A Near-Process 2D Heat-Transfer Model for Continuous Slab Casting of Steel

Market requirements on steel products with the highest surface and internal quality demand stimulate a systematic control of the steel solidification behavior during continuous casting (CC). Computational process modeling is increasingly applied to optimize casting practices and calibrate the soft reduction to guarantee the required product quality. Here in this study, an overview of m2CAST as a “development platform” for the CC process is presented. This platform consists of a numerical heat‐transfer model, considers results of laboratory experiments in the calculations, for example, thermal analysis and nozzle measuring stand (NMS), and provides the option to use relevant process data. Two case studies on a continuous slab caster at voestalpine Stahl Linz GmbH are investigated. In doing so, thermal boundary conditions obtained by the NMS are implemented, and the simulation trials are validated with temperature measurements of the dragged thermocouple method installed during the casting process. The temperature distribution over the strand width is measured additionally with two pyrometers placed in the straightening zone. Excellent agreement between the calculated strand surface temperature and the measured temperature is obtained. Furthermore, the relevance of considering the roller‐bearing areas in defining the boundary conditions is presented to accurately predict the shape of the crater end.

DOI: 10.1002/srin.202200089 Market requirements on steel products with the highest surface and internal quality demand stimulate a systematic control of the steel solidification behavior during continuous casting (CC). Computational process modeling is increasingly applied to optimize casting practices and calibrate the soft reduction to guarantee the required product quality. Here in this study, an overview of m 2 CAST as a "development platform" for the CC process is presented. This platform consists of a numerical heat-transfer model, considers results of laboratory experiments in the calculations, for example, thermal analysis and nozzle measuring stand (NMS), and provides the option to use relevant process data. Two case studies on a continuous slab caster at voestalpine Stahl Linz GmbH are investigated. In doing so, thermal boundary conditions obtained by the NMS are implemented, and the simulation trials are validated with temperature measurements of the dragged thermocouple method installed during the casting process. The temperature distribution over the strand width is measured additionally with two pyrometers placed in the straightening zone. Excellent agreement between the calculated strand surface temperature and the measured temperature is obtained. Furthermore, the relevance of considering the roller-bearing areas in defining the boundary conditions is presented to accurately predict the shape of the crater end.
The mold's thermal boundary conditions and secondary cooling conditions are given as temperature-dependent heat-transfer coefficients (HTC). The selected nozzle parameters in the secondary cooling zone of the casting machine are characterized in the nozzle measuring stand (NMS), [9] providing information on the water impact density (WID) and the resulting HTC in the spray zone. A 2D finite volume (FV) calculates the temperature distribution over the whole strand length. Only half of the slab's width is calculated by assuming geometrical symmetry. However, this approach enables different HTC and cooling strategies on the fixed and loose sides of the strand to be taken into account.
The calculated thermal history then represents, for example, the basis for the experimental simulation of surface defect formation in the in situ materials characterization by bending (IMC-B) test, [10] the in situ observation of grain growth and phase transformations by high-temperature laser scanning confocal microscopy (HT-LSCM), [5,11] or the simulation of surface oxidation by thermal gravimetry. [12] The experimental results indicate if the formation of defects in the product for the given process conditions is likely or not.
Finally, the results are reversely transferred to the model and used in computationally adjusting the process parameters, for example, cooling strategy, to avoid internal or surface defects in the process. The solver behind m 2 CAST is a 2D-FV solution of the enthalpy formulation of the heat conduction equation (Equation (1)).
Here, ρ denotes the steel density, H is the enthalpy, T is the temperature, and κ is the thermal conductivity of steel. The effective heat capacity method is applied (Equation (2)) to account for the heat released during the solidification of the steel.
Here, C p eff denotes the effective specific heat (Equation (3)).
C p denotes the heat capacity, L H is the latent heat of fusion, and f L is the liquid fraction. The effective heat capacity includes the latent heat, the heat of transformation of the peritectic reaction, and the ferrite to austenite transformation. The resulting C p eff peaks may cause convergence problems. The integration of C p eff around the peaks helps heat conservation and stabilizes the solver. [13] Finally, this system is solved by the alternating direction implicit (ADI) method described in refs. [2,14] Splitting each computational step into semi-implicit time steps helps reduce the required CPU time. The implicit time half-steps make an iteration to adjust the temperature-dependent physical properties necessary. An additional improvement in the convergence is achieved by implementing an empirical relaxation factor ω, according to Equation (4).
The index "n" denotes the time step of the iteration, T n Calc stands for the calculated temperature and T n Iter for the iterated temperature. Due to these convergence and stabilization Figure 1. The basic structure of the "casting development platform" m 2 CAST includes preprocessing, heat-transfer solver, and post-processing modules. Reproduced with permission. [7] Copyright 2021, ASMET. interventions, a fixed grid with a minimum size of a few millimeters and a nevertheless good calculation time in the order of magnitude of minutes (30-40 min) become possible.

Thermal Boundary Conditions: Mold and Secondary Cooling Zone
For the modeling of heat removal along with the mold/strand interface, a temperature-dependent local HTC is similarly applied, as proposed by Hietanen et al. [15] and Preuler et al. [16] The temperature of the mold is assumed to be 150°C. The total heat withdrawal in the mold is adjusted to the measured integral heat flux.
In the case of secondary cooling, four zones of heat removal can be distinguished, [16] shown in Figure 2 and summarized in Table 1: The heat removal at the strand/roll contact is defined with an HTC of 700 Wm À1 K À1 and a roll temperature of 150°C; [17] the contact length is fixed at 20 mm. [17] Bearing blocks interrupt the guiding rolls, and in these areas, heat removal is currently simplified as radiation only. The HTC caused by the spray/strand interaction is measured by the NMS, [9] described in Section 3.2. The air/mist sprays also influence the heat removal between the direct strand/spray contact zone and roll/strand contact. Depending on their position, these zones are called the "pre-cone zone" and the "post-cone zone." It must be noted that the HTC in these zones depends on the amount of water per spray nozzle V (H 2 O). Heat removal in the pre-spray cone area is influenced by convection and radiation, whereas the post-spray zone accounts for convection, the so-called "pocketwater" and radiation.

Material Data: Microsegregation, High-Temperature Phase Transformations and Thermophysical Properties
The present simulation platform includes a material module (m 2 MAT) to calculate solidification temperatures and hightemperature phase transformations and to estimate the temperature-dependent material properties of steel, which is required to solve the heat conduction equation (Equation (1)). m 2 MAT comprises an analytical microsegregation model to characterize the solidification under nonequilibrium conditions. The calculation principle can be summarized as follows. [18] As input data, the definition of the initial chemical composition of the steel grade and the cooling rate is mandatory. The algorithm is based on Ohnaka's approach, [19] later modified to a semi-integrated form by You et al. [20] The concentration of the residual liquid (C L ) is calculated by step-wisely increasing the solid fraction ( f S ) according to Equation (5). [20] In Equation 5, k denotes the equilibrium partition coefficient of an alloying element between the liquid and the solid solution. The value of k is directly related to the respective phase diagram. It is defined by the ratio of the concentration of the alloying element in the solid phase and the liquid phase (k ¼ C S /C L ). k γ/L and k δ/L were adjusted to DSC measurements of binary phase diagrams, for example, Fe-P, [4] and ternary Fe-C-X (X ¼ C, Si, Mn, P, …) systems, [3,4] as well as to results of extensive thermodynamic calculations using FactSage 8.0 software and its databases. [8] Δf S is the finite increase of f S during solidification, and Γ is related to the back diffusion tendency of alloying elements from the liquid to the solid phase. Γ is given for hexagonally shaped domains by Equation 6. [19] Figure 2. a) Different heat removal zones assumed along the strand axis and b) symbolic representation of contact zones involved in a secondary cooling zone (1 ¼ contact strand/roll, 2 ¼ pre-cone zone, 3 ¼ spray cone zone, 4 ¼ post-cone zone). Reproduced with permission. [7] Copyright 2021, ASMET. 2 Pre-spray cone The length depends on the spray cone Heat-transfer coefficient: Radiation, emissivity: 0.84 3 Spray cone According to measurement or formula 4 Post-spray cone (pocket water) First 10 m inner bow and first 1 m outer bow The length depends on the slope of the strand surface Heat-transfer coefficient: In Equation 6, α is the dimensionless Fourier number, defined in Equation (7): D S is the diffusion coefficient of a solute element in the solid phase; t f is the local solidification time. The characteristic length x is usually represented by the half of the secondary dendrite arm spacing (x ¼ λ 2 /2). For α ! 0, the Ohnaka model reduces to the Scheil-Gulliver equation [21,22] with no back diffusion of a solute element from the liquid to the solid; if α ! ∞, the lever rule applies.
The temperature dependence of solute diffusion coefficients follows an Arrhenius approach (Equation (8)), where R ¼ 8.314 J mol À1 K À1 is the gas constant and T is the temperature in kelvin.
All currently used values for partition coefficients of the alloying elements and the diffusion parameters (Q and D 0 ) can be found in the work of M. Bernhard. [18] Coarsening of the dendritic microstructure during solidification is considered by the relation between t f and λ 2 as published by Pierer and Bernhard (Equation (9)). [23] Note that %C C,0 is the initial carbon content in mass percent.
The actual temperature at the corresponding solid fraction is obtained by inserting the interdendritic concentration in mass percent into analytical liquidus equations. [24,25] Estimating the start of the peritectic transformation during solidification is essential for steels with compositions inside the peritectic range. Below this temperature, the transformation from liquid to γ-Fe and δ-Fe to γ-Fe is mainly controlled by the austenite layer formed between the ferrite and liquid phases. The change of k δ/L to k γ/L significantly influences the microsegregation and hence, the calculated solidus temperature. In m 2 MAT, a simple but thermodynamically consistent strategy is employed: The two separated equations for the liquidus temperature (T L ) of ferrite and austenite are used according to Miettinen and Howe [24] and slightly modified by M. Bernhard et al. [25] The T L of both phases, γ-Fe and δ-Fe, is calculated from C L ( f S ) within the calculation procedure. The phase with the higher value of T L is considered the primary phase stable at the current temperature or solid fraction, respectively. If the argument T L δ (C L , f S ) < T L γ (C L , f S ) is valid during freezing, the peritectic reaction starts. Finally, the model accurately predicts liquidus, solidus (T S ), and peritectic (T P ) temperatures, considering the formation of nonmetallic compounds, for example, MnS and TiN. Nonmetallic inclusions will form if the interdendritic concentration exceeds the temperature-dependent solubility product. [26,27] Selected examples of improving the microsegregation model by laboratory techniques will be presented in Section 3.1.
A simple austenite decomposition (ADC) module simulates the phase transformations below 1000°C. First, the isothermal time-temperature-transformation (TTT) diagram is calculated for ferrite, pearlite, bainite, and martensite. [28] To obtain the continuous cooling diagram (CCT), the TTT is transformed by the Scheil-Avrami, or additivity, rule. [28] At this stage, all thermophysical material data is calculated as a function of steel composition and cooling rate: thermal conductivity and density, [29] effective heat capacity, [8,30] and phase fractions of solid, liquid, ferrite, austenite, and pearlite versus temperature. Please note that within the current CCT calculation, the cooling rate is fixed at 0.1°Cs À1 . The bainite and martensite fractions are neglected in the case of very slow surface cooling in the casting process. Only fractions of ferrite and pearlite are considered depending on their multicomponent equilibrium phase fractions. [28] 3. Improvements of Metallurgical Models by Selected Laboratory Techniques

DSC: Thermodynamics and Phase Diagrams
Thermal analysis techniques, such as heat-flux DSC or differential thermal analysis (DTA), [31] are valuable tools to measure phase diagrams of steel [4,5] and observe phase stabilities in the peritectic range. [3,4] The present chapter briefly introduces the strategy of incorporating DSC data and thermodynamic optimizations into the microsegregation module of m 2 MAT, taking the example of the determination of the equilibrium partition coefficient of phosphorus between liquid/ferrite and liquid/austenite.
In a recent study, [4] some of the authors experimentally reassessed the binary Fe-P and ternary Fe-C-P systems using the DSC method. The focus was placed on high-temperature phase equilibria in the Fe-rich region. The new data was then utilized to thermodynamically optimize the Fe-P and Fe-C-P systems within the CALPHAD (calculation of phase diagrams) framework. [6] Figure 3a shows the binary Fe-P phase diagram calculated up to 12 mass percent according to M. Bernhard et al. [6] using the modified quasichemical model (MQM) [32,33] for the liquid phase. Experimental data from literature [34][35][36][37][38][39][40][41][42][43][44][45][46][47] is plotted for comparison. The system consists of the liquid phase, ferrite, austenite, and the Fe 3 P stoichiometric compound in this composition range. A eutectic phase equilibrium exists between ferrite, liquid, and Fe 3 P; ferrite and austenite form a closed "γ-loop" in the solid state.
Concerning the relevant solidus and liquidus lines, the most recent data of Morita and Tanaka, [47] obtained from equilibration experiments and electron probe microanalysis (EPMA), is highlighted together with the DSC results. [4] Both datasets are generally in reasonable agreement over the whole composition range investigated. In the present study, the DSC data are considered more accurate [4] as in the data of Morita and Tanaka, [47] the measured P content in the ferrite and liquid phases shows increased deviations.
Detailed discussion on the partition coefficient of P between liquid/ferrite have been reported in previous experimental work, [47][48][49][50] typically studied through zone-melting techniques [51] www.advancedsciencenews.com l www.steel-research.de steel research int. 2022, 93, 2200089 and equilibration trials. [47] However, based on the newly gained information on the Fe-P phase diagram, [4] it is possible to accurately validate the partition coefficient k P δ over the whole temperature range. By considering the presently reconstructed progress of the liquidus and solidus lines, the final results are shown in Figure 3b; the proposed values of Morita and Tanaka [47] are plotted for comparison. The melting temperature of pure Fe (1538°C) represents the dilute solution, whereas the lower limit of 1100°C is already close to the eutectic temperature in the binary Fe-P phase diagram. Since the solidus and liquidus lines in Figure 3a do not decrease perfectly linearly, the partition coefficients of P depend on the temperature. Again, the present results of k P δ correlate well with the data of Morita and Tanaka, [47] with a slightly higher value of k P δ ¼ 0.17 at infinite dilution; below 1350°C both datasets become very similar (k P δ % 0.25).
In m 2 MAT, the value of k P δ ¼ 0.18 is set as constant in the microsegregation module, which agrees excellently with the data of zone melting experiments carried out by Fischer et al. [49,50] (k P δ ¼ 0.18, [49] k P δ ¼ 0.16 AE 0.04 [50] ). As the value of k P γ in binary Fe-P alloys cannot be directly derived from the phase diagram data or zone-melting experiments, k P γ was estimated using thermodynamic software FactSage 8.0 [8] and the CALPHAD optimization of M. Bernhard et al. [6] The ferrite phase was suppressed in the calculations. The final partition coefficient of P between liquid and austenite is k P γ ¼ 0.12. In contrast to k P δ , k P γ does not show a pronounced temperature dependence. Again, the value obtained is compared with the experimental data of Morita and Tanaka. [47] It has to be noted that in their study, a significant amount of C was added (> 2 mass percent) to stabilize the austenite phase up to the liquid/solid phase equilibrium. Their recalculated value for the dilute solution was given by k P γ ¼ 0.06. [47] Compared to the present result of k P γ ¼ 0.12, the low value enhances the interdendritic enrichment of P in the microsegregation calculation. It will lead to a significant drop in the model's solidus temperature.

NMS: WID and Local HTC
To characterize the local heat transfer caused by the impingement of the air/mist spray on the strand surface, Montanuniversitaet Leoben developed a NMS. Details on the testing setup and the measurement procedures can be found in previous work. [9] It should be noted that the secondary cooling zone of a casting machine is divided into subzones. The total amount of cooling water is adjusted according to so-called "cooling tables" and individual types of nozzles are in use for each cooling zone. The characterization of the thermal boundary condition in a casting machine requires measuring every kind of nozzle for a wide variety of parameters (water flow rate, air pressure, distance from surface).
The setup for the determination of the 2D water distribution is shown in Figure 4. [9] The nozzles are installed inside an experimental chamber at a defined distance to a measuring grid. The grid consists of 7 Â 100 cells with a 10 Â 10 mm inlet area. The number of cells can be changed according to the spray dimensions. While the nozzle is operated at specific flow rates and pressures, these cells collect the water. After a defined period, the measuring grid is removed, and the WID is determined using image processing. [9,52] Each experiment is performed with two nozzles by including effects of spray cone overlapping on the water distribution.
The nozzles are reversely positioned to measure the HTC; see Figure 5a. [9] A movable and instrumented steel cylinder is placed on the top of the cooling chamber, heated up by an inductive heating unit. The temperature is limited to 1200°C due to the melting point of the heat-resistant, austenitic sample material. Inside the specimen, thermocouples record the change in temperature at different distances to the surface (Figure 5b), [9] enabling the calculation of the HTC by an inverse heat conduction model. The cylinder moves through the spray with adjustable speed and preselected x-position. The measurements are usually carried out in different positions along the spray axis to achieve the 2D view of the cooling characteristic of a nozzle. [9,52] Figure 3. a) Binary Fe-P system up to 12 mass percent P according to M. Bernhard et al. [6] Results of previous experimental work are plotted as a comparison. [34][35][36][37][38][39][40][41][42][43][44][45][46][47] The most recent data of Morita and Tanaka [46] and M. Bernhard et al. [4] is highlighted. Adapted under the terms of the Creative Commons CC BY license 4.0. b) Partition coefficients of P between ferrite/liquid and austenite/liquid in the temperature range of 1100-1535°C.

Plant Measurements: Dragged Thermocouple Method and Traverse Pyrometers
The measurement of the slab surface temperature along the casting direction was carried out using a technique known as "dragged thermocouples." In the process, a guide tube is installed through the spray chamber. The thermocouple is inserted into the guide tube at a distance of about 4 m below the mold level. The plate is rolled into the strand surface after the first contact with the support roller. Subsequently, the strand pulls the embedded thermocouple along the way through the caster. A high-speed computer records the temperature. Figure 6a [7] schematically shows the first roll-in of the thermocouple plate by the support roller and Figure 6b [7] shows the wire on the slab surface on the roll-out table.   During straightening, the slab surface temperature is a critical parameter regarding the formation of surface cracks and final product quality. Two pyrometers are installed permanently on both sides of the strand on modern casting machines to control the temperature distribution along the broad face. During the casting process, the instruments continuously scan the slab surface from the narrow side over the edges in the direction of the center.

Case Studies and Discussion of Simulation Results
The case studies investigated (chemical composition, casting parameters, and slab geometries) are summarized in Table 2. Please note that the steel grade in case II is alloyed with minor amounts of Cr, Ni, and Mo (<0.30 mass percent). Both steel grades were cast on the same continuous slab caster but with different thickness s (225 and 355 mm), width w (1640 and 1760 mm), and casting speed v c (1.30 and 0.70 m min À1 ). A very soft secondary cooling strategy (A and B) was selected in both cases. Figure 7 graphically summarizes the calculation results using the m 2 MAT module; note that a local cooling rate of 1°C s À1 was assumed. In Figure 7a, the fractions obtained of the liquid phase, ferrite (δ-Fe) and austenite (γ-Fe) are plotted versus temperature. At T L ¼ 1529.8°C, the first fraction of ferrite forms. During further cooling, the solidification to ferrite progresses and is completed by reaching the solidus temperature T S ¼ 1488.4°C. A narrow, single ferrite region exists in the temperature range  of 1488.4-1483.9°C, before the solid/solid phase transformation of δ-Fe to γ-Fe starts. The δ-Fe/γ-Fe transformation terminates at 1441.2°C. The phase fractions calculated indicate that the chemical composition of steel grade I is very close to the critical hypoperitectic range in the multicomponent Fe-C system. Even small changes in the alloying content may cause typical peritectic behavior in the casting process. Possible consequences are the nonuniform heat withdrawal and shell formation during the initial solidification process, pronounced oscillation marks, or, in the worst case, a breakout if the casting parameters are not adequately adjusted. [53] Figure 7b-d shows the temperature dependence of the effective heat capacity, density, and thermal conductivity in 1600-1200°C. In the case of the heat capacity (Figure 7b), two separated deviations from the baseline are visible, which correspond to the heat released during solidification (liquid ! δ-Fe) and the solid/solid phase transformation (δ-Fe ! γ-Fe). Figure 8 shows the local WID distribution for each cooling zone of the casting machine (cooling strategy A). The NMS characterized the spraying patterns considering the adjusted nozzle parameters (water flow rate and air pressure) in nine zones. The WID is highest close to the mold exit (zone I) with values up to 30 kgm À2 s À1 and continuously decreases in the subsequent cooling zones. In zones II-IX, the nozzle operation parameters are typically adjusted to 0.5-6 L min À1 water flow rate and air pressure of P > 1 bar; the maximum WID varies in the range of 0.30-7 kgm À2 s À1 . Similarly, the measured local HTCs were, along with radiation, applied as thermal boundary conditions.

Simulation Results and Discussion: Case I
In Figure 9a, the calculated strand surface temperature on the loose side (middle of broad face) for case I is compared with two independently performed thermocouple measurements. The numerical simulation fits very well with the experimentally determined temperatures. In the 7.5-12 m strand length range, the simulated profile is partly lower, indicating a maximum deviation of ΔT % 20°C. Excellent agreement was found from 12 to 20 m below the meniscus. In the final section (>20 m), the supporting rollers are cooled softly by water under real casting   conditions to guarantee a long lifetime of the material. In m 2 CAST, only radiation is assumed as boundary condition. This simplified approach leads to a slight reheating of the slab surface and, consequently, to higher deviations between the calculation results and the plant data.
The thermocouple data makes it possible to roughly estimate the strand/roll contact by analyzing the steep decrease in the measured surface temperature when the thermocouple passes the rolls. [17] The strand/roll contact for case I varies in the range of 20-25 mm, confirming that the contact length defined in the thermal boundary conditions of m 2 CAST (see Table 1) is reasonable.
The crater end calculated for case I is shown in Figure 9b, plotted as a slice through the strand centerline between the broad faces. The contour of the final solidification point corresponds to f L ¼ 0 and is not uniform over the slab width. The formation of a maximum and minimum point of solidification can be assigned to the lower heat transfer in the area of the bearing blocks. The missing strand/roll contact and the absence of pocket water in these regions lead to a reduced solidification rate apart from the slab axis. The maximum solidification length is obtained at a distance of 400 mm from the slab center with a value of 24.65 m, whereas in the center, the slab solidifies earlier at 23.80 m. Therefore, the difference between the maximum and minimum final solidification point is 0.85 m and has to be considered mainly for improving the internal quality by soft reduction positioning.

Simulation Results and Discussion: Case II
For case II, the corresponding material data and WID profiles were similarly calculated as demonstrated in Section 4.2.1. The slab's temperature measurements on both sides, loose and fixed side, were available in case II. Due to the specially adjusted cooling strategy in the first cooling zones, the temperatures measured on the loose and fixed sides slightly differ between 5 and 10 m from the meniscus. The comparison with the calculation results is shown in Figure 10a,b, respectively. For both datasets, excellent agreement is obtained with the m 2 CAST simulation. Similar to case I, a temperature drop on the loose side is measured at 20 m strand length. The slab/roll contact shows a mean value of 22 mm, which is in reasonable agreement with case I. The transversely measured temperatures by the pyrometers are plotted in Figure 10c along with the results of m 2 CAST at 18.60 m strand length. Both datasets are perfectly matched at the slab corner (T ¼ 750°C) and for the reheating at the narrow side. m 2 CAST predicts slightly higher temperatures in the middle of the broad face (970°C) compared to the measurements (%960°C). However, the slight deviation is acceptable for optical measurements as the surface scale will have a minor impact on the accuracy of the experimental data. The enlarged crater end section can be seen in Figure 10d. Due to the different slab geometry (thickness and width) and the cooling program selected, the solidus isotherm is shaped differently as in the case I; see Figure 9b.The plotted points of final solidification (33.75 and 33.0 m) were confirmed by the industrially observed values of 33.22 and 32.50 m. The results clearly show that the bearing blocks have to be considered in the numerical simulation as the reduced heat transfer in this region will influence the shape of the crater end. In the present study, the simplified boundary condition of radiation demonstrates the first step to reproduce real situations in this region.

Conclusion and Outlook
This work introduces the 2D heat-transfer model "m 2 CAST" for the continuous slab casting of steel. In addition to the algorithm, implementing laboratory results, for example, local HTCs, thermodynamics, and phase diagrams, are described. Two case studies on a continuous slab caster at voestalpine Stahl Linz GmbH were investigated to validate the model. Temperature measurements were performed at the slab surface with the dragged thermocouple method and two transverse pyrometers. The industrial data is in excellent agreement with the results calculated using m 2 CAST. Potential applications of m 2 CAST are predicting local hot spots on the strand surface and improving certain process parameters in the secondary cooling system to guarantee a more uniformly distributed temperature profile.
Future research is intended particularly to focus on the following: 1) the modeling of the heat removal along with the mold/ strand surface, 2) the heat removal in the region of the rolls bearing blocks, and 3) the further speedup of the CPU time for calculating the temperature field.