Macroexothermic phenomena in exothermic additions: mathematical and physical modelling

In this paper a mathematical model is presented to predict the macroexothermic phenomena occurring when exothermic additions in lump form are assimilated in ferrous metals. The macroexothermic phenomena take place during the free assimilation period of exothermic additions in ferrous metals. These phenomena are characterized by unique coupled heat, mass and momentum transport phenomena. The presence of a moving boundary complicates further these phenomena. The model uses the Simpler algorithm to solve numerically the pertinent partial differential equations. The extensive verification of the model was carried out in two contexts. The first was, in a low temperature physical model consisting of ice immersion in different sulfuric acid solutions. The melting of ice in these solutions is extremely exothermic. In this physical model, both temperature and velocity measurements were carried out. The model results were compared with experimental measurements and they were found to be in excellent agreement. The second context employed high temperatures, involving the assimilation of silicon in high carbon liquid iron. The model was also applied to predict the position of the moving boundary for these high temperature experiments and a good agreement was obtained. In addition new dimensionless convective heat transfer correlations that quantify these complex phenomena are presented.     

The interaction of a DC transferred arc with a melting metal: Experimental measurements and mathematical description

Experimental measurements are reported on the transient development of temperature profiles in a hemispherical metal anode onto which a DC plasma jet is impinging. The main process variables were the arc current, the electrode separation, and the argon flow rate. These experimental measurements were compared with the predictions of a mathematical model, which involved the statement of the turbulent heat and fluid flow equations in the plasma, coupled to the heat flow in the testpiece through the boundary conditions. The experimental measurements were in reasonable agreement with the predictions, and convective heat transfer was found to be dominant in the heat exchange between the plasma and the anode.     

Heat transfer and fluid flow in plasma spraying

A mathematical model is developed to describe the plasma spray process in which particular attention is paid to the fluid flow and temperature fields in the plasma jet, the plasma/particle interaction, and the heat transfer phenomena associated with the deposition process. On the basis of the heat transfer analysis it was possible to define the limiting conditions for satisfactory operation of the deposition process in terms of basic process variables. For high deposition rates, high levels of superheat, and low thermal conductivity of the deposit, the limiting condition is set by the rate at which heat may be removed by the substrate. For large particle sizes and materials with high melting points the limiting condition is determined by the need to transfer sufficient thermal energy to the particles so that they arrive at the substrate in a fully molten state. Wherever possible, the model predictions were compared with experimental measurements and good agreement was obtained.     

The influence of temperature gradient zone melting on microsegregation

Adding an algorithm for considering temperature gradient zone melting (TGZM) to an existing numerical model for predicting microstructure and microsegregation allows the prediction of migration distances of dendrite arms and asymmetric concentration distributions in the arms. Provided that detailed information on the time dependence of the temperature gradient as well as the cooling rate is available from heat flow calculations, accurate predictions of the type and amount of secondary phases or dendrite arm spacings are possible for cooling conditions at which TGZM is active. Parameter studies are performed to investigate the influence of TGZM for typical temperature gradients (0.01 to 10 K/mm). Sawtoothlike concentration distributions are predicted for high-temperature gradients. A binary Al-6.8 wt pct Cu alloy is solidified unidirectionally and asymmetrical concentration profiles are measured. Considering TGZM in the simulation results in good agreement of model predictions with experimental measurements in the position of the minimum concentration and the asymmetric shape of the concentration profile as well as dendrite arm spacings and amount of second phase.     

On the calculation of the free surface temperature of gas-tungsten-arc weld pools from first principles: Part I. modeling the welding arc

A mathematical formulation has been developed and computed results are presented describing the temperature profiles in gas tungsten arc welding (GTAW) arcs and, hence, the net heat flux from the welding arc to the weld pool. The formulation consists of the statement of Maxwell's equations, coupled to the Navier-Stokes equations and the differential thermal energy balance equation. The theoretical predictions for the heat flux to the workpiece are in good agreement with experimental measurements — for long arcs. The results of this work provide a fundamental basis for predicting the behavior of arc welding systems from first principles.     

A model for predicting the yield stress of AA6111 after multistep heat treatments

A model has been developed to predict the yield stress (YS) of the aluminum alloy AA6111 after multistep heat treatments that involve combinations of ambient-temperature aging and high-temperature artificial aging. The model framework follows the internal state variable framework where the two principal state variables are the volume fraction of clusters that form at ambient temperature and the volume fraction of metastable phases that form during high-temperature aging. The evolution of these state variables was modeled using a set of coupled differential equations. The mechanical response (the YS) was then formulated in terms of the state variables through an appropriate flow stress addition law. To test the model predictions, a series of experiments were conducted that examined two scenarios for multistep heat treatments. In general, good agreement was observed between the model predictions and the experimental results. However, for the case where a short thermal excursion at 250 °C was applied immediately after the solution treatment, the results were not satisfactory. This can be understood in terms of the importance of the temperature dependence for the nucleation density of metastable precipitates.     

Numerical and Experimental Investigation on Heat Propagation Through Composite Sinter Bed With Non-Uniform Voidage. Part I Mathematical Model and Its Experimental Verification

The mechanisms of heat transfer through the sinter beds of the MEBIOS process are discussed and their comprehensive mathematical model is proposed. The MEBIOS process, the concept of which has been proposed earlier by the authors, allows using both coarse and fine particles of iron ores in the same sinter bed. The study includes two parts. The first part describes the model content and the results of its experimental verification. The model accounts for coal combustion, limestone decomposition, moisture evaporation/condensation, and melting/solidifying of solid phases. The model predictions are in good agreement with the experimental data. Typical numerical results of the sintering process and the key parameters influencing the process efficiency are discussed in the second part of the study.     

Electromagnetic and fluid flow phenomena in a mercury model system of the electromagnetic casting of aluminum

A mathematical formulation has been developed to represent the electromagnetic force field and the velocity field in the melt for the electromagnetic casting of aluminum. The theoretical predictions based on fundamental considerations are compared with experimental measurements obtained on a physical model system. The measurements and predictions were found to be in good agreement, regarding both the velocity fields and the electromagnetic force fields. The principal conclusion emerging from this work is of critical importance in achieving the dual objective, that is providing a restraining force, while minimizing the melt velocity perpendicular to the free surface. The mathematical formulation presented in the paper provides the theoretical framework for quantitatively defining these conditions in terms of the coil and the shield parameters. J.L. Meyer, formerly with the Department of Materials Science and Engineering at Massachusetts Institute of Technology N. EL-KADDAH, formerly with the Department of Materials Science and Engineering at MIT     

On the Mechanism of Natural Convection and Equiaxed Structure during Dendritic Solidification Processes

A new technique has been developed to generate dendritic‐equiaxed structures in aluminium alloy casting processes, not only to improve the mechanical properties but also to study the effect of crystal structure on the chemical and physical properties of alloys to be cast. The investigation combined laboratory experimental work, metallographic examination and mathematic modelling. The laboratory experimental work involved different superheats for Al‐4.5%Cu alloy in cast ingots. Measurements of temperature distributions were conducted to verify the solidification model. A metallographic study combined macro and micro structural evolution of cast ingot samples. Two‐dimensional mathematical models of fluid flow and heat transfer were developed to characterise the natural convection streams and thermal fields. The model predictions were compared to temperature and isotherms measurements where a good agreement was found. The formation of cast structure and columnar, equiaxed transition (CET) and macro segregation phenomena were studied and discussed, based not only on the theories of nucleation but also on the thermal effects in the mushy and liquid zones.     

The Dissolution of titanium in liquid steel

The dissolution kinetics of solid cylinders of titanium in liquid steel has been studied. Two separate dissolution periods were identified: asteel shell period anda free dissolution period. During thesteel shell period a customary shell of frozen steel encased the cylinder following its initial immersion. Premature internal dissolution then began as a result of liquid eutectic forming at the inner steel shell boundary.This phenomenon triggered an exothermic dissolution of the inner surface of the steel shell. The net result was to shorten considerably shell melting times. In the second,or free dissolution period it was found that the surface temperature of the exposed titanium cylinder rose above the bath temperature as a result of continued exothermic dissolution phenomena. This caused the dissolution process to become self-accelerating. A simplified mathematical model of the process has been developed to describe the complex coupled heat and mass transfer phenomena involved.     

Decarburization of stainless steel: Part I. A mathematical model for laboratory scale results

A mathematical model is presented for describing the reaction of iron-chromium-carbon melts with pure oxygen, air and oxygen-argon mixtures. The model is based on the generalization of the Asai-Muchi model for oxygen steelmaking to systems containing chromium and where the oxygen blown is diluted by an inert gas. The predictions based on the model were compared to the experimental measurements of Barnhardt obtained with heats of about 1.2 to 1.5 kg, having carbon contents ranging from 0.3 to 0.6 wt pct and chromium contents of 0.0 to 21.0 wt pct. The agreement between measurements and predictions was quite good for a variety of blowing arrangements which included top blowing with pure oxygen or air and bottom blowing with air. The fact that this good agreement was obtained by using a single value of the adjustable parameter in the model for all runs, renders very promising the extension of the model to more complicated systems. On leave from Department of Iron and Steel Engineering, Nagoya University, Nagoya, Japan.     

Kinetics of scrap melting in liquid steel

The melting rate of steel bars with various sizes, shapes, and initial temperatures in a 70 kg liquid steel bath (1650 °C) was measured to investigate the kinetics involved in steel scrap melting. Our measurements revealed that a solidified shell was formed around the original bar immediately after it was immersed into the liquid steel. This shell and an associated interfacial gap generated between it and the original bar were found to be critical to the melting kinetics. We also found that the total melting time decreased linearly with increasing initial bar temperature. The melting process was simulated using a two-dimensional phase-field model that considered heat convection with a constant heat-transfer coefficient. Our simulations were in good agreement with our experiments and showed that the heat conduction associated with the interfacial gap was one of the most important physical aspects controlling the melting of steel scrap.     

Numerical and Experimental Investigation on Heat Propagation through a Composite Sinter Bed with Non-uniform Voidage. Part 1 : Mathematical Model and Its Experimental Verification.

The present study discusses mechanisms of heat transfer through sinter beds of the MEBIOS process and offers its comprehensive mathematical model. The MEBIOS process, the concept of which has been proposed earlier by the authors, allows using both coarse and fine particles of iron ores in the same sinter bed. The study includes two parts. The first part describes the model content and the results of its experimental verification. The model accounts for coal combustion, limestone decomposition, moisture evaporation/condensation, melting/solidifying of solid phases. Good agreement was obtained between the model predictions and experimental data. Typical numerical results of the sintering process and the key parameters influencing the process efficiency will be discussed in the second part of the study.     

Heat flux transients at the solder/substrate interface in dip soldering

In the present work an experimental set-up was designed to simulate the dip soldering conditions. The set-up was used to estimate heat flux transients at the solder/substrate interface during solidification of Sn 3.5Ag solder against copper substrate. The inverse heat conduction problem (IHCP) was solved in the substrate region for estimation of heat flux transients. A reasonably good agreement between measurements and model predictions was achieved validating the inverse model. The computer code developed in the present study could be used to assess the effect of process variables on contact heat flux at the solder/substrate interface.     

Fluid velocities in induction melting furnaces: Part I. Theory and laboratory experiments

Fluid flow in an induction furnace due to electromagnetic stirring forces is predicted theoretically from furnace design parameters by the simultaneous solution of the Maxwell and Navier Stokes equations. Streamline plots and velocity profiles are obtained and compared with surface velocities measured experimentally. The measurements were made on a mercury pool stirred inductively by a Tocco 30 kW 3 kHz induction melting unit. The agreement between the experimental measurements and theoretical predictions was good considering that no curve fitting by manipulation of adjustable parameters was involved. It is believed that such a model would be of value in the design and development of induction furnaces.     

Mathematical modeling of an exothermic leaching reaction system: pressure oxidation of wide size arsenopyrite participates

In the design of processes involving exothermic reactions, as is the case of several sulfide leaching systems, it is desirable to utilize the energy liberated by the reaction to drive the reactor toward autogenous operation. For optimal reactor design, models which couple leaching kinetics and heat effects are needed. In this paper, the principles of modeling exothermic leaching reactions are outlined. The system investigated is the high-temperature (160 °C to 200 °C) pressure (O2) oxidation of arsenopyrite (FeAsS). The reaction system is characterized by three consecutive reactions: (1) heterogeneous dissolution of arsenopyrite particles, (2) homogeneous oxidation of iron(II) to iron(III), and (3) precipitation of scorodite (FeAsO4-2H2O). The overall kinetics is controlled by the arsenopyrite surface reaction. There was good agreement between laboratory-scale batch tests and model predictions. The model was expanded to simulate the performance of large-scale batch and single-stage continuous stirred tank reactor (CSTR) for the same rate-limiting regime. Emphasis is given to the identification of steady-state temperatures for autogenous processing. The effects of operating variables, such as feed temperature, slurry density, and retention time, on reactor operation and yield of leaching products are discussed.     

Least-squares adjustment of mathematical model of heat and mass transfer processes during solidification of binary alloys

Classical boundary and initial-boundary value problems in heat and mass transfer are generally formulated in a mathematically unique way. Boundary and initial conditions together with physical properties of the thermodynamic system are treated as exactly known. The influence of different kinds of mathematical model simplifications on the accuracy of solution and reliability of the model are not usually analyzed. The problems become more complicated when inverse ill-posed initial-boundary problems are considered. The widely used procedure of model validation is based on direct comparison of analytical or numerical solution, unique in a mathematical sense, with measurement results. The main feature of the method presented in this article is that all experimental results are included into the mathematical model. Thus, because of the inevitable errors of measurements, the system of model equations becomes internally contradicted as the number of unknown variables is less than the number of equations. In consequence, basic laws of energy and mass conservation are not satisfied. To adjust the experimental data to the mathematical model, an orthogonal least-squares method is proposed. Special attention has been paid to the coupling of experimental data with the nucleation and grain growth models formulated by Rappaz and co-workers. Theoretical considerations are illustrated with experimental data for an Al-Si alloy.     

A model for accurate predictions of self-diffusivities in liquid metals, semimetals, and semiconductors

By combining the modified Stokes-Einstein formula with the authors’ model for the melting-point viscosity, the authors present a model for accurate predictions of self-diffusivity of liquid metallic elements. The model is expressed in terms of well-known physical quantities and has been applied to various liquid metallic elements for which experimental data are available. The results of calculations show that agreement with experimental data is excellent; the uncertainties in the calculations of the self-diffusivities in various liquid metallic elements are equal to the uncertainties associated with experimental measurements. Also, the authors propose an expression for the temperature dependence of self-diffusivity in liquid metallic elements in terms of melting-point temperature. Using the model, self-diffusivity data are predicted for liquid iron, cobalt, nickel, titanium, aluminum, magnesium, silicon, and so forth.     

Measurement of velocity in high-temperature liquid metals

There is a paucity of methods available for the measurement of velocity in high-temperature liquid metals. This is due to the hostile environmental conditions which characterize liquid metals. This article proposes and appraises a new velocity measurement technique for liquid metal flows at high temperatures. The melting rates of metallic spheres in metal baths of the same chemical composition as the spheres are studied under isothermal conditions. It is dem-onstrated that the metallic sphere can be used as a probe for measuring the average velocity in a metal flow system over a distance equivalent to the diameter of the sphere. The system that was chosen for study is the commercial purity aluminum bath. The experimental calibration setup examined three different elements: (a) it introduced a stationary sphere in a metallic bath of a given temperature and compared its melting rate with that of a moving sphere with known external velocity along the periphery of a circle in a metallic bath of the same temperature; (b) three different sphere diameters were used; and (c) a range of bath temperatures was investi-gated. By studying the effect of these three elements concurrently, it was possible to determine the interplay of these elements. Results showed that the sphere melting time was related linearly to the flow velocity for the range of velocities of 0 to 40 cm/s and for bath superheat up to 100 °C. In order to verify the accuracy of the results obtained by the proposed technique, a comparison was undertaken between mathematical predictions and experimental results of a fluid flow field obtained in an AC induction furnace with molten aluminum. These predictions were made by solving numerically the relevant differential equations under the appropriate boundary conditions. The experimental results attained using the proposed technique were in close agreement with those from the mathematical predictions. A.C. MIKROVAS, formerly Graduate Student, Department of Metallurgy and Materials Science, University of Toronto.