Systematic Study on the Thermal Performance of Casting Slab Under Varying Environmental Conditions

Abstract

Accurate prediction of slab temperature during the continuous casting and rolling process is essential for optimizing reheating furnace scheduling and achieving energy savings and emission reductions in steel production. However, because of the dynamic boundary conditions caused by the complex transport processes, obtaining precise temperature data for slabs remains challenging. These difficulties lead to issues such as low hot charging rates, mixing of hot and cold slabs in reheating furnaces, and excessive heat loss from slabs after cutting. To address these challenges, this study develops a mathematical model to calculate slab temperatures during the continuous casting and rolling process, providing a foundation for production scheduling optimization. The model accounts for the coupled heat transfer effects induced by dynamic slab stacking and the stacking heat transfer effects resulting from slabs with varying cross-sectional dimensions. Validation against experimental data demonstrated the model’s accuracy and reliability. Key findings highlighted that neglecting dynamic stacking effects or simplifying slab dimensions introduces errors. These results enhance slab temperature tracking in complex processes and advance related theoretical understanding.

1. Introduction

The steel production process involves a series of meticulously controlled stages to ensure quality and efficiency. An essential intermediary stage between continuous casting and hot rolling is the slab cooling process, where steel slabs cool in stacked arrangements before entering subsequent production phases. Managing the cooling rate and temperature uniformity during this stage is critical, as these factors significantly influence the structural integrity, surface quality, and macroscopic properties of the steel slabs [1,2,3,4]. Past research has examined various aspects of slab cooling, from the impacts of stacking arrangements to the application of computational models in heat transfer analysis [5,6,7,8]. However, the challenge remains to account for the variability in environmental conditions, which can alter the cooling rates and temperature distributions across stacked slabs.

The slab yard is the scheduling core in the process of casting and rolling connection, as an important buffer to ensure a normal production process [9,10,11]. Zhao et al. [12] developed a model for scheduling multiple double-load cranes in steel slab yards, a previously unexplored problem. The results demonstrated improved scheduling performance, especially for large-scale, practical applications. Peng et al. [13] introduced a multiobjective mixed-integer programming model for optimizing crane scheduling and storage assignments in slab yards. Using an improved nondominated sorting genetic algorithm III (NSGA-III) algorithm, the model reduced relocation rates from 30% to 10%, outperforming the multiobjective evolutionary algorithm based on decomposition (MOEA/D). The results underscored efficiency gains in logistics for unmanned slab yards, enhancing production balance. Shi et al. [14] addressed the hot rolling scheduling (HRS) in steel production, focusing on reducing slab shuffling during scheduling. The results demonstrated the approach’s effectiveness, enhancing scheduling performance across varying scales. Wang et al. [15] developed a mixed-integer programming model for multicrane scheduling in slab yards, aiming to minimize travel distance and balance workload. Real-data experiments confirmed the approach’s efficiency over that of exact methods, improving operational performance in steel slab yards. Rajabi et al. [16] developed integer programming models for minimizing slab stack shuffles in steel yards. Addressing two problem variants, the models achieved optimal solutions efficiently, reducing shuffling by 22.7% on average in practical tests, improving scheduling alignment with hot rolling processes. These studies have laid a solid foundation for the study of slab storage in the jointing process of casting and rolling.

In recent years, with the gradual deepening of research, more advanced scheduling and the study of slab details have gradually come into focus. Ai et al. [17] developed a predictive model for crack-sensitive temperature ranges and phase fractions in slabs during continuous casting. The model accurately identified crack-sensitive regions, guiding temperature control to minimize deformation risk and improve slab quality. Bruno et al. [18] addressed the slab stack shuffling problem (SSSP) for efficient slab retrieval, focusing on minimizing shuffles and expired slabs. A biobjective multiperiod mathematical model was developed and validated through simulations, highlighting storage strategies and trade-offs that enhanced retrieval efficiency in steel production and shipbuilding supply chains. Cho et al. [19] applied reinforcement learning to optimize steel plate stacking in shipyard stockyards, aiming to reduce crane use by aligning stacking order with fabrication schedules. Using a Markov decision process and an asynchronous advantage actor–critic algorithm, the model minimized sorting work, outperforming conventional heuristics and metaheuristics in dynamic, real-time conditions. Cho et al. [20] developed a machine learning model optimized by a deep neural network. The model accurately forecast impact absorption energy (E_IA), helping prevent thermal cracking by identifying safe temperature and composition thresholds, thus guiding improved cooling and scarfing practices postcasting. Kovacic et al. [21] optimized slab casting by abandoning cooling under hoods and heat treatment for certain grades, guided by defect prediction models using linear regression and genetic programming. This approach increased casting sequence lengths, reduced slab stock, and maintained internal quality, significantly enhancing productivity and resource efficiency. Wang et al. [22] examined the impact of partial information on inventory stacking in steel plants, considering both inbound and outbound factors. A time window allocation approach was proposed to improve stacking efficiency and reduce retrieval uncertainty. Wang et al. [23] developed a stochastic programming model to optimize slab yard crane assignment and scheduling (SYCAS) under uncertain conditions, such as fluctuating arrival times.

In practice, from after flame cutting to reheating furnace entry, continuous casting slabs undergo complex transitions involving specific intermediate stages and continuously changing boundary conditions. However, previous studies on slab heat transfer under variable boundary conditions have been limited in scope and depth, failing to fully reflect the complexities of actual production scenarios. These studies often simplified dynamic boundary variations and neglected key factors such as sequential slab stacking, which significantly impacts heat transfer. Furthermore, existing research has not addressed the heat transfer conditions of stacked slabs with varying cross-sectional dimensions, a common occurrence in industrial production. Dynamic stacking processes, characterized by the addition of slabs over time and the consequent evolving thermal interactions, have been overlooked in previous models, which typically assume static or uniform stacking configurations. This oversimplification has resulted in significant discrepancies between model predictions and real-world observations.

To bridge these gaps, this study investigates the heat transfer process of continuous casting slabs under dynamic boundary conditions, incorporating realistic factors such as a dynamic stacking process and variations in slab cross-sections. The main scientific objective of this study is to develop a heat transfer model for continuous casting slabs that accurately reflects industrial conditions. The model accounts for dynamic boundary conditions, incorporating both the evolving stacking process and the thermal interactions among slabs with varying cross-sectional dimensions. This approach aims to enhance temperature prediction accuracy and provide a robust theoretical and methodological foundation for optimizing heating furnace operations and improving energy efficiency.

2. Mathematic Model

2.1. Grid Division and Establishment of the Difference Equation

This research was accomplished through the self-programming of Python 3.11. Therefore, the basic equations and discretization methods employed in this study are exactly as described in Section 2.1. The program was written based on these equations. The contours and some line charts were implemented through the dependency package matplotlib. The data in these charts all came from the results of the iterations. The control equation of heat transfer during continuous casting slab transportation is as follows:

$$\rho c{\displaystyle \frac{\partial T}{\partial t}}={\displaystyle \frac{\partial }{\partial x}}\left(\lambda {\displaystyle \frac{\partial T}{\partial x}}\right)+{\displaystyle \frac{\partial }{\partial y}}\left(\lambda {\displaystyle \frac{\partial T}{\partial y}}\right)$$

(1)

In this paper, the explicit finite difference method was used to discretize the two-dimensional differential equation of thermal conductivity. To solve the mathematical model, the sections of the slab are divided into grids, and certain space and time steps are set. The difference and grid of the slab section are shown in Figure 1.

According to the heat transfer characteristics of different nodes on the cross-section and the heat balance method, the differential differentiation of the equation is obtained as follows:

$$t_{i,j}^{k+1}=t_{i,j}^k+{\displaystyle \frac{\Delta \tau }{\rho c}}\left[{\displaystyle \frac{{\lambda }_1\left(t_{i-1,j}^k-t_{i,j}^k\right)}{{\left(\Delta \mathrm{x}\right)}^2}}+{\displaystyle \frac{{\lambda }_2\left(t_{i+1,j}^k-t_{i,j}^k\right)}{{\left(\Delta \mathrm{x}\right)}^2}}+{\displaystyle \frac{{\lambda }_3\left(t_{i,j-1}^k-t_{i,j}^k\right)}{{\left(\Delta \mathrm{y}\right)}^2}}+{\displaystyle \frac{{\lambda }_4\left(t_{i,j+1}^k-t_{i,j}^k\right)}{{\left(\Delta \mathrm{y}\right)}^2}}\right]$$

(2)

$$t_{i,j}^{k+1}=t_{i,j}^k+{\displaystyle \frac{\Delta \tau }{\rho c}}\left[{\displaystyle \frac{2{\lambda }_1\left(t_{i-i,j}^k-t_{i,j}^k\right)}{{\left(\Delta \mathrm{x}\right)}^2}}+{\displaystyle \frac{{\lambda }_3\left(t_{i,j-1}^k-t_{i,j}^k\right)}{{\left(\Delta \mathrm{y}\right)}^2}}+{\displaystyle \frac{{\lambda }_4\left(t_{i,j+1}^k-t_{i,j}^k\right)}{{\left(\Delta \mathrm{y}\right)}^2}}-{\displaystyle \frac{2q_2}{\Delta \mathrm{x}}}\right]$$

(3)

$$t_{i,j}^{k+1}=t_{i,j}^k+{\displaystyle \frac{\Delta \tau }{\rho c}}\left[{\displaystyle \frac{{\lambda }_1\left(t_{i-1,j}^k-t_{i,j}^k\right)}{{\left(\Delta \mathrm{x}\right)}^2}}+{\displaystyle \frac{{\lambda }_2\left(t_{i+1,j}^k-t_{i,j}^k\right)}{{\left(\Delta \mathrm{x}\right)}^2}}+{\displaystyle \frac{2{\lambda }_4\left(t_{i,j+1}^k-t_{i,j}^k\right)}{{\left(\Delta \mathrm{y}\right)}^2}}-{\displaystyle \frac{2q_3}{\Delta \mathrm{y}}}\right]$$

(4)

$$t_{i,j}^{k+1}=t_{i,j}^k+{\displaystyle \frac{\Delta \tau }{\rho c}}\left[{\displaystyle \frac{{\lambda }_1\left(t_{i-1,j}^k-t_{i,j}^k\right)}{{\left(\Delta \mathrm{x}\right)}^2}}+{\displaystyle \frac{{\lambda }_2\left(t_{i+1,j}^k-t_{i,j}^k\right)}{{\left(\Delta \mathrm{x}\right)}^2}}+{\displaystyle \frac{2{\lambda }_3\left(t_{i,j-1}^k-t_{i,j}^k\right)}{{\left(\Delta \mathrm{y}\right)}^2}}-{\displaystyle \frac{2q_4}{\Delta \mathrm{y}}}\right]$$

(5)

$$t_{i,j}^{k+1}=t_{i,j}^k+{\displaystyle \frac{2\Delta \tau }{\rho c}}\left[{\displaystyle \frac{{\lambda }_1\left(t_{i-1,j}^k-t_{i,j}^k\right)}{{\left(\Delta \mathrm{x}\right)}^2}}+{\displaystyle \frac{{\lambda }_4\left(t_{i,j+1}^k-t_{i,j}^k\right)}{{\left(\Delta \mathrm{y}\right)}^2}}-\left({\displaystyle \frac{q_2}{\Delta \mathrm{x}}}+{\displaystyle \frac{q_3}{\Delta \mathrm{y}}}\right)\right]$$

(6)

$$t_{i,j}^{k+1}=t_{i,j}^k+{\displaystyle \frac{2\Delta \tau }{\rho c}}\left[{\displaystyle \frac{{\lambda }_2\left(t_{i+1,j}^k-t_{i,j}^k\right)}{{\left(\Delta \mathrm{x}\right)}^2}}+{\displaystyle \frac{{\lambda }_4\left(t_{i,j+1}^k-t_{i,j}^k\right)}{{\left(\Delta \mathrm{y}\right)}^2}}-\left({\displaystyle \frac{q_1}{\Delta \mathrm{x}}}+{\displaystyle \frac{q_3}{\Delta \mathrm{y}}}\right)\right]$$

(7)

$$t_{i,j}^{k+1}=t_{i,j}^k+{\displaystyle \frac{2\Delta \tau }{\rho c}}\left[{\displaystyle \frac{{\lambda }_1\left(t_{i-1,j}^k-t_{i,j}^k\right)}{{\left(\Delta \mathrm{x}\right)}^2}}+{\displaystyle \frac{{\lambda }_3\left(t_{i,j-1}^k-t_{i,j}^k\right)}{{\left(\Delta \mathrm{y}\right)}^2}}-\left({\displaystyle \frac{q_2}{\Delta \mathrm{x}}}+{\displaystyle \frac{q_4}{\Delta \mathrm{y}}}\right)\right]$$

(8)

$$t_{i,j}^{k+1}=t_{i,j}^k+{\displaystyle \frac{2\Delta \tau }{\rho c}}\left[{\displaystyle \frac{{\lambda }_2\left(t_{i+1,j}^k-t_{i,j}^k\right)}{{\left(\Delta \mathrm{x}\right)}^2}}+{\displaystyle \frac{{\lambda }_3\left(t_{i,j-1}^k-t_{i,j}^k\right)}{{\left(\Delta \mathrm{y}\right)}^2}}-\left({\displaystyle \frac{q_1}{\Delta \mathrm{x}}}+{\displaystyle \frac{q_4}{\Delta \mathrm{y}}}\right)\right]$$

(9)

The above eight difference equations require convergence. In the eight difference equations, the stability of the diagonal nodal difference equation is very strict. The corner difference equation is as follows:

$${\mathrm{t}}_{i,j}^{k+1}=\left[1-{\displaystyle \frac{2\Delta \tau }{\rho \mathrm{c}}}\left({\displaystyle \frac{{\lambda }_1}{{\left(\Delta \mathrm{x}\right)}^2}}+{\displaystyle \frac{{\lambda }_3}{{\left(\Delta \mathrm{y}\right)}^2}}+{\displaystyle \frac{{\mathrm{h}}_2}{\Delta \mathrm{x}}}+{\displaystyle \frac{{\mathrm{h}}_4}{\Delta \mathrm{y}}}\right)\right]{\mathrm{t}}_{i,j}^k+{\displaystyle \frac{2\Delta \tau }{\rho \mathrm{c}}}\left[{\displaystyle \frac{{\lambda }_1{\mathrm{t}}_{i-1,j}^k}{{\left(\Delta \mathrm{x}\right)}^2}}+{\displaystyle \frac{{\lambda }_3{\mathrm{t}}_{i,j-1}^k}{{\left(\Delta \mathrm{y}\right)}^2}}+{\displaystyle \frac{{\mathrm{h}}_2{\mathrm{t}}_0}{\Delta \mathrm{x}}}+{\displaystyle \frac{{\mathrm{h}}_4{\mathrm{t}}_0}{\Delta \mathrm{y}}}\right]$$

(10)

If the difference equation is to converge, the coefficient of the term above is required to be greater than or equal to zero, that is:

$$1-{\displaystyle \frac{2\Delta \tau }{\rho \mathrm{c}}}\left({\displaystyle \frac{{\lambda }_1}{{\left(\Delta \mathrm{x}\right)}^2}}+{\displaystyle \frac{{\lambda }_3}{{\left(\Delta \mathrm{y}\right)}^2}}+{\displaystyle \frac{{\mathrm{h}}_2}{\Delta \mathrm{x}}}+{\displaystyle \frac{{\mathrm{h}}_4}{\Delta \mathrm{y}}}\right)\ge 0$$

(11)

Therefore, the stability condition is:

$${\displaystyle \frac{\Delta \tau }{\rho \mathrm{c}}}\left({\displaystyle \frac{{\lambda }_1}{{\left(\Delta \mathrm{x}\right)}^2}}+{\displaystyle \frac{{\lambda }_3}{{\left(\Delta \mathrm{y}\right)}^2}}+{\displaystyle \frac{{\mathrm{h}}_2}{\Delta \mathrm{x}}}+{\displaystyle \frac{{\mathrm{h}}_4}{\Delta \mathrm{y}}}\right)\le {\displaystyle \frac{1}{2}}$$

(12)

2.2. Boundary Conditions

During the roller conveyor transport process, the primary heat transfer mechanisms for continuous casting slabs include convective heat transfer with ambient air, conductive heat transfer between the bottom surface of the slab and rollers, and radiative heat transfer to the surrounding environment. Because of the minimal heat exchange between the slab and the rollers, this heat transfer component was neglected to simplify the model.

Clamp crane transport represents a crucial component of the slab handling system, characterized by the ability to lift two slabs simultaneously, thereby enhancing operational efficiency. In this phase, heat transfer mechanisms include convection and radiation from the top surface to ambient air, conduction from the slab’s side surface to the clamp crane, and both convection and radiation between the slab and the surrounding environment. Additionally, the slab’s bottom surface undergoes convection and radiation with the air, and conduction occurs between adjacent slabs. However, because of the negligible conduction between the slab and the clamp crane, this component was excluded from the model for simplicity.

To address lateral transport requirements in a two-strand system, transfer cars are employed. During transport by transfer cars, heat transfer mechanisms encompass convection and radiation from the top surface to the air, convection and radiation from the side surfaces to the surrounding environment, and conduction between the bottom surface and the transfer car.

Slabs transported by overhead vehicles are typically cold slabs inspected or trimmed upstream and stacked vertically for maximum space utilization. Overhead vehicles are generally loaded to their capacity before storage. Heat transfer mechanisms in this stage include convection and radiation from the top surface to the air, convection and radiation from the side surfaces to the surrounding environment, conduction between the bottom surface and the vehicle, and conduction between adjacent slabs.

The stacker platform, an essential piece of equipment in postcasting operations, facilitates slab stacking and unstacking to align with the operational rhythms of roller conveyors and clamp cranes. While slabs remain on the stacker platform, heat transfer includes convection and radiation from the top surface to the air, convection and radiation from the side surfaces to the environment, conduction from the bottom surface to the platform, and conduction between adjacent slabs.

In the slab stacking process, radiative and convective boundary conditions were applied to the top and side surfaces of the stack, while conduction occurred between adjacent slabs. For the bottom slab in contact with the ground, the lower surface was treated as a semi-infinite solid, simplifying the model as a transient heat conduction problem in a semi-infinite domain. The summary of the boundary conditions for all the above processes is presented in

Table 1. Boundary conditions in transportation.

Transportation	Boundary Condition
Roller	$q_{r,top}=\epsilon \sigma \left(T_s^4-T_a^4\right)+h_r(T_s-T_a)$ $q_{r,side}=\epsilon \sigma \left(T_s^4-T_a^4\right)+h_r(T_s-T_a)$ $q_{r,bottom}=\epsilon \sigma \left(T_s^4-T_a^4\right)+h_r(T_s-T_a)$
Crane	$q_{c,top}=\epsilon \sigma \left(T_s^4-T_a^4\right)+h_c(T_s-T_a)$ $q_{c,side}=\epsilon \sigma \left(T_s^4-T_a^4\right)+h_c(T_s-T_a)$ $q_{c,bottom}=\epsilon \sigma \left(T_s^4-T_a^4\right)+h_c(T_s-T_a)$
Traverse	$q_{t,top}=\epsilon \sigma \left(T_s^4-T_a^4\right)+h_t(T_s-T_a)$ $q_{t,side}=\epsilon \sigma \left(T_s^4-T_a^4\right)+h_t(T_s-T_a)$ $q_{t,bottom}={\displaystyle \frac{T_s-T_e}{R_t+rt{h}_t}}$
Over span	$q_{o,top}=\epsilon \sigma \left(T_s^4-T_a^4\right)+h_o(T_s-T_a)$ $q_{o,side}=\epsilon \sigma \left(T_s^4-T_a^4\right)+h_o(T_s-T_a)$ $q_{o,bottom}={\displaystyle \frac{T_s-T_e}{R_o+rt{h}_o}}$
Stacker platform	$q_{sp,top}=\epsilon \sigma \left(T_s^4-T_a^4\right)+h_{sp}(T_s-T_a)$ $q_{sp,side}=\epsilon \sigma \left(T_s^4-T_a^4\right)+h_{sp}(T_s-T_a)$ $q_{sp,bottom}={\displaystyle \frac{T_s-T_e}{R_{sp}+rt{h}_{sp}}}$ $q_{sp,contact}={\displaystyle \frac{T_{s,upper}-T_{s,lower}}{R_{sp}+rt{h}_{sp}}}$
Stack	$q_{s,top}=\epsilon \sigma \left(T_s^4-T_a^4\right)+h_s(T_s-T_a)$ $q_{s,side}=\epsilon \sigma \left(T_s^4-T_a^4\right)+h_s(T_s-T_a)$ $q_{sp,contact}={\displaystyle \frac{T_{s,upper}-T_{s,lower}}{R_s+rt{h}_s}}$ $q_{s,bottom}=B\epsilon \sigma \left(T_s^4-T_g^4\right)$

.

2.3. Optimization of the Calculation Method of the Convective Heat Transfer Coefficient

Although different convection heat transfer coefficients were set for various operating conditions, these were fixed values that could not accurately reflect real-world scenarios. To address this limitation, the convection heat transfer coefficient was determined using the Grashof number (Gr), and a program was developed to update it in real time, ensuring greater solution accuracy. Natural convection heat transfer can be categorized into large-space natural convection and confined-space natural convection. Large-space natural convection refers to scenarios where the development of the thermal boundary layer is not hindered or obstructed, regardless of geometric constraints such as infinite dimensions. In contrast, confined-space natural convection involves cases where either the boundary layer development is interfered with or the fluid flow is restricted, resulting in heat transfer behaviors distinct from those in large-space conditions. In the case of heat transfer during slab stacking, the process aligns with large-space natural convection. Therefore, experimental correlations for large-space natural convection could be employed, as described by the following formulation:

$$Gr={\displaystyle \frac{g{\alpha }_v\Delta t{L}^3}{v^2}}$$

(13)

$${Nu}_m=C{(Gr\mathrm{P}\mathrm{r})}_m^n$$

(14)

$$\mathrm{P}\mathrm{r}={\displaystyle \frac{c\mu }{\lambda }}$$

(15)

$$h={\displaystyle \frac{\lambda }{l}}Nu$$

(16)

The Grashof number (Gr) plays a role in natural convection phenomena like the role of the Reynolds number in forced convection. The Nusselt number (Nu), represented as Nu_m, is composed of the average surface heat transfer coefficient. The subscript “m” refers to the arithmetic mean temperature of the boundary layer, which is used for qualitative temperature calculations. The Prandtl number (Pr) compares a fluid’s momentum diffusion with its thermal diffusion capabilities. In the Grashof number equation, the temperature difference is the difference between the wall temperature and the fluid temperature. For gases that exhibit ideal gas properties, the Grashof number incorporates the reciprocal of the qualitative temperature as the volumetric expansion coefficient. The constant C and the exponent n are determined from

Table 2. Constants of C and n [24].

Heating Surface Shape and Position	Flow Regime	C	n	Gr Number Applicable Range
Vertical plate and vertical cylinder	Laminar	0.59	1/4	1.43 × 104–3 × 109
	transition	0.0292	0.39	3 × 109–2 × 1010
	turbulence	0.11	1/3	>2 × 1010

[24]. The calculation process is as follows: First, the Grashof number (Gr) is calculated using Equation (13). Then, using Equation (14) and Table 2, the Nusselt number (Nu_m) is derived. Finally, once Nu_m is determined, the appropriate convection heat transfer coefficient (h) can be calculated according to Equation (15), thereby solving the related problems. This approach provides a dynamic and accurate representation of the heat transfer behavior in natural convection scenarios.

For the convective heat transfer coefficient of the upper part of the continuous casting slab, the formula for the convective heat transfer coefficient of the horizontal heat side upward can be adopted:

$$Nu=0.54{(GrP\mathrm{r})}^{1/4},{10}^4\le GrP\mathrm{r}\le {10}^7$$

(17)

$$Nu=0.15{(GrP\mathrm{r})}^{1/4},{10}^7\le GrP\mathrm{r}\le {10}^{11}$$

(18)

For the convection heat transfer coefficient of the lower part of the continuous casting slab, the formula of convection heat transfer coefficient under the horizontal heat side can be adopted:

$$Nu=0.27{(GrP\mathrm{r})}^{1/4},{10}^5\le GrP\mathrm{r}\le {10}^{10}$$

(19)

In the following formula, the characteristic length is L, while A_p and P are the heat transfer area and perimeter length of the plate, respectively.

$$L={\displaystyle \frac{A_p}{P}}$$

(20)

2.4. Optimization of Initial Temperature

To enhance the accuracy of the initial temperature field in numerical simulations, an optimization process was performed for the continuous casting slab’s temperature distribution. We took the section temperature of the slab after flame cutting as the initial temperature. This initial temperature was calculated as follows. First, we assumed that the temperature of the slab section is uniformly 1200 °C. Then, through calculation, the process continued until the center point of the upper surface reached the actual measured temperature of the upper surface center point after the flame cutting. The resulting temperature field at this moment was used as the initial condition for subsequent computations. The temperature distribution of slab cutting is displayed in Figure 2.

Figure 3 shows the thermal history of a specific continuous casting slab. The dashed portion of the curve represents the rate of temperature change with time before flame cutting, while the solid portion indicates the rate of temperature change after cutting. The dashed lines in the figure represent the gradual cooling of the slab from 1200 °C until it reaches the temperature for flame cutting. In the subsequent figures, the initial temperature optimization process of the slab is illustrated using dashed lines. By capturing these thermal transitions, the initial temperature field more accurately reflects the slab’s actual conditions, thereby improving the reliability of the computational model in subsequent analyses. This approach ensures that the model accounts for significant thermal events, such as flame cutting, that induce spatial and temporal variations in temperature distribution. Such considerations are critical for improving the fidelity of simulations in thermal engineering applications involving continuous casting slabs.

2.5. Characteristics of the Stacking Model

2.5.1. Dynamic Stacking Process of Slabs Within the Stack

In previous studies, the temperature field of slabs within the stack did not consider the dynamic superposition process of the slabs, treating all slabs within the stack as a single entity for heat dissipation research. However, in practical scenarios, slabs are sequentially placed into the stack, and the temperature changes of slabs within the stack are different from those during transportation. Therefore, neglecting the stacking sequence may introduce errors in thermal simulation.

2.5.2. Heat Transfer During the Stacking Process of Slabs with Different Cross-Sections

Previous studies simplified the heat transfer process in stacked continuous casting slabs by assuming that all slabs within a stack had identical cross-sectional dimensions. However, field observations revealed that in practical scenarios, slabs with varying cross-sectional dimensions are often stacked together at the same location. Simplifying such a scenario by treating all slabs as having identical dimensions introduces errors in the heat transfer analysis, particularly at the ends of longer slabs. These ends, which have three surfaces exposed to the air, experience a faster temperature drop than those in stacks with uniform cross-sections. To improve the accuracy of heat transfer modeling, a new model was developed to account for the stacking of slabs with varying cross-sectional dimensions. This enhanced model considers the distinct thermal behaviors arising from these variations, particularly the differential heat loss at exposed areas.

3. Results and Discussion

3.1. Model Validation

3.1.1. Validation of the Roller Transport Process

Multiple temperature measurements were performed on continuous casting slabs during roller conveyor transport. Measurements were taken from several slabs under varying conditions, and the results are summarized in

Table 3. Roller transport temperature measurement.

Slab Identification Number	Time	Temperature (°C)	Test Position
42A01971B02	16:20	833.6	Top surface center
	16:24	803.3
42A01971A03	16:28	830.2
	16:34	805.4
42A01971A04	16:36	846.6
	16:42	808.4
42B02421C08	15:12	760.0
	15:26	650.0
	15:35	620.0

. From these measurements, the average rate of temperature drop was calculated as 6.08 K/min. To conduct a more precise comparison, we selected the slab with the identification number 42B02421C08. For the heat transfer simulation, the slab cross-sectional dimensions were specified as 1200 mm × 237 mm, consistently with those of the selected slab. The flame-cutting temperature was set to 760 °C, as determined through on-site measurements, and the roller conveyor transport duration was 23 min. The simulated temperature evolution over time is presented in Figure 4, where the center point of the slab’s top surface reached a temperature of 619.9 °C, differing by less than 1 °C from the measured value of 620 °C. The simulated cooling rate of 6.06 K/min closely matched the experimentally determined rate of 6.08 K/min, demonstrating the high accuracy of the model.

Figure 4 also illustrates the experimental validation of the slab cooling process during transport. In the figure, the x-axis origin (time = 0) represents the moment of flame cutting. Negative x-axis values correspond to the cooling phase before cutting, starting from the initial uniform slab temperature of 1200 °C. The simulation began at this uniform temperature and was calculated until the flame-cutting surface center-point temperature was obtained, which served as the initial temperature field for subsequent calculations. In the graph, the dashed curve represents the simulated temperature drop before flame cutting, while the solid curve represents the temperature drop after cutting. A schematic of the slab is included in the lower-left corner of the figure, with key points marked, including the center point of the top surface (black dot) and the center point of the cross-section (red dot). These points were used as reference locations for validation and comparison.

This comprehensive approach ensured that the model accurately reflected the thermal behavior of the slab during both pre- and postcutting phases. The high degree of agreement between the simulation and experimental data highlights the robustness and reliability of the proposed heat transfer model for continuous casting slab transport. Such precision is critical for applications requiring detailed thermal analysis in industrial slab handling processes.

3.1.2. Validation of Slabs in Stacks

In the casting-to-rolling transition zone, continuous casting slabs (CC slabs) are typically air-cooled by stacking after being flame-cut. This practice is a standard cooling and management method that serves as an essential buffer unit between the continuous casting and rolling operations. Stacking provides sufficient time for hydrogen to diffuse within the slabs, reducing internal defect rates. Additionally, production schedules, hot transport surface cracks, and the necessity of hydrogen diffusion often led to extended stacking durations in storage yards. During this period, significant heat energy is lost, and slab temperatures experience notable reductions. For special steel grades, the slab temperature at the end of stacking often determines whether direct reheating in the furnace is feasible. Considering these factors, establishing an accurate heat transfer model for stacked slabs is critical to improving operational efficiency and process reliability.

To analyze the cooling behavior of stacked CC slabs, a stacking test was conducted for 180 min. Measurements were taken at 10 min intervals to record the temperature drop of slabs at different positions within the stack. The temperature drop rates across the slabs from top to bottom during the stacking period were as follows: 0.63 K/min, 0.44 K/min, 0.17 K/min, 0.15 K/min, 0.16 K/min, 0.30 K/min, 0.38 K/min, and 0.33 K/min. For practical analysis, these results were grouped into three distinct regions: top slabs, with an average cooling rate of 0.63 K/min; middle slabs, with an average cooling rate of 0.27 K/min; and bottom slabs, with an average cooling rate of 0.33 K/min. These temperature drop rates provided a benchmark for validating numerical simulation results. The simulation began with a uniform slab temperature of 1200 °C, decreasing to the flame-cutting temperature of 760 °C after 23 min of roller transport. The stacking process was then simulated for 3 h, yielding the following results: top slab temperature, 395.9 °C; middle slab temperature, 459.5 °C; bottom slab temperature, 443.6 °C. Experimental measurements, taken under identical conditions, yielded the following corresponding temperatures: top slab temperature, 409.1 °C; middle slab temperature, 464.8 °C; bottom slab temperature, 446.9 °C. The simulation errors for the top, middle, and bottom slabs were 3.4%, 0.7%, and 0.7%, respectively, demonstrating high model accuracy.

Figure 5 illustrates the cooling process of the stacked slabs, focusing on the top, middle, and bottom layers. The x-axis origin (time = 0) represents the moment of flame cutting. Dashed lines denote the temperature evolution before flame cutting, while solid lines represent the postcutting cooling process. A schematic diagram in the lower-left corner highlights the measurement points, the center point of the top surface (black dot) and the center point of the slab cross-section (red dot). The simulation captured critical thermal phenomena, including position-dependent cooling rates influenced by slab arrangement and ambient air interactions. The high level of agreement between the simulated and experimental results confirmed the reliability of the proposed heat transfer model for stacked slabs. This model serves as a valuable tool for predicting thermal profiles during storage and optimizing thermal management strategies in industrial processes. It also supports decision-making related to slab reheating and subsequent rolling operations, particularly for specialized steel grades.

3.2. Temperature Field During the Transportation and Stacking Process

3.2.1. Continuous Casting Slab Process Tracking

Figure 6 illustrates the temperature drop curves at characteristic points of a continuous casting slab during transport and stacking. These points included the center of the upper surface, the center of the left side, the slab’s geometric center, and the top-left corner. The temperature variation over time is depicted, with gray dashed lines marking the moment of flame cutting. Additional colored dashed lines represent the temperature changes at each characteristic point prior to flame cutting, while solid lines correspond to the temperature changes after flame cutting up to the final recorded time. The figure reveals that the temperature drops at the slab’s center proceeded at a slower rate than those at the other characteristic points. Following this, the upper surface center exhibited a moderate cooling rate, and the left side center displayed a slightly faster rate. The top-left corner experienced the most rapid temperature drop. Interestingly, the temperatures at the upper surface center and the top-left corner showed significant increases around 1380 s. This phenomenon occurred because of the placement of a hot continuous casting slab on top of the existing slab stack at that time. The direct contact between the newly added hot slab and the target slab led to heat transfer, resulting in a noticeable temperature rise. With the high effective heat transfer coefficient and the large temperature difference between the two slabs, the heat transfer caused a sharp temperature rise in a short time.

Figure 7 presents the temperature distribution diagrams of a stack consisting of six continuous casting slabs, each with cross-sectional dimensions of 1550 mm × 237 mm, at 1980 s and 5580 s into the stacking process. The simulation was configured so that after the slabs underwent transport, one slab was added to the stack every two minutes, culminating in a total of six slabs. The time stamp of 1980 s corresponds to the moment when the last slab was added to the stack, while 5580 s represents the total stacking duration set by the program. The diagrams reveal that during the early stages of stacking, the high-temperature zones within the stack were concentrated in the middle to upper regions. This phenomenon occurred because, during the initial stacking phase, high-temperature slabs were continuously added to the stack, maintaining elevated temperatures in the upper sections. However, as stacking progressed, the high-temperature zones gradually shifted downward, and the upper sections became low-temperature regions. This transition can be attributed to rapid heat loss from the upper surfaces of the top slabs, which led to faster cooling rates for these slabs compared with those located deeper within the stack.

Figure 8 displays the temperature variation curves of the center points on the upper surfaces of different continuous casting slabs within the stack, providing clear information on the number of slabs in the stack, their stacking order, and the time intervals at which they were added. The newly added slabs had just completed the transportation phase, and because the transportation time was significantly shorter than the stacking phase, they retained relatively high temperatures. Consequently, their heat loss during transportation was minimal. In contrast, slabs already in the stack experienced rapid heat loss from their upper surfaces due to faster heat transfer, resulting in a more significant cooling rate. When a new slab was added, its hot lower surface contacted the much cooler upper surface of the top slab in the stack. The large temperature difference caused rapid heat transfer, raising the top slab’s surface temperature. This effect appeared in the temperature curves, showing the dynamic heat transfer during slab addition.

3.2.2. Effect of Stacking Sequence and Varying Dimensions

Figure 9 displays the temperature distribution cloud map of the stack, considering the stacking sequence. As shown in Figure 9a–d, the number of slabs within the stack increased dynamically, reaching a total of four slabs. With each addition, the heat transfer conditions within the stack evolved, altering the heat transfer dynamics. For example, initially, the upper surface of the first slab was in contact with the air. As the second slab was stacked on top, the boundary condition of the upper surface of the first slab then changed to contact between solids. These changes were reflected in the temperature distribution, underscoring the importance of considering the dynamic stacking process in thermal models. As shown in Figure 9e–h, without considering dynamic stacking, the entire stack would be regarded as a single entity. Because of the failure to consider the changes in the slab boundary conditions, the accuracy of the temperature prediction was consequently reduced. By incorporating these dynamic boundary conditions, the simulation provided a more accurate representation of the thermal behavior, enhancing the reliability of predictions in industrial stacking operations.

Figure 10 illustrates the temperature distribution for eight slabs with varying dimensions stacked under identical conditions, with a total cooling duration of 5000 s. The time when the last slab was added to the stack was 2220 s. The simulation also incorporated the stacking sequence. The results revealed that areas with greater protrusion, such as the sides of larger slabs, exhibited higher cooling rates due to increased exposure to ambient air. This highlights the importance of considering slab dimensional variability in heat transfer modeling to accurately capture localized cooling effects and improve the reliability of thermal predictions in industrial stacking operations.

3.3. Temperature Comparison Under Different Paths

The common flow of the casting and rolling connection process is shown in Figure 11. In continuous casting–hot rolling integrated production, direct hot-charge rolling (DHCR), hot-charge rolling (HCR), and cold-charge rolling (CCR) are all connection modes between the casting and rolling processes. The common goal of these processes is to ensure the quality of the steel while achieving smooth process integration and temperature control, thereby enhancing energy utilization and production efficiency. DHCR refers to the direct transfer of the cast slab to a heating furnace for heating before rolling. This process makes full use of the residual heat during the casting process, achieving the lowest energy consumption and the highest operational efficiency. However, it requires strict equipment layout, production rhythm, and scheduling accuracy. HCR refers to the process where the temperature of the continuous casting slab is below the A1 temperature, above 400 °C, and the slab is not allowed to cool down before being sent to the reheating furnace for heating and subsequent rolling. HCR retains a portion of the sensible heat generated during continuous casting, and its energy consumption and connection efficiency are moderate. CCR refers to the cold loading rolling process. Slabs from the continuous casting machine are first cooled, usually to below 400 °C. They are then reheated in the furnace to the rolling temperature and finally sent to the hot rolling mill. This results in the highest energy consumption and the greatest scheduling flexibility for CCR, but with the lowest operational efficiency.

Figure 12 and Figure 13 demonstrate a comparison of temperatures along different paths. In industrial production, the direct hot charging time for continuous casting slabs is generally 30 min. This simulation also assumed a direct hot charging time of 30 min for the slabs, with the slab specifications being 1200 mm by 237 mm.

The slab storage area acts as a buffer and coordinator between the continuous casting machine and the reheating furnace, providing greater flexibility in matching production processes and managing logistics. The microstructure of the slab is essentially the same as that of conventional cold-charged furnace slabs. However, for certain low-alloy steels and medium- to high-carbon steels, cracks are prone to forming during cooling, and hot charging can lead to surface deterioration. The transportation process for the slab includes the following: flame cutting–roller conveyance–stacking platform–clamshell crane–top of the stack–clamshell crane–stacking platform–roller conveyance–heating furnace. The simulation was conducted when the center point temperature of the top surface of the top slab was 500 °C and it was removed from the stack for simulation. When the slab entered the furnace, most of the slab’s temperature was above 400 °C, falling within the category of hot charging. At this time, the center of the slab was still at a relatively high temperature, which, compared with cold charging, reduced the energy consumption of the reheating furnace.

Generally, 400 °C is taken as the low-temperature limit for hot charging. The transportation process for the slab includes the following: flame cutting–roller conveyance–stacking platform–clamshell crane–top of the stack–clamshell crane–stacking platform–roller conveyance–heating furnace. The simulation was conducted when the center point temperature of the top surface of the top slab was 300 °C and it was removed from the stack for simulation. As the time in the stack increased, the overall temperature of the slab tended to become uniform. The temperature drop rate of the slab within the stack was relatively low, and before being sent to the heating furnace, the overall temperature of the slab was basically below 400 °C.

4. Conclusions

(1)

The proposed model accurately predicted slab temperature evolution during transportation and stacking. For 1200 mm × 237 mm slabs, the center point temperature on the upper surface after 23 min of transportation was 619.9 °C, differing by less than 1 °C from experimental data. After 180 min of cooling within the stack, predicted temperatures for top, middle, and bottom slabs were 409.1 °C, 464.8 °C, and 446.9 °C, with errors of 3.4%, 0.7%, and 0.7%, respectively, confirming its reliability under dynamic stacking conditions.

(2)

During the stacking process, the addition of a new slab created a pronounced temperature gradient between the upper surface of the top slab (already cooled to some extent) and the lower surface of the newly placed slab, which retained a much higher temperature. This sharp thermal contrast drove intense heat conduction from the hotter new slab to the cooler top slab, resulting in rapid reheating of the latter’s surface. Such transient thermal interactions underscore the critical influence of dynamic stacking on the overall heat transfer behavior within the stack.

(3)

Neglecting the variations in slab geometry can lead to substantial prediction errors, particularly at slab ends, where thermal gradients are more pronounced. The proposed model explicitly incorporates both the dynamic feature of stacking and the variability in slab cross-sections, allowing for a more faithful representation of real-world heat transfer phenomena.