Abstract Several numerical methods and algorithms, for solving the mathematical model of a continuous casting process, are presented, and theoretically studied, in this work. The numerical algorithms can be divided in to three different groups: the Schwarz type overlapping methods, the nonoverlapping Splitting iterative methods, and the Predictor-Corrector type nonoverlapping methods. These algorithms are all so-called parallel algorithms i.e., they are highly suitable for parallel computers. Multiplicative, additive Schwarz alternating method and two asynchronous domain decomposition methods, which appear to be a two-stage Schwarz alternating algorithms, are theoretically and numerically studied. Unique solvability of the fully implicit and semi-implicit finite difference schemes as well as monotone dependence of the solution on the right-hand side are proved. Geometric rate of convergence for the iterative methods is investigated. Splitting iterative methods for the sum of maximal monotone and single-valued monotone operators in a finite-dimensional space are studied. Convergence, rate of convergence and optimal iterative parameters are derived. A two-stage iterative method with inner iterations is analyzed in the case when both operators are linear, self-adjoint and positive definite. Several new finite-difference schemes for a nonlinear convection-diffusion problem are constructed and numerically studied. These schemes are constructed on the basis of non-overlapping domain decomposition and predictor-corrector approach. Different non-overlapping decompositions of a domain, with cross-points and angles, schemes with grid refinement in time in some subdomains, are used. All proposed algorithms are extensively numerically tested and are founded stable and accurate under natural assumptions for time and space grid steps. The advantages and disadvantages of the numerical methods are clearly seen in the numerical examples. All of the algorithms presented are quite easy and straight forward, from an implementation point of view. The speedups show that splitting iterative method can be parallelized better than multiplicative or additive Schwarz alternating method. The numerical examples show that the multidecomposition method is a very effective numerical method for solving the continuous casting problem. The idea of dividing the subdomains to smaller subdomains seems to be very beneficial and profitable. The advantages of multidecomposition methods over other methods is obvious. Multidecomposition methods are extremely quick, while being just as accurate as other methods. The numerical results for one processor seem to be very promising.

Abstract In today's world, enormous amounts of data are being collected everyday. Thus, the problems of storing, handling, and utilizing the data are faced constantly. As the human mind itself can no longer interpret the vast datasets, methods for extracting useful and novel information from the data are needed and developed. These methods are collectively called knowledge discovery methods. In this thesis, a novel combination of feature selection and data modeling methods is presented in order to help with this task. This combination includes the methods of basic statistical analysis, linear correlation, self-organizing map, parallel coordinates, and k-means clustering. The presented method can be used, first, to select the most relevant features from even hundreds of them and, then, to model the complex inter-correlations within the selected ones. The capability to handle hundreds of features opens up the possibility to study more extensive processes instead of just looking at smaller parts of them. The results of k-nearest-neighbors study show that the presented feature selection procedure is valid and appropriate. A second advantage of the presented method is the possibility to use thousands of samples. Whereas the current rules of selecting appropriate limits for utilizing the methods are theoretically proved only for small sample sizes, especially in the case of linear correlation, this thesis gives the guidelines for feature selection with thousands of samples. A third positive aspect is the nature of the results: given that the outcome of the method is a set of conditional probabilities, the derived model is highly unrestrictive and rather easy to interpret. In order to test the presented method in practice, it was applied to study two different cases of steel manufacturing with hot strip rolling. In the first case, the conditional probabilities for different types of retentions were derived and, in the second case, the rolling conditions for the occurrence of wedge were revealed. The results of both of these studies show that steel manufacturing processes are indeed very complex and highly dependent on the various stages of the manufacturing. This was further confirmed by the fact that with studies of k-nearest-neighbors and C4.5, it was impossible to derive useful models concerning the datasets as a whole. It is believed that the reason for this lies in the nature of these two methods, meaning that they are unable to grasp such manifold inter-correlations in the data. On the contrary, the presented method of conditional probabilities allowed new knowledge to be gained of the studied processes, which will help to better understand these processes and to enhance them.

In this thesis an equation of heat conduction including phase and structural changes is derived, involving various boundary conditions. It seems to be the most suitable to calculate the equation by enthalpy method. In this equation not only enthalpy occurs, but also the temperature, and in consequence the relationship between these variables is quite complicated. In this paper I use the values measured or calculated using solidifcation models. The calculation is implemented in Matlab Simulink, which is a very popular blocks scheme in common practice of regulation. The calculation is based on steady state set up with help of experts and as a result, the program could be put into practice. The program calculates the intensity of cooling according to the initial casting speed, casting inlet temperature and the desired temperature curves. The rate of inuence of cooling can be changed according to the given criteria. The thesis compares the surface temperatures and cooling in the case of a predictive controller is or is not applied in the program.

Due to the high efficiency and productivity of continuous casting process, this method has been the most employed process to produce steel in past decades. The need to improve and optimize the finished product made it essential to gain more knowledge about the process, types of defects that may occur and the reasons for them. Moreover, the solutions for reducing the shortcomings in continuous casting process have been an intriguing subject to study. Many attempts have been done in order to reach this goal. Understanding, determining and optimizing the mould flux slag properties is especially important, since it plays an important and significant role in this process. For this, it is of outmost importance to acquire more knowledge about different properties of mould flux powders. Hence, there has been a world wide effort to measure and model the properties of mould flux properties, such as liquidus and solidus temperatures, heat capacity, enthalpy, thermal expansion, density, viscosity, electrical conductivity, surface tension and thermal conductivity. This thesis presents a brief review on continuous casting process, mould flux powder and its properties and characteristics. Furthermore, it focuses on the thermo-physical properties of mould fluxes. In present work, different industrial mould flux powders have been analyzed to measure their viscosity, break temperature, physical properties such as density, flowablity of powder, slag structure and chemical composition. The experimental data have been compared to some of the most commonly used models such as Riboud model, Urbain model, Iida model and KTH model.

Mould powders are synthetic slags that contain mixtures of silica (SiO2), lime (CaO), sodium oxide (Na2O), fluorspar (CaF2), and carbon (C). When heated to elevated temperatures these powders liquefy and float on the liquid steel in the mould. Mould oscillation helps the liquid flux to penetrate the tiny gap between the mould and the newly formed solid steel shell. In this position the liquid flux partially solidifies against the water cooled mould, while a small portion of the flux remains liquid next to the steel shell to provide lubrication between the moving parts. Effective horizontal heat transfer in the mould is critical for solidifying the liquid steel inthe mould. This process is largely influenced by the thickness and the nature of the flux layer that infiltrates the mould/shell gap. When casting titanium stabilised stainless steels the alloying element reacts with the molten flux, ultimately changing the behaviour of the flux. During the casting process, titanium from the liquid steel reacts with the molten flux producing solids at high temperatures known as perovskite (CaTiO3). Research has shown that perovskite reduces the lubrication capabilities of casting fluxes leading to detrimental effects on product quality while posing a serious threat of machine damage (breakout). The focus of this study is to investigate the effect of titanium pickup on the solidification nature of mould flux and the consequences on horizontal heat transfer. To achieve this, an experimental setup was constructed to simulate the behaviour of mould flux during continuous casting. Analyses of the test flux indicated that the liquid flux closest to the cold side (mould) instantly froze to produce a glassy solid structure. Closer to the hot side (steel shell), solid particles such as perovskite, cuspidine (Ca4Si2O7F2), olivine (Ca,Mg,Mn)2SiO4 and nepheline (Na2O.Al2O3.(SiO2)2) could be identified. Similar solid particles were also found in a slag rim sample taken during the industrial casting of 321- titanium stabilised stainless steel using SPH-KA1 mould powder. Further investigations of the crystalline flux layers showed the entrapment of many tiny gas bubbles during solidification. This porous structure acted as a thermal heat barrier limiting horizontal heat transfer. Experimental testing on 3.0 and 6.0mm flux thickness revealed that the overall thermal conductivity of mould flux decreased as the flux porosity increased. Larger amounts of gas entrapment (in the solid flux structure) resulted in higher thermal resistances which ultimately reduced the heat transfer capabilities of the flux. A second heat barrier, which has a far more dominating effect on the overall heat transfer, is created on mould surface during flux solidification. This thermal contact resistance is also found to be the result of entrapped gas bubbles. Experimental results concluded that the effect of titanium pickup on heat transfer is primarily overshadowed by the larger effect of the thermal contact resistance that is formed during mould flux solidification. The contact resistance in combination with gas entrapment in the solid crystalline structure is considered to be the key factors preventing horizontal heat transfer during continuous casting. / Dissertation (MEng (Metallurgical Engineering))--University of Pretoria, 2006. / Materials Science and Metallurgical Engineering / MEng / unrestricted

Scania has experienced difficulties with large variations of the slope of the back plane after press quenching of case hardened crown wheels of especially type R780 Steg supplied from ingot cast material. This leads to that a large number of crown wheels has to be remeasured and sorted according to back slope which is time consuming for operators. Also, after sorting of the crown wheels, hard machining has to be adjusted according to the different slopes of the back plane of the crown wheels. In some cases, it also leads to scrapping of the crown wheels.This master’s thesis was divided in two parts. The aim of the first part was to confirm that the crown wheel type and casting technique that exhibits the largest variations in slope of the back plane is the R780 Steg originating from ingot cast material. The crown wheel types that were compared were the R780 Steg, R780 Slät and R885 Slät. Crown wheels manufactured from ingot cast material and from continuous cast material were compared. Hence, 6 combinations were examined. The slope of the back plane was measured with the measuring probe FARO after press quenching. The slope of the crown wheels was found to depend on both casting technique and the geometry of the crown wheel. The results confirmed that the crown wheel type and supplier combination that by far yields the largest variations in slope of the back plane is the R780 Steg supplied by Steel Plant A who uses the ingot casting technique. For this combination the variation exceeds 0,1 mm. All other combinations of crown wheels and suppliers yield acceptable variations.The second part of this master’s thesis was composed of determining if segregations in the cast ingot are the cause of the variations in slope of the back plane of the crown wheel type R780 Steg. This was done by measuring if there is a correlation between the slope of the back plane of the crown wheel after press quenching, the chemical composition and the original position of the crown wheel in the ingot. As in the first part of the study, the distortion was measured by the measuring probe FARO. The samples were sent to Degerfors Laboratorium for chemical analysis. Analyses of C, S and N were made by using combustion analyses. For As, P, B and Al optic spectrometry (spark) was used. All other elements were analysed by x-ray fluorescence. Segregations were found to be present and in combination with the geometry of R780 Steg to be the cause of the large variations in slope of the crown wheels.The results of this thesis show that, for the crown wheel type R780 Steg, Scania should not use suppliers that employ the ingot casting technique. Instead, only suppliers using the continuous casting technique should be used. However, for the other crown wheel types ingot or continuously cast material can be used.

<p>Continuous casting is a vital process in the production of semi-finished steel, converting molten metal into solid form. Primary cooling, a critical stage of this process, uses water to cool the solidifying shell as it descends through the mold. The quality of the final cast product is significantly influenced by the fluid flow characteristics during this phase. Given the inherent complexities and costs associated with experimental studies in this area, numerical modeling has emerged as a crucial tool for understanding, predicting, and optimizing fluid flow dynamics within the mold. This research focuses on the implications of argon injection within the mold during the primary cooling stage of the continuous casting process.</p><p>In this thesis, a comprehensive computational investigation of the transportation, entrapment, and fluid dynamic effects of argon injection is presented. Through an exploration of bubble sizes, SEN submergence depths, and slide gate openings, the study reveals how these parameters can significantly influence the casting process.</p><p>The research finds that argon bubble size plays a critical role in determining bubble trajectories and residence times, with smaller bubbles showing a longer residence time and increased boundary interaction due to the dominance of drag forces. The submergence depth of the submerged entry nozzle (SEN) also influences these factors, with deeper submergence resulting in longer bubble trajectories and greater residence times. The study highlights how bubble diameter impacts their entrapment probability, with bubbles ranging from 0.3mm to 0.6mm being most prone to entrapment.</p><p>The effects of argon injection on fluid flow within the SEN demonstrate an enhancement of turbulence, thus promoting a uniform outflow. However, excessively high argon flow rates risk a critical reduction in meniscus velocity, which could lead to overcooling. The research further elucidates the influence of argon on X-velocity near the mold's narrow faces, indicating a potential method for controlling dendritic growth and enhancing the final product quality.</p><p>This work underlines the complex and multifaceted impacts of argon injection on the continuous casting process. It suggests that through careful manipulation of argon bubble size, SEN submergence depth, and slide gate opening, it is possible to optimize the transportation and entrapment of argon bubbles, manage fluid flow dynamics, and ultimately, improve the quality of the cast product.</p>

The thesis work consists of two parts. First, the development of two-dimensional numerical models of a Bridgman unidirectional solidification furnace, and second, the construction work of the furnace at KTH. The aim is to build a Bridgman furnace which is capable of close control over temperature gradient and growth rate such that the solidification structures of a duplex stainless steel (SAF2507) could be replicated at a laboratory scale for different cooling rates.Two numerical models of Bridgman furnace are created using COMSOL Multiphysics. The models are used as predictive tools to simulate the locations of solidification front and the temperature gradients at the solidification fronts, which are parameters difficult to access during experiments. Different hot zone temperatures of the furnace (1500~1550 °C) and different sample pulling rates (0.5~10 mm/s) are studied in simulations. The major finding from modeled results is that the temperature gradient of the sample at the solidification fronts range from 5 ~ 17 K/mm, which are lower than the furnace temperature gradient of ~50 K/mm. The corresponding steady-state cooling rates range between 5 ~ 85 K/s. The next step is to validate the models with experimental temperature profiles of the furnace, and decide whether the furnace design should be modified to achieve the cooling rates of interests. / Examensarbetet består av två delar. Först utvecklingen av tvådimensionella numeriska modeller av en Bridgman enkelriktad stelningsugn, och för det andra konstruktionsarbetet för ugnen vid KTH. Syftet är att bygga en Bridgman-ugn som har förmåga att kontrollera temperaturgradienten och tillväxthastigheten så att stelningsstrukturerna i ett duplex-rostfritt stål (SAF2507) skulle kunna replikeras i laboratorieskala för olika kylningshastigheter. Två numeriska modeller av Bridgman-ugnen skapas med COMSOL Multiphysics. Modellerna används som prediktiva verktyg för att simulera placeringen av stelningsfronten och temperaturgradienterna vid stelningsfronterna, vilket är parametrar som är svåra att komma åt under experiment. Olika varmzonstemperaturer i ugnen (1500~1550 °C) och olika provdragningshastigheter (0.5~10 mm/s) studeras i simuleringar. Det viktigaste fyndet från modellerade resultat är att provets temperaturgradient vid stelningsfronterna sträcker sig från 5 ~17 K/mm, vilket är lägre än ugns temperaturgradient på ~ 50 K/mm. Motsvarande stabilitetskylningshastigheter varierar mellan 5 ~ 85 K/s. Nästa steg är att validera modellerna med experimentella temperaturprofiler för ugnen och bestämma om ugnsutformningen ska modifieras för att uppnå intressens kylningshastigheter.

It is well known that REMs are strong oxide and sulphide formers that can easily form large clusters which have harmful effect on the casting process as well as the quality of the final steel product. By adding these elements right before casting, the number of narrow transfer parts are eliminated (compared to if added in ladle) Also, the REM inclusions has less time to sinter together to form large clusters, preventing clogging. The general idea behind this alloying method in the continuous casting machine is to feed a wire of FeSiRE powder blend, coated with a metal strip, into the melt in the chilled Cu mold (CC mold) Adding REMs to the steel, in particular Ce, can increase the resistance to oxidation at high temperatures by improving the properties of the chromia layer. This is of big interest for SANDVIK as it can improve their corrosion resistant grades and may also, in the future, enable alloying in with other volatile elements such as Zr. This master thesis has the objective to find out the dissolution time of the wire, coated with three different metal strips; Steel, Cu and Al. The experiments were performed with steel grade 316L, provided by SMT, in a 2kg melt in a lab scaled induction furnace at 1500oC, 1510oC and 1530oC. The operations were performed both with and without FeSiRE powder inside. The results obtained with powder inside at 1500oC showed that the Al wire experienced the shortest dissolution time ( 0,5 to 1s) followed by Cu (≤10s) and Steel (18 to 20s). In addition to this, sampling procedures was implemented ( 1, 1, 3, 5 10 and 29mins after wire addition) in a depth of 40mm for each wire. Here, the yield of Ce 1 minute after wire addition was highest for the Steel wire (41.9%) followed by Cu wire (25%) and Al wire (&lt;14.8%). From samples taken 1 and 5mins after wire addition in the Al wire experiment, inclusions were extracted and collected on a film filter after electrolytic extraction and filtration. The film filter was observed in SEM. The morphology and compositions were analysed and compared. It was found that Ce and La was present as Ce La oxy sulfides both individually and on Al Mg O clusters.

Surface defects in steel slabs during continuous casting pose significant challengesfor quality control and product integrity in the steel industry. Predicting and classifyingthese defects accurately is crucial for ensuring product quality and minimizing productionlosses. This thesis investigates the effectiveness of machine learning models in predictingsurface defects of varying severity levels (ordinal classes) during the primary coolingstage of continuous casting. The study evaluates four machine learning algorithms,namely, XGBoost (main and baseline models), Decision Tree, and One-vs.-Rest SupportVector Machine (O-SVM), all trained with imbalanced defect class data. Model evaluationis conducted using a set of performance metrics, including precision, recall, F1-score,accuracy, macro-averaged Mean Absolute Error (MAE), Receiver Operating Characteristic(ROC) curves, Weighted Kappa and Ordinal Classification Index (OCI). Results indicatethat the XGBoost main model demonstrates robust performance across most evaluationmetrics, with high accuracy, precision, recall, and F1-score. Furthermore, incorporatingtemperature data from the primary cooling process inside the mold significantly enhancesthe predictive capabilities of machine learning models for defect prediction in continuouscasting. Key process parameters associated with defect formation, such as tundish temperature,casting speed, stopper rod argon pressure, and submerged entry nozzle (SEN) argonflow, are identified as significant contributors to defect severity. Feature importance andSHAP (SHapley Additive exPlanations) analysis reveal insights into the relationship betweenprocess variables and defect formation. Challenges and trade-offs, including modelcomplexity, interpretability, and computational efficiency, are discussed. Future researchdirections include further optimization and refinement of machine learning models andcollaboration with industry stakeholders to develop tailored solutions for defect predictionand quality control in continuous casting processes.