## 4 Results and Discussion  
The kinetics study results of the flotation process are presented in Table 2. A graphical interpretation is shown in Figures 3-5. The equation of the flotation process kinetics of the second order was used for a mathematical description of the obtained results. The equation has the form:  
\[\varepsilon(t)=\frac{\varepsilon_{0}t}{t+\frac{\varepsilon_{0}}{k}}\]  
where: \(\varepsilon_{0}\)--theoretical recovery limit, \(k\)--fotation rate constant, min\({}^{-1}\), \(t\)--fotation time, min. \begin{table}
\begin{tabular}{l l c c c} \hline \hline  & \multicolumn{1}{c}{**Valuable**} & \multicolumn{1}{c}{**Time Intervals**} & \(\varepsilon_{0}\)**, \%** & \(k\)**, min\({}^{-1}\)** \\  & **Component** & & & \\ \hline Flotation without collectors & Pyrite & - & 15.04 & 14.45 \\ Arsenopyrite & - & 9.67 & 8.10 \\ Potassium butyl xanthate & Pyrite & - & 46.73 & 100.00 \\ Arsenopyrite & - & 38.61 & 36.10 \\ Potassium butyl xanthate and sodium & Pyrite & I & 70.42 & 166.67 \\ dibutyldithiophosphate & Arsenopyrite & I & 46.51 & 58.48 \\  & Pyrite & II & 90.91 & 103.09 \\  & Arsenopyrite & II & 76.92 & 29.76 \\ \hline \hline \end{tabular}
\end{table} 
Table 2: The kinetics study results of flotation enrichment. Figure 2: The algorithm of the experimental series. Figure 4: Recovery profiles for flotation with PNBX application. Figure 3: Recovery profiles for flotation without collectors. On the basis of the obtained results, it was established that the maximum possible recovery of valuable components increases with the addition of sodium dibutyldithio-phosphate (SBDTP) to the collectors mixture. This is likely caused by the growth of the hydrophobizing ability of the mixture, which allows us to recover mineral particles of pyrite and arsenopyrite with a lower degree of liberation into the froth product. For the second time interval, a lower value of the rate constant for arsenopyrite flotation compared to the flotation with potassium butyl xanthate (PNBX) was established. The recovery values for pyrite and arsenopyrite, established during a series of experiments of flotation without collectors, characterize the valuable component recovery in the foam product due to the mechanical removal and flotation of the maximum opened aggregates that are capable of flotation without the collector. Based on the ratio, the values of flotation rate constants were established and the values of specific flotation rate constants for each factor were calculated. The results are presented in Table 3.  
For the dynamic simulation, a pneumococcal flotation machine with one cell was selected. The characteristics of the flotation machine are presented in Table 4.  
Figure 5: Recovery profiles for flotation with PNBX and SDBTP application. \begin{table}
\begin{tabular}{l c c c c} \hline \hline
**Value** & \(\mathbf{k_{\mathit{frotther}}}\)**, \(\mathbf{\min^{-1}}\) & \(\mathbf{k_{\mathit{PNBX}}}\)**, \(\mathbf{\min^{-1}}\) & \(\mathbf{k_{\mathit{SBDTP}}}\) (I), \(\mathbf{k_{\mathit{SBDTP}}}\) (II), \(\mathbf{\min^{-1}}\) & \(\mathbf{\min^{-1}}\) \\ \hline Pyrite & 14.45 & 6.92 & 1.67 & 1.03 \\ Arsenopyrite & 8.10 & 4.46 & 1.62 & 0.82 \\ \hline \hline \end{tabular}
\end{table} 
Table 3: Established kinetic constants for simulation. The determining of the surface area bubble flux value was carried out on the basis of determining the size of bubbles in the process of air dispersion in the flotation machine chamber with the addition of a frother. The results of the calculations are presented in Table 5.  
The analysis of the results shows that the relative velocity of the rise of bubbles grows with an increasing air flow rate, which leads to an increase in the value of the specific aeration intensity. Additionally, there is a growth in the size of the bubbles, which is caused by an increase in the specific volume of air at the points of bubble formation. This leads to an increase in the capillary pressure inside the bubbles and, as a consequence, an increase in the stability of larger bubbles.