A simple Lagrangian travelling slice model has been successfully used to predict the relations between the process parameters and the strand temperatures in the continuous casting of steel. The present paper aims to include a simple macrosegregation, grain structure and mechanical stress and deformation model on top of the thermal slice framework. The basis of all the mentioned models is the slice heat-conduction model that considers the complex heat extraction mechanisms in the mould, with the sprays, rolls, and through radiation. Its main advantage is the fast calculation time, which is suitable for the online control of the caster. The macroscopic thermal and species transfer models are based on the continuum mixture theory. The macrosegregation model is based on the lever rule microsegregation model. The thermal conductivity and species diffusivity of the liquid phase are artificially enhanced to consider the convection of the melt. The grain structure model is based on cellular automata and phase-field concepts. The calculated thermal field is used to estimate the thermal contraction of the solid shell, which, in combination with the metallostatic pressure, drives the elastic-viscoplastic solid-mechanics models. The solution procedure of all the models is based on the meshless radial basis function generated finite difference method on the macroscopic scale and the meshless point automata concept on the grain structure scale. Simulation results point out the areas susceptible to hot tearing.

A turbulent flow of Newtonian incompressible fluid in two dimensions is considered. The partial differential equations describing the conservation of mass and momentum are solved by a strong-form meshless method using five-noded local collocation with multiquadric radial basis functions and explicit time-stepping. The fractional step method is used for the pressure-velocity coupling. The large-eddy simulation (LES) model, the Smagorinsky-Lilly model for calculating turbulent viscosity, and the Van Driest damping function describe the turbulent motion of the fluid flow. The novel application of the described numerical approach is thoroughly tested on a channel flow and backward-facing step flow. The simulations are systematically assessed by comparing the meshless results with standard finite-volume method large-eddy simulation results for Re=5000. Additionally, the parametric study and the influence of the filter width, effective distances used to calculate filter width, and the Smagorinsky constant were assessed for the meshless solution procedure. The obtained results agree with the reference results, approving the suitability of the developed meshless solution for solving turbulent fluid flow problems based on LES formulation. The study has opened the way for using the described approach in more complicated industrial fluid flow problems.

ŠTORE STEEL Ltd. is one of the three steel plants in Slovenia. Continuous cast 180 mm × 180 mm billets can undergo cooling to room temperature using a turnover cooling bed. They can also be cooled down under hoods or heat treated to reduce residual stresses. Additional operations of heat treatment from 36 h up to 72 h and cooling of the billets for 24 h, with limited capacities (with only two heat treatment furnaces and only six hoods), drastically influence productivity. Accordingly, the casting must be carefully planned (i.e., the main thing is casting in sequences), while the internal quality of the billets (i.e., the occurrence of inner defects) may be compromised. Also, the stock of billets can increase dramatically. As a result, it was necessary to consider the abandoning of cooling under hoods and heat treatment of billets. Based on the collected scrap data after ultrasonic examination of rolled bars, linear regression and genetic programming were used for prediction of the occurrence of inner defects. Based on modeling results, cooling under hoods and heat treatment of billets were abandoned at the casting of several steel grades. Accordingly, the casting sequences increased, and the stock of billets decreased drastically while the internal quality of the rolled bars remained the same.

Abstract The present work aims to solve the continuous casting benchmark problem in axisymmetry with turbulence in the melt by the meshless method. The physical model of the liquid-solid system that involves mass, momentum and energy conservation is formulated in the mixture continuum approximation. The melt is assumed Newtonian and incompressible. A k-epsilon Reynolds averaged Navier-Stokes turbulence model is implemented with Abe-Kondoh-Nagano closures. The mushy region is modelled as a Darcy porous media with the Kozeny-Carman permeability model. The solution of partial differential equations is implemented locally using collocation with radial basis functions and explicit time-stepping. The velocity pressure coupling is performed using the fractional step method. The validation of the model is assessed by comparing its outcomes with the results of the finite volume method. A sensitivity study of the varying casting speed and temperature on the velocity, temperature, and solid fraction fields is shown.

Abstract This paper describes the development of meshless simulation of thermomechanics of steel in a continuous hot rolling process using a novel meshless solution procedure. During the process, steel billets at high temperatures are transferred through multiple roll passes, each with a specific groove shape and roll gap. The major outcome of this simulation system is to design the rolling schedule and confirm whether the final shape is inside the acceptable range. For this purpose, a perpendicular 2D slice model approach is used to obtain fast and reasonably accurate simulation results. A return mapping algorithm solves the elasto-plastic material model with linear hardening to obtain displacements, strains and stresses. They are calculated by the local radial basis function collocation method (LRBFCM), which uses the local interpolation of the strong form of partial differential equations. The predefined slice positions are entered into the rolling system, and for each position, the slice contact line and amount of reduction are predicted and used as boundary conditions. Thermal and mechanical models are run sequentially, which is repeated until the final slice position reaches the exit from the last roll. The simulation system was validated with multiple rolling schedules provided by the industrial partner and was successfully used in the steel production plant afterwards. The meshless LRBFCM successfully solves a complex elasto-plastic large deformation problem of hot rolling for the first time.

Abstract With the water model, we experimentally confirmed the possibility of using a suspension to imitate the molten casting powder (slag) at the upper edge of the melt during the continuous casting of steel. The project requires that the slag particles do not leave the suspension layer during operation or that they return to it from the molten steel. It turns out that this is possible if the selected suspension particles mimic sufficiently large inclusions in the real system. The imitation of slag particles in a real system with suspension particles in water model is ensured by the equal terminal flotation velocities of both. Among the discussed polyethylene and cork particles, only cork particles, with a size of 0.2 mm to 0.5 mm, and a density of around 400 kg/m3, meet this condition. It was shown that the presented methodology could be used to estimate the proper position of the submerged entry nozzle.

This paper aims to systematically assess the local radial basis function collocation method, structured with multiquadrics (MQs) and polyharmonic splines (PHSs), for solving steady and transient diffusion problems. The boundary value test involves a rectangle with Dirichlet, Neuman, and Robin boundary conditions, and the initial value test is associated with the Dirichlet jump problem on a square. The spectra of the free parameters of the method, i.e., node density, timestep, shape parameter, etc., are analyzed in terms of the average error. It is found that the use of MQs is less stable compared to PHSs for irregular node arrangements. For MQs, the most suitable shape parameter is determined for multiple cases. The relationship of the shape parameter with the total number of nodes, average error, node scattering factor, and the number of nodes in the local subdomain is also provided. For regular node arrangements, MQs produce slightly more accurate results, while for irregular node arrangements, PHSs provide higher accuracy than MQs. PHSs are recommended for use in diffusion problems that require irregular node spacing.

Macrosegregation presents a considerable defect in the continuous casting of billets and can critically affect the final properties of the product. The numerical modelling can help to predict and better understand the segregation and flow patterns inside the mould. The process is modelled with a physical model described by a set of conservation equations describing the t heat transfer, turbulence, fluid flow, solidification and segregation. A two-equation low-Re k-epsilon model and Abe-Kondoh-Nagano closures are used to close governing equations in this incompressible fluid flow example. The Boussinesq approximation is applied to account for the thermo-solutal buoyancy effects, and the Darcy approximation is applied for the description of the flow through the porous mushy zone. On a microscale, a lever rule solidification model is used to couple liquid fraction, temperature and concentration. The three-dimensional model is solved with the method based on local collocation with multiquadric radial basis functions on seven-nodded subdomains. The aim of this contribution is to explore the three-dimensional macrosegregation patterns of 0.51 wt% carbon steel in the solidified shell of the steel in the mould.

The formation of macrosegregation pattern in steel is explored for the continuous casting process. The related solidification problem is described by incompressible turbulent fluid flow, governed by mass, momentum, energy, and species conservation equations. The solid-liquid system is designed in a mixture continuum model formulation, and the mushy region is approximated with the Darcy model. The lever rule describes the formation of microsegregation of carbon. Turbulence is modelled with a large-eddy simulation approach implemented with the Smagorinsky model and van Driest correction. The synthetic isotropic fluctuations of the inlet velocity are generated by the von Karman Pao spectrum. The solution of partial differential equations is implemented locally by using collocation with radial basis functions for spatial discretization and the explicit Euler method for time discretization. The two-dimensional large eddy simulation model solutions are compared to the previously calculated examples obtained with the low-Re k − ε turbulence model.

A large-eddy simulation (LES) based meshless model is developed for the three-dimensional (3D) problem of continuous casting (CC) of steel billet. The local collocation meshless method based on radial basis functions (RBF) is applied in 3D. The method applies scaled multiquadric (MQ) RBF with a shape parameter on seven nodded local sub-domains. The incompressible turbulent fluid flow is described using mass, energy, and momentum conservation equations and the LES turbulence model. The solidification system is solved with the mixture continuum model. The Boussinesq approximation for buoyancy and the Darcy approximation for porous media are used. Chorin’s fractional step method is used to couple velocity and pressure. The microscopic model is closed with the lever rule model. The LES model is compared to the two-equation Low Re k−ε turbulence Reynolds Averaged Navier–Stokes (RANS) model in terms of temperature, velocity and computational times. The LES model resolves transient character of vortices which RANS-type turbulence models are unable to tackle. The computational cost of LES models is considerably higher than in RANS. On the other hand, it results in a much lower computational cost than the direct numerical simulation (DNS). The paper demonstrates the ability of the method to solve realistic industrial 3D examples. Trivial adjustment of nodal densities, high accuracy, and low numerical diffusivity are the main advantages of this meshless method.

The paper discusses the numerical solution of the large-eddy formulation for modelling the turbulent fluid flow with solidification in the continuous casting of steel. This industrially relevant problem is solved by a meshless method for the first time. The solid-liquid system is formulated within the continuum mixture formulation of the governing, mass momentum and energy equations. The Darcy porous media model describes the mushy region. The large-eddy formulation is based on the Smagorinsky model for sub-grid scale viscosity and the van Driest damping function. The synthetic turbulence is prescribed at the molten steel inlet. The generated turbulent fluctuations are isotopic and correlated in time through an asymmetric time filter. Explicit time discretisation, combined with meshless local radial basis function collocation method, is employed. Five-noded subdomains, non-uniform node arrangement and multiquadrics shape functions are used for solving a two-dimensional model. The sensitivity to different node arrangements, time steps, inlet temperature, and velocity boundary conditions is elaborated and successfully verified by comparison with our previously published meshless k -epsilon turbulence model of the same process.

The travelling slice approximation of continuous casting process is widely used in industrial practice to model the heat transfer and solidification. Short computational times that are possible due to the reduction from three to two dimensions compensate for lower accuracy introduced by the approximation. In the same spirit, the travelling slice approximation is also used for the thermomechanical model in the present paper. The model includes contributions from metallostatic pressure, thermal contraction and viscoplastic deformation to describe the stress in the material slice. It incorporates simple models to predict the hot tearing and cracking. The model is solved with a novel strong-form meshless method. It allows flexible adaptive node distribution to accurately describe the behaviour in the solidifying region. In this paper we focus on the two-way coupling of the heat transfer and deformation of the strand to model the air-gap formation in the initial stage of the casting process and to study the emergent behaviour that is the result of this coupling. We find that the inclusion of the air-gap model significantly changes the stress distribution in the corners of the strand because of the reheating caused by the reduced heat-transfer.

This paper explores, for the first time, the application of the novel mesh-free local radial basis function collocation method (LRBFCM) to the solution of a mull-physics problem in three dimensions. A related benchmark problem is solved by considering the natural convection of an incompressible Newtonian fluid in a differentially heated cubic cavity with and without the application of a magnetic field. The research is limited to typical magnetic fields used in the magnetohydrodynamic processing of liquid metals. For this purpose the assumption of small magnetic Reynolds numbers Re-m << 1 is made. Spatial discretization is performed by local non-uniform collocation with scaled multiquadrics radial basis functions (RBFs) with the shape parameter set to a constant value and the explicit Euler formula used to perform the time stepping. The involved temperature, velocity and pressure fields are represented on overlapping seven-nodded sub-domains. The pressure-velocity coupling is resolved by the fractional step method. The originality of the contribution represents LRBFCM solution of the classic three-dimensional steady natural convection benchmark for Rayleigh numbers from 10(5) to 10(7) and Prandfi number 0.71, and its extension to Prandtl number 0.1, and Hartman numbers 0, 10, 50 and 100. The accuracy of the LRBFCM is found to be comparable with the published benchmark results obtained using established numerical methods.

This paper represents an overview of the elements of the user-friendly simulation system, developed for computational analysis and optimization of the quality and productivity of the electromagnetically direct-chill cast semi-products from aluminium alloys. The system also allows the computational estimation of the design changes of the casting equipment. To achieve this goal, the electromagnetic and the thermofluid process parameters are coupled to the evolution of Lorentz force, temperature, velocity, concentration, strain and stress fields as well as microstructure evolution. This forms a multi-physics and multi-scale problem of great complexity, which has not been demonstrated before. The macroscopic fluid mechanics, solid mechanics, and electromagnetic solution framework is based on local strong-form meshless formulation, involving the radial basis functions and monomials as trial functions, and local collocation or weighted least squares approximation. It is coupled to the micro-scale by incorporating the point automata solution concept. The entire macro-micro solution concept does not require meshing and space integration. The solution procedure can be easily and efficiently automatically adapted in node redistribution and/or refinement sense, which is of utmost importance when coping with fields exhibiting sharp gradients, which occur in the phase-change problems. The simulation system is coded from scratch in modern Fortran. The elements of the experimental validation of the system and the demonstration of its use for round billet casting in IMPOL Aluminium Industry are shown.

This paper demonstrates the development of meshless Local Radial Basis Function Collocation Method (LRBFCM) for solution of three dimensional (3D) turbulent molten steel flow and solidification under the influence of electromagnetic stirring (EMS) and its application to continuous casting process of steel billets. A mixture continuum approach is used to formulate the coupled set of macroscopic equations for mass, momentum, energy, turbulent kinetic energy, and dissipation rate in Cartesian coordinates. The mushy zone is treated as a Darcy porous media. The explicit Chorin fractional step method is used to resolve the pressure-velocity coupling. The LRBFCM uses explicit time discretization and spatial discretization with shape-adaptive multiquadrics radial basis functions collocation on non-uniform seven-noded influence domains, and displacement-adaptive upwind scheme. The Lorentz force due to EMS is provided by a one-way coupling by Elmer software. The advantages of the meshless method are trivial adaptation of the nodal densities, simple upgrade to 3D from previous two-dimensional models, and high accuracy. A study on the influence of changing the EMS parameters on the calculated temperature and velocity fields is performed. The paper confirms the ability of LRBFCM meshless solution of a realistic 3D multiphysics industrial problem with complicated swirling flow pattern.

The electric arc furnace operation at the Štore Steel company, one of the largest flat spring steel producers in Europe, consists of charging, melting, refining the chemical composition, adjusting the temperature, and tapping. Knowledge of the consumed energy within the individual electric arc operation steps is essential. The electric energy consumption during melting and refining was analyzed including the maintenance and technological delays. In modeling the electric energy consumption, 25 parameters were considered during melting (e.g., coke, dolomite, quantity), refining and tapping (e.g., injected oxygen, carbon, and limestone quantity) that were selected from 3248 consecutively produced batches in 2018. Two approaches were employed for the data analysis: linear regression and genetic programming model. The linear regression model was used in the first randomly generated generations of each of the 100 independent developed civilizations. More accurate models were subsequently obtained during the simulated evolution. The average relative deviation of the linear regression and the genetic programming model predictions from the experimental data were 3.60% and 3.31%, respectively. Both models were subsequently validated by using data from 278 batches produced in 2019, where the maintenance and the technological delays were below 20 minutes per batch. It was possible, based on the linear regression and the genetically developed model, to calculate that the average electric energy consumption could be reduced by up to 1.04% and 1.16%, respectively, in the case of maintenance and other technological delays.

Numerical model of controlled cooling in the production of steel hot rolled bars was developed. The numerical model and algorithm are completed to solve problems in controlled cooling of hot rolled bars in cooling beds. The controlled cooling is performed by special placement of hot rolled bars on cooling beds. By numerical model of controlled cooling is possible to predict a transient temperature field, microstructure evolution and hardness of round steel bars during their cooling in cooling beds. The numerical model of transient temperature field is based on control volume method. The hardness and microstructure distribution in steel bars has predicted by using equations of austenite decomposition kinetic. The algorithm for prediction is based the real chemical composition. Numerical model and computer program were experimentally verified by experimental work in a real industrial production of low-alloyed steel bars. The verification of developed numerical model was performed by comparison of simulated with experimentally evaluated results hardness and cooling curve.

Abstract Numerical model of controlled cooling in production of steel hot rolled bars was developed. By numerical model of controlled cooling is possible to predict a transient temperature field, microstructure evolution and hardness of rectangular steel bars during their cooling in cooling beds. The numerical model of transient temperature field is based on control volume method. The algorithm for prediction of hardness and microstructure distribution in steel bars is based on continues cooling transformation, (CCT) diagrams and real chemical composition. The numerical model and algorithm is completed to solve problems in controlled cooling of hot rolled bars in cooling beds. The controlled cooling are performed by special placement of hot rolled bars on cooling beds. Numerical model and computer program was experimentally verified by simulation of real industrial production of low alloyed steel bars. The verification of developed numerical model was performed by comparison of simulated hardness with experimentally evaluated results.

The purpose of this paper is a multiphysics simulation of 3D temperature and velocity fields in continuous casting of steel under the influence of electromagnetic stirring by a combined meshless - finite element method approach. The transport phenomena are calculated by a meshless local radial basis function collocation technique and the magnetic force by the finite element method solver Elmer. The electromagnetic stirring increases the mixing in the molten steel. The thermal gradient is sharper and solidification is faster along the strand. The results are similar to other publications in the field. The local radial basis function collocation method is for the first time applied to 3D continuous casting problem with mold electromagnetic stirring.