Equiaxed dendritic solidification with convection: Part III. Comparisons with NH4Cl-H2O experiments

This third article on equiaxed dendritic solidification is intended to provide experimental validation of the multiphase model developed in part I. Numerical and experimental results are presented for the solidification of a NH₄C1-70 wt pct H₂O solution inside a square cavity cooled equally from all sidewalls. The numerical simulations were performed using the numerical procedures developed in part II. The experiments were conducted to measure the temperature historiesvia thermocouples and to record the images of the solidification process using a shadowgraph system. Preliminary validity of the multiphase model is demonstrated by the qualitative agreement between the measurements and predictions of cooling curves as well as of the evolution of the crystal sediment bed. In addition, several important features of equiaxed dendritic solidification are identified through this combined experimental and numerical study, including the grain generation and growth behaviors in the presence of liquid flow, the sedimentation of equiaxed crystals, the formation of a crystal sediment bed, and a bottom zone of negative segregation resulting from the countercurrent solid-liquid multiphase flow. Quantitative comparisons between the numerical simulation and experiment reveal several areas for future research.     

Equiaxed dendritic solidification with convection: Part II. Numerical simulations for an Al-4 Wt pct Cu alloy

The multiphase model developed in part I for equiaxed dendritic solidification with melt convection and solid-phase transport is applied to numerically predict structural and compositional development in an Al-4 wt pct Cu alloy solidifying in a rectangular cavity. A numerical technique combining a fully implicit control-volume-based finite difference method with a multiple time-step scheme is developed for accurate and efficient simulations of both micro- and macroscale phenomena. Quantitative results for the dendritic microstructure evolution in the presence of melt convection and solid movement are obtained. The remarkable effects of the solid-liquid multiphase flow pattern on macrosegregation as well as the grain size distribution are illustrated.     

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.     

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).     

A unified microscale-parameter approach to solidification-transport phenomena-based macrosegregation modeling for dendritic solidification: Part I. Mixture average-based analysis

Based on the scale-length comparisons of the microscale heat or solutal-mass transfers with dendrite structures, a general form for the time-differential mixture-averaged composition (TDMAC) term for a macroscale metallic dendrite solidification model with any finite solid backdiffusion (SBD) was shown to be necessary and valid. Starting from such an integral term-included TDMAC expression with a general dendrite geometric modeling, Fick’s second diffusion law and the species massconservation principle were applied to confirm the equivalency of a unified Φ-parameter-involved all-differential TDMAC form to the original TDMAC term. Through the modeling, the unified microscale parameter Φ was found to be a function of the Fourier diffusion number with dendrite geometrical modifications (ϕ) and of another nondimensional parameter, θ, representing the sensibility of the interdendritic-liquid-concentration variation in response to the SBD inside the growing dendrites. In the solved ϕ-parameter function of ϕ= (D _S(T)/R _f)ζ·A _2N , the scalar vector product defines the geometric modification accounting for a general growing dendrite morphology with five basic units of spherical, cylindrical, platelike, inward cylindrical, and inward spherical shapes. Through an approximate solution to the integral equation with respect to the solutal-mass solid/liquid (S/L) interface flux, (Eq. [19]), the unified Φ parameter was proposed to take a function form of Φ=θ·ϕ/(1+θ · ϕ), where θ=(1+β) · k · f _S/f _L ² . The result discussions and ϕ-f _S, and θ-f _S, and Φ-f _S curve sample calculations on an Al-4.5 pct Cu alloy with different constant alloy properties, solidifying phase morphologies, and solidification parameters were carried out to investigate the behaviors of various influential factors on the extent of SBD in a dendrite solidification process.     

Directional dendritic solidification of a composite slurry: Part I. Dendrite morphology

A simple analytical model to describe the morphology of a growing dendrite in the presence of an inert particle has been presented. The presence of a particle changes the solute concentration gradient at the tip of a growing dendrite and this, in turn, affects the dendrite tip radius. Results of the analysis show that the dendrite tip radius decreases at a high growth velocity due to the presence of a particle, while there is no influence on the tip radius at both low and intermediate growth velocities. Lower thermal conductivity of the particle decreases the tip radius, while a higher thermal conductivity increases the radius. However, the effect of thermal conductivity on the tip radius is only significant in the cellular growth regime. The analysis shows that the presence of SiC particles in Al-Cu alloys reduces the cell to dendrite transition velocity. Results of directional solidification experiments carried out on an Al-4.5Cu-SiC composite system agree with our model.     

Modeling of interdendritic strain and macrosegregation for dendritic solidification processes: Part I. Theory and experiments

In the first part of this two-part article, mathematical models have been developed to characterize temperature, interdendritic stain, and segregation distributions during dendritic solidification. This aims to predict the effect of interdendric strain associated with sudden changes in the cooling conditions on the macrosegregation distributions, i.e., the combined effect of interdendric strain and macrosegregation on the dendritic structure. These theoretical models were verified on a laboratory scale. Four laboratory ingots of 0.53 and 0.9 wt pct C steels were cast horizontally and unidirectionally in a static mold under cooling conditions designed to approximate those in the continuous-casting process. Thermocouples recorded temperatures in the ingot at different locations from copper chill. The ingots were examined for macro-microstructure, and the extent of carbon macrosegregation was determined by wet chemical analysis. The experimental results indicate that static mold with sudden changes in the cooling conditions on the copper chill provides an approximately similar structure and macrosegregation profiles to those in a continuous-casting process. It is concluded that these cooling conditions have a significant effect on the fluctuated macrosegregation phenomenon. The sudden drop in the heat flux on the chill causes a positive segregation, whereas a sudden increase in heat flux results in a negative segregation. Also, the metallographic examination shows that there is high inelastic deformation of the dendrites due to the sudden drop in heat flux on the chill.     

Macrotransport-solidification kinetics modeling of equiaxed dendritic growth: Part I. Model development and discussion

An analytical model that describes solidification of equiaxed dendrites has been developed for use in solidification kinetics-macrotransport modeling. It relaxes some of the assumptions made in previous models, such as the Dustin-Kurz, Rappaz-Thevoz, and Kanetkar-Stefanescu models. It is assumed that nuclei grow as unperturbed spheres until the radius of the sphere becomes larger than the minimum radius of instability. Then, growth of the dendrites is related to morphological instability and is calculated as a function of melt undercooling around the dendrite tips, which is controlled by the bulk temperature and the intrinsic volume average concentration of the liquid phase. When the general morphology of equiaxed dendrites is considered, the evolution of the fraction of solid is related to the interdendritic branching and dynamic coarsening (through the evolution of the specific interfacial areas) and to the topology and movement of the dendrite envelope (through the tip growth velocity and dendrite shape factor). The particular case of this model is the model for globulitic dendrite. The intrinsic volume average liquid concentration and bulk temperature are obtained from an overall solute and thermal balance around a growing equiaxed dendritic grain within a spherical closed system. Overall solute balance in the integral form is obtained by a complete analytical solution of the diffusion field in both liquid and solid phases. The bulk temperature is obtained from the solution of the macrotrasport-solidification kinetics problem.     

Fracture in Equiaxed Two Phase Alloys: Part I. Fracture in Alloys with Isolated Elastic Particles

Fracture in equiaxed two phase alloys containing isolated elastic particles has been analyzed from the viewpoint of a recently proposed model for fracture initiation and propagation in such materials. This model predicts fracture toughness parameters in terms of the microstructural geometry, relative phase volume fractions, and tensile properties of the materials. Predictions of the model are tested experimentally for two phase Co-CoAl alloys over a wide range of compositions, and the results indicate good agreement between predicted and observed fracture toughnesses. M. A. PRZYSTUPA, formerly Graduate Research Assistant, Department of Metallurgical Engineering, Michigan Technological University.     

A unified microscale parameter approach to solidification-transport process-based macrosegregation modeling for dendritic solidification: Part II. Numerical example computations

This article, as the second part of the present work, applied the unified Φ-parameter-based micro/macrosegregation modeling, which was proposed and described in the first part, to two groups of alloy solidification systems. The first group is on closed simultaneous solidification systems of an Al-4.5 pct Cu alloy and a Fe-0.5 pct C based plain carbon steel to investigate the microscale solute-redistribution behaviors with different influential factors including the solid phase morphologies. A new solute-redistribution equation was derived from the present microscale unified Φ-parameter-based modeling, which includes more microscale dendrite-solidification features while preserving the simple function form equivalent to the well-known lever rule and the Scheil, Flemings-Brody, Clyne-Kurz, and Ohnaka models. In the second example computation group on a directionally solidified, bladelike casting system, a previous continuum model and the related PC-based codes developed by the present authors were extended and modified by incorporating the present microscale Φ-parameter approach to any extent of solid back-diffusion effects. Numerical simulations on the directional-solidification transport phenomena and micro/macrosegregation formations in a bladelike Al-4.5 pct Cu casting show the feasibility and the efficiency of the present solidification-transport process (STP)-based two-scale segregation model and the numerical methods.     

Directional dendritic solidification of a composite slurry: Part II. Particle distribution

The present work attempts to form an understanding of particle distribution during dendritic solidification of a composite slurry. It is shown that the magnitude and nature of the forces involved between a foreign particle and a growing dendrite are distinctly different from those during plane front solidification. A particle distribution map, based on theoretically evaluated forces, is proposed to predict the conditions under which engulfment, entrapment, or pushing of particles occurs during dendritic solidification. Directional solidification experiments have been conducted to study particle distribution, and comparison of these results with theoretical predictions based on the map agree well with a large number of experimental observations reported to date.     

Cellular and dendritic growth: Part I. Experiment

An Al-4.1 mass pct Cu alloy was unidirectionally solidified under various growth rates. Features, such as tip radius of curvature, primary arm spacing, and tip concentration were measured as functions of growth rate. Dependence of tip radius on growth rate was different between cells and dendrites. Measured tip radius and primary arm spacing were maximum at the cellular-dendritic transition. Tip concentration, however, monotonously decreases with growth rate. Linear relationships between tip radius and characteristic dimensions of dendrites like core diameter, half length of tip arc, and the first secondary arm spacing are obtained to determine what affects growth rate, convection, and gravity segregation. Experimental results are compared with current theoretical models for dendrite growth under controlled solidification. It was determined that the measured tip radius is larger than that of theoretical predictions at fast growth rate, but the measured tip concentration is in good agreement with theoretical predictions.     

Dendritic solidification of Cu-Ni alloys: Part I. Initial growth of dendrite structure

Small Cu-Ni melts were furnace cooled at different rates and with different "nucleation" temperatures. Thermal analysis showed that the thermal arrests associated with the formation of the dendrite skeletons showed a significant supercooling below the liquidus temperatures. This supercooling increased with increased rates of heat extraction,i.e. higher cooling rates and lower "nucleation" temperatures; and was associated with higher dendrite growth velocities. The solute content of the dendrites measured on samples quenched during the thermal arrest quantitatively supported these observations. The magnitude of the supercooling for a given rate of heat extraction varied directly with the freezing range of the alloy but remained finite (although small) for unalloyed copper. The approximate measurements of dendrite growth velocities for one alloy (40 wt pet Ni) at different supercoolings agreed well with the predictions of Trivedi’s theory of dendrite growth. E. A. FEEST, formerly Graduate Student, University of Sussex K. HOLM, formerly Visiting Research Student, University of Susses     

Dimensionless correlations for forced convection in liquid metals: Part I. single-phase flow

Two main objectives were addressed in this article. First, a dimensionless heat-transfer correlation for single-phase flow forced convection in liquid aluminum has been derived using a novel experimental method. An aluminum sphere was rotated with a specific tangential velocity in liquid aluminum. Its melting time was measured and correlated with the convective heat-transfer characteristics. The resulting correlation has the following form:

Investigation on dendritic solidification and microsegregation in presence of melt convection using phase field simulation

Abstract

A phase field model has been used to simulate dendritic solidification of a binary alloy in the presence of forced melt convection. The influence of melt flow on morphology and solute distribution was investigated for various conditions. The results showed that incorporation of fluid flow causes asymmetric dendritic growth which is amplified by increasing fluid velocity. Moreover, it has been found that the effects of melt flow on the growth of different arms depend on the preferred growth orientation of the dendrite with respect to flow direction. Solid microsegregation study of the growing dendrite arm perpendicular to the flow direction indicated that the position of the arm axis varied almost linearly with flow velocity. Introducing an adjusting term called an antitrapping current in the concentration equation prevents solute from being highly trapped in the solid phase and causes the phase field simulations to be more realistic, especially for high undercoolings.

On a utilisé un modèle de champ de phase pour simuler la solidification dendritique d’un alliage binaire en présence de convection forcée du bain. On a examiné l’influence de l’écoulement du bain sur la morphologie et la distribution de soluté sous des conditions variées. Les résultats ont montré que l’incorporation de l’écoulement du fluide produisait une croissance dendritique asymétrique qui est amplifiée par l’augmentation de la vitesse du fluide. De plus, on a trouvé que l’effet de l’écoulement du bain sur la croissance de différentes branches dépendait de l’orientation préférée de la croissance de la dendrite par rapport à la direction de l’écoulement. L’étude de la microségrégation solide de la branche de dendrite en croissance perpendiculaire à la direction de l’écoulement indiquait que la position de l’axe de la branche variait presque linéairement avec la vitesse de l’écoulement. L’introduction d’un terme d’ajustement, appelé courant de contre-piégeage, dans l’équation de concentration empêche le soluté d’être trop piégé dans la phase solide et rend les simulations de champ de phase plus réalistes, particulièrement dans les cas de surfusions élevées.     

The transition from gray to white cast iron during solidification: Part I. Theoretical background

The present work is based on a heat balance during the solidification of cast iron. Accordingly, an analytical expression was derived to relate the chilling tendency (CT) of cast iron to nucleation and growth processes associated with the eutectic graphite and cementite constituents. A relationship is found between the CT and factors such as the physicochemical state of liquid, the distribution of nucleation sites, and the density of nucleation sites for eutectic cells. In particular, it is found that the CT can be related to the critical casting modulus (M _cr ), enabling determinations of minimum wall thickness for chilled castings or chill widths in wedge-shaped castings. Finally, the present work provides a rational for the effect of technological factors such as the chemistry of the melt, inoculation practice, holding temperature, and time on the resultant CT and chill of the cast iron.     

Numerical models for casting solidification: Part I. The coupling of the boundary element and finite difference methods for solidification problems

An investigation on the coupling technique for boundary element and finite difference methods was carried out to simulate solidification processes. In the coupling model, solidification problems in casting and unsteady-state heat conduction problems in mold regions were analyzed by explicit finite difference and boundary element methods, respectively. A comparison was made between the coupling model and the finite difference method on the solidification of castings in metal and sand molds. The proper range of time steps for boundary elements in transient problems was presented for a simple geometry. And two types of time marching schemes were proposed for application of the boundary element method to solidification processes. On leave from Kyung-Pook National University, Taegu, Korea.     

Analysis of thin-slab casting by the compact-strip process: Part I. Heat extraction and solidification

This article reports on an extensive experimental and modeling study undertaken to elucidate the thermal evolution of thin slabs during their passage through the mold and secondary cooling system of a compact-strip process (CSP) caster. In industrial trials covering a wide range of casting conditions, temperature measurements were carried out at (1) the copper plates of an operating mold and (2) the stainless steel frame of an operating grid. Separately, water-flux and heat-flux distributions generated by the several water and air-mist sprays produced by the different nozzles used in the process were determined in the laboratory. The analysis of these pieces of information, together with a detailed consideration of the geometry of the mold and the arrangement of the rolls and spray nozzles, were used to establish appropriate boundary conditions for a two-dimensional, curvilinear-coordinate, unsteady-state heat-conduction model for predicting the solidification rate of thin slabs. The predicted slab surface temperatures show very good agreement with corresponding measured values taken in plant tests at several locations along and across the secondary cooling system. The validation trials involved a wide range of low- and medium-carbon steel grades, casting speeds, slab widths, and secondary cooling strategies. The second part of this article combines the solidification model with a creep model of the shell to yield useful information about design parameters and casting conditions associated with undesirable bulging behavior of the slab after the last support roll, which causes stoppage of the process by slab clogging at the pinch rolls.     

Modeling of gas-liquid reactions in ladle metallurgy: Part I. Physical modeling

A physical model of a ladle degassing operation was developed to simulate the reactions at rising bubbles and at the free surface. Carbon dioxide desorption from a sodium hydroxide solution was used to simulate the liquid-phase diffusion-controlled decarburization of liquid steel. It was found that under reduced pressure, the reactions were faster than attributable to solely the increase in volumetric flow rate. It was possible to separate the reactions with the bubbles from the free surface reactions; 20 to 40 pct of the reactions occurred at the free surface, depending on injection conditions. The free surface desorption rate depended on the gas flow rate and the number of injectors. The mass transfer coefficients to the bubbles were in reasonable agreement with previous work. Plume bending was observed when small bubbles were influenced by the bulk liquid flow patterns.