Numerical Simulation of In Situ Reaction Synthesis of TiC-Reinforced Steels from Elemental Fe-Ti-C Powders

Abstract

This research presents a study of the thermal synthesis of titanium carbide (TiC) within the Fe-Ti-C system, utilizing a physically grounded semi-empirical mathematical model. The model enables estimation of TiC formation, including nucleation, growth, and the associated time-temperature regime. It incorporates numerical descriptions of thermal, diffusion, and chemical processes, accounting for phase transformations and their dependencies on chemical composition. The study’s key focus is on assessing the impact of system parameters, including chemical composition and furnace temperature, on the kinetics of TiC thermal synthesis. Simulation results demonstrate the profound influence of Fe content on combustion temperature and particle sizes, aligning closely with experimental data. The research not only examines average size values but also delves into carbide particle size distribution. The findings provide valuable insights into optimizing Fe-TiC composites production processes and the developed computer model serves as a useful tool for preliminary investigations, and it complements experimental research, offering a holistic understanding of system behavior and responses to parameter variations.

Appendices

Appendix 1. Heat Transfer Calculation Data and Equations

Α coefficient dictated by the Nusselt number (Nu):

$$\alpha = \frac{{{\text{Nu}} \cdot \lambda }}{l}$$

(11)

where λ is thermal conductivity coefficient of the environment, W/m·K; l is a characteristic size, m.

The empirical equations used for the Nusselt number estimation:

For the flat horizontal surfaces:

$${\text{Nu}}_{flat} = m \cdot \left( {\Pr \cdot {\text{Gr}}} \right)^{n}$$

(12)

For the cylindrical side surfaces

$${\text{Nu}}_{{{\text{side}}}} = m \cdot \left( {\Pr_{{T - {\text{env}}}} \cdot {\text{Gr}}} \right)^{n} \cdot \left( {\frac{{\Pr_{{T - {\text{env}}}} }}{{\Pr_{{T - {\text{surf}}}} }}} \right)^{p}$$

(13)

where Pr_T-env is Prandtl number of the environment at the common environment temperature; Pr_T-env is Prandtl number of the environment at the heat transfer surface temperature.

Values of the empirical parameters in Eq 12 and 13 depending on the Pr × Gr product are given in Table

Table 2 Empirical parameters for the Nusselt number calculation

2.

The Grashof number:

$${\text{Gr}} = \frac{{g \cdot l^{3} \cdot \beta \cdot \Delta T}}{{\upsilon^{2} }}$$

(14)

where g is gravitation acceleration, m/s²; l is a characteristic size, m; β is the coefficient of volume expansion of the environment, K^-1; ΔT is the difference between the object surface and the environment temperatures, K; υ is the kinematic viscosity of the environment, m2/s.

The effective approximate value of volume expansion coefficient β:

$$\beta = \frac{{T_{1} + T_{2} }}{{2 \cdot T_{1} \cdot T_{2} }}$$

(15)

where T₁ and T₂ are, correspondently, the bulk and near the surface (≈ the sample surface temperature) temperatures of the environment.

The temperature dependences of the kinematic viscosity υ and thermal conductivity coefficient of the environment λ were given through a second-degree polynomial with empirical coefficients (16):

$$\nu ,\lambda = a_{0} + a_{1} \cdot T + a_{2} \cdot T^{2}$$

(16)

The following values of the factors a₀, a₁ and a₂ were used:

For kinematic viscosity: a₀ = 1.31 × 10⁻⁵, a₁ = 9.97 × 10⁻⁸, a₂ = 6.49 × 10⁻¹¹

For thermal conductivity: a₀ = 0.0246, a₁ = 7.57 × 10⁻⁵, a₂ = − 2.02 × 10⁻⁸

For the radiation heat transfer the heat flux from/to the surface:

$$q = \varepsilon_{{{\text{np}}}} \cdot C_{0} \cdot \left( {T_{{{\text{out}}}}^{4} - T_{{{\text{surf}}}}^{4} } \right)$$

(17)

where ε_np is emissivity factor; C₀ is black body emissivity, W/m²·K⁴ C₀ = 5.67 × 10⁻⁸; T_out is the temperature at the muffle wall surface (≈ inner environment temperature), K; T_surf is the temperature of the sample surface, K.

The reduced emissivity factor ε_np depends of the ones of the sample and the muffle wall:

$$\varepsilon_{{{\text{np}}}} = \frac{1}{{\frac{1}{{\varepsilon_{1} }} + \frac{1}{{\varepsilon_{2} }} - 1}}$$

(18)

where ε₁ and ε₂ are correspondently the emissivity factors of the sample and the muffle wall.

The true values of the emissivity factors are not precisely known and could somewhat differ in real systems. For the modeling purpose they assumed to be dependent on temperature according to the following proposed empirical formulas for their estimation:

$${\text{For}}\;{\text{the}}\;{\text{sample}}:\varepsilon = 1.13 \cdot e^{{ - \frac{977}{T}}}$$

(19)

$${\text{For}}\;{\text{the}}\;{\text{muffle}}\;{\text{wall}}:\varepsilon = 1.56 \cdot e^{{ - \frac{1479}{T}}}$$

(20)

So the resulting emissivity factor ε_np varied from ~ 0.05 to ~ 0.6 in the considered temperature range.

Appendix 2. Thermophysical Properties Used in the Simulation

Reactions heat effects estimation equations (grounded on the data from Ref 46, 47):

$$\begin{aligned} - \Delta H_{TiC} = & 186.7 + (15.86 + 12.03 \times 10^{ - 3} \cdot T + 9.67 \times 10^{ - 6} \cdot T^{2} - 6051.5 \times T^{ - 1} ) \\ & \; + ( - 22.80 + 6.17 \times 10^{ - 3} \cdot T - 3.09 \times 10^{ - 6} \cdot T^{2} + 6326.2 \times T^{ - 1} ) \cdot x_{C} + \\ & \; + ( - 5.680 + 19.06 \times 10^{ - 3} \cdot T) \cdot x_{C}^{2} \;\;\;\;\;\;\;{\text{kJ}}/{\text{mol}} \\ \end{aligned}$$

(21)

$$- \Delta H_{{{\text{TiFe}}_{2} }} = 7.8125 + 0.002136 \cdot T + 2{9}{\text{.375}} \cdot x_{{{\text{Fe}}}} \;\;\;{\text{kJ}}/{\text{mol}}$$

(22)

where x_C and x_Fe are, correspondently, atomic fractions of C and Fe in the formed compound assuming that TiC and TiFe₂ have wide homogeneity regions, and the value of enthalpy depends on the exact composition.

The temperature dependences of the specific heat (C_p) and density (ρ) values were approximated by second degree polynomial equations like 23:

$$C_{p} ,\rho = a_{0} + a_{1} \cdot T + a_{2} \cdot T^{2}$$

(23)

The values of the empirical factors a₀, a₁ and a₂ are presented in Table

Table 3 Empirical parameters for the specific heat and density values calculation by Eq 23

3. The presented values are the result of our approximations made based on data collected from different resources such as internet searches, paper reference books, software databases, etc.

Appendix 3. Equations Used for Particles Nucleation Description

The excess phases’ particles nucleation is described by the equation from Ref 48:

$$\frac{{{\text{d}}N}}{{{\text{d}}t}} = N_{n} \cdot F_{{\text{Z}}} \cdot \beta \cdot \exp \left( { - \frac{{\Delta G_{C} }}{{K_{{\text{B}}} \cdot T}}} \right)$$

(24)

where N_n is a volumetric density of the potential nucleation centers, m-⁻³; F_Z is the Zeldovich factor [49]; β is a factor corresponding to the diffusive accession of the atoms to the nuclei of critical size; ΔG_C is a thermodynamic barrier of nucleation, J/mol.

The Zeldovich factor F_Z allows to conceder the possibility of the fluctuation dissolution of the nuclei with a size more than the critical one. It can be estimated as follows:

$${\text{F}}_{{\text{Z}}} = \sqrt {\frac{{\Delta G_{C} }}{{3 \cdot \pi \cdot R \cdot T \cdot n_{C}^{2} }}}$$

(25)

where n_c is the number of the effective elemental volumes of the compound in the nuclei.

If to assume a spherical shape of the nuclei n_c could be calculated by the following formula 26:

$$n_{c} = \frac{4}{3} \cdot \pi \cdot \frac{{R_{{\text{C}}}^{3} }}{{V_{a} }}$$

(26)

where R_C is the critical radius of the nuclei, m; V_a—the conventional effective elemental volume of the compound.

The conditional elementary volume V_a is calculated as the ratio of the molar volume of a given compound to the Avogadro’s number.

The critical radius R_C is calculated according to Ref 50 by Eq 27:

$$R_{{\text{C}}} = - \frac{{2 \cdot \gamma_{{{\text{p}}/\gamma }} }}{{\Delta G_{p} }}$$

(27)

where γ_p/γ is the specific surface energy on the border particle/matrix, J/m²; ΔG_p is the volumetric Gibbs energy change, J/m³.

The value of ΔG_p was estimated in the way like in [51]:

$$\Delta G_{p} = - \frac{R \cdot T}{{V_{{\text{m}}} }} \cdot \ln \frac{{X_{{{\text{Me}}}} \cdot X_{C}^{*} }}{{X_{{{\text{Me}}}}^{{{\text{eq}}}} \cdot X_{C}^{{*{\text{eq}}}} }}$$

(28)

where V_m is the molar volume of the compound, m³/mol; X_Me and X_C^* are molar parts of the components (if it is a carbide X_C^* is the part of carbon); X_Me^eq and X_C^*eq are the equilibrium molar parts of the components in the solid solution.

It was assumed that the nuclei appear at the dislocations grid modes. So the potential nucleation sites density was estimated according to Ref 48 by formula 29:

$$N_{0} = 0.5 \cdot \rho_{{\text{d}}}^{1.5}$$

(29)

where ρ_d is dislocation density, m⁻¹.

According to the considerations from Ref 50 and 52 the dislocation density was estimated as follows (30):

$$\rho_{{\text{d}}} = \left( {\frac{{\sigma_{{\text{Y}}} }}{\alpha \cdot \mu \cdot b}} \right)^{2}$$

(30)

where σ_Y is the yield strength of the phase, Pa; μ is the shear modulus, Pa; b is the Burgers vector modulus value, m; α is an empirical factor.