On an inverse problem arising in continuous casting of steel billets

In continuous casting of steel, the control of the solidification front by means of the amount of water sprayed onto the strand is of great practical interest. We study the thermal history in a continuously cast cylindrical billet. The mathematical model is a two-dimensional nonlinear heat equation div[k(u)gradu] = ut subject to water-cooling and heat radiation boundary conditions. We establish existence, uniqueness and stability results for both the temperature field and the solidification front. We study the monotonicity behaviour of the temperature field and show that certain technically easy-to-realize cooling-strategies may generate double liquid fingers at the final stage of solidification. The inverse problem of determining the cooling strategy is an ill-posed problem. We therefore use Tikhonov regularization as a stable and convergent methodfor treating this problem.     

A three-dimensional steady state thermal fluid model of jumbo ingot casting during electron beam re-melting of Ti–6Al–4V

A 3-D coupled thermal-fluid model describing mass, momentum and energy transport within a Ti–6Al–4V rolling ingot cast in an (Electron Beam Cold Hearth Remelting) EBCHR process has been developed to describe steady state casting conditions. The model incorporates a number of the physical phenomena inherent to the industrial process, including a metal inlet in the center of one of the narrow faces, complex boundary conditions based on industrial practice, buoyancy driven flow within the liquid and flow attenuation using a Darcy momentum source term within the mushy zone. The model ignores turbulence in the liquid pool and Marangoni (surface tension) driven surface flows. The model has been validated against liquid pool depth and profile measurements made on an experimental casting seeded with insoluble dense markers and doped with dense alloy additions. Comparisons have also been made to video images taken of the top surface during casting. The results indicate that the model is able to quantitatively predict the steady state sump depth and profile and is able to qualitatively predict aspects of the top surface temperature distribution. The model has also been used to conduct a process heat balance and sensitivity analyses. The process heat balance conducted on the model domain indicates that at steady state the liquid metal inlet contributes 88% of the total power input, while the electron beam provides net 12% after accounting for radiation losses from the top surface; 62% of the heat is lost through the ingots sides and the balance is lost via bulk transport of sensible heat through the bottom of the domain. The results of the sensitivity analysis on pool depth indicate that casting rate has the largest effect followed by metal inlet superheat. The thermal, flow and pressure fields predicted by the steady state model serves as the initial conditions for a transient hot-top model, which is the subject of a forth-coming paper.     

凝固过程的温度场数值模拟

通过对铸件凝固过程中各换热边界条件的研究,建立了凝固过程的二维非稳态温度场计算数学模型;并运用了有限差分方法对模型进行离散,得到大型方程组,并利用超松驰迭代法(即SOR法)解该方程组,据此,利用Turbo C编制了计算机程序.上机运行结果表明,可较满意地模拟凝固过程温度场的分布.     

The use of artificial intelligence technique for the optimisation of process parameters used in the continuous casting of steel

The productivity and quality of a continuous caster depend mainly on process parameters, i.e. casting speed, casting temperature, steel composition and cleanliness of the melt, water flow rates in the different cooling zones, etc. This work presents the development of an algorithm, which incorporates heuristic search techniques for direct application in metallurgical industries, particularly those using continuous casting process for the production of steel billets and slabs. This is done to determine the casting objectives of maximum casting rate as a function of casting constraints. These constraints are evaluated with the aid of a heat transfer and solidification model based on the finite difference technique, which has been developed and integrated with a genetic algorithm. The essential parts of continuous casting equipment, which must be subjected to monitoring, as well as a methodology of mathematical model and physical settlements in each cooling region, are presented. The efficiency of the intelligent system is assured by the optimisation of the continuous casting operation by maximum casting rate and defect-free products. This approach is applied to the real dimension of a steel continuous caster, in real conditions of operation, demonstrating that good results can be attained by using heuristic search, such as: smaller temperature gradients between sprays zones, reduction in water consumption and an increase in casting speed.     

Transient thermal model for the hot chamber injection system in the pressure die casting process

This paper describes a three-dimensional numerical model that is used to predict the transient thermal behaviour of the metal injection system of a hot chamber pressure die casting machine. The behaviour of the injection system is considered in conjunction with that of the die. The Boundary Element Method (BEM) is used to model the transient thermal behaviour of the injection system elements and the die blocks. A perturbation approach is adopted. By adopting this approach, only those surfaces over which a significant transient variation in temperature occurs need be considered. The model assumes that a corresponding steady-state analysis has first been performed so that time-averaged thermal information is available. A finite element based technique is used to model the phase change of the liquid metal in the die cavity and in the injection system. At injection the nozzle and die are assumed to be instantly filled with liquid metal, however, a procedure is presented that attempts to model the heat transfer associated with the flow through the nozzle, gate, and runner regions during injection. Model predictions are compared against thermocouple readings and thermal images obtained from experimental tests. Good agreement is obtained between predicted and measured temperatures. The transient thermal behaviour of an existing hot chamber injection system is investigated in detail and recommendations for improved performance are made. In an attempt to improve the solidification pattern of the casting and the thermal behaviour of the injection system, a redesign of the experimental die is considered. The numerical predictions indicate that the redesign will have a beneficial effect on the solidification pattern of the casting, and on the performance of the injection system.     

Simulation of direct chill casting under the influence of a low-frequency electromagnetic field

A comprehensive, multiphysics, meshless, numerical model is developed for the simulation of direct chill casting under the influence of a low-frequency electromagnetic field. The model uses mixture-continuum-mass, momentum and energy-conservation equations to simulate the solidification of axisymmetric aluminium-alloy billets. The electromagnetic-induction equation is coupled with the fluid flow and used to calculate the Lorentz force. The involved partial-differential equations are solved with the meshless-diffuse-approximate method by employing second-order polynomial shape functions and a 13-noded local support. An explicit time-stepping scheme is used. The boundary conditions for the heat transfer involve the effects of hot-top, mould chill and direct chill. The use of a meshless method and the automatic node-arrangement generation made it possible to investigate the complicated flow structures in geometrically complex inflow conditions, including sharp and curved edges, in a straightforward way. A time-dependent adaptive computational node arrangement is used to decrease the calculation time. The model is demonstrated by casting an Al-5.25wt%Cu aluminium alloy billet with a radius of 120 mm. Results on simplified and realistic inflow geometry are considered and compared. The effect of the low-frequency electromagnetic force on the temperature, liquid fraction and fluid flow are investigated under different current densities and frequencies.     

Verified reduction of a model for a continuous casting process

The casting of metals is known to involve the complex interaction of turbulent momentum and heat transfer in the presence of solidification, and it is believed that computational fluid dynamical (CFD) techniques are required to model it correctly. Here, using asymptotic methods, we demonstrate that the key quantities obtained in an earlier CFD model for a particular continuous casting process – ostensibly for a pure metal, but equally for an alloy of eutectic composition – can be recovered using a much simpler model that takes into account just the heat transfer, requiring the numerical solution of a two-phase Stefan problem. Combining this with a more recent asymptotic thermomechanical model for the same continuous casting process, we postulate that it should be possible, with the additional help of algebraic manipulation, to reduce a model that takes into account turbulent momentum and heat transfer in the melt and the thermomechanics in the solid shell to one formulated in terms of only heat transfer, without adversely affecting model predictions.     

An analytical solution for the unidirectional solidification problem

The extraction of heat from a molten casting is resisted by an imperfect thermal contact at the mold-casting interface. The nature of the contact varies throughout the casting process and has the effect of increasing the thermal resistance at the interface. This can be modelled by incorporating a gaseous gap at the mold-casting interface that grows with increasing time.

This paper is concerned with an analytical solution of the unidirectional solidification problem, which incorporates movement of the casting at the interface. The derivation of the analytical solution requires the simultaneous solution of the transient heat equations, for the mold, gaseous gap, and solid and liquid parts of the melt. The analytical solution is extended so that contamination layers on the mold and casting can be incorporated as well as an initial gap. This is achieved by introducing virtual layers of mold, gas, and casting. Using the extended solution, the effects of interfacial resistance, air conductivity, and gap variation on solidification rates are examined.     

A three-dimensional fluid flow model for puddle formation in the single-roll rapid solidification process

The single-roll rapid solidification process (SRRSP) is considered to be a process of perspective to produce a Fe-Si-B ribbon of amorphous microstructure and near net shape products such as thin strips of stainless steel. The condition of a melt puddle between the nozzle and rotating wheel in the single-roll rapid solidification process significantly affects the quality and dimensional uniformity of the products as well as the smoothness of the operation. The purpose of this study was to develop a three-dimensional fluid flow analysis system to model the formation of puddle and flow conditions of molten metal in the puddle for the single-roll rapid solidification processes which include the planar flow casting (PFC) process and the single-roll strip casting process. The model is based on a computational fluid dynamics technique called the SOLA-VOF scheme, which possesses the capability of treating transient fluid flow problems with the evolution of free boundaries. Furthermore, the SOLA-VOF scheme is extended from two dimensions to three dimensions. The simulated results reveal how the melt puddle is formed between the nozzle and the rotating substrate and its corresponding fluid flow behavior for the PFC process as well as the single-roll strip casting process. The test results also demonstrate that two-dimensional analysis cannot properly consider the actual flow condition in the puddle.     

Solidification of castings with easily separable runner

The article formulates a mathematical model of hydrodynamic and thermophysical processes during the solidification of a casting with easily separable runner that makes it possible to determine the optimal dimensions of the feeder channel. It was established by a numerical experiment that the introduction of a separating baffle reduces the intensity of movement of the melt, and considerably reduces the heat exchange between the runner and the body of the casting. The article recommends the optimal dimension of the feeder channel ensuring effective feed to the casting up to its complete solidification.Translated from Teoreticheskaya i Prikladnaya Mekhanika, No. 20, pp. 103–106, 1989.     

Short shots and industrial case studies: Understanding fluid flow and solidification in high pressure die casting

The geometric complexity and high fluid speeds involved in high pressure die casting (HPDC) combine to give strongly three dimensional fluid flow with significant free surface fragmentation and splashing. A simulation method that has proved particularly suited to modelling HPDC is Smoothed Particle Hydrodynamics (SPH). Materials are approximated by particles that are free to move around rather than by fixed grids, enabling more accurate prediction of fluid flows involving complex free surface motion. Three practical industrial case studies of SPH simulated HPDC flows are presented; aluminium casting of a differential cover (automotive), an electronic housing and zinc casting of a door lock plate. These show significant detail in the fragmented fluid free surfaces and allow us to understand the predisposition to create defects such as porosity in the castings. The validation of flow predictions coupled with heat transfer and solidification is an important area for such modelling. One powerful approach is to use short shots, where insufficient metal is used in the casting or the casting shot is halted part way through, to leave the die cavity only partially filled. The frozen partial castings capture significant detail about the order of fill and the flow structures occurring during different stages of filling. Validation can occur by matching experimental and simulated short shots. Here we explore the effect of die temperature, metal super-heat and volume fill on the short shots for the casting of a simple coaster. The bulk features of the final solid castings are found to be in good agreement with the predictions, but the fine details appear to depend on surface behaviour of the solidifying metals. This potentially has significant implications for modelling HPDC.     

Mathematical modeling of coupled turbulent flow and solidification in a single belt caster with electromagnetic brake

A mathematical model has been developed to simulate turbulent fluid flow and solidification in the presence of a DC magnetic field in an extended nozzle for metal delivery to a single belt caster. This paper reports on predicted effects of DC magnetic field conditions in modifying flows and solidification behavior in the metal delivery system. It is shown that the application of a DC magnetic brake to the proposed system can result in a reasonably uniform feeding of melt onto the cooled moving belt. This, in turn, optimises the rate of even shell growth along the chilled substrate. In order to account for the effects of turbulence, a revised low-Reynolds k–ε turbulent model was employed. A Darcy-porosity approach was used to simulate fluid flow within the mushy solidification region. Simulations were carried out for plain carbon steel strip casting. The fully coupled transport equations were numerically solved using the finite volume method. The computed flow patterns were compared with those reported in the literature. The performance of the magnetic flow control device proposed in this work is evaluated and compared with flow modifications obtained by inserting a ceramic filter within the reservoir.     

Influence of electromagnetic stirrer position on fluid flow and solidification in continuous casting mold

A computational study of the effect of stirrer position on fluid flow and solidification in a continuous casting billet mold with in-mold electromagnetic stirring has been carried out. The numerical investigation uses a full coupling method in which alternating magnetic field equations are solved simultaneously with the governing equations of fluid flow and heat transfer. An enthalpy-porosity technique is used for the solidification analysis while the magnetohydrodynamics technique is used for studying the fluid flow behavior under the electromagnetic field. The streamline, liquid fraction, and solid shell thickness at the mold wall have been predicted with and without EMS application at different positions along the length of the mold. Recirculation loops are seen to be formed above and below the stirrer position when fluid flow and electromagnetic field equations were solved, without incorporating the solidification model. Application of the solidification model interestingly resulted in the reduction of the size of the recirculation loops formed. The tangential component of velocity of the fluid near the solidification front, stirring intensity and the effective length of stirring below the stirrer decrease as the stirrer position is moved downwards. Significant changes in characteristics of solid shell formation like delay in initiation of solidification at the mold wall and formation of a gap in the re-solidified shell have been observed with change in stirrer position.     

Modelling and validation: Casting of Al and TiAl alloys in gravity and centrifugal casting processes

Components which best utilise the properties of high temperature titanium alloys are characterised by thin sections of a few millimetres thickness and hundreds of millimetres length. These alloys however are difficult to work with, being highly reactive in a molten state, necessitating a low superheat during processing. Centrifugal casting is therefore utilised as a candidate production method, as under the centrifugal force, metal can rapidly fill thicknesses substantially less than a millimetre before it solidifies. However, due to the high liquid metal velocity developed there is a high risk of turbulent flow and of the trapping of any gas present within the liquid metal.This challenging application involves a combination of complex rotating geometries, significant centrifugal forces and high velocity transient free surface flows, coupled with simultaneous heat transfer and solidification. Capturing these interacting physical phenomena, free surface flows, trapped air and associated defects is a complex modelling task.Building upon earlier work on computational modelling the authors have previously described to capture and validate the fluid dynamics behaviour of rotating systems, this contribution considers the modelling and validation of such systems to capture the coupled flow and thermal solidification behaviour and associated defect development.A bench-mark test case is employed to validate the effect of solidification on the fluidity of an aluminium alloy. Validation is also performed against a series of casting experiments to establish the models ability to capture the filling process and predict defects due to air entrapment within the solidified metal.     

A Heat Balance Integral Method for Estimating Practical Solidification Parameters

A heat balance integral technique based on an enthalpy formulationof a metallurgical solidification problem is presented. Unlikethe majority of previous heat balance integral methods the newtechnique can analyse situations in which the phase change takesplace over a temperature range. This means that solidificationparameters of practical significance may be efficiently estimated.An application of the new technique is made to a problem ofbinary alloy solidification. The results obtained are comparedwith existing numerical models.     

Discretisation procedures for multi-physics phenomena

Procedures are described for solving the equations governing a multi-physics process. Finite volume techniques are used to discretise, using the same unstructured mesh, the equations of fluid flow, heat transfer with solidification, and solid deformation. These discretised equations ofe then solved in an integrated manner. The computational mechanics environment, PHYSICA, which facilitates the building of multi-physics models, is described. Comparisons between model predictions and experimental data are presented for the casting of metal components.     

A new coupled model for alloy solidification

A new coupled model in the binary alloy solidification has been developed. The model is based on the cellular automaton (CA) technique to calculate the evolution of the interface governed by temperature, solute diffusion and Gibbs-Thomson effect. The diffusion equation of temperature with the release of latent heat on the solid/liquid (S/L) interface is valid in the entire domain. The temperature diffusion without the release of latent heat and solute diffusion are solved in the entire domain. In the interface cells, the     

A new coupled model for alloy solidification

A new coupled model in the binary alloy solidification has been developed. The model is based on the cellular automaton (CA) technique to calculate the evolution of the interface governed by temperature, solute diffusion and Gibbs-Thomson effect. The diffusion equation of temperature with the release of latent heat on the solid/liquid (S/L) interface is valid in the entire domain. The temperature diffusion without the release of latent heat and solute diffusion are solved in the entire domain. In the interface cells, the energy and solute conservation, thermodynamic and chemical potential equilibrium are adopted to calculate the temperature, solid concentration, liquid concentration and the increment of solid fraction. Compared with other models where the release of latent heat is solved in implicit or explicit form according to the solid/liquid (S/L) interface velocity, the energy diffusion and the release of latent heat in this model are solved at different scales, i.e. the macro-scale and micro-scale. The variation of solid fraction in this model is solved using several algebraic relations coming from the chemical potential equilibrium and thermodynamic equilibrium which can be cheaply solved instead of the calculation of S/L interface velocity. With the assumption of the solute conservation and energy conservation, the solid fraction can be directly obtained according to the thermodynamic data. This model is natural to be applied to multiple (< 2) spatial dimension case and multiple (< 2) component alloy. The morphologies of equiaxed dendrite are obtained in numerical experiments.     

Mathematical and physical modelling of steel flow and solidification in twin-roll/horizontal belt thin-strip casting machines

Near-net-shape casting technology is one of the most important research areas in the iron and steel industry today. Driving forces for the development of this technology include a reduction in the number of operations needed for conventionally produced strip. This is especially true of hot rolling operations. The consequent reduction in investment cost when considering new industrial facilities, makes near-net-shape casting operations extremely attractive from a commercial standpoint. Various processes for near-net-shape casting of steel are currently being developed around the world. Of these processes, twin-roll casting machines represent a major area of concentration. We believe that one of the main issues concerning the design of twin-roll casters is the metal delivery system and its effect on the homogeneity of solid shell formation, segregation and surface quality. In the present work, computational fluid dynamics has been used to study different metal delivery systems for twin-roll casting (TRC) and horizontal belt casting (HBC) operations. The METFLO code has been adapted to simulate three-dimensional turbulent fluid flows, heat transfer and solidification in these types of machines. The enthalpy–porosity technique was used to couple fluid flow and solidification phenomena. Two configurations for metal delivery system have been studied to date for TRC: one is a conventional tubular nozzle with horizontal outlets in the directions of the side dams; the other is a slot nozzle with a vertical inlet stream. These simulations have been applied to a pilot caster being studied in Canada, with a roll radius of 0.30 m, producing steel strips with thicknesses ranging from 4 to 7 mm, at relatively low roll speeds ranging between 4 and 12 m/min. Different positions and penetrations of the nozzles in the liquid pool have also been analysed. It has been shown that a tubular nozzle leads to the formation of a non-uniform solid shell along the roll width. In both configurations, a thicker solid shell is formed close to the roll edges, due to the presence of the side dams. In the case of HBC, computations have been made for an extended nozzle metal delivery system, and preliminary water modelling tests carried out to confirm the flow delivery concepts proposed. In addition, instantaneous heat flux measurements to simulated belt substrates have been performed for the horizontal casting of aluminum strip that show somewhat similar characteristics to those measured for steel in the pilot TRC, in terms of transient peaks and decays.     

Existence and uniqueness of a classical solution for the coupled heat and mass transfer during the freezing of high‐water content materials

A recent model for the coupled problem of heat and mass transfer during the solidification of high‐water content materials like soils, foods, tissues and phase‐change materials was developed. This model takes into account the role played by material properties and process variables on the advance of freezing and sublimation fronts, temperature and water vapour profiles and weight loss. The goal of this paper is to determine the existence of a unique local classical solution for the corresponding two‐phase coupled free boundary problem in an adequate functional space. Copyright © 2011 John Wiley & Sons, Ltd.