*The following article has been republished from Concentration of Minerals, an excellent resource compiled by Dr Natalia Petrovskaya.*

** Flotation kinetics** (from the Greek

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).

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** 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.

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** Flotation rate (velocity)** for the i-th component is the ratio of the recovery

**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 *R*_{max} – 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**

[1] (the differential equation in the operator form of Laplace)*Equation (inertia model) of flotation kinetics of the first order with delay*

where *R*_{max} – 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).

[1] (the differential equation in the operator form of Laplace)*Equation (inertia model) of flotation rate of the first order with delay*

where *R*_{max} – 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).

*Equation (inertia model) of flotation kinetics of second order with delay*

where *R*_{max} – is ultimate recovery of the valuable component is set from 0 to 100% based on the experimental data of the flotation process, %; *T*_{2} – is time constant of flotation (constant of inertia), secs; τ – is time of pure transport delay, secs; τ_{tr} – is time of transition delay, secs; *s* – is complex variable (on Laplace).

Parameters *T*_{2}, τ_{tr} are determined by known analytical tables.

So in our case for the second-order accuracy the parameters of inertia model are *T*_{2} = *T* / 2.72, τ_{tr} = 0,107 · *T* .

The parameters *T* and τ are experimentally determined graphically on the experimental time curve (see. Fig. 2, Fig. 3).

*Equation (inertia model) of flotation rate of the second order with delay*

where *R*_{max} – is ultimate recovery of component is set from 0 to 100 % based on the experimental parameters of the flotation process, %; *T*_{2} – is time constant of flotation (constant of inertia), secs; τ – is time of pure transport delay, secs; τ_{tr} – 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**

*Ph.D. Igor Bobin, Ph.D. Natalia Petrovskaya. ***December 19, 2015**