INTRODUCTION
Column flotation in mineral processing industries for the
concentration of ore is essentially a counter-current bubble column. For
the maximum recovery from the ore, the hydrodynamics of the flotation
column must be clearly understood. In a typical flotation column, a
slurry feed is introduced against a plume of rising gas bubbles. The
portion of the column below the slurry feed point operates in the bubbly
flow (homogeneous) regime and is referred to as collection zone (Finch
and Dobby, 1990). The froth which is formed in the column above the
slurry feed point serves the important purpose of cleaning and
transporting the concentrate mineral to overflow. The design and
operating parameters must be maintained such that the bubbly flow regime
and froth regime co-exist in a flotation column.
The height of the froth zone can vary considerably with the gas and
liquid flow rates and the nature and the amount of surface-active
components (frothers) present in the liquid phase. A small amount of
water is sprinkled on the top of the froth and it is referred to as wash
water. A part of wash water overflows with the froth bubbles and the
remaining part flows down the froth counter-current to the gas phase and
is referred to as bias water. A net downward flow of water through the
froth is referred to as "positive bias" in the flotation
literature (Finch and Dobby, 1990). The height of the froth zone
increases considerably due to the practice of addition of wash water and
maintenance of a positive bias. The column froths are relatively wet
(higher liquid content) compared to conventional foams. They contain
spherical or deformed gas bubbles as against the cellular structure of
polyhedral bubbles observed in conventional foams (Bhakta and
Ruckenstein, 1996).
The schematic of a column froth is shown in Figure 1. The two
distinct zones, namely the homogeneous bubbly zone and the froth zone,
are separated by an interface. Typically, the gas holdup in the
collection zone can be 10-20% depending upon the superficial gas
velocity and bubble size. At the interface, there is an abrupt increase
in the gas holdup to about 60% (Yianatos et al., 1986). The gas holdup
can increase along the froth height from about 60% (at the interface) to
about 80% (near the wash water addition point). Mean bubble size at any
axial location in the froth can be measured by photographic analysis.
Figure 2 shows the axial profile of mean bubble size along the froth
height (Finch and Dobby, 1990). Typically, the sauter mean diameter
increases from 1 mm to 3 mm along the froth height due to bubble
coalescence (Yianatos et al., 1986; Ata et al., 2003).
[FIGURE 1 OMITTED]
[FIGURE 2 OMITTED]
Various investigators have modelled the gas holdup in the froth.
Finch and Dobby (1990) have carried out drift flux analysis. This
methodology relies on the drag law used to describe the hindered motion
of the bubbles in the swarm. Yianatos et al. (1986) obtained the gas
holdup as a function of bubble diameter in the froth. They have
considered the froth to be made up of two different sections viz:
expanded bubble bed and packed bubble bed. Separate models were
developed for the two sections. While the model for the packed bubble
bed was based on a foam drainage equation, the model for the expanded
bed was based on the analysis of frictional pressure drop. However,
their expression for the frictional pressure gradient needs careful
attention. Further, the assumption of laminar flow is restrictive for
typical bubble Reynolds numbers encountered in flotation columns. In
this work, we present the analysis of frictional pressure gradient based
on energy balance. Both laminar and turbulent contributions to the
frictional pressure gradient based on Ergun equation are considered.
Most importantly, the effect of superficial gas and liquid velocities on
the gas holdup is clearly brought out.
MATHEMATICAL MODEL
One-dimensional steady-state axial flow of the gas and liquid is
considered. The hydrodynamic variables like gas holdup, bubble diameter
and flow rates, etc., can be considered to be averaged over the column
cross-section. We do not consider the presence of a solid phase in the
froth, although it is known to affect bubble coalescence and froth
stability (Ata et al., 2003). The bubble coalescence is not addressed
directly. Instead, the variation of mean bubble diameter along the froth
height is assumed or taken from the literature. The axial profile of gas
holdup is obtained accordingly. Thus, the total froth height is already
assumed. The model is based on the mass and energy balances for the gas
and liquid flowing in the froth. Since a positive bias is assumed in the
column, the net liquid flow is counter-current to the rising bed of
bubbles.
Consider a cylindrical column of diameter D in which the froth is
present. The axial location z = 0 corresponds to the froth-liquid
interface (shown in Figure 1). The bubbles continuously enter the froth
at this location and they exit the froth bed from the top (z = H) either
due to coalescence with air or the movement into the concentrate. Steady
state is assumed in the analysis and hence the total height of the froth
is constant. Imagine a small volume element of diameter D and thickness
[DELTA]z at any location z in the froth. The gas and liquid flow
countercurrent to each other in this volume element. The thickness of
the volume element, [DELTA]z is small enough to consider uniformity of
gas holdup over it and let this gas holdup be represented as [[member
of].sub.G] (z). Typically [DELTA]z can be of the order of ten times the
bubble diameter. An energy balance is considered over this volume
element. The kinetic energies associated with the incoming and outgoing
gas have been neglected. The same is the case with the kinetic energy of
the liquid. The gas entering the volume element must overcome the
pressure head acting on it and hence, it possesses higher-pressure
energy than the gas leaving the control volume. However, the gas leaving
the control volume has higher potential energy due to an extra
elevation, [DELTA]z. Hence, the energy dissipated per unit time by the
gas in the control volume is given by:
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.] (1)
Similarly, the energy dissipated per unit time by the liquid in the
control volume is given by:
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.] (2)
The net energy dissipated per unit time in the control volume due
to counter-current flow of gas and liquid is obtained from Equations (1)
and (2):
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.] (3)
Here, we have utilized the fact that the slip velocity for the
counter-current gas-liquid flow is given by:
[V.sub.S] (z) = [V.sub.G]/[[member of].sub.G](z) + [V.sub.L]/
[[member of].sub.L](z) (4)
The frictional pressure drop and the net energy dissipated per unit
time in the control volume are related to each other through the
following equation:
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.] (5)
From Equations (3) and (5), the frictional pressure gradient is
given by:
[[DELTA]P.sub.F]/[DELTA]z = ([[rho].sub.L] - [[rho].sub.G])[[member
of].sub.G] (z)g (6)
The pressure gradient given by Equation (6) is analogous to the one
in the case of fluidized beds. The friction at the column wall and the
friction at the gas-liquid interface both contribute to the pressure
gradient. Since the total gas-liquid interfacial area is much higher
than the surface area of the wall, the friction at the wall can be
neglected. The frictional pressure drop in the packed bed of bubbles can
be described by Ergun equation.
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.] (7)
The constants A and B have the values of 150 and 1.75,
respectively, for the packed bed of solids (Ergun, 1952). The two terms
on the RHS of Equation (7) correspond to the viscous and inertial
contributions to the frictional pressure gradient. When the Reynolds
number is less than 10, the viscous contribution to the pressure drop
dominates and the second term on RHS on Equation (7) can be neglected.
On the other hand, when the Reynolds number is greater than 1000, the
inertial contribution to the pressure drop dominates and the first term
on the RHS of Equation (7) can be neglected. Typically in the column
froth with bubble diameter about 2 mm and bubble slip velocity about 30
mm/s, the Reynolds Number is about 60 (Yianatos et al., 1986). Thus, the
froth flow is in the transition regime and the viscous as well as
inertial contributions are significant.
The values of constants A and B, as they appear in the Ergun
equation, are for packed bed of solids. The solid-fluid interface is
rigid. With this condition and the energy balance, Pandit and Joshi
(1998) have provided the justification for the values of the constants
(A = 150 and B = 1.75). Although the froth can be treated analogous to a
packed bed, there is an important difference between the solid particles
and bubbles. The gas-liquid interface is generally partially mobile. The
extent of mobility of the interface depends upon the bubble Reynolds
number, [Re.sub.B] and the presence of surface-active agents at the
interface. Hence, the values of constants A and B, in case of a froth,
need a modification. The greater the mobility of the interface, the
smaller will be the velocity gradients in the liquid surrounding the
bubble. This would lead to relatively smaller values of pressure drop
and hence the constants A and B are expected to be smaller in case of
packed bed of bubbles. In this work, we have found that A = 65 and B =
0.8 best describe the available experimental data of the gas holdup in
the froth (as shown in Figure 3). These constants are obtained from the
regression analysis and have been maintained throughout the work.
Langberg and Jameson (1992) have also modified the values the constants
appearing in Ergun equation while analyzing the conditions for the
coexistence of froth and liquid phases in a flotation column. It is
instructive to note that these constant can vary with the nature and the
concentration of the frother added in the flotation column.
Equations (6) and (7) can be combined to obtain the following
equation for the gas holdup:
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.] (8)
The slip velocity in the above equation is to be obtained from
Equation (4). Equation (8) represents a non-linear equation in the gas
holdup. Knowing the superficial gas and liquid velocities and the bubble
size at any axial location, the gas holdup at that location can be
obtained. The z-coordinate does not appear explicitly in Equation (8).
In other words, we assume that the froth height is already given and
then predict the gas holdup at various axial locations. Obviously,
Equation (8) predicts a constant gas holdup for the entire froth bed if
the bubble size is the same for all z's. Whether or not the bubble
size remains the same over the entire froth height depends upon the
bubble coalescence, which in turn, is dependent upon the nature and
amount of surface-active components present in the froth. In this
analysis, we have assumed that the bubbles are perfectly spherical.
However, the bubbles in the froth are deformed. They can be oblate
ellipsoidal as observed by Pal and Masliyah (1989). If the aspect ratio
of the ellipsoidal bubbles is known, its shape factor can be calculated
and appropriately included in Equation (8). The aspect ratio of a single
bubble in an infinite liquid medium can be obtained as a function of
bubble diameter (Clift et al., 1978). However, with the present status
of knowledge, it is very difficult to obtain the aspect ratio of a
bubble in the presence of other bubbles in the froth. Hence, we have
analyzed the problem with the assumption of spherical bubbles only.
[FIGURE 3 OMITTED]
RESULTS AND DISCUSSION
A case of constant bubble diameter for the entire froth bed is
considered first. In this case, Equation (8) predicts the average gas
holdup in the froth when the values of constants A and B are known. An
extensive experimental database of the mean bubble size and gas holdup
for various combinations of superficial gas and liquid velocities in a
flotation column froth has been obtained by Pal and Masliyah (1989).
They have conducted the two sets of experiments employing two different
surfactants to generate the froth. In set I, 30 mg/L of Dowfroth 250-C
was used whereas in set II, 15 mg/L of Triton x-100 was used. Utilizing
their experimental data in Equation (8), the constants A and B were
evaluated. It was found that A = 65 and B = 0.8 allows a reasonable fit
for the gas holdup data in the froth as shown in Figure 3. As noted
earlier, these constants are smaller than the standard constants for a
packed bed of spherical solids (i.e., A = 150 and B = 1.75). A
frictional pressure loss which occurs at the gas-liquid interface in a
packed bed of bubbles is smaller due to partial mobility of the
interface. Hence, the constants A and B for a froth are naturally
expected to be smaller. Although a surfactant is present at the
gas-liquid interface in a typical froth, it is not sufficient to impart
the complete rigidity to the interface.
We now consider the case of axially varying bubble size. In this
case, the gas holdup also varies along the froth height. Employing A =
65 and B = 0.8 in Equation (8), the gas holdup at various axial
locations (z) can be obtained when the bubble diameter as a function of
z is known from Figure 2. Typical values of the superficial gas and
liquid velocities prevalent in flotation columns are considered for the
predictions. At any axial location in the froth, the gas holdup is seen
to decrease with an increase in the superficial gas velocity as well as
superficial liquid velocity. This result is in agreement with the
experimental findings of Yianatos et al. (1986). It is observed that the
effect of superficial gas velocity on the gas holdup is different in the
homogeneous bubbly flow and froth flow. In case of homogeneous bubbly
flow, the gas holdup increases almost linearly with an increase in the
superficial gas velocity due to concomitant increase in the number of
bubbles per unit volume (Joshi et al., 1998). However, in case of froth,
the number density of bubbles cannot increase in this manner. Froth bed,
being a packed bed of bubbles, the number density depends upon the
arrangement of bubbles and the gas holdup can change only when the
nature of packing of bubbles changes. In fact, with the increase in the
superficial gas velocity, more amount of liquid is entrained in the
froth zone through the interface and hence the gas holdup decreases
(Finch and Dobby, 1990). This effect is captured in Figure 4.
Similarly, the effect of superficial liquid velocity is also
different in the case of column froths and homogeneous bubbly regime.
The froth becomes wetter and hence the gas holdup decreases with the
increase in the superficial liquid velocity as shown in Figure 5.
However, in case of counter-current bubble column operating in the
homogeneous regime the gas holdup increases with the increase in
superficial liquid velocity as observed by Finch and Dobby (1990) due to
increase in the residence time of bubbles in the column. Thus, the
hydrodynamics of the column froths is significantly different from the
homogeneous bubbly regime.
[FIGURE 4 OMITTED]
[FIGURE 5 OMITTED]
Yianatos et al. (1986) have measured the bubble size distribution
as well as gas holdup at various axial locations in a froth in a
continuous flotation column. Bubble sizes were obtained by analysis of
photographic images of the froth whereas the gas holdup was obtained by
pressure sensors or conductivity measurements. Utilizing their
experimental data for sauter mean bubble diameter, the gas holdup at
various axial locations in the froth is obtained and it is compared with
the experimental gas holdup values in Figure 6. A good agreement between
the two essentially highlights the utility of the model.
The model developed in this work is applicable for expanded bubble
bed. Considering an arrangement of mono-sized spherical gas bubbles in
hexagonal close packing, the upper limit on the gas holdup for an
expanded bubble bed is 74%. However, this upper limit on the gas holdup
can increase when a bubble size distribution exists and the bubbles are
deformed. It is in this spirit, we have utilized the model even for the
gas holdup higher than 74%. However, we note that the model becomes
increasingly inapplicable at very high gas holdup (above 85%) where the
polyhedral bubbles structure exists. In such a case, the analysis of
liquid drainage through the plateau borders under the influence of
gravity and capillary forces becomes imperative (Bhakta and Ruckenstein,
1996; Neethling et al., 2000; Stevenson et al., 2003).
[FIGURE 6 OMITTED]
CONCLUSION
A one-dimensional steady-state model developed in this work is
based on the idea that the froth can be treated as the inverse fluidized
bed of bubbles. It can be taken as a first approximation to the analysis
of two-phase froths and its predictions are reliable as compared with
the laboratory data ([+ or -] 5% from the experimental results).
Further, it correctly captures the effect of superficial gas and liquid
velocities and bubble diameter on the axial variation of the gas holdup
in the froth. A mathematical model that describes accurately the froth
behaviour in flotation columns is important because it gives the
possibility to understand the relationship between the hydrodynamics of
the froth and the hydrodynamics of collection zone, both as a unit. In
this regard, the model developed here serves as a point of reference for
the development of elaborate representations that may include the effect
of bubble shape and even the presence of a solid phase.
Manuscript received October 10, 2006; revised manuscript received
February 10, 2007; accepted for publication March 16, 2007
REFERENCES
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in Bubble Column Reactors," Proc. Indian Natl. Sci. Acad. 64A,
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Approach," Rev. Chem. Eng. 14, 321-371 (1998).
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(2003).
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Manish R. Bhole and Jyeshtharaj B. Joshi * Institute of Chemical
Technology, University of Mumbai, N. P. Marg, Matunga, Mumbai,
Maharashtra, India 400019
* Author to whom correspondence may be addressed. E-mail address:
jbj@udct.org
NOMENCLATURE
A proportionality constant for viscous term in
Ergun equation
B proportionality constant for inertial term in
Ergun equation
[d.sub.B] bubble diameter (m)
D column diameter (m)
g acceleration due to gravity (m/[s.sup.2])
[[DELTA]P.sub.F] frictional pressure drop (N/[m.sup.2])
[Re.sub.B] bubble Reynolds number, [d.sub.B][[rho].sub.L]
[V.sub.S]/[[micro].sub.L]
[V.sub.G] superficial gas velocity (m/s)
[V.sub.L] superficial liquid velocity (m/s)
[V.sub.S] slip velocity, i.e., relative velocity between
the bubbles and liquid (m/s)
z axial coordinate in the froth column (m)
Greek Symbols
[[member of].sub.G] fractional gas holdup at an axial location z in
the froth
[[member of].sub.L] fractional liquid holdup at an axial location z
in the froth
[[member of].sub.G] density of gas (kg/[m.sup.3])
[[rho].sub.L] density of liquid (kg/[m.sup.3])
[[micro].sub.L] viscosity of liquid (kg/m.s)