(70n) Application of the Theory of Markov Chains to Model Non-Linear Phenomena in Comminution

One of the basic mathematical models of grinding kinetics is the population balance model, which can be used in the form of equation continuous with time and particle size, as well as in the discrete form for finite time steps and finite size fractions. It allows presenting the model in matrix notation, and using the number of loading as a conditional time. The matrix of grinding is supposed energy dependent, and, if the energy is kept constant at every step of loading, it is also kept constant at these steps of loading. Thus, the specific energy to material is taken into account implicitly in these models, and the energy balance is even out of observation.

Another group of models deal namely with the energy balance. These models are associated with names of Rittinger, Bond, and Kick. In these models the energy balance is the most important objective of the analysis, but the particle size distribution is used implicitly just to determine the mean particle size before and after grinding, and the way of averaging depends on the energy low, which is in use.

The objective of the study is to couple the equations of fraction mass and energy balance, and to find out what kind of limitations it imposes onto the matrix of grinding, or onto the selection and breakage matrices presenting it. The process of grinding is presented as a random markovian process, which is discrete in time and in the sample space — contents of finite size fractions. The transformation of PSD can be described by the matrix of transition probabilities, which is to meet the mass balance equation and the energy balance equation at every time step of loading.

However, the total energy, absorbed at every step of loading, directly depends on the specific surface before and after loading. It is shown that a constant matrix of grinding never gives a linear grows of the specific surface with the number of loading. In particular, this means that at equal energy consumption at every step of loading we get different calculated increase of specific surface. It follows from that that the constant matrix and the energy balance cannot exist together in principle. A model, in which the mass balance equation and the energy balance equation are combined together, was proposed, and its solution on the basis of the principle of maximum entropy was proposed. According to this solution, if P(E) is the matrix of grinding (E is the specific energy), then P(2E)P(E)P(E), i.e., the process is a non-linear in principle.

The conclusion from the analysis is that no one linear model of grinding using a constant matrix of grinding can meet the condition of energy balance in principle. In order to make a model of grinding kinetics, which meets the both balance equations, it is necessary to have the matrix of grinding varying from one step of loading to another. One of these models is the entropic model, which takes the energy balance into account in advance as a natural constraint to the optimisation problem.