940 XXXI International Mineral Processing Congress 2024 Proceedings/Washington, DC/Sep 29–Oct 3
flotation (Zn rougher froth grade) in Figure 11. The final
model shown in Figure 11 is:
..03
..66h^x .64h
..64h2
y x1 41 52 0 2.59x2
0 19 500 0
47 32^x 0
1 2
2
=--
---
--
^x
where y is rougher concentrate Zn grade, x1 is CuSO4 addi-
tion rate (x-axis in Figure 11), and x2 is mill feed Zn grade
(y-axis in Figure 11). The dots on the contour lines show the
direction of increasing Zn rougher froth grade in Figure 11,
so not every contour line is paired with Zn rougher froth
grade to have adequate contour lines to show model details.
Table 5 reports ANOVA and Table 6 parameter estimates,
with p0.05 signifying significant whole model and all
model parameters.
CuSO4 is added to activate zinc sulfide for collec-
tion by xanthate, more with lower Zn in feed as shown
in Figure 11. It is important to note that the model is not
only statistical and numerical, but also thermodynamic and
kinetic since more CuSO4 is needed and therefore con-
sumed to find and activate Zn (sphalerite) when Zn is low
(0.5%). The same thermodynamic and kinetic fundamen-
tals are true, for example, in copper stockpile leach—when
copper ore grade is low or as copper is leached out, more
sulfuric acid is needed and consumed, and it’s taking longer
to find and leach out copper.
Lead Recovery—Introducing Full RSM with Stepwise
Regression
Pb recovery in flotation of complex Pb/Zn/Cu ores is
affected by many factors, and full RSM with stepwise
regression is the ideal, and maybe the only effective model-
ing approach, as shown here.
The final model statistics for fitting Pb recovery are
shown in Figure 12 (predicted vs actual regression plot),
Figure 13 (studentized residuals), Table 7 (ANOVA), and
Table 8 (parameter estimates), all indicating an excellent
model. In the parameter estimates Mill Feed Rate is insig-
nificant, but second order terms involving it are significant,
a common phenomenon in multivariate regression.
The variance inflation factors (VIF) measure the vari-
ance inflation due to the collinearity of the variables, all
below 10 indicating the model does not overrepresent col-
linear terms.
This model shows the complexity of flotation, with
16 parameters between first-order terms (x1), inter-term
Figure 11. Zn Rougher froth grade versus CuSO
4 addition
to the Zn roughers and Mill Feed Zn grade. This model
contains no additional dimensional parameters not shown in
the contour
Table 5. ANOVA statistics for model in Figure 11
Source Degrees of Freedom Sum of Squares Mean Square F-Ratio
Model 4 97,214.5 24,303.6 1,925.1
Error 13,277 167,617.6 12.6
C. Total 13,281 264,832.1 p 0.0001
Table 6. Parameter statistics for model in Figure 11
Term Estimate Std. Error t Ratio p Value
Intercept 41.52 0.2810 147.75 0.0001
x
1 – CuSO
4 addition –0.03 0.0004 –69.71 0.0001
x2 – Mill Feed Zn% 2.59 0.2332 –11.10 0.0001
(x1 – 500.657)(x2 – 0.63833) –0.19 0.0028 –67.05 0.0001
(x
2 – 0.63833)2 –47.32 0.8666 –54.60 0.0001
flotation (Zn rougher froth grade) in Figure 11. The final
model shown in Figure 11 is:
..03
..66h^x .64h
..64h2
y x1 41 52 0 2.59x2
0 19 500 0
47 32^x 0
1 2
2
=--
---
--
^x
where y is rougher concentrate Zn grade, x1 is CuSO4 addi-
tion rate (x-axis in Figure 11), and x2 is mill feed Zn grade
(y-axis in Figure 11). The dots on the contour lines show the
direction of increasing Zn rougher froth grade in Figure 11,
so not every contour line is paired with Zn rougher froth
grade to have adequate contour lines to show model details.
Table 5 reports ANOVA and Table 6 parameter estimates,
with p0.05 signifying significant whole model and all
model parameters.
CuSO4 is added to activate zinc sulfide for collec-
tion by xanthate, more with lower Zn in feed as shown
in Figure 11. It is important to note that the model is not
only statistical and numerical, but also thermodynamic and
kinetic since more CuSO4 is needed and therefore con-
sumed to find and activate Zn (sphalerite) when Zn is low
(0.5%). The same thermodynamic and kinetic fundamen-
tals are true, for example, in copper stockpile leach—when
copper ore grade is low or as copper is leached out, more
sulfuric acid is needed and consumed, and it’s taking longer
to find and leach out copper.
Lead Recovery—Introducing Full RSM with Stepwise
Regression
Pb recovery in flotation of complex Pb/Zn/Cu ores is
affected by many factors, and full RSM with stepwise
regression is the ideal, and maybe the only effective model-
ing approach, as shown here.
The final model statistics for fitting Pb recovery are
shown in Figure 12 (predicted vs actual regression plot),
Figure 13 (studentized residuals), Table 7 (ANOVA), and
Table 8 (parameter estimates), all indicating an excellent
model. In the parameter estimates Mill Feed Rate is insig-
nificant, but second order terms involving it are significant,
a common phenomenon in multivariate regression.
The variance inflation factors (VIF) measure the vari-
ance inflation due to the collinearity of the variables, all
below 10 indicating the model does not overrepresent col-
linear terms.
This model shows the complexity of flotation, with
16 parameters between first-order terms (x1), inter-term
Figure 11. Zn Rougher froth grade versus CuSO
4 addition
to the Zn roughers and Mill Feed Zn grade. This model
contains no additional dimensional parameters not shown in
the contour
Table 5. ANOVA statistics for model in Figure 11
Source Degrees of Freedom Sum of Squares Mean Square F-Ratio
Model 4 97,214.5 24,303.6 1,925.1
Error 13,277 167,617.6 12.6
C. Total 13,281 264,832.1 p 0.0001
Table 6. Parameter statistics for model in Figure 11
Term Estimate Std. Error t Ratio p Value
Intercept 41.52 0.2810 147.75 0.0001
x
1 – CuSO
4 addition –0.03 0.0004 –69.71 0.0001
x2 – Mill Feed Zn% 2.59 0.2332 –11.10 0.0001
(x1 – 500.657)(x2 – 0.63833) –0.19 0.0028 –67.05 0.0001
(x
2 – 0.63833)2 –47.32 0.8666 –54.60 0.0001