Two-dimensional Solidification Research Articles

Simulation Analysis of the Freezing Process For Achieving Bidirectional Freezing Coating Via the Freeze-Casting Technique

To delve deeper into the shapes and positions of the solidification front, as well as the detailed temperature distributions under various solidification conditions, this study employed the commercial software ANSYS FLUENT to conduct a meticulous numerical simulation of the two-dimensional solidification process in freeze casting. As part of the research, the enthalpy-porous media solidification/melting model was chosen as the core theoretical framework, and the accuracy of the numerical calculations was verified by comparing them with locally measured temperature changes from experiments. Subsequently, we further examined the influence mechanisms of factors such as cold source temperature, solid content, mold depth, and mold thermal conductivity on the speed of the solidification front. This is crucial because, in the freeze casting process, the speed of the solidification front directly determines the formation and evolution of macropore structures. The results of the simulation analysis indicate that an increase in mold thermal conductivity, a decrease in cold source temperature, a shallower mold depth, and a higher solid content all lead to a corresponding increase in the movement speed of the solidification front. These simulation outcomes not only provide strong theoretical support for optimizing the process parameters of freeze casting but also aid in achieving more precise control over the microstructure of materials, thus enabling a more efficient and accurate material preparation process.

Measurement of shrinkage void and identification of solid-liquid phases in phase change materials: Ultrasound-based approach and simulated predictions

Phase change materials in thermal energy storage systems undergo solidification and melting during cyclic operations. During solidification, internal shrinkage voids are formed randomly in phase change materials due to density differences in the solid and liquid phases. These shrinkage voids can severely affect the heat transfer performance and storage capacity of thermal energy storage systems. Additionally, the containers in which phase change materials are kept are usually metallic, making them opaque. Therefore, identification of the solid–liquid phase by direct visualisation during the phase change is limited. The present work aims to develop a non-destructive ultrasonic scanning approach to detect and measure internal shrinkage voids in solid phase change materials and the solid–liquid phases during melting and perform multi-phase numerical simulations that simultaneously capture solidification/melting and the associated shrinkage/expansion. The most commonly used phase change materials, n-eicosane, and n-octadecane, are used in this study. The effectiveness of the ultrasonic technique is evaluated by measuring cavities of known sizes. Further, three cases are considered for examining and measuring the shrinkage void. Case A involves two-dimensional solidification with a surface void, while Case B and C involve three-dimensional solidification with an internal shrinkage void. The numerically obtained shrinkage cavity profile shows a maximum deviation of 7.8% with experimental results for case C. Furthermore, the developed approach is implemented to uniquely detect and identify the solid and liquid phases during melting in an opaque container.

Quantitatively characterizing the evolution of hole defects between overlapped aluminum droplets by a two-dimensional solidification model

Quantitatively characterizing the effect of process parameters on the evolution of holes in metal droplet 3D printing enables effective prediction and elimination of hole defects in printed parts. In this work, a two-dimensional solidification model is proposed to quantitatively characterize the influence of substrate temperature on hole morphologies between overlapped aluminum droplets. First, a phenomenon is observed in experiments that the surfaces of overlapped droplets show typical "L"-shaped solidification ripples, which indicates that the solidification of overlapped droplets involves two-dimensional solidification. Hole defects are formed in the transition area of two-dimensional solidification. In addition, the appearance of the first L-shaped ripple signifies that the hole is no longer changing. Based on this discovery, a two-dimensional solidification prediction model for the hole size is constructed by combining a hyperbola and the one-dimensional heat transfer model to trace the ripple, which can accurately demonstrate the relationship between the hole size and the parameters (i.e., the thickness of solidified layer X(t) , the two-dimensional solidification angle β , the solidification angle θ ). The hole formed between the two overlapped droplets can be divided into two parts: the hole Ⅰ caused by the two-dimensional solidification and the hole Ⅱ caused by the morphology of the previously deposited droplet. When X(t) ≤ 0 at the end of the spreading of the overlapping droplets on the substrate, the hole can be completely eliminated. This work provides effective theoretical guidance for the prediction and elimination of hole defects in metal droplets printed parts.

PRISMS-PF: A general framework for phase-field modeling with a matrix-free finite element method

A new phase-field modeling framework with an emphasis on performance, flexibility, and ease of use is presented. Foremost among the strategies employed to fulfill these objectives are the use of a matrix-free finite element method and a modular, application-centric code structure. This approach is implemented in the new open-source PRISMS-PF framework. Its performance is enabled by the combination of a matrix-free variant of the finite element method with adaptive mesh refinement, explicit time integration, and multilevel parallelism. Benchmark testing with a particle growth problem shows PRISMS-PF with adaptive mesh refinement and higher-order elements to be up to 12 times faster than a finite difference code employing a second-order-accurate spatial discretization and first-order-accurate explicit time integration. Furthermore, for a two-dimensional solidification benchmark problem, the performance of PRISMS-PF meets or exceeds that of phase-field frameworks that focus on implicit/semi-implicit time stepping, even though the benchmark problem’s small computational size reduces the scalability advantage of explicit time-integration schemes. PRISMS-PF supports an arbitrary number of coupled governing equations. The code structure simplifies the modification of these governing equations by separating their definition from the implementation of the numerical methods used to solve them. As part of its modular design, the framework includes functionality for nucleation and polycrystalline systems available in any application to further broaden the phenomena that can be used to study. The versatility of this approach is demonstrated with examples from several common types of phase-field simulations, including coarsening subsequent to spinodal decomposition, solidification, precipitation, grain growth, and corrosion.

Phase-field model for growth and dissolution of a stoichiometric compound in a binary liquid.

We propose a simple formulation of the phase-field model for a stoichiometric compound growing in a binary liquid. In previous models, chemical free energies of stoichiometric compounds have been approximated by parabolic functions of composition; however, the curvature has been determined arbitrarily in spite of the fact that the stoichiometric composition was undesirably modified depending on the curvature. To avoid this uncertainty, we supposed that the chemical free energy of the stoichiometric compound is represented by a single value at a given temperature and derived the phase-field equations without the parabolic free-energy approximation. The phase mobility was derived both for diffusion- and interface-controlled solidification based on a thin interface limit analysis. We carried out numerical simulations of one-dimensional calculations both in diffusion-controlled and interface-controlled cases and found that the growth velocities agreed with the analytic predictions. We also examined two-dimensional solidification of a circular crystal and confirmed that the equilibrium state was shifted, as suggested by the Gibbs-Thomson effect. This study is an important step for phase-field modeling that includes stoichiometric compounds with their accurate thermodynamic properties.

Refined energy-conserving dissipative particle dynamics model with temperature-dependent properties and its application in solidification problem.

It has been observed previously that the physical behaviors of Schmidt number (Sc) and Prandtl number (Pr) of an energy-conserving dissipative particle dynamics (eDPD) fluid can be reproduced by the temperature-dependent weight function appearing in the dissipative force term. In this paper, we proposed a simple and systematic method to develop the temperature-dependent weight function in order to better reproduce the physical fluid properties. The method was then used to study a variety of phase-change problems involving solidification. The concept of the "mushy" eDPD particle was introduced in order to better capture the temperature profile in the vicinity of the solid-liquid interface, particularly for the case involving high thermal conductivity ratio. Meanwhile, a way to implement the constant temperature boundary condition at the wall was presented. The numerical solutions of one- and two-dimensional solidification problems were then compared with the analytical solutions and/or experimental results and the agreements were promising.

Analysis of soft reduction on bloom internal crack by a strain model

ABSTRACTSoft reduction (SR) can effectively improve the strand centre quality by decreasing its porosity and segregation. However, it tends to cause internal crack defects in the bloom if an improper reduction amount or improper reduction zone is implemented on the strand. In this work, a two-dimensional non-steady solidification model and a strain model are established for 350 mm × 470 mm bloom casting. Using the strain model, the strain field for GCr15 bloom upon SR is calculated. Through the online SR test and macrostructure inspection of the cast blooms, the minimum centre solid fraction suitable for SR of the bloom is determined. The results show that the established strain model calculation agrees well with the practical bloom macrostructure inspection. Thus, with the strain model, one can calculate whether internal crack defects would occur, as well as determine the right reduction amount or proper reduction zone.

Numerical simulation of macrosegregation during solidification of WE43 (Mg–4.3 wt-% Y–2.0 wt-% Nd–0.6 wt-% Gd) magnesium alloy

The formation of macrosegregation driven by multicomponent thermosolutal convection during solidification of WE43 magnesium alloy is numerically studied. A two-dimensional solidification model that describes the conservation of mass, momentum, energy and solute is presented. The experiments are performed in the resin sand mould with a graphite chill placed on one side of the cavity. The validation of computer codes is carried out by comparisons with the experimentally measured cooling curves and concentrations. The influences of mould material and characteristic grain size on macrosegregation are numerically investigated. For a total sand mould casting, the magnitude of macrosegregation is reduced, although the solidification rate is slower than that cooled with a graphite chill. Increasing the characteristic grain size not only causes a serious positive segregation but also leads to the formation of channel segregates. The channel segregation occurs thanks to the weak instabilities of the growing front triggered by the fluid flow.

Percolation and disorder-resistance in cellular automata

We rigorously prove a form of disorder-resistance for a class of one-dimensional cellular automaton rules, including some that arise as boundary dynamics of two-dimensional solidification rules. Specifically, when started from a random initial seed on an interval of length $L$, with probability tending to one as $L\to\infty$, the evolution is a replicator. That is, a region of space-time of density one is filled with a spatially and temporally periodic pattern, punctuated by a finite set of other finite patterns repeated at a fractal set of locations. On the other hand, the same rules exhibit provably more complex evolution from some seeds, while from other seeds their behavior is apparently chaotic. A principal tool is a new variant of percolation theory, in the context of additive cellular automata from random initial states.

Numerical Solution of Heat Transfer Process in PCM Storage Using Tau Method

Thermal energy storage units that utilize phase change materials have been widely employed to balance temporary temperature alternations and store energy in many engineering systems. In the present paper, an operational approach is proposed to the Tau method with standard polynomial bases to simulate the phase change problems in latent heat thermal storage systems, that is, the two-dimensional solidification process in rectangular finned storage with a constant end-wall temperature. In order to illustrate the efficiency and accuracy of the present method, the solid-liquid interface location and the temperature distribution of the fin for three test cases with different geometries are obtained and compared to simplified analytical results in the published literature. The results indicate that using a two-dimensional numerical approach can predict the solid-liquid interface location more accurately than the simplified analytical model in all cases, especially at the corners.

A comparative study of dendritic growth by using the extended Cahn–Hilliard model and the conventional phase-field model

An extended Cahn–Hilliard model (ECHM) was compared with the conventional phase-field model (CPFM) for simulating the operating state of a dendrite tip during the two-dimensional solidification of pure undercooled melts over a wide range of interfacial energy anisotropy. ECHM differs from CPFM in terms of how interfacial energy anisotropy is introduced. In ECHM, anisotropy comes solely from the anisotropic nature of the fourth-rank tensor terms included in free energy density, and not from assuming an orientation-dependent gradient energy coefficient ɛ(θ), which is the case in CPFM. ECHM resulted in dendrites growing with a rounded tip, even when anisotropy (δ) was greater than its critical value (δc), but the tip radius at large anisotropy (δ⩾δc) was limited by the interface width. In contrast to CPFM, ECHM did not engender an anomalous increase in the tip radius with bulk undercooling at small anisotropy (δ<δc). In the simulation by ECHM, the tip velocity increased continuously with increasing δ beyond δc. When compared in terms of the selection parameter σ∗ of the dendrite tip, data obtained from ECHM fitted better to the σ∗∝δ7/4 relationship over a wider range of δ than those obtained from CPFM. The present comparative study suggests that ECHM hinders the transition of the dendritic growth kinetics from diffusion-limited to interface-kinetic-limited, which occurs in the case of CPFM as the tip velocity increases with an increase in either undercooling or anisotropy.

Growth of Mushy Zone during Two-dimensional Solidification with Supercooling （Theoretical model for selective growth of primary dendrite arms）

Growth of Mushy Zone during Two-dimensional Solidification with Supercooling （Theoretical model for selective growth of primary dendrite arms）

Study on Prediction of Reduction Position Based on Solidification and Heat Transfer Model

As one of the important parameters of dynamic soft reduction technology, reduction position seriously affects the using effect of this technology to decrease center segregation and porosity of continuous casting slabs. According to the actual production conditions, a two-dimensional solidification and heat transfer model was established. In order to obtain more accurate model, the heat transfer coefficient was amending by the pin shooting technique in measuring shell thickness. The laws of the effect of casting speed, superheat degree and specific water flow on reduction position were analyzed respectively and reduction positions were predicted in different working conditions based on the model. The experimental results show that the center segregation and porosity decreased obviously when it produces at the reduction position predicted by the model.

Evolution of the solidification front of a binary mixture

We perform a numerical study of two-dimensional solidification of a binary alloy. We employ a front-fixing transformation and develop a finite-difference numerical scheme, which is then used to simulate the evolution of an initially planar solidification front. The self-similar solution is taken as a base state for numerical investigation; the parameter choice corresponds to the case when the melt is constitutionally supercooled and the linear instability is expected. The perturbed interface takes the form of traveling waves with nonlinear growth rate, with the increased Stefan number causing the slow-down of solidification. Another important feature is the decay of perturbation when the time t0 at which the perturbation is imposed to the self-similar base solution is large enough, despite the fact that the system is expected to be linearly unstable. Finally, we provide a numerical investigation of the lowest value of t0 for which the perturbation decays in time and its dependence on Stefan number.

Numerical study of effect of sulphur element on mesosegregation by thermosolutal convection in Iron-Carbon-Sulphur system

The effects of a sulphur element on the mesosegregation formation by the multicomponent thermosolutal convection in a Fe-0.21wt pct C-X wt pct S system studied numerically. A two-dimensional solidification model that describes conservations of mass, momentum, energy and solute is presented. Simulations are performed on the Hebditch-Hunt casting. The validation of codes is carried out by the comparison with a consensus of previous numerical simulation for the Sn-5 wt pct Pb alloy. Three cases, encompassing variations of the initial concentration of the sulphur (0, 0.05, 0.1 wt pct), are studied. Simulated results show that no mesosegregates form for S = 0.0 wt pct. The mesosegregates are triggered by the weak instabilities of the solidification front growth for S = 0.05 wt pct. The increased fluid flow induces the local remelting, and results in the mesosegregation for S = 0.1 wt pct.

An implicit lattice Boltzmann model for heat conduction with phase change

In this work, a new variation of the lattice Boltzmann method (LBM) was developed to solve the heat conduction problem with phase change. In contrast to previous explicit algorithms, the latent heat source term was treated implicitly in the energy equation, avoiding iteration steps and improving the formulation stability and efficiency. The Bhatnagar–Gross–Krook (BGK) approximation with a D2Q9 lattice was applied and different boundary conditions including Dirichlet and Neumann boundary conditions were considered. The developed model was tested by solving a one-dimensional melting problem for a pure metal, and one and two-dimensional solidification problems for a binary alloy. The results of the LBM solution were compared with analytical and finite element solutions and a good consistency was observed. Considering the special capabilities that LBM offers, like local characteristic, and inherent parallel structure, the developed model is an interesting alternative to traditional continuum models.

Growth of Mushy Zone during Solidification of Supercooled Alloy Melts: Experiment for primary arm spacing selection during two-dimensional solidification

Growth of Mushy Zone during Solidification of Supercooled Alloy Melts: Experiment for primary arm spacing selection during two-dimensional solidification

Migration-collision scheme of Lattice Boltzmann method for heat conduction problems involving solidification

The phase-change problem is solved by the migration-collision scheme of lattice Boltzmann method. After formula derivation, we find that this method can give a rigorous numerical value for the phase-change temperature, which is of crucial importance. One-dimensional solidification in half-space and two-dimensional solidification in a corner are simulated. The phase change temperature and the liquid-solid interface are both obtained, and the results conform to the analytical solution.

A Mathematical Modeling of Heat Transfer in Continuous Casting Slab

A two-dimensional (2-D) heat transfer and solidification model has been established and applied to calculate the temperature distribution and solid shell thickness profile of a continuous casting slab in a steel plant. A finite difference method was used for the numerical simulation. For thermal analysis, the 2-D slice unsteady-state heat conduction equation with enthalpy convention was used. Meanwhile, non-linear material properties of specific heat and thermal conductivity as well as phase changes during solidification were considered in the model. The temperature distribution and solid shell thickness calculated by mathematical model agree with those predicted by industrial and experimental measurements. The model could also be used to predict the optimum process parameters on casting speed, heat removal rates and the water distribution of secondary cooling zone.

Lattice Boltzmann Simulation of Incompressible Transport Phenomena in Macroscopic Solidification Processes

A lattice Boltzmann (LB) simulation strategy is proposed for the incompressible transport phenomena occurring during macroscopic solidification of pure substances. The proposed model is derived by coupling a passive scalar-based thermal LB model with the classical enthalpy–porosity technique for solid–liquid phase-transition problems. The underlying hydrodynamics are monitored by a conventional single-particle density distribution function (DF) through a kinetic equation, whereas the thermal field is obtained from another kinetic equation which is governed by a separate temperature DF. The phase-changing aspects are incorporated into the LB model by inserting appropriate source terms in the respective kinetic equations through the most formal technique following the extended Boltzmann equations along with an appropriate enthalpy updating scheme. The proposed model is validated extensively with one- and two-dimensional solidification problems for which analytical and numerical results are available in the literature, and finally, it is used for solving a benchmark problem, the Bridgman crystal growth in a square crucible.