UNIVERSAL EQUATION OF GRINDING KINETICS

By Ph.D. Igor Bobin and Ph.D. Natalia Petrovskaya

Today unconventional types of ores are involved in the processing in large quantities. The main trend now is to obtain ultra fine grinding of ores. This has led to the fact that crushing and grinding operations are the most costly at the plant. Costs for grinding can reach 60% of the total cost of ore processing.

Now we can’t say that there is enough knowledge about the problem of the development and the use of mathematical models in relation to the grinding and crushing processes. Currently, there are no universal models for crushing and grinding processes. Throughout the world there is an objective need for the development of mathematical models of crushing and grinding.

Thus, the universal mathematical model of grinding kinetics and grinding velocity are very relevant and in demand. So, Igor Bobin first proposed a single equation (1) that with the same accuracy can describe the kinetics of crushing and grinding processes with time delay and without time delay. For the first time Igor Bobin created a mathematical model that can be used to represent 100 % of the finished product and less (see Fig.1).

Fig. 1. The universal equation of comminution kinetics and its analytical curves

The figure 1 demonstrates analytical grinding kinetics curves. These curves were obtained using a single equation (1) of comminution kinetics.

However, it is not always necessary to obtain 100 % of the finished product. For example, very often placed next task: to receive 80 % of the size class -0,071 mm, receive 50 % of class -0.044 mm size, and so on.

Example

There is a mineral ore with its experimental dependence of the grinding kinetics (see Fig.2).

In this case, parameter values of ​​approximating equation (1) are amounted to: c₀ =3.4 %, τ=40 secs, T=1065 secs.

Let's substitute these parameter values ​​in the Bobin Equation (1), and we get the mathematical model of the grinding kinetics C₁(t)

The analytical dependence and experimental of grinding kinetics for the ore obtained by simulation at MATLAB are shown on Figure 2.

Fig. 2. Grinding kinetics curves

In this case, the grinding process has a high inertia, so it can’t be completed in the allotted time in the mill. The size class content of the finished product doesn’t reach its steady-state value of 100%. Such a case often occurs in the production. Also, the grinding process has a marked delay within one minute.

The Bobin Equation can significantly improve the quality of grinding while reducing grinding costs. Ultimately, it leads:

· To a perfection in grinding technology;

· To an elevation in recovery and (or) content of valuable mineral;

· To a significant reduction in energy consumption at ore grinding.

The Bobin Equation can be used with success for modeling and optimization of crushing process and grinding.

© Ph.D Igor Bobin, Ph.D. Natalia Petrovskaya

January 15, 2017

bobin.igor@yahoo.com

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