Numerical Investigation for the Influence of Turbulent Heat Transfer of Mushy Zone on Shell Growth in the Slab Mold

Abstract

The turbulent heat transfer in the mushy zone caused by the impinging of molten steel jets in the slab mold has a significant effect on the growth of the solidified shell and the formation of impinging zone, as well as the quality of steel products. In the present work, a Peclet number of the mushy zone (Pe_m) is proposed to analyze the influence of turbulent heat transfer on the shell growth in the mushy zone and define the boundary of the impinging zone. A three-dimensional mathematical model based on the enthalpy-porosity method has been established to predict the flow, heat transfer and solidification processes in the slab mold. The influences of casting speed and secondary dendrite arm spacing (SDAS) on the turbulent heat transfer behavior are also investigated. The results indicate that the turbulent heat diffusion has a great effect on the formation of the impinging zone and is the only source of energy to maintain the boundary of the impinging zone. The Pe_m can evaluate the turbulent heat transfer in the mushy zone and is a more reasonable method to obtain a clearer boundary of the impinging zone. The impinging zone is formed when the Pe_m reaches a critical value, been dependent on the casting speed and SDAS.

1. Introduction

In 2019, global crude steel production has exceeded 1.8 billion tons, 95% of it was produced through the continuous casting process.¹⁾ As an important part of the continuous casting process, the flow, heat transfer and solidification in the mold have significant influences on the quality of the steel production. It is well known that the double roll flow pattern is a typical and suitable flow pattern to control the meniscus status in the mold.²⁾ However, in the formation process of double roll flow pattern, as shown in Fig. 1, the molten steel hits on the shell and the impinging zone is formed, which causes the local shell-thinning effect and increases the risk of defects or even breakout.

Fig. 1.

Flow and impinging of steel stream in the mold region of a slab caster. (Online version in color.)

In the past few decades, many researchers have studied the growth of the shell by numerical simulations. However, several researchers^{3,4,5,6,7,8,9)} did not consider the influence of the jet in the mold on the growth of the shell, and a parabolic curve of shell growth was obtained. Huang et al.¹⁰⁾ and Yu et al.¹¹⁾ simulated the superheat removal in the slab mold respectively, and the results showed that the heat flux was highest at the impact point, which could affect the growth of the shell. Unfortunately, these models did not consider the effect of solidification on the flow. Garcia-hernandez et al.¹²⁾ studied the influence of mold curvature on the shell growth and simulated the uniformity of the shell. The results indicated that the shell growth is not a typical parabolic shape and depended on the impinging of molten steel jet. Ito et al.¹³⁾ developed a real-time calculation of the solidified shell thickness by thermocouple, and found that the shell would be re-melting under the condition of high speed casting and narrow width slabs. Xu et al.¹⁴⁾ studied the transport phenomenon in a beam-blank mold by physical and numerical simulation, and it was pointed out that the improper combination of operating parameters and the nozzle structure could cause the breakout accident induced by the impinging of the molten steel jet. Kim et al.¹⁵⁾ investigated the effect of electromagnetic fields on the flow, heat transfer and solidification in the mold, and observed that electromagnetic braking could reduce the impact strength of the jet on the shell and increase the shell thickness. Wu et al.¹⁶⁾ researched the characteristic of shell thickness in a slab mold by numerical simulation and the result indicated that the x-velocity component perpendicular to the narrow face in the liquid zone was relatively larger, inducing the formation of the impinging zone. Vakhrushev et al.¹⁷⁾ numerically studied the shell formation in a thin slab casting mold and indicated that the turbulent heat transfer plays an important role in the local shell-thinning. In a word, the above studies focused on the heat transfer behavior in the liquid zone or outside of the mold, and the heat transfer behavior in the mushy zone was not studied. Recently, the flow, heat transfer and macrosegregation in the mushy zone during solidification have been studied using numerical simulation.^{4,18,19,20,21,22,23,24,25)} However, the influence of turbulent heat diffusion in the mushy zone on the formation of the impinging zone has been rarely reported.

In the present paper, a three-dimensional model based on the enthalpy-porosity method is developed for describing the flow, heat transfer and solidification processes in the slab mold, and a Peclet number of the mushy zone is proposed to analyze the effect of turbulent heat transfer on the shell growth in the mushy zone and define the boundary of impinging zone. The influences of some key variables such as casting speed and secondary dendrite arm spacing (SDAS) are also investigated.

2. Mathematical Model

2.1. Model Assumption

In the continuous casting mold, the interacting phenomena are complex, such as fluid flow, heat transfer, solidification, multiphase turbulence flow, interfacial behavior, segregation and so on.²⁶⁾ To simplify the simulations, the following assumptions are introduced in the modeling:

(1) The flow pattern in the mold is assumed to be at a stable state.

(2) The influence of the strand curvature and mold oscillation on the fluid flow is ignored.

(3) The molten steel is assumed to be Newtonian incompressible fluid and the steel density is assumed to be constant due to a fixed grid.

(4) The latent heat of solid phase transition compared to the latent heat of solidification is considered to be negligible.

(5) The chemical reaction, segregation and liquid level fluctuation in the mold are also negligible.

(6) The mushy zone is treated as a porous medium, where the flow obeys Darcy’s law.

In this work, the most limiting factor is the fixed grid, which requires the use of a constant density and the oscillation cannot be taken into account. However, this model does include the coupled effects of fluid flow, heat transfer and solidification, which is sufficiently accurate to investigate the formation of the impinging zone in the mold.^{12,14,15,16,17)}

2.2. Governing Equations

Continuity equation:   

$$\frac{\partial u_i}{\partial x_i}=0$$(1)

where the x_i is the direction along different axis of Cartesian coordinates, u_i is the velocity component in x_i direction (m/s).

Navier–Stokes equation:   

$$\frac{\partial \left(\rho u_j{u}_i\right)}{\partial x_j}=-\frac{\partial P}{\partial x_j}+\frac{\partial }{\partial x_j}{\mu }_{\mathrm{e}\mathrm{f}\mathrm{f}}\left(\frac{\partial u_j}{\partial x_i}+\frac{\partial u_i}{\partial x_j}\right)+\rho g+S_m$$(2)

  

$${\mu }_{eff}=\mu +{\mu }_t$$(3)

where the ρ is the density of molten steel (kg/m³), P is the pressure (Pa), g is the acceleration of gravity (m/s²), μ_eff is the effective viscosity coefficient (kg/(m·s)), the μ is the laminar fluid viscosity of molten steel (kg/(m·s)), μ_t is the turbulent fluid viscosity (kg/(m·s)). The k-ε turbulent model has been provided acceptable results in previous studies and it was used in this work.^{12,14,15,16,17)} The corresponding governing equations can be written as follows:   

$$\frac{\partial \left(\rho u_{j}k\right)}{\partial x_j}=\frac{\partial }{\partial x_j}\left(\frac{{\mu }_{eff}}{{\sigma }_k}\frac{\partial k}{\partial x_j}\right)+G-\rho \epsilon +S_k$$(4)

  

$$\frac{\partial \left(\rho u_j\epsilon \right)}{\partial x_j}=\frac{\partial }{\partial x_j}\left(\frac{{\mu }_{eff}}{{\sigma }_{\epsilon }}\frac{\partial k}{\partial x_j}\right)+C_1\frac{\epsilon }{k}G-C_2\frac{{\epsilon }^2}{k}+S_{\epsilon }$$(5)

  

$$G={\mu }_t\frac{\partial u_i}{\partial x_j}\left(\frac{\partial u_i}{\partial x_j}+\frac{\partial u_j}{\partial x_i}\right)$$(6)

  

$${\mu }_t=\rho C_{\mu }\frac{k^2}{\epsilon }$$(7)

where k and ε are the turbulent kinetic energy (m²/s²) and its rate of dissipation (m²/s³), respectively. The constants of C₁=1.44, C₂=1.92, C_μ=0.09, σ_k=1.0 and σ_ε=1.3 are used in the k-ε turbulent model.

In the model, the mush zone is treated as porous media, and the adding source term S_m, S_k and S_ε are used to accounts for this damping effect. According to Darcy’s law, it can be expressed as:^27,28)   

$$S_m=\frac{{\left(1-f_l\right)}^2}{f_l^3+\xi }\frac{180\mu }{{\lambda }_2^2}\left(u_j-u_c\right)$$(8)

  

$$S_k=\frac{{\left(1-f_l\right)}^2}{f_l^3+\xi }\frac{180\mu }{{\lambda }_2^2}k$$(9)

  

$$S_{\epsilon }=\frac{{\left(1-f_l\right)}^2}{f_l^3+\xi }\frac{180\mu }{{\lambda }_2^2}\epsilon$$(10)

where ξ is a small constant (0.001) to prevent division by zero, λ₂ is the secondary dendrite arm spacing (m), u_c is the casting speed (m/s) and the f_l is the liquid fraction and is defined as follow   

$$f_l=\{\begin{array}{ll}1\hfill & \mathrm{\hspace{0.17em} }\mathrm{\hspace{0.17em} }\mathrm{\hspace{0.17em} }T>T_l\hfill \\ \frac{T-T_s}{T_l-T_s}\hfill & T_s\le T\le T_l\hfill \\ 0\hfill & \mathrm{\hspace{0.17em} }\mathrm{\hspace{0.17em} }\mathrm{\hspace{0.17em} }T<T_s\hfill \end{array}$$(11)

where T is the temperature of molten steel (K), T_s is the solidus temperature (K), T_l is the liquidus temperature (K).

Energy equation:   

$$\frac{\partial \left(\rho u_{j}H\right)}{x_j}=\frac{\partial }{\partial x_j}\left({\lambda }_{eff}\frac{\partial T}{\partial x_j}\right)+S_H$$(12)

where H is the enthalpy (J/kg) and the λ_eff is the effective thermal conductivity (W/(m·K)), and they were calculated as follow:   

$${\lambda }_{eff}=\lambda +\frac{c_p{\mu }_t}{\mathrm{P}{\mathrm{r}}_{\mathrm{t}}}$$(13)

  

$$H=H_0+{\int }_{T_0}^T{C}_{p}dT+f_l\cdot L$$(14)

where λ is the thermal conductivity (W/(m·K)), c_p is the specific heat (J/(kg·K)), Pr_t is the turbulent Prandtl number and L is the latent heat (J/kg).

2.3. Boundary Conditions and Numerical Details

The backflow is avoided due to the computational domain extending to 3000 mm. Previous studies have proved that the flow field similar to that in this work is symmetrical and stable.^29,30) To reduce the computational cost, a quarter of the geometrical model is built as the computational domain considering the symmetry of the model. The computational domain is meshed into 324826 cells, and the finest mesh used to calculate the mushy zone and solidification near the mold wall is 1 mm. Moreover, tetrahedral meshes are used around the nozzle and hexahedral meshes are used in other regions to ensure the accuracy of the geometric model and the efficiency of modeling. The thermo-physical properties and operating conditions used in the numerical simulation are listed in Table 1.

Table 1. Thermo-physical, geometrical properties, and operational conditions used for numerical simulation.

Parameter	value
Density, kg/m3	7000
Dynamic viscosity, kg/(m·s)	0.0055
Specific heat, J/(kg·K)	680
Thermal conductivity, W/(m·K)	34
Liquidus temperature, K	1804
Solidus temperature, K	1769
Superheat, K	25
Latent heat, J/kg	270000
Mold width, mm	1540
Mold thickness, mm	150
Casting speed, m/min	1.2–2.0
SEN submergence depth, mm	120
Effective length of mold, mm	1100
Angle of side port, °	15
Area of side port, mm2	2870
Diameter of bottom port, mm	38

A uniformed velocity boundary condition is applied at the inlet of the SEN and the bottom of the calculation domain based on the casting speed. The inlet and initial temperature are both assumed to be the sum of the liquidus temperature and the superheat. Considering that the heat loss on the top free surface is much less than that on the mold wall,³¹⁾ so the top free surface is defined as the adiabatic condition with zero shear force.¹⁴⁾ Following the previous approach,³²⁾ the heat flux from the surface of solidified shell to the mold is assumed to be a function of the casting speed and the distance below the meniscus, and a square-root function simplification of local heat profile is loaded into mold:   

$$q=a-b\sqrt{t}$$(15)

  

$$a={\overline{Q}}_2+\frac{{\overline{Q}}_1-{\overline{Q}}_2}{\left(1-\sqrt{\frac{V_2}{V_1}}\right)}$$(16)

  

$$b=\frac{3\left({\overline{Q}}_1-{\overline{Q}}_2\right)}{2\left(\sqrt{\frac{h_m}{V_2}}-\sqrt{\frac{h_m}{V_1}}\right)}$$(17)

where q is the local heat flux down the mold length (W/m²), and t is the residence time (min) of the strand below the meniscus in the mold. Q₁ and Q₂ are the average heat flux (W/m²) from casting speed V₁ and V₂ (m/min), respectively. h_m is the effective mold length (m).

The heat transfer coefficients h loaded into secondary cooling zone is calculated by Eq. (18).¹⁴⁾   

$$h=\frac{1.57\times W^{0.55}\times \left(1.0-0.0075\times T\right)}{\alpha }$$(18)

Where W is the water flow rate (L/min), T is the temperature of the spray cooling water (K), and α is the machine-dependent calibration factor.

The SIMPLE algorithm is applied to discrete the governing equations, and the discretization scheme of momentum is second order upwind and the scheme for the pressure algorithm is PRESTO. When the residual for energy equation was smaller than 10⁻⁶ and others were smaller than 10⁻⁴, the converged solution was assumed to be achieved.

2.4. Model Validation

Figure 2 shows the comparison of flow pattern obtained by numerical and physical simulation. The physical simulation often is widely used to replicate the fluid pattern observed for the molten steel. Based on the Froude similarity number, the physical result was obtained by a 0.5: 1 scale model mold at the prototype casting speed of 2.0 m/min and the flow pattern in the physical experiment was shown in the left part of Fig. 2. Since the structure of SEN, the molten steel is divided into two streams. One stream directly flows to the mold exit, and the other flows to the narrow face and forms impinging zone on the narrow face. Numerical simulation results by the present mathematical model at the casting speed of 2.0 m/min and the SDAS of 10⁻⁴ m are given in the right part of Fig. 2. The numerical result shows that the impinging point near the narrow face is located at 0.35 m from the top free surface, which is in good agreement with the experimental result. It is indicated that the numerical simulation results agree well with the experimental results in terms of the flow pattern of molten steel and the location of impinging zone. Figure 3 shows the measured and calculated shell thickness distribution at the narrow face under the same operating parameters of Fig. 2. The measured data obtained from a breakout shell in the practical production experiment show that the solidified shell in the range of 0.2 m to 0.45 m is thinner, which indicated that the molten steel jet could affect the growth of the shell. The simulated results of the iso-surface of f_l=0.7 agree well with the measured results in terms of impinging zone formation. The calculated shell thickness has the same growth trend and is a few millimeters different. Especially, the impinging zone on the shell was captured by this model. Therefore, both Figs. 2 and 3 confirm the validity of the present mathematical model.

Fig. 2.

Comparison of numerical and physical simulation results. (Online version in color.)

Fig. 3.

Calculated and measured shell thickness at the narrow face.

3. Results and Discussion

3.1. Characterization of Impinging Zone

Figure 4 shows the simulation results for the velocity distribution at the symmetry plane of the mold. The shell thickness and velocity distribution on the shell are shown in Fig. 5. For this case, the casting speed is 1.6 m/min and the SDAS is 10⁻⁴ m. After the molten steel flows out of the SEN side port, it directly impinges on the shell. Due to the adverse pressure gradient at the shell, the velocity of molten steel dramatically slows down and splits into two parts. Figure 5(a) shows the shell growth at the narrow face. It can be seen that the solidified shell thickness stops growing in the region of 0.25 m to 0.49 m. The reason is that the heat transferred through the mold is almost equal to the sum of the heat carried by the molten steel jet and the heat released by the solidification of molten steel. In other regions, the solidification of molten steel is the only source of heat, hence the solidified shell keep growth to ensure the heat balance. Figure 5(b) shows the velocity distribution on the shell. Duo to the molten steel jet is stagnant at the impinging point, so the velocity distribution on the shell tends to be “M-shaped”. As such, the variation tendency of velocity magnitude is different from that of shell growth, especially in region A and B. Meanwhile, no clear boundary of impinging zone defined by velocity magnitude can be observed. Since the molten steel jet leaves the mushy zone after impinging on the shell, the X-velocity distribution tends to be “U-shaped”, which has the similar variation tendency with the impinging zone, as shown in Fig. 5(c). However, when the X-velocity is higher than 0 in region C, it indicates that the molten steel is flow away from the mushy zone, which is inconsistent with the stop of the shell growth. This means that it is unreasonable to explain the formation of impinging zone using the velocity and the X-velocity. Hence, based on the original physical meaning of Pe number, a new defined Peclet number of mushy zone (Pe_m) is proposed to characterize the effect of turbulent heat diffusion/convective heat transfer combinations on the turbulent heat transfer in the mushy zone and identify the boundary of impinging zone by adding the turbulent kinetic energy into the equation and the Pe_m is calculated as shown in Eq. (19). Meanwhile, since the research region is in the mushy zone, the characteristic length is set as the SDAS, which has an important impact on the flow in the mushy zone.   

$$P{e}_{\mathrm{m}}=Pe\cdot Fr=\frac{\lambda }{\rho v{c}_{\mathrm{p}}{\lambda }_2^{}}\cdot \frac{v^2}{\mathrm{g}{\lambda }_2^{}}=\frac{\lambda k}{\rho v{c}_{\mathrm{p}}\mathrm{g}{\lambda }_2^2}$$(19)

ρ, k, v, c_p, g and λ is density (kg/m³), turbulent kinetic energy (m²/s²), velocity (m/s), specific heat (J/(kg·K)), acceleration of gravity (m/s²) and thermal conductivity (W/(m·K)) of mushy zone respectively.   

$$Q_{\mathrm{c}\mathrm{m}}=\rho \sqrt{u_{\mathrm{x}}^2+u_{\mathrm{y}}^2+u_{\mathrm{z}}^2}H$$(20)

  

$$Q_{\mathrm{d}\mathrm{m}}={\lambda }_{\mathrm{e}\mathrm{f}\mathrm{f}}\sqrt{{\left(\frac{\partial T}{\partial x}\right)}^2+{\left(\frac{\partial T}{\partial y}\right)}^2+{\left(\frac{\partial T}{\partial z}\right)}^2}$$(21)

  

$$Q_{\mathrm{c}\mathrm{x}}=\rho u_{\mathrm{x}}H$$(22)

  

$$Q_{\mathrm{d}\mathrm{x}}={\lambda }_{\mathrm{e}\mathrm{f}\mathrm{f}}\frac{\partial T}{\partial x}$$(23)

Fig. 4.

Velocity distribution at the symmetry plane. (Online version in color.)

Fig. 5.

Shell thickness and velocity distribution at the narrow face: (a) shell thickness distribution (f_l=0.7) on the narrow face; (b) velocity on the shell; (c) X-velocity on the shell. (Online version in color.)

Figure 6(a) shows the shell thickness distribution and the profile of Pe_m on the shell. It can be noted that the“inverted V-shaped” distribution of the Pe_m facilitates the identification of the boundary of the impinging zone and the location of the impinging point. To evaluate the heat transfer capabilities of the resultant convective heat transfer and the resultant turbulent heat diffusion, Fig. 6(b) shows the profile of Q_cm and Q_dm on the shell, and they were calculated by Eqs. (20) and (21). Affected by the jet-induced turbulence, the Q_dm is higher in the impinging zone. The Q_cm in the impinging zone is not significantly higher than that outside the impinging zone. In the impinging zone, the value of Q_dm/Q_cm is in the range of 7.0%–91.1%. It means that although more heat is transferred by the convection, the turbulent heat diffusion is the main feature of the impinging zone.

Fig. 6.

(a) Shell thickness (f_l=0.7) at the narrow face and Pe_m on the shell; (b) heat transfer on the shell; (c) heat transfer of x-direction on the shell. (Online version in color.)

According to the value of Pe_m, the impinging zone can be divided into two parts. In the Region I (Pe_m> 150), which also can be called the stagnation zone, as shown in Fig. 6, the velocity of molten steel is lowest, and the heat transfer rate is affected by the convection and turbulent diffusion. In the Region II (47.1 < Pe_m< 150), which also can be called the wall jet zone, as shown in Fig. 6, the flow direction of the molten steel gradually changes to a direction parallel to the shell, inducing a significant reduction in the turbulent heat diffusion. Outside the impinging zone (Pe_m< 47.1), the molten steel jet gradually flows away from the shell, which further reduces the capacity of turbulent heat diffusion. This means that in the mushy zone, the Pe_m is an effective method to evaluate the characterization of turbulent heat transfer and identify the boundary of the impinging zone.

To evaluate the heat transfer capabilities of the convective heat transfer and the turbulent heat diffusion of the x direction, Fig. 6(c) shows the profile of Q_cx and Q_dx on the shell, and they were calculated by Eqs. (22) and (23). It can be seen that in the impinging zone, the heat transfer capacity by turbulent heat diffusion is much higher than that by convection. Especially near the boundary of the impinging zone, the molten steel jet is away from the impinging zone, which means that the convective heat transfer promotes shell growth. The turbulent heat diffusion is the only heat source that maintains the boundary of the impinging zone. Therefore, the turbulent heat diffusion has a significant effect on the formation of the impinging zone.

3.2. Effect of Casting Speed

The effects of the casting speed on the impinging zone and Pe_m distribution on the shell with the SDAS of 10⁻⁴ m are investigated, respectively, as shown in Fig. 7. The decrease of shell growth rate and the change of impinging zone are noted with the increase of the casting speed in the mold, as shown in Fig. 7(a). With the increase of the casting speed, the impinging zone of the shell in the center plane of the mold extends along the forward and reverse direction of the gravity and the thickness direction of the shell. The Pe_m also increases and the impinging point slightly moves downward with the increase in casting speed, as shown in Fig. 7(b). It can be considered that with the increase of casting speed, the momentum of molten steel increases, as well as the turbulent heat diffusion increases greatly in the impinging zone, leading to a size increase of impinging zone along the forward and reverse directions of gravity. Meanwhile, the increase in the casting speed reduces the cooling time of the shell in the mold, which leads to a decrease in the shell thickness of the impinging zone. Due to the higher heat flux of the shell caused by the higher casting speed, more heat transfer by turbulent heat diffusion is used to maintain the boundary of the impinging zone, inducing the critical value of Pe_m increasing from 34.0 to 68.3. Therefore, the higher casting speed can increase the Pe_m and enlarge the impinging zone, which has a negative effect on reducing the breakout rate and improving steel quality.

Fig. 7.

(a) Shell thickness (f_l=0.7) at the narrow face and (b) Pe_m distribution on the shell for different casting speeds. (Online version in color.)

3.3. Effect of SDAS

As it is discussed in the previous sections of this work, the fluid flow in the mushy zone, which also depends on the SDAS,^27,33) strongly affects the turbulent heat transfer and the formation of the impinging zone. Based on the previous study,²⁷⁾ the SDAS, which is determined by cooling rate, composition and other factors, is of the order of 10⁻⁴ m. Thus, three SDAS values of 5×10⁻⁵ m, 1×10⁻⁴ m and 2×10⁻⁴ m are used to provide suggestions for its control. The predicted impinging zone on the narrow face is presented in Fig. 8(a) for different values of SDAS at the casting speed of 2.0 m/min.

Fig. 8.

(a) Shell thickness distribution (f_l=0.7) at the narrow face and (b) Pe_m distribution on the shell for different SDASs. (Online version in color.)

With the growth of SDAS, the impinging zone of the shell in the center plane of the mold sequentially increases in the size and enlarges to the direction of mold exit; the thickness at the mold exit decreases from 23.96 mm to 22.99 mm. Especially, the growth rate of shell thickness and the upper boundary of impinging zone remains mostly unchanged. The effect of SDAS on the Pe_m along the centerline of the shell at the narrow face is shown in Fig. 8(b). It is observed that with an increase of SDAS, there occurs a decrease of Pe_m and shift upward of the impinging point. One can see that, with the biggest SDAS, the flowing resistance is decreased, and more energy is transferred by the convection process. For the smallest SDAS, the flow of molten steel is damped in the mushy zone, so more energy transferred by turbulent heat diffusion is used to maintain the boundary of the impinging zone, which increases the critical value of Pe_m from 38.4 to 167.3 and decreases the size of impinging zone. Thereby, reducing SDAS is an effective way to improve the quality of steel products and promote the increase of casting speed. Moreover, the SDAS is very important for accurate prediction in the mushy zone. However, more details require further research considering the influences of turbulence models and relevant key variables such as SDAS distribution in the mold, mold width, etc.

4. Conclusions

In the presented study, a three-dimensional model is developed to describe the flow, heat transfer and solidification processes in the mold, and the Pe_m is proposed to study the turbulent heat transfer behavior in the mushy zone and define the boundary of the impinging zone. The effects of casting speed and the SDAS of the shell are investigated. The conclusions obtained are summarized as follows:

(1) In the impinging zone, the turbulent heat diffusion is much higher and the ratio of the resultant turbulent heat diffusion to the resultant convective heat transfer can be as high as 91.1%. On the x-direction, the turbulent heat diffusion is higher than the convective heat transfer and the turbulent heat diffusion is the only source of energy to maintain the boundary of the impinging zone.

(2) Compared with the “M-shaped” velocity distribution and the “U-shaped” X-velocity distribution, the “inverted V-shaped” Pe_m distribution is more reasonable and has the advantages of obtaining the clearer boundary of impinging zone and the location of the impinging point.

(3) The increase in the casting speed can increase the Pe_m and its critical value, and enlarge the impinging zone along the forward and reverse directions of gravity and the thickness direction of the shell.

(4) SDAS is a key parameter for the accurate prediction of turbulent heat transfer in the mold. The investigation shows that, as the SDAS drops, the Pe_m and its critical value increase, and the size of impinging zone decreases and the impinging point moves upward. The growth rate of shell thickness is barely influenced by the SDAS.

Acknowledgements

The authors gratefully acknowledge the National Natural Science Foundation of China (No. 52174301, 51974080) which has made this research possible.

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