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EQUATION OF COMMINUTION KINETICS WITH DELAY
By Ph.D. Igor Bobin, Ph.D. Natalia Petrovskaya
Rock is a variety of different minerals united together. A comminution process is used to liberate one mineral from another. The size of the pieces of rock decreases with the comminution and mineral particles are released.
The purpose of the comminution is to obtain the finished product of a certain size. The finished product is directed to the separation of one mineral from another.
Crushing and grinding operations are expensive, so the kinetics of grinding (сrushing, comminution) is very important for the management.
The comminution kinetics reflects the change of the content of the size fraction (size class) of finished product in time: C = f(t).
The main objective of the comminution kinetics is a quantitative description of the comminution process. Many comminution processes can be described by an exponential function. Experience shows that almost all comminution systems have an inertia. Often, however, besides an inertia, comminution processes occur with a some delay of time. It should be accounted in modeling.
Ph.D. Igor Bobin has proposed the following his own equation with delay for analytical calculations of the comminution kinetics C(t) [1]
where С(t) – is the content of the size fraction of ready product, %; c0 – is initial level of content, which is set from the experimental data of the comminution process, % (see. Fig. 1); T – is the time constant of comminution (the constant of inertia), secs; τ – is the time delay of comminution, secs; t – is the time, secs.
The parameters c0, τ and T are determined graphically on the experimental kinetic curve of the comminution process by known metod.
Example
Example
There are mineral ores of two kinds with their experimental dependences of the comminution kinetics.
In this case, parameter values of approximating equation (1) for first ore is amounted to: c0 =9.54 %, τ =60 secs, T=180 secs.
Let's substitute these parameter values in the Bobin’s equation (1), and we get the mathematical model of the comminution kinetics C1(t)
The analytical dependence and experimental of comminution kinetics for first ore obtained by simulation at MATLAB are shown on Figure 1.
Fig. 1. The dependences of the comminution kinetics for first ore
Parameter values of approximating equation (1) for second ore is amounted to: c0 =3.97 %, τ =0 secs, T=630 secs.
Let's substitute these parameter values in the Bobin’s equation (1), and we get the mathematical model of the comminution kinetics C1(t)
Actually the equation (3) is converted to the equation (4) in this case
Actually the equation (3) is converted to the equation (4) in this case
The analytical dependence and experimental of comminution kinetics for second ore obtained by simulation at MATLAB are shown on Figure 2.
Fig. 2. The dependences of the comminution kinetics for second ore
Fig. 2. The dependences of the comminution kinetics for second ore
Bobin equation (1) is well-suited for modeling the kinetics of comminution C(t) of processes with delay. Thus, the proposed method of modeling is convinient and it has enough accuracy for engineering calculations and scientific.
© Ph.D. Igor Bobin, Ph.D. Natalia Petrovskaya
December 18, 2016
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