数学模型及其在钢铁工业中的应用

1背景介绍 高校数学系的教师和钢铁公司的科研人员有着悠久的合作历史,最明显的证据就是由牛津工业与应用数学研究中心发起的、由高校与企业界联办的讨论班提出的各种数学问题.     

Melting and dissolution of high‐melting alloys in steel melts

About one year ago a booklet by Liuyi Zhang and Franz Oeters titled Melting and mixing of alloying agents in steel melts. Methods of mathematical modelling was published [1]. On pages 45, 46‐48, 49‐51 of this booklet numerical calculations are presented, inter alia, for the simultaneous melting and dissolution of high‐melting alloys in steel melts, and from these calculations certain conclusions are drawn. Caused by a programming error, these calculations and in consequence, also the conclusions are wrong. Meanwhile, the error has been eliminated, and new calculations were done. Because the results are considerably different from those before, we would like to publish the altered results in this communication. In the report, a short introduction into the problem is presented first, then some mathematical equations are explained, and finally, the new calculation results and their discussion are given. Reference is made in detail to the publication [1].     

Laboratory experiments and process modelling of the melting and dissolution of low‐density ferro‐molybdenum in steel melts

A low‐density sintered ferro‐molybdenum is presented as a new alloying agent. The paper describes laboratory studies about the dissolution in liquid iron of briquettes of this alloy and of classical high density ferro‐molybdenum pieces. It presents further a mathematical model of the melting and dissolution process. During dissolution, an approximately 1mm thick layer infiltrated by melt forms on the particle surface. The infiltrated melt solidifies in the plane where the temperature has reached the eutectic temperature in the system iron‐molybdenum. Internal dissolution of alloy material in the layer is weak, which means that the dissolution proceeds almost exclusively from the outer surface of the briquette. Dissolution rate increases with sinking briquette density. The lower molybdenum content per volume in the briquettes which is proportional to the density has the effect that the liquidus concentration and the liquidus temperature at the solid‐liquid interface decrease in comparison with compact material. This reduces the mass transfer rate and increases the heat transfer rate. The effect is a faster movement of the interface. Below a critical density of approximately 5200 kg/m³ for the alloy considered, the molybdenum concentration on both sides of the interface becomes equal. From this moment, the alloy is liquefied solely by melting of the moving interface. Mass transfer from the interface gets negligible and distribution of the molten alloy into the bulk melt takes place only by the outer mixing process. From the described behaviour it follows, that below the critical density the melting rate is solely determined by heat transfer. Since heat transfer is faster than mass transfer, melting of alloys with reduced density is correspondingly accelerated. The extremely slow dissolution rates of high melting alloys can thus, be overcome by giving the alloy a certain porosity. The mathematical process model describes the phenomena quantitatively.     

合金熔体结构和原子- 分子热力学模型

对金属熔体结构的研究证实熔体中存在短程有序结构,而含金属间化合物的合金熔体中发现了原子- 分子结合的团簇结构,即固态下的金属间化合物在熔体中并没有消失,而是以团簇结构（实际上就是液态的分子）存在,并与熔体中的自由原子存在化学平衡。从熔体结构出发,提出了适用于合金熔体的原子- 分子热力学模型。然后以Ca- Mg合金熔体为例,介绍了该模型的构建和求解方法,并计算了1010K下的Ca- Mg合金熔体中各物质的摩尔分数。最后,将计算得到的钙、镁摩尔分数与各自的实测活度值进行比较,发现两者吻合得较好。这从计算上证明了合金熔体中同时存在着金属原子和金属间化合物分子,两者处于动态化学平衡之中。且达到平衡时,金属原子的摩尔分数实际上就是各自的活度。因此,对于合金熔体而言,活度并不存在。     

Mathematical modelling of molten steel flow in a whole degasser during RH refining process

Abstract

A three-dimensional mathematical model for molten steel flow in a whole degasser during the RH (Ruhrstahl–Heraeus) refining process is proposed. The model has been developed considering the physical characteristics of the process, particularly the behaviour of gas–liquid two phase flow in the up snorkel and the momentum exchange between the two phases. The fluid flow fields and gas holdups of liquid phases, among other parameters, in a 90 t RH degasser and a water model unit of one-fifth linear scale have been computed using this mathematical model. The results show that the flow pattern of molten steel in a whole RH degasser can be well represented by the mathematical model. Apart from the area close to the free surface and the zone between the two snorkels in the ladle, the molten steel in an RH degasser, especially in the vacuum vessel, is reasonably fully mixed during the refining process. However, there is a boundary layer between the descending liquid stream from the down snorkel and the surrounding liquid, which is typical liquid–liquid two phase flow, and the molten steel in the ladle is not perfectly mixed. The blown lifting gas ascends mostly near the up snorkel wall, which is more obvious under the conditions of an actual RH degasser, and the flow pattern of bubbles and molten steel in the up snorkel is closer to annular flow. Calculated circulation rates for the water model unit at various lifting gas rates are in good agreement with values determined by means of water modelling experiments.     

The melting and dissolution of low-carbon steels in iron-carbon melts

Direct experimental proof is presented in the paper for the role played by the mass transfer of carbon in accelerating or facilitating the melting and dissolution of pure iron specimens in iron-carbon melts. It is shown that pure iron may readily melt in iron-carbon melts even under conditions where the temperature of the molten phase is considerably below the melting point of the pure iron or low-carbon specimen. A mathematical interpretation is developed for these experimental results that includes the mass transfer of carbon and the unsteady state heat transfer within a moving boundary system. The results of the analysis were found to agree with the experimental data thus providing a basis for further calculations aimed at predicting the melting of scrap in the basic oxygen furnace. These calculations show that scrap melting, facilitated by carbon diffusion from the melt to the scrap surface, begins very early during the process and that melting is retarded and even terminated during the blow when the bath has insufficient superheat to provide the necessary sensible and latent heat required for melting. It follows therefore that the rate of scrap melting in steelmaking processes is accelerated if the removal of carbon in the bath is retarded or if the temperature of the bath is increased rapidly in order to maintain a high level of superheat during the refining process.     

Mathematical modelling of fluid flow,heat transfer and solidification phenomena in continuous casting of steel

A steady state, two-dimensional mathematical model for continuous billet casting operations has been developed. Towards this, governing equations of fluid flow and heat transfer together with their appropriate set of boundary conditions were derived and solved numerically via a control volume based implicit finite difference procedure (e.g., SIMPLE). The effect of various assumptions and procedures applied to modelling of turbulence phenomena, thermal buoyancy, flow through the mushy zone, free surface conditions etc., on the sensitivity of the computed results was investigated computationally. Of all these, modelling of heat and fluid flow phenomena in the mushy region was found to have relatively more effect on the predicted results. In addition to these, a set of three different billet casting operations reported in literature were simulated mathematically and direct comparisons were made between predicted and observed solidified shell profiles. Such comparisons demonstrated reasonable to excellent agreement between theory and experiments.     

Mathematical modelling of avascular-tumour growth

A system of nonlinear partial differential equations is proposed as a model for the growth of an avascular-tumour spheroid. The model assumes a continuum of cells in two states, living or dead, and, depending on the concentration of a generic nutrient, the live cells may reproduce (expanding the tumour) or die (causing contraction). These volume changes resulting from cell birth and death generate a velocity field within the spheroid. Numerical solutions of the model reveal that after a period of time the variables settle to a constant profile propagating at a fixed speed. The travelling-wave limit is formulated and analytical solutions are found for a particular case. Numerical results for more general parameters compare well with these analytical solutions. Asymptotic techniques are applied to the physically relevant case of a small death rate, revealing two phases of growth retardation from the initial exponential growth, the first of which is due to nutrient-diffusion limitations and the second to contraction during necrosis. In this limit, maximal and "linear' phase growth speeds can be evaluated in terms of the model parameters.     

Electrochemical reduction of niobium from oxide-fluoride melts upon electroslag melting

Some regularities of electrochemical reduction of niobium from oxide-fluoride melts upon electroslag melting (ESM) have been established for various electrical regimes. It has been found that the use of an asymmetric electric current during ESM substantially improves the technological parameters of the electrolysis process. Based on the experimental data obtained, a special program has been developed for correcting the electrical regime of melting depending on the Nb₂O₅ content in the oxide-fluoride melt, which permits one to obtain a desired niobium concentration in the metal ingot.     

Kinetics of steel melting in carbon-steel alloys

For the melting of scrap metal in baths with a high carbon content, in which the melting temperature lies below the bath temperature, a model is being developed which is based on the coupling of mass and heat transfer. It will be shown that the mass transport determines the melting rate only for very low bath temperatures. Furthermore, it will be discussed how the melting efficiency is influenced by the supplied energy flow and mass transfer in the bath.     

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.     

Microstructure and properties of ZrO2-reinforced 24CrNiMoY alloy steel prepared by selective laser melting

ABSTRACT

The process parameters of selective laser melting (SLM) fabricated 24CrNiMoY alloy steel were optimised. Different contents of zirconia powder were added to the 24CrNiMoY alloy steel powder. With the optimum process parameters, the fabricated samples were characterised using various modern analysis methods. The results show that the main phase composition of 24CrNiMoY alloy steel is proecctectoid ferrite (PF) and granular bainite (GB). After the addition of zirconia, Lath bainite (LB) begins to appear, and with the increase of zirconia, the content of LB first increases and then decreases. When the zirconia content is 0.3%, the average micro hardness of the 24CrNiMoY alloy steel reaches the maximum and is 374?HV. In addition, the tensile strength is 811?MPa, the elongation is 6.7%, and the fracture surface shows ductile fracture.     

Preparation and printability of 24CrNiMo alloy steel powder for selective laser melting fabricating brake disc

Spherical 24CrNiMo alloy steel powder used for selective laser melting (SLM) fabricating high-speed train brake disc was prepared by the vacuum induction melting gas atomisation (VIGA) method. Powder morphology, particle size, flowability and microstructure were measured. Part properties fabricated by SLM were investigated via some modern analysis method. The experimental results showed that powder mean particle size D₅₀ was 75?μm, flowability was 16.69?s/50?g and apparent density was 4.71?g?cm^?3. 24CrNiMo alloy steel specimen microstructures prepared by SLM consisted of proeutectoid ferrite and granular bainite. Average microhardness was 346?HV, tensile strength was 1223?MPa, extensibility was 13.1% and the product of strength and elongation was 16.1?GPa%. 24CrNiMo alloy steel powder prepared by the VIGA method had good laser printability and huge potential application value for SLM-fabricated brake disc.     

Physical refining of steel melts by filtration

The removal of nonmetallic inclusions from steel melts prior to casting has significant merit. Laboratory prepared steel melts containing carefully prepared alumina inclusions have been successfully filtered at 1600 °C. Two distinct types of filters were used: (i) tabular alumina packed bed (0.2 to 0.5 cm nominal diameter) and (ii) extruded monolithic alumina (400 cells per square inch). The kinetics of the filtration process have been modeled, and inclusion removal efficiency of up to 96 pct has been achieved in laboratory melts. The results show that inclusion removal efficiency is a strong function of melt velocity in the range of 0.08 to 0.68 cm per second and is weakly dependent on filter length. The type of filter utilized affects inclusion removal efficiency significantly. The inclusion capture kinetics and the filtration characteristics of the filter media tested are discussed. Formerly with Drexel University, Philadelphia, PA.     

Rapidly solidified surface melts of Ni-B-Si-Cr brazing alloy

Sintered powder layers of a Ni-based brazing alloy were consolidated by scanned electron beam radiation to produce a continuous fused coating. The surface of this coating was then remelted by laser and electron beams under differing conditions, resulting in a variety of resolidification structures. Alloy BNi2 was chosen for these studies because it exhibits substantial hardening on grain refinement and because it can be prepared in the glassy state relatively easily. Surface microhardness for BNi2 reaches a maximum of about 1200 DPH at a cooling rate approaching 10⁵ K/s. For higher quench rates, hardness decreases and ductility increases. As the cooling rate approaches 10⁷ k/s, overlapping beam scans produce an extended amorphous surface. A solidification rate higher than that needed to produce an amorphous structure in a single melt pass is necessary to avoid surface cracking or crystallization when overlapping melt passes are employed. formerly at the Naval Research Laboratory, Washington, DC     

Study on the melting characteristics of steel scrap in molten steel

Steel scrap is of great benefit for environmental protection. In converter steelmaking, bottom carbon injection was applied to enhance the scrap ratio and in EAF steelmaking, submerged carbon powder injection was used to accelerate the smelting of scrap. In these two cases, carbon powder is directly injected into molten metal to improve the scrap melting with effective carburization capacity and intense stirring effect. In this study, the induction furnace experiments were carried out to study the melting characteristics of steel scrap with different carbon contents and bottom-blowing gas flow rates. The results show that larger carbon content and faster fluid flow can promote scrap melting because the carburizing reaction can be accelerated by larger carbon concentration gradient and the heat transfer can be enhanced by larger stirring intensity. Finally, the convective mass transfer coefficient and heat transfer coefficient between steel bar and molten metal were also calculated.