## 2 Materials and Methods
### Model Development  
Several types of flotation models are proposed in flotation theory [40]: kinetic models; mass-balance models; and probabilistic models. The most common are kinetic flotation models. In these models, the first order kinetic equation is widely used to determine the kinetic parameters of flotation. This equation is based on the idea of an elementary flotation act as an analogue of a chemical process, where a bubble and a mineral particle interact. It follows that the rate of the flotation process is directly proportional to the number of elementary flotation acts. The coefficient of proportionality in this case is the flotation rate constant. Thus, the basis for the development is the method for determining the flotation constant [41]. However, the practice of applying these models to describe real technological processes has shown that, due to the complexity of the process, the modelling results using a single equation of flotation kinetics do not have sufficient accuracy. The solution to this problem was found in the description of the process by a system of equations with different flotation rate constants, which represents the flotation process as a set of several sub-processes. In the literature, there are two principal approaches in determining the value of flotation rate constants [42]:  
* Dividing the flotation process into sub-processes according to the phase where the process takes place and determining the rate constant for each of these sub-processes;
* Dividing all material into flotation classes based on its hydrophobicity, and establishing the rate constant for each of these classes. Partitioning into phases has proven to be a good idea for models using the principles of computational fluid dynamics. Kinetic laws of transition of mineral particles between \"pulp phases\" and \"froth phases\" allow us to establish correlations with calculation of particle movement trajectories in the field of velocities of the medium. By dividing the whole material into different hydrophobic fractions, the concept of \"floatability index\" is introduced. The flotation index is a dimensionless value, reflecting the relative probability of particle attachment to the bubble and its further removal into the foam product by true flotation or mechanical removal. The concept is based on the principle of classifying the entire mass of flotation material into classes for particles whose difference in attachment probability is insignificant. This is necessary in order to compensate for the difference between the recovery values determined in laboratory experiments and theoretical calculations [43]. To simulate the process, AVEVA used an approach to describe flotation as analogous to a chemical process. It is based on an equation describing the change in mineral particle concentration with flotation time. \[\frac{dN}{dt}=kN^{m}\]  
where: \(N\)--particle concentration, \(k\)--flotation rate constant, \(t\)--flotation time. The physical meaning of the kinetic equation's order is conditioned by the specific number of mineral particles per air bubble. Thus, the main task is to determine the flotation rate constant. One approach for determining the rate constant is to estimate the partial constants of the effect of the different factors on flotation efficiency. In this approach, all flotation material is divided into classes, with respect to component and size class.