XXXI International Mineral Processing Congress 2024 Proceedings/Washington, DC/Sep 29–Oct 3 935
flotation. This development may lead to widespread appli-
cations and profound changes. Comprehensive analysis
and modeling of both bench-scale flotation test data and
plant-scale industrial flotation process data is key to this
achievement. Three well known kinetic models reveal fun-
damental laws governing flotation, from bench test data.
Three widely used multivariant modeling methodologies
lead to identifying, defining, controlling, and optimizing
flotation fundamentals. Understanding the dynamic and
complex behavior of flotation allows Digital One to be
developed into an expert control system which can remain
reliable in the face of complex ores.
Digital One expands beyond controlling just the froth
and has been able to take over almost every part of the flo-
tation plant at Buick Mill. Digital One achieves automatic
control of the pseudo-stochastic flotation process with
machine learning.
DATA SOURCE
Doe Run maintains a thorough historical PI database stor-
ing nearly all data from DCS. Advanced data analytics
using this data played a critical role in the development of
Digital One, as generally stated (Mang et al., 2024b). The
models presented best illustrate the fundamentals of Digital
One, using data over random 2-day periods of representa-
tive mill operation, unless otherwise noted.
Analysis of the data was performed using the latest ver-
sion of JMP from SAS Institute. The lead author has over
20 years of experience in JMP data modeling for mining
and processing.
FLOTATION KINETICS
Bench scale kinetic flotation tests of twelve runs were per-
formed for an independent investigation by a contracted
firm to evaluate grinding phenomena. The tests used a
sample of rod mill feed from Buick Mill, with 2.55% Pb,
0.44% Zn, and 0.56% Cu and followed Doe Run flotation
test protocol. Kinetic test data were taken at 0.5, 1.5, 3
and 5 minutes and Pb, Zn and Cu percent recoveries deter-
mined. Firstly, test data was fitted against an exponential
model:
exp^cth R a b =+
where R is recovery, a is the asymptote, b is a scale, c is the
growth rate, and t is the flotation time. One such model is
shown in Figure 1, with model statistics given in Table 1.
The asymptote is the maximum recovery, the growth rpoint.
Secondly, the data was also fitted using Michaelis
Menten model:
R b t
at =+
where a is the maximum reaction rate and b is the inverse
affinity. An example is shown in Figure 2, with model sta-
tistics given in Table 2. Maximum reaction rate is also the
maximum recovery and inverse affinity is also the time to
get halfway to the maximum recovery.
Thirdly, the Gompertz model as follows:
exp R a b chhh =-exp^- -^^t
where R is again the recovery, a is the maximum recovery,
b is the growth rate, and c is the inflection point. An exam-
ple fit is shown in Figure 3, with model statistics given in
Table 1. Parameter estimate statistics for exponential model in Figure 1
Parameter Estimate Std Error Wald Chi2 p Value 95% Lower Bound 95% Upper Bound
Asymptote 86.24 0.24 131863.87 0.0001 85.78 86.71
Scale 18.97 2.06 84.94 0.0001 –23.01 –14.94
Growth Rate –1.61 0.22 52.20 0.0001 –2.04 –1.17
Figure 1. Exponential model fit of one kinetic flotation test,
R2 =0.998, all model parameters statistically significant
flotation. This development may lead to widespread appli-
cations and profound changes. Comprehensive analysis
and modeling of both bench-scale flotation test data and
plant-scale industrial flotation process data is key to this
achievement. Three well known kinetic models reveal fun-
damental laws governing flotation, from bench test data.
Three widely used multivariant modeling methodologies
lead to identifying, defining, controlling, and optimizing
flotation fundamentals. Understanding the dynamic and
complex behavior of flotation allows Digital One to be
developed into an expert control system which can remain
reliable in the face of complex ores.
Digital One expands beyond controlling just the froth
and has been able to take over almost every part of the flo-
tation plant at Buick Mill. Digital One achieves automatic
control of the pseudo-stochastic flotation process with
machine learning.
DATA SOURCE
Doe Run maintains a thorough historical PI database stor-
ing nearly all data from DCS. Advanced data analytics
using this data played a critical role in the development of
Digital One, as generally stated (Mang et al., 2024b). The
models presented best illustrate the fundamentals of Digital
One, using data over random 2-day periods of representa-
tive mill operation, unless otherwise noted.
Analysis of the data was performed using the latest ver-
sion of JMP from SAS Institute. The lead author has over
20 years of experience in JMP data modeling for mining
and processing.
FLOTATION KINETICS
Bench scale kinetic flotation tests of twelve runs were per-
formed for an independent investigation by a contracted
firm to evaluate grinding phenomena. The tests used a
sample of rod mill feed from Buick Mill, with 2.55% Pb,
0.44% Zn, and 0.56% Cu and followed Doe Run flotation
test protocol. Kinetic test data were taken at 0.5, 1.5, 3
and 5 minutes and Pb, Zn and Cu percent recoveries deter-
mined. Firstly, test data was fitted against an exponential
model:
exp^cth R a b =+
where R is recovery, a is the asymptote, b is a scale, c is the
growth rate, and t is the flotation time. One such model is
shown in Figure 1, with model statistics given in Table 1.
The asymptote is the maximum recovery, the growth rpoint.
Secondly, the data was also fitted using Michaelis
Menten model:
R b t
at =+
where a is the maximum reaction rate and b is the inverse
affinity. An example is shown in Figure 2, with model sta-
tistics given in Table 2. Maximum reaction rate is also the
maximum recovery and inverse affinity is also the time to
get halfway to the maximum recovery.
Thirdly, the Gompertz model as follows:
exp R a b chhh =-exp^- -^^t
where R is again the recovery, a is the maximum recovery,
b is the growth rate, and c is the inflection point. An exam-
ple fit is shown in Figure 3, with model statistics given in
Table 1. Parameter estimate statistics for exponential model in Figure 1
Parameter Estimate Std Error Wald Chi2 p Value 95% Lower Bound 95% Upper Bound
Asymptote 86.24 0.24 131863.87 0.0001 85.78 86.71
Scale 18.97 2.06 84.94 0.0001 –23.01 –14.94
Growth Rate –1.61 0.22 52.20 0.0001 –2.04 –1.17
Figure 1. Exponential model fit of one kinetic flotation test,
R2 =0.998, all model parameters statistically significant
