## 2 Materials and Methods
### Model Development
\[k_{ij}=k_{j}^{frother}*k_{j}^{coll}*k^{air}*k^{imp}*k_{i}^{PSD}\]  
where: \(k_{ij}\)--overall flotation rate constant for specific \(i\)--particle size class and \(j\)--reagent regime, \(k_{i}^{PSD}\)--flotation rate constant component that correlates with particle size distribution, \(k_{j}^{fronther}\)--flotation rate constant component that correlates with frontier concentration in pulp for specific reagent regime, \(k_{j}^{coll}\)--flotation rate constant component that correlates with collector concentration in pulp for specific reagent regime, \(k^{air}\)--flotation rate constant component that correlates with air flow rate, \(k^{imp}\)--flotation rate constant component that correlates with impeller speed. This approach has several key advantages. Firstly, it allows us to more accurately account for the influence of various technological factors on the efficiency of the flotation process, based on the results of flotation tests on ores. Secondly, the approach allows for scaling the results of experimental studies from laboratory installations to industrial ones with higher values of adequacy. One of the main issues, scaling the results of laboratory studies to industrial plants, stems from the influence of flotation cell hydrodynamics on the measure of air dispersion in the chamber. The approach to the issue is to keep the parameter of surface area bubble flux (\(S_{b}\)) and the superficial gas velocity (\(J_{g}\)) equal. The resulting parameters are the starting point for scaling the results. The parameter \(S_{b}\) relates the amount of air supplied to the flotation process to the geometric parameters of the bubbles, as well as to the geometric characteristics of the machine chamber [44]. \[S_{b}=\frac{6*Q_{A}}{A_{cell}}*\frac{\sum_{i=1}^{n}{d_{i}}^{3}}{\sum_{i=1}^{n}{ d_{i}}^{2}}\]  
\[J_{g}=\frac{Q_{A}}{A_{cell}}\]  
\[d_{32}=\frac{\sum_{i=1}^{n}{d_{i}}^{3}}{\sum_{i=1}^{n}{d_{i}}^{2}}\]  
where: \(d_{i}\)--bubble diameter, \(Q_{A}\)--air flow rate, \(A_{cell}\)--fotation cell's cross-sectional area. The geometric size distribution parameter of the bubbles was determined by analyzing the image of the air bubbles in the cell. This approach is based on the assumption that, when mineral particles attach to bubbles, the size deformation of the latter is minor. The sizing was achieved from the captured images of the dispersion process. Only the foaming agent was used. Development of a Methodology to Assess the Suitability of the Model for Use in a Computer-Based Digital Learning Simulator  
Currently, there are many technologies related to digital ones in one way or another. The work [45] investigated technologies used in the mining industry. Among them, the most in demand are: advanced process control, mine nerve center, virtual and augmented reality, automation, digital twins, artificial intelligent, robotics, big data, cyber security, etc. Figure 1 shows a simplified diagram of the computer simulator under development. This diagram demonstrates some of its features, namely:  
* The control and power systems are separated into separate units (this will make it possible to understand whether to virtualize the functions of these systems or to leave the real ones in place);
* The physical and dynamic models are connected together.his solution allows the user to practice the functionality on physical simplified models and to see how they scale to the size of the entire enterprise;
* A separate user interface is allocated for the administrator and the instructor. This will give the instructor the ability to develop training tasks, and will give the administrator the ability to configure the system, etc. ;
* Models and systems are communicated using the OPC protocol. This protocol is basic and can be changed to any other (e.g., MQTT etc.). The study of the model jointly in terms of different technologies has its own characteristics. For example, the dynamic model can be used to investigate the performance of digital twin technology as a tool for changing process conditions.