A Hydrodynamic Simulation of Mineral Flotation. Part I- The

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 ] ,


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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.


3/14

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


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 ,


5/14

K . D a r c o vi c h / P o w d e r Te c h n o lo g y 8 3 1 9 9 5 ) 2 1 1 - 2 2 4 2 1 5

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.


6/14

216

K. Darcovich / Powder Technology 83 1995) 211-224

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


7/14

K. Darcovich / Powder Technology 83 1995) 211-224 2 1 7

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 .


8/14

2 1 8

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-


9/14

K. Darcovich / Powder Technology 83 1995) 211-224 2 1 9

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.


10/14

2 2 0

K. Darcovich / Powde r Technology 83 1995) 211- 224

( 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 .


11/14

K . D a r c o v i c h / owder Technology 8 3 1 9 9 5 ) 2 1 1 - 2 2 4 221

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


12/14

2 2 2 K. Darcovich / Powder Technology 83 1995) 211-224

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


13/14

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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A Hydrodynamic Simulation of Mineral Flotation. Part I- The

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