936 XXXI International Mineral Processing Congress 2024 Proceedings/Washington, DC/Sep 29–Oct 3
Table 3. The inflection point describes when diminishing
returns on the recovery begin.
The exponential model and the Michaelis Menten
model correspond with first (n=1) and second order (n=2)
kinetic behavior, following the differential form:
dt
dC kC n =
where C is the concentration of a given species, k is a rate
constant, and n is the reaction order.
The 4-point kinetic flotation test data generated 3
model parameters with one degree of freedom and diag-
nostic statistics for all 108 models -3 models for 3 metal
recoveries over 12 different tests each. In each of 108 cases,
model fits are statistically significant, as shown in the prior
3 model figures, randomly chosen from 108 models, one
for each metal recovery.
These models inspired us to investigate controlling and
optimizing hydrodynamics, kinetics, and thermodynam-
ics of complex industrial flotation like Buick Mill flotation
plant. Very importantly, these models gave us confidence
that flotation would follow fundamental laws which could
be exploited. Thermodynamics inspired us to explore
reagent possibilities and kinetics to understand the impact
of bubble formation and froth transport, and hydrodynam-
ics to adjust air and MIBC. These approaches became the
focus of the Digital One system and proved to be applicable
and successful, leading to ever more applications.
Figure 2. Michaelis Menten model for another kinetic
flotation test, R2=0.997, all parameters statistically
significant
Table 2. Parameter estimate statistics for Michaelis Menten model in Figure 2
Parameter Estimate Std Error Wald Chi2 p Value 95% Lower Bound 95% Upper Bound
Max Reaction
Rate 83.61 0.23 133498.89 .0001* 83.16 84.06
Inverse Affinity 0.057 0.003 367.551 .0001* 0.051 0.063
Figure 3. Gompertz model fit of another flotation test,
R2=0.998, all model parameters statistically significant
Table 3. Parameter estimate statistics for Gompertz model in Figure 3
Parameter Estimate Std Error Wald Chi2 p Value 95% Lower Bound 95% Upper Bound
Asymptote 91.44 0.22 180264.39 0.0001 91.01 91.86
Growth Rate 1.71 0.25 47.87 0.0001 1.22 2.19
Inflection Point –0.94 0.20 21.26 0.0001 –1.34 –0.54
Table 3. The inflection point describes when diminishing
returns on the recovery begin.
The exponential model and the Michaelis Menten
model correspond with first (n=1) and second order (n=2)
kinetic behavior, following the differential form:
dt
dC kC n =
where C is the concentration of a given species, k is a rate
constant, and n is the reaction order.
The 4-point kinetic flotation test data generated 3
model parameters with one degree of freedom and diag-
nostic statistics for all 108 models -3 models for 3 metal
recoveries over 12 different tests each. In each of 108 cases,
model fits are statistically significant, as shown in the prior
3 model figures, randomly chosen from 108 models, one
for each metal recovery.
These models inspired us to investigate controlling and
optimizing hydrodynamics, kinetics, and thermodynam-
ics of complex industrial flotation like Buick Mill flotation
plant. Very importantly, these models gave us confidence
that flotation would follow fundamental laws which could
be exploited. Thermodynamics inspired us to explore
reagent possibilities and kinetics to understand the impact
of bubble formation and froth transport, and hydrodynam-
ics to adjust air and MIBC. These approaches became the
focus of the Digital One system and proved to be applicable
and successful, leading to ever more applications.
Figure 2. Michaelis Menten model for another kinetic
flotation test, R2=0.997, all parameters statistically
significant
Table 2. Parameter estimate statistics for Michaelis Menten model in Figure 2
Parameter Estimate Std Error Wald Chi2 p Value 95% Lower Bound 95% Upper Bound
Max Reaction
Rate 83.61 0.23 133498.89 .0001* 83.16 84.06
Inverse Affinity 0.057 0.003 367.551 .0001* 0.051 0.063
Figure 3. Gompertz model fit of another flotation test,
R2=0.998, all model parameters statistically significant
Table 3. Parameter estimate statistics for Gompertz model in Figure 3
Parameter Estimate Std Error Wald Chi2 p Value 95% Lower Bound 95% Upper Bound
Asymptote 91.44 0.22 180264.39 0.0001 91.01 91.86
Growth Rate 1.71 0.25 47.87 0.0001 1.22 2.19
Inflection Point –0.94 0.20 21.26 0.0001 –1.34 –0.54