A generalized Stefan model accounting for system memory and non-locality

The Stefan problem, involving the tracking of an evolving phase-change front, is the prototypical example of a moving boundary problem. In basic one- dimensional problems it is well known that the front advances as the square root of time. When memory or non-locality are introduced into the system however, this classic signal may be anomalous; replaced by a power-law advance with a time exponent that differs from n = 1/2. Up to now memory treatments in Stefan problem models have only been able to reproduce sub-diffusive front movements with exponents n 1/2. In the present paper, using a generalized Caputo fractional derivative operator, we introduce new memory and non-local treatment for Stefan problems. On considering a limit case Stefan problem, related to the melting problem, we are able to show that, this gen- eral treatment can not only produce arbitrary power-law in time predictions for the front movement but, in the case of memory treatments, can also produce non-power-law anomalous behaviors. Further, also in the context of the limit problem, we are able to establish an equivalence between non-locality and a space varying conductivity and memory and a time varying conductivity

Dynamic Analysis of Unidirectional Pressure Infiltration of Porous Preforms by Pure Metals

Unidirectional pressure infiltration of porous preforms by molten metals is investigated numerically. A phenomenological model to describe fluid flow and transport phenomena during infiltration of fibrous preforms by a metal is formulated. The model describes the dynamics of the infiltration process, the temperature distribution, and solid fraction distribution. The numerical results are compared against classical asymptotic analyses and experimental results. This comparison shows that end effects may become important and render asymptotic results unreliable for realistic samples. Fiber volume fraction and initial temperature appear as the factors most strongly influencing infiltration. Metal superheating affects not only the length of the two-phase zone but also the solid fraction distribution in the two-phase zone. The effect of constant applied pressure, although significant on the infiltration velocity, is almost negligible on the two-phase zone length and on solid fraction distribution. When the initial preform temperature is below the metal melting point, and constant pressure is applied under adiabatic conditions, the flow ceases when sufficient solidification occurs to obstruct it. A comparison with literature experiments proves the model to be an efficient predictive tool in the analysis of infiltration processes for different preform/melt systems

A Comprehensive Case Study of Macrosegregation in a Steel Ingot

This is the author accepted manuscript. The final version is available from Springer via http://dx.doi.org/10.1007/s11663-015-0386-yA case study is presented that examines the macrosegregation and grain structure present in a 12-tonne steel ingot, which was cast for experimental purposes. Details of the casting procedure were well documented and the resulting ingot was characterized using a number of techniques that measured chemical segregation, shrinkage, and porosity. The formation of the porosity and segregation patterns is discussed in reference to the particular grain structure observed in the ingot. It is hoped that this case study can be used as a tool for the validation of future macromodels.This work was undertaken as part of a Project sponsored by Rolls-Royce Power Nuclear plc in collaboration with Sheffield Forgemasters International

Simulation of Channel Segregation During Directional Solidification of In—75 wt pct Ga. Qualitative Comparison with In Situ Observations

International audienceFreckles are common defects in industrial casting. They result from thermosolutal convection due to buoyancy forces generated from density variations in the liquid. The present paper proposes a numerical analysis for the formation of channel segregation using the three-dimensional (3D) cellular automaton (CA)—finite element (FE) model. The model integrates kinetics laws for the nucleation and growth of a microstructure with the solution of the conservation equations for the casting, while introducing an intermediate modeling scale for a direct representation of the envelope of the dendritic grains. Directional solidification of a cuboid cell is studied. Its geometry, the alloy chosen as well as the process parameters are inspired from experimental observations recently reported in the literature. Snapshots of the convective pattern, the solute distribution, and the morphology of the growth front are qualitatively compared. Similitudes are found when considering the coupled 3D CAFE simulations. Limitations of the model to reach direct simulation of the experiments are discussed

Experimental and Numerical Modeling of Segregation in Metallic Alloys

International audienceElectromagnetic levitation (EML) has been used as an experimental technique for investigating the effect of the nucleation and cooling rate on segregation and structure formation in metallic alloys. The technique has been applied to aluminum-copper alloys. For all samples, the primary phase nucleation has been triggered by the contact of the levitated droplet with an alumina plate at a given undercooling. Based on the recorded temperature curves, the heat extraction rate and the nucleation undercooling for the primary dendritic and the secondary eutectic structures have been determined. Metallurgical characterizations have consisted of composition measurements using a scanning electron microscope (SEM) equipped with energy dispersive X-ray spectrometry and the analysis of SEM images. The distribution maps drawn for the composition, the volume fraction of the eutectic structure, and the dendrite arm spacing (DAS) reveal strong correlations. Analysis of the measurements with the help of a cellular-automaton (CA)-finite-element (FE) model is also proposed. The model involves a new coupling scheme between the CA and FE methods and a segregation model accounting for diffusion in the solid and liquid phases. Extensive validation of the model has been carried out on a typical equiaxed grain configuration, i.e., considering the free growth of a mushy zone in an undercooled melt. It demonstrates its capability of dealing with mass exchange inside and outside the envelope of a growing primary dendritic structure. The model has been applied to predict the temperature curve, the segregation, and the eutectic volume fraction obtained upon single-grain nucleation and growth from the south pole of a spherical domain with and without triggering of the nucleation of the primary solid phase, thus simulating the solidification of a levitated droplet. Predictions permit a direct interpretation of the measurements

Moore's law and numerical modeling

An estimate of the rate of increase in numerical simulation grid sizes with time is obtained by counting the grids (measured in terms of number of node points) reported in the nine volumes of an established proceedings on the numerical modeling of solidification phenomena dating back to 1980. It is shown that the largest grids used in a given year increase at a rate consistent with the well-known Moore's law on computing power, i.e., the number of nodes in the grids double every 18 months. Front this observation, approximate bounds on the available grid size in a current year are established. This approximation is used to provide projections as to when, assuming Moore's law continues to hold, direct simulations of physical phenomena, which resolve to the smallest scale present, will be achievable

An explicit scheme for coupling temperature and concentration fields in solidification models

A numerical scheme for coupling temperature and concentration fields in a general solidification model is presented. A key feature of this scheme is an explicit time stepping used in solving the governing thermal and solute conservation equations. This explicit approach results in a local point-by-point coupling scheme for the temperature and concentration and avoids the multi-level iteration required by implicit time stepping schemes. The proposed scheme is validated by predicting the concentration field in a benchmark solidification problem. Results compare well with an available similarity solution. The simplicity of the proposed explicit scheme allows for the incorporation of complex microscale models into a general solidification model. This is demonstrated by investigating the role of dendrite coarsening on the concentration field in the solidification benchmark problem