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Mathematical model for the heating of the slab in the walking beam furnace


In the steel industry, the walking beam furnace is used to heat the slabs before rolling. In order to achieve the high quality requirements, a very uniform temperature profile is required in the slabs. With the help of Computational Fluid Dynamics (CFD), a model for calculating and analyzing the heat flows in a walking beam furnace was created in ANSYS Fluent 15. The model includes the modeling of combustion, radiation, turbulence as well as slab movement and scale build-up on the slab surface. The results from the simulation and a drag test were then compared, and it can be seen that the model is able to correctly map the heat flows in the furnace. Subsequently, the developed model was used to determine the causes of temperature inhomogeneities in the slabs. Using a series of simulations, it was then possible to find a furnace program that enabled a significant reduction in temperature differences in the slabs by shifting the output of the burners.


In the steel industry, the walking beam furnace is used to heat up and reheat steel slabs for the rolling process. In order to meet the demands for high quality steel, a very homogeneous temperature profile in the slabs is required. Using Computational Fluid Dynamics (CFD) in ANSYS Fluent 15, a walking beam furnace has been investigated to analyze the causes of inhomogeneous temperature distribution in the steel slabs. The model includes the description of combustion, radiation turbulence as well as slab movement and scale formation on the steel. The results out of a simulation and a towing test have been compared and it has turned out that the model is able to predict the heat fluxes in the furnace very precisely. Next the model has been used to find the reasons of inhomogeneous temperature distribution in the steel slabs. With a certain number of simulations, a furnace program has been found which reduces the temperature differences in the steel slabs just by adjusting the power of the burners.


A homogeneous temperature distribution in the slabs during the rolling process is of great importance for the production of high-quality steel sheets. In order to guarantee this homogeneity or to enable it at all, a very precise knowledge of the heat flows in the upstream process unit, the walking beam furnace, is required. As part of this work, a numerical flow simulation (CFD model) was created for a furnace of this type. In addition to the beam movement, the model developed takes into account, among other things, the effects of radiation and heat losses on the furnace walls as well as the insulating effect of the beam scaling. Due to the high temperatures in the furnace chamber, exact temperature measurements of the furnace atmosphere are very difficult and, if so, can only be carried out selectively and with great effort. With the help of the developed mathematical model, it is now possible to calculate the heat flows in the entire furnace unit and to carry out case studies without affecting ongoing operation of the furnace.

Model description

Furnace geometry

The walking beam furnace shown in the model consists of 48 burners, which are operated with top gas and natural gas. Since the furnace is operated symmetrically, i.e. the burners on opposite sides have the same output, only one half of the furnace was simulated. This reduces the number of cells in the overall model from 440 million tetrahedra to 220 million. In order to prevent opposing burners from influencing each other, a test case was created prior to creating the geometry, which shows the mutual flow deflection of two opposing burners. It was found that, due to the large distance of 12 meters, there is no mutual influence of the burner and the use of a plane of symmetry is therefore possible and legitimate.

Thermal resistance model of the wall

The structure of the furnace wall is quite complex. This consists of several layers of different refractory materials, some of which themselves have a high dependence of the thermal conductivity on the temperature. The furnace wall also exchanges radiation with the furnace space atmosphere and plays a major role in the introduction of heat into the slabs. In order to enable a precise temperature prediction, these aspects must be taken into account. For this purpose, a thermal conductivity model of the furnace wall was developed, which is implemented efficiently. With the help of Matlab, the furnace wall, as shown in Fig. 1, was discredited one-dimensionally.

The resolution took place in 100 cells, and the heat conduction equation was iteratively solved using finite differences in Matlab. The temperature-dependent thermal conductivity in each cell was taken into account. In addition to the specification of the layer structure, the surface temperature of the surroundings was set as constant boundary conditions for the calculation. The temperature of the furnace chamber atmosphere was then gradually increased and the heat flow through the wall structure was calculated. A polynomial can be generated from this data, which describes the heat flow through the furnace wall as a function of the furnace wall temperature in the interior, taking into account the wall structure and the temperature-dependent thermal conductivity of the refractory materials used. With the help of a UDF (User Defined Function), this heat flow is calculated at each furnace wall cell of the simulation and taken into account in the energy balance.

Determination of the emission coefficients of the refractory materials and the steel slab

As the results show, approx. 92% of the energy introduced into the slab is transferred in the form of thermal radiation and only 8% by convection. Therefore, the exact knowledge of the emission coefficients of the refractory materials used and the steel slab is of particular importance. As part of this work, the values ​​were determined in the laboratory. The sample was drilled and one thermocouple was placed in the middle of the sample and one in the furnace chamber. A thermogram was recorded through a small hole in the furnace using a thermal imaging camera, which enables conclusions to be drawn about the emission coefficient based on the temperatures of the thermocouples. The measurement was carried out in steps from 300 to 1100 ° C in 200 ° C steps. After changing the temperature at the furnace, it was ensured that the temperature of the middle of the sample and the temperature of the furnace chamber were matched. Only in this way can it be assumed that the surface temperature of the sample also corresponds to the core temperature, that there is no more heat flow over the surface and that a steady state is present. There is a slightly increasing trend in the emission coefficients of all samples measured from 0.90 at 300 ° C to 0.95 at 1100 ° C.

Slab movement

A so-called MRF (Multi Reference Frame) is used to take the slab movement into account in the model. In the area of ​​the slab movement, this adds a convective part to the static network, which represents the movement of the slabs. In contrast to a dynamic network, you can avoid problems caused by constant re-networking and numerical instabilities while maintaining the same accuracy.

Turbulence modeling

The description of the turbulence is very important for modeling the gas phase kinetics. In this case the realizable k ‑ ε model was used. As suggested in the literature, the constant C was used for a better description of the flame acceleration reduced from 1.9 to 1.8 [1, 2]. An isotropic turbulent viscosity was used in the model. Thus the turbulence model consists of two transport equations, the k-equation for the turbulent kinetic energy of the eddies in the flow and the ε-equation for the dissipation of k.

Combustion modeling

The flamelet model is the most efficient solution for describing the combustion in a model size of this furnace [3]. The entire combustion chemistry is solved before the actual simulation and stored in tables. For this purpose, a one-dimensional countercurrent diffusion flame is released with the help of a detailed chemical model and the so-called flamelets are generated. Then the PDF (Probability Density Function) integration takes place in order to take the turbulence fluctuations into account. The data generated in the process are stored in the PDF table, which contains the tabulated combustion values. This data includes, but is not limited to, reaction rates, heat of combustion, and species information. On the basis of the molecular composition of the gas phase, the temperature and turbulence, the substance conversions and heat tones are read from the PDF table for each cell. In contrast to a continuous calculation of the detailed chemistry, this accelerates the calculation by a factor of 100.

Radiation modeling

The Discrete Ordinate Model (DOM) in combination with the Weighted Sum of Gray Gases Model (WSGGM) was used to describe the heat flow through radiation [4, 5]. In combination, these two models enable a very precise description of the radiation. The DOM forms a certain number of rays from each edge cell into the hemispherical space and describes the radiation transport in the geometry, taking shadowing effects into account. The WSGGM forms an emission and absorption coefficient that depends on the gas mixture and is required for radiation transport.

Scaling model

The scaling of the slabs plays a very important role in the transport of heat. The formation of scale on the slab surface forms a layer with poor thermal conductivity. This increasingly insulates the slab and reduces the heat input into the steel. The scale formation is described using a model from the literature [6]. The model describes the build-up of scale on the slab over time and takes into account the transport resistance of iron through the scale layer. The model requires the oxygen concentration on the steel surface, the temperature of the steel and the material data of the gas phase and the steel. The scale build-up is calculated using a stationary 1D model using finite differences. The thickness of the scale layer is the only transport variable that is transported at the slab speed. The mass transfer coefficients and the oxygen concentration on the slab surface are read out from the CFD model. The temperature of the slabs is either used from the measurement data for the drag test or iteratively determined with the simulation data in the event that no slab temperature is available. The data are averaged over the slab width and discredited in the correct manner of the slab movement. This creates the model that is one-dimensionally discretized in the feed direction. At each node the reaction rate for the scale build-up is calculated and added to the transport variable scale thickness. The thickness of the scale is transported through the furnace and, as a result, provides the thickness of the scale layer on the slab, which is dependent on the furnace coordinate. When calculating the thickness of the scale layer, the diffusion control of iron through the scale layer and the diffusion control of oxygen to the surface are taken into account. The diffusion of iron in the scale is given by Eq. 1 determined from the literature [6].

$$ D \ left (T \ right) = 1,21e ^ {- \ frac {1,24E5} {RT}} $$

The increase in scale is calculated using a stationary diffusion approach based on Eq. 2 calculated. In order to avoid division by zero, a negligible scale thickness of 1 µm is specified as a starting value at the furnace entry.

$$ \ frac {ds} {dt} = \ frac {D \ left (T \ right)} {s + \ text {small}} \ frac {MM _ {\ text {Tinder}}} {\ rho _ {\ text {Tinder}}} $$

But not all of the scale remains on the slab. During furnace operation, for example, it can be seen that the scale flakes off in the last 20% of the furnace length and then comes to rest on the furnace floor. The build-up of scale is therefore stopped from this furnace position and it is assumed that the newly created scale will flake off the slab.

A polynomial is then formed from this data, which describes the thermal resistance of the scale layer as a function of the furnace coordinate. In addition to the thickness of the scale layer, the temperature-dependent thermal conductivity of the scale is also taken into account.

Model validation

To validate the model, a comparison with a tow test was carried out. For this purpose, a measuring slab, which is provided with thermocouples, was transported through the furnace and the temperature profile of the slab and the gas atmosphere were recorded. A spatially resolved temperature profile can be created using the feed speed and the temperature recording over time. Fig. 2 shows this comparison between the simulation and the towing test. It turns out that a precise prediction of the temperature profile is possible by means of simulation and that the model can be used for further case studies.


After the successful model validation, the model was used to study the causes of inhomogeneities in the slab temperature distribution. It can be seen in operation that a sagging temperature profile is created, especially with high slab throughput and high furnace performance. This means that the steel in the middle of the furnace is cooler than on the edge. This behavior should be avoided, so it was carefully examined using a CFD model. It turned out that the cause of the temperature differences lies in the mode of operation and the flow profiles of the burners and that the thermal radiation is responsible for the overheating at the edge. Fig. 3 shows this undesirable burner behavior very well. In the lower burner level, an almost even heat input is made possible up to the middle of the slab (right edge of the picture), since the radiation intensity below the slab is almost evenly distributed over the entire width. The behavior on the upper side of the slab is different, which is responsible for the higher temperature at the edge of the slab (left). A hotspot with a high thermal radiation density forms between the first and second bars in front of the burner. This heat is transferred directly to the edge of the slab and heats it up more strongly there than in the zones with lower radiation density further to the right.

As part of the study, various burner programs were tested and the effects on slab heating were investigated. Ultimately, it was possible to find a furnace program by shifting the load and adjusting the output of the burners very precisely, which avoids the formation of hotspots and enables a more homogeneous temperature profile. It is important to note that the burners must never be viewed individually. The analyzes show that neighboring burners definitely influence each other and changes in one burner level also lead to changes in other burner levels. Therefore, a holistic view of the furnace is absolutely necessary in order to be able to make precise statements.


In order to study the cause of inhomogeneities in the temperature distribution of slabs, a detailed furnace model for the walking beam furnace was created. A detailed description of turbulence, combustion, radiation and slab movement as well as scaling are implemented in this CFD model. The development of the model showed that a precise temperature prediction is only possible if all these factors are taken into account. After successful model validation based on a drag test, the model was used to study various power settings on the burners and their effects on the slab heating. It was shown that the formation of hot spots in the burners resulted in very different power inputs into the slab at different locations. By changing the furnace program and alternative power settings on the burners, these effects could be avoided, which leads to a significantly more uniform slab temperature at the furnace discharge.


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Author information


  1. K1-MET GmbH, Stahlstrasse 14, 4020, Linz, Austria

    Dipl.-Ing., Dipl.-Ing. Werner Pollhammer BSc

  2. Chair for Thermal Process Technology, Montanuniversität Leoben, Franz Josef-Strasse 18, 8700, Leoben, Austria

    Dipl.-Ing., Dipl.-Ing. Werner Pollhammer BSc, Christoph Spijker & Harald Raupenstrauch

  3. voestalpine Stahl GmbH, voestalpine-Strasse 3, 4020, Linz, Austria

    Jakob Six & Daniel Zoglauer

Corresponding author

Correspondence to Dipl.-Ing., Dipl.-Ing. Werner Pollhammer BSc.

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Pollhammer, W., Spijker, C., Six, J. et al. Mathematical model for the heating of the slab in the walking beam furnace. Mountain Huettenmaenn monthly165, 315-319 (2020).

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  • Walking beam furnace
  • CFD model
  • simulation
  • Slab heating


  • Walking beam furnace
  • CFD
  • Model
  • simulation
  • Steel slabs heating