Prediction of Final Solidification Position in Continuous Casting Bearing Steel Billets by Slice Moving Method Combined with Kobayashi Approximation and Considering MnS and Fe₃P Precipitation

Abstract

Control of MnS and Fe₃P precipitate are of vital importance for the quality of the bearing steels. The precipitation behavior is not only related to shortening the bearing steel’s fatigue life, but also to another serious engineering problem i.e. changing the billet final solidification position. In order to distinguish the different precipitate behavior on the influencing of the final solidification position, a slice moving method combined with Kobayashi approximation and the MnS and Fe₃P precipitation is developed. The continuous casting billet of seven-component bearing steel, i.e. Fe–C–Cr–Mn–Si–P–S system, is considered as the raw material. Upon the present chemical composition of 0.004 to 0.007 mass% S and 0.011 to 0.012 mass%P in 100Cr6 (DIN-Norm) and RAD1 (GB-Norm) alloy, the MnS but not Fe₃P precipitate covers the billet cross section. The onset of Fe₃P precipitation is at 0.019 mass% P in 100Cr6 alloy and 0.021 mass% P in RAD1 alloy. The distribution of the maximum amount of MnS and Fe₃P precipitate is similar, i.e. concentrating at the billet center. The increase of P composition, besides accelerating the precipitation of MnS and Fe₃P, elongates the liquid core length. In contrast, the increase of S composition and the precipitation of MnS greatly shortens the liquid core length. Thus it is vital to control composition of S, P solute to a low level in bearing steels in order to stabilize the final solidification position.

1. Introduction

Better abrasion and fatigue resistance against the long-time alternated load are always expected for bearing steels such as RAD1 (GB-Norm) and 100Cr6 (DIN-Norm) series. It runs as rolling bearing at a high speed, however, the central segregation, porosity and shrinkage, etc. internal defects always influence the alloys’ mechanical properties, thus reduce the life time of this type of engineering materials. For instance, the solute segregation and subsequent carbide eliquation are reported to be responsible for the accumulation of local stress and the consequent micro-cracks under the alternated loads.¹⁾

In order to reduce the internal defects, the electromagnetic stirring and soft reduction are frequently applied to the continuous casting bearing steel billets. The electromagnetic stirring at the final-stage of the solidification aims to uniform the solute distribution as well as break up the network-like bridging in dendrites at mushy zone. The impose of soft reduction ahead of the final solidification regime is proposed to reduce the central segregation and shrinkage. Both of the techniques need to be positioned properly and accurately upon the final solidification zone to obtain a sound performance.

The final solidification position in continuous casting billet is mainly influenced by alloy composition and cooling conditions and is always clarified through the numerical modeling such as slice moving method combined with experimental measurements like nail-shooting test. However, in the method, the microsegregation approximation is simplified using a linear dependence of the solid fraction on temperature, i.e. $f_s=\frac{T_{\text{L}}-T}{T_{\text{L}}-T_{\text{S}}}$, as solving the unsteady heat conduction equation, which usually limits the accuracy of the results.

The microsegregation approximations embedded in Fluent etc. commercial software are usually for binary alloys, in which the solidification path, i.e. the dependence of solute composition, phase fraction etc. on the temperature, is described by Lever rule (LR),²⁾ assuming the infinite diffusion of solutes, or Gulliver-Scheil (GS) approximation,^3,4,5,6) assuming no diffusion of solutes in solids. Although Brody-Flemings,⁷⁾ Clyne-Kurz⁸⁾ and Ohnaka⁹⁾ etc. considered the finite diffusion of solutes in solid, their models are valid still for binary alloys and the solidification between liquid and a single solid phase. Since the actual alloy is a multi-component system, for example, the bearing steel contains Cr, Mn, P, S, Si and Ti, etc. solutes besides the fundamental elements of Fe and C, the influence of multiple components’ interaction on the solidification path needs to be considered. Thus Kobayashi’s exact analytical solution of Brody-Flemings approximation^10,11,12,13) has gained favor in proposing the analytical expression of the solidification path for a multi-component system with considering the finite diffusion of solutes in solids. Moreover, upon the RAD1 and 100Cr6 alloy, besides the primary austenite phase, the containing of the Mn, P, S solute leads to the precipitation of MnS and Fe₃P particles, which changes the solute composition in residual melt and furthermore alters the alloy solidification path and the final solidification position.

The tabulation of the phase transformation path through prescription with a thermodynamic database is another efficient way to obtain the phase transformation path in multicomponent alloys. However, since the calculation cost for coupling of the tabulated phase transformation path with the macroscopic transport modeling is increasing with the increase of the solute number, the tabulation is just applied in the macroscopic transport calculation of a binary or ternary alloy system at present.¹⁴⁾

In this paper, the Kobayashi’s microsegregation approximation for multi-component alloy combining with the prediction of MnS and Fe₃P precipitation was adopted to obtain the dependence of phase fraction, solute composition and temperature. They are associated with the unsteady heat transfer equation and the slice moving method to predict the solidification of continuous casting RAD1 and 100Cr6 bearing steel billets. The solidified shell thickness distribution along the casting direction and the final solidification position were calculated and verified with the nail-shooting experiments. Subsequently the influence of S, P composition on the distribution of MnS and Fe₃P precipitate and the sequent final solidification position were discussed. This work aims to shed light on establishing a new simulation methodology to predict the final solidification position considering the precipitate influence. Benchmark of the proposed method in this work also aims for a wide application in comprehensive grades of steels and alloys.

2. Methodology

The continuous casting is a complex process combining of heat transfer, fluid flow and solute transport. In the steel manufacturing research, the solidification of the billet is always simplified and modeled through the slice moving method in which an unsteady heat conduction in billet’s cross-section (i.e. a two-dimensional slice moving with the casting) is calculated. It means that the heat transfer along casting direction is neglected and an enlarged heat conductivity (i.e. equivalent heat conductivity) for the enhancement of heat transfer via melt flow is adopted.

The following assumptions are made in the model:

1) The slice moves downward from the meniscus with the billet casting speed and the surface cooling intensity is altered as passing through each chilled zone of the continuous caster.

2) A quarter of the rectangular billet is selected for the current study, assuming the symmetric billet structure and cooling condition.

3) The heat conductivity in solid billet is a function of temperature, and other physical properties are set as constant.

4) The influence of fluid flow on the solidification and heat transfer is considered by using an equivalent thermal conductivity.

2.1. Heat Conduction Model

The two-dimensional unsteady heat conduction equation is written as:   

$$\frac{\partial H}{\partial t}=\nabla \cdot (\lambda \nabla T)$$(1)

Where, H is the average enthalpy per unit volume, J/m³; T is the temperature, °C; λ is the heat conductivity, W/m·K.

The enthalpy field is then obtained through the explicit discretization of Eq. (1),   

$$H_{n+1}=H_n+\nabla \cdot (\lambda \nabla T_n)\mathrm{\Delta }t$$(2)

2.2. Temperature and Solid Fraction Through Kobayashi Approximation

A kinetic microsegregation model, i.e. Kobayashi approximation¹⁰⁾ was adopted here to obtain the temperature and solid fraction from the local enthalpy in the RAD1 and 100Cr6 bearing steels. Kobayashi approximation is aimed to predict the microsegregation in the scale of secondary dendrite arm spacing. It concerns the cooling rate and the finite diffusion of solutes including carbon in solids in a multi-component alloy system. The concerned compositions of the RAD1 and 100Cr6 bearing steels are listed in Table 1.

Table 1. Composition of RAD1 and 100Cr6 bearing steels (mass%).

Sample No	Alloy	C	Cr	Mn	Si	P	S
No. 380	RAD1	0.99	1.45	0.30	0.26	0.011–0.026	0.004–0.015
No. 382	100Cr6	0.94	1.48	0.39	0.28	0.012–0.026	0.007–0.015

The equilibrium temperature T in °C of each node is calculated through   

$$\frac{\text{1}}{\text{(}T+\text{273)}}=\frac{\text{1}}{\text{(}{T}_{\text{L}}^{\text{0}}+\text{273)}}+\frac{R}{\mathrm{\Delta }{H}_{\text{LS}}}{\displaystyle \sum }_j(\text{1}-k_j^{\text{S/L}})\frac{C_{j\text{,n}+\text{1}}}{\text{100}{W}_j}$$(3)

where $T_{\text{L}}^{\text{0}}$ is the melting temperature of pure solvent, here is Fe, °C; R is the gas constant, J/(mol·K); ΔH_LS is the latent heat of liquid/solid transformation, J/kg; $k_j^{\text{S/L}}$ is the partition coefficient of solute j at solid/liquid interface, $k_j^{\text{S/L}}=C_j^{\text{S}}/C_j^{\text{L}}$; W_j is the atomic weight of solute j, kg/mol; C_j,n+1 is expressed as:   

$$\begin{array}{l}{C}_{j\text{,n}+\text{1}}=\\ (C_{j\text{,n}}-\mathrm{\Delta }{C}_{j\text{,n}}){\left(\frac{P_{j\text{,n}+\text{1}}}{P_{j\text{,n}}}\right)}^{{\zeta }_j}\left[\text{1}+\frac{k_{\text{j}}^{\text{S/L}}(\text{1}-k_j^{\text{S/L}}){\beta }_j^{\text{3}}}{\text{2}{\gamma }_{\text{j}}{(\text{1}-{\beta }_j{\text{k}}_j^{\text{S/L}})}^{\text{3}}}\left(Q_{j\text{,n}+\text{1}}-Q_{j\text{,n}}\right)\right]\end{array}$$(4)

whereas, subscript n and n + 1 represent two sequent time interval; C_j,n+1 and C_j,n are the liquid composition of solute j in mass% at time interval n + 1 and n; ΔC_j,n is the composition change of solute j due to the phase precipitation; other parameters are defined as,   

$${\gamma }_j=\frac{\text{8}{D}_j\mathrm{\Delta }t}{{\lambda }^{\text{2}}(f_{\text{s,n}+\text{1}}^{\mathrm{\hspace{0.17em} }\text{2}}-f_{\text{s,n}}^{\mathrm{\hspace{0.17em} }\text{2}})}$$(5)

  

$${\beta }_j={\gamma }_j/(\text{1}+{\gamma }_j)$$(6)

  

$$P_{j\text{,n}}=\text{1}-(\text{1}-{\beta }_j{k}_j^{\text{S/L}})f_{\text{s,n}}$$(7)

  

$${\zeta }_j=\frac{k_j^{\text{S/L}}-\text{1}}{\text{1}-{\beta }_j{k}_j^{\text{S/L}}}$$(8)

  

$$\begin{array}{l}{Q}_{j\text{,n}}=\left(\text{1}-\frac{\text{1}+{\beta }_j}{\text{2}}{k}_j^{\text{S/L}}\right)\frac{\text{1}}{P_{j\text{,n}}^{\text{2}}}-\left(\text{5}-(\text{2}+\text{3}{\beta }_j)k_j^{\text{S/L}}\right)\frac{\text{1}}{P_{j\text{,n}}}\\ \hspace{1em}\hspace{1em}-\left(\text{3}-(\text{1}+\text{2}{\beta }_j)k_j^{\text{S/L}}\right)\text{ln}{P}_{j\text{,n}}\end{array}$$(9)

where, f_s,n and f_s,n+1 are the solid fraction at n and n + 1 time interval; D_j is the diffusion coefficient of solute j in solid, $D_j=D_{\text{0}}\text{exp}\left(-\frac{Q}{R\text{(}T+\text{273)}}\right)$, the values of D₀ and Q for solute j in γ (austenite) phase of the alloy are shown in Table 2, which are obtained by nonlinear curve fitting of the D_j in γ from MOB2 database.¹⁵⁾

Table 2. Solute diffusion coefficients in γ phase, retrieved from MOB2 database.¹⁵⁾

Solute	Phase γ
	D0 × 104 (m2/s)	Q × 10−5 (J/mol)
C	0.295	1.49
Cr	1.76	2.87
Mn	0.154	2.61
Si	0.0694	2.42
P	0.0627	1.93
S	1.71	2.22

The secondary dendrite arm spacing (SDAS) λ is determined by the experimental correlation:   

$$\lambda =\text{A}\cdot {\dot{T}}^{-\text{m}}$$(10)

whereas the constant A and m for Fe-1 mass% C-1.5 mass% Cr based alloy are chosen as A = 5.502 × 10⁻⁵ m, m = 0.55;¹⁶⁾ the cooling rate $\dot{T}$ between liquidus and solidus is approximated by   

$$\dot{T}=\frac{T_{\text{L}}-T}{t-t_{\text{L}}}\text{,}\mathrm{\hspace{0.17em} }\mathrm{\hspace{0.17em} }\mathrm{\hspace{0.17em} }t>t_{\text{L}}$$(11)

Here, T_L and T are the liquidus and local temperature; t_L and t are the solidification start and local time, respectively.

Solution of solid fraction and temperature from enthalpy is as follows: Firstly, calculate the enthalpy from Eq. (2); then estimate a value of f_S,n+1 at time interval n + 1 to obtain the liquid composition (Eq. (4)) and temperature (Eq. (3)); adjust f_S,n+1 to get the balance among enthalpy, temperature and solid fraction through Eq. (12); thus obtain the converged solid fraction and temperature.   

$$H_{\text{n}+\text{1}}=\rho \mathrm{\Delta }{H}_{\text{LS}}(\text{1}-f_{\text{s,n}+\text{1}})+\rho c_{\text{p}}{T}_{\text{n}+\text{1}}$$(12)

whereas c_p is the specific heat, J/(kg·K).

2.3. MnS Precipitation

With solidification, the composition of solute Mn and S enriches in inter-dendritic melt. Once their local solubility product exceeds the equilibrium one, K_MnS (=[%Mn][%S]), MnS can precipitate as   

$$[\text{Mn}]+[\text{S}]=\text{MnS}$$(13)

K_MnS is expressed as¹⁷⁾   

$$\text{log(}{K}_{\text{MnS}}\text{)}=-\frac{\text{8}\mathrm{\hspace{0.17em} }\text{220}}{\text{(}T+\text{273)}}+\text{5}\text{.022}-\mathrm{\Delta }{G}_{\text{MnS}}$$(14)

  

$$\mathrm{\Delta }{G}_{\text{MnS}}={\displaystyle \sum }_j(e_{\text{Mn}}^j+e_{\text{S}}^j)C_j$$(15)

where T is the temperature in °C; ΔG_MnS is the change of Gibbs free energy; $e_{\text{Mn}}^j$ and $e_{\text{S}}^j$ are the interaction coefficient of solute j to Mn and S, written as $e_m^j=\frac{a_m^j}{\text{(}T+\text{273)}}+b_m^j$ (m = Mn or S); the values of $a_m^j$ and $b_m^j$ are listed in Table 3.

Table 3. Interaction coefficients of Mn, S and P in molten steel at 1873 K.^17,18,19)

Solute m	Solute j	$a_m^j$	$b_m^j$
Mn	C	−1371	0.690
	Cr	6.7418)	0.0
	Mn	0.0	0.0
	Si	−1838	0.964
	P	−6.56	0.0
	S	−89.9	0.0
S	C	206	0.0
	Cr	−20.618)	0.0
	Mn	−48.7	0.0
	Si	134.9	0.0
	P	52.4	0.0
	S	233	−0.153
P	C	3070.0	−1.57
	Cr	0.0	0.0
	Mn	0.0	0.0
	Si	206.0	0.0
	P	0.0	0.0
	S	48.7	0.0

The equilibrium solute composition product K_MnS could also be expressed as   

$$K_{\text{MnS}}=(C_{\text{Mn}}-\mathrm{\Delta }{C}_{\text{Mn}})(C_{\text{S}}-\mathrm{\Delta }{C}_{\text{S}})$$(16)

where C_Mn and C_S are the liquid composition of solute Mn and S; ΔC_Mn and ΔC_S represent the change of composition Mn and S in liquid due to MnS precipitation.

As indicated in Eq. (13), the amounts of [Mn] and [S] consumed in mole fraction are same in the reaction, denoted as X, i.e.,   

$$X=\frac{\mathrm{\Delta }{C}_{\text{Mn}}}{W_{Mn}}\text{=}\frac{\mathrm{\Delta }{C}_{\text{S}}}{W_S}$$(17)

whereas, W_Mn and W_S are the atomic weight of solute Mn and S, kg/mol, respectively.

X is calculated through Eqs. (14), (15), (16), (17) as   

$$X=\frac{PP-\sqrt{P{P}^2-4(QQ-K_{\text{MnS}})/W_{\text{Mn}}/W_{\text{S}}}}{2}$$(18)

where $PP=\frac{C_{\text{Mn}}}{W_{\text{Mn}}}+\frac{C_{\text{S}}}{W_{\text{S}}}$ and QQ = C_Mn·C_S. Then ΔC_Mn and ΔC_S could be obtained from Eq. (17). The amount of MnS precipitate in mass fraction is calculated through Eq. (19),   

$$\mathrm{\Delta }{f}_{\text{MnS}}=(\text{1}-f_{\text{s}})\cdot \mathrm{\Delta }{C}_{\text{S}}\cdot \left(\text{1+}\frac{{\text{W}}_{\text{Mn}}}{W_{\text{S}}}\right)\cdot \left(\frac{\rho }{\text{100}{\rho }_{\text{MnS}}}\right)$$(19)

where, ρ_MnS is the density of MnS, a value of 4000 kg/m^{3 20)} is used here.

2.4. Fe₃P Precipitation

Fe₃P precipitating¹⁹⁾ with solidification is divided into two steps: Once the liquid composition of solute P exceeds the equilibrium composition, Fe₂P first precipitates at the temperature among 1365 to 1160°C; subsequently, Fe₃P precipitates below 1160°C. The precipitation reactions are,   

$$\text{2[Fe]}+[\text{P}]={\text{Fe}}_{\text{2}}\text{P}$$(20)

with equilibrium solubility product,   

$$K_{{\text{Fe}}_{\text{2}}\text{P}}=C_{\text{P,}\mathrm{\hspace{0.17em} }\text{eq}}=\left(\text{21}\text{.7}-\sqrt{\frac{\text{1}\mathrm{\hspace{0.17em} }\text{365}-T}{\text{5}\text{.0}}}\right)\times {\text{10}}^{-\mathrm{\Delta }G}$$(21)

And   

$$\text{3[Fe]}+[\text{P}]={\text{Fe}}_{\text{3}}\text{P}$$(22)

with equilibrium solubility product,   

$$K_{{\text{Fe}}_3\text{P}}=C_{\text{P,}\mathrm{\hspace{0.17em} }\text{eq}}=\left(\text{15}\text{.3}-\sqrt{\frac{\text{1}\mathrm{\hspace{0.17em} }\text{160}-T}{\text{4}\text{.77}}}\right)\times {\text{10}}^{-\mathrm{\Delta }G}$$(23)

  

$$\mathrm{\Delta }G={\displaystyle \sum }_j{e}_{\text{p}}^j{C}_j$$(24)

where T is the temperature in °C; ΔG is the change of free energy; $e_{\text{p}}^j$ is the interaction coefficient of solute j to P, written as $e_m^j=\frac{a_m^j}{\text{(}T+\text{273)}}+b_m^j$ (m = P); the values of $a_m^j$ and $b_m^j$ are listed in Table 3.

Then the change of liquid composition of P due to precipitation is given by:   

$$\mathrm{\Delta }{C}_{\text{P}}=C_{\text{P}}-C_{\text{P,}\mathrm{\hspace{0.17em} }\text{eq}}$$(25)

The amounts of Fe₂P and Fe₃P precipitate in mass fraction are calculated through the following expressions,   

$$\mathrm{\Delta }{f}_{{\text{Fe}}_{\text{2}}\text{P}}=\left[(\text{1}-f_{\text{s}})\cdot \mathrm{\Delta }{C}_{\text{P}}\cdot \left(\text{1}+\frac{\text{2}{W}_{\text{Fe}}}{W_{\text{P}}}\right)\right]\cdot \left(\frac{\rho }{\text{100}{\rho }_{{\text{Fe}}_{\text{2}}\text{P}}}\right)$$(26)

  

$$\mathrm{\Delta }{f}_{{\text{Fe}}_{\text{3}}\text{P}}=\left[(\text{1}-f_{\text{s}})\cdot \mathrm{\Delta }{C}_{\text{P}}\cdot \left(\text{1}+\frac{\text{3}{W}_{\text{Fe}}}{W_{\text{P}}}\right)\right]\cdot \left(\frac{\rho }{\text{100}{\rho }_{{\text{Fe}}_{\text{3}}\text{P}}}\right)$$(27)

where, W_P and W_Fe are the atomic weight of solute P and solvent Fe, kg/mol. ${\rho }_{{\text{Fe}}_{\text{2}}\text{P}}$ and ${\rho }_{{\text{Fe}}_{\text{3}}\text{P}}$ are the densities of Fe₂P and Fe₃P, chosen as 6560 and 6740 kg/m³,²⁰⁾ respectively.

2.5. Equivalent Thermal Conductivity

The intrinsic thermal conductivity of the Fe-1 mass%C-1.5 mass%Cr alloy is expressed as a two-part function of the temperature according to the data in Ref. 21.   

$${\lambda }_{\text{s}}=\{\begin{array}{cc}-\text{0}\text{.0295}\mathrm{\hspace{0.17em} }T+\text{48}\text{.7}& \text{(}T\le \text{760}°\text{C)}\\ \text{0}\text{.00986}\mathrm{\hspace{0.17em} }T+\text{16}\text{.6}& \text{(}T>\text{760}°\text{C)}\end{array}\mathrm{\hspace{0.17em} }\mathrm{\hspace{0.17em} }\mathrm{\hspace{0.17em} }(\text{W}/\text{m}\cdot \text{K})$$(28)

During continuous casting process, the melt flow in liquid and mushy zone could accelerate the heat transfer and, in turn, the growing dendrites in mushy zone could also block the flow. Thus, an equivalent thermal conductivity, i.e. M times of the intrinsic one, is adopted to reflect this effect.   

$${\lambda }_{\text{eff}}={\lambda }_{\text{s}}{f}_{\text{s}}+(\text{1}-f_{\text{s}})\text{M}{\lambda }_{\text{s}}$$(29)

whereas, the amplified coefficient M is selected as 3 in mold as well as in water-cooled foot roller zone (z ≤ 1.28 m), 1.5 is selected in the air-cooled zone (1.28 m < z ≤ 28 m) through several tests.

2.6. Initial and Boundary Conditions

One quarter of the rectangular billet was chosen for the calculation assuming the symmetry of the cooling conditions. The billet surface is chilled with the following condition.   

$$-\lambda \frac{\partial T}{\partial \text{n}}=h\text{(}{T}_{\text{s}}-T_{\text{w}}\text{)}$$(30)

where, T_s is the billet surface temperature, °C; T_w is the temperature of cooling water, °C, and a value of 25°C is used.

With casting down from the meniscus, the heat transfer coefficient, h (W/m²K), varies as follows, according to Ref. 22.

At mold zone, h = 580;

At the foot-roller water-chilled area,   

$$h=\text{1}{\text{.165W}}^{\text{0}\text{.67}}{T}_{\text{s}}^{-\text{0}\text{.95}}\left(\text{1}-\text{0}\text{.004}(T_{\text{w}}-\text{40})\right)\times {\text{10}}^{\text{4}}$$(31)

At the spray and air-cooled area,   

$$h=h_{\text{s}}+h_{\text{rad}}$$(32)

  

$$h_{\text{s}}=\text{5}\text{.717}{T}_{\text{s}}^{\text{0}\text{.12}}{W}^{\text{0}\text{.52}}{V}_a{}^{\text{0}\text{.37}}$$(33)

  

$$h_{\text{rad}}=\text{5}\text{.693}\epsilon \left[{\left(\frac{T_{\text{s}}+\text{273}}{\text{100}}\right)}^{\text{4}}-{\left(\frac{T_{\text{w}}+\text{273}}{\text{100}}\right)}^{\text{4}}\right]/({\text{T}}_{\text{s}}-{\text{T}}_{\text{w}})$$(34)

where, h_s is the heat transfer coefficient at spray zone, W/m²K; h_rad is the radiative heat transfer coefficient, W/m²K; W is the flux density of cooling water, l/(m²·min), its value at each part of the spray zone is shown in Table 4; V_a is the impact velocity of the mist, the value is selected as 15 m/s; ε is the emissivity, 0.85 is used here.

Table 4. Flux density of cooling water at spray zone for RAD1 and 100Cr6 billets.

Section No.	Distance from meniscus (m)	Water flux density (l/(m2·min))	Chilling condition
0	0–0.8	–	Mold, water-cooled
1	0.8–1.28	50	Foot-roller, water-cooled
2	1.28–2.95	45	Spray zone, water-cooled and air-cooled
3	2.95–6.02	30	Water-cooled and air-cooled
4	6.02–28.0	0	Air-cooled

2.7. Finishing Criterion of Calculation

According to the operation condition, the billet length from the meniscus to the flame cutting position is 28 m, then the calculation finishes once the slice moving reaches this limit.

3. Results and Discussion

The solidification of RAD1 and 100Cr6 alloys with a size of 280 × 250 mm, a casting speed of 0.52 to 0.57 m/min, overheat of 24 to 41.25°C was simulated by the slice moving method. The RAD1 and 100Cr6 alloy belong to the same bearing steel grade with a little composition difference. Specifically, there is a slightly lower C content but a slightly higher alloying elements (Cr, Mn, Si) in 100Cr6 alloy. They follow the same continuous casting process. These two alloys are selected for a more general benchmark for the modelling results. Thus, the influence of the composition difference on the final solidification position can be evaluated. The in-situ surface temperature measurement were performed on RAD1 alloy billet while the nail-shooting experiments were operated on both of the alloys.

Due to symmetry, the 1/4 section of the rectangular billet, i.e. a 140 mm × 125 mm region with a 2.5 mm × 2.5 mm grid size, was studied. The time step was chosen as 0.002 sec according to the stability analysis.

The continuous casting process parameters and physical properties are shown in Tables 5, 6.

Table 5. Continuous casting process parameters of bearing steels.

Steel type	RAD1	100Cr6
Casting speed (m/min)	0.52	0.57
Initial temperature (°C)	1495.25	1481
Overheat (°C)	41.25	24
Cooling -water temperature (°C)	25	25

Table 6. Physical properties of RAD1 and 100Cr6 alloys.

Parameters	Unit	RAD1	100Cr6	Note
Liquidus temperature	TL (°C)	1454	1457	by GS
Solidus temperature	TS (°C)	938	949	GS
Melting point of pure Fe	$T_{\text{L}}^{\text{0}}$(°C)	1525.63 at T > 1154°C	1525.63 at T > 1154°C	Eq. (3)
		1416.1 at T ≤ 1154°C	1382.13 at T ≤ 1154°C
Specific heat	cp (J/(kg·K))	684	684	GS
Density	ρ (kg/m3)	7314	7314	GS
Latent heat of fusion	ΔHLS (J/Kg)	213.0–129.35 × 103 corresponding to T∈ [1454, 1154°C), linear relation	217.0–130.55 × 103 corresponding to T∈ [1457, 1154°C), linear relation	Eq. (3)
		129.35–105.1 × 103 corresponding to T∈ [1154, 957°C], linear relation	130.55–110.012 × 103 corresponding to T∈ [1154, 949°C], linear relation	Eq. (3)
C composition in liquid at eutectic point	CE (mass%)	2.1	1.95	GS

3.1. Phase and Thermodynamic Properties in RAD1 and 100Cr6 Alloys

With Kobayashi approximation, the solidification path of one solid precipitating from the liquid or the peritectic transformation can be predicted,²³⁾ but the precipitation of other phases like the MnS and Fe₃P etc. can not be evaluated directly without the help of the additional precipitation models.

Thus, the solidification path was predicted beforehand by Gulliver-Scheil approximation with the TCFE6 database²⁴⁾ to confirm the stable phases and phase properties, such as the phase precipitation sequence and the precipitation amount, as well as the thermodynamic properties, for instance the solute partition coefficient at the S/L interface etc., in RAD1 and 100Cr6 alloys.

Gulliver-Scheil approximation is a thermodynamic approximation. The solidification path predicted through Gulliver-Scheil approximation is fulfilled with the thermodynamic equilibrium calculation at each temperature step. Following a global minimization technique of the Gibbs energy of a system, a thermodynamic equilibrium calculation is performed providing the conditions to zero degree of freedom. The c + 2 conditions need to be set for a c-component system, including the c − 1 component compositions, the amount of the system, the temperature and the pressure. With temperature decreasing, the thermodynamic equilibrium calculation under Gulliver-Scheil approximation is performed by setting the liquid composition and amount as the system ones. It implies that the already formed solids remain frozen and their compositions and amounts are unchanged under the assumption of negligible diffusion in solids (Ds → 0), the new solids precipitating from the residual liquid are just adhered to the already formed ones.²⁵⁾

As seen from Fig. 1, regarding the RAD1 alloy, the γ, MnS, Cementite and M₃P phases precipitate respectively at 1454, 1147, 1154 and 964°C with the volume fraction approaching to 0.960, 1.30 × 10⁻⁴, 0.038 and 3.71 × 10⁻⁴ at the solidus temperature 938°C; and for 100Cr6, the γ, MnS, Cementite and M₃P phases precipitate at 1457, 1172, 1154 and 970°C with volume fraction approaching 0.964, 2.27 × 10⁻⁴, 0.035 and 4.07 × 10⁻⁴ at solidus 949°C. From the predicted phase precipitation sequence and the precipitation amount, it is confirmed that the main solid is γ phase, follows the MnS and Fe₃P precipitation during solidification.

Fig. 1.

Solidification path predicted by Gulliver-Scheil approximation for two bearing steels. (Online version in color.)

Figure 2 shows the liquid composition evolution at the interface with temperature decrease by Gulliver-Scheil prediction. As can be seen, the interface compositions of C and Cr in the residual liquid decrease obviously, due to the precipitation of cementite around 1154°C. Alternatively, the interface compositions of Mn, Si and P increase, and the interface composition of S varies little. Thus, the partition coefficients of solute j in Kobayashi approximation (Eq. (3)), $k_j^{\gamma /\text{L}}$, adopt two groups of values at above and below the turning-point 1154°C, as listed in Table 7. The values of ΔH_LS and $T_{\text{L}}^{\text{0}}$ in Table 6 are correspondingly adjusted via Eq. (3).

Fig. 2.

Interface liquid composition and partition coefficient evolution with temperature during solidification of RAD1 and 100Cr6 alloys, predicted by Gulliver- Scheil approximation. (Online version in color.)

Table 7. Average solute partition coefficients in RAD1 and 100Cr6 alloys, predicted by Gulliver- Scheil approximation.

Solute	Partition coefficient of solute j at γ/liquid interface, $k_j^{\gamma /\text{L}}$, -
	RAD1		100Cr6
	T > 1154°C	T ≤ 1154°C	T > 1154°C	T ≤ 1154°C
C	0.4174	0.4803	0.4171	0.5081
Cr	0.7502	0.6525	0.7535	0.6376
Mn	0.6136	1.1951	0.6146	1.3682
Si	0.8322	2.7200	0.8350	2.8510
P	0.1525	0.0871	0.1496	0.0735
S	0.0072	0.0035	0.0070	0.0031

3.2. Temperature History along Billet Casting Direction

The temperature evolution along casting direction at the billet cross section are predicted and shown in Fig. 3(a). The curves denoting as seg.A, seg.B and seg.C in Fig. 3(a) represent the average temperature along 1/4 Lx, 1/2 Lx and 3/4 Lx segment. Lx denotes the half length of the long side in billet cross section, which is set along x axis. Ly is the half length of the short side setting along y axis. The positions of spot No. 1 to 4 in cross section are shown in Fig. 3(b). The temperatures are measured at the centerline (point 3) and near edge (upper- spot 2’ and lower- spot 2) along the short-side part of the billet in spray zone, as shown in Fig. 3(c).

Fig. 3.

Temperature history on 2D billet slice along casting direction. (Online version in color.)

It is seen that the predicted temperature in billet cross section decreases with the downward movement of casting billet. At the first segment, i.e. from the meniscus to around 6 m below, the temperatures at the spot No. 1 to 3 decrease greatly due to the strong chilling of the cooling water. Among them, the largest decrease of the temperature is at the vertex spot 2. Subsequently, it is the spot 3 locating at the center of the short side, and the last sequence is the spot 1 at the center of the long side. The temperature decrease of segment C (closer to the billet surface) is greater than those of segment B (middle of Lx in billet) and segment A (closer to the billet center). The temperature decrease at billet center spot 4 is slowest. Around 6 m below the meniscus, there is an obvious temperature recovery at the curves of spot 1 to 3. This is because the chilling changed from water-cooled to air-cooled mode at the billet surface below 6.015 m slows down the heat transfer. The heat accumulates and leads to the rise of the temperature. Downwards from this regime, the temperature continues to decrease with the billet movement. The temperature was measured among 10 to 28 m below the meniscus at the position 2, its symmetrical spot 2’ and 3 using PT120 infrared thermometer with a range −25°C to 1200°C, an accuracy of ±1%. Its emissivity, ε, was calibrated as 0.85 by checking the measured temperature with the experiential data at both the dummy bar disconnecting position and the flame cutting position. As seen, the measured temperature at the upper part of the billet, i.e. spot 2’, is a little larger than the one at its symmetrical lower part, i.e. spot 2. It verifies the fact that the centerline of the liquid zone in a billet through vertical-bending caster is a little upward shifted. From Fig. 3(a), the predicted temperature curves at position 2 and 3 fit with the measured data.

3.3. Solidified Shell Thickness Distribution along Casting Direction

Heat is extracted from billet center to surface with casting and is taken away continuously by cooling water, resulting in the temperature at billet cross section decreasing with the distance along casting direction. Figure 4 shows the temperature (Fig. 4(a)) and solid fraction (Fig. 4(b)) evolution at several typical positions. Under the symmetrical boundary conditions, at the mold exit, the solidified shell thickness is around 11 mm for 100Cr6 and 10 mm for RAD1 from both 280 mm- and 250 mm- side. With the rectangular cross section configuration, the elliptical- shaped isotherm and iso- solid fraction curves are kept in each cross section till the calculation finishes.

Fig. 4.

Predicted temperature and solid fraction evolution in cross section of bearing steel billets along casting direction. (Online version in color.)

Figure 5 shows the solidified shell thickness distribution along casting direction from 280 mm-side for RAD1 and 100Cr6 alloys. The solidified shell thickness is determined as the thickness designated by fs = 0.7 curve in engineering.²⁶⁾ And the region between fs = 0.01 and fs = 1 curves represents the mushy zone in billet. As seen, the solidified shell thickness increases gradually along the casting direction until the final solidification position, i.e. 10.74 m from meniscus for 100Cr6 and 9.82 m for RAD1. From the nail-shooting experiments combined with the sulfur print/electrolytic etching examination (as shown in Fig. 6), the solidified shell thickness are determined as 0.11 m at z = 9.64 m for 100Cr6 (denoted as hollow square in Fig. 5) and 0.14 m at 10.64 m for RAD1 (denoted as x-centered hollow square). As shown in Fig. 5, the predicted shell thicknesses for these two alloys fit with the measurements.

Fig. 5.

Predicted solidified shell thickness along casting direction (shown from 280 mm side). (Online version in color.)

Fig. 6.

Nail-shooting experiments with sulfur print/electrolytic etching examination to determine final solidification position. (Online version in color.)

The liquid core length or the metallurgical length can be roughly estimated by the empirical formula, according to Ref. 27,   

$$L=\frac{T{h}^{\text{ 2}}{\text{v}}_{\text{cast}}}{\text{4}{K}^{\text{2}}}$$(35)

where L is the liquid core length, m; Th is the thickness of billet, a value of 280 mm is used here; v_cast is the casting speed, here a value of 0.52–0.57 m/min is used. K is the solidification coefficient, mm/min^1/2, which is 28 to 35 for square billets. Thus the estimated liquid core length through Eq. (35) is 8.32 to 13.00 m corresponding to a casting speed of 0.52 m/min, and 9.12 to 14.25 m for a casting speed of 0.57 m/min. The predicted length is 10.74 m for 100Cr6 (a casting speed of 0.57 m/min) and 9.82 m for RAD1 (a casting speed of 0.52 m/min) by the present model, which is within this range. It is more accurate through considering the influence of alloy composition, phase precipitation and the actual cooling conditions.

3.4. MnS Precipitation

Figure 7 shows the MnS precipitate distribution in (a) 100Cr6 and (b) RAD1 billets with the solidification process. Similar to each other, MnS distributes all over the cross section of the two alloys. The original precipitating zone is near the vertex of the billet, as shown in Figs. 7(a1) and 7(b1). It is the original cooled spot 2 in Fig. 3(c). Thereafter, the precipitation zone extends in a belt profile towards the billet center. The precipitation amount approaches the larger one near billet vertex (x is around 0.1 m and y is around 0.09 m) in a belt shape and then gradually decreased towards one half of the region in the diagonal direction, which are shown from Figs. 7(a2) to 7(a3), Figs. 7(b2) to 7(b3) as well as Fig. 8. Then from there towards the billet center the precipitation amount is becoming increased due to the gradual accumulation of Mn and S solutes in residual liquid with the solidification towards the billet center, which again rises up the solute composition product and favors the precipitation of MnS, thus around the billet center, the MnS precipitation approaches largest (Figs. 7(a3) to 7(a4), Figs. 7(b3) to 7(b4) and 8).

Fig. 7.

Predicted distribution of MnS in cross section of (a) 100Cr6 and (b) RAD1 billet along casting direction. (Online version in color.)

Fig. 8.

Predicted distribution of MnS amount along centerline and diagonal in cross section of 100Cr6 and RAD1 billet at z = 28 m. (Online version in color.)

The final solidification position determined from fs = 0.7 curve in Fig. 5 is z = 10.74 m for 100Cr6, while near the billet center there still exists a certain of mushy zone. Thus, the MnS continues precipitating in the central mushy zone, as seen from Fig. 7(a2) at z = 10.74 m and Fig. 7(a3) at z = 11.37 m, until approaching the fully solidified position at billet center (i.e. z = 12.61 m for 100Cr6, fs = 1 curve in Fig. 5), as illustrated in Fig. 7(a4). MnS precipitate approaches its maximum amount f_MnS,max = 4.34 × 10⁻⁴. Afterwards, the billet is fully solidified and the MnS stops precipitating. Same cases are shown for RAD1, its final solidification position (from fs = 0.7 curve) is z = 9.82 m, and the fully solidified position at billet center (fs = 1 curve) is z = 11.37 m. So in Fig. 7(b2) at z = 10.74 m, MnS continues to precipitate towards the billet center. Till Fig. 7(b3), at the fully solidified position z = 11.37 m, the MnS finishes precipitation.

As seen in Figs. 7(a4) and 7(b4), the position for maximum MnS precipitation is approaching to the billet center, and the junior one is near the billet vertex in a belt shape. This tendency is a combined result of the precipitation and accumulation of the solutes. It can also be found that the amount of MnS in cross section of 100Cr6 billet is always larger than that in RAD1 (Figs. 7 and 8), which is due to the higher nominal composition of Mn and S in 100Cr6, as shown in Table 1. The nominal compositions of Mn and S in 100Cr6 are higher by 0.09 mass% Mn and 0.003 mass% S than in RAD1. Especially the ratio of S composition in 100Cr6 and RAD1 is 1.75, resulting in the maximum MnS amount in mass fraction in 100Cr6 (i.e. 4.3462 × 10⁻⁴) is 2 times of that in RAD1 (i.e. 2.1934 × 10⁻⁴).

Comparing Fig. 8 with Fig. 1, the predicted MnS precipitation amounts in the two alloys are in the same order with the ones by GS solidification path prediction. The predicted precipitation tendency with the Mn and S composition in the two alloys also fit with the GS prediction.

3.5. Fe₃P Precipitation

With the present nominal composition of P, there is no Fe₃P precipitation predicted in RAD1 and 100Cr6 alloy, although the GS approximation has shown its existence. By GS, with the present low composition of P in the alloy, the predicted precipitation temperature 970°C for 100Cr6 and 964°C for RAD1 are much closer to the solidus point 949°C (for 100Cr6) and 938°C (RAD1). While the actual solidification finishes above this temperature, the P composition accumulated in residual liquid does not exceed the composition limit for Fe₃P precipitation. Through increasing the P composition by 0.001 mass% increment to maximum 0.026 mass%, which is limited according to the in situ composition fluctuation range, in these two alloys and keeping the other compositions and the corresponding thermodynamic parameters unchanged, the Fe₃P precipitate is obtained at 0.019 mass% P for 100Cr6 and 0.021 mass% P for RAD1 alloy, as seen in Fig. 9. Figure 10 illustrates the Fe₃P precipitation distribution. It is the same for both alloys that the maximum Fe₃P precipitation amount is positioned at the billet center, which is just like the MnS precipitation distribution. However, the Fe₃P precipitation is not distributed all over the region, but concentrates merely at the central regime. The maximum precipitation amount of Fe₃P is one or two order less than that of MnS, compared to Figs. 7 and 10. This is because the partition coefficient of P at γ/L interface is one or two order bigger than that of S, as shown in Table 7. It results in less P composition accumulated in the residual liquid, and that the precipitation temperature of Fe₃P is much lower than that of MnS (illustrated in Fig. 1).

Fig. 9.

Predicted maximum Fe₃P precipitate amount in billet changing with P composition. (Online version in color.)

Fig. 10.

Predicted distribution of Fe₃P in cross section of (a) 100Cr6 and (b) RAD1 billet cross section at final calculation section (z = 28 m). (Online version in color.)

As seen in Figs. 9 and 10, the amount of Fe₃P is increasing with the increased P composition for both alloys. Comparing their amount with the same P composition, for instance, the maximum Fe₃P precipitate amount is 3.3983 × 10⁻⁶ in 100Cr6 versus 1.1283 × 10⁻⁶ in RAD1 for the case of 0.021 mass%P, i.e. the amount in the former is over 3 times of that in the latter. The reason for this tendency can be evaluated from the contributions of the factors influencing the length of the mushy zone such as the casting speed and overheat, etc. and the ones directly related to the precipitation of Fe₃P like the nominal composition etc. From the viewpoint of Fe₃P precipitation kinetics, as seen in section 2.4, the Fe₃P precipitation amount is dependent on the equilibrium composition C_P,eq, obtained from Eqs. (21) and (23). It is proportional to the item 10^−ΔG with $\mathrm{\Delta }G=\underset{}{{\displaystyle {\mathrm{\Sigma }}_j}}{e}_{\text{p}}^j{C}_j$. As shown from Table 3, ΔG for Fe₃P precipitate is mainly related to the composition of solute C, Si, S and is dominated by the contribution of C. Since the nominal composition of C in 100Cr6 is less than the one in RAD1, the sum of the product $\underset{}{{\displaystyle {\mathrm{\Sigma }}_j}}{e}_{\text{p}}^j{C}_j$, i.e. ΔG, for 100Cr6 is less than the one for RAD1. Subsequently the value of 10^−ΔG as well as the equilibrium composition C_P,eq for 100Cr6 is always larger than RAD1 at the same depth from the meniscus or at the same temperature. This is shown from the equilibrium P composition evolution in liquid with solidification at billet center in Fig. 11. At the same time, the accumulated P composition in residual liquid is keeping the increasing tendency, and then catches up with the equilibrium composition C_P,eq firstly in RAD1 billet. This leads to the precipitation of Fe₂P. As seen in the figure, this catching-up stage is shorter for RAD1, lasting from 11.28 m to 11.37 m at the final stage above the solidus temperature. While it is much longer for 100Cr6, lasting from 12.34 to 12.61 m. In a word, the larger casting speed in the 100Cr6 alloy (i.e. 0.57 m/min, as seen in Table 5) overcomes the influence of overheat, and it leads to a longer mushy zone and further a larger precipitation amount of Fe₂P in 100Cr6 alloy although its Fe₂P precipitation stage is much delayed compared with the one in RAD1.

Fig. 11.

Evolution of local P composition and equilibrium P composition in liquid with solidification at billet center with same nominal 0.021 mass% P composition for RAD1 and 100Cr6 alloy. (Online version in color.)

3.6. Liquid Core Length Variation with P and S Compositions

The P and S compositions are strictly controlled in bearing steels for the final properties, while there are still some fluctuations. According to the in situ composition variation, the composition of P is ranged from the present value in the two alloys to maximum 0.026 mass% and the composition of S is ranged up to 0.015 mass% in the present calculation.

The increase of P composition promotes a little the precipitation amount of Fe₃P. As shown in Fig. 9, with the increase of P from 0.02 to 0.026 mass%, the precipitation amount of Fe₃P increases from 1.9279 × 10⁻⁶ to 1.2540 × 10⁻⁵, increased by 1.0612 × 10⁻⁵ for 100Cr6. It also increases from 0 to 9.5116 × 10⁻⁶, increased by 9.5116 × 10⁻⁶ for RAD1 alloy.

The increase of P composition also influences the precipitation amount of MnS, as seen from Eq. (15) and interaction coefficients in Table 3. The P is the influencing solute on the precipitation amount of MnS, and the increase of P leads to a decrease of the equilibrium solute composition product K_MnS. Thus, it promotes the MnS precipitation although the effect is also little with such a little P composition. As shown in Fig. 12(a), with the increase of P from 0.012 to 0.026 mass%, the maximum MnS precipitate is increased from 4.3462 × 10⁻⁴ to 4.3709 × 10⁻⁴ for the case of 100Cr6. Also, this value increases from 2.1954 × 10⁻⁴ to 2.2314 × 10⁻⁴ for RAD1 alloy.

Fig. 12.

Predicted maximum MnS precipitate amount in billet changing with (a) P and (b) S composition. (Online version in color.)

In contrast, the increase of S composition greatly increases the precipitation of MnS but does not lead to the Fe₃P precipitation at present nominal composition of P. As shown in Fig. 12(b), the maximum MnS precipitate amount increases from 4.3462 × 10⁻⁴ to 6.1495 × 10⁻⁴ with the increase of S from 0.007 to 0.015 mass% for 100Cr6 alloy. It is also increased from 2.9731 × 10⁻⁴ to 4.3262 × 10⁻⁴ for RAD1 alloy.

With the precipitation of MnS and Fe₃P, which changes the solute composition and their interaction in residual liquid, the liquid core length in 100Cr6 and RAD1 alloys is also influenced. As shown in Fig. 13(a), with the increase of P from 0.012 to 0.026 mass%, the liquid core length increases a little in both alloys. It means that the length increases from 10.7409 to 10.7414 m for 100Cr6 alloy and increases from 9.8181 to 9.8195 m for RAD1 alloy. While with the increase of S from 0.007 to 0.026 mass%, the liquid core length decreases much for both alloys. The length is decreased from 10.7409 to 10.7341 m for 100Cr6 alloy and is decreased from 9.8152 to 9.8087 m for RAD1 alloy. To sum up, the liquid core length is shortened with the increase of S composition and the precipitation of MnS, while is elongated a little with the increase of P composition by the mutual contributions of MnS and Fe₃P precipitation.

Fig. 13.

Predicted liquid core length change with (a) P and (b) S composition. (Online version in color.)

It should be noticed that the present calculation doesn’t consider the corresponding change of other parameters with the composition, while some thermodynamic properties such as the partition coefficient and phase precipitation temperature etc. do change with the composition variation from 100Cr6 to RAD1 , as illustrated in Fig. 1 and Table 7.

4. Conclusions

(1) The final solidification position in a Fe–C–Cr–Mn–Si–P–S bearing steel continuous casting billet is predicted by the slice moving method combined with Kobayashi approximation. The precipitation of MnS and Fe₃P is considered. The predicted solidified shell thickness and the billet surface temperature are verified with the nail-shooting experiments as well as the surface temperature measurement.

(2) Regarding the composition of 0.004 to 0.007 mass%S in 100Cr6 and RAD1, the MnS distributes all over the billet cross section, mainly concentrates at the center and around the belt-shape regime near vertex. The maximum amount of MnS in 100Cr6 and RAD1 alloy is proportional to the individual nominal S composition.

(3) The onset of Fe₃P precipitate is at 0.019 mass%P for 100Cr6 and 0.021 mass%P for RAD1 alloy. It precipitates mainly around the billet center. The larger casting speed for 100Cr6 alloy leads to a longer mushy zone and further a larger precipitation amount of Fe₃P, overcoming the influence of the overheat on the length of the mushy zone and the nominal composition on the precipitation kinetics of Fe₃P.

(4) It is vital to control composition of S, P solute to a low level in bearing steels. This will decrease the formation of MnS and Fe₃P and stabilize the final solidification position. Since the increase of P composition not only accelerates the precipitation of MnS and Fe₃P but also elongates the liquid core length. In contrast, the increase of S composition and the precipitation of MnS greatly shortens the liquid core length.

Acknowledgements

The work is supported by National Natural Science Foundation of China [51574074] and National Natural Science Foundation of China and Shanghai Baosteel [U1460108]. Swedish iron and steel research office (Jernkontoret), in particular Prytziska fonden 2 is also acknowledged by WM and KN.

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