Development and Theory of Centrifuga1 Flotation Cells of rotating flows on flowfields have...
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Development and Theory of Centrifuga1 Flotation Cells by Jun-Xiang Guo A thesis submitted to the Department of Mining Engineering in conformity with the requirements for the degree of Doctor of Philosophy Queen's University Kingston, Ontario, Canada December, 200 1 Copyright O Jun-Xiang Guo, 200 1
Transcript of Development and Theory of Centrifuga1 Flotation Cells of rotating flows on flowfields have...
Development and Theory of Centrifuga1 Flotation Cells
by
Jun-Xiang Guo
A thesis submitted to the Department of Mining Engineering
in conformity with the requirements for
the degree of Doctor of Philosophy
Queen's University
Kingston, Ontario, Canada
December, 200 1
Copyright O Jun-Xiang Guo, 200 1
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Abstract
Rotating fluids occur in a very wide range of applications. The dramatic, large-scaie
effects of rotating flows on flowfields have stimulated flotation cell designers to make use
of them for his particular purpose: improve the processing of fine particles. In this thesis,
the author oners a personal view of the applicability of various rotating fluids to flotation
ce11 and mechanism design. Three types of rotating flows, namely, Rankine's combined
vortex, externaily-forced vortex, and internally-forced vortex, are considered to be
applicable to flotation and their features of interest are illustrated and discussed. As a
result, two new types of centrifùgd flotation cells, CFC-QI and CFC-Q2, were developed.
Both models introduce the reactodseparator concept in their designs and use an extemai
in-line air sparger made of porous stainless steel to aerate the pulp. The operating
mechanism of the CFC-QI ceIl uses a rotating drum or bowl to create a centrifugai force
on the feeding slurry. Froth concentrate is collected on the central top of the ce11 and
tailing is discharged dong the top edge of the rotating drum or bowl. The CFC-Q2 ce1
utilizes a vertical rotating feeder to introduce the feeding slurry into a stationary structure.
A centrifiigai force is created by the rotating feeder for the feeding siurry. Froth
concentrate floats over the ce11 and tailing is discharged through the bonom of the cell.
The major operating variables of the CFC-QI and CFC-Q2 models include the rotating
speed, the air flow rate into the air sparger, and the slurry feeding rate. The metallurgical
performances of the CFC-QI and CFC-Q2 cells were compared with that of a
conventional laboratory Denver mechanical cell. Fiotation tests were c&ed out with two
artincial mixtures and two natural base metal ore samples. The test results clearly indicate
the advantages possessed by the CFC-QI and CFC-Q2 cells in the processing of fine
particles. The grade-recovery curves produced from the CFC-QI and CFC-Q2 cells are
generally better than or quivalent to that from the Denver cell. The study coincides with
the pnor theories that strong force fields such as centrikgal force fields would increase the
particle-bubble collision efficiencies in the notation process and consequently the recovery
of fine particles would be improved.
Acknowledgements
My sincerest thanks are due to dl faculty, staff and colleagues in the Mining Engineering
Department, who instructed me, advised me a d helped me throughout my entire graduate
study at Queen's University. Above di, 1 wish to thank my supervisor, Dr. W. T. Yen, for
his valuable guidance and financial support.
1 would also like to thank Mr. A. Pindred who helped with the developrnent of
centrifuga1 flotation cells and with the flotation expenments.
The financial support for the project, "The Modification of Centrifugai notation
CelP*, undertaken in collaboration with InterCitic-Envirotec Inc., and the financial suppon
from Queen's University, specifically, G.C. Bateman Fellowship, J.J. Demy Memonal
Fellowship, Dean's Award, Graduate Awards, tuition and thesis bursaries, are gratefûlly
acknowledged.
Table of Contents
Abstract ........................................................................................................................ ii
......................................................................................................... Acknowledgements iv ...
List of Tables ................................................................................................................ viir
........................................................................................ Lia of Figures and illustrations x
Chaptcr 1 . 1.1.
1.2.
Chapter 2 . 2.1.
2.1.1.
2.1.2.
2.1.3.
Introduction .............................................................................................. 1
Generd ........................................................................................................ 1
............................................................................ Objective of Investigation 6
Literature and Basic Theory Review ........................................................ 8
Rotating Flows Applicable to Flotation ........................................................ 8
Guidelines for Selection of Rotating Flows for Flotation Purpose .............. 8
Generation and Classification of Rotating Flows ..................................... 10
Rotating Flow Type 1: Rankine's Combined Vortex ................................ 14
2.1.3.1. Flow processes in a cyclone separator ............................................... 15
2.1.3.2. Fluid-flow patterns ............................................................................ 17
.................................................... 2.1.3.3. Application to flotation ce11 design 24
2.1.4. Rotating Flow Type II: Extemally-forced Vortex .................................... 29
2.1.4.1. Flow processes in a rotating cylinder filled with water ....................... 29
2.1.4.2. Fluid-flow patterns .................. .. .................................................. 3 1
2.1 .4.3. Application to notation ce11 design ................................................ 3 5
2.1.5. Rotating Flow Type m: Intemally-forced Vortex .................................... 40
2.1 . 5 . 1 . Flow processes in a stationary fluid with a rotating disk .................... 41
2.1 -5 .2 . Fluid-flow patterns ............................................................................ 42
2.1 S . 3 . Application to flotation ceIl design .................................................... 47
...................................................................... 2.2. Basic Rotating Fiuid Theory 50
2.2.1. Rotating Coordinate Systems ...................~............................................. 5 0
2.2.2. Fiuids in Rotating Systems ....................~................................................. 52
......................................................... 2.2.3. Solid Particles in Rotating Systems 55
Table 6-2. Estimated bubble terminal velocity in pure water for db= 1.5 mm . . . . . .. . . . -165
Table 6-3. Bubble terminal velocity for db= 1.5 mm in surfactant solution at 30 O C . .166
Table 6-4. Flow regime around an air bubble ................................................. 166
List of Figures and Illustrations
Figure 2- 1 .
Figure 2.2 .
Figure 2.3 .
Figure 2 4 .
Figure 2.5 .
Figure 2.6 .
Figure 2.7 .
Figure 2.8 .
Figure 2.9 .
Figure 2-1 0 .
Figure 2- 1 1 .
Figure 2.12 .
Figure 2- 13 .
Figure 2- 14 .
Figure 3- 1 .
Figure 3.2 .
Figure 3.3 .
Figure 3.4 .
Figure 4- 1 .
Figure 4.2 .
Figure 4.3 .
Figure 4.4 .
Figure 4.5 .
Figure 4 6 .
Figure 4.7 .
Figure 4.8 .
. . Cylindncal polar coordinates .................................................................. 11
Cutaway view of cyclone classifier ....................................................... 16
..................................................................... Rankine's combined vortex 22
Perspective of the air-sparged hydrocylone ............................................. 27
............................................................................. The cyclo-colurnn cell 28
Spin-up of water in a cylinder ................................................................ -30
................................................................... A rotating container of fluid -33
Centrifugai notation ce11 developed by Clean Earth Technologies ........... - 37
Modified centrifugai notation ce11 CFC-Q 1 .......................................... -39
Conventional subaeration mechanical flotation ce11 ................................. -40
A rotating disk in a stationaiy fluid ......................................................... 42
Dimensiodess velocity components of flow ( U: . u:. U: ) induced by a spinning disk, in a fluid othenvise at rest. are a function of dimensionless distance (z) frorn the disk ............... ... ................. 46
........................................................... Centrifùgal notation ce11 CFC-Q2 49
Reference h e b has an arbitrary time-vaiying position and orientation relative to fiame n .......................................................... 50
Perspective view of CFC-QI-2 .............................................................. -63
Cross-sectional view of CFC-Q 1-3 ........................................................ 6 4
Perspective view of CFC-Q2- 1 ............................................................... 69
Cross-sedonal view of CFC-Q2-I ........................................................ -70
.............. Size distribution of galena-silica mixture with 20 minutes grind -83
..................................... Sire distribution of galena with 20 minutes grind 83
............... Size distribution of galena-silica mixture with 30 minutes grhd 84
..................................... Size distribution of galena with 30 minutes grhd 84
............... Size distniution of galena-silica mixture with 60 minutes grind 85
..................................... Sire distribution of galena with 60 minutes grind 85
Size distribution of c halcopyrite-silica mixture with 3 0 minut es grind ..... -87
Size distribution of chalcopyrite with 30 minutes grind ............................ 88
Figure 4.9 .
Figure 4- 10 .
Figure 4- 1 1 .
Figure 4- 12 .
Figure 4- 13 .
Fi-gure 4- 14 .
Figure 4- 1 5 .
Figure 4- 16 .
Figure 5-1 .
Figure 5.2 .
Figure 5.3 .
Figure 5-4 .
Figure 5.5 .
Figure 5.6 .
Figure 5.7 .
Figure 5.8 .
Figure 6- 1 .
Figure 6.2 .
Figure 6.3 .
Figure 6-4 .
Figure 6.5 .
Figure 6.6 .
Figure 6.7 .
Figure 6.8 .
Figure 6.9 .
...... Size distribution of chaicopyrite-silica mixture with 60 minutes @nd 88
Size distribution of chdcopyrites with 60 minutes grind .......................... 89
Size distribution of Gaspe copper ore with 30 minutes grind ................... 90
Size distribution of copper sulfides fiom Gaspe copper ore with 30 minutes Grind ......................................................................... 9 0
Size distribution of Gaspe copper ore with 60 minutes grind ................... 91
Size distribution of copper sufides from Gaspe ore with 60 mintes grind -91
........................... ............................... Fraunhofer diffraction pattern .. 101
...................... Lead recovery from sire fraaions (gdena-silica mixture) 134
.................... Lead recovery fiom size fractions (galena-silica mixture) .. 135
Copper recovery from size fractions (chalcopyrite-silica mixture) .......... 136
Copper recovery fiom sire fractions (chalcopyrite-silica mixture) .......... 137
Copper recovery fiom size fractions (Gaspe copper ore) ....................... 138
Grade-recovery relationship for galena-silica mixture (30 min . grind) .... 139
Grade-recovery relationship for chacopyrite-silica mixture (30 min . grind)
. ........ Grade-recovery relationship for Gaspe copper ore (60 min grind) 141
............. Zones of interaction forces between the bubble and the particle 146
....................................... Notations for liquid flow around an air bubble 161
Collision of particles with a bubble ........................................................ 171
... Effect of bubble size on collision efficiency under different conditions 173
............................... Geometry of a single.bubble. single-particle systern 178
Collision efficiency as a tùnction of Stokes' number and nondimensional force for particle-to-bubbie radius ratio of 0.0 with various terminal settiing velocity OK) ............................................................................. 184
Collision efficiency as a fùnction of Stokes' number and nondimensionai ........... force for viscous flow with particle-to-bubble radius ratio of 0.0 184
Collision efficiency as a function of Stokes number and nondimensional .... force for potentiai flow with particle-to-bubble radius ratio being 0.1 185
Collision efficiency as a hnction of Stokes number and nondimensional force for viscous flow with change in particle-to-bubble radius ratio ...... 185
Chapter 1
Introduction
1.1. Ceneral
Flotation is a widely used method in the rnining industry to concentrate the valuable
minerals prior to the metals extraction phase. Unlike physical methods of minerai
processing such as gravity, magnetic and high tension separation, froth flotation is a
complex physico-chernical process taking place in a pulp which comprises three phases:
one solid and two fluids (usually water and air). The solid, suspended by a turbulent flow,
is a mixture of finely divided mineral particles that may be separated if their surfaces are
either hydrophobic or hydrophilic, regardless of their bodily compositions. The separation
uses air bubbles as a Iifting vehicle. Solid particles collide with air bubbles in the turbulent
flow. Characteristically those with hydrophobic surfaces tend to adhere to air bubbles
enabling them to be floated to the pulp surface, whereas the hydrophilic particles remain
wetted by water and do not float.
In their natural state most minerals are wettable and therefore non-floatable. By
adding to the pulp a smali quantity of collectors the surfaces of the desired minerais are
given an air-avid, water-repellent coating that readily adheres to an air bubble. Sometimes
the pulp must be pretreated before the desired minerals will adsorb the collector
satisfactorily. Under normal circumstances, the desired rninerals oniy account for a very
small percentage of al1 solid particles in the pulp. Therefore the judicious use of a wide
variety of chernical reagents that make the desired minerals hydrophobic while maintainhg
the hydrophilic character of the other rninerals is the key to the fiotation process.
The success of the flotation process also depends to a significant degree on the
efficiency of the attachent of solid particles to air bubbles in a turbulent flow. In
modeling the attachment process, it is usually divided into two distinct and sequential sub-
processes: collision and adhesion. The collision of solid particles with air bubbles is largely
independent of reagent adsorption at their surfaces, but is strongly idluenced by the
bubble properties and hydrodynamics in a flotation cell. The process of adhesion, on the
other hand, is essentially controiied by the surface chemistry of the air bubbles and mineral
particles present in a Botation cell (Yoon, 2000).
The puticle-bubble collision process is one of the most important elementary steps
in flotation. Although most of the flotation machines are operated under intensely agitated
conditions, not al1 solid particles will coilide with air bubbles. In other words, the
probability of collision is not 100%. To understand this, consider the collision of a single
solid particle with a single air bubble. When viewed simply as the collision between two
spheres, the fioth flotation case diffen from the raindrop coalescence and dust and mia
collection in that the two spheres are of markedly diEerent densities and sizes, and they
move in opposite directions (Flint and Howarth, 1971). Both develop tluid flow patterns
with diverted streamlines around themselves. The air bubble is usually much larger than
the solid particle, hence the fluid flow pattern in the particle-bubble system can be
characterizai by the flow around the bubble. In approaching the bubble surface, the
particle is deflected away by the fluid streamlines around the bubble and under certain
conditions the particle would simply skirt around the bubble without effecting any contact.
The collision characteristics of a particular particle-bubbie pair have generally been
studied probabilisticaily. The probability of collision, or ofien called "collision efficiency",
is defined as the ratio of the number of solid particles that actually collide with the bubble
to the number that would collide if the fluid streamlines were not diverted by the bubble.
The study of collision efficiency is heuristic in developing new technologies to
improve the flotation of tine particles. It is well known in the mining industry that
flotation's effectiveness is limited to a relatively narrow particle size range. Recovery is
usually best for particles of an intermediate size, roughly in the range of 1û-150 Pm.
Particles finer than 10 pm are generally defined as "the fine particles" or "the slimes" in
flotation, although this definition can be flexible to some extent because al1 minerals have
different densities. Fine particles can be detrimental to flotation due to reduced selectivity
of collector adsorption, entrainment of fine gangue particles, reduced rate of flotation etc.
However, the processing of fine particles is tiindarnentally irnponant to the mining
industry. Faced with the depletion of many easy-to-process ores, miners are delving
deeper for seams that are harder to exploit. Most mine output today is the hard-to-treat
hely disseminated ore of lower grade and more cornplex composition that requires
extensive grinding to liberate the valuable minerals from the gangue. As a result, fine
particles are produced in quantities. Inefficiencies in fine particle flotation translate into
both an enormous loss of revenue and an unnecessary waste of natural resources.
Over the yean many attempts have been made to improve the flotation of fine
particles. An excellent review on the major developments up to the mid-1970's was
complied by Trahar and Warren in 1976 (Trahar and Warren, 1976). Some of the methods
are still being used in practice, some are being revived due to the introduction of new
technologies. Two main lines have been followed. One line is to improve old processes by
adding chemisorbing collectors, polymeric electrolytes, or neutral oils with the aim to
enhance the agglomeration of fine particles. The probability of collision between air
bubbles and mineral particles increases when the latter are present as agglomerates or are
attached to large (hydrophobic) particles which act as carrier particles (Fuerstenau, 1995).
The other line is to develop new cell and mechanism with the aim to create more favorable
hydrodynamic conditions for fine particle flotation. The mechanical cells continue to
dominate the flotation machinery market, a situation unchanged for nearly a century. Part
of their success is related to the processing of relatively rich, coarse mineral particles.
Faced with the increasing demands to improve the processing of fine particles, the
mechanical cell suppliers have shown their resilience by increased ce11 size, the use of
adjustable fkoth crowders and new agitation mechanisans (Finch, 1998). The largea cells
now in operation are the 160 m3 Outokumpu's ~ank~ells*. S k t y years ago at Phelps
Dodge's largest operating mil1 plant at Morenci, AZ., the size of the Fagergren cells was
oniy 1 -7 m3 (Arbiter, 2000).
Alternative flotation machinery has always been around, and some of them have
gained solid ground in the mining industry. Major advances in flotation technologies
during recent years include flotation colurnns and the Jameson cell. The former is being
increasingly used in the cleaner stage (Sanchez et al., 1997) and the latter has become the
preferred method for fine coal recovery at more than half of Australia's coal flotation
plants (CWord, 1998). Another highly promising class of flotation machinery is known as
the "centfigal flotation cells" in which a rotating Buid is the principal phenomena of
interest. Conventional flotation is conducted in the earth's gravitational field. In the case
of fine particle flotation, the weight of fine particles is so smdl that their inertia could be
neglected under the action of gravity. Fine particles follow the fluid streamlines largely
with no slip and thus can easily be diverted £tom the strearnlines around the bubble. A
rotating fluid, however, creates a centrifiigal force field in which centnfugal forces are
developed and they change the balance of various forces acting on the fine particles.
Consequently the bubble-particle collision efficiency in a centnfugal force field would be
higher than that in a gravitational force field (Karr et al., 1990).
Rotating masses of Buid exhibit some unusual properties. Conventional mechanical
flotation cells are basicdly stirring machines. The operation takes place in a highly
turbulent flow. Such random motions are not possible in a rotating fluid; instead, the
permissible flows have a distinctly two-dimensional property. Depending on how a
rotating fluid originates and runs, the phenomena involved can be vastly varied. Some are
conducive to flotation, some not. Due to the absence of mechanical agitation, the
conventional bubble generation and dispersion mechanisrns used by the mechanical ceiis
are no longer applicable. In a rotating fluid, air bubbles do not move outward but toward
the core of the vortex. Therefore, the fioth collection in a rotating pulp will be different
from the conventional skimming. It is clear that the entire flotation system has to be re-
designed if a rotating fluid is to be used.
1.2. Objective of Investigation
Centrifuga1 flotation technology has been developed since the 1980's, but so far none of
centrifbgal flotation cells have been incorporated into industrial-level production, owing to
the natural cornpetition for a market that tends to favour those technologies that have been
applied over decades. The hydrodynarnics of centrifuga1 flotation cells, while understood
by some, were not well communicated to ce11 designers or users. In this thesis, the author
offers a personal view of what centrifuga1 flotation holds for minerai processing, especially
for the processing of fine particles.
The principal objective of this investigation is to modiQ the design of the new
centrifuga1 Botation ce11 (CFC) proposed in 1995 by Clean Earth Technologies, a wholly-
owned subsidiary of Inter-Citic Envirotec Inc. One of the persistent problems has been the
rapid blockage of the tailing discharge ports in the outer rim of the rotating drum. If the
feed slurry has a pulp density higher than 30% solids, these tailing ports would be easily
plugged up, resulting in the disruption of the entire flotation process. Beside, in actual
practice, it is very common that the ore pulp carries wood, debris, or lime scale. They can
accumulate and become a nuisance to the operation of this centfigal flotation cell.
To permit the rationai modification of the original CFC, it is critical that various
rotating fluid phenomena, especially their implications to flotation cell and mechanism
design are well understood. Detailed studies on such subjects are scarce in the flotation
literature. The first part of this thesis (Chapter 2) systematically analyses the potentialities
of different types of rotating fluids for fhth flotation. Various flow phenomena are
exarnined for ideas, which result in two new designs CFC-QI and CFC-Q2. The CFC-QI
ce11 is the offspnng of the original CFC. while the CFC-Q2 ce11 represents a new
generation of centrifiigal flotation cells. Each model has several variations that confonn to
some general principles.
In order to evaluate the efficiencies of the new designs, flotation tests were canied
out using both artificial ore mixtures and natural ore samples. The operating variables
associated with both models were studied. The metallurgical efficiencies were compared
with that of the standard Denver laboratory flotation cell (Mode1 D-12). notation
methodology and test results are summanzed in Chapter 4 and Chapter 5.
Chapter 6 relates the macroscopic rotating fluid effects to the microscopic
elementary stages of the flotation and in doing so, justifies the flotation in rotating fluids
by some of the pnor work conceming the froth flotation collision efficiencies in centrifuga1
force fields. Although the establishment of a comprehensive mathematical model for the
d e - u p of the new designs is beyond the scope of this investigation, it is believed that
Chapter 6 will help the future work in this field by providing a better understanding of
hydrodynamics in centrifbgal flotation ceiis.
Chapter 2
Literature and Basic Theory Review
2.1. Rotating Flows Applicable to Flotation
2.1.1. Guidelines for Selection of Rotating Flows for Flotation Purpose
Rotating fluids occur in many areas of engineering and in geophysics, padcularly in the
atmosphere and the oceans. Depending on how rotating flows originate, the phenornena
involved are so varied that, in order to illustrate some of the principal design features
common to most rotating machinery, we shall confine our attention to single phase,
homogeneous fluids and to their curved motions, boundary Iayers, and secondary flows.
As far as notation is concemed, the design of the ce11 in which the concentrate is floated
must facilitate as many as possible of the following duties (Pryor, 1974):
" 1. Reception and aeration of the pulp without allowing settlement of solids.
2. Discharge of impoverished tailings aller aeration has removed a mineraiised
fioth.
3. Avoidance of short circuiting of pulp fiom entry to discharge without
being worked in the cell.
Search of the hl1 pulp volume with Wbubbles' of suitable sizes, thoroughly
dispersed, and in adequate quantity for fioth-colurnn concentration.
Provision of a zone where a quiet blanket of mineralised froth cm fonn and
from which gangue can drop back into the pulp.
Disc harge of "float s" and "sinks" by separate channels.
Controllability for pulp level and height of froth column.
Aeration without letting too large bubbles, or "bursts" of air create
disorder.
Provision for easy re-start afler mechanical failure, without "sanding up" of
mechanical parts and discharge orifices.
Efficient use of power, mil1 space, and impellers.
Easy maintenance with no odd corners where wood, debns, or lime scale
can accumulate and become a nuisance.
Provision for quick and easy changing of feed and intercirculation channels
in the line of which the ce11 forms a unit.
Ability to cope with maximum-sized sinking particles in the feed without
risk of accumulation and choking.
Working adjustment between new feed circulating past the entry-point of
air and of pulp recirculating inside the cell.
' The term "N4ubblem (an abbreviation of "N&ubblem) is applied to an immersed bubble of air (or any other gas) which bas the essential characteristic that at the moment of its arrivai in water the surface tension at the air-water interface is at its maximum for the qstem From uiat moment until its energence from the water as an independent bubble the d a c e tension at the interface is progressively reduced by se& from the aqwus phase of any moldes which will Iower the surface tension (Pxyor, 1974).
15. Arrangement for periodic bonom discharge of accumulated sand too
coarse to flow over the tailing weir."
These are very practical considerations and various points are usPd as a basis to
determine which type of rotating tlows can be used in flotation ce11 and mechanism design.
2.1.2. Generation and Classification of Rotating Flows
In general, rotating flows can be generated by three principal methods:
1. tangentid entry into a stationary round structure
2. direct rotation of a cylinder, a tnincated cone, or a bowl-shaped container
3. guided vane system consisting of a fixed set of vanes at angle ) to the
axial incident stream, deflecring the stream into rotation in a stationary
round structure
Rotating flows generated by the above methods can be classified into three
categories accordingiy:
1. Rankine's combined vortex
2. extemally-forced vortex
3. intemally-forced vortex
Each has its own features of interest, including secondary flows resulting from the
wbtle fluid-structure interactions, that can be emphasized by good design. To appreciate
these rotating flows, it is convenient to look specifically at a fluid in cyiindrical polar
coordinates, rotating about the axis. Cylindricd polar coordinates are denoted by (r, 0, Z)
which are show in Fig. 2-1. The coordinate r is the distance (perpendicular) fiom the z-
axis. The point P in Fig. 2-1 has coordinates (r, O, z) using r = O on the z-axis and 8 = O on
the x-axis.
Figure 2- 1. C y lindrical polar coordinat es
Accordingly, fluid velocity at point P has three components:
1. axial velocity component, denoted by u f
2. radial velocity component, denoted by ~f
3. tangentid veiocity component, denoted by U:
Another important physical property in the study of fluid mechanics is called
"vorticity". Denoted by 5, vot-ticity is defined as the curl of the velocity field and is a
measure of how rnuch curl or rotation the flow has at each point. Except for a factor of
112, vorticity is identical to the angular velocity of fluid elements. Its mathematical
expression is given by
where V is the del operator and Ur is the fluid velocity vector. In a cylindrical (r, 8, z)
coordinate system,
where r, 0, z are unit vecton. The unit vector r at any point has magnitude equal to 1, and
points in the direction of increasing r at that point; 0 has magnitude 1 and points in the
direction of increasing 0; z has magnitude 1 and points in the direction of increasing z.
Note that only z is a constant unit vector. r and 0 are constant in magnitude only (= one
unit) but their directions Vary fiom point to point.
In a simple rotating fluid with no anal or radial velocity components (u: =u: =
O), the tangentid velocity U: is the only velocity component which is a finaion merely of
the perpendicular distance r from the z-ais (refer. Fig. 2-1). Asniming axisymmetry, then
the calculation of vorticity is reduced to:
Rotating flows with
are calledjbee vorrices where K is a constant (Gupta et al., 1984). Substituting Eq. (2.4)
into Eq. (2.3). clearly, the vorticity vanishes (5 = O). Thus a fiee vortex is characterized by
irrotational flow in which individual fluid elements do not rotate about their own axis as
they move in circular paths. Small objects placed in a perfect such flow would move in
circular paths but would not rotate.
Rotating flows with solid-body rotation
are cailed forced vortices where r' is a constant (Gupta et ai., 1984). The vorticity does
not vanish in this case (5 F O). Therefore the forced vortex displays rotationai flow in
which each individual fluid element rotates about its own axis. Smail objects placed in
such a flow would aiso rotate as they move in cùcular paths.
For rotating fluid applications in fkoth flotation, it is important to understand that
the basic theory of fluid rotation and vorticity distinguishes between vorticity and curved
(e.g., circular) translation of fluid elements. Generally, viscous or frictional effects in a
fluid give rise to vorticity. For example, laminar flow of a viscous fluid in a straight
channel rnoves along straight streamlines. The fluid elements and small objects placed in
the flow would rotate, however, because ofviscosity, as they translate along straight lines.
On the other hand, in a flow field called an inviscid fke vortex (e.g. a whirlpool fonned in
a sink or bathtub) al1 fluid elements move in circular paths. However, small objects here
would not rotate, indicating a fluid that is not rotating, but merely translating in circular
paths. These two flows illustrate two extremes, one that has straight pathlines but fluid
element rotation, and the second that has circular pathlines but fluid elements which do
not rotate. Viscosity in the first flow produces the fluid element rotation called vorticity,
which is absent in the second flow.
2.1.3. Rotating Flow Type 1: Rankine's Combined Vortex
Rankine's combined vortex (or 6ee-forced vortex) consists of a central forced vortex core
of radius R rotating as a solid with angular velocity CO, in which the vorticity is nonzero,
surrounded by an inviscid 6ee vortex in which the vorticity is zero. The tangential velocity
component U: is zero on the axis of symmetry. The free and forced vortices can be
distinguished by the radial position of the maximum value of the tangential veiocity
component U; . in the fkee vortex the maximum is found near the axis of symmetry while
in the forced vortex the maximum is found at the outer edge of the vortex. The tangential
velocity component U; and the pressure p are continuous at the radius R.
The relevant phenomena in this type of rotating fluid are best iilustrated by a
cyclone separator, which is widely used in the mining industry as classifiers and
dewatering or dust collection devices.
2.1.3.1. Flow processa in a cyclone separator
Rotating fluid can be easily developed by a tangential entry into a stationary round
structure like a cyclone separator. A typical configuration is show in Fig. 2-2. It consists
of a cylindrical section (A) mounted on a tmncated cone (B) with an inlet noule (C) that
directs flow into the innet cylindrical section tangentially. The opening at the apex of the
tmncated cone serves as the underflow noule @), and a tube (E) called "vortex finder" is
extended partially into the center of the cylindrical section as an overflow noule.
In the basic operation of a cyclone classifier, the ore pulp is pumped into the
cylinder at high velocity. The nonrnoving structure then deflects the incoming fluid into a
circular path, forming a vortex that develops centritùgal force that acts to throw the
mineral particles outward toward the wail and creating a whirling action to push the pulp
spirais downward. The coarser, heavier particles rnove preferentially toward the waü of
the cyclone. Secondary flows on the conicai section help to carry the heavier particles to
the bonom outlet and exit as underflow. The iighter, finer particles also spiral downward
but, king less affected by centrifuga1 force, do not reach the wall. A choking action
created by the coarse particles at the bottom of the cone pushes fine particles into an
upward-moving inner spiral, rotating in the sarne direction as the outer one. The inner
spiral exits through the vortex finder as overtlow.
fine
y -- fraction
Figure 2-2. Cutaway view of cyclone classifier (Boldt Ir., 1967)
2.1.3.2. Fluid-flow patterns
In cyclone chambers, the ultimate destiny of any solid particle entrained in the fluid
depends on the balance of field (gravitational, centrifbgal) force, buoyant force, drag force
and additional hydrodynamic or inertial forces that are operative. Among them the drag
force and centrifbgal force are directly associated with the fluid-flow patterns. For this
reason, the velocity characteristics of the fluid must be studied separately from those of
the solid particles.
The fluid flow inside a cyclone closely approximates Rankine's combined vortex. It
consists of a ngidly rotating core (forced vortex) with radius R surrounded by an inviscid
vortex with matching tangentid velocity U: and pressure p at R. An inviscid vortex
(viscosity p = O) dissipates no energy and requires no driving force for the steady-state
flow. Also, the flow is irrotational (vorticity = O), which means fluid elements and
infinitesimaily small objects (e.3. very fine mineral particles) do not rotate about their own
avis in this flow but merely translate in a circle. In the forced vortex, the fluid rotates
about the vertical axis like a rigid body, i.e. with angular velocity o independent of the
radial distance r from the axis of symmetry.
In the case of a cyclone, the fluid e n t e ~ g the device is accelerated by the pressure
differential existing across the inlet nozzie. The ingressing fluid is introduced in a slightly
downward direction and tangentially to the cylinder wail. The geometry of the cyclone
then creates a strong vortical (swirling) flowfield which possesses pressure gradients to
accelerate the fluid in both radial and axial directions. In addition, the swirling fluid,
characteristic of a vortical field, passes from a free vortex to a forced vortex condition.
In cylindrical polar coordinates (r, 8, z) (refer. Fig. 2 4 , the velocity of the fluid in
a cyclonic fiow has three components: axial u:, radial LJf and tangential U: respectively.
The fluid enters a cyclone in a slightly downward direction and the axial velocity ~f of
the fluid adjacent to the cone wall continues to increase due to the geometry of the cone.
The axial velocity U: of the fluid moving towards the center of the cone experiences a
transition from a downward direction (in fiee vortex) to an upward direction (in forced
vortex). The magnitude of the upward axial velocity within the forced vortex is many
times greater than the cone wall axial velocities.
The radial velocity ~f of the fluid in the vortical field also continues to increase
toward the apex of the cone. At a given horizontal position, the radial velocity is greatest
at the cone wall and approaches zero somewhere between the cone wall and the axis of
the cone. In the vicinity of the vortex finder, the radial velocity is reversed in sign and a
circulation pattern or eddy curent appears. The high radiai velocities at the cone wall are
attrîbuted to the axial velocity U: being deflected by the cone wall and added to the radial
veiocity component uf .
The tangential velocity U: of fluid in the vortical flow increases with decreasing
radius across a fiee vortex and decreases with decreasing radius in a forced vortex. The
transition zone between the free and forced vortices occurs at the point of maximum
tangential velocity of the fiee vortex and is the shearing section of the vortical field
supplying the torque to the forced vortex.
The tangential velocity of a fluid particle in a steady free vortex is given by
where K is a constant which is the inviscid vortex strength, and r is the radial distance of
the point fiom the axis.
The tangential velocity for fluid inside a forced vortex is given by
where o is the angular velocity of the forced vortex and is the axial component of
vorticity (k = 20). o is independent of r. By Stokes's theorem circulation r is the
integral of l;z over the area of the circle of R, so r = x ~ ~ b (Vanyo, 1993). R is the radius
of the forced vortex.
The inviscid vortex strength equivalent to this r is then
giving vorticity
as a fùnction of an equivalent vortex with strength K. Substituting Eq. (2.9) into Eq. (2.7)-
the tangential velocity for fluid inside a forced vortex is obtained as
where K is a constant known as the inviscid vortex strength (the same as the one used in
Eq. (2.6)), r is the radial distance of the point from the axis, and R is the radius of the
rigidly rotating core (forced vortex).
K At r = R, both Eq. (2.6) and Eq. (2.10) become U: = - indicating that the
R'
tangential velocity inside the forced vortex is continuous with the tangential velocity inside
the fiee vortex.
Pressure is also continuous over r as follows. Inside the core (forced vortex)
pressure is (Vanyo, 1993):
where p, is the ambient pressure at intinity, R is the radius of the rotating core (forced
vortex), pl is the fiuid density and r is the radiai distance of the point fiom the axis.
For pressure field p(r) in the extemal free vonex outside the core (forced vortex),
pressure is given by (Vanyo, 1993):
At r = R, both pl and p2 becorne
Fig. 2-3 provides a graphitai representation of ~ é ( r ) and p(r) in Rankine's
cornbined vortex. It shows that the tangentid velocity component of fluid, U: (r),
increases as a tluid element in the free vortex moves closer to the forced vortex, reaches
the maximum value at the boundary between the free vortex and the forced vortex, and
then decreases as the fluid element moves hrther to the centre and in the end, becornes
zero at the mis of symmetry. Pressure, p(r), however, continuously decreases with
decreasing radial distance of any point in the flow field.
Rigid rotation
Ug - r vortex
Figure 2-3. Rankine's combined vortex (Vanyo, 1 993)
The three components of the fluid velocity exen dEerent influences on the
effiiveness of separation of the cyclone. The axial velocity ~f of the fluid at the cone
wall helps to carry the heavier solid particles to the bottom outlet. The radial velocity U:
of the fluid, however, reduces the separation efficiency as the drag force resulting fiom the
tluid tends to drag the solid particles dong with the moving Stream toward the center of
the cone. The tangentid velocity U; of the fluid dominates much of the flowfield,
developing centrifuga1 forces on the solid particles. The centrifugai force is an inertia
effect created by the mass of the moving particles desiring to travel in straight paths. This
centrifugaf force in a fiee vortex can be expressed as
where d, is the diameter of the solid particle, and p, and pl are the density of the solid
particle and the fluid, respectively, U: is the tangential velocity component of fluid at the
point if no solid particle was there, and r is the radial distance of the solid particle from the
axis of symmetry at the centre of cyclone.
It is clear fiom Eq. (2.14) that high centrifuga1 forces will result from large (4)
and dense (p,) particles at small radial distance (r) with low specific gravity fluid (pf) at
high tangential velocities ( U: ).
The centrifbgal force on the solid particle opposes the radiai drag effects of the
fluid and tends to restrain the particles from moving towards the axis of the cone. The
drag force can be approximated by Stokes's equation
where p is the vixosity of the fluid and U: is the radial velocity of the fluid.
Cleariy, the size of the solid particle has a greater effect (third power) on the value
of the centrifuga1 force than on the drag force (first power). Therefore, the larger the
particle, the greater the separation efficiency. The effectiveness of vortical separation can
also be enhanced by the reduction of the drag forces - both radiai and axial achieved by
small particles in low vixosity fluid (Gupta et al., 1984).
2.1.3.3. Application to flotation ceIl design
Flotation is a heterocoagulation process in which the attachment of hydrophobic particles
to air bubbles results in aggregates (mineralized bubbles) of lower density which float to
the pulp surface. The hydrophilic particles remain in the solid-liquid-air dispersion. Thus
the separation process takes place essentially according to the density difference. When a
mineral particle adheres to an air bubble, the density of this particle-bubble pair is lower
than the rest of the minera1 particles. Thus the particle-bubble pairs cm be viewed as the
lighter but "coarser" particles, whereas the remaining minerals become the heavier but
"finer" particles. Since the separation of heavier particles from lighter particles is exactly
what the cyclone type of classifier does, the cyclone type of flotation ceU appears to be a
reasonable idea.
Compared with the conventionai sub-aeration mechanical flotation cells,
hydrocyclones have the following main characteristic features of interest that can be
emphasized or abated for the sake of flotation:
1. Long residence times, particularly with long cyclones (Gupta et al., 1984).
Solid particles cm be separated or suspended for very long periods by the
centrifuga1 force field generated by the swirling motion of the fluid. At a
glance, it seems to suggest that the rate of notation in a cyclone would be
very slow. However, when viewed simply as the separation of lighter
particles from heavier particles, the froth flotation case difYers fiom the
classification case in that air bubbles are involved in the process. The
particle-bubble aggregates have a density not only smaller than that of the
non-floatable particles, but also smaller than that of the fluid. They do not
move outward in the vortex but toward the core. Therefore, the residence
time of those floatable particles would be much shoner than that of the
non-tloatable particles.
2. The fiee vortex is an irrotational flow in which an immersed solid particle
will not rotate about its own axis while it moves in circular paths. This
would stabilize the adhesion of solid particles to air bubbies
3. Large radial boundary layer flows can develop close to the cone wails due
to the reduced centrikgal force field in this region. This will reduce the
separation effectiveness of the cyclone since the radial flows tend to drag
the particles toward the axis of the cone.
Solid particles are subjected to hcreasing centrifùgal forces as they move
towards the centre of the cyclone because the tangential velocity
component U; of the fluid contii?uously increases with decreasing radial
distance r (from the axis of the cyclone) across the free vortex. Once the
solid particle enters the forced vortex in which the upward axial velocity
U: of the fluid dominates the flow field, the centrifbgai forces the solid
particle would experience will decrease continuously and become
insignificant. The solid particles in the forced vortex exit through the
vortex finder.
An application example of cyclone-type rotating fluids is Professor Miller's Air-
Sparged Hydrocyclone (ASH), shown in Fig. 2-4, which has evolved as a fast flotation
device after a decade of continued research at the University of Utah @as and Miller,
1996). The concept of ASH for fine particle flotation is based on the proposition that the
energy for the inertial collision between a fine particle and an air bubble will be increased
sufficiently in a arong centrifuga1 force field to achieve film rupture, bubble attachent
and flotation.
The ASH design has a cylindrical geometry with a tangentid or involute feed entry
at the top. It consists of two concentric right-vertical tubes, a conventional cyclone header
at the top, and a froth pedestal at the bottom. The i ~ e r tube is a porous tube through
which air is sparged. The outer nonporous tube serves as an air jacket to provide for the
even distribution of air thrwgh the imer porous tube.
The ore pulp enters through the tangentid inlet at the top of the ASH and follows
a hdical path before it exits in swirl flow through the underflow opening. During passage,
collision between centrifbged particles and air bubbles takes place, bubble attachent to
hydrophobic particles occurs, and the particle-bubble aggregate is transponed dong with
the fioth towards the vonex finder into the overtlow Stream. The high-speed swirl flow
exerts a considerable shear force at the inner porous tube wall. This, coupled with the fact
that the air is introduced through mal1 pores, results in the generation of a large nurnber
of small air bubbles wtiich facilitates the flotation of fine particies.
Figure 2-4. Perspective of the air-sparged hydrocyclone @as and Miller, 1996)
Anothet application example, known as the cyclo-column flotation ceIl, rnay also
be placed in the same category for centrifuga1 flotation cells utilizing the principles of
Rankine's cornbineci vortex. Fig. 2-5 shows the basic structure of a laboratory sale cyclo-
column cell (Yakin, 1995).
Figure 2-5. The cyclo-column ce11 (Yalcin, 1995)
The principal element is a Qrcular, centrifiigal column inside which a cenuifiigal
force field is generated by pumping a pre-aerated flotation feed tangentially Uito the
wlumn at its lower end. ïhe column is closed at the bottom and open at the top. As a
result of tangentid entry, the feed swirls inside the column as it moves upwards. During
the process, bubble-particle attachent takes place and the resulting bubble-particle
aggregates, being lighter than the rest of the pulp, collect around the centrai axis of the
column. Inside the centritiigal column, there is a second column, called the bubble column
which captures the bubble-particle aggregates and transports them, dong with some pulp,
into the fioth column above. The 60th column is wider enabiing the entrained pulp to slow
d o m and drain back as the Eoth expands, resulting in a clean froth product. The material
that does not go into the bubble column cornes out of the centrifuga1 column at the top
and flows into a pulp collecter that surrounds it.
2.1.4. Rotating Flow Type II: Extemally-forced Vortex
A stationaiy fluid in a container can become a rotating fluid by spiming the container.
Termed as externaIlj-jiorced vortex in this thesis, this type of rotating fluid presents some
unique phenomena that can be used for flotation.
2.1.4.1. Flow processa in a rotatiog cylinder filled with water
Take a liquid spin-up process for example (Vanyo, 1993). Fig. 2-6 shows water in a
cylinder. In the lefi photograph, both the cylinder and the water are stationary. Note that
in the top one-eighth of the cylinder is a layer of colored water which is slightly less dense
than the clear water. In the right photograph, the cylinder has been impulsively accelerated
to a constant angular velocity, and the water is gradually being "spun-up" to the angular
velocity of the cylinder. Since the bottom and periphery of the cylinder have only
tangentid motions relative to the fluid, viscosity must be relied on to transfer momentum
fiom the cylinder surfaces to the water.
Figure 2-6. Spin-up of water in a cylinder (Vanyo, 1993)
Two mechanisms assist viscous d e r of momentm. One is the boundary hyer
at the cylinder wal1 which grows with the. With this mechanism, the entire mass of fluid
rotates only d e r the bounâaiy layer thickness has grown to the radius of the cylinder. The
second mechanism. which is much more efficient, involves secondary flow at the bottom
end sunace. A boundary layer fonns here in the same way as at the cylinder wall.
Centrifùgal force inside t his very t hin (almost invisible) spinning bottom boundary layer
moves clear water outward and then up dong the outside cyiinder wall displacing the top
(ail! nonspinning) colored water inward and dom. Colored water in the interior is drawn
downward until dl the water will be pwnped outward through the very thin, bottom
boundary layer. A 2% buoyancy of the colored water is opposing the pumping action and
causes the boundary between the clear and colored water to be tapered rather than
cylindrical. As more fluid is pumped outward, the cylindrical layer grows in thickness,
finally filling the entire volume and rotating at the same angular velocity as the cylinder.
In this experiment, liquid spin-up is achieved about 1% by viscous interaction at
the cylinder side walls and about 99% by the viscous secondary flow at the cylinder
bottom (Vanyo, 1993).
An extemally-forced vortex in equilibrium state rotates about the vertical axis like
a rigid body with no relative motion between fluid elements. If there was no colored water
in the cylinder, the k e fluid surface will be a typical paraboloid under which the pressure
varies with depth in the same way as the pressure distributions in the fluids at rest. Note
that in cyclone separators the core of Rankine's combined vortex is a forced vonex which
draws the torque from the tiee vortex.
2.1 A.2. Fiuid-llow patterns
Consider a fluid body rotating unifomly as a whole without relative motion of its parts.
The angular velocity a, about the fixed vertical axis is independent of r, the radial distance
of the point fkom the ais. A fiuid element describes a circular path in a horizontal plane
with constant speed rcu; its acceleration is radial and inward and of magnitude 02r. The
pressure is the same as in a stationary fluid with a body force (force per unit mas) having
the constant component g vertically downward and a radial outward component 02r.
Working in polar coordinates with z and r as shown in Fig. 2-7, the pressure satisfies the
following equations (Duncan et al., 1970):
where is the fluid density. Integrating Eq. (2.16) with o independent of r, the pressure
distribution is obtained as
where p, is the pressure at the origin of coordinates. The surfaces of constant pressure are
therefore the paraboloids of revolution described by
Note that for constant z, p increases as 8. A centrifuga1 pump and a centfige
make use of this principle. An enclosed mass of liquid is whirled rapidly to create a great
difference in pressure between its center and its periphery.
Figure 2-7. A rotating container of fluid. Gravity acts in the negative z direction. (Hughes and Brighton, 1999)
If a solid puticle of mas m is revolving at a radial distance r from the axis of
symmetry with an angular velocity a, it is aibjected to a centrifiigal force Fe = m 2 r in a
radial direction and to a gravitationai force F, = mg in a vertical direction. The ratio of the
centrifiigal to the gravitational force is oAen used to masure the separating power in a
centrihige. This ratio, 2, is hown as the centrifuge effêct, or relative centrifiigal force:
Since angular velocity o is constant throughout the entire container, it is clear that
the separating power increases with increasing radial distance r fiom the axis of symmetry
in a forced vortex. The tangentid velocity of fluid, U: = or, therefore increases with
increasing radial distance r across a forced vortex Note that this is opposite to the
situation in a free vortex where the tangentid velocity of fluid U: decreases with
increasing radiai distance r.
The centrifuga1 force on the solid particle in the rotating fluid is given by
where d, is the diarneter of the solid particle, p, and pf are the density of the solid particle
and the fluid respectively, o is the angular velocity about the fixed vertical axis, and r is
the radiai distance of the point from the axis.
Again high centnfbgal forces occur at large radial distance fiom the axis of
rotation in a forced vortex, while in a free vortex high centrifùgal forces will result fiom
small distance. Using Stokes's equation for the drag force (Fd = 311pd~UI ) which is
opposite to the centrifuga1 force, the radiai terminal velocity or sedimentation speed that a
solid particle can reach at certain distance from the axis is
where p is the fluid viscosity.
2.1.4.3. Application to notation ce11 design
For flotation ce11 design, an extemally-forced vortex has the following features of interest:
1. Centritiigal forces increase with increasing radial distance (r) of the point
fiom the axis. Solid particles are subjected to increasing centrifbgal forces
as they move to the wall of the fluid container.
2. In a steady rigid rotation flow, a fluid has only tangential velocity. There is
no radial secondary flow that would reduce the separation effectiveness.
3. The ore pulp needs to be introduced into the rotating container through a
central hollow shaft. As the pulp is gradually being "spun up", the same
rotating fluid phenornena involved in the liquid spin-up process (refer. Fig.
2-6) occur too. The secondary flow at the bottom moves the pulp ouward
and then up dong the outside vessel wall. Thus individual grains do not
settIe at the bottom of the vessel.
4. Injecting air bubbles at the bonom of the vessel would double the function
of the secondary flow. This type of bubble generation mechanism would
bnng about as many contacts as possible between the solid particles and the
air bubbles. The secondary flow moves the particle-bubble aggregates
outward and then upward into the rotating flow. Because the density of
the aggregates is smaller than that of the fluid, they do not move outward
in a vortex but toward the core.
5. Rigid rotation fiow has vorticity at each point, which means a solid particle
CO-existing with the fluid will spin about its own ais . The effect of self-
spinning on particle-bubble collision may be insignifiant, but as to the
subsequent adhesion sub-process, the effect could be disadvantageous.
6. The fluid container, Le. flotation cell, has to be spun up. Energy
consumption would be much higher than cyclone separators which have no
moving parts. For this reason, d e - u p of such equipment seems to be
prohibitive.
7. The disposai of tailings. One design is to open some orifices along the
periphery at some distance away fiom the bottom surface. Another way is
to simply let the tailing "climb higher and exit f'rom the edge of the
rotating vessel. Both tailing disposai mechanisms make use of the
secondary flow that forms at the bottom and moves outward and then up
along the vessel wall.
An application example is the Centrifugai notation Ce11 (CFC), which was
originally developed by Clean Earth Technologies to separate oil from water and was
nibsequently applied to froth flotation (Lakefield Research, 1995). Fig. 2-8 illustrates the
design of the original CFC, which consists of a rotating drum (A) where separation
occurs, a pulp feeding pipe (B) and an air injector (C).
Figure 2 4 . Cemrifùgal flotation ceil (CFC) developed by Clean Earth Technologies (Lakefield Reseafch, 1995)
In practice, the pulp enten at the boaom of the spinning drum near the center. The
centrifugai force causes the sluny to migrate outward across a screen of bubbles that are
injected at high velocity through a jet at the bottom of the cylinder. W~th the ce11 rotating,
angular momentum is continuously being tmsferred to the fluid and in a short time, the
fluid inside the cell will becorne a rigidly rotating body, creating a centrifbgai force field in
the range of 50-1 50g. The mineralized froth has a density smaller than that of the fluid so
it will move to the inner surface, where it overflows a weir at the top and is coiiected in
the froth launder. The tailing slurry is discharged through tailing ports (orifices) dong the
outer rim of the dmm.
The major advantage of the original CFC over conventional ce11 appears to be a
greater rate of mineral flotation, based on the evaluation of results obtained fiom the
flotation of metal values fiom massive sulphide and porphyry ores (Lakefield Research,
1995). It was also noted that the fine particles behave differently under the combining
effects of gravitational and centrihgal forces.
However, the original CFC possesses certain disadvantages. One of the persistent
problems has been the rapid blockage of the tailing discharge pons in the outer nm of the
drum. If the feed slurry has a pulp density higher than 30% solids, these tailing ports
would be easily plugged up, resulting in the disruption of the entire flotation process. In
actual practice, wood, debns, or lime scale in the ore pulp can accumulate and would
quickly become a nuisance in the operation of the original CFC.
The original CFC was modified and A-desiped at Queen's University (Yen et al.,
1998). The result, CFC-QI, has a much different look with significant changes in the ceIl
configuration and the bubble generation method. The CFC-QI ce11 consists of a feeding
pipe inserted with an air sparger for producing fine bubbles, a rotating bowl (a drum, or
cell) where separation occurs, a weir centered inside the ceil to direct the mineralized
bubbles and a charnber nirrounding the ceU to collect the taiIing discharge. The ore slurry
is pumped through a feeding pipe where it is in contact with air bubbles, then travels
through a horizontal pipe and is finally discharged to the spinning bowl through a vertical
pipe. As the bowl is rotating, a centrifuga1 force is created, forcing the non-bubble-
attachable particles towards the side of the bowl wall. The particles then flow upwards
dong the wall to the top edge of the drum and enter the collecting charnber. Minerals
attachable to the bubbles will accumulate in the froth and rnove to the center where it
overflows the top of weir for discharging. A detailed description of the CFC-QI cell is
given in the Chapter 3.
Figure 2-9. Modified centrifuga1 notation cell CFC-QI (Yen et al., 1998)
2.1.5. Rotating Flow Type III: Internally-forced Vortex
The third type of rotating Ruids. temed internaIly-forced vortex here, uses an interior
rotating device to transfer energy to a fluid. The vesse1 (cell) is stationary and its shape
must be cylindrical in order to maintain the rotation motion of the fluid inside it. At a
glance, the conventional mechanical subaeration flotation cells (Fig. 2-10) can be easily
converted into a rotating fluid apparatus by changing the shape of the ceIl fiom
rectangular to cylindfical. However, since the principal flow region of interest will be the
rotating flows, the design of the rotating device should be different from the impeller
system in conventional mechanicd cells. In the case of flotation of fine particles using
conventional mechanical ceUs the impeller qstem's ability to create a maximum shear
unie between the rotor and stator is usuaily a n p h a s i d regardless the shape of the tank.
To condua flotation in rotating fluids, it is the rotating device's abiüty to spin up the fîuid
that matters moa.
Figure 2-1 O. Conventional subaeration flotation ce11 (Boldt Jr., 1967)
2.1.5.1. Flow processes in a stationary nuid with a rotating disk
To begin with, consider an infinite Bat disk rotating at constant rate in its plane in an
invixid, constant density and nonrotating fluid which extends to infinity as shown in Fig.
2-1 1. Assuming no-slip boundary conditions at the rotating sunace, fluid in contact with
the disk sunace rotates with the same angular velocity as the sufiace and experiences the
sarne centripetal acceleration.
At the start of motion, viscosity diffises angular momentum of the surface into the
fluid and a boundary layer begins to form in the tangential U: direction. Fluid in this
boundary layer oust above the surface) begins to spin but cannot maintain the same
centnpetal acceleration as the surface does. It acquires an outward radial velocity
component LJf . As the radial velocity component LJf increases in magnitude, a secondary
boundary layer develops in the radial direction with stresses centraily directed. These
stresses do provide a centrai force and a centripetal acceleration greater than zero, but less
than that of the surface. At distances greater than a characteristic depth S from the disk
surface, the tangential boundary layer thickness is exceeded. The fluid above 6 has no
rotation and there is no mechanism available to continue the radial flow. Due to the
continuity requirement, a downward flow occurs to match the outward flow in volume.
The net efféct is that zero angular momentum fluid is drawn axially dong the axis of
rotation from intinity, given angular momentum in the boundary layer, and then pumped
radially outward as high angular momentum fluid (Vanyo, 1993).
Briefly the tangent id velocity, associateci centrifiigal forces and pressure gradients
dominate the flowfields. The secundary flow is significant within a boundary layer lining
the surface of the disk, but decays far fiom the disk. The thickness of the boundary layer is
a hnction of angular velocity C2 of the disk.
~ d a l outf iow Surface (di&) rotnes at n = constant
Figure 2- 1 1. A rotating disk in a stationary fluid (Vanyo, 1993)
2.1.5.2. Ruid-flow patterns
The velocity characteistics of the fluid in this problem can be studied by solving the
complete set of Navier-Stokes equations which can be reduced by a transformation to an
exact set of ordinary differential equations for solution by numencal or other means. Mer
omitting ternis that are identicaily zero, the equations of motion in physid variables and
in an inertial reference system are (Vanyo, 1993):
where v is the kinematic viscosity ( v = dynamic viscosity / density = p / ).
Omission of the 3/80 tems implies rotational symrnetry for the solution. The
assurnptions made in a.rriving at Eqs. (2.22a-d) are (a) steady flow, (b) flows remain
laminar, (c) density constant. For higher angular velocities, flows that are periodic in 0
andlor turbulence occur, invalidating this solution.
Boundary conditions consistent with no-slip at the rotating surface are
The equations are made dimensionless using
After substituting Eqs. (2.23a-e) into Eqs. (2.22a-d), the following set of four
coupled, dimensionless, ordinary differential equations is obtained. Asterisks are omitted
for clarity. The physical parameters v and R do not appear in the new equations nor in the
boundary conditions, and consequently do not predetermine the solution.
d ' ~ : -- du' dz'
d2u; ,: du: -- -- 2u:'u; = 0 dz' dz
Boundary conditions, also in dimensionless variables, are now
Solutions are shown pictonally in Fig. 2-12, graphically in Fig. 2-13, and
numerically in Table 2-1 (Vanyo, 1993). Clearly, at certain z, the tangential velocity
component U: of the fluid increases with increasing radial distance from the axis of
rotation. Centrifugai forces are directly associaied with the magnitude of the tangentid
velocity component u:, therefore solid particles moving above the rotating disk would
experience increasing centrifuga1 forces as they move away fiom the axis of rotation. The
outward, radial velocity component ~f would also push the solid particles further away
fiom the axis of rotation.
On the other hand, at certain radial distance r, the tangential velocity U: decreases
with increasing axial distance z from the surface of the rotating disk. Therefore, as solid
particles or bubble-particle aggregates move upward, they will be less afEected by the
centrifugai forces. Instead, the downward axial velocity component U: of the fluid will
dominate the flow field above the characteristic depth 6 and from the point of view of
froth flotation, the downward flow may act as wash water that cleans the unwanted
minerals in the fiot h.
Assuming edge effbcts on the finite disk are negligible, the moment required to
maintain constant fl in dimensional variables is
The quantity of fluid pumped outward dong one side of a disk at a radius R in
dimensional variables is
z (normal to disk)
Figure 2-12. Dimensioniess velocity components of flow (u: , U: , u:) Ïnduced by a spinning disk, in a fluid otherwise at rest, are a fùnction of dirnensionless distance (2) from the disk (Vanyo, 1993)
Table 2-1 Dimensioniess flow solution of a rotating disk in a fluid at rest (cornpiled by Vanyo, 1993)
2.1.5.3. Application to flotation cell design
As far as flotation is concemed, an intemally-forced vortex has the following main
characteristic features of interest:
1 . Much lower energy consumption is expected in cornparison to an
extemally-forced vortex. In the original CFC and the CFC-Q1, the rotating
fluid results from the spinning of the whole flotation cell. An interndly-
forced vortex only requires an interior rotating device to generate it.
2. There exists a characteristic depth 6 above which fluid has no rotation.
Instead, zero angular momentum fluid in this region is drawn axially
downward to compensate the outward flow pumped radially by the rotating
disk. This phenornenon is beneficid to flotation in that the downward flow
is like a washing water that could reduce the entrainment.
3. The charactenstic depth 6 = 4 I/= . The relation indicates that the higher
the rotating speed of the disk, the shorter the characteristic depth. This
limits the disk's ability to develop a strong rotating flow and thus a
supplementary mechanism is needed to increase the energy transferred to
the fluid. This can be achieved by a good design of the feeder. Details will
be covered in the next chapter.
An application example is the CFC-Q2 cell, shown in Fig. 2-13, developed at
Queen's University (Yen et al., 1998). In cornparison with the CFC-QI ceil, the CFC-Q2
ce11 has a completely different structure except for part of the feeding pipes and the air
sparger. The key part is a belt-driven joint comecting a stationary vertical pipe on the top
section and a rotating vertical pipe on the lower section. The flotation ce11 itself is a
cylindncal vessel with a conical section comected to the bottom of the cylindrical section.
The rotating pipe extends to the boundary of the conical and cylindrical sections, and is
attached there to a circular platform. There are four square holes between the lower end of
the rotating vertical pipe and the platform. As the ore slurry flows through the rotating
pipe, the slurry will tût the platform and exit through the holes. A centrifuga1 force is
created which forces the rotating slurry to the wall of the flotation ceIl. The minera1
particles anachable to the bubbles will be wried by the froth up to the upper collecting
area while the unattachable particles will be discharged to the lower collecting area. A
more detailed description of the invention will be provided in the following chapter.
Figure 2- 13. Centrifuga1 flotation cell CFC-Q2 (Yen et al., 1998)
2.2. Basic Rotating Fluid Theory
2.2.1. Rotating Coordinate Systems
This section presents a review of basic equations describing the velocity characteristics of
the rotating fluid and the behaviour of solid particles in the rotating fluid. These equations
underlie the analysis of the notation process camed out in rotating fluids, which is
discussed in detail in Chapter 6.
Consider in Fig. 2-14 a contained quantity of fluid defined as the flow region of
interest. The container rnay be translating and rotating relative to inertial space.
Particle, Region b for use in Eulerian rnechanics
Body b for use in y rigid body rnechanics
Newtonian (inertial) frame n
Figure 2-14. Reference h e b has an arbitrary time-varying position and orientation relative to h e n (Vanyo, 1993)
Assume that an Eulenan grid is fixed in the moving container reference hune b
that is not inertial. R, U, and A are position, velocity, and acceieration of point P relative
to the inertial frame n, respectively, while r. u, and a are equivdent tenns but relative to
the b frame. R, U, and A. define position, velocity, and acceleration of the origin of the b
frame relative to the n b e , respectively. The derivative of R relative to the n ûame
gives U relative to the n b e . In the same way the b fkme denvative of r gives u relative
to the b M e .
The position of point P in Fig. 2-14 relative to the origin point N of an inertial
b e n is given by
Assume that frame b is rotating at angular velocity relative to M e n. The
differential operator "d/dt represents the changes relative to the space-fixed coordinate
system n* which c m be wrinen in terms of a differential operator bd/dt representing
changes relative to the rotating coordinate systern b as follows (Vanyo. 1993):
Applying this operator to the position vector R (relative to n), one obtains
u = u + [Uo + o x r ] Inertial= Local + Origin + Tangential
Here the cross product O x r is the tangentid velocity of the location of point P
in the b fiame due to the rotation of b relative to n.
Applying the operator (2.28) to the velocity vector U (relative to n), one obtains
A = a + [ & + ~ x ( o x r ) + 2 t ~ x u + a x r ] (2.30) Inertial= Local + Origin + Centripetai + Coriolis + Tangentid
Here a = bdo/dt = "dddt since CO x OI = 0.
2.2.2. Fluids in Rotating Systems
In the Eulerian fonnulatim, fluid acceleration relative to the b frame is the substantive
derivative taken in the b fiarne
DU, - au, b h f + ( u ~ . ~ V ) U ~ = - ar =--- + ~ ( u , ' 12) - u, x (V x u,) (2.31)
Dt ct Ct
Subaituting Eq. (2.3 1) into Eq. (2.30), one obtains
where & a, and a define motion of the b m e ; r represents positions of Eulenan grid
points in the b frame; and uf, V, and a / i3 are all relative to the b frame.
The Navier and Stokes' formulation of Newton's second law is
Here the n in "DUr/Dt is used to make explicit that the derivative must be taken
relative to an inertial (Newtonian) M e . The total force F is made up of the total surface
force F, (pressure and shear) and a body force Fb which is a force per unit volume.
Assuming the Buid has constant density and constant viscosity, and with gravity as the
only body force, the Navier-Stokes vector equation is
"DU, -- 1 3- - Vp + -v2u,
Dt Pt- Pr
The forces and the vector Laplacian " v'U, are invariant to coordinate
transformations. Assuming A, and a are zero or negligible, the Navier-Stokes vector
equation written for use in a reference hune rotating at constant angular velocity becomes
(Vanyo, 1993):
Assuming the fluid is inviscid, and shifing the centnpedai acceleration and Coriolis
acceleration to the right-hand side so they become inertial forces, the equation takes the
fonn:
Noting further that gravity can be expressed as the gradient of some potential
1 . 9
2 O -r- function(@,),and-o x (a x r)=o r = - V ( - 2
) , the equation becomes
Relative acceleration Pressure Gravity Centrifuga1 Coriolis
where a, is the gravitational potential, the velocity vector uf is measured relative to the
O 'r' rotating fiame. - V ( - ) is the centrifuga1 force which, since it is a function of relative
2
position only, can be cornbined with the gravity term to give an apparent gravitational
force. - 20 x Ur is the Coriolis force, which is responsible for many of the unfamiliar
features of rotating flows (Fultz, 1988).
2.2.3. Solid Particles in Rotating Systems
In this thesis, the term "particle" is reserved for a solid particle or mineral particle of
macroscopic dimensions. In fluid mechanics, the terni "fluid particles" is used frequently as
a synonym for fluid molecules that are microscopie. They are differentiated here by
subscripts "f' or "p" whenever necessary to avoid confusion. In flotation, if the bubble is
covered by an adsorption Iayer and is not too large it is often treated as a d i d particle in
the analysis of elemeniary processes, e.g. collisions between solid particles and air bubbles.
The equation of motion of the particle in rotating systems differs tiom the fluid
equation of motion in that the acceleration of the particle is not a substantive denvative
and the forces acting on the particle are different. Starting from Eq. (2.32), assuming &
au, and a are zero, and ap = 7
By Newton's second law
where mp is the mass of the particle, up is the particle velocity measured relative to the
rotating coordinate system b and is time independent.
The force F consists of two parts: FI from the drag on the particle, Stokes'
resistance; and F2 from the buoyancy of the particle. The drag force as given by Stokes'
resistance law for a spherical particle is
where p is the local viscosity of the fluid and dp is the particle diameter. The buoyance
force on the particle is given by
where p is the pressure in the fluid, n is the outward-directed normal at the particle
surface, and the integral is to be evaluated over the surface of the particle. By Green's
theorern, the surface integral can be converteci into an integral over the volume of the
part icle
Assuming that the particle is sufficiently small so that the pressure gradient Vp
does not change in the region of space occupied by the particle
where Vp is the particle volume. Note fùrther that fiom the equation of motion for the
fluid rotating as a rigid body
Substituting the volume of the particle and its mass by X D & ~ a d pp XD:/~, one
obtains the equztion of motion of a spherical particle in a rotating coordinate syaem (Hsu,
198 1):
The terms on the right-hand side of Eq. (2.45) represent the particle acceleration,
the Coriolis effect, and the effect of the centritiigal field.
Chapter 3
Development of Centrifuga1 Flotation Cells
3.1. Mode of Introduction of Cas
This chapter surnmarizes the development of two new types of centrifùgal flotation cell
that utilizes rotating fluids to conduct âoth flotation. One uses the extemally-forced
vortex (CFCQI), the other adopts the intemally-forced vonex (CFC-Q2). Basic theory
and features of interest have been reviewed in Chapter 2.
At a glance, the unusual properties exhibited by the rotating masses of fluid seem
to de@ some general pnnciples for flotation ce11 design. In a rotating Eiuid, e.g., water
rotating with constant angular velocity, random motions are not possible and any
turbulence induced would be severeiy conarained. Therefore, there is no way that a
centrifùgd flotation ce11 can hnction as a stimng vessel. However, this does not mean that
flotation cm not proceed in the flow field dorninated by rotating flows. In mechanical
flotation cells, the flow processes in the machine comprise directional and turbulent flows.
These flows carry the solid particles with the aim to bnng about as many contacts as
possible with air bubbles and to transport the particle-bubble aggregates into the tioth
layer, which is essential for flotation. A minimum velocity of the circulating flow is
required to disperse the solid particles in a state of suspension. The so-cailed one-second
cnterion is used to characte& the date of suspension: individual grains do not remain
settled at the bottom of the vessel for more than 1 second (Schulze, 1984).
Rotating flows cm achieve these goals easily. Later in Chapter 6, we will see that
the collision efficiencies of small particles with air bubbles are actually higher in a
centrifugal force field than in a gravitational field. As to the suspension of solid particles,
rotating flows cm carry the particles dong with them just like other moving flows. In an
extemally-forced vortex, overweight particles falling to the bottom of the vessel are
subjected to the extremely high centrifuga1 forces that can prevent them from settling. In
an intemally-forced vortex or a Rankine's combined vortex, the vessel (cell) itself is
stationary. The impovenshed tailings including the overweight particles can be discharged
through the bottom of the vessel, thus avoiding the settling problem altogether.
Like al1 other flotation cells, the mode of introduction of air, i.e. the aeration
method, is essential to the centrifugai flotation cell and mechanism design. In the
mechanical type of flotation cells such as the Denver cell, the horizontal impeller near the
bottom of the ce11 acts as a centrifuga1 pump. As the impeller is rotated, pulp is expelled
outward leaving a void around the impeller hub. This allows air to be drawn d o m through
the air standpipe to the impeller blades where it mixes with the circulating pulp. This
mixture is then subjected to the intense pressure and vacuum of the rotating impeller
blades, and is expelled against the diffuser blades, which further mix and shear the air and
the pulp. In another mechanical type of flotation cells such as the Agitair cell, pulp enten
below the hub of the impeller, which is an agitating device ody. The irnpeiler rotates
inside a bafne systern. Air is blown in at low pressure through a hollow sh& and sheared
into bubbles as it enters the pulp (Pryor, 1974).
In the flotation columns bubbles are generated using air spargers which are placed
intemally near the boaom of the column. This type of aeration method has been plagued
by the rapid clogging of air spargers. Alternative techniques include the use of compressed
air to aerate the pulp before it is pumped imo the cyclo-column cell (Yalcin, 1995). and
the use of an external flow-through spargerlcontactor (TortoreIli et al., 1997).
in the pneumatic ce11 such as the early Southwestern cell, air is blown d o m
vertical pipes from a distributing header, and agitates the pulp in a long trough (Pryor,
1974). A newcomer in this category is the EKOFLOT pneumatic flotation systern. The
cells mn with a self-aspirating aerator guaranteeing clean and cost-fiee aeration. The flow
of the pulp through the "venturi nozzle" pulls the necessary air into pulp thus a
compressed air supply is not needed for operation (Sanchez et ai., 1997).
A rarely used aeration method involves setting up a vacuum above the pulp
surface. Air precipitates from the aqueous phase on to the moa hydrophobie particles with
sufficient lifting power to buoy them to the surface. Some measure of precipitation is aiso
thought to occur in the Iow-pressure zone swept out behind a fast-spinning impeller. In
this comection the precipitation of dissolved air may play an important role for the so-
called combined bubble-particle attachent. The combined attachment is an event which
occurs when, &er the precipitation of microbubbles on a particle, the attachent of a
separate larger bubble happens (Schubert, 1999).
Air spargers were selected as the aeration method for the CFC-QI and CFC-Q2
cells. Sparging can be accomplished in many dEerent process variations. In-tank
applications, with the sparging elements located in the tank, can be either batch or
continuous flow, with or without agitation. Continuous pipe line sparging, which was
adopted in the CFC-QI and CFC-Q2 flotation systems, can be in-line using intrusive pipe
line mounted sparger elements. In an air-sparged hydrocyclone (Fig. 2-4) the whole ce11 is
a porous tube through which air is sparged. In the original centrifuga1 flotation cell
developed by Clean Earth Technologies (Tig. 2-8). air is injected at high velocity through
a jet at the bottom of the rotating cylinder.
The most interesting feature of air spargers, especially the ones made of porous
stainless steel, is the generation of a large number of smail air bubbles. This is especiaily
important for fine particle flotation. Studies dealing with the collision efficiency of small
particles with air bubbles have confirmed the practicai experience that the recovery of very
fine particles can be improved by flotation with finer bubbles (Flint & Howarth, 1971;
Schulze, 1984; Karr et al., 1990). For most sas sparging applications, spargers with a 2.0-
micrometer pore size are recommended and were used in the CFC-QI and CFC-Q2
flotation systems (descnied next). The compressed air flows through the pores and
nucleates on the wall of the sparger tube. The slurry then shears off the bubbles that have
formed on the wall.
3.2. Modified Centrifugal Flotation Cells
In Chapter 2, the original centrifugai flotation ceIl (CFC) developed by Clean Earth
Technologies (a whollyswned subsidiary of Inter-Citic Envirotec inc .) was introduced
(refer. Fig. 2-8). The ceil uses the direct rotation method to generate rotating flows,
creating extremely high centrifuga1 acceleration fields that are 50-150 times the
acceleration due to gravity. However, one of the persistent problems with the original
CFC has been the rapid plugging-up of the tailing discharge orifices. Any couse substance
(e.g. wood chip), or high pulp density (e.g. 30% solids or higher) in the feed slurry would
block the tailing discharge onfices easily and cause the interruption of continuous
operation. Thus the initial objective of this investigation was to build a laboratory scde
modified CFC with a better tailing discharge system. This has resulted in the CFC-QI cell,
a centrifugai flotation ceIl with rotating dmm or bowl. As the project went on, another
type of centrifbgal flotation cell, the CFC-Q2 cell. was built, with a completely different
design. Both CFC-QI and CFC-Q2 models have several variations that conform in the
main to a few general principles.
3.2.1. Centrifuga1 Notation Cell with Rotatiag Vesse1 (CFC-QI)
There are three versions of CFC-QI. The fira mode1 CFC-QI-1 was abandoned due to
the problem of fioth removal in its triai. The next mode4 referred to as CFC-QI-2, uses a
rotating drum to carry out tlotation. The CFC-Q 1-2 was tùrther modified by replacing the
drum with a bowl, rewlting in a new mode1 referred to as CFC-QI-3. Fig. 3-1 provides a
perspective view of CFC-QI-2. Fig. 3-2 is a cross-sectional view of CFC-QI-3 taken
Figure 3-1. Perspective view of CFC-Q 1 -2 (üS Patent #5,928,125)
Fi y rc 3-2. Cross-seaional view of CFC-Q 1-3 dong line 2-2 of Fig. 3- 1 (US Patent 5,928,125)
Basically the CFC-Q 1 ceU introduces operational desegregaiion for flotation, that
is to Say, one system for controlling the feed conditions, another for the bubble-particle
reaction (flotation reactor), and another controlling system for the separation of
concentrates fiom tailings. The sarne design principle was used in the EKOF pneumatic
flotation technology (Sanchez et al., 1997).
The ore pulp is conditioned with the necessary chernical reagents in a stimng tank
mounted on a Moyno progressing cavity pump (not show in Fig. 3-1 and Fig. 3-2). The
pump speed can be changed to give the required slurry feed rate. The conditioned pulp is
pumped into the flotation reactor (14) comprising of a vertical sparger section (16), a
horizontal section (18), and a downcomer (20). A Mott 2.0-micrometer tube sparger (22)
is placed inside the vertical sparger section (16). The tube sparger is made of 316L
stainless steel, which provides good corrosion resiaance. A compressed air line in the lab
is comected to the tube (22), providing the air that flows through the pores and nucleates
on the outside wall of the tube (22).
The sluny Stream, flowing outside the tube (22) at high superficial velocities,
shean off the bubbles that have formed on the outer wail. The aerated slurry continues to
flow horizontally through the transverse section (18) of the sluny feed line and
downwardly throogh the downcomer (20) into the bottom (38) of the rotating vesse1 (12).
The downcomer is a fixed stationary vertical pipe.
An elongated shaft (24) extends vertically through a bearing housing and collar
(26) mounted on the upper portions of the downcomer (20). The upper end of the shaft
(24) is co~ec ted to a variable speed motor (28). The lower end of the sh& is securely
and concentrically attached to the bottom (38) of the rotating vessel (12). The CFC-QI-2
ceU uses a drum (12) with a sharp bottom edge. The CFC-Ql-3 ce11 uses a bowl(36) with
a concave bottom (38) and curved rounded sidewalls (40) as depicted in Fig. 3-2. The
rnotor drives and rotates the sh& and the vessel (12 or 36).
A fioth column (42) with an upright annular wall (44) is placed between the
downcomer (20) and the sidewalls of the rotating vessel. Although the fioth column's
location and dimensions fiord some control of the density at which separation takes
place, its principal function is to prevent short-circuiting between the waste Stream and the
concentrate froth. The upright annular wall (44) provides a vertical weir which extends to
a height above the sidewalls of the rotating vessel. The weir is spaced away Eom the
sidewalls of the rotating vessel to provide an annular passageway (46) for upward exit of
the waste stream. A fioth launder with an inclined discharge chute (48) is mounted on the
top of the weir. The chute extends outwardly and downwardly from the weir to discharge
the fioth continuously. A top rail (49), which provides a flange, is positioned dong the top
of the chute and weir.
A housing (50) provides an extenor shell with an inclined floor (52) that extends
downwardly to a waste discharge outlet (54). The floor and outlet are positioned at a level
below the bottom of the rotating vessel (drum or bowl) to discharge the waste stream. The
housing (shroud) has upright vertical housing walls (56) which are positioned
concentrically about and are spaced outwardly from the sidewalls of the rotating vessel to
provide an annular gangue-receiving chamber (58). An annular containment plate (62)
provides a bamier to contain the waste Stream in the annular gangue-receiving chamber.
The rotating vessel is the principal element in the CFC-QI cell. It acts as a
centrifuge inside which the separation of concentrates from tailings takes place. As the
vessel rotates, the froth containing air bubbles with attached minera1 particles moves
toward the downcomer (ZO), rises to the pulp surface, and exits as overtlow through the
froth column (42). The oveflow flows down the launder where it is collected as the
concentrate. On the other hand, the waste Stream containing non-floatable particles moves
outward and upward, passes the annular passageway (46), falls into the housing (50), exits
through the waste outlet (54) by gravity, and flows back into the conditioning tank.
The slurry feed rate in the CFC-QI cell ranges from 1-5 liters per minute. The air
flow rate fiom the tube sparger ranges from 2-10 liten per minute. The rotating vesse1
(drum or bowl) can spin at a speed of 100-400 rpm.
3.2.2. Centrifuga1 Notation Cell with Rotating Feeder (CFC-Q2)
The CFC-Q2 ceIl is a thoroughly different design compared to the CFC-Q 1 cell. It utilkes
an intemal rotating device to transfer energy to a fluid. The rotating flows thus created fali
into the category of "internally-forced vortex", which is to be distinguished fiom the
"externally-forced vortex" in the CFC-QI ceii. Nevenheless, both of them adopt the same
design principle: operational desegregation for flotation.
Fig. 3-3 provides a perspective view of CFC-Q2-I and Fig. 3-4 is a cross-sectional
view of CFC-Q2- 1 taken dong line 2-92 of Fig. 3-3.
The CFC-Q2 cell consists essentially of a rotating feeder (12) placed inside a
stationary cylinder (10) which is mounted on a stationary cone section (70). The ore pulp
is conditioned with the necessary chemicd reagents in a stirring tank mounted on a Moyno
progressing cavity pump (not shown in Fig. 3-3 and Fig. 3-4). The pump speed cm be
changed to give the required slurry feed rate. The conditioned pulp is pumped into the
flotation reactor (14) comprising of a venical sparger section (1 8) and a horizontal section
(20). A Moa 2.0-micrometer tube sparger (26) is placed inside the venical sparger section
(18). The tube sparger is made of 3 16L stainless steel, which provides good corrosion
resistance. A compressed air line in the lab is connected to the tube (26). providing the air
that tlows through the pores and nucleates on the outside wall of the tube (26).
The slurry aream, flowing outside the tube (26) at high superficial velocities,
shears off the bubbles that have formed on the outer wall. The aerated slurry continues to
flow through the horizontal section (20) into a stationary vertical downcomer (22). This
stationary downcomer (22) is positioned dong a vertical axis and communicates with a
rotatable downcomer (12) through an annular collar (36).
Fipm 3-3. Perspective view of CFC-Q2- 1 (US Patent # 5,9 14,034)
Figure 3-4. Cross-sectional view of CFC-Q2-1 dong line 2-2 of Fig. 3-3 (US Patent # 5,914,034)
The core of the CFC-Q2 ceii is the rotating feeder comprishg of the rotatable
downcomer (12) and a circular array (40, 42). The circular array is made up by two flat
imperforate plates between which four apertures provide exit ports (37, 38, 39) for radiai
discharge of the feed sluny. The upper annular plate (42) is welded to the outer wall
surface of the rotatable downcomer (12), providing an upper annular barrier to prevent
upward discharge of the k d slurry above the exit ports (37, 38, 39). The lower circular
dix. (40) is welded to the bottom end of the rotatable downcomer (12) below the exit
ports (37, 38, 39). The functions of the lower circular disc (40) are two-fold: It closes the
rotatable downcomer (12) to prevent downward vertical discharge of the feed slurry.
Secondly, it blocks the upward-moving spiral formed at the bottom of the cone (70). The
lower and upper circular plates (40, 42) are parallel and cooperate with each other to
provide baffles to enhance radial discharge of the slurry, waste Stream, froth and bubbles
fiom the exit ports.
The annular collar (36) provides a dnven pulley that is rotatably coupled. The
collar is welded to the rotatable downcomer (12). The driven pulley consists of a collared
rim with a belt-receiving grooved central portion (46) to snugly receive a drive belt (48).
The drive belt operatively comects and rotatably couples the driven pulley (collar) (36)
with a drive pulley (50). The drive pulley can be smaller, larger, or the sarne size as the
driven pulley (collar) to decrease, increase, or to be the same rotational speed (rpm),
respectively, as the driven pulley. The drive pulley comprises an outer rim with a belt-
receiving grooved central portion (52) to snugly receive the drive belt. The drive pulley is
comected by an upright rotatable vertical shaft (54) to an overhead variable speed motor
(56). The shaft (54) is welded to the top of the drive pulley. The motor (56) rotates the
shafl (54), drive pulley (50)' drive belt (48)' driven pulley (coilar) (36). and downcomer
(12) with d c i e n t speed (rpm). The rotating flows thus generated separate the slurry in
the flotation charnber into a waste Stream comprising non-floating gangue material and a
particle-enriched froth comprising air bubbles carrying a substantial portion of the valuable
particles. The waste Stream and fkoth are discharged and propelled radially and outwardly
h m the exit ports at the lower end of the rotatable downcomer (12).
A notation chamber (58) provides a housing that is concentricdly positioned about
the rotating feeder. The flotation charnber has an annular circular vertical wall (64) with
upright wall portions having an intenor inner surface (62) and an extenor outer surface
(64). The upright wall ponions of the flotation chamber's annuiar vertical wall comprise
an upper overtlow portion providing an upnght vertical overflow weir (66) and a lower
portion (68) connected to an upwardly diverging, conical portion (70). The conical
portion (70) is inclined and extends downwardly and inwardly from the uprighi wall
portions to provide an inclined floor. A discharge conduit provides a tailing outlet (72)
that is spaced at a ievel below the lower circular disc (40). The tailing outlet (72) is
positioned dong the vertical axis and is concentnc to the rotatable downcomer (12).
The upright annular wall of the Dotation charnber provides a vertical weir that
extends to a height slightly below the collar. The weir is spaced away from and cooperates
with the rotatable downcomer (12) to provide an annular passageway (74) for upward
passage of the concentrate froth. A froth launder with an inclined overflow discharge
chute (76) is connected to the top of the weir. The chute (76) extends outwardly and
downwardly at an angle of inclination from the top portion of the weir of the flotation
chamber to discharge the concentrate froth. A top rail (78), which provides a flange, is
positioned dong the top of the chute and weir.
Compared with the CFC-QI cell, the CFC-Q2 ce11 econornizes on power input.
Instead of rotating the entire vessel, the CFC-Q2 ceil introduces an intemal rotating device
to generate the rotating flows. The ore pulp is conditioned and pumped into the flotation
reactor (14) where it is injected and aerated with air bubbles fiom the sparger (26). The
flotation reactor (14) used in this work ailows a large nurnber of high energy
bubble/particle collisions due to the highly turbulent conditions in which the bubbles and
particles are brought into contact. The contacted slurry, containing a large number of
bubble-particle aggregates and air bubbles, flows through the horizontal section (20) of
the slurry feed Iine and the stationary downcomer (22) until it enters the rotating feeder.
At the bottom of the spinning circular disc (40), the centrifuga1 force acts to throw the
feed slurry outward into a rotating flow field generated mainly by the spinning upper
annular plate (42) and to a less degree by the rotating downcomer (12). Note that when
the feed slurry leaves the edge of the spinning circular disc (40) it has acquired the highest
achievable tangentid velocity. Since the feed slurry rotates in the same direction as the
bulk of the pulp inside the cylinder (10) does, it helps to maintain the rotation of the pulp
inside the cylinder.
The bubblelparticle attachrnent process continues in the rotating flows.
Theoretically speaking, the collision efficiencies appear to be higher in the centritùgal
force fields than in the gravitational force field. Therefore, if the desired minerals failed to
attach to the air bubbles inside the Rotation reactor (14), they stiU have a second chance
once they enter the flotation cell. The bubble-particle aggregates move upwardly and rises
to the pulp surface forming a fkoth. The fioth then flows over the top of the overfiow weir
and down the launder where it is coliected as a concentrate. The waste stream on the other
hand spirals downward because of gravity and exits as a tailing undedlow at the bottom of
the cone (72). In practice, the underflow is retumed to the conditioning tank and
recycled .
The sluny feed rate in the centrifuga1 flotation ce11 with the rotating feeder ranges
fiom 1-3 liters per minute. The air flow rate (sparger air injection rate) ranges fiom 2 - 10
liters per minute. The rotating feeder rotates at a speed of 100 - 800 rpm.
3.3. Discussion
The CFCQl and CFC-Q2 centnfbgal flotation cells belong to the
reactor/separator class of flotation machine. The notion is introduced that the attachrnent
of minera1 particles to air bubbles is a physico-chernical process that "there is no reason
why the reaction could not occur in a pipe, for example." (Finch, 1998).
The design of both models adopts an extemal flow-through spargedcontaaor as
the flotation reactor. The sparger, which is made of Mott 2.0-micrometer porous media, is
piaced in pipe through which a slurry Stream is directed. The air flows through the pores
and nucleates on the outside wall of the sparger tube. The size of the bubbles is
determined by the velocity of the slurry Stream because a faster flowing slurry will shear
off bubbles while they are still small. This is a great technical advantage possessed by the
reactor/separator class of flotation machine. It is believed that the recovery of very fine
particles can be improved by flotation with finer bubbles (Flint and Howarth, 1971;
Schulze, 1984; Karr et al., 1990). In conventional flotation machines the size of the
bubbles is lefi uncontrolled.
Another advantageous property of the bubbles generated in the flotation reactor of
the CFC-QI and CFC-Q2 cells is that the bubble surface is mobile. In pure liquids the
phase boundary of the bubble is movable. If the bubble is covered by an adsorption layer,
the bubble surface becomes rigid. The bubbles that are sheared off the sparger surface
collide with the passing solid particles in such a short time that their surfaces should
remain mobile at the moment of collision. The implication of this feature is that the
mobility of a bubble surface has been found to be evidently associated with the bubble-
particle attachent efficiency (Nguyen, 1999). To achieve approximately the same
attachment efficiency, the bubble-particle system with a mobile bubble sufiace requires a
shorter induction time, by an order, compared with the induction time of the system with
an immobile bubble surface. Bubbles with a mobile surface are hydrophobie and favour
flotation.
Thus it can be s m that the bubbles generated in the flotation reactor of the CFC-
QI and CFC-Q2 cells possess important advantages which are especiaiiy useful in the
flotation of fine particles: the smaller bubble size and the mobile bubble surface.
The flotation reactor of the CFC-QI and CFC-Q2 cells creates highly turbulent
conditions almost cost-fiee, except that an air compressor is needed to produce a
compressed air for the sparger. The intense turbulence allows fine particles to acquire
energy to compensate for their low momentum. This means higher collision efficiencies
when these fine particles collide with air bubbles.
The physico-chemical reaction taking place in the flotation reactor results in the
bubble-particle aggregate with a density smaller than that of the fluid. This product must
be separated as a froth product from the unreacted slurry, which is essential for flotation.
The separation however, can not be achieved in the pipe reactor due to the high energy
dissipation there. Therefore, a separator is required to complete the flotation process.
To achieve a satisfactory separation, entrainment has to be minimized. Entrainment
is a microprocess by which slurry enters the lamellae of the flotation fioth, moves upward
with the froth, and finally leaves the flotation ce11 with the concentrate. This slurry carries
particles that are more or less homogeneously ~spended regardless of whether the
particles are hydrophobie or hydrophilic. Despite its possible contribution to increase the
overall recovery, entrainment must be reduced as much as possible because it directly
contradicts the technological aim of flotation separations.
h mechanical and pneumatic flotation cells including flotation columns,
entrainment is controlled under quiescent conditions with wash water in order to promote
the drainage of entrained pmicles from the fioth.
The CFC-QI and CFC-Q2 cells possess superior advantages in minimïzing the
degree of entrainment. Fim, the turbulence is severely constrained by the characteristic
two-dimensional nature of a rotating fluid. Since the extent of the entrainment is
determined considerably by the turbulent flow conditions in the flotation cell, the use of
rotating flows to condua flotation vinually eliminates the root cause of entrainment.
Secondly, in a sluny placed in a centrikgal force field far stronger than gravity, the solid
particles in the entrained sluny between the lamellae of the froth will behave like large,
coarse particles and would drop out of the entrained slurry easily. Thirdly, aside ftom the
rotating flows that dominate the flow field in the centnhgal flotation cells, there are
secondary flows produced by the Buid-structure interactions. Some secondary flows can
fùnction as the washing water due to their downward moving direction, providing one
more means to reduce the entrainment.
The most important feature of centrifuga1 flotation technology lies in the fact that
the collision efficiency of fine particles with small air bubbles is higher in the centrifugai
force field created by the rotating fiows than in the gravitational force field (Karr et al.,
1990). Therefore, in the operation of the CFC-QI and CFC-Q2 cells, the collision of solid
particles with air bubbles takes place not only in the reactor, but also in the separator.
In summary, a well-designed centrifugai flotation ce11 based on a good
understanding of rotating fluid phenornena and flotation hydrodynamics, would provide a
superior environment to conduct fioth flotation, especially fine particle flotation. Both of
the designs of the CFC-Ql and CFC-Q2 centrifbgal Rotation systems are theoretically
tenable.
Chapter 4
Experimental
Guidelines for Evaluating Flotation Cell Efliciency
The CFC-QI and CFC-Q2 centrifuga1 flotation cells are the latest contributions to the
centrifuga1 flotation technology. In the cornpetition for the market, a comparison between
mechanical cells and centrifugai cells is perforce necessary. A Denver laboratory flotation
cell (Model D-12) was chosen for this purpose. The Model D-12 incorporates the same
basic principles of operating as the commercial size Sub-A Flotation Machines. Fumished
with a diffiser and impeller, the machine has a square tank and produces its own air. The
aeration intensity is controlled by the air valve at the top of the standpipe.
Faced with so many differences in the ce11 and mechanism design, a scientific basis
for comparison of performance between the CFC-Q 1, the CFC-Q2, and the Denver D- 12,
with varied settings and working conditions, is apt to be misleading. Operating skill is by
no means a negligible factor. Therefore, in order to evaluate the efficiency of any flotation
machine, a set of guidelines must Eïrst be established.
In their book of 1955 Sutherland and Wark described the situation in these words:
"We can accurately define machine efficiency ONLY with respect to one particular aspect
of the process. We may with Fahrenwald (1 944) detennine the efficiency of a flotation ce1
in terms of its ability to grind air into smaii bubbles; or we may determine its efficiency in
keeping sand suspended; or judge its efficiency by its capacity to float so many tons of
minerd per hour, or determine its efficiency in making a clean separation; or in floating
large particles of ore; or, again, we may assess the efficiency by the power required to
treat a ton of ore. None of these methods are satisfactory; the metallurgist must decide for
himself what he needs from the process and machine. and his final critenon is that of cost.
He must take into account the amount of power required, the capital cost and maintenance
charges of the cell, as well as its ability to meet a vaqhg market by alteration in the
composition of the produa recovered."
This is a very practical definition and various points were used as a basis for
evaluating the CFC-QI and CFC-Q2 cells. Cornparison between the CFC-QI, the CFC-
42, and the D-12 is based on their metallurgicai performance in overall recovery, size
fraction recovery, concentrate grade and flotation kinetics. Since rotating fluid flows
constitute a major feature of interest in centrifuga1 flotation cells, the author was naturally
curious to l e m the fine particle behaviour in a centrifuga1 force field. This is done by
calculating the recovery in the minus 10 microns s ix fiaction.
Great care should be exercised to interpret the results of notation tests, since much
depends on the operator's skiil and on his prejudices. The method adopted in this research
is to compare the BEST results obtained from the CFC-QI, the CFC-Q2 and the D-12
when treating the pulp prepared by the same grinding.
4.2. Ore Samples for Testing
The flotation tests were carried out on two types of artincial ore mixtures and two types
of natural ore samples.
4.2.1. Galena-Silica Mixture
Galena is the moa common lead mineral. As galena is a soft, high specific gravity mineral,
sliming due to overgrinding of the galena is a persistent problem in actual practice. To
alleviate this problern, unit cells in the grinding circuit, or stage ginding with flotation
between stages, is practiced at some operaiions (Mning Chernicals Handbook, 1986).
Galena is thus chosen as one of the flotation test samples for reasons outlined
above. A high punty natural galena mineral sample was purchased from WARD's Natural
Science Establishment Inc. The label on the study pack indicates that the source of galena
is Brushy Creek, Missouri, USA.
The sample was hand-picked and crushed to al1 minus 48 mesh. Clean silica sand
was screened to remove ail plus 48 mesh portion and leave only the minus 48 mesh
fiaction to mix with the prepared galena sarnple. One-kilogram charges, comprising 50
grarns galena and 950 grarns silica, of the galena-silica d u r e were prepared for al1 test
work. The test charge thus prepared contains about 4 wt.% Pb.
The one kilogram test charge was ground in a 20cm x 35cm laboratory rod mil1 to
minimke tramp oversize and sliming. The pulp density was 600/0 solids and the grinding
tirnes were 20, 30 and 60 minutes. The size distribution analyses of the ground pulps at
various grinding times are plotted as in Fig. 4-1 (20 min.), Fig. 4-3 (30 min.), and Fig. 4-5
(60 min.), while Fig. 4-2 (20 min.), Fig. 4-4 (30 min.), and Fig. 4-6 (60 min.) show the
size distributions of the corresponding flotation concentrates which were taken as the
high-grade galena. The screen sizes through which 80 W.% and 90 W.% of particles pass
are summarized in the Table 4- 1.
The results of size analysis indicate clearly that in the galena-silica mixture, galena
was ground much finer than the silica sand.
Flotation tests were carried out with the CFC-Q 1-2, the CFC-Q 1-3, the CFC-Q2- 1
and the Denver D-12 cells under natural pH. Sodium ethyl xanthaie (SEX) was chorn as
the collecter due to its maximum selectivity. In practice, SEX is most frequently used to
float galena with lead-zinc ores (Mining Chernicals Handbook, 1986).
Table 4-1. Ground produas of galena-silica mixture
Grind Time
(minut es)
Head
8û% Passing
Head
90% Passing
Galena
80% Passing
1
Gaiena
900/0 Passing
1 1 I 1 ï I I I 1 I 1 I 1 I I I I .--,-I,,I-A-I-LLLLL----L--L-l-IJ-LU
I I 1 I l I I I 1 1 I 1 I 1 I I I +--+-4-+4+w
1 I 1 1 I I l I I I 1 I 1 1 I I I .---- I 1 - 1 I I I I I I I 1-1 1-III
+7-1 ~-rrrn----- r--T-7 ~1 r m I 1 1 I 1 I I I 1 I I I 1 I \ I I
.----I--l-J-I-LLLLL__----L--+-J-IJ-LU t 1 I I 1 I l I I I I I 1 I I I
I I I I I I l I I l 1 1 1 I I I I .---- I I i I I I I I 1 1 1 1 I 1 I I I
+--t-1-t-tttn----- t-- t- t- t l -Pl-t 1 I t I l t I I I I I 1 l 1 I I I 1 I I I I I I I I I I I 1 I I I I I I I I l
I I I I I I I I I I I I I I I 1 I I I I f I I I I I I l I l I I
1 O0
Saeerr Ske (microns)
I I I I I I I 1 I I I I J-IJ-LU I 1 1 I I I I I I I I I I 1 I I I I 1 I 1 I I I I I t I I I 1 - 1 I l I 1 7 T l rrr
1 1 I I I 1 I I I I I 1 I I I I I I I I I l i I I I I I I I 1 1 I I I I 1 1 I I I I I I I l l 1 I l I I 1 -
Figure Cl. Sue distribution of galena-silica mixture with 20 minutes grind (Test No. D- 1)
I I I 1 I I I I I 1 I I I 1 I I I I I l I I
Partide Sue, pm
Figure 4-2. S k e distribution of galena with 20 minutes grind (Test No. D-1)
I I I I I I I I I I I i I i I I I ---- 1 I 1 I l 1 1 1 1 I 1 1 I I I I I T--7-7-T-rrr,,-----T-- T-~~-t-rrr I I 1 1 1 1 1 1 1 1 I I 1 I I l l
----L--l-J-l-LLLL----L--L-J-LA-LU I I I I l 1111 1 I I I l I I I
I i i i i I 1 I I I
--tl-ttt I I I I 1 I I I l l 1 I I I I
I I I I I I I I I I i i i ~ t t i I I l 1 1 I I I 1 1 1 1 I I I I I I I 1 I l I I ----- l 1 1 I t I I I T--T-7-Tl-TIT 1 I I I I I I I I 1 I I I I I I I I I 1 I l I I I 1 1 I I I 1 1 1 I 1 I t I I I 1 I I 1 I l I I
I I l 1 1 1 1 I I I I 1 I l l I I I 1 I I I I I I I 1 I I I 1
I 1 I 1 I I I I I I I 1 I I I t l 1
10 100 1 O00
Screen Sue (microns)
Figure 4-3. Size distriiution of gaiena-s0ilica mixture with 30 minutes grind (Test NO. D-2)
Figure 4-4. Size distribution of galena with 30 minutes grind (Test No. D-2)
I 1 I I l I I I t--t-1-t-l-Ptt I I I I 1 I I I I I 1 1 1 1 1 1
1 I 1 I l l 1 1 1 I I I I I I I I I I I I I 1 1 1 1 I I I I 1 I l l I I t I I 1 1 1 1 I 1 1 I l I l l t I I I 1 1 1 1 1 I I I I I l I l 1 1 I I l 1 1 1 1 l 1 I 1 I I I I ----- 1 --7-7- 1 I I l -Trr,,----- 1 1 1 1 1 1 I 1 I l I I
T--T-7-Tl'-TTT I l 1 I l I l l 1 I 1 1 I I I I I I t 1 1 1 1 1 1 1 I I I 1 1 I I I
10 1 00 1000
Scteen Size (microns)
Figure 4-5. Size distribution of galena-silica mixture with 60 minutes grind (Test No. D-3)
4 5 6 7 8 9 1 0 20 JO 40 506070ôûOaOO
Particle Size, pm
Figure 4-6. Size distribution of galena with 60 minutes grind (Test No. D-3)
Chalcopy rite-Silica Mixture
Chalcopyrite is one of the predominant copper suüides in porphysr copper sulfide ores
which represent the largest tonnage of copper ore treated. A high purity natural
chalcopyrite minerai sample was purchased from WARD's Naturai Science Estabiishment
inc. The label on the study pack indicates that the source of chalcopyrite is Ontario,
Canada. The sample was hand-picked and crushed to ail minus 48 mesh. Clean silica sand
was screened to remove al1 plus 48 mesh portion and leave oniy the minus 48 mesh
W i o n to mix with the prepared chaicopyrite sample. One-kilogram charges, comprising
50 grarns chalcopyrite and 950 gram silica, of the chalcopyrite-silica mixture were
prepared for al1 test work.
The head grade is 0.7 wt.% Cu. The one-kilogram test charge was ground in 20cm
x 35cm laboratory rod mil1 at 60% solids for 30 and 60 minutes. The size distribution
analyses of the ground pulps at various grinding times are plotted as in Fig. 4-7 and Fig. 4-
9, while Fig. 4-8 and Fig. 4-10 show the size distributions of the corresponding flotation
concentrates which were taken as the high-grade chalcopyrite.
The screen sizes through which 80 W.% and 90 W.% of particles pass are
summarized in the Table 4-2.
The results of size analysis indicate clearly that in the chalcopyrite-silica mixture,
chalcopyrite was ground much finer than the silica wid.
Flotation tests were carried out with the CFC-Q 1-3, the CFC-Q2-I and the Denver
ce11 (D-12) under natural pH. Since chalcopyrite is not as readily floatable as galena,
potassium arnyl xanthate (PAX) was chosen as the collector in the Botation tests. PAX is
the most powerful and least selective xanthate which is often used as a scavenger collector
following a more selective rougher collector (Mining Chernicals Handbook, 1986).
t i 1 I 1 I I I I 1 I 1 I I I I 1 1 1 I 1 I I I 1 1 1 - 1 I l I I r--T-7 ,--,m 1 I I l I I I 1 I 1 1 I I I I I I I I l I I I 1 I 1 I I I I I I I I I 1 i l 1 I 1 1 i 1 I I I
l I 1 I l 1 1 1 1 1 f I 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 I I 1 1 I I I I I 1 1 1 I l 1 1 1 1 t I I I 1 1
1 1 l 1 1 I l l 1
Table 4-2. Ground products of chalcopyrite-silica mixture
100
Screen Sie (microris)
Figure 4-7. Size distribution of ~halcopy~te-silica mixture with 30 minutes grind (Test No. 41-3-8)
Gfind Time
(mi nu t es) L
30
60
Head
90% Passing
- 65 W
Head
8% Passing
- 55 pm
Chalcopyrite
80% Passing
44 pm
30 pm
Chalcopyrite
90% Passing
52 Cim
40 Cun
Figure 48.
I I I I I
-rni~~'----- -TTlll-:----- 'TT117 -----
I I I I l I I I I I I I I I I 1 1 I l l l
Partide Sue, pm
Sire distribution of chaicopyrite with 30 minutes grind (Test No. Q 1-3-8)
I l I 1 1 1 1 1
I L I 1 I I l l I I I 1 1 1 1 1
I 1 1 1 l l l l I 1 I I l I l l I I I l l l l i I I 1 1 l I l l
I I 1 1 1 1 1 1 1 I 1 I I I t l l I I I 1 1 1 1 1 1 I 1 1 1 1 1 1 1 1 I I 1 1 1 1 1 f 1 I I 1 1 1 1 1 I 1 1 1 I l l l l I 1 1 I l I l l l t 1 I l l l l l I I I 1 1 1 1 1 ----- 1 I I 1 1 I I 1 1 I I I 1 1 1 1 1 T--7- l-TTTTr--- 'T--T-T-TlTm 1 1 I 1 1 1 1 1 1 I 1 I I I I 1 1 I I I 1 1 1 1 1 1 I I I I t I l l t l t 1 1 1 1 l 1 I I l 1 1 1 1 1 I I 1 1 1 1 1 1 1 I 1 I l 1 1 1 1 1 I I l I I l I I 1 I I 1 1 1 1 1 I I I t 1 I l I I I I 1 1 ! 1 1 1 I I I 1 1 1 1 1 1 1 I I l 1 1 1 1 I 1 I 1 I I l I l 1 I I I l I l l 1 I 1 I t I l I l 1 t I I l l f l I I 1 1 1 1 1 1 1 1 1 1 I I l l I
1
100 1 O00
Screen Sie (microns)
Figure 4-9. Size distribution of chalcopyrite-silica mixture with 60 minutes grind (Test No. D-4)
5 6 7 d 910 20 30 $0 50 6070ôûûûiW
Partide Sue, pm
Figure 4-10. Size distribution of chalcopyrite with 60 minutes grind (Test No. D-4)
4.2.3. Gaspe Copper Ore
One-kilogram test charges were prepared for the Gaspe copper ore containhg 0.7 W.%
Cu. The ore sample was ground for 30 and 60 minutes. The size distributions of the
ground head samples and concentrates (copper sulfides) are shown in Fig. 4-1 1 to Fig. 4-
14. The xreen sizes through which 80 W.% and 90 W.% of particles pass are
surnrnarized in the Table 4-3. The results of size analysis demonstrated again that the
valuable minerals (copper sulfides) were reduceâ to h e r size range &er grinding.
Rotation tests were canied out with the CFC-Q 1 3 , the CFC-Q2-1 and the Denver D-12.
Potassium amyl xanthate (PAX) was selecîed as the collecter.
I I I I I I 1 l I l I t I I I ----- 1 1 - 1 - 1 1 T--7 7 77- I I I I I I I I I I I I I I I I
' I I I l I I 1 1 1 1 1 I I I I I I 1 1 1 1 1 I I I I I I 1 1 1 1 1 I I I I 1 - 1 - I l - l i l
-TT---- r--T 7 TT r l I l I I I l 1 1 1 1 t I I I I 1 1 1 1 1 1 I I I I 1 1 1 1 1 1 1 I I I I 1 1 1 1 1 1 1 -1
Figure 4-11. Size distribution of Gaspe ore with 30 minutes grind (Test No. D-6)
Figure 4-12.
1 1 1
-i-+llT - r7 i i7 - -TT111- -ttlli-
I I I l l i I I l l I I I I I -f-Lul
f 1 I I I l I I I I I I I I I I ----- C--+-4-4-"CC I I l I I I I I I I f I I l i f i I I I l I I I I I I I l I I I
Size distribution of copper sulfides fiom Gaspe ore with 30 minutes grind (Test No. D-6)
I I I I I I I I ----L--L--!-LA
l 1 I 1 I I I I
.---4
i i i I I I I I I I I 1 I I I I I t--*-1-tt+tw---- t--t-t-tlt I I I 1 t t I l l I I I I I l I I I I I I I I I I ! I I I I I I I I 1 I I I t I l t l I I I I I l I l 1 I I 1 1 1 1 1 1 1 I I 1 I 1 I l I l I I I I I l I I I I 1 I t f l l l 1 1 I I I I I I 1 I I 1 1 1 1 1 1 1 I I 1 1 1 1 1 1
.---- I 1 - 1 I l l l l t--7-7 ?-mm----- I I 1 1 1 1 1 1 1 l l I i 1 1 1 1 1 I I 1 l I I I I
! : ; : t I l I l i l ; Ml I 1 I I I I
1 1 1 1 1 1 1 I I I I I I t I 1 l 1 1 1 1 1 1 I 1 I I I I t I I 1 1 1 1 1 1 I t I t I I I I 1 I I 1 1 1 1 1 I I I I I I I
10 100
Screen Sue (microns)
Figure 4-13. Size distribution of Gaspe ore with 60 minutes grind (Test No. D-5) - -rf 111' -rni7- -rmiT-
I I I I I I I I I I I I I I I I I I I I u
- - - I I I I I I I I t--+--+-+-kt+ I I I I I l l I I I I I l I I +i ----- t--+-4-+-CC+ J I I I l I I I I I I I I I I I I I I i I I I I I I I I I I I I 1
Figun 4-14. Size distribution of copper sulfides frorn Gaspe ore with 60 minutes grind (Test No. D-5)
Tabk 4-3. Ground products of Gaspe copper ore
4.2.4. Cominco Lead-Zinc Ore
Grind Time
(minutes)
One kilogram test charges were prepared for a natural lead-zinc ore provided by Cominco.
The head grade is 5.0 W.% Pb and 7.7 W.% Zn. The ore sample was ground for 20
minutes. Screen analysis was not conducted on this sample. Flotation tests were carried
out with the CFC-Q 1-3, the CFC-Q2- I and the Denver cell @- 12). Reagent regime will
be introduced in the next chapter.
4.3. Testing Procedure
Head
80% Passing
Al1 the flotation tests which were carried out with the CFC-QI and CFC-Q2 used a
standard procedure as follows: first one kilogram ore sample was ground and pulped with
4 liters tap water to make a 20% solids in the conditioning tank mounted on a Moyno
progressing cavity pump. A mechanical mixer is used to keep the solids in suspension in
the conditioning tank. Collecter and frother were added to the conditioning tank. Air flow
is tumed on and the slurry is pumped to the flotation cell. The flotation pulp was
recirculated through the cell. Both concentrates and tailing were collected at specifïc t h e
Head
90% Passing
Copper sulfides
80% Passing
Copper sulfides
90% Passing
intervais for assay. At the end of flotation, the matenal left in the cell was flushed out and
collected into a separate pail.
The amounts of reagent additions, conditioning time, flotation time, and the CFC
operating variables were varied for the specific wnples. Similar conditions were employed
for each flotation device when testing the same ore sarnple.
For flotation tests which were carried out with the Denver cell @-12), one
kilogram ore sample was pulped with 4 liten water to prepare a 20% solids slurry in a I
kilogram cell. The agitator speed was 1800 rpm for al1 the flotation tests. The reagent
additions are similar to the CFC tests.
4.4. Particle Size Analysis in the Sub-Sieve Range
Size analysis of the various products of flotation tests conaitutes a fundamental part of the
test work. Several methods of size analysis were combined in order to cover a wide size
range of particle size. Dry sieving with sieves no finer than 75 pm (200 mesh) aperture,
was used on the original head samples. Wet sieving with sieves no finer than 38 pxn (400
mesh) aperture was used on the ground materid and mil1 products. Both dry and wet
sieving were facilitated by using a vibratory sieve shaker. Particles finer than about 38 pm
are referred to as being in the "sub-sieve" range and a Warman Cyclosizer and a Fritsch
Laser Particle Sizer were used to determine the size distribution.
4.4.1. Warman Cyclosùer
The W m a n Cyclosizer is an elutnator that can separate a sample into specitic size
fiactions on the basis of resistance to motion in a fluid. This resistance to motion is
charactented by the free falling velocity which the particle attains as it is alloweâ to fa11 in
a fluid under the influence of gravity. As the particle increases in speed, the viscous
resistance force increases until it just balances the effective weight' of the particle. At this
point the acceleration is zero and the speed increases no further. This maximum speed,
~ f ; , is called the terminal velocity or sedimentation velocity. For particles within the sub-
sieve range, assuming they are spherical, the terminal velocity is given by the well-known
Stokes' Equation:
where d, is referred to as the Stokes' diameter, p, and pl are the density of the particle and
fluid, respectively, p is the fluid viscosity, and g is the acceleration due to gravity.
Elutnation is a reverse of gravity sedimentation by using an upward current of
fluid, and Stokes' equation still applies in the sub-sieve range. At the fine end of the scale,
separations become impracticable below about 10 pm, as the material tends to
' The effective weight of the prticle in the fluid is equal io the weight of the panide minus th weight of fiuid displad.
agglomerate, or extremely long separating times are required. Separating times can be
considerably decreased by utilkation of centritiigal forces and one of such applications is
the Warman Cyclosizer.
Cyclosizer differs from conventional elutriation, however, in that the elutriating
action takes place in a fast rotating fluid (water) hstead of Ui an upward cunent of fluid.
The e f f ' of a rotating fluid is to exert a force on a particle as if the acceleration of
gravity were increased to a value of 02r; here, o is the angular velocity of the particle it
obtains from the rotating fluid and r is the distance of the panicle from the ê u s of rotation.
Thus, we c m still use Eq. (4.1) if we substitute the effective acceleration of gravity equal
to 02r, for g:
Hence the particle diarneter d, is given as
Aside from the centrifùgal forces many times those due to gravity, the high
shearhg forces which are developed in a rotating tluid overcome any natural tendency for
the fine materiai to agglomerate thus excellent dispersion of the particles is ensured.
Figure 4-15. Wamuui Cyclosizer
The cyclosizer unit consists of five inverted cyclones (see Chapter 2 for a fUU
description of the principle of the hydrocyclone) arrangeci in series such that the overtlow
of one unit is the feed to the next unit (Fig. 4-15). There is a successive decrease in the
idet area and vortex outlet diameter of each cyclone in the direction of the flow, resulting
in a corresponding increase in d e t velocity and an increase in the centrifùgal forces within
the cyclone and, consequently, there is a successive decrease in the limiting particle
separation sizes of the cyclones. In a sizing determination, therefore. the coarsest fraction
is collected in the No. 1 cyclone and the finest fraction in the No.5 cyclone.
There are four important operating variables which determine the effective particle
wparation sizes for the five cyclones: (i) water flowrate, (ii) water temperature, (iii)
particle density, (iv) time of elutnation. The standard levels of these operating variables
are:
(i) Water flowrate: 1 1.6 liters per minute
(ii) Water temperature: 20 O C
(iii) Particle density: 2.65 gram I cm3
(iv) Time of elutriation: infinite
At the standard levels of the operating variables the five cyclones in Our Cyclosizer
have Iimiting particle sepuation sizes as shown:
Cyclone 1
Cyclone 2
Cyclone 3
Cyclone 4
Cyclone 5
Since practical operation of the Cyclosizer is rarely carried out at standard
operating conditions, a correction factor for each variable within the specified operating
range is required in the relationship between the specified limiting particle separation size
and the eflective part icle separation size:
where de is the effective particle separation size of a cyclone, dl is the limiting particle
separation size of the sarne cyclone, andJ , & , , jr are the separate correction factors
for water flowrate, water temperature, particle density, and time of elutriation,
respectively. In some cases, the correction factors5 ,f2 , f3 , fr are chosen before running
the cyclosizer in order to set the operating conditions under which the effective particle
separation size of Cyclone 5 can be reduced to minus 10 microns. This can be done by
using warmer water, higher water flowrate, and longer elutriation time. A set of correction
graphs can be found in the cyclosizer instruction manual.
Al1 samples were wet screened on a 200 mesh sieve (or 400 mesh Peve in some
cases) before being sent to the Cyclosker. The weight of the test sample should be such
that not more than 15 gram collects in any one of cyclones. The initial distribution of the
sarnple is an approxirnate size separation with each cyclone and apex chamber containhg
an excess of undersize material. Controlled elutriation is then effected by reducing the
water flow to a pre-determineci figure and holding it constant for a specified time while
particles smaller than the limiting particle separation size of each cyclone are gradually
elutriated to the vortex outlet. Solids smaller than the limiting size of the final cyclone pass
out with water to the collection buckets. Mer the elutriation time has elapsed, the water
flow is increased again and, as soon as practical thereafter, the solids which have collected
in the five cyclones are dischargecl into separate beakers by opening the cyclone apex
valves. The solids (including the finest solids in the collection buckets) are settled, the
water is decanted and the solids drkd and weighed.
Altogether six size fractions in the sub-sieve range can be obtained: -75 + 37, -37
+ 26, -26 + 19, - 19 + 14, -14 + 10, and - 10 pm, respectively2. Solids in each ab-sieve
size fraction are assayed to determine metal contents so that recovery of each size fiaction
c m be calculated.
4.4.2. Fritsch Laser Particle Sizer
The Fritsch Laser Particle Sizer ("Analysette 22" ECONOMY-version) is another rnethod
that is used for particle size andysis in the sub-sieve range. It is simple to use and
extremely fast. The principle is based on the diffraction of a monochromatic Iight beam by
-
' caicuiated using 0.877 (= 10 pm 11 1.4 pm) as the overall correction factor.
the grains of powder to be analysed, which results in difiaction patterns that can be used
to calculate the particle diameter.
For example, if a sphencal particle is illuminated by parallel, monochromatic light,
a di5action pattern referred to as the "Fraunhofer difiaction pattern" (Fig. 4-16) after the
person who discovered it is produced in the focal plane of a lens in the beam of light
behind the particle due to interference of the light waves ditfracted at the edge. The
difiaction pattern of a round dix or a sphencal particle which is visible in the focal plane
consists of aitemately light and dark concentric rings. The diameter of the particle can be
calculated by the following equation:
where & is the radius of the first circle of light produced due to difiaction, fis the focal
length of the imaging lens and )c is the wavelength of the light.
In practice, the powder to be examined is dispersed in a liquid (water) which is
then circulated through a glass cell. A parallel beam from a low-power laser lights up the
cell, and then the beam which leaves is focused by means of a convergent optical system.
The values of illumination with and without sarnple are read by an electronic detector and
fed into a prograrnmed processor, which then displays the results as cumulative percentage
undersize.
Figure 4-16. Fraunhofer difiaction pattern
4.5. Determination of Metal Contents
To determine metal contents in sulphide minerais, ores and flotation produas, the sample
is pulverwd for 1 - 3 minutes in a laboratory BlGû vibrating cup miil to analytical
fineness. 0.1 - 1 .O grams of powdered sample. depending on the expeaed content of the
element(s) to be determined. is then transferred to a 250 ml giass beaker. Add 30 ml of
concentrated hydrochlonc acid, cover the beaker with a glass cover, and boil for 30
minutes on the hot plate. Remove the beaker from the hot plate, add 30 ml concentrated
nitric acid into the beaker, cover the beaker with a glass cover and boil for another 30
minutes on the hot plate. Cool, wash down the sides of the beaker with approximately 30
ml de-ionized water, and heat the beaker again to dissolve the soluble salts. Remove the
beaker from the hot plate, filter it into a 250 ml volumetric flask, again depending on the
expected content of the element(s), and wash the beaker, paper and residue with de-
ionized water. Cool the filtrate to room temperature, dilute to volume with de-ionired
water and mix thoroughly.
Measure the lead, copper, and zinc absorbantes of the blank and simple solutions,
or suitable aliquots diluted to appropnate volumes with 2% nitric acid, on a Perkin Elmer
Atomic Absorption Spectrometer 3300 at 283.3, 324.8, 21 3.9 nm, respectively, in an air-
acetylene fiarne. The content of the desired element(s), in the solutions or aliquotq is
obtained directly fiom the computer screen.
4.6. Recovery Formulas
Flotation tests run by the centnfbgal flotation cells CFC-QI and CFC-Q2 are semi-
continuous. Computations of recovery are entirely dependent on the assays and weights,
where known, of the process feed and products of separation. Any increase in the number
of separations and minera1 components to be accounted for greatly increases the
complexity of the computations.
Recovery (%) represents the ratio of the weight of metal or minerai value
recovered in the concentrate to IOPh of the same constituent in the heads or feed to the
process, expressed as a percentage.
4.6.1. Two Product Formulas
For the simplest separation where only one concentrate and one tailing result fiom a given
ore feed, we have the following notations (Mining Chemicais Handbook, 1986):
-- - - - - -- - - -
Product Weight or Wt.% Sarnple Assay % Calculated
Feed
Concentrate
Tailing
Recovery, %
By assays f, c and t only:
c(f - t) R=- x 100 = the recovery, % f(c - 0
By weights F and C, plus assays c and t:
R = 100- 100p - C)t
= the recovery, % (C x C) + (F - C)t
4.6.2. Three Product (Bi-Metallic) Formulas
Frequently, a concentrator will mil1 a complex ore requinng the production of two
separate concentrates, each of wbich is enriched in a diEerent metai or valuable mineral,
plus a 6nai tailing acceptably low in both constituents. Formulas have been developed
which use the feed tonnage and assays of the two recovered values to obtain the ratios of
concentration, the weights of the three products of separation and the recoveries of the
values in their respective concentrates. For illustrative purposes data fiom a lead-zinc
separation is assumed (Mining Chernicals Handbwk, 1986):
Product Weight or Wt.% % Pb Assay % Zn Assay Calculated
Feed F Pi i
Pb Concentrate P Pz 22
Zn Concentrate Z P3 2 3
Tailing T P d Z,
Recovery, % Ra, Rzn
Z = F x ( ~ 2 -p4Xzi -'4)-(pi -p4Xz? -4) = tonnes Zn concentrate (4.9) (P? -p4Xz3 -z4)-(zI -z4Mp3 -P,)
- x 100 = lead recovery, % Rpb -
L x Z, x 100 = anc recovery, %
Chapter 5
Flotation Test Results and Discussion
5.1. Operating Variables
The artificial galena-silica d u r e was used to eaablish the operating parameters for the
new CFC-QI and CFC-Q2 centrifuga1 flotation cells. The test charge has a head grade of
4% Pb. The operating variables examined included the following: fineness of grind,
rotating speed of the rotating vesse1 in the CFC-QI cell, rotating speed of the rotating
feeder in the CFC-Q2 cell, and air flow rate through the sparger.
5.1.1. Effects of Openting Variables on CFC-QI-2 Performance
The galena-silica mixture was ground at a pulp density of 60% solids and then poured into
the conditioning tank mounted on a Moyno progressing cavity pump. The pulp density
was reduced to 20% solids in the conditioning tank by adding tap water. The ground pulp
was conditioned with 100 g/t sodium ethyl xanthate (SEX) for 5 minutes, then 25 m g L
DF-250 for 1 minute. The compressed air valve was opened and tuned to supply the
sparger in the flotation "reactor" with air at a flowrate of 6 Umin. The Moyno pump
started up, pumping the slurry out of the conditioning tank at a rate of 2.7 - 2.8 Umin.
The conditioned slurry flowed to the flotation "reactor" where it was aerated by the
sparger. The aerated sluny then entered the rotating flotation "separator" (the drum)
through the downcomer. The flotation process commenced. The froth product was
removed from the fioth colurnn and the tailings flowed back into the conditioning tank by
gravity. The flotation time was 6 minutes. The concentrate and tailing were collected
separately and prepared for metal assay. At the end of flotation, the speed of the
"separatot' was maximized to get rid of the matenal left in the dmm. The whole system
was flushed out by tap water and collected into a separate pail as the "remains". Note that
the "remains" are not the flotation tailings.
Table 5-1 summarizes the effect of flotation drum rotating speed on the gaiena
recovery and the concentrate grade. The one kilogram sample was ground for 10 minutes.
The rotating speed varied tiom 100 rpm to 200 rpm. The results show that the lead
recovery was 80.8% with the concentrate grade of 59.5% Pb by rotating the drum at 100
rpm. Increasing the drum rotating speed to 150 rpm, the Pb recovery was increased to
92.6% with a higher concentrate grade of 62.1% Pb. Funher increasing of the drum speed
to 200 rpm, the concentrate grade increased to 83.6% Pb, but the Pb recovery was
reduced to 84.7%.
Table 5-1. Effect ofdrum rotating speed on galena flotation with CFC-QI-2 (air flow at 6 Umin., gind 10 min., Boat 6 min.)
Test No. ' Drum Speed Conc. Grade
(% Pb)
59.5
62.1
3
2
4
100 rpm
150 rpm
200 rpm
Tail Assay
(% Pb)
0.70
0.3 1
Pb Recov.
(%) 1
80.8 J
92.6
83.6 1 0.70 84.7
The results clearly identiS, the rotating speed of the flotation "separator" (the
dmm) as a signifiant operating factor influencing flotation results. The higher the rotating
speed, the higher the concentrate grade. This can be weU explained by the fact that the
rotating speed is directly related to the centrifugai force effects. The viscosity of the slurry
with a pulp density of 20% solids is much higher than that of pure water. Therefore, with
the "separator" (the drum) rotating, the pulp will be spun-up to the angular velocity of the
drum in a very short time. Thus the rotating speed of the drum can be taken as the rotating
speed of the pulp. The faster the pulp rotates, the stronger the centrifuga1 force it creates.
The centrifuga1 force acts to throw the non-floatable particles outward toward the wail of
the dmrn where they enter an upward-moving secondary flow that will be discharged into
the tailing charnber through the nm of the drum. The mineralized froth moves toward the
core of this "extemally-forced vortex" because its density is less than that of the fluid.
High rotating speeds create a aronger centrifugai force that enhances the separation of
bubble-particle aggregates from the unreacted slurry and also reduces the "mechanical
carry-out" of fine gangue particles in the fioth, thereby producing a higher grade
concentrate.
The results in Table 5-1 also suggest that the rotating speed of the flotation
"separator" (the drum) can be optimized to produce the highest recovery with an
acceptable concentrate grade. At 150 rpm, the CFC-Q I I ceIl achieved the highest Pb
recovery 92.6% with the lowest tailing grade 0.3 1% Pb. The concentrate grade was
62.1%, which is perféctly acceptable. One reason for this optimum rotating speed is
related to the effect of centrifuga1 force on the bubble-particle adhesion. Very possibly the
centrifugal force becomes a disruptive force associated with the weight of the particle
which would be signincantly increased in a centrifuga1 force field, thereby advetsely
affecting recovenes. if the rotating speed exceeded a certain limit, the centrifuga1 force
will drive everything out of the rotating vessel.
The effect of grind on galena flotation with the CFC-QI-2 ceIl is summarized in
Table 5-2. The rotating drum speed was maintained at 150 rpm and the air flow rate was
maintained at 6 liters per minute. With 10 minutes grind, the Pb recovery was 92.6% with
the concentrate grade of 62.1% Pb. With 30 minutes grind, the Pb recovery dropped to
80.4%. the concentrate grade dropped to 39.0% Pb, and the tailing grade was tnpled to
0.93% Pb.
Table 5-2. Effet of gind on galena flotation with CFC-Q 1-2 (air flow at 6 Wmin., drum speed 150 rpm, float 6 min.)
The results indicate that centrifbgal ceils are susceptible to fineness of grind just
like mechanical cells. Note that the samples used in this group of tests were the synthetic
mixtures of galena and silica sand, that is to Say, the galena particles were l W ? liberated.
The total flotation time was the sarne too. Thus, it is reasonable to attribute the poor
Test
No. 1
2
5 1
6
Grind Time
(min.)
10
20
30
Galena
80% passing
- 47 CUn
3 3 ~
Conc. Grade
(% Pb)
62.1
64.0
, 39.0
Tail Assay
(Y0 Pb)
0.3 1
0.65
I 0.93
r
Pb Recov.
(%)
92.6
85.0
80.4
flotation efficiency in the Test No.6 to the centrihgal force effects. With decreasing
particle sue and the same rotating speed, the separation of bubble-particle aggregates
from the unreacted slurry will be slower. resulting in a lower recovery. Besides, the
drainage of entrained particles from the froth will become more difficult. resulting in a
lower concentrate grade and higher tailing grade.
The sliming problem may be overcome with the use of faster rotating speed.
Longer flotation circuit residence time would cenainly improve the recovery.
5.1.2. Effects of Operatiog Variables on CFC-QI3 Performance
The pulp with a pulp density of 20% solids was conditioned with 50 g/t sodium ethyl
xanthate (SEX) for 4 minutes and then with 25 m g 5 DF-250 for 1 minute. The pulp was
pumped to the flotation "reactor" (the sparger) and then entered the flotation "separator"
(the bowl) at a rate of 2.7-2.8 liters per minute. Flotation was conducted for 3 minutes.
The concentrate and a portion of tailing were colleaed for assay. An additional 50 g/t
SEX was then added and flotation was continued for a fiirther 5 minutes. Both final
concentrate and tailing were collected for metal assay. In al1 these series of tests, the air
flowrate was maintained at 6 liters per minutes.
The effects of the rotating speed of the flotation "separato?' (the bowl) and the
fineness of grind on the flotation of galena with the CFCQI-3 ce11 were sumrnarized in
Table 5-3.
Table 5.3. Effects of rotating speed and grind on galena flotation with CFC-QI-3 (air flow at 6 Umin., float 8 min.)
The results of Test No.2, No. l and No.3 show that, while there was no significant
variation in the overail recovery, the highest grade concentrate was produced at the lowest
rotating speed. At 100 rpm, the concentrate grade was 72.W Pb and the Pb recovery was
89.6%. As the rotating speed increased to 150 rpm, the concentrate grade decreased to
36.7% Pb and the Pb recovery increased to 92.1%. At 175 rpm, the concentrate grade was
60.7% Pb and the Pb recovery was 90.1 %.
These results turned out contrary to the results of the CFC-Q 1-2 ceU (refer. Table
5-1). The effect of the drum rotating speed on the concentrate grade produced by the
CFC-QI-2 cell was: the higher the rotating speed, the higher the concentrate grade. This
difference was attributed to the stage addition of collecter and longer flotation tirne in the
flotation tests run with the CFC-QI-3 ceU. These two factors apparently boosted the
overall recovery of galena, thereby overshadowing the effect of the bowl rotating speed.
Tai1 Assay
(% Pb)
0.45
0.34
0.43
0.62
O. 84
Conc. Grade
(% Pb)
72.0
36.7
60.7
36.1
41.4
Pb Recov.
(%)
89.6
92.1
90.1
86.5
80.2
Test
No. 1
2
1
3
5
4
Rotating
Speed
100 rpm
150rpm
175rpm
175 rpm
175rpm
Grind
Time
20 min.
20Mn.
20min.
40 min.
60rnin.
Galena
80% Passing
47 W
47 Cun
47 pm
29 pm
26 W
The results of Test No.3, No.5 and No.4 show that both the concentrate grade and
the Pb recovery were decreased with increasing grind time. By increasing the grind time
fiom 20 minutes to 60 minutes, the concentrate grade decreased from 60.7% Pb to 4 1.4%
Pb, respectively, and the Pb recovery also reduced fiom 90.1% to 80.2% respectively. It is
seen that both models, CFC-QI -2 and CFC-Q 1-3, are susceptible to the particle size.
Table 5-4 compares the results of two flotation tests, one was carrieci out with the
CFC-QI-2 cell, the other with the CFC-QI-3 cell. The particle size of the feed to the
CFC-QI-3 ceIl was finer than that of the feed to the CFC-QI-2 cell, but the CFC-QI -3
ceIl outperfomed the CFC-QI-2 cell by a considerable amount. The concentrate grade
was 72.0% Pb with the CFC-QI-3 ce11 as compared to 59.5% Pb with the CFC-QI-2 cell.
The overall recovery was 89.6% with the CFC-QI-3 ceIl as compared to 80.8% with the
CFC-Q 1-2 cell.
The flotation time in the Test No. Q 1-3-2 is longer than that in the Test No. Q 1-2-
3 (8 min. vs. 6 min.). The collector was aage added to the conditioning tank in the
flotation test with the CFC-QI-3 cell, while it was added at the very begi~ing in the
flotation test with the CFC-Q 1-2 cell. Longer flotation time and stage addition of collector
usually would improve the recovery but they may reduce the concentrate grade. However,
both indexes (recovery and grade) produced by the CFC-QI-3 ceIl were higher than those
produced by the CFC-QI-2 cell. Therefore, it must be the shape of the flotation
"separatot' that made the difference. The CFC-QI-2 ce11 uses a dnim, and the CFC-QI-3
cell uses a bowl which appean to be a better design.
Ta blc 5-4. Comparison between CFC-Q 1-2 and CFC-Q 1-3
5.1.3. E ffects of Operating Variables on CFC-Q2- I Performance
The flotation conditions on the CFC-Q2-1 cell were slightly different from the CFC-QI
cell. The variables studied included the following: the rotating speed of the feeder, the
fineness of grind and the air flowrate. The pulp with a pulp density of 20% solids was
conditioned with 50 g/t sodium ethyl xanthate (SEX) for 3 minutes and then with 50 m@
DF-250 for 1 minute. The pulp was then pumped at 3 liters/rnin. into the "reactor" (the
sparger) and then entered the "separatoi' (the stationary vessel) through the rotating
feeder. The flotation time was 1 minute. Both concentrate and taiiing were collected for
metal assay.
In the first series of tests (Test No. 10, No. Il, No. 12). the sample was mt ground
(-48 mesh) except that Test No.21 was ground for 30 minutes and floated at 6 literslmin
air rate. For al1 other tests. the air flowrate was 12 literdmin. Without reducing the air
flow rate, the strong froth was not controllable in the Test No.21. The effect of the
rotating speed of the feeder is summarized in Table 5-5.
Conc. Grade
('Xi)
59.5
72.0
CFC
Mode1
41-2
41-3
Test
No.
3
2
Float
Time
6 min.
%min.
Tai1 Assay
(y&)
0.70
0.15
Gnnd
Time
10 min.
20min.
Rotating
Speed
100 rpm
100rpm
Pb Recov.
(%)
80.8
89.6
Galena
800h Passing
- 1 7 p n
The results in the Table 5-5 indicate that the low Pb recovery was produced under
two extreme conditions: very high speed or no rotation at d l . At 8 15 rpm, 89.2% of Pb
was recovered with a concentrate grade of 54.2% Pb. Without rotating the feeding pipe (O
rprn), 88.1% of Pb was recovered with a concentrate grade of 60.4% Pb. With the feeder
rotating at a speed between those two extreme conditions, 93.3% and 94.3% of Pb were
recovered respectively at the rotating speed of 440 rpm and 220 rpm. The concentrate
grade was 82.7% Pb and 58.% Pb respedvely at the speed of 440 rpm and 220 rpm.
Table 5-5. Effect of feeder rotating speed on galena flotation with CFC-Q2-1
It is clearly evident that the rotating speed of the feeder in the CFC-Q2-1 cell, like
the rotating speed of the vesse1 in the CFC-QI cell. cm be optimized. Within a pennissible
limit, the faster the feeder rotates, the higher the concentrate grade will be.
Test No.
L
21 . 10
r
11 L
12
The second senes of tests was to determine the effect of grind on the galena
flotation. The rotating speed of the feeder was maintained at 440 rpm and the air flow rate
was set at 12 iiters per minute. The results summarized in Table 5-6 show that the gaiena
Pb Recov.
('w 89.2
93.3
94.3
88.1
Tai1 Assay
(% Pb)
0.4 1
0.28
0.27
O. 58
Feeder Speed
815 rpm
440 rpm
220 rpm
O rPm
Conc. Grade
(% Pb)
54.2
82.7
58.0
60.4
recovery was decreased with increasing fineness of grind. Under the same other
conditions, about 93.3% of galena was recovered with no grind (O minute) and 74.7%
galena recovery was obtained with 30 minutes grind.
Table 56. Efféct of grind on galena flotation with CFCQZ-1
The third series of tests focus on the effect of the air flow rate on the galena
flotation. The sample was ground for 30 minutes and the rotating speed of the feeder was
440 rpm. Results sumrnarized in Table 5-7 indicate that the galena recovery increased with
higher air flow rate. The concentrate grade, however, responded to the change of the air
flow rate differently. With the air flowrate set at a low nurnber (3 Urnin.), the concentrate
grade was 77.9%. The concentrate grade was reduced to 60.9% Pb at a high air flow rate
(12 Uminl). The best setting was found to be in the middle (6 Uminl), which produced
the highest concentrate grade (82.3% Pb).
The results in the Table 5-7 indicate that the air flow rate also needs to be
optimized. High air flowrates produce many more air bubbles that may coalesce and form
large bubbles in the pipe reactor. Once the contacted slurry enters the separator, the f d in
Test
No.
10
15 I
16
Grind Time
O min.
15 min.
30 min.
Galena
800h Passing
- 55 pm
33 pm
Conc. Grade
(% Pb)
82.7
65.5
60.7
Tai1 Assay
(% Pb)
0.28
0.87
0.78
Pb Recov.
(%)
93.3 1
75.5 1
74.7
pressure (atmospheric pressure in the separator) will cause an increase in the size of
bubbles, therefore more bubbles d l coalesce. The "bursts" of air create disorder in the
rotating fluids and dismpt the separation of the bubble-particle aggregates from the
u~eacted slurry. As a result, the recovery may increase but the concentrate grade would
certainly be compromised because more gangue particles will be mechanicaiiy carried over
into the froth by the excessive amount of bubbles. Therefore, it is important not to over
aerate a pulp in the operation of the CFC-QI and CFC-Q2 cells.
Table 5-7. Effect of air flow rate on galena flotation with CFC-Q2-1
5.2. Cornparison Between CFC-QI, CFC-Q2 and Denver Cell
The efficiencies of the CFC-QI-2, CFC-QI -3, and CFC-Q2-I cells were compared with
that of the Denver D- 12 cell. Two artificial mineral mixtures (galena-silica, chalcopyrite-
silica) and two naturai ore samples (Gaspe copper ore, Corninco lead-zinc ore) were used
for the tests. Each test charge was one kilogram, ground for 20, 30 and 60 minutes. The
optimum operating conditions for the CFC-QI and CFC-Q2 cells were determined fiom
the previous results surnmarized in Section 5.1. The designs of CFC-QI and CFC-QZ
Test
No.
16
17 L
18 A
Air
Flow Rate
12 Wrnin.
6 Umin.
3 Wmin.
Conc. Grade
(% Pb)
Tai1 Assay
(% Pb)
Pb Recov.
(%)
74.7
71.0
59.6
60.9
82.3
77.9
O. 78
0.90
1.47
centrifuga1 flotation systems are completely dflerent from the conventional mechanically
agitated cells like the one used in the test work (the Denver D-12 cell), therefore the
cornparison between them was based on metaiiurgical performance when treating the same
pulp. The reason behind this rnethodology is outlined in Section 4.1.
5.2.1. Results of Galena Flotation
The galena-silica mixture with a head grade of 4% Pb was ground for 20, 30 and 60
minutes. Sodium ethyl xanthate (SEX) with a total amount of 100 g/t was stage added to
the pulp twice. The dosage of DF-250 was 25 mgK. Depending on the type of ce11 and the
fineness of grind, the concentrate and the tailing were collected at 1, 3, 6, 9, 13, 19
minutes. The flotation results are summarized in the Table 5-8, 5-9, 5- 1 O and 5- 1 1.
Table 5-8. Galena flotation with CFC-QI-2 (dmm speed 150 rpm, feed at 2.8 Urnin., air flow at 6 Urnin.)
Test N0.8
20 min. Gnnd
Test No. 7 t
30 min. Grind
Float Time
(minutes)
1
Conc. Grade
(% Pb) 67.0
Pb Recov.
(W 82.2
3
6
L---.--,
85.5
88.3
60.8
59.0
Conc. Grade - - - -
Pb Recov.
47.5
47.6
88.4
90.7
(% Pb) 1 (W 47.8 85.8
Table 5-9. Galena flotation with CFCQI -3 (dnim speed 150 rpm, feed at 1 -8 Umin., air flow at 4 Urnin.)
1 Fioat Time
1 Tail Assay
.
Test No.6 1 Test No.7 1 30 min. Grind 1 60 min. Grind 1
Table 5-10. Galena flotation with CFC-Q2-1 (feeder speed 440 rpm, feed at 2.8 Urnin., air flow ai 6 Umin.)
Conc. Grade (Yo Pb)
74.2
Pb Recov.
(%)
87.6
Test No. 1 9
FIoat Time (minutes)
Conc. Grade (Yo Pb)
57.4
Test No.20
Tai1 Assay
Pb Recov.
(W 72.9
Test No. 20a
20 min. Gnnd
0.25
Conc. Grade
(% Pb)
30 min. Gnnd
Pb Recov.
(%)
O
Conc. Grade
(% Pb)
60 min. Grind
Pb Recov.
(%)
Conc. Grade
(% Pb)
0.27
Pb Recov.
(%)
L
O 0.4 1 -
Tabk 5-1 1. Galena flotation with Denver D-12 (1 800 rpm)
1 1 Test No. 1 1 Test No. 2 1 Test No. 3 1 I 1 20 min. Gnnd 1 30 min. Grind ( 60 min. Grind 1
Comparing the results of flotation tests using the CFC-QI-2 ce11 (Table 5-8) and
the Denver D-12 cell (Table 5-1 1). it is seen that while the concentrate grades obtained
from the CFC-Q 1-2 ce11 were significantly lower than those obtained from the Denver D-
12 ceIl, a big improvement is achieved in the initial flotation rate by the CFC-QI-2 cell.
From the 20 minutes grind sample, in the first minute, the CFC-QI-2 ceIl had recovered
82.2% Pb while the Denver D-12 ce11 had recovered 749% Pb. The advantage is more
evident in the case of 30 minutes grind sample: in the first minute, the CFC-QI-2 ceIl had
recovered 85.8% Pb while the Denver D-12 ce11 had only recovered 45.3% Pb. M e r 6
minutes flotation (30 min. grind), the CFC-QI-2 had recovered 90.7% Pb while the
Denver D-12 ce1 had recovered 86.8% Pb.
Float T i e (minutes)
1
3
6
9
19
Tai1 Assay
Conc. Grade
(% Pb)
83.5
80.9
76.9
74.8
- 0.24
Pb Recov.
(%)
74.9
89.7
93 -2
94.3
- -
Conc. Grade
(% Pb)
80.1
79.9
74.1
72.1
O
0.42
Pb Recov.
(%)
45.3
69.8
86.8
91.4
- -
Conc. Graâe
(% Pb)
- 64.1
70.4
63 -9
61 -9
0.34
Pb Recov.
(%)
- 46.3
82.0
87.6
92.5
O
Nevertheless, the Denver D-12 cell outperfonned the CFC-QI-2 &er 3 minutes in
the case of 20 minutes grind time (89.7% Pb recovery vs. 85.5% Pb recovery), 9 minutes
in the case of 30 minutes grind t h e (91.4% Pb recovery vs. 90.7% Pb recovery).
The faa that at the early stage of flotation, the CFC-QI-2 ce11 floated faster than
the Denver D-12 ce11 is consistent with the hdings from the prior work concerning the
flotation kinetics in other types of centrihgal flotation cells wch as the air-sparged
hydrocyclone and the original centrifùgai flotation cell. The lower concentrate grade
yielded by the CFC-QI-2 may be attnbuted to the intensity of the aeration in the flotation
"reactor", which is controlled by the air flowrate. The extenial fiow-through
bubble/particle contact apparatus (the "reactor") allows for both a large number of high
energy bubbldparticle collisions, and the creation of fine bubbles. On the one hand, the
high energy bubbldparticle collision would result in a weak attachent of Hlica particle to
an air bubble. This bubble-silica aggregate may survive for a very short period of time in
the rotating fluids so it will move toward the flotation column with the fkoth, and finally
leaves the flotation "separator" with the concentrate, causing a decrease in the concentrate
grade. On the other hand, high air flowrates may create an excessive amount of fine
bubbles that would block the paths of hydrophilic particles being thrown away toward the
dnim wall by the centrifùgal force as these fine bubbles move in opposite directions. Thus
some of hydrophilic particles are heId up in the kath, causing a fùnher decrease in the
concentrate grade.
It was clearly evident during these exploratory tests that the air flowrate was
playing an important role in the operation of the CFC-QI-2 cell, affe*ing both the
flotation rate and the quality of the 60th product. The concentrate grade could be
improved by reducing the air flowrate andor increasing the rotating speed of the
"separator" (the drum). Nevertheless, an increase of the entrainment can be advantageous
in some special cases. This is when a maximum of recovery, above al! in the range of very
fine particles, is airned ai, especialiy in the rougher and scavenger flotation (Mïtrofanov et
al, 1985). The CFC-QI-2 ce11 may therefore float reluctant and slightly hydrophobie
particles of the son which slip back fiom most fioths.
Comparing the results of flotation tests using the CFC-QI-3 ce11 (Table 5-9) and
the Denver D-12 ce11 (Table 5-1 i), it is seen that while the concenirate grades obtained
fiom the CFC-Q 1-3 cell were generally lower than those obtained h m the Denver D-12
cell, the fiotation recovery by the CFC-QI3 ce11 for the 30 minutes grind sarnple was
better than that by the Denver D-12 ce11 (95.6% recovery vs. 91 -4% recovery in 9
minutes). For the 60 minutes grhd sarnple, the flotation recovery was much faster fiom
the CFC-QI-3 ceIl than the Denver D- 12 ce11 at initial 3 minutes (72.9% recovery vs.
46.3% recovery). Mer 9 minutes flotation, the Denver P l 2 ce11 outperformed the CFC-
41-3 cell(87.6Y0 recovery vs. 78.6% recovery).
From Table 5-8 and Table 5-9, it is seen the CFC-QI -3 ce11 outperformed the
CFC-QI-2 cell by a considerable amount in tems of concentrate grade and recovery. For
the 30 minutes grind sarnple, in 3 minutes the CFC-QI-3 celi had recovered 87.6% Pb and
the concmtrate grade was 74.2% Pb. The CFC-QI-2 ce11 had recovered 88.4% Pb in 3
minutes but its concentrate grade was only 47.5% Pb. In 9 minutes, the recovery by the
CFC-Q 1-3 ce11 was increased to 95.6% Pb and the concentrate grade was 65.8% Pb. The
rewvery by the CFC-QI-2 ce11 was also increased to 90.7% Pb in 9 minutes, but the
concentrate grade was essentially unchanged (47.6% Pb).
The big improvements in the metallurgical performance in the CFC-QI-3 were
attributed to the change in the shape of the flotation "separator", the air flowrate, and the
slurry flowrate. The CFC-QI -3 uses a bowl instead of the drum in the CFC-QI-2. The
bowl has a smoother transition edge between the bottom and the wall, thus making the
discharge of the tailing more easily. The air flowrate is related to the intensity of aeration
in the flotation "reactor" (the sparger in the pipe). Its effect on the flotation process has
been discussed earlier. The slurry flowrate determines the slurry velocity at which it flows
through the sparger. The slower the slurry velocity, the larger the size of the bubble
sheared off the sparger surface. In the flotation tests of the 30 minutes grind sarnple, the
rotating speeds of the bowl in the CFC-QI -3 cell and of the drum in the CFC-QI-2 ce11
were the same. However, in the test work, the CFC-QI-3 ce11 used less air (4 UMin.)
compared to the 6 Urnin. used by the CFC-QI-2 cell. The sluny speed in the CFC-QI-3
ce11 was also slower than that in the CFC-QI-2 ce11 (1 -8 Vrnin vs. 2.8 Vmin). Both factors
should result in a significant reduction in the arnount of fine bubbles generated by the
sparger, which evidently improves the quality of the 60th product.
Comparing the results of flotation tests using the CFC-Q2- 1 ce11 (Table 5-10) and
the Denver D l 2 cell (Table 5-1 l), it is seen that the former generaily outperformed the
latter. In the case of 20 minutes grind, the CFC-Q2- I had recovered 75.1% Pb in 1 minute
and the concentrate grade was 82.6% Pb. The Denver D-12 ceil produced similar results
(749% recovery. concentrate grade 83.5% Pb). In 3 minutes, the recovery by the CFC-
42-1 ce11 increased to 94.0%, as compared to 89.7% by the Denver D-12 cell. As a result,
the concentrate grade dropped to 78.7% Pb (CFCQ2-1), as compared to 80.m Pb
(Denver D-12). At the end of 6 minutes flotation, the recoveries achieved by the CFC-Q2-
1 ce11 and the Denver D- 12 ceIl were close to identical (93.5% vs. 93.2%).
In the case of treating the 30 minutes grind sample, the CFC-Q2-1 ce11 floated
much faster with acceptable concentrate grades throughout the entire flotation process.
For the 60 minutes grind sample, the grade and recovery from the CFC-Q2-1 ce11 was
better than the Denver D-12 ce11 at initial 3 minutes (68.2% Pb with 76.9% recovery vs.
64.1% Pb with 46.3% recovety). At 6 minutes flotation, the flotation recovery was
essentiaily the same for both cells (81 -7% recovery vs. 82.W recovery). At 19 minutes
flotation, the Denver D-12 ce11 recovery was slightly better than the CFC-Q2-1 ce11
(92.5% recovery vs. 90.0% recovery).
The galena-silica flotation test results consistently show that one of the distinctive
advantages of the CFC-QI cell and the CFCQ2 ce11 over the Denver D-12 ce11 is the
faster rate of mineral flotation.
A substantiai improvement was also achieved in the quaiity of the concentrate by
the changes in the design and operating variables associated with the CFC-QI and CFC-
42 centrifuga1 flotation cells. Regiouping the data in Table 5-8, 5-9, 5-10, and 5-1 1 into
Table 5-12, 5-13, and 5-14, it is seen bat, while al1 three centrifuga1 cells o u t p e r f o d
the Denver D- 12 ceil with significantly higher recoveries,
1. CFC-Q-2-1 outperformed CFCQ-1-2 with higher concentrate grades and
recoveries in the flotation of 20, 30 minutes grind samples (Table 5-12 and
5-13). The results provide strong evidence that the design of the CFC-QZ-1
ce11 is superior to that of the CFC-Q 1-2 cell;
2. CFC-QI-3 outperformed CFC-QI-2 with a higher concentrate grade in the
flotation of 30 minutes grind sarnple (Table 5- 13). Possible reasons include
the different shape of the rotating vessel, a lower air flowrate, and a slower
slurry flowrate. The results indicate that several design and operating
parameters can significantly influence the performance of the CFC-Q 1 cell;
3. CFC-Q2- 1 outperformed CFC-Q 1-3 and Denver D-12 with both a higher
concentrate grade and a higher recovery in the flotation of 60 minutes grind
sample (Table 5-14). The results ciearly indicate the CFC-Q2-1 ce1 is
effective for fine particle flotation.
Table 5-1 2. Cornparison between CFC-Q 1-2, CFC-Q2- 1 and D- 12 (grind 20 min., float 3 min-)
Table 5-13. Cornparison between CFC-Q 1-2, CFC-Q 1-3, CFC-Q2- 1 and D-12 (grind 30 min., float 3 min.)
Cell
Mode! 1
QI-2 . Q2-1
D-12
Tabk 5-14. Cornparison between CFC-Q 1-2, CFC-QI -3, CFC-Q2- 1 and D-12 (grind 60 min., float 3 min.)
Test
No.
8
19
1
CeIl
Model
41-2
41-3 1
42-1
D-12
Feed
(Umin.)
2.8
2.8
O
Test
No.
7
6
20
2
Ceii
Mode1
4 1 - 3
Feed
(Urnin.)
2.8
1.8
2.8
-
Air
(Urnin.)
6
6
O
7
Air
(Urnin.)
6
4
6
Rotation
(rprn)
150
150
440
1800
Test
No.
7
Rotation
(rpm)
150
440
1800
Conc. Grade
(% Pb)
47.5
74.2
66.3
79.9
Feed
(UMn.)
1.8
1
Pb Recov.
(%) 1
88.4
87.6
90.7
69.8
Conc. Grade
(% Pb)
60.8
78.7
80.9
Air
(Umin.)
4
Pb Recov.
(%)
85.5 l
94.0
89.7
Rotation
(rpm)
150
Conc. Grade
(% Pb)
57.4
Pb Recov.
(%)
72.9
5.2.2. Results of Chalcopyt-ite Flotation
The artificial chalcopyrite-silica mixture at a head grade of 0.7% Cu was ground for 30
and 60 minutes. Potassium amyl xanthate (PAX) was used as the collecter and stage
added twice at a rate of 30 g/t and 20 g/t respectively. The flotation results fiom the CFC-
41-3 cell, the CFC-QZ-1 ceIl and the Denver D- 12 cell are summarized in Table 5-1 5 , 5-
16 and 5-17.
The 30 minutes grind samples were tested on the CFC-QI-3 ce11 and the CFC-Q2-
1 cell. The results in Table 5-15 and 5-16 show that the CFC-Q2-1 ce11 outperformed the
CFC-Q 1-3 by considerable amounts both in recovery and concentrate grade. In 3 minutes,
the CFC-QZ- 1 had recovered 85.3% Cu in a concentrate containing 17.1% Cu, while it
took the CFC-QI-3 five minutes to recover 68.7% Cu with a concentrate grade 12.8%
Cu. The gap was substantially widened as the flotation process continued. Mer 9 minutes,
the CFC-QZ-1 had recovered 92.8% Cu with a concentrate grade 17.29% Cu, as
compared to the results of the CFC-QI-3 which had only recovered 73.3% Cu in 15
minutes with a concentrate of lower grade (10.47% Cu).
Such a big difference in the flotation of chalcopyrite-siiica mixture may be
attributed to the air flowrate. The CFC-Q2-1 used 8 - 10 Urnin. air, which was twice the
arnount of air used in the operation of the CFC-QI -3 (4 Umin.). Note that chaicopyrite's
specific gravity is 4.1 - 4.3, while gaiena's specific gravity is 7.58. Therefore, chalcopyrite
is much more "lighter " than galena. The need for energy to compensate for their low
momentum increases. This can be done either by increasing the velocity at which the slurry
passes through the "reactor", or by using higher air flowrates which wiii increase the
turbulence inside the reactor. Thus in the flotation of chalcopyrite-silica mixture by the
CFC-Q 1-3 and CFC-Q2-1 cells, more air appeared to have resulted in good performance
by the CFC-Q2- 1.
Tabk 5-1 5. Chalcopyrite flotation with CFC-Q 1-3 (bowl speed 150 rpm, feed at 1.8 Ymin., air flow at 4 Umin.)
Float Test No. 8
1 Tai1 Assay ( O. I9 1 -
Test No. 9 1 60 min. Grind I
Conc. Grade, %Cu 1 Cu Recov, % 1
Table 5-16. Chalcopyrite flotation with CFC-Q2- 1 (feeder speed 440 rpm, feed at 1 -8 Umin., air flow at 8-10 Umin.)
1 minutes 1 Conc. Grade, %Cu 1 Cu Recov.% 1 Conc. Grade, %Cu 1 Cu Recov.,%
#
Test No. 22 4
60 min. Grind
Float
Time
Test Na25
30 min. Grind
19
Tail Assay
8.3
0.04
94.4
- 8.7
0.08
88.6 1 -
Table 5-17. Chalcopyrite flotation with Denver D- 12 (1800 rpm, air valve fiilly open)
1 Float 1 Test No.4
The 60 minutes flotation data were available for each of the CFC-QI-3, the CFC-
42-1 and the Denver D-12. The results in Table 5- 15, 5- 16 and 5- 17 indicate that, while
the Denver D l 2 ce11 outperformed the CFC-QI-3 ce11 by higher recoveries (70.9% as
compared to 59.2% in 9 minutes flotation) and much better concentrate grades (9.27% Cu
as compared to 2.82% Cu), the CFC-Q2-1 ce11 outperformed the Denver D-12 ce11
substantiall y.
Time
minutes
For a 60 minutes grind sample, in 9 minutes, the CFC-Q2-1 produced a
concentrate with a grade at 11 -49% Cu and the resulting recovery was 82.5%, while the
concentrate fiom the Denver D-12 was 9.27% Cu with the recovery of 70.9?/0. As the
notation time was extended to 19 minutes, 88.6% of copper was recovered fiom the CFC-
42- 1 and 79.4% of copper recovered from the Denver D- 1 2.
60 min. Grind
Conc. Grade, %Cu Cu Recov.%
Results of Gaspe Copper Ore Flotation
One-kilogram charges of Gaspe copper ore sample were ground for 30 and 60 minutes.
notation tests were carrieci out with the CFC-Q 1-3, the CFC-Q2-1 and the Denver D-12.
Potassium amyl xanthate (PAX) was used as collecter.
The results in Table 5-1 8 and 5-20 show that the CFC-Q1-3 ce11 ha the best copper
recovery in the fine grind. For 60 minutes grind sarnple, the CFC-QI-3 ce11 recovered
8 8 . m copper at 19 minutes flotation, while the Denver D-12 ce11 recovered 86.1% under
the same conditions.
The results in Table 5-19 and 5-20 indicated that the CFC-Q2-1 ce11 has fister
notation kinetics at initial (3 minutes) stage. But, the flotation performance of the Denver
D-12 cell was better than that of the CFCQ2-1 ceIl &er 6 minutes flotation.
Table 5-18. Gaspe ore Rotation with CFC-Q 1-3 (bowl speed 125 rpm, feed at 1.8 Umin., air flow at 6-7 Urnin.)
Test No. IO Float
Time
minutes
5
Test No. 1 1
30 min, Wnd 60 min. Gnnd
Conc. Grade, %Cu
7.63
Conc. Grade, %Cu
3.19
Cu Recov.%
66.1
1
Cu Recov.%
64.4
Tabk 5-19. Gaspe ore flotation with CFC-Q2-1 (feeder speed 440 rpm, feed at 1.8 Urnin., air flow at 8-10 Umin.)
Float 1 Test No.24 I Test No. 21a
Time
&nutes
Table 5-20. Gaspe ore flotation with Denver D- 12 (1 800 rpm, air valve fully open)
Tai1 Assay
1 Float 1 Test No.6 1 Test No. 5 1
30 min. (Jrind
Conc. Grade, %Cu 1 Cu Recov.,%
1 Time 1 30 min. Grind I 60 min. Grind 1
60 min, Grind
Conc. Grade, %Cu ( Cu Recov.,%
O. 10
1 minutes 1 Conc. Grade, %Cu 1 Cu Recov.,% 1 Conc. Grade, %Cu 1 Cu Recov.% 1
-
5.2.4. Results of Lead-Zinc Ore Supplied by Cominco
6
9
19
Tai1 Assay
One kilogram Comùico ore was ground in a 20cm x 35cm Iaboratory rod miii at 60%
solids for 20 minutes with 2 kg/t soda ash, 400 glt N a m 800 g/t ZnSo4 and 800 g/t
0.23 -
8.48
5.94
5.02
0.04
90.4
93.6
95.6
-
5.64
3.95
3.16
O. 13
66.6
80.3
86.1
-
Cao. The ground slurry was pulpeci to 200/0 solids with pH adjusted to 9.5 by lime. The
slurry was then conditioned with 10 g/t NaCN and 20 glt sodium ethyl xanthate (SEX) for
3 minutes, followed by lead (Pb) flotation (7 minutes in the Denver ceIl and 32 minutes in
the CFC cells - due to the sluny flowrate used in the CFC units, the 32 min. flotation time
produces an actud retention time in the flotation ce11 of about 7 min.).The lead rougher
tailing was adjusted to pH 11.0 with lime and conditioned with 5 kglt CuS04 and 50 g/t
sodium isopropyl xantbte (SIX) for 5 min. Zinc (Zn) flotation was conducted for 26 min.
in the CFC cells and 6 min. in the Denver cell.
The operating conditions for the CFC-Q 1-3 celi and the CFC-Q2- 1 cell were: the
speed of rotating bowl 200 rpm, the rotating pipe speed 440 rpm, 1.8 literdmin. feeding
rate, 20% solids, 5-7 literslmin. air flow rate.
The results (Table 5-21, 5-22, 5-23) show that the Denver D-12 ce11 gave better
Pb flotation while both CFC cells produced a better Zn flotation. The overali Pb recovery
was 94.5% corn the CFC-QI -3 cell, 94.8% from the CFC-Q2-1 cell, and 97.0% corn the
Denver D- 12 cell. The overall Zn recovery was 96.0% from the CFC-Q 1-3 cell, 93 -5%
from the CFC-Q2- 1 cell, and 92.9% €rom the Denver D-12 cell.
The Pb concentrate grade was higher fiom the Denver ce11 (32.9%j and lower
from the CFC cells (23%). The Zn conc. grade was higher fiom the CFC cells (28.9% - 36.1%) and lower from the Denver ceIl (24.8%).
Table 5-21. Flotation o f Corninco ore with CFC-Q 1-3 (bowl speed 150 rpm, feed at 1.8 Urnin., air flow at 5-7 Umin.)
Test No. 14
Product
Pb Ro. Conc.
Table 5-22. Flotation of Corninco ore with CFC-Q2-1 (feeder speed 440 rpm, feed at 1.8 Umin., air flow at 5-7 Urnin.)
ZnRo.Conc.
Ro. Tailing
Calc. Head
Wt,%
18.17
13.80
68.03
100.00
Test No. 27
Table 5-23. Flotation of Cominco ore with Denver ce11 (1 800 rprn, air valve hlly open)
L
Product
PbRo.Conc.
Zn Ro. Conc,
Ro. Tailing
Calc. Head
1 TestNo.8 1 1 Assay 1 % Distribution 1
A S ~ Y
3.54
0.41
5.07
Assay
%Pb
23-70
% Distribution
% Distribution
Wt, %
18.65
1 1.23
70.1 1
100.00
%Zn
16.09
Pb
84.88
28.93
0.42
7.20
Product 1
Zn
40.61
9.62
5.50
100.00
% Pb
23.23
4.53
0.38
5.1 1
Wt, %
55.43
3.97
100.00
%Zn
15.91
36.14
0.70
7.52 &
Pb
84.82
9.96
5.52
100.00
Pb Ro. Conc.
Zn Ro. Conc.
Ro. Tailmg
Calc. Head
Zn
39.47
54.00
6.53
100.00
%Pb
14.27
17.64
68.09
100.00
%Zn
32-86
1.80
0.23
5.16
Pb
17.03
24.83
0.76
7.33
1
Zn 1
90.82
6.15
3 .O3
100.00
33-17
53.77 1
7.06
100.00
5.3. Discussion
Further methods of analysis were also ernployed to compare the flotation performance
between the CFC-Q1 cell, the CFC-Q2 ce11 and the Denver D-12 celi. Several size
fractions of the concentrates and tailings were assayed for metal content and the metal
recovery by size was determlned. As well, conventional GradefRecovery relationships
were also developed for cornparison of the diferent ceIl performances.
5.3.1. Size Fraction Recovery
Fig. 5-1 shows the lead recovery by size from the galena-silica mixture. It is obvious that
the galena particles coarser than 10 Pm are recovered better fiom the Denver D- 12 cell.
But, Fig. 5-2 indicates that the recovery of the galena particles finer than 10 p are better
fiom the CFC-QI-3 and CFCQ2- 1 cells than fiom the Denver D- 12 cell.
Fig. 5-3 and Fig. 5-4 show the copper recovery frorn the size fiactions of
chalcopyrite-silica mixture. The results show that the copper particles coarser than 60 pm
are recovered better €tom the CFC-QI-3 ce11 and the CFC-Q2-lcell than fiom the Denver
D- 12 celi. A h , the flotation recovery of copper particles finer than 10 pm are better Erom
the CFC-QI-3 cell than fiom the Denver D-12 cell. Fig. 5-5 shows the copper recovery
fiom the size fiactions of Gaspe copper ore sarnple. For the flotation recovery of finer
than 10 pm particles in these conditions, the CFC-Q 1-3 ce11 did better than the Denver D-
12 cell, but the Denver D-12 ce11 did better than the CFC-Q2-I cell.
5.3.2. G rade-Recovery Relatioasbip
Fig. 5-6, Fig.5-7, ând Fig. 5-8 show the grade-recovery relationship for the flotation of the
galena-silica mixture, the chaicopyrite-silica mixture, and the Gaspe copper ore. Usually
the curve in the upper part of the graph gives the better flotation performance than the
curve in the lower part of the graph.
Fig. 5-6 shows the galena flotation in the CFC-QI-3 ce11 follows the sarne curve as
the Denver D-12 cell initially, but continues on to produce higher recovery values. The
galena flotation in the CFC-Q2-I ce11 is not as good as in the Denver D-12 cell. The
galena flotation perfomance is the worst in the CFC-Q 1-2 cell.
Fig. 5-7 shows that CFC-Q2-1 ce11 produced the much better copper flotation than
the Denver D-12 ce11 from the chalcopyrite-silica mixture, showing both higher grades at
equivalent recovery and also higher overall recovery. The CFC-QI-3 ce11 gives the wora
copper fiotation results. The copper flotation in the Denver D-12 ce11 is just between the
CFC-Q2- 1 and CFC-Q 1-3.
Fig. 5-8 shows the copper flotation of Gaspe copper ore, which indicates that the
flotation in the Denver D-12 ce11 outperforms the CFC-Q2- 1 ce11 and the CFC-QI-3 cell.
+ QI-3 -t Q2-1 4 - Denver
Particle Size, um
Figure 5-1. Lead recovery from size biens (galena-silica mixture)
-
+ QI-3 - Q2-1 + Denver
Particle Size, um
Figure 5-2. Lead recovery iiom size t'ractions (galena-silica mixture)
4 QI-3 + Q2-1 + - Denver
Figure S3. Copper recovery fiom size fiactions (chalwpyrite-silica mixture)
Figure 5-4. Copper recovery from size fiactions (chdcopyrite-silica mixture)
Particle Size, um
Figure 5-5. Copper recovery front size fiaaions (Gaspe copper ore)
+ QI-3
6 42-1
-F Denver
Figun 5-6. Grade-recovery relationship for galena-silica mixture (30 min. grind)
I l l I [ l l r l ~ r l r ( - t 1 1 ï 1 1 1 1 K I T I I T I I - - - - - - - - - - - - - œ
- - - - a
- - *
- - C1
r -
e
w
L
C
d
C Denver d
b
- - - ; 3 - - - - - - - - œ - - - - - - C
w
L
- - - - - - - w Q1-3 9
- - a
- - C
- m
I
- - 1 1 1 1 1 1 1 1 1 1 1 1 I l l 1 1 1 1 1 1 1 1 1 I l f i l -
9
% Cu Recovery
Figure 5-7. Grade-recovery relationship for chalcopyrite-siiica mixture (30 min. grind)
F u 8 . Grade-recovery relationship for Gaspe copper ore (60 min. grind)
Chapter 6
Hydrodynamics of Centrifuga1 Flotation Cells (CFC)
6.1. Guidelines for Modeling the Flotation Process in CFCs
The historical evolution of centrifuga1 flotation started at the University of Utah. In
particular there was, firstly the air-sparged hydrocyclone dunng the 1980's. then the cyclo-
column cell, the Clean Earth Technologies ceIl during the nineties, up to the new designs
CFC-QI and CFC-Q2. As seen in Chapter 2, the type of flow occurring in a centrifugai
tlotation ce11 depends on the way that rotating fluids originate and the geometry of the
vessel. Without doubt, the complex rotating flow features generated within the ce11 play a
very important role in the flotation process carried out in this device. Therefore, to permit
the rational design of a centntùgal flotation machine, it is important that the velocity
characteristics of the flowfield can be measured instrumentally or predicted through
numencal solution of the Navier-Stokes equations and the continuity equation.
Regardless of many subtle fluid-structure interactions that produce vorticity and
secondary flows, the rotating fluid is the principal phenornena of interest. In order to
evaluate correctly the influence of this rotating fluid on the flotation process, it is
necessary to retum to the fundamentai theory of flotation, specificaüy, the physico-
chernical elementary processes.
In Chapter 5, the metallurgical peflomance of CFC-QI and CFC-Q2 has been
presented. It was generaiiy observed that the flotation recoveries by both models were
higher and faster than that by the Denver D-12 cell. In many cases the concentrate grades
produced by these two models were also higher than that by the Denver D-12 cell. The
size -ion recovery results showed that the recovery of fine particles less than 10 pm by
CFC-Q1 and CFC-Q2 were higher than that by the Denver D-12 cell. The flotation test
results provide indirect but strong evidence that the centrifuga1 force created by the
rotating fluid can influence the hydrodynamic particle-bubble interaction in flotation.
Flotation is a rate process accomplished in a sequence of subprocesses (or
microprocesses). Trahar and Warren (1976) suggested that flotation be divided into four
main steps and each of these four steps be further subdivided, as shown in Table 6-1, to
give a total of ten subprocesses. Among them Step n, bubble-particle coilection (or
attachent), is the central process in flotation compnsing collision, adhesion and
detachment subprocesses. Each has a probability associated with them, namely, collision
probability P,, adhesion probability P. and detachment probability Pd. It is generally
accepted that these three probabilities sufficiently descnbe the probability of a particle
being collected by an air bubble in the pulp phase of a flotation ce11 in the following way:
where P, is the probability of a successfiil collision, P. is the probability of a collided
particle attaching (adhering) to a bubble, and Pd is the probability of a particle being
detached fiom a bubble. For fine particles, Pd can be negligibly small because of the low
inertia, in which case Eq. (6.1) becomes (Yoon, 2090):
In generai, the probability is defined as the ratio of the real to ideal rate of the
respective microprocess (Nguyen, L 999).
Table 6-1. A scheme of sub-processes in Botaiion (Trahar and Warren, 1976)
Step 1. The introduction of feed materials Introduction of pulp Introduction of air
Step II.
Step DI.
Step IV.
The bubble-particle collection Collision between particles and bubbles Attachent of collided particles to bubbles Detachment of attached particles fiom bubbles
The transport processes between pulp and froth Transport of mineralised bubbles into froth Direct entrainment of particles into 60th Retum of particles from froth to pulp
The removal of flotation products Removal of fioth Removal of tailings
The probability P of floating a particle is directly related to the first order flotation
rate constant k by the following relationship (Ralston et ai.. 1999a; Yoon, 2000):
where Q is the gas volumetric flow rate, Vr is the reference volume of height h through
which bubbles of diameter db rise, A (=VJh) is the cross-sectional area of the flotation cell,
and V, (= Q/A) is the superficial gas rate.
In Eq. (6.3), Q and db are the operating variables that can be predetermined, Vr, h
and A are the geometrical dimensions of the flotation cell. Therefore, if P can be
calculated from the detailed knowledge of its subprocesses, then the rate of flotation can
be predicted. Suppose that in a batch flotation ce11 the rate of flotation is studied with
time. Let C (mass per unit volume) be the concentration of floatable minera1 remaining in
the pulp at time t, the rate of flotation is then given by:
where k is a 'rate constant'. Integrating between the limits O and t:
where Co is the initiai concentration of the minerai. Further the cumulative recovery R
after time t cm be predicted by:
Thus the probability P is connected with the recovery of particles. Obviously If the
probability P could be increased by conducting flotation in centrifuga1 force fields or using
fine bubbles, then the recovery of particles should be improved. Besides, to permit a
rational scale-up of a flotation machine such as centrifuga1 flotation cells, it is crucial that
the rate of flotation can be predicted by the equations above.
6.2. Physical Model for Bubble-Particle Attachrnent Process
In order to calculate P. and Pa, the physical mode1 for the bubble-particle attachent
process must be established first. Froth flotation may be simplified as the relative motion
of a single particle and a single bubble. The two spheres are of matkedly different densitks
and they move in opposite directions. The approach of the minera1 particle to the bubble
surface is regardai as taking place in three stages corresponding to movement of the
particles through three zones (see Fig. 6 1 ) which are mainiy characterized by the kinds of
forces involved (Derjaguin and Dukhin, 1960):
Figure 6-1. Zones of interaction forces between the bubble and the pariide suggested by Deryaguin and Dukhin (Ralston et al., 1999b)
Zone 1: The region far fiom the bubble surface where particles are abject
to viscous, inenid and field forces (gravitational, centnfugal). The
presence ofa bubble can act on a mineral particle in this zone ody
through hydrodynamic effects, due primarily to distortion of the
liquid streamlines flowing past the bubble. Anaiysis of the
movement of the particles under the action of these forces gives the
collision efficiencyl (E or P,), which is defined as the ratio of the
number of particles in the path of a bubble that actually collide with
it to the number that would collide if the fluid streamlines were not
diverted by the bubble.
Zone 2: The region of the diffusion boundary layer of the bubble, caused by
the liquid flow around each bubble which disturbs the equilibnum
distribution of adsorbed ions there. Charged mineral particles
entering this layer would expenence specific forces (termed
'difisiophoretic forces') similar to electrophoretic forces and
would be either attracted towards, or repelled fkom the bubble
surface. However, the existence of the 'difisiophoretic forces' is of
a purely hypothetical nature (Schulze, 1984) and has not been
confirmed yet (Ralston et ai., 1999b). For this reason, they have
been generaily excluded from the study of bubble-particle
collection.
The terms "efficienq" and "pmbability" are used synonymousiy and altemately in the literature.
Zone 3 : The thin wetting tüm in which surface forces are operative. The
main components are van der Waals, electrostatic and structural
forces. These forces may accelerate, or slow d o m and even
prevent the thinning of the liquid film between the particle and the
bubble. Among them, the van der Waals and electrostatic forces are
hydrophilic and thus repulsive, except in rare circumstances
(Ralston et al., 1999b). The structural force, which was onginally
considered as 'hydrophilic or repulsive force' by Dejaguin and
Duhkin, may in fact be the 'hydrophobie or attractive force' that
can accelerate the thinning of the wetting film between the particle
and the bubble (Yoon, 2000). Analysis of the interparticle
interaction in this zone gives nse to the attachment efficiency (E. or
Pa), which rnay be defined as the ratio of the induction tirne to the
sliding tirne. The induction time is nonnally taken as the time
required for bubble-particle attachent to occur, once the two are
brought into contact. The sliding time is the time taken for the
particle to slide around the bubble surface, until it moves away fiom
the bubble surface (Ralston et al., 1999b). If the sliding time is
longer than the induction time, the particle wii! have long enough
contact time to thin and rupture the disjoining film between the
particle and bubble (Yoon, 2000).
The estimation of collision efficiency E. for conventional flotation has k e n well
addressed in the literature and some results have received general acceptance. The
subprocess of adhesion, on the other hand, is least understood because it is essentially
controlled by the surface chemistry of the syaem, which is complex and difficult to mode1
mathematicdiy (Yoon, 2000). Adhesion efficiency models that have been developed so far
are al1 based on the relative magnitude of the induction time and the sliding time and have
been cnticized for neglecting the fundamental issues of thin film drainage and three phase
contact line (TPCL) movement which are the major contributors to E, (Ralston et al.,
1999b).
Particle Trajec tory Equation
In calculating the collision efficiency of small particles with air bubbles, the particle
trajectory equation must be established first. We stan from the motion of a small solid
particle in a fluid at rest, adding complications step by step until we reach the general
equation that describes the motion of a solid particle translating in a uniform unaeady
fluid flow. For simplicity, we assume the solid particle (and later, an air bubble) is
spherical throughout t his chapter.
A solid particle of mass m, falling through a fiuid at rest under the action of gravity
has several forces acting on it:
- The gravitational force (F,)
where V, is the volume of the soiid particle, p,, is the density of the sotid panicle,
& is the radius of the solid particle, and g is the acceleration due to gravity.
- The buoyant force (F'b), antiparallel to the gravitational force, arises fiom the fact
that the pressure in a fluid increases with depth.
where Vp is the volume of the solid particle (and hence the volume of the fluid
displaced by the solid particle), and pf is the density of the Buid.
- The hydrodynamic drag force (Fd), a hydrodynamic force due to the viscosity of
the fluid and also, at high speeds, to turbulence behind the moving solid particle.
For a non-rotating solid particle the drag force (Fd) is given by (Schulze, 1984):
where Cd is the drag coefficient, 4 is the cross-sectional area of the particle, pr is
the density of the fluid, Up is the particle velocity vector, Uf is the velocity vector
the fluid would have at the position of the solid particle if no soiid particle were
there. For our present case, Ur is zero.
The drag coefficient (Cd) is strongly dependent on the Reynolds number of the
solid particle @+), which is defined to characterize the motion of the solid particle
relative to a fluid:
where dp is the diameter of the solid particle, pf and p are the density and wscosity of the
fluid, and U is the solid particle's velocity relative to the fluid. This Reynolds number ,
R%, must be clearly diainguished fiom Reynolds number of an air bubble, Ra, although
the form is similar:
db ' ~ f Re, = - CL
where db is the diameter of the air bubble and U is the air bubble's rising velocity relative
to the fluid.
When the Reynolds number of the particle Re,, < 0.5 - 1, the fiow around the
settling solid particle in an otherwise stationary fluid is essentiaily laminar, and the drag
coefficient Cd = 24 / R+ hence
which is known as Stokes's Law.
Working in an inertial h e of reference in which the fluid far away from the
moving solid particle is and remains stationary at al1 times. we apply Newton's second law
for the motion of the solid particle, stating that the rate of change of linear momentum of
the solid sphere balances the vector sum of the gravitationai force (FE), buoyant force
(Fb), and drag force (Fd):
When the solid particle reaches its temiinal velocity (U:), the acceleration is zero
so Eq. (6.1 3) becomes
Rearranging, we obtain the terminal velocity (UT) of a solid particle settling
under the action of gravity in an infinite fluid at rest:
This terminal (settling) speed of the solid particle, UT, divided by the termimai
(rising) speed of an air bubble, Ue, is equal to an important dimensiodess parameter G:
Eq. (6.7) can also be written in the form
du* The vector -m, - is referred to as the inertiai force vector. If it is treated in dt
the same way as a "force vector", then the state of "equilibnum" created is referred to as
dynamic equilibrium. This method for application of the equation of motion is ofien
referred to as the D' Alembert principle, named after the French mathematician Jean le
Rond d'Alembert (Hibbeler, 1995). Note that Eq. (6.16) is still written in an inertial fiame
of reference.
Sometimes it is convenient to place ourselves (in theory, if not physically) into a
reference h e that accelerates linearly or rotates. This type of reference frame is not an
inertial reference fhne in which Newton's laws of motion do not hold. However, working
in a noninenial reference fiame, we can still apply Newton's Iaws by introducing some
pseudoforces or fictitious forces which are ofien called inertial forces too. In general, in a
noninertial reference fiame we can write Newton's second Iaw as
where is the sum of al1 real forces and m, is the sum of the inertial forces.
In a linearfy accelerating reference frame with constant acceleration a, the inertial
force (Fi ) acting on a moving body (solid particle) is given by
Substituting Eq. (6.18) into Eq. (6.17) and rearranging, we find
In a reference frame rotating at constant angular velocity R , there are two inertial
forces which act on a moving body (solid particle). One is the well-known "centrifbgal
forceT' which acts outward, the other is known as the "Coriolis forceT' which acts to deflect
the body sideways:
where r and Up are position and velocity of the body relative to the noninertial fiame,
respectively. Note that the Coriolis force acts on a body in a rotating system only if the
body is moving.
Substituting Eq. (6.20) and Eq. (6.21) into Eq. (6.17) and rearranging, we find
Thus, we can see that so-called inertiai forces, if shified to the lefi-hand side, can
be interpreted as parts of absolute acceleration of the body moving in a noninertial
reference fiame. Note that Eq. (6.19) and Eq. (6.22) can be rmanged into the "dynamic
equilibrium" form like Eq. (6.16) by shifting everything on the lefi-side to the right-side
and interpreting them as inertial forces. Therefore, we can say that inertial force exists in
inertial reference h e s or noninertid reference fiames.
We concentrate now on the force exerted on a solid particle that translates without
rotation with time-dependent velocity Up(t) in an othedse quiescent ideal fluid (and
hence viscosity p = O) of infinite expanse, in the absence of any extenor or interior
boundaries. In general,
where p is the pressure in the fluid, a is the unit normal vector pointing »ito the fluid at the
particle surface, and the integral is to be evaluated over the surface of the particle.
Using Green's theorern and Bernoulli's equation, Eq. (6.23) obtains the following
form (Po~kidis, 1 997):
where plis the density of the fluid, V, is the volume of the soiid particle, U, is the velocity
of the solid particle, and a is the coefficient of Wtual inertia or added mass.
The two terms on the nght-hand side of Eq. (6.24) represent, respectively, the
adrhd mars fwce, and Archimedes' buoyantforce. Physically the added mas force results
from the fact that in an acceleration not only the particle but together with it a certain
mass of fluid adjacent to the particle has to be accelerated. In the case of a spherical solid
particle, it is equal to the force necessary to accelerate half the arnount of fluid displaced
by the solid particle.
Next we consider the force exerted on a solid particle that is held stationary in an
incident time-dependent uniform potential flow (and hence viscosity 1 = O) with velocity
Udt). If we work in a frame of reference in which the Far flow is and remains stuionary at
al1 times (and hence it is a noninertial reference fiame), then the solid particle appean to
translate with velocity U,(t) = - Udt) in an otherwise quiescent fluid. Adding to the right-
hand side a ficticious inertial force due to the distnbuted body force field per unit mass
To cornpute the force exerted on a body that translates with velocity Udt) and
without rotation in a uniform unsteady potential flow (and hence viscosity p = O) with
veiocity Udt), we still choose a h e of reference in which the far flow is and rcmains
stationary at ali times (and hence the h e of reference is noninenial), in which case the
solid particle appears to translate with velocity Up(t)- Udt) in an othenuise quiescent fluid.
Repeating the arguments preceding Eq. (6.25). we find (Pomkidis, 1997)
Adding the gravitational force and the drag force, we obtain a more general
particle trajectory equation
Forces = Gravity + Drag + Added Mass + Inertid + Buoyant
Finally there exists a Basset force F- that acts on a solid particle moving in a
turbulent flow. Known as the "hiaory" term, the Basset force takes into account the effect
of the deviation in the flow pattern from steady state. Physically the Basset force
constitutes an instantaneous flow resistance due to the energy expended in setting the
liquid itself in motion and is given by (Soo, 1967; Schulze, 1984; Nguyen, 1999).
where ( t - t ) is the time elapsed since the past acceleration
Adding the Basset force into Eq. (6.28). we obtain a general particle trajectory
equation known as the extended Basset-Boussinesq-Oseen (BBO) equation, modified by
Tchen (Soo, 1967; Schulze, 1984; Nguyen, 1999):
The simplifjing assumptions made by Tchen in various stages of formulation and
solution are: 1. the particle is sphericai and is so smail that its relative motion to the fluid
gives rise to resistance according to Stokes's law; 2. the particle is small when cornpared
to the smailest wavelength of the turbulence; the Magnus effect' of particle motion due to
shear flow is neglected; 3. the flow field is not perturbed by the presence of the solid
particle; 4. during the motion of the solid particle, the sarne fluid element remains in its
neighborhood ; 5. the turbulence of the fluid is homogeneous and steady; 6. the domain of
turbulence is infinite in extent (Soo, 1967). Moreover, any particle-particle interaction or
particle-wall (of a bubble surface) interaction, which is characteristic of particle collection
in flotation, are neglected in the BBO equation and its modification by Tchen (Nguyen,
2999).
' Magnus effkt refers to a phenomenon encounrered in a flow over a sphere in a tube of finite radius. Rotation of a solid particle may arise due to the presenœ of a velocity gradient in the fluid such as rhe shear Iayer near a wall. At low Reynolds numbers, rotation causes fluid entrainmen& increasing the fluid velocity on the one side of the body and lowenng the Quid velocity on the other side. The particle tends to move towatd the region of higher fluid velocity (Soo, 1%7)-
Solutions to Eq. (6.30) are not easy to obtain. in developing the theory to deal
with particle-bubble interaction in flotation, certain assumptions and modifications of Eq.
(6.30) are necessary. For exarnple, we may neglect the integral (or "hiaory") terrn in Eq.
(6.30) since it becomes substantiai only when the solid particle is accelerated at high rate,
we may mode Eq. (6.30) by neglecting the added mass and the integral, or we may
ignore the pressure gradient effect of the tiuid acceleration (the/icticious inertial force) in
a steady flow, in addition to neglecting the added mass force and the integral. Regardless
of the particular approximation adopted, the liquid velocity Ur around an air bubble in
flotation must be soived first.
6.4. Liquid Velocity around an Air Bubble in Flotation
During the particle collection process in fioth flotation, both particles and bubbles rnoving
in the fluid develop some types of flow field around their contours. Since the bubble is
much larger than the particle, especially in the case of fine particle flotation, consideration
of the interaction between flow fields of particle and bubble is not necessary and the fluid
flow pattern is charactenzed by the flow around the bubble.
In theory, the continuity equation (the mass conservation) and the Navier-Stokes
equation (the momentum conservation) together with appropnate boundary conditions
completely specify the velocity field due to a nsing air bubble. To descnbe liquid flow
around an air bubble, it is convenient to use a rotationally symmetric (axisymmetric)
spherical coordinate system (Fig. 6-2) moving with the terminal velocity (Ue) of the
bubble and having its ongin at the center of the bubble, in which case the bubble appears
to be held stationary in an incident flow with velocity UB:
Figure 6 2 . Notations for liquid flow around an air bubble (Nguyen, 1999)
Wtth the notation given in Fig. 6-2, the axisymmetric condition yields d I &p = 0.
Hence, the velocity field about the bubble under steady-state conditions is descr i i by the
following simplified differential equations (Nguyen, 1999):
The continuity equation:
Navier-Stokes equations:
where the velocities and the radiai distance are made dimensionless by dividing with the
terminal velocity (Ue) of the bubbie, and with the bubble radius &), respectively. 8 is the
polar angle measured fiom the frontal stagnation point. P is the dimensionless pressure,
made by dividing the difference between the dynamic and static pressure with the pressure
head, (l/2)paa where pf is the liquid density. Re = 2U&, pf I p and is the bubble Reynolds
number, p is the liquid viscosity. A is the Laplace operator, which is defined in the
axisymrnetric spherical CO-ordinate system (r, 0) by
1 d A = $4[ri $1 + r- Gr r' sine i3l Ce
To simplifi the solving process, a Stream hnction Y is introduced. In a two-
dimensional flow, the lines of constant Y are the streamlines, and the difference between
the numerical values of two streamlines is equai to the flow raie between the streamlines.
The strearn fiindon Y is a scalar iùnction which is related to the fluid velocity in the
rotationally symmetric flow by :
where y is made dimensionless by dividing with U , R ~ . Substituting Eq. (6.34a) and Eq.
(6.34b) into Eq. (6.3 1), Eq. (6.32a) and Eq. (6.32b), yields:
E'W = r sine
where is the dimensionless vorticity made by dividing with U A .
J is the Jacobian operator defined as
(6.3 Sa)
(6.35b)
and E* is the differential operator defined as
ô' . si:0 8 { 1 ô } E' E-
Cr' r- iB sineCl3
The generic definition of vorticity, 6 = V x U, , under the conditions of rotationdly
symmetric flow, simplifies to
The partial differential Eqs. (6.35a) and (6.35b) can be solved for Stream funaion
and vorticity. Analytical solutions to these equations are available for Re -. O and Re -.
m. For the Reynoids number of the intermediate range, both Eqs. (6.35a) and (6.35b) have
to be simultaneously solved by a numerical method (Nguyen, 1999).
The Stream fiinction Y may take different forms, depending on the value of the
Reynolds number (Reb) of the bubble defined as
where db is the bubble diameter, pf is the fluid density, p is the dynamic viscosity of the
fluid and Ue is the terminal velocity of rising bubble. At hi@ Reynolds numbers (of the
bubble), the magnitude of the teminal velocity is given by (Pozrikidis, 1997, p.334):
For the Stokes regime, Ue d i (Flint and Howarth, 1971).
According to Zhang and Finch (1999), a free-rising single bubble experiences three
stages: acceleration to a maximum followed by deceleration to a constant velocity. For
pure water, the maximum can be equated with the terminal velocity. For surfactant
solutions, the terminal velocity is equated with the constant velocity. Table 6-2 and 6-3
summarize their experimentally measured bubble terminal velocities in pure water and
sufiactant solutions.
Table 6-2. Estimated bubble terminal velocity in pure water for db = 1.5 mm (based on maximum velocity in profile for given media) (Zhang & Finch, 1999)
Temperature, OC Medium UB, crnis
20 Tap water 37.0 (0.7)'
30 Tap water 37.7 (1.7)
DF 250 solution (0.06 ppm) 37.6 (0.2)
Ethanol solution (17 ppm) 38.4 (0.7)
KCl solution (10 ppm) 37.6 (O. 1)
*Standard devïation in parenthesis
Tabk 6-3. Bubble terminal velocity for db = 1.5 mm in sufiactant solution at 3 0 O C
Surfactant UB, c d s (M.5)
DF 250 (= 30 ppm)
MIBC (= 30 ppm)
Pine oil (= 30 ppm)
DF 1263 (= 30 ppm)
Ethanol (= 460 ppm)
Octanol (= 30 ppm)
Dodecylamine (= O. 5 5 ppm)
Triton X-100 (= 0.5 ppm)
The relationship between the bubble diameter and the corresponding Reynolds
number of the bubbfe is summarked in Table 6-4.
Table 6-4. Fiow regime around an air bubble (Jiang and Holtham, 1986)
Bubble diameter, pm Reynolds number Flow regime
< 80 < 0.2 Very low Ra 80 - 260 0.2 - 5.0 LOW Reb
260 - 1000 5.0 - 100.0 Intermediate Ra > 1000 > 100.0 High Reb
Consideration of the range of Ra from zero to infinity separates fluid flows into
manageable regions for analysis (Vanyo, 1993). Consider
- Ra is qua1 to zero if pt = O or db = 0, but then there is no problem. Reb = O also
if U = O or if p + a. If U = O the fluid analysis is a hydrostatic problem, and if p
+ a, the material is a rigid body.
- When Ra « 1, the Bow pattern is well known as the Stokes flow and the
character of the steady Bow of a Iiquid past a fixed spherical obstacle (Le., the
bubble in our case) is most concisely expressed in terms of Stokes' stream function
y (in sphericai polar coordinates ) (Lamb, 1932; Jiang & Holtham, 1986):
- The case Ra - 1 represents many real flows in mechanically agitated ceiis. Exact
analytic solutions are impossible to obtain, but certain approximate solutions can
be generated as a trial strearn fiinction. The trial solution, such as the one derived
by Kawaguti (1955), presents as a polynomial fonn with undetermined constants
that may be determined by variational or error-distribution methods (Jiang and
Holtham, 1986; Nguyen, 1999):
is used if Reb < 2.0 and ~2 is used for R a in the range of 10-80. Constants Ai
and Bi are determined by satisQing boundary conditions and the Navier-Stokes equations.
Yoon and Luttrell (1989) also derived a Stream function for the intermediate
Reynolds number range (Yoon, 2000):
- When Ra » 1, the potentid flow (see below) can be a useful reference point
for real Buid flows. The 80w region around an air bubble may be divided into three
distinct regions. At high Reynolds number, far ftom the bubble surface the flow is
essentially ideal, with viscosity insignificant and thus can be treated as the potential
flow. Near the bubble the fluid develops a boundq layer where viscosity and/or
turbulence is important. Behind the bubble a wake develops and is generaily a
region of high turbulence and low pressure. For the study of three microprocesses
(coiiision, attachent and detachment), we are concemed only with the flow fields
in the forward region ahead of the bubble equator. The region behind the bubbie,
as related to fiotation, is significant for the research of fine particle entraiment due
to the vortices fonned in the wake (Nguyen, 1999).
- Reb can approach infinity if pf , Rt, , or U approach infinity, none of which are
redistic, or if p = O. The case p = O represents an ideal or prefect fluid and if only
conservative forces exist and the fluid density is constant, the flow is known as
potential fiow. The stream fùnction for inviscid liquid flows past a single sphere
(i.e., the bubble in Our case) in an unbounded volume is given by (Po~kidis,
1997):
Once a particular flow pattern around the bubble has been specified, the liquid
velocity components can be obtained fkom the seiected strearn fùnction, thus the particle
trajectory equations become uniquely specified. We are then ready to calculate the
probabilities of microprocesses (collision and adhesion).
6.5. Probability of Collision
6.5.1. Collision Efficiencies in Gravitational Force Field
The nature of the collision is given by the collision efficiency E, (or probability of collision
P,), which is defined as the number of particles that actually collide with the bubble
divided by the number of particles that would collide if the fluid streamlines were not
diverted by the bubble. Assume that particle concentration (particle number per unit
volume) is uniforni, then Ec is equal to the ratic of the cross-sectional area of the original
stream from which particles of a given size are collided because the trajectories intersect
the bubble surface, to the projected area of the bubble in the direction of flow (refer. Fig.
6-3). Introducing y, (or its nondimensional form y:) as the maximum initial displacement
of the particle that allows the particle to just g m e the bubble, E, is given by
where R. and & are the raâii of the bubble and the soiid sphere, respectively. If the
particle is small relative to the bubble (% cc ), then F may be simplified as
D z
Puiicles striking bubbics when stramlina are w
P h c l a siriking bubblcs whcn ~ c r d i n c s u c divcrtcd. whac R, is the radius of the bubMç
no( divaicd w k c It is the radius of the bibblc, R, is Ihc radius of the puiicle. and y: is the R, is Lhe radius of the puriclc.
nondimensionri initial position of lhc parliclc
Figure 6-3. Collision of particles with a bubble (Karr et al., 1990)
Theoretically E is calculated by determining the grazing trajectory y: fiom
Eq.(6.46) for the particular hydrodynamic conditions. In reality the set of equations are so
difncult that they have never been d v e d for general problems where each tenn is
nonnegligible. y: has to be found iteraîively by numerical integration of Q(6.30).
However, there are rnany praaical problems where some tems can be assume- to be
zero, or s d l enough to be neglected. In these cases, approximate solutions are available.
Yom sumarizes the expressions for probability of collision derived fiom strearn
fùnctions under different flow conditions (Yoon, 2000):
For the Stokes flow condition (R-I), by Gaudin in 1957, applicable only for
very small bubbles (Yoon, 2000):
For the intermediate Reynolds number range, by Weber and Paddock in 1983
(Yoon, 2000):
For the intemediate Reynolds number range, by Yoon and Luttrell in 1989 (Yoon,
For the potential flow condition (R-l), valid only for bubbles that are much
larger than those used in Rotation practice (Sutherland, 1949):
Fig. 6-4 shows the values of P, calculated for d,=11.4 pm using the above four
different expressions (Yoon, 2000). Note that for the bubble sizes used in flotation
practice, Sutherland's equation (Eq. (6.5 1)) overestimates P,, while Gaudin's (Eq. (6.48))
underestimates it. However, Gaudin's equation can still be usefui for bubbles smailer than
approximately 100 Pm. Beyond this limit, the two equations derived by Weber and
Paddock (Eq. (6.48)) and Yoon and Luttrell (Eq. (6.49)) may be usefùl.
O 0.1 0.2 0.3 0.4 0.5 0.6
BUBBLE DIAMETER (mm)
Figure 6-4. Efféct of bubble site (d,) on collision efficiency (P,) under different conditions. Experimental data by Yom (Yoon, 2000)
The above P, expressions (Eq. 6.48-6.5 1 ) are based on the interceptional collision
model, which may be usehl for notation under relatively quiescent conditions. They may
be applicable to flotation colurnns with large length-to-diameter ratios or for columns with
sufficient baffles (Yoon, 2000). However, in mechanically agitated cells, which still
prevails in flotation practice, a bubble and a particle approach each other in a highly
turbulent field of fiow. The Reynolds number of the machine, defined as (Schulze, 1984)
nd$, Re, = - P
are in the range of 1 o6 - 7 x 10' in industrial-scale machines (Schubert, 1999). Here d2 is
the diameter of the rotor or impeller and n is its speed. Therefore the modelling of a
flotation process necessitates the application of essential results of the statistical
turbulence theory. Three effects of turbulence are important in flotation: the turbulent
transport phenomena (suspension of particles), the turbulent dispersion of air and the
turbulent particle-bubble collisions. While the turbulent transport phenomena are mainly
caused by the macroturbulence, the microturbulence controls the two last-named
microprocesses. According to Schuben (1 999), the particle-bubble attachent almoa
exclusively occurs in the zone of high energy dissipation rates, i.e., in the impeller stream.
Outside of the impeller stream the local energy dissipation rate is so far reduced below the
mean dissipation rate that the preconditions for r d i n g these microprocesses, which are
controlled by the microturbulence, are no longer met. Based on their findings, Schubert
and Bischofberger suggested that the wide-spread and too simplistic ideas on the course of
the particle-bubble attachent need substantial adjustments andlor completions (Schubert
and Bischofberger, 1998; Schubert, 1999). Their expression for predidng the collision
rate Zn, Le., the number of collisions per unit volume of slurry and the, is given by:
where: Np and NB are the number of particles and bubbles per unit volume, respectively,
v - and v,- are the root-mean-square values of the turbulent velocity fluctuations of the c IF pariicles and bubbles, respeaively, relative to the turbulent Buid velocity. In some cases,
v; 1s approximately given by (Schubert, 1999): F*
where the subxnpt i refers to bubble or particle, e the specific energy dissipation, pr the
density of the medium, and Ap is the density difference between i and the medium, q is the
dynarnic viscosity of the medium.
In centriîùgal notation machines (including ASHC), the operation takes place in a
rotating flow. Turbulent, random fluid motions are not possible in a rotating fluid.
Therefore, it is the interceptional collision model, not the turbulence model, that is
applicable here. It is towards the prediction of collision efficiencies in rotating fluids that
this thesis is principally directed.
6.5.2. Collision Elficiencies in Strong Force Fields
So fjir there are three hypotheses of particle collection on gas bubbles. The first
mechanism is the precipitation of dissolved gas on the hydrophobic surface of the mineral.
In the subaeration and agitation machines there are regions of high pressure and low
pressure before and &er the impeller blades. These pressures produce super- and under-
saturation, respectively, of the pulp. The dissolved air bubbles can then separate fiom the
supersaturateci liquid and precipitate on the hydrophobic solid surfaces where the work of
nucleation for the formation of bubble nuclei is lowest. The probability of the generation
of such adhering microbubbles increases with the hydrophobicity of a solid surface. After
the precipitation of microbubbles on a particle the attachment of a separate larger bubble
happens, resulting in a greater reduction of the free energy than an attachent event which
is not assisted by one or more adhering microbubbles. The microprocesses (collision and
adhesion), which are decisive for the kinetics of the macroprocess, almost exclusively
occur in the zone of high energy dissipation (Le., in the zone of the impeller stream).
Outside of the impeller stream the local energy dissipation rate is so far reduced below the
rnean dissipation rate that the preconditions for realizing these microprocesses, which are
controlled by the microturbulence. are no longer met (Schubert. 1999).
Cruising bubbles collision (CBC) occurs under gravitationai acceleration. This is
the main mechanism present in flotation columns and in the quiescent zone of mechanical
flotation ceils. However, this type of mechanism is known to lose its efficiency for fine
particles. Because of their small m a s . fine particles have low inertia and insufficient
momentum to resist the tendency to follow the fluid strearnlines around the bubble surface.
High acceleration collision (HAC) occurs in stronger force fields than
gravitational. Such fields enhance the inertia of small particles, allowing hem to lave the
fluid strearnlines and collide with the movhg bubble.
To study the efféct of the centrifuga1 force field on the collision efficiencies, take
the case of collision of a solid sphere of radius & with a spherical bubble of radius &
with single scattenng, i.e. the solid sphere &er a collision does not retum for another
collision within a finite amount of time. Working in the CO-ordinate system descnbed in
Fig. 6-5, the equations of motion of the solid sphere in x-direction and y-direction can be
represented by (Flint and Howarth, 1971; Karr et al., 1990)
where R,, is the radius of the particle, p, and pf are mass densities of the particle and the
fluid, respectively, U: and UF are velocity components of the particle in respective
directions. ~f and U: are velocity cornponents of the fluid in respective directions that
would exist at the position of the particle if no particle were there. g, is the strength of the
force field. In the case of the gravitational field, g, = acceleration of gravity g. In a
centrifugai force field, g, = r oz, in which r stands for the radial distance fiom the center of
the centrifuge, o for the angular velocity of fluid in the centrifùgal field.
Figure 6 5 . Geumetry of a single-bubble, single-particle system, where g is the force acting on the system, v is the velocity of the particle, y. is the initial position of the particle, and &, is the radius of the bubble (Karr et al., 1990)
These four equations have four variables (y, U! . U. and U: ), three independent
variables (x, y. t) , and m e n parameten (p, B, , & , % ,UB, &). The preceeding
equations cm be made dimensionless in the following way:
In the process the seven parameters are reduced to two parameters:
where p is the dynamic viscosity, UB is the velocity of the rising bubble, and & and & are
particle and bubble radii, respectively.
The parameter G represents the dimensionless terminal settling velocity of the
particle in an undisturbed fluid when g is the extemal force due to gravity; in a centrifùgal
force field, G represents the terminal radial migration veiocity of the particle.
The parameter K is known as "Stokes' number" or the "inertia number", which is
the ratio of inertial force to viscous force. Depending on the Stokes number K, four
distinct regions of particle-bubble behaviour can be defined (Ralston et ai., 1999a):
(1) K « 1: hertial forces have practidly no e8éct on the motion of the
particles, which can be considered as inertia-free. However, according to
Flint and Howarth (197 l), collision efficiencies Ec = G/(l+G) in this region
are never zero in flotation systems since for this case the value of G can
never be zero.
(2) K s 0.1: Inertial forces can impede particle deposition on a bubble.
Collision efficiency is virtually independent of K but is strongly dependent
on G and can be calculated fiom Ec = G/(l+G). Since G decreases with
increasing bubble size, the collision efficiency of a particular sized particle
is increased by a reduction in the bubble size.
(3) 0.1 < K < 1: An inelastic inertial impact of particles on a bubble surface is
characteristic of this region. A major portion of kinetic energy of the
particle is lost both during the approach to the bubble and at the impact
itself, when a liquid interlayer is formeci between the surfaces of a particle
and a bubble.
(4) K > 1-3: The trajectory of a particle deviates very slightly from the linear
and the energy of the particle as it approaches the bubble and on collision
changes so little that the impact can be considered as being quasi-elastic.
For coarse particles, Le. characterized by K greater than about 1.0,
collision efficiency depends most strongly upon inertial forces. Since bubble
nsing velocity is a function of bubble radius, K increases with bubble size.
Consequently, in this coarse particle region collision efficiency is increased
by increasing the bubble sire.
Thus Eqs. (6.55-58) in their dimensionless fonns become:
duP' KL = G + u ~ -u:' dt'
dx' -= q' dt'
dy' - -- dt'
Eq. (6.61) and Eq. (6.63) become uniquely specified once the components
( ~ f ' and ut' ) of liquid velocity Ur have been derived. For very small values of the
Reynolds number of the bubble (RQ l), the flow pattern is well described by the
Stokes' fiow solution (Ref. Eq. (6.41)) and in diensionless form:
Likewise, for very high Reynolds number of the bubble (Ra ml) the flow pattern
is described by a potential flow solution (ReE Eq. (6.45)):
Recalling the equation to calculate the collision efficiency:
Ft& is predetermineâ, therefore the rest of the problem is to determine maximum
initial position of the particle, y:, that aiiows the particle to just grare the bubble. The
criterion for this is that the final values of y;,, and x, satise the condition
y h , and x;,, are functions of K, G and y: . During numerical integration of Eqs.
(6.61 -64), K and G are set as constant, predetermined values. Therefore, y&, and x, are
fiinctions of y: only. By varying the value ofyz arbitrarily, the integration ends at the time
the function F( y: ) = x;, + y;:, - (1 + ?] becorner zero.
The above approach is followed by Karr et al. (1990) in their investigation of the
collision efficiency for use in a comprehensive air-sparged hydrocyclone (ASHC) mode1
(see Fig. 2-4). Collision efficiencies are detennined from the calculated particle trajectories
for values of Stokes' number K in the range of 0.01-100 and the nondimensional force,
W, between 0.0 and 100. The values considered are much larger than those exhibited in
conventional notation machines but are representative of the forces experienced in
ASHC's, in which the bubble-particle system is wbjected to strong force fields. Effects of
changes in particle-to-bubble radius ratio on collision efficiencies were also considered for
both viscous and potentid flows. The results, which are surnmarized in Fig. 6-6 to 6-9,
support the observation of better flotation recovery rates in strong force fields and also
help explain the improved recovery in flotation of fine particles using smaller bubbles.
Figure 6-6. Coilision efficiency as a function of Stokes' number and nondimemional force for particle-to-bubble radius ratio of 0.0 with various t e r d settling velocity (W). (Karr et ai .. 1 990)
Figure 6-7. Collision efficiency as a function of Stokes' nurnber and nondimensional force for viscous flow with particle-to-bubble radius ratio of 0.0 (Karr et al., 1990)
Figure 6-8.
r O-' t o0 t Q I
STOKES' NUMBER (K)
Collision efficiency as a fundon of Stokes number and nondimensional force for potential flow with particle-to-bubble radius ratio king O. 1. (Karr et al., 1990)
Figure 6 9 . Collision efficiency as a ttnction of Stokes number and nondimensional force for viscous flow with change in particle-to-bubble radius ratio (Karr et al., 1990)
6.6. Probability of Adhesion
Not al1 the particles colliding with air bubbles result in flotation. Characteristically only the
hydrophobic particles adhere to the surfiace of air bubbles. The time required for bubble-
particle adhesion to occur once the two are brought into contact is called the Uiduction
time ( tlnd ) and is synonymous with the adhesion time. The surface forces can accelerate,
retard or even prevent the draining of the liquid film between the collided particle and the
bubble. Therefore, it is possible that the collided particle may move away from the bubble
surface after spending some time sliding around the bubble surface. The time is cailed
sliding time ( ts, ). If the sliding time is longer than the induction time, the particle will have
long enough contact time to thin and rupture the disjoining film between the particle and
bubble (Yoon, 2000). Most adhesion efficiency models are based on the relative
magnitude of the induction time and the sliding time.
The basic equation of the mode1 is expressed as (Ralston et al., 1999):
sin' 0, E, =-
sin' O-
where 8,. termed the adhesion angle, is a specific collision angle where if a particle couides
at this angle, its sliding time is just equai to the induction time; 8, is the maximum possible
collision angle. 0, and 8, are measured from the front stagnation point of the bubble, or the
north pole in the case of a rising bubble.
The maximum collision angle (8,) is a complex funetion of the bubble Reynolds
numb2r and satisfies the equation denved by Dai et al. in 1998 (Ralston, et al., 1999):
where p is a dimensionless number defined as
where dp and dB are particle and bubble diameters, p, and p~ are particle and fiuid
densities, respectively, K is the Stokes number (Eq. 6.60).
Dobby and Finch in 1986 derived an expression for E. under potential flow
conditions (Ralston et al., 1999):
where tus is the induction tirne, VP is the particle veiocity and Ue is the bubble rising
velociîy.
The Yoon-Luttrell adhesion efficiency mode1 is proposed for fine particles which
do not rebound from the bubbie surfaces. For the case of potential flow conditions,
= sin2 1 2 arctanexp Lzt]} The 'sin290"' in the Eq. (6.75) implies the assumption that particle-bubble collision
occurs uniformly over the entire upper half of the bubble surface. This has been shown to
be incomezt by some researchers (Ralston et al., 1999). If Eq. 6.72 is substituted for 8c,
Eq. 6.75 becomes
For flow patterns at the intermediate Reynolds numbers (Yoon, 2000):
The induction time (b) is strongly a function of particle hydrophobicity and can
be routinely detemiined in the laboratory.
Neither of these models address the findamental issues of thin film drainage, and
three phase contact line movement which are the major contributors to E,, nor is there any
indication that field forces (gravitational, centrifugai) may affect E.. Yoon and Luttrell's
work in 1989 showed that as the particle size decreases E. increases, and E, increases with
decreasing bubble size until it decreases again as the bubble size becomes too smail (Yoon,
2000).
Cbapter 7
Summary, Conclusions and Recommendations
The sustainable development of the mining industry depends to a signifiant degree on its
ability to develop new technology to lirnit the mpense. Faced with the depletion of high
grade, easy-to-process ores, miners are delving deeper for seams that are harder to
exploit. Most mine output today is the finely disserninated ores of low grade and complex
composition that require extensive grinding to obtain adequate liberation. Processing the
resultant fine particles would require new flotation technology which are effective below
the 10 pm particle size range. Fundamental flotation theory suggests that the recovery of
very fine particles could be improved by flotation with finer bubbles in strong force fields.
The most important outcome of this study has been the development of the new
generation of centrifuga1 flotation cells, CFC-Q2 - a system that integrates the
fundamental research results and reflects today's philosophy of squeezing costs to the
minimum. The design is based on the solid appreciation of the rotating fluid phenomena
that, while understood by some, were not well communicated to flotation ce11 designers or
users before. Major points are summarized below.
(1) Three types of rotating fluids, namely, Rankine's combined vortex,
extemally-forced vortex and internally-forced vortex, are applicable for
froth flotation. Rankine's combined vonex consists of a rigidly rotating
core with radius R surrounded by an hviscid free vortex with matching
tangentid velocity component U: and pressure p at R Externally-forced
vortex refers to the fluid which rotates about the vertical axis like a rigid
body. Intemally-forced vortex is a combination of tangential (O), radial (r),
and axial (2) flows.
(2) The effectiveness of centrifuga1 flotation cells depends on the generation of
strong centriftgal forces. Centrifugd force acting on a suspended solid
particle in a rotating fluid is directly related to the tangential velocity
component of the fluid (u:). The higher the u:, the aronger the
centrifuga1 force.
(3) ln Rankine's combined vortex, the tangential velocity of fluid iacreases with
decreasing radial distance from the central vertical axis across a free vonex
and decreases with decreasing radial distance from the central vertical axis
in the forced vortex core. Therefore, a solid particle will experience
increasing centrifuga1 force across the free vortex as it moves imvardly to
the forced vortex core-
(4) In extemally-forced vortex, the angular velocity o of fluid is independent
of the radial distance r &om the central vertical axis. Therefore, the
tangential velocity of fluid increases with increasing radial distance r from
the central vertical axis. Solid particles suspended in it will experience
increasing centrifiigal forces as they move ou~urd& away from the centrai
vertical axis.
( 5 ) In intemaily-forced vortex, there exists a characteristic length 6 above
which fluid has no rotation but a downward axial (2) fiow. Within the 6, the
tangentid velocity of fluid increases with increasing radial distance r from
the central vertical axis but decreases with increasing axial distance z from
the rotating disk. Solid particles suspended in such a vortex will experience
strongest centrifuga1 force near the disk and centrifugai force increases
with increasing radial distance r from the centrai vertical axis.
(6) Secondaq flows resulting from the fluid-structure interactions are cornmon
to most rotating rnachinery including centrifugai flotation cells. Some
effects are favorable such as the downward axial flow in the intemally-
forced vortex which can be used as wash water. Other effects are
undesirable and can be curtailed by weful design, e.g., the air-sparged
hydrocyclone has only a cylindrical flotation ceIl so the radial flow common
in a conventional hydrocylone is minimized.
(7) Special attention must be given to the formation process of rotating fluids
because flotation itself is a continuous process with slurry in and out of the
ce11 al1 the the . Therefore, rotating fluids (except Rankine's combined
vortex) can hardly reach the steady state of motion.
(8) Rankine's combined xrtex can be generated by direct tangentid entry into
a cyiindrical structure. Extedly-for& vortex can be generated by direct
rotation of the container. Intedly-forced vortex can be generated by
transfemng energy âom an intemally placed rotating device to a fluid.
Energy needed to generate and maintain the rotation of fluids varies and it
is postulated that a centrifugai flotation ceU using Rankine's combined
vortex would consume the least amount of energy as the structure has no
moving parts. The CFC-QI ce11 using externally-forced vortex needs to
rotate the whole ceIl therefore it seems to be an expensive option. The
CFC-Q2 ceIl using intemally-forced vortex provides an econornic
alternative by rotating an intemal device instead of the whole ceil to
generate rotation of fluid.
Centrifuga1 force fields created by rotating fluids have an effect on the efficiency of
flotation. Collision efficiency of solid particles with air bubbles are reportedly increased
with increasing strength of the force field. Use of small shed bubbles are reportedly
beneficial in terms of increased collision efficiency. The in-line air sparger system adopted
in the CFC-QI and CFC-Q2 centnfbgal flotation systems provides a superior mechanism
to generate a large number of fuie bubbles.
The CFC-QI and CFC-Q2 centrifbgal flotation cells belong to the
reactorlseparator class of flotation machine. They could continuously operate for houn
without slurry plugging problems as occurred in the original CFC. The CFC-QI ce11 uses
the rotating drum (or bowl) to prodiice the centrifugai forces on the feeding slurry. The
tailing is discharged fiom the open top edge of the dmm. The CFC-Q2 ce11 uses the
rotating vertical feeding pipe to produce the centrifuga1 force on the sluny. The tailing is
discharged fiom the bottom of the flotation column.
The new designs have been compared with a standard Denver laboratory flotation
cell for the flotation of four ore samples: galena-silica mixture, chahpyrite-silica mixture,
Gaspe copper ore and Cominco lead-zinc ore. The results clearly show that the new CFC
cells can recover more particles finer than 10 as compared with the Denver cell.
Galena-silica mixture: 42- 1 > Q 1-3 > Denver
Chalcopyrite-silica mixture: 42- 1 > Denver > Q 1-3
Gaspe copper ore: 41-3 > Denver > 42-1
notation results for the Cominco lead-zinc ore sarnple indicated that both the
CFC-QI -3 and the CFC-Q2- 1 have a better zinc recovery than that of the Denver cell,
while the Denver ce11 gives better lead flotation recovery.
FIotation performance compared by the grade-recovery curves are summarized as
follows:
Galena-silica mixture: Q 1-3 = Denver
Chaicopyrite-silica mixture: Q2-1 > Denver > 41-3
Gaspe copper ore: Denver > 42-1 and 41-3
Flotation kinetics of modified CFC models are much faster than for the Denver
ceIl. The volume of modified models are smailer than that of the Denver ce11 for the same
operating capacity.
The new generation of centrifuga1 flotation cells, namely, CFC-QI and CFC-Q2,
did not corne into being as the result o f an intensive fundamentai research effort but, in a
manner comon to the development of other flotation technology, they were deveioped
large1 y through intuitive work. As flotation results carne out, it becarne increasingly clear
that the success or failure of centrifùgal flotation cells depend criticaily on understanding
and predicting rotating fluid phenornena.
Economicai design of flotation-usable apparatus can be greatly facilitated by
estimates made from complementary expenmentation and modeling studies. Without
doubt, the proven success of CFCQi and CFC-Q2 centrifuga1 flotation technology would
be fùrther enhanced through such work that combines expenmental and theoretical
hydrodywnics with sophistiutted computational fluid dynamics.
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