We present a bibliography on moving and free boundary problems for the heatdiffusion equation, particularly regarding the Stefan and related problems. It contains 5869 titles referring to 588 scientific Journals, 122 books, 88 symposia (having at least 3 contributions on the subject), 30 collections, 59 thesis and 247 technical reports. It tries to give a comprehensive account of the western existing mathematicalphysical-engineering literature on this research field. RESUMEN Se presenta una bibliografía sobre problemas de frontera móvil y libre para la ecuación del calor-difusión, en particular sobre el problema de Stefan y problemas relacionados. Contiene 5869 títulos distribuidos en 588 revistas científicas, 122 libros, 88 simposios (teniendo al menos 3 contribuciones en el tema), 30 colecciones, 59 tesis y 247 informes técnicos o prepublicaciones. Se da un informe amplio de la bibliografía matemática, física y de las ingenierías existente en occidente sobre este tema de investigación. Primary Mathematics Subject Classification Number (*): 35R35, 80A22 Secondary Mathematics Subject Classification Number (*): 35B40, 35C05, 35C15, 35Kxx, 35R30, 46N20, 49J20, 65Mxx, 65Nxx, 76R50, 76S05, 76T05, 93C20. (*) Following the 1991 Mathematics Subject Classification compiled by Mathematical Reviews and Zentralblatt fur Mathematik. Primary key words: Enthalpy formulation or method, Filtration, Free boundary problems, Freezing, Melting, Moving boundary problems, Mushy region, Phase-change problem, Solidification, Stefan problem. Secondary key words: Continuous mechanics, Diffusion process, Functional analysis, Heat conduction, Mathematical methods, Numerical methods, Partial differential equations, Variational inequalities, Weak solutions. Palabras claves primarias: Método o formulación en entalpía, Filtración, Problemas de frontera libre, Congelación, Derretimiento, Problemas de frontera móvil, Región pastosa, Problema de cambio de fase, Solidificación, Problema de Stefan. Palabras claves secundarias: Mecánica del continuo, Procesos difusivos, Análisis funcional, Conducción del calor, Métodos matemáticos, Métodos numéricos, Ecuaciones diferenciales a derivadas parciales, Inecuaciones variacionales, Soluciones débiles. El manuscrito fue recibido y aceptado en octubre de 1999. D.A. Tarzia, A bibliography on FBP. The Stefan problem, MAT Serie A, # 2 (2000). 3

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A bibliography on moving-free boundary problems for the heat-diffusion equation. The stefan and related problems

Many of the properties of austempered ductile iron depend on the austenite which is retained following the bainite reaction. A neural network model within a Bayesian framework has been created using published data to model the retained austenite content. The model allows the quantity of retained austenite to be estimated as a function of the chemical composition and heat treatment parameters. The computer programs associated with the work have been made freely available ( http //www.msm.cam.ac.uk/map/mapmain.html )

Estimation of the amount of retained austenite in austempered ductile irons using neural networks

The heat-transfer coefficient was measured during the unidirectional solidification of Al-7 wt pct Si alloy castings against a water-cooled Cu chill. Heat-transfer coefficients in the range of 2.5 to 9 kW m−2 K−1 were obtained with solidification vertically upward associated with higher values than solidification vertically downward. Horizontal solidification was associated with intermediate values. Profiles taken across the diameters of the casting surfaces at the interface with the chill showed them to be convex toward the chill by an amount which would produce a mean gap between the casting and the chill that would account for a significant proportion of the measured heat-transfer coefficient. The convex casting surfaces were attributed to the deformation of the initial casting skin by the thermal stress produced during solidification. Heat transfer during casting solidification is shown to be a complex mechanism controlled by the microscale surface roughness of the respective surfaces, mesoscale deformation of the casting skin by thermal stress, and macroscale movements of the casting and the chill due to their relative thermal expansion and contraction.

The Heat-Transfer Coefficient during the Unidirectional Solidification of an Al-Si Alloy Casting

A model is presented for the prediction of the interfacial heat-transfer coefficient during the unidirectional solidification vertically upward of an Al-7 wt pct Si alloy cast onto a water cooled copper chill. It has been experimentally determined that the casting surfaces were convex toward the chill, probably due to the deformation of the initial solidified skin of the casting. The model was, therefore, based upon a determination of the (macroscopic) nominal contact area between the respective rough surfaces and, within this region, the actual (microscopic) contact between the casting and the chill surfaces. The model produced approximate agreement with both experimentally determined values of the heat-transfer coefficient and the measured curvature of the casting surface and showed a reasonable agreement with measured temperatures in the casting and the chill also. A common experimental technique for the experimental determination of the heat-transfer coeffcient involves the assumption of one-dimensional heat transfer only. An implication of the approach adopted in this model is that the heat transfer in the region of the casting-chill interface may be two-dimensional, and the subsequent error in the experimentally determined values is discussed.

A model of the interfacial heat-transfer coefficient during unidirectional solidification of an aluminum alloy

Interfacial heat-transfer coefficients were measured during the solidification of Al-Si alloys against coated die steel chills with varying chill temperature, coating thickness and coating type. Two principal resistances to heat transfer across the casting-chill interface were identified, namely, (1) the resistance to heat transfer of the coating itself and (2) the resistance to heat transfer of a layer of gas, (assumed to be air), trapped between the coating and casting surfaces by virtue of their roughness. These thermal resistances were evaluated by measurement of the coating thermal conductivity and determination of the thickness of the applied coatings and the thickness of the layer of air between the coating and casting surfaces. This produced a simple equation to predict the interfacial heat-transfer coefficient during the solidification of Al alloy die castings, which produced values that were found to agree well with the experimentally determined results. This equation was used to interpret the experimentally measured heat-transfer coefficients and to explain their variation with the different experimental conditions employed. A simple modification of the equation can also take into account the formation of an air gap, where the casting locally retreats away from the die surface, leading to a local reduction in the heat-transfer coefficient.

A model of the interfacial heat-transfer coefficient for the aluminum gravity die-casting process

The deformation of initial solid shell has important effects on the formation of surface defects of continuously cast slabs of steel. In order to understand the thermal deformation behavior of initial solid shell of steel, droplets of Fe-C alloys and commercial steels were solidified on a chill plate and the bottom shape of solid shell was measured. The solid shell always deforms during solidification and cooling to give a spherical bottom shape. This deformation of solid shell shows a very strong dependence on carbon content. When carbon content is lower than about 0.6 mass% or higher than 2.5 mass%, solid shells show positive deformation, i.e. convex towards the chill. When crbon content lays between 0.6-2.5 mass%, solid shells show negative deformation, i.e. concave towards the chill. Two peaks of positive deformation were found as carbon contents come near to 0 mass% (pure iron) or 0.12 mass%. These two peaks of positive deformation of solid shell correspond to the reported peaks of frequencies of surface defect occurrence such as longitudinal cracks and deep oscillation mark of continuously cast steel slabs. The deformation of pure iron solid shell was accurately pridicted with a deformation model of initial solid shell previously derived by the authors. An explanation for the maximum deformation at 0.12 mass% carbon conent was also made by the model using a numerical heat transfer calculation with a coupling of deformation and solid shell-chill interface heat transfer coefficient.

Free Deformation of Initial Solid Shell of Fe-C Alloys

In the austempering of ductile cast iron, it is necessary to alloy the metal to prevent pearlite transformation. The addition of alloying elements delays the pearlite transformation and at the same time, causes a similar delay in the beginning of the stage I reaction. In general, alloying addition levels greater than a specified amount are not desirable in the conventional austempering process due to the decrease in toughness and ductility. In this experiment, 1% manganese alloyed ductile iron was used in the successive-stage, High-Low Austempering Temperature Process (HLAT Process) in different time combinations of high and low temperature austempering to improve the toughness of high manganese ductile cast iron. The impact strength provided by the HLAT Process is a function of high temperature austempering time as well as low temperature austempering time. The impact strength at the best process combination is much higher than conventional high manganese ADI and well comparable to low alloy aust...

Effects of Successive-stage Austempering on the Structure and Impact Strength of High-Mn Ductile Iron

Droplets of several non-ferrous metals and alloys were solidified on a chill plate and the shape of the bottom after cooling was measured to study the basic deformation behaviour of the initial sol...

https://www.jstage.jst.go.jp/article/tetsutohagane1955/79/9/79_9_1060/_pdf

Free Deformation of the Initial Solid Layer of some Non-ferrous Alloys

Manganese, which may be used in ductile cast iron as a potent hardenability promoter, segregates in the intercellular region. This segregation becomes more severe as a consequence of poor inoculation, low cooling rate, or increasing of nominal Mn content in the alloy. In severely Mn-segregated regions, Mn eutectic carbide may be formed, which has a deteriorating effect on the mechanical properties of casting. In this study, a 1 Pct Mn ductile iron was used to investigate the chemical composition and crystal structure of the Mn eutectic carbide by electron microscopy (scanning and transmission), X-ray, and an electron probe microanalyzer (EPMA). Transmission electron microscope (TEM) and X-ray studies show that the crystal structure of carbide is orthorhombic with lattice parameters ofa = 14.825,b = 11.415, andc = 8.880 (A). The concentrations of Mn, Si, and Cr in carbide, analyzed by scanning electron microscope-energy-dispersive X-ray spectrometer (SEM-EDX) and transmission electron microscope-energy-dispersive X-ray spectrometer (TEM-EDX), were 5.0 to 7.0, 0.5 to 2.8 and 1.5 to 2.2 (wt Pct), respectively. The ratio of Fe plus other metal atoms to C was calculated from EPMA experiments to be 2.5-2.9. It was shown that by diminishing Mn segregation, precipitated eutectic carbide can be reduced. It is expected that this can be achieved by reducing nominal content of Mn or by increasing nodule count.

Chemical composition and structural identification of eutectic carbide in 1 pct Mn ductile iron

A 1 wt.% Mn ductile iron austenitized at 900 °C for 90 min and austempered at 375 °C for different periods was used to study some aspects of bainitic reaction in high Mn austempered ductile iron with reference to carbide precipitation in bainitic ferrite. Transmission electron microscopy (TEM) energy-dispersive X-ray analysis (EDXA) study shows that precipitation of carbide in the ferritic component of bainite is a function of the local concentration of alloying elements. In other words, in the region near graphite where si segregates and there is negative Mn segregation as well as carbon, the bainitic ferrite is carbide free. However, in the intercellular region where Mn segregates and Si is depleted, the ferritic component of bainite occurs together with very fine and almost uniformly distributed carbide. Furthermore, TEM-EDXA results show that the increase in Mn content not only delays stage I (the initial transformation of austenite to ferrite and retained austenite) of the bainitic reaction, but also delays stage II (decomposition of retained austenite to ferrite and carbide).

Eisuke Niyama

Papers

Free Deformation of Initial Solid Shell of Fe-C Alloys

Effects of Successive-stage Austempering on the Structure and Impact Strength of High-Mn Ductile Iron

Free Deformation of the Initial Solid Layer of some Non-ferrous Alloys

Chemical composition and structural identification of eutectic carbide in 1 pct Mn ductile iron

Transmission electron microscopy study of high temperature bainitic transformation in 1 wt.% Mn ductile iron