*3.4. Use of the Ohnaka Model without Considering the E*ff*ect of Carbon on SDAS L*

As mentioned above, when considering the effect of carbon on SDAS *L*, TiN will not precipitate during the solidification process. However, in other references [13,22,30] on analyzing the segregation ratio of different solute elements, the SDAS *L* (herein, the unit of *L* calculated by Equation (31) is μm) was calculated as a function of the cooling rate *R*<sup>C</sup> (K/s) only, see Equation (31),

$$L = 688 \cdot RC^{-0.36} \tag{31}$$

Similarly, by substituting Equations (27), (28), (30), and (31) and the equilibrium distribution coefficients and diffusion coefficients of different solute elements into Equation (24), the corresponding segregation ratios of the solute elements N and Ti during the solidification process can be obtained, as shown in Figure 7. From Figure 7 it can be seen that both N and Ti show a strong segregation tendency especially in the latter period and in comparison, the segregation ratio of Ti is much bigger than that of N. In addition, the effect of cooling rate on the segregation ratio can be nearly ignored (the plot is not given in the current paper). Therefore, it is possible for TiN to precipitate even though the initial concentrations of N and Ti are very low, as shown in Figure 8, and in which the critical solid fraction is a little smaller than that obtained by the LRSM model (0.98 vs. 0.9966). That is to say, TiN can precipitate a little earlier. Anyhow, TiN is only generated at the very late stage closing to the complete solidification of the molten steel.

**Figure 7.** Segregation ratio of solute elements N and Ti during the solidification process when Equation (31) was used with a cooling rate of 10 K/s.

**Figure 8.** Comparison of equilibrium solubility product with the calculated values obtained by the Ohnaka model (without considering the effect of carbon on SDAS *L*).

Similar to the previous calculation, another surprising phenomenon is also found in this case. The segregation ratios of solute elements N and Ti are nearly the same as those obtained by the Scheil model, as shown in Figure 9. As mentioned above, when the inverse diffusion coefficient φ equals zero, Equation (24) will turn into the Scheil model. In the current discussion, φ nearly equals zero for both N and Ti, as seen in Figure 10. That is to say, both N and Ti almost completely diffuse in the liquid and have no diffusion in the γ-Fe phase in this situation. The possible reason for this result may be due to characteristics of the microstructure (mostly likely due to the much larger SDAS *L*) between the liquid and solid phases, which makes the inverse diffusion of solute elements insufficient.

**Figure 9.** Comparison of segregation ratio of solute elements N and Ti when adopting the Ohnaka (without considering the effect of carbon on SDAS *L*) and Scheil models.

**Figure 10.** Inverse diffusion coefficients of solute elements N and Ti when adopting the Ohnaka model (without considering the effect of carbon on SDAS *L*).

#### **4. Conclusions**

According to the calculated results, it is possible to summarize the precipitation behavior of TiN inclusion in SWRH 92A tire cord steel during the whole solidification process, as follows:


stage of the solidification process, with solid fractions larger than 0.9966 and 0.98, respectively. When considering the effect of carbon on SDAS *L* for the Ohnaka model, TiN will not precipitate in both the liquid phase and mushy zone.


**Author Contributions:** Writing-original draft, methodology and review, L.W.; Methodology, visualisation and investigation, Z.-L.X.; Formal analysis, Y.-L.C.; Writing-review and editing, X.-G.B. All authors have read and agreed to the published version of the manuscript.

**Funding:** This research received no external funding.

**Acknowledgments:** The authors gratefully acknowledge the financial support from the National Natural Science Foundation of China, and the Special Project of Central Government for Local Science and Technology Development of Hubei Province (51874214 and 2019ZYYD076, respectively).

**Conflicts of Interest:** The authors declare no conflict of interest.
