# Understanding TiN Precipitation Behavior during Solidification of SWRH 92A Tire Cord Steel by Selected Thermodynamic Models

^{1}

^{2}

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## Abstract

**:**

## 1. Introduction

_{x}N

_{1-x}, x represents the molar ratio of TiC in TiC

_{x}N

_{1-x}), a continuous solid solution formed via replacing partial moles of N in TiN crystal with C has similar properties to those of TiN. It also has a detrimental effect on the fatigue performance and, as a result, leads to wire breaking during the drawing and stranding processes [7]. It has been reported [8] that the molar ratio of TiC increases with increasing strength of tire core steel, which would cause a more seriously destructive effect. However, the value of x in TiC

_{x}N

_{1-x}is still very small [8]. In other words, the main composition of Ti-bearing inclusion precipitated in tire cord steel is still TiN. Thus, it is important to control TiN inclusion to improve the performance of SWRH 92A tire cord steel.

## 2. Material and Equilibrium Solubility Product

_{L}) and solidus temperature (T

_{S}) as well as the equilibrium solubility product of N and Ti were first calculated.

_{L}and T

_{S}[15], respectively,

_{Fe}was the melting point of pure Fe, 1811 K; $\mathsf{\Delta}{t}_{\mathrm{L}}$ and $\mathsf{\Delta}{t}_{\mathrm{S}}$ were the reduced temperature values for element i when the mass fraction was 1 mass%, K, the corresponding values can be acquired from Table 2 [15]; w

_{[i]}represented the mass fraction of element i, 1 mass% was considered as the unit. Combining Table 1, Table 2, Equations (1) and (2), the values of T

_{L}and T

_{S}can be calculated, i.e., T

_{L}= 1748 K, T

_{S}= 1636 K.

## 3. Thermodynamic Analysis

#### 3.1. Segregation Models

_{C}= 0.34, k

_{N}= 0.48, and k

_{Ti}= 0.30 [18,19,20]; g represents the solid fraction; ϕ (in the range of 0–1) denotes the inverse diffusion coefficient and α is the Fourier parameter.

#### 3.2. Usage of the LRSM Model

_{L-S}) and solid fraction (g) can be expressed by Equation (23) [22],

_{L-S}= 1637 K) can be easily deduced, which is almost the same as the theoretical solidus temperature (T

_{S}= 1636 K) of the studied tire cord steel. This result suggests that TiN will not precipitate in the mushy zone until nearly close to complete solidification.

#### 3.3. Usage of Ohnaka Model on Considering the Effect of Carbon on SDAS L

^{2}/s), SDAS L [cm, the unit of L was converted from μm (calculated by Equations (29) and (31)) to cm for the calculation in Equation (28), corresponding to the unit of ${D}_{i}^{\gamma}$], and the local solidification time τ (s), as seen in Equation (28) [27],

_{C}(K/s) and the carbon concentration ${w}_{[\mathrm{C}]}$, as expressed by Equation (29) [28]; the local solidification time τ (s) was calculated by Equation (30) [21,29].

#### 3.4. Use of the Ohnaka Model without Considering the Effect of Carbon on SDAS L

_{C}(K/s) only, see Equation (31),

## 4. Conclusions

- (1)
- Precipitation of TiN will not occur in the liquid phase region regardless of the selected micro- segregation models.
- (2)
- When adopting the LRSM and Ohnaka (without considering the effect of carbon on secondary dendrite arm spacing (SDAS L)) models, TiN will precipitate in the mushy zone at the very late 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.
- (3)
- Results of different segregation models show that the Ohnaka model (considering the effect of carbon on SDAS L) is similar to the Lever-rule model at the very late stage during the solidification process; however, for the case without considering the effect of carbon on SDAS L, the result is similar to the Scheil model.
- (4)
- Due to the fact that different segregation models may lead to different results, more attention should be paid to selecting the appropriate model or developing new models to analyze the actual segregation phenomena. Besides, further experimental work and theoretical analysis to understand in depth the precipitation behavior of TiN in SWRH 92A tire cord steel will be required in the future.

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

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**Figure 1.**Required solubility product of N and Ti for the precipitation of TiN inclusion in SWRH 92A tire cord steel.

**Figure 2.**Diffusion coefficients of solute elements N and Ti in the γ-Fe phase at different temperatures.

**Figure 3.**Comparison of equilibrium solubility product with the calculated value obtained by the LRSM (Lever-rule model was applied for the N and Scheil model for Ti) model.

**Figure 4.**Effects of cooling rates on the segregation ratios of solute elements N and Ti during the solidification process (the results obtained by the Lever-rule model are given as a reference).

**Figure 6.**Comparisons of equilibrium solubility product with the calculated value obtained by the Ohnaka model (considering the effect of carbon on SDAS L).

**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).

**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).

Elements | Si | P | S | O | Mn | N | Ti |
---|---|---|---|---|---|---|---|

Content/mass% | 0.18 | 0.018 | 0.0064 | 0.0018 | 0.51 | 0.0043 | 0.0005 |

**Table 2.**Values of $\mathsf{\Delta}{t}_{\mathrm{L}}$ and $\mathsf{\Delta}{t}_{\mathrm{S}}$ in Equations (1) and (2) [15], respectively.

Elements | C | Si | P | S | O | Mn | N | Ti |
---|---|---|---|---|---|---|---|---|

$\mathsf{\Delta}{t}_{L}$ | 65 | 8 | 30 | 25 | 80 | 5 | 90 | 20 |

$\mathsf{\Delta}{t}_{\mathrm{S}}$ | 175 | 20 | 280 | 575 | 160 | 30 | - | 40 |

${\mathit{e}}_{\mathit{j}}^{\mathit{i}}(\mathit{i}\to )$ | C | Si | P | S | O | N | Mn | Ti |
---|---|---|---|---|---|---|---|---|

${e}_{\mathrm{Ti}}^{i}$ | −0.165 | 0.05 | −0.0064 | −0.11 | −1.8 | −1.8 | 0.0043 | 0.013 |

${e}_{\mathrm{N}}^{i}$ | 0.13 | 0.047 | 0.045 | 0.007 | 0.05 | 0 | −0.021 | −0.53 |

**Table 4.**Micro segregation models for solute elements during the solidification process [14].

Models | Equation | Conditions | No |
---|---|---|---|

Lever-rule | ${w}_{[i]}/{w}_{[i]}^{0}={[1-(1-{k}_{i})g]}^{-1}$ | Complete diffusion both in liquid and γ-Fe phase | (1) |

Scheil model | ${w}_{[i]}/{w}_{[i]}^{0}={(1-g)}^{{k}_{i}}{}^{-1}$ | Complete diffusion in liquid and no diffusion in γ-Fe phase | (2) |

Basic equations | ${w}_{[i]}/{w}_{[i]}^{0}={[1-(1-\varphi {k}_{i})g]}^{({k}_{i}-1)/(1-\varphi {k}_{i})}$ | Complete diffusion in liquid and finite diffusion in γ-Fe phase | (3) |

Brody–Fleming model | $\varphi =2\alpha $ | (4) | |

Ohnaka model | $\varphi =4\alpha /(1+4\alpha )$ | (5) | |

Clyne–Kurz model | $\varphi =2\alpha (1-{e}^{-\frac{1}{\alpha}})-{e}^{-\frac{1}{2\alpha}}$ | (6) |

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**MDPI and ACS Style**

Wang, L.; Xue, Z.-L.; Chen, Y.-L.; Bi, X.-G.
Understanding TiN Precipitation Behavior during Solidification of SWRH 92A Tire Cord Steel by Selected Thermodynamic Models. *Processes* **2020**, *8*, 10.
https://doi.org/10.3390/pr8010010

**AMA Style**

Wang L, Xue Z-L, Chen Y-L, Bi X-G.
Understanding TiN Precipitation Behavior during Solidification of SWRH 92A Tire Cord Steel by Selected Thermodynamic Models. *Processes*. 2020; 8(1):10.
https://doi.org/10.3390/pr8010010

**Chicago/Turabian Style**

Wang, Lu, Zheng-Liang Xue, Yi-Liang Chen, and Xue-Gong Bi.
2020. "Understanding TiN Precipitation Behavior during Solidification of SWRH 92A Tire Cord Steel by Selected Thermodynamic Models" *Processes* 8, no. 1: 10.
https://doi.org/10.3390/pr8010010