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.     

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.     

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.     

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.     

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.     

An approximation of the thermal field in a continuous casting process of a thin metal layer

We consider the problem of a phase change in a continuous casting process: molten bronze is poured on to a moving steel strip cooled from below, in order to solidify the bronze. An estimate of the width of the solidification zone, depending on the thickness of the strip and on the casting velocity, is obtained, neglecting conduction in the direction of the strip motion.     

Boundary element and finite element analysis for the efficient simulation of fluid-structure interaction and its application to mold filling processes

Metal casting and polymer molding are widely used for the economical shape processing of complex geometries. In these manufacturing processes, a liquid melt (metal, mineral or synthetic) is filled into a mold with a cavity of the desired shape. Cooling and solidification of the melt results in a product with almost the same shape as the cavity. Numerical simulations can be employed to increase the accuracy of the process. To this end, boundary element method for Stokes flow and a finite element formulation for liquid membranes are investigated in this work. (© 2017 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

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.     

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.     

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.     

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.     

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.     

Flow of polymer melts in a rotary extruder head

The flow of a non-Newtonian fluid in an annular extruder channel formed by two coaxial cylinders is investigated theoretically. Complex shear conditions are created by the pressure difference between the ends of the channel and the rotation of the cylinders in opposite directions at constant speeds. An expression is obtained for the rate of flow of a polymer melt regarded as a non-Newtonian fluid. The flow of a polyethylene melt at 140° C through an extruder head with a rotating mandrel has been experimentally investigated. The experimental data serve to confirm the theoretical conclusions.Mekhanika Polimerov, Vol. 4, No. 3, pp. 531–539, 1968     

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.     

Computed oscillations of a confined submerged liquid jet

The jet oscillation observed in thin slab continuous casting is studied numerically by modelling the flow of liquid injected through a submerged entry nozzle and into a cavity. The oscillation relies on the exchange of fluid between recirculation cells on each side of the jet via a cross-flow through the gap between the nozzle shaft and the broad face of the cavity wall. Features of the oscillating jet are investigated by varying the resistance to cross-flow. This resistance occurs naturally since the nozzle obstructs cross-flow. The predicted oscillation can be manipulated by altering the cross-flow (through the use of an effective resistance force in the model) or stopped altogether to form a static asymmetrical flow pattern. Flow calculations are performed using a transient, two-dimensional, turbulent, fluid flow model.     

Finite volume simulation framework for die casting with uncertainty quantification

The present paper describes the development of a novel and comprehensive computational framework to simulate solidification problems in materials processing, specifically casting processes. Heat transfer, solidification and fluid flow due to natural convection are modeled. Empirical relations are used to estimate the microstructure parameters and mechanical properties. The fractional step algorithm is modified to deal with the numerical aspects of solidification by suitably altering the coefficients in the discretized equation to simulate selectively only in the liquid and mushy zones. This brings significant computational speed up as the simulation proceeds. Complex domains are represented by unstructured hexahedral elements. The algebraic multigrid method, blended with a Krylov subspace solver is used to accelerate convergence. State of the art uncertainty quantification technique is included in the framework to incorporate the effects of stochastic variations in the input parameters. Rigorous validation is presented using published experimental results of a solidification problem.     

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.     

二元拉伐尔喷管不可压缩流动精确解

在物理平面上,仔细分析沿拉伐尔喷管中心线和喷管型线的流动,可以发现拉伐尔喷管流动的上下两半部分在速度平面中是两个相同的具有尾缘点前后错开的双尾的裂缝厚翼型。该两个翼型处在不同的黎曼面内。翼型的内部在复位势平面中可转绘成无限长的条带。利用这些结果得到了二元拉伐尔喷管内不可压缩位势流动的精确解。精确解对任意给定的收缩比n₁、扩张比n₂和喉部壁面曲率半径R*都适用。作为应用的举例,给出了一些典型的喷管型线,喷管内的流速分布以及不同瞬间流体质点的所在位置。     

Oscillation of Confined Jets in Continuous Casting Mold Flow

In continuous slab casting, the liquid steel is introduced into the mould via a submergered entry nozzle. This nozzle usually has two opposed orifices on its side walls, generating two diametrically opposed turbulent jets that are declined about 20° to the horizontal axis. These jets interact with the surrounding walls of the mould, which leads to an unstable flow situation and a self induced oscillation of the jets. Although both mould and nozzle geometry have two perpendicular symmetry planes, the oscillations are asymmetric. The fluid flow inside the mold is calculated with a 3D finite volume solver using turbulence models based on Reynolds-averaging. The massflow of the jets and the mould extensions are varied, and the numerical results are partially compared with PIV-measurements at a 1:1 scaled watermodel of the mould. (© 2006 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Efficiency of a DC magnetic field used for braking the flow within a continuous casting mould

The continuous casting technology provides about 90 percent of the world steel production. The application of DC magnetic fields in form of so-called electromagnetic brakes is considered for an effective flow control with substantial capabilities to improve the product quality or to enhance the productivity of the process. The main effect of the DC magnetic field is supposed to result in a uniform reduction of the maximum velocities in the discharging jet from the submerged entry nozzle and to damp violent turbulent fluctuations. However, the electromagnetic braking of such highly turbulent and complex flows is complicated phenomenon and has not been understood fully until now. We present numerical and experimental investigations focusing on the fluid flow in the continuous casting mould under the influence of a transverse magnetic field. Numerical calculations were performed using the software package CFX with an implemented RANS-SST turbulence model. the non-isotropic nature of the MHD turbulence was taken into account by specific modifications of the turbulence model. Corresponding experimental investigations were carried out at the mock-up LIMMCAST at HZDR. The comparison between our numerical calculations and the experimental results display a very well agreement. An important outcome of this study is the feature that the magnetic field does not provide a continuous reduction of the velocity fluctuations at the nozzle port. (© 2011 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)