A 3-D Numerical Prediction of Turbulent Flow,Heat Transfer and Solidification in a Continuous Slab Caster for Steel

Abstract

This study describes the numerical modeling of three-dimensional coupled turbulent flow, heat transfer, and solidification in a continuous slab caster for stainless steel. The model uses generalized transport equations which are applicable to the liquid, mushy and solid regions within the caster. The turbulent characteristics in the melt pool and mushy region are accounted for using the low-Reynolds number k–ε turbulence model by Launder and Sharma. This version of the low-Reynolds number turbulence model is found to be more easily adaptable to the coupled flow and mushy region solidification caster problem compared to the standard high-Reynolds number and other low-Reynolds number turbulence models. The macroscopic solidification process itself is based on the enthalpy-porosity scheme. The governing transport equations are solved employing the primitive variables and using the control volume based finite-difference scheme on a staggered grid. The process variables considered are the casting speed and the inlet superheat of the melt. The effects of these process variables on the velocity and temperature distributions and on the extent of the solidification and mushy regions are reported and discussed. The numerical predictions of solidification profile are compared with the limited experimental data available in the literature, and very good agreement was found. © 1998 Canadian Institute of Mining and Metallurgy. Published by Elsevier Science Ltd. All rights reserved.

Résumé

Cette étude decrit la modélisation numerique de l'écoulement turbulent couple, à trois dimensions, du transfert de chaleur et de la solidification, lors du coulage en continu de dallé d'acier inoxydable. Le modéle utilisé des equations generalisées de transport qui sont applicables aux regions liquides, pateuses et solides à l'interieur de l'equipement de coulage. On tient compte des caractéristiques turbulentes du bain fondu et de la région pateuse en utilisant le modéle de turbulence a petit nombre de Reynolds k–ε, de Launder et Sharma. Cette version du modèle de turbulence à petit nombre de Reynolds est plus facilement adaptable à l'écoulement couple et la solidification de la région pateuse du problème de coulage que la version utilisant un nombre de Reynolds normal, élevé, ou que d'autrés modèles de turbulence à petit nombre de Reynolds. Le procédé meme de solidification macroscopiqué est base sur la combinaison enthalpié-porosite. On resout les équations de transport qui gouvernent en employant les variables primitives et enutilisant la mèthode des différences finies basée sur le volume de contrôle, sur une grille alterne. Les variables considerées dans ce procédé sont la vitesse de coulage et la surchauffe du liquide a l'entrée. On rapporte et discute de l'effet de ces variables du procédé sur la distribution de velocité et de température et sur l'étendue des régions solides et pateuses. On compare les prédictions numériqués du profil de solidification avec les données expérimentales limitées disponibles dans la litterature et on trouvé une très bonne concordance. © 1998 Canadian Institute of Mining and Metallurgy. Published by Elsevier Science Ltd. All rights reserved.     

Coupled turbulent flow,heat, and solute transport in continuous casting processes

A fully coupled fluid flow, heat, and solute transport model was developed to analyze turbulent flow, solidification, and evolution of macrosegregation in a continuous billet caster. Transport equations of total mass, momentum, energy, and species for a binary iron-carbon alloy system were solved using a continuum model, wherein the equations are valid for the solid, liquid, and mushy zones in the casting. A modified version of the low-Reynolds numberk-ε model was adopted to incorporate turbulence effects on transport processes in the system. A control-volume-based finite-difference procedure was employed to solve the conservation equations associated with appropriate boundary conditions. Because of high nonlinearity in the system of equations, a number of techniques were used to accelerate the convergence process. The effects of the parameters such as casting speed, steel grade, nozzle configuration on flow pattern, solidification profile, and carbon segregation were investigated. From the computed flow pattern, the trajectory of inclusion particles, as well as the density distribution of the particles, was calculated. Some of the computed results were compared with available experimental measurements, and reasonable agreements were obtained.     

A numerical and experimental study of the solidification rate in a twin-belt caster

A numerical and experimental study was carried out to investigate the solidification process in a twin-belt (Hazelett) caster. The numerical model considers a generalized energy equation that is valid for the solid, liquid, and mushy zones in the cast. Ak-ε turbulence model is used to calculate the turbulent viscosity in the melt pool. The process variables considered are the belt speed, strip thickness, nozzle width, and heat removal rates at the belt-cast interface. From the computed flow and temperature fields, the local cooling rates in the cast and trajectories of inclusions were computed. The cooling rate calculations were used to predict the dendrite arm spacing in the cast. The inclusion trajectories agree with earlier findings on the distribution of inclusion particles for near horizontally cast surfaces. This article also reports the results of an experimental study of the measurement of heat flux values at the belt-cast interface during the solidification of steel and aluminum on a water-cooled surface. High heat fluxes encountered during the solidification process warranted the use of a custom-made heat flux gage. The heat flux data for the belt surface were used as a boundary condition for the numerical model. Objectives of the measurements also included obtaining an estimate of the heat-transfer coefficient distribution at the water-cooled side of the caster belt. Y.G. KIM, formerly Graduate Student, Materials Engineering Department, Drexel University.     

连铸流动与凝固耦合模拟中糊状区系数的表征及影响

分析提出了连铸流动与凝固耦合数值模拟中, 钢液在两相区流动时的糊状区系数(A_mush)与渗透率的关系; 通过建立大方坯连铸结晶器三维耦合数值模型, 揭示了不同糊状区系数对钢液流动、传热与凝固进程的影响, 以及早期相关研究结果差异的源头.结果表明: 糊状区系数越大, 钢液在糊状区内的流动阻力越强, 凝固时钢液流动速度降低越快.采用较大的糊状区系数时, 糊状区呈较窄的"带状"分布在固液相之间; 当糊状区系数较小时, 糊状区范围变大, 钢液在结晶器内温降过快, 自由液面处出现过冷现象, 凝固坯壳局部发生重熔.结合实验数据验证与模型分析, 认为糊状区系数取值1×108~5×108 kg·m-3·s-1可以较可靠地揭示连铸结晶器内的实际凝固现象.      

A two-dimensional heat and fluid-flow model of single-roll continuous-sheet casting process

Single-roll continuous-sheet casting process has been simulated using a mathematical model based on considerations of fluid flow, heat transfer, and solidification. The principal model equations include momentum and energy balances which are written for various zones comprising the process. The flow of liquid metal in the pool is taken to be a two-dimensional recirculatory flow. The concepts of vorticity and stream function are used to reduce the number of equations and number of unknowns, respectively. Model equations and boundary conditions are written in terms of dimensionless variables and are solved, using an implicit finite difference technique, to give stream functions and velocity fields in the metal pool, temperature fields in the metal pool, sheet, and caster drum, and the final sheet thickness for various operating parameters. The parameters examined are: (1) rotational speed of the caster drum, (2) liquid metal head in the tundish, (3) superheat of the melt, (4) caster drum material, and (5) cooling conditions prevailing at the inner surface of the caster drum. The final sheet thickness decreases with increasing rotational speed of the caster drum and melt superheat, but it increases with increasing standoff distance and metal head in the tundish.     

Modelling of Melt Flow and Solidification in the Twin‐Roll Strip Casting Process

A three‐dimensional mathematical model has been developed to simulate turbulent fluid flow, heat transfer and solidification in the pool of a twin‐roll strip caster. A Darcy‐porosity approach was used to study the fluid flow within the mushy solidification zone in the pool. The effect of the heat transfer coefficient and permeability constant on the flow and solidification was also predicted. It was shown that an even flow and temperature distribution of the pool can be obtained by using a suitable feeding system. The heat transfer between the rolls and the solidifying metal has a big influence on the location of the solidification end point. The permeability of the mushy zone is a key factor which affects the flow and solidification in the twin‐roll strip casting process.     

Solidification and Particle Entrapment during Continuous Casting of Steel

Avoiding particle entrapment into the solidifying shell of a steel continuous caster is important to improve the quality of the continuous cast product. Therefore, the fluid flow dynamics in the steel melt and mushy zone, heat transfer and solidification of the steel shell, as well as the motion and entrapment of inclusion particles during the casting process were investigated using computational models. Solidification of the strand shell is modelled with an enthalpy‐formulation by assuming a columnar morphology in the mushy zone. The motion of particles is tracked with a Lagrangian approach. When the particles reach the solidification front, they can be entrapped/engulfed into the solid shell or pushed away from the solidification front, depending on the mushy zone morphology and the forces acting on them. The current paper focuses on the mould region at a steel continuous caster, including the submerged entry nozzle (SEN) and 1.2 m length of the strand. The results are validated with plant measurements and demonstrate the potential of the model to predict fluid flow, shell growth and the positions and the amount of entrapped/engulfed particles in the solidifying strand.     

An Improved Model for the Flow in an Electromagnetically Stirred Melt during Solidification

A mathematical model for simulating the electromagnetic field and the evolution of the temperature and velocity fields during solidification of a molten metal subjected to a time-varying magnetic field is described. The model is based on the dual suspended particle and fixed particle region representation of the mushy zone. The key feature of the model is that it accounts for turbulent interactions with the solidified crystallites in the suspended particle region. An expression is presented for describing the turbulent damping force in terms of the turbulent kinetic energy, solid fraction, and final grain size. Calculations were performed for solidification of an electromagnetically stirred melt in a bottom chill mold. It was found that the damping force plays an important role in attenuating the intensity of both the flow and turbulent fields at the beginning of solidification, and strongly depends on the final grain size. It was also found that turbulence drops significantly near the solidification front, and the flow becomes laminarized for solid fraction around 0.3.     

连铸凝固传热时液相有效导热系数的定量化

在已验证的电磁-热-溶质传输耦合模型的基础上,以某钢厂同时装配有M-EMS和F-EMS的方、圆坯先进铸机为研究对象,对二维切片凝固传热模型中液相有效导热系数的放大倍数m值进行了定量化研究。结果表明,溶质再分配作用下,方、圆坯凝固终点处的钢液液相线温度较浸入式水口入口处的分别约下降了23.27和5.54℃;与二维切片模型相比,采用耦合模型计算时,铸坯凝固终点位置分别后移了1.8和0.9m;为保证同时准确获取铸坯表面温度分布状态及其内部凝固终点位置,在本方、圆坯工况下,二维切片模型中纯液相和糊状区内液相有效导热系数放大倍数的推荐值范围分别为2.2~2.4和1.1~1.2。     

Numerical Investigation of Shell Formation in Thin Slab Casting of Funnel-Type Mold

The key issue for modeling thin slab casting (TSC) process is to consider the evolution of the solid shell including fully solidified strand and partially solidified dendritic mushy zone, which strongly interacts with the turbulent flow and in the meantime is subject to continuous deformation due to the funnel-type mold. Here an enthalpy-based mixture solidification model that considers turbulent flow [Prescott and Incropera, ASME HTD, 1994, vol. 280, pp. 59–69] is employed and further enhanced by including the motion of the solidifying and deforming solid shell. The motion of the solid phase is calculated with an incompressible rigid viscoplastic model on the basis of an assumed moving boundary velocity condition. In the first part, a 2D benchmark is simulated to mimic the solidification and motion of the solid shell. The importance of numerical treatment of the advection of latent heat in the deforming solid shell (mushy zone) is specially addressed, and some interesting phenomena of interaction between the turbulent flow and the growing mushy zone are presented. In the second part, an example of 3D TSC is presented to demonstrate the model suitability. Finally, techniques for the improvement of calculation accuracy and computation efficiency as well as experimental evaluations are also discussed.     

Solidification of a Vacuum Arc-Remelted Zirconium Ingot

As the quality of vacuum arc-remelted (VAR) zirconium ingots is directly linked to their chemical homogeneity and their metallurgical structure after solidification, it is important to predictively relate these factors to the operating conditions. Therefore, a detailed modeling study of the solidification process during VAR has been undertaken. To this purpose, the numerical macromodel SOLAR has been used. Assuming axisymmetrical geometry, this model is based on the solution of the coupled transient heat, momentum, and solute transport equations, under turbulent flow conditions during the remelting, hot-topping, and cooling of a cylindrical ingot. The actual operating parameters are defined as inputs for the model. Each of them, mainly the melting current sequence, melting rate sequence, and stirring parameters (current and period), is allowed to vary with time. Solidification mechanisms recently implemented in the model include a full coupling between energy and solute transport in the mushy zone. This modeling can be applied to actual multicomponent alloys. In this article, the macrosegregation induced by solidification in a zirconium alloy ingot is investigated. In order to validate the model results, a full-scale homogeneous Zy4 electrode has been remelted, and the resulted ingot has been analyzed. The model results show a general good agreement with the chemistry analyses, as soon as thermosolutal convection is accounted for to simulate accurately the interdendritic fluid flow in the central part of the ingot.     

A Multiscale 3D Model of the Vacuum Arc Remelting Process

A three-dimensional, transient, multiscale model of the VAR process is presented, allowing novel simulations of the influence of fluctuations in arc behavior on the flow and heat transfer in the molten pool and the effect this has on the microstructure and defects. The transient behavior of the arc was characterized using the external magnetic field and surface current measurements, which were then used as transient boundary conditions in the model. The interactions of the magnetic field, turbulent metal flow, and heat transfer were modeled using CFD techniques and this “macro” model was linked to a microscale solidification model. This allowed the transient fluctuations in the dendritic microstructure to be predicted, allowing the first coupled three-dimensional correlations between macroscopic operational parameters and microstructural defects to be performed. It was found that convection driven by the motion of the arc caused local remelting of the mushy zone, resulting in variations in permeability and solute density. This causes variations in the local Rayleigh number, leading to conditions under which freckle solidification defects will initiate. A three-dimensional transient tracking of particle fall-in was also simulated, enabling predictions of “white spot” defects via quantification of the trajectory and dissolution of inclusions entering the melt.     

Numerical analysis of fluid flow and heat transfer in pool of twin roll strip caster

Abstract

Twin roll strip casting is regarded as a prospective technology offering many economic benefits. The control of fluid flow in the pool is, however, particularly difficult due to the high casting speed and small pool volume. In the present study, a three-dimensional mathematical model has been developed for the coupled analysis of fluid flow, heat transfer and solidification in the pool using the finite difference method. The characteristics of transport phenomena in the pool of a twin roll strip caster using a wedge type melt delivery system were analysed by numerical simulation. The results show that it is desirable for the wedge melt delivery system to provide the uniformity of flow and temperature in the pool to maintain the casting process and improve the strip quality.     

A Comprehensive Model of the Electroslag Remelting Process: Description and Validation

Electroslag remelting (ESR) is widely used for the production of high-value-added alloys such as special steels or nickel-based superalloys. Because of high trial costs and the complexity of the mechanisms involved, trial-and-error-based approaches are not well suited for fundamental studies or for optimization of the process. Consequently, a transient-state numerical model has been developed that accounts for electromagnetic phenomena and coupled heat and momentum transfers in an axisymmetrical geometry. The model simulates the continuous growth of the electroslag-remelted ingot through a mesh-splitting method. In addition, solidification of the metal is modeled by an enthalpy-based technique. A turbulence model is implemented to compute the motion of liquid phases (slag and metal), while the mushy zone is described as a porous medium the permeability of which varies with the liquid fraction, thus enabling accurate calculation of solid/liquid interaction. The coupled partial differential equations (PDEs) are solved using a finite-volume technique. The computed results are compared to the experimental observation of an industrial remelted ingot; the melt pool depth and shape, in particular, are investigated, in order to validate the model. These results provide valuable information about the process performance and the influence of the operating parameters. In this way, we present an example of a model used as a support in analyzing the influence of the electrode fill ratio. This article is based on a presentation given at the International Symposium on Liquid Metal Processing and Casting (LMPC 2007), which occurred in September 2007 in Nancy, France.     

Numerical Simulation of Filling Process During Twin-Roll Strip Casting

The modeling and controlling of flow and solidification of melt metal in the filling process is important for obtaining the optimal pool level and the formation of the solidified metal layer on the surface of twin-rolls during the twin-roll strip casting. The proper delivery system and processing parameters plays a key role to control flow characteristics in the initial filling stage of the twin-roll strip casting process. In this paper, a commercial CFD software was employed to simulate the transient fluid flow, heat transfer, and solidifications behaviors during the pouring stage of twin-roll strip casting process using different delivery systems. A 3D model was set up to solve the coupled set of governing differential equations for mass, momentum, and energy balance. The transient free-surface problem was treated with the volume of fluid approach, a k–? turbulence model was employed to handle the turbulence effect and an enthalpy method was used to predict phase change during solidification. The predicted results showed that a wedge-shaped delivery system might have a beneficial impact on the distribution of molten steel and solidification. The predicted surface profile agreed well with the measured values in water model.     

Effect of Shrinkage on Primary Dendrite Arm Spacing during Binary Al-Si Alloy Solidification

Upward and downward directional solidification of hypoeutectic Al-Si alloys were numerically simulated inside a cylindrical container. Undercooling of the liquidus temperature prior to the solidification event was introduced in the numerical model. The finite-volume method was used to solve the energy, concentration, momentum, and continuity equations. Temperature and liquid concentrations inside the mushy zone were coupled with local equilibrium assumptions. An energy equation was applied to determine the liquid fraction inside the mushy zone while considering the temperature undercooling at the solidifying dendrite/liquid interface. Momentum and continuity equations were coupled by the SIMPLE algorithm. Flow velocity distribution at various times, G, R, λ ₁, and solidification time at mushy zone/liquid interface during solidification were predicted. The effect of shrinkage during solidification on these solidification parameters was quantified. Transient temperature distribution, solidification time for the mushy zone/liquid interface, and λ ₁ were validated by laboratory experiments. It was found that better agreement could be achieved when the fluid flow due to solidification shrinkage was considered. Considering shrinkage in upward solidification was found to only have a marginal effect on solidification parameters, such as G, R, and λ ₁; whereas, in the downward solidification, fluid flow due to shrinkage had a significant effect on these solidification parameters. Considering shrinkage during downward solidification resulted in a smaller R, stronger fluid flow, and increased solidification time at the mushy zone/liquid interface. Further, the flow pattern was significantly altered when solidification shrinkage was considered in the simulation. The effect of shrinkage on G and λ ₁ strongly depended on the instantaneous location of the mushy zone/liquid interface in the computational domain. The numerical results could be validated by experimental data only when both the undercooling of the liquidus temperature prior to solidification and fluid flow in the liquid caused by the effect of shrinkage during solidification were included in the model.     

Effect of Turbulence Modeling on the Melt Flow and Inclusions Transport in a Steel Filtration Experiment

The current article deals with the effect of turbulence modeling on inclusions transport and melt flow in an induction crucible furnace (ICF), which was employed in order to investigate the efficiency and the performance of a ceramic filter. Furthermore, the influence of the discrete random-walk dispersion model on the behavior of inclusions in the ICF was investigated. Different turbulence models were employed in order to predict the turbulent melt flow. The numerical results show that the flow field is affected by the turbulence modeling method. Moreover, the distribution of the turbulent kinetic energy depends considerably on the choice of the turbulence model. In addition, the turbulence model and dispersion model also affect the inclusion transport in the melt. The filtration rate is also affected by the choice of the turbulence model.     

Physical and Numerical Simulation of Fluid Flow and Solidification at the Twin‐Roll Strip Casting Process

The fluid flow in a twin‐roll strip caster is investigated by physical and numerical simulation on a 1:1‐scale water model. A laser‐optical measurement technique (Laser Doppler Anemometry ‐ LDA) is used to validate the numerical results for the water flow. The numerical simulations are then transferred to the melt flow in the strip caster. The investigations are focused on different SEN concepts (submerged entry nozzle), a single‐nozzle system with two outlet ports and a double‐nozzle system with one outlet port each. The Influence of these concepts on the velocity, turbulence, and temperature distribution inside the liquid pool between the casting rolls and on the solidification and growth of the strip shells are investigated by numerical simulations (Computational Fluid Dynamics ‐ CFD). The non‐isothermal melt flow is calculated considering the solidification enthalpy as well as the behaviour of the solidifying melt. In addition to the numerical simulations of the melt flow inside the pool the temperature distribution in the cast strip is simulated. The SEN concept directly correlates with the temperature distribution Inside the strip. Furthermore, the surface temperature of the strip below the outlet of the roll gap is measured using a line‐scanner and is compared with the CFD simulation. In order to simulate the shape of the free surface in the liquid pool, CFD simulations of the water flow in the physical model are carried out using a Volume of Fluid model (VoF). This two‐phase model is able to reproduce free surface waves.     

Application of Three-Dimensional Hydrodynamic Model for Lake Okeechobee

A 3D hydrodynamic and heat transport model was developed for Lake Okeechobee. Continuity, momentum, and temperature transport equations were solved. Dynamically coupled transport equations for turbulent kinetic energy and turbulent scale also were solved. The numerical scheme used spatial finite differencing and a three-time-level, external-internal mode splitting procedure. A 28-day calibration was conducted, using measured bathymetry, rainfall, relative humidity, total solar radiation, wind velocity, inflow, and outflow data. During the calibration period, little rainfall occurred, and lake water levels receded. Water surface elevation, horizontal velocities, and temperature were computed. Agreement between observed and simulated values was based on graphical comparisons, minimizing mean absolute and root-mean-square errors, and spectral analysis. Comparisons showed that the model reproduced general observed trends and short-term fluctuations. The model's heat transport and turbulence closure schemes behaved as expected with regard to water column stratification and mixing. Simulation accuracy may potentially be improved by adding wind-wave and vegetation resistance algorithms to the model.     

耦合枝晶凝固微观特性的连铸宏观传输模型

在建立的连铸三维宏观传输数学模型中,对枝晶凝固微观结构参数的影响效果进行了充分地耦合研究。采用微观偏析半解析模型对浇铸钢种的非平衡凝固路径进行近似确定;采用复合理论方法对两相糊状区的渗透特性加以描述;在确定多孔介质的渗透率时,考虑了枝晶凝固模式的影响效果。应用该三维耦合模型,并结合早前建立的连铸二维传热模型,针对某钢厂方坯连铸机进行了复合数值模拟研究,且研究成果已投入到铸机的实际生产中。现场生产状况表明,仿真结果具有较好的合理性和实用性,说明耦合模型具有良好的仿真精度,可广泛应用于实际连铸过程的数值仿真研究中。