Flotation kinetics (from the Greek Kinētikós
- driving) is studying the regularities of flotation process at the time, the rate
and the flotation mechanism.
Kinetics of flotation reflects the flotation results in variable states and is
characterized by dependency of the recovery R of floatable mineral in
concentrate from time t, i.e. R = f (t). It allows
a quantitative description of the flotation process in time.
Kinetic curve of
floatability is a graph showing the change in concentration of substances in
time. At Fig. 1 there are two of the most common type of flotation kinetic
dependences R = f (t): exponential function and sigmoidal function.
The exponential dependence is characteristic for hydrophobic and most easily
floatable mineral particles and particles whose surface has time to react
quickly and easily with flotation reagents (primarily activators and collectors).
The sigmoid dependence resembling the letter S, is often called S-shaped
dependence. Sigmoid dependence is characteristic for the less hydrophobic and
difficult floatable minerals, the surface of which are not rapidly react with
flotation reagents, and hydrophobic surfaces that require a longer contact with
the flotation reagents.
Fig. 1. Types of flotation kinetic dependences: 1 - exponential function;
2 - sigmoidal function
At first glance, many processes can be described by an
exponential function. Experience shows that almost all flotation systems have
an inertia from a few seconds up to several hours. The exponential dependence
allows sufficient accuracy to describe the processes of inertia, and it is a
standard mathematical model of the kinetics of flotation. Often, however,
besides an inertia, flotation processes occur with a some delay of time. It should be accounted
in modeling.
Intensity of the
process of flotation is characterized by a rate of flotation. Because the
feedstock for the flotation process can content more that one valuable
component, in practice the flotation rate is determined for each component
separately.
Flotation rate (velocity) for the i-th component is the ratio of the recovery R
of i-th valuable component in the foam product to the unit of time t.
Flotation rate is derivative of time v
= dR/dt (%/secs).
Ph.D. Igor Bobin has
proposed and published the following his own equations of flotation kinetics:
1. Improved equation of flotation kinetics of the traditional kind, but with
delay
where Rmax – is
ultimate recovery of the valuable component, which is set from 0 to 100% based
on the experimental data of the flotation process, %; T – is time
constant of flotation (constant of inertia), which is determined by the
graphical method (see Fig. 2, Fig. 3) on an experimental curve of the kinetics of flotation, secs; τ– is time delay of
flotation, which is determined by the graphical method (see Fig. 2, Fig. 3) on an experimental curve of the kinetics of flotation, secs; t –
is time, secs.
Improved equation (1) is well-suited for modeling the kinetics of flotation R(t)
of processes with exponential kinetics curves with delay. Equation (1) is
devoid of the drawbacks of conventional flotation kinetics equation, it does
not require the selection of parameters, all the parameters of the flotation
kinetics model (1) is conveniently determined by standard graphic method on the
schedule of the experimental curve of kinetics (see Fig.
2, Fig. 3)
[1].
Fig.
2. Approximation of dependence R = f (t) for copper
Fig.
3. Approximation of dependence R = f (t) for zinc
2. Equation (inertia model) of flotation kinetics of the
first order with delay [1] (the differential equation in the operator form of
Laplace)
where Rmax – is
ultimate recovery of the valuable components in the concentrate is set from 0
to 100% based on the experimental data of the flotation process, %; T –
is time constant of flotation (constant of inertia), which is determined by the
graphical method (see Fig.
2, Fig. 3) on an
experimental curve of the kinetics of flotation, secs; τ– is time delay, which
is determined by the graphical method (see Fig. 2,
Fig. 3) on an experimental curve of the kinetics of
flotation, secs; s – is complex variable (on Laplace).
Equations (2) and (1) are identical in nature, but (2) is written in the
operator form and it is convenient for
modeling (using MATLAB).
3. Equation (inertia model) of flotation rate of the
first order with delay [1] (the differential equation in the operator form of
Laplace)
where Rmax – is
ultimate recovery of the valuable component is set from 0 to 100% based on the
experimental data of the flotation process, %; T – is time constant of
flotation (constant of inertia), secs; τ – is time delay, secs; s – is
complex variable (on Laplace).
4. Equation (inertia model) of flotation kinetics of
second order with delay
where Rmax – is ultimate recovery of
the valuable component is set from 0 to 100% based on the experimental data of
the flotation process, %; T2 – is time constant of flotation
(constant of inertia), secs; τ – is time of pure transport delay, secs; τtr
– is time of transition delay, secs; s – is complex variable (on
Laplace).
Parameters T2, τtr are
determined by known analytical tables.
So in our case for the second-order accuracy the
parameters of inertia model are T2 = T / 2.72,
τtr = 0,107 · T .
The parameters T and τ
are experimentally determined graphically on the experimental time curve (see. Fig. 2, Fig. 3).
5. Equation (inertia model) of flotation rate of the second order with delay
where Rmax – is ultimate recovery of component is set from
0 to 100 % based on the experimental parameters of the flotation process, %; T2
– is time constant of flotation (constant of inertia), secs; τ - is time of
pure transport delay, secs; τtr – is time of transition delay, secs;
s – is complex variable (on Laplace).
Fig. 4. Experimental dependence of the kinetics of zinc flotation 1
and analytical dependence of zinc flotation kinetics of the first order of
accuracy 2 and of the second order of accuracy 3 obtained by
simulation
Fig. 5. Analytical dependence of the zinc flotation rate of the first
order of accuracy 1 and of the second order of accuracy 2
obtained by simulation
Formulas
(2) and (4) are transfer functions W(s) for model of the flotation kinetics R
(t). Formulas (3) and (5) are transfer functions W(s) for model of the
flotation rate (velocity) v(t). For practical modeling of flotation kinetics
(e.g. using Simulink) enough to apply the unit constant value 1(t) to the input
of a system described by transfer function (2), (3), (4) or (5); then from the
output of the system, we obtain the corresponding kinetics curve R(t) or
flotation velocity curve v(t) (See. Fig. 6). It is the most convenient and
simple way of modeling of the flotation kinetics and the flotation rate. Also,
the expressions (2), (3), (4) and (5) allow to model the dynamics of flotation.
Fig.
6. Practical simulation of the flotation kinetics and the flotation rate using equations
(2), (3), (4) and (5)
The requirement for
further enhancing the accuracy of modeling necessitates use in a model
increasingly complex differential equations. For example, in applications
requiring higher accuracy of calculations, for mathematical description of the
flotation kinetics and the flotation rate can be used the inertial models of
second or higher order with delay. Especially recommended to use similar models
(4) and (5) to describe the kinetics and dynamics of the flotation of minerals
with S-shaped experimental curves of the time. In this case, in the simulation,
the shape of the analytical curves of kinetics and dynamics corresponds to the
shape of the experimental curves. What could be more important in the
optimization of the ore flotation process. Practical simulation of flotation
kinetics with Bobin’s equations (2), (3), (4) and (5) may be performed using
any system of computer mathematics. For example MATLAB. Bobin’s equation (1) is
less demanding on the researcher's toolbox, you'll only have possession of MS
Excel or even a manual calculation.
References
1. I. Bobin, N. Petrovskaya. USAGE OF INERTIAL MODELOF 1ST ORDER WITH DELAY FOR ANALYSIS OF KINETICS OF FLOTATION, 2008.
2. MINERAL MODELING
Ph.D. Igor Bobin, Ph.D. Natalia Petrovskaya. December 19, 2015
![Ph.D. Igor Bobin has proposed and published the following his own equations of flotation kinetics: 1. Improved equation of flotation kinetics of the traditional kind, but with delay where Rmax – is ultimate recovery of the valuable component, which is set from 0 to 100% based on the experimental data of the flotation process, %; T – is time constant of flotation (constant of inertia), which is determined by the graphical method (see Fig. 2, Fig. 3) for the experimental curve of the kinetics of flotation, secs; τ– is time delay of flotation, which is determined by the graphical method (see Fig. 2, Fig. 3) for the experimental curve of the kinetics of flotation, secs; t – is time, secs.Improved equation (1) is well-suited for modeling the kinetics of flotation R(t) of processes with exponential kinetics curves with delay. Equation (1) is devoid of the drawbacks of conventional flotation kinetics equation, it does not require the selection of parameters, all the parameters of the flotation kinetics model (1) is conveniently determined by standard graphic method on the schedule of the experimental curve kinetics (see Fig. 2, Fig. 3) [1].](https://sites.google.com/site/concentrationofminerals/how-to-choose-a-method-for-the-separation-of-minerals/flotation-separation/kinetics-of-flotation/Ph.D._I.Bobins_equation_of_flotation_kinetics-1.jpg?attredirects=0)
