Integrated CFD Modeling of Combustion, Heat Transfer, and Oxide Scale Growth in Steel Slab Reheating

Abstract

In this study, a three-dimensional simulation of a walking-beam reheating furnace was developed to improve the steel slab reheating process and reduce surface oxidation kinetics using computational fluid dynamics (CFD). Combustion, heat transfer, fluid dynamics, and chemical reaction models were integrated into the numerical framework of this study. In addition, dynamic mesh remeshing was coupled through user-defined functions (UDFs), enabling the simultaneous simulation of slab movement and evolution of the involved transport phenomena. Turbulence was modeled with the realizable k-ε formulation, combustion with the Eddy Dissipation model, and radiation with the P-1 model coupled with WSGGM to include CO₂ and H₂O gas radiation. Scale formation was modeled using customized functions based on Arrhenius-type kinetics and Wagner’s oxidation model, evaluating its growth as a function of time, temperature, and furnace atmosphere. The predicted thermal evolution inside the furnace was validated using industrial data, yielding an average deviation of 5%. Furthermore, the proposed operating conditions led to an average slab temperature of 1289.77 °C at the exit of the homogenization zone, which was 16 °C higher than that under the current operation but still within the target range (1250 ± 50 °C). The reduction in combustion air decreased energy losses and improved product quality, lowering the molar oxygen content in the furnace atmosphere from 4.9 × 10² mol to 6.7 × 10¹ mol. Additionally, annual savings of 4,793,472 kg of natural gas and 13,884 tons of steel were estimated owing to reduced oxidation losses. The proposed air–fuel adjustment led to estimated annual energy savings (equivalent to 4,793,472 kg of natural gas) and a reduction in material loss due to oxidation from 4.5% to 3.75% (an absolute reduction of 0.75 percentage points; relative reduction ≈ 16.7%), which has a significant industrial impact on metal conservation and descaling cost reduction. Although there are CFD studies on plate overheating and scale growth separately, this work presents three main contributions: (1) the integration, within a single numerical framework, of combustion, radiation, species transport, oxidation kinetics, and actual plate movement using a dynamic mesh; (2) validation against continuous industrial records (16 thermocouples) and quantification of operational benefits such as fuel savings and reduced material loss; and (3) a comparative analysis between actual and optimized conditions, which standardize the air–methane ratio.

1. Introduction

The reheating of steel slabs is a key stage in the production chain of hot-rolled steels, as it determines the mechanical properties and surface quality of the final product [1,2,3]. In industrial walking-beam furnaces, steel slabs must reach temperatures between 1050 and 1200 °C to ensure a homogeneous thermal distribution that promotes plastic deformation without generating structural or surface defects [4,5,6]. During heating, the steel comes into direct contact with hot combustion products (CO₂, H₂O, N₂, and O₂), which creates a strongly oxidizing atmosphere [7]. This phenomenon leads to the formation of iron oxides on the material surface, commonly known as scales [8,9]. This layer predominantly consists of wüstite (FeO), magnetite (Fe₃O₄), and hematite (Fe₂O₃), with relative proportions that depend on the thermal regime, residence time, and furnace gas composition [3]. Excessive scale formation results in a direct loss of metallic material and impairs both the surface quality and mechanical performance of rolled steel. Furthermore, its removal during the descaling stage requires additional resources and may generate defects if it is performed only partially [3,7]. The precise control of the thermal profile and internal furnace atmosphere has been shown to significantly reduce scale formation and consequently improve overall process efficiency [9,10,11]. Traditionally, oxidation analysis has been addressed using experimental or semi-empirical models based on the parabolic law of oxide growth [3]. However, these approaches do not account for the complex interactions among heat transfer, gas dynamics, oxidation kinetics, and the actual motion of slabs inside the furnace [12,13,14,15]. This multiscale coupling requires sophisticated tools, such as computational fluid dynamics (CFD) [16]. Specialized CFD software (ANSYS-Fluent 2025 R2) enables the high-resolution modeling of heat fluxes, species transport, and momentum within reheating furnaces [14]. In particular, the integration of combustion, thermal radiation, and mass transfer models enables a detailed description of realistic operating conditions of the burner [3,5,6,7]. In this context, ANSYS Fluent^® has established itself as a robust platform for solving these phenomena using the finite volume method and implicit schemes for transient simulations [17,18]. Unlike previous studies, this study solves the coupled phenomena by incorporating a dynamic mesh, which enables the simulation of the actual slab displacement on walking beams. This technique, implemented through user-defined functions (UDFs), assigns controlled motion to solid bodies within the domain, allowing for the thermal coupling of residence time effects and local heat exchange between the gases and the moving metal surface [19,20,21,22,23,24,25]. The implementation of a dynamic mesh is particularly valuable for representing transient conditions in which the slab position modifies the temperature and velocity fields within the furnace [21]. This methodology has been applied in other areas, such as internal combustion engines, oscillating valves, and forging processes; however, its use in modeling reheating furnaces with explicit scale formation remains limited and novel [26,27,28,29]. Regarding the numerical models employed, the realizable k–ε model was selected to describe turbulence, as it substantially improves predictions for flows with strong curvatures and recirculation zones, which are common in industrial combustion chambers [30,31,32,33]. This model provides an adequate description of the turbulent kinetics and energetic flow structures [34,35,36]. For the combustion simulation, the Eddy Dissipation model was employed, which assumes that the turbulent mixing of air and fuel controls the reaction rate [37]. This hypothesis is valid for non-premixed flames, as found in industrial furnaces, and enables a realistic prediction of the temperature and combustion product concentration profiles. Radiative heat transfer, which is dominant in reheating furnaces (contributing up to 90% of the heat transferred to the slabs), was resolved using the P-1 model [35,36]. This simplified model, derived from the spherical harmonics’ expansion, provides an efficient solution to the radiative transfer equation at low computational cost [38]. To account for the effects of emitting gases such as H₂O and CO₂, the weighted sum of gray gases model (WSGGM) was used, which approximates the total emissivity as a weighted sum of gray gases [19,20,21,39,40]. The surface oxidation of steel was modeled using a Custom Field Function (CFF) implemented on the slab surfaces. This function incorporates Arrhenius-type oxidation kinetics, which are sensitive to the local temperature, enabling the real-time calculation of the scale layer growth under dynamic operating conditions [19]. The combination of these numerical techniques enables a highly accurate coupled reproduction of the thermal, fluid-dynamic, and kinetic-chemical behavior of the furnace. In addition, key industrial parameters, such as the scale thickness, slab temperature distribution, and gas velocity profiles in the different furnace zones, were quantified [40,41,42,43]. This level of detail is not attainable with traditional experimental methods, because of the extreme operating conditions and the impossibility of directly instrumenting the slabs during the process [44,45,46,47]. Previous investigations reported data that closely matched the experimental results, validating the proposed approach [48,49]. Dynamic meshing in CFD simulations of processes involving moving parts has been demonstrated to be a powerful tool for optimizing complex thermal industrial processes, particularly for understanding and minimizing surface-scale formation on slabs in reheating furnaces [19,49,50,51]. In previous investigations, including those by the present authors, results were reported that examined thermal evolution, fluid dynamics, or scale formation individually, without considering slab motion under the full furnace load. In contrast, this study couples slab motion under full load, oxidation kinetics, temperature, fluid dynamics, and chemical species, explicitly including the effect of the oxidizing atmosphere on the evolution of the generated FeO scale thickness, to reduce production costs and greenhouse gas fuel consumption while maintaining process productivity. During the reheating of steel slabs (900–1250 °C), oxide layers form on the surface, whose evolution depends on temperature, residence time, and oxidizing species, following a parabolic growth law controlled by diffusion. Fixed-mesh approaches assume static slabs, neglect the actual movement, and therefore limit the ability to capture the thermal history and the spatially varying radiative heat flux. In contrast, a dynamic mesh couples slab displacement with combustion and heat transfer, enabling a more faithful representation of oxidation kinetics as a function of temperature, time, and oxidizing atmosphere. Previous CFD studies have analyzed combustion and heat transfer in reheating furnaces and, in some cases, slab motion; however, oxidation kinetics and combustion phenomena have typically been treated separately. Such simplifications reduce computational cost but restrict the model’s capability to capture the transient coupling between slab displacement, radiative flux redistribution, local oxygen availability, and temperature-dependent scale growth. In this work, a fully transient CFD model is proposed that integrates turbulent combustion, radiative heat transfer, dynamic meshing, and parabolic oxidation kinetics within a unified computational domain, allowing for a more comprehensive and realistic representation of the industrial process. Furthermore, the influence of operational adjustments on the thermal field and scale growth is evaluated, providing a practical tool to optimize energy consumption and reduce metal losses.

2. Methodology

The plant furnace has a production capacity of 210 tons of steel per hour and reaches a rolling temperature of 1250 °C. The product with the highest production volume complies with the ASTM A36/A36M [52] (C ≤ 0.26%, Mn 0.80–1.20%, P ≤ 0.04%, S ≤ 0.05%, Si ≤ 0.40%): To perform the process simulation, certain simplifications were established to reduce the computational cost as much as possible, while obtaining an approximate yet realistic solution. The vertical movement of the slab is implicit in the total residence time. In continuous industrial furnaces, vertical displacement is significantly lower compared to longitudinal advancement, and its influence on global radiative heat exchange is limited. It is acknowledged that these simplifications may slightly affect instantaneous local thermal distributions; however, they do not alter the comparative trends or the overall conclusions of this study. Notably, the primary purpose of a numerical simulation is to predict system behavior rather than reproduce it exactly; its validity is assessed by comparison with physical experiments or real measurement data. Commercial computational fluid dynamics (CFD) software was used for the model development. Modern CFD tools enable the modeling of various combustion aspects and incorporate diverse thermal radiation models, which are the predominant mechanisms in direct-flame slab heating. The software was selected based on the finite volume technique, which ensured proper domain discretization. ANSYS Fluent^® was employed because it integrates a comprehensive set of models and capabilities for accurately representing the transport phenomena involved in slab heating for rolling, including dynamic meshes using user-defined functions (UDFs). Two case studies were conducted: Case A (current plant conditions) and Case B (improved conditions, with a 10% fuel reduction in heating zone 1 and a standardized air-to-fuel ratio of 10:1 in all the zones). The methodology accounted for all the variables influencing scale formation. The customized UDF for controlling the slab displacement was programmed based on real plant data with a velocity of 0.004 m·s⁻¹. Convergence criteria for the continuity and momentum equations were set below 10⁻⁴, while that for energy was set to 10⁻⁶. A time-step size of 1 × 10⁻³ s was used, considering the actual furnace residence time of 120 min. The methodology involves the cyclic resolution of slab motion based on the simulation time, as summarized in Figure 1.

The geometry of the virtual model was developed at a 1:1 scale based on a reheating furnace used by a steel manufacturing company. The simulated physical phenomenon corresponds to the combustion process that occurs inside the furnace, whose oxidation reaction directly contributes to heating the slab. The gases involved in the combustion are air, which is considered to have a composition of 21% oxygen and 79% nitrogen, and methane gas as fuel. The geometric dimensions of the furnace and its operating conditions were provided by the company. Figure 2 shows the boundary conditions and locations of the 16 thermocouples, numbered from the inlet to the outlet of the slabs and distributed along the length and width of the furnace. The furnace geometry was modeled in AutoCAD^® 2024 version and divided into four functional zones, each with a specific burner configuration. The first zone corresponds to the preheating zone, which does not have burners, because the gas extraction systems are located in this section. The second zone is heating zone 1, which is equipped with 12 burners distributed along both sides of the furnace. The third zone is heating zone 2, which consists of a row of eight lower burners and six upper burners. Finally, the fourth zone is the homogenization zone, which contains 24 burners at the top and eight at the bottom. The furnace has 58 burners, which are strategically distributed to ensure efficient heat transfer throughout the slab reheating process, as reported in previous studies [19,20,21]. The boundary conditions and locations of the 16 thermocouples are also shown, numbered from the slab inlet to the outlet and distributed both longitudinally and transversely within the furnace. Details of the furnace geometry and extension volume are also included; this extension volume is necessary to allow for slab movement during the operation of the dynamic mesh, as described in [19]. The slabs considered in this study had dimensions of 9.76 m in length, 1.27 m in width, and 0.20 m in thickness. The 16 thermocouples are distributed throughout the furnace and are numbered. These dimensions were selected because they correspond to the product with the highest production volume in the kiln, representing approximately 15% of the total production.

For greater clarity and reproducibility, the main boundary conditions used in the CFD model are presented in

Table 1. Boundary conditions.

Location	Boundary Condition	Value
Wall furnace	Temperature	1200 °C
Extractor hoods	Constant pressure output	94,859.94 Pa
Wall slab	Couple	-
Burners	Mass flow inlet	See Table 2

. The data are based on industrial operation, and heat losses to the surroundings were not considered, as the losses reported by the plant are less than 5%. The walls of the slabs were considered as a coupling condition between the surface layer and the adjacent fluid. The gas outlet was modeled with a constant-pressure outlet condition according to the information provided by the plant.

The investigation cases were carried out using variations in the air and fuel flows and ratios, as shown in

Table 2. Actual and proposed operating conditions.

	Current Conditions (Case A)			Proposed Conditions (Case B)
Burner’s Location	Air, kg·s1	Methane, kg·s−1	Ratio A:M	Air, kg·s−1	Methane, kg·s−1	Ratio A:M
North side burners	8.699	0.473	10:1	7.829	0.426	10:1
South side burners	8.699	0.473	10:1	7.829	0.426	10:1
Top burners	5.353	0.291	10:1	4.818	0.262	10:1
Rear burners	5.687	0.309	10:1	5.119	0.278	10:1
North front lower burners	1.172	0.053	12:1	0.976	0.053	10:1
South front lower burners	1.184	0.053	12:1	0.987	0.053	10:1
North soaking burners	0.917	0.038	13:1	0.705	0.038	10:1
South soaking burners	0.943	0.042	12:1	0.786	0.042	10:1

.

The solid materials used were alumina as insulation for the walls and steel for the charge material. Their properties were sourced from the CFD software database and are listed in

Table 3. Material properties and fuel mixture.

Property	Units	Air–Methane Mix	Steel	Alumina
Specific heat	J·kg−1·K−1	1000	480	880
Density	kg·m−3	0.657	7850	3950
Thermal conductivity	W·m−1·K−1	0.0454	Polynomial *	18
Emissivity	-	WSGGM	0.6	0.75
Viscosity	kg·m−1·s−1	1.72 × 10−5	-	-
Diffusivity	m2·s−1	2.879 × 10−5	-	-
Scattering coefficient	1·m−1	0	-	-
Refractive index	-	1	-	-

* K_s(T) = a₀ + a₁T + a₂T² + a₃T³ + a₄T⁴ + a₅T⁵ + a₆T⁶ + a₇T⁷ [36].

.

Figure 3 shows the mesh generated for the two analyzed cases, which differed only in their operating conditions. The mesh was composed of approximately 1,250,000 elements, with a maximum deformation of 0.85. The mesh was generated using the finite volume technique in ANSYS^® Meshing 2025 R2 software, as illustrated in Figure 2. It is essential that the mesh is adequately adapted to the physical phenomenon being simulated. Furthermore, the available computational capacity for solving governing equations and models must be considered. A tetrahybrid configuration was used for all volumes of the geometry, including the planks, to facilitate the mesh deformation process in the adjacent regions. This mesh included an extended displacement volume highlighted in yellow, as described previously, in which the dynamic mesh model was solved. This model was implemented using a user-defined function (UDF) that specified the velocity and direction of the motion of the solid.

2.1. Governing Equations

Table 4. Equations and models used.

Governing Equations	Equation
Continuity Equation [19,22]	${\displaystyle \frac{\mathsf{\partial }\rho }{\mathsf{\partial }t}}+\nabla \cdot (\rho \cdot \overrightarrow{\nu })=0$
Momentum Equation [19,20]	${\displaystyle \frac{\mathsf{\partial }\left(\rho \overrightarrow{\nu }\right)}{\mathsf{\partial }t}}+\nabla \cdot \left(\rho \overrightarrow{\nu }\overrightarrow{\nu }\right)=-\nabla p+\nabla \cdot \left(\overline{\tau }\right)+\rho \overrightarrow{g}+\overrightarrow{F}$
Energy Equation [19,20]	${\displaystyle \frac{\mathsf{\partial }\left(\rho E\right)}{\mathsf{\partial }t}}+\nabla \cdot \left(\overrightarrow{\nu }\left(\rho E+p\right)\right)=-\nabla \cdot \left(k_{eff}\nabla T-{\displaystyle \sum _j}{h}_j{\overrightarrow{J}}_j+\left({\overline{\tau }}_{eff}\cdot \overrightarrow{\nu }\right)\right)+S_h$
Numerical Models	Equation
$\mathsf{\kappa }-\mathsf{\epsilon }$ realizable model [19,20]	For $\kappa$: ${\displaystyle \frac{\mathsf{\partial }}{\mathsf{\partial }t}}\left(\rho \kappa \right)+{\displaystyle \frac{\mathsf{\partial }}{\mathsf{\partial }t}}\left(\rho \kappa {\nu }_i\right)={\displaystyle \frac{\mathsf{\partial }}{\mathsf{\partial }{x}_j}}\left(\left(\mu +{\displaystyle \frac{{\mu }_t}{{\sigma }_{\kappa }}}\right){\displaystyle \frac{\mathsf{\partial }\kappa }{\mathsf{\partial }{x}_j}}\right)+G_{\kappa }+G_b\rho \epsilon -Y_M+S_{\kappa }$
	For $\epsilon$: ${\displaystyle \frac{\mathsf{\partial }}{\mathsf{\partial }t}}\left(\rho \epsilon \right)+{\displaystyle \frac{\mathsf{\partial }}{\mathsf{\partial }{x}_i}}\left(\rho \epsilon {\nu }_i\right)={\displaystyle \frac{\mathsf{\partial }}{\mathsf{\partial }{x}_j}}\left(\left(\mu +{\displaystyle \frac{{\mu }_t}{{\sigma }_{\epsilon }}}\right){\displaystyle \frac{\mathsf{\partial }\epsilon }{\mathsf{\partial }{x}_j}}\right)+C_{1\epsilon }{\displaystyle \frac{\epsilon }{\kappa }}+\left(G_{\kappa }+C_{3\epsilon }{G}_b\right)-C_{2\epsilon }{\displaystyle \frac{\rho {\epsilon }^2}{\kappa }}+S_{\epsilon }$
Species Transport Model [19,20]	${\displaystyle \frac{\mathsf{\partial }}{\mathsf{\partial }t}}\left(\rho \kappa \right)+{\displaystyle \frac{\mathsf{\partial }}{\mathsf{\partial }t}}\left(\rho \kappa {\nu }_i\right)={\displaystyle \frac{\mathsf{\partial }}{\mathsf{\partial }{x}_j}}\left(\left(\mu +{\displaystyle \frac{{\mu }_t}{{\sigma }_{\kappa }}}\right){\displaystyle \frac{\mathsf{\partial }\kappa }{\mathsf{\partial }{x}_j}}\right)+G_{\kappa }+G_b\rho \epsilon -Y_M+S_{\kappa }$
P-1 Radiation Model [19,20]	$q_r=-{\displaystyle \frac{1}{3\left(\alpha +{\sigma }_s\right)-C{\sigma }_s}}\nabla G$
WSGGM [19,20]	$\epsilon ={\displaystyle \sum _{i=0}^I}{\alpha }_{\epsilon ,i}\left(T\right)\left(1-e^{-{\kappa }_{i}ps}\right)$

lists the equations and models used to solve the fluid dynamics and thermal fields. It also includes turbulence, species transport, radiation, and gray gas models, which were previously reported in prior work [19,20] to solve the numerical system more accurately.

2.2. Radiation Model

The P-1 radiation model was employed due to its optimal balance between accuracy and computational cost in industrial furnaces, where it delivers satisfactory results. Although more detailed combustion models such as EDC and DO exist, their high computational expense and complexity render them impractical when coupled with dynamic meshing, transient states, and surface oxidation; hence, a robust and extensively validated model was adopted. Overall, the model selection reflects a trade-off between physical fidelity and computational feasibility under realistic industrial conditions [3,44].

2.3. Oxidation Model

To model the total thickness of the oxide scale, the oxidation time (t) [24,47] is considered, assuming a parabolic oxide growth mechanism.

$$x^2=k_{p}t$$

(1)

where k_p is the parabolic oxidation rate constant represented as the Arrhenius equation (Equation (2)).

$$k_p=k_0{·p_{O_2}·e}^{{\displaystyle \frac{-Ea}{RT}}}$$

(2)

where E_a is the activation energy, R is the universal gas constant and k₀ is a temperature-dependent parameter; E_a will depend on the reaction mechanism. The oxide layers on iron, using air as the oxidizing agent, and we determined the kinetic equations for wüstite, magnetite, and hematite. The scale growth kinetics, considering the three oxides generated on the top side, can be represented by the following equation, where PO₂ is the partial pressure of oxygen [24]:

$$k_p=6.1e^{{\displaystyle \frac{-\mathrm{40,500}}{RT}}}{\displaystyle \frac{\mathrm{c}{\mathrm{m}}^2}{s}}$$

(3)

2.4. Dynamic Mesh Model

In the dynamic mesh model, a defined volume moves directly within the mesh domain to represent the motion during the simulation. To control this displacement, a user-defined function (UDF) specifies both the velocity and direction of the moving volume. The dynamic mesh model implemented in ANSYS Fluent 2025 R2 enabled automatic remeshing based on the shape and trajectory of the volume. One main advantage of this approach is its ability to perform three-dimensional simulations, allowing for a more accurate representation of heat absorption at each volume position. In this study, only the heat transfer phenomenon was solved, excluding mass transfer, which contributes to greater precision in location-specific thermal calculations. However, a key limitation of the dynamic mesh model is that it cannot be applied in the steady-state regime. Thus, a reheating furnace simulation must first be run transiently without motion until convergence, and then dynamic meshing is activated, which increases the computational cost. During the transient simulation, the steel slabs moved from the charging door to the discharge door, following a horizontal trajectory in the +X direction. Although real operations may include vertical motion, it was neglected in this study. To control this displacement, a custom UDF defined the motion velocity of the slabs based on real data, which was set to 0.004 m/s. In industrial operations, slab positions are monitored every minute using an operational control model. In this simulation, the positions were updated every 30 s. Regarding dynamic meshes, the integral form of the general scalar conservation equation within an arbitrary control volume V with a moving boundary can be expressed as

$${\displaystyle \frac{d}{dt}}\underset{V}{\int }\rho \varphi dV+\underset{\mathsf{\partial }V}{\int }\rho \varphi \left(\overrightarrow{u}-{\overrightarrow{u}}_g\right)·d\overrightarrow{A}=\underset{\mathsf{\partial }V}{\int }\mathsf{\Gamma }\nabla \varphi ·d\overrightarrow{A}\underset{V}{\int }{S}_{\varphi }dV$$

(4)

where ρ is the fluid density, $\overrightarrow{u}$ is the flow velocity vector, ${\overrightarrow{u}}_g$ is the mesh velocity of the moving mesh, $\mathsf{\Gamma }$ is the diffusion coefficient, and $S_{\varphi }$ is the source term of $\varphi$. Here, $\mathsf{\partial }V$ represents the boundary of the control volume V.

And using a first-order differential regression formula, the time derivative in terms of the following equation can be written as

$${\displaystyle \frac{d}{dt}}\underset{V}{\int }\rho \varphi dV={\displaystyle \frac{{\left(\rho \varphi V\right)}^{n+1}-{\left(\rho \varphi V\right)}^n}{∆t}}$$

(5)

where n and n + 1 denote the respective quantity at the current time level and at the next one, respectively. The (n + 1) time-level volume, Vⁿ⁺¹, is computed from

$$V^{n+1}=V^n+{\displaystyle \frac{dV}{dt}}∆t$$

(6)

where ${\displaystyle \frac{dV}{dt}}$ is the temporal derivative of the control volume. To satisfy the mesh conservation law, the temporal derivative of the control volume is calculated from

$${\displaystyle \frac{dV}{dt}}=\underset{\mathsf{\partial }V}{\int }{\overrightarrow{u}}_g·d\overrightarrow{A}=\sum _j^{n_f}{\overrightarrow{u}}_{g,j}·{\overrightarrow{A}}_j$$

(7)

where n_f is the number of faces in the control volume and ${\overrightarrow{A}}_j$ is the face area vector. The dot product ${\overrightarrow{u}}_{g,j}·{\overrightarrow{A}}_j$ is calculated over each control volume face from

$${\overrightarrow{u}}_{g,j}·{\overrightarrow{A}}_j={\displaystyle \frac{{\delta V}_j}{∆t}}$$

(8)

where ẟV_j is the volume swept by the control volume face j in the time step ∆t.

The model captures the coupling between combustion, radiation, dynamic billet movement, and scale growth under representative industrial conditions, but its direct application is limited to furnaces with similar geometry and operation, potentially requiring recalibration for other designs. Dynamic meshing entails high computational cost and assumes uniform longitudinal displacement, while oxidation is modeled as continuous parabolic growth controlled by diffusion, without accounting for spallation or heterogeneities. Consequently, although comparative trends are robust within the same numerical framework, experimental or industrial validation of scale thickness is required to establish its absolute predictive capability. Considering the assumptions associated with the oxidation kinetics, numerical discretization, and industrial-scale CFD modeling, some uncertainty in the precise magnitude of scale growth is expected. Nevertheless, the model reliably reproduces the relative variation in scale formation under the analyzed operating conditions.

The simulation was solved on a computer equipped with an Intel Core i9 12th-generation processor (Intel Corporation, Santa Clara, CA, USA), 32 GB RAM, and an NVIDIA K1000 graphics card (NVIDIA Corporation, Santa Clara, CA, USA).

3. Results

To perform the grid sensitivity analysis (GCI, Grid Convergence Index), three distinct meshes were generated, with details presented in

Table 5. Mesh sensitivity analysis.

Number of Elements	Size of Elements (mm)	Variable Velocity (m·s−1)	Geometry Volume m3
1,550,349	102	4.85	1750
1,250,000	113	4.75	1750
1,100,000	117	4.69	1750

. Richardson extrapolation was applied to assess the influence of element size and number on the simulation results [22]. This method verifies whether reducing the element size—and consequently increasing their number—significantly affects the obtained results, assuming that the discretization error follows a power-law dependence on the mesh size. Mesh 2 was selected based on the convergence analysis using Richardson extrapolation.

When two numerical solutions are available ϕ₁ and ϕ₂, calculated with mesh sizes h₁ and h₂, when h₁ < h₂ and the refinement ratio r = h₂·h₁⁻¹ is constant, the following can be used:

$${\varphi }_{ext}={\varphi }_1+{\displaystyle \frac{{\varphi }_1-{\varphi }_2}{r^p-1}}$$

(9)

where Φ_ext is the extrapolated value (estimate of the true value), ϕ₁ is the solution on the fine mesh, ϕ₂ is the solution on the coarser mesh, r is the refinement ratio between meshes, and p is the order of the numerical method accuracy. From the processed data, the following results were obtained: GCI_1,2 = 0.2224; GCI_2,3 = 0.187; r^p = 1.88; and GCI_1,2/GCI_2,3 = 1.88. For this analysis, the relation GCI_1,2/GCI_2,3 ≈ r^p must hold.

Figure 4 shows the Richardson extrapolation results, confirming that reducing the element size to an infinitesimally small value does not significantly alter the average velocity results used in the mesh sensitivity analysis. Therefore, a mesh with 1,255,349 elements was selected because it provided reliable results without excessive computational cost.

To validate the numerical model (Figure 5), the predicted gas temperatures were compared with thermocouple measurements obtained from the industrial reheating furnace. The model predictions show good agreement with the experimental measurements, with deviations within an acceptable range for industrial-scale CFD simulations. The results were obtained through simulation with actual plant data to evaluate the model’s accuracy. The company provided the thermal histories of 16 thermocouples, each with a continuous 24 h record, and the average value was compared with the simulation. These sensors are strategically located in the furnace structure and distributed along the side and top walls (roof) [19,20,21,22]. The maximum error calculated using Equation (10) between the numerical model and plant data was approximately 7.6%, while the minimum error was approximately 0.6%, with an average of approximately 2.55%. Thus, the model conditions satisfactorily predicted the thermal behavior inside the furnace.

$$\%\mathrm{E}\mathrm{r}\mathrm{r}\mathrm{o}\mathrm{r}={\displaystyle \frac{T_{plant}-T_{simulation}}{T_{plant}}}\times 100$$

(10)

In addition to the validation described above, Figure 6 presents a comparison between the temperatures measured on-site and the simulation results for each zone of the furnace during a 20 min operating period. The first observation in this figure is the oscillatory behavior of the temperature, regardless of the measurement technique used. This oscillation is the result of convective phenomena, primarily from the movement of slabs and jets driven by each burner. Figure 6a shows the temperature corresponding to the preheating zone, which represents the initial thermal contact of the slab upon entering the furnace. The side burners, which operate with the highest methane and air flows, are located in this zone as well as in the other zone. From the first few seconds of the simulation, the temperature in these zones reached a state of near-stability, with the predicted thermal evolution curve from the numerical model remaining slightly below that of the on-site measurements. In heating zones 1 and 2 (Figure 6b,c), the thermal evolution predicted by the numerical model closely approximated the plant data. The simulation showed thermal fluctuations for zone 1 until approximately 12 min and for zone 2 until approximately 7 min, after which they tended to stabilize. Regarding the homogenization zone (Figure 6d), a slight thermal variation was observed in the simulation at approximately 7 min, where the numerical model predictions evolved slightly above the data obtained from the plant. Overall, the simulated temperatures were in good agreement with the current data, remaining within an acceptable margin of error.

Figure 7a illustrates the burner lines of each zone in the plans, and Figure 7b shows the differences between both cases, using open circles for case A and closed circles for case B. At point ①, the thermal plume rises with a greater inclination and shows greater extension, indicating a jet with higher energy and greater interaction with the surrounding flow. In contrast, at point ❶, the thermal plume was shorter and slightly more contained, suggesting less penetration or a more stable jet. Case A shows greater dispersion and reaches of the hot jet, while case B shows a more compact jet. At point ②, higher-temperature zones are observed, with greater expansion toward the lower region of the plane. However, point ❷ shows smaller, more uniform zones and less thermal dispersion. Case A presented greater turbulence and lateral propagation, whereas case B was more uniform and contained. At point ③, the jet structure showed greater variability in height and shape, with more pronounced maximum temperature zones, whereas at point ❸, the burner plumes were more homogeneous among themselves and slightly less intense at the tip. Case A exhibited workshop and irregular jets, whereas case B showed more uniform and less elongated patterns. At point ④, the thermal plume region shows greater lateral extension compared to that at point ❹, which shows a more compact hot zone with less dispersion. Case A exhibited a hotter, more open, and more turbulent jet than case B, which exhibited a cooler, narrower, and more stable jet. Point ⑤ illustrates that the thermal columns are more intense and longer vertically, indicating more penetrating combustion compared to point ❺, which shows shorter, more regular columns with a slightly lower core temperature. At point ⑥, greater lateral dispersion was observed, with a wider pattern than at point ❻, where the jet pattern was finer and more uniform. Case A showed a greater jet opening than case B, which was more concentrated. Point ⑦ shows more separated hot structures, with more pronounced intermediate cold zones than its corresponding point ❼, which has a greater continuity between hot zones and a more homogeneous distribution. Case A generated colder spaces between the jets than case B, which reduced the cold spaces and improved thermal homogeneity. At point ⑧, the jet impact region was wider, with a more marked hot core, whereas at its corresponding point ❽, it showed a smaller hot core with less lateral expansion. Case A presents a stronger impact on the thermal field, and case B generates more moderate heating. Point ⑨ demonstrated greater irregularity in the plumes, with size differences between the jets compared to point ❾, which showed more uniform plumes with very similar intensities. Case A has greater variability between burners, while case B offers thermal uniformity and stability.

Figure 8 shows the temperature analysis in the linear sensors, displaying the thermal profile for each burner line at a fixed simulation time of 1000 s. Figure 8a shows the geometry and strategic location of the sensors placed in front of each burner line. It also shows the locations of the two endpoints of each analysis line. Figure 7b shows the temperature and length graphs. Sensor L1 shows how the temperature progressively decreases toward the sensor center and then increases again toward the furnace center; case B maintains a slightly lower temperature than case A. Sensor L2 showed a higher temperature for case A, oscillating between 1200 and 1400 °C, with well-defined peaks slightly higher than those in case B, showing two consecutive hot zones clearly associated with burners located along the line. Sensor L3 presented very high peaks of 1400 °C, followed by sharp drops to 1150 °C, which were very similar for both cases owing to the burner distribution and fluid inlet velocity. Sensor L4 presented strong oscillations between 1000 and 1380 °C; for both cases, it was the most thermally unstable line, and case B presented higher peaks in the central sensor zone. Sensor L5 maintained values between 1000 and 1360 °C, with a smoother curve compared to L4, indicating a zone where the gas flow was more stable and the profile was more homogeneous. This line appeared to be in a region with more uniform mixing and better thermal control for cases A and B, maintaining the same trend. There was a significant thermal gradient depending on the position within the furnace. The central zones (L2 and L3) exhibited higher and more stable temperatures, indicating greater radiation and convection.

Figure 9 presents the transient fluid dynamic field inside the furnace for case A. In the Figure 9a was captured at 1000 s, whereas Figure 9b was obtained at 1210 s, both corresponding to Case A. This temporal variation highlights the displacement of the slabs along the interior of the furnace and its direct influence on the flow structure, resulting in oscillatory variations in the temperature measurements, as previously discussed. Fluid dynamics plays a decisive role in the slab heating process because convective mechanisms enhance heat transfer, particularly in regions where recirculation zones develop. In the preheating zone, point ① exhibited a well-defined recirculation, characterized by a large vortex and reduced velocities near the furnace inlet. In contrast, at point ❶, corresponding to the subsequent instant, the recirculation maintained a similar size, but the streamlines appeared more compact and less dispersed in the lower region, suggesting a more localized and potentially more efficient heat transfer. At point ②, located at the end of heating zone 1 and before entering heating zone 2, recirculation can be observed, which promotes the retention of hot gases, thereby enhancing the convective heating of the slabs. At a later time, point ❷ shows a change in the size of this recirculation, which is directly associated with the displacement of the slab. Point ③, corresponding to heating zone 2, exhibits a larger-scale recirculation that is distorted by the geometry of the slab arrangement, which divides the lower and upper furnace regions. At the subsequent time, point ❸ shows a more uniform recirculation owing to the new positions of the slabs within the section. Finally, in the homogenization zone, point ④ displays a noticeably distorted recirculation, whereas point ❹ reveals a more uniform flow structure, promoting more homogeneous heating of the lower part of the slab. Overall, these variations indicate that the furnace fluid dynamics and temperature fields experience cyclic changes induced by the continuous movement of slabs throughout the process.

Figure 10 illustrates the thermal evolution of the slab from its entry into the furnace until its exit after a total residence time of 120 min. The calculation was performed by averaging three diagonal measurement positions across both the width and thickness, as indicated in Figure 10a. During the first 60 min, a rapid temperature increase was observed, driven by the high thermal gradient between the furnace atmosphere and slab surface. During this period, the slab moves through the first three zones of the furnace. Upon reaching 80 min, the slab transitioned from heating zone 2 to the homogenization zone, where the temperature experienced a slight decrease owing to the specific operating conditions of that zone. The function of this zone is not only to stabilize the temperature but also to contribute to the microstructural homogenization of the metallic matrix of the steel. The graph in Figure 10b shows that in both simulated cases, a very similar heating trend is maintained; only in heating zone 1 and approximately three-quarters of heating zone 2 does the slab heating rate for Case B shift slightly below that of Case A. This behavior is expected, given the changes in the operating conditions. Notably, in Case B, applying the optimized operating conditions achieved a substantial reduction in fuel consumption and standardization of the air–natural gas ratio without negatively affecting the thermal efficiency of the process. The final slab temperature in both cases reached approximately 1150 °C under dynamic mesh solution conditions. A relevant aspect considered in this investigation was the inclusion of a new variable: the furnace exit gate was kept permanently open, which enabled a more realistic simulation of the thermal behavior observed in the plant. This is important because the gate opening generates fluctuations in the furnace thermal profile. Under real conditions, the exit temperature of the slabs varies between 1050 and 1250 °C, which is attributed to the constant variation in the dimensions and mass of the slabs fed into the furnace. This variability directly affects the internal thermal behavior and, consequently, the final temperature achieved by each slab upon its exit.

The chemical species generated in the methane oxidation reaction with air were analyzed, as they are the main agents responsible for scale formation on the steel surface. This is particularly relevant because the oxygen-enriched furnace atmosphere maintains direct contact with the slabs during heating. Figure 11 shows the distribution of the oxygen mass fraction obtained from simulations. Figure 11A, corresponding to Case A, shows that a significant accumulation of oxygen was observed in the homogenization zone. This behavior is associated with the air-to-gas ratio used by the company, which, in that zone, reaches values up to 13:1, despite the manufacturer recommending an ideal ratio of 10:1. This elevated ratio promotes excess oxygen within the furnace volume, generating a highly oxidizing atmosphere at the operating temperature. However, in Figure 11B, corresponding to Case B, a distinctly different oxygen behavior is observed across the furnace zones. In this case, a controlled air-to-gas ratio of 10:1 was used, resulting in a significant reduction in free oxygen and thus decreasing the oxidizing nature of the atmosphere. The combustion ratio is one of the primary factors influencing oxide formation. An excess of air promotes surface oxidation. However, the chemical composition of the steel also plays a role, as a high carbon content in the material can lead to losses of this element from the surface during heating [3]. In quantitative terms, the simulation under the current operating conditions (Case A) showed an accumulated volume of free oxygen of approximately 4.9 × 10² mol over a period of 1000 s. In contrast, the simulation with optimized operating conditions (Case B) exhibited a significantly lower volume of only 6.7 × 10¹ mol, demonstrating the positive impact of adjusting the air-to-gas ratio on reducing the oxidizing atmosphere within the furnace.

Figure 12a shows the oxide scale growth rate (left Y-axis), which was calculated using the Wagner oxidation model described previously. To accurately predict the oxidation kinetics, the local temperature and oxygen partial pressure must be known. From these parameters, kinetic values are obtained, which, when multiplied by the slab’s residence time in the furnace, allow the estimation of the oxide layer thickness. The evolution of scale growth on the slab surface is illustrated with solid lines for Case A, showing accelerated growth in the first three zones of the reheating furnace until it begins to slowly continue advancing in the homogenization zone, where the temperature decrease plays a key role in the scale growth rate. Case B shows a trend similar to that of Case A; however, it remains below Case A, indicating a reduction in the thickness of the oxide scale formed on the surface. The comparison between Cases A and B demonstrates that, under optimized operating conditions, a 0.75% reduction in the oxide layer thickness is achieved starting from heating zone 1. The parabolic oxidation model accurately predicted the growth behavior, which was initially proportional to time until the slab entered heating zone 2, where the lower temperature reduced the oxidation rate and, consequently, the scale growth. This translates to a reduction in material loss due to oxidation, decreasing from 4.5% under real conditions to 3.75% under the proposed conditions. Since oxidation kinetics are parabolic and temperature-dependent, small thermal variations can amplify the estimated oxide thickness. Thus, the 0.75% reduction should be interpreted as a comparative trend under specific operating conditions, rather than a universally absolute model value. The order of magnitude of the thickness aligns with data reported in the literature for similar temperature ranges. This improvement is primarily attributed to the reduced gas flow and standardization of the air–fuel ratio, which minimizes the oxidizing atmosphere inside the furnace, thereby enhancing material conservation. Likewise, Figure 12a illustrates the oxidation kinetics (right Y-axis) using a validated methodology [19] in the same graph using dashed lines. The oxidation kinetics increased rapidly until reaching heating zone 2, the hottest zone of the furnace, and then decreased as it progressed to the homogenization zone, maintaining the same trend for both cases, but with Case B consistently below Case A due to the proposed operating conditions. Figure 12b shows the profile of a moving slab during its passage through the furnace and the oxide formation on its surface. The color change corresponds to the studied slab at various times, with each contour captured between time intervals representing 7 min of furnace residence time and obtained using the dynamic mesh model. Point ① is located in heating zone 1 for Case A, highlighting the evolution of the scale formation. This zone exhibits one of the highest thermal gradients of the furnace and free oxygen accumulation, with a slight mark visible at the center of the slab, which is the farthest area from the burners. Compared to point ❶ in Case B, the scale generation was reduced because of the lower free oxygen levels under the proposed operating conditions. Points ② and ❷ illustrate the scale formation differences in heating zone 2, where the highest oxidation kinetics occur, showing that the central mark becomes increasingly homogeneous across the slab surface. Points ③ and ❸ depict entry into the homogenization zone, where Case A slabs exhibit greater scale homogenization under the current conditions, whereas Case B shows reduced scale formation overall. Points ④ and ❹ were located just prior to the exit in both cases, where the differences in the slab egress were minimal. Case B exhibited a slower scale generation and a reduced amount of scale owing to the proposed operating conditions. These results are beneficial for the process, as the subsequent descaling of the slabs would be simpler, with lower material loss, representing millions of dollars saved in material losses. The obtained thicknesses fall within the order of magnitude reported in the literature for similar temperature ranges in industrial reheating furnaces. However, it is important to note that the model does not account for effects such as local chemical variability, microstructural heterogeneities, or partial scale spallation [33].

The simulations of the slab heating process revealed similar thermal profiles up to heating zone 2, validating the effectiveness of the optimized operating conditions, such as the reduced gas flow rate and adjusted air–fuel ratio. These changes, applied primarily in heating zone 1, did not affect the final slab temperature or residence time, demonstrating enhanced energy efficiency without compromising the production output. However, overloading the furnace with slabs can disrupt the residence time and thermal uniformity, negatively impacting the final product quality. Therefore, stricter operational control is recommended to maintain process stability. Temperature distributions exhibit thermal gradients across all furnace zones, with temperature being a critical factor in scale formation, particularly in heating zone 2, where temperatures exceed 1860 °C near the burners, leading to maximum scale accumulation and underscoring the need for thermal level control to minimize oxidation losses. Validation simulations confirmed high free oxygen concentrations under real conditions, promoting oxidation, whereas optimized conditions reduced oxygen levels by up to eightfold without impairing combustion efficiency, enabling target temperatures before the homogenization zone, and significantly curbing scale formation. An increase in CO₂ and H₂O under the new conditions reflected more efficient combustion. Although these products may contribute to oxidation, their chemically bound nature results in a lower impact than that of free oxygen. Overall, adjusting the air–fuel ratio and optimizing the flow rates reduced energy consumption and scale formation, yielding a more efficient and sustainable thermal process without sacrificing productivity.

4. Conclusions

This study integrated thermal, fluid dynamics, kinetic phenomena, and real slab movement within a single numerical model using CFD and dynamic meshing to analyze the behavior of an industrial reheating furnace. The simulation enabled highly accurate reproduction of the furnace thermal profile and oxidizing atmosphere, as well as the prediction of scale growth under both real and optimized operating conditions. The results demonstrated that appropriate adjustment of the air–fuel ratio and reduction in flow rates significantly decreased surface oxidation and energy consumption while maintaining the target temperature and process productivity. This numerical approach provides a solid foundation for optimizing furnace operations and improving the quality of reheated steel. The conclusions derived from this study are as follows:

• The implementation of the CFD model, integrating the realizable turbulence model, P-1 radiation model, and Eddy-Dissipation combustion model, enabled the validation of the simulation with high accuracy, achieving an average error below 2.5% in the measured temperatures. This low deviation supports the robustness and reliability of the model, making it applicable for the optimization of thermal processes in industrial furnaces.
• Under the optimized conditions (Case B), the thermal gradient across the slab width decreased by up to 12–18 °C compared with Case A. This is critical for ensuring steel quality, as a homogeneous temperature distribution minimizes defects and microstructural variations in the steel. The average exit temperature improved to 1289.77 °C, 16 °C above the current case but still within the target range (1250 ± 50 °C).
• The simulation reproduced the plant-measured temperatures with an average error of 2.5%. Adjusting the air–fuel ratio by reducing the air flow rate by 10% and standardizing the mixture at a 10:1 ratio resulted in more efficient combustion. Under these conditions, the target temperature was achieved without altering the production time, yielding significant energy savings and improved fuel consumption management.
• The optimized conditions enabled a considerable reduction in scale formation on the steel surface. Specifically, a 0.75% decrease in the oxide layer thickness was observed, representing a reduction in the material losses from 4.5% to 3.75%. This benefit enhances the final product quality and reduces additional slag removal processes.
• The reduction in excess oxygen in the furnace atmosphere contributed to decreased surface oxidation. Even with a lower airflow rate, the required homogenization temperature was achieved in a less oxidizing environment, favoring slab surface integrity and minimizing defects. The excess free oxygen decreased from 4.9 × 10² mol to 6.7 × 10¹ mol, reducing the oxidative intensity and stabilizing the convective heat transfer.

Model Limitations

The movement of slabs assumes stable operating patterns, which may overlook minor industrial fluctuations. The oxidation model uses the kinetics described in the literature, which has been validated with scale-up experiments under controlled atmospheric and temperature conditions [44]. The results represent trends and approximations, not absolute values. The mesh resolution and time interval introduce small numerical uncertainties despite mesh sensitivity studies. Therefore, the validation strategy focused on the predicted temperature field, which governs the oxidation kinetics. Sensor temperatures were compared with plant measurements and showed good agreement. Since oxidation rates are highly dependent on temperature, this agreement supports the reliability of the predicted oxidation trends. Furthermore, the oxidation model used in this work is based on widely documented kinetic formulations for high-temperature steel oxidation. The plant reported average losses of 4% in material loss, compared to 2–3% reported in the literature [3,18,24,33,44].