Mathematical modeling and computer simulation of the rotating impeller particle flotation process: Part I. Fluid flow

Removal of unwanted particles from molten metal by flotation is one of the most useful melt cleansing techniques used by the foundry industry. An effective way of flotation of particles in a melt relies on purging a gas into the molten metal through holes in a rotating impeller. Impeller rotation creates turbulence inside the melt, which helps agglomerate the impurity particles and, thereby, enhances their removal from the melt. In addition, turbulence increases the probability of particles attaching to the rising gas bubbles and, therefore, enhances the chance of their removal from the molten metal. A mathematical model has been developed to simulate the turbulent multiphase flow field inside the flotation treatment furnace. Simulations based on the model were used to demonstrate the effect of the various process parameters on the performance of a batch-type rotating impeller particle flotation process.     

Mathematical modeling of sulfide flash smelting process: Part II. Quantitative analysis of radiative heat transfer

A mathematical model has been developed to describe the rate processes in an axisymmetric copper flash smelting furnace shaft. A particular feature of the model is the incorporation of the four-flux model to describe the radiative heat transfer by combining the absorbing, emitting, and anisotropic scattering phenomena. The importance of various subprocesses of the radiative heat transfer in a flash smelting furnace has been studied. Model predictions showed that the radiation from the furnace walls and between the particles and the surrounding is the dominant mode of heat transfer in a flash smelting furnace. Formerly Graduate Student, Department of Metallurgical Engineering, University of Utah.     

Mathematical modeling of the argon-oxygen decarburization refining process of stainless steel: Part I. Mathematical model of the process

Some available mathematical models for the argon-oxygen decarburization (AOD) stainless steelmaking process have been reviewed. The actual situations of the AOD process, including the competitive oxidation of the elements dissolved in the molten steel and the changes in the bath composition, as well as the nonisothermal nature of the process, have been analyzed. A new mathematical model for the AOD refining process of stainless steel has been proposed and developed. The model is based on the assumption that the blown oxygen oxidizes C, Cr, Si, and Mn in the steel and Fe as a matrix, but the FeO formed is also an oxidant of C, Cr, Si, and Mn in the steel. All the possible oxidation-reduction reactions take place simultaneously and reach a combined equilibrium in competition at the liquid/bubble interfaces. It is also assumed that at high carbon levels, the oxidation rates of elements are primarily related to the supplied oxygen rate, and at low carbon levels, the rate of decarburization is mainly determined by the mass transfer of carbon from the molten steel bulk to the reaction interfaces. It is further assumed that the nonreacting oxygen blown into the bath does not accumulate in the liquid steel and will escape from the bath into the exhaust gas. The model performs the rate calculations of the refining process and the mass and heat balances of the system. Also, the effects of the operating factors, including adding the slag materials, crop ends, and scrap, and alloy agents; the nonisothermal conditions; the changes in the amounts of metal and slag during the refining; and other factors have all been taken into account. []—metal phase; ()—slag phase; {}—gaseous phase; and 〈〉—solid phase     

Fluid dynamics in bubble stirred ladles: Part II. Mathematical modeling

A mathematical model which describes the fluid flow in a bubble stirred ladle is presented. The model predicts mean flow, turbulent characteristics, bubble dispersion, and gas-liquid interaction from fundamental principles. Numerical predictions for a water model of a ladle show very satisfactory quantitative agreement with experimental results for all regions of the ladle. The model is applied to the study of refractory wear and yields results that are in qualitative agreement with practical experience. Formerly with the Department of Chemical Engineering and Fuel Technology, Sheffield, United Kingdom     

Fluid dynamics in bubble stirred ladles: Part II. Mathematical modeling

A mathematical model which describes the fluid flow in a bubble stirred ladle is presented. The model predicts mean flow, turbulent characteristics, bubble dispersion, and gas-liquid interaction from fundamental principles. Numerical predictions for a water model of a ladle show very satisfactory quantitative agreement with experimental results for all regions of the ladle. The model is applied to the study of refractory wear and yields results that are in qualitative agreement with practical experience.     

Mathematical modeling of the argon-oxygen decarburization refining process of stainless steel: Part II. Application of the model to industrial practice

The mathematical model proposed and presented in Part I of the present work has been used to deal with and analyze the austenitic stainless steel making (including ultralow-carbon steel) and has been tested on data of 32 heats obtained in producing 18Cr9Ni-grade steel in an 18-t argon-oxygen decarburization (AOD) vessel. The results indicated that the carbon concentrations and bath temperatures at the endpoints of blowing periods, calculated by the model, are in excellent agreement with the determined data, and the Cr content after the predeoxidization, obtained from the model predictions, also agrees very well with the observed value. The Gibbs free energies of the oxidation reactions of elements can be used to characterize fully the competitive oxidation among the elements during the refining process and to determine reasonably the corresponding distribution ratios of oxygen. The critical carbon concentration of decarburization (after which the decarburization changes to become controlled by the mass transfer of carbon in molten steel) for the AOD refining process of austenitic stainless steel in an 18-t AOD vessel is in the range of 0.25 to 0.40 mass pct. The model can provide some very useful information and a reliable basis for optimization of the technology of the AOD refining process of stainless steel and control of the process in real time and online.     

Transport phenomena in electric smelting of nickel matte: Part II. Mathematical modeling

A three-dimensional mathematical model was developed to simulate the distributions of electrical potential, heat release, temperature, and velocity in the slag and matte in a six-in-line 36 MVA capacity furnace for smelting nickel calcine. From Part I of this series, it was found that there was a substantial electrical potential drop at the electrode surface, likely due to arcing through evolved carbon monoxide. The incorporation of this phenomenon into the model permitted accurate calculation of the current, power, and temperature distributions in the slag and matte. The slag was found to be thermally homogenized due to the evolved gas, and to a lesser extent by natural convection. In contrast, the matte was thermally stratified; this finding was attributed to poor momentum transfer across the slag/matte interface. Ninety percent of the electrical energy was used in smelting reactions in the calcine; to simulate the heat transfer from the slag to the calcine, a heat transfer coefficient was deduced from plant data. The implications of these findings for stable furnace operation are discussed.     

Mathematical modeling of sulfide flash smelting process: Part III. Volatilization of minor elements

The mathematical model described in Part I^[14] was extended to include the minor element behavior inside a flash-furnace shaft during flash smelting of copper concentrate. The volatilization of As, Sb, Bi, and Pb was computed, and experiments were carried out for Sb and Pb in a laboratory flash furnace. Satisfactory agreement between the predicted and measured results was obtained for antimony and lead. From the computational results, the behavior of each minor element was predicted for various target matte grades. The model predictions show that the elimination of As and Bi to the gas phase increases sharply at about 0.3 m from the burner; however, that of the Sb increases gradually along the flash-furnace shaft, and that of lead occurs suddenly at about 0.6 m from the burner. The predicted results also show that the elimination increases for Bi and Pb as the target matte grade increases; however, it is relatively independent of the target matte grade between 50 and 60 pet Cu for As and Sb. At higher target matte grades above 60 pet Cu, the elimination of As and Sb decreases as matte grade increases. formerly Graduate Student, Department of Chemical Engineering, University of Utah,     

Tree-ring formation during vacuum arc remelting of INCONEL 718: Part II. Mathematical modeling

Tree-ring grain formations, a common microstructural feature found in vacuum arc remelted (VAR) ingots of nickel-based superalloys, were characterized experimentally in Part I. The experimental observations led to the conclusion that tree rings are chains of fine-equiaxed grains interrupting a predominately columnar-dendritic structure. Several possible mechanisms for their formation were considered, and their implications correlated with experimental observations. The most likely mechanism was determined to be that process perturbations cause changes in the thermal (or solutal) fields ahead of the columnar-dendrite tips, temporarily altering the conditions to increase grain nucleation and, hence, forming fine-equiaxed grains. In this article, Part II, a multiscale mathematical model of the VAR process is presented that simulates the macroscopic heat and momentum transport and combines it with a mesoscopic model of the nucleation and growth of grains. Using this multiscale model, the transient development of the VAR grain structure was simulated with varying levels and durations of fluctuations in the principal process parameters: power supply, arc focus, melt rate, and the ingot-crucible heat-transfer coefficient. The simulations were shown to agree with optical and electron back-scattered diffraction (EBSD) measurements of grain morphology and crystallographic orientation. The model results predict that tree-ring structures (consistent with those observed experimentally) can be formed by process perturbations that alter the thermal field conditions at the solidification front. A sensitivity study of the effect of the different process fluctuations on the microstructure formation was performed, providing process maps predicting the range of conditions where tree rings will not form.     

Object-oriented simulation of hydrometallurgical processes: Part II. Application to the bayer process

The steady-state process simulator developed by Kiranoudis et al. has been used for the detailed simulation of the Bayer process flowsheet. It consists of the digestor/flash section as well as the precipitation and washing circuits. Advanced hydrometallurgical process models for the specific unit operations involved were developed and are appropriately described. The simulation studies mainly focus on studying the overall effects of certain design parameters on the entire plant efficiency. The overall performance of the bauxite digestion section is greatly and positively affected by the free-soda concentration of the feed liquor, the corresponding concentration of solid particles, and the operation temperature of the circuit. Precipitation of alumina in crystallizers is greatly affected by the corresponding soda concentration of the washing unit product stream, indicating the importance of this section. Furthermore, ambient temperature is important to the precipitation kinetics, influencing negatively the quantitative precipitation but resulting in particle populations of higher mean diameter.     

Simulation of die filling for the wax injection process: Part II. Numerical simulation

This article is the second of two articles dedicated to the modeling of wax injection for the investment casting (or lost wax) process. This article presents the experiments and numerical simulations carried out in order to validate the models developed and presented in a previous article.^[1] Three different experiments have been used. The first was to characterize the flow of the wax during the filling and the capacity of the models developed to describe it accurately, in the liquid state. The second experiment was to test the capacity of the models to predict the apparition of filling defects. The third was to compare the predictions in liquid and paste or semisolid state. A good agreement between experiments and simulations has been found, showing that the models are able to represent the behavior of the wax used.     

Study of transient flow and particle transport in continuous steel caster molds: Part II. Particle transport

Particle motion and capture in continuous steel casters were simulated using a Lagrangian trajectory-tracking approach, based on time-dependent flow fields obtained from large-eddy simulations (Part I of this article). A computation was first conducted on a water model of a full-scale standard slab caster, where measurements were available. It simulated the transport of 15,000 plastic particles and their removal by a screen positioned near the mold top surface. The computation shows the screenremoval fractions to be 27±5 pct for 0 to 10 seconds and 26±2 pct for 10 to 100 seconds, which agrees with previous measurements. The flow exiting the nozzle was relatively uniform, and turbulent motion in the domain was very chaotic, so particle removal did not depend on the initial location of particles introduced in the nozzle port. A computation of motion and capture of 40,000 small inclusions (10 and 40 μm) was then performed in an actual thin-slab steel caster. The particles moved through the mold region with an asymmetrical distribution, which was caused by transients in fluid turbulence in the lower recirculation region, rather than by inlet variations at the nozzle port. Only about 8 pct of these small particles were removed to the top surface. This removal fraction was independent of both particle size and density, likely because all the simulated particles were too small to deviate significantly from the surrounding fluid flow. Finally, the computational results were further processed to predict the ultimate distribution of impurity particles in the solid thin slab after a short burst of inclusions entered the mold. They were reprocessed to reveal the distribution of total oxygen content for a steady inclusion supply from the nozzle. The results of this work confirm the important role of flow transients in the transport and capture of particles during continuous casting and can serve as a benchmark for future simplified models.     

Fluid flow in pachuca (Air-Agitated) tanks: Part II. Mathematical modeling of flow in pachuca tanks

A generalized mathematical model that combines Bernoulli’s equation and thek-ε model of turbulence, using only the gas flow rate and tank geometry as inputs, has been formulated to predict the fluid flow pattern in industrial-scale full-center-column Pachuca tanks. Predictions from this model reproduce reasonably well the trends that were observed in experiments when design and operating parameters, such as superficial air velocity and tank height/tank diameter ratio, were varied. Results from these calculations indicate that design parameters, such as tank height/tank diameter and draft tube/tank diameter ratios, have a significant effect on the flow pattern in full-center-column Pachuca tanks at large tank diameters or small tank height/tank diameter ratios.     

Mathematical modeling of the zinc pressure leach process

A comprehensive mathematical model is described for the zinc pressure leaching process. Generic kinetic expressions were derived from experimental data found in the literature for the following reaction events: (1) dissolution of marmatite, (Zn,Fe)S, (2) oxidation of ferrous to ferric, and (3) precipitation of lead jarosite. Aqueous solution properties, oxygen solubility, density, enthalpy, and vapor pressure, were correlated with solution composition and temperature. Subsequently, a kineticsbased model for simultaneous sulfide dissolution and iron precipitation in a multistage, three-phase reactor was developed. The population balance method was used for sulfide mineral material balances, and apparent equilibrium was assumed for iron precipitation. A gas-phase material balance was included, which allows for prediction of oxygen utilization. The model was solve for a particular operation of the Cominco Ltd. (Trail, BC) autoclave, and prediction results were shown to be in very good comparison with actual plant performance.     

Mathematical model for computer simulation and control of steelmaking

A dynamic model for computer simulation and control of steelmaking has been developed. It is essentially based on multicomponent mixed transport control theory with the incorporation of energy balance calculations. The model is applicable to both steelmaking in electric furnaces as well as in oxygen steelmaking converters. The adjustable parameters of the model for simulation of oxygen steelmaking are gas evolution rate (G_co). oxygen flux factor (F_o) and emulsification factor (EM). These simulation parameters, when combined with on-line measurement of off-gas composition and temperature, enable complete dynamic control of the process. The model developed is applied, as an example, to an industrially produced heat in a top blown oxygen steelmaking converter and the results of simulation are discussed.     

Mathematical model of COREX melter gasifier: Part II. Dynamic model

A dynamic model of the COREX melter gasifier is developed to study the transient behavior of the furnace. The effect of pulse disturbance and step disturbance on the process performance has been studied. This study shows that the effect of pulse disturbance decays asymptotically. The step change brings the system to a new steady state after a delay of about 5 hours. The dynamic behavior of the melter gasifier with respect to a shutdown/blow-on condition and the effect of tapping are also studied. The results show that the time response of the melter gasifier is much less than that of a blast furnace.     

Fundamental Mathematical Model for AOD Process. Part II: Model validation

A fundamental mathematical model for AOD process has been developed and proposed in “Fundamental Mathematical Model for AOD Process. Part I: Derivation of the Model” 1 . Validation of the model with process data, measured from full scale AOD process, is presented in this paper. A broad selection of input data for the model was exported from various types of full scale industrial AOD heats. In this study 6 different types of heats were studied and simulated. Process data was measured from two AOD converters (95 t, 150 t). Validation of the model was then done by comparing simulated and measured values for carbon and chromium content, carbon release rate, melt composition, slag composition and bath temperature during final stages of carbon removal. The validation results showed that the model was in good agreement with the measured process data, and same model parameters were valid in all of the simulated heats.     

Directional dendritic solidification of a composite slurry: Part II. Particle distribution

The present work attempts to form an understanding of particle distribution during dendritic solidification of a composite slurry. It is shown that the magnitude and nature of the forces involved between a foreign particle and a growing dendrite are distinctly different from those during plane front solidification. A particle distribution map, based on theoretically evaluated forces, is proposed to predict the conditions under which engulfment, entrapment, or pushing of particles occurs during dendritic solidification. Directional solidification experiments have been conducted to study particle distribution, and comparison of these results with theoretical predictions based on the map agree well with a large number of experimental observations reported to date.     

Mathematical model of the thermal processing of steel ingots: Part II. Stress model

A mathematical model has been developed to predict the internal stresses generated in a steel ingot during thermal processing. The thermal history of the ingot has been predicted by a finite-element, heat-flow model, the subject of the first part of this two-part paper, which serves as input to the stress model. The stress model has been formulated for a two-dimensional transverse plane at mid-height of the ingot and is a transient, elasto-viscoplastic, finite-element analysis of the thermal stress field. Salient features of the model include the incorporation of time-temperature and temperature-dependent mechanical properties, and volume changes associated with nonequilibrium phase transformation. Model predictions demonstrate that the development of internal stresses in the ingot during thermal processing can be directly linked to the progress of the phase transformation front. Moreover, the low strain levels calculated indicate that metallurgical embrittlement must be very important to the formation of cracks in addition to the development of high tensile stresses. B. G. THOMAS, formerly a Graduate Student at the University of British Columbia