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By J. W. Evans

Mathematical modeling has now become a standard technique in the development of new technology and the improvement of existing technology in the processing industries. Physical modeling has a longer history but a smaller following at present. The paper reviews some examples of the application of both types of modeling in non-ferrous metal extraction, particularly within the copper and aluminum industries. The objective is to show that well founded models can lead to results that have practical consequence in scaleup, design and operation.

Jan 1, 1993

By Pinczewski WV, Nowak LP, Tanzil WBU, Jenkins DR, McCarthy MJ

An experimental and theoretical study has been made of the drainage behaviour of two-dimensional packed beds drained from an outlet in the sidewall of the bed in order to identify the fundamental me- chanisms involved. Under these conditions a grad- ient develops in the air-liquid interface in the direction of flow with the interface tilting towards the outlet. This results in a significant volume of liquid being retained in the packing on completion of draining when the interface first reaches the bed outlet. Residual liquid volume and the shape of the air-liquid interface as a function of time were det- ermined experimentally in a two-dimensional packed bed where the packing consisted of glass spheres ranging in size from 0.6 to 10 mm dia. and the bed was drained at superficial liquid velocities rang- ing from 0.007 to 0.32cm/s. The test liquids used were water, glycerine/water solutions and kerosene resulting in a viscosity range of 1 to 10 mPa.s and a surface tension range of 0.0243 to 0.072 N/m. Assuming laminar flow within the bed, a compu- tational model was developed to predict both the liquid residual ratio and the temporal evolution of the drainage profile within the bed. The computed results were in good agreement with the experimental data over the entire range of conditions invest- igated. the computational model was used to estimate the relative importance of the effects of changes in hearth conditions on residual slag volume for typ- ical blast furnace operating conditions. On the basis of the computed results it is concluded that coke quality is a major parameter affecting hearth drainage performance.

Jan 1, 1981

By Y Li, D J. Williams, E Topal

A mine waste rock dump does not have any intrinsic value, yet is an inevitable structure in surface mining operations. Since the purpose of mining is to maximise profit, possible expenditure reduction in hauling waste rock to the dump, which can represent up to 50 per cent of the total operating cost, is highly desirable. Apart from cost, potential acid mine drainage (AMD), due to seepage from surface waste rock dumps containing reactive waste rock, could threaten the ecological balance, drinking water supplies and irrigation systems downstream from a mine site. In order to minimise these potential risks, selective placement of benign and reactive waste rock, and the encapsulation of reactive waste rock by benign waste rock, need to be carried out during the construction of the rock dump.In this paper, a new methodology, based on mixed integer programming (MIP) is developed to manage waste rock dumping. The model aims to minimise waste rock haulage costs by locating the optimum destination(s) for each waste rock block, while at the same time controlling the placement of reactive waste rock to minimise the potential for AMD. Tests on small-scale virtual data demonstrated that the model is capable of dealing with selective placement of different waste rock types, encapsulating reactive, potentially acid forming (PAF) waste rock, and stockpiling benign waste rock for encapsulation and rehabilitation purposes. The solution time for the MIP model is far less than the traditional manual approach, and the resulting schedule is up to 46 per cent more cost effective.CITATION:Li, Y, Topal, E and Williams, D J, 2012. Mathematical approach for better mine waste rock dumping management, in Proceedings Life-of-Mine 2012 , pp 207-214 (The Australasian Institute of Mining and Metallurgy: Melbourne).

Jul 10, 2012

By Marius B. Wechsler

There are considered deep piles which are fixed in blocks of rotating machinery foundations. A mathematical background is developed for determining the behavior of this kind of piles for motions caused by horizontal and vertical oscillations, as well as by roc-king and twisting. It is assumed that the piles come across a top soil layer of depth c, not capable of lateral support, and penetrate over a length b, in a bottom firm soil layer, or in a bottom soft soil layer in which vibrations can produce voids around the piles. With respect to b and c, as well as to the elastic properties of piles and soils, the analysis leads to the maximum bending moments and shear forces subjecting the piles, to the maximum lateral bearing on the bottom soil layer, as well as to the deflection of the pile and the displacement of the foundation block. The results of the performed analysis can be applied to seismic loads too, and they are illustrated by a numerical example.

Jan 1, 1987

By H. H. Kellogg

The thermochemical properties of molten solutions in the binaries Cu-Bi, Cu-Sb and Bi-Sb, as well as the ternary Cu-Bi-Sb, have been correlated and described in mathematical form by use of the associated species concept combined with the three-suffix Margules equations to account for the non-idea 1 behavior of the solution species. Agreement of the correlations with experimental data is excellent (relative deviation of less than 2.5% in the values of the activity coefficients) in most cases. The correlations are applicable over the entire composition range of each system, and in the temperature range 700°-1250°C.

Jan 1, 1985

As a bubble travels through the pulp region of a flotation cell, it will collect the most hydrophobic particles of an intermediate particle size range, this being the primary source of true flotation selectivity. Despite its role on bubble-particle attachment, however, particle hydrophobicity is still missing a mathematical definition that can be used for plant simulation. In this paper, particle hydrophobicity is defined in terms of the induction time necessary for attachment and then linked to the mass of collector adsorbed per particle. This allows particle hydrophobicity to be written as a function of density of collector adsorption as well as particle size and mineralogy. The model is calibrated from the experimental measurement of the flotation rate constant. Liberation data from an industrial flotation rougher is used for model validation. It is demonstrated that the model can describe a broad experimental dataset by means of a minimal number of parameters. Main industrial application is in simulating the effect of the chemical environment upon plant metallurgical performance.

Jan 1, 2006

By W. G. Davenport

The remainder of this book develops and uses mass and heat balances to describe steady"state flash smelting. Its objectives are to: a) describe the many ways a flash furnace operator can produce specified products, e,g. 65% eu matte, 35% SiOz slag, 1500 K (b) determine the oxygen, air, flux and hydrocarbon fuel requirements of each method (c) show how the flash furnace can be controlled to give specified products with varying feed compositions (d) suggest how the flash furnace operator can maximize profitability while producing specified products. This chapter develops the basiC heat and mass balance equations of steady-state ?flash smelting. It then adapts the equations to industrial feed materials .and products. Lastly it determines the amounts of oxygen and flux needed to produce molten matte and 35% SiOz slag from chalcopyrite concentrate in an Inco oxygen flash furnace,

Jan 1, 2001

By Jing Zhong, Yugang Li, Chen Yan, Shaoda Zhang, Qian Wang

Online surface quenching technology has been developed for the hot charging process to prevent the surface cracks in high strength low-alloy steel slabs. In this paper, a two-dimensional heat transfer model of surface quenching process was presented. This finite element model includes nonlinear thermodynamic properties, by which the slab temperature distributions were computed. The model predicted temperatures show reasonable agreement with the measurements. The effects of the water flow rate and slab movement velocity on temperature variation during the quenching and subsequent tempering process were investigated. The result shows that the temperature drop increases but the tempering temperature changes slightly with increasing water flow rate and decreasing slab velocity. Keeping the slab movement velocity at 1.2-2.1m/min and the water flow rate at 55-70m3/h, the slab surface experiences a temperature drop of 400-600 firstly, then recovers above 650, the quenching and energy-saving effect are remarkable.

Jan 1, 2015

By Gennady I. Gulyayev

A mathematical model of plastic deformation occurring in cold plug drawing of mother tubes with initial cross-sectional wall thickness variations (CSWTV) has been developed. The main provision of the model is that eccentricity em is of such value that projection of all forces acting upon the plug in the deformation zone along the axis perpendicular to the drawing axis equals to zero. Calculations have proven that with other conditions being equal increase in the die cone angle and the deformation cycle number reduces CSWTV. Theoretical findings were confirmed by practical results. Keywords: cold, drawing, plug, die cone, plastic deformation.

Jan 1, 2003

By T. Fabritius, L. Hekkala, J. Härkki

The Outokumpu steel belt sintering technology is used for manufacturing chromite pellets that are charged into smelting furnace for ferrochromium production. Hot air is blown through the moving bed of raw pellets. Direct temperature and flow measurements are difficult to do in the bed of a real scale belt sintering process. Therefore, a tested and reliable mathematical model would be a valuable help for optimising operational parameters of the process and designing new constructional improvements. The aim of this work is to present a mathematical model that calculates the gas flow and temperature distribution in the bed of pellets and the atmosphere. Computational fluid dynamics program FLUENT is used to calculate the gas flow, composition and temperature of the gas in the sintering bed during processing. Energy equation for the bed, containing the effects of evaporation of water and oxidation of carbon and iron oxide, will be added to the model to obtain the temperature of the pellets. The porous media model used for pressure drop calculations in the bed is shortly discussed. A comparison between CFD-calculations and experiments with a pilot scale batch-sintering reactor are also presented. The final model can be used as an engineering tool when developing the process and the equipment further. Different designs can be easily tested with the model and the effects of the changes to the flow pattern and temperature distribution can be studied.

Jan 1, 2004

By H. Saint-Raymond, F. Gruy, P. Gardin, M. Cournil

"The control of inclusion elimination is getting more and more important to obtain clean steel. This paper presents a methodology under development for predicting cluster growth and elimination in ladle - RH system. The first key point is to express the collision efficiency, which is very specific for alumina cluster due to the non-wetting fluid. Second, a description of the two-phase flow in the system is necessary. In our calculations with Fluent CFD package and Lagrangian approach for bubbles, bubble growth is implemented, which improves the predictions. Using global reactor decomposition makes possible the prediction of the time evolution of cluster distribution. Important effect of fractal dimension is demonstrated.IntroductionElaboration of new steel grades requires not only to have a high purity in terms of C, O and N content but also to eliminate particles such as alumina or slag droplets. Flow optimisation making easier inclusion elimination is then a key figure to satisfy cleanliness requirements. Mathematical models of flow and inclusion behaviour are widely used for this purpose (1, 2). But having a predictive tool is still a challenge, because the number of phases is important in steelmaking industry: liquid steel, slag layer, bubbles and inclusions (with a large range of composition and rheology). The paper presents the methodology which is developed at IRSID to predict oxygen content evolution during RH degassing.The main mechanisms which have to be considered are :-inclusion growth by turbulent aggregation of elementary inclusions (keeping in mind that liquid steel is a non-wetting medium for inclusions); the difficulty is to express collision efficiency for alumina particles,-inclusion removal by flotation; the difficulty for alumina clusters stems from the complex morphology of particle; fortunately, the use of fractal concept makes it possible to cope with this problem.The paper describes the general modelling of inclusion removal taking into account the previous mechanisms. Hydrodynamic parameters are obtained by means of Fluent CFD package and a specific coding is developed for cluster growth."

Jan 1, 2001

By P. J. Baguley

The model has been developed on the basis of the observation that the partitioning behaviour of a particle is related to its calculated terminal velocity. It utilises a theoretical calculation of the terminal velocity of each particle class. This is then correlated with the observed partition data for a plant separator using empirically-derived functions of operating conditions which take into account effects such as bath turbulence, holes in drum lifters and feed arrangements. The model successfully decouples the effects of medium density and viscosity and is appropriate for any bath-type separator. The correlation between calculated terminal velocity and actual partitioning behaviour was established and modelled using data obtained from two Wemco drums treating Fe and Mn ores. The model can be used to predict the prevailing size-by-size partition curves and the separation (product grades/recoveries) given the washability of the feed and operating conditions. Simulations are used to show that, for the treatment of Fe ore, there is a maximum medium density beyond which product grades fall and that this maximum decreases as the proportion of magnetite in the medium increases. This is attributed to the dominant effect of viscosity at higher medium solids concentrations

Jun 18, 1905

By Pietro Navarra

Cooling of critical furnace equipment to produce freeze lining is of particular interest as many metallurgical processes are intensified while attempting to prolong campaign life. This idea was combined with high-capacity heat pipes in the design of a copper slag launder. The design methodology of the launder and heat pipe cooling system is examined in this paper. With reliable material properties and operating conditions, CFD was used to quantitatively predict the skull thickness and the corresponding heat load applied to the slag launder. The operational limitations of heat pipe technology were considered and an appropriate cooling system was incorporated into the model. Parametric modeling was used to simulate several launder and heat pipe configurations. The generated results were then used to optimize the design of the launder, which was built and tested in August of 2005. The experimental performance of the cooling system is evaluated and compared with the model results.

Jan 1, 2006

By Roberto Parreiras Tavares, Frederico da Costa Fernandes, Guilherme Antonio Defendi, Vangleik Ferreira da Cruz, Júlio César de Sousa Pena

"Near-net-shape casting is an important area of research in the iron and steel industry today. Among the near-net-shape casting processes, the single-belt caster seems to be the most promising, since it is the only one that can achieve levels of productivity similar to those obtained with conventional continuous casting machines. One of the most important aspects in the single-belt casting is the feeding system used to deliver liquid metal over the belt. This system determines the velocity distribution and the levels of turbulence of the liquid metal and affects the quality of the strips. In the present work, a metal delivery system having a simple geometry was proposed. Using the CFD code CFX 5.6, a mathematical model to simulate three-dimensional turbulent fluid flow and heat transfer in that system has been developed. This model was used to analyze different configurations of the metal delivery system and the free surface shape of the fluid was also determined. A physical model of the feeding system was built and used to validate the predictions of the mathematical model. The fluid flow calculations have been validated by dye injection experiments and laser doppler anemometry. Measurements of the free surface levels have also been performed. These experiments provided supporting evidence for the predictions of the mathematical model.IntroductionTo maintain its competitiveness in the present market, the metallurgical sector, particularly the steelmaking companies, are developing new techniques of production. Some of these new processes are focussed on the strip casting technologies. Among these technologies, the SingleBelt process is considered the most interesting option, reducing the number of rolling steps and, consequently, the final cost of the product, especially the part associated to the energy consumption (fuel and/or electric energy). This process can also reach levels of productivity that can not be matched by other strip casting methods(1). One of the most important aspects concerning these new processes is related to the liquid metal delivery system. This system can have a significant effect on the final quality of the strips, since it determines the velocity distribution and the levels of turbulence of the liquid metal."

Jan 1, 2004

By Naiyi Li, Henry Hu, Alfred Yu

"In the past few years, various squeeze cast aluminum components have been developed and successfully implemented in various vehicles by the automotive industry because of their superior engineering performance. However, fundamental understanding of the formation of air gap during squeeze casting processes is still very limited despite of its significant influence on the extent of heat transfer between the casting and mould. In this paper, a mathematical model has been developed to simulate phenomena of air gap between the casting and mold' during squeeze casting of aluminum alloys. The model considers the effect of process parameters such as applied pressures and mold temperatures on the events of air gap formation. The predication indicates that the occurrence of highly enhanced heat transfer in squeeze casting of aluminum alloys primarily results from the presence of excess applied pressures.IntroductionA number of squeeze cast Al components has been developed and used in mass-produced vehicles in the past few years, such as steering knuckles, Road wheel, Engine Block etc. This is attributed to their superior engineering performance. In squeeze casting, the cooling rate can be increased by applying high pressure during solidification which results in the noted refinement in the micro structure scale leading to better casting properties. However, the excessive pressure increased interfacial heat transfer coefficient is affected during the solidification by air gap formation and die surface condition etc., which is believed to increase thermal resistance at the casting/die interface [l - 4]."

Jan 1, 2004

By C. Hennesmann, D. Maijer, S. L. Cockcroft, P. Yo

"Die cast aluminum wheels are one of the most difficult automotive castings to produce because of stringent cast surface and internal quality requirements. As part of a collaborative research agreement between researchers at the University of British Columbia and Canadian Auto Parts Toyota Incorporated, work has been underway to predict heat transport and porosity formation in die cast A356 wheels. The basic heat transfer equations have been solved using the commercial finite element package ABAQUS to which specialized routines have been added to predict porosity. Preliminary work on predicting porosity formation focused on using the Niyama parameter as a measure of the probability of porosity. The latest version of the model incorporates a new fundamentally based porosity criterion, which takes into account the effect of hydrogen concentration. The development of this model together with its application to a series of cylindrical test castings as well as a production cast wheel are presented.I. IntroductionDie cast aluminum wheels are one of the most difficult automotive castings to produce because of stringent cast surface and internal quality requirements. Microporosity, in particular, is a common defect that can adversely affect wheel quality. The mechanism of microporosity formation in aluminum alloys is complex owing to its dependence on the interaction of a number of phenomena occurring during solidification including the decrease in hydrogen solubility, the decrease in volume, the development of the solidification structure, and the increase in difficulty in interdendritic feeding [1]. As a result, the amount, size, and distribution of microporosity is affected by a large number of casting parameters including thermal variables, melt composition, grain structure, modification, and inclusion content, making optimization by trial-and-error methodologies difficult.The use of criteria functions, based on thermal conditions during casting, offers a relatively simple approach to predicting microporosity. In practical terms, they are useful because they provide a straightforward method of assessing the impact of varying casting parameters on porosity through various thermal parametersl21 such as thermal gradient, cooling rate, solidification time, and solidus velocity. Drawbacks of criteria functions include insensitivity to varying hydrogen concentration and the limitation to qualitative predictions. Consequently, in aluminum alloys, attempts have been made to develop models incorporating the effects of hydrogen gas and microstructure evolution. However, these types of models represent a significant increase in complexity and sacrifice the simplicity and ease of application of the thermally based criteria functions."

Jan 1, 2001

By M. E. Huckman

In this paper, we review the equations needed to construct an ion exchange column model. These models contain three major components: 1) a differential material balance around the colunm, 2) a differential material balance around an ion exchange particle, and 3) a constitutive equation for diffusive flux We then examine the ability of a fairly simple mathematical model to predict ion exchange column experimental data. This model contains the traditional absorption theory equations and is used to predict the behavior of a column filled with a hydrous crystalline silico-titanate and which is removing trace amounts of Cs+ from a high pH, complex electrolyte solution. Finally, we propose replacing Fick's law with the Nernst-Planck equation as the constitutive equation for diffusive flux We use a Nernst-Planck type model to predict batch experimental data, then evaluate it by comparing it's ability to predict experimental data to that of a more traditional Fick's law type model? The data from this comparison suggests that the Nemst-Planck type model yields better results.

Jan 1, 1996

By D. Gu, R. G. Anthony, C. V. Philip, M. E. Huckman

"In this paper, we review the equations needed to construct an ion exchange column model. These models contain three major components: 1) a differential material balance around the column, 2) a differential material balance around an ion exchange particle, and 3) a constitutive equation for diffusive flux. We then examine the ability of a fairly simple mathematical model to predict ion exchange column experimental data. This model contains the traditional absorption theory equations and is used to predict the behavior of a column filled with a hydrous crystalline silico-titanate and which is removing trace amounts of Cs+ from a high pH, complex electrolyte solution. Finally, we propose replacing Fick's law with the Nemst-Planck equation as the constitutive equation for diffusive flux. We use a NemstPlanck type model to predict batch experimental data, then evaluate it by comparing it's ability to predict experimental data to that of a more traditional Fick's law type model. The data from this comparison suggests that the Nemst-Planck type model yields better results.IntroductionMillions of gallons of strongly basic, radioactive waste are currently stored in underground storage tanks at Hanford and Savannah River and several locations within the United States. Due to the limited life expectancy of the tanks, there is a pressing need to remediate these wastes. This need has prompted the recent development of new ion-exchange materials for removing radioactive elements from aqueous solutions. A novel material called TAM-5 has been developed jointly by Texas A&M University and Sandia National Laboratories. TAM- 5, a crystalline silico-titanate, is highly selective for Cs + in the complex electrolytic solutions typical of these wastes. It is stable in very basic solutions and in radioactive environments [1, 2 ,3]. In laboratory testing to determine the selectivity of various materials, TAM-5 was superior to other candidate ion exchangers [4, 5]. The selectivities were determined by measuring a distribution coefficient, Kd, which is defined as the concentration, gig or mmols/g of Cs in the solid phase divided by the concentration, mg/L or mmols/mL, of Cs in the liquid phase. The units of Kd are ml/g or L/kg. Table 1 shows Kd values for TAM-5 compared to 9 other candidate materials in a typical waste solution [6]."

Jan 1, 1996

By Luís Marcelo Tavares, Henrique Dias Gatti Turrer, Luciana Pereira Alves

Desliming is a method of classifying mineral particles, which aims at removal of slimes so as to prepare the slurry to downstream concentration processes. It is a common item in mineral processing flowsheets, being mostly carried out in hydrocyclones, in which separation is result of complex fluid flow and its interaction with the mineral particles. When dealing with particles contained in size ranges relevant to desliming, the challenge in describing the process is even greater. The work deals with the fitting of the Narasimha-Mainza model of classification in hydrocyclones to data from desliming an itabirite iron ore, for prediction of feed slurry flowrate, corrected cut size and short-circuit to the underflow. As such, tests were conducted at both pilot and industrial plants in KREBS 2.7” and 4” cyclones operating under a variety of conditions. In the validation of the model, deviations between the predicted and measured values were lower than 34% for feed slurry flowrate, less than 22% for corrected cut size and less than 18% for short-circuit for underflow.

Oct 3, 2019

By A. McLean, L. Gao, Z. Shi, Y. D. Yang, K. Chattopadhyay, G. F. Zhang

Levitation melting of metals is one of the emerging applications of magnetic and electric fields in liquid metal refining. The varying magnetic field generates induced current inside the metal droplet and leads to the Joule heating effect and Lorentz force against gravity. This contactless process can avoid contamination from the crucible and can be used to produce high-purity materials. Visualization of the levitation melting process is difficult, and thus generally investigated by mathematical modeling. In the present study, a three-dimensional mathematical model of metal levitation was developed using finite element method for proper understanding of the levitation process. The effects of harmonic magnetic field frequency, coils power input, and sample size on levitation nickel droplet were investigated. Temporal evolution of temperature fields were predicted directly though the model. Levitation and balance condition of the droplet were investigated as well.

Jan 1, 2015