A. Mathematical formulation

Transport phenomena occurring during steel solidification for slab CC processes are complex. Reasonable assumptions are necessary to simplify the process and derive a mathematical model. The assumptions are summarized as follows:

1. (i) The transport phenomena occurring during slab CC processes are treated as steady state processes, i.e., the casting speed and casting temperature remain unchanged.

2. (ii) The influences of strand curvature, mold oscillation, mold powder, mold taper, and gap occurring in the mold were ignored.

3. (iii) The heat transfer conditions in the secondary cooling region were simplified, i.e., the cooling intensity in the same cooling zone (cooling circuit) is assumed to be the same, and the integrated heat transfer coefficient was used to delineate the heat transfer conditions at the strand surface.

4. (iv) The latent heat released through solid phase transformation was ignored.

5. (v) Solute redistribution is related to the solidified rate, local solidification time, etc. To more easily describe the phenomenon of solute redistribution at microscale, a local thermodynamic equilibrium was assumed.

The formation of centerline segregation is closely related to the transport phenomena of flow, heat, and composition. The numerical model for centerline segregation in slab CC processes includes three parts: the fluid and turbulent model, the heat transfer and solidification model, and the solute transfer model. The governing equations for each model are described as follows:

1. (i) Fluid and turbulent model

The continuity equation is defined as

(1)$${{\partial \rho } \over {\partial t}} + {{\partial \left( {\rho {u_i}} \right)} \over {{\partial x_i}}} = 0\quad ,$$

and the momentum equation defined as

(2)$${{\partial \left( {\rho {u_i}} \right)} \over {\partial t}} + {{\partial \left( {\rho {u_i}{u_j}} \right)} \over {\partial {x_j}}} = - {{\partial P} \over {\partial {x_i}}} + {\partial \over {\partial {x_i}}}\left( {{\mu _{{\rm{eff}}}}{{\partial {u_i}} \over {\partial {x_j}}}} \right) + {\partial \over {\partial {x_i}}}\left( {{\mu _{{\rm{eff}}}}{{\partial {u_j}} \over {\partial {x_i}}}} \right) + \rho {g_i} + {s_{i,{\rm{mom}}}} + {s_{i,{\rm{P}}}}\quad ,$$

where S _i,mom is the sink term of momentum due to the reduced porosity in the mushy zone and S _i,P is the thermosolutal buoyancy, as defined as

(3)$${S_{i,{\rm{mom}}}} = {{{{\left( {1 - {f_{\rm{L}}}} \right)}^2}} \over {\left( {{f_{\rm{L}}}^3 + 0.01} \right)}}{A_{{\rm{mushy}}}}\left( {{u_i} - {v_{i,{\rm{p}}}}} \right)\quad ,$$

(4)$${S_{i,{\rm{P}}}} = \rho {g_i}{\beta _{\rm{T}}}\left( {T - {T_{\rm{L}}}} \right) + \sum\limits_m {\rho {g_i}{\beta _{{\rm{c}},m}}\left( {{C_{{\rm{L}},m}} - {C_{{\rm{L}},m,0}}} \right)} \quad .$$

The standard K–ε equations are defined as

(5)$${{\partial \left( {\rho k} \right)} \over {\partial t}} + {{\partial \left( {\rho k{u_i}} \right)} \over {\partial {x_i}}} = {\partial \over {\partial {x_i}}}\left( {{{{\mu _{{\rm{t}}}}} \over {{\sigma _{\rm{k}}}}}{{\partial k} \over {\partial {x_j}}}} \right) + G - \rho \varepsilon + {s_{\rm{k}}}\quad ,$$

(6)$${{\partial \left( {\rho \varepsilon } \right)} \over {\partial t}} + {{\partial \left( {\rho \varepsilon {u_i}} \right)} \over {\partial {x_i}}} = {\partial \over {\partial {x_i}}} {\left( {{{\mu _{\rm{t}}}} \over {{\sigma _\varepsilon }}}{{\partial \varepsilon} \over {\partial {x_j}}}} \right) + {c_1}{\varepsilon \over k}G} - {c_2}{{{\varepsilon ^2}} \over k}\rho + {s_\varepsilon }\quad ,$$

(7)$${\mu _{{\rm{eff}}}} = \mu + {c_\mu }\rho {{{k^2}} \over \varepsilon }\quad ,$$

(8)$${S_{\rm{k}}} = {{{{\left( {1 - {f_{\rm{L}}}} \right)}^2}} \over {\left( {{f_{\rm{L}}}^3 + 0.01} \right)}}{A_{{\rm{mushy}}}}k\quad ,$$

(9)$${S_\varepsilon } = {{{{\left( {1 - {f_{\rm{L}}}} \right)}^2}} \over {\left( {{f_{\rm{L}}}^3 + 0.01} \right)}}{A_{{\rm{mushy}}}}\varepsilon \quad ,$$

where the standard values of σ_k, σ_k, c ₁, c ₂, and c _μ recommended by Launder and Spalding are 1.0, 1.3, 1.44, 1.92, and 0.09, respectively.^{Reference Launder and Spalding27}

1. (ii) Heat transfer and solidification model

The energy equation is defined as

(10)$${{\partial \left( {\rho H} \right)} \over {\partial t}} + {{\partial \left( {\rho {u_i}H} \right)} \over {\partial {x_i}}} = {\partial \over {\partial {x_i}}}\left( {\lambda {{\partial T} \over {\partial {x_i}}}} \right)\quad ,$$

where

(11)$$H = h + \Delta H = {h_{{\rm{ref}}}} + \int_{{T_{{\rm{ref}}}}}^T {{c_{\rm{p}}}{\rm{d}}T + {f_{\rm{L}}}\Delta {H_{\rm{f}}}} \quad .$$

The solidus and liquidus temperatures in a species mixture are calculated as

(12)$${T_{\rm{S}}} = {T_{{\rm{melt}}}} - \sum\limits_m {{{{G_m}{C_m}} / {{k_m}}}\quad ,}$$

(13)$${T_{\rm{L}}} = {T_{{\rm{melt}}}} - \sum\limits_m {{G_m}{C_m}} \quad .$$

1. (iii) Solute transfer model

Here, the diffusion equation is defined as

(14)$${{\partial \left( {\rho {C_m}} \right)} \over {\partial t}} + {{\partial \left( {\rho {u_i}{C_m}} \right)} \over {\partial {x_i}}} = {\partial \over {\partial {x_i}}}\left( {\rho {D_{{\rm{L}},m}}{{\partial {C_m}} \over {\partial {x_i}}}} \right) + {s_{i,{\rm{dif}}}} + {s_{i,{\rm{con}}}}\quad ,$$

where s _i,dif and s _i,con are the diffusion source item and convection item and are defined as

(15)$${S_{i,{\rm{dif}}}} = {\partial \over {\partial {x_i}}}\left[ {\rho {f_{\rm{S}}}{D_{{\rm{S}},m}}{{\partial \left( {{C_{{\rm{S}},m}} - {C_m}} \right)} \over {\partial {x_i}}}} \right] + {\partial \over {\partial {x_i}}}\left[ {\rho {f_{\rm{L}}}{D_{{\rm{L}},m}}{{\partial \left( {{C_{{\rm{L}},m}} - {C_m}} \right)} \over {\partial {x_i}}}} \right]\quad ,$$

(16)$${S_{i,{\rm{con}}}} = {\partial \over {\partial {x_i}}}\left[ {\rho \left( {{u_i} - {u_{i,{\rm{S}}}}} \right)\left( {{C_{{\rm{L}},m}} - {C_m}} \right)} \right]\quad .$$

To solve the above system of equation, expressions relating the liquid and solid fractions (f _L and f _S) and the phase concentrations (C _L,m and C _S,m) are required.

The liquid fraction is updated as

(17)$$f_{\rm{L}}^{n + 1} = f_{\rm{L}}^n - \beta {{{\alpha _{\rm{P}}}\left( {T - {T^ * }} \right)\Delta t} \over {\rho V\Delta {H_{\rm{f}}} - {\alpha _{\rm{P}}}\Delta t\Delta {H_{\rm{f}}}{{\partial {T^ * }} \over {\partial {f_{\rm{L}}}}}}}\quad .$$

In the present calculation, the local thermodynamic equilibrium is assumed and the Lever rule is selected for species segregation at microscale. According to the Lever rule, the interface temperature T* is calculated as

(18)$${T^ * } = {T_{{\rm{melt}}}} + \sum\limits_m {{G_m}\left( {{{{C_m}} \over {{k_m} + {f_{\rm{L}}}\left( {1 - {k_m}} \right)}}} \right)} \quad .$$

The liquid and solid fractions must add to unity, defined as

(19)$${f_{\rm{S}}} + {f_{\rm{L}}} = 1\quad .$$

Based on the classic mixture theory, the mixture concentration of the solute m (C _m) is defined as

(20)$${C_m} = {C_{{\rm{L}},m}}{f_{\rm{L}}} + {C_{{\rm{S}},m}}{f_{\rm{S}}}\quad ,$$

then

(21)$${C_{{\rm{L}},m}} = {{{C_m}} \over {1 + \left( {1 - {f_{\rm{L}}}} \right)\left( {{k_m} - 1} \right)}}\quad ,$$

(22)$${C_{{\rm{S}},m}} = {{{k_m}{C_m}} \over {1 + {f_{\rm{S}}}\left( {{k_m} - 1} \right)}}\quad .$$

B. Computational domain

For numerical modeling of centerline segregation in a continuously cast slab, the computational domain includes the region from the meniscus to the solidification end, and the calculation cost increases exponentially with the number of grid points.^{Reference Li, Wu, Ludwig and Kharicha21} To decrease the amount of calculations and improve calculation efficiency, the computational domain is treated as follows. First, according to the previous nail shooting test,^{Reference Long, Chen, Wang, Luo, Han, Liu and Gao28} the strand in the present case solidifies completely in the secondary cooling zone 8. Thereby, the computational domain only includes the region from the meniscus to the end of secondary cooling zone 8. Second, a quarter symmetry strand computational domain is chosen to calculate in this study. Third, according to the flow characteristics of the molten steel in the liquid pool, the computational domain is reasonably divided into two parts: 3-D turbulent flow region and natural convection region. This is the most effective way to improve the calculation efficiency. The division method is illustrated in Fig. 1(a). In the turbulent flow region, the molten steel in the liquid pool exhibits reflux in the casting direction. The turbulent fluxion of the molten steel has a great impact on the solute distribution in the liquid pool. Hence, the flow states of the molten steel in three dimensions must be included in the coupling calculation of the flow, heat, and solute transport in the turbulent flow region. Thus, a 3-D coupling model of the flow, heat, and solute transport is applied in the turbulent flow region to investigate the transport phenomena during the CC process. However, when the z-component of velocity of the molten steel is equal to the casting speed, plug flow occurs in the liquid pool, and the local molten steel and shell exist in a relatively static state. Therefore, we deduce that there is no convection of the molten steel in the casting direction, meaning that the molten steel flow in the casting direction has no effect on the solute distribution. Therefore, a 2-D slicing model for the centerline segregation prediction can be applied to this natural convection region. In the 2-D slicing model, the natural convection (except the flow in the casting direction), the heat, and the solute transport in the transverse section are coupled. Based on the above analysis, we believe that the first transverse section where the z-component velocities are equal to the casting speed is the interface between the turbulent flow region and the natural convection region. The data of interface from the calculation of the 3-D turbulent flow region are transmitted to the initial 2-D slice applied in the natural convection region by a coordinate interpolation algorithm. Using such a process, the 3-D coupling model is combined with the 2-D slicing model to predict the centerline segregation during the slab CC process. The geometry and working parameters of the CC process are listed in Table I.

FIG. 1.

(a) Division of the computational domain for the simulation of the centerline segregation during the slab CC process, (b) meshes of the 3-D turbulent flow region, and (c) meshes of the 2-D natural convection region.

TABLE I.

The geometry and working parameters of the slab CC process.

To determine the interface between the 3-D turbulent flow region and the 2-D natural convection region, a part of the computational domain (from the meniscus to the secondary cooling zone 4) was built and calculated. The position of this interface is obtained by analyzing the variation of z-component of the velocity of molten steel at the central symmetry plane of the strand narrow face. The meshes adopted for the 3-D turbulent flow region and the 2-D natural convection region, which were generated by ANSYS ICEM CFD 14.0, are shown in Figs. 1(b) and 1(c). The maximum size of the grid is 5 mm, which is sufficient to determine a precise solute distribution and centerline segregation profile.^{Reference Li, Wu, Ludwig and Kharicha21}

C. Boundary conditions and physical properties

The inlet velocity of the submerged entry nozzle (SEN) was calculated based on the mass conservation. The K and ε were estimated using semi-empirical relations. The values of these parameters are v _in = 1.502 m/s, k = 0.003 m²/s², ε = 0.0134 m²/s³, and T = 1811 K. The initial concentrations of each solute elements (C-initial) listed in Table II are the mean values of 60 heats of steel Q345 from plant records.

TABLE II.

Initial concentrations and species-dependent thermophysical properties.

A wall condition was used at the strand surface. The heat flux at the mold surface was calculated by Eq. (23),

(23)$${q_{\rm{m}}} = 2,680,000 - 276,000 \times \sqrt {{{60{L_{\rm{Z}}}} \over {{V_{\rm{c}}}}}} \quad .$$

In the secondary cooling zone, the heat transfer coefficient was calculated using Eq. (24). More details about the boundary conditions for heat transfer in the secondary cooling zone are available in our previous publications.^{Reference Long and Chen17,Reference Long, Dong, Chen, Liao and Ma18} In every cooling zone (cooling circuit), the integrated heat transfer coefficient was used.

(24)$$h = 2950.190 \times T_{\rm{w}}^{ - 0.235} \times {w^{0.805}}\quad .$$

The partition coefficient (k _m) is a critical parameter for segregation simulation and modeling accuracy. It is highly dependent on the phase constitution of the mushy zone, the temperature, and the alloy components. Therefore, k _m should be determined according to the initial concentrations in the steel (listed in Table II) by thermodynamic calculation prior to the macrosegregation simulation. Additional details on the thermodynamic calculation for k _m in a multicomponent system are available in our previous publication.^{Reference Chen, Long, Cao, Chen, Liu and Dong26}

Figure 2 shows the evolution of k _m with the phase constitution of the mushy zone during the solidification process for steel Q345. Figure 2(a) indicates that the mushy zone of steel Q345 experiences three difference phase constitutions during solidification. First, as the temperature decreases to 1786 K (liquidus temperature), liquid phase (L) and ferrite (δ) coexist. In this coexistence region, the solutes redistribute between δ and L. As the solidification proceeding, the temperature of the system decreases and L transfers into δ, which results in an increase in the amount of δ. Second, when the temperature of the system falls below 1755 K, the peritectic reaction occurs. The L and δ phases transform into austenite (γ), meaning that the amount of L and δ decrease gradually, and the amount of γ from the peritectic reaction increases. In this region, where the temperature ranges from 1755 to 1753 K, three phases (L, δ, and γ) coexist, and the solutes redistribute among L, δ, and γ until exhausting the amount of δ. When the temperature of the system reaches 1753 K, the peritectic reaction terminates. Third, after the peritectic reaction, as the temperature of the system further decreases, L transforms into γ gradually until solidification is complete. In this region, where temperature ranges from 1753 to 1737 K, L and γ coexist, and solutes distribute between γ and L.

FIG. 2.

The evolution of equilibrium partition coefficients with solidification path during steel Q345 solidification: (a) solidification path and (b) equilibrium partition coefficient.

Figure 2(b) illustrates the evolution of k _m during solidification of steel Q345. For the solute C, from 1786 to 1755 K, k _C between δ and L remains almost constant at 0.148. From 1755 to 1753 K, the amount of γ generates from the peritectic reaction increases gradually. Over this temperature range, k _C among δ, γ, and L increases rapidly as k _C between γ and L, which is roughly twice of that between δ and L. From 1753 to 1737 K, k _C between γ and L further increases as the temperature decreases. For the solute Si, k _Si increases as the temperature decreases, although it experiences a slight reduction when the phase constitution of the mushy zone changes from δ and L to δ, γ, and L. In contrast to the solute Si, k _Mn decreases with the temperature, but it experiences an increase in the three-phase (δ, γ, and L) coexistence region. The reason for this is that k _Mn between γ and L is greater than that between δ and L. In addition, the values of k _P and k _S exhibit similar variation tendencies. First, k _P and k _S slightly increase as the temperature decreases and then experience a sharp reduction in the three-phase coexistence region. Finally, k _P and k _S further decrease with the temperature until solidifying completely.

According to the above analysis, it is evident that the effect of phase constitution on k _m is significant. This effect was considered in the present study. However, due to the slight difference in the range of the mushy zone between the macrosegregation numerical model and the thermodynamic model during solidification, it can cause numerical errors and convergence difficulties when the evolution curves of k _m are applied directly to the macrosegregation simulation. Instead, the mean values of k _m for difference solutes (listed in Table II) over the entire mushy zone (from 1786 to 1737 K) were applied in the centerline segregation model.

The diffusion coefficients for different solutes were calculated using Eq. (25),

(25)$$D = {D_0} \cdot \exp \left( { - {Q \over {RT}}} \right)\quad ,$$

where the constants D ₀ and Q are listed in Table II, T is the temperature, and R is the gas constant.

The other species-dependent thermophysical properties are listed in Table II.

The physical properties of steel Q345 used in the calculation are listed in Table III.

TABLE III.

Physical properties of steel Q345.

D. Solution procedure

The centerline segregation simulation model was solved using the SIMPLE algorithms with ANSYS Fluent 14.0. The calculation was performed on a computer with a 3.50-GHz Intel core i7-4770k processor and 16.0-GB RAM. In the present simulation, the time step size was set at 0.05 s. The residuals of continuity, velocity components, solutes, and K were set 0.001, whereas the residual for ε was set at 1 × 10⁻⁶. For each time step, if the residual is less than the specified value, the particular equation is deemed to have converged.

E. Model validation

To verify the accuracy of the centerline segregation model, first, the solute distribution in the slab transverse section predicted by the present model was compared with the morphology of strand transverse section. Figure 3(a) shows the distribution of the solute C in the slab transverse section after complete solidification. The solute elements are rejected from the solid phase to the liquid phase during steel solidification. With the effect of molten steel convection, the solute elements are transported to the liquid pool, leading to the solute concentration of solutes in the liquid pool increasing as solidification proceeding. Finally, after complete solidification, the solute elements are found to enrich the centerline region of the slab transverse section. This centerline segregation morphology of the slab is indirectly in accordance with the segregation line exhibited in the morphology of the strand transverse section obtained by acid etching [shown in Fig. 3(b)]. Furthermore, as the results shown in Fig. 3(a), we found that the solutes enrich more seriously at the front-end of the triangle region. And this characteristic pattern is also detected in the morphology of the strand transverse section, as shown in Fig. 3(b). These comparisons demonstrate that the distribution pattern of solute C in the slab transverse section agrees with the morphology of the slab transverse section.

FIG. 3.

(a) The distribution of C predicted by the hybrid simulation model in half of the slab transverse section. (b) Morphology of the centerline segregation in half of the slab transverse section. (c) The comparison between the calculated and measured profiles of the solute C for 1/2 line (Y = 0 mm) of the strand.

To further verify the accuracy of the centerline segregation model, the chemical element C across the 1/2 line (Y = 0 mm) of the strand was evaluated using a carbon and sulfur analyzer (CS-8820, Wuxi Jiebo Electrical Technology Co., Ltd., Wuxi, China). Details regarding the location of the sample drillings and the test results are shown in Fig. 3(c). The diameter of the drill is 5 mm and the distance between two drilled holes is 15 mm. The results shown in Fig. 3(c) indicate that the segregation degree of C from the present numerical model of centerline segregation exhibits the same tendencies as the data from the experiment. From the slab surface to the slab center, the concentration of C increases gradually and reaches a maximum in the center. The calculated and experimental centerline segregation degrees of C are 1.135 and 1.15, respectively, corresponding to a deviation of 1.3%. Furthermore, the average deviation (taken using seven sampling points) between the calculated and experimented results reaches as low as 0.5%. The comparisons between the simulation and the experiment indicate that the results from the present simulation model feature relatively high reliability. Thus, the 3-D combined with the 2-D hybrid simulation model of centerline segregation was used for the subsequent numerical calculations. The characteristics of solute distribution in the entire strand and the macrosegregation profile in the transverse section of the slab during the CC process were obtained by numerical modeling using this hybrid model.