948 XXXI International Mineral Processing Congress 2024 Proceedings/Washington, DC/Sep 29–Oct 3
the flotation of coarse particles, whose kij values are low due
to low composite contact angles (i
ij )(Gupta et al., 2022).
A solution to his problem may be to use stronger collectors
to increase contact angles.
It is, therefore, the objective of the present investiga-
tion to test stronger collectors, known as Super Collectors,
developed at Virginia Tech to increase
ij i of composite
particles and, hence, to improve the recovery of coarse par-
ticles. These reagents are designed to enhance the hydro-
phobicity of a target mineral (e.g., chalcopyrite) well
beyond the level that can be achieved using a conventional
collector (e.g., xanthate) alone. The benefits of using the
new reagents have been determined using a flotation simu-
lator based on first principles (Gupta &Yoon, 2024). The
simulator allows one to determine flotation rate constants
(kij) of composite particles from the values of
ij i deter-
mined from liberation analysis. The kij values determined
in this manner may be referred to as intrinsic rate constants,
as opposed to those determined experimentally in labora-
tory flotation experiments using 10–15 times higher energy
dissipation rates (f) but at substantially shorter flotation
times. A distinct advantage of using intrinsic kij for plant
design is that there is no need to assume scale-up factors.
Various design parameters, e.g., retention time, cell volumes
and numbers, f ,froth height, etc., are input variables for
designing a plant by simulation. A user-friendly simulator
has been designed to make it easier for process engineers to
design a flotation circuit and for plant operators to opti-
mize plant operations by varying both hydrodynamic and
surface chemistry parameters as input.
MODEL
Flotation begins in the pulp phase where air bubbles are
introduced to selectively collect hydrophobic particles and
enter the froth phase due to increased buoyancy. The pro-
cess can be modeled using a second-order rate equation as
follows,
dt
dN1 N N
Z P
p 1 2
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 the prduct of three sub-processes,
i.e., (),P P P P 1
c a d =-in which the subscripts c, a, and d
represent the collision, attachment, and detachment sub-
processes, respectively. Abrahamson (1975) derived an
expression for predicting collision frequency (Z12) under
extremely turbulent conditions as follows,
Z N d u u 5N
12 1 2 12
2
1
2
2
2 =+_i [2]
in which d12 is collision diameter, and u
1
2 and u
2
2 are
the RMS velocities of particles and bubbles, respectively.
The RMS velocities were calculated by assuming that the
energy dissipation rate near the impeller zone was 15-times
higher than the collection zone of the pulp phase and that
the volumetric fraction of the impeller and collection zones
were o.1 and 0.9, respectively.
From Eqs. [1] and [2], one obtains kp as follows,
(k N N
Z
P Pa Pd) 1
p c
1 2
12 =-[3]
which may be simplified to obtain,
()k Z P P 1 *
p a d 12 =-[4]
The first term of Eq. [4], i.e., /N Z Z P N *
c 12 12 1 2 =,repre-
sents the collision frequency normalized by N N
1 2 ,while
the second term, i.e., P P 1
a d -^h, represents a selection
function, representing the fraction (Pa) of particles collid-
ing with a bubble and a fraction (1 – Pa) of the attached
particles not being detached in the pulp phase. Pc (1)
is effectively a correction factor for Z12 which overpredicts
collision in a flotation cell.
Substituting the Pa and Pd functions, both in exponen-
tial forms (Luttrell and Yoon, 1992 Yoon and Mao, 1996),
into Eq. [4], one obtains kp as follows,
exp k Z El
E E W
*
p
k
h a
12
1 =-
+-^h F [5]
which is of the same form as the simple Arrhenius equa-
tion. In Eq. [5], E1 is the energy barrier for bubble-particle
attachment, which can be determined from the surface
forces present in wetting films, and Eh is the hydrodynamic
resistance to film thinning. Both E1 and Eh present sig-
nificant energy barriers to the particle-bubble attachment
step, while Wa (=cos R 1
LV 1
2 r ih2) -^represents the work
required to detach a particle of radius R1 and contact angle
θ from an air bubble with a surface tension γLV (Yoon
and Mao, 1996). One can see that Wa should increase
substantially with θ, which provides a rationale for using
Super Collectors to increase the recovery of coarse particle
flotation.
The bubble-particle aggregates formed in the pulp
phase of a flotation cell enter the froth phase. The lamella
films between two air bubbles drain and rupture at a criti-
cal rupture thickness to become a larger bubble, causing
less hydrophobic particles to drop off from the bubble due
to the limited ‘parking’ area. Thus, froth phase recoveries
can be determined one can predict the bubble coarsening
the flotation of coarse particles, whose kij values are low due
to low composite contact angles (i
ij )(Gupta et al., 2022).
A solution to his problem may be to use stronger collectors
to increase contact angles.
It is, therefore, the objective of the present investiga-
tion to test stronger collectors, known as Super Collectors,
developed at Virginia Tech to increase
ij i of composite
particles and, hence, to improve the recovery of coarse par-
ticles. These reagents are designed to enhance the hydro-
phobicity of a target mineral (e.g., chalcopyrite) well
beyond the level that can be achieved using a conventional
collector (e.g., xanthate) alone. The benefits of using the
new reagents have been determined using a flotation simu-
lator based on first principles (Gupta &Yoon, 2024). The
simulator allows one to determine flotation rate constants
(kij) of composite particles from the values of
ij i deter-
mined from liberation analysis. The kij values determined
in this manner may be referred to as intrinsic rate constants,
as opposed to those determined experimentally in labora-
tory flotation experiments using 10–15 times higher energy
dissipation rates (f) but at substantially shorter flotation
times. A distinct advantage of using intrinsic kij for plant
design is that there is no need to assume scale-up factors.
Various design parameters, e.g., retention time, cell volumes
and numbers, f ,froth height, etc., are input variables for
designing a plant by simulation. A user-friendly simulator
has been designed to make it easier for process engineers to
design a flotation circuit and for plant operators to opti-
mize plant operations by varying both hydrodynamic and
surface chemistry parameters as input.
MODEL
Flotation begins in the pulp phase where air bubbles are
introduced to selectively collect hydrophobic particles and
enter the froth phase due to increased buoyancy. The pro-
cess can be modeled using a second-order rate equation as
follows,
dt
dN1 N N
Z P
p 1 2
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 the prduct of three sub-processes,
i.e., (),P P P P 1
c a d =-in which the subscripts c, a, and d
represent the collision, attachment, and detachment sub-
processes, respectively. Abrahamson (1975) derived an
expression for predicting collision frequency (Z12) under
extremely turbulent conditions as follows,
Z N d u u 5N
12 1 2 12
2
1
2
2
2 =+_i [2]
in which d12 is collision diameter, and u
1
2 and u
2
2 are
the RMS velocities of particles and bubbles, respectively.
The RMS velocities were calculated by assuming that the
energy dissipation rate near the impeller zone was 15-times
higher than the collection zone of the pulp phase and that
the volumetric fraction of the impeller and collection zones
were o.1 and 0.9, respectively.
From Eqs. [1] and [2], one obtains kp as follows,
(k N N
Z
P Pa Pd) 1
p c
1 2
12 =-[3]
which may be simplified to obtain,
()k Z P P 1 *
p a d 12 =-[4]
The first term of Eq. [4], i.e., /N Z Z P N *
c 12 12 1 2 =,repre-
sents the collision frequency normalized by N N
1 2 ,while
the second term, i.e., P P 1
a d -^h, represents a selection
function, representing the fraction (Pa) of particles collid-
ing with a bubble and a fraction (1 – Pa) of the attached
particles not being detached in the pulp phase. Pc (1)
is effectively a correction factor for Z12 which overpredicts
collision in a flotation cell.
Substituting the Pa and Pd functions, both in exponen-
tial forms (Luttrell and Yoon, 1992 Yoon and Mao, 1996),
into Eq. [4], one obtains kp as follows,
exp k Z El
E E W
*
p
k
h a
12
1 =-
+-^h F [5]
which is of the same form as the simple Arrhenius equa-
tion. In Eq. [5], E1 is the energy barrier for bubble-particle
attachment, which can be determined from the surface
forces present in wetting films, and Eh is the hydrodynamic
resistance to film thinning. Both E1 and Eh present sig-
nificant energy barriers to the particle-bubble attachment
step, while Wa (=cos R 1
LV 1
2 r ih2) -^represents the work
required to detach a particle of radius R1 and contact angle
θ from an air bubble with a surface tension γLV (Yoon
and Mao, 1996). One can see that Wa should increase
substantially with θ, which provides a rationale for using
Super Collectors to increase the recovery of coarse particle
flotation.
The bubble-particle aggregates formed in the pulp
phase of a flotation cell enter the froth phase. The lamella
films between two air bubbles drain and rupture at a criti-
cal rupture thickness to become a larger bubble, causing
less hydrophobic particles to drop off from the bubble due
to the limited ‘parking’ area. Thus, froth phase recoveries
can be determined one can predict the bubble coarsening