## 3 Results and Discussion of the Research Results  
An example of the structure of non-metallic inclusions in two heats is shown in Figure 1.  
In the analyzed heat, the largest share of inclusions, amounting to approx. 40%, was found for Al\({}_{2}\)O\({}_{3}\) of all inclusions constituting the oxide phases (Figure 1). Another oxide inclusion observed in the amount of approx. 13% is SiO\({}_{2}\). Inclusions of FeO, CaO, and MgO compounds occur in amounts of about 10%, while Cr\({}_{2}\)O\({}_{3}\) and MnO are slightly below 10%. The relative volume of inclusions for individual dimensional ranges of impurities is shown in Figure 2.  
Figure 1: Non-metallic inclusions for two heats. Figure 2: The relative volume of inclusions for different dimensional ranges of impurities. The average impurity diameter for individual dimensional ranges of impurities is shown in Figure 3.  
Impurity spacing \(\lambda\) for individual dimensional ranges of impurities is shown in Figure 4.  
Impurity spacing \(\lambda\) for the volume of non-metallic inclusions is shown in Figure 5.  
The share of inclusions in other tested heats may slightly differ from the values presented for heat no. 2. This is due to the individual course of the process on an industrial  
Figure 4: Impurity spacing \(\lambda\) for different dimensional ranges of impurities. Figure 5: Impurity spacing \(\lambda\) for volume of non-metallic inclusions. Figure 3: Average impurity diameter for different dimensional ranges of impurities. scale and, among others, the difference in the charge of steel scrap, which may introduce exogenous impurities. Analyzing the relative volume of inclusions for individual dimensional ranges of impurities (Figure 2), it was found that the largest volume is occupied by fine inclusions with a diameter of less than 2 \(\upmu\)m, followed by inclusions with a diameter of 10 \(\upmu\)m and more, which account for about 60% of the volume of fine inclusions with a diameter of below 2 \(\upmu\)m. The average impurity diameter of inclusions from 10 \(\upmu\)m is close to about 11 \(\upmu\)m (Figure 3). Thus, inclusions with a diameter significantly exceeding 10 \(\upmu\)m are very rare. Inclusions with a diameter of 10 \(\upmu\)m occur at an average distance of \(\lambda\) = 170 \(\upmu\)m from each other (Figure 4). Inclusions with a diameter smaller than 2 \(\upmu\)m have an average diameter of about 1.8 \(\upmu\)m (Figure 3) and are spaced about 12 \(\upmu\)m apart. Inclusions with diameters of 2 to 5 \(\upmu\)m and 5 to 10 \(\upmu\)m occur in a similar amount. This is evidenced by the volume ratio (Figure 2) for both of these impurity diameter ranges. For this reason, in Figures 3 and 4, both of these diameter ranges have been merged into one, creating a range from 2 to 10 \(\upmu\)m. The average size of inclusions in the diameter range of 2-10 \(\upmu\)m is about 3.7 \(\upmu\)m. They are separated by an average distance of about 2.3 \(\upmu\)m. Comparing Figures 2 and 4, it was found that the steel contains the largest number of fine inclusions with a diameter of less than 2 \(\upmu\)m. Non-metallic inclusions with a diameter of 10 \(\upmu\)m, commonly recognized in the literature as harmful [44], are present in the tested steel in a very small amount. Impurity spacing \(\lambda\) for all volumes of non-metallic inclusions (Figure 5) shows the proportional relationship described by the first degree line. Its analysis showed a decrease in the distance between the inclusions and an increase in their share of the steel volume. Which is consistent with the known theory and needs no comment. Impurity spacing \(\lambda\) for the volume of non-metallic inclusions and the value of the correlation coefficient r are presented in Equation (5). Figure 2.0  
The bending fatigue strength coefficient k of tested steel after hardening and tempering at 200 \({}^{\circ}\)C depends on impurity spacing \(\lambda\) and is presented in Figure 2.1  
The regression equation and the value of the correlation coefficient r are presented in Equation (6).