GROWTH CONDITIONS FOR STABILITY OF A CELLULAR SOLID-LIQUID INTERFACE

1957 ◽  
Vol 35 (10) ◽  
pp. 1223-1227 ◽  
Author(s):  
E. L. Holmes ◽  
J. W. Rutter ◽  
W. C. Winegard
Keyword(s):  
Freezing Temperature ◽  
Growth Conditions ◽  
Solute Concentration ◽  
Critical Value ◽  
Liquid Interface ◽  
Cellular Solid ◽  
First Approximation ◽  
Comparison Of The Results ◽  
Solid Liquid Interface ◽  
Solid Liquid

Samples of zone-refined lead containing various amounts of silver as solute were solidified under well-controlled conditions to study the transition from cellular to dendritic freezing as a function of composition, speed of freezing, temperature gradient in the melt during freezing, and crystallographic orientation of the solidifying crystal. A comparison of the results of this investigation with those of Tiller and Rutter (1956) on alloys of tin in lead shows that to a first approximation the onset of dendritic freezing under any given growth conditions occurs at a critical value of the average solute concentration in the liquid at the solid–liquid interface, independent of whether the solute present is tin or silver.

Transient Freezing of Liquids in Turbulent Flow inside Tubes

1979 ◽  
Vol 101 (3) ◽  
pp. 465-468 ◽  
Author(s):  
Chul Cho ◽  
M. N. O¨zis¸ik
Keyword(s):  
Heat Flux ◽  
Prandtl Number ◽  
Turbulent Flow ◽  
Circular Tube ◽  
Freezing Temperature ◽  
Liquid Interface ◽  
Wall Heat Flux ◽  
Uniform Temperature ◽  
Solid Liquid Interface ◽  
Solid Liquid

The problem of freezing of a liquid in turbulent flow inside a circular tube whose wall is kept at a uniform temperature lower than the freezing temperature of the liquid is solved. The radius of the solid-liquid interface and the local wall heat flux are determined as a function of time and position along the tube for several different values of the Prandtl number and the freezing parameter.

Transient Freezing of Liquids in Forced Flow Inside Circular Tubes

1969 ◽  
Vol 91 (3) ◽  
pp. 385-389 ◽  
Author(s):  
M. N. O¨zis¸ik ◽  
J. C. Mulligan
Keyword(s):  
Solid Phase ◽  
Freezing Temperature ◽  
Local Heat ◽  
Liquid Interface ◽  
Forced Flow ◽  
Transfer Rates ◽  
Tube Wall Temperature ◽  
Temperature Constant ◽  
Solid Liquid Interface ◽  
Solid Liquid

The transient freezing of a liquid flowing inside a circular tube is investigated analytically under the assumption of a constant tube wall temperature which is lower than the freezing temperature, constant properties, a slug-flow velocity profile and quasisteady state heat conduction in the solid phase. The variation of the local heat flux and the profile of the solid-liquid interface during freezing has been determined as a function of time and position along the tube. The analysis produced steadystate heat transfer rates and profiles for the solid-liquid interface which agreed well with experiments.

A New Growth Kinetics in Simulation of Dendrite Growth by Cellular Automaton Method

2007 ◽  
Vol 26-28 ◽  
pp. 957-962 ◽  
Author(s):  
Bo Wei Shan ◽  
Xin Lin ◽  
Lei Wei ◽  
Wei Dong Huang
Keyword(s):  
Cellular Automaton ◽  
Growth Kinetics ◽  
Solute Concentration ◽  
Dendrite Growth ◽  
Liquid Interface ◽  
Interface Temperature ◽  
Algebraic Equations ◽  
Interface Growth ◽  
Solid Liquid Interface ◽  
Solid Liquid

A modified cellular automaton model was proposed to simulate the dendrite growth of alloy. Different from previous models, this model used neither an analytical equation(such as KGT model) nor an interface solute gradient equation to solve the velocity of solid-liquid interface, but used the interface solute and energy conservation and thermodynamic equilibrium condition to describe the solid/liquid interface growth kinetics process. In present model, once the temperature field and solute field were solved by finite different method in the entire domain, the material thermodynamic properties can be substituted into four algebraic equations to easily determine the variation of solid fraction, interface temperature and solute concentration, instead of calculating interface moving velocity. As a result, the complexity of the calculation can be largely reduced. The simulated dendrite growth was in a good agreement with the Lipton–Glicksman–Kurz (LGK) model for free dendritic growth in undercooled melts.

The Solution of a Heat Transfer Problem With Change of State Inside a Spherical Shell

2012 ◽  
Author(s):  
Danillo Silva de Oliveira ◽  
Fernando Brenha Ribeiro
Keyword(s):  
Heat Transfer ◽  
Heat Conduction ◽  
Spherical Shell ◽  
Outer Surface ◽  
Transfer Problem ◽  
Freezing Temperature ◽  
Liquid Interface ◽  
Inner Surface ◽  
Solid Liquid Interface ◽  
Solid Liquid

The heat transfer problem is solved for the case of cooling, below the freezing temperature, an initially liquid material inside a spherical shell. The shell is limited by a fixed inner surface and by an outer surface, free to radially expand or contract. As boundary conditions it is imposed that the inner surface is kept constant below the freezing temperature of the liquid and the outer surface is maintained constant above it. The solution represents the formation of a solidified layer that expands outward, separated from the liquid by a spherical surface kept constant at the freezing temperature. The problem is solved in the form of two closed form solutions, written in non-dimensional variables: one for the heat conduction equation in the solid layer and the second for the heat conduction – advection equation in the liquid layer. The solutions formally depend of and are linked by the time dependent radius of the spherical solid–liquid interface and its time derivate, which are, at first, unknown. A differential equation describing the solid–liquid interface radius as function of time is obtained imposing the conservation of energy through the interface during the heat exchange process. This equation is non-linear and has to be numerically solved. Substitution of the interface radius and its time derivate for a particular instant in the heat transfer equations solutions furnishes the temperature distribution inside the spherical shell at that moment. The solution is illustrated with numerical examples.

THE EFFECT OF GROWTH CONDITIONS UPON THE SOLIDIFICATION OF A BINARY ALLOY

1956 ◽  
Vol 34 (1) ◽  
pp. 96-121 ◽  
Author(s):  
W. A. Tiller ◽  
J. W. Rutter
Keyword(s):  
Binary Alloy ◽  
Impurity Content ◽  
Growth Conditions ◽  
Orientation Dependence ◽  
Liquid Interface ◽  
Critical Ratio ◽  
Liquid Lead ◽  
Careful Study ◽  
Solid Liquid Interface ◽  
Solid Liquid

This investigation provides both a theoretical and an experimental analysis of the factors which affect the mode of solidification of a binary alloy. These factors are: (i) the concentration of solute in the melt (C0); (ii) the rate of solidification (R); (iii) the temperature gradient in the melt at the solid–liquid interface (G). Extremely high purity lead was produced by zone refining and, from this material, crystals containing known concentrations of tin, silver, and gold were grown under a range of well-controlled growth conditions. The mode of solidification was investigated by a careful study of the change in appearance of the solid–liquid interface with a change in growth conditions. For a crystal containing a specific C0 of solute it was observed that (a) the transition from a smooth interface to a cellular interface occurred at a critical ratio of G to R; (b) the width of the cells varied inversely as G and inversely as R; (c) the transition from a cellular interface to a dendritic interface exhibited a large orientation dependence, and for a constant orientation breakdown occurred at a critical ratio of G to [Formula: see text]. The experimental observations confirm both the existence of a solute-rich layer of liquid adjacent to the solid-liquid interface and its quantitative features. From this agreement with theory the diffusion coefficients of tin, silver, and gold in liquid lead at 327 °C. are determined. This work serves to illustrate the effect of extremely small amounts of particular solutes upon the development of substructures during solidification. A technique is proposed for obtaining a measure of the purity of low impurity content alloys.

CELL TO DENDRITE TRANSITION IN TIN BASE ALLOYS

1960 ◽  
Vol 38 (8) ◽  
pp. 1077-1088 ◽  
Author(s):  
T. S. Plaskett ◽  
W. C. Winegard
Keyword(s):  
Temperature Gradient ◽  
Distribution Coefficient ◽  
Cell Size ◽  
Dendritic Growth ◽  
Solute Concentration ◽  
Cellular Growth ◽  
First Approximation ◽  
High Concentrations ◽  
Solid Liquid Interface ◽  
Solid Liquid

The transition from cellular growth to dendritic growth was investigated for dilute binary alloys of zone-refined tin with lead, bismuth, and antimony in terms of the rate of growth R, the temperature gradient in the liquid ahead of the solid-liquid interface G, and the solute concentration C0. It was found that a previous relationship describing the transition for lead base alloys applied, at least to a first approximation, for the low solute concentration tin results. At the high concentrations of solute, however, it was necessary to introduce another variable, namely the cell size at transition dt. The transition relationships only applied for alloy systems with a distribution coefficient k0, less than unity. For systems with k0 > 1 (antimony in tin) a k0 < 1 was used which was equivalent to the k0 > 1. A method is described for determining this "equivalent k0".

scholarly journals Concentration Micro-Field for Lamellar Eutectic Growth

2008 ◽  
Vol 272 ◽  
pp. 123-138 ◽  
Author(s):  
Waldemar Wołczyński
Keyword(s):  
Boundary Conditions ◽  
Mass Balance ◽  
Structure Formation ◽  
Concentration Field ◽  
Eutectic Growth ◽  
Solute Concentration ◽  
Liquid Interface ◽  
Eutectic System ◽  
Solid Liquid Interface ◽  
Solid Liquid

A possibility of a modification of the Jackson-Hunt theory of an oriented structure formation is analysed. A new model for the formation of a concentration field ahead of growing regular lamellae with respect to the solid / liquid interface shape is presented. A coordinate system applied in the model is attached to the solid / liquid interface to be advancing in the z - direction, identically with interface moving at a constant velocity, v . The solution to a diffusion equation is given for the improved formulation of the boundary conditions. The boundary conditions are related to the interplay between the diffusion required for phase separation and the formation of the interphase between both lamellae. The boundary conditions are formulated to establish the stability of lamellar structure formation under steady-state conditions. It is assumed that stable growth of the lamellae is ensured by the separation of concentration fields within a boundary layer ahead of the solid / liquid interfaces of both the α and β " phases. Coupled lamellar growth with the presence of a leading phase protrusion is defined. The general mass balance is analysed for a solute concentration in the liquid, taking into account a planar solid / liquid interface. A local mass balance is also ensured but it requires envisaging a protrusion of the minor eutectic phase. The existence of a lead distance is confirmed experimentally for the (Pb)-(Cd) eutectic system. The difference in undercooling is also considered as a phenomenon associated with the separation of concentration fields and the existence of a protrusion to relax the assumption of an isothermal interface (ideally coupled growth) given by the Hunt and Jackson theory.

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