2568 XXXI International Mineral Processing Congress 2024 Proceedings/Washington, DC/Sep 29–Oct 3
conventional sulfide mineral collector, e.g., potassium amyl
xanthate (KAX), for the flotation of a low-grade porphyry
copper ore. The results were used to predict the performance
of the reagents in plant operations using a flotation model
derived from first principles. The model can predict grades
vs. recovery curves that are useful for evaluating plant per-
formances (Huang et al., 2022 Gupta et al., 2022, 2023).
THEORY
Flotation begins with air bubbles colliding with mineral
particles to form bubble-particle aggregates in the pulp
phase of a flotation cell. The rate of the aggregate formation
may be represented as follows,
dt
dN1
N1 N2
Z P
p
12
=-k
=-
[1]
in which N1 and N2 are the number densities of particles
and bubbles in the pulp phase, respectively, Z12 is the col-
lision frequency, P is the probability of flotation, and kp
is the rate constant. P is known as the flotation probabil-
ity and is given by (),P P P P 1
c a d =-where Pc, Pa, and
Pd represent the probability of collision, attachment, and
detachment, respectively.
Abrahamson (1975) derived a model for the collision
frequency (Z12) under turbulent conditions as follows,
Z N d u u 5N
12 1 2 12
2
1
2
2
2 =+_i [2]
where, d12 is the collision diameter and the u
1
2 and
u
2
2 are the RMS velocity fluctuations of particles and
bubbles, respectively. Pc is used to as a correcting factor
for the cell less turbulent hydrodynamics and thus, colli-
sion frequency for flotation conditions can be defined as
Z Z P o
c 12 12 =.Therefore, the pulp phase rate constant (kp)
can be described as,
()k N N
Z
P P 1
p
o
a d
1 2
12 =-[3]
The model for bubble-particle attachment (Pa) was
given by Luttrell and Yoon (1992) as follows,
expc- Pa E
E
k
1 =m [4]
in which E1 is the energy barrier that is a function of surface
forces acting in wetting films and Ek is the kinetic energy
for bubble-particle interaction at the critical rupture thick-
ness of the wetting film (hc). The probability of detachment
(Pd) was suggested by Yoon and Mao (1996) as follows,
expd- P
E
W E
'd
k
a 1 =
+
n [5]
where, Wa is the work of adhesion for a bubble and a par-
ticle of radius R1.
One can also rewrite Eq. [3] as (Gupta and Yoon,
2024),
exp k Z El
E Ehh Wa
*
p
k
12
1 =-
+-^F [6]
in which, Eh is the hydrodynamic resistance to film thin-
ning. Ek is the kinetic energy due to the turbulence in the
flotation cell.
The process of bubble-particle attachment step involves
thinning of the wetting film formed when an air bubble
approaches a mineral surface. Initially, the film thinning is
driven by the capillary pressure (pc) created by the changes
in the bubble curvature. As the film thins further to
~250 nm, film thinning is controlled by the surface forces
in the wetting film. The surface forces operating in flotation
include the van der Waals (vdW), electrical double-layer
(EDL), and hydrophobic (HP) forces. The balance between
these forces controls the disjoining pressure (P(h)) in the
wetting film as follows,
()((h h) h)
vdW EDL HP P P P P =++[7]
which can be used to determine change in the free energy
(DG) of bubble-particle interaction,
()2rrdr G h) R R
R R2
h 2r
1
1h
r
r
LV
1 2
1
0
c i
P( =
+
=-
3
=
=
^cos
c m #
[8]
The G(h) isotherm varies with h, as shown, and the film
should rupture at P =0, one can determine the energy bar-
rier (E1) for bubble-particle attachment as follows (Gupta
et al., 2022),
)E G( h 2rrdr
r
r
1 0 0
=3
P ==
=#[9]
As evident from the Eqs. [8]-[9], the necessary
condition for bubble-particle attachment is q 0.
Thermodynamically, the larger the contact angle, the bub-
ble-particle interaction becomes more favoured. One way
of achieving a higher contact angle is to modify the interfa-
cial tensions in Young’s equation,
cos 13 12 i c23
c c
=
-
[10]
in which g13, g12, and g23 are the interfacial tensions
between solid 1, liquid (water) 2, and air 3. In the pres-
ent work, novel reagents known as Super Collectors (SC)
have been developed to modify the interfacial tensions and
improve the contact angles to q 150° (Gupta and Yoon,
conventional sulfide mineral collector, e.g., potassium amyl
xanthate (KAX), for the flotation of a low-grade porphyry
copper ore. The results were used to predict the performance
of the reagents in plant operations using a flotation model
derived from first principles. The model can predict grades
vs. recovery curves that are useful for evaluating plant per-
formances (Huang et al., 2022 Gupta et al., 2022, 2023).
THEORY
Flotation begins with air bubbles colliding with mineral
particles to form bubble-particle aggregates in the pulp
phase of a flotation cell. The rate of the aggregate formation
may be represented as follows,
dt
dN1
N1 N2
Z P
p
12
=-k
=-
[1]
in which N1 and N2 are the number densities of particles
and bubbles in the pulp phase, respectively, Z12 is the col-
lision frequency, P is the probability of flotation, and kp
is the rate constant. P is known as the flotation probabil-
ity and is given by (),P P P P 1
c a d =-where Pc, Pa, and
Pd represent the probability of collision, attachment, and
detachment, respectively.
Abrahamson (1975) derived a model for the collision
frequency (Z12) under turbulent conditions as follows,
Z N d u u 5N
12 1 2 12
2
1
2
2
2 =+_i [2]
where, d12 is the collision diameter and the u
1
2 and
u
2
2 are the RMS velocity fluctuations of particles and
bubbles, respectively. Pc is used to as a correcting factor
for the cell less turbulent hydrodynamics and thus, colli-
sion frequency for flotation conditions can be defined as
Z Z P o
c 12 12 =.Therefore, the pulp phase rate constant (kp)
can be described as,
()k N N
Z
P P 1
p
o
a d
1 2
12 =-[3]
The model for bubble-particle attachment (Pa) was
given by Luttrell and Yoon (1992) as follows,
expc- Pa E
E
k
1 =m [4]
in which E1 is the energy barrier that is a function of surface
forces acting in wetting films and Ek is the kinetic energy
for bubble-particle interaction at the critical rupture thick-
ness of the wetting film (hc). The probability of detachment
(Pd) was suggested by Yoon and Mao (1996) as follows,
expd- P
E
W E
'd
k
a 1 =
+
n [5]
where, Wa is the work of adhesion for a bubble and a par-
ticle of radius R1.
One can also rewrite Eq. [3] as (Gupta and Yoon,
2024),
exp k Z El
E Ehh Wa
*
p
k
12
1 =-
+-^F [6]
in which, Eh is the hydrodynamic resistance to film thin-
ning. Ek is the kinetic energy due to the turbulence in the
flotation cell.
The process of bubble-particle attachment step involves
thinning of the wetting film formed when an air bubble
approaches a mineral surface. Initially, the film thinning is
driven by the capillary pressure (pc) created by the changes
in the bubble curvature. As the film thins further to
~250 nm, film thinning is controlled by the surface forces
in the wetting film. The surface forces operating in flotation
include the van der Waals (vdW), electrical double-layer
(EDL), and hydrophobic (HP) forces. The balance between
these forces controls the disjoining pressure (P(h)) in the
wetting film as follows,
()((h h) h)
vdW EDL HP P P P P =++[7]
which can be used to determine change in the free energy
(DG) of bubble-particle interaction,
()2rrdr G h) R R
R R2
h 2r
1
1h
r
r
LV
1 2
1
0
c i
P( =
+
=-
3
=
=
^cos
c m #
[8]
The G(h) isotherm varies with h, as shown, and the film
should rupture at P =0, one can determine the energy bar-
rier (E1) for bubble-particle attachment as follows (Gupta
et al., 2022),
)E G( h 2rrdr
r
r
1 0 0
=3
P ==
=#[9]
As evident from the Eqs. [8]-[9], the necessary
condition for bubble-particle attachment is q 0.
Thermodynamically, the larger the contact angle, the bub-
ble-particle interaction becomes more favoured. One way
of achieving a higher contact angle is to modify the interfa-
cial tensions in Young’s equation,
cos 13 12 i c23
c c
=
-
[10]
in which g13, g12, and g23 are the interfacial tensions
between solid 1, liquid (water) 2, and air 3. In the pres-
ent work, novel reagents known as Super Collectors (SC)
have been developed to modify the interfacial tensions and
improve the contact angles to q 150° (Gupta and Yoon,