The improved equation of flotation kinetics with delay
THE IMPROVED EQUATION OF FLOTATION KINETICS WITH DELAY
Ph.D. Igor Bobin
Flotation is widely used in the enrichment of non-ferrous metals, non-metallic minerals and man-made materials. Recently flotation is also used in wastewater treatment. When flotation achieved a high degree of concentration of valuable components. Results of flotation depend on the mineral composition and granulometric characteristics of the feedstock, the density, temperature, and degree of aeration of the pulp, the ionic composition of the liquid phase and other factors [1].
All variety of technological factors affecting the efficiency of the flotation is reflected by kinetics of flotation.
Kinetics of flotation reflects the flotation results in the transitive states and is characterized by dependency of the recovery R of floatable mineral in concentrate from time t, i.e. R=f(t). The main purpose of the flotation kinetics is a quantitative description of the flotation process in time. The flotation kinetics of monomineral particles is usually described in mineral processing by the K.F. Beloglazov’s equation of R(t)=1–e– k·t or by other similar equations.
However, the equation of K. Beloglazov (as other flotation kinetics equations, too) has a number of disadvantages [2]:
1. Limit of the recovery of valuable component is always taken as 100 %, which is not true. With flotation, the actual recovery of valuable component never reaches 100 %.
2. In practical calculations, the slope of curve R=f(t) is taken into account rather rudely: by selection of the parameter k, which gives a considerable error in the calculation of flotation kinetics.
3. For many minerals the flotation kinetics can’t be described only by an exponential dependence, since there is a delay due to the surface properties of minerals. For such minerals dependence R=f(t) has a pronounced S-shaped form with a delay (Fig. 2).
Thus, the limiting factor of the theory and practice of the ore flotation is still imperfection of the used models of the flotation kinetics.
Taking into account these shortcomings of known kinetic equations, Ph.D. Igor Bobin has proposed and published the following his own improved equation with delay for analytical calculations of ore flotation kinetics R(t) [3]
where Rmax – is the ultimate recovery of the valuable component, which is set from 0 to 100% from the experimental data of the flotation process, % (see. Fig. 3); T – is the time constant of flotation (the constant of inertia), secs; τ – is the time delay of flotation, secs; t – is the time, secs.
The parameters Rmax, τ and T are determined graphically on the experimental kinetic curve of the flotation process (see. Fig. 3).
Flotation kinetics equation (1) of I. Bobin was obtained on the basis of the graphic-analytical method of approximation of experimental time characteristic by the operator equation of the first order with delay [2].
Implementation of this modeling method is as follows.
We need to get an approximate Transfer Function W(s) of the dynamics of the flotation process for an interesting channel "Input X(s) – Output Y(s)" (Fig. 1). For example, the channel "Change in pH of pulp – Recovery of valuable component".
Fig. 1. The Transfer Function of element "Flotation process"
The approximate transfer function of the flotation system of the first order with delay (2) is a universal dynamic model of the system, which allows us to describe the statics, kinetics and dynamics of the flotation process
where k – is the transfer coefficient of the dynamic system; T – is the time constant of the inertia of the dynamic system, secs; τ – is the time delay, secs; s – is the complex variable on Laplace.
The parameters k, τ and T are determined graphically on the experimental kinetic curve of the flotation process (see. Fig. 3).
If we need the equation of the flotation kinetics only, then we can get the Transient Function (Transient Response) h(t) of our dynamic system "Flotation process".
Transient Response h(t) is a reaction of a dynamic system on the unit step influence of 1(t).
Then (L-1– is inverse Laplace transform) it implies from (2) that the kinetic equation for the output variable (dependent variable) Y(s) is
Also we need to take into account that the value of the input variable (independent variable) X(s) – is the unit step influence 1(t). Then we obtain the following expression
Let’s substitute the expression (4) in the expression (3)
Let’s substitute the expression (2) in the expression (5), then we get
Next, we carry the inverse Laplace transform L-1 and we go from the operator form of the expression to the usual form
Let’s consider the example of the usage of Ph.D, I. Bobin’s equation (1) for modeling the kinetics of flotation of copper-zinc ore.
Example
In the laboratory, we conducted studies on the kinetics of flotation of copper-zinc ore. Grade of copper of 1.55%, grade of zinc of 0.83%. Weight content of the class of less than 0.071 mm was of 80%. Ratio of liquid to solid was L:S=3:1. Consumption of reagents: С4Н9ОСSSК of 5 gram/tonne, T-80 of 15 gram/tonne. Flotation process was carried in a neutral environment. Flotation time was of 10 minutes. The experimental curves of the flotation kinetics for copper and zinc are presented on Fig.2. We need to obtain the analyticaldependences of the flotation kinetics for copper and zinc.
Decision
Figure 2 shows the flotation kinetics R(t) for copper and zinc obtained from experiments in [2].
If we take into account that function y(t) represents the kinetic dependence R(t) and the transfer coefficient k =Rmax then equation (7) is transformed to equation (1) in this case. As a result, we have the improved equation of flotation kinetics (1) of the traditional kind, but with a delay.
The improved equation of flotation kinetics (1) proposed by Ph.D. I. Bobin is a simple and clear solution of the problem of modeling the kinetics of the ore flotation. Equation (1) is well suited for modeling of flotation kinetics R(t) of the processes with exponential kinetic dependence and S-shape kinetic dependence, with delay or without delay, too. The equation (1) of I. Bobin is devoid of these drawbacks of the equation of K. Beloglazov. The equation (1) is fully taken into account the upper limit of recovering a valuable component as a percentage, the delay and the slope of kinetic curve R(t); model parameters Rmax, τ and T are conveniently determined according to the schedule of the experimental curve of flotation kinetics using the standard graphical method (see. Fig. 3), there is no need for the selection of model parameters.
Fig. 2. The experimental dependences of the flotation kinetics
of copper 1 and zinc 2
On the shape of the experimental curves R(t) we can conclude that the dependence R(t) of copper has an exponential form without delay. The dependence R(t) of zinc is S-shaped with a delay.
For a practical simulation of the flotation kinetics using I. Bobin’s equation (1) we need to first identify the model parameters Rmax, τ and T. Let’s use the graphical method to solve this task (see. Fig. 3).
Fig. 3. The approximation of the experimental dependence R=f(t) [2]:
a - for copper; b - for zinc
Method of the approximation is the following. For exponential dependences, we need to construct a tangent to the portion of the curve with a constant slope on the schedule R(t) to determine the time constant of inertia T (Fig. 3, a). In this case, the time delay τ, usually is zero. For S-shaped dependences, we need to construct a tangent at the inflection point of the S-shaped curve on the schedule R(t) to define parameters τ and T (Fig. 3, b). In both cases, the parameter Rmax is determined by the steady-state value of the curve R(t) (Fig. 3, a, b).
Parameter values of approximating equation (1) for copper in this case amounted to: Rmax=62%, τ =0 secs, T=69 secs.
Let's substitute these parameter values in the Bobin’s equation (1), and we get the mathematical model of the copper flotation kinetics RCu(t)
Actually for copper the equation (8) is converted to the equation (9)
Parameter values of approximating equation (1) for zinc in this case amounted to: Rmax=29 %, τ =50 secs, T=65 secs.
Let's substitute these parameter values in the Bobin’s equation (1), and we get the mathematical model of the zinc flotation kinetics RZn(t)
Practical simulation of the flotation kinetics using the formula (1) is as follows. One need to obtain values of function R(t) in the interesting range of the argument t. Mathematical modeling can be performed using a manual calculation with engineering calculator, MS Excel, MATLAB or other software.
Sample program to calculate the flotation kinetics of copper (9) and zinc (10) by the formula (1) of Ph.D. I. Bobin for MATLAB as follows:
script
RmaxCu=62; TCu=69; tauCu=0; tCu=[tauCu:0.01:600];
xCu=(tCu-tauCu)./TCu; RCu=RmaxCu.*(1-exp(-xCu));
RmaxZn=29; TZn=65; tauZn=50; tZn=[tauZn:0.01:600];
xZn=(tZn-tauZn)./TZn; RZn=RmaxZn.*(1-exp(-xZn));
plot(tCu,RCu,tZn,RZn); title('Flotation kinetics curves R(t)');
xlabel('Time, secs'); ylabel('Recovery, %'); axis([0 600 0 65]);
The analytical dependences of the flotation kinetics for copper and zinc obtained by simulation at MATLAB are shown in Figure 4.
Fig. 4. The analytical dependences of the flotation kinetics for copper and zinc obtained by simulation using formula (1) of Ph.D. I. Bobin
Thus, the proposed method of modeling the flotation kinetics, which based on the Ph.D. I. Bobin equation (1) is convinient, intuitive and not difficult, it does not require any special skills or knowledge and it has enough accuracy for engineering calculations.
References
1. Petrovskaya, N.I. (2007). Fundamentals of the theory of flotation, Izd. Ajur, Yekaterinburg.
2. Bobin, I.S., Petrovskaya, N.I. (2008). Usage of inertial model of 1st order with delay for analysis of kinetics of flotation, Tsvetnie Metaly, № 10, Moscow, pp. 30-33.
3. Bobin, I., Petrovskaya, N. (2015). Kinetics of ore flotation, Web Resource “CONCENTRATION OF MINERALS” https://sites.google.com/site/concentrationofminerals/home
Ph.D. Igor Bobin, April 3, 2016