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8/19/2019 A Hydrodynamic Simulation of Mineral Flotation. Part I- The
1/14
E L S E V I E R P o w d e r T e c h no l o gy 8 3 1 99 5) 2 1 1- 2 24
POWDER
TE HNOLOGY
A h y d r o d y n a m i c s i m u l a t io n o f m i n e r a l f l o t a t i o n . P a r t I: t h e
n u m e r i c a l m o d e l
K D a r c o v i c h
Ins t i tu te for Env i ronm e nta l Re se arc h and Te chnology Nat io nal Re se arc h Coun c i l o f Canada Ot tawa Ont . KI A OR 6 Can ada
Received 14 February 1994; revised 12 December 1994
bs t rac t
Minera l f l o t a t ion i s a p rocess wh ereby pa r t i cu la t e s conta in ing mine ra l be a r ing con s t i tuen t s p re fe ren t i a l ly adhe re to gas
bubbles in a l i qu id medium. Thi s i s a way to sepa ra t e and upgrade ores o r o the r mine ra l ma t t e r . In o rde r t o s imula t e th i s
un i t ope ra t ion hydrodynamica l ly , a t h ree -phas e sys t em for t he l iqu id phase and d i spe rsed so lids and bubbles must be employed .
Addi t iona l s imula t ion fea tures a re the ce l l geomet ry , t he boundary condi t ions , d rag t e rms, va r i ab le v i scous e f fec t s , and the
t rea tment o f t he adhes ion of pa r t i c l e s t o bubbles . Thi s pape r , t he f i r s t o f a se r i e s o f two, de t a i l s t he formula t ion of t he
govern ing equa t ions , t he numer i ca l implementa t ion and shows some re su l t s fo r a two-d imensiona l , i ncompress ib l e case . The
formulat ion and implementat ion are suff ic ient ly general and this simulat ion was appl ied to the f lota t ion of coal-oi l agglomerates
as deta i led in Part I I of this ser ies.
Ke y words :
Flotation; Th ree-pha se flow; Turbulence; Finite-volume method; Interphase mass transfer
1.
I n t r o d u c t i o n
M i n e r a l f l o t a t i o n i s a u n i t o p e r a t i o n w h e r e l a r g e
t a n k s o r c e l l s c o n t a i n i n g p a r t i c l e - b e a r i n g s l u r r i e s a r e
s u b j e c t e d t o t u r b u l e n t m i x in g w i t h g a s e o u s b u b b l e
s t r e a m s t o p r o v i d e s u f f ic i en t c o n t a c ti n g b e t w e e n t h e
d i s p e r s e d p h a s e s . F u n d a m e n t a l l y , m i n e r a l f l o t a t i o n i n -
v o l v e s t h e c o l l is i o n a n d s u b s e q u e n t a t t a c h m e n t o f a
p a r t i c le w i th a b u b b l e i n a n a q u e o u s m e d i u m . O f t e n ,
t h i s o p e r a t i o n i n v o l v e s se l e c t iv e a d h e s i o n o f a m i n e r a l
b e a r i n g p a r t i c l e r a t h e r t h a n a g a n g u e p a r t i c l e , s o i n
e f f e c t t h e m i n e r a l c o n t e n t i s r e c o v e r e d i n a n e n r i c h e d
s t a t e . T h i s e v e n t is h ig h l y i n f l u e n c e d b y t h e o v e r a l l
h y d r o d y n a m i c s o f a ll t h r e e p h a s e s i n a f l o t a ti o n c e ll .
I n a d d i t i o n t o t h e s u r f a c e c h e m i c a l f a c t o r s w h i c h e s-
s e n t ia l ly i n f lu e n c e t h e p a r t i c l e - b u b b l e a t t a c h m e n t , h y d r o -
d y n a m i c s s i m u l t a n e o u s l y a s s i s t i n d e t e r m i n i n g s u c h a n
i n t e r a c t i o n , p o t e n t i a l l y g i v i n g r i s e t o a c o l l i s i o n , a t-
t a c h m e n t a n d s u c c e ss f u l f l o ta t i o n .
O f i n t e r e s t i n t h is s t u d y a r e h o w t h e p r e v a i l i n g
h y d r o d y n a m i c s i n th i s t h r e e - p h a s e s y s t e m p r o d u c e t h e
s u m o f i n t e r a c t i o n s w h i c h d e f i n e t h e e f f ic i en c y o f t h e
u n i t o p e r a t i o n . S i n c e t h e f lo w p h e n o m e n a i n v o l v ed c a n
b e c o m p l e x , r e s ol v in g t h e h y d r o d y n a m i c fo r c e s w o u l d
¢ NRCC No. 37582.
Elsevier Science S A
S S D I 0 0 3 2 - 5 9 1 0 ( 9 4 ) 0 2 9 5 9 - R
p r o v i d e a b as i s f o r d e c o u p l i n g t h e s u r f a c e c h e m i c a l
c o n t r i b u t i o n t o t h e k i n e t ic s o f m i n e r a l f l o t a ti o n .
T h e r e h a s b e e n c o n s i d e r a b l e r e s e a r c h i n t h e p a s t
o n t h e r o l e o f h y d r o d y n a m i c s in t h e e l e m e n t a r y
b u b b l e - p a r t i c l e c o l l i s i o n i n f l o t a t i o n , a s w e l l a s t o
c h a r a c t e r i z e t h e m a c r o s c o p i c b e h a v i o u r o f a f l o ta t io n
c e ll . E a r l y w o r k b a s e d o n t h e i n it ia l m o d e l b y S u t h e r l a n d
[ 1] f o c u s e d o n h o w a p a r t i c l e f o l l o w i n g a f l u i d s t r e a m l i n e
a r o u n d a b u b b l e i n a p o t e n t i a l f lo w r e g im e c o u l d c o n t a c t
t h e b u b b l e . L a t e r r e s e a r c h b y F l i n t a n d H o w a r t h [ 2] ,
D e r j a g u i n e t a l. [3 ] , S c h u l z e a n d G o t t s c h a l k [ 4] , W e b e r
[ 5] , W e b e r a n d P a d d o c k [ 6] a n d D o b b y a n d F i n c h [ 7]
d e v e l o p e d h y d r o d y n a m i c c o ll is i o n m o d e l s w h e r e i n e r ti a l
e f f e c t s f r o m t h e p a r t i c l e a n d f l o w r e g i m e s i n a n i n -
t e r m e d i a t e r a n g e b e t w e e n S t o k e s a n d p o t e n t i a l f l o w
w e r e c o n s i d e r e d . T h u s , t h e u n d e r s t a n d i n g o f h o w o n e
b u b b l e i n t e r a c t s w i t h o n e p a r t i c l e h a d b e e n r e f i n e d .
T h e o t h e r p r i n c i p a l m e t h o d t o m o d e l f l o t a t i o n , h a s
b e e n t o a p p l y s o m e b a s i c h y d r o d y n a m i c - r e l a t e d t r e a t -
m e n t t o s i m u l a t e t h e e n t i r e u n i t o p e r a t i o n o f a fl o t a ti o n
c e ll . S e v e r a l p a p e r s o n t h is t o p i c h a v e b e e n s u m m a r i z e d
i n a r e v i e w b y M a v r o s [ 8] . T y p i c a l l y s o m e m i x i n g p a t t e r n
w a s a s s u m e d a n d a p p l i e d t o a r e s i d e n c e t i m e d i s tr i b u t i o n
f o r k i n e ti c m o d e l s s c a l e d u p f r o m b u b b l e - p a r t i c l e i n -
t e r a c t i o n a n a l y s e s . A c o m p a r a t i v e l y i n - d e p t h s t u d y b y
D o b b y a n d F i n c h f o r c o l u m n f l o t a t i o n s y s t e m s [ 9 , 1 0 ] ,
8/19/2019 A Hydrodynamic Simulation of Mineral Flotation. Part I- The
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2 1 2
K. Darcovich Powder Technology 83 1995) 211-224
made use of experimental residence time measurements
to model the unit under a modified plug flow regime,
without any mechanically introduced turbulence. The
particulate concentration was determined in the axial
dimension only via regular differential equations using
an axial dispersion coefficient to account for some
turbulent action.
The computational fluid dynamics (CFD) approach
for the present work was adopted as a means to enable
flotation to be modeled in an integrated sense. This
work is part of an on-going project on the recovery
and beneficiat ion of waste coal fines by oil agglomeration
[11]. Part II of this series [12] describes in detail the
surface chemical aspects of the project. The objective
of this paper (Part I) is to detail the formulations for
simulating the steady-state motion of the liquid, gas
and solid phases in a continuous float cell, such that
the contacting patterns can be resolved. That is, to
provide a mechanistically based model of flotation, the
necessary steps of collision and adhesion must be ac-
counted for. By determining a three-phase flow field
solution, a turbulence based collision model based on
local relative velocities and dispersed phase concen-
trations can be used to illuminate how the particulate
surface properties lead to successful bubble-particle
adhesion. Part II in this series will then apply this
simulation to flotation recovery data for coal-oil ag-
glomerates in order to demonst rate the surface chemical
influence on bubble-particle adhesion in flotation. With
a resolved flow field, the influence of the particulate
surface properties can be decoupled from the hydro-
dynamics of the system, which in previous studies, had
to be considered simultaneously.
2 M a t h e m a t i c a l m o d e l
The general partial differential equations which gov-
ern fluid transport were employed in this simulation.
Modification of these equations from their basic forms
was required to incorporate an interdependent multi-
phase system.
2 1 System va riables
The system to be simulated in this project required
the resolution of twelve different field variables. Since
the simulation was modeled as incompressible and in
two-dimensions, the field variables were the two di-
rectional components of the liquid, gas and solid phase
velocities, the pressure, two turbulence parameters, the
gas and solid phase volume fractions and the extent
of particle adhesion onto the bubbles of the gas phase.
The present model allows for the distinct flow paths
and local velocities of each of the three phases. The
flow field solution produces a two-dimensional velocity
vector field and a scalar field of all other variables.
Each phase is treated as a continuum and they share
the domain according to the local volume fractions.
The dimensions of the interfaces between the liquid
and dispersed phases are prescribed by the volume
fraction of each dispersed phase subdivided into par-
ticles or bubbles of each input diameter. The interfacial
area thus determined reflects the multi-phase nature
of the model.
2 2 Governing equations
The finite-volume formulation used in this work
employed the basic mass and momentum transport
equations with the turbulence features worked into
them expressed as statistical averages of the random
fluctuations. The turbulence in the flows is accounted
for by the turbulent variables k and E, from the two-
equation model developed by Launder and Spalding
[13]. The problem is fully defined with the specification
of boundary conditions.
2 2 1 Transport equations
The instantaneous properties of a flow are given by
the mass and momentum conservation equations [14].
The variables, u, p and P are respectively velocity,
density and pressure. For brevity, the Einsteinian indicial
notat ion is used, subscripts i, j and k referring to the
spatial directions.
ap
Mass: ~ + (pu) j.j=0 (1)
Momentum: a(pu)i + [(pu)jul].j = - P i + ~ij j (2)
~ t '
Above, ~-~j is the shear stress tensor given by:
2
7 i j = ] . Z ( U i , j t - U j , i ) - - 3 ~ ' £ U k . k ~ i j ( 3 )
Here, /x is the laminar viscosity of the fluid.
2 3 Turbulence mo del
Via dimensional analysis, two parameters are required
to evaluate /z,. The k - E model [15] has proven to be
a suitable method to model turbulent flows. The pa-
rameters have a physical significance, and transport
equations may be obtained for them in a straightforward
manner. For high Reynolds number flows, the parameter
e is defined as,
Here, u; is the mean velocity fluctuation and e defines
the dissipation rate of the turbulent kinetic energy.
8/19/2019 A Hydrodynamic Simulation of Mineral Flotation. Part I- The
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K . D a r c o v ic h P o w d e r T e c h n o l o g y8 3 1 9 9 5 ) 2 1 1 - 2 2 4 2 1 3
From dimensional analysis, the relation between the
turbulent viscosity/xt and the two turbu lence paramete rs
is,
C ~ , p ) k 2
u
= (4)
E
In Eq. (4), C~, is found empirically, and for high Reynolds
numbers, has a value of about 0.09. Fr om these expres-
sions, equations entirely analogous to Eqs. (1) and (2)
can be written [16]. These equations rely on a density-
weighted ensemble-averaged (DWEA) form for the
field variables. That is, the field terms will be treated
with a mean value, but a non-zero time-average for
turbulent effects is included.
2 .3 .1 . Tu rb u len t k in e t ic en erg y t ra n sp o r t eq u a t io n
The variable k is solved for in the flow domain with
its behaviour defined by a transport equation. The
DWEA turbulence energy transport equation was de-
rived by Watkins [17] by manipulating the momentum
equation and making use of the relations between the
fluctuating and mean flow variables. The form of the
equation that is used in this work is,
i ) P ) k + p ) t ~ i k - l I . ~ + I z t ) k . j )
at ~r , J
2
= ] J , t( /, ~ i( /, ~ i, ~ - /,~ j , ) ) . i - ~ ( / ~ j [~ / , t/ ~ i , i [ - ( p ) k ] ) , j - ( p ) E
5 )
The nota tions (p) and fi~ refer to, respectively, an
averaged value and a density-weighted ensemble-av-
eraged value. In Eq. (5), irk is an empirical constant
of value 1.0.
2.3.2. T u r b u l e n c e e n e r g y d i s s i p a t i o n r a t e e q u a t i o n
The ensemble-averaged energy dissipation equa-
tion is given below, in terms of DWEA variables, as
this form [17,18] is considered the most correct,
of the available relevant formulations. It is written
a s ,
i ) p ) , ) _ b p ) / ~ j , _ L 1 ~ . k _ /. /, t) ,. .i )
bt o-, J
e { c [ t x t ~ t i , . i + d j ) )
k 1 t i , i , j
~ ( / ~ j ( l / ' t / '~ i , ~ - P ) k ) ) , i ] - C 2 p ) e ]
+ C 3 p ) a ~ j . j 6 )
The model constants, cr,, C1 , Cz and C3 are given in
Table 1.
T a b l e 1
T u r b u l e n c e m o d e l c o n s t a n t s
Co nst ant C~, CI ?2 K E ~r o- , Ca
Va lu e 0 .0 9 1 . 4 4 1 . 9 2 0 .4 1 8 7 9 .7 9 3 1 .0 ~ 1 .0
C 2 - C l ) C ~ 5
T a b l e 2
I n t e r p h a s e d r a g t e r m s f o r t h e m o m e n t u m e q u a t i o n s
P h a s e s D r a g t e r m
9 ~ G / .t ,L .
l i q u i d - g a s F L G i = T
[ U G i - - U lA )
l i q u i d - s o li d F L si = B u u - - U s i )
2 . 4 . T h r e e - p h a s e m o d e l
2 .4 .1. G a s -p h a se eq u a t io n s
Continuity: a o p o u o i ) . i = O (7)
Momentum: aG pGUGiU~i-- p.GeffUGi.j) . j
= - - o tGP, i + o tGPGgi+FGLi + a o 3 r i p c , U G , P-ceff)
8 )
Above, aG is the gas-phase volume fraction, and the
term FGL is the interphase drag term for the gas phase
in relation to the liquid phase [19,20]. The terms
indicated by 9-~(pc, uG,/zG~fr) arise from the curvature
of the coordinate system selected. For this work, a
Cartesian system was used, and these terms drop out.
The interphase drag terms are summarized in Table
2. The effective viscosity term, /zceer is the sum of
laminar and turbulent contributions to the viscosity. In
this simulation,/zG, is not considered as the turbulence
is restricted to the main transporting phase, which is
the liquid.
Based on assumptions made by Lai and Salcudean
[19], the gas-phase momentum equations may be sim-
plified. The gas phase is assumed to be dispersed in
the liquid phase. Further, in typical mineral flotation
applications, the presence of chemical frothers and
conditioners causes the bubbles to act as rigid spheres
[5]. The gas bubbles are also modeled as incompressible
which is a simplifying assumption; in this way, a constant
system volume may be obtained. Further, inherent
instabilities such as oscillations at the free surface of
the vessel are neglected. The bubbles are treated as
having a constant given diamete r which therefore does
not allow for coalescence. Naturally, bubble coalescence
is a very real phenomena, but incorporating it into this
model would introduce a great deal of complexity to
a system which is a first step in flotation modeling by
the solution of field equations. Coalescence would serve
to increase the mean bubble size and reduce the number
8/19/2019 A Hydrodynamic Simulation of Mineral Flotation. Part I- The
4/14
2 1 4 K . D a r c o v ic h / P o w d e r T e c h n o lo g y 8 3 1 9 9 5 ) 2 1 1 - 2 2 4
d e n s i t y o f b u b b l e s t h e r e b y c o n t r i b u t i n g t o a l e s s e r
f lo t a t ion recovery .
D e s p i t e t h e s e s h o r t c o mi n g s , t h e s i mu l a t i o n c a n d e m-
o n s t r a t e t h e b a s i c p h y s i c a l b e h a v i o u r o f t h e s y s t e m. I n
a n y c a s e , s i mp l i f i c a t i o n s o f t h e p h y s i c a l g e o me t r y o f
t h e f l o t a t i o n u n i t u s e d i n t h e s i mu l a t i o n a r e ma d e ,
s u c h t h a t a n o v e r l y c o mp l e x a n d c o mp l e t e l y p h y s i c al l y
c o r r e c t t r e a t m e n t o f t h e g a s- p h a s e b e h a v i o u r w o u l d b e
s u p e r f l u o u s .
I f t h e a c t i o n o f t h e l i q u i d p h a s e i s a s s u me d t o b e
p r i ma r i l y r e s p o n s i b l e f o r t h e g a s - p h a s e t r a n s p o r t , t h e n
o n l y t h e p r e s s u r e , d r a g a n d b u o y a n c y t e r m s a r e r e t a i n e d .
In th i s way , the gas ve loc i t i es may be exp l i c i t l y
d e t e r m i n e d f r o m t h e g a s v e l o c i t y t e r m i n t h e d r a g
e x p r e ss i on . T h e r e s u l ti n g m o m e n t u m e q u a t i o n b e c o m e s ,
0 =
- - aoP ~+ aGpG gi+ F GL~+ aGSr~( pG, UG,
/ZG.,,) (9)
2.4.2. Liquid-phase equations
Co ntinu i ty: (aLPLUL~). = 0 (10)
M o m e n t u m :
aL (pL ULjUL~ - /ZL~ffUL~. ). i
= - -aLP .~+FL GI+FL si+aLSrI (PL, UL,
/ZLaf) (11)
A b o v e , a L i s t h e l i q u i d - p h a s e v o l u me f r a c t i o n . I n E q .
( 1 1 ) , t h e t e r m
aLPLg~
i s o mi t t e d s i n c e t h e e q u a t i o n
m a k e s u s e o f a
corrected
p r e s s u r e P , w h e r e ,
P ~ i - - P , i p l _ g i
H e n c e , i t c a n b e s e e n t h a t u s i n g
P
i n s t e a d o f P t a k e s
c a r e o f t hi s t e r m . T h e i n t e r p h a s e f r i c t i o n t e r ms a r e
t h e s a me o n e s a s g i v e n i n T a b l e 2 . I t s h o u l d b e n o t e d
t h a t i n g e n e r a l , w h e n a p h a s e 1 i n t e r a c t s w i t h a p h a s e
2 in a j -d i rec t ion , i t can b e sa id ,
F 1 2 = - F 2 1 j
H e n c e , w i t h r e s p e c t t o e q u a t i o n s f o r o t h e r p h a s e s , t h e
s i g n s a r e r e v e r s e d o n t h e i n t e r p h a s e d r a g t e r ms w h e n
i n s e r t e d i n t o t h e l i q u i d - m o m e n t u m e q u a t i o n .
2.4.3. Solid-phase equations
Cont inu i ty :
(aspsUs,),
~= 0 (12)
M o m e n t u m :
as(psUsjUs~).j
= - o t s P i + a s P s g ~ + F s L i + a s S r s i ( P s , Us) (13)
A b o v e , a s i s t h e s o l i d - p h a s e v o l u me f r a c t i o n . Fo r t h e
so l id phase , t he v i scou s t e rm s wi ll be o f a neg l ig ib le
ma g n i t u d e i n c o mp a r i s o n w i t h t h e i n e r t i a l t e r ms , a n d
a r e t h u s d r o p p e d f r o m E q . ( 1 3 ) .
2.4. 4. Interphase drag
Ba s e d o n r e s u l t s f r o m So o [ 21 ] a n d L a i a n d S a l c u d e a n
[ 1 9 ] , t h e l i q u i d - g a s a n d l i q u i d - s o l i d i n t e r p h a s e d r a g
t e r ms a r e g i v e n i n T a b l e 2 . A l s o i n T a b l e 2 w e h a v e
t h e c o e f f i c i e n t B f o r t h e l i q u i d - s o l i d i n t e r p h a s e d r a g
c o e f f i c i e n t . T h i s t e r m i s b a s e d o n d e r i v a t i o n s b y E r g u n
[22] , W en and Y u [23] and L yczkow sk i and W ang [20] .
W e u s e s a n d ~s t o r e f e r t o n o r ma l i z e d ( g a s p h a s e
exc luded ) l i qu id and so l id vo lu m e f rac t ions . Th i s ap -
p r o x i ma t i o n i s ma d e a s i t i s a s s u me d t h a t t h e l i q u i d
p h a s e i s r e s p o n s i b l e f o r t r a n s p o r t i n g t h e d i s p e r s e d
p h a s e s , a n d t h a t g a s - s o l i d a n d s o l i d - s o l i d i n t e r a c t i o n s
a r e n e g l e c t e d .
O~L
O~L --1- O~S
A l s o ,
a s
~ s - - - -
Ot~L Ot~S
A s s h o w n i n [ 2 0 ] , t w o c a s e s a r e c o n s i d e r e d f o r B . W e
have ,
P L l u L - - U s I I s
for ~:s>0.2 (14)
B = 150 ~ + 1.75 2rp
I n d e n s e s l u rr i e s, t h e f l o w fi e ld d i s t u r b a n c e s a r o u n d
s p h e r e s b e g i n t o o v e r l a p a n d h e n c e a s e p a r a t e e x p r e s s i o n
i s n e e d e d f o r s u c h a c a s e . Fo r mo r e d i l u t e l i q u i d - s o l i d
sys tems , a d i f fe ren t reg im e ex i s ts and i s mod ele d [20,23]
a s ,
B = 0 . 7 5 C D
U26 slu -u I
f o r ~ s < 0 . 2 ( 1 5 )
rp
Ro w e [ 2 4 ] r e l a t e d t h e d r a g c o e f f i c i e n t t o t h e Re y n o l d s
n u m b e r b y ,
t 24
D = Re~ (1 + 0.15Re~ 687) for Res < 1000
0.44 for
Re s >~
1000
w h e r e ,
R e s
2~pL(U L-- US)rp
I~L
2.4.5. Multi-phase mass balance
T h r o u g h o u t t h e f lo w d o m a i n , a c h e c k m u s t b e m a d e
o n t h e s u m o f t h e v o l u me f r a c t i o n s o f t h e v a r i o u s
phases . At a l l l oca t ions , t he fo l lowing cond i t ion mus t
be sa t i s f i ed ,
E a M = 1 ( 1 6)
M
Continuity of pha ses
I n t h is s i mu l a t io n , t h e r e a r e t h r e e p h a s e s c o n s i d e r e d ,
wa ter , bu bb l es o f vary ing par t i c l e load ing , and pa r t i c l es .
E a c h p h a s e i s s u b j e c t t o a m o m e n t u m b a l a n c e w h i c h
i s em ploy ed fo r the ca l cu la t ion o f it s respe c t ive ve loc i t i es ,
w h i c h i n t u r n mu s t b e u s e d t o d e t e r mi n e l o c a l v o l u me
f r a c t i o n s t o e n s u r e p r o p e r c o n t i n u i t y f o r e a c h p h a s e ,
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and taken all together, to conserve an overall, multi-
phase mass balance.
Liquid phase
As described by Lai [16], the ma in transpor t phase
velocities are determined by solution of the momentum
equation, and then the conservation of mass is ensured
by coupling the velocity determination with a pressure
correction, which diminishes with convergence. The
volume fraction for the liquid phase is not explicitly
calculated, rather it is simply calculated as the remaining
volume fraction, once gas and solid phase volume
fractions have been calculated. That is, the liquid volume
fraction, aL is calculated as,
aL = 1 - a c - as (17)
If at any point in the numerical computation of the
phase volume fractions, the local sum of them exceeds
unity, then they are all simultaneously normalized fol-
lowing,
O/i
ai = 3 (18)
~ a j
j - 1
Gas phase
A local flux balance at all nodes in the grid is employed
to determine the gas phase volume fraction at all points
in the flow domain. This is based on an upwind scheme
presented by Spalding [25].
We can express the flux Gi, for the ith phase, as
G=p./lul ,
where A is the area of the face through
which this flux takes place. A local flux balance for
any phase can be written as,
E a iG i = ~o t i G i + R i
(19)
C U T I N
The term R~ refers to a mass generation or consumption
in a finite-volume cell. The adhesion of particles to
bubbles represents a mass transfer from the solid to
gas phase. The derivation of this term is discussed in
section 2.4.7. Any outgoing flux from a cell will represent
the volume fraction oqtp in that cell. The incoming
fluxes will contain the volume fract ions in the respective
neighbouring cells. We can rewrite Eq. (19) in view of
the upwind differencing scheme as,
a~tP ~ Gi =
ZoqG,+R~
(20)
O U T I N
Now, we have a finite volume expression for o~ijpl which
is the unknown volume fraction at the point P, or cell
where this flux balance is applied.
~ a i G i - I- -R i
IN
°t IPl-
~ Gi
C U T
Substituting the above result into Eq. (16) gives,
~aGGG+RG ~aLGL
N N
+ - - + a s = l (21)
c o e L
O U T O U T
Rearranging gives,
~ G L ( ~ a G G ~ + R c ) + ~ G o ~_~C tLG L
OUT OUT N
Now, substituting
(F.IN~aGG+RG)/aGtp
for ~ouTGc
on the right hand side of the above expression, then
rearranging gives,
22)
Eq. (22) may be used in the finite volume scheme for
solving the gas phase volume fraction, in an iterative
fashion, using current values of aL and as to update
aGtPl- Upon inspection, it may be seen that Eq. (22)
is inherently stable, as it has been constructed to
calculate values of aG~Pl which are between 0 and 1.
Solid phase
In a fashion analogous to the previous development
for the gas phase, the mass balance for the solid phase
was derived.
2.4.6. T urbu lent collision mo del
Successful mineral flotation depends on having suit-
able hydrodynamic conditions, or mixing to bring the
bubbles and particles into contact with one another,
as well as the necessary particulate surface properties
to create a lasting adhesion between these particles
and bubbles.
With a statistical approach to the collection of par-
ticles by bubbles, the micro-kinetics of a flotation system
may be expressed. This must be done in view of the
prevailing hydrodynamics of the system [26], which can
be either laminar or turbulent.
Based on Abrahamson s model [27] for parti-
cle-particle collisions in a turbulent fluid, Bischofberger
and Schubert [28] derived the following expression for
the number of collisions per unit volume per unit time
ZpB, in a turbulent flotation system.
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2 2 0.5
Z p B = r , , n p n B d e a V p B + g~B) (23)
H e r e , n p a n d n B a r e r e s p e c t i v e ly t h e n u m b e r c o n c e n -
t r a t i o n o f p a r t i c le s a n d b u b b l e s . T h e t e r m d p B is d e f i n e d
a s ,
d p B = r p + r B
T h e t e r m s V p a n d v B a r e g i v e n b y ,
E4 /9 • d 7 /9 ~ [ A p ~ 2 /3
vi =0 4~ ~ ]kP--LLJ
A b o v e , A p = I P i - P L I , a n d VL i s t h e k in e m a t i c v i s c o s it y
o f t h e l i q u i d p h a s e .
T h e t e r m r , r e p r e s e n t s t h e a d h e s i o n a l p r o b ab i li t y
p a r t o f t h e c o l l i s i o n e x p r e s s i o n . T h e m e c h a n i s t i c b a s i s
o f r~ is r e l a t e d t o t h e s u r f a c e p r o p e r t i e s o f t h e s o l id
p a r t i c l e s a n d i s d i s c u s s e d i n P a r t I I o f t h i s s e r i e s [ 1 2 ] .
T h e r e a r e a n u m b e r o f l i m i t a t i o n s w h i c h a p p l y t o t h e
u s e o f E q . ( 2 3 ).
T h i s e q u a t i o n w a s d e r i v e d f r o m t h e S m o l u c h o w s k i
e q u a t i o n w h i c h p r e d i c t e d t h e c o l l i s i o n r a t e o f t w o
p a r t i c le s u n d e r s h e a r f l o w c o n d i t i o n s . T h e e q u a t i o n i s
u s e d h e r e o n l y i n s o f a r as i t p r o v i d e s a b a s i c t r e n d f o r
t h e c o l l i si o n r a t e o c c u r r i n g a t t h e e a c h n o d e i n t h e
c e l l . I t d o e s n o t e x p l i c i t l y a c c o u n t f o r t h e v a r i o u s p h y s i c o -
c h e m i c a l p r o c e s s e s a s s o c i a t e d w i t h t h e c o l l i s i o n s s u c h
a s f i l m - t h i n n i n g a n d s u r f a c e d e f o r m a t i o n a n d b o u n c e .
T h e t u r b u l e n c e m i x i n g l e n g t h l, o f t h e s y s t e m i s d e f i n e d
as , l = 0 .025D~. W ith a 20 cm s lur ry in le t we hav e l =
0 . 0 0 5 m , a n d t h e b u b b l e d i a m e t e r i s 0 . 0 0 2 m , t h e r e b y
s a t is f y i n g t h e r e q u i r e m e n t t h a t l > d B f o r E q . ( 2 3 ) t o
b e v a l i d . T h e B i s c h o f b e r g e r a n d S c h u b e r t m o d e l w a s
s e l e c t e d f o r t h i s s i m u l a t i o n s i n c e i t c a n b e c o n v e n i e n t l y
a p p l i e d t o a p o p u l a t i o n o f p a r t ic l e s a n d b u b b l e s , m a k i n g
d i r e c t u s e o f t h e s i m u l a t i o n v a r i a b le s w h i c h h a v e b e e n
c a l c u l a t e d t h r o u g h o u t t h e d o m a i n . T h e f l o t a t i o n c o l -
l i s i o n m o d e l i s a p p l i e d t o t h e s i m u l a t i o n i n a f a s h i o n
a n a l o g o u s t o h o w a d r o p c o l l i s i o n a n d c o a l e s c e n c e
m o d e l w a s u s e d f o r s t u d y in g l i q u id s p r a y s [ 2 9,3 0 ]. I n
t h e n u m e r i c a l s i m u l a t i o n , t h e d i s p e r s e d p h a s e s w e r e
m o d e l e d w i t h t h e i r a v e r a g e d i a m e t e r s , s o t h e c o l l i s i o n
r a t e s f r o m E q . ( 2 3 ) w e r e u s e d d i r e c t l y . N a tu r a l l y , i t i s
a s im p l i f i c a ti o n t o u s e m e a n d i a m e t e r s r a t h e r t h a n t o
p e r f o r m t h e s i m u l a t i o n a c co u n t i n g f o r th e b u b b l e a n d
a g g l o m e r a t e s i z e d i s t r i b u t i o n s , b u t f o r t h e p u r p o s e s o f
d e m o n s t r a t i n g t h e i n f l u e n c e o f t h e a g g l o m e r a t e s u r f a c e
p r o p e r t i e s u n d e r p r e v a i l in g h y d r o d y n a m i c s o n o b s e r v e d
f l o t a t i o n t re n d s , t h e p r e s e n t t r e a t m e n t i s s u f fi c ie n t . E q .
( 2 3 ) w a s f o u n d t o g i v e p l a u s i b l e r e s u l t s a n d e q u a l l y
i m p o r t a n t l y , p r o v i d e d a m e a n s t o d e c o u p l e t h e c o l l i -
s i o n a l a n d a t t a c h m e n t p r o b a b i l i ti e s i n v o l v e d in f l o t a ti o n .
A n i m p r o v e d o r m o r e s o p h i s t i c a t e d c o ll i si o n m o d e l
c o u l d b e v e r y e a s i l y s u b s t i t u t e d i n t o t h i s s i m u l a t i o n ,
a n d t h e r e s u l t i n g f u n c t i o n a l r e l a t i o n s h i p s c o u l d b e
r e s o l v e d i n t h e s a m e m a n n e r a s d o n e i n t h i s w o r k .
2 . 4 . 7 . I n t e r p h a s e m a s s t r a n s f e r
T h e c o n s e r v a t i o n o f t h e m a s s o f a p h a s e i s g o v e r n e d
b y t h e t r a n s p o r t E q . ( 2 4 ). F o r t h i s w o r k , i t i s w r i t t e n
in t e r ms o f t h e s o l i d p h a s e , a s t h i s i s u s e d a s t h e
r e f e r e n c e p h a s e w h e r e m a t e r i a l i s l o s t t h r o u g h m a s s
t r a n s f e r t o t h e g a s p h a s e . B e l o w f o s i s t h e d e n s i t y
f r a c t i o n o f t h e s p e c i e s i n t h e p a r t i c u l a t e p h a s e w h i c h
i s p a r t i c i p a t i n g i n t h e m a s s t r a n s f e r [ 3 1] . B y d e f in i t i o n ,
f p s = 1 a t t h e s t a r t o f a n y i t e r a t i o n , a s t h e e n t i r e p a r -
t i c u l a t e p h a s e i s c o n s id e r e d t o b e c o a l - - o i l a g g lo me r a t e s
w h i c h a r e i n v o l v e d i n t h e m a s s t r a n s f e r .
- - + ( f~suj +Js j ) . j Rs (24)
at
A b o v e , J sj i s t h e d i f f u s io n f lu x v e c to r f o r t h e s o l i d
p h a s e , a n d R s i s t h e r a t e o f c o n s u m p t i o n o f t h e s o l id
p h a s e , i n u n i t s o f k g / s. T h e t e r m J sj r e f e rs t o m o l e c u l a r
d i f fu s i o n , a n d c a n b e n e g l e c t e d f o r t h i s s y s t e m w h i c h
m o d e l s l a r g e d i s p e r s e d b o d i e s . S i n c e a s t e a d y - s t a t e
s y s t e m is b e i n g s i m u l a t e d , t h e t i m e d e r iv a t iv e t e r m m a y
a l s o b e d r o p p e d . T h u s E q . ( 2 4 ) s imp l i fi e s t o ,
Rs =fpsuj.~ (25)
T h e r a t e t e r m R s i s c a l c u l a t e d u s i n g t h e r a t e o f
c o l l i s i o n t e r m z p B f r o m E q . ( 2 3 ) , w h ic h g iv e s a v a lu e
in u n i t s o f [ c o l li s i o n s s - 1 m- 3 ] . T h u s ,
R s = r ~ N e m M A x - - N e m ) Z K B VC E L Lm S (26)
T h e r e a r e a n u m b e r o f a d d i t i o n a l f a c t o r s i n t h e a b o v e
e x p r e s s io n f o r R s . T h e t e r m ,
Np~MA×-Np~)
e x p r e s s e s t h e f r a c t i o n o f t h e b u b b l e s u r f a c e w h ic h i s
s t il l a v a i l a b l e t o c o n t a c t a p a r t i c l e . V C EL L s t h e v o lu me
o f t h e l o c a l c e l l w h e r e t h i s c o l l i s i o n r a t e i s a p p l i e d ,
a n d ms i s t h e m a s s o f o n e s o l i d p a r t i c l e .
S o l u t i o n o f E q . ( 2 5 ) w il l p r o d u c e a n a r r a y o f u p d a t e d
s o l id p h a s e d e n s i t y f r a c t i o n s f ~ s, w i t h v a l u e s b e t w e e n
0 a n d 1 , w h i c h r e p r e s e n t s t h e f r a c t i o n o f t h e s o l i d
p h a s e r e m a i n i n g a f t e r s o m e o f i t h a s t r a n s f e r r e d i n t o
t h e g a s p h a s e u n d e r t h e p r e v a i li n g t r a n s p o r t c o n d i t i o n s .
W e s e t ,
R =fk~
w h e r e R i i s t h e m a s s g e n e r a t i o n o r c o n s u m p t i o n t e r m
f o r t h e i t h p h a s e , u s e d i n t h e e q u a t i o n s i n s e c t i o n 2 . 4 . 5
w h i c h d e t e r m i n e t h e l o c a l v o l u m e f r a c t i o n s o f t h e
d i s p e r s e d p h a s e s a c c o u n t i n g f o r t h e t r a n s p o r t c o n d i t i o n s
w i t h s i m u l t a n e o u s m a s s t r a n s f e r .
E f f e c t i v e m a s s t r a n s f e r
T h e p r o b l e m b e i n g m o d e l e d i n t h i s w o r k i n v o l v e s
t r a n s f e r o f m a s s f r o m t h e s o l i d p h a s e t o t h e g a s p h a s e .
T h i s r e q u i r e s t h a t t h e e f fe c ti v e b u b b l e d i a m e t e r m u s t
i n c r e a s e a s ma s s i s t r a n s f e r r e d . A n a v e r a g e b u b b l e s i z e
c a n b e c o m p u t e d f o r e a c h c e l l i n t h e f l o w f i e l d . T o
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d o t h i s , w e e mp l o y t h e me t h o d o f Sp a l d i n g [ 2 5 ] k n o w n
a s t h e
shadow
so lu t ion .
T h e s h a d o w v o l u m e f r a c t io n &c , is t h e v o l u me f r a c t i o n
t h e g a s p h a s e w o u l d h a v e p o s s e s s e d , a t e a c h n o d e , i f
t h e i n t e r p h a s e m a s s t r a n s f e r h a d n o t t a k e n p l ac e . T h a t
is , E q . ( 1 9 ) is e mp l o y e d w i t h o u t t h e R i te r m . H o w e v e r ,
s in c e th e s e e q u a t i o n s r e p r e s e n t a c o n ti n u u m w h e r e t h e
o u t g o i n g f l u x e s mu s t b a l a n c e t h o s e c o m i n g in , E q . ( 2 2 )
i s mo d i f i e d t o th e f o r m b e l o w f o r d e t e r mi n i n g t h e
s h a d o w v o l u me f r a c t i o n .
, o P , o o o )
27)
T h e d i f f e r e n c e b e t w e e n a o a n d & c c a n b e a t t r i b u t e d
t o d i a me t e r g r o w t h v i a ma s s t r a n s f e r . T h u s ,
dB [ \ 113
A b o v e , d s 0 is th e d i a m e t e r o f t h e b u b b l e i n t h e p a r t i c u l a r
c e l l a t t h e s t a r t o f t h e i t e r a t i o n . Fu r t h e r , i t c a n b e
d e d u c e d t h a t ,
Vo - Voo
MVpm (29)
s
s i n c e t h e g a i n i n g a s p h a s e v o l u me i s a t t h e e x p e n s e
o f t h e s o l i d p h a s e . T h e t e r m V~ r e f e r s t o t h e v o l u m e
o f o n e b o d y ( b u b b l e o r p a r t i c l e ) f r o m e i t h e r t h e g a s
o r s o l i d p h a s e .
Fina l ly , as wi l l be d i scussed in the nex t sec t ion , t he
e x t e n t o f s o l i d a d h e s i o n t o b u b b l e s w a s c o n s t r a i n e d .
T h u s , i t w a s a s s u me d t h a t t h e g a s - p h a s e s o l i d l o a d i n g
a n d c o r r e s p o n d i n g d e n s i ty i n c r e a se w e r e s m a l l e n o u g h
t o a l l o w t h e s i mp l i f i e d g a s p h a s e t r a n s p o r t e q u a t i o n
(Eq . 9 ) to s t i l l app ly .
2 4 8 Drag on a loaded bubble
Bubble loading
T h i s s i mu l a t i o n h a s i n c o r p o r a t e d a mo d i f i e d d r a g o n
a b u b b l e w h i c h h a s p a r t i c l e s a d h e r i n g t o i t . T w o c o n -
s t r a i n ts w e r e i m p l e m e n t e d t o c o n t r o l th e l e v e l o f p a r -
t i c le s a d h e r i n g t o a s in g l e b u b b l e . T h e m a x i mu m b u b b l e
l o a d i n g i s c a l c u l a t e d b y a s s u mi n g t h a t c i r c u l a r p r o j e c -
t i o n s o f t h e a r e a s o f t h e p a r t i c l e s ma y o c c u p y th e
s u r f a c e o f th e b u b b l e . T h e p a r t i c le s a r e m o d e l e d t o
p a c k i n a t w o - d i me n s i o n a l h e x a g o n a l l a t t i c e , t h e r e b y
c o v e r in g a m a x i m u m o f 9 0 .7 o f t h e b u b b l e ' s g e o -
m e t r i c a l a r e a . T h e m a x i m u m n u m b e r o f p a rt ic l e s p e r
bubble, NpmM,,×, i s thus ,
NemM,,X = 0-907(r4~2B)
30)
A s e c o n d c o n s t r a i n t a r i s e s f r o m c o n s i d e r i n g t h e r e l -
a t i v e f l u x e s o f t h e g a s a n d p a r t i c u l a t e p h a s e s i n t h e
f l o a t c e ll . N u m e r i c a l i n s ta b i li t y c o u l d a r i s e i f t h e b u b b l e s
b e c a m e t o o h e a v i ly l o a d e d a n d g a v e o u t l e t p a r t ic u l a t e
f luxes g re a te r than the i r i n l e t f luxes . Assu min g a l l t he
i n l e t g a s e x it s in t h e p r o d u c t s t r e a m ( n o t v i a t h e l i q u id
e x it ) th e n t h e m a x i m u m n u m b e r o f p a r ti c le s p e r b u b b l e
b e c o m e s ,
gs PG Vo
N~/BM.x = - - 31)
gopsVs
Ab ov e g i re fe rs to th e f lux o f the i th ph ase , and V~
r e f e r s t o t h e v o l u me o f o n e d i s p e r s e d b o d y . I n p r a c t i c e ,
i f a m ass - f lux ins t ab i l i t y a rose , NemM^x was rep la ced
by N~,mM^x in the algori thm .
Particle patches
Pa r t i c l e s a r e mo d e l e d t o f o r m c o n c e n t r i c h e x a g o n s
o n t h e b a c k s i d e o f mo v i n g b u b b l e s a s s h o w n i n F i g .
1 . T h e s e h e x a g o n s a r e d e n o t e d a s l e v e l s , a n d t h e t o t a l
num ber o f par t i c l es a t each l eve l , n i s,
N I = I
n
N , = 1 + ~ ] 6 q - 1 ) ( 32 )
j 2
Particle patch drag model
T h e p a r t i c l e p a t c h w a s t h e n a p p l i e d t o a mo d e l
d e v e l o p e d b y Pe n d s e e t a l . [ 32 ], w h i c h w a s f o r d r a g
f o r c e c a l c u l a t i o n s o f s p h e r i c a l c o l l e c t o r s w i th d e p o s i t e d
par t i c l es in var iou s conf igura t ions . Fo r cases wi th severa l
p a r t i c l e s a t t a c h e d , t h e h y d r o d y n a m i c i n t e r a c t i o n a m o n g
t h e a t t a c h e d p a r t i c l e s mu s t b e c o n s i d e r e d .
F o r o u r c a s e , w e m a k e t h e a s s u m p t i o n t h a t l o a d e d
b u b b l e s t r a n s p o r t e d b y f a s t - mo v i n g w a t e r w i l l o r i e n t
N : N i n t e r a c t i o n s
. .. .. .. .. .. .. N : N 1 i n t e r a c t i o n s
F i g . 1 . S c h e m a t i c p a r t i c l e p a t c h .
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K. Dar cov ich / P ow der Techno logy 83 ( 1995) 211 - 224
themselves with the particle patch at the rear of the
bubble, in line with the velocity vector of the bubble.
The drag effects are a function of the position of the
particle relative to the bubble's line of motion, as well
as the product of all the interaction terms from all the
attached particles.
Pendse et al. [32] give the following expression for
a multi-particle interaction which augments the drag
force.
AF D .M j=A Fo .o f(PM ,.,, PM 2.,, PM2.2, ...PMN .j)
(33)
Above, AFD.~ represents the increment of the jth
particle from the Mth level on the drag of a bubble
carrying such a number of particles. This increment is
normalized with respect to the drag of an unloaded
bubble. The function
( P M , , a , P M 2 , , , P M 2 . 2 , . P M , ~ . i )
rep-
resents the interaction of all attached particles with
each other. Thus, the overall change in drag for a
loaded bubble o f N levels, with j particles a t this level,
AFD[N:j], will be,
N L
~FD [N. . jl=Fo. , , + ~ ~ AFo.~,
(34)
M = 2 i ~ l
L = 16 (M - 1) for
M < N
where,
t
for
M = N
The interaction term for only two particles was given
in a derivation by Happe l and Brenner [33]. The
complete derivation of the expressions used to arrive
at this loaded bubble drag modifier is given in [34].
As an example, for a bubble of 2.0 mm diameter
and a particle of 60/zm diameter, the calculated drag
coefficients are illustrated in Fig. 2. These calculations
agree with the basic results of Pendse et al. [32] which
they only calculated for cases with one and two attached
particles.
2.4.9. Viscosity effects
The presence of up to three phases in variable
concentration throughout the flow domain necessitates
a means for specifying a basic laminar viscosity in each
2 . 0 m m B u b b l e 6 0 / ~ m P a r t i c l e s
1 . 0 1 0
1 . 0 0 8
G
1 . 0 0 e
e
c
¢ ~ 1 . 0 0 4
CIt
m
a
1 . 0 0 2
1 . 0 0 @ ~
2 4 e 8
A t t a c h e d P a r t i c l e s
F i g . 2. D r a g f o r c e c o e f f i c i e n t s v s. b u b b l e l o a d i n g .
1
cell. The viscosity of a slurry is known to increase with
solids concentration. The presence of air tends to give
a much re duced viscosity, along the lines of a weighted
average of the slurry and air viscosities. Since the
viscosity of air is about three orders of magni tude below
that of water, its effect is neglected [35]. That is,
/ ' / TOTAL = (1 - aO)/XSLURRV
A model was incorporated into the simulation to
account for variations in the laminar viscosity arising
from local conditions. The two effects considered were
those of particle size and particle concentration. Data
from Borghesani [36] was fitted with a multivariate
linear response surface. The response function was
reduced
(or normalized) viscosity, which is the viscosity
of the slurry, divided by the viscosity of the pure
suspending liquid. Hence, this provides a local coef-
ficient, k,,, for the laminar liquid viscosity which is used
in the program.
The analysis of the residuals showed an average error
of about 8% over the entire range of the model, which
extended from 0 to 40 v/o solids concentrations and
particle diameters from 1 to 450/zm. A model of this
nature was considered adequate to convey any trends
associated with the rheological properties of the 3-
phase system.
3 N u m e r i c a l i m p l e m e n t a t i o n
The solution of the discretized differential equations
is done via a hybrid of central and upwind differencing
in the spatial dimensions, thereby permitting accurate
computation over an extended range of Reynolds and
Peclet numbers. Each finite volume equation is solved
sequentially, in an iterative manner. All of the previous
equations which have been expressed as transport equa-
tions have assumed a similar format. In fact, any of
the transport variables may be represented by a gen-
eralized transport variable, ~b. q5 can be interpreted as
either a time-average or a density-weighted average.
In any case, the form of the equations are all identical.
The different averages need not be indicated, as tur-
bulence parameters along with the mean flow variables
account for the entire flow field description.
In our general form, we have,
a(p4,) + (pu, 4, - F,4,. ,). ,= S, (35)
8t
where F6 and 6 are coefficients respectively known
as the effective exchange coefficient and volumetric
source rate of the variable .~b.
The computer code developed to implement this
finite volume formulation is quite modular in structure,
allowing simple specifications to dictate steady or un-
steady, laminar or turbulent, compressible or incom-
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pressible, reacting or stable states as well as heat and
mass transfer conditions. Grids defining the flow domain
may be prescribed in either Cartesian or cylindrical
coordinates.
The formulations given above, as well as the numerical
devices detailed below, were incorporated into a com-
puter code known as MO-PrIASE (multiple, dispersed
phases), which was based on previous I~MA and rOR-
COM codes [16].
3 1 The grid system
The flow domain is divided into control volumes, or
for this two-dimensional case, control regions, defined
by the coordinate-based grid lines. Fig. 3 illustrates
this finite volume grid system. Based on the method
of Gosman and Ideriah [37], a staggered grid is used
to enhance numerical stability. Scalar qualities are
determined at the grid points, and the nodes for the
velocities (or vector quantities in general) are displaced
in their respective directions to the mid-point of the
scalar nodes along the grid lines. Scalar control regions
are centered about the point P as indicated in Fig. 3,
while the momentum control regions are centered about
the points U and V for the respective velocity direction.
This particular grid arrangement is well suited for
imposing accurate boundary conditions since the mid-
point locations for vector variables coincide with those
of the normal velocity components, and thus prescribed
boundary values and/or fluxes need not be modified.
3 2 Discretization proced ure
The differential equations which govern the physics
of the flow are integrated over the volume of the flow
domain. The entire domain is partitioned into cells
delimited by the grid, and the unknown quantities are
taken to be constant within each cell, with this mean
value assigned to the node point.
P V g r i d l i n e U g r i d l i n e
~ / /U
c e l l b o u n d a r y
- i i
i l l i
i i :
c e b o u n r y
P , V c e l l bound ry
c o n t r o l v o l u m e f o r n o d e
P U c e l l b o u n d a r y
g r i d l i n e
P U c e l l b o u n d a r y
V c e l l b o u n d a r y
r i d l i n e
V c e l l b o u n d a r y
F i g . 3 . T h e f i n i t e - v o l u m e c o n t r o l r e g i o n f o r a t w o - d i m e n s i o n a l C a r t e -
s i a n s y s t e m .
The general transport equation (Eq. (35)), can be
written for the generalized variable, ~b, over one cell
volume, Vc, with a time step of &. We can thus write,
d-- (P4')dV+ f (Ou '4'-r*4 3nidA= S*
(36)
Vc A c Vc
Above, nl is the unit outward normal from the control
surface, Ac. The upwind differencing technique is then
applied to this integral form of the transport equation
to produce a system of linear equations for the field
variables at each node in the domain. The details of
such procedures are given explicitly in [16].
3 3 Equ ation solution algorithm
Beginning with the initial values for all the field
variables, the governing equations are solved in sequence
until the convergence criteria are satisfied. These are
referred to as the outer iterations.
Each linear equation, which is formulated over the
domain for each variable, is solved by an inner iteration,
an efficient scheme for treating a large matrix.
3 3 1 Nu mer ical solution procedure
The matrix which is assembled for each field variable
equation is solved by a block iteration technique. This
procedure sweeps across planes defined by coordinate
index. A tri-diagonal matrix form is assembled locally
as in the Gauss-Seidel method. This matrix can be
solved implicitly. Details are given in [16]. The plane
is solved grid line by grid line, and is repeated in
alternating directions towards convergence. Since this
is the inner iteration, full convergence is not required,
as coupled variables are being simultaneously modified.
The outer iterations ensure full final convergence, so
usually no more than three inner sweeps are performed
for each variable.
3 3 2 Execu tion sequence
As seen, the governing finite difference equations
are coupled and non-linear, thereby requiring an it-
erative method of solution. In each outer iteration, the
equations are individually solved with inner iterations
in a sequential fashion, adjusting each variable by its
own equation, while holding others constant. The steps
comprising the numerical algorithm are itemized below.
(i) The fields of all variables were initialized either
by guess or calculation.
(ii) The liquid phase momentu m equation was solved.
(iii) The pressure equation was solved, and pressure
corrections (to ensure mass balance) were applied to
liquid velocity field.
(iv) The turbulence parameters were determined.
(v) Gas-phase momentum equation was solved, and
gas volume fractions were calculated.
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( vi ) S o l i d - p h a s e m o m e n t u m e q u a t i o n w a s s o lv e d , a n d
s o li d v o l u m e f r a ct i o n s a n d s h a d o w s o li d v o l u m e f ra c t i o n
w e r e c a l c u l a t e d .
( v ii ) T h e e x t e n t o f s o l id s c o n v e r s i o n w a s c a l c u l a t e d
a n d u s e d t o u p d a t e t h e g a s a n d s o li d v o l u m e f r a c t i o n s
a n d t h e g a s p h a s e s o l i d s l o a d i n g a n d n e t d i a m e t e r .
( v ii i) S t e p s 2 t h r o u g h 7 w e r e r e p e a t e d u n t i l th e
c o n v e r g e n c e c r i t e r i a w e r e s a t i s f i e d .
3 . 3 . 3 . C o n v e r g e n c e c r i t e r i a
T h e r e s i d u a l s o u r c e s o f e a c h f i n i t e v o l u m e e q u a t i o n
f o r e a c h v a r i a b le w e r e c o m p a r e d t o a r e f e r e n c e t o l e r a n c e
t o a s se s s t h e c o n v e r g e n c e o f e a c h e q u a t i o n . T h i s r e f -
e r e n c e t o l e r a n c e R , , R E F, i s s e l e c t e d a s t h e i n l e t f lu x
o f th e d e p e n d e n t v a r i a b l e ¢ . T h i s c o n v e r g e n c e c r it e r i o n
i n d i c a t e s h o w w e l l t h e c u r r e n t v a l u e s o f ¢ s o l v e t h e
e q u a t i o n i n q u e s t i o n , r a t h e r t h a n h o w m u c h t h e v a l u e s
o f ~b h a v e c h a n g e d o v e r s u c c e ss i v e i t e r a t io n s .
R e f e r r i n g t o E q . 3 5 , w h i c h i s t h e g e n e r a l i z e d f i n i t e
v o l u m e e q u a t i o n , t h e l o c a l r e s id u a l f o r t h e ¢ e q u a t i o n
f o r t h e i t h i t e r a t i o n i s d e f i n e d a s ,
0 p ~ ) + p U i ~ - - F ~ b ~ , i ) , i - - S ~ b 3 7 )
R = a t
I t c a n b e s e e n t h a t w h e n t h e s o l u t i o n i s o b t a i n e d , R ,
g o e s t o z e r o . C o n v e r g e n c e i s c o n s i d e r e d s a t i s f a c t o r y i f
t h e s u m o f th e n o r m a l i z e d a b s o l u t e r e s i d u a l s h a s f a ll e n
b e l o w a s p e c i f i e d v a l u e A , i n t h i s c a s e s e l e c t e d a s 1 0 - 1 .
T h a t i s f o r e a c h ¢ e q u a t i o n ,
E EIR , l , . J l l < XR, , ~ (38)
i j
3 .4 . B o u n d a r y c o n d i t io n s a n d s im u l a t i o n i n p u ts
F o r t h e f l o t a t i o n s i m u l a t i o n t h e b o u n d a r y c o n d i t i o n s
a r e s h o w n i n F i g . 4 . T h e c e l l i s m o d e l e d a f t e r a
c o m m e r c i a l c e ll m a d e b y M i n p r o o f M i ss is s au g a , C a n a d a
a n d h a n d l e s a s o li d s th r o u g h p u t o f 4 .3 2 t o n n e s / d a y a n d
a n a i r r a t e o f 1 . 5 m 3 / m i n . I t s d i m e n s i o n s a r e g i v e n i n
T a b l e 3 . T h e t o p f a c e o f t h e f l o a t c el l is c o n s i d e r e d
f la t a n d o p e n t o t h e a t m o s p h e r e . A z e r o r e d u c e d
p r e s s u r e is im p o s e d h e r e a l o n g w i th t h e c o n d i t io n t h a t
t h e g a s p h a s e m a y e x i t i n t h e v e r t i c a l d i r e c t i o n , w h i l e
U L a n d U s w e r e s e t t o z e r o .
T u r b u l e n c e c o n d i t i o n s a t in l e ts w e r e m o d e l e d a f t e r
e q u a t i o n s g i v e n i n [ 1 6 ] . C o n c r e t e l y ,
I = 0 . 0 0 5 l = 0 . 0 2 5 D i
kl.5
k = I u 2 e =
l
A b o v e , I is t h e t u r b u l e n c e i n t e n s i t y , a n d l i s t h e t u r -
b u l e n c e m i x i n g l e n g t h , g iv e n i n t e r m s o f t h e i n l e t
d i a m e t e r D i .
O v j = 0 . 0 u L = 0 . 0 , u ~ = 0 . 0 , O u ~ = 0 . 0
0y 0x
U ~ , v j = 0 . 0
o n a l l w a l l s
v ~ = 0 . 0 5
v s = 0 . 0 . 5
a s = 0 . 1 0
= u G = 0 . 0 5
v c = 0 . 5 0
F ig . 4 . B o u n d a r y c o n d i t i o n s fo r t h e M D - P H A S E f l o t at i o n s i m u l a t io n .
T a b l e 3
F l o a t c e l l d i m e n s i o n s
P a r a m e t e r S i z e
W i d t h 1 . 4 2 m
H e i g h t 1 . 6 m
S p a r g e r w i d t h 5 2 . 7 c m
S l u r r y i n le t d i a m e t e r 2 0 c m
W e i r d r o p 5 - 1 5 c m
T a b l e 4
P h y s i c a l p r o p e r t i e s o f th e t h r e e p h a s e s
P h a s e D e n s i t y V i s c o si t y D i a m e t e r
L i q u i d 9 9 7 . 1 8 . 9 4 × 1 0 - 4 N/A a
G a s 1 . 2 1 . 4 2 × 1 0 - 6 2 . 5 m m
S o l i d 1 2 0 0 . 0 N / A ~ 2 0 - 6 0 / ~m
N/A n o t a p p l i c a b l e .
T h e r e a r e s o m e a d d i t i o n al b o u n d a r y t r e a t m e n t s w h i c h
a r e i n c l u d e d i n t h e f o r m u l a t i o n t o p r o d u c e m o r e s t a b le
n u m e r i c a l b e h a v i o u r . T h e s e a r e d i s c u s s e d b y L a i a n d
S a l c u d e a n [ 1 9 ] .
T h e t h r e e p h a s e s i n t h i s s i m u l a t i o n w e r e i n p u t t h e
p h y s i c a l p r o p e r t i e s l i s t e d i n T a b l e 4 . T h e u n i t s a r e
b a s e d i n t h e m k s s y s t e m .
T h e g r i d u s e d f o r t h e s i m u l a t i o n i s s h o w n i n F i g . 5 .
T h e r e g i o n s o f h i g h s w i r l o r w h e r e s t r e a m s m i x h a v e
m o r e g r i d l in e s to e n h a n c e t h e r e s o l u t i o n . G r i d s e n s it i v it y
w a s n o t i n v e s t i g a t e d i n t h i s w o r k . T h e i n h e r e n t e r r o r
a n d l i m i t a t io n s o f t h i s s im u l a t i o n ( i e, s i m u l a t i o n i n 2 D
r a t h e r t h a n 3 D , w i t h s i m p l i f i e d g e o m e t r y ) r e n d e r t h e
r e s u l ts u s e f u l i n t h e s e n s e o f h o w t h e y i l l u m i n a t e t h e
s i m u l t a n e o u s i n t e r p l a y o f t h e v a r i o u s f l o t a t io n m e c h -
a n is m s . T h e p r e s e n t s i m u la t io n w o u l d n e e d t o b e r u n
i n t h r e e d i m e n s i o n s f o r a c t u a l e n g i n e e r i n g d e s i g n w o r k .
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F i g . 5 . F i n i te v o l u m e g r i d u s e d f o r f lo t a t i o n c e l l.
I : : : : : : : : : : :
~ - ~ ' ~ - - ~ ~ :~
• r
t ~ t / ¢ t / t I / I [
t i t ~ t y ~ / l , ' / t l
~
F i g . 6 . T h r e e - p h a s e f l o t a ti o n v e l o c i t y a n d p r e s s u r e f i e ld s . dp= 43 .6
p m a n d r~,=0.20.)
4 . R es u l t s
A d a ta s e t fo r th e s tu d y o f th e f l o ta t i o n p a ra meters
w a s co n s tru c ted b y ru n n i n g th e th ree -p h a s e s i mu l a t i o n
fo r a co mb i n a t i o n o f p a r t i c l e d i a meters a n d v a l u es fo r
th e a t ta ch m en t e f f i c ien cy , r , , w h i ch w a s i n tro d u ced i n
E q . (2 6 ) . T h i s d a ta s e t w a s th en u s ed to d ed u ce a
re l a t i o n b e tw een th e p ro p er t i e s o f th e f l o a ted ma ter i a l
a n d i t s r eco v ery . T h e s u r fa ce ch emi ca l a s p ec t o f th i s
project i s deta i led in Part I I o f th i s paper .
R u n - t i me co n tro l w a s a ch i ev ed b y ru n n i n g a l l th ree
p h a s es a t o n ce , b u t w i th th e f req u en cy o f i t e ra t i o n fo r
th e d i s p ers ed p h a s es red u ced u n t i l a s ta b l e b a s i c f l o w
fie ld pattern based on l iquid transport was es tabl i shed.
T h e f req u en cy o f i t e ra t i o n fo r th e d i s p ers ed p h a s es
was gradual ly increased unt i l a ra t io o f 1 :1 was s table ,
a n d co n v erg en ce co u l d b e a ch i ev ed .
F i g s. 6 -8 s h o w a fu l l th ree -p h a s e f l o ta t i o n f l o w f ie l d
s i mu l a t i o n re s u l t . T h i s ca s e h a d d p = 4 3 .6 / ~ m co rre -
s p o n d i n g to a n co a l -o i l a g g l o mera te ma d e w i th 2 w t .
o i l and the a ttachm ent ef f ic iency , ra = 0 .20 . The pressure
f i e l d s h o w n i n F i g . 6 s h o w s th e d y n a mi c p res s u re .
For the example i l lus trated in these f igures , the f lux
o f s o l i d s a cco mp a n y i n g th e g a s p h a s e f ro m th e to p ex i t
was 0 .447 kg /s , g iv ing a f lo ta t ion recovery for th is case
o f 3 7 . 2 .
For the o ther cases run, the graphica l output i s very
s i mi l a r . T h e s a me b a s i c t ren d s a re fo l l o w ed . W h a t ca n
be not iced i s s l ight s treaml ine di f ferences for heav ier
p a r t i cl e s. T w o ca s e s a re s h o w n i n F i g s. 9 a n d 1 0. A n o th e r
feature ev ident in F igs . 9 and 10 i s that for the 60 / .~m
case , the s treaml ines travel a t the top o f the ce l l in
th e y -d i rec t i o n a s th ey ex i t o v er th e w e i r . T h e i r g rea ter
s e t t l i n g t en d en cy d o es n o t en a b l e th e f l u x i n th i s ca s e
to t ra n sp o r t th em u p o v er th e w e i r a s w i th th e 2 0 / ~ m
part ic les . A s tronger c ircula t ion i s es tabl i shed in the
top h a l f o f the ce l l to create a current o f suff ic ient
f lux to carry the unf loated so l ids out over the weir .
Add i t iona l ly , F igs . 11 and 12 sho w s treaml ines ca l -
cu l a ted fo r b u b b l e d i a m eters o f 2 .5 a n d 1 .0 mm re -
s p ec t iv e l y. T h e p a r t i c le d i a m eter i n th i s ca s e i s 6 0 / ~ m .
T h e g rea ter b u o y an cy fo rce o n th e l arg er b u b b l e s ren d ers
them less subject to the trajectory dev ia t ions brought
about by the s lurry motion.
g a s f r a c t i o n
K e Y TO C O N T O U R V A L U e 8
0 . 6 0 0
0 4 0 0
0 8 0 0
O.JO0
0 1 0 0 0
0.0000
p a r t i c l e f r a c t i o n
E l Y ? 0 C O N T O U R V A L U B I
0 8 0 0
0 1 | 0
o o s o o
o o 8 0 o
O.OlO00
0 . 0 0 0 0 0
o o e t o o
b u b b l e l o a d
I C gY T O C O N T O U R V A L U E S
0 0 0 0
4 0 0 0
0 0 0 0
0 0 0 0
1 ooo
o . s e o
Q
I
\ \
\ J
/ /
F i g . 7 . T h r e e - p h a s e f l o t a ti o n c o n c e n t r a t i o n a n d m a s s t r a n s f e r c o n t o u r s . d p = 4 3 . 6 ~ m a n d r a = 0 . 2 0 . )
_o
e~
.Q
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F i g. 8. T h r e e - p h a s e f l o t at i o n t u r b u l e n c e c o n t o u r s. d p = 4 3 .6 ~ m a n d
r o = 0 . 20 . )
F i g . 1 1 . B u b b l e s t r e a m l i n e , d B = 2 . 5 m m .
F i g . 9 . P a r t i c l e s t r e a m l i n e , dp=20 ixm.
F i g . 1 0 . P a r t i c l e s t r e a m l i n e , dp=60 lzm.
4 1 In t eg r a t e d f l o t a t io n mo d e l
T h e C F D w o r k c a r r i e d o u t a s p a r t o f t hi s p r o j e c t
e n a b l e d a n i n t e g r a t e d s t u d y o f m i n e r a l r e c o v e r y by
f l o t a ti o n t o b e a c h i e v e d . T h a t i s t h e o u t p u t f r o m t h e
u n i t o p e r a t i o n w a s d e t e r m i n e d a s a s u m o f a ll l o c a l
m a s s t r a n s f e r o c c u r r i n g t h r o u g h o u t t h e d o m a i n . I n t h is
w a y t h e e f f e c t s w h i c h w e r e o f a s p e ci f ic a l ly h y d r o -
d y n a m i c n a t u r e s u c h a s d i s p e r s e d p h a s e d i a m e t e r s a n d
d e n s i t i e s t h e p r e v a i l in g fl o w p a t t e r n s a n d l o c a l t u r -
b u l e n c e a c c o u n t e d f o r t h e c o l l i s i o n s l e a d i n g t o p o s s i b l e
F i g . 1 2 . B u b b l e s t r e a m l i n e , d B = 1 .0 m m .
s u b s e q u e n t m i n e r a l c o l le c t i o n . T h u s t h i s p r o v i d e d a
m e a n s t o d e c o u p l e s u r f a c e c h e m i c a l e f f e c ts w h i c h w e r e
r e s p o n s i b l e f o r t h e a d h e s i o n o f t h e p a r t i c l e s t o th e
b u b b l e s o n c e a c o l l i s i o n h a d t a k e n p l a c e . T h i s i s f u l l y
d i s c u s s e d i n P a r t I I o f t h i s p a p e r .
5 C o n c l u s i o n s
A t w o - d i m e n s i o n a l f in i te v o l u m e c o d e f o r t h r e e -p h a s e
f lo w w a s f o r m u l a t e d a n d i m p l e m e n t e d t o s i m u l a te t h e
p r o c e s s o f m i n e r a l f l o ta t io n . T h i s m o d e l i n c o r p o r a t e d
a t u r b u l e n t b u b b l e - p a r t i c l e c o l l i s i o n m o d e l g i v in g r i s e
t o i n t e r p h a s e m a s s t r a n s f e r w h i c h d e f i n e d t h e f l o t a t i o n
p r o c e s s r e c o v e r y . T h e s e e f f e c t s w e r e s u c c e s s f u ll y c o u p l e d
w i t h t h e m a s s b a l a n c e e q u a t i o n s f o r e a c h p h a s e . A l s o
i n c l u d e d w e r e t h r e e - p h a s e r h e o l o g i c al e f f ec t s a n d a
m o d i f i e d d r a g c o e f f i c ie n t f o r t h e g a s p h a s e w h e n s m a l l
s o l i d p a r t i c l e s a d h e r e t o b u b b l e s u r f a c e s .
T h e n o v e l u se o f a C F D f l o ta t i o n m o d e l p r o v i d e d
a n i n t e g r a t e d s i m u l a t i o n a l lo w i n g r e c o v e r y p r e d i c t i o n s
o n f u n d a m e n t a l m a t e r i a l p r o p e r t ie s a n d u s i n g a m i c r o -
f l o t a t i o n c o l l is i o n m o d e l i n a m a c r o s c o p i c h y d r o d y n a m i c
8/19/2019 A Hydrodynamic Simulation of Mineral Flotation. Part I- The
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K. Darcovich / Powder Technology 83 1995) 211-22 4 223
context where all local variables are prescribed. This
ensured soundness of results and an adaptability to
various behaviour regimes. The present simulation has
been shown to be suitable for making a parametric
study of flotation. Part II of this series makes use of
resolved hydrodynamics to illuminate attachment phe-
nomena for coal-oil agglomerate flotation.
The flow model itself is in a generalized form, so
the basic framework is in place to convert to three
dimensions. In three dimensions, the full effects of cell
geometries come into resolution, and as such, the cell
configuration and the turbulence parameters would be
highly useful for actual engineering design work.
S ~
u i
l) B
l )p
Vc
V ~
Vo
V ~
s
Z p B
ubscripts
volumetric source rate
time
velocity
bubble turbulent collision velocity
particle turbulent collision velocity
finite difference cell volume
volume of local cell
volume of one bubble
volume of one bubble at start of iteration
volume of one particle
collisions per unit volume per unit time
6 L i s t o f s y m b o l s
A
B
C o
C,
C 1 C 2 C 3
dB
d B o
dp
deB
D i
E
3ri
Gi
I
j ~
k
k,
1
m s
n i
nB
n p
Npm
P
P
rB
rp
ra
R t
R i
R,b
R,[,, Jl
R ~ b R E F
Re
Res
area of cell face for flux
coefficient for liquid-solid drag
drag coefficient
k-E model constant
k--E model constants
bubble diameter
bubble diameter at start of iteration
particle diameter
rp+ra
inlet diameter
turbulence model wall velocity constant
dispersed phase-liquid interphase drag term
curvature related source terms
flux of ith phase
turbulence intensity
diffusion flux vector for Jth phase
turbulence kinetic energy
laminar slurry viscosity coefficient
turbulence length scale
mass of one solid particle
unit outward normal
number concentration of bubbles
number concentration of particles
bubble loading, particles per bubble
maximum bubble loading under mass-flux
constraint
pressure
finite-volume node point
bubble radius
particle radius
attachment efficiency
rate of flotation
mass generation or consumption of ith
phase
residual for ~b equation
local nodal residual for 4)
reference residual for 4 equation
Reynolds number
particle Reynolds number
G , L , S
i , j , k
[p]
t
gas, liquid or solid phase
tensor spatial indices
finite volume node point
turbulent quantity
Superscripts
I
m
mean turbulent fluctuation
density-weighted ensemble-averaged
shadow volume fraction quantity
Greek letters
li
AFD
~ k F D 0
E
r~
/ (
A
/z
/d,eff
VL
P
Ap
(rk
r~j
ith phase volume fraction
increment on drag due to jth particle in
Mth level
overall change in drag for a loaded bubble
of N levels, with j particles
bubble drag increase due to one adhering
particle
turbulence energy dissipation
effective exchange coefficient
turbulence kinetic energy
convergence coefficient
laminar viscosity
effective viscosity, composed of sum of lam-
inar and turbulent contributions
kinematic viscosity of the liquid phase
normalized liquid and solid phase volume
fractions
density
Ip -pLI
k - e
model constant
k - e
model constant
shear stress tensor
generalized transport variable
e f e r e n c e s
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