2442 XXXI International Mineral Processing Congress 2024 Proceedings/Washington, DC/Sep 29–Oct 3
established a first-order exponential relationship for the
recovery increase (cumulative) as a function of time, t.
exp^-kth@ R th R 1 =-^6 (1)
Within Equation (1), R∞ is the maximum achievable recov-
ery, k is the flotation rate constant (deterministic), and t is
the cumulative flotation time.
Although the Single Flotation Rate (SFR) model of
Equation (1) has been claimed as a good approximation to
represent kinetic responses, its applicability has proven to
be limited due to the heterogeneity of particle properties
in flotation (Ofori et al., 2014). A distribution of rate con-
stants, f(k), was proposed by Imaizumi and Inoue (1963) as
a generalization of Equation (1), from which different prob-
ability density functions, such as Rectangular, Triangular,
Gamma, Exponential, Normal, among others, have been
proposed. The former corresponds to a uniform f(k) in the
range 0 to kmax (Huber-Panu et al., 1976 Klimpel, 1980),
and has proven to have satisfactory results, keeping only
two parameters. The Rectangular model for R(t) is given by
Equation (2):
exp^-k
R R k t
th
1
1
max
max =-
-
3 ^th F (2)
The Gamma distribution proposed by Imaizumi
and Inoue (1963) has proven to be suitable in describ-
ing the heterogeneity of the flotation process (Murhula
et al., 2022). This distribution generalizes Normal and
Exponential density functions. The Gamma model has an
additional parameter with respect to the Rectangular and
SFR models, which allows for higher flexibility to simulta-
neously represent particles with high and low floatability.
Equation (2) represents the Gamma f(k), in which aG and
kG are shape and scale parameters, respectively, and Γ is the
Gamma function.
f k
k e
G G
a
a
k
k 1
G
G
G C^a =
-
-^kh
h
(3)
Equation (4) represents the R(t) model, assuming a Gamma
f(k).
R R k thaGE 1 1
G
=-+3 ^th ^1 (4)
Although higher reactions orders have also been proposed
to justify deviations with respect to Eq. (1) (Rao, 2023),
this approach has proven to be equivalent to first-order
systems with an apparent f(k), as reported by Vinnett &
Waters (2020).
Even though kinetic modelling may exhibit suitable
fitting under specific experimental conditions, uncertain-
ties may occur due to non-standard flotation protocols and
inadequate experimental designs (Sandoval-Zambrano &
Montes-Atenas, 2012 Napier-Munn, 2012). Improvements
in kinetic characterization have been continuously reported
in literature, with focus on sample representativeness, auto-
mation in the concentrate removals, decrease in data vari-
ance, and others (Amelunxen et al., 2018 Napier-Munn,
2012 Ross, 2019). These advancements have been relevant
to define standard flotation tests, a topic that has been
extensively discussed among practitioners. However, the
determination of flotation intervals for the concentrate
removals and its effect on the R∞-f(k) estimations have not
received sufficient attention, only being assessed under ideal
conditions (Xiao &Vien, 2004). This paper evaluates the
sensitivity of kinetic characterizations to different selections
of flotation intervals in batch tests. From kinetic responses
with n time-recovery data, the R∞-f(k) pairs obtained from
all possible responses of (n – 1) datasets are evaluated using
the R(t) models of Equations (1), (2) and (4). The R∞-f(k)
estimations from the sub-sampled kinetic responses are
compared with those obtained from the entire dataset (n
datapoints). This sensitivity analysis for batch tests, under
moderate changes in the flotation intervals, will aid deter-
mining opportunities for improvement in kinetic charac-
terizations at lab scale.
EXPERIMENTAL PROCEDURE
Samples of a copper porphyry ore were characterized from
batch flotation tests. The 95% confidence interval for the
mean of the Cu grade was 0.92 ± 0.08%. The samples were
previously crushed, and after quartering, homogenization
and splitting procedures, 1-kg batches were prepared for
grinding. Dry grinding was performed in a Joy Denver® rod
mill (Joy Manufacturing Company, Denver Equipment
Division, USA). A grinding time of 5.5 min was set to
obtain a P80 ≈ 212 μm.
The ground product was poured into a 2.7 L EDEMET
flotation cell (EDEMET Ltda., Chile). Tap water was
added to obtain a solid percentage of 33% w/w, and pH
was adjusted and manually controlled at 9.5, using a lime
slurry. Hostaflot 18275 (Clariant Mining Solutions, USA)
at 22.5 g/ton was employed as a collector, with a condi-
tioning time of 5.0 minutes. Aerofroth 70 (Cytec, USA)
was used as a frother at 24 mg/L and conditioned for 2.5
minutes. The flotation cell was operated at 700 rpm, with
a constant air flowrate of 8 L/min. Concentrates were sam-
pled at cumulative flotation times of 0.5, 1.5, 4, 10, 16, 24,
and 32 minutes. Three flotation tests were conducted.
All flotation products were filtered and dried at 90°C.
For chemical assaying, representative samples of the feed
and products were digested on a hot plate using nitric
established a first-order exponential relationship for the
recovery increase (cumulative) as a function of time, t.
exp^-kth@ R th R 1 =-^6 (1)
Within Equation (1), R∞ is the maximum achievable recov-
ery, k is the flotation rate constant (deterministic), and t is
the cumulative flotation time.
Although the Single Flotation Rate (SFR) model of
Equation (1) has been claimed as a good approximation to
represent kinetic responses, its applicability has proven to
be limited due to the heterogeneity of particle properties
in flotation (Ofori et al., 2014). A distribution of rate con-
stants, f(k), was proposed by Imaizumi and Inoue (1963) as
a generalization of Equation (1), from which different prob-
ability density functions, such as Rectangular, Triangular,
Gamma, Exponential, Normal, among others, have been
proposed. The former corresponds to a uniform f(k) in the
range 0 to kmax (Huber-Panu et al., 1976 Klimpel, 1980),
and has proven to have satisfactory results, keeping only
two parameters. The Rectangular model for R(t) is given by
Equation (2):
exp^-k
R R k t
th
1
1
max
max =-
-
3 ^th F (2)
The Gamma distribution proposed by Imaizumi
and Inoue (1963) has proven to be suitable in describ-
ing the heterogeneity of the flotation process (Murhula
et al., 2022). This distribution generalizes Normal and
Exponential density functions. The Gamma model has an
additional parameter with respect to the Rectangular and
SFR models, which allows for higher flexibility to simulta-
neously represent particles with high and low floatability.
Equation (2) represents the Gamma f(k), in which aG and
kG are shape and scale parameters, respectively, and Γ is the
Gamma function.
f k
k e
G G
a
a
k
k 1
G
G
G C^a =
-
-^kh
h
(3)
Equation (4) represents the R(t) model, assuming a Gamma
f(k).
R R k thaGE 1 1
G
=-+3 ^th ^1 (4)
Although higher reactions orders have also been proposed
to justify deviations with respect to Eq. (1) (Rao, 2023),
this approach has proven to be equivalent to first-order
systems with an apparent f(k), as reported by Vinnett &
Waters (2020).
Even though kinetic modelling may exhibit suitable
fitting under specific experimental conditions, uncertain-
ties may occur due to non-standard flotation protocols and
inadequate experimental designs (Sandoval-Zambrano &
Montes-Atenas, 2012 Napier-Munn, 2012). Improvements
in kinetic characterization have been continuously reported
in literature, with focus on sample representativeness, auto-
mation in the concentrate removals, decrease in data vari-
ance, and others (Amelunxen et al., 2018 Napier-Munn,
2012 Ross, 2019). These advancements have been relevant
to define standard flotation tests, a topic that has been
extensively discussed among practitioners. However, the
determination of flotation intervals for the concentrate
removals and its effect on the R∞-f(k) estimations have not
received sufficient attention, only being assessed under ideal
conditions (Xiao &Vien, 2004). This paper evaluates the
sensitivity of kinetic characterizations to different selections
of flotation intervals in batch tests. From kinetic responses
with n time-recovery data, the R∞-f(k) pairs obtained from
all possible responses of (n – 1) datasets are evaluated using
the R(t) models of Equations (1), (2) and (4). The R∞-f(k)
estimations from the sub-sampled kinetic responses are
compared with those obtained from the entire dataset (n
datapoints). This sensitivity analysis for batch tests, under
moderate changes in the flotation intervals, will aid deter-
mining opportunities for improvement in kinetic charac-
terizations at lab scale.
EXPERIMENTAL PROCEDURE
Samples of a copper porphyry ore were characterized from
batch flotation tests. The 95% confidence interval for the
mean of the Cu grade was 0.92 ± 0.08%. The samples were
previously crushed, and after quartering, homogenization
and splitting procedures, 1-kg batches were prepared for
grinding. Dry grinding was performed in a Joy Denver® rod
mill (Joy Manufacturing Company, Denver Equipment
Division, USA). A grinding time of 5.5 min was set to
obtain a P80 ≈ 212 μm.
The ground product was poured into a 2.7 L EDEMET
flotation cell (EDEMET Ltda., Chile). Tap water was
added to obtain a solid percentage of 33% w/w, and pH
was adjusted and manually controlled at 9.5, using a lime
slurry. Hostaflot 18275 (Clariant Mining Solutions, USA)
at 22.5 g/ton was employed as a collector, with a condi-
tioning time of 5.0 minutes. Aerofroth 70 (Cytec, USA)
was used as a frother at 24 mg/L and conditioned for 2.5
minutes. The flotation cell was operated at 700 rpm, with
a constant air flowrate of 8 L/min. Concentrates were sam-
pled at cumulative flotation times of 0.5, 1.5, 4, 10, 16, 24,
and 32 minutes. Three flotation tests were conducted.
All flotation products were filtered and dried at 90°C.
For chemical assaying, representative samples of the feed
and products were digested on a hot plate using nitric