FORMULA FOR THE DIRECT CALCULATION OF THE COMMINUTION VELOCITY By Ph.D. Igor Bobin

In previous articles [1-3] we have talked about the modeling of comminution kinetics (crushing and grinding kinetics) C = f (t), where C – is the content of the size fraction (size class) of final product, %;  t – is time, secs.

Now we are acquainting with сomminution velocity v_c. 
 

Comminution velocity (rate) v_c is ratio of the size fraction content C to unit of time t. Comminution velocity is derivative of time  v_c = dC/dt  (% / secs).

Our modeling approach to comminution kinetics allows to carry direct calculation of сomminution velocity, pioneering move. A visual representation of the сomminution velocity in time is very important for solving the optimization problem of the mineral processing technology.

I have proposed the following my own formula with delay for the direct analytical calculations of the comminution velocity v_c
(the differential equation in the operator form of Laplace)

where   c₀ – is initial level of content, which is set from the experimental data of the comminution process, %; T – is time constant of comminution (constant of inertia), secs; τ – is time delay, secs; s – is Laplace complex variable.

 

The parameters c₀, τ and T are determined graphically on the experimental kinetic curve of the comminution process by known method. The mathematical model (1) of the comminution velocity v_c of the first order with delay is Transfer Function W(s), which is convenient for modeling using MATLAB. The mathematical model (1) has enough accuracy for engineering calculations.
 
Example
There are mineral ore with its experimental dependence of the comminution kinetics with delay (see Fig.1).
We have mathematical model of the comminution kinetics C = f(s) in the operator form of Laplace, a similar the equation in the usual form [1]

In this case, parameter values of approximating equation (2) is amounted to:  c₀=9.54 %, τ =60 secs, T=180 secs.

Let's substitute these parameter values in the Bobin’s equation (2), and we got the mathematical model of the comminution kinetics C(s)

The analytical dependence and experimental of comminution kinetics obtained by modeling at MATLAB are shown on Figure 1.

 

 Fig. 1. Curves of the comminution kinetics C(t)

 
Now let's substitute parameter values in the Bobin’s equation (1), and we got the mathematical model of the comminution velocity v_c(s)

The analytical dependence of comminution velocity obtained by modeling at MATLAB is shown on Figure 2.

 

 Fig. 2. The analytical curve of the comminution velocity v_c(t),  (% / secs)

Thus, we can clearly calculate the comminution velocity (comminution rate) and use it to optimize the process. In our case, modeling of comminution kinetics is an indispensable tool for analisis of the mineral technology. The use of this tool (amongst other things) by the best expert on minerals concentration allows us to reach previously unattainable level of the mineral production efficiency.
 

References

1. Ph.D. Igor Bobin, Ph.D. Natalia Petrovskaya. «EQUATION OF COMMINUTION KINETICS WITH DELAY» Web resurs "CONCENTRATION OF MINERALS". December 18, 2016
2. Ph.D. Igor Bobin, Ph.D. Natalia Petrovskaya. «EQUATION OF COMMINUTION KINETICS WITH DELAY» News aggregation app Linkedin Pulse.  December 18, 2016
3. Ph.D. Igor Bobin, Ph.D. Natalia Petrovskaya. «EQUATION OF COMMINUTION KINETICS WITH DELAY» Open publishing platform Scribd. December 19, 2016

© Ph.D. Igor Bobin

December 27, 2016

bobin.igor@yahoo.com

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