A cell-by-cell thermally driven mushy cell tracking algorithm for phase-change problems

The finite volume based numerical approach is used to simulate phase-change processes including natural convection. This approach is based on a cell-by-cell, thermally driven mushy cell tracking equation, developed in Part I [20], to trace the front at which phase-change occurs. A mushy cell is a specialized cell where the interface between liquid and solid phases is located. In this paper, the mushy cell tracking equation and the associated boundary condition around the mushy cells are derived in a general manner and shown to have the same form as that used in Part I. The SIMPLE algorithm is adopted to solve the flow, including pressure field, as well, in the liquid phase and a conjugate gradient method is used when solving the system of discretized equations. To reduce computational time, an acceleration technique, based on a justified quasi-steady state assumption, is adopted. The proposed numerical method is applied to simulate the solidification and melting of Tin with natural convection. The numerical predictions are compared well with the available experimental data and previously published numerical results. Specifically, these comparisons demonstrate that the proposed methodology is capable of predicating the location of moving fronts and the temperature distributions for phase-change processes with natural convection.     

Modified enthalpy method for the simulation of melting and solidification

Enthalpy method is commonly used in the simulation of melting and solidification owing to its ease of implementation. It however has a few shortcomings. When it is used to simulate melting/solidification on a coarse grid, the temperature time history of a point close to the interface shows waviness. While simulating melting with natural convection, in order to impose no-slip and impermeability boundary conditions, momentum sink terms are used with some arbitrary constants called mushy zone constants. The values of these are very large and have no physical basis. Further, the chosen values affect the predictions and hence have to be tuned for satisfactory comparison with experimental data. To overcome these deficiencies, a new cell splitting method under the framework of the enthalpy method has been proposed. This method does not produce waviness nor requires mushy zone constants for simulating melting with natural convection. The method is then demonstrated for a simple one-dimensional melting problem and the results are compared with analytical solutions. The method is then demonstrated to work in two-dimensions and comparisons are shown with analytical solutions for problems with planar and curvilinear interfaces. To further benchmark the present method, simulations are performed for melting in a rectangular cavity with natural convection in the liquid melt. The solid–liquid interface obtained is compared satisfactorily with the experimental results available in literature.     

A finite element model for liquid phase electroepitaxy

A finite element numerical simulation model for the liquid phase electroepitaxial growth process of gallium arsenide is presented. The basic equations obtained from the fundamental principles of electrodynamics of continua, the constitutive equations for the liquid and solid phases derived from a rational thermodynamic theory, and the associated interface and boundary conditions are presented for a two-dimensional axisymmetric growth cell configuration. The field equations are solved numerically by an adaptive finite element procedure. The effect of moving interfaces is taken into account. Numerical simulations are carried out for different convection levels by changing the value of the gravitational constant. Results show that convection has significant effect on the growth process under normal gravity conditions and results in thickness non-uniformity of the grown layers. The thickness non-uniformity leads to curved interfaces of growth and dissolution, which enhance convection.     

水平冷表面上结霜发生随机性研究

水蒸气在冷壁面上的结霜是制冷、低温领域普遍存在的问题。过饱和的水蒸气有两种相变途径，一种是气液固相变；一种是直接气固相变。气液固相变经历气液相变和液固相变两个阶段，实验研究表明其发生具有随机性。在前人实验研究的基础上，从理论上给出了液珠半径随时间的增长规律，并进一步得到了冷壁面自然对流情况下结霜发生概率随时间的变化关系式。     

Computational model of pulsed laser-induced melting, evaporation and solidification of CdZnTe

A model of melting, evaporation and solidification of CdZnTe pseudobinary alloy due to pulsed laser irradiation is formulated using the mass and internal energy balances in the liquid and solid phases, the vapor being assumed to be removed from the sample surface into vacuum instantaneously. The interface between the solid and liquid phases is modeled as a discontinuity surface where, in addition to balance conditions, an interface response function is formulated on the basis of the Wilson–Frenkel theory so that both melting and solidification are treated as nonequilibrium processes. The liquid/vapor interface is modeled in a similar way, with an interface response function defined on the basis of the kinetic theory of gases. The numerical solution of the mathematical model is done using the Galerkin finite element method combined with a front-fixing technique. In the numerical simulations of pulsed laser-induced phase change processes in CdZnTe, the temperature and concentration fields, the positions and velocities of the solid/liquid and liquid/vapor phase interfaces, and the time-resolved incident laser reflectivities are calculated as functions of the laser energy density for two types of lasers, namely the Nd:YAG laser and ruby laser. The results obtained for both the lasers are discussed and recommendations for the optimization of experimental setups are given.     

Numerical Prediction of Eutectic Temperature Using a Multi-phase-Field Model

To evaluate the reliability of metal-carbon eutectic systems as fixed points for the next generation of the international temperature scale, the effect of the eutectic microstructure on the temperature at the solid/liquid (s/l) interface during solidification and melting is preliminarily investigated using a multi-phase-field model. First, the effects of furnace temperature, lamellar spacing, and interface energy on the average temperature of the s/l interface are studied in the solidification process. With increased furnace undercooling, the s/l interface temperature was found to decrease. Calculated eutectic microstructures are then adopted as initial conditions for a melting simulation. The interface undercooling during melting is observed to be smaller than that observed during solidification. This difference in interface undercooling is attributed to the solute/solvent concentration profiles in the liquid phase near the s/l interface being different for melting and solidification.     

Experimental study of thermal conductivity of solid and liquid phases at the phase transition

An experimental method for the determination of the thermal conductivity of a pure material at its freezing melting point has been developed. In the present investigation. this method is discussed further by studying the effective thermal conductivity of the frozen as well as the thawed state of a wet porous material. and of solid and liquid benzene at the interface. The method involves the study of the transient propagation of the freezing or melting zone through the specimen as well as the steady state of the freezing melting process. The method makes use of the heat of transition as the heat flow, and it enables the interlace to act as a heat transfer surface, thus incorporating realistic features of the phase change process. Compared with data from the literature. the experimental results for benzene agree within 2% for the solid phase and within 3% for the liquid phase. when some precautions are made to avoid free convection. The experimental effective thermal conductivity of the packed bed is compared with data from a numerical analysis: the results agree within 4%, indicating that the method is applicable also for measurements on heterogeneous materials.Paper presented at the Twelfth Symposium on Thermophysical Properties, June 19–24, 1994, Boulder, Colorado, U.S.A.On leave from Central South University of Technology, 410083 Changsha, P.R. China.     

A least-squares front-tracking finite element method analysis of phase change with natural convection

This paper focuses on the numerical modelling of phase-change processes with natural convection. In particular, two-dimensional solidification and melting problems are studied for pure metals using an energy preserving deforming finite element model. The transient Navier–Stokes equations for incompressible fluid flow are solved simultaneously with the transient heat flow equations and the Stefan condition. A least-squares variational finite element method formulation is implemented for both the heat flow and fluid flow equations. The Boussinesq approximation is used to generate the bulk fluid motion in the melt. The mesh motion and mesh generation schemes are performed dynamically using a transfinite mapping. The consistent penalty method is used for modelling incompressibility. The effect of natural convection on the solid/liquid interface motion, the solidification rate and the temperature gradients is found to be important. The proposed method does not possess some of the false diffusion problems associated with the standard Galerkin formulations and it is shown to produce accurate numerical solutions for convection dominated phase-change problems.     

Control of the freezing interface morphology in solidification processes in the presence of natural convection

This paper presents a methodology for the solution of an inverse solidification design problem in the presence of natural convection. In particular, the boundary heat flux q₀ in the fixed mold wall, δΩ₀, is calculated such that a desired freezing front velocity and shape are obtained. As the front velocity together with the flux history q_ms on the solid side of the freezing front play a determinant role in the obtained cast structure, the potential applications of the proposed methods to the control of casting processes are enormous. The proposed technique consists of first solving a direct natural convection problem of the liquid phase in an a priori known shrinking cavity, Ω_L(t), before solving an ill-posed inverse design conduction problem in the solid phase in an a priori known growing region, Ω_S(t). The direct convection problem is used to evaluate the flux q_ml in the liquid side of the freezing front. A front tracking deforming finite element technique is employed. The flux q_ml can be used together with the Stefan condition to provide the freezing interface flux q_ms in the solid side of the front. As such, two boundary conditions (flux q_ms and freezing temperature θ_m) are especified along the (known) freezing interface δΩ_I(t). The developed design technique uses the adjoint method to calculate in L₂ the derivative of the cost functional, ∥θ_m – θ( x , t; q₀)∥, that expresses the square error between the calculated temperature θ( x , t; q₀) in the solid phase along δΩ_I(t) and the given melting temperature. The minimization of this cost functional is performed by the conjugate gradient method via the solutions of the direct, sensitivity and adjoint problems. A front tracking finite element technique is employed in this inverse analysis. Finally, an example is presented for the solidification of a superheated incompressible liquid aluminium, where the effects of natural convection in the moving interface shape are controlled with a proper adjustment of the cooling boundary conditions.     

Vibrational Coupling and Kapitza Resistance at a Solid–Liquid Interface

The rapid development and application of nanotechnologies have promoted an increasing interest in research on heat transfer across the solid/liquid interface. In this study, molecular dynamics simulations are carried out to elucidate the effect of vibrational coupling between the solid and the liquid phases on the Kapitza thermal resistance. This is accomplished by altering the atomic mass and interatomic interaction strength in the solid phase (thus, the vibrational properties), while keeping the solid–liquid interfacial interaction unchanged. In this way, the Kapitza resistance can be altered with a constant work of adhesion between the solid and the liquid phases. The simulation results show that the overlap degree between the vibrational density of states profiles of the interfacial liquid layer and the outermost solid layer, which measures the degree of interfacial vibrational coupling, increases with larger atomic mass and weaker inter-atomic interaction in the solid phase. An inverse relation exists between the Kapitza resistance and the overlap degree of the vibrational density of states profiles. It means that the Kapitza resistance decreases with better interfacial vibrational coupling. The simulations show that the Kapitza resistance is not only affected by the interfacial bonding strength but also the vibrational coupling between the solid and the liquid atoms. The interfaces with better thermal transport efficiency should be the ones with stronger interfacial interaction and preferable vibrational coupling between solid and liquid phases.     

Phase Dependence (Liquid/Solid) of Normal Spectral Emissivities of Noble Metals at Melting Points

Normal spectral emissivities of liquid and solid Cu, Ag, and Au have been determined at their melting (freezing) points in the visible region using a cold crucible as the heating method. The use of the cold crucible enables the solidification front to be moved on the molten metal surface slowly enough to measure the emissivities of liquid and solid phases separately at the freezing point. Combined standard uncertainties of the spectral emissivities and wavelengths have been estimated. In silver, the spectral emissivity obtained for the liquid is systematically larger than that for the solid over the visible region, which is consistent with the prediction from a classical free-electron model. In copper and gold, the spectral emissivities at wavelengths around their absorption edges do not change for the solid-to-liquid transition. The wavelength range where the emissivity of copper is independent of the phase is unexpectedly broad (the width is greater than 40 nm), which differs significantly from classical experimental studies on the so-called X-point in the emissivity of copper. A qualitative explanation is provided for the difference in the phase dependence (liquid/solid) of the emissivity between copper and gold.     

A finite element method for the study of solidification processes in the presence of natural convection

A new numerical technique has been developed for the analysis of two-dimensional transient solidification processes in the presence of time-dependent natural convection in the melt. The method can cope with irregular, transient morphologies of the solid—liquid interface using a new Galerkin formulation for the energy balance on the solid—liquid interface. The finite element solution to the Galerkin formulation yields the displacement of individual nodes on the solid—liquid interface. The displacement of the nodes is expressed by uncoupled components in the x and y directions. The fluid flow problem was solved using a ‘penalty’ formulation. Numerical experiments were performed for Rayleigh numbers as high as 10⁶ to demonstrate the method and to indicate the effect of natural convection on the solid—liquid interface morphology.     

Effects of ultrasonic field on the Pb‐Sn alloys during heating process

In this work, the effects of ultrasonic field on the Pb‐Sn alloys during heating process have been discussed using electrical resistivity and thermal curves. The semi‐solid and liquid alloy have different resistivity responses to ultrasonic irradiation, due to ultrasound irradiation only available to liquid phase. Sensitive structural change by ultrasound can be only found in the liquid state, according to the resistivity drop and thermal curves. It might be due to the low volume fraction of liquid phase at initial stage and the fast solid‐to‐liquid transitions at the final stage. Ultrasound can also trigger the nucleation and growth of liquid phase, through driving the melting of solid phase at solid/liquid interface and transferring into the liquid phase. The concerned mechanisms have been discussed in detail.     

Determination of interfacial energies of solid Sn solution in the In–Bi–Sn ternary alloy

The equilibrated grain boundary groove shapes of solid Sn solution (Sn-40.14 at.% In-16.11 at.% Bi) in equilibrium with the In–Bi–Sn liquid (In-21.23 at.% Bi-19.04 at.% Sn) were observed from the quenched sample at 59 °C. Gibbs–Thomson coefficient, solid–liquid interfacial energy and grain boundary energy of the solid Sn solution have been determined from the observed grain boundary groove shapes. The thermal conductivity of solid phase for In-21.23 at.% Bi-19.04 at.% Sn alloy and the thermal conductivity ratio of liquid phase to solid phase at the melting temperature have also been measured with radial heat flow apparatus and Bridgman type growth apparatus, respectively.     

Iron Burning in Pressurised Oxygen Under Microgravity Conditions

An investigation of cylindrical iron rods burning in pressurised oxygen under microgravity conditions is presented. It has been shown that, under similar experimental conditions, the melting rate of a burning, cylindrical iron rod is higher in microgravity than in normal gravity by a factor of 1.8 ± 0.3. This paper presents microanalysis of quenched samples obtained in a microgravity environment in a 2.0 s duration drop tower facility in Brisbane, Australia. These images indicate that the solid/liquid interface is highly convex in reduced gravity, compared to the planar geometry typically observed in normal gravity, which increases the contact area between liquid and solid phases by a factor of 1.7 ± 0.1. Thus, there is good agreement between the proportional increase in solid/liquid interface surface area and melting rate in microgravity. This indicates that the cause of the increased melting rates for cylindrical iron rods burning in microgravity is altered interfacial geometry at the solid/liquid interface.     

MOVING BOUNDARY COMPUTATION OF THE FLOAT-ZONE PROCESS

A computational capability has been developed to predict the free surface shape, heat transfer and melt–crystal interface shapes in float-zone processing. A moving boundary, second order, finite volume, incompressible Navier–Stokes solver has been developed for the fluid flow and heat transfer calculations. The salient features of the approach include solving the dynamic form of the Young–Laplace equation for the free surface shape, dynamic remeshing to fit the free boundary, a flexible, multi–block, grid generation procedure and the enthalpy method to capture the melt–crystal and the melt–feed interfaces without the need for explicit interface tracking. Important convective heat transfer modes; natural convection and thermocapillary convection have been computed. It is shown that, whereas the overall heat transfer is not substantially affected by convection, the melt–crystal interface shape acquires significant distortion due to the redistribution of the temperature field by the thermocapillary and buoyancy-induced convective mechanisms. It is also demonstrated that the interaction of natural and thermocapillary convection can reduce the melt–crystal interface distortion if they act in opposing directions. It is found that the meniscus deformation can cause the height of the zone to increase but the qualitative nature of the melt–solid interface shapes are not significantly affected. Results are compared with literature to validate the predictive capability developed in this work. © 1997 by John Wiley & Sons, Ltd.     

Measurements on Melting Pressure,Metastable Solid Phases,and Molar Volume of Univariant Saturated Helium Mixture

A concentration-saturated helium mixture at the melting pressure consists of two liquid phases and one or two solid phases. The equilibrium system is univariant, whose properties depend uniquely on temperature. Four coexisting phases can exist on singular points, which are called quadruple points. As a univariant system, the melting pressure could be used as a thermometric standard. It would provide some advantages compared to the current reference, namely pure $^3$ He, especially at the lowest temperatures below 1 mK. We have extended the melting pressure measurements of the concentration-saturated helium mixture from 10 to 460 mK. The density of the dilute liquid phase was also recorded. The effect of the equilibrium crystal structure changing from hcp to bcc was clearly seen at $T=294$  mK at the melting pressure $P=2.638$  MPa. We observed the existence of metastable solid phases around this point. No evidence was found for the presence of another, disputed, quadruple point at around 400 mK. The experimental results agree well with our previous calculations at low temperatures, but deviate above 200 mK.     

Convective Auto-Oscillations Near a Drop–Liquid Interface in a Horizontal Rectangular Channel

The concentration convection in an isothermal liquid near a drop (or an air bubble) clamped between the vertical walls of a horizontal channel is studied numerically within the framework of two simple mathematical models: with and without the surface phase at the drop–liquid interface formed by adsorption/desorption process. The interaction between the buoyancy and the Marangoni convective flows is responsible for the onset of auto-oscillation regime. Such oscillations have been experimentally investigated in other works. In our numeric experiments, more than 20 outbursts of the Marangoni convection were observed. The surfactant distributions obtained numerically at different oscillation phases agree well the experimen tal data.     

Experimental study of laminar natural convection heat transfer from a vertical cylinder to slush nitrogen under constant heat flux conditions

Natural convection heat transfer from a vertical cylinder immersed in slush and subcooled liquid nitrogen and subjected to constant heat fluxes was investigated in order to determine the relative merits of slush nitrogen (SlN₂) for immersion cooling. A glass dewar was used as a test vessel in which a cylindrical heater was mounted vertically, and heat transfer measurements were carried out for SlN₂ and subcooled liquid nitrogen (LN₂) in the laminar flow range. The results revealed advantages of SlN₂ over subcooled LN₂ in natural convection cooling. The local temperatures of the heated surface surrounded by solid nitrogen particles are measured to increase at much slower rates than in subcooled LN₂, which is due to the latent heat of fusion of solid nitrogen. Even after the solid nitrogen particles surrounding the heater are apparently depleted, the average heat transfer coefficients for SlN₂ are still found to be greater than those for LN₂ with the improvement in heat transfer being larger for lower Grashof number regime. Our analysis also indicates that solid nitrogen particles in close proximity to heated surface do not discourage local convection due to the porous nature of SlN₂, making the heat transfer in SlN₂ more effective than in the case of solid–liquid phase change of nitrogen involving melting and conduction processes.     

利用焓-多孔介质法对垂直Bridgman生长CdTe的数值模拟

利用数值模拟研究了碲化镉在垂直Bridgman炉中生长时的固-液界面的形状．采用焓-多孔介质法,在固定网格上对碲化镉的固液两相用统一的控制方程进行了整场求解,用一特征参数确定界面的位置和形状．结果表明,当晶体的生长速率较低时,界面的形状与物质在固态和液态两相下的热扩散率有关．如果两种热扩散率的数值相近,界面的形状是平坦的．液态区自然对流是界面形状的影响因素之一,而积聚在固态区的结晶潜热是形成弯曲固-液界面的主要原因．