2726 XXXI International Mineral Processing Congress 2024 Proceedings/Washington, DC/Sep 29–Oct 3
air bubble 2 and water 3, and θ is the contact angle. From
Eq. [10], one finds that
E G^ hh
h 1 0 =
=
which represents the kinetic energy barrier for contact angle
formation.
A Simplified Flotation Model
Flotation is a kinetic process, which can be represented as
follows (Gupta and Yoon, 2024), dN1
dt
dN
k N N
p
1
1 2 =-(13)
in which N1 and N2 are the number densities of particles
1 and bubbles 2, respectively, and kp is the rate constant,
expc- k Z El
W
E
E
*
p
k
a
k
12
1 =m (14)
in which Z12 *representing a corrected bubble-particle col-
lision frequency and Wa/E'
k a simplified detachment prob-
ability, while the exponential term is the probability of
attachment.
As an air bubble approaches a mineral surface, the
potential energy (G) changes as a function of the closest
separation distance (h) between the two macroscopic sur-
faces as shown in Figure 3. The changes in the potential
energy may be given as follows,
1h
h
23 13 12
23
0
c c
c i
DG
P^hhdh
=-+
=-
=
3
^cos
^c h
#
(15)
Figure 2. (a) Force vs. time curves obtained in the TLFs of water confined between air bubble and MTCS-coated silica surface
with θr =40° (b) corresponding spatiotemporal film profiles (c) Π(h) isotherms extracted from force measurement Пd, Пe,
Пh and П represent the disjoining pressures due to the van der Waals, EDL, hydrophobic, and the sum of the three forces,
respectively -d. Free energy isotherm using Eq. [12] showing that E1 occurs at Π(h) =0
air bubble 2 and water 3, and θ is the contact angle. From
Eq. [10], one finds that
E G^ hh
h 1 0 =
=
which represents the kinetic energy barrier for contact angle
formation.
A Simplified Flotation Model
Flotation is a kinetic process, which can be represented as
follows (Gupta and Yoon, 2024), dN1
dt
dN
k N N
p
1
1 2 =-(13)
in which N1 and N2 are the number densities of particles
1 and bubbles 2, respectively, and kp is the rate constant,
expc- k Z El
W
E
E
*
p
k
a
k
12
1 =m (14)
in which Z12 *representing a corrected bubble-particle col-
lision frequency and Wa/E'
k a simplified detachment prob-
ability, while the exponential term is the probability of
attachment.
As an air bubble approaches a mineral surface, the
potential energy (G) changes as a function of the closest
separation distance (h) between the two macroscopic sur-
faces as shown in Figure 3. The changes in the potential
energy may be given as follows,
1h
h
23 13 12
23
0
c c
c i
DG
P^hhdh
=-+
=-
=
3
^cos
^c h
#
(15)
Figure 2. (a) Force vs. time curves obtained in the TLFs of water confined between air bubble and MTCS-coated silica surface
with θr =40° (b) corresponding spatiotemporal film profiles (c) Π(h) isotherms extracted from force measurement Пd, Пe,
Пh and П represent the disjoining pressures due to the van der Waals, EDL, hydrophobic, and the sum of the three forces,
respectively -d. Free energy isotherm using Eq. [12] showing that E1 occurs at Π(h) =0