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NASA NTRS1993

The effect of gravitational modulation on convection in vertical Bridgman growth

During vertical directional solidification of a binary alloy at constant velocity, buoyancy-driven solutal convection may occur due to the solute gradient associated with the solidification process. This problem is further complicated if time-periodic forcing is considered, which is relevant to materials processing in a microgravity environment or as a means of dynamic control of flow instabilities. The effect of time-periodic modulation is studied by introducing a gravitational acceleration which is a sinusoidal function of time. The onset of solutal convection is treated by a stability analysis of the linearized governing equations and boundary conditions. Solutions are obtained numerically employing two distinct computational implementations of Floquet theory. Results are presented for materials with large Schmidt number, and an analysis for large frequency and large Schmidt number yields a more complete description of the behavior in this relevant limit.

Murray, B. T.↗

NASA NTRS1992

Convective and morphological instabilities during crystal growth: Effect of gravity modulation

During directional solidification of a binary alloy at constant velocity in the vertical direction, morphological and convective instabilities may occur due to the temperature and solute gradients associated with the solidification process. The effect of time-periodic modulation (vibration) is studied by considering a vertical gravitational acceleration which is sinusoidal in time. The conditions for the onset of solutal convection are calculated numerically, employing two distinct computational procedures based on Floquet theory. In general, a stable state can be destabilized by modulation and an unstable state can be stabilized. In the limit of high frequency modulation, the method of averaging and multiple-scale asymptotic analysis can be used to simplify the calculations.

Coreill, S. R.↗

NASA NTRS1983

Analytical and Numerical Solution for a Solidifying Liquid Alloy Slab

Numerical and analytical solutions are presented for the temperature and concentration distributions during the solidification of a binary liquid alloy slab. The slab is taken to be of a finite depth but infinite in the horizontal direction. The solidification process is started by withdrawing a fixed amount of heat from the lower surface of the slab. The upper surface of the slab is subjected to both radiation and convective conditions. The solution gives the concentration and temperature profiles and the interface position as a function of time. Due to the smallness of the mass diffusion coefficient in the solid, the numerical solution method breaks down whenever the ratio of the diffusivities in the solid and the liquid falls below a certain value. An analytical method is developed which gives accurate solution for any value of the diffusivity ratio.

Antar, B. N.↗

NASA NTRS1980

Analysis and calculation of macrosegregation in a casting ingot. MPS solidification model. Volume 1: Formulation and analysis

The physical and numerical formulation of a model for the horizontal solidification of a binary alloy is described. It can be applied in an ingot. The major purpose of the model is to calculate macrosegregation in a casting ingot which results from flow of interdendritic liquid during solidification. The flow, driven by solidification contractions and by gravity acting on density gradients in the interdendritic liquid, was modeled as flow through a porous medium. The symbols used are defined. The physical formulation of the problem leading to a set of equations which can be used to obtain: (1) the pressure field; (2) the velocity field: (3) mass flow and (4) solute flow in the solid plus liquid zone during solidification is presented. With these established, the model calculates macrosegregation after solidification is complete. The numerical techniques used to obtain solution on a computational grid are presented. Results, evaluation of the results, and recommendations for future development of the model are given. The macrosegregation and flow field predictions for tin-lead, aluminum-copper, and tin-bismuth alloys are included as well as comparisons of some of the predictions with published predictions or with empirical data.

Maples, A. L.↗

NASA NTRS1988

Heat of mixing and morphological stability

A mathematical model, which incorporates heat of mixing in the energy balance, has been developed to analyze the morphological stability of a planar solid-liquid interface during the directional solidification of a binary alloy. It is observed that the stability behavior is almost that predicted by the analysis of Mullins and Sekerka (1963) at low growth velocities, while deviations in the critical concentration of about 20-25 percent are observed under rapid solidification conditions for certain systems. The calculations indicate that a positive heat of mixing makes the planar interface more unstable, whereas a negative heat of mixing makes it more stable, in terms of the critical concentration.

Nandapurkar, P.↗

NASA NTRS1987

Directional solidification in an imperfect furnace

The upward directional solidification of a binary alloy in which the rejected solute is more dense than the solvent is considered. The system is stably stratified both thermally and solutally. If heat is lost through the sidewalls, horizontal thermal gradients drive a steady double-diffusive convection. Solidification cells with small aspect ratios (width/length) A are considered, and asymptotic solutions to the steady double-diffusive system are obtained as A tends to 0 under the assumption heat losses are also small. When the thermal Grashof number is large, buoyancy boundary layers are at the sidewalls. The shape of the crystal-melt interface is determined by the coupled effects of local thermodynamic equilibrium, diffusion and convection.

Brattkus, K.↗

NASA NTRS1985

Dendritic solidification. III - Some further refinements to the model for dendritic growth under an imposed thermal gradient

Some further refinements to a simple model for dendritic solidification in a binary alloy melt under an imposed positive thermal gradient are presented. Two new expressions for the dendrite tip undercooling have been obtained and shown to yield a limiting value of Delta T sub 0 and very small growth rates. Here Delta T sub 0 is the equilibrium solidification range of the alloy. At very large growth rates, all three tip undercooling expressions reach the same limiting value depending on the value of a dimensionless parameter lambda which is related to the effective diffusion distance ahead of the dendrite tip. The predicted tip undercoolings are, however, somewhat lower at intermediate growth rates. An improved calculation for the solute buildup at the dendrite tip due to curvature effects is also included.

Laxmanan, V.↗

NASA NTRS1989

Thermosolutal convection during dendritic solidification

This paper presents a mathematical model for directional solidification of a binary alloy including a dendritic region underlying an all-liquid region. It is assumed initially that there exists a nonconvecting state with planar isotherms and isoconcentrates solidifying at a constant velocity. The stability of this system has been analyzed and nonlinear calculations are performed that show the effect of convection in the solidification process when the system is unstable. Results of calculations for various cases defined by the initial temperature gradient at the dendrite tips and varying strength of the gravitational field are presented for systems involving lead-tin alloys. The results show that the systems are stable for a gravitational constant of 0.0001 g(0) and that convection can be suppressed by appropriate choice of the container's size for higher values of the gravitational constant. It is also concluded that for the lead-tin systems considered, convection in the mushy zone is not significant below the upper 20 percent of the dendritic zone, if al all.

Heinrich, J. C.↗

NASA NTRS1992

Boundary-conforming mapping applied to computations of highly deformed solidification interfaces

A new boundary-conforming mapping is developed for the calculation of highly deformed cellular solidification interfaces in a model of directional solidification of a binary alloy. The mapping is derived through a variational formulation that is designed so that the grid penetrates the grooves between cells along the interface without causing a loss of ellipticity of the mapping equations. A finite element/Newton method is presented for simultaneous solution of the free boundary problem described by the solutal model of directional solidification and the mapping equations. Results are compared to previous calculations and demonstrate the importance of accurate representation of the interface shape for understanding the solution structure.

Tsiveriotis, K.↗

NASA NTRS1994

Numerical formulation of composition segregation at curved solid-liquid interface during steady state solidification process

The lateral solute segregation that results from a curved solid-liquid interface shape during steady state unidirectional solidification of a binary alloy system has been studied both analytically and numerically by Coriell, Bosivert, Rehm, and Sekerka. The system under their study is a two dimensional rectangular system. However, most real growth systems are cylindrical systems. Thus, in a previous study, we have followed Coriell etc. formalism and obtained analytical results for lateral solute segregation for an azimuthal symmetric cylindrical binary melt system during steady state solidification process. The solid-liquid interface shape is expressed as a series combination of Bessel functions. In this study a computer program has been developed to simulate the lateral solute segregation.

Wang, Jai-Ching↗

NASA NTRS1988

Microsegregation in directionally solidified Pb-8.4 at. pct Au alloy

The dependence of microsegregation behavior on growth rate and thermal gradient has been examined in a Pb-8.4 at. pct Au alloy material partially directionally solidified and quenched. The composition of the quenched 'liquid' at the dendrite tip (Ct), that of the eutectic-like solid phase freezing from the interdendritic liquid at the base of dendrite (Cse), the volume fraction of this eutectic-like region (fe), and solute profiles in the interdendritic quenched liquid and ahead of the dendrite have been measured. Two dendritic growth models for solidification of a binary alloy melt in a positive thermal gradient at the liquid-solid interface, one for dendrites with 'minimum undercooled dendrite tip' and the other for an Ivantsov type of dendrite with 'marginally stable tip', have been examined for a quantitative comparison with measured values of Ct, Cse, and fe. Convection in the melt, possibly due to horizontal density gradients, is found to be a serious limitation for theoretical understanding of the observed experimental behavior and meaningful comparison of theories.

Tewari, S. N.↗

NASA NTRS1984

Thermosolutal convection during directional solidification

During solidification of a binary alloy at constant velocity vertically upward, thermosolutal convection can occur if the solute rejected at the crystal-melt interface decreases the density of the melt. We assume that the crystal-melt interface remains planar and that the flow field is periodic in the horizontal direction. The time-dependent nonlinear differential equations for fluid flow, concentration, and temperature are solved numerically in two spatial dimensions for small Prandtl numbers and moderately large Schmidt numbers. For slow solidification velocities, the thermal field has an important stabilizing influence: near the onset of instability the flow is confined to the vicinity of the crystal-melt interface. Further, for slow velocities, as the concentration increases, the horizontal wavelength of the flow decreases rapidly - a phenomenon also indicated by linear stability analysis. The lateral inhomogeneity in solute concentration due to convection is obtained from the calculations. For a narrow range of solutal Rayleigh numbers and wavelengths, the flow is periodic in time.

Mcfadden, G. B.↗

NASA NTRS1997

Thermal Problems in Material processing of High-Temperature Materials

In this final report axial segregation in unsteady diffusion dominated directional solidification of a binary alloy with a planar solid-liquid interface profile is considered. The analysis is an extension of the work of Smith et al. to account for an arbitrary initial axial concentration profile. This situation is important in the experimental studies of growth of crystals in a microgravity environment in which the growth is from melts formed by re-melting crystals that have a known axial impurity profile. Three situations are discussed. There are the initial transient, a re-solidified crystal, and multipass solidification.

Korpela, Seppo A.↗

NASA NTRS1979

Influence of the space environment on some materials processing phenomena

The influence of the space environment on materials processing phenomena is studied by applying the principles of physical similarity to a system E on Earth and a system S in the microgravity environment of space. If these systems can be characterized by a set of dimensionless groups, then they are similar if corresponding members of the set are equal for E and S. Similarity is often impractical or impossible if Q is larger than a few samples. For example, there is a simple law of similarity for an isothermal liquid zone floating between inert solids; however, similarity is lost if the zone becomes nonisothermal. A molten zone during crystal growth is so complex that a complete set cannot be identified with certainty. A second example is double-diffusive convection during unidirectional solidification of a binary alloy. Results show that buoyancy driven fluid dynamical instabilities couple with constitutionally related instabilities quite differently in E and S.

Sekerka, R. F.↗

NASA NTRS1984

Macrosegregation in aluminum alloy ingot cast by the semicontinuous direct chill method

A theoretical model of the semicontinuous DC casting method is developed to predict the positive segregation observed at the subsurface and the negative segregation commonly found at the center of large commercial-size aluminum alloy ingot. Qualitative analysis of commercial-size aluminum alloy semicontinuous cast direct chill (DC) ingot is carried out. In the analysis, both positive segregation in the ingot subsurface and negative segregation at the center of the ingot are examined. Ingot subsurface macrosegregation is investigated by considering steady state casting of a circular cross-section binary alloy ingot. Nonequilibrium solidification is assumed with no solid diffusion, constant equilibrium partition ratio, and constant solid density.

Yu, H.↗

NASA NTRS1985

Double-diffusive convection with sidewalls

Stommel et al. (1956) have first described an instability, known as thermosolutal convection, thermohaline convection, or double-diffusive convection. This instability may occur in the case of a fluid in a gravitational field with two diffusing components present. The present study is concerned with the effect of sidewalls on flow in the fingering regime in the absence of applied horizontal gradients. The work was motivated by numerical results obtained on the basis of a simulation of thermosolutal convection occurring during the unidirectional solidification of a binary alloy. In this case, the unperturbed solute field in the liquid ahead of the solidifying planar interface has an exponential vertical profile because of the rejection or preferential incorporation of solute by the solid phase.

Mcfadden, G. B.↗

NASA NTRS1985

Convection During Unidirectional Solidification

The fluid flow and solute segregation which occur during directional solidification, including effects of gravity and microgravity were studied. The nature of the fluid flow, its effects on the shape of the crystal-metal interface and the resulting distribution of solutes are examined. During solidification of a binary alloy at constant velocity vertically upwards, thermosolutal convection occurs if the solute rejection or preferential incorporation at the crystal-melt interface decreases the density of the melt. Numerical calculations of the solute, temperature, and flow fields are carried out by using previously developed algorithms for Schmidt numbers Sc = 1, 10, 81. The time periodic flow, which occurs over a narrow range of parameters, is investigated.

Coriell, S. R.↗

NASA NTRS1987

Thermosolutal convection during directional solidification. II - Flow transitions

The influence of thermosolutal convection on solute segregation in crystals grown by vertical directional solidification of binary metallic alloys or semiconductors is studied. Finite differences are used in a two-dimensional time-dependent model which assumes a planar crystal-melt interface to obtain numerical results. It is assumed that the configuration is periodic in the horizontal direction. Consideration is given to the possibility of multiple flow states sharing the same period. The results are represented in bifurcation diagrams of the nonlinear states associated with the critical points of linear theory. Variations of the solutal Rayleigh number can lead to the occurrence of multiple steady states, time-periodic states, and quasi-periodic states. This case is compared to that of thermosolutal convection with linear vertical gradients and stress-free boundaries.

Mcfadden, G. B.↗