Solidification of binary aqueous solution cooled from above

Freezing and melting phenomena are important in many different fields, including crystal growth, casting, metallurgy, geophysics, and oceanography. Solidification of a multi‐component solution is the one often observed in nature. In order to investigate basic features of the freezing processes of binary systems, we conducted a series of laboratory experiments in a rectangular box cooled from above using aqueous NaNO₃ solution. During the freezing, the solid phase always grows into many needle‐like crystals called the mushy layer. We measured the growth of the mushy layer thickness, the solid fraction, the temperature, and the concentration distributions. The average solid fraction is found to increase with time in the mushy layer. This causes a slow descent of the released solute in the mushy layer and its eventual fall into the liquid region below because of gravity. We propose a one‐dimensional model to explain the horizontally‐averaged mushy layer growth. In the model, the estimate of a heat flux at the mushy‐liquid interface due to natural convection is found essential for a correct prediction. The proposed theory predicts well the growth of the mushy‐layer and the average solid fraction, once the convective heat flux is properly given. © 2009 Wiley Periodicals, Inc. Heat Trans Asian Res; Published online in Wiley InterScience ( www.interscience.wiley.com ). DOI 10.1002/htj.20278     

The steady-state solidification scenario of ternary systems: Exact analytical solution of nonlinear model

A mathematical model describing the steady-state solidification of ternary systems with mushy layers (primary and cotectic) is formulated: solidification along a liquidus surface is characterized by a primary mushy layer, and solidification along a cotectic line is characterized by a secondary (cotectic) mushy layer. Exact analytical solutions of the model under consideration are found in a parametric form (thicknesses of mushy layers, growth rate of their boundaries, temperature and composition fields, solid fractions are determined in an explicit form). The velocity of solidification is completely determined by temperature gradients in the solid and liquid phases. This velocity coincides with similar expressions describing binary melt solidification with a planar front or a mushy layer. It is shown that the liquid composition of the main component decreases in the cotectic and primary layers, whereas the second (cotectic) composition increases in the cotectic layer, attains a maximum point and decreases in the primary layer.     

氯化铵水溶液定向凝固实验的传热分析

选用类合金NH4Cl-H2O二元溶液进行垂直定向凝固实验研究,再现过共晶合金结晶过程,测量记录凝固过程中的温度场和固、液相界面位置;重点分析了两相区的传热特性,包括局部热流和释放潜热,并尝试用实验数据与数值计算相结合的方法确定两相区局部固相分数与温度的关系曲线。研究表明:在结晶过程中,各点温度呈线性下降,局部热流在进入两相状态后达到峰值;各相区内温度梯度恒定,但相界面附近温度梯度变化显著。两相区凝固过程中,先期潜热释放总量大,总凝固分数大,两相区厚度迅速增长;随后总凝固分数随相界面迅速上移而急剧下降,经历准稳态过程后再缓慢上升。溶液沿凝固方向分层,NH4Cl质量分数逐渐增大,相应结晶温度逐渐升高。     

Nonlinear dynamics of directional solidification with a mushy layer. Analytic solutions of the problem

We present new analytic results relating to the nonstationary Stefan-type problems for the unidirectional solidification of binary solutions or melts with a mushy layer. Our detailed analysis of the field data is based on the classical model of a mushy layer, which is modified in order to obtain explicit solutions (solid phase thickness and growth rate, temperature distributions, conductive and latent heat fluxes are determined). Predictions for the growth rate and temperature profiles of the mixed-phase and solid regions agree well with existing observations on young sea ice dynamics.     

Natural convection suppression and crystal growth during unidirectional solidification of a binary system

We studied experimentally the interaction between natural convection and dendritic growth in the mushy layer during unidirectional solidification of aqueous ammonium chloride solutions cooled from below. Small amounts of hydroxyethylcellulose were added to the solution to increase its viscosity, leading to the suppression of convection. Natural convection consists of salt fingers in the liquid phase and plumes in the mushy layer for a low‐viscosity solution, but the onset of plumes is suppressed for a high‐viscosity solution. The mushy layer becomes sparsely packed, and the primary and secondary arms of the dendrites grow to noticeable sizes with increasing its viscosity, which yields a low solid fraction, such as 1% on average in the mushy layer for a viscosity ratio of 25.5. This demonstrates that natural convection strongly affects the morphology of dendrites. © 2000 Scripta Technica, Heat Trans Asian Res, 29(2): 120–131, 2000     

In-flight oxidation of composite powder particles during thermal spraying

This paper presents a mathematical model of the in-flight thermal behavior and oxidation of powder particles during thermal spray deposition. In particular, stainless steel and Cr₃C₂–NiCr particles are investigated. The in-flight model accounts for the acceleration and deceleration of the particles during flight under variable fluid velocity, while the thermal model takes into account heating, melting, cooling and possible solidification as the particle travels towards the substrate. A finite difference method is used to solve the thermal energy conservation equation of the particles. The model includes non-equilibrium calculations of the phase change phenomena in the liquid–solid (mushy) zone and dissolution of the ceramic reinforcement in the composite particles. The growth of the oxide layer at the particle surface is represented by a modified boundary condition, which includes finite-rate oxidation. The results obtained give the interrelations between various process parameters and the oxidation phenomenon.     

Solidification of a ternary alloy liquid layer on a substrate subject to self-consistent melting

Micro-electro-mechanical (MEMS), plasma or powder spray deposition, surface coating, semiconductor technology, splat cooling, single and twin-roller melt-spinning, strip and slab casting, melt-extraction, etc. are usually characterized by solidification of a thin liquid layer on a cold substrate. A one-dimensional model of enthalpy formation of the energy and species conservation equations with thermodynamic relationships from ternary equilibrium diagram are solved to study the solidification processes for ternary alloys molten liquid layer on the ternary eutectic solid substrate. The solidification path of the liquid layer may pass through the primary, cotectic and eutectic solidification regions. The melting and re-solidification of the substrate happens at the ternary eutectic point. The thermal physical properties of the splat and substrate are identical and imperfect contact of contact surface between the splat and substrate is considered. The temperature functions as compositions are assumed as linear along the liquidus surface and cotectic curves. The temperature distributions of the solidified splat and the melted, re-solidified substrate, the thicknesses of the different mushy layers of splat and melting of substrate subject to different process parameters and thermal physical properties are quantitatively and extensively investigated. The initiation times for primary, cotectic mushy and eutectic solid fronts of splat and the complete re-solidification times of the substrate are affected by different parameters, these are also investigated. Results of this study are compared with experimental data provided by Aitta et al. The growth rates of the cotectic and eutectic fronts are found to agree well with experimental data. The effects of initial solute concentrations of liquid layer, solute concentrations and temperatures at the binary and ternary eutectic points on the thicknesses of different mushy layers are important and presented.     

MICROSOLIDIFICATION PROCESS IN MULTICOMPONENT SYSTEM

In conventional solidification of multicomponent mixtures, a mushy zone appears between the pure solid and liquid regions and promotes stable solidification by accepting the rejected solute regionally. From the standpoint that the fineness of inhomogeneity influences the mechanical properties in material processing, the linking of macro heat transfer and microsolidification in the mushy zone was studied. First, the crystal growth and its accompanying concentration field near the advancing front of the mushy zone were observed precisely by using the light absorption method. It was clarified that the mushy zone consisted of the leading front in which the frame structure formed with an accompanying concentration boundary layer and a growing region where the solidification proceeds by fattening of the crystals. Second, the mechanism of side-branch evolution was studied in conjunction with interfacial instability due to constitutional supercooling and curvature supercooling around the primary arm surface. Summarizing these results, the microsolidification process is discussed quantitatively in relation to macro heat transfer.     

A Semi-Exact Solution for Solidification of a Binary Solution on a Cold Isothermal Surface below Eutectic Temperature

The solidification of a binary solution on a cold horizontal surface below eutectic temperature is solved using a semi-exact method. The temperature distributions in the solid and liquid zones are obtained by exact solutions, while heat transfer in the mushy zone is obtained by an integral approximate method. The locations of the interface between solid and mushy zones and interface between mushy and liquid zones are obtained by coupling the temperature distributions in the three regions. The effects of initial temperatures, wall temperatures, and initial concentrations on the solidification of the binary solution are investigated.     

Flow-induced morphological instability and solidification with the slurry and mushy layers in the presence of convection

The linear analysis of convective morphological instability of the planar liquid–solid phase transition boundary is developed. The new stability criterion, dependent on the main parameter–extension rate (proportional to the vertical derivative of the fluid velocity), is deduced. This criterion generalizes analytical results of the recent works [H. Shimizu, J.P. Poirier, J.L. Le Mouël, Phys. Earth Planet. Inter. 151 (2005) 37–51; R. Deguen, T. Alboussière, D. Brito, Phys. Earth Planet. Inter. 164 (2007) 36–49], where convective mechanisms were only partially introduced in the model equations and stability analysis. The convective stability criterion demonstrates that the neutral stability curve divides two possible domains of morphologically stable and unstable solidification. These domains existing in the constitutionally supercooled conditions lead to two different crystallization scenarios “constitutional supercooling + morphological stability” and “constitutional supercooling + morphological instability”, which are described by idealized nonlinear slurry and mushy layer models with convection. Analytical solutions of these models taking into account nucleation and kinetic mechanisms of the growing solid phase are constructed for the steady-state solidification conditions.     

Solidification of a ternary melt from a cooled boundary, or nonlinear dynamics of mushy layers

We present a mathematical model and its analytical solution describing directional solidification of a ternary (three-component) system cooled from below. We focus on the solidification theory in the presence of two distinct mushy layers: (1) solidification along a liquidus surface is characterized by a primary mushy layer, and (2) solidification along a cotectic line is characterized by a secondary (cotectic) mushy layer. We consider the case when the phase transition temperatures in two mushy layers represent arbitrary functions of the compositions. We obtain an exact analytical solution of the nonlinear set of equations and boundary conditions in the case of a self-similar solidification scenario. Model predictions are in good agreement with existing experimental data.     

Convective instability of directional crystallization in a forced flow: The role of brine channels in a mushy layer on nonlinear dynamics of binary systems

The linear analysis of morphological instability of the crystallization process with a mushy layer is carried out. The instability is caused by the influence of a forced oceanic flow leading to the formation of brine channels in the anisotropic and heterogeneous phase transition domain. Two instability criteria are found for the cases of non-turbulent and turbulent boundary conditions at the mushy layer–ocean interface. The theory under consideration shows that the instability domain will depend highly on the physical and operating parameters of the system.     

MODELING OF DIELECTRIC FLUID SOLIDIFICATION WITH CHARGED PARTICLES IN ELECTRIC FIELDS AND REDUCED GRAVITY

A mathematical model and an explicit finite-difference iterative integration algorithm for two-dimensional laminar steady flow and solidification of an incompressible, viscous, electrically conducting but neutrally charged melt containing electrically charged panicles and exposed to an externally applied electrostatic field were developed. The system of governing electrohydrodynamic equations was derived from a combination of Maxwell's equations and the Navier-Stokes equations, including thermally induced buoyancy, latent heat release, and Joule heating, while accounting for the mushy region. Physical properties were treated as arbitrarily temperature-dependent. Numerical results demonstrate the existence of strong electrothermoconvective motion in the melt and quantify its influence on the amount of accrued solid, deposition pattern of the electrically charged particles inside the accrued solid, and the melt/solid interface shape.     

Self-similar solidification of an alloy from a cooled boundary

The self-similar solidification process of an alloy from a cooled boundary is studied on the basis of two models with a planar front and mushy layer. Approximate and exact analytical solutions of the process, which demonstrate unusual dynamics near the point of constitutional supercooling, are found. The rate of solidification and front position of the solid/mush boundary (parabolic growth rate constant) are expressed in an explicit form in the case of slow dynamics of this boundary. The theory under consideration is in a good agreement with experimental and numerical studies carried out by Huppert and Worster for ice growing from aqueous salt solutions.     

Application of a hybrid model of mushy zone to macrosegregation in alloy solidification

Solidification of an aqueous ammonium chloride (NH₄Cl-H₂O) solution inside a two-dimensional cavity is numerically investigated using a continuum mixture mathematical model. The mushy region where solid and liquid phases co-exist is considered a non-Newtonian fluid below a critical solid fraction, and a porous medium thereafter. This critical solid fraction is chosen as that corresponding to the coherency point, where a solid skeleton begins to form. The numerical results show that the solidification of a hypereutectic NH₄Cl-H₂O solution is mainly characterized by the rejection of solute at the mushy region and double diffusive convection induced by the opposing solutal and thermal buoyancy forces. The mathematical model agrees satisfactorily with the available experimental and numerical data.     

Effect of macro scale phenomena on microsegregation

A metal solidification system consists of solid, mushy and liquid regions. In many systems the two phase mushy region has a fine scaled columnar dendritic morphology. Microscopic models of metal solidification systems focus on the mass diffusion (microsegregation) and movement of the solid/liquid interface in the dendrite arm spaces. In this paper the effects of macroscopic variables on the microsegregation predictions are studied. In particular, the effect of cooling and macrosegregation histories on the solid solute profile, the eutectic fraction formed and solid/liquid interface movement in the arm spacing will be investigated.     

The structure of plumes generated in the unidirectional solidification process for a binary system

The present article describes the structure of plumes generated during solidification of a binary system. A transparent aqueous ammonium chloride solution is employed for super-eutectic growth in a Hele-Shaw cell. The velocities of plume convection in the melt layer and interstitial fluid flow within the mushy layer are measured by the particle tracking and dye tracing methods, respectively. Several important features are identified for each convective flow. In particular, the plume convection is found to consist of the upward flow enveloped in the downward flow, i.e., double flow structure. The downward flow enhances the solidification in the neighborhood of the exit of the channel emanating the plume, like a volcano. Interstitial fluid within the mushy layer is observed to move downward uniformly, which is induced by the plume convection.     

Mushy zone equilibrium solidification of a semitransparent layer subject to radiative and convective cooling

Equilibrium solidification in a semitransparent planar layer is studied using an isothermal mushy zone model. The layer is made up of a pure material being emitting, absorbing and isotropically scattering and is subject to radiative and convective cooling. The model involves solving simultaneously the transient energy equation and the radiation transport equation. An implicit finite volume scheme is employed to solve the energy equation, with the discrete ordinate method being used to deal with the radiation transport. A systematical parametric study is performed and the effects of various materials optical properties and processing conditions are investigated. It is found that decreasing the optical thickness and increasing the scattering albedo both lead to a wider mushy zone and a slower rate of solidification.     

Melting in a vertical cylindrical tube: Numerical investigation and comparison with experiments

The present work numerically investigates melting of a phase-change material (PCM) in a vertical cylindrical tube. The analysis aims at an investigation of local flow and thermal phenomena, by means of a numerical simulation which is compared to the previous experimental results .The numerical analysis is realized using an enthalpy–porosity formulation. The effect of various parameters of the numerical solution on the results is examined: in particular, the term describing the mushy zone in the momentum equation and the influence of the pressure–velocity coupling and pressure discretization schemes. PISO vs. SIMPLE and PRESTO! vs. Body-Force-Weighted schemes are examined. No difference is detected between the first two. However, considerable differences appear with regard to the last two, due to the mushy zone role.Image processing of experimental results from the previous studies is performed, yielding quantitative information about the local melt fractions and heat transfer rates. Based on the good agreement between simulations and experiments, the work compares the heat transfer rates from the experiments with those from the numerical analysis, providing a deeper understanding of the heat transfer mechanisms. The results show quantitatively that at the beginning of the process, the heat transfer is by conduction from the tube wall to the solid phase through a relatively thin liquid layer. As the melting progresses, natural convection in the liquid becomes dominant, changing the solid shape to a conical one, which shrinks in size from the top to the bottom.     

Mathematical modelling of surface segregation in aluminum DC casting caused by exudation

A mathematical model for the development of a segregated layer of exudated droplets during DC casting of aluminum ingots is established. The model accounts for the metallostatic pressure driven interdendritic melt flow through the mushy zone by a Darcy type equation, the surface segregation due to this melt flow, and the decrease of the total solute concentration in different positions of the mush as a result of the exudation. The solution domain for the governing differential equations is constituted by the mushy zone of the cast. The main physical phenomena included in the model have been studied in a simple one dimensional case study.