2650 XXXI International Mineral Processing Congress 2024 Proceedings/Washington, DC/Sep 29–Oct 3
y R UF F
d UF d d d
rec
d
rec
i i i i 1 1/2
=--
++1 +1
^UF h_F i
/
/
F OF OFd
R UF UF
100)(UFd
100)(OF
d
rec
d
OF d d d
i i i i
i i i
=-+
=-+
^RUF
^h
h
where y
d1 1/2 +
is the probability of a particle of size di+1/2
to be recovered in the underflow, RUF and ROF are the
mass recoveries in the underflow and overflow respectively,
UF
di and OF
di are the retained proportion for the mesh
di for the underflow and the overflow respectively, finally
F
di
rec is the retained proportion of the feed for the mesh
di which is computed from measurement of the underflow
and overflow.
All partition curves are computed from both the total
mass of sample recovered in the underflow and overflow
and the PSD computed using laser light scattering method.
Flotation
Design of experiments methodology
The design of experiments (DOE) methodology studies the
influence of k parameters at l levels for a desired number of
response(s). These responses are modelled using the stud-
ied parameters. It is common that the number of levels is
set to 3 or higher so that non-linear phenomena can be
further investigate. Designs were specifically built for this
issue, such as Box-Behnken, Central Composite Design,
etc. They are all based on the following equation, which
allows to investigate the interaction parameters between the
variables studied and the quadratic terms:
y a a x a x x
a x x x x
i
k
i i
i
k-
j i
k
ij i j
i
k-
j i
k-
l j+
k
ijl i j l
i
k
ii i
0
1 1
1
1
1
2
1
1
1 1
2 g f
=++
++++
===+
==+==
///
////a
where y is the studied response, a0 is a constant, ai is the
linear coefficient, aij the interaction coefficient, aii the qua-
dratic coefficient and ε is a residual.
All coefficients are modelled using least square method
based on the experimental results. This was done using the
JMP ® statistical software. Statistical analysis of the DOE
was performed by computing a Student test on the calcu-
lated coefficient of the model (confidence level of 95% to
select the significant coefficient). The selection between
significant and non-significant coefficients was carried out
using analysis of variance (ANOVA). Finally, for statistical
analysis of the model, both the correlation coefficient (R2)
and the root mean squared error (RMSE) were studied.
They are defined in the following equations.
R
y
yih2
1
i i i
i 2
1
1 =-
-
-
=
=/n
/n ^yi
^y h2
RMSE n y y 1
i
n
i i
1
=-
=
/^h2
where y represents the observed value, y the value pre-
dicted by the model and y the mean of the observed values.
For the purpose of this study, two optimal Central
Composite Designs were used with the parameters
described in Table 2 and Table 3, namely 3 factors studied
at 3 levels. This led to a total of 16 tests with a duplicated
test on the center of DOE for both designs.
For the second DoE, the pH value was set at 2 and the
collector dosage at 150 g/ton for the rougher stage and 75g/
ton for the scavenger stage.
Tests
As highlighted by Korbel et al., in order to recover all prod-
ucts from the Beauvoir granite, gravity concentration steps
should be added prior flotation tests (C. Korbel et al., 2023).
Thus the sample used for flotation tests was submitted first
Table 2. Factors and corresponding levels for the physico-chemical conditions design of experiments
Factor pH Value Amine Dosage, g/ton Desliming Size, µm
Levels
3 3 3
1.5 2.0 2.5 100 250 400 10 40 63
Table 3. Factors and corresponding levels for the hydrodynamic conditions design of experiments
Factor Rotor Speed Air Flow Rate, u.a. Desliming Size, µm
Levels
3 3 3
600 800 1,000 20 40 (=0.27 m3/h) 60 10 40 63
y R UF F
d UF d d d
rec
d
rec
i i i i 1 1/2
=--
++1 +1
^UF h_F i
/
/
F OF OFd
R UF UF
100)(UFd
100)(OF
d
rec
d
OF d d d
i i i i
i i i
=-+
=-+
^RUF
^h
h
where y
d1 1/2 +
is the probability of a particle of size di+1/2
to be recovered in the underflow, RUF and ROF are the
mass recoveries in the underflow and overflow respectively,
UF
di and OF
di are the retained proportion for the mesh
di for the underflow and the overflow respectively, finally
F
di
rec is the retained proportion of the feed for the mesh
di which is computed from measurement of the underflow
and overflow.
All partition curves are computed from both the total
mass of sample recovered in the underflow and overflow
and the PSD computed using laser light scattering method.
Flotation
Design of experiments methodology
The design of experiments (DOE) methodology studies the
influence of k parameters at l levels for a desired number of
response(s). These responses are modelled using the stud-
ied parameters. It is common that the number of levels is
set to 3 or higher so that non-linear phenomena can be
further investigate. Designs were specifically built for this
issue, such as Box-Behnken, Central Composite Design,
etc. They are all based on the following equation, which
allows to investigate the interaction parameters between the
variables studied and the quadratic terms:
y a a x a x x
a x x x x
i
k
i i
i
k-
j i
k
ij i j
i
k-
j i
k-
l j+
k
ijl i j l
i
k
ii i
0
1 1
1
1
1
2
1
1
1 1
2 g f
=++
++++
===+
==+==
///
////a
where y is the studied response, a0 is a constant, ai is the
linear coefficient, aij the interaction coefficient, aii the qua-
dratic coefficient and ε is a residual.
All coefficients are modelled using least square method
based on the experimental results. This was done using the
JMP ® statistical software. Statistical analysis of the DOE
was performed by computing a Student test on the calcu-
lated coefficient of the model (confidence level of 95% to
select the significant coefficient). The selection between
significant and non-significant coefficients was carried out
using analysis of variance (ANOVA). Finally, for statistical
analysis of the model, both the correlation coefficient (R2)
and the root mean squared error (RMSE) were studied.
They are defined in the following equations.
R
y
yih2
1
i i i
i 2
1
1 =-
-
-
=
=/n
/n ^yi
^y h2
RMSE n y y 1
i
n
i i
1
=-
=
/^h2
where y represents the observed value, y the value pre-
dicted by the model and y the mean of the observed values.
For the purpose of this study, two optimal Central
Composite Designs were used with the parameters
described in Table 2 and Table 3, namely 3 factors studied
at 3 levels. This led to a total of 16 tests with a duplicated
test on the center of DOE for both designs.
For the second DoE, the pH value was set at 2 and the
collector dosage at 150 g/ton for the rougher stage and 75g/
ton for the scavenger stage.
Tests
As highlighted by Korbel et al., in order to recover all prod-
ucts from the Beauvoir granite, gravity concentration steps
should be added prior flotation tests (C. Korbel et al., 2023).
Thus the sample used for flotation tests was submitted first
Table 2. Factors and corresponding levels for the physico-chemical conditions design of experiments
Factor pH Value Amine Dosage, g/ton Desliming Size, µm
Levels
3 3 3
1.5 2.0 2.5 100 250 400 10 40 63
Table 3. Factors and corresponding levels for the hydrodynamic conditions design of experiments
Factor Rotor Speed Air Flow Rate, u.a. Desliming Size, µm
Levels
3 3 3
600 800 1,000 20 40 (=0.27 m3/h) 60 10 40 63