Column Flotation Dynamic Model

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Chapter 4

Column Flotation Two-Phase Dynamic Model

4.1 Introduction

In an operational flotation column, two types of flows move countercurrently

throughout the collection zone: an air stream, as small bubbles which rise up the column,

and slurry flowing downwards. Certain elements that characterize the air phase play a

very important role in the flotation process. They include the bubble size (or size

distribution), the bubble rise velocity and the number of bubbles in the column cell. For

the analysis of such variables, as well as the study of flow behavior, a two-phase model is

often a preliminary stage (Yianatos et al., 1986; Dobby, Yianatos and Finch, 1988; Pal and

Masliyah, 1989; Langberg and Jameson, 1992; Ityokumbul, 1995). In this way, the air-

slurry system is initially approximated by a column with an aqueous surfactant solution

and air bubbles.

Although the particles are left out, a two-phase model can be a tool to investigate

the interactions between countercurrent liquid and gas flows, conditions for bubbly flow

regime, and bubble expansion and coalescence. In this work, formulation of a two-phase

dynamic model was undertaken first in order to evaluate a coalescence representation

based on a coalescence-efficiency-rate parameter. This approach follows the method used

in the development of pelletization models (Sastry, 1981). The process of solving the air

phase equations, before introducing the solid phase, also provided some insight on the

numerical stability of the model. Of particular interest were the conditions under which a

numerically stable solution can be achieved for the froth region. Meanwhile, it should be

borne in mind that solid particles seem to have an effect on air fraction (Banisi et al., 1995;

Ityokumbul et al., 1995). Such effect may have to be taken into consideration later on,

when representing the mineralized process.

4.2 Background

The air-phase transport equations are based on drift flux theory, which relates thetwo-phase flow parameters in the following way:

Vg VlUgs

ε ε +

−=

1[1]

where Vg and Vl are the gas and liquid superficial velocities respectively, ε is the air

volume fraction, and Ugs is the bubble slip velocity. The slip velocity Ugs is defined as





108

The set of equations presented above has to be solved iteratively since they

constitute a system of nonlinear simultaneous equations. A different approach to estimate

the bubble slip velocity is to utilize the general form

Ugs Ut F = * ( )ε [10]

where the functional forms F(ε ) that are most commonly used are the Richardson-Zaki

relationship and the Marrucci's equation (Marrucci, 1965). Other forms for this function

have been estimated empirically to obtain a better fit of air fraction data in a specific range

(Lockett and Kirkpatrick, 1975; Pal and Masliyah, 1989). The relationships between the

slip velocity and the air fraction for the various expressions reported in the literature are

indicated in Figure 4.1. The evaluations were made assuming an average bubble size of

one millimeter and the functions plotted are:

Slip Equation No.1:

Ugs Ut m= − −( )1 1ε (non-iterative solution with Richardson-Zaki equation) [11]

Slip Equation No.2:

Ugs Ut =−

−

1

1

5

3

ε

ε (Marrucci's expression); [12]

Slip Equation No.3:

( )[ ]Ugs Ut = −0 8 2 9 2 1. exp . .ε (Pal and Masliyah's empirical expression)[13]

Slip Equation No.4:

( )Ugs Ut m= − +−( ) .1 1 2 551 3ε ε (Lockett and Kirkpatrick's empirical equation) [14]

Slip Equation No.5:

Equation [2], along with [3] and [4] (iterative solution according to Masliyah (1979))

Lockett and Kirkpatrick (1975) built a similar plot for bubble columns, but they

used the characteristic or drift velocity instead of the bubble rise velocity. They compared

the Richardson-Zaki expression in Equation [11], for m=2.39, and the Marrucci's

equation, along with other equations proposed by Turner (1966) and Davidson and

Harrison (1966), with some experimental data. They found that the Richardson-Zaki

espression fitted the data quite well for air fractions up to 20%. When evaluating the

relatioships available at the time, Davidson and Harrison (op.cit) also determined that the



109

Richardson-Zaki equation could represent fairly well the relationship between air fraction

and superficial gas velocity in a bubbling fluidized bed. As the plot in Figure 4.1 suggests,

at air fractions greater than 20%, the Marrucci's equation predicts much higher velocities

than the other functional forms. This is attributed to the fact that this expression was

derived from a mechanistic analysis which is valid at low air holdups. The equations

derived by either Lockett and Kirkpatrick or by Pal and Masliyah were intended to fit airfraction experimental data greater than 30% and 70% respectively. As to the two

remaining relationships, which incorporate the Richardson-Zaki empirical form, the one

derived from iterating on Equations [2]-[4] always provides significantly higher velocities.

It approaches the Marrucci's equation at air fractions lower than 20%.

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.80

2

4

6

8

10

12

Air Fraction

B u b b l e S l i p

V e l o c i t y ( U g s ) - c m / s e c

+++ Eq.No.1

ooo Eq.No.2

----- Eq.No.3

_._. Eq.No.4

**** Eq.No.5

Figure 4.1: Comparison of the Bubble Slip Velocity Predicted by Several Expressions

(Eq.1: Richardson-Zaki equation; Eq.2: Marrucci's; Eq.3: Pal and Masliyah's;

Eq. 4: Lockett and Kirkpatrick's; Eq. 5: iterative solution with Richardson-

Zaki relationship)



110

There are generally two possible solutions to the drift flux model for particular

values of gas velocity, liquid velocity and average bubble size. The solutions are given by

the two points at which the curves determined by the left-hand side and right-hand side of

the following equation intercept (using Equation [2] to describe the bubble slip velocity):

( )

( )Vg Vb

gDb

k k

k k

ave susp b k k

m

l bε ε

ρ ρ ε

µ ∑ ∑∑

+−

=− −

+

−

1

1

18 1 015

2

2

0 687. . Re[15]

The two solutions are illustrated in Figure 4.2 for Vg = 1.0 cm/sec, Vl = 0.1 cm/sec and

Db = 1 mm. In a flotation column, these solutions would correspond to the zones right

below and above the interface. If the right-hand-side of Equation [15] is given by

Equation [11] instead, the pulp air fraction predicted by this model is higher while the

froth phase solution decreases.

0 0.2 0.4 0.6 0.8 1

0

5

10

15

20

25

Air Fraction

B u b b l e S l i p V e l o c i t y

( U g s ) - c m / s e c

___ Left-Hand Side Drift Flux Eq.

----- Iterative Eq. for Ugs (Masliyah, 1979)

Figure 4.2: Solutions of the Drift-Flux Model Corresponding to the Pulp and Froth

Phases (Interception Points)



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There is a maximum gas velocity for a particular bubble size as well as a minimum

bubble size for a given gas rate, so that a solution can be found to the previous equation.

Such limits correspond to the case when the values of air fraction on both sides of the

interface are equal and, therefore, there is only one solution for the drift flux equation. In

Figure 4.3, the air fraction solutions for a range of average bubble sizes and for several gas

velocities are shown. The liquid velocity is assumed constant and equal to 0.1 cm/sec.The tip of each of the parabolic curves corresponds to the minimum bubble size for that

particular air rate. The maximum theoretical pulp air fraction appears to be between 0.3

and 0.4 for the range of air velocities shown, and the model indicates that it is lower for

smaller bubble sizes.

0.04 0.06 0.08 0.1 0.12 0.14 0.16 0.18 0.20

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

Bubble Diameter (cm)

A i r F r a c t i o n

Vg=0.8

1.0

1.2

1.41.6

1.8

2.0

Vg in cm/sec

Figure 4.3: Solutions of the Drift Flux Model for Different Gas Superficial Velocities

Expressed in cm/sec.

The interaction between the air and fluid phases in the column stabilized froth has

been estimated to be analogous to an expanded bubble bed (Yianatos et al. 1986; Goodall

and O'Connor, 1991b). For an expanded bubble bed, the expressions generally used to

describe the two-phase flow include the Richardson-Zaki equation and Ergun's equation

(Ergun, 1952).



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The latter states that:

( )

( )U gs K

u

K D b

K gDb

K u

l

l

l

l

=−

+

−

−

1

2

2

2 3 3

1

2 2 2

1

2

2 11

4 11

ε ε ρ

ρ ε ε [16]

where K 1 was estimated to be around 150 for a granular bed, K 2 was approximately 1.75,

µ l is the liquid viscosity, ρ l is the liquid density, Db is the bubble diameter, g the

gravitational constant, and ε the air volume fraction. Yianatos et al. (1986a) applied a

mechanistic approach to describe the liquid drainage along an expanded bed and

developed a relationship that relates the relative bubble velocity to the air holdup:

( )

( )

( )

Ugs

DbdP

dL

l

=−

− −

1

72 1 05 1

2 2

2

ε

ε µ ε . log

[17]

dP

dLg

V

V V l

bias

bias G

= − ++

ρ ε 1 [18]

In the pressure gradient expression (Equation [18]), V bias and V G are the superficial bias

and gas velocities. The other symbols have been introduced previously. An empirical

correlation was found by Pal and Masliyah (1990) for the froth, relating the bubble slip

velocity in the froth to the terminal velocity. The expression is:

UgsUt f = − +exp( . . )2 4 2 5ε [19]

with ε f representing the liquid holdup.

Figure 4.4 illustrates the differences in the predictions obtained from Equations

[2],[16],and [17] with Vg=1.0 cm/sec, Vl=0.1 cm/sec and Db=0.1 cm, for values of air

fraction greater than 50%. Among the functions displayed in Figure 4.4, Masliyah's

expression predicts considerably greater relative velocities than the others, particularly in

the air fraction range between 0.6 and 0.8. The equation derived by Yianatos et al. seems

to provide high velocity values when the air fraction is lower than 50%, but it falls below

the others for air fractions greater than about 60%. In spite of its empirical nature, the

Richardson-Zaki equation follows a trend very similar to that of the Ergun's equation,although the velocities are slightly higher. The differences between the values predicted

by these relationships decrease as the air fraction approaches 90%. Despite its

mechanistic origin, the utilization of the Ergun's equation does not seem to lead to

substantial differences from the Richardson-Zaki expression, particularly for air fractions

greater than 0.7. In addition, the presence of two empirical constants K 1 and K 2introduces more unknown parameters.



113

0.5 0.6 0.7 0.8 0.9 10

1

2

3

4

5

6

7

Air Fraction

B u b b l e S l i p V e l o c i t y ( U g s ) - c m / s e c

1

2

3

4

1 - Masliyah's Iterative Eq. (1979)

2 - Yianatos et al. (1986)

3 - Ergun's Eq.

4 - Richardson-Zaki Eq.

Figure 4.4: Bubble Slip Velocity-Air Fraction Relationships for Expanded Bubble Beds



114

4.3 Model Development

For the modeling task, column flotation can be regarded as a multiphase system

where there is a continuous liquid phase and a set of discrete components characterized by

their size and composition. In a two-phase operation, the air bubbles constitute the onlydiscrete phase. Because of the particulate character of this process, population balance

modeling techniques can be applied to determine the changes in each of the phase

concentrations in the various column regions. The general macroscopic population-

balance-model equation is:

( ) ( )1 1

1 1V

d V

dt

d v

d D A

V Q

z

z i

ii

N

pp pp

z

k k k

K ϕ ϕ

ζ ϕ + + − = −

= =∑ ∑ , [20]

where V z is the zone volume, ϕ is the particle concentration in volume V z for a specific

component characterized by property ζ ,and vi is the continuous change with time of the

particle property ζ . The parameter Qk stand for the k-th flow exiting the zone. Also, in

Equation [20] D pp is the disappearance term, which quantifies the particles belonging to

the component class under consideration that disappear from the column zone due to rate

phenomena. The remaining term, A pp , refers to the particles that appear in the zone as a

result of rate events.

The macroscopic population balance model does not account for spatial changes in

concentration inside the zone. If the spatial concentration gradients need to be calculated,

the microscopic version of the population balance model should be applied, whose general

equation is:

( ) ( ) ( ) ( )d

dt

d v

dx

d v

dy

d v

dz

d v

d D A

x y z i

ii

N

pp pp

ϕ ϕ ϕ ϕ ϕ

ζ + + + + + − =

=∑

1

0. [21]

In this case, ϕ is the local concentration of each component with property ζ i , v x,v y and v z

are the average transport velocities in the x-,y-,and z-directions, and the appearance and

disappearance terms are local quantities.

A flotation column can be divided into three main regions characterized by

different flow regimes and where different types of interactions take place. These zones

are known as the pulp, stabilized froth, and draining froth. In two-phase operation, three

additional transition regions can be identified (as illustrated in Figure 4.5):

- the aeration zone, where the gas inlet is located;

- the interface, defined as the section along the column height where the sharp

transition between pulp and froth occurs; and

- the wash water zone, where the wash water distributor is situated.





116

4.3.1 Pulp Region

Model Assumptions

The assumptions made for the derivation of the air-phase equations in the pulp

region can be summarized as follows:

a) Mixing in the collection zone can be represented by the tanks-in-series model. The

higher the number of zones, the closer to plug flow the pulp behavior is considered to be.

b) All the air entering the column leaves with the concentrate (air in the tailings

negligible).

c) The bubbles have a size distribution which can be represented by a discrete density

function f v,d,k defined for a number N b of size classes.

d) Each bubble in size class k has a relative rise velocity Ugsk and the average slip velocityis given by

UgsUgs

ave

k k

k

=∑

∑ε

ε . [22]

e) The slip velocity for each bubble size class can be estimated iteratively from the solution

to the multiphase Navier-Stokes equation presented by Masliyah (1979). Equation [2]

was therefore selected to be incorporated into the air-phase equations. The validity of this

equation will be confirmed during the model verification phase.

f) Bubble coalescence and bubble breakage are negligible in the pulp.

g) Bubble expansion due to the reduction in hydrostatic pressure with height was

disregarded.

Air-Phase Dynamic Equations

The general form of the macroscopic population balance model for the perfectly

mixed regions forming the pulp is

( )( )

d n f

dt Rate terms

Q Q

Vz

z t n d t d in d out , , , + =−

, [23]

where n z,t is the total number of bubbles in volume Vz at time t , f n,d,t is the fraction of

bubbles of size class d at time t , and Qd in and Qd out are the net flows into and out of the



117

column zone. The rate terms are not part of the equation models for the pulp in the two-

phase system (no coalescence).

After transforming from a number-based population balance to a volume balance,

the dynamic conservation equations for each perfectly mixed tank in the collection zone of

the column are as follows:

• For the aeration zone, the air-phase balance equation is:

d

dt

k ε=

( ) f Qg Qg Qt Ugs A A Ugs

Vz

v d k k

z

k

z

k

z

k

z

k

z

k k

z

, ,− − − +

∑ε ε ε ε

; [24]

• for each of the zones between the gas inlet and the interface, it is

d

dt

k ε=

Qg Qt A Ugs Ugs A Qg Qt A Ugs Ugs A

Vz

k

z

k

z

k k

z

k

z

k

z

k

z

k

z

k k

z

k

z

k

z− −

+ − − −

−

− − − − −∑ ∑1 1 1 1 1ε ε ε ε ε ε

[25]

In Equations [24] and [25], f v,d,k is the volume-based size distribution of the generated

bubbles, Qg is the gas rate, Qt is the tailings rate, A is the column cross-sectional area,

Ugsk is the slip velocity for bubbles in size class k , and ε k is the volumetric fraction of

bubbles in size class k . The superscripts z and z-1 refer to the zone under consideration

and the one immediately below it.

4.3.2 Froth Regions

Model Assumptions

The air-phase equations for the stabilized froth and the draining froth are based on

the following premises:

a) In the stabilized froth region, the flow behavior is assumed to be plug-flow. The mean

bubble velocity is given by

UbVg

k

= ∑ε , [26]

b) In the froth zones, the bubbles are assumed to remain spherical, which eliminates the

need for using a mathematically complex froth model based on pentagonal dodecahedral

bubbles, as appears widely in the literature. This assumption seems to be more valid in the



118

stabilized froth than in a draining froth. In any case, the use of detailed geometry in the

froth model introduces several parameters that cannot yet be determined experimentally.

c) In the froth regions, coalescence is treated as a rate process, where the rate of

occurrence is a function of the sizes of the two bubbles involved. Moreover, the

coalescence rate is expected to be dependent on other factors as well, such as the solids inthe films between bubbles and the surfactant adsorbed on the bubble surfaces. At the

present moment, however, investigations into the effects of such factors on bubble

coalescence have fallen short of establishing mathematical relationships.

Coalescence Representation

It has been observed that the air fraction increases very rapidly close to the

interface and that any further increase along the stabilized froth is of relatively small

magnitude. The reason for the stability of the froth has been attributed to the downward

flow of bias water, which maintains a liquid film between the bubbles and, consequently,

reduces coalescence. In an effort to represent mathematically such behavior, severalapproaches were explored.

The first assignment was to find an adequate representation of the coalescence

events that could be applied to the entire froth region. The procedure for developing a

mathematical description of bubble coalescence in column flotation was based on previous

studies of coalescence in granulation and pelletization (Sastry, 1981; Sastry and

Fuerstenau, 1973; Kapur, 1972). Sastry (op.cit.) developed a model of the pelletization

process which incorporates a phenomenological description of pellets coalescence. The

change in mass within a pellet size class is considered to be proportional to the number of

collisions between those pellets with any other in the system, and to an efficiency

parameter. This parameter provides a measure of how many interactions result incoalescence.

In the present work, Sastry’s approach was adapted to describe bubble

coalescence in a froth. Instead of calculating the change in mass, the froth model

determines the changes in air volume (air fraction) for each bubble size class due to

coalescence. From a statistical analysis, the total number of collisions between bubbles of

size classes i and j is given by

N n n

N collisions

i j

a= [27]

where ni and n j are the numbers of bubbles of size class i and j in the region, respectively.

N is the total number of bubbles, and a is a parameter dependent on how closely packed

the bubbles are.



119

Assuming that the bubbles are restricted but not fixed in space (a = 1), the possible

number of interactions between bubbles in size class i and bubbles in size class j (i,j =

1..Nk ) is given by:

N

n n

ncollisions

i j

k k

Nk = ∑ [28]

where Nk is the total number of discrete size classes in the distribution. In terms of

volume fractions, the total number of possible interaction can also be expressed as:

N V

Db Db Db

V

Db Db

Db

collisions

z i j

i j

k

k k

Nk

z i j

k i j

k k

Nk = =

∑ ∑

ε ε

π ε

ε ε

π ε 6 6

3 3

3

3 3

3

[29]

where ε i and ε j are the air fractions for bubbles in size classes i and j, Db is the bubble

diameter and V z is the zone volume. An efficiency rate parameter λ (i,j,t) can be defined,

which represents the fraction of the total number of collisions that result in coalescence

per unit time. The number of new bubbles of size class l, created as a result of the

coalescence of bubbles in size classes i and j, is then:

( )n i j t V

Db Db

Db

l

z i j

k i j

k k

Nk =

∑

1

2

6

3 3

3

λ ε ε

π ε , , [30]

where λ (i,j) is the coalescence efficiency rate parameter for the i-j pair, Db is bubblediameter, and ε k is the air fraction corresponding to each bubble size class k .

The increase in volumetric air fraction corresponding to size class l is thus given

by:

( )∆ε λ ε ε

ε l

i j

k i j

k l

k k

Nk i j t

Db Db

Db Db

=

∑

1

2 3 3

3

, , [31]

Likewise, the decrease in the air holdup corresponding to size class l, due to thecoalescence of those bubbles with bubbles in size class j, is given by the following

equation:

( )∆ε λ ε ε

ε l

l j

k j

k k

Nk l j t

Db

Db

=

∑, , 3

3

[32]







122

The system of equations formulated above was solved using a finite-difference numerical

method for a superficial gas velocity equal to 1 cm/sec and the following average bubble

diameters in each class: [0.08 cm, 0.08*√ 2 cm, 0.16 cm, 0.16*√ 2 cm]. Each of the

approximate algebraic equations for the air fraction components ε j at each height interval

dz is of the following form:

( )ε ε ε

ε ε

ε ε ε ε ε

ε ε j

z i j

z i

j z i

k j

z i

k z i j z i

k z i j z i

k

z

k j

z

j j z

z z

Vg

A D+ +

+ += +

−

−

− −

+

−

−

∑

∑ ∑ ∑

∑1 1

1

4

11

4

1 1

4

1

4 2

1

4, ,

,

,

, , , ,

∆∆

[44]

where i refers to the iteration step. In solving the equations, the initial boundary condition

(at the base of the froth) was a typical pulp air fraction (ε ≈ 0.15). The values assigned tothe coalescence efficiency rate parameters were: λ 11=0.05/sec, λ 12=0.10/sec, λ 13=0.20/sec.

These values were initially selected through a trial-and-error approach, using as a criterion

the prediction of air fraction values between 0.6 and 0.8 at the top of the froth.

The air fraction profile predicted by this set of equations (Figure 4.6) does not fit

the normally observed profile in the stabilized froth, characterized by a jump in air fraction

at the interface and little increase above it. However, when the coalescence-efficiency-rate

parameters were given new values so that the rate of coalescence was significantly higher

for small bubbles, the shape of the profile changed greatly. For λ 11=0.70/sec,

λ 12=0.50/sec, λ 13=0.15/sec, the calculated profile was the one depicted in Figure 4.7,

which resembles the widely reported air fraction profile shape for the column stabilizedfroth (Yianatos, Finch, Laplante, 1986; Finch and Dobby, 1990).

A serious difficulty with the application of these equations is that they are ill-

posed. Consequently, the shape of the predicted profile is very sensitive to the individual

values of λ . A small change in one of the coalescence parameters can result in calculated

air fractions greater than one, or in a drastic change in the profile to an unfeasible form.

A different technique was also explored for the solution of the system of

differential equations. Equation [38] was written in another form to yield the following

new finite-difference approximation:

( )( ) ( )d

dz zF

F F j j

z j

z

j Nk z

j Nk z

j Nk z

ε ε ε ε ε

ε ε ε ε =

−= =

++

+

+1

1 1

2

1 11

2∆...

... ...[45]



123

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.80

5

10

15

20

25

30

35

40

45

Air Fraction

F r o t h H e i g h t ( c m )

Figure 4.6: Air Fraction Profile in the Stabilized Froth Calculated with the Set of

Simultaneous Equations [40]-[43] (λ 11=0.05/sec, λ 12=0.10/sec, λ 13=0.20/sec)

Using the first two terms of the Taylor series expansion of the righ-hand-side term

in the previous equation, and writing the air fraction components in vector form, the

estimate of the air fraction at position z+1 is given by:

( )ε ε ∂

∂ ε ε ε

z z z

z

z z z F z

F + +

= + + −1 1

1

2∆ ∆ [46]

After passing the unknown terms to the left-hand-side of the equation, the final expression

for the air fraction components is:

ε ∂

∂ ε ε

∂

∂ ε ε

z

z

z z

z

z I z

F z F z

F +

−

= −

+ −

1

1

1

2

1

2∆ ∆ ∆ [47]

where I is the identity matrix.



124

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.80

10

20

30

40

50

60

70

80

Air Fraction


Figure 4.7: Air Fraction Profile in the Stabilized Froth Calculated with the Simultaneous

Equations [40]-[43] (λ

11=0.70/sec,λ

12=0.50/sec,λ

13=0.15/sec)

For a system with three size classes, the vector-matrix form of the set of

simultaneous equations is:

( ) ( ) ( )

( ) ( ) ( )

( ) ( ) ( )

ε

ε

ε

ε

ε

ε

∂

∂ ε

∂

∂ ε

∂

∂ ε ∂

∂ ε

∂

∂ ε

∂

∂ ε ∂

∂ ε

∂

∂ ε

∂

∂ ε

ε

ε

ε

ε

ε

ε

1 1

2

1

3 1

1

2

3

1

2

3

1

1

1

2

1

3

2

1

2

2

2

3

3

1

3

2

3

3

1 1

2

1

3 1

1

2

3

1

2

z

z

z

z

z

z

z

z

z

z

z

z

z

z

z

z

z F

zF

z F

z

F F F

F F F

F F F

+

+

+

+

+

+

==

==

+

+

−

∆

∆

∆

∆ *

[48]

The equations were normalized using the transformation shown below (Equation [49]) so

that the froth height is made equal to one ( L is the actual froth height). Several air

fraction profiles were calculated using the normalized equations for different initial air

fraction values, gas velocities and froth heights.



125

′ = z z

L[49]

When λ 11 >> λ 12, the profile obtained is like the one shown in Figure 4.8, which

corresponds to the following parameter values in the equations:Vg=1.0 cm/sec, Vl=0.1 cm/sec, λ 11 =0.75/sec, λ 12 = 0.075/sec, froth length=100 cms, pulp

air fractions in each class: [0.10 0.02 0.02], and Db=[0.08, 0.08*√ 2, 0.16 ] cm.

0 0.2 0.4 0.6 0.8 10

10

20

30

40

50

60

70

80

90

100

Air Fraction


Figure 4.8: Air Fraction Profile in the Stabilized Froth Calculated with the Taylor Series

Approximation (Equation [48]) (λ 11=0.75/sec, λ 12=0.075/sec)

The profiles corresponding to Vg=1.0 cm/sec and Vg=1.5 cm/sec are compared in

Figure 4.9. The graph indicates that the liquid content in the froth increases for higher air

velocities, which is the normal response in an operating column because of the increase in



126

the water entrained with the bubbles. The effect of increasing the coalescence efficiency

rate parameters can be seen in Figure 4.10. The value of λ 11 was increased from 0.75/sec

to 0.95/sec, and λ 12 was raised to 0.095/sec from its previous value of 0.075/sec. As

expected, the higher coalescence rate parameters resulted in a higher average froth air

fraction.

0 0.2 0.4 0.6 0.8 10

10

20

30

40

50

60

70

80

90

100

Air Fraction


1 - Vg=1.5 cm/sec

2 - Vg=1.0 cm/sec

21

Figure 4.9: Comparison of the Air Fraction Profiles in the Stabilized Froth Calculated

with the Taylor Series Approximation for Two Different Gas Velocities (Vg

in cm/sec)(λ 11=0.70/sec, λ 12=0.075/sec)



127

0 0.2 0.4 0.6 0.8 10

10

20

30

40

50

60

70

80

90

100

Air Fraction


lambda_1

lambda_2

lambda_2 > lambda_1


with the Taylor Series Approximation for Two Different Sets of Coalescence

Parameters (Fromλ

11=0.70/sec andλ

12=0.075/sec toλ

11=0.95/sec andλ 12=0.095/sec)

Finally, the air fraction components at the base of the froth zone (initial boundary

condition) were varied from: ε f =[0.10, 0.02, 0.02] to ε f =[0.1023, 0.0204, 0.0073]. The

new boundary values resulted in a higher average air fraction in the froth region, as shown

in Figure 4.11. However, this is not an indication of an established trend. The effect of the

boundary condition on the froth profile appears to be dependent on the particular values of

the pulp air fraction components, in combination with the values of the λ 's.

It was realized that, when λ 11 < λ 12, the calculated profile does not resemble the

profile shape measured using conductivity electrodes in the stabilized froth. Nonetheless,

it looks similar to some of the experimental two-phase draining froth profiles, which were

also obtained through conductivity measurements.



128

0 0.2 0.4 0.6 0.8 10

10

20

30

40

50

60

70

80

90

100

Air Fraction


12

1 - Initial Pulp Air Fraction:

e1=0.1, e2=0.02, e3=0.02

2 - Initial Pulp Air Fraction:

e1=0.1023, e2=0.0204, e3=0.007


with the Taylor Series Approximation, with λ 11=0.70/sec and λ 12=0.075/sec,

for Two Different Boundary Conditions: (Pulp Air Fraction Components)

Since the differential equations are in open form, which means that a boundary

condition at z=L is not specified, the air fraction values can mathematically exceed unity.

It is possible to solve the equations (for λ 11 < λ 12) in closed form, by introducing a new

parameter, V cw, which represents the liquid velocity at the top of the froth. This velocity is

assumed to be known and an iteration is performed by considering that the froth height

can change until the equation below is satisfied.

( )V Vg

cw

z L

z L= −

==ε

ε 1 [50]

The predicted profiles (for each air fraction component as well as the total air fraction) for

the following conditions:



129

{Vg = 1.0 cm/sec, Db: [0.08 cm, 0.08*√ 2 cm, 0.16 cm], pulp air fraction: [0.10, 0.02,

0.02], V cw=0.1 cm/sec, efficiency rate parameters: λ 11=0.025/sec, λ 12=0.075/sec },

are depicted in Figure 4.12. As shown, the air fraction component corresponding to the

smallest size class (ε 1) decreases along the froth because of coalescence, while the

volumetric fraction of the largest bubbles (ε 3) increases the most.

0 0.2 0.4 0.6 0.8 10

20

40

60

80

100

120

Air Fraction


e1e2e3

Total e

Figure 4.12: Predicted Air Fraction Profiles in the Froth Corresponding to Each

Bubble Size Class, Calculated with the Taylor-Series Aproximation and

Iterating on the Froth Height (λ 11=0.025/sec, λ 12=0.075/sec)



130

Quasi-Steady-State Approach for Representing Process Dynamics

In a truly dynamic model it would be necessary to solve the complete set of partial

differential equations in order to account for the variations in air fraction with time and

with froth height. However, the solution of the dynamic equation (Equation [35]) isalways unstable. An alternative procedure for describing the dynamics of a froth like the

one depicted in Figure 4.6 was explored, which consists of a quasi-steady-state technique.

First, the pulp air fraction was defined in terms of a dynamic equation rather than a

constant set of values. For a cocurrent system:

( )d

dt

f Vg Vg Vl Ugs Ugs

L

k v Db ave total k k k

pulp

ε ε ε ε =

− + − −,, k = 1...Nk [51]

In Equation [51], f v,Db is the discrete size distribution of the bubbles in the pulp on a

volume basis, and L is the length of the pulp region.

The simultaneous steady-state equations were then solved using the pulp air

fraction at each time interval (calculated by solving Equation [51]) as the boundary value

at z=0. In this way, the changes in the pulp air fraction were assumed to propagate

through the froth at each time step. The froth height was varied at each time step until the

liquid velocity in equation [51] (Vl) was within a tolerance value away from the calculated

V cw. The changes in time of the pulp air fraction components (given by the numerical

solution of equation [51]) are represented in Figure 4.13. Meanwhile, the net change in

the calculated froth profile from the beginning of the simulation (t=0) to the last time

interval (t =140 secs) is shown in Figure 4.14.

Taking as an initial steady-state condition the results of the previous simulation,

the effects of varying the gas and liquid velocities, as well as the coalescence rate

parameters, were then determined. In Figure 4.15, the predicted steady-state profile for

V l=0.1 cm/sec (initial condition) and the steady-state profile obtained after V l was

increased to 0.2 cm/sec are compared.

It is observed that since an increase in liquid velocity resulted in a small decrease

in pulp air fraction, the froth profile for V l =0.1 cm/sec reaches any particular air fraction

value at a shorter froth length than the one for V l=0.2 cm/sec. Since the liquid flow was

assumed to be cocurrent to the bubbles, the top air fraction was therefore smaller for the

condition of higher liquid velocity.

The corresponding variation in froth height during the simulation interval is shown

in Figure 4.16. These graphs appear to indicate that, by using this technique, the shape of

the profile is maintained in spite of changes in the simulation conditions.



131

0 20 40 60 80 100 120 1400

0.02

0.04

0.06

0.08

0.1

0.12

Time (sec)

P u l p A i r F r a c t i o

n

e1

e2

e3

Figure 4.13: Dynamic Changes in the Pulp Air Fraction Predicted by the Dynamic Pulp

Equation (Equation [51] ) with λ 11=0.025/sec, λ 12=0.075/sec



132

0 0.2 0.4 0.6 0.8 10

20

40

60

80

100

120

140

Air Fraction


t=0

t=140 sec

Figure 4.14: Predicted Change in the Overall Air Fraction Profile in the Froth

Corresponding to the Dynamic Change in Pulp Air Fraction Depicted in

Figure 4.13 (Quasi-Steady-State Technique)



133

0 0.2 0.4 0.6 0.8 10

20

40

60

80

100

120

140

Air Fraction


t=0

t=140 sec

Figure 4.15: Predicted Change in the Overall Air Fraction Profile in the Froth from t=0 to

t=140 sec After an Increase in the Liquid Velocity Vl from 0.1 cm/sec to 0.2

cm/sec (Quasi-Steady-State Technique)



134

0 20 40 60 80 100 120 140110

112

114

116

118

120

122

124

126

128

130

Time (sec)


Figure 4.16: Time Variation in Froth Height for an Increase in Liquid Velocity from

Vl=0.1 to Vl=0.2 cm/sec (Quasi-Steady-State Technique)



135

When the gas velocity was the parameter varied from 1.0 cm/sec to 1.5 cm/sec,

with V l=0.1 cm, the effect was similar to the previous case because the higher air rate also

causes an increase in pulp air fraction. However, the decrease in froth height was much

more pronounced in this situation. The liquid velocity and, therefore, the air fraction at

the top of the froth remained constant. The two calculated profiles are those shown in

Figure 4.17, while the corresponding change in total froth height is illustrated in Figure4.18.

0 0.2 0.4 0.6 0.8 10

20

40

60

80

100

120

140

Air Fraction


Vg = 1.0 cm/sec

Vg = 1.5 cm/sec

Figure 4.17: Predicted Change in the Overall Air Fraction Profile in the Froth After an

Increase in the Superficial Gas Velocity Vg (Quasi-Steady-State Technique)



136

0 20 40 60 80 100 120 14050

60

70

80

90

100

110

120

130

Time (sec)


Figure 4.18: Predicted Variation in Froth Height for an Increase in Gas Superficial

Velocity from Vg=1.0 cm/sec to Vg=1.5 cm/sec (Quasi-Steady-State

Technique)

The effect of decreasing the values of the coalescence parameters fromλ 11=0.025/sec, λ 12=0.075/sec to λ 11=0.015/sec, λ 12=0.030/sec can be seen in Figures 4.19

and 4.20, which suggest that a deeper froth is then needed to maintain the same liquid rate

at the top of the froth. At a constant froth depth, the reduction in the values of the

coalescence parameters would result in a smaller gas holdup at the top of the froth (less

coalescence) and, therefore, a larger calculated liquid rate.





138

0 20 40 60 80 100 120 140120

140

160

180

200

220

240

260

280

Time (sec)


Figure 4.20: Change in the Froth Height Required to Maintain the Same Overflow Rate

after a Decrease in the Values of the Coalescence Rate Parameters in

Equation [48] (Quasi-Steady-State Technique)





140

( )∂ε ∂

∂ ε

∂ k k

k k t

v

z D A+ + − = 0 [55]

where the subscript k refers to the bubble size class, v is the average bubble rise velocity,

and the appearance and disappearance terms ( Dk and Ak ) are defined according to

equations [34] and [33], respectively. Since the average bubble rise velocity is given by

vVg

k k

= ∑ε , [56]

the space derivative can be expanded in the following manner:

( )d v

dzv

d

dz

dv

d

d

dzv

dv

d

d

dz

Vg Vg d

dz

k k

k

k

k

k

k

k

k k

k

k k

k ε ε

ε ε

ε ε

ε

ε

ε

ε

ε

ε = + = +

= +−

∑ ∑[57]

Substituting in the general equation, the changes in air fraction with time and position

along the froth are represented by:

d

dt

Vgd

dz D A

k

k k

k

k k

k

k k

ε ε ε

ε

ε +

−

+ − =∑

∑2 0 [58]

Air-Phase Dynamic Equations

• Interface

An aspect of the behavior of a flotation column at the pulp-froth interface which

cannot be reproduced using the froth model equations developed thus far is the

interdependence between the air fraction values at both sides of the interface. The sharp

transition that takes place at the pulp-froth interface indicates that two flow regimes

actually coexist in the zone. The air fractions are the solutions to the drift flux equation(Equation [1]).

In order to represent this phenomenon with the column dynamic model, the

interface zone is defined as a transition region where the air fraction undergoes a rapid

change from a lower value, characteristic of the collection region, to a higher value which

signals the onset of the stabilized froth. The air fraction at the interface, at each time

interval, is calculated by obtaining the highest viable solution to the nonlinear equation





142

Estimation of the Coalescence Efficiency Rate Parameter

By definition, the coalescence rate parameters introduced earlier are

analogous to the probability of collection in the case of bubble-particle interaction, with

the exception that they have units of [number/time], as a rate constant. The factors that

determine the occurrence of bubble coalescence in a flotation froth include the frotherconcentration, the presence of solids in the film between adjacent bubbles, and the surface

characteristics of such solid particles. Therefore, the values of the coalescence rate

parameters are ultimately affected by those variables.

From the experimental air fraction profiles, it has been observed that the

coalescence phenomena in the stabilized froth and the draining froth have to be explained

by different mechanisms. In the stabilized froth, bubbles are relatively small and stable due

to the countercurrent wash water. On the other hand, liquid drains rapidly in the draining

froth, which result in bubble deformation and growth. Furthermore, the shape of the

profiles appear to suggest that the values of the coalescence rate parameters vary along

the froth. Coalescence in the stabilized froth occurs mainly close to the interface anddecreases with height, while in the draining froth the rate of coalescence increases rapidly

with froth height. Such dependence on the position in the froth can be mathematically

expressed as a relationship with bubble size. Consequently, the coalescence rate

parameters are expected to decrease as the bubble size increases, but the opposite applies

to the draining froth, that is, coalescence rate seems to increase with bubble size.

The dependence of the parameter λ on the bubble sizes (d i and d j) can be expressed

through a functional form such that

( )λ = f d d p pi j n, , ,...1 [61]

The fitting parameters p1...pn establish the connection between the coalescence rate

parameters and the presence of surfactants and solids. Their values can be estimated by

fitting the experimental air fraction profiles to the general steady-state equation below:

V d

dz A D pp pp

ε = − [62]

The left hand side of Equation [62] can be approximated by

( )V

d

dz

Vg

z z

z zε ε

ε ε =

− −1

∆ [63]

and the terms on the right-hand-side of the equation are replaced with Equations [33] and

[34]. The number of parameters (n) in equation [61] is best limited to 1 or 2.



143

Two-Phase:

a)Stabilized Froth

Several functional forms were used for fitting the experimental air holdup profiles

to Equation [62] while backcalculating the values of the λ 's and the correspondingestimation errors. The functions are listed below, while Figure 4.21 illustrates how the

parameter changes with bubble size for the first four functions examined. In all the plots

shown, λ decreases with the mean of the diameters of the two interacting bubbles. The

functions were selected that way because, in the stabilized froth, coalescence appears to

decrease rapidly with height until a stable average bubble size is reached. To represent

that behavior, the coalescence parameter is then assumed to decrease as the bubbles

become larger.

a)

( )

λ d d

a

d d

1 2

1 2

2,=

+

[64]

b) ( )λ d d a d d b1 2 1 2, *= − + + [65]

c) ( )λ d d a d d b1 2 1 2

2

, *= − + + [66]

d)( )

λ d d ab

d d 1 2

1 2

, *exp=+

[67]

e)( )

λ d d bad d

1 2

1 2

, =+

[68]

The four size classes used in the fitting procedure were:

Dbk = [0.2000 0.2828 0.4000 0.5657 ].

The errors from the optimization procedure along with the corresponding equations,

constants and estimated coalescence rates are listed in Table 4.1.

Equation [65] resulted in the smallest error for the experimental profile used, but

the values of λ given by this equation can be greater than one or even negative. Since the

coalescence-efficiency-rate parameter is by definition a fraction, it can only be betweenzero and one. Equation [64] has only one fitting parameter and the error is not

significantly higher. In Figure 4.22, the air fraction profile calculated on the basis of the

parameters given by Equation [64] are shown, along with the empirical data points.

In their mathematical representation of coalescence in gas fluidized beds, Argyriou,

List and Shinna (1971) proposed a one-parameter model, where the parameter was a

measure of the difference in the velocities of bubbles of unequal sizes. In Equation [64],



144

the parameter a is expected to vary with any of the operating conditions that can have an

effect on the concentration of frother in the region. An increase in frother concentration

should therefore be reflected in a lower value of a.

0.05 0.1 0.15 0.20

0.2

0.4

0.6

0.8

Average Bubble Size (cm)

1 / m i n

0.05 0.1 0.15 0.20

0.2

0.4

0.6

0.8


1 / m i n

0.05 0.1 0.15 0.20

0.2

0.4

0.6

0.8


1 / m i n

0.05 0.1 0.15 0.20

0.2

0.4

0.6

0.8


1 / m i n

a) b)

c) d)

Figure 4.21: Functions Used for Fitting the Coalescence Parameter to Two-Phase

Experimental Profiles in the Stabilized Froth



145

Table 4.1: Estimated Coalescence Rate Parameters in a Two-Phase Stabilized Froth

Using Several Functions Relating the Rate Parameter to Average Bubble

Diameter

Equation Constants Rate Parameters Fitting Error

( )λ d d

a

d d 1 2

1 2

2, =+

a=0.0680 λ1,1=0.425/sec

λ1,2=0.292/sec

λ1,3=0.189/sec

λ2,2=0.213/sec

0.0185

( )λ d d a d d b1 2 1 2, *= − + +a=4.5507

b=4.0015λ1,1 > 1/sec

λ1,2 > 1/sec

λ1,3 > 1/sec

λ2,2 > 1/sec

0.0137

( )λ d d a d d b1 2 1 2

2

, *= − + + a=1.1020

b=0.9000λ1,1=0.724/sec

λ1,2=0.643/sec

λ1,3=0.503/sec

λ2,2=0.547/sec

0.0153

( )λ d d a

b

d d 1 2

1 2

, *exp=+

a=0.0244

b=1.1399λ1,1=0.422/sec

λ1,2=0.259/sec

λ1,3=0.163/sec

λ2,2=0.183/sec

0.0186

( )λ d d b

a

d d 1 2

1 2

, =+

a=0.0754

b=1.2020λ1,1=0.227/sec

λ1,2=0.181/sec

λ1,3=0.139/sec

λ2,2=0.150/sec

0.0185



146

0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

15

20

25

30

Air Fraction

D i s

t a n c e B e l o w C o l u m n T o p L i p

( c m )

Figure 4.22: Comparison of the Backcalculated and Empirical Air Fraction Profiles in a

Two-Phase Stabilized Froth

A similar procedure was followed to estimate the coalescence-efficiency-rate terms

for one of the experimental profiles obtained through conductivity measurements in the

region above the wash-water addition point, as explained next.

b) Draining Froth

In a draining froth, it seems likely that the coalescence rate would increase along

the froth height since the liquid film between the bubbles thins due to drainage. Thebubble size also increases rapidly with height. Accordingly, a different type of

mathematical function is proposed for relating the coalescence rate parameters to the

bubble sizes than the one employed for the stabilized froth. The coalescence parameter

can be linked to the bubble size using a general relationship of the form

( )λ d d

c

a d d b1 2 1 2, *= + + [69]



147

The experimental data from the conductivity tests in a column draining froth were used to

determine the values of constants a and b, for c equal to 1 and for c equal to 2, and of a

and c, for b equal to zero. A bubble size distribution with four size classes was defined

initially. Once again, it was assumed that the volume fraction of air in the smallest size

class decreased gradually with froth height while the volume air fraction in the largest size

class increased. The results are summarized in Table 4.2, which provides the errors andcalculated constants for each of the mathematical relationships that were tested in the

determination of the coalescence parameters.

Table 4.2: Estimated Coalescence Rate Parameters in a Two-Phase Draining Froth

Using Several Functions Relating the Rate Parameter to Average Bubble

Diameter

Equation Constants Rate Parameters Fitting Error

( )λ d d a d d b1 2 1 2,

*= + + a=0.2975

b=0.6889λ1,1=0.808/sec

λ1,2=0.833/sec

λ1,3=0.867/sec

λ2,2=0.857/sec

0.0274

( )λ d d a d d b1 2 1 2

2

, *= + + a=0.5555

b=0.8458λ1,1=0.935/sec

λ1,2=0.975/sec

λ1,3 > 1/sec

λ2,2 > 1/sec

0.0273

( )λ d d

b

a d d 1 2 1 2, *= + a=0.4502b=0.3306 λ1,1=0.333/sec

λ1,2=0.354/sec

λ1,3=0.380/sec

λ2,2=0.373/sec

0.0274

Calculation of λ by substituting in any of the first two equations may result in

values greater than one, which, as mentioned previously, are in conflict with its definition.

The third equation provides reasonable values for the bubble sizes assumed during the

fitting task.

A number of investigations about the stability of cellular foams have established a

relationship between coalescence and liquid film thickness, surface tension, liquid density

and viscosity (Barber and Hartland, 1975; Steiner, Hunkeler and Hartland, 1977). Allak

and Jeffreys (1974) also correlated the probability of drop coalescence in dispersion bands

to the size of drops, the surface tension, band thickness, and dispersed-phase flow rate.



148

The fitting parameters a nd b in the third equation in Table 4.2 should then be associated

with the amount of liquid in the froth (in a 'wet' froth, drainage occurs more slowly), froth

depth, liquid properties such as density and viscosity, and amount of frother in solution.

The air fraction profile calculated using the third equation in Table 4.2 can be

observed in Figure 4.23. The air fractions from the conductivity data are also shown.

0.5 0.6 0.7 0.8 0.9 1

0

1

2

3

4

5

6

Air Fraction

D i s t a n c e B e l o w C o l u m n T o p

L i p ( c m )

Figure 4.23: Backcalculated and Empirical Air Fraction Profile of a Two-Phase Draining

Froth

Three-Phase:

a) Stabilized Froth:

In this investigation, it was assumed that λ is related to the bubble sizes by the

functional form:



149

( )λ 1 2

1 2

2, =+

a

Db Db, [70]

and, using the experimental air fraction profile, the values of λ that provided the best fit to

the balance equation were estimated. A discrete bubble size distribution was also assumedwith four size classes:

Dbk = [0.2 cm, 0.282 cm, 0.4 cm, 0.566cm].

The calculated and empirical air fraction profiles corresponding to Tests 1-3 are

compared in Figures 4.24-4.26 respectively. The values of the coalescence rate parameters

obtained through the fitting procedure, as well as the estimation error, are provided in

Table 4.3. The figures show that, for the profiles corresponding to Tests 1 and 3, the fit

obtained with the calculated λ 's is not satisfactory.

0.6 0.65 0.7 0.75 0.8 0.85 0.9 0.95 1

10

12

14

16

18

20

22

24

26

28

Air Fraction

D i s t a n c e B e l o w C o l u m

n T o p L i p ( c m )

*** Experimental Profile

__ Backcalculated Profile

Figure 4.24: Backcalculated and Experimental Air Fraction Profile in Stabilized Froth

with Solids (Test No.1)



150

0.6 0.65 0.7 0.75 0.8 0.85 0.9 0.95 1

10

12

14

16

18

20

22

24

26

28

Air Fraction









151

0.6 0.65 0.7 0.75 0.8 0.85 0.9 0.95 1

15

20

25

30

Air Fraction

D i s


( c m )





Table 4.3: Estimated Coalescence Rate Efficiency Parameters for Each of the Measured

Air Fraction Profiles in the Three-Phase Stabilized Froth

Test No. λ1,1 λ1,2 λ2,2 λ1,3 error1 0.420/sec 0.288/sec 0.210/sec 0.187/sec 0.036

2 0.234/sec 0.161/sec 0.117/sec 0.104/sec 0.016

3 0.600/sec 0.412/sec 0.300/sec 0.267/sec 0.031



152

b) Draining Froth:

Estimates of coalescence rate efficiency parameters for a three-phase draining froth were

also obtained using the air fraction profiles derived from conductance measurements. The

functional form that was used to relate the parameters λ to the bubble sizes was the

following:( )λ

1 21 2

2

,= +

aeb Db Db

[71]

It is not well understood how the coalescence rate parameter should relate to bubble size

in the solids-laden draining froth. However, it does seem that the presence of hydrophobic

material makes the froth more stable. One mechanism that would explain this effect is the

increase in the viscosity of the liquid film when solid particles are present. The particles

can also be viewed as barriers that prevent the thinning of the films to the critical rupture

point.

The constants a and b were estimated using the experimental profiles, Equations[62] and [63], and by assuming a bubble size distribution with the following size classes:

Dbk = [0.2 cm, 0.282cm, 0.4cm, 0.566cm]. The resulting coalescence rate parameters,

for three profiles corresponding to different feed rates, are given in Table 4.4. It was

observed that the backcalculated coalescence parameters decreased as the average size of

the pair of coalescing bubbles increased. The calculated coalescence rates for conditions

when the froth is well loaded with solids turned out to be much lower than those for a

two-phase operation. The three experimental profiles are compared to the ones

determined by the calculated coalescence rate parameters in Figure 4.27.

Analyzing how the constants a and b could be related to the froth characteristics, it

is observed that the parameter a determines the extent of the increase in air fraction along

the froth, and it could be therefore associated with the fractional liquid content. On the

other hand, the value of b determines the shape of the profile, which suggests that it could

account for the presence of solid material.

Table 4.4: Estimated Coalescence Rate Efficiency Parameters Corresponding to Draining

Froth Profiles Obtained with Three Different Feed Rates

Feed Velocity λ1,1 λ1,2 λ2,2 λ1,3 error

0 >1/sec 0.707/sec 0.245/sec 0.151/sec 0.0172

0.06cm/sec 0.229/sec 0.101/sec 0.038/sec 0.025/sec 0.0032

0.08cm/sec 0.062/sec 0.056/sec 0.049/sec 0.046/sec 0.0107



153

0.5 0.6 0.7 0.8 0.9 1

0

1

2

3

4

5

6

Air Fraction

D i s


( c m )

* - Vfeed=0 cm/sec

+ - Vfeed=0.06 cm/sec

o - Vfeed=0.08 cm/sec

Figure 4.27: Backcalculated and Experimental Air Fraction Profiles in Draining Froth for

Three Distinct Feed Rates

4.4 Model Solving

The model equations were solved using MATLAB. This made the programming

task faster since MATLAB incorporates built-in functions for solving differential

equations as well as algebraic equations. The capabilities for handling matrices and

vectors can make the code simpler as well.

The model is set up so that a tailings flowrate would be assumed initially. The

dynamic equations are then solved simultaneously for all the column zones using a finite-

difference approximation. At the column top, the product flowrate is determined. From

this point on, the iterations can proceed in either of the following ways, according to

which parameters are known or constant:

i) By assuming a constant pulp level, the tailings flowrate is adjusted at the end of each run

until the mass balance of the flows into and out of the column is satisfied;



154

ii) or, assuming that the flowrates are known, an iteration is done adjusting the froth depth

until the flow balance converges.

The information needed in order to run a simulation includes the bubble size

classes, the gas rate, wash water rate, the initial bubble size distribution, the coalescence

efficiency rate terms, the number of perfectly mixed regions in the collection zone, and theheight of the zones. Of all these variables, the coalescence rate parameters are the only

ones that have to be estimated on-line. The other parameters are set during operation or

can be measured.

4.5 Simulations

Simulation No. 1

The model equations were solved first for a set of typical operating conditions.

The simulation results were then analyzed based on actual column responses.The initial operating conditions were the following:

• Vg=1.0 cm/sec;

• initial Vb=0.22 cm/sec;

• Number of bubble size classes Nb=6 ;

• In the pulp zones, initial ε k (k=1..6) = 0.01;

• In the froth intervals, initial ε k (k=1..6) = 0.70/Nb;

• wash water velocity Vw=0.4 cm/sec;

• constant stabilized froth depth;

• Bubble diameters representative of the six size classes:

[0.07 cm, 0.099 cm, 0.14 cm, 0.198 cm, 0.28 cm, 0.396 cm] ;• Volume fraction of bubbles of each size class generated at the bottom of the column:

[0.1, 0.7, 0.2, 0, 0, 0];

• The coalescence rate parameters were calculated using Equation [69], where the value

of the constant a varies for each region in the froth: in the stabilized froth, a=6e-5; in

the wash water zone, a=1e-3; in the draining froth, a=3e-3.

At the end of the run, the product water is calculated from the gas velocity and the

predicted air fraction at the top of the column so that

( )Vc Vg

top

top

=−1 ε

ε [72]

The bias water is estimated using the water balance equation

Vb Vw Vc= − [73]

and this value is compared to the assumed bias velocity at the start of the simulation. If

the difference between them is greater than a previously defined tolerance, a new bias

velocity is calculated according to the following equation and the simulation is repeated.



155

new Vb VbVw Vc Vb

= ±− −

2[74]

The + sign applies if the calculated product water Vc is less than the difference Vw-Vb

(assumed bias is too low). On the other hand, if the assumed bias velocity is higher than

the one calculated with the balance equation, the - sign is used in the equation above. Thisiterative procedure is repeated until the difference between the previous bias velocity and

the value calculated from the model solution converge.

The predicted dynamic changes in air fraction in all column regions, for the

operating conditions already described, are shown in Figure 4.28. Each curve represents a

column zone. The shape of the steady-state air fraction profile predicted during this

simulation can be appreciated better in Figure 4.29. The final bias velocity, after four

iterations, was Vb=0.281 cm/sec. The shift in the froth air fraction profile during

iterations is represented in Figure 4.30.

0 1 2 3 4 5 6

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

Time (min)


Collection Region

Froth Zones

Figure 4.28: Dynamic Solution to Two-Phase Model for Each of the Column Zones

(Vg=1.0 cm/sec; Vb=0.281 cm/sec)



156

0 0.2 0.4 0.6 0.8 10

50

100

150

200

250

300

Air Fraction

D i s t a n c e f r o m B

o t t o m o f

t h e C o l u m n ( c m )

Interface

--->--->

DrainingFroth

Figure 4.29: Predicted Air Fraction Profile Along the Full Column Length for

Each Iteration (Vg=1.0 cm/sec; Vb=0.281 cm/sec)



157

0 0.2 0.4 0.6 0.8 1180

190

200

210

220

230

240

250

260

Air Fraction


o t t o m o f


<---FirstIteration

--->LastIteration

Figure 4.30: Close View of the Predicted Air Fraction Profile Along the Froth for Each

Iteration (Vg=1.0 cm/sec; Vb=0.281 cm/sec)



158

Simulation No.2

Another simulation was performed assuming an initial bias velocity Vb equal to

0.28 cm/sec and a gas velocity Vg= 1.2 cm/sec. The predicted air fraction at the top, with

Vg=1.2 cm/sec, was lower than the value obtained in the previous run (Vg=1.0 cm/sec).

This indicates that the product liquid velocity increased with gas rate, so the bias wasreduced with respect to the one in Simulation No.1 (new Vb=0.225 cm/sec). This is in

agreement with the behavior observed in operating flotation columns, where a higher gas

rate normally results in a reduction in bias water and a wetter froth due to increased slurry

entrainment. The average air fraction in the pulp increased slightly, which is the normal

response after an increase in gas rate during column operation. Meanwhile, the air

fraction at the base of the froth decreased. The profile obtained after two iterations is

shown in Figure 4.31 and, in a bigger scale, in Figure 4.32, while the dynamic responses

are depicted in Figure 4.33.

0 0.2 0.4 0.6 0.8 10

50

100

150

200

250

300

Air Fraction


o t t o m o

f t h e C o l u m n ( c m )

Pulp

Interface

StabilizedFroth

--->

DrainingFroth

Figure 4.31: Predicted Air Fraction Profile Along the Whole Column Length (Vg=1.2

cm/sec; Vb=0.225 cm/sec)



159

0 0.2 0.4 0.6 0.8 1180

190

200

210

220

230

240

250

260

Air Fraction


o t t o m o f


Figure 4.32: Close View of the Predicted Air Fraction Profile Along the Froth (Vg=1.2

cm/sec; Vb=0.225 cm/sec)





161

Simulation No. 3

For the next simulation, the average bubble size was reduced. The new discrete

distribution, with six size classes, is represented by:

Db = [0.0600 0.0850 0.1200 0.170 0.2400 0.3390] cms.

The superficial gas velocity was Vg=1.0 cm/sec, and the initial bias velocity wasVb=0.281 cm/sec. The reduction in bubble size caused a small increase in the predicted

air fraction in the pulp, while at the base of the froth it decreased, as calculated with the

drift-flux model. This result can be easily explained since, when smaller bubbles are

generated, the rise velocity decreases and air fraction increases. The final result was a

small reduction in the bias water (Vb=0.275 cm/sec), which can be justified by the higher

surface area crossing the interface, which translates into more entrained water. The profile

is shown in Figure 4.34, and a close view of the froth region is presented in Figure 4.35.

By comparing Figure 4.35 with Figures 4.30 or 4.32, it can be observed that the net

increase in air fraction in the stabilized region was greater when smaller bubbles were

assumed. This effect is a consequence of the changes in the coalescence rate parameters,

which were are automatically adjusted based on Equation [69].

0 0.2 0.4 0.6 0.8 10

50

100

150

200

250

300

Air Fraction


o t t o m o

f t h e

C o l u m n ( c m )

Figure 4.34: Predicted Air Fraction Profile Along the Column Length (Smaller Db ave;

Vg=1.0 cm/sec; Vb=0.275 cm/sec)



162

0 0.2 0.4 0.6 0.8 1180

190

200

210

220

230

240

250

260

Air Fraction


o t t o m o


Figure 4.35: Close View of the Predicted Air Fraction Profile Along the Froth (Smaller

Db ave; Vg=1.0 cm/sec; Vb=0.275 cm/sec)

Simulation No. 4

Finally, the position of the interface was raised by decreasing the froth depth from

50 cm to 30 cm and increasing the pulp height proportionally. The profiles are provided inFigures 4.36 and 4.37. Use of a more shallow froth in the model equations resulted in a

lower bias velocity (Vb=0.26 cm/sec) and a smaller air fraction at the top of the froth.

Such result appears reasonable since a deeper froth would give the bubbles more

opportunity to coalesce. In addition, raising the interface level requires decreasing the

tailings rate, so the bias water decreases while the draining froth becomes wetter.



163

0 0.2 0.4 0.6 0.8 10

50

100

150

200

250

300

Air Fraction


o t t o m o


Figure 4.36: Predicted Air Fraction Profile Along the Column Length (Froth Depth = 30

cm; Vg=1.0 cm/sec; Vb=0.260 cm/sec)



164

0 0.2 0.4 0.6 0.8 1180

190

200

210

220

230

240

250

260

Air Fraction


o t t o m o


Figure 4.37: Close View of the Predicted Air Fraction Profile Along the Froth (Froth

Depth=30 cm; Vg=1.0 cm/sec; Vb=0.260 cm/sec)



165

4.6 Model Validation

Verification of the prediction capabilities of the model was carried out as follows:

• Air fraction profiles in a two-phase column were recorded using a conductivity probe.

Since any change in the frother concentration in the froth would alter the coalescence

rate parameters significantly, the aeration rate was the only parameter varied betweentests. The underlying assumption is that, if the changes are small, the air rate has a

lesser effect on the dilution of frother concentration than other operating parameters.

• The values of the coalescence rate parameters corresponding to a particular superficial

gas velocity were estimated from the experimental data using the steady-state model

equations.

• Setting as an initial condition the profile used in the previous step along with the

calculated coalescence rate parameters, the simulated profile for a new aeration rate

was obtained.

• The simulated profile was compared to the experimental profile corresponding to the

same gas rate.

Figure 4.38 shows the experimental profiles corresponding to three different gas

rates, obtained while operating at the conditions listed in Table 4.5.

Table 4.5: Operating Conditions Set During the Measurement of the Conductivity

Profiles Employed for Model Verification

Test No. Superficial Air

Velocity

Frother Rate Wash Water

Rate

Froth Depth Measured

Pulp ε g1 1.35 cm/sec 0.034 ml/min 400 ml/min 33 cms 10%

2 1.5 cm/sec 0.034 ml/min 400 ml/min 33 cms 12%

3 1.65 cm/sec 0.034 ml/min 400 ml/min 33 cms 13%



166

0 0.2 0.4 0.6 0.8 1

15

20

25

30

35

40

45

50

Air Fraction

D i s t a n c e B e l o w C o l u m n T o p L i p ( c m )

1

2

3

1 - Vg = 1.3 cm/sec2 - Vg = 1.5 cm/sec3 - Vg = 1.65 cm/sec

Figure 4.38: Experimental Air Fraction Profiles Obtained Through Conductivity

Measurements at Several Gas Velocities and Used for Model Verification

In the first validation test, the coalescence rate parameters for the profile

corresponding to a gas superficial velocity Vg= 1.35 cm/sec were estimated from the

empirical values. A bubble size distribution was assumed with the following size classes:

Db=[0.2cm 0.2*√ 2 cm 0.4cm 0.4*√ 2 cm] and an average bubble size of about 0.25 cm.

Figure 4.39 shows the measured air fractions along the stabilized froth in contrast with the

values backcalculated using the estimated coalescence parameters.





168

0.5 0.6 0.7 0.8 0.9 1

15

20

25

30

Air Fraction

D i s t a n c e B e l o w C o l u m n T o p L i p ( c m )

Figure 4.40: Predicted Profile (solid line) Versus Experimental Air Fractions for Vg=1.5

cm/sec

When the validation procedure was repeated using the profile obtained at Vg=1.5

cm/sec for estimation of the coalescence rate parameters and the profile at Vg=1.65

cm/sec for evaluating the model prediction , a similar situation was encountered. Figure

4.41 shows the results of fitting the air fraction data to the steady-state model equation,while Figure 4.42 compares the predicted and empirical profiles at Vg=1.65 cm/sec. Once

again, the predicted profile is slightly more stable.



169

0.5 0.6 0.7 0.8 0.9 1

15

20

25

30

Air Fraction



Figure 4.41: Backcalculated and Experimental Profile in the Stabilized Froth

Corresponding to Vg=1.5 cm/sec



170

0.5 0.6 0.7 0.8 0.9 1

15

20

25

30

Air Fraction

D i s t a n

c e B e l o w C o l u m n T o p L i p ( c m

)

Figure 4.42: Predicted Profile (solid line) Versus Experimental Air Fractions for Vg=1.65

cm/sec

Next, it was examined how the calculated profile along the pulp and froth would

agree with actual column profiles. The steady-state solution for Vg=1.35 cm/sec and the

steady-state profile predicted by the model after an increase to Vg=1.5 cm/sec were

compared to empirical profiles (Figure 4.38), which were recorded before and after an

equivalent air rate increment. An average bubble size of 0.7 cm in the pulp was utilized

for the model calculations. As shown in Figure 4.43 , the model predictions are very close

to the measured profiles. When the gas rate was further increased to 1.65 cm/sec, a

similar result was obtained, as indicated by the comparison in Figure 4.44 of the measuredand calculated profiles before and after the increase (from Vg=1.5 to 1.65 cm/sec). In

both plots, the calculated air fractions in the froth are a little higher than the measured

values. However, the agreement is very good, particularly since the empirical Richardson-

Zaki relationship (Equation [11]) was employed to calculate the froth air fraction for each

pulp air fraction solution. Nevertheless, a more extensive examination is required to

determine which equation for the bubble rise velocity is appropriate for most situations.



171

0 0.2 0.4 0.6 0.8 1

15

20

25

30

35

40

45

50

Air Fraction

D i s t a n c e f r o m C

o l u m n T o p L i p ( c m )

* Vg=1.35 cm/sec (Measured)

o Vg=1.50 cm/sec (Measured)

1 - Vg=1.35 cm/sec (Predicted)


1 2

Figure 4.43: Measured and Predicted Steady-State Profiles for an Increase in Gas Rate

from Vg=1.35 cm/sec to Vg=1.5 cm/sec



172

0 0.2 0.4 0.6 0.8 1

15

20

25

30

35

40

45

50

Air Fraction

D i s t a n c e f r o m C

o l u m n T o p L i p ( c m )

1 2* Vg=1.50 cm/sec (Measured)

o Vg=1.65 cm/sec (Measured)



Figure 4.44: Measured and Predicted Steady-State Profiles for an Increase in Gas Rate

from Vg=1.5 cm/sec to Vg=1.65 cm/sec

The ability of the model to approximate the dynamic responses of a real two-phase

operation was verified by comparing the actual time that lapsed between a change in air

rate and the reaching of steady state with times predicted by the model. During the

experiments, air fraction was measured in the collection region utilizing two pressure

transducers. First, the aeration rate was decreased from 1.65 cm/sec to 1.35 cm/sec, while

keeping other operating conditions constant. The measured dynamic variation of the pulp

air fraction is presented in Figure 4.45 along with the dynamic response predicted by the

model equations for a zone located halfway down the collection region. The time constantof the simulated system turned out to be very close to that of the actual column used in the

experiments. In another test, the air rate was increased from Vg=1.5 cm/sec to Vg=1.65

cm/sec. The measurement from one conductivity electrode in the stabilized froth was

recorded to establish the dynamic variations in air fraction at a a particular froth height.

Figure 4.46 shows how the froth air fraction reaches steady-state in an interval which is

similar to the time required in the simulated responses.



173

0 0.5 1 1.5 2 2.5 30

0.05

0.1

0.15

0.2

0.25

Time (min)

P u l p A i r F r a c t i o

n2

1

1 - Measured Response

2 - Simulated Response

Figure 4.45: Comparison of Measured and Simulated Time Responses for the Pulp Air

Fraction



174

0 50 100 150 200 250 3000.5

0.55

0.6

0.65

0.7

0.75

0.8

0.85

0.9

0.95

1

Time (sec)


Measured Response

Predicted Dynamic Responses

Lowest Height Intervals in Stabilized Froth

Figure 4.46: Comparison of Measured and Simulated Time Responses for Air Fraction in

the Stabilized Froth



175

4.7 Solution Using Quasi Steady-State Technique for the Draining

Froth

The quasi-steady-state solution of the froth equations is not likely to provide a

good representation of the process dynamics unless the froth time constant is muchsmaller than that of the collection region. This requirement is not expected to be satisfied

in the stabilized froth, especially because of the large froth depths that are normally

involved. It was investigated, however, if the quasi-steady state technique could be

applied for finding an approximate dynamic solution to the draining froth equations. A

new air fraction value at each time interval is calculated using the steady-state equation

below:

Vg Vg d

dz D Ak k

k

k k

k

pp ppε ε ε

ε

∑ ∑−

+ − =2 0 [75]

After applying a finite difference approximation, the air fraction at the region is given by

ε ε

ε

ε

ε ε

ε

ε

k

z

k

z

k

z

k

k

z

k z

k

pp

k

z

k

k

z

k z

k

ppdz

Vg Vg A dz

Vg Vg D= + −

− −

−−

−

−

−

−−

−

−

∑ ∑ ∑ ∑

1

1

1

1

2

1

1

1

1

2

1

[76]

Figure 4.47 compares the time response for the top draining froth zone obtained

with this approximation with the response derived with the truly dynamic equations, for

Vg=1.2 cm/sec and the same bubble size distribution and column dimensions (depth of

draining froth equal to 10 cms). The plot indicates that both models behave almost

exactly alike. The steady-state values are slightly different because of discretization

errors introduced by the finite difference approximation.



176

0 1 2 3 4 50

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

Time (min)

A i r F r a c t i o n a t T o p

o f C o l u m n

1

2

1 - Time Response from Dynamic Equations

2 - Time Response from Quasi S-S Equations

Figure 4.47: Comparison of the Solutions Provided by the Quasi-Steady-State

Approximation and the Dynamic Equations at the Top Column Zone



4.8 Summary

• A population balance model that describes the dynamic behavior of a two-phase

flotation column has been derived. The degree of mixing in the collection zone was

represented by dividing the region in a series of perfectly mixed tanks, while the

behavior of the froth was assumed to be mainly plug flow.

• The model incorporates a technique to represent bubble coalescence in the froth

regions by introducing a coalescence-efficiency-rate parameter. This coalescence

model was based on the work done by several investigators on the mechanisms of

granulation and pellet growth. Several numerical techniques were investigated for the

approximate solution of the froth equations. The shape of the profiles was found to be

very sensitive to the values of the coalescence parameters. Using an iteration

technique, it was possible to obtain a closed-form solution that satisfies a mass balance

around the column flows.

• Empirical profiles were used to calculate approximate values of the coalescence rate

terms by fitting them to the steady-state population balance model. A few

mathematical functions were suggested to express their dependence on bubble size.

The estimates of the coalescence-efficiency-rate parameters were calculated by

assuming a bubble size distribution and an air fraction distribution over the range of

bubble size classes.

• For the stabilized froth, it was assumed that smaller bubbles at the bottom of the zone

are more likely to coalesce than the larger bubbles. There were no significant

differences between the calculated coalescence rates for a two-phase and three-phase

froth. • The steady-state population balance model was also employed to calculate typical

values of coalescence rate parameters in the draining froth. The estimated coalescence

rate parameters turned out to be higher in a 'wet' froth than in a 'dry' one. The presence

of solids appears to have a pronounced effect since, in a well-loaded froth, the

estimated air fractions were normally higher than in a two-phase froth.

• A series of simulations were performed in order to compare the model predictions to

known column behavior. In a laboratory column, the gas rate was increased in small

increments, and the initial and final steady-state profiles were measured with a

conductivity probe extending from the pulp to stabilized froth These profiles along

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