Boundary layer correlation for dendrite tip growth with fluid flow (original) (raw)
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Three Dimeasional Computer Model of Dendrite Growth in Tertiary Al-Cu-Si Alloys
Form casting, precision casting, soldering, welding are al1 manufacturing processes that involve solidification. In casting, solid nuclei appear in the liquid, leading to the formation of columnar or dendritic (tree-like structures) crystal structures. The type of microstructures that forms, whether dendritic or columnar, will influence such important material properties as strength and toughness. In this paper a computer simulation of a dendrite in three dimensions (3-D) based on the cellular model is described. Our model assumes constant temperature and the dendrite develops due to concentration gradients. The simulation is carried out for Al-Si-Cu alloys. Concentration profiles in three dimensions are obtained fiom the transport equation of solute diffusion and changes in concentration due to phase field changes. Concentration at each ce11 with a liquid fiaction is calculated for every tirne step. The interface velocity is calculated fiom the cwature at the interface, concentrations of each alloying component, undercooling and a kinetic coefficient constant. The time step and the velocity provide a new position of the solid-liquid interface for each iteration. Ln this model, curvature at each ce11 is calculated using an averaged phase field. The average phase field in 3-D is obtained by multipiying the solid fiaction of each ce11 and its neighbors by weight factors. This model takes anisotropy effects into account in order to avoid splitting of the dendrite tips. The data obtained tiom the model allow us to anaiyze the dendrite morphology and parameters such as tip radius, spacing between secondary arms, growth velocity and their dependence on undercooling. It is expected that the dendrite tip will grow with a stable parabola-like shape while the growth at the sides of the dendrite will be unstable. This instability at the solid liquid interface cm result in the growth of secondary arms. The results are compared with analytical mode1 data.
Three Dimeasional Cornputer Model of Dendrite Growtb in Tertiary Al-Cu-Si Alloys
1999
Form casting, precision casting, soldering, welding are al1 manufacturing processes that involve solidification. In casting, solid nuclei appear in the liquid, leading to the formation of columnar or dendritic (tree-like structures) crystal structures. The type of microstructures that forms, whether dendritic or columnar, will influence such important material properties as strength and toughness. In this paper a computer simulation ofa dendrite in three dimensions (3-D) based on the cellular model is described. Our model assumes constant temperature and the dendrite develops due to concentration gradients. The simulation is carried out for Al-Si-Cu alloys. Concentration profiles in three dimensions are obtained fiom the transport equation of solute diffusion and changes in concentration due to phase field changes. Concentration at each ce11 with a liquid fiaction is calculated for every tirne step. The interface velocity is calculated fiom the cwature at the interface, concentrations of each alloying component, undercooling and a kinetic coefficient constant. The time step and the velocity provide a new position of the solid-liquid interface for each
Parameter-free test of alloy dendrite-growth theory
Physical Review B, 1999
In rapid alloy solidification the dendrite-growth velocity depends sensitively on the deviations from local interfacial equilibrium manifested by kinetic effects such as solute trapping. The dendrite tip velocityundercooling function was measured in dilute Ni͑Zr͒ over the range 1-25 m/s and 50-255 K using electromagnetic levitation techniques and compared to theoretical predictions of the model of Trivedi and colleagues for dendritic growth with deviations from local interfacial equilibrium. The input parameter to which the model predictions are most sensitive, the diffusive speed V D characterizing solute trapping, was not used as a free parameter but was measured independently by pulsed laser melting techniques, as was another input parameter, the liquid diffusivity D L . Best-fit values from the pulsed laser melting experiment are V D ϭ26 m/s and D L ϭ2.7ϫ10 Ϫ9 m 2 /s. Inserting these values into the dendrite growth model results in excellent agreement with experiment with no adjustable parameters.
Model of Dendrite Growth in Metallic Alloys
Metallurgy and Foundry Engineering, 2010
Constantly increasing demand for high quality products as tendency to reduction production costs impose the use of innovative and advanced computer modelling techniques. Today's technician is unable to provide a high quality product in a demanded time without the use of computer aided tools. However, increasing competition between manufacturers causes use of basic algorithms becomes insufficient. That is why developers of new technology and computer intended algorithms more often take into account processes that runs at micro-and nano-scale. While computing power is increasing it is possible to get results of good accuracy after reasonable time, even though improved models need much more computational effort. Predicting the microstructure of modern alloys and composite materials requires a knowledge of differential equations that describe the rate of nucleation, dendrite growth and the alloying elements concentration in the grain and the fluid surrounding it. Attempts to develop such models have already been undertaken by numerous authors [1-8]. Mentioned authors attempt to analyze the process of nucleation. It is the stage, which has the greatest importance to the formation of casted component microstructure and its properties. Models built by them took into account the various processes that may take place in the metal during the initial stage of crystallization. They try to find the link between the effect of the nucleation rate and growth of existing grains on the latent heat of crystallization release. Partitioning durig solidification leads to build-up or depletion of solute in the liquid adjected to the solid. Mentioned process effects the crystals growth rate because the diffusion in the liquid is main phenomenon that controls the growth rate. As the grains radiuses grows the solid volume fraction grows. This leads to release the latent heat of solidification which affects the rate of undercooling change.
Sedimentation speed of a free dendrite growing in an undercooled melt
Computational Materials Science, 2010
Free equiaxed dendrites in solidifying alloy melts are subjected to hydrodynamic effects as a result of gravity. The sedimentation of dendrites is one such effect and believed to be a cause of macro segregation in partitioning alloys. A novel computational model is proposed to estimate the settling speed of free dendrites at moderate Reynolds numbers. Growth of the dendrite, momentum changes, internal solid fraction evolution within a spherical dendrite envelope of changing diameter, and surface morphology of the dendrite while settling are taken into account in the development of the model. Comparison with results from a series of equiaxed dendrite settling experiments, on solidifying transparent alloy analogues to metals, shows good agreement between predicted and experimental settling speeds. The correlation between surface morphology of the dendrite which affects drag force and the physical parameters of the settling dendrite is studied. The feasibility of applying the proposed model to metallic systems is also explored and the outlook is positive.
Non-steady 3D dendrite tip growth under diffusive and weakly convective conditions
Materialia, 2019
Three dimensional α-Al dendrite tip growth under varying solute gradients in an Al-Cu-Si alloy melt has been studied using real time synchrotron X-ray imaging and mathematical modelling. X-radiographic image sequences with high temporal and spatial resolution were processed and analysed to retrieve three-dimensional spatial details of the evolving dendrite and the solute concentration field, providing vastly improved estimates for the latter, in particular for the melt regions adjacent to the dendrite tips. Computational results obtained from an extended Horvay-Cahn dendrite tip model, capable of taking into account the effects of sample confinement, showed good agreement with the experimental data, and can be taken to verify the robustness of the 3D data extraction protocol.
Study of dendrite growth in a rotational flow field
2011
A numerical model to study the growth of dendrites in a pure metal solidification process with an imposed rotational flow field is presented. The micro-scale features of the solidification are modeled by the well-known enthalpy technique. The effect of flow changing the position of the dendrite is captured by the Volume of Fluid (VOF) method. An imposed rigid-body rotational flow is found to gradually transform the dendrite into a globular microstructure. A parametric study is carried out for various angular velocities and the time for merger of dendrite arms is compared with the order estimate obtained from scaling.
Metallurgical and Materials Transactions B, 2019
The interaction between convection and solute transport during solidification has significant influence on the dendritic evolution. By employing the phase-field lattice-Boltzmann approach together with the parallel and adaptive-mesh-refinement algorithm, the dendritic evolution under convection is simulated in both 2D and 3D cases. The flow-induced redistribution of the solute alters both tip velocity and the development of dendritic arms. The effect of both convection and undercooling is quantified and compared using the length ratio of the dendritic arms. The effect of convection behavior (i.e., natural and forced) and domain dimension (i.e., 2D and 3D) on dendritic growth is discussed. Results show that the convection effect is mainly dominated by the convection mode, and the melt flow in 2D can produce biased results comparing with those in 3D.
General evolution equation for the specific interface area of dendrites during alloy solidification
Acta Materialia
The specific area of the solid-liquid interface of an assembly of dendrites is an important integral measure of the morphology of the microstructure forming during alloy solidification. It represents the inverse of a characteristic length scale and is needed for the prediction of solidification defects and material properties. In the present study, the evolution of the interfacial area of dendrites is analysed using 3D phase-field simulations. A general evolution equation is developed for the specific interface area as a function of time and solid volume fraction that accounts for the effects of growth, curvaturedriven coarsening and interface coalescence. The relation is validated using data from previously performed synchrotron X-ray tomography and isothermal coarsening experiments. It is found to be valid for arbitrary and even varying cooling rates and for a wide range of binary alloys. The rate constant in the evolution equation is successfully related to alloy properties.