Comprehensive Evaluation Method for Cooling Effect on Process Thermal Dissipation Rate during Continuous Casting Mold

Abstract

Cooling in the continuous casting mold is the essential process of the molten steel solidifying into a slab shell. The synergistic relationship of casting state, process operation, continuous casting equipment, and other factors is complex and has a significant influence on thermal transfer in the mold. Therefore, a concept of “process thermal dissipation rate” defined by mold system thermal input and output was proposed in this work. The thermal input of molten steel was calculated through the casting temperature, and the slab residual thermal at the outlet of the mold was calculated by the solidification heat transfer model. Consequently, the thermal dissipation rate was calculated to quantify the multi-factor cooperative relationship of mold. The industrial case reflected that the thermal dissipation rates of three stable castings were 12.5%, 14.3%, and 18.8%, respectively, and all of them were obviously abnormal in unsteady casting such as start casting, changing tundish, and end casting. The results above indicated that the thermal dissipation rate could characterize the mold cooling target under the cooperation of complex factors and provide a new method for the dynamic evaluation of the mold system cooling effect with different casting states. Accordingly, the correlation analysis between superheat, casting speed, cooling water flow, and thermal dissipation rate revealed the synergistic influence law of multi-operation on mold cooling effect, which provided a new idea for the precise control of multi-process collaboration in continuous casting.

1. Introduction

Efficient and stable cooling is the key to ensuring slab forming and quality in the continuous casting process, which promotes the whole steelmaking process running dynamically, orderly, synergistically, and continuously.^1,2) The thermal transfer and dissipation in mold, which has been called the “heart” of continuous casting, has a profound influence on molten steel solidification and the shell grows evenly, dominating both the slab quality^3,4,5) and process anterograde.^6,7,8) The quantitative description and modeling of cooling effect for multi-state (stable casting, begin casting, end casting, etc.), multi-input (molten steel, superheat), multi-operation (casting speed, cooling water flow, superheat, etc.) is one of the important methods to improve the collaborative optimization in the continuous casting mold. Accordingly, a comprehensive evaluation method is necessary.

Some energy field researchers proposed a lot of factors and methodologies from the perspective of energy conversion, such as process thermal consumption, thermal analysis, energy level analysis, thermo-economic analysis. Process thermal consumption refers to energy consumption minus energy recovery per ton of steel produced in a statistical period.⁹⁾ Based on the law of energy conservation and transformation, thermal analysis investigates the external energy loss caused by the discharge of an energy system to the outside world from the perspective of a quantitative relationship.¹⁰⁾ Based on thermal analysis, the difference between available and unusable energy is considered in energy level analysis. This theory “weighed” energy, and the more available energy, the heavier the energy.^11,12) The thermo-economic analysis, which introduces the concept of capital, converts the available energy into capital cost, building a bridge between the use-value and economic value of energy.^13,14,15) The energy factors above comprehensively evaluate the steel manufacturing process from the perspective of energy conversion. Unfortunately, their statistical periods are years, months, days, and furnace times, which is suitable for energy management but difficult to meet the requirement of process evaluation and optimization. Additionally, the purpose and advantages of the factors above are to save energy and reduce emissions rather than provide operational guidance for process anterograde and slab quality control, which are the most important functions of the mold. Therefore, the methods and factors in the energy field mentioned above just can provide a reference for the evaluation of the molten steel cooling process in continuous casting mold instead of applying to evaluate directly.

To establish the evaluation factor of the cooling process in mold, thermal transfer mechanisms are necessary. Many remarkable studies in this field have been reported by former researchers.^{16,17,18,19,20,21,22,23,24,25,26,27,28)} Zhang et al.¹⁷⁾ established a 2D numerical model by Fluent software to simulate the fluid flow, heat transfer, and solidification of the steel in the mold. Suzuki et al.²⁰⁾ studied the slab shell deformation in the mold, and it was related to the initial abnormal thermal transfer. Liu et al.²²⁾ simulated the solidification for continuous casting of duplex stainless steel by ProCAST software, reflecting that the surface temperature distribution was more sensitive to the casting speed rather than superheat. In more detail, the thermal of molten steel in the mold is transferred following the path of shell-air gap/liquid slag layer/solid slag layer-mold copper, removed from the mold by cooling water eventually. Cho et al.^26,27,28) found that the liquid slag layer, solid slag layer, and air gap between the slab and the copper as thermal resistance played an important role in the heat transfer of the mold, which means the heat transfer condition in mold could be improved by adjusting the properties of the mold flux.

The studies above researches have made a further understanding of the thermal transfer microscopic mechanism in mold and provided suggestions on optimization. Nevertheless, the solidification and thermal transfer of molten steel in the mold is the result of the coupling effect of several factors, such as molten steel properties, mold operations, mold state, etc. The single factor optimization guidance based on micro-mechanism is not practical in the complex continuous casting process with multi-state and multi-operation cooperative control. Thomas²⁹⁾ reviewed the modeling of thermal transfer in the continuous casting process and pointed out that future advances to the real commercial processes will require a combination of all factors available. As consequence, a comprehensive evaluation method which can characterize the multi-factor synergistic relationship of mold and dynamically evaluate the cooling effect with different casting state is particularly necessary.

In this work, the definition and calculation method of “process thermal dissipation rate” was proposed, which is based on the reference to thermal consumption factors, to describe the multi-factor synergistic relationship on mold abstractly and evaluate the comprehensive mold cooling effect. Additionally, the single and synergistic effects of multi-condition on process thermal dissipation rate were discussed based on industrial data, which could provide a theoretical target of mold cooling operation and a method on multi-operation collaborative effect evaluation for the continuous casting mold.

2. Theoretical Fundamental

2.1. Problem Description

The timely and sufficient dissipation of molten steel thermal in the mold has a significant effect on the process running and slab quality. The typical mold thermal transfer behavior is shown in Fig. 1. The thermal is input by molten steel from SEN (submerged entrance nozzle). After tempestuously cooling in the mold, the shell slab is formed and output. Different cooling effects are required under different continuous casting conditions, steel grades, slab size, and even superheat. Since the cooling performance is controlled by multiple operations such as casting speed, cooling water, electromagnetic field, etc. The mold thermal transfer is a systematic and complex problem influenced by multiple factors such as working state, input, operation, output, and equipment, which can be abstractly described as Formula (1). To achieve Y_output aim at the target G (qualified slab quality, continuous casting efficiency), it is necessary to define synergy relation F and evaluate the cooling effect in the mold through systematic comprehensive indicators, to provide a theoretical basis for multi-operation collaborative intelligent control corresponding to multi-state.   

$$Y_{\text{output}}=\underset{system\to G}{\text{optimal}}f(X_{\text{state,i}},X_{\text{input,i}},X_{\text{operation,i}},X_{\text{equipment,i}})$$(1)

where, Y_output represents system output factors, the slab; G represents the continuous casting targets, including qualified slab quality and casting efficiency; F represents the nonlinear synergistic relationship of X_state,i, X_input,i, X_operation,i, and X_equipment,i; X_state,i represents continuous casting state factors, including normal state and abnormal casting (start, end, tundish change, SEN change, steel grade change, etc.); X_input,i represents system input factors, such as superheat and steel grade. X_operation,i represents system operation factors, such as casting speed, cooling water flow rate, cooling water temperature, etc. X_equipment,i represents system equipment factor, such as mold width, mold thickness, etc.

Fig. 1.

Thermal transfer diagram of the continuous casting mold. (Online version in color.)

In this problem, X_state,i and X_equipment,i are the objective conditions of continuous casting. and X_input,i is the control by the pre-process. All of them cannot be adjusted in the process of continuous casting. Consequently, X_operation,i is the only one that can be controlled to achieve casting goal. Therefore, it is necessary to clarify the cooperative relationship between different working conditions and operation factors such as casting speed and cooling water, so as to improve the theoretical basis for accurate control of continuous casting process.

The comprehensive evaluation method of mold cooling effect based on “process thermal dissipation rate” is to analyze the thermal exchange relationship in the mold from the perspective of the system. Based on the energy conservation principle and thermal transfer mechanism, the difference of thermal dissipation of mold under a different state, steel, and operation conditions can be calculated. Simultaneously, the influence of various system factors on thermal dissipation can be analyzed to provide a theoretical basis and optimization direction for multi-operation cooperative control modeling of mold. Based on this analysis, the evaluation and optimization of the cooling effect of the mold system can be realized.

2.2. Process Thermal Dissipation Principle and Calculation Method

The systematic thermal transfer behavior in the mold can be sketched out in Fig. 2. The molten steel and slab, which mainly contributes physical heat such as overheating, latent heat, and sensible heat, are regarded as thermal input and output for the system. The consumption of electricity, Ar, cooling water, etc. are regarded as energy supply for the system. The thermal absorbed by mold flux, cooling water, air, etc. is regarded as thermal dissipation for the system. Therefore, the thermal transfer process and process thermal dissipation rate in the whole mold system can be expressed as follows respectively:   

$$Q_{\text{in}}+Q_{\text{s}}=Q_{\text{out}}+Q_{\text{d}}$$(2)

  

$$\phi =\frac{Q_{\text{in}}-Q_{\text{out}}}{Q_{\text{in}}+Q_{\text{s}}}\times 100\%$$(3)

where, Q_in is thermal input for the system, J; Q_s is the energy supply for the system, J; Q_out is thermal output for the system, J; Q_d is thermal dissipation for the system, J; φ is process thermal dissipation rate, %.

Fig. 2.

Schematic of thermal transfer behaviors in continuous casting mold system. (Online version in color.)

Among the Q_s, the electricity drive casting operation and complex electromagnetic field action, the Ar is used to improve the flow field in the mold, neither of them is involved in the thermal transfer in the mold. Among the Q_d, the energy dissipated to air is much less than that absorbed by mold flux melting and cooling water. Therefore, the effect of the above thermal medium on the process thermal transfer could be ignored, and φ can be simplified as follows:   

$$\phi =\frac{Q_{\text{in}}-Q_{\text{out}}}{Q_{\text{in}}}=\left(1-\frac{Q_{\text{slab}}}{Q_{\text{molten}}}\right)\times 100\%$$(4)

where, Q_molten and Q_slab are the thermal brought in and kept in mold, J. Both of them can be calculated by the slab temperature in the mold. Moreover, the temperature in the mold can be calculated by establishing a two-dimensional solidification heat transfer model using the finite thickness slicing method. As shown in Fig. 3, the mold can be spatially discretized, and each slice in the computational domain space is numerically discretized to obtain the temperature information.

Fig. 3.

Schematic of the calculation area in mold. (Online version in color.)

The model solving process made the following appropriate simplifications or assumptions:

(1) Only one fourth of the slab transverse section was required for simulation, and the other part were obtained by the method of symmetry.

(2) This work focused on the input and output thermal of the whole mold system. Therefore, the influence of the mold taper, mold oscillation, and surface fluctuation on the temperature field was negligible.

(3) It was assumed that the meniscus temperature was uniform and it was the casting temperature.

(4) Only the thermal transfer of slab thickness and width direction was considered in this work.

(5) The influence of molten steel convection on the temperature was processed become effective thermal conductivity of steel.

Based on the above assumptions, the thermal conductivity differential equation of two-dimensional solidification heat transfer in the mold can be described as follows:   

$$\rho C_{\text{e}ff}=\frac{\partial T}{\partial \tau }=\frac{\partial }{\partial x}\left({\lambda }_{eff}\frac{\partial T}{\partial x}\right)+\frac{\partial }{\partial y}\left({\lambda }_{eff}\frac{\partial T}{\partial y}\right)$$(5)

where, T is the temperature, °C; τ is the time, s; x is the distance in the slab width direction, m; y is the distance in the slab thickness direction, m; ρ is the density of steel, 7.4×10³ kg/m³ was adopted in this work; C_eff is the effective specific heat, J/(kg·°C); λ_eff is the effective thermal conductivity, J/(m·s·°C). The solidification process of molten steel is accompanied by the latent heat, which was calculated into the specific heat capacity in this work:   

$$C_{eff}=\{\begin{array}{l}{C}_p+\frac{L_f}{T_L-T_S}\hfill \\ C_p\hfill \end{array}\hspace{1em}\begin{array}{r}\hfill T_S\le T\le T_L\\ \hfill T>T_{L}orT<T_S\end{array}$$(6)

where, C_p is the actual specific heat capacity of steel, 0.84×10³ J/(kg·°C) was adopted in this work; L_f is the solidification latent heat, 2.6×10⁵ J/kg was adopted in this work; T_S and T_L are the solidus and liquidus temperature of steel, °C, respectively.

In this work, the thermal conductivity of solid slab was simplified as a constant, 30 J/(m·s·°C). Since the flow of molten steel in the mold could enhance the heat transfer effect, the thermal conductivity of molten steel was treated as a multiple of that in the solid slab. Moreover, the presence of dendrites in the two-phase region slows down the flow of liquid steel to some extent, and the thermal conductivity was considered to be between the solid slab and the liquid steel. Therefore, the effective thermal conductivity can be calculated by the following:   

$${\lambda }_{eff}=\{\begin{array}{l}3\lambda \left(T-T_L\right)/\left(T_C-T_L\right)+4\lambda \hfill \\ 3\lambda \left(T-T_S\right)/\left(T_L-T_S\right)+\lambda \hfill \\ \lambda \hfill \end{array}\hspace{1em}\begin{array}{r}\hfill T>T_L\\ \hfill T_L>T>T_S\\ \hfill T<T_S\end{array}$$(7)

At τ=0, the slicing element of the slab is just at the meniscus of the mold. According to the assumption (3), the initial conditions can be obtained as follows:   

$$T(x,y,\tau )|{}_{\tau =0}=T_C$$(8)

where, T_C is the casting temperature, °C.

At τ>0, the central and surface boundary conditions of the slab are:   

$$-{\lambda }_{eff}\frac{\partial T}{\partial x}|{}_{\begin{array}{l}x=0\\ \tau >0\end{array}}=0$$(9)

  

$$-{\lambda }_{eff}\frac{\partial T}{\partial y}|{}_{\begin{array}{l}y=0\\ \tau >0\end{array}}=0$$(10)

  

$$-{\lambda }_{eff}\frac{\partial T}{\partial x}|{}_{\begin{array}{l}x=\frac{a}{2}\\ \tau >0\end{array}}=q_n$$(11)

  

$$-{\lambda }_{eff}\frac{\partial T}{\partial y}|{}_{\begin{array}{l}y=\frac{b}{2}\\ \tau >0\end{array}}=q_w$$(12)

where, q_n and q_w are the heat flux in the direction of mold narrow and wide side, respectively, w/m².

Since this work focused on the input & output rather than the temperature field inside the mold, the average heat flux can be used instead of the instantaneous heat flux to simplify the calculation:   

$$q_{\text{n}}=60\frac{C_{\text{w}}{V}_{\text{w,}\mathrm{\hspace{0.17em} }\text{n}}{\rho }_{\text{w}}\mathrm{\Delta }{T}_{\text{w,}\mathrm{\hspace{0.17em} }\text{n}}}{A_n}$$(13)

  

$$q_{\text{w}}=60\frac{C_{\text{w}}{V}_{\text{w,}\mathrm{\hspace{0.17em} }\text{w}}{\rho }_{\text{w}}\mathrm{\Delta }{T}_{\text{w,}\mathrm{\hspace{0.17em} }\text{w}}}{A_w}$$(14)

where, C_w is the specific heat capacity of water, 4.2×10³ J/(kg·°C) was adopted in this work; V_w,n and V_w,w are the cooling water flow on the narrow and wide sides of the mold, respectively, L/min; ρ_w is the density of water, 0.98×10³ kg/m³ was adopted in this work; ΔT_w,n and ΔT_w,w are the temperature difference of narrow and wide side of cooling water, respectively, °C; A_n and A_w are the narrow and wide surface areas of the mold, respectively, m².

From the meniscus to the outlet of the mold, the slab was discretized into n slice elements of finite thickness. The 1/4 area of the slice unit was taken as the calculation object, and this area of each slice unit was evenly meshing, as shown in Fig. 4. In the calculation region, a finite number of grid discrete nodes were used to replace the continuous points, and then the temperature of each offline node was numerically solved.

Fig. 4.

Schematic of discretization of slice unit grid. (Online version in color.)

In this work, the display finite difference method was adopted to discretize the differential equations of the solidification heat transfer model. According to Taylor series, Eq. (5) can be expressed as follows:   

$$\begin{array}{l}{T}_{i,j}^{k+1}=T_{i,j}^k+{\displaystyle \frac{{\lambda }_{eff}\mathrm{\Delta }\tau }{C_{\text{e}ff}\rho }}\left[{\displaystyle \frac{(T_{i+1,j}^k-T_{i,j}^k)-(T_{i,j}^k-T_{i-1,j}^k)}{\mathrm{\Delta }{x}^2}}\right]\\ +{\displaystyle \frac{{\lambda }_{eff}\mathrm{\Delta }\tau }{C_{\text{e}ff}\rho }}\left[{\displaystyle \frac{(T_{i,j+1}^k-T_{i,j}^k)-(T_{i,j}^k-T_{i,j-1}^k)}{\mathrm{\Delta }{y}^2}}\right]+o(\mathrm{\Delta }x\mathrm{\Delta }y+\mathrm{\Delta }\tau )\end{array}$$(15)

where, i, j and k are the width, thickness and casting direction nodes, respectively; T_i,j^k is the temperature of node (i,j,k); Δx, Δy and Δτ are the space step in the width, thickness and casting direction, respectively. o(ΔxΔy+Δτ) is the truncation error of the difference equation. Ignore the truncation errors of (15) and combined Eqs. (9), (10), (11), (12), the temperature calculated at various points in Fig. 5 difference equation are as follows:   

$$T_{i,j}^{k+1}|{}_{\begin{array}{l}i=0\\ j=0\end{array}}=T_{i,j}^k+\frac{2{\lambda }_{eff}\mathrm{\Delta }\tau }{C_{\text{e}ff}\rho }\left[\frac{(T_{i+1,j}^k-T_{i,j}^k)}{\mathrm{\Delta }{x}^2}+\frac{(T_{i,j-1}^k-T_{i,j}^k)}{\mathrm{\Delta }{y}^2}\right]$$(16)

  

$$\begin{array}{l}{T}_{i,j}^{k+1}|{}_{\begin{array}{l}i=0\\ 0<j<{\displaystyle \frac{b}{2}}\end{array}}=\\ T_{i,j}^k+{\displaystyle \frac{{\lambda }_{eff}\mathrm{\Delta }\tau }{C_{\text{e}ff}\rho }}\left[{\displaystyle \frac{2(T_{i+1,j}^k-T_{i,j}^k)}{\mathrm{\Delta }{x}^2}}+{\displaystyle \frac{(T_{i,j+1}^k-T_{i,j}^k)-(T_{i,j}^k-T_{i,j-1}^k)}{\mathrm{\Delta }{y}^2}}\right]\end{array}$$(17)

  

$$\begin{array}{l}{T}_{i,j}^{k+1}|{}_{\begin{array}{l}i=0\\ j={\displaystyle \frac{b}{2}}\end{array}}=\\ T_{i,j}^k+{\displaystyle \frac{2{\lambda }_{eff}\mathrm{\Delta }\tau }{C_{eff}\rho }}\left[{\displaystyle \frac{\left(T_{i+1,j}^k-T_{i,j}^k\right)}{\mathrm{\Delta }{x}^2}}+{\displaystyle \frac{\left(T_{i,j+1}^k-T_{i,j}^k\right)}{\mathrm{\Delta }{y}^2}}-{\displaystyle \frac{q_w}{{\lambda }_{eff}\mathrm{\Delta }y}}\right]\end{array}$$(18)

  

$$\begin{array}{l}{T}_{i,j}^{k+1}|{}_{\begin{array}{l}0<i<{\displaystyle \frac{a}{2}}\\ j={\displaystyle \frac{b}{2}}\end{array}}=\\ T_{i,j}^k+{\displaystyle \frac{{\lambda }_{eff}\mathrm{\Delta }\tau }{C_{eff}\rho }}\left[{\displaystyle \frac{\left(T_{i+1,j}^k-T_{i,j}^k\right)-\left(T_{i,j}^k-T_{i-1,j}^k\right)}{\mathrm{\Delta }{x}^2}}+{\displaystyle \frac{2\left(T_{i,j+1}^k-T_{i,j}^k\right)}{\mathrm{\Delta }{y}^2}}-{\displaystyle \frac{2q_w}{{\lambda }_{eff}\mathrm{\Delta }y}}\right]\end{array}$$(19)

  

$$\begin{array}{l}{T}_{i,j}^{k+1}|{}_{\begin{array}{l}i={\displaystyle \frac{a}{2}}\\ j={\displaystyle \frac{b}{2}}\end{array}}=\\ T_{i,j}^k+{\displaystyle \frac{2{\lambda }_{eff}\mathrm{\Delta }\tau }{C_{eff}\rho }}\left[{\displaystyle \frac{\left(T_{i-1,j}^k-T_{i,j}^k\right)}{\mathrm{\Delta }{x}^2}}+{\displaystyle \frac{\left(T_{i,j+1}^k-T_{i,j}^k\right)}{\mathrm{\Delta }{y}^2}}-{\displaystyle \frac{q_n}{{\lambda }_{eff}\mathrm{\Delta }x}}-{\displaystyle \frac{q_w}{{\lambda }_{eff}\mathrm{\Delta }y}}\right]\end{array}$$(20)

  

$$\begin{array}{l}{T}_{i,j}^{k+1}|{}_{\begin{array}{l}i={\displaystyle \frac{a}{2}}\\ 0<j<{\displaystyle \frac{b}{2}}\end{array}}=\\ T_{i,j}^k+{\displaystyle \frac{{\lambda }_{eff}\mathrm{\Delta }\tau }{C_{eff}\rho }}\left[{\displaystyle \frac{2\left(T_{i-1,j}^k-T_{i,j}^k\right)}{\mathrm{\Delta }{x}^2}}+{\displaystyle \frac{\left(T_{i,j+1}^k-T_{i,j}^k\right)-\left(T_{i,j}^k-T_{i,j-1}^k\right)}{\mathrm{\Delta }{y}^2}}-{\displaystyle \frac{2q_n}{{\lambda }_{eff}\mathrm{\Delta }x}}\right]\end{array}$$(21)

  

$$\begin{array}{l}{T}_{i,j}^{k+1}|{}_{\begin{array}{l}i={\displaystyle \frac{a}{2}}\\ 0<j<{\displaystyle \frac{b}{2}}\end{array}}=\\ T_{i,j}^k+{\displaystyle \frac{2{\lambda }_{eff}\mathrm{\Delta }\tau }{C_{eff}\rho }}\left[{\displaystyle \frac{\left(T_{i-1,j}^k-T_{i,j}^k\right)}{\mathrm{\Delta }{x}^2}}+{\displaystyle \frac{\left(T_{i,j-1}^k-T_{i,j}^k\right)}{\mathrm{\Delta }{y}^2}}-{\displaystyle \frac{q_n}{{\lambda }_{eff}\mathrm{\Delta }x}}\right]\end{array}$$(22)

  

$$\begin{array}{l}{T}_{i,j}^{k+1}|{}_{\begin{array}{l}0<i<{\displaystyle \frac{a}{2}}\\ j=0\end{array}}=\\ T_{i,j}^k+{\displaystyle \frac{{\lambda }_{eff}\mathrm{\Delta }\tau }{C_{eff}\rho }}\left[{\displaystyle \frac{\left(T_{i+1,j}^k-T_{i,j}^k\right)-\left(T_{i,j}^k-T_{i-1,j}^k\right)}{\mathrm{\Delta }{x}^2}}+{\displaystyle \frac{2\left(T_{i,j-1}^k-T_{i,j}^k\right)}{\mathrm{\Delta }{y}^2}}\right]\end{array}$$(23)

  

$$\begin{array}{l}{T}_{i,j}^{k+1}|{}_{\begin{array}{l}0<i<{\displaystyle \frac{a}{2}}\\ 0<j<{\displaystyle \frac{b}{2}}\end{array}}=\\ T_{i,j}^k+{\displaystyle \frac{{\lambda }_{eff}\mathrm{\Delta }\tau }{C_{eff}\rho }}\left[{\displaystyle \frac{\left(T_{i+1,j}^k-T_{i,j}^k\right)-\left(T_{i,j}^k-T_{i-1,j}^k\right)}{\mathrm{\Delta }{x}^2}}+{\displaystyle \frac{\left(T_{i,j+1}^k-T_{i,j}^k\right)-\left(T_{i,j}^k-T_{i,j-1}^k\right)}{\mathrm{\Delta }{y}^2}}\right]\end{array}$$(24)

Fig. 5.

Original industrial data abnormal: (a) ladle-time-scale; (b) real-time-scale. (Online version in color.)

In this work, “constant spacing method” was adopted to divide the resulting slice elements along the casting direction. The slice thickness was assumed to an infinitesimal σ, and the spacing was constant at 0.1 m. When the calculated signal was received, the slice was generated from the meniscus and moves forward at the current casting speed. When the distance from the meniscus exceeds the outlet of the mold, it disappeared. The temperature at the meniscus surface was uniform as the casting temperature, then the system input thermal of the mold was:   

$$\begin{array}{l}{Q}_{\text{molten}}=\left(C_{\text{p}}\times (T_{\text{C}}-T_L)+C_p+\frac{L_f}{T_L-T_S}+C_{\text{p}}\times (T_{\text{C}}-T_E)\right)\\ \hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\times a\times b\times \sigma \times \rho \end{array}$$(25)

While the temperature at the mold outlet was non-uniform and multi-phase coexists, so the system output thermal should be calculated by micro-element accumulation:   

$$\begin{array}{l}{Q}_{\text{slab,}\mathrm{\hspace{0.17em} }\text{L}}=\\ 4\sigma \rho \sum _{i,j}^{\frac{a}{2},\frac{b}{2}}\left(C_{\text{p}}\times (T_{i,j}^k-T_L)+C_p+\frac{L_f}{T_L-T_S}+C_{\text{p}}\times (T_{i,j}^k-T_E)\right)\hspace{1em}{T}_{i,j}^k>T_L\end{array}$$(26)

  

$$\begin{array}{l}{Q}_{\text{slab,}\mathrm{\hspace{0.17em} }\text{L-S}}=\\ 4\sigma \rho \sum _{i,j}^{\frac{a}{2},\frac{b}{2}}\left(C_p+\frac{L_f}{T_L-T_S}+C_{\text{p}}\times (T_{i,j}^k-T_E)\right)\hspace{1em}{T}_L>T_{i,j}^k>T_S\end{array}$$(27)

  

$$Q_{\text{slab,}\mathrm{\hspace{0.17em} }\text{S}}=4\sigma \rho \sum _{i,j}^{\frac{a}{2},\frac{b}{2}}(C_{\text{p}}\times (T_{i,j}^k-T_E))\hspace{1em}{T}_{i,j}^k<T_S$$(28)

  

$$Q_{\text{slab}}=Q_{\text{slab,}\mathrm{\hspace{0.17em} }\text{L}}+Q_{\text{slab,}\mathrm{\hspace{0.17em} }\text{L-S}}+Q_{\text{slab,}\mathrm{\hspace{0.17em} }\text{S}}$$(29)

where, T_E is the environment temperature, 20°C was adopted in this work; Q_slab,L, Q_slab,L-S, Q_slab,L are the heat retained in liquid phase, two-phase, and solid phase of mold outlet, respectively, J. According to the above formula and the parameters in Table 1, the thermal dissipation rate can be calculated.

Table 1. Industrial continuous casting production data.

Categories	Type	Scale	Value*
Xstate,i	Text	/	stable, start, end, SEN change, etc.
	Steel Grade/#	LTS	Sample 1 (75, 61.5%), Sample 2 (21, 19.3%), Sample 3 (8, 7.3%)
Xinput,i	Liquidus Temperature/TL	LTS	1512°C (75, 68.8%), 1526°C (21, 19.3%), 1490°C (13, 11.9%)
	Solidus Temperature/TS	LTS	1477°C (75, 68.8%), 1494°C (21, 19.3%), 1450°C (13, 11.9%)
	Superheat/ΔTC	LTS/RTS	26°C [23°C–29°C]
	Casting Steel Amount/M	LTS	206 t [203 t–208 t]
Xequipment,i	Mold/Slab Width/a	LTS	1850 mm (57, 52.3%), 2050 mm (10, 9.2%), 2200 mm (42, 38.5%)
	Mold/Slab Thickness/b	LTS	200 mm (109, 100%)
	Stable Liquid Surface Level/Z	LTS	800 mm (109, 100%)
Xoperation,i	Casting Period/P	LTS	11 h [9 h–13 h]
	Casting Speed/VC	RTS	1.21 m/min [1.205 m/min–1.21 m/min]
	Cooling Water Flow of Wide Face/Vw, w	RTS	3950 L/min [3870 L/min–3980 L/min]
	Cooling Water Temperature Difference of Wide Face/ΔTw, w	RTS	7.5°C [7°C–8.5°C]
	Cooling Water Flow of Narrow Face/Vw, n	RTS	441 L/min [416 L/min–450 L/min]
	Cooling Water Temperature Difference of Narrow Face/ΔTw, n	RTS	7.0°C [6.5°C–7.5°C]

*  The values of all dates have been preprocessed.

3. Industrial Data and Preliminary Application

3.1. Data Collection and Category

The industrial continuous casting process was affected by many factors that have complex mechanisms of cooperation. The production data of industrial continuous casting for 7 days were collected in this work to analyze the thermal dissipation behaviors in the mold under the actual continuous casting production. Among them, more than 5000 groups of data were collected in real-time-scale (RTS), is the minimum scale of continuous casting process operation record. The data in RTS change dynamically over time. Simultaneously, 109 groups of data were collected with each ladle ID as the scale, that is ladle-time-scale (LTS), which is the minimum scale of continuous casting production management. The data in LTS were consistent when casting molten steel from the same ladle ID. The major data with detailed information is shown in Table 1. In this work, the whole continuous casting mold was regarded as a holistic system, and all of the collected data were divided into four categories according to X_state,i (such as start, end, stable, etc.), X_input,i (such as steel grade, liquidus temperature, superheat, casting steel amount, etc.), X_operation,i (casting period, casting speed, cooling water, etc.) and X_equipment,i (mold/slab size, submerged entry nozzle grade, mold grade, etc.). Among them, the attributes data, such as steel grade, were expressed in percentage. The variables data, such as casting speed, were expressed in quartile, which is a method of sorting data from small to large, and choosing the 50%th [25%th–75%th] data to reflect the dispersion of the middle accurately.

3.2. Data Preprocessing

Due to the harsh and complex condition, the original industrial data appeared missing, abnormal fluctuation, timing mismatch, etc. For example, as shown in Fig. 5(a) for ladle-time-scale data, several slab width went 200 mm, most of casting steel amounts were closed to 0, and even the superheat went negative, which were incredible continuous casting process. Besides, as shown in Fig. 5(b) for real-time-scale data, each type of data was collected with random time intervals, resulting in the missing and timing mismatch for casting speed, superheat, cooling water temperature, etc. Therefore, it was difficult to describe the thermal dissipation behaviors veritably by analyzing original industrial data without any preprocessing.

To preprocess the original industrial data, the analysis of the abnormal reasons would be a priority. As for slab width abnormal at ladle-time-scale, it was found that the data of 200 mm slab width were familiar with that of 2200 mm for the same steel grade, implying that “200” was just a sensor recording error and it should be “2200”. It was also be verified by the response from the continuous casting workshop. As for the superheat at ladle-time-scale, it was derived from the average temperature of the 10 sensors in tundish. It was the first sensor anomaly that caused the superheat to go negative. And the solution was to remove the data from the first sensor and calculate the average temperature of the remaining 9 sensors for superheating. According to the continuous casting workshop, the abnormal steel casting amount at the ladle-time-scale was caused by the damaged sensor on the steel ladle turret. Due to too much abnormal data, the steel casting amount was integrated by real-time casting speed, slab size, and slab density instead of the original one. As for data missing and timing mismatch in real-time-scale, a data packing method was adopted by MATLAB software. The time interval of real-time data was set to minutes at first. Since the industrial continuous casting process was relatively stable and only a small amount of data was missing, the missing data at a certain time could be regarded as the average of the collected data at the previous and last minute. In this way, every minute and real-time data correspond exactly and the timing mismatch problem is solved.

3.3. The Application of Process Thermal Dissipation

The dynamic real-time process thermal dissipation rate of 3 different casting processes were calculated as shown in Fig. 6. It is obvious that the thermal dissipation rate existed the stable level at during continuous casting. Due to the different casting conditions, such as steel grade, cooling intensity, slab size, the dynamic thermal dissipation rate of each casting process maintained different stable levels, 12.5%, 14.3%, and 18.8% for casting 1, 2, and 3. It is worth noting that the thermal dissipation rate of each casting process fluctuated 3 times. The smaller one was in the middle of the casting process, while the other two were bigger and were close to the start and end, respectively.

Fig. 6.

Real-time thermal dissipation during four casting processes. (Online version in color.)

Due to the rapid solidification of molten steel in contact with the dummy bar at the beginning of pouring, the cooling requirement was strong, resulted that the thermal dissipation rate was higher than that of stable state. Except for that, the thermal dissipation rate gradually increased from 0 to stable level. Moreover, it was found that the middle fluctuated one had to do with the change of tundish, which led to the decease of casting speed. And the terminal one also increased with the casting speed decreasing rapidly. The above phenomenon indicated that the cooling intensity of mold was not adjusted in time according to the abnormal events, which led to the sharp increase of process thermal dissipation rate, deteriorating the slab quality eventually.

It is not difficult to the summary that compared with the thermal analysis method, the process thermal dissipation rate can dynamically reflect the thermal transfer behaviors in the mold and the slab quality, even guiding the real-time adjustment of cooling intensity, casting speed, and other parameters. Besides, compared with local characteristic factors such as liquid level fluctuation and mold copper temperature, thermal dissipation rate can reflect the thermal dissipation in mold systematically, especially for the cooperative relationship of multiple control parameters. Therefore, the thermal dissipation rate proposed in this work could characterize the mold cooling target under the cooperation of complex factors and provide a new method for the dynamic evaluation of the mold system cooling effect with different casting states.

4. Results

4.1. Process Thermal Dissipation with Different Input Conditions

Figure 7 shows the dynamic real-time process thermal dissipation rate of 3 grades steel samples, which was mainly reflected in the liquidus temperatures, with the same slab size and stable casting speed. It can be found that the thermal dissipation trend of the three steel samples was the same, and the dissipation rate had a certain fluctuation when changed tundish. The difference was that as the liquidus temperature increased, the stable process thermal dissipation decreased significantly. The reason might be that a higher liquidus temperature resulted in a higher casting temperature, that is, a more thermal input to the system. However, the cooling intensity of the mold changed a little, implying that thermal dissipation was almost stable. Therefore, the process thermal dissipation rate decreased with the liquidus temperature increased. Another phenomenon also supported this conclusion. As for steel sample 1, the casting speed was increased at the second half of the casting period, and the cooling intensity increased accordingly. As the result, the process thermal dissipation rate was increased.

Fig. 7.

Real-time thermal dissipation with different steel grades. (Online version in color.)

Whereas, the different phenomenon was detected in the real-time process thermal dissipation rate with different superheat at the same slab size and stable casting speed for one steel grade. As the superheat increased, which means more thermal input to the mold system, the process thermal dissipation was supposed to decreased. As shown in Fig. 8, by contrast, the thermal dissipation rate increased weakly. Simultaneously, the variation trend of superheat was consistent with that of the cooling water temperature difference. Therefore, it could be inferred that for the same steel casting, although the increase of superheat brought more input thermal to the mold system, the temperature difference of cooling water would also increase correspondingly, leading to a stronger cooling intensity. On the one hand, the superheat had little influence on the input thermal, and the thermal dissipation rate increases slowly with stronger cooling intensity. On the other hand, the liquidus temperature had a greater influence on the input thermal, and the thermal dissipation rate still decreased significantly even with stronger cooling intensity.

Fig. 8.

Furnace-time thermal dissipation with different superheat: (a) 2200 mm×200 mm; (b) 1850 mm×200 mm. (Online version in color.)

4.2. Process Thermal Dissipation with Different Casting Operating Conditions

In the industrial continuous casting, it is required to reduce the casting speed from slab level to 0.6–0.8 m/min for tundish replacement and other operations to guarantee the slab quality. The dynamic real-time process thermal dissipation rate with different casting speed during continuous casting, except beginning and ending, were calculated as shown in Fig. 9. And it decreased significantly with the casting speed increased from 0.6 m/min to 1.2 m/min. The reason was that the faster casting speed led to the short thermal transfer time for molten steel in mold. The mold system input thermal was stable while the transferred part decreased. As a result, the process thermal dissipation rate decreased.

Fig. 9.

Real-time thermal dissipation with different casting speed. (Online version in color.)

Another casting operating condition is cooling water flow, which affects the cooling intensity in mold directly. As shown in Fig. 10, the thermal dissipation rate increased slightly with the higher cooling water flow during all the slab continuous casting processes. The reason was just the opposite of casting speed. The higher cooling water flow led to the more efficient thermal transfer for molten steel in mold. The mold system input thermal was stable while the transferred part increased. As a result, the process thermal dissipation rate increased.

Fig. 10.

Real-time thermal dissipation with different cooling water flow. (Online version in color.)

4.3. Process Thermal Dissipation with Different Mold Conditions

The ladle-time-scale process thermal dissipation rate with different mold/slab widths was calculated showing in Fig. 11. As the slab width increased from 1850 mm to 2200 mm, the process thermal dissipation rate decreased step by step. The wider slab brought more molten steel, which means more thermal input, to the system, it also expanded the thermal transfer area, which means the system need stronger cooling intensity. According to the industrial data, however, the actual cooling water flow and temperature difference did not increase significantly. In other words, according to Eq. (12), the heat flux of the mold decreased with the expansion of the heat transfer interface. Therefore, under the condition that the cooling water flow was basically stable, so was the thermal dissipation of the system, while the wider slab brought more thermal input in the mold, which led to the thermal dissipation rate decreased eventually. This also suggested that the stronger cooling intensity was required to ensure stable thermal dissipation when casting a wider slab.

Fig. 11.

Furnace-time thermal dissipation with different slab size. (Online version in color.)

5. Discussion

From the above analysis, each factor of the mold system has a different influence on the process thermal dissipation. For real industrial continuous casting process, X_state,i, and X_equipment,i in Formula (1) are objective factors, X_input,i is controlled by tundish. Neither of them can realize the adaptability adjustment in the mold during continuous casting. As a consequence, X_operation,i, which includes casting speed and cooling water, is the only controllable factor.

For stable casting state, the casting speed tends to be constant while the cooling water flow is the key adjustment means to deal with the X_input,i variation. As shown in Fig. 12, most of the samples were clustered at different levels according to the cooling targets of steel or slab size. And they were distributed along the longitudinal section approximately. It indicated that in response to the rising superheat, the cooling water flow rate should be increased to achieve the target of thermal dissipation. Additionally, there were some outlier samples with similar superheat but significant differences in cooling water flow, such as “A” in Fig. 12(a) and “B” in Fig. 12(b). Their cooling water flow did not follow the synergistic relationship with superheating. Even in position “B”, the cooling water flow decreased with the increase of superheat. Homoplastically, industrial production results also confirm this phenomenon. The slab corresponding to outlier samples had a higher risk of surface defects, indicating that the mold cooling water should be adjusted timely according to the superheat to achieve a reasonable thermal dissipation level under stable continuous casting state.

Fig. 12.

Real-time thermal dissipation with multiple factors in stable state: (a) different steel grade; (b) different slab size. (Online version in color.)

For unstable states such as start and end, it has the characteristics of relatively short duration and great variation of casting speed, which has a strong effect on thermal dissipation. Accordingly, the influence of X_input,i is reduced, and cooling water is the key adjustment means. As shown in Fig. 13, the stable state samples were relatively concentrated, and the corresponding casting speed and cooling water flow were relatively fixed. In the case of the start casting, the end casting, and abnormal casting (change tundish in this work), the samples were distributed along 3 longitudinal sections based on stable state samples. Detailly, most of the start casting samples were concentrated in the lower-left position, indicating that the casting speed gradually increases while the cooling water flow did not increase correspondingly, resulting in an insufficient cooling effect. In contrast, end and abnormal casting samples were concentrated in the upper left position, where the casting speed decreases while the cooling water flow was almost unchanged, resulting in excessive cooling intensity. Theoretically, the thermal dissipation should be adjusted through the operation of casting speed and cooling water to meet the stable thermal dissipation rate as soon as possible in the unstable casting. In the case of this work, unfortunately, the cooling water of the unstable casting samples was not sensitive to the casting speed variation, which led to the increased risk of slab surface defects.

Fig. 13.

Real-time thermal dissipation with multiple factors in unstable state. (Online version in color.)

As consequence, although in the independent effect on the thermal dissipation, the casting speed was stronger than the cooling water. Due to the stable casting speed regulation, it did not have the condition to meet the dynamic variation of continuous casting factors in time. In the contrast, the cooling water flow of mold was relatively flexible. It can be adjusted not only according to superheat adaptively in stable state but also according to casting speed in unstable state. Accordingly, cooling water would be the bridge method to coordinate all factors of the mold system to control and evaluate the cooling effect.

6. Conclusions

A concept of “process thermal dissipation rate” defined by mold system thermal input and output was proposed. And its relationship between the system state, system input, and system operation was discussed to provide a theoretical basis for controlling cooling effect on mold from the perspective of system. The specific conclusions are as follows:

(1) The factors related to mold cooling, such as casting state, steel grade, slab size, casting speed, superheat, etc. were deconstructed into system state, system input, system operation, and system equipment to describe mold thermal transfer behavior. Furthermore, the process thermal dissipation rate was defined to evaluate the cooling effect on mold based on the energy conservation principle and the thermal transfer relationship in the mold.

(2) Based on the two-dimensional heat transfer differential equation, the temperature prediction model was established to calculate the slab temperature field at the mold outlet. Then, combined with the molten steel temperature at the meniscus, the thermal dissipation rate was calculated. The results indicated that the thermal dissipation rates of the three industrial cases were almost concentrated on 12.5%, 14.3%, and 18.8%, respectively. Furthermore, they were obviously abnormal at unstable casting, which reflected that the thermal dissipation rate could evaluate the cooling effect on mold for different casting state under the coordination of complex system factors.

(3) It is possible to achieve the reasonable thermal dissipation rate by coordinating the superheat, casting speed, cooling water flow, and other system factors. The cooling water flow and the molten superheat were positively correlated with process thermal dissipation, while steel liquids temperature, casting speed, and slab size were negative. Among them, the cooling water could be adjusted according to the superheat in stable state and the casting speed in unstable state, which should be the key and bridge for controlling the cooling effect on the mold by all factors cooperative system during continuous casting.

Acknowledgment

The authors are grateful for support from the National Key R&D Program of China (No. 2021YFE0113200), the National Natural Science Foundation of China (No. 51734004), and the Chongqing Postdoctoral Science Foundation project (No. cstc2020jcyj-bshX0104)

Interest Conflict Statement

On behalf of all authors, the corresponding author states that there is no conflict of interest.

Symbol

Y_output: system output factors

G: the continuous casting targets

F: the nonlinear synergistic relationship of X_state,i, X_input,i, X_operation,i, and X_equipment,i

X_state,i: continuous casting state factors

X_input,i: system input factors

X_operation,i: system operation factors

X_equipment,i: system equipment factor

Q_in: thermal input for the system

Q_s: energy supply for the system

Q_out: thermal output for the system

Q_d: thermal dissipation for the system

φ: process thermal dissipation rate

Q_molten: thermal brought in mold

Q_slab: thermal kept in mold

T: temperature

τ: the time

x: the distance in the slab width direction

y: the distance in the slab thickness direction

ρ: the density of steel

C_eff: the effective specific heat

λ_eff: the effective thermal conductivity

C_p: the actual specific heat capacity of steel

L_f: the solidification latent heat

T_S: the solidus temperature of steel

T_L: the liquidus temperature of steel

T_C: the casting temperature, °C.

q_n: the heat flux in the direction of mold narrow side

q_w: the heat flux in the direction of mold wide side

C_w: the specific heat capacity of water

V_w,n: the cooling water flow on the narrow sides of the mold

V_w,w: the cooling water flow on the wide sides of the mold

ρ_w: the density of water

ΔT_w,n: the temperature difference of narrow side of cooling water

ΔT_w,w: the temperature difference of wide side of cooling water

A_n: the narrow surface areas of the mold

A_w: the wide surface areas of the mold

i: the width direction nodes

j: the thickness direction nodes

k: the casting direction nodes

T_i,j^k: the temperature of node (i,j,k)

Δx: the space step in the width direction

Δy: the space step in the thickness direction

Δτ: the space step in the casting direction

o(ΔxΔy+Δτ): the truncation error of the difference equation

σ: slice thickness

T_E: the environment temperature

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