Interdendritic Liquid Flow Research Articles

Effective evaluation of permeability for interdendritic fluid flow using adaptive mesh refinement: Phase-field lattice Boltzmann study

An effective permeability prediction method was developed by applying an adaptive mesh refinement (AMR) scheme to the lattice Boltzmann (LB) simulations to compute interdendritic liquid flow. The developed method can automatically create adaptive meshes for LB simulations based on dendrite morphologies obtained by phase-field (PF) simulations. The AMR scheme drastically reduces the number of grid points and accurately computes the liquid flow around dendrites. We found that the reduction rate of the number of meshes was remarkable when using coarser meshes compared to those used in the PF simulation. This method will enable the systematic evaluation of the permeability for liquid flow in various dendritic structures.

Numerical Studies on the Effects of Axial Magnetic Fields and Cooling Intensities on the Macrosegregation of Directionally Solidified Ni-Based Alloys: Elimination of Thermal Convection and Stabilization of Interdendritic Liquid Flow

Numerical Studies on the Effects of Axial Magnetic Fields and Cooling Intensities on the Macrosegregation of Directionally Solidified Ni-Based Alloys: Elimination of Thermal Convection and Stabilization of Interdendritic Liquid Flow

Phase-field study on an array of tilted columnar dendrites during the directional solidification of a binary alloy

An array of columnar dendrites is important to the microsegregation and permeability of interdendritic liquid flow. In this study, an array of tilted columnar dendrites growing during the directional solidification of a binary alloy was investigated via large-scale phase-field simulation. The main conclusion is that the hexagonal array is the dominant array for reasonably tilted dendrites as well as dendrites that grow along the temperature gradient. It is also concluded that the array ordering gradually improves with growth for all tilt angles, and the ordering rate is faster for a small tilt angle.

Quantifying the effects of cooling rates and alloying additions on the microporosity formation in Al alloys

Microporosity in aluminum alloy castings is detrimental to the final mechanical properties such as fatigue life. Previous studies have shown that alloying additions in sand casting have a large impact on the percentage of porosity but no systematic studies on the effects of cooling rates and alloying additions have been reported. In this study, we investigate the porosity formation in three alloys with different cooling conditions using X-ray computed tomography (XCT). The microporosities of the three aluminum alloys at different cooling rates were visualized and statistically analyzed. When the cooling rate increased from 0.7 to 4.2 °C/s, the average equivalent diameter of microporosity in Al-7.3Si-1.5Cu decreased from 149.8 μm to 61.3 μm, with a size reduction of about 60 %; At a cooling rate of 0.7 °C/s, the average equivalent diameter of microporosity increased nearly three times with the increase of Si content to 7.3 wt.%. Comparing to three-dimensional (3D) cellular automata (CA) model predictions of porosity evolution in Al-Cu and Al-Si-Cu alloys, it was found that high cooling rates can increase the nucleation rate of microporosity and adding Cu and Si will give rise to the low solubility of hydrogen in the interdendritic liquid region during solidification. Therefore, robust casting components can be achieved by controlling the chemistry and applying non-equilibrium solidification conditions which change the interdendritic liquid flow and hydrogen supersaturations.

A review on the porous medium approaches to model the flow of interdendritic liquid during the solidification of alloys in casting processes: theory and experiments

The porous medium approach is one of the most popular methods to model the flow of liquid alloys through the solid interdendritic channels in the mushy zone, which is crucial to predict casting defects. In particular, characterizing the permeability of the mushy zone is of vital importance to reliably predict shrinkage porosity and blowholes. Nevertheless, this is not as simple task as permeability continuously evolves through the different stages of the solidification process. For these reasons, a great effort has been made in the past to obtain relationships relating the microstructural parameters to the permeability of the mushy zone. Also, some researchers have taken advantage of the advances in pore network modeling to predict permeability, obtaining promising results. Although a number of numerical experiments have been performed, their reliability has traditionally been hampered by the technical difficulties in reproducing real process conditions. The recent development of imaging techniques with improved spatial and temporal resolution, such as synchrotron 4D (3D + time) X-ray microtomography, has marked a turning point in the experimental validation of the models and is an efficient means to explore the solidification mechanisms. This context of prolific theoretical modeling and major technical achievements provides an excellent opportunity for the materials processing research community to apply the recent progress in fluid flow modeling through complex porous media to the field of casting processes. The aim of this article is to review the state of the art in modeling and experimental evaluation of the permeability of semi-solid zone of casting alloys during solidification so as to smooth the way to future studies. The main stumbling blocks in the field are also presented.

Simulation of macrosegregation in a 36-t steel ingot using a multiphase model

Macrosegregation is the major defect in large steel ingots caused by solute partitioning and melt convection during casting. In this study, a three-phase (liquid, columnar dendrites, and equiaxed grains) model is proposed to simulate macrosegregation in a 36-t steel ingot. A supplementary set of conservation equations are employed in the model such that two types of equiaxed grains, either settling or adhering to the solid shell, are well simulated. The predicted concentration agrees quantitatively with the experimental value. A negative segregation cone was located at the bottom owing to the grain settlement and solute-enriched melt leaving from the mushy zone. The interdendritic liquid flow was carefully analyzed, and the formation of A-type segregations in the mid-height of the ingot is discussed. Negative segregation was observed near the riser neck due to the specific relationship between flow direction and temperature gradient. Additionally, the as-cast macrostructure of the ingot is presented, including the grain size distribution and columnar-equiaxed transition.

The Influences of Grain Size and Morphology on the Hot Tearing Susceptibility, Contraction, and Load Behaviors of AA7050 Alloy Inoculated with Al-5Ti-1B Master Alloy

The influences of grain size and morphology on the hot tearing susceptibility of AA7050 alloy inoculated with Al-5Ti-1B master alloy were investigated by the authors. It was found that with the optimal addition of Al-5Ti-1B, coarse columnar grains were transformed into fine globular equiaxed grains. Moreover, due to the changes of grain size and morphology, the hot tearing susceptibility was decreased remarkably, which was attributed to the lower mechanical coherency temperature, better feeding ability, lower strain, and strain rate imposed to the mushy zone and more meandering propagation paths of hot tears. But the excess Al-5Ti-1B additions did not affect the grain structure, and greatly promoted hot tearing susceptibility due to the agglomerations of secondary phase particles from Al-5Ti-1B master alloy. The formation of massive secondary phases at grain boundaries hindered the interdendritic liquid flow and substantially deteriorated the feeding ability in the last stage of solidification. Meanwhile, TiB2 agglomerates would also act as stress raisers and cause the formation of voids. The contraction and load behaviors of AA7050 alloy influenced by grain size and morphology would be explored and connected with the hot tearing occurrence in this study.

Modeling of Heat Transfer and Interdendritic Strain for Exuded Surface Segregation Layer in the Direct Chill Casting of Aluminum Alloys

This investigation on the formation of exuded surface segregation layer “ESSL” is intended to provide experimental and simulated comparison to verify the model developed previously by El-Bealy. Preliminary verification and calibration of the previous 2D mathematical model are demonstrated by quantitative errors between the previous measurements and predictions of temperature and macrosegregation. Also, the results from these comparisons reveal that the errors are in the reasonable and within allowable limits. These comparisons lead to the fact that the exuded surface segregation layer mostly forms on the middle slice of broad sheet ingot face and in the early stages of mold zone. The model predictions point out also that the different interdendritic strain hypotheses associated with fluctuations of mold cooling conditions. This affects the interdendritic liquid flow between the equiaxed crystals which influences the severity of ESSL formation and its macrosegregation level. The mechanism of ESSL with heat flow and interdendritic strain generation has been analyzed and discussed. The quantitative comparisons between the pervious experimental results and numerical simulation in this investigation reveal also several solutions to prevent this defect for future work.

Simulation of unconstrained solidification of A356 aluminum alloy on distribution of micro/macro shrinkage

In the condition of Newtonian heat transfer, A356 aluminum alloy is solidified with randomly distributed equiaxed dendrites. Ability of interdendritic liquid flow is described by permeability parameter using Darcy's law and this parameter is used to predict the micro-shrinkages. In this study the interdendritic liquid flow during nucleation and grain growth are simulated in a 1mm×1mm domain. Temperature gradient is zero in the initial condition of the unconstrained solidification. The numerical simulation procedure includes two stages; first, numerical evolution of the shape, number, size, and distribution of dendrites during solidification using a novel Cellular Automation Finite Volume (CA-FV) method, and second, numerical determination of the micro-permeability by a Computational Fluid Dynamics (CFD) technique. Subsequently, the effect of Reynolds number, cooling rate and solidification rate on a critical permeability range was investigated in order to predict the micro/macro shrinkage distribution. Results showed that it is possible to propose a mathematical model to relate the Reynolds number and liquid flow rate, in the creeping flow range, on the micro-permeability during unconstrained solidification.

Evaluation of Permeability for Columnar Dendritic Structures by Three-dimensional Numerical Flow Analysis

In order to evaluate the permeability of columnar dendritic structures, three-dimensional (3-D) flow simulations of interdendritic liquid were carried out. The 3-D columnar dendrite morphologies, created by means of a computer-aided design (CAD) software, were based on two-dimensional dendrite morphologies calculated by a phase-field method. The artificial 3-D columnar dendrites were regularly arranged, and six kinds of 3-D columnar dendritic structures were observed, each with different liquid volume fractions between 0.56 and 0.95. For these 3-D columnar dendritic structures, the flow parallel and normal to the primary arms were calculated using FLUENT, and the permeability of six 3-D columnar dendritic structures for both flow directions were determined using the Darcy law. The values of our simulated permeability were compared with those values of permeability obtained experimentally [K. Murakami, A. Shiraishi and T. Okamoto: Acta Metall., 31 (1983), p. 1417, 32 (1984), p. 1423, C. Y. Liu, K. Murakami and T. Okamoto: Mater. Sci. Tech., 5 (1989), p. 1148]. For both flow directions, our simulated permeability for high liquid volume fractions complemented their experimental permeability for low liquid volume fractions. Therefore, we confirmed the consistency of simulated reading with extrapolations of experimental values of low volume fractions to high volume fractions. In addition, we discussed the limitation of flow within the mushy region, and found that defining the limiting permeability of interdendritic flow, in order to evaluate the relationship between dendritic morphology and the solid volume fraction where interdendritic liquid flow ceases, was effective.

3次元数値流体解析による柱状デンドライト組織の透過率評価

In order to evaluate the permeability of columnar dendritic structures, three-dimensional (3-D) flow simulations of interdendritic liquid were carried out. The 3-D columnar dendrite morphologies, created by means of a computer-aided design (CAD) software, were based on two-dimensional dendrite morphologies calculated by a phase-field method. The artificial 3-D columnar dendrites were regularly arranged, and six kinds of 3-D columnar dendritic structures were observed, each with different liquid volume fractions between 0.56 and 0.95. For these 3-D columnar dendritic structures, the flow parallel and normal to the primary arms were calculated using FLUENT, and the permeability of six 3-D columnar dendritic structures for both flow directions were determined using the Darcy law. The values of our simulated permeability were compared with those values of permeability obtained experimentally [K. Murakami, A. Shiraishi and T. Okamoto: Acta Metall., 31 (1983), p. 1417, 32 (1984), p. 1423, C. Y. Liu, K. Murakami and T. Okamoto: Mater. Sci. Tech., 5 (1989), p. 1148]. For both flow directions, our simulated permeability for high liquid volume fractions complemented their experimental permeability for low liquid volume fractions. Therefore, we confirmed the consistency of simulated reading with extrapolations of experimental values of low volume fractions to high volume fractions. In addition, we discussed the limitation of flow within the mushy region, and found that defining the limiting permeability of interdendritic flow, in order to evaluate the relationship between dendritic morphology and the solid volume fraction where interdendritic liquid flow ceases, was effective.

Alignment and Permeability of Al-7Si Alloy Directional Solidification with the Application of a Pulsed Magnetic Field

An experiment on the directional solidification of Al-7Si alloys under the influence of a pulsed magnetic field (PMF) is performed. Maximal peak value of PMF intensity up to 0.226 T between two iron cores was generated when a capacitor bank of 120 µF capacitance with initial voltage of 1000 V was triggered to a pair of solenoids. The PMF decreases the temperature gradient ahead of the liquidus front and increases the width of the mushy zone. The convection of symmetrical vortices seems to charge the intersectional growth of primary dendrite stems and the decrease of dendrite arms spacing under a PMF with higher magnetic intensities. Using Darcy's permeability law and Lehmann's model, the order of magnitude of the interdendritic liquid flow velocities with the application of PMF is evaluated. A strong damped effect of the mushy zone in the bulk flow of the melt is revealed.

On the Formulation of a Freckling Criterion for Ni-Based Superalloy Vacuum Arc Remelting Ingots

A criterion for freckling prediction that includes the effect of a tilted solidification front was proposed and evaluated with experimental data available in the literature. The criterion is based on the maximum local Rayleigh number in the mush layer and was developed using Flemings’ criterion and assuming that the interdendritic liquid flow is governed by the Darcy law. The proposed form preserves the anisotropic nature of the permeability tensor throughout the derivation and provides improved resolution on freckle prediction. A clear separation between the freckled and nonfreckled experiments was obtained for all compositions. The effect of the tilted solidification front over the freckling potential was corroborated, and the results suggested that the directionality of permeability affects the location within the mush layer of the potential nucleation sites for the channels leading to freckles. A threshold zone was determined from the enclosing experiments data, and the range contained one of the proposed critical values for superalloys, which previously was developed by a completely different method.

Permeability in the Mushy Zone

When modeled at macroscopic length scales, the complex dendritic network in the solid-plus-liquid region of a solidifying alloy (the “mushy zone”) has been modeled as a continuum based on the theory of porous media. The most important property of a porous medium is its permeability, which relates the macroscopic pressure gradient to the throughput of fluid flow. Knowledge of the permeability of the mushy zone as a function of the local volume-fraction of liquid and other morphological parameters is thus essential to successfully modeling the flow of interdendritic liquid during alloy solidification. In current continuum models, the permeability of the mushy zone is given as a deterministic function of (1) the local volume fraction of liquid and (2) a characteristic length scale such as the primary dendrite arm spacing or the reciprocal of the specific surface area of the solid-liquid interface. Here we first provide a broad overview of the experimental data, mesoscale numerical flow simulations, and resulting correlations for the deterministic permeability of both equiaxed and columnar mushy zones. A extended view of permeability in mushy zones which includes the stochastic nature of permeability is discussed. This viewpoint is the result of performing extensive numerical simulations of creeping flow through random microstructures. The permeabilities obtained from these simulations are random functions with spatial autocorrelation structures, and variations in the local permeability are shown to have dramatic effects on the flow patterns observed in such microstructures. Specifically, it is found that “lightning-like” patterns emerge in the fluid velocity and that the flows in such geometries are strongly sensitive to small variations in the solid structure. We conclude with a comparison of deterministic and stochastic permeabilities which suggests the importance of incorporating stochastic descriptions of the permeability of the mushy zone in solidification modeling.

Properties of stochastic permeability

When modeled at macroscopic length scales, the complex dendritic network in the solid-plus-liquid region of a solidifying alloy (the “mushy zone”) has been modeled as a continuum based on the theory of porous media. The most important property of a porous medium is its permeability, which relates the macroscopic pressure gradient to the throughput of fluid flow. Knowledge of the permeability of the mushy zone as a function of the local volume-fraction of liquid and other morphological parameters is thus essential to successfully modeling the flow of interdendritic liquid during alloy solidification. Permeability is usually treated as a deterministic function of parameters that can be calculated by the model (e.g., local solid fraction, dendrite arm spacing). However, recent results show that the length scales that must be resolved are too small for the assumption of deterministic behavior to be valid, and investigators must confront the stochastic behavior of the permeability field. We describe early work to investigate the spatial structure of the stochastic permeability at these small scales, with a view to develop a comprehensive treatment of stochastic permeability to enable improved modeling.

Modelling and simulation of equiaxed dendritic structures permeability for Pb–Sn alloys

In this investigation, the permeability of interdendritic liquid flow through the equiaxial mushy zone has been modelled for Pb–Sn alloys based on experimental measurements. In the present work by solving Navier–Stokes equation, the flow pattern around the equiaxed dendrite has been obtained and then permeability has been determined by applying Darcy's law. Numerical determined values of permeabilities have been analysed by the use of Statistical Package for the Social Sciences (SPSS) statistical software. Then an experimental method has been used to measure the permeability for flow through equiaxial mushy zone of Pb–Sn alloys. Results show that increasing the solid fraction and perturbation decreases the permeability, but unlike the solid fraction, perturbation has less effect on permeability. It seems that the empirical relationship presented in this article could be used to estimate permeability in the mushy Pb–Sn alloys.

Prediction of Macrosegregation in Steel Ingots: Influence of the Motion and the Morphology of Equiaxed Grains

Although a significant amount of work has already been devoted to the prediction of macrosegregation in steel ingots, most models considered the solid phase as fixed. As a result, it was not possible to correctly predict the macrosegregation in the center of the product. It is generally suspected that the motion of the equiaxed grains is responsible for this macrosegregation. A multiphase and multiscale model that describes the evolution of the morphology of the equiaxed crystals and their motion is presented. The model was used to simulate the solidification of a 3.3-ton steel ingot. Computations that take into account the motion of dendritic and globular grains and computations with a fixed solid phase were performed, and the solidification and macrosegregation formation due to the grain motion and flow of interdendritic liquid were analyzed. The predicted macrosegregation patterns are compared to the experimental results. Most important, it is demonstrated that it is essential to consider the grain morphology, in order to properly model the influence of grain motion on macrosegregation. Further, due to increased computing power, the presented computations could be performed using finer computational grids than was possible in previous studies; this made possible the prediction of mesosegregations, notably A segregates.

Modeling of permeability with the aid of 3D interdendritic flow simulation for equiaxed dendritic structures

In this study, the permeability for interdendritic liquid flow through equiaxial mushy zone has been modeled and experimentally measured. In the present work, by applying a virtual dendrite in a micro-domain and solving Navier–Stokes equation, flow pattern around the dendrite has been obtained and then by applying Darcy's law to this 3D domain the permeability has been determined. In this micro-model the influence of solid fraction and geometry of dendrites have been assessed. Numerically determined values of permeabilities have been analyzed by the use of SPSS statistical software. Then an experimental method is used to measure the permeability for flow through equiaxial mushy zone of Pb–Sn alloys in high solid fractions. Results show that increasing the solid fraction and perturbation has been led to decreasing the permeability but unlike the solid fraction, perturbation has less effect on the permeability. The results are in good agreement with the available experimental data and analytical solutions. The empirical relationship presented in this article could be used to estimate permeability in the mushy Pb–Sn alloys.

Simulation of interdendritic liquid permeability for low and high solid fractions during the solidification of mushy alloys

A numerical model has been developed for the determination of liquid flow permeability through columnar dendrite during growth and segregation in the Al–Si alloys. Therefore, in the present work two separate computational models of the grain growth and interdendritic liquid flow are coupled for modelling of the permeability in partly solid alloy. The grain growth is simulated using a cellular automation finite difference (CAFD) for a 2D dendrite, and fluid flow by using a computational fluid dynamic (CFD) model for determining permeability. A new model has been presented for calculation and modification of the dendrites permeability in the high solid fractions. Simulation results show which Si concentration variations in each time step could transform of dendrite shape. Also dendrites morphology could alter the interdendritic permeability factor.

Permeability measurements of the flow of interdendritic liquid in equiaxed aluminum-silicon alloys

A method developed recently for measuring instantaneous permeability in the mushy zone of aluminum-copper alloys is used to measure permeability in aluminum-silicon alloys. Permeability was also measured in packed glass beads using water, because measurements could be conducted in the absence of microstructural coarsening. Experimental data are compared with the Kozeny-Carman equation to determine the value of kC, the Kozeny-Carman constant. The results indicate that when kC=5.0, all the instantaneous permeability data agree well with the Kozeny-Carman equation. Factors affecting the accuracy of permeability measurements, such as surface oxidation of the sample, flow channeling during the test, microstructural coarsening, and liquid fraction reduction due to back diffusion in the solid are discussed. An experimental technique that minimizes errors due to these factors is outlined.