Enthalpies of a binary alloy during solidification

The purpose of the paper is to present a method of calculating the enthalpy of a dendritic alloy during solidification. The enthalpies of the dendritic solid and interdendritic liquid of alloys of the Pb-Sn system are evaluated, but the method could be applied to other binaries, as well. The enthalpies are consistent with a recent evaluation of the thermodynamics of Pb-Sn alloys and with the redistribution of solute in the same during dendritic solidification. Because of the heat of mixing in Pb-Sn alloys, the interdendritic liquid of hypoeutectic alloys (Pb-rich) of less than 50 wt pct Sn has enthalpies that increase as temperature decreases during solidification. For some concentrations of Sn, the enthalpy of the dendritic solid at the solid-liquid interface also increases with decreasing temperature during solidification. Of particular concern, in formulating the energy equation, is the fact that the heat of fusion during solidification increases as much as 80 pct for hypoeutectic alloys and decreases as much as 25 pct for hypereutectic alloys. Thus the often applied assumptions of a constant specific heat and/or a constant heat of solidification could lead to errors in numerical modeling of temperature fields for dendritic solidification processes.     

Thermosolutal convection during dendritic solidification of alloys: Part II. Nonlinear convection

A mathematical model of thermosolutal convection in directionally solidified dendritic alloys has been developed that includes a mushy zone underlying an all-liquid region. The model assumes a nonconvective initial state with planar and horizontal isotherms and isoconcentrates that move upward at a constant solidification velocity. The initial state is perturbed, nonlinear calculations are performed to model convection of the liquid when the system is unstable, and the results are compared with the predictions of a linear stability analysis. The mushy zone is modeled as a porous medium of variable porosity consistent with the volume fraction of, interdendritic liquid that satisfies the conservation equations for energy and solute concentrations. Results are presented for systems involving lead-tin alloys (Pb-10 wt pct Sn and Pb-20 wt pct Sn) and show significant differences with results of plane-front solidification. The calculations show that convection in the mushy zone is mainly driven by convection in the all-liquid region, and convection of the interdendritic liquid is only significant in the upper 20 pct of the mushy zone if it is significant at all. The calculated results also show that the systems are stable at reduced gravity levels of the order of 10⁻⁴ g ₀ (g ₀=980 cm·s⁻¹) or when the lateral dimensions of the container are small enough, for stable temperature gradients between 2.5≤G _l≤100 K·cm⁻¹ at solidification velocities of 2 to 8 cm·h⁻¹.     

Thermal strain in the mushy zone for aluminum alloys

The parameters in a recently developed constitutive equation for macroscopic thermal strain in the mushy zone have been determined for the commercial alloys A356, AA2024, AA6061, and AA7075 in addition to an Al-4 wt pct Cu alloy. The constitutive equation for macroscopic thermal strain in the mushy zone reflects that there is no thermal strain in the solid part of the mushy zone at low solid fractions and that the thermal strain in the mushy zone approaches thermal strain in the fully solid material as the solid fraction increases toward 1. The development of thermal strain in the mushy zone is determined by combining experimentally measured contraction of a cast sample with thermomechanical stimulations. Experiments were performed at cooling rates in the range from 2 to 5.5 °C/s. The solid fractions when the tested alloys start to contract,g _s ^th, are in the range from 0.63 to 0.94. Grain refinement increasesg _s ^th for all the tested alloys. For most of the tested alloys the thermal strain in the mushy zone increases rapidly to the same level as thermal strain in fully solid material once the solid fraction becomes higher thang _s ^th.     

Solute diffusion model for equiaxed dendritic growth

A new approach to the modeling of the equiaxed solidification of dendritic alloys is proposed. It is assumed that, in metallic alloys, microstructure formation is primarily controlled by solute diffusion (i.e. there is complete “thermal mixing” at the scale of one grain), and that the dendrite interface is an iso-concentrate at all times. The evolution of one dendritic grain is therefore modelled as follows: (i) complete mixing of solute within the interdendritic liquid; (ii) no back-diffusion in the solid; (iii) spherical solute diffusion in the liquid around the grain envelope; (iv) overall solute balance; (v) overall thermal balance; (vi) growth velocity, υ_g, of dendrite tips governed by the kinetic equation derived for the isolated dendrite case. By using an explicit finite difference scheme to solve these coupled equations, the concentration profiles, cooling curve, fraction of solid and evolution of dendritic grain envelope can be calculated. The intitial conditions used to start the calculation are provided by two parameters related to nucleation: the initial undercooling and the density of gains. The effect of nucleation and thermal conditions on equiaxed growth are studied. The theoretical predictions of recalescence and of the distribution of an interdendritic eutectic phase are in good agreement with experimental observations.     

Solutal convection induced macrosegregation and the dendrite to composite transition in off-eutectic alloys

The effect of solute gradient induced convection during vertical solidification on the macrosegregation of Pb-rich Pb-Sn off-eutectic alloys is determined experimentally as a function of composition and growth rate. In many cases macrosegregation is sufficient to prevent the plane front solidification of the alloy. The transition from dendritic to composite structure is found to occur when the composition of the solid is close enough to the eutectic composition to satisfy a stability criterion based onG _L /V (liquid temperature gradient/growth rate). A vertical or horizontal magnetic field of 0.1 T (1 kilogauss) does not reduce macrosegregation, but downward solidification (liquid below solid) virtually eliminates macrosegregation in small (∼3 mm) diameter samples.     

Thermosolutal convection during dendritic solidification of alloys: Part i. Linear stability analysis

This paper describes the simulation of thermosolutal convection in directionally solidified (DS) alloys. A linear stability analysis is used to predict marginal stability curves for a system that comprises a mushy zone underlying an all-liquid zone. In the unperturbed and nonconvecting state .e.}, the basic state), isotherms and isoconcentrates are planar and horizontal. The mushy zone is realistically treated as a medium with a variable volume fraction of liquid that is con-sistent with the energy and solute conservation equations. The perturbed variables include tem-perature, concentration of solute, and both components of velocity in a two-dimensional system. As a model system, an alloy of Pb-20 wt pct Sn, solidifying at a velocity of 2 X 10^-3 cm s^-1 was selected. Dimensional numerical calculations were done to define the marginal stability curves in terms of the thermal gradient at the dendrite tips,G _L ,vs the horizontal wave number of the perturbed quantities. For a gravitational constant of 1g,0.5 g, 0.1g, and 0.01g, the marginal stability curves show no minima; thus, the system is never unconditionally stable. Nevertheless, such calculations quantify the effect of reducing the gravitational constant on reducing convection and suggest lateral dimensions of the mold for the purpose of suppressing convection. Finally, for a gravitational constant of 10^-4 g, calculations show that the system is stable for the thermal gradients investigated (2.5 ≤G _L ≤ 100 K-cm^-1).     

Advanced Solute Conservation Equations for Dendritic Solidification Processes Part I: Experiments and Theory

The macrosegregation formed in dendritic equiaxed structure during early stages of solidification of Al‐4.5%Cu alloy has been studied by experimental work and by metallurgical study of cast samples taken from the experimental work. An experimental work was conducted to study the coupled effect of natural convection streams, interdendritic strain and mushy permeability of Al‐4.5%Cu aluminum alloy solidified in horizontal rectangular parallelepiped cavity at different superheats. The metallurgical study includes macro‐microstructure evaluation, measurements of grain size of equiaxed crystals and macrosegregation analysis. This study shows that the level of surface segregation exhibiting as positive segregation varies with superheat whereas the rest of inner ingot areas show the light fluctuation in segregation values. In addition to experimental work, there is a mathematical study which contains a complete derivation of local solute redistribution equations based on Fleming's approach under different solute diffusion mechanisms in the dendritic solid. This derivation includes also the effects of interdendritic strain and mushy permeability on the local solute redistribution distribution. Owing to the length of the study, it is presented in two parts. The first part describes the experimental work and its results as well as a detail derivation of solute conservation equations. This part also involves comparison and discussion between existing and proposed solute conservation equations. The second part contains the mathematical analyses of a two dimensional mathematical model of fluid flow, heat flow, solidification, interdendritic strain and macrosegregation. Also, this part also contains the numerical simulations by using finite difference technique “FDT” to create convection patterns, heat transfer, interdendritic strain, and macrosegregation distributions. This part also includes comparisons between the available measurements and model predications as well as full discussion of different model simulations. The mechanism of interdendritic strain generation and macrosegregation formation during solidification of dendritic equiaxed structure under different diffusion mechanisms in dendritic solid has also been explained and discussed.     

Specific permeability of partially solidified dendritic networks of Al-Si alloys

Porous dendritic networks of Al-4 pet Si and Al-4 pct Si-0.25 pet Ti alloys with volume fraction solid,g _sA > 0.628 were prepared by removing the segregated interdendritic liquid from partially solidified samples of the alloys. In the equiaxed samples, available channels for flow were predominately between the grains. Specific permeabilities of the porous dendritic networks were measured with a triaxial cell permeameter. Measured values of specific permeability were 1 × 10^-9 to 3.5 × 10^-11 cm² in the Al-4 pet Si alloy for volume fractions solid of 0.655 to 0.94, respectively. Specific permeabilities in Al-4 pct Si-0.25 pct Ti alloy were 4.82 × 10^-10 to 7.6 × 10^-11 cm² for volume fractions solid of 0.628 to 0.837, respectively. For equivalent volume fractions solid, the measured specific permeabilities were consistently lower for the grain refined samples. Flow through the porous dendritic networks obeys D’Arcy’s law and equations derived from the capillaric flow model for volume fraction liquid less than ∼0.35. Formerly a graduate student in the Department of Metallurgy and Materials Science, M.I.T.     

The origin of freckles in unidirectionally solidified castings

The origin of freckles during unidirectional solidification is studied in a transparent, low melting model system, 30 wt pct NH₄C1-H₂O. In 30NH₄Cl-H₂O, freckles are caused by upward flowing liquid jets in the mushy zone. The jets erode the mushy zone causing localized segregation and start new grains by producing dendritic debris. It is shown that the jets observed in 30NH₄C1-H₂O are free convection resulting from a density inversion in the mushy zone. A comparison of driving force, thermal transport effects and solute transport effects in 30NH₄C1-H₂O and metallic systems shows that jets are possible in metallic alloys where light elements segregate normally or heavy elements segregate inversely. It is concluded that freckles in unidirectionally solidified castings and vacuum consumable-electrode ingots are caused by convective jets. It is shown that the tendency to freckle is greatest in alloys with a large density inversion, high thermal diffusivity, low solute diffusivity, and low viscosity. For a given alloy, the driving force for freckling is proportional to the inverse square of the thermal gradient. Erosion by the jets is decreased by increasing the thermal gradient and growth rate. The location of freckles is influenced by mushy zone curvature. Formerly Research Assistant at the Advanced Materials Research and Development Laboratory     

Solute redistribution during and after solidification of Ni-Al-Ta dendritic monocrystals

Dendritic Monocrystals of Ni-Al-Ta alloys were grown at 0.05, 0.25, and 2.00 m/h and in some cases at other intermediate rates, under thermal gradients of 8 × 10³ and 18 × 10³ K/m. The growth of such monocrystals provides a rapid and easy way for:a) establishing the distribution of solute during and after solidification, as well as its dependence on local cooling rate; b) determining the effect of dendritic coarsening on this distribution and; c) studying the solution kinetics of the nonequilibrium interdendritic γ′ phase. Back-diffusion in the solid rather than dendritic coarsening was found to control the evolution of the solute distribution profile across the dendritic structure during solidification. With increasing local cooling rate the maximum solute concentration,C _M, remained practically unchanged, the minimum solute concentration,C _m, slightly decreased, the segregation ratio,S = C _M/C_m, increased and so did the volume fraction of nonequilibrium interdendritic γ′ phase. This phase dissolved during crystal pulling much faster at higher crystal growth rates. Solution kinetics were found to depend on the dimensionless parameterDθ/L ², whereD is diffusivity of solute at a given temperature at which a given transverse cross-section of the crystal remains for a timeθ andL is half the primary dendrite arm spacing.     

Mushy zone morphology during directional solidification of Pb-5.8 wt pct Sb alloy

The Pb-5.8 wt pct Sb alloy was directionally solidified with a positive thermal gradient of 140 K cm⁻¹ at a growth speed ranging from 0.8 to 30 μm s⁻¹, and then it was quenched to retain the mushy zone morphology. The morphology of the mushy zone along its entire length has been characterized by using a serial sectioning and three-dimensional image reconstruction technique. Variation in the cellular/dendritic shape factor, hydraulic radius of the interdendritic region, and fraction solid along the mushy zone length has been studied. A comparison with predictions from theoretical models indicates that convection remarkably reduces the primary dendrite spacing while its influence on the dendrite tip radius is not as significant.     

Macrosegregation during dendritic arrayed growth of hypoeutectic pb- sn alloys: Influence of primary arm spacing and mushy zone length

Thermosolutal convection in the dendritic mushy zone occurs during directional solidification of hypoeutectic lead tin alloys in a positive thermal gradient, with the melt on the top and the solid below. This results in macrosegregation along the length of the solidified samples. The extent of macrosegregation increases with increasing primary dendrite spacings for constant mushy zone length. For constant primary spacings, the macrosegregation increases with decreasing mushy zone length. Presence of convection reduces the primary dendrite spacings. However, convection in the interdendritic melt has significantly more influence on the spacings as compared with that in the overlying melt, which is caused by the solutal buildup at the dendrite tips. Formerly Graduate Student, Chemical Engineering Department, Cleveland State University     

A Mathematical Model of Interdendritic Thermometallurgical Strain for Dendritic Solidification Processes

A mathematical model of interdendritic thermometallurgical strain (ITM) generated during dendritic solidification of steel alloys has been developed. The model consists of two main parts in which the first part represents the derivation of ITM strain based on the concept of thermal storage energy. Consequently, the second one represents the volume changes associated with dendritic solidification phenomena during the interdendritic coherent region in the mushy zone. Calculations for Fe-C binary alloys show that the straining criteria such as strain rate and accumulated strain are sensitive to the solidification behavior of different steel alloys. The results also point out that good predications of different phases, accurate alloy properties (especially thermal expansion coefficient), and precise enthalpy functions during dendritic solidification are extremely necessary. Also, the predications show that the dendrite coherency criterion is considered an essential parameter to define ITM strain of particularly long solidification interval alloys. Model predications of ITM strain are compared and discussed with other established theoretical approaches and previous modeling work.     

Modeling of solute redistribution in the mushy zone during solidification of aluminum-copper alloys

A mathematical model has been established to predict the formation of macrosegregation for a unidirectional solidification of aluminum-copper alloys cooled from the bottom. The model, based on the continuum formulation, allows the calculation of transient distributions of temperature, velocity, and species in the solidifying alloy caused by thermosolutal convection and shrinkage-induced fluid flow. Positive segregation in the casting near the bottom (inverse segregation) is found, which is accompanied by a moving negative-segregated mushy zone. The effects of shrinkage-induced fluid flow and solute diffusion on the formation of macrosegregation are examined. It is found that the redistribution of solute in the solidifying alloy is caused by the flow of solute-rich liquid in the mushy zone due to solidification shrinkage. A higher heat-extraction rate at the bottom increases the solidification rate, decreasing the size of the mushy zone, reducing the flow of solute-rich liquid in the mushy zone and, as a result, lessening the severity of inverse segregation. Comparisons between the theoretical predictions from the present study and previous modeling results and available experimental data are made, and good agreements are obtained.     

Macrosegregation during steady-state arrayed growth of dendrites in directionally solidified Pb-Sn alloys

Macrosegregation along the length of the directionally solidified samples is produced when Pb-Sn alloys (10 to 58 wt pct Sn) are directionally solidified in a positive thermal gradient (melt on top, solid below, and gravity pointing down) with steady-state dendritic arrayed morphology (the length of the mushy zone, much smaller than the initial length of the melt column, remaining nearly constant during growth). The extent of the macrosegregation increases with increasing tin content, becomes maximum for 33.3 wt pct Sn, and decreases with further increase in tin content. The intensity of the interdendritic thermosolutal convection responsible for the longitudinal macrosegregation can be represented by the effective partition coefficient(k _E), anempirical parameter obtained from the dependence of the longitudinal macrosegregation on fraction distance solidified. The extent of the macrosegregation appears to be related to a parameter,x03BB; ² ₁f_E(C_E−C_t)}, where A, is the primary dendrite spacing,f _Eis the volume fraction of the interdendritic melt, andC _EandC _tare the eutectic composition and the melt composition ahead of the dendrite tips, respectively.     

Measurement of liquid permeability in the mushy zones of aluminum-copper alloys

Measurements of liquid permeability in the mushy zones of Al-15.42 pct Cu and Al-8.68 pct Cu alloy samples were performed isothermally just above the eutectic temperature, using eutectic liquid as the fluid. A modified method was developed to determine the specific permeability as a function of time (K _s) during the test from the data collected on these alloys. Factors affecting permeability measurements are discussed. The permeabilities are observed to vary throughout the experiment. This is attributed to microstructural coarsening and channeling that occur in the sample during the experiment. Coarsening rates are determined for the isothermal coarsening tests without fluid flow, and the results are observed to be less than the rates indicated from permeability tests where fluid flow is present. Careful measurement of the volume fraction of liquid (g _L) shows that g _L decreases during the test. The permeability is then related to the microstructure of the sample using the Kozeny-Carman equation. The correlation between the measured K _S, g _L, and specific solid surface area (S _V) improves markedly when compared to previous studies, when microstructural parameters at the initial stage of the test are used.     

Conservation of mass and momentum for the flow of interdendritic liquid during solidification

In this paper, mass and momentum conservation equations are derived for the flow of interdendritic liquid during solidification using the volume-averaging approach. In this approach, the mushy zone is conceived to be two interpenetrating phases; each phase is described with the usual field quantities, which are continuous in that phase but discontinuous over the entire space. On the microscopic scale, the usual conservation equations along with the appropriate interfacial boundary conditions describe the state of the system. However, the solution to these equations in the microscopic scale is not practical because of the complex interfacial geometry in the mushy zone. Instead, the scale at which the system is described is altered by averaging the microscopic equations over some representative elementary volume within the mushy zone, resulting in macroscopic equations that can be used to solve practical problems. For a fraction of liquid equal to unity, the equations reduce to the usual conservation equations for a single-phase liquid. It is also found that the resistance offered by the solid to the flow of interdendritic liquid in the mushy zone is best described by two coefficients, namely, the inverse of permeability and a second-order resistance coefficient. For the flow in columnar dendritic structures, the second-order coefficient along with the permeability should be evaluated experimentally. For the flow in equiaxial dendritic structures(i.e., isotropic media), the inverse of permeability alone is sufficient to quantify the resistance offered by the solid.     

Fluid flow through a solid-liquid dendritic interface

Fluid flow through a dendritic solid-liquid interfacial zone, during solidification, has been observed in a series of Sn-Pb alloys. It was found that liquid penetrates only a short distance into the zone, relative to the total thickness of the zone. The amount of solid present at the point of maximum penetration varies from 12 to 22 pct, and depends on the alloy concentration. Flow due to volume shrinkage, thermal convection, or solute convection well inside the zone, was not detected. Fluid flow through a wire mesh model of a thin section of the solid-liquid dendritic zone was examined. The results are not in agreement with that predicted for flow through a porous barrier, which had been found applicable to interdendritic fluid flow. Formerly Graduate Student, University of British Columbia, Vancouver, British Columbia, Canada     

Dendrite coherency during equiaxed solidification in binary aluminum alloys

Dendrite coherency, or dendrite impingement, is important to the formation of the solidification structure and castability of alloys. Dendrite coherency in the systems Al-xMn, Al-xCu, Al-xFe, and Al-xSi(x = 0 to 5 wt pct) has been studied by continuous torque measurement in solidifying samples. The fraction solid at the dendrite coherency point, fs*, varies with the alloy system and the solute concentration in the alloy, from 18 to 56 pct for the present alloys investigated. An increase in solute concentration decreases the coherency fraction solid,fs*. An alloy system with a large slope of the liquidus line has a high coherency fraction solid. A theoretical approach has been developed to account for the effects of the alloy system and solute concentration on the dendrite coherency in the alloy. The grain sizes of the alloys were evaluated using the parameters at coherency point.