tlie late of flotation nith suitably defined mean size of bubble; w.perimenta1 method6 of determining bubble size and number are dehcribed;
424
I
