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 sulfide flash smelting process: Part I. Model development and verification with laboratory and pilot plant measurements for chalcopyrite concentrate smelting

A mathematical model has been developed to describe the various processes occurring in a flash furnace shaft. The model incorporates turbulent fluid dynamics, chemical reaction kinetics, and heat and mass transfer. The key features include the use of thek-ε turbulence model, incorporating the effect of particles on the turbulence, and the four-flux model for radiative heat transfer. The model predictions were compared with measurements obtained in a laboratory flash furnace and a pilot plant flash furnace. Good agreement was obtained between the predicted and measured data in terms of the SO₂ and O₂ concentrations, the amount of sulfur remaining in the particles, and the gas temperature. Model predictions show that the reactions of sulfide particles are mostly completed within about 1 m of the burner, and the double-entry burner system with radial feeding of the concentrate particles gives better performance than the singleentry burner system. The model thus verified was used to further predict various aspects of industrial flash furnace operation. The results indicate that from the viewpoint of sulfide oxidation, smelting rate can be substantially increased in most existing industrial flash furnaces. Formerly Graduate Student, Department of Metallurgical Engineering, University of Utah.     

镍闪速熔炼过程的模糊建模

针对冶金工业中镍闪速熔炼复杂工艺过程,提出了利用模糊理论建立镍闪模型的方法。一种方法是利用专家知识和操作经验（即ＩＦ－ＴＨＥＮ规则）建立闪速炉的先验模型;另一种方法是利用自适应模糊神经网络方法建立闪速炉的学习模型。综合考虑两种模型的建模结果,建立整个的模型。经过两个月的现场离线指导表明,这种建模方法能够较为准确地反映镍闪速炉的运行过程。     

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 minor-element behavior in flash smelting of copper concentrates and flash converting of copper mattes

A mathematical model has been developed to describe the behavior of minor elements during flash smelting and flash converting. The model incorporates equations describing volatilization of minor elements from the molten particles and distribution of these elements between the molten phases in the settler. The basic premise of the volatilization model is that at the surface of the molten particle, the partial pressures of the minor-element species are those at equilibrium. Transport of the minor-element species to the gas then is described by external mass transfer. Good agreement has been obtained between observed and predicted behaviors. The effects of oxygen enrichment, matte grade, and wall temperature, as well as the bath temperature, on minor-element behavior have been elucidated. Formerly Assistant Professor, Department of Metallurgy and Metallurgical EngineeringUniversity of Utah, Salt Lake City, UT Formerly Metallurgical Engineer Kennecott, Salt Lake City, UT     

脆硫铅锑矿闪速熔炼过程热力学分析

基于最小吉布斯自由能原理,采用元素势法,研究建立了脆硫铅锑矿闪速熔炼过程多相平衡热力学模型,考察了吨矿氧量（OVPTC）、富氧浓度（OG）和熔炼温度（T）对铅与锑在各平衡产物中分配比的影响。结果表明,对一定成分的脆硫铅锑矿,铅在合金中的分配比随OVPTC的增大不断下降,随T的升高略有上升;锑在合金中的分配比随OVPTC的增大有所下降,随T的升高大幅上升;熔炼烟气量随OG的增大而大幅减少。综合考虑铅和锑的直收率,脆硫铅锑矿闪速熔炼应控制一定的OVPTC,适当提高OG和T。      

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     

Thermochemical nature of minor elements in copper smelting mattes

Equilibrium distribution measurements of As, Sb, and Bi between molten copper and white metal were conducted at 1473 K using a static system. The results of this investigation have been used to evaluate the activity of those elements in white metal, and the results are compared to other data in the literature. An empirical model is presented and used to correlate the activity of As and Sb in matte as a function of the number of vacant electronegative sites in the matte, VS, and a parameter, ψ, used to represent the strength of the bond between Fe and the minor element in comparison to that which occurs with Cu. The activity coefficients of As and Sb in matte were found to be represented by the following equations:

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.     

高铁硫化锌精矿冶炼工艺研究进展

叙述了近年来针对高铁锌精矿冶炼工艺所开展的研究工作,详细介绍了高铁锌精矿中铁的自催化—高温加压浸出工艺。工业试验表明,该工艺锌浸出率达98.05%,铁浸出率为29.22%,可有效地实现锌的选择性浸出。     

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.     

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     

Mathematical modeling and computer simulation of the rotating impeller particle flotation process: Part II. Particle agglomeration and flotation

Effective removal of unwanted particles from a molten metal alloy by flotation relies on purging a gas into the melt through a rotating impeller. This device is commonly known as a rotary degasser. Unwanted particles in the melt attach to the rising gas bubbles and rise to the slag layer where they are removed from the metal bulk. In addition, the turbulence created by the rotating impeller causes the randomly distributed solid particles to agglomerate into relatively large clusters. These clusters float up or settle down due to the difference between their density and that of the melt. A mathematical model has been developed to describe the particle dynamics and particle agglomeration that occur during the rotary degassing of aluminum melts. While previous investigations addressed particle collisions in low intensity turbulent fields where the size of the colliding particles is smaller than the Kolmogorov length scale, this model is more encompassing as it considers both low intensity and high intensity turbulence. Consequently, this model is more representative of a typical industrial rotary degassing operation.     

Mechanical behavior of Al-Li/SiC composites: Part III. Micromechanical modeling

A micromechanical model is developed to compute the stress-strain curve of particle-reinforced metal-matrix composites under monotonic and cyclic deformation. The composite was modeled as a three-dimensional array of hexagonal prisms, each containing an intact or fractured reinforcement. The average stresses acting on the intact and damaged cells — as well as on the ceramic particles — were computed from the finite-element analysis of axisymmetric cylindrical cells, and the overall composite response was then calculated through an isostrain approach. The model was validated against the experimental results, reported in Parts I and II of this article, for an 8090 Al alloy reinforced with 15 vol pct SiC particles,^[1,2] where the matrix and reinforcement properties were obtained from mechanical tests on the unreinforced alloy and from quantitative microscopy analyses of the fraction of broken reinforcements in the composite. The critical mechanisms which controlled the deformation and damage processes in the composite during monotonic and cyclic deformation are discussed in light of the model results.     

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.     

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.     

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.     

Kinetics of the zinc slag-Fuming process: Part III. model predictions and analysis of process kinetics

In the final part of this paper the mathematical model of the slag fuming process, developed in Part II based on the analysis of industrial measurements from Part I, has been subjected to a sensitivity analysis, then employed to elucidate the rate limiting steps and to predict the influence of process variables on fuming. The kinetics analysis has been based on model predictions of fuming efficiency (Zn/coal) of the coal particles injected into the slag. The model predicts that fuming efficiency passes through a maximum with increasing residence time of coal particles in the slag. At shorter times, the zinc reduction kinetics are governed by the Boudouard Reaction, but at longer times, beyond the time at which the peak fuming efficiency is reached, the diffusion of ferric iron to the interface between the secondary bubbles containing the coal and the slag is rate determining. The level of ferric iron in the slag, which depends on ferrous iron oxidation rate, melting/freezing of slag at the water-cooled jacket, and ferric iron reduction by coal entrained in the slag, is therefore an important variable affecting the fuming kinetics. With respect to the influence of manipulable process variables, the model predicts that zinc fuming can be enhanced by increasing the fraction of coal entrained by the bath up to an optimum value at a fixed coal rate. An increase in entrainment could be achieved by injecting the desired portion of the coal at high pressure and solids loading through a small number of tuyeres. This strategy is preferable, from the standpoint of fuming efficiency, to simply increasing the rate of coal injection at normal pressures. Similarly, there is an optimum charge weight and bath height for a given furnace size. The best coal for zinc fuming, according to the model, has the following properties: low moisture and ash content, high fixed carbon (or volatiles), and high reactivity. Model predictions also suggest that there are advantages to fuming in a continuous operation rather than in a batch mode. Formerly Graduate Student     

Mechanical behavior of Al−Li/SiC composites: Part III. Micromechanical modeling

A micromechanical model is developed to compute the stress-strain curve of particle-reinforced metal-matrix composites under monotonic and cyclic deformation. The composite was modeled as a three-dimensional array of hexagonal prisms, each containing an intact or fractured reinforcement. The average stresses acting on the intact and damaged cells—as well as on the ceramic particles —were computed from the finite-element analysis of axisymmetric cylindrical cells, and the overall composite response was then calculated through an isostrain approach. The model was validated against the experimental results, reported in Parts I and II of this article, for an 8090 Al alloy reinforced with 15 vol pct SiC particles,^[1,2] where the matrix and reinforcement properties were obtained from mechanical tests on the unreinforced alloy and from quantitative microscopy analyses of the fraction of broken reinforcements in the composite. The critical mechanisms which controlled the deformation and damage processes in the composite during monotonic and cyclic deformation are discussed in light of the model results.     

Measurement of magnitude and direction of velocity in high-temperature liquid metals. Part I: Mathematical modeling

This article describes the development of a mathematical model that predicts the time required for a metal sphere to melt in a metal bath under different fluid flow conditions. The sphere is made from the same metal as the bath. The model solves numerically the pertinent momentum and energy equations in three dimensions, employing the SIMPLER algorithm. For the case of a pure metal, the model uses the heat integration algorithm to account for the latent heat of fusion. For the situation of a metal alloy with long freezing range, it incorporates the enthalpy method to account for the latent heat of fusion. The model is validated extensively: first, by using Paterson’s analytical solution; second, by using the experimental results of Gallium melting in a rectangular enclosure; and third, by using experimental results involving ice spheres melting in water. The practical use of this model is to study the influence of various parameters in the sphere melting system. This study facilitates the detection of liquid metal velocity using the sphere melting technique.