Scheil Equation Research Articles

Fe-C-Si系球状黒鉛鋳鉄の凝固組織予測のためのコンピュータ・シミュレーション

A model to predict the solidification of spheroidal graphite (SG) cast iron during the cooling process was developed. Using the model, we treated the problem of the ternary Fe-C-Si system and aimed at simulating the evolution of the solidifying microstructure. When the eutectic temperature was reached, eutectic grains were formed. The rate of nucleation depended on the melt characteristics and supercooling. The nucleation rate was assumed to be presented by power law function of undercooling. The growth of the austenite grain was controlled by carbon diffusion through austenite. Carbon and silicon concentrations at different interfaces were calculated from the Fe-C-Si equilibrium ternary phase diagram and the Scheil equation was applied to calculate the silicon content in the bulk liquid and the concentration of carbon in the liquid was determined to keep the eutectic reaction.The model was applied to a cylindrical sand casting with four-step diameters of 10, 20, 30 and 40 mm. The simulated results were compared with the experimental ones using a post processing program which was also developed to simulate the microstructural evolution graphically. The results showed relatively good agreement and the usefulness of the proposed model was confirmed.

Analysis on the non-equilibrium dendritic solidification of a binary alloy with back diffusion

Micro-Macro approach is conducted for the mixture solidification to handle the closely linked phenomena of microscopic solute redistribution and macroscopic solidification behavior. For this purpose, present work combines the efficiency of mixture theory for macro part and the capability of microscopic analysis of two-phase model for micro part. The micro part of present study is verified by comparison with experiment of Al-4.9 mass% Cu alloy. The effect of back diffusion on the macroscopic variables such as temperature and liquid concentration, is appreciable. The effect, however, is considerable on the mixture concentration and eutectic fraction which are indices of macro and micro segregation, respectively. According to the diffusion time, the behavior near the cooling wall where relatively rapid solidification permits short solutal diffusion time, approaches Scheil equation limit and inner part approaches lever rule limit.

Effect of laser processing parameters on the structure of ductile iron

Laser processing of structure sensitive hypereutectic ductile iron, a cast alloy employed for dynamically loaded automative components, was experimentally investigated over a wide range of process parameters: from power (0.5–2.5 kW) and scan rate (7.5–25 mm s −1) leading to solid state transformation, all the way through to melting followed by rapid quenching. Superfine dendritic (at 10 5 °C s −1) or feathery (at 10 4 °C s −1) ledeburite of 0.2–0.25 μm lamellar space, γ-austenite and carbide in the laser melted and martensite in the transformed zone or heat-affected zone were observed, depending on the process parameters. Depth of geometric profiles of laser transformed or melt zone structures, parameters such as dendrile arm spacing, volume fraction of carbide and surface hardness bear a direct relationship with the energy intensity P/ UD b 2 (10–100 J mm −3). There is a minimum energy intensity threshold for solid state transformation hardening (0.2 J mm −3) and similarly for the initiation of superficial melting (9 J mm −3) and full melting (15 J mm −3) in the case of ductile iron. Simulation, modelling and thermal analysis of laser processing as a three-dimensional quasi-steady moving heat source problem by a finite difference method, considering temperature dependent energy absorptivity of the material to laser radiation, thermal and physical properties (κ, ρ, c p) and freezing under non-equilibrium conditions employing Scheil's equation to compute the proportion of the solid enabled determination of the thermal history of the laser treated zone. This includes assessment of the peak temperature attained at the surface, temperature gradients, the freezing time and rates as well as the geometric profile of the melted, transformed or heat-affected zone. Computed geometric profiles or depth are in close agreement with the experimental data, validating the numerical scheme.

Analysis of Dopant Distributions in LEC‐InP

AbstractIt is often important to be able to estimate the concentration of dopant atoms incorporated into InP crystals grown from InP melt of given composition. In this paper we present a simple parameter (G) to revise the commonly used effective distribution coefficient (keff) and the Scheil equation. The results obtained for various dopants and different initial concentrations in LEC‐grown InP ingots are discussed. It is shown that the revised dopant concentration curves tally with the real distributions.

Modeling solidification of turbine blades using theoretical phase relationships

The solidification of hot-stage turbine blades made from Rene N4 nickel-base superalloy has been modeled to show the morphology of porosity and the local changes in solute concentration. The key task of the present study was the calculation of the solid-liquid phase equilibria of this 9-component nickel-base superalloy from the thermodynamic values of these phases. The Gibbs energies of the solid and liquid phases were obtained from those of the 36 binaries using the Muggianu and Kohler methods of extrapolation. The phase equilibrium data were then used to compute the change in fraction solid with temperature, initially using the complete mixing approximation (Scheil equation). The predicted freezing range was somewhat longer than measured. A modified Scheil equation was derived assuming incomplete mixing. Assuming 60 pct mixing of the solute, the calculated freezing range agreed with experiments. Fraction solid temperature allowed the detailed morphology of the“mushy”zone to be predicted. Using measured dendrite spacings and assuming the crystals to grow in a cubic array, the shape of the crystals and, consequently, the size of the liquid channels were predicted as a function of position. Hence, computation of the rate of fluid flow in the channels (from the known changes of temperature with time) allowed the pore morphology to be inferred.

Morphology and Macrosegregation in Continuously Cast Steel Billets.

In the transverse sections of low carbon continuously cast steel billets, samples for analyses of carbon and sulphur were collected by drilling at the billet centres as well as at the columnar-equiaxed transition (CET) boundaries, as revealed in etched macrostructures. Correlation of degrees of segregation for carbon (rc) with those for sulphur (rs) agreed closely with that predicted by Scheil's or modified Scheil's equation at the CET boundaries, but not at the billet centres. Variations of rc and rs with fractional solidification, although did not agree with predictions of the above equations, were in qualitative agreement with actual macrosegragation data reported in literature. Attempts were also made to correlated the experimental macrostructural and macrosegragation data with predictions based on conjugate fluid flow-heat transfer model developed earlier. Results of this exercise demonstrated close agreement of the predicted locations of CET boundaries with the measured values. This is being taken as additional confirmation of the reliability of the conjugate fluid flow-heat transfer model. Also, it seems that this model may be employed to estimate the size of equiaxed zone in continuous casting of steel.

A deforming finite element method for analysis of alloy solidification problems

A better understanding of the heat transfer associated with an alloy solidification process may lead to precise control of the solidification process and improvement of the quality of castings. This requires accurate modeling of heat flux discontinuities and the multiple phases (solid, liquid, and mush) inside solidifying bodies. To deal with the complexity of the problem and improve the accuracy of numerical results, a deforming finite element method for alloy solidification is developed in this paper. In this method, the velocities of the solid/mush and mush/liquid interfaces are primary variables. Special procedures are developed to calculate these interface velocities. Therefore, both solid/mush and mush/liquid interfaces can be tracked continuously. This method assumes complete solute mixing with no mass diffusion and require knowledge of the relationship between the solid fraction and the temperature in the early mushy region. In the example problems of solidification of binary alloys, Scheil's equation and the lever rule are used to calculate the solid fraction in the mushy region. This method is accurate and can be easily implemented.

Mathematical modeling of microsegregation in binary metallic alloys

A mathematical model was developed to calculate microsegregation in binary metallic alloys. This model utilized the mathematical techniques of the method of lines combined with invariant imbedding (MOL/II) to solve the problem of combined heat and mass transfer during and after solidification. Model predictions were compared to experimental measurements in the Al-Cu system and to other microsegregation models. The MOL/II model predicted nonequilibrium second-phase contents within ±3 pct at low and intermediate cooling rates, when dendrite-arm coarsening was included in the model. It also was able to reproduce concentration profiles reasonably well. The analytical models commonly used (equilibrium cooling, Scheil equation, Brody/ Flemings model, Clyne/Kurz model, Solari/Biloni model, and Basaran equation) in micro-segregation calculations were shown to be considerably less accurate than the numerical models (MOL/II and the Ogilvy/Kirkwood model).

Macrosegregation during convection-free plane front solidification: II. Cylindrical and spherical geometries

The diffusion equation has been solved by numerical methods for cylindrical and spherical geometries in which the volume of the fluid phase decreases nonlinearly with time. Solute profiles were determined for conditions of constant velocity, convection-free plane front solidification with negligible solid state diffusion. The profiles depend upon two parameters, the equilibrium distribution coefficient, k, and a Peclet number, Pe, defined as the ratio of sample radius, R, to a liquid diffusion length, D/V, where D is the diffusion coefficient and V the velocity of the freezing interface. It is found that significant differences occur in the solute profiles for inward versus outward solidification, as well as for rectilinear versus cylindrical versus spherical geometries, and these are discussed. At large Pe values the solid concentration achieves a near constant pseudo-steady state value which is greater than 1 for inward solidification, but less than 1 for outward solidification. For values of k(1 − k)Pe ⩽ ≈ 0.1 the profile follows the Scheil equation for both the cylindrical and spherical cases.

Modeling of multidimensional solidification of an alloy

The solidification of an alloy and pure metal is distinct because of the morphological differences between the respective interfaces. For certain conditions of imposed heat extraction, this dis-tinction can lead to large differences in the times required for complete solidification of the alloy over the pure metal. These conditions are examined in this paper. To do this, a powerful numerical technique to model alloy solidification is introduced which allows for the precise integration of the Scheil equation. The model takes into account the nonlinearity of the fraction liquid with temperature as well as the interface nonlinearity in the heat flow. As a test for the model, accurate temperature profiles from a previously published experiment are numerically simulated. The numerical results are noted to closely match experimental values, and as a con-sequence, contact heat transfer values are tabulated. One- and two-dimensional solidification of aluminum-copper alloys are now simulated for different cooling conditions (i.e., varying Biot numbers). The results obtained indicate the essential differences for alloy solidification at low and high Biot numbers and highlight the importance of properly accounting for the mushy zone.

Directional solidification cells with grooves for a small partition coefficient.

Using the asymptotic matching procedure of Dombre and Hakim, we determine properties of the steady-state cellular structures with deep narrow grooves observed in directional solidification experiments. The method is valid in the experimentally relevant region of small partition coefficient and finite P\'eclet number. An extension of the Scheil equation for the grooves is given, and the importance of conservation in determining the groove closing is pointed out.

Macrosegregation during convection-free plane front solidification: I. Rectilinear geometry

The diffusion equation has been solved by numerical methods to determine the solute profiles produced in convection-free plane front solidification for finite rectilinear geometries with negligible solid state diffusion. It is shown that the profiles depend upon two parameters, the equilibrium distribution coefficient, k, and a Peclet number, Pe, defined as the ratio of sample length, L, to a liquid diffusion length D / V. It is found that for Pe > Pe∗ the solute profile follows the initial and final transient solutions of Smith . [Can. J. Phys. 33 (1955) 723], where data for Pe∗ versus k are presented graphically. For k(1−k)Pe⪅0.1 the Scheil equation [Z. Metallk. 34 (1942) 70] is followed. For intermediate values of k(1−kPe the solute profile follows the initial transient equation of Smith et al. out to some value of fraction solidified, f∗. Values of f∗ are presented for Pe from 0.1 to 1000 and k from 0.01 to 1.90.

An analysis of the applicability of the Scheil equation in the determination of segregation in monocrystals of the manganese‐zinc ferrites Mn1xZnxFe2O4

AbstractTheoretical assumptions concerning the segregation phenomenon of alloying components during formation of monocrystals of the manganese‐zinc ferrite Mn1xZnxFe2O4 have been presented. Taking into account the case of the unidirectional solidification of the above ferrite by means of the Bridgman furnace, which is working in: Closed system. Open system, with adding of the pure component (melting at an elevated temperature). Open system, with adding of ferrite of a nominal composition. The method for calculation of alloying components segregation has been analysed. A possibility of modelling the segregation by means of the choice of the size of the melting zone has been considered for a given monocrystal.

Phenomenon of the maximum inhomogeneity in solidification of alloys from the peritectic‐eutectic phase systems

AbstractAn application of the Scheil equation with the purpose to: 1 describe the maximum dendritic microsegregation phenomenon, i.e. amount of the solid solution in function of the initial composition of alloys from the AgSb phase system, 2 determine the nature of the maximum microsegregation in the Ag–30 Sb and Ag–10 Sb alloys is presented.A new relation, supplementary to the Scheil equation is proposed.

Nature of the Maximum Microsegregation of Binary Alloys from Peritectic Phase Systems

AbstractA value of the distribution coefficient is of a significant importance for the microsegregation phenomenon. Moreover, the nature of the microsegregation is due to a given phase system, and the Sb–30 Sn alloy has been set as an example to show the maximum microsegregation which can form during solidification of this alloy. A new relation, supplementary to the Scheil equation has been proposed.

The incorporation of tin in indium phosphide

The incorporation of tin in LEC-grown InP has been studied using the gamma emitting Sn 113 isotope. Quantitative measurements of tin content by radiotracer counting, as a function of fraction of melt solidified, were compared with similar measurements of carrier concentration as deduced by Hall effect assessment. Close agreement between measured total tin concentration and the free electron concentration in a series of samples shows that practically 100% of the tin atoms occupy indium lattice sites. This is believed to be the first direct measurement showing near zero compensation in tin doped InP. The macro-segregation of the tin is shown to obey the Scheil equation when this is modified to take account of the dependence of the segregation coefficient on the melt stoichiometry. Observed mobilities are in good agreement with previously published values; however, they fall short of those expected for zero compensation from theoretical considerations. This discrepancy is discussed but not resolved.

Solute redistribution in a duplex dendritic structure

A duplex dendritic structure consists of co-existing regions of coarse and fine dendrites. Such a structure in aluminium-4,5 wt % copper alloy was produced by rapid quenching from the partially solid-partially liquid state. The concentration of solute was found to be low and constant within the coarse structure, which contains a smaller amount of non-equilibrium secondary phase than does the fine dendritic structure. The solute distribution profiles within both the coarse and fine dendritic structures were found to be fairly accurately described by the Scheil equation for non-equilibrium solidification. Complete dissolution of the secondary phase occurred within the coarse structure after a shorter solutionizing time than it did within the fine dendritic structure. A spherical diffusion model accurately described solution kinetics within the coarse structure and a cylindrical diffusion model within the fine structure.

Ostwald ripening during solidification of nondendritic spherical structures

A theoretical model was introduced for the study of Ostwald ripening during solidification of the nondendritic spherical structure observed in Mg-Zn alloys grain-refined with Zr. This model assumed that: (1) particles are spherical; (2) each particle receives during solidification a mass increment proportional to its volume fraction with respect to the total volume of the particles in the system; (3) the Scheil equation applies. Experimental results agreed with analytical predictions on the time variation of the total solid-liquid interface area per unit volume and of the relative number of particles per unit volume.

Dendritic coarsening during solidification

A model for dendritic coarsening during solidification was developed, assuming: dissolution and shrinkage of small dendrite arms; lateral growth of large dendrite arms; applicability of the Scheil equation to solidification; deposition of solidified material on large arms. A reasonable agreement was found between calculated and measured values of surface-to-volume ratio, S v, versus time. It was concluded that: solidification growth contributes more to the decrease of S v than does coarsening; coarsening contributes to the disappearance of dendrite arms and, hence, enhances the effect of solidification growth on S v.