8/6/2019 Column Flotation Dynamic Model http://slidepdf.com/reader/full/column-flotation-dynamic-model 1/72 106 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 the two-phase flow parameters in the following way: Vg Vl Ugs 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
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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-
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
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.
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.
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
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
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
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
F r o t h H e i g h t ( c m )
Figure 4.8: Air Fraction Profile in the Stabilized Froth Calculated with the Taylor Series
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.
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.
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.
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],
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
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.
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
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.
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)
A i r F r a c t i o n
Collection Region
Froth Zones
Figure 4.28: Dynamic Solution to Two-Phase Model for Each of the Column Zones
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
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 )
Pulp
Interface
StabilizedFroth
--->
DrainingFroth
Figure 4.31: Predicted Air Fraction Profile Along the Whole Column Length (Vg=1.2
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
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 )
Figure 4.34: Predicted Air Fraction Profile Along the Column Length (Smaller Db ave;
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.
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
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.
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.
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.
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.