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Virginia Tech | minutes. The receding contact angle of the hydrophobic gold surface in water is 53.5 ± 2.0o. A
close fit was obtained between the experimental data and numerical results, in which the
hydrophobic interaction was included in the extended DLVO theory with K = 12 x 10−18 J for
132
the estimation of the surface forces. A comparison of the results obtained at 10 minutes and at 30
minutes hydrophobization time showed that the hydrophobic force became more attractive when
the gold surfaces were immersed in a 10-5 M KEX solution for a longer immersion time. It was
shown in the green line that the overall total force increased slowly and became attractive when
the film ruptured. Note that, the deviation was found at t > 2 s between the experimental data
and the theoretical prediction. The result showed that the theoretical prediction with an inclusion
of the hydrophobic interaction of a power law overestimated the overall interaction force. At t >
2 s, the film thickness was below 100 nm. It appears that the hydrophobic force of a power law
might overestimate the force at a short-range distance.
Figure 10.9 shows the interaction force in wetting films formed on the gold surfaces
hydrophobized in the 10-5 M KEX solution for 120 minutes. At 120 minutes, the contact angle of
water on the gold surface decreased slightly, and θ = 51.0 ± 1.2o. We have shown that the force
r
constant (K ) for the hydrophobic force decreased to 10.4 x 10−18 J at a hydrophobizaiton time
132
of 120 minutes. It is clearly shown that the use of a power law for the hydrophobic interaction
might overestimate the hydrophobic interaction at short-range separation distance. However, it
quantitatively predicted the surface force at a long-range distance.
Figure 10.10 shows the disjoining pressure in a wetting film between an air bubble and a
hydrophobic gold surface treated in a 10-5 M KEX solution for different immersion time. The
black curve shows the van der Waals dispersion force (Π ) in a wetting film formed on a gold
d
surface with the Hamaker constant (A ) of −14.8 x 10−20 J. The curve labeled “0 min”
132
represents the disjoining pressure between an air bubble and a bare gold surface. A repulsion was
found at a long-range distance, while an attraction was found at a short-range distance as
predicted by the HHF theory. When the gold surfaces were rendered hydrophobic in the 10-5 M
KEX solutions, the disjoining pressure became increasingly attractive. It was shown that the
hydrophobic force constant (K ) in wetting films increased when the gold surfaces were
132
rendered hydrophobic in KEX solutions from 10 minutes to 30 minutes, and decreased at 120
minutes. An increase of the hydrophobic force at short hydrophobizaiton time might be attributed
to a formation of the hydrophobic monolayer on gold surfaces, while a decrease of the
227 |
Virginia Tech | Figure 10.10 Disjoining pressure isotherms in TLFs between air bubbles and gold surfaces
treated in KEX solutions for the varying hydrophobization time. The inclusion of
hydrophobic force in the extended DLVO theory is used to explain the negative
disjoining pressure in a wetting film between an air bubble and a hydrophobic
gold surface.
hydrophobic force at longer time might be attributed to the formation of a multilayers above the
monolayers. The adsorption of the multilayers might render the hydrophilic head group of
xanthate towards the water phase, resulting in a decrease of solid hydrophobicity. As shown from
the contact angle measurement, the contact angle decreased slightly when leaving the gold
surfaces in a KEX solution for a longer hydrophobization time.
10.5 Summary
The interaction forces between air bubbles and gold surfaces were studied in water using the
force apparatus for deformable surfaces (FADS). The new apparatus is capable of directly
measuring the interaction force between a solid surface and an air bubble across a thin liquid film,
while monitoring the bubble deformation during the course of the interaction. In the present
work, we have studied the effect of the solid hydrophobicity on the forces acting between an air
228 |
Virginia Tech | Chapter 11. Conclusions and Recommendations for Future
Research
11.1 Conclusions
The major findings and contributions from this work may be summarized as follows.
1. When an air bubble is pressed against a flat substrate immersed in water, the bubble
flattens and creates a thin liquid film (TLF) of water between the bubble and the substrate.
The curvature change associated with the formation of the wetting film creates an excess
pressure (p) in the film, which in turn causes the film of water to drain. The film drainage
continues until the excess pressure becomes equal to the disjoining pressure () in the
film. The TLF is stable when the disjoining pressure is positive (repulsive) and is
unstable when the disjoining pressure is negative (attractive). One can readily determine
the positive disjoining pressures when the wetting films are stable. However, no one has
ever been able to measure the negative disjoining pressures as the films drain too fast to
do meaningful measurements.
2. In the present work, the thin film pressure balance (TFPB) technique, originally
developed for the study of foam films, has been modified to measure the negative
disjoining pressures in wetting films. It is equipped with a high-speed video camera to
record the interference patterns (Newton rings) of the fast-evolving wetting films. The
interference patterns were then analyzed offline to reconstruct the spatial and temporal
(spatiotemporal) profiles of the wetting films with a nano-scale resolution. The
experimental data obtained in this manner were analyzed using the Reynolds lubrication
theory to determine the changes in disjoining pressure () with time and film thickness
(h). The results showed that long-range hydrophobic forces were present in the wetting
233 |
Virginia Tech | films formed on the gold surfaces hydrophobized by short-chain alkylxanthates.
According to the thermodynamic analysis based on the Frumkin-Derjaguin isotherm, the
kinetics of film thinning is expedited by the long-range hydrophobic forces, while the
short-range hydrophobic forces are responsible for the rupture of the wetting film formed
on a hydrophobic surface.
3. The role of collector on flotation has been studied by measuring the disjoining pressures
of the wetting films with and without hydrophobization of the substrate. The results
showed that collector coating increases the hydrophobicity of minerals and thereby
creates a negative disjoining pressure, so that wetting films thin faster and ruptures.
4. The kinetics of film thinning has been studied by using the different sizes of the bubbles
to form wetting films. The results showed that the wetting films formed with smaller air
bubbles thin faster due to the larger curvature pressures.
5. The negative disjoining pressures observed in the wetting films formed on hydrophobic
surfaces are the consequence of asymmetric hydrophobic interactions between the
air/water and solid/water interfaces in wetting films. It has been found that the
hydrophobic force constant (K ) for the asymmetric hydrophobic interactions can be
132
predicted from the hydrophobic force constant (K ) for the symmetric hydrophobic
131
interactions between hydrophobic surfaces and the symmetric hydrophobic force constant
(K ) between air bubbles using the geometric mean combining rule. In view of the fact
232
that the geometric mean combining rule is used for the van der Waals force, the
hydrophobic force may be considered a molecular force.
6. It has been shown in the present work that wetting films can be destabilized by the
attractive double-later interactions in wetting films. In the presence of 3x10-5 M Al3+ ions,
the silica/water interface is positively charged while the air/water interface is negatively
charged. The double-layer interaction between the two oppositely charged surfaces
created a negative disjoining pressure. As a consequence, the wetting film thinned fast
234 |
Virginia Tech | and ruptured with a finite contact angle. The measured contact angle is in agreement with
the prediction from the Frumkin-Derjaguin isotherm.
7. It has been a challenge to directly measure the interaction forces involved in bubble-
particle interactions. The main reason was that air bubbles deform during the interaction,
which makes it difficult to determine the separation distances between the two
macroscopic surfaces. As a result, much of the data reported in the literature are
inconsistent with flotation practices. The force apparatus for deformable surfaces (FADS)
developed in the present work is capable of measuring both hydrodynamic and surface
forces involved in bubble-particle interactions. The system has two optical systems, one
to directly measure the deflection of the cantilever spring using a fiber optic system, and
the other to record the interference patterns (Newton rings) of the fast-evolving wetting
films using a high-speed video camera. The former is used for direct force measurement,
and the latter is used to reconstruct the spatial and temporal film profiles of the wetting
films offline.
8. Analysis of the FADS data show that an air bubble approaching a solid surface deforms
in response to both the hydrodynamic and surface forces in the system. In the presence of
a strong repulsive disjoining pressure, the wetting film becomes flat. In the presence of a
strong attractive force, a concave (pimpled) wetting film is formed when the approach
speed is slow, while a convex (dimpled) wetting film is formed when the approach speed
is high. An unstable, either pimpled or dimpled, film was developed under an attractive
force. It has been shown in the present work that bubble-particle interactions are
controlled initially by hydrodynamics, followed by surface forces. Unlike the interactions
between rigid particles, the energy associated with bubbles (or other soft matter) is
conserved by shape changes when subjected to an external force.
9. The direct force measurements conducted with the FADS showed that hydrophobic
forces are present in wetting films. The measured hydrophobic forces increase with
increasing surface hydrophobicity. This finding is consistent with the thermodynamic
235 |
Virginia Tech | prediction from the Frumkin-Derjaguin isotherm that a wetting film ruptures only when
its disjoining pressure is negative.
11.2 Recommendation for Future Research
On the basis of the present work, future research directions under the topic of the wetting film
are recommended as follows.
1. FADS developed in the present work is a scientific breakthrough for the study of the TLF
formed on a solid surface. However, FADS requires the upgrades for better performances.
First of all, the current design is capable of measuring the force in a thin liquid film
formed on a solid surface with a resolution of 5 nN. A better mechanical design with low
drift and background noise is essential. Secondly, the spatiotemporal thickness profiles of
the liquid films were determined from the interference fringes of the fast-evolving
wetting films by means of a high-speed camera. The reconstruction of the spatiotemporal
profiles was done using the monochromatic interferometry technique. However, such
technique naturally lacks of the capability of determining the order of the fringes.
Additionally, the use of the monochromatic interferometry is particularly limited in
determining the film thickness below 20 nm. Future research will focused on developing
a multi-wavelength interferometry technique to monitor the separation distance with a
sub-angstrom resolution. Thirdly, FADS was specifically designed for the force
measurement between an air bubble and a solid surface, with a lack of the accessories to
study other soft matters, such as oil droplets, supercritical CO . A multifunctional
2
platform will be constructed with an environment chamber for dust control and pressure
regulation.
2. The results in Chapter 10 showed that xanthate adsorption on gold surfaces in an open
circuit created a strong hydrophobic attraction in a wetting film between an air bubble
and a hydrophobic gold surface. The attraction might be the hydrophobic interaction due
to a rise of the surface hydrophobicity. It has been well documented that xanthate
236 |
Virginia Tech | Figure A.2 Spatial profiles of the wetting films obtained at t = 1340 ms. Solid line represents
a fitting curve using a six-order polynomial fitting method.
where p is the curvature pressure due to the surface tension, p the hydrodynamic pressure and
cur
Π the disjoining pressure. By definition, the interaction force, F(t), exerting on the cantilever
surface can be obtained from the integral of the hydrodynamic pressure and the disjoining
pressure,
F(t) 2 p(r,t)(r,t) rdr (A.2)
r0
in which r = 0 represents the symmetric axis in a cylindrical coordinate, i.e., the center of the
film. By substituting eq. (B.1) to eq. (B.2), one can obtain an alternative expression for the
interaction force between an air bubble and a solid surface,
F(t) 2 p (r,t)rdr (A.3)
cur
r0
in which the interaction force, F(t) is expressed in term of p . In a thin film, the curvature
cur
pressure, p , can be estimated from the profiles of the thin liquid film using the following
cur
equation.
2 h
p r (A.4)
cur R r r r
239 |
Virginia Tech | Figure A.3 Curvature pressure vs. radial position of the film at t = 1340 ms. The profiles of
the p was obtained from the numerical analysis of the thickness profiles shown
cur
in Fig. B.2 using the eq. (B.4).
where R is the radius of the bubble in the far field, and γ is the interfacial tension of air/water
interface. In eq. (B.4), the second term of p is the local Laplace pressure evaluated at radial
cur
position of the film. At flat film, ∂h/∂r is equivalent to zero at flat film, and thus p = 72 N/m2
cur
for 2 mm radius of an air bubble. Thus, the interaction force can be determined when the spatial
thickness profiles of the TLF is known.
The calibration is carried out by pressing an air bubble towards a hydrophilic cantilever
surface using a piezo actuator. Initially, a manual micrometer was used to lower the position of
the cantilever until an equilibrium film is formed. Afterwards, the position of a bubble is
controlled by means of a piezo actuator. When the piezo actuator extends by applying the voltage,
the thin film expands at equilibrium film thickness, resulting an increase of the total force. Thus,
the spring constant can be calibrated from a linear fit between the interaction force and the
deflection of the cantilever by increasing the size of the flat film.
240 |
Virginia Tech | Figure A.4 A linear fit between the interaction force and the deflection. It shows that spring
constant k = 3.24 N/m.
Fig. (B.1) shows the deflection vs. time by manually elevating an air bubble towards a
cantilever in steps. It was shown that the interaction force increased when the piezo extended,
while remained constant when the piezo stopped. The film profiles were tracked simultaneously
by capturing the interference fringes by means of a high-speed camera. Fig. (B.2) shows the
spatial thickness profiles of the wetting film at t = 1340 ms. The profiles were fitted using a six-
order polynomial fitting method. Using the eq. (B.4), the curvature pressure can be obtained, as
shown in Fig. (B.3).
From the integral of curvature pressure using eq. (B.3), we obtained that F = 525.5 nN. In a
same manner, we determined the force at other time. Table B.1 shows the force and deflection at
different time. A plot of the force vs. deflection is shown in Fig. (B.4). From a best linear fit, we
obtain that the spring constant k = 3.24 N/m.
241 |
Virginia Tech | where the hydrodynamic pressure was obtained by integrating the velocity from the infinity to
the local radial position. By integrating the hydrodynamic pressure over the entire thin film, one
can obtain an expression for the hydrodynamic force,
R
F 2 prdr
(B.5)
r0
Fig. (B.1) shows the effect of approaching speed on the hydrodynamic force in a thin liquid
film of water between two solid surfaces. The results were shown between a 2 mm sized particle
and a flat solid surface. It was found that the numerical results covering 300 and 500 µm radii
of the film area were in a good agreement with the lubrication theory, while those covering 100
µm radii of area underestimated the hydrodynamic force. Thus, all the calculations for the
hydrodynamic force in this work was done by the integral of the hydrodynamic pressure over the
film areas at r = 0 - 300 µm.
Figure B.1 Effect of approaching velocity on the hydrodynamic force exerting on the solid
surface when a 2 mm radii particle is used. The solid lines and the circle points
represent the hydrodynamic force predicted using the eq. (B.1) and the numerical
analysis using the eqs. (B.2)-(B.5), respectively. The hydrodynamic forces
obtained from the integral of the hydrodynamic pressure over the radial distance
of (a) 100 µm, (b) 300 µm and (c) 500 µm were compared.
244 |
Virginia Tech | Modeling Bubble Coarsening in Froth Phase
from First Principles
Seungwoo Park
ABSTRACT
Between two neighboring air bubbles in a froth (or foam), a thin liquid film (TLF) is formed.
As the bubbles rise upwards, the TLFs thin initially due to the capillary pressure (p ) created by
c
curvature changes. As the film thicknesses (H) reach approximately 200 nm, the disjoining
pressure (П) created by surface forces in the films also begins to control the film drainage rate and
affect the waves motions at the air/water interfaces. If П < 0, both the film drainage and the
capillary wave motion accelerate. When the TLF thins to a critical film thickness (H ), the
cr
amplitude of the wave motion grows suddenly and the two air/water interfaces touch each other,
causing the TLF to rupture and bubbles to coalesce.
In the present work, a new model that can predict H has been developed by considering
cr
the film drainage due to both viscous film thinning and capillary wave motion. Based on the H
cr
model, bubble-coarsening in a dynamic foam has been predicted by deriving the geometric relation
between the thickness of the lamella film, which controls bubble-coalescence rate, and the Plateau
border area, which controls liquid drainage rate.
Furthermore, a model for predicting bubble-coarsening in froth (3-phase foam) has been
developed by developing a film drainage model quantifying the effect of particles on p . The
c
parameter p is affected by the number of particles and the local capillary pressure (p ) around
c c,local
particles, which in turn vary with the hydrophobicity and size of the particles in the film. Assuming
that films rupture at free films, the p corrected for the particles in lamella films has been used to
c
determine the critical rupture time (t ), at which the film thickness reaches H , using the Reynolds
cr cr
equation. Assuming that the number of bubbles decrease exponentially with froth height, and
knowing that bubbles coalesce when film drains to a thickness H , a bubble coarsening model has
cr
been developed. The model predictions are in agreement with the experimental data obtained using
particle of varying hydrophobicity and size. |
Virginia Tech | Acknowledgement
Foremost, I would like to express my sincerest gratitude to my research advisor, Dr. Roe-
Hoan Yoon, for providing me an opportunity to challenge such an attractive research area and
teaching me how to generate new ideas from fundamental studies.
I am also grateful to Dr. Gerald Luttrell for introducing me hydrodynamic theories, Dr.
Gregory Adel for teaching me how to make a professional presentation through his seminar
courses, Dr. Saad Ragab for his critical comments on this dissertation, and Dr. Sunghwan Jung for
his continued academic advice and valuable classes.
The completion of this work would not have been possible without the support and
encouragement of all of the staff members and students at Center for Advanced Separation
Technologies. Particular appreciations are expressed to Gaurav Soni and Kaiwu Huang for
incorporating the theoretical models developed in this study into a simulator, Dr. Lei Pan for useful
discussions and innovative suggestions, and Whiusu Shim for helping me prepare samples and
assisting me with image-processing.
I also have been blessed with Dr. Jai-Koo Park and Dr. Sungsoo Cha, who helped me
achieve career planning during their visit.
I am also grateful to FLSmidth for funding continuously and providing me with necessary
facilities for this project.
I wish to express my sincere thanks to Prof. Siyoung Jeong for propelling me forward since
I was an undergraduate.
Finally, I want to acknowledge my family for their love, sacrifice, and patience.
iii |
Virginia Tech | Chapter 1. Introduction
1.1 Preface
Froth flotation is a method for selectively separating a particulate material from one another
by means of the air bubbles dispersed in liquid medium. The flotation technology has been widely
used for upgrading pulverized ores in the mining industry, removing contaminants from waste
water in the water treatment industry, capturing printing ink from paper fibers in the paper
recycling industry, and extracting bitumen form oil sands.
In the minerals separation process, the surface of desired mineral particles is rendered
hydrophobic by the absorption of collector (hydrophobizing reagent), whereas that of undesirous
minerals remain hydrophilic. The Gibbs energy change (∆G) associated with the bubble-particle
attachment is given by the following form [1],
G(cos1) (1.1)
where γ is the interfacial tension between liquid and air phases and θ is the water contact angle.
Eq. (1.1) indicates that only the attachment between a bubble and a particle with θ > 0° is
thermodynamically favorable. Hence, only the hydrophobized particles selectively can attach to
air bubbles, form particle-bubble aggregates, which rise upward due to buoyancy, and form a froth
layer at the surface of the pulp phase.
True recovery in the froth phase occurs when the particles attached to bubbles reaches the
top of the froth and finally flow into a launder without detachment. As the particle-coated bubbles
rise upward along the froth height, they coalesce with each other and become larger. As the bubbles
become coarser, the bubble surface area decreases, thereby detaching desired hydrophobic
particles from bubbles and causing the true recovery to decrease. However, less hydrophobic
particles tend to be detached preferentially, so the froth phase contributes to enhance the grade of
a froth product [2].
1 |
Virginia Tech | On the other hand, particles suspended in the pulp can also be recovered without adhesion
to bubbles through the entrainment mechanism. The froth recovery due to entrainment occurs
when interstitial liquid between bubbles drag particles from the pulp and allow them to reach the
launder. The dragging effect may stem from the agitation in the pulp (mechanical entrainment),
the wake behind a rising bubble, or bubble swarm crowding effect (hydraulic entrainment). Since
entrainment is non-selective, not only hydrophobic but also hydrophilic particles can be recovered,
lowing the grade product [3].
As stated, flotation is a multi-phase process involving a variety of physicochemical
phenomena and those are interacting each other. As a result, quantitative flotation modelling and
simulations are highly required to explore the flotation science and optimize the operation of
overall process. In the following section, literatures associated with the two distinct phases, pulp
and froth phases, respectively, are briefly reviewed, particularly in modelling perspective. Next,
the objective of the present doctoral research is presented.
1.2 Literature Review
1.2.1 Pulp
A. Bubble Generation Model
In a flotation cell, bubbles are generated by splitting injected air into tiny air bubbles and
the splitting pressure is originated from rotational motion of a rotor. The splitting pressure is
opposed by the surface tension force tending to resist deformation and minimize the surface area
of the air/water interface. The ratio of splitting pressure to capillary pressure is defined as Weber
number (We). Hinze [4] suggested that if We reaches a critical value, bubble breaks. The critical
Weber number (We ) is obtained from the following expressions,
cr
Splitting pressure u2d
We 2,max (1.2)
cr Capillary pressure
where ρ is the liquid density, u2 is the mean square velocity difference between two points, d
2,max
is maximum stable bubble size, and γ is the surface tension. The value of u2 is given by [5],
2 |
Virginia Tech | B. Drift-Flux Model
The bubbles generated in the pulp phase of a flotation cell rise upwards due to buoyancy,
and the bubble velocity relative to water, or the bubble slip velocity (U ), is expressed as [10],
slip
V V
U g l (1.8)
slip 1
where V and V are the superficial gas and liquid (slurry) velocities, respectively, and ε is the liquid
g l
volume fraction. The +/- signs indicates counter-current and-co-current flows of gas and liquid,
respectively.
Empirically, it was found that U is a function of the terminal rise velocity (U) of a single
slip t
bubble in an infinite pool and ε [11],
U Um1 (1.9)
slip t
where m is the empirical parameter that depends on flow patterns. In using Eq. (1.9), the value of
U is given by [12],
t
gd 2
U 2 (d 1.5 mm) (1.10)
t 18(10.15Re0.687) 2
where g is the gravitational , d is the bubble diameter, μ is the liquid viscosity, and Re is the
2
Reynolds number, given by d ρU / μ . In the case of a coarse bubble1(1.5 mm ≤ d ≤ 10 mm), U
2 t 2 t
is independent of d and reaches approximately 21 cm/s [13].
2
By combining Eqs. (1.8) ~ (1.10), one can predict one of unknown parameters if the others
are measurable. Banisi and Finch [14] and Ityokumbul et al. [15] used the drift-flux analysis to
predict the average bubble size dispersed in the pulp in a flotation column. Stevenson et al. [16]
used the theory to predict liquid overflow rate from a foam column.
1.2.2 Froth
In the Section A below, the bulk motion of a froth (or a foam) in a flotation cell is described.
The froth is composed of a cloud of air bubbles. Between three bubbles, a triangular channel called
4 |
Virginia Tech | a Plateau border (PB) is formed, through which liquid drains, as will be described in Section B
below. Although liquid drainage models neglect the water presence in the thin film (lamella)
formed between two bubbles, the lamella films play an importance role forth and foam stability
since bubbles coalesce when the thickness of a lamella film reaches a critical value (h ). In Section
c
C, the thinning kinetics of lamella films is discussed. In section D, a number of methodology to
quantify froth stability is reviewed.
A. Bulk Motion of a Froth
Based on continuum-flow approach, the bulk motion of a froth has been described by the
Laplace equation [17-19],
20 (1.11)
where Ψ is the stream function. As is well known, Eq. (1.11) requires incompressible and
irrotational flow conditions. The incompressible condition has been assumed due to the fact that
the water content in flotation froth may not be enough to compress bubbles and change the volume
of them. On the other hand, the irrotational flow assumption has been made based on the premise
that the geometry of a flotation cell is generally simple and thus swirling flow cannot be created.
B. Foam Drainage
Verbist et al. [20] calculated the liquid drainage velocity through a Plateau border (PB)
based on the assumption of Poiseuille-type flow and no-slip boundary condition on air/water
interface. Consider a single vertical PB with a cross-sectional area A. From the Young-Laplace
equation, one can obtain the pressure difference across the PB,
pp p (1.12)
g R g A/C
PB
where p is the pressure in the PB, p is the gas pressure in a bubble, and R is the radius of the
g PB
PB, given by A/C . Here C is the dimensionless geometric parameter. The pressure gradient
along the vertical direction x is written as,
5 |
Virginia Tech | p A
0.5CA3/2 (1.13)
y y R x
PB
The pressure gradient should be balanced by gravity, ρg, and dissipative force,
A fu
0.5CA3/2 g (1.14)
x A
where u is the liquid drainage velocity averaged over A and f is the PB drag coefficient, can vary
with geometry of PB and boundary condition of PB wall. Hence, the drainage rate u is
gA C A
u (1.15)
fu 2fA1/2 x
The first and the second term on the right side represent the contributions from gravity and
capillary suction, respectively.
Eq. (1.15) have some limitations for describing the liquid drainage in a flotation froth. In a
flotation froth, air bubbles are not stationary and moves relatively to the liquid. The rising bubble,
therefore, could may affect the liquid flow by dragging, but the dragging effect is neglected.
Inertial effect is also neglected in Eq. (1.15), but in a froth consisting of large bubbles, inertia may
become significant. Moreover, it assumes that viscous loss occurs only in the PBs (channel-
dominated hypothesis). However, several researchers have recently reported that most viscous loss
occurs at the node, where four PBs meet [21-24]. In a dry foam, the liquid content in the node is
quite negligible compared to the same in the PB, thus the channel-dominated hypothesis may be
applicable.
C. Film Drainage
Reynolds [25] used simplified Navier–Stokes equations with lubrication approximation to
describe the squeezing liquid flow between two flat immobile surfaces. Similarly, the drainage of
a lamella film can be described by the Reynolds lubrication equation. The pressure in the lamella
film can be calculated by the following equation,
6 |
Virginia Tech | p 2u
(1.16)
r z2
where p is the excess pressure in the lamella, u is the axial liquid velocity along the radial direction
r, and z is the vertical coordinate. By substituting Eq. (1.16) into continuity equation, one can
deduce the film thinning rate,
dH 1 p
rH3 (1.17)
dt 12r r r
where H is the film thickness and t is the drainage time. By considering the averaged p within the
entire film, one can obtain the Reynolds lubrication equation,
dH 2H3p
(1.18)
dt 3R2
f
where R is the film radius. Here the average pressure p is given by,
f
p p (1.19)
c
where p is the capillary pressure and П is the disjoining pressure incorporating intermolecular
c
forces. The capillary pressure p is given by,
c
2
p (1.20)
c R
where R is the bubble radius. In calculating П, in the original model, only the contributions from
the electrostatic and van der Waals forces, according to the classical Derjaguin-Landau-Verwey-
Overbeek (DLVO) theory. However, in the present study, according the extended DLVO theory
[26], the contribution from the hydrophobic force is considered additionally,
e A K
64C RTtanh2 s exp(H) 232 232 (1.21)
el vw hp el 2 4kT 6H3 6H3
where П , П , and П represent the contribution from the electrostatic, van der Waals,
el vw hp
hydrophobic forces, respectively, C the electrolyte concentration, R the gas constant, T the
el 2
7 |
Virginia Tech | absolute temperature, e the electronic charge, ψ the surface potential at the air/water interfaces, k
s
the Boltzmann’s constant, and κ the reciprocal Debye length.
D. Froth Stability
The froth (or foam) stability, by definition, is the ability of air bubbles in froth phase to
resist coarsening and bursting. As is well known, froth stability has a significant effect on
determining product grade and recovery in flotation. Therefore, various criteria have been
proposed to quantify the froth stability. In general, they can be categorized into two subgroups, i.e.
static tests and dynamic tests.
In static tests, froth is freely allowed to coalesce without creating additional bubbles.
Iglesias [27] used a froth decay rate as an indicator of froth stability. When froth height (h) reaches
f
an equilibrium, the air supply to the cell was discontinued and immediately started to measure the
h as a function of time t, as follows,
f
dh(t)
Froth decay rate = f (1.22)
dt
Instead of monitoring a decaying foam continuously, Zanin [28] simply measured the time need
for a foam height to drop to one half of its initial height and the half-life time was used as a measure
of froth stability.
On the other hand, in dynamic tests, the froth stability is measured while bubbles are being
generated continuously. Therefore, during the tests, the bubbles at the base of the froth are allowed
to experience coalescences while moving upwards along the froth height. Due to these similarities
between the dynamic tests and real flotation tests, as compared to static tests, dynamic tests may
provide more reliable froth stability criterion in flotation applications [27]. Bikerman [29] was the
first to propose a methodology for a dynamic test. He introduced the concept of the dynamic
stability factor (Σ), which is the ratio of the maximum volume of froth to the air flowrate,
h A
f,max (1.23)
Q
8 |
Virginia Tech | where A is the cross-sectional area of a cell and Q is the volumetric air flowrate. Barbian [30]
suggested the maximum froth depth h can be a froth stability indicator at equilibrium status.
f,max
In the case of growing froth, he monitored h as a function of time t and the froth rising velocity,
f
u (t) = dh(t)/dt, was used as an indicator. The model suggests that as froth becomes stable, u (t)
f
approaches the superficial gas velocity (V ). The air recovery has been used by several researchers
g
and it is defined as the fraction of air injected into a flotation cell that overflows the cell lip as
unburst bubbles [31-33]. The primary limitation of the air recovery concept is that it is volume-
based approach though the flotation performance relies mainly on the bubble surface area. Ata [34]
measured the bubble size distribution along froth height and the froth stability was quantified in
terms of the bubble growth rate. Laplante [35] presented the froth retention time (FRT), which is
given by the following equation,
h
FRT f (1.24)
V
g
Eq. (1.24) can also be regarded as the measure of average life time of bubble. More recently, Hu
[36] monitored the change in electrical impedance by means of a pair of electrodes immersed in
froth phase. It was found that bubble coalescence or froth structure variation can vary the value of
electrical impedance sensitively. Based on this finding, the degree of impedance variation was
used as a froth stability criteria in his study.
1.2.3 Flotation Model
The flotation as a first-order rate process can be represented as follows [37, 38],
dN
1 kN Z P (1.25)
dt 1 12
where N is the number of particles in the cell at time t, k is the flotation rate constant, P is the
1
probability of flotation, and Z is the collision frequency between particle 1 and bubble 2, given
12
by the following relation [7],
Z 23/21/2N N d2 u2 u2 (1.26)
12 1 2 12 1 2
9 |
Virginia Tech | where N is the number of bubbles on the cell, d is the collision diameter (sum of radii of bubbles
2 12
and particles), and u2 and u 2 are root mean square velocities of the particles and bubbles,
1 2
respectively. Here, the values of u2 and u 2 can be determined as presented previously by Schubert
1 2
[7] and Lee et al. [39], respectively.
In using Eq. (1.25), P may be expressed in the form,
PPP(1P)P (1.27)
c a d f
where P is the probability of collision, P the probability of attachment, P the probability of
c a d
detachment, and P the probability of particles surviving froth phase.
f
In using Eq. (1.27), P can be determined from stream functions for water around air
c
bubbles [40],
3 3 Re d
P
c
tanh2
2 1 16
10.249Re0.56
d1
2
(1.28)
where Re is the Reynolds number, d and d the diameters of a particle and a particle, respectively.
1 2
In using Eq. (1.27), P can be readily calculated using the following equation [40, 41],
a
E
P exp 1 (1.29)
a E
k
where E is the energy barrier and E is the kinetic energy of a particle. E is equal to the maximum
1 k 1
potential energy between a bubble and a particle, which can be obtained from the extended DLVO
theory and E is calculated using the following relation,
k
E 0.5m(U )2 (1.30)
k 1 Hcr
where m is the mass of a particle, and U is the velocity of a particle approaching a bubble at
1 Hcr
the critical rupture thickness (H ) of the wetting film in between. The value of U is inversely
cr Hcr
proportional to the hydraulic resistance force against the thinning.
10 |
Virginia Tech | In using Eq. (1.27), P is given as follows [26],
d
W E
P exp a 1 (1.31)
d E
k
where W is the work of adhesion and E ’is the kinetic energy that detaches a particle from a
a k
bubble surface. Here W is obtained as follows,
a
W r2(1cos)2 (1.32)
a 1
where γis the surface tension of water, r is the particle radius, and θ is the water contact angle. In
1
using Eq. (1.31), E’ may be found by the following relation [26],
k
2
E' 0.5m (d d ) /v (1.33)
K 1 1 2
where is the energy dissipation rate and is the kinematic viscosity.
Once particles enter the froth phase, only a part of them survives the froth phase and
reaches the launder. In determining P using Eq. (1.27), the probability of particles surviving froth
phase (P) is given as [42, 43],
f
P P P R exp()R exp(0.03250.063d) (1.34)
f fa fe max f FW,max 1
in which P and P are the froth recovery due to attachment and entrainment, respectively, R
fa fa max
the maximum recovery factor, α fitting parameter, τ the froth retention time, R themaximum
f FW,max
feedwater recovery to froth, and ∆ρ the specific gravity difference between particle and water.
Here R is given by,
max
S d
R t 6V /d / 6V /d 2,b (1.35)
max S g 2,t g 2,b d
b 2,t
where S and S are the surface area rates of the bubbles at the top and bottom of the froth phase;
t b
d and d are bubble diameters at the top and bottom of the froth phase, respectively; and V is
2,t 2,b g
the superficial gas velocity.
11 |
Virginia Tech | 1.3 Research Objective
As shown in Eqs. (1.34) and (1.35), the froth recovery depends strongly on the bubble-
coarsening factor (d /d ), which is the bubble size ratio between the bottom and the top of a froth
2,b 2,t
phase. As a result, it is of critical importance to understand the mechanisms of bubble-coarsening
phenomena in a flotation froth. In determining the bubble-coarsening factor, the bubble size at the
bottom of a froth (d ) may be considered to be the same as the bubble size in the pulp phase,
2,b
which can be readily predicted from bubble generation model. However, at present, there are no
models that can predict the bubble size at the top of a froth (d ) from d . The primary objective
2,t 2,b
of the present study is, therefore, to develop a model for predicting the bubble-coarsening in a
froth phase from first principles.
For this to be possible, it is essential to know the critical thickness (H ) of lamella films at
cr
which two neighboring bubbles in a froth coalesce and bubble-coarsening occurs. Therefore, the
present work is also aimed to develop a predictive model for the value of H .
cr
1.4 Organization
The body of this dissertation consists of six chapters.
Chapter 1 provides a comprehensive review of flotation models. Some of them are used to
develop theoretical models presented in the following chapters. This chapter also introduces the
goal of the present work.
Chapter 2 introduces a new predictive model for the critical rupture thickness (H ) of a
cr
foam film. The model has been developed on the base of the capillary wave model and the extended
Derjaguin-Landau-Verwey-Overbeek (DLVO) theory. By considering the contribution from
hydrophobic disjoining pressure, this model can predict H reasonable well.
cr
Chapter 3 describes a new model for predicting the bubble-coarsening in a dynamic foam
as a function of aeration rate, foam height, and surface tension (frother dosage). This foam model
is on the base of the H model developed in Chapter 2. The model has been validated using a foam
cr
column equipped with a high-speed camera.
12 |
Virginia Tech | Chapter 4 presents a new first principle froth model for predicting the bubble-coarsening
in a froth (3-phase foam) as a function of aeration rate, froth height, surface tension (frother
dosage), particle hydrophobicity (collector dosage), and particle size. This chapter is focused on
the effect of particle hydrophobicity on bubble-coarsening. This model has been validated in a
series of flotation tests using monosized spheres of varying hydrophobicity.
Chapter 5 is the same as Chapter 4 except that it is mainly focused on the effect of particle
size on bubble-coarsening. The froth model developed in Chapter 4 is capable of predicting the
bubble-coarsening as a function of particle size. The model predictions are in good agreement with
the changes in bubble sizes measured experimentally in the presence of different sizes of particles.
Chapter 6 summarizes the key findings and accomplishments presented in the foregoing
chapters and suggests future research topics.
References
[1] Laskowski, J., The relationship between floatability and hydrophobicity, in Advances in
mineral processing: A half century of progress in application of theory and practive. 1986.
[2] Seaman, D.R., E.V. Manlapig, and J.P. Franzidis, Selective transport of attached particles
across the pulp–froth interface. Minerals Engineering, 2006. 19(6–8): p. 841-851.
[3] George, P., A. Nguyen, and G. Jameson, Assessment of true flotation and entrainment in
the flotation of submicron particles by fine bubbles. Minerals Engineering, 2004. 17(7): p.
847-853.
[4] Hinze, J.O., Fundamentals of the hydrodynamic mechanism of splitting in dispersion
processes. AIChE Journal, 1955. 1(3): p. 289-295.
[5] Batchelor, G. Pressure fluctuations in isotropic turbulence. in Mathematical Proceedings
of the Cambridge Philosophical Society. 1951. Cambridge Univ Press.
[6] Schulze, H.-J., Physico-Chemical Elementary Processes in Flotation: Analysis from the
Point of View of Colloid Science Including Process Engineering Considerations 1984:
Elsevier Science Ltd
[7] Schubert, H., On the turbulence-controlled microprocesses in flotation machines.
International Journal of Mineral Processing, 1999. 56(1–4): p. 257-276.
13 |
Virginia Tech | Chapter 2. Prediction of the Critical Rupture Thickness of
Foam Films
ABSTRACT
In flotation, a froth recovery depends critically on bubble-coarsening. The bubble-
coarsening occurs when a thin liquid film (TLF) confined between two bubbles breaks. As a
precursor to developing a model that can predict the bubble-coarsening in a froth (or foam), we
have developed a model for predicting the critical film thickness (H ) at which a foam film
cr
ruptures. The model has been derived by considering the film drainage due to viscous film thinning
and the capillary wave motion at air/water interfaces. This approach is based on the premise that
if the disjoining pressure (П) created by surface forces in the TLF is negative (attractive), both the
film drainage and the capillary wave motion accelerate and consequently the TLF ruptures. The
model predictions are consistent with the H values measured experimentally. It has been found
cr
in the present study that at a relatively low frother (or surfactant) concentration, corrugation of
air/water interfaces grow faster in amplitude and thereby the TLF ruptures at a larger film thickness.
At a high frother concentration, hydrophobic force is dampened, causing the grown of fluctuation
to decrease and hence causing the H to decrease.
cr
17 |
Virginia Tech | 2.1 Introduction
In froth (or foam) phase, two air bubbles are in close contact with each other and a thin
liquid film (TLF) is formed in between. As the bubbles rise along the height of a froth phase, the
water in the TLF drains initially due to the capillary pressure (p ) created from the changes in
c
curvature. As the film thickness (H) reaches approximately 150-200 nm, the process begins to
slow down due to the repulsive electrical double-layer forces between the two air/water interfaces
facing each other. When the p becomes equal in magnitude to the disjoining pressure ( ) due to
c e
double-layer repulsion, the film thinning stops at an equilibrium film thickness (H ). If there exists
e
an attractive force in the TLF, the film thinning continues and the film ruptures and the two bubbles
become one at a critical film rupture thickness (H ).
cr
Scheludko [1, 2] was the first to present a theoretical model to predict H . The model was
cr
derived based on the premise that the air/water interfaces of TLFs are thermally fluctuated and that
the TLF becomes unstable and ruptures if the negative disjoining pressure gradient along film
thickness H exceeds the capillary pressure gradient along H,
A 2
H 232 (2.1)
cr 128
where A is the Hamaker constant of water between two air bubbles and is the wavelength of
232
the surface fluctuations. The primary limitation of this model is that the value of is undefined.
Based on Scheludko’s theory [1, 2], Vrij and his co-workers [3-5] developed a more
advanced H model in that not only the model defined the values of by calculating the Gibbs
cr
energy change associated with fluctuation growth, but also it included the role of film drainage in
determining film stability. The following is the H model developed by the authors,
cr
A 2R2
H 0.207 232 f (2.2)
7
cr p
c
where R is the radius of contact area and γ is the surface tension. Unfortunately, Eq. (2.2) fails to
f
predict H accurately, particularly at low surfactant (frother) concentrations, where air bubbles
cr
become more hydrophobic [6]. The failure of Eq. (2.2) may arise from the fact that at the time the
18 |
Virginia Tech | classical Derjaguin-Landau-Verwey-Overbeek (DLVO) theory was developed and no one knew
that air bubbles in water are hydrophobic. Further, no one thought about the possibility of
hydrophobic force playing a role in foam stability. Recently, it has shown that the hydrophobic
force is also present in foam films in addition to the van der Waals force, and that the former is
larger than the latter at low frother concentrations [7, 8].
More recently, Do [9] modified Eq. (2.2) by considering the possibility that the TLF
between air bubbles drains faster due to slip on the hydrophobic air/water interfaces. In addition,
he introduced the variation of film thinning velocity from the difference between the wavy film
radius and the flat film radius by considering the geometric relation,
2
H
18
4K
232
1 3 1
42 /2 1
1 42 /2 2 sin2d3
cr 3 A 0 42 /2 1
232 (2.3)
A 2R2 1/7
0.207 232 f
p
c
where K is the hydrophobic force constant of water between two air bubbles, η is the amplitude
232
of the wave at the air/water interfaces in a foam film, and λ is the wavelength. However, in this
model, the possibility that the amplitude of capillary waves grows faster in the presence of stronger
attractive hydrophobic force was not considered. Moreover, this model describes the film thinning
process using the classical DLVO theory, which neglects the contributions from the hydrophobic
force. Also, the fact that value of / in Eq. (2.3) needs to be empirically determined is problematic.
In the present work, we have derived a model for predicting H by incorporating the
cr
hydrophobic force in the capillary wave model developed by Vrij. This new model has been
developed by considering the effects of hydrophobic force on the film drainage and the wave
motion.
2.2 Model Incorporating Hydrophobic Force
The capillary wave model developed by Vrij et al. [3-5] described the film thinning process
and the capillary wave motion to predict the critical rupture thickness (H ) of a foam film. In
cr
19 |
Virginia Tech | Section 2.2, we incorporate the hydrophobic force into the Vrij el al.’s model to predict H at low
cr
frother concentrations.
2.2.1 Film Thinning
The kinetics of thinning of a horizontal thin liquid film (TLF) between two air bubbles can
be described in view of the Reynolds lubrication equation,
dH 2H3p
(2.4)
dt 3R2
drain f
where H is the TLF thickness, t the drainage time, μ the dynamic viscosity, R the film radius,
drain f
and p the driving force for TLF thinning. In Eq. (2.4), the driving force p is given by the expression,
p p (2.5)
c
where p is the capillary pressure and П is the disjoining pressure. The capillary pressure p is
c c
given by,
2
p (2.6)
c R
where R is the bubble radius and γ is the surface tension of water. In calculating П, in the original
Vrij’s model, only the contributions from the electrostatic and van der Waals forces are included,
according to the classical DLVO theory. However, in the present study, according the extended
DLVO theory, the contribution from the hydrophobic force is considered additionally,
e A K
64C RTtanh2 s exp(H) 232 232 (2.7)
el vw hp el 2 4kT 6H3 6H3
where П , П , and П represent the contribution from the electrostatic, van der Waals,
el vw hp
hydrophobic forces, respectively, C the electrolyte concentration, R the gas constant, T the
el 2
absolute temperature, e the electronic charge, ψ the surface potential at the air/water interfaces, k
s
the Boltzmann’s constant, and κ the reciprocal Debye length.
20 |
Virginia Tech | 2.2.2 Growth of Surface Fluctuation
Figure 2.1 shows one of corrugating components of air/water interfaces of the TLF due to
thermal motion. Corrugation due to thermal motion increases the surface area of the air/water
interface and hence the free energy. However, the corrugation causes the film thickness at the
valleys to decrease as the inverse curve of the film thickness due to the van der Waals attraction,
which in turn causes the free energy to decrease and the film thinning kinetics to accelerate. By
considering film thinning due to the growth of fluctuation, Vrij estimated the time it takes for a
film to thin due to fluctuation (t ) as follows,
fluct
2
d
t 24fH3 (2.8)
fluct dH
HH
where f is the adjustable parameter. f = 6 was assumed in the original model. We also use the same
value in the present study. In using Eq. (2.9), we recognize the contributions from П in addition
hp
to П and П , as presented in Eq. (2.7).
el vw
Figure 2.1 One of the capillary wavy patterns at the facing air/water interfaces confining
the foam film with the average film thickness H and the local thickness h.
represents the wavelength and represents the amplitude of the wave motion.
21 |
Virginia Tech | 2.2.3 Calculation of H
cr
Vrij assumed that a TLF ruptures when two surfaces touch each other. This condition can
be represented by the following relation,
dt dt
drain fluct 0 (2.9)
dH dH
HHm HHm
where H is the mean film thickness at which satisfies the above requirement.
m
Substituting Eqs. (2.4) and (2.8) into Eq. (2.9), one can obtain H as follows,
m
2 3 2 3H 3R2
1443H 4 2H 3 m f 0 (2.10)
m H HHm m H HHm H2 HHm 2(p c )
Vrij recognized the fact that the critical film rupture thickness (H ) may be slightly smaller than
cr
H due to the fact that the viscous film thinning continues while the wave is growing. Accordingly,
m
he suggested the following relation,
H 0.845H (2.11)
cr m
2.3 Model Validation
To validate the model developed in the present work, the model predictions were compared
with experimental H values reported by Wang [10]. Wang measured H values of the foam films
cr cr
stabilized with methyl isobutyl carbinol (MIBC) in the presence of 0.1 M NaCl, where П ≈ 0 due
el
to double-layer compression. In his study, foam films of small film radius (R < 50 μm) were tested
f
to avoid dimple formation.
In the present work, H values have been predicted from Eqs. (2.10) and (2.11). In using
cr
them, the values of K were obtained from Wang and Yoon’s results [11]. They determined the
232
K values at different concentrations of MIBC by fitting experimental film thinning data to the
232
Reynolds equation coupled with the extended DLVO theory. The values of A γ, and ψ were
232, s
used as reported by Wang [10], Comley et al. [12], and Srinivas et al. [13], respectively. Only the
22 |
Virginia Tech | As shown in Figure 2.2 (a), there is an excellent agreement between the experimental (filled
circles) and the model predictions (line). The model include the contributions from the
hydrophobic disjoining pressure, which is the reason for the excellent fit between theory and
experiment. If the hydrophobic force is not considered, the model predicts substantially smaller
H values as shown by the bottom most (green) line in Figure 2.2 (a).
cr
Figure 2.2 (b) shows the H values (the topmost blue line) predicted using Eqs. (2.10) and
cr
(2.11) at different concentrations of MIBC in the absence of NaCl. As shown, the model gives
substantially higher H values than those obtained at 0.1 M NaCl, which can be attributed to the
cr
fact that the hydrophobic force (or K ) increases with decreasing NaCl concentration. It is
232
believed that the hydrophobic force originating from the cohesive energy of water is compromised
in the presence of electrolytes.
2.4 Summary and Conclusions
In this study, we have developed a predictive model for the critical rupture thickness (H )
cr
of foam films. This new model has been derived on the basis of the capillary wave model and the
extended DLVO theory, which includes the contributions from the attractive hydrophobic force.
It has been found in the present work that at lower frother concentration, where the hydrophobic
force is stronger, a foam film ruptures at higher film thickness due to faster growth rate of surface
wave. It has also been found that H decreases with increasing frother concentration, which may
cr
be attributed to a decrease in the hydrophobic force, resulting in retarded wave motion. As is well
known, bubble-coarsening in a froth (or foam) occurs when thin liquid film confined between two
bubbles ruptures. This H model will be useful for developing a model describing bubble-
cr
coarsening phenomena in a froth.
References
[1] Scheludko, A., Sur certaines particularités des lames mousseuses. Proc. Konikl. Ned. Akad.
Wet. B, 1962. 65: p. 86-99.
[2] Sheludko, A., Thin liquid films. Advances in Colloid and Interface Science, 1967. 1(4): p.
391-464.
24 |
Virginia Tech | Chapter 3. Development of a Bubble-Coarsening Model in a
Dynamic Foam
ABSTRACT
In a flotation froth (and foam), air bubbles become larger due to coalescence, causing
bubble size to increase, bubble surface area to decrease, and hence causing less hydrophobic
particles to drop off to the pulp phase below. Thus, bubble coarsening provides an important
mechanism by which product grades are increased. On the other hand, excessive bubble coarsening
results in low recoveries. In the present work, a model describing the process of bubbles becoming
coarser in a foam as they rise to the top by deriving a mathematical relation between the Plateau
border area, which controls liquid drainage rate, and the lamella film thickness, which controls
bubble-coalescence rate. The model developed in the present work can predict the bubble size ratio
between the top and bottom of a foam as functions of aeration rate, foam height, and surface
tension (frother dosage). The model predictions are in good agreement with the changes in bubble
sizes measured using a high-speed camera.
26 |
Virginia Tech | 3.1 Introduction
Froth flotation is the most widely used method of upgrading pulverized ores in the minerals
industry. In this process, hydrophobic particles selectively attach to air bubbles in a pulp phase,
forming particle-bubble aggregates, which rise upward due to reduced buoyancy and form a froth
phase. In the froth phase, the bubbles coalesce with each other and become larger as they rise. As
the bubbles become larger, the bubble surface area decreases, thereby restricting the number of
hydrophobic particles that can be carried upward and subsequently flow into a launder. Therefore,
in flotation modelling, froth recovery due to attachment (Ra) is given as a function of surface area
f
as follows,
R a R exp( ) (3.1)
f max f,p
where R is the maximum froth recovery factor representing the carrying capacity limit, α is a
max
fitting parameter, τ is the retention time of particles in the froth phase. In using Eq. (3.1), R
f,p max
can be expressed as follows,
S 6V /d d
R t g 2,t 2,b (3.2)
max S 6V /d d
b g 2,b 2,t
where S and S is the bubble surface area flux at the top and base of a froth, respectively, d and
t b 2,t
d are the corresponding bubble sizes, and V is the superficial gas velocity.
2,b g
On the other hand, the bubble coarsening helps increase the grade of a froth product, as
less hydrophobic particles selectively drop off the coarsening bubbles [1]. Thus, it is of critical
importance to understand and control bubble coarsening in the froth phase. Despite of its
significance, the bubble coarsening in the froth phase has not been studied widely until now, due
to the structural complexity and the opaqueness of a mineral-coated froth. Recently, Barbian et al.
[2, 3] conducted dynamic froth stability tests in a flotation column and observed experimentally
that increasing gas flow rate benefits the froth stability. Tao et al. [4] found that lowering froth
height help increase froth stability by measuring the water recovery in a flotation column, as a
measure of froth stability. Wang and Yoon [5] reported that the disjoining pressure (П) created by
surface forces in the lamella films controls the foam stability. More recently, the first theoretical
27 |
Virginia Tech | model that can predict bubble-coarsening or bubble size distribution in foam has been developed
by coupling a liquid drainage model with a population balance model [6]. The authors employed
a single fitting parameter associated with the frequency of film rupture and assumed that it is a
function of fluid viscosity and surface chemistry. The model was validated by a single experiment
in a surfactant solution of a fixed concentration.
In the present work, we have developed a bubble-coarsening model that can predict the
bubble coarsening (d /d ) by deriving a mathematical relation between the Plateau border area,
2,t 2,b
which controls liquid drainage rate, and the lamella film thickness, which controls bubble-
coalescence rate. The model is based on the capillary wave model [7, 8], in which air/water
interfaces in form films oscillate due to thermal motions such that the instantaneous thickness of a
foam film becomes smaller than the average thickness. Because the van der Waals force varies as
H-3, where H is film thickness, a small change in film thickness can greatly decrease the free energy
of film rupture. The bubble-coarsening model developed in the present work is also based on the
recognition that the air bubbles dispersed in water are hydrophobic [9], which will in turn increase
drainage rate and hence facilitate bubble coarsening.
The objective of the present work is to develop a model for predicting bubble-coarsening
in foams and verify it in experiment. We have measured the bubble size ratio (d /d ) of a foam
2,t 2,b
stabilized with Methyl isobutyl carbinol (MIBC) at varying frother concentrations, foam heights,
and gas rates.
3.2 Model Development
As is well known, bubbles become larger as they rise along the height of a foam mainly
due to bubble coalescence. The bubble coarsening can be controlled by frother addition.
Thermodynamically, frother decrease the free energy of bubble coalescence, i.e., G = -2, by
c
decreasing the surface tension, , at the air/water interface. Kinetically, frother decreases the rate
of film thinning by decreasing capillary pressure, P = 2/R, where R is the Plateau boarder radius.
It has been shown also that frother dampens the hydrophobic force in the lamellar (or foam) films,
which has been shown to accelerate the rate of film thinning at film thicknesses (H) below
approximately 250 nm [10]. As film thinning continues, H reaches a critical thickness (H ), where
cr
28 |
Virginia Tech | the film ruptures catastrophically. Thus, it is necessary to model the kinetics of film thinning at the
Plateau border and predict H for developing a mathematical for bubble coarsening.
cr
In a dry foam, when two bubbles face each other, they form a lamella, a thin liquid film
(TLF). When three TLFs meet together, they form a Plateau border (PB). A PB is a channel in
which liquid flows though. Four PB forms a vertex, where liquid meet together. In the absence of
external water supply to a foam (e.g. wash water in column flotation), the base of the foam is
relatively wet, whereas the top of the foam is dry due to the downward liquid drainage in the foam
phase. In general, water amount in the TLF is small compared to the same in the PB, especially in
the case of dry area. Therefore, for the modeling purpose of a foam or liquid drainage in the foam,
most studies have been neglected water in the TLF.
3.2.1 Foam Drainage
Now consider liquid drainage in a PB in the vertical direction. On the base of consideration
of the force balance and the mass conservation, the drainage of liquid through a PB can be derived
as follows [11, 12],
1 1 A
U gA (3.3)
A x
where U is the drainage rate; g is the gravitational acceleration; µ, ρ, and γ are the kinematic
viscosity, density, and surface tension of water, respectively; A is the cross-sectional area of the
Plateau border (PB), and x is the distance from the top of a foam. For the force balance, the balance
between gravitational force (forcing downward), capillary force (forcing upward), and viscous
force (forcing opposing direction of the liquid flow) is considered. In terms of viscous loss, there
are two controversial hypotheses.
3.2.2 Steady State
At a steady state, the downward liquid drainage velocity (U) is equal to the upward
superficial gas velocity (-V ),
g
U V (3.4)
g
29 |
Virginia Tech | Combining Eqs. (3.3) and (3.4) and then integrating from the top to the bottom of a foam,
one obtains,
V gA h gV
A g tantan1 b f g (3.5)
t g V 2
g
where A is the PB area at the top of a foam and A is the same at the bottom, and h is the foam
t b f
height.
3.2.3 Bubble Coarsening Model
In a dry foam, each PB is formed by the intersection of three lamellar films, and each corner
of the surrounding polyhedron is the intersection of four PBs. Thus, it would be reasonable to
assume that the number of PBs (N ) is proportional to the number of bubbles, which can be
pb
calculated by dividing the cross-sectional area (S) of a foam (or froth) column by bubble diameter,
i.e., 4S/πd2. One can then write the following relation,
N 4S/d 2 d 2
pb,t 2,t 2,b (3.6)
N 4S/d 2 d
pb,b 2,b 2,t
where N and N are the PB numbers at the top and bottom of a foam, respectively, and d and
pb,t pb,b 2,t
d are the corresponding bubble sizes.
2,b
As a foam drains, A becomes smaller, or the foam becomes drier. As a foam becomes drier,
the thickness (H) of the lamella films will also become thinner. As H becomes thinner, there will
be a point where, a lamellar film will rupture catastrophically, which is referred to as critical
rupture thickness (H ). Likewise, it may be reasonable to assume that bubbles begin to coalesce
cr
when A reaches A , provided that there is a mathematical relation between A and H . As bubbles
cr cr cr
coalesce, N will decrease. In the present work, N may be related to A as follows [11],
pb pb cr
A
N N expC cr (3.7)
pb 0 A
30 |
Virginia Tech | where N is the number of PBs at the base of a foam, and C is an adjustable parameter. Eq. (3.7)
0
suggests that N increases exponentially with the square root of A , which can be related to H
pb cr cr
analytically as will be shown later.
Substituting Eq. (3.7) into Eq. (3.6), one obtains the following relation,
d 1 1
2,b expC A (3.8)
d cr A A
2,t b t
which will allow prediction of bubble size ratio (or bubble coarsening) if the values of A , A and
b t
A are known.
cr
According to Eq. (3.5), one can determine the value of A if A is known. One can determine
t b
A by considering the geometrical relationships between bubble size, lamellar film thickness, and
b
Plateau border radius [13, 14],
A 0.124(d )2 1.52d H (3.9)
b 2,b b 2,b b b
where d is the bubble size at the base of a foam (or froth), which may be considered the same as
2,b
the bubble size in the pulp phase; ε is the liquid fraction at the base of the foam under
b
consideration; and H is the lamella film thickness at the base.
b
One can determine d using a simple bubble generation model [15],
2,b
0.6
2.11
d (3.10)
2,b 0.66
bg
where ε is the energy dissipation rate in the bubble generation zone of a flotation cell.
bg
In using Eq. (3.9), one can obtain ε using a drift-flux analysis [16],
b
31 |
Virginia Tech | Figure 3.1 A model for the unit cell of the bubbles packed at the base of a foam (or froth).
V V gd 2
U g l 2 b (3.11)
slip 1 18(10.15Re0.687)
b b
where U is the slip velocity of bubbles, V and V are the superficial velocities of gas and liquid,
slip g l
respectively, and Re is the Reynolds number.
By the geometrical considerations depicted in Figure 3.1, the H of Eq. (3.9) was
b
determined combining Eqs. (3.12) and (3.13),
3
4 d
2,b N
3 2 2,b
1 (3.12)
b S(d H )
2,b b
where N is the number of bubbles of size d in a foam column of cross-sectional area of S.
2,b 2,b
Assuming that the bubbles at the base are spherical and form a two-dimensional monolayer as
depicted in Figure 3.1, one calculate N as follows,
2,b
S
N 2N 2 (3.13)
2,b t,b 0.5(d H )2cos30
2,b b
32 |
Virginia Tech | Figure 3.2 A geometrical relationship between bubble size (R ), Plateau boarder radius
2
(R ), the critical Plateau boarder area (A ), and the critical lamella film
pb cr
thickness (H ) in a foam.
cr
Since each triangular unit contains one half of a bubble, N is twice the number of the triangles
2,b
at the base (N ), which can be determined simply by dividing S with the area of a triangle
t,b
In calculating the bubble size ratio using Eq. (3.8), it is also necessary to know the value
of A . In the present work, the values of A were calculated from those of H based on the
cr cr cr
geometric relation between A and H [14], as shown in Figure 3.2.
A ( 3 )R 2 3R H (3.14)
cr 2 pb pb cr
where R is the radius of curvature of PB. In Eq. (3.14), H can be determined from the
pb cr
methodology presented in Chapter 2.
3.3 Experimental
3.3.1 Materials
In this study, the model presented in the foregoing section was validated using a specially
designed foam column shown in Figure 3.3. The column (15cm wide × 15cm deep × 47 cm high)
33 |
Virginia Tech | All experiments was conducted using a Millipore pure water with a resistivity of
18.2MΩ·cm at 25°C. Methyl isobutyl carbinol (MIBC, 98% purity, Aldrich) was chosen as a
frother since it is the most widely used in industry [17].
Prior to each test, the column was placed in a base solution (distilled water, hydrogen
peroxide and, ammonium hydroxide at a ratio of 4:1:1) for 10 min. to clean the surface of the glass
wall. The column was then thoroughly rinsed with distilled water and allowed to be dried.
3.3.2 Experimental Procedure
The cell was first filled with aqueous solutions of methyl isobutyl carbinol (MIBC).
Bubbles were generated by pumping the nitrogen gas through an air sparger located at the bottom
of the foam column. The flow rate of gas was accurately controlled by adjusting the valve
connected to a flow meter (Aalborg). The foam generated in this manner was allowed to freely
overflow, mimicking flotation. The overflowed solution was recycled continually. The
recirculation help keep the amount of MIBC in the solution, thus the value of surface tension may
be kept. The foam height was monitored using a graduated scale embedded on the front wall of
the column and was manually controlled by adjusting the level of pulp/froth interface. Behind the
column, a 250 W light source was placed to illuminate the observation region with plastic board
to obtain clear images. Once a steady state condition was reached, the images of the bubbles in a
foam column were recorded by means of a high-speed camera (Fastec imaging).
3.3.3 Bubble Size Measurement
The average bubble diameters at the base of the froth (d ) and the same at the top (d )
2,b 2,b
were then calculated by analyzing the images offline using the BubbleSEdit, image-analysis
software. The average bubble diameters were given by Sauter mean diameter (d ),
32
nd3
d i i (3.15)
32 nd2
i i
where n is the number of bubbles with diameter d. In measuring the average bubble size in a foam
i i
or a froth, d is widely used since it can represent the average suitably [18] and also the flotation
32
35 |
Virginia Tech | Figure 3.4 Bubble size ratio (d /d )between the top and bottom of a foam as a function
2,t 2,b
MIBC concentration at different gas rates at a foam height of 4 cm. The lines
drawn through the experimental data points represent the model predictions
obtained using Eq. (3.7) on the basis of the critical film rupture thicknesses (H )
cr
calculated from Eqs. (2.11) and (2.12).
rate is critically related with surface area of bubbles [19]. In the study, the values of bubble size
ratio (d /d ) were used as a measure of froth stability.
2,t 2,b
3.4 Results and Discussion
3.4.1 Effect of Frother Concentrations
Figure 3.4 shows the effect of frother (MIBC) concentration on bubble size ratio (d /d ),
2,t 2,b
a measure of bubble coarsening. While the foam height was kept constant at 4 cm, the experiments
were conducted at 51, 71, and 102 ppm MIBC, which are much higher than industrial flotation
dosage ranging between 1 and 10 ppm [17]. Such a high dosage was employed because at lower
36 |
Virginia Tech | concentrations, stable foam phase was not generated, mostly due to the absence of particles. It is
generally known that particles are a solid-state surfactant that can enhance the form stability [20,
21]. The values of the C parameter were obtained by fitting the experimental data to the model and
summarized in Table 3.1. The fit between the experimental data and simulation results (curves)
were excellent. Both the model prediction and experimental data showed that the bubble
coarsening decreased with increasing MIBC concentration, indicating that the frother stabilized
the foam and retarded bubble coarsening. In effect, MIBC caused a decrease in A or H .
cr
cr
As suggested previously, the hydrophobic force present in foam films may be the major
driving force for foam film rupture and that the hydrophobic force decreases with increasing
frother concentration [9, 22]. Therefore, the decrease in hydrophobic force with increasing MIBC
concentration can provide an explanation for the decrease in both d /d and H .
2,t 2,b cr
3.4.2 Effect of Foam Height
Figure 3.5 compares the values of d /d at the foam heights of 2 and 4 cm. As expected,
2,t 2,b
it was observed that the bubble size ratio increased with increasing foam height at a given
superficial gas rate (V ). The reason is simply that bubbles will have a longer time to coalesce in a
g
taller froth height. The experiments were conducted at three different gas rates, i.e., V = 2, 3, and
g
4 cm/s. At a higher gas rates, the froth/pulp interface was lost due to flooding [23]. As shown, the
bubble size ratio decreased most probably due to shorter residence times of bubbles [4].
Table 3.1 Values of fitting parameter C of Eqs. (3.7) and (3.8) used to predict bubble
coarsening.
Superficial Gas MIBC (ppm)
Rate (cm/s) 51 71 102
2 16.60 10.01 3.33
3 8.44 5.75 1.65
4 5.30 2.29 0.36
37 |
Virginia Tech | 3.4.3 Effect of Gas Flow Rate
As shown in Figure 3.5, the experiments were conducted at three different gas flow rates,
i.e., V = 2, 3, and 4 cm/s. A lower gas rate was not enough to create a foam phase. On the other
g
hand, at a higher gas rates, the froth/pulp interface was lost due to flooding [23]. As shown, the
bubble size ratio decreased with increasing gas rate. It may be due to the less chances for the bubble
to be coalesce, most probably due to shorter residence times of bubbles in the foam phase [4].
3.5 Summary and Conclusions
A bubble-coarsening foam model has been developed. It is capable of predicting the bubble
size at the top of a foam from the bubble size at the bottom. The input parameters of the bubble-
coarsening model include froth height, frother dosage, gas flow rate, and the bubble size in the
pulp phase. The model predictions show that bubble coarsening increases with decreasing frother
dosage, which may be attributed to the increased hydrophobic force at lower frother dosages. It is
also shown that bubble coarsening increases with increasing foam height and decreasing aeration
rate, which can be attributed to longer residence times of the bubbles in the foam (or froth) phase.
The model predictions are consistent with the experimental results. At this point, however, we are
not considering the effects of particle size and particle hydrophobicity, which is known to affect
froth stability [20, 21]. Thus, the model presented in this communication is for foam rather than
for froth.
References
[1] Schwarz, S., et al., Water behaviour within froths during flotation. 2001.
[2] Barbian, N., K. Hadler, and J.J. Cilliers, The froth stability column: Measuring froth
stability at an industrial scale. Minerals Engineering, 2006. 19(6–8): p. 713-718.
[3] Barbian, N., et al., The froth stability column: linking froth stability and flotation
performance. Minerals Engineering, 2005. 18(3): p. 317-324.
[4] Tao, D., G.H. Luttrell, and R.H. Yoon, A parametric study of froth stability and its effect
on column flotation of fine particles. International Journal of Mineral Processing, 2000.
59(1): p. 25-43.
39 |
Virginia Tech | Chapter 4. Modeling Froth Stability: Effect of Particle
Hydrophobicity
ABSTRACT
As bubbles rise in a froth phase, a thin liquid film (TLF) confined between bubbles thins
by the capillary pressure (p ) and the disjoining pressure (П). Once the intervening TLF ruptures,
c
bubble-coarsening occurs. In the present work, we have studied the effect of particle
hydrophobicity (or water contact angle θ) on the bubble-coarsening (or froth stability). The study
was conducted by measuring the bubble size ratio between the top and bottom of a forth in the
presence of monosized particles of varying hydrophobicity. The experimental results showed that
the froth stability increases with θ up to θ = 70° and decreases with further increase in θ. We have
also developed a model for predicting the bubble-coarsening in a froth by deriving a film drainage
model quantifying the effect of θ on p . The model shows that with increasing θ up to θ = 70°, p
c c
may decrease and thereby the film thinning is decelerated and the froth becomes more stable. At
θ > 70°, however, the froth becomes less stable due to increased p and, hence, increased drainage
c
rate.
42 |
Virginia Tech | 4.1. Introduction
Froth flotation is the most widely used method of upgrading pulverized ores in the minerals
industry [1]. In this process, hydrophobic particles selectively attach to air bubbles in a pulp phase,
forming particle-bubble aggregates, which subsequently rise upward due to reduced buoyancy and
form a froth phase. In the froth phase, the bubbles coalesce with each other and become larger as
they rise. As the bubbles become larger, the bubble surface area decreases, thereby restricting the
number of hydrophobic particles that can be carried upward and subsequently flow into a launder.
On the other hand, the bubble coarsening helps increase the grade of a froth product, as less
hydrophobic particles selectively drop off the coarsening bubbles [2]. Thus, it is of critical
importance to study and control bubble coarsening in the froth phase.
It is known that bubble coarsening occurs when a thin liquid film (TLF) formed between
two bubbles in a froth phase breaks as a result of thinning. The kinetics of thinning of a TLF can
be described by the Reynolds lubrication equation [3],
dH 2H3p
(4.1)
dt 3R 2
f
where H is the TLF thickness, t the drainage time, μ the dynamic viscosity, R the film radius, and
f
p the driving force for TLF thinning. The equation is derived based on the Navier-Stokes equations
under the assumptions of plane-parallel films and no-slip boundary conditions at the air/water
interfaces.
In using Eq. (4.1), the driving force is given by the following relation,
p p (4.2)
c
which shows that the driving force is the sum of capillary (or curvature) pressure (p ) and
c
disjoining pressure (П).
During the initial film thinning, the drainage of a TLF is governed by the p whose
c
magnitude is given by the Laplace equation [4],
43 |
Virginia Tech | 2
p (4.3)
c R
where R is the bubble radius and γ is the surface tension of water.
When H reaches 200 nm, surface forces or disjoining pressure (П) between two air/water
interfaces begin to act and control the thinning of the TLF. In Eq. (4.2), П is determined according
to the extended DLVO theory [5],
e A K
64C RTtanh2 s exp(H) 232 232 (4.4)
el vw hp el 2 4kT 6H3 6H3
where П is the disjoining pressure due to electrostatic interaction, П is the disjoining pressure
el vw
due to the van der Waals dispersion force, П is the disjoining pressure due to hydrophobic force,
hp
C is the electrolyte concentration, R the gas constant, T the absolute temperature, e the electronic
el 2
charge (e = 1.6 ×10-19 C), ψ the surface potential at the air/water interfaces, k the Boltzmann’s
s
constant, κ the reciprocal Debye length, A the Hamaker constant, and K the hydrophobic
232 232
constant.
When the H finally reaches a critical thickness (H ), the TLF ruptures and thereby bubble
cr
coalesces. The value of H was theoretically predicted by Vrij and his co-workers based on
cr
capillary wave theory [6-8],
2 3
2 3H 3R 2
1443H 4 2H 3 m f 0 (4.5)
m H m H H2 2(p )
HH m HH m HH m c
where H 0.845H . The model assumes that the air/water interfaces of a foam film fluctuate
cr m
due to thermal or mechanical motion, creating corrugation. The amplitude of the wave motion
grows if the disjoining pressure of the film () is negative, and the film ruptures when the two
interfaces touch each other. In their original model, only the van der Waal disjoining pressure ( )
vw
and the electrostatic disjoining pressure ( ) were considered, which makes it difficult to predict
el
H at low frother concentration, at which air bubbles become strongly hydrophobic [9]. Park et al.
cr
[10] predicted the values of H of foam films containing methyl-isobutyl carbinol (MIBC) using
cr
44 |
Virginia Tech | the extended DLVO theory, which considers the contributions from the hydrophobic force ( )
hp
[11]. The hydrophobic force constant (K ) determined by Wang [12] were used for this approach.
232
The model predictions are in agreement with the H values measured by Wang [12], and the results
cr
show that foam film stability increases with increasing frother dosage due to dampening of the
hydrophobic force in the presence of a frother.
Froth is a 3-phase foam, and its behavior is different from foam due to the presence of
particles. As is well known, froth becomes more stable in the presence of particles [13-16]. Ata et
al. [17] experimentally observed that the froth stability depends critically on particle
hydrophobicity (or water contact angle θ). They measured the bubble size distribution within froth
phase along the height in the presence of glass particles of different contact angles (θ = 50°, 66°,
and 82°). The bubble size increased with froth height owing to increased bubble coalescence. They
found also that the bubble growth rate was sensitive to θ. The bubble growth rate was lowest in
the presence of the intermediate hydrophobic particles (66°) and highest in the presence of the
weekly hydrophobic particles (50°). Experimentally, their results indicated that there is an
optimum particle hydrophobicity (~66°) for the maximum froth stability.
Binks [18] derived a model that can calculate the energy required to detach a particle from
an air/water interface,
G r2(1cos)2 (4.6)
d 1
where r is the particle radius. For the TLF of a froth to rupture and disappear, the particle located
1
at the interface should be removed. Eq. (4.6) suggests that less hydrophobic particles can be readily
washed off the TLF, whereas more hydrophobic particles will be more difficult to be washed off
and thereby help stabilize the froth. If this was the case, the maximum froth stability should be
achieved at θ = 90°. However, the Binks’ model alone cannot successfully explain the Ata et al.’s
observations [17]. The discrepancy may be originated from the limitation that the Binks’ model is
derived on the basis of only thermodynamic aspect, though the kinetics of the TLF is a more
dynamic process including the hydrodynamic effect.
Denkov et al. [19] proposed a stabilization mechanism of emulsions by absorbed particles.
The mechanism was based on the recognition that the particles create changes in the curvature of
45 |
Virginia Tech | Figure 4.1 Sketch of an absorbed particle in the thin liquid film between two air bubbles.
The curvature of the air/water interface around the particle creates the
difference between the pressure inside of the bubble (p ) and the same in the
air
film (p ).
1
the air/water interface and hence the local capillary pressure (p ), which in turn affect thin film
c,local
rupture. Due to the similarity between a foam and an emulsion, their model may possibly be used
to explain the froth stability. As shown in Figure 4.1, the p (≡ p - p ) arises from the local
c,local air water
curvature changes around the particle and it can be given by the Young-Laplace equation,
d
p (rsin) (4.7)
c,local r dr
where r is the radial position, and ϕ is the running slope angle. In their paper, the p was
c,local
calculated numerically by combining the following geometrical relationships,
sinr
p 2 c c (4.8)
c,local b2 r2
c
b2 r2
z b dr (4.9)
rc
2
2
r2 (b2 r2)2
p
c,local
46 |
Virginia Tech |
H 2 acosz (4.10)
r asin() (4.11)
c c
where ϕ is the slope of the curvature at contact line, r the radial position of contact line, α the
c c
angle between the vertical line and contact line, ∆z the depth between the contact line and the
interface, θ the contact angle, b the radius of the cell, and H the TLF thickness at the boundary of
the cell.
Figure 4.2 shows the p vs. H plots predicted from Denkov’s model. It indicates that
c,local
p increases slightly with decreasing H and reaches a maximum at H = 0. The model assumed
c,local
that the TLF ruptures at H = 0 and the maximum values of p were used as a criteria for
c,local
evaluating the emulsion stability. The higher the maximum values of p , the more stable
c,local
emulsions. Thus, the calculation results shown in Figure 4.2 suggest that the froth stability
Figure 4.2 Local capillary pressure (p ) arising from a particle with contact angle (θ)
c,local
of 40°, 55°, 70° and 85°, respectively, as a function of the film thickness (H).
The plots are obtained using Eqs. (4.8) ~ (4.11) with a = 70 μm, b = 210 μm, γ
= 0.0724 N/m.
47 |
Virginia Tech | decreases with increasing θ. However, this trend is not compatible with previous experimental
observations mentioned above [17]. The discrepancy may be due to several limitations of
Denkov’s model. First, the model did not consider the whole area of a film. Only the force balance
around particles was considered. Second, surface forces were neglected although recently it has
shown that they are important in determining the stability of foam films [5, 20]. Finally, the model
assumes that films rupture when film thickness is reduced to zero. Many investigators showed,
however, that films rupture at critical rupture thicknesses, which are non-zero and vary
significantly with surfactant type and dosage [5, 7, 8].
More recently, Morris et al. [21-23] conducted computational simulations to estimate the
TLF stability by calculating the maximum values of p in the TLF containing multiple spherical
c,local
particles. Their approach was a step forward from Denkov et al.’s work in that the whole area of
a film was modelled. However, their simulation result showing that the maximum values of p
c,local
and the TLF stability decrease with increasing θ still do not agree well with the Ata et al.’s
observations [17]. The discrepancy may be attributed to the still unresolved limitations that surface
forces and H were not considered and that the possible variation of the number of particles in a
cr
TLF with changes in θ.
In the present work, we have conducted laboratory-scale flotation testes to measure froth
stability in the presence of silica particles of varying surface hydrophobicity (θ = 40°, 55°, 70°,
85°). In each experiment, using a high-speed camera and image-analysis software, we have
measured the bubble size ratio (d /d ), the ratio of average bubble diameter at the top of the froth
2,t 2,b
(d ) and to the same at the base of the froth (d ), as a measure of froth stability. We have also
2,t 2,b
developed a theoretical model that can explain the effects of θ on bubble coarsening (or froth
stability). The model was derived by modelling the whole area of a TLF by considering both the
capillary force and the surface forces. Furthermore, the model developed here enables us to predict
d /d as a function of frother dosage, froth height, gas rate, specific power, particle size, particle
2,t 2,b
hydrophobicity (collector dosage), particle concentration, etc.
48 |
Virginia Tech | 4.2. Experiment
4.2.1 Materials and Hydrophobization of Silica Surfaces
In the present study, silica spheres (Potters industries) of 35 μm diameter (d ) were used
1
for flotation test. The spheres along with a referential silica plate were firstly cleaned in Piranha
solution (H O /H SO 3: 7 by volume) for 1 h at 120 °C, rinsed with ultrapure water for 10 min,
2 2 2 4
and then dried at 160 °C in an oven for 24 h. The silica surface is naturally hydrophilic; therefore,
its surface was rendered hydrophobic using octadecyltrichlorosilane (OTS, 95% purity, Alfa
Aesar). The spheres were immersed in a 10-4 M OTS-in-toluene (99.9% purity, Spectrum Chemical)
solution along with a referential silica plate, so that both could have identical hydrophobicity. The
hydrophobized surfaces were then rinsed with chloroform (99.9% purity, Fisher Chemical),
acetone (99.9 % purity, Aldrich), and ultrapure water sequentially. The hydrophobicity of the
particle surfaces, as measured by water contact angles (θ), was controlled by varying the
Figure 4.3 Effects of immersion time on the contact angles of silica in 10-4 M OTS-
in-toluene solutions.
49 |
Virginia Tech | a fixed gas rate of 6 L/min to obtain a superficial gas velocity of 1cm/s. The slurry was kept in
suspension by agitating it at an impeller speed of 900 rpm.
The froth was allowed to freely overflow, let to destabilize, and then recycled back to the
cell by means of a peristaltic pump. The froth height was monitored using a graduated scale
embedded on the front wall of the cell. Once a steady state condition was reached, the images of
the bubbles in the froth were recorded by means of a high-speed camera. The average bubble
diameters at the base of the froth (d ) and at the top (d ) were then obtained by analyzing the
2,b 2,b
images offline using the BubbleSEdit, image-analysis software. The average bubble diameters
were given by calculating the Sauter mean diameter (d ),
32
nd3
d i i (4.12)
32 nd2
i i
Figure 4.5 The bubble size ratio (d /d ) measured in the presence of particles with
2,t 2,b
contact angle() of 40°, 55°, 70°, and 85°, respectively. The dotted line
represents d /d in the absence of particles.
2,t 2,b
51 |
Virginia Tech | where n is the number of bubbles with diameter d. In measuring the average bubble size in a foam
i i
or a froth, d is most widely used in flotation literature. The number-mean diameter gives
32
excessive weights on fine bubbles, while the volume-mean diameters does son on coarse bubbles.
On the other hand, flotation rate is shown is critically related to surface area of bubbles [17].
Therefore, the Sauter mean diameter may be considered most suitable for flotation studies [24].
4.3 Experimental Results
Laboratory flotation tests were conducted both in the absence and presence of monosized
(35μm) silica particles. To investigate the effect of particle hydrophobicity (θ) on the bubble size
ratios (d /d ), which is considered as a measure of froth stability in the present work, particles
2,t 2,b
of different contact angles (θ = 40°, 55°, 70°, and 85°, respectively) were tested. In all tests, the
frother dosage (10-5 M MIBC), gas flow rate (V = 1 cm/s), froth height (h = 4 cm), agitation speed
g f
(900 rpm), and the particle concentrations (5% w/w) were carefully kept constant.
Figure 4.5 shows the experimentally measured values of d /d . As shown, the d /d
2,t 2,b 2,t 2,b
ratios becomes smaller in the presence of particles. As shown, air bubbles becomes stable in the
presence of particles of θ < 90°. Note also that d /d decreased with increasing θ, reaching a
2,t 2,b
minimum at θ = 70°, and the increased with further increase in θ. The observed trends are
consistent with Ata et al.’s results [17]. It was shown that bubbles in the froth phase immediately
collapsed when very hydrophobic particles (θ > 90°) were added to a forth phase.
4.4 Model Development
In section 4.4, we present a new methodology that can predict the bubble size ratios (d /d )
2,t 2,b
in a froth phase. In Subsection 4.4.1, we first calculate the driving force (p) for the thinning of a
froth film containing particles with different contact angles (θ). To calculate p, we need to know
N (the number of particles in a single film). Subsection 4.4.2 introduces a new model that can
1,film
predict N as a function of θ. In Subsection 4.4.3, substituting the p values into the Reynolds
1,film
equation, the thinning rates of the froth films are determined as a function of θ. After that, the
critical rupture thickness (H ) model developed in Chapter 2 is used to predict the critical rupture
cr
time (t ) of the froth film as a function of θ. In Subsection 4.4.4, a model that can predict d /d
cr 2,t 2,b
from t is developed.
cr
52 |
Virginia Tech | 4.4.1 Driving Force for a Froth Film Thinning
a. Foam Film
Prior to modelling a froth film, consider a foam film stabilized in the presence of 10-5 M
MIBC. As shown in Eq. (4.2), the driving force (p) for film thinning is the sum of the capillary
pressure (p ) and the disjoining pressure (П). These parameters are functions of film thickness (H)
c
as shown in Eqs. (4.2), (4.3), and (4.4). The model parameters were calculated using the values of
R = 0.44 mm and T = 298 K in accordance to the actual bubble sizes measured at the base of a
foam and the temperature at which the experiments were conducted. The values of γ = 0.0724 N/m
and ψ = -30 mV were also used as reported by Comley et al. [25] and Srinivas et al. [26]. Also
s
the values of K = 2.3×10-19 J and A = 4×10-21 J were used as reported by Wang [12]. Only the
232 232
value of κ-1= 30 nm was assumed. The calculation results are shown in Figure 4.6. As shown, p
c
is constant during film thinning due to the assumption of flat lamella film. On the other hand, П
becomes more negative as the film becomes thinner, which can be attributed to the presence of
Figure 4.6 The changes in driving pressure (p), capillary pressure (p ), and disjoining
c
pressure (П) as a function of a MIBC foam film thickness (H). The plots are
drawn from Eqs. (4.2) ~ (4.4) with K = 2.3×10-19 J, A = 4×10-21 J, R = 0.44
232 232
mm, γ = 0.0724 N/m, T = 298 K, e = 1.6 ×10-19 C, ψ = -30 mV, and κ-1= 30
s
nm.
53 |
Virginia Tech | attractive force in the film that varies as H-3. Note here that as the film drains, p increases mainly
due to the increase in the negative disjoining pressure ( < 0).
The critical rupture thickness (H ) of the foam film was determined using Eq. (4.5). In
cr
using Eq. (4.5), we needed to know the film size (R). Assuming that a bubble has a dodecahedron
f
structure, consisting of 12 films, the surface area of a single film (πR2) should be the same as
f
4πR2/12. Thus, R can be determined from R as follows,
f
R R/ 3 (4.13)
f
in which R = 0.44 mm as measured at the base of the froth (or foam) in this study. Eq. (4.13) gives
the value of R to be 250 μm. In the present study, R was assumed to be independent of H in
f f
accordance to the previous experimental observations [20]. Figure 4.7 shows the model prediction
of the values of H using Eq. (4.5). As shown, H decreases with MIBC concentration, which can
cr cr
be attributed to the effect of dampening of hydrophobic force in the presence of a surfactant [27].
At 10-5 M MIBC, where tests were conducted, H was found to be 171 nm.
cr
Figure 4.7 Plots of the critical film rupture thicknesses (H ) of a foam film predicted using
cr
the H model developed in Chapter 2 vs. MIBC concentration.
cr
54 |
Virginia Tech | Figure 4.8 Sketch of a layer of particles located in the thin liquid film between two air
bubbles. Near the particle a curvature change occurs (Section I), while the free
film is formed away from the particle (Section II). Section III denotes a Plateau
boarder area.
b. Froth Film
The presence of particles in froth films should cause the local curvatures at the air/water
interface to change, which should in turn cause the capillary pressure (p ) to change from that of
c
free films. The presence of particles may also change the disjoining pressure (П) and hence the
hydrodynamic pressure (p). Figure 4.8 represents a lamella film, in which three spherical particles
are embedded. The film around each particle, may be subdivided into Sections I, II and III,
representing the area in the vicinity of a particle, the area away from the particle, and the outside
the film (Plateau border), respectively.
The pressure (or force) balance in each section may be give as follows,
p p p (4.14)
air I c,local
55 |
Virginia Tech | p p 0 (4.15)
air II
2
p p (4.16)
air III R
where p is the pressure in the air bubble, p and p are the those in Sections I and II, respectively,
air I II
p is the local capillary pressure that may be calculated using Eqs. (4.7)-(4.11), and R is the
c,local
bubble radius. Assuming that p ≈ 0, one obtains the capillary pressures in Sections I and II as
III
follows,
2
p p (4.17)
I R c,local
2
p (4.18)
II R
The capillary pressure (p ) of the film as a whole may then be calculated as follows,
c
Overallcapillaryforce p AN p A
p I I 1,film II II (4.19)
c Filmarea AN A
I 1,film II
where A and A is the film areas occupied by Section I and II, respectively, and N is the
I II 1,film
number of particles present in the single film. The values of N may vary with operating
1,film
conditions. A model for predicting N is presented in Section 4.4.2.
1,film
Likewise, one can calculate the overall disjoining pressure (П) using the following relation,
Overall surfaceforce AN A
I I 1,film II II (4.20)
Filmarea A
II
in which and are the disjoining pressures in Sections I and II, respectively. Section II is a
I II
free film; therefore, its disjoining pressure can be given as follow,
e A K
64C RT tanh2 s exp(H) 232 232 (4.21)
II el vw hp el 4kT 6H3 6H3
56 |
Virginia Tech | On the other hand, the disjoining pressure in Section I (П) may be obtained by integrating the
I
local disjoining pressure along the radial direction,
b
(r)(2r)dr
r c (4.22)
I A
I
In the vicinity of a particle, (r) is small because the film thickness (H) increases sharply with
decreasing r, approaching the length scale of the particle. At such large film thicknesses, disjoining
pressures should be close to zero. Thus, 0. Substituting this into Eq. (4.20), , which
I II
means that particles should not seriously affect П.
4.4.2 Number of Particles in a Froth Film
In using Eqs. (4.19) and (4.20), we need to know the value of N (the number of particles
1,film
in a froth film). This section shows a new model that can predict N theoretically. N may
1,film 1,film
change continually along the froth height due to the continuing detachment of particles resulting
Figure 4.9 Probability of particles surviving in a pulp phase P P (1-P ) vs. particle contact
c a c
angles (). The inset shows the probability of collision (P ), the probability of
c
attachment (P ), and the probability of detachment (P ), respectively.
a d
57 |
Virginia Tech | from bubble coarsening. This model is aimed to model the base of a froth. Thus, N at the base
1,film
of a froth is need to be predicted. The model for predicting N at the base of a froth was derived
1,film
based on the assumption that the value of N may be proportional to the probability of particles
1,film
reaching a froth phase.
The probability of particles reaching a froth phase (P) is given by,
PPP(1P) (4.23)
c a d
where P is the probability of collision, P the probability of attachment, and P the probability of
c a d
detachment. The probability functions are given as follows, respectively,
3 3 Re d
P
c
tanh2
2 1 16
10.249Re0.56
d1
2
(4.24)
E
P exp 1 (4.25)
a E
k
Table 4.1 Input parameters for the simulation shown in Figure 4.9.
Frother (MIBC) concentration (M) 10-5
Solids concentration (% w/w ) 5
Specific power (kW/m3) 22.8
Superficial gas rate (cm/s) 1.0
Froth height (cm) 4
Particle size (μm) 35
Particle Zeta potential (mV) -0.08
Bubble Zeta potential (mV) -0.03
58 |
Virginia Tech | W E
P exp a 1 (4.26)
d
E
k
where Re is the Reynolds number, E is the energy barrier, E is the kinetic energy available during
1 k
attachment process, W is the work of adhesion, and E’ is the kinetic energy of detachment. The
a k
plots of probability functions (P , P , and P ) are shown in Figure 4.9 and input parameters for this
c a d
simulation is summarized in Table 4.1. In the simulation, the bubble zeta potential was obtained
from Comley et al.’s study [25], the particle zeta potential was assumed, and the other parameters
were measured from experiments. Note in Figure 4.9 that the value of 1-P increases with
d
increasing θ, which is mainly due to the increase in adhesion force (W ) between a bubble and a
a
particle. It is also noteworthy that P is close to 1 even for hydrophilic particles. This is mainly due
a
to the relatively higher power dissipation rate (22.8 kW/m3) of laboratory flotation cells as
compared to industrial flotation machines, resulting in higher kinetic energy.
In the present study, N may be related to P (=P P (1 - P )) as follows,
1,film c a d
N N PP(1P) (4.27)
1,film 1,seg c a d
Figure 4.10 The number of particles residing in a froth film (N ) predicted as a function
1,film
of particle contact angles ().
59 |
Virginia Tech | where N is the number of the particles having a chance to collide with a bubble surface segment
1,seg
in pulp phase ending up a single lamellar film in the froth phase. The value of N may be
1,seg
proportional to the number of particles suspending in the pulp. In each experiment, we added same
amount of particles, so the constant N value (N = 2667) was assumed. After that, as shown
1,seg 1,seg
in Figure 4.10, we finally predicted the values of N as a function of θ. It was found that N
1,film 1,film
critically depends on θ. Note here that more particles can locate in a film at higher θ, most probably
due to lower P values.
d
Figure 4.11 The changes in driving pressure (p), capillary pressure (p ), and disjoining
c
pressure (П) as a function of the froth film thickness (H) containing particles
with contact angle () of (a) 40°, (b) 55°, (c) 70°, and (d) 85°. The values of
K = 2.3×10-19 J, A = 4×10-21 J, R = 0.44 mm, γ = 0.0724 N/m, T = 298 K, e
232 232
= 1.6 ×10-19 C, ψ = -30 mV, and κ-1= 30 nm were used.
s
60 |
Virginia Tech | 4.4.3 The Critical Rupture Time (t ) of a Froth Film
cr
Substituting the N values predicted in Section 4.4.2 into Eq. (4.19), one can determine
particle
the values of p . On the other hand, П can be determined using Eq. (4.21) using the values of K
c 232
= 2.3×10-19 J, A = 4×10-21 J, R = 250 μm, T = 298 K, γ = 0.0724 N/m, ψ = -30 mV, and κ-1= 30
232 f s
nm were used. Substituting the values of p and П obtained in this manner into Eq. (4.2), one can
c
obtain p as a function of particle hydrophobicity (θ). Figure 4.11 shows the calculated values of
p , П, and p. It is noteworthy that as compared to the foam film (shown in Figure 4.6), the absorbed
c
particles decreased p and p during the process of film drainage, which can be attributed to the
c
local capillary pressure (p ) created by the particles. However, П was not changed, as
c,local
mentioned above. More importantly, Figure 4.11 shows that the values of p and p change with .
c
It was found also that with increasing up to = 70°, p decreased and, therefore, p decreases. On
c
the other hand, at > 70° p and p increased with further increase in .
c
Figure 4.12 Effect of particle hydrophobicity on film thinning rate. The red line represents
the critical rupture thickness (H ) of a 10-5M MIBC foam film predicted from
cr
the H model developed in Chapter 2.
cr
61 |
Virginia Tech | Figure 4.13 Rupture mechanism of a thin liquid film in the presence of a particle. The film
rupture may occur in the free film at H .
cr
Figure 4.12 shows the film thinning kinetics of a foam film and froth films containing
particles with θ = 40°, 55°, 70°, and 85°, respectively. We obtained the film thinning rates by
using the calculated p values shown in Figures 4.6 and 4.11 into the Reynolds equation shown in
Eq. (4.1). Note in Figure 4.12 that the foam film thins fastest when p is large. In the presence of
particles, the thinning rate is retarded due to relatively smaller p. In the case of the foam film, it is
obvious that it ruptures at its critical rupture thickness (H ), which was predicted to 171 nm in
cr
Section 4.4.1. The foam film, therefore, ruptures in 2.8 s. A question that may be raised here is
how to predict the H value for froth films and how to relate H of a foam film to that a froth film.
cr cr
Until now, there has been no model for predicting H of a froth film.
cr
Figure 4.13 shows a proposed rupture mechanism of a froth film. It may be reasonable to
assume that the film rupture occurs in the free film section of a froth film. Since the free film
section is exactly same as a foam film, we assume in the present work that H of a foam film and
cr
that of a froth film are identical. Accordingly, we can determine the critical rupture time (t ) of a
cr
Table 4.2 The critical rupture time (t ) of a froth film predicted as a function of .
cr
(°) t (s)
cr
40 3.1
55 3.3
70 3.7
85 3.2
62 |
Virginia Tech | froth film at different , as shown in Table 4.2. It has been found that t increases with , reaching
cr
a maximum at θ = 70°, and then decreases with further increase in θ.
4.4.4 Prediction of Bubble Size Ratio (d /d ) from t
2,t 2,b cr
Figure 4.14 shows the schematic representation of a model for predicting the bubble size
ratio (d /d ) from t . As a bubble rises along the y direction in a froth phase, the number of
2,t 2,b cr
bubbles decreases due to bubble coarsening. The number of bubbles can be calculated by dividing
the cross-sectional area (S) of a flotation cell by bubble diameter, i.e., 4S/πd2. One can then write
the following relation,
2
N S/(d /2)2 d
bubble,t 2,t 2,b (4.28)
N S/(d /2)2 d
bubble,b 2,b 2,t
Figure 4.14 A model for predicting d (bubble size at the top of the froth) from d (the
2,t 2,b
same at the base). A bubble created in the pulp arrives at the base at the terminal
velocity of U. Then it starts to rise along the y direction at the velocity of U
t froth
and thin liquid films between bubbles starts to drain. At a critical rupture time
(t ), the film ruptures and the rising bubbles coalesce.
cr
63 |
Virginia Tech | where N and N are the bubble numbers at the top and bottom of a froth, respectively.
bubble,t bubble,b
Eq. (4.28) indicates that the bubble size ratio can be obtained if the bubble number ratio is
predicted.
We assume that the number of bubbles decreases exponentially with froth height,
y
N (y) N exp(C ) (4.29)
bubble bubble,b h
f
where C is the decay constant, h is the froth height, and y is the distance from the base of a froth,
f
which can be given by,
yV t (4.30)
g
where V is the superficial gas velocity and t is the time. Knowing that bubbles coalesce at the
g
critical rupture time (t ), at which the film thickness reaches its H , Eq. (4.29) can be rewritten as
cr cr
follows,
y
N (y ) N exp(C ttcr ) (4.31)
bubble ttcr bubble,b h
f
In the process of bubble coalescence, the total volume of bubbles should be conserved. If
a single thin liquid film between two bubbles of initial diameter d ruptures and thereby one bubble-
i
coarsening event occurs, the diameter of the final bubble (d) should be 21/3 d. Therefore, one can
f i
obtain a more generalized relationship,
d
Nrupture
f 2 3 (4.32)
d
i
in which d is the initial bubble diameter, d is the final bubble diameter, and N is the number
i f rupture
of film rupture for coarsening the bubble diameter from d to d.
i f
Combining the relation shown in Eqs. (4.28) and (4.32), one can obtain the following
equation,
64 |
Virginia Tech | Figure 4.15 Bubble size ratio (d /d )as a function of particle contact angle (). The lines
2,t 2,b
drawn through the experimental data points represent the model predictions.
N (y ) d
2
Nrupture2
bubble ttcr 2,b 2 3 (4.33)
N d (y )
bubble,b 2 ttcr
where N is the number of the films that rupture at t = t among 12 films consisting a
rupture cr
dodecahedron-shaped bubble.
Then, by combining Eqs. (4.28) ~ (4.33), finally one can deduce the following model
predicting the bubble size ratio,
0.5
d 0.46h N
2,t exp f rupture (4.34)
d 2,b t crV g
In using Eq. (4.34) note that the values of h, t , and V are known, while one needs to determine
f cr g
N . The values of N were back-calculated by fitting the model predictions using Eq. (4.34)
rupture rupture
to experimental results shown in Figure 4.5. In the present study, N = 5 was used at θ < 70°,
rupture
65 |
Virginia Tech | Figure 4.16 Effect of particle contact angle () on bubble size ratio (d /d ), the number of
2,t 2,b
particles (N ) in a froth film, and the local capillary pressure (p ) around
1,film c,local
a particle. At <70° the increase in froth stability with increasing may be
mainly due to the increase in N whereas at >70° the decrease in froth
1,film,
stability with increasing may be mainly due to the decrease in p .
c,local
while N = 7 was assumed at θ = 85°. It indicates that more films ruptured at θ = 85°. This may
rupture
be probably due to the possible existence of very hydrophobic particles (θ > 90°), which act as a
foam breaker.
In Figure 4.15, the dots represent the experiential results shown in Figure 4.5 and the line
represents the model predictions from Eq. (4.34). As shown, the model predictions are in good
agreement with the experimentally measured d /d values.
2,t 2,b
The model presented in the present work suggests that two independent parameters govern
bubble–coarsening in a froth. One parameter is the number of particles (N ) in a film, and the
1,film
66 |
Virginia Tech | Figure 4.17 Effect of particle contact angle () on bubble size ratio (d /d ) and the
2,t 2,b
capillary pressure (p ) at the critical film thickness (H ).
c cr
other is the local capillary pressure (p ) created by a single particle. The model shows that the
c,local
higher the N or the p , the lower the overall capillary pressure (p ), resulting in a slower
1,film c,local c
drainage rate and hence a higher froth stability. Figure 4.16 shows the changes in N and p
1,film c,local
with θ. Note here that at θ < 70°, the decrease in p with increasing θ can partially cause p to
c,local c
increase, but the effect of increasing N may overcome the p and hence decreases p . On
1,film c,local c
the other hand, at θ > 70°, with increasing θ, the decrease in p , countering the N effect,
c,local 1,film
causes p to decrease
c
As a result, as shown in Figure 4.17, it was found that the overall capillary pressure p
c
significantly decreases with θ and begins to increase at θ = 70°.
Figure 4.18 shows the effect of θ on driving force p for film drainage. This trend can give
the explanation for the effect θ of on forth stability in terms of the drainage rate according to the
Reynolds equation.
67 |
Virginia Tech | Figure 4.18 Effect of particle contact angle () on bubble size ratio (d /d ) and the
2,t 2,b
driving pressure (p) for the film thinning at the critical film thickness (H ).
cr
4.5 Summary and Conclusions
The effect of particle hydrophobicity (or water contact angle θ) on the bubble-coarsening
(or froth stability) has been studied by measuring the bubble size ratio between the top and bottom
of a forth in the presence of monosized (35μm) silica particles of varying hydrophobicity (θ = 40°,
55°, 70°, and 85°, respectively). It has been found that the froth stability increased with increasing
θ, reached a maximum at θ = 70°, and decreased with further increase in θ.
In addition, we have developed a model for predicting the bubble-coarsening in a froth by
deriving a film drainage model quantifying the effect of θ on the capillary pressure (p ), which
c
drives the drainage process. The parameter p is affected by the number of particles (N ) in a
c 1,film
film and the local capillary pressure (p ) around particles, which in turn vary with θ. The model
c,local
shows that as θ increases, p decreases but N increases sharply, causing p to decrease. As
c,local 1,film c
68 |
Virginia Tech | Chapter 5. Modeling Froth Stability: Effect of Particle Size
ABSTRACT
A lamella film formed between two bubbles in a flotation froth drains due to the capillary
pressure (p ) and the disjoining pressure (П). When the film breaks, the two bubbles become one
c
and the bubble size becomes coarser. In the present work, the effect of particle size (d ) on the
1
bubble-coarsening (or froth stability) was investigated. The study was conducted by measuring the
bubble size ratio between the top and bottom of a forth in the presence of different sizes of particles
(d = 11, 35, 71, and 119 μm). It was found that the froth stability decreases considerably as particle
1
size becomes coarser from 11 to 71 μm. However, as particle size increases further to 119 μm, the
froth stability changes little.
In the present work, a model for predicting the bubble-coarsening in a froth has also been
developed by deriving a film drainage model quantifying the effect of d on p . The model indicates
1 c
that as d increases, p increases and thereby the film thins faster and the froth becomes unstable.
1 c
72 |
Virginia Tech | 5.1 Introduction
As bubbles migrate upward in a froth phase, they coalesce with each other and become
larger. As bubbles become larger, bubble surface area becomes smaller, restricting the number of
hydrophobic particles that can be carried upward and flow into a launder. On the other hand, in
the process of the bubble coarsening, less hydrophobic particles tend to be removed more easily,
it can contribute to increase the grade of a froth product [1]. Thus, the throughput of a flotation
cell depends on bubble coarsening. Therefore, it is important to understand the basic mechanisms
of bubble coarsening and froth (or foam) stability.
A foam is thermodynamically unstable due to the large surface area, which means high
surface energy. Therefore, in pure water air bubbles collapse immediately to reduce the free energy.
It is generally known that surfactants enhance the foam stability. When a surfactant adsorbs at
air/water interface, it can reduce the thermodynamic instability by lowering the surface energy. In
the case of an ionic surfactant, it can also enhance the foam stability by increasing the electrostatic
repulsion force acting between two air/water interfaces and thereby regarding the film thinning.
It is known that a solid particle can also act as a surfactant. A froth is a three-phase foam,
where particles are present and a froth is generally more stable than a foam due to the presence of
particles [2-5]. It has been reported that the froth stability depends on particle properties, including
surface hydrophobicity [2, 3, 5-7], shape [8], concentration [2], and size [2-4].
With regard to particle size (d ) effect, several experiential studies have been reported.
1
Tang et al. [4] have found that smaller particles can increase the froth stability. They used silica
particles of d < 770 nm, which is relatively fine as compared to flotation practice. Johansson and
1
Pugh [5] conducted static and dynamic froth stability tests in the presence of fine (26 ~ 44 μm)
and coarse (74 ~ 106 μm) quartz particles with alcohol type frothers. They also found that generally
fine particles can cause higher stabilizing effect, but how the change in d affect the froth stability
1
was explained. Ip et al. [2] have measured the froth life time, as indicator of the froth stability, in
a flotation column in the presence of silica particles (40 μm < d < 150 μm) while the volume
1
faction of particles in the slurry was kept constant. They also observed that the froth stability
increases as d decreases. It was suggested that at a constant volume faction, as d decreases, the
1 1
number of particles in the slurry and the collection efficiency can increases, which will result in
73 |
Virginia Tech | the high surface coverage of a bubble by particles. They assumed that the higher surface coverage
by finer particles can enhance the froth stability. More recently, Aktas and his co-workers [3] also
have shown that fine particles benefit the froth stability through dynamic stability tests. However,
up to now, there is no theoretical model that can explain the influence of particle sizeon the froth
stability.
Fortunately, we have recently developed a froth model for predicting bubble coarsening in
a froth phase as a function of particle size. The model is based on the recognition that particles can
reduce the capillary pressure (p ), which contribute to thin a liquid film formed between bubbles
c
in a froth. As a result, particles can decrease the film drainage rate and thereby enhance the froth
stability. The model is also based on the premise that parameter p is affected by the number of
c
particles and the local capillary pressure (p ) around particles, which vary with particle size in
c,local
the film.
In the present work, the model was verified by comparison with our experiments testing
the influence of particle size on froth stability. We have conducted a series of laboratory-scale
flotation experiments with 10-5 M methyl-isobutyl carbinol (MIBC) solutions in the presence of
silica particle of different sizes (d = 11, 35, 71, and 119 μm). In each experiment, the bubble size
1
ratio (d /d ), the ratio of average bubble diameter at the top of the froth (d ) and to the same at
2,t 2,b 2,t
the base of the froth (d ), as a measure of the froth stability, was calculated.
2,b
5.2 Experiment
5.2.1 Materials and Hydrophobization of Silica Surfaces
In the present study, to investigate the influence of particle size on froth stability, four
different diameters (d = 11, 35, 71, and 119 μm) of silica spheres (Potters industries) were treated,
1
respectively. First, silica particles of identical size and a reference silica plate were cleaned by
immersing them in boiling Piranha solution (H O /H SO 3: 7 by volume) for 1 h. Then they were
2 2 2 4
rinsed in ultrapure water for 10 min and subsequently allowed to dry thoroughly in an oven at
160 °C overnight. After that, the cleaned surfaces of both the particles and the plate were
simultaneously hydrophobized by soaking them together in 10-4 M octadecyltrichlorosilane (OTS,
95% purity, Alfa Aesar)-in-toluene (99.9% purity, Spectrum Chemical) solution. After
74 |
Virginia Tech | hydrophobization, they were washed with chloroform (99.9% purity, Fisher Chemical), acetone
(99.9 % purity, Aldrich), and ultrapure water, sequentially. Due to the same conditioning time of
the particles and the plate, it may reasonable to assume that they exhibit the identical water contact
angles (θ). Hence, the θ value of the particles were simply estimated by measuring that of the plates
using a goniometer (Rame-hart instrument co.). By varying the immersion time in 10-4 M OTS-in-
toluene, the θ value was controlled. Until θ reaches 40°, by repeating the procedure stated above,
four different diameters (d = 11, 35, 71, and 119 μm) of silica spheres with θ = 40° were prepared
1
prior to experiments.
5.2.2 Experimental Procedure
In the present study, we have carried out flotation tests by means of a Denver laboratory
flotation machine. A 1.5 L of transparent glass cell was specially made to observe bubble-
coarsening phenomena in a froth phase clearly. The cell was built with flat plates to reduce optical
distortion. Prior to each experiment the cell was cleaned thoroughly with distilled water. The cell
was, then, filled with 10-5 M methyl-isobutyl carbinol (MIBC, 98% purity, Aldrich) aqueous
solution. Before generating air bubbles, the 60 g silica particles (solids concentration of 5% w/w)
of an identical size were added to the MIBC solution and stirred at the rotation speed of 900 rpm
for 5 min for wetting. After that, a froth phase was created by injecting air to the cell at the
superficial gas velocity of 1 cm/s. The formed froth was allowed to overflow and then recirculated
to the cell using a peristaltic pump. The froth height was adjusted to be 4 cm by controlling the
pulp-froth level. During the experiment, the bubble images in a froth were recorded using a high-
speed camera (Fastec imaging). We repeated the experiments with changing particle sizes.
Additionally, an experiment was conducted in the absence of particles for comparison with foam
stability.
After the flotation tests, the Sauter mean bubble diameter at the base of the froth (d ) and
2,b
the same at the top (d ) were calculated by analyzing the acquired images by means of
2,b
BubbleSEdit, image-analysis software. Finally, the bubble size ratio (d /d ) of each particle size
2,t 2,b
was obtained.
75 |
Virginia Tech | Figure 5.1 The bubble size ratio (d /d ) measured in the presence of different sizes of
2,t 2,b
particles at θ = 40°. The dotted line represents d /d in the absence of
2,t 2,b
particles.
5.3 Experimental Result
Fig ure 5.1 shows the values of experimentally measured bubble size ratio (d /d ) in the
2,t 2,b
absence of particles and in the presence of particles of different sizes at θ = 40°. It should be
reasonable to assume that the higher d /d indicates the lower froth stability. As shown, it was
2,t 2,b
observed that froth stability decreases considerably as particle size becomes coarser from 11 to 71
μm. As particle size increases further to 119 μm, the froth stability changes little. It is also
noticeable that in the present experimental range the d /d values of the froth were lower as
2,t 2,b
compared to the foam phase. It indicates that particles at θ = 40° can stabilize bubbles or thin liquid
films between the bubbles over wide range of particle sizes.
76 |
Virginia Tech | Table 5.1 Input parameters for the simulation shown in Figure 5.3.
Frother (MIBC) concentration (M) 10-5
Solids concentration (% w/w ) 5
Specific power (kW/m3) 22.8
Superficial gas rate (cm/s) 1.0
Froth height (cm) 4
Particle contact angle (°) 40
Particle Zeta potential (mV) -0.08
Bubble Zeta potential (mV) -0.03
where ρ is the density of the particle. In the present experiment, m was set to 60 g and ρ of silica
1 1 1
was 2.65 g/cm3. As shown in Figure 5.2, with increasing d , N sharply decreases as d -3.
1 1,pulp 1
The plots of probability functions (P , P , and P ) were then drawn as a function of d in
c a d 1
Figure 5.3 and input parameters used for the calculations were listed in Table 5.1. While only the
values of particle zeta potential was assumed, the bubble zeta potential was obtained from Comley
et al.’s study [9] and the others were measured in experiments.
Figure 5.4 shows the values of P P (1-P ) obtained from Figure 5.3. It was found that fine
c a d
particles are less unlikely to survive in the pulp due to lower P and P , while coarse particles are
c a
less unlikely to survive in the pulp due to higher P .
d
Finally, as shown in Figure 5.5, the values of N were obtained by substituting Eqs. (5.2)
1,film
and (5.3) into (5.1). In using (5.2), the value of c was arbitrary chosen to 384244. It was found that
N decreases as d increases. The N values shown in Figure 5.5 were used to calculate the
1,film 1 1,film
driving force for froth films in the following section.
79 |
Virginia Tech | where П and П are the disjoining pressure arising in section I and II, respectively. The values of
I II
П and П were expressed as, respectively,
I II
b
(r)(2r)dr
rc 0 (5.10)
I A
I
e A K
64C RT tanh2 s exp(H) 232 232 (5.11)
II el vw hp el 4kT 6H3 6H3
where П is the disjoining pressure due to electrostatic interaction, П is the disjoining pressure
el vw
due to van der Waals dispersion force, П is the disjoining pressure due to hydrophobic force, C
hp el
is the electrolyte concentration, R the gas constant, T the absolute temperature, e the electronic
2
charge (e = 1.6 ×10-19 C), ψ the surface potential at the air/water interfaces, k the Boltzmann’s
s
Figure 5.7 Effect of particle size on film thinning rate. The red line represents the critical
rupture thickness (H ) of a 10-5M MIBC foam film predicted from the H model
cr cr
developed in Chapter 2.
83 |
Virginia Tech | constant, κ the reciprocal Debye length, A the Hamaker constant, and K the hydrophobic
232 232
constant.
Figure 5.6 shows the effect of d on p, p , and П. In the calculations, T = 298 K was used
1 c
from the measurement in the experiment. The values of γ = 0.0724 N/m and ψ = -30 mV were
s
used as reported by Comley et al. [9] and Srinivas et al. [12], respectively. Also the values of K
232
= 2.3×10-19 J and A = 4×10-21 J were used as reported by Wang [13]. Only the value of κ-1= 30
232
nm was assumed. As shown, p increases as d becomes larger from 11 to 71μm. As d increases
c 1 1
further to 119 μm, p changes little. Since the value of П is independent of d , the change in p
c 1 c
dominate the variation of p. Note in Figure 5.5 that the variation trend of p with d is similar with
1
that of d /d .
2,t 2,b
5.4.3 The Critical Rupture Time (t ) of a Froth Film
cr
As shown in Figure 5.7, substituting the p values obtained in Figure 5.6 into Eq. (5.4), we
calculated the film thinning rates of films containing particles of d = 11, 35, 71, and 119 μm,
1
respectively. In the calculation, assuming that a bubble has a dodecahedron structure, R was
f
calculated to 250 μm from the following geometrical considerations,
R R/ 3 (5.12)
f
in which R was 0.44 mm as measured at the base of the froth (or foam) in this study. It is noticeable
that the thinning velocity of the film containing particles of d =11 μm is the slowest, which is
1
attributed to the smallest p values. As d gets coarser to 35 μm, the thinning rate becomes
1
significantly slower, mostly due to the significant decrease in p . As d becomes coarser to 71 μm,
c 1
the thinning rate becomes little slower, mostly due to small decrease in p . It was also found that
c
the thinning rates of d = 71 μm and 119 μm are almost same due to similar p values.
1
As presented in Chapter 2, in the present work, the critical rupture thickness (H ) of the
cr
foam film were predicted by incorporating the hydrophobic force into the capillary wave theory,
84 |
Virginia Tech | 2 3 2 3H 3R 2
1443H 4 2H 3 m f 0
m H HHm m H HHm H2 HHm 2(p c ) (5.13)
H 0.845H
cr m
The model prediction from Eq. (5.13) showed that H of the 10-5 M MIBC foam film of R = 250
cr f
μm is 171 nm. In the present study, it is assumed that the film in the presence of particles has the
identical H values as compared to the foam film, due to the hypothesis that the film rupture of
cr
the froth film may happen in the free film section, which is similar with a foam film. Consequently,
the values of the critical rupture time (t ) of the froth film was obtained as a function of d , as
cr 1
shown in Figure 5.8.
Figure 5.8 The critical rupture time (t ) of a froth film predicted as a function of particle
cr
size.
85 |
Virginia Tech | 5.4.4 Prediction of Bubble Size Ratio (d /d ) from t
2,t 2,b cr
In Chapter 4, a theoretical model for predicting bubble size ratio (d /d ) from t was
2,t 2,b cr
derived, as follows,
0.5
d 0.46hN
2,t exp f rupture (5.14)
d t V
2,b cr g
where h is the froth height, V is the superficial gas rate, and N is the number of the films
f g rupture
that rupture at t among 12 films forming a dodecahedron-shaped bubble. In using Eq. (5.14), the
cr
values of h, t , and V are known, N needs to be determined. N was assumed to be 5. at
f cr g rupture rupture
d = 11 and 35 μm. At d =71 and 119 μm, however, based on the recognition that possibly more
1 1
films could rupture at t due to the increase of the area occupied by free film with small particle
cr
Figure 5.9 Bubble size ratio (d /d )as a function of particle size (d ). The lines drawn
2,t 2,b 1
through the experimental data points represent the model predictions.
86 |
Virginia Tech | Figure 5.10 Effect of particle size (d ) on bubble size ratio (d /d ) and the number of
1 2,t 2,b
particles in a froth film (N ). The increase in d /d with increasing d may
1,film 2,t 2,b 1
be partially attributed to the decrease in N .
1,film
number, slightly higher value (N = 5.5) was used. As shown in Figure 5.9, the model
rupture
predictions agree with the experimental results.
In Chapter 4, it was found that two independent parameters determine the value of d /d .
2,t 2,b
One parameter is the particle number in a film (N ) and the other is the local capillary pressure
1,film
(p ) around a particle. The increase in both parameters causes the overall capillary pressure p
c,local c
to decrease, resulting in decrease in driving force p and drainage rate. It was found in the present
work that the two parameters vary with particle size as shown in Figure 5.10 and 5.11, respectively.
Note in Figure 5.10 that N decreases as a particle become coarser, which may partially attribute
1,film
to the increase in overall capillary pressure p . Note in Figure 5.11 that p decreases as a particle
c c,local
become coarser, which may also partially attribute to the increase in p .
c
87 |
Virginia Tech | Figure 5.11 Effect of particle size (d ) on bubble size ratio (d /d ) and the local capillary
1 2,t 2,b
pressure (p ) when the film ruptures. The increase in d /d with increasing
c,local 2,t 2,b
d may be partially attributed to the decrease in p .
1 c,local
As shown in Figure 5. 12, since both N and p cause p to increase with increasing
1,film c,local c
particle size, it was found that p increases with particle size
c
Figure 5.13 shows the effect of particle size on driving force p for filming drainage. This
trend can give the explanation for the effect of on particle size forth stability in terms of the
drainage rate according to the Reynolds equation.
5.5 Summary and Conclusions
In the present study, the effect of particle size (d ) on the bubble-coarsening (or froth
1
stability) has been studied by measuring the bubble size ratio between the top and bottom of a forth
in the presence of different sizes of particle (d = 11, 35, 71, and 119 μm). We found that the froth
1
88 |
Virginia Tech | Chapter 6. Conclusions and Recommendations for Future
Research
6.1 Conclusions
The primary findings and contributions presented in the dissertation are summarized as
follows.
1. A thin liquid film (TLF) confined between two bubbles in a froth phase (or in a foam)
drains by the capillary pressure (p ) created from the changes in curvature and the disjoining
c
pressure (П) created by surface forces in the films. If П is negative (attractive), the film drainage
rate and the wave motions at the air/water interfaces accelerate. When the TLFs thin to a critical
film thickness (H ), the TLF ruptures and the two bubbles become one. The capillary wave model
cr
describes the film thinning process and the wave motions using the classical DLVO theory, which
considers the repulsive double-layer force and the attractive van der Waals forces only. It has been
found in the present work that the H values predicted from the capillary wave model are
cr
substantially smaller as compared to the experientially measured values in the case of the foam
films stabilized by with weak surfactants, e.g., MIBC.
2. Based on the recognition that attractive hydrophobic force is also present in foam films
in addition to the double-layer force and the van der Waals force, by incorporating the hydrophobic
force in the capillary wave model, the author has developed a model that can predict H more
cr
accurately. The model shows that the hydrophobic force contributes to accelerate both film
thinning rate and surface corrugation growth rate and thereby may cause the film to rupture at
higher film thickness.
3. Based on the new H model, a model for predicting bubble-coarsening in a dynamic
cr
foam has been developed in the present work. The model was developed by deriving a
92 |
Virginia Tech | mathematical relation between the Plateau border area, which controls film drainage rate, and the
lamella film thickness, which controls bubble-coalescence rate. The model is able to predict the
bubble size ratio between the top and bottom of a foam as a function of surface tension (frother
dosage), aeration rate, and foam height. The model was validated using a specially designed foam
column equipped with a high-speed camera. It has been found that bubble-coarsening increases
with decreasing frother dosage and aeration rate and increasing foam height.
4. In the present study, a model for predicting bubble-coarsening in a froth (3-phase foam)
has been developed for the first time. The model was developed by deriving a film drainage model
quantifying the effect of particles on p . The parameter p is affected by the number of particles
c c
and the local capillary pressure (p ) around particles, which in turn vary with the
c,local
hydrophobicity and size of the particles in the film. Assuming that films rupture at free films, the
p corrected for the particles in lamella films has been used to determine the critical rupture time
c
(t ), at which the film thickness reaches H , using the Reynolds equation. Assuming that the
cr cr
number of bubbles decrease exponentially with froth height, and knowing that bubbles coalesce
when film drains to a thickness H , a bubble coarsening model has been developed. This first
cr
principle model is capable of predicting the bubble size ratio between the top and bottom of a froth
phase as a function of frother dosage, collector dosage (contact angle), particle size, aeration rate,
and froth height.
5. The bubble-coarsening froth model developed in the present study has been verified
using a Denver laboratory flotation cell using spherical particles of varying hydrophobicity and
size. It has been found in the present study that bubble-coarsening decreases with particle contact
angle (θ) up to θ = 70o due to a decrease in p , resulting in retarded film drainage rate. At θ > 70o,
c
on the other hand, the bubble-coarsening increases due to increased p and, hence, increased
c
drainage rate. In addition, it has been shown that bubble-coarsening decreases with particle size,
which may be attributed to decreased p and drainage rate.
c
6.2 Recommendations for Future Research
Finally, the author of this dissertation recommends the following works for future research.
1. In a flotation model, a froth recovery is given by a function of the bubble-coarsening
factor, which is the bubble size ratio between the top and bottom of a froth phase. By incorporating
93 |
Virginia Tech | VALIDATION AND APPLICATION OF A FIRST PRINCIPLE
FLOTATION MODEL
KAIWU HUANG
ABSTRACT
A first principle flotation model has been derived from the basic mechanisms involved in
the bubble-particle and bubble-bubble interactions occurring in flotation. It is a kinetic model
based on the premise that the energy barrier (E ) for bubble-particle interaction can be reduced
1
by increasing the kinetic energy (E ) for bubble-particle interaction and by increasing the
k
hydrophobic force in wetting films. The former is controlled by energy dissipation rate (), while
the latter is controlled by collector additions. The model consists of a series of analytical
equations to describe bubble generation, bubble-particle collision, attachment and detachment,
froth recovery, and bubble coalescence in froth phase. Unlike other flotation models that do not
consider role of hydrophobic force in flotation, the first principle model developed at Virginia
Tech can predict flotation recoveries and grades from the chemistry parameters such as ζ-
potentials, surface tension (), and contact angles () that may represent the most critical
parameters to control to achieve high degrees of separation efficiencies.
The objectives of the present work are to i) validate the flotation model using the
experimental data published in the literature, ii) incorporate a froth model that can predict bubble
coarsening due to coalescence in the absence of particles, iii) develop a computer simulator for a
froth model that can predict bubble coarsening in the presence of particles, and iv) study the
effects of incorporating a regrinding mill and using a stronger collector in a large copper
flotation circuit.
The model validation has been made using the size-by-class flotation rate constants (k )
ij
obtained from laboratory and pilot-scale flotation tests. Model predictions are in good agreement
with the experimental data. It has been found that the flotation rate constants obtained for
composite particles can be normalized by those for fully liberated particles (k ), which opens
max
the door for minimizing the number of flotation products that need to be analyzed using a costly
and time-consuming liberation analyzer.
A bubble coarsening froth model has been incorporated into the flotation model to predict
flotation more accurately. The model has a limitation, however, in that it cannot predict bubble-
coarsening in the presence of particles. Therefore, a new computer simulator has been developed
to predict the effects of particle size and particle hydrophobicity on bubble coarsening in froth
phase. In addition, the first principle flotation model has been used to simulate flotation circuits
that are similar to the Escondida copper flotation plant to study the effects of incorporating a re-
grinding mill and using a more powerful collector to improve copper recovery. The flotation
model developed from first principles is useful for predicting and diagnosing the performance of
flotation plants under different circuit arrangements and chemical conditions. |
Virginia Tech | ACKNOWLEDGEMENT
My most sincere thanks go to my advisor, Dr. Roe-Hoan Yoon, who introduced me to the
wonders and frustrations of scientific research. I thank him for his guidance, encouragement and
support throughout this project. I would also like to thank Dr. Gerald Luttrell for his great help in
dealing with problems in Excel-VBA. Finally, I would like to thank Dr. Greg Adel for serving on
my committee.
I would like to express my deepest appreciation to Professor Jean-Paul Franzidis, SA
Research Chair, Department of Chemical Engineering, University of Cape Town, South Africa,
for his permission to use the size-by-class flotation rate constants and liberation data presented in
the Ph.D. thesis authored by his former student Dr. Simon Welsby at the University of
Queensland, Australia. I would also like to express my appreciation to Dr. Jaakko Leppinen,
Technology Director – Mineral Processing, Outotec, for providing experimentally determined
flotation rate constants along with relevant liberation data.
I am also grateful to FLSmidth for funding continuously for this project.
I also want to express my sincere gratitude to Dr. Lei Pan, Dr. Seungwoo Park, Dr. Juan
Ma, Dr. Aaron Noble, Gaurav Soni, Zhenbo Xia, and Biao Li for helping me from time to time
by giving me valuable advice.
Finally, I would like to express my eternal gratitude to my parents for their everlasting
support and love.
iii |
Virginia Tech | Nomenclature and Symbols1
DLVO- Derjaguin and Landau, Verwey and Overbeek
MIBC- Methyl Isobutyl Carbinol
a & b- Fitting parameters in the normalized rate constant (k/k ) model
max
A- Cross-sectional area of the plateau border
A - Plateau border area at the bottom of a foam
b
A - Critical rupture PB area
cr
A- Plateau border area at the top of a foam
t
A - Hamaker constant for van der Waals interaction between two air/water interfaces
232
b- Fitting parameters in the contact angle calculation
i
C- Fitting parameter in the bubble coarsening foam model
C - Electrolyte concentration
el
d - Particle diameter
1
d - Bubble diameter
2
d - Collision diameter
12
d - Diameter of bubbles entering the froth phase
2,b
d - Diameter of bubble at the top of froth phase
2,t
e- Electronic charge
E - Energy barrier
1
E - Kinetic energy of attachment
k
E’ - Kinetic energy of detachment
k
g- Gravitational acceleration constant
ΔG- Change of Gibbs free energy
h- Froth height
f
H- Thin liquid film thickness
H - Critical rupture thickness
cr
H - Medium thickness of a thin liquid film
m
H - The closest separation distance between two bubble surfaces
0
k- Overall flotation rate constant
k - Overall rate constant for the fully-liberated particles
max
k - Flotation rate constant in the pulp phase
p
k’- Boltzmann’s constant
K - Hydrophobic force constant between bubble and particle
132
K - Hydrophobic force constant between two particles
131
K - Hydrophobic force constant between two bubbles
232
L- Rate constant ratio
m - Mass of the particle
1
m - Mass of the bubble
2
n- Number of flotation cells
N - Number of plateau borders at the base of a foam
0
1 Symbols in red are the fitting parameters in the model.
x |
Virginia Tech | N - Number of the particles
1
N - Number of the bubbles
2
N - Number of the plateau borders
pb
N - Number of films that rupture in the bubble coalescence process
rupture
p - Capillary pressure
c
P- Flotation probability
P - Probability of attachment
a
P - Probability of collision
c
P - Probability of detachment
d
P- Probability of bubble-particle aggregate surviving the froth phase
f
P- Probability of bubble-particle aggregates transferring from the pulp to the froth
t
r - Radius of the particle
1
r - Radius of the bubble
2
R- Overall flotation recovery
R - Froth recovery due to entrainment
e
R - Pulp phase recovery
p
R- Froth phase recovery
f
R - Film radius
film
R - Maximum fraction of the particles entering the froth phase
max
R - Maximum theoretical water recovery
w
R’- Gas constant
Re- Reynolds number
S - Bubble surface area flux
b
t- Retention time
t - Critical rupture time of the thin liquid film
cr
t - Drainage time
d
T- Absolute temperature
𝑢̅ - Particle RMS velocity
1
𝑢̅ - Bubble RMS velocity
2
U- Liquid drainage velocity
U - Radial velocity of the particle approaching a bubble
1
U - Velocity of a particle approaching a bubble at the critical rupture distance
Hc
V - van der Waals interaction energy
D
V - Electrostatic interaction energy
E
V - Hydrophobic interaction energy
H
V - Superficial gas rate
g
W - Work of adhesion
a
x- Fractional surface liberation
Z - Collision frequency between particles and bubbles
12
α- Fitting parameter in froth recovery calculation
β- Drag coefficient
ε- Energy dissipation rate
ε - Energy dissipation rate at bubble generation zone
b
ε ’- Liquid fraction at the base of a foam (or froth)
b
xi |
Virginia Tech | Chapter 1: INTRODUCTION
1.1 Background
1.1.1 Flotation History and Application
Flotation is undoubtedly the most important and versatile industrial process for the
separation and concentration of minerals [1]. It is an amazing separation process that enables
minerals denser than water to float to the top as bubble-particle aggregates for collection. This is
achieved by exploiting the differences in surface properties between valuable minerals and
gangue.
In 1860, the first hint that minerals can be separated from each other according to the
differences in their surface properties appeared in a patent awarded to William Haynes [2], who
claimed that sulfides could be floated by oil and non-sulfide minerals could be removed by
washing in a powdered ore.
Bessel brothers built the first commercial flotation plant in Dresden, Germany, which was
used to purify the graphite ore. The CO bubbles were applied in their plant, which were
2
generated by the reaction of lime with acid. The first flotation plant to process sulfide ores was
based on Carrie Everson’s patent. The year 1885 was important in the flotation history due to the
patents by the Bessel brothers and Carrie Everson.
True industrialization of the flotation process, from being a research topic in the lab to a
more commercially valuable tool, occurred in the early twentieth century [3]. In 1901, the
immediate problem occurred at Broken Hill, Australia, which was finding a way to recover
sphalerite fines from the waste dumps. Several flotation processes and machines were studied
there by engineers in different programs. The results of these programs were that froth flotation
was developed as an industrial process for concentrating sulfides and was used to recover zinc
from millions of tons of slime tailings.
Froth flotation was used in the United States for the first time in 1911. The first flotation
plant in the US was installed by James M. Hyde in Basin, Montana [4], who understood and
verified that the use of rougher-cleaner closed circuits could remove entrained gangue particles
from concentrates. The success of this plant was a milestone which represented flotation was
poised to take off [5].
During 1925-1960, the introduction of chemical reagents and the trend of selective
flotation brought about more widespread application of flotation process as an economic tool. To
meet the increased demand for minerals by flotation, the capacity of flotation plant increased a
lot. In the following decades, with the improvement of flotation cells and development of on-
stream analysis (OSA) systems, the mineral production from flotation increased rapidly and
accurate control of flotation circuits was achieved.
1 |
Virginia Tech | Over the past decades, flotation has been used not only in the mineral processing
industries, but also in the food industries, e.g., removing solids in butter and cheese. It is also
commonly used for removing the contaminant from water so that purification can be achieved.
Other areas in which flotation can be applied are de-inking of recycling paper, paint
manufacturing, and paper industry [5].
The flotation process as it exists today remains essentially the same as it was in Broken
Hill [6]. A feed of slurry is pre-treated with suitable reagents in a tank where it is agitated to keep
the solids in suspension before being pumped into a series of flotation cells. In the flotation cell,
the slurry is agitated by an impeller, where air is also injected to generate fine bubbles. As
bubbles rise in the slurry, they collect hydrophobic particles selectively and enter the froth phase
on the top. The forth laden with hydrophobic particles overflows the cell lip and recovered as a
concentrate.
1.1.2 Flotation Process
Flotation is a process for separating finely divided solids from each other using air
bubbles under hydrodynamic environment. The process is based on separating hydrophilic
particles from hydrophobic ones in the slurry by attaching the latter selectively onto the air
bubble surfaces [3]. Specific chemicals, which are known as collectors, are added to the slurry
before flotation to increase the differences in hydrophobicity of the minerals to be separated. In
general, the recovery and selectivity of flotation increases with increasing hydrophobicity
difference.
Thermodynamically, for bubble-particle attachment to occur, the change of Gibbs free
energy (ΔG) must be less than zero. The changes of free energy in bubble-particle attachment
can be described as the changes in the interfacial tensions at the solid-liquid, solid-air and air-
liquid interfaces [7]. By applying Young’s equation, one can obtain the following relationship
for the change of Gibbs free energy,
2
c o s 1
G
lv
(1)
where γ is the interfacial tension and θ is the contact angle at the three phase contact point. Eq.
lv
(1) shows that G < 0 when > 0, and that the higher the contact angle, the more negative ΔG
becomes.
Bubble-particle aggregates rise through the pulp since the overall density of the
aggregates are lower than the density of the slurry. At the pulp/froth interface, some of the air
bubbles loaded with various particles enter the froth phase, while others may drop off depending
on the bubble size, particle size, and bubble loading. Froth is a complex three-phase system,
which contains air bubbles, particles, and liquid films. At the bottom of a froth phase, bubbles
are separated by thick water films. The liquid films become thinner as the bubbles rise in the
froth phase, creating thin liquid films (TLFs) (or lamellae films). Three lamellae films meet at a
plateau border (PB), through which water drains. As the thickness of the lamella film between
two bubbles reach a critical thickness (H ), the TLF ruptures and two bubbles become one,
cr |
Virginia Tech | which is referred to as bubble coarsening. As bubbles become larger, the surface area on which
hydrophobic particles are attached (or ‘parked’) become smaller, forcing less hydrophobic
particles to detach and return to the pulp phase [8]. Thus, the bubble coarsening provides a
mechanism by which product grade improves.
Two mechanisms, i.e., recovery due to attachment and recovery due to entrainment,
contribute to the recovery in froth phase. The former represents true recovery based on
hydrophobic interactions, while the latter represents unwanted recoveries due to hydraulic
entrainment associated with the recovery of water or water split [9]. Fine particles with low
inertia are prone to the hydraulic entrainment.
1.1.3 Role of Modeling Flotation
Flotation is a complex physiochemical process involving solid, liquid, and gas phases;
therefore, the number of parameters affecting the process is large. These parameters can be
subdivided broadly into two groups, i.e., hydrodynamic and surface chemistry parameters. The
former includes particle size, bubble size, energy dissipation rate, etc., while the latter includes
contact angle (θ), -potential, Hamaker constants, and surface tension (γ). Many investigators
developed flotation models in the past, most of which are based on the hydrodynamic
parameters. On the other hand, the separation efficiencies of flotation are determined by control
of surface chemistry parameters rather than hydrodynamic parameters, particularly the
hydrophobicity of the particles to be separated, as has already been noted in the foregoing
section. For this reason, Virginia Tech has been developing a flotation model using both the
hydrodynamic and surface force parameters. The first principle model can, therefore, predict
both the recovery and grade for the first time.
Having a first principle model has many advantages, the most important aspects
including predictive and diagnostic capabilities. There are some parameters that are difficult to
be tested in experiment without affecting other parameters. For example, changing the pH to
study the effect of -potentials of particles also affect the -potentials of air bubbles as well as
the collector adsorption and hence the particle contact angles. A first principle model can easily
study the effects of isolated process variables one at a time and, thereby, optimize a flotation
circuit. In addition, the model-based simulator can be used design flotation plants with minimal
input from experiment.
1.2 Literature Review
1.2.1 Flotation Modeling
Flotation is a 3-phase separation process. Therefore, modelling flotation is difficult
simply because of the large number of parameters involved. Furthermore, a bubble-particle
interaction involves several different sub processes, which need to be modeled separately. A
large number of factors and interactions between them need to be considered in the flotation
model, since all of these can affect the flotation results in different ways.
3 |
Virginia Tech | Currently, there are several academic or commercial flotation models and simulators to
predict the performance of flotation circuits or a flotation unit, such as the P9 Flotation Model,
limn, USIM PAC simulator and SUPASIM flotation simulation program.
The P9 Flotation Model has been developed at the Julius Kruttschnitt Mineral Research
Center (JKMRC) over the past twenty years [6]. Both entrainment and true flotation are
considered when it comes to simulating the recovery of particles. The recovery is given as [10]:
4
R
i, j
1
P
i,
P
Sj
i,
b
Sj
R
b
fi,
R
j
fi,
1
1j R
w
R
w
E
N
E
T
N
Ri
T
i
w
R
w
(2)
where subscript i represents a particle size class, and subscript j represents a surface liberation
class. P is ore floatability, R is overall recovery, τ is residence time, R is water recovery, S is
w b
the bubble surface area flux in the pulp zone and ENT is degree of entrainment.
Drawback of this model is that the parameters used in the model must be acquired from
flotation tests data and surveying. Therefore, the collection of representative samples and good
experimental data can determine that whether a simulation of a flotation circuit/unit is a
successful one or not.
Limn software is an Excel-based application that allows the user to draw and model a
circuit. Limn software incorporates partition models for gravity separation and size separation,
which is powerful in stream simulation. In the Limn, however, the flotation recovery is also
simulated by the partition model, in which the Ep and Rho50 values are entered manually to fit
the yield and grade data from the flotation tests. The lack of effective flotation model is the main
disadvantage of the Limn software.
SUPASIM flotation simulation model was developed in the mid-1980s by Eurus Mineral
Consultants to predict plant performance from standard laboratory flotation tests [11]. The model
is based on Kelsall’s unmodified equation in which two rate constants appear,
R 1 0 0 1 e x p k
f
t 1 e x p k
s
t (3)
where Θ is slow floating fraction, t is flotation time, R is percentage recovery at time t, k is fast
f
floating rate constant and k is slow floating rate constant. Θ, k, and k are estimated from the
s f s
laboratory batch flotation tests. Based on these parameters, the continuous flotation process can
be modelled by applying the scale-up algorithms.
USIM PAC simulator is an easy to use steady-state simulation software developed by
BRGM since 1986 [12]. It contains several flotation models which can be classified as
“performance” models and “predictive” models [13]. Performance models are made for material
balance calculation and definition, while predictive models are based on kinetic approach. |
Virginia Tech | A Model with two kinetic rate constants considers that the feed is composed of three
“sub-populations”, non-floating, fast floating and slow floating. Assuming that each cell is
represented as a perfectly mixed reactor, flotation can be described as,
1 1
F F Rinf 1 1 1 (4)
fj j j j 1ks j j 1kf j
where F is flow rate of mineral j in the froth, F is flow rate of mineral j in the feed, Rinf is
fj j j
maximum possible recovery of j in the froth, φ is proportion of mineral j capable of floating and
j
which shows slow floating behavior, and τ is mean residence time.
Another predictive model incorporates a distribution of kinetic constants according to
particle size. The kinetic rate constant is calculated for each mineral j and each size class i as
below,
5
k
i, j
x 0i .5
1
x
x l
i
j
1 .5
e x p
x
2
e
x
j
i
2
(5)
where x is average size in size fraction i, α is adjustment parameter for mineral j, xl is the
i j j
largest floating particle size for mineral j, and xe is the easiest floating particle size for mineral j.
j
In perfectly mixed condition, flotation can be described as,
F f
i, j
F
i, j
R i n f 1
1
1
k
i, j
(6)
where Ffi,is flow rate of mineral j and size class i in the froth and Fi,j is flow rate of mineral j
j
and size class i in the feed.
USIM PAC also includes an entrainment model, which is based on the reference [14].
Recovery due to entrainment is shown below,
R
i, j
P
i, j
R
w
(7)
where P is recovery of mineral j in size class i in one cell and R is water recovery.
i,j w
The flotation models or simulators mentioned above all require basic input data from
flotation tests, which causes the limitation of applying these models to predict flotation
performance. However, the flotation model developed from first principles helps better
understand and predict flotation process. A flotation model considering both surface chemistry
parameters and hydrodynamic conditions was first proposed by Yoon and Mao, 1996. The model
was further improved by other researchers at the Center for Advanced Separation Technologies |
Virginia Tech | at Virginia Tech [3, 15, 16]. The first principle flotation model will be presented in the following
chapter.
1.3 Research Objectives
The objectives of the present research are to verify and to improve the first principle
flotation model developed at Virginia Tech. The main focus of the improvement will be to
incorporate the bubble coarsening model to the model, so that it can predict bubble size ratio in
the froth phase. To validate the model and simulator, the results of a series of flotation tests
conducted by other researchers and reported in the literature will be used as data base for model
verification and simulation using the first principle flotation model. The model parameters will
be adjusted so that the model prediction and simulation results will be in close agreement with
the flotation test results. Once the model has been verified, the computer simulator based on the
first-principle model will be used to predict the effects of various parameters that are critically
important in industry. The parameters to be studied will include particle size, degree of mineral
(or surface) liberation, circuit arrangement, and others.
1.4 Organization
The body of this thesis consists of five chapters:
Chapter 1 provides background information of flotation and flotation modeling. Several
flotation models or simulators are introduced in this chapter that have been developed and
applied in the mineral processing industry. This chapter also introduces the research objective of
the present work.
Chapter 2 presents the model equations for flotation and bubble coarsening in froth
phase. The models are developed from first principles, which can help understand the various
sub processes occurring in flotation.
Chapter 3 presents the results of model validation. A computer simulator is used to
validate the model against the results of a series of pilot-scale continuous flotation tests reported
in the literature and against a laboratory-scale flotation tests.
Chapter 4 presents the simulation results without experimental validation. The flotation
model is used to study the effects of various process parameters such as particle size, contact
angle, forth height, ζ-potential and others. Furthermore, the simulator is used to see the effect of
changing flotation circuits as a means for optimization.
Chapter 5 summarizes the results of the work presented in the foregoing chapters and
suggests future work.
6 |
Virginia Tech | Chapter 2: FLOTATION MODEL BASED ON FIRST PRINCIPLES
2.1 Framework
2.1.1 Pulp Phase Recovery
Flotation Kinetics
Flotation process can be modeled as a first-order rate equation [17, 18],
7
d N
d t
1
k N
1
(8)
in which k is the rate constant and N is number of particles in a cell. Under a steady condition, k
1
can be determined by [19],
k
1
4
S
b
P (9)
where P is probability of flotation and S is bubble surface area flux.
b
Generally, P is composed of four parts as shown below,
P P P 1P P (10)
a c d t
where P represents the probability of attachment, P the collision probability, P the probability
a c d
of detachment in pulp phase, and P represents the probability of bubble-particle transfer at the
t
pulp-froth interface.
In the past, flotation processes were often modeled as a first-order process with a single
rate constant for the sub-processes of pulp and froth phase recoveries, which is overall rate
constant (k). In the current model, however, the two sub-processes are considered separately and
subsequently combined to obtain an overall flotation rate.
Basically, flotation process should be considered a second-order process in that its rate
should depend on concentration of particles (N ) and bubbles (N ). If one assumes N >>N or N
1 2 2 1 2
remains constant during flotation, the process may be considered a pseudo first-order process and
may be represented as
d N
d t
1 k
p
N
1
Z
1 2
P (11)
where k is the rate constant in the pulp phase, and Z is the collision frequency. Z can be
p 12 12
calculated as below [20], |
Virginia Tech | 8
Z
1 2
2 3 / 2 1 / 2 N
1
N
2
d 21
2
u 21 u 22 (12)
which was derived originally by Abramson [21] based on the assumption that particle velocities
were independent of fluid flow. In Eq. (12), d is the collision diameter (sum of radii of bubbles
12
and particles), and 𝑢̅ and 𝑢̅ are the root-mean-square (RMS) velocities of the particles and
1 2
bubbles, respectively. Substituting Eq. (11) into Eq. (12), one can obtain,
d N
d t
1 2 3 / 2 1 / 2 N
1
N
2
d 21
2
u 21 u 22 P (13)
From Eqs. (11) and (13), one can obtain,
k
p
Z
1
N
P2
1
(14)
Bubble Generation Model
The diameters of bubbles (d ) were calculated using the bubble generation model derived
2
by Schulze [22],
d
2
2 .1
3
1
0b
lv.6
6
0 .6
(15)
where γ is the surface tension of the water in a flotation cell, ρ is the density of the water, and
lv 3
ε is the energy dissipation rate in the bubble generation zone. In the present work, it is assumed
b
that air bubbles are generated at the high energy dissipation zone in and around the rotor/stator
assembly, which has 15-times larger energy dissipation rate than the mean energy dissipation
rate (ε) of a flotation cell [22].
RMS Velocities
The RMS velocity of the particles is calculated using the following empirical relation
[20],
2/3
4/9d7/9
u 0.4 1 1 3 (16)
1 1/3
3
where ε is energy dissipation rate, d particle diameter, ν kinematic viscosity of water, ρ is
1 1
particle density, and ρ is the density of water.
3
On the other hand, the RMS velocity for bubbles is calculated using the following
equation [23], |
Virginia Tech | K rr
V 132 1 2 (20)
H 6H r r
0 1 2
where H is the closest separation distance between bubble of radius r and particle of radius r
0 2 1
in water, and K is the hydrophobic force constant between the bubble and the particle [27]. It
132
has been shown that hydrophobic interaction between hydrophobic solid surface and bubble
surface with different contact angles can be predicted using the combining law [19],
K K K (21)
132 131 232
where K is the hydrophobic force constant between two particles in a medium and K is the
131 232
hydrophobic force constant between two air bubbles in a medium [28]. Figure 2.1 [29] shows the
relationship between K and particle advancing contact angle, from which one can clearly see
131
that K increases with the increase of the contact angle. In the present work, the values of K
131 131
has been determined using the data presented in Figure 2.1.
To calculate P using Eq. (18), it is necessary to know the value of E . The kinetic energy
a k
may be calculated using following relation,
10
E
k
0 . 5 m
1
U
H c
2 (22)
where m is the mass of the particle, and U is the velocity of the particle approaching a bubble
1 Hc
surfaces at the critical rupture distance (H ). In the present work, the following equation has been
c
used to calculate U ,
Hc
U
H c
U
1
/ (23)
where 𝑈 is radial velocity of the particle approaching a bubble and is the drag coefficient in
1
the boundary layer of the bubble [30]. The values of U and have been determined as described
1
previously [31],
b) Probability of Collision (P )
c
In the present work, Eq. (24) is used to determine the probability of collision, which is
shown below,
3 3 Re d
P c tanh2
2 1 16 10.249Re0.56 d1
2
(24)
where d and d are bubble and particle diameters, respectively, and Re is the Reynolds number.
1 2
Eq. (12) represents a hard-core collision model, that is, bubble-particle collision occurs when the
two macroscopic spheres approach each other within the collision radius (r ), which is effective
12
only under extremely turbulent conditions. For quiescent flow, however, the collision is affected |
Virginia Tech | by the streamlines around bubbles. The truth may lie in between. Therefore, one may get correct
Z by multiplying P [31].
12 c
c) Probability of Detachment (P )
d
The probability of detachment is calculated using the following expression [19]
11
P
d
e x p
W
a
E
'k
E
1
(25)
where W is the work of adhesion, and 𝐸′ is the kinetic energy of detachment. W can be
a 𝑘 a
obtained from the following relation,
21 1 c o s 2 W
a
lv
r (26)
where γ is the surface tension of water, r is the radius of the particle, and θ is the contact angle.
lv 1
By using Eq. (25), 𝐸′ can be calculated using the following relation [15],
𝑘
' 0 .5
1
1 2
/ 2 E
k
m d d (27)
where is the energy dissipation rate and is the kinematic viscosity.
Pulp Recovery Calculation
In a mechanically-agitated cell, the pulp phase recovery, R , can be calculated as below,
p
k t
R p (28)
p 1k t
p
in which k is the flotation rate constant in the pulp phase. Eq. (28) is applicable for perfectly
p
mixed flotation cells as is the case with a mechanically-agitated individual cell in a flotation
bank.
For plug-flow reactors, one may use the relation below to calculate R ,
p
R
p
1 e k tp (29)
2.1.2 Froth Phase Recovery
It is well kwon that the forth recovery accounts for two independent mechanisms, i.e.
recovery due to attachment and recovery due to entrainment. Fine particles are recovered by
entrainment, while coarse and hydrophobic particles are recovered by attachment. Thus, the
overall froth recovery (R) can be written as follows [16],
f |
Virginia Tech | 12
R
f
R
f
R
e
m ax
e x p (30)
in which R is the maximum fraction of the particles entering the froth phase that is recovered
max
into a launder, is the retention time of air in the froth, and R is the recovery of particles due to
f e
entrainment. Former researchers [15] have developed equations for calculating of R and τ. In
e f
addition, it can be readily seen that R should vary with bubble coalescence as follows,
max
R
m ax
S
S
t
b
d
d
2
2
,b
,t
(31)
where S and S are the bubble surface areas on the top and bottom of a froth phase, respectively,
t b
while d and d are the bubble diameters at the top and bottom, respectively.
2,t 2,b
2.1.3 Overall Recovery
Figure 2.2 shows the diagram to calculate the overall flotation recovery [3], in which R
p
is the recovery in the pulp phase and R is the recovery in the froth phase. According to the
f
diagram, one can readily find that the overall recovery, R, can be calculated using the equation
below,
R
R
p
R
R
f
p
R
1
f
R
p
(32)
Figure 2.2: Diagram of mass balance of materials around a flotation cell. Soni, G.,
Development and Validation of a Simulator based on a First-Principle Flotation
Model. 2013, Virginia Tech. Used under fair use, 2015. |
Virginia Tech | Note here that one can calculate the overall rate constant (k) using the equation below
under a perfectly mixed flotation cell,
13
k k
p
R
f
(33)
By combining Eqs. (28), (32) and (33), one can get the relationship between overall
flotation rate constant (k) and overall flotation recovery (R),
k t
1
R
R
(34)
which is the same as the overall rate constant equation under perfectly mixed condition
developed by Levenspiel [32].
2.2 Bubble Coarsening Model
In the flotation froth, air bubbles become larger and larger with the increase of froth
height, which is due to bubble coalescence phenomenon. Bubble coalescence has significant
impacts on the flotation recoveries and grades, as it causes the bubble surface area to decrease
along the vertical direction, which forces some particles to drop back to the pulp phase. In the
calculation of froth recovery due to attachment, one needs to determine R , which depends on
max
the bubble size ratio (d /d ). In the past, an assumed value of bubble size ratio was used in Eq.
2,t 2,b
(31). At present, a bubble coarsening model has been incorporated in the current flotation model
to predict the bubble size ratio accurately.
2.2.1 Bubble Coarsening Foam Model
In this section, a 2-phase bubble coarsening model will be introduced, which does not
consider the effects of particle size and particle hydrophobicity on the stability of the froth.
Therefore, the model to be described below may be applicable for foam rather than froth.
Eq. (35) describes the drainage of liquid in a foam,
1 1 A
U gA (35)
A x
where U is the drainage rate; g is the gravitational acceleration; µ, ρ, and γ are the dynamic
viscosity, density, and surface tension of water, respectively; A is the cross-sectional area of the
plateau border (PB), and x is the distance from the top of the foam.
At a steady state, the downward liquid drainage velocity (U) should be equal to the
upward superficial gas velocity (-V ),
g
U V
g
(36) |
Virginia Tech | Combining Eqs. (35) and (36) and then integrating from the top to the bottom of a foam,
one can obtain,
14
A
t
V
g
g
t a n
t a n 1
V
g A
g
b
h
2
f
g V
g (37)
where A and A are PB areas at the top and bottom of a foam, respectively, and h is the foam
t b f
height.
In a dry foam, it is reasonable to assume that the number of PBs (N ) is proportional to
pb
the number of bubbles, which can be calculated by dividing the cross-sectional area (S) of a foam
(or froth) column by bubble size, i.e., 4s/πd2. One can then write the following relation,
N
N
p
p
b ,t
b ,b
4
4
s
s
/
/
d
d
22
,t
22
,b
d
d
2
2
,b
,t
2
(38)
where N and N , are the numbers of PB at the top and bottom of a foam, respectively, and d
pb,t pb b 2,t
and d are the corresponding bubble sizes.
2,b
As a foam drains, A decreases with time, or the foam becomes drier. At the same time,
the thickness (H) of the lamella films will also become thinner. As H becomes smaller, there will
be a critical point where a lamellar film will rupture instantaneous, which is referred as critical
rupture thickness of the liquid film (H ). Accordingly, it may be reasonable to suggest that
cr
bubbles begin to coalesce when A reaches A . As bubbles coalesce, N will decrease. In the
cr pb
present work, the changes of N is represented by the following relation,
pb
N
p b
N
0
e x p
C
A
A
c r
(39)
where N is the number of PBs at the base of a foam, and C is an adjustable parameter. Eq. (39)
0
shows that N decreases exponentially with the square root of A .
pb cr
Substituting Eq. (39) into Eq. (38), one obtains the following relation,
d 1 1
2,b expC A (40)
d cr A A
2,t b t
which shows that bubble size ratio, or bubble coarsening, can be predicted if the values of A , A
b t
and A are known.
cr |
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