The modeling of pool profiles,temperature profiles and velocity fields in ESR systems

Through the simultaneous statement of Maxwell’s equations, the turbulent Navier-Stokes equations and the differential thermal energy balance equations a mathematical model has been developed to represent the pool profiles, the velocity fields and the temperature profiles in an ESR system. The major advance over earlier modeling efforts is the fact that the model is capable of predicting the pool profiles from first principles. The theoretically predicted pool profiles and temperature fields were found to be in reasonable agreement with experimental measurements reported by Mellberg for a laboratory scale system. The model is used to investigate the interdependence of key process parameters, with the power input, fill ratio, amount of slag used and the position of the electrode as the independent variables and the casting rate, pool depth, velocity and temperature fields as the dependent variables.     

A mathematical model for the dynamic behavior of melts subjected to electromagnetic forces: Part I. model development and comparison of predictions with published experimental results

A mathematical model was developed to predict electromagnetically driven flow and (particularly) free surface behavior in melts subject to electromagnetic forces. Such melts appear in electromagnetic casters, induction furnaces, and other metal processing units. The calculations started with Maxwell’s equations and Ohm’s law, which were solved by a novel “modified hybrid technique.” The instantaneous continuity and Navier-Stokes equations (rather than their time-averaged versions) were then solved with electromagrretic forces as input. The calculations allowed for the dynamic behavior of the free surface of the melt, and electromagnetic fields were recomputed as the free surface changed. In this first part of a two-part article, the model predictions are compared with the experimental measurements of induced current, magnetic field, melt velocity, and free surface deformation reported by others.     

Experimental measurement and numerical computation of velocity and turbulence parameters in a heated liquid metal system

Experimental measurements are reported describing the velocity field and the turbulence parameters in molten Woods metal due to the passage of a DC current between two electrodes immersed into the melt. The measurements were made using a hot film anemometer. A mathematical model has been developed to represent these measurements; in essence, this relied on the solution of Maxwell’s equations to represent the electromagnetic force field and turbulent Navier-Stokes equations to represent the fluid flow field. Three different turbulence models were examined; two were variants of thek-ε model, while the third was mixing length model type. In general, the velocities were well predicted by all three of these models, but there were significant discrepancies as far as the turbulence parameters were concerned. In the paper, a new criterion was suggested to predict the onset of turbulence in systems of this type.     

Experimental measurement and numerical computation of velocity and turbulence parameters in a heated liquid metal system

Experimental measurements are reported describing the velocity field and the turbulence parameters in molten Woods metal due to the passage of a DC current between two electrodes immersed into the melt. The measurements were made using a hot film anemometer. A mathematical model has been developed to represent these measurements; in essence, this relied on the solution of Maxwell’s equations to represent the electromagnetic force field and turbulent Navier-Stokes equations to represent the fluid flow field. Three different turbulence models were examined; two were variants of thek-ε model, while the third was mixing length model type. In general, the velocities were well predicted by all three of these models, but there were significant discrepancies as far as the turbulence parameters were concerned. In the paper, a new criterion was suggested to predict the onset of turbulence in systems of this type.     

A mathematical model of slag and metal flow in the ESR Process

Through the statement of the turbulent Navier-Stokes equations and Maxwell’s equations a mathematical representation is developed for the electromagnetic force field and the velocity field in the slag phase and the metal pool of cylindrical ESR units. Computed results are presented for both industrial scale (0.5 m electrode diameter) and laboratory scale (0.05 m electrode diameter) units operating with direct currents. It was found that for industrial scale units, the computed slag velocities ranged from 5 to 10 cm/s, while the velocities in the metal pool were substantially lower, except at the slag-metal interface. At a given spatial position, the velocity was found to increase in an almost linear fashion with the current density. The flow was found to be predominately laminar in the laboratory scale units and for comparable current densities the melt velocities were very much smaller. Some 600 to 900 s were required on a CDC 6400 digital computer for the solution of each case involving turbulent flow. Presently on leave from Institute of Chemical Engineering and Technology, Punjab University, Lahore-1, Pakistan.     

Turbulent recirculating flow in induction furnaces: A comparison of measurements with predictions over a range of operating conditions

Experimental measurements and theoretical predictions are presented concerning the velocity fields, the maps of the turbulent kinetic energy, and the turbulent kinetic energy dissipation in an inductively stirred mercury pool. A single coil arrangement was used, and the frequencies examined ranged from 50 to 5000 Hz. A hot film anemometer and a direction probe were employed for characterizing the velocity fields. The theoretical predictions were based on the numerical solution of the turbulent Navier-Stokes equations. The technique of mutual inductances was employed to compute the magnetic field, while thek-ε model was used for calculating the turbulent viscosity. Overall, the theoretical predictions were in reasonable agreement with the measurements both regarding the velocities and the turbulence parameters. By presenting the results in a normalized, dimensionless form these findings were given a rather broader applicability than the actual numerical range explored. Formerly of the Department of Materials Science and Engineering at MIT     

Flow pattern velocity and turbulence energy measurements and predictions in a water model of an argon-stirred ladle

Experiments were carried out using a simplified water model of an argon-stirred ladle system. The flow patterns were determined by a flow visualization technique and the velocity and turbulence energy fields were quantitatively measured using hot-film anemometry. The latter quantities were predicted by solving the turbulent Navier-Stokes equations using Spalding’sk-W model for the turbulence viscosity. There is semiquantitative agreement between predictions and measurements. Mixing lengths also were computed. This agreement between measurements and predictions provides further evidence that modeling is a promising approach for the study of recirculating turbulent flows in steel processing operations. J. SZEKELY, formerly of the State University of New York at Buffalo.     

The measurement and prediction of the melt velocities in aturbulent,electromagnetically driven recirculating low melting alloy system

Experimental measurements are reported describing the velocity field in an inductively stirred low melting alloy. The molten metal was held in a cylindrical container, 500 mm high and having an inside diameter of 250 mm; induction stirring was supplied by a three phase coil, which provided a maximum field strength of 350 Gauss (0.035 Wb/m²). The velocities in the melt were measured by a mechanical force reaction probe and were found to range up to about 0.5 m/s. Theoretical prediction of the melt velocities was made by solving Maxwell’s equations, together with the turbulent Navier-Stokes equations, using a digital computer. The experimentally measured and theoretically predicted velocities were found to agree within about 30 pct, thus providing direct experimental proof for the validity of modelling electromagnetically driven flows using this technique.     

Modeling of materials synthesis in hybrid plasma reactors: Production of silicon by thermal decomposition of SiCI4

A mathematical model for the analysis and design of a combined direct current (DC) and radio frequency (RF) plasma reactor for advanced materials synthesis has been developed. The RF electromagnetic field is calculated by solving Maxwell’s equations expressed in terms of the vector potential, which permits great flexibility in the specification of the RF coil design. The velocity and temperature fields in the plasma are calculated by solving the turbulent Navier-Stokes equations and the thermal energy balance equation. The model is used to study the thermal decomposition of silicon tetrachloride through the solution of the mass diffusion equation arid the associated reaction kinetics. It is found that the coupling between the flow and temperature fields generated by the RF and DC components significantly improves the silicon production and recovery in the reactor as compared with either DC or RF systems alone. In addition, the conversion efficiency of the hybrid system is found to depend on both the design and the operating parameters of the RF coil.     

Heat transfer and fluid flow phenomena in electroslag refining

A mathematical formulation has been developed to represent the electromagnetic force field, fluid flow and heat transfer in ESR units. In the formulation, allowance has been made for both electromagnetically driven flows and natural convection; furthermore, in considering heat transfer the effect of the moving droplets has been taken into account. The computed results have shown that the electromagnetic force field appears to be the more important driving force for fluid motion, although natural convection does affect the circulation pattern. The movement of the liquid droplets through the slag plays an important role in transporting thermal energy from the slag to the molten metal pool, although the droplets are unlikely to contribute appreciably to slag-metal mass transfer The for-formulation presented here enables the prediction of thermal and fluid flow phenomena in ESR units and may be used to calculate the electrode melting rates from first principles. While a detailed comparison has not yet been made between the predictions based on the model and actual plant scale measurements, it is thought that the theoretical predictions are consistent with the plant-scale data that are available.     

Fluid flow and mass transfer in an inductively stirred four-ton melt of molten steel: A comparison of measurements and predictions

Experimental measurements are reported on melt velocities and on the rate at which immersed carbon rods dissolve in a 4-ton induction furnace, holding a low carbon steel melt. These measurements are compared with theoretical predictions, based on the numerical solution of Maxwell’s equations and the turbulent Navier-Stokes equations. In general, good agreement has been obtained, both regarding the absolute values of the velocities and the mass transfer coefficients and the trends predicted by the theoretical analysis. In addition to providing further proof regarding the applicability of the mathematical modeling technique, the principal contribution of the work is that it provides an improved insight into the behavior of inductively stirred melts. In particular it was found that for an inductively stirred melt both the velocities and the rate of turbulence energy dissipation are relatively uniform spatially, in contrast to bubble stirred systems, where most of the agitation is confined to the jet plume and to the near surface region. It was found, furthermore, that the mass transfer coefficient characterizing the rate of dissolution of immersed carbon rods depends both on the absolute values of the melt velocity and on the local values of the turbulence intensity; thus significant mass transfer will occur in the region of the eye of the circulation, where the absolute value of the mean velocity is small. On leave at Massachusetts Institute of Technology     

The mathematical and physical modeling of turbulent recirculating flows

A physical model has been constructed to represent turbulent recirculating flows that occur in argon-stirred ladles. By using a mechanically driven circulating system including a moving tube it was possible to generate flow fields such that all the boundary conditions could be defined unambiguously. The velocity fields developed in the system and the spatial distribution of the turbulent kinetic energy were measured experimentally using a laserdoppler anemometer. The experimental measurements were found to be in good agreement with predictions based on theK-W model for turbulent recirculating flows, provided appropriate wall functions were used. A simplified model was also described in the paper, for representing the transient decay of turbulence in teemed systems or in bubble stirred vessels after the agitation had been terminated. This model, which in essence involved the use of a simple algebraic relationship, gave semiquantitative agreement with measurements. R.METZ, formerly Graduate Student Department of Chemical Engineering, State University of New York at Buffalo,     

Modeling Natural Convection in Copper Electrorefining: Describing Turbulence Behavior for Industrial-Sized Systems

A computational fluid dynamics (CFD) model of copper electrorefining is discussed, where natural convection flow is driven by buoyancy forces caused by gradients in copper concentration at the electrodes. We provide experimental validation of the CFD model for several cases varying in size from a small laboratory scale to large industrial scale, including one that has not been compared with a CFD model. Previously, the large-scale systems have been thought to be turbulent by some workers and modeled accordingly with k-ε type turbulence models, but others have not considered turbulence effects in their modeling. We find that the turbulence model does not predict turbulence exists; however, we analyze carefully the fluctuation statistics predicted for a transient model, finding that most cases considered do exhibit a type of turbulence, an instability related to the interaction between velocity and copper concentration fields. We provide a comparison of the extent of turbulence for various electrode heights, and gap widths, and we emphasize industrial-sized electrorefining cells.     

Heat transfer and fluid flow phenomena in electroslag refining

A mathematical formulation has been developed to represent the electromagnetic force field, fluid flow and heat transfer in ESR units. In the formulation, allowance has been made for both electromagnetically driven flows and natural convection; furthermore, in considering heat transfer, the effect of the moving droplets has been taken into account. The computed results have shown that the electromagnetic force field appears to be the more important driving force for fluid motion, although natural convection does affect the circulation pattern. The movement of the liquid droplets through the slag plays an import-ant role in transporting thermal energy from the slag to the molten metal pool, although the droplets are unlikely to contribute appreciably to slag-metal mass transfer. The for-formulation presented here enables the prediction of thermal and fluid flow phenomena in ESR units and may be used to calculate the electrode melting rates from first principles. While a detailed comparison has not yet been made between the predictions based on the model and actual plant scale measurements, it is thought that the theoretical predictions are consistent with the plant-scale data that are available. Presently on leave from Institute of Chemical Engineering and Technology, Punjab University, Lahore-1, Pakistan.     

Magnetohydronamic and thermal behavior of electroslag remelting slags

The paper is based on the development and use of a mathematical model that simulates the electroslag remelting (ESR) operation. The model assumes axisymmetrical geometry and steady state. Maxwell equations are first solved to determine the electromagnetic forces and Joule heating. Next, coupled fluid flow and heat transfer equations are written for the two liquids (slag and liquid metal). Thek-ε model is used to represent turbulence. The system of coupled partial differential equations is then solved, using a control volume method. Using the operating parameters as inputs, the model calculates the current density, velocity, and temperature throughout the fluids. This paper is concerned with fluid flow and heat transfer in the slag phase. After being validated by comparing its results with experimental observation, the model is used to evaluate the influence of operating variables, such as the fill ratio, and the thermophysical properties of the slag.     

Measurements of magnetic fields and electromagnetically driven melt flow in a physical model of a hall-héroult cell

A physical model (approximately one-tenth scale and operated at 1000 A) was constructed to simulate the electromagnetically driven flow occurring in Hall cells. The model contained Wood’s metal, in which magnetic fields and velocities were measured, as the single liquid. Data have been generated for (future) comparison with mathematical models of Hall cells. The model has also proved useful in examining the effects of cell changes and upsets long thought by operators to have an influence on cell performance. Effects of current maldistribution in the “collector bars,” “cold” anodes, “muck,” and alternative bus-bar arrangements have been observed. In many cases, these effects can be predicted qualitatively from an examination of the model’s magnetic field. S.K. Banerjee, having received his Ph.D. from the Department of Materials Science and Mineral Engineering at the University of California.     

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.     

A 3-D numerical prediction of turbulent flow,heat transfer and solidification in a continuous slab caster for steel

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.     

Turbulent Fluid Flow Phenomena in a Water Model of an AOD System

Experimental measurements are reported regarding fluid flow and turbulence property measurements in a water model of an AOD vessel. Laser velocimetry was used to determine the time smoothed velocities, the turbulent kinetic energy, and the Reynolds stresses in the system; in addition, the rate of melting of immersed ice rods was also measured to determine the local heat transfer rates. The measurements have shown that for the model AOD studied both the velocity fields and the distribution of the turbulent kinetic energy were quite uniform; the absence of inactive or dead zones would render these systems ideal for mixing and for a range of ladle metallurgical operations. The rate at which immersed ice rods dissolved depended on both the local velocities and on the turbulence levels; a previously developed correlation could be employed to predict the appropriate heat transfer coefficients. Finally, the rate of turbulent energy dissipation per unit volume in real industrial AOD vessels was found to be much higher than in any other ladle metallurgy operations. This could raise interesting possibilities regarding the more widespread use of these systems for molten metals processing.     

The trajectories and distribution of particles in a turbulent axisymmetric gas jet injected into a flash furnace shaft

Numerical computations have been performed for the behavior of a vertical turbulent particle-laden gas jet exemplified by the shaft region of a flash-smelting furnace. The two-equation(k-ε) model was used to describe turbulence. Model predictions for the gas and solid flow fields give a satisfactory representation of experimental data taken from the literature. The predictions of flow properties of the two phases under flash-smelting conditions have been obtained for various inlet conditions, particle sizes, particle loading, and oxygen enrichment. Model predictions show that the axial velocity of the particle phase is substantially higher than that of the gas phase. The presence of solid particles causes the axial velocity of the gas phase to be greater near the centerline and lower in the outer region than in a single-phase gas jet. A more uniform distribution of particles was obtained by introducing a strong radial velocity of the distribution air at the inlet. The implications of the behavior of a particle-laden gas jet on flash-smelting processes arc discussed.