One-dimensional Solidification Research Articles

Development and validation of a one-dimensional solidification model — Part II: Model validation for molten lead-bismuth

The lead-bismuth eutectic (LBE) cooled fast reactor (LFR) has shown promising applications due to its excellent neutron economy and thermal-hydraulic characteristics. Nevertheless, the high-freezing point of LBE arises a risk of coolant solidification, which is identified as a design basis accident for LFRs. Accurate one-dimensional (1D) solidification models are critical for analyzing the coolant solidification behavior. However, the development and evaluation of solidification models are hindered by the lack of experimental data. In this study, a series of molten LBE horizontal penetration solidification experiments were designed and performed with different experimental conditions. The final penetration distances were measured, and the experimental conditions were used as boundary and initial conditions of 1D simulations based on the ASYST-SF code. The simulation results showed the relative error of the final penetration distance is within 12%. This study provides a series of LBE solidification experimental data and demonstrates the reliability of the ASYST-SF code.

Development and validation of a one-dimensional solidification model — Part I: Analytical model

The utilization of high-freezing point coolants in Gen IV reactors offers advantages in terms of high thermodynamic efficiency and inherent safety. However, it also introduces the risk of coolant solidification, which has been identified as a design basis accident for these reactor types. Accurate one-dimensional (1D) solidification models play a crucial role in reactor licensing and operation. Nevertheless, existing 1D models primarily focus on the growth of the solidified layer thickness, neglecting the important mushy zone effects that occur at the boundary layer overlap. This paper presents a novel general solidification model that incorporates the mushy zone effects and is applicable to high-freezing point coolants. The model has demonstrated favorable accuracy, with a maximum relative error of 10% when compared to experimental data. The findings highlight that the omission of mushy zone effects can lead to significant deviations from the actual scenario.

Numerical Study of Grain Growth in Laser Powder Bed Fusion Additive Manufacturing of Metals

Laser Powder Bed Fusion (LPBF) is an additive manufacturing method that manufactures high density and quality metal products. We present a coupled grain growth and heat transfer modeling technique to understand the materials microstructure evolution in metals during the cooling process of LPBF. The phase-field model is combined with a transient heat transfer equation to simulate the solidification and crystallization of the melt pool simultaneously. Specifically, the variable domain and driving force of the order parameters in the phase-field calculation are defined using current temperature distribution. Additionally, the latent heat generated by crystallization is introduced as a heat source to affect temperature evolution in the cooling process. The finite element method with a staggering strategy is employed to solve the coupled governing equations on an irregular computational domain. The computational framework is verified in a one-dimensional solidification problem by comparing the velocity of the fluid-solid interface. The two-way coupling solution of solidification and crystallization is studied in an example of LPBF of Aluminum alloys.

On small-time similarity-solution behaviour in the solidification shrinkage of binary alloys

In the one-dimensional solidification of a binary alloy undergoing shrinkage, there is a relative motion between solid and liquid phases in the mushy zone, leading to the possibility of macrosegregation; thus, the problem constitutes an invaluable benchmark for the testing of numerical codes that model these phenomena. Here, we revisit an earlier obtained solution for this problem, that was posed on a semi-infinite spatial domain and valid for the case of low superheat, with a view to extending it to the more general situation of a finite spatial domain, arbitrarily large superheat and both eutectic and non-eutectic solidification. We find that a similarity solution is available for short times which contains a boundary layer on the liquid side of the mush–liquid interface; this solution is believed to constitute the correct initial condition for the subsequent numerical solution of the full non-similar problem, which is deferred to future work.

Solidification dynamics of an impacted drop

This paper is dedicated to the solidification of a water drop impacting a cold solid surface. In the first part, we establish a one-dimensional (1-D) solidification model, derived from the Stefan problem, that aims at predicting the freezing dynamics of a liquid on a cold substrate, taking into account the thermal properties of this substrate. This model is then experimentally validated through a 1-D solidification set-up, using different liquids and substrates. In the second part, we show that during the actual drop spreading, a thin layer of ice develops between the water and the substrate and pins the contact line at its edge when the drop reaches its maximal diameter. The liquid film then remains still on the ice and keeps freezing. This configuration lasts until the contact line eventually unpins and the liquid film retracts on the ice. We measure and interpret this crucial time of freezing during which the main ice layer is built. Finally, we compare our 1-D model prediction to the thickness of this ice pancake and we find a very good agreement. This allows us to provide a general expression for the frozen drop’s main thickness, using the drop’s impact and liquid parameters.

Visualization of phase distribution in lead–bismuth eutectic during one-dimensional solidification process

Solidification process of lead–bismuth eutectic (LBE) is one of the key phenomena to prevent flow channel blockage accident in an LBE-cooled accelerator-driven system. However, the solidification of liquid metal cannot be observed optically and it is difficult to detect noninvasively. In this study, the one-dimensional solidification process of the LBE was visualized by pulsed neutron transmission imaging. Neutrons have higher transmittivity to the LBE than X-ray and neutron transmission spectrum of the LBE sample can be obtained by pulsed neutron imaging technique. The solid and liquid phases of the LBE were identified during the solidification process by the presence or absence of Bragg edge in the measured neutron transmission spectrum, and the transient behavior of the solid–liquid interface could be visualized. In addition, the characteristic spatial distribution of the crystalline structure was found in Bragg-edge transmission image after the solidification.

An Optimum Enthalpy Approach for Melting and Solidification with Volume Change

Classical numerical methods for solving solid–liquid phase change assume a constant density upon melting or solidification and are not efficient when applied to phase change with volume expansion or shrinkage. However, solid–liquid phase change is accompanied by a volume change and an appropriate numerical method must take this into account. Therefore, an efficient algorithm for solid–liquid phase change with a density change is presented. Its performance for a one-dimensional solidification problem and for the quasi two-dimensional melting of octadecane in a cubic cavity was tested. The new algorithm requires less than 1/9 of the iterations compared to the source based method in one dimension and less than 1/7 in two dimensions. Moreover, the new method is validated against PIV measurements from the literature. A conjugate heat transfer simulation, which includes parts of the experimental setup, shows that parasitic heat fluxes can significantly alter the shape of the phase front near the bottom wall.

Temperature dependent properties of heat treated aluminium alloys

The mechanical properties of different aluminium alloys used in the manufacture of power train automotive parts were studied. The alloys were cast in wedge shape ingots that promote a one-dimensional solidification gradient. Tensile samples were machined from bars that were cast with different degrees of microstructural refining and were heat treated at times and temperatures that depended on their composition. The mechanical properties were measured from samples that had been held for 200 h within the temperature range of 25 to 300°C. It was found that the increase of Cu in alloys of the Si–Mg type alloys enhanced the mechanical properties above room temperature. The best properties were found in the Al–Cu alloy.

One-dimensional solidification of supercooled melts

In this paper a one-phase supercooled Stefan problem, with a nonlinear relation between the phase change temperature and front velocity, is analysed. The model with the standard linear approximation, valid for small supercooling, is first examined asymptotically. The nonlinear case is more difficult to analyse and only two simple asymptotic results are found. Then, we apply an accurate heat balance integral method to make further progress. Finally, we compare the results found against numerical solutions. The results show that for large supercooling the linear model may be highly inaccurate and even qualitatively incorrect. Similarly as the Stefan number β→1+ the classic Neumann solution which exists down to β=1 is far from the linear and nonlinear supercooled solutions and can significantly overpredict the solidification rate.

Phase-Separated CsI-NaCl Scintillator With Optical Guiding Function

We demonstrate a new scintillator material that has a one-dimensional eutectic structure and optical light-guiding properties. Our light propagation calculations predicted that a structure where high refractive index luminescence cylinders are embedded in a matrix should show effective light-guiding properties. An inverse structure where a high refractive index luminescence matrix contains low refractive index cylinders was also predicted to possess light-guiding properties. A CsI-NaCl:Tl eutectic phase, which contained NaCl cylinders in a CsI matrix, was grown by a one-dimensional solidification method. This material showed high spatial resolution because of the difference in the refractive indices of NaCl and CsI.

Onsager approach to the one-dimensional solidification problem and its relation to the phase-field description

We give a general phenomenological description of the steady-state 1D front propagation problem in two cases: the solidification of a pure material and the isothermal solidification of two-component dilute alloys. The solidification of a pure material is controlled by the heat transport in the bulk and the interface kinetics. The isothermal solidification of two-component alloys is controlled by the diffusion in the bulk and the interface kinetics. We find that the condition of positive-definiteness of the symmetric Onsager matrix of interface kinetic coefficients still allows an arbitrary sign of the slope of the velocity-concentration line near the solidus in the alloy problem or of the velocity-temperature line in the case of solidification of a pure material. This result offers a very simple and elegant way to describe the interesting phenomenon of a possible non-single-value behavior of velocity versus concentration that has previously been discussed by different approaches. We also discuss the relation of this Onsager approach to the thin-interface limit of the phase-field description.

One-dimensional solidification of pure materials with a time periodically oscillating temperature boundary condition

A finite difference method is used to solve a one-dimensional solidification problem with a periodic boundary condition prescribed at the bottom of the mold of finite thickness. The temperature distributions in the solidified shell and mold, the position of the moving freezing front, and its velocity are evaluated. Analytical results are obtained for the limiting cases and then compared with the numerical predictions to establish the validity of the model and the numerical approach. Interactive effects of the process parameters such as Stefan number of the solidified shell material, the mold thickness, the thermal conductivity and thermal diffusivity between the shell and mold materials on the evolution of the freezing front and its velocity are investigated in detail. The results show that the solidified materials with larger Stefan number grow slower than those with relatively smaller Stefan number. The impact of oscillating mold temperature boundary on the growth of shell thickness is particularly significant at earlier stages of the process and more pronounced for smaller Stefan numbers. Increasing mold thickness or thermal conductivity ratio between the shell and mold materials slows down the evolution of the shell thickness.

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.

Explicit solutions for one-dimensional two-phase free boundary problems with either shrinkage or expansion

Abstract We consider a one-dimensional solidification of a pure substance which is initially in liquid state in a bounded interval [ 0 , l ] . Initially, the liquid is above the freezing temperature, and cooling is applied at x = 0 while the other end x = l is kept adiabatic. At the time t = 0 , the temperature of the liquid at x = 0 comes down to the freezing point and solidification begins, where x = s ( t ) is the position of the solid–liquid interface. As the liquid solidifies, it shrinks ( 0 r 1 ) or expands ( r 0 ) and appears a region between x = 0 and x = r s ( t ) , with r 1 . Temperature distributions of the solid and liquid phases and the position of the two free boundaries ( x = r s ( t ) and x = s ( t ) ) in the solidification process are studied. For three different cases, changing the condition on the free boundary x = r s ( t ) (temperature boundary condition, heat flux boundary condition and convective boundary condition) an explicit solution is obtained. Moreover, the solution of each problem is given as a function of a parameter which is the unique solution of a transcendental equation and for two of the three cases a condition on the parameter must be verified by data of the problem in order to have an instantaneous phase-change process. In all the cases, the explicit solution is given by a representation of the similarity type.

A modified truncation error-based adaptive time-step control strategy for fully- and semi-implicit simulation of isothermal solidification processes

When analyzing the transient characteristics of solidification processes, choosing appropriately-sized time steps is difficult. Accordingly, the current study develops a modified local time truncation error (LTE)-based strategy designed to adaptively adjust the size of the time step during the simulated solidification procedure in such a way that the time steps can be adapted in accordance with the local variations in latent heat released during phase change or the effects of pure conduction in a single solid or liquid phase. In the approach presented in this work, the LTE-based time-step evaluation procedure is applied not only after a convergent temperature field is obtained at each time step, but also during the nonlinear iterations performed at each time step whenever a convergence problem is encountered. The computational accuracy and efficiency of the proposed method are demonstrated via the simulation of the one-dimensional and two-dimensional solidification problems and compared with those of other adaptive time step and the uniform time step methods. Furthermore, the performance of these approaches has also been demonstrated using fully-implicit and semi-implicit schemes.

Numerical Analysis of Coalescence Characteristics of Low Melting Point Alloy Fillers Using a Non-Equilibrium Phase Field Model

A non-equilibrium phase field model (NPFM) is proposed to examine the three-dimensional coalescence characteristics of low melting point alloy (LMPA) fillers, widely used for self-organized interconnection applications. This model sufficiently considers the non-equilibrium process during phase transitions, resulting in accurate prediction of interfaces between two different phases with a high density ratio. The preliminary simulation is carried out for validation of the present model and numerical predictions are in good agreement with analytical solutions for a one-dimensional solidification problem. The proposed NPFM successfully predicts the phase transition, the heat transfer from solid to liquid, and the coalescence behaviors of LMPA fillers, whereas the conventional enthalpy method fails to describe the non-equilibrium phase transition. Moreover, the results show that there exists grid dependency in the mush zone, suggesting that special care should be taken of the mush zone for better prediction of interfaces between different phases.

Approximate solution methods for one-dimensional solidification from an incoming fluid

This paper concerns a one-dimensional model for solidification due to incoming supercooled liquid impacting on a substrate that is maintained at a fixed temperature. Using a boundary immobilisation method, and assuming that both the solid and liquid layers remain thin throughout the process, a second-order accurate perturbation expansion is determined. An alternative approximate solution, found using the heat balance integral method, is also described to analyse the problem, and the liquid height and temperatures in the solid and liquid are subsequently found for both approximate solutions. These are then compared with a numerical scheme which solves the full Stefan problem. The perturbation solution is shown to be more accurate, but the HBI method is simpler to implement and avoids complications that arise in the ordering of terms in the perturbation expansion as the difference between the substrate and melting temperature changes.

Structure and internal dynamics of a side chain liquid crystalline polymer in various phases by molecular dynamics simulations: A step towards coarse graining

Side chain liquid crystalline polymer with relatively long spacer was modeled on a semiatomistic level and studied in different liquid crystalline phases with the aid of molecular dynamics simulations. Well equilibrated isotropic, polydomain smectic and monodomain smectic phases were studied for their structural and dynamic properties. Particular emphasis was given to the analysis on a coarse-grained level, where backbones, side chains, and mesogens were considered in terms of their equivalent ellipsoids. The authors found that the liquid crystalline phase had a minor influence on the metrics of these objects but affected essentially their translational and orientational order. In the monodomain smectic phase, mesogens, backbones, and side chains are confined spatially. Their diffusion and shape dynamics are frozen along the mesogen director (the one-dimensional solidification) and the reorientation times increase by one to one-and-half orders of magnitude. In this phase, besides obvious orientational order of mesogens and side chains, a stable detectable order of the backbones was also observed. The backbone director is confined in the plane perpendicular to the mesogen director and constantly changes its orientation within this plane. The backbone diffusion in these planes is of the same range as in the polydomain smectic phase at the same temperature. A detailed analysis of the process of field-induced growth of the smectic phase was performed. The study revealed properties of liquid crystalline polymers that may enable their future fully coarse-grained modeling.

Modeling the Effect of Finite-Rate Hydrogen Diffusion on Porosity Formation in Aluminum Alloys

A volume-averaged model for finite-rate diffusion of hydrogen in the melt is developed to predict pore formation during the solidification of aluminum alloys. The calculation of the micro-/macro-scale gas species transport in the melt is coupled with a model for the feeding flow and pressure field. The rate of pore growth is shown to be proportional to the local level of gas supersaturation in the melt, as well as various microstructural parameters. Parametric studies of one-dimensional solidification under an imposed temperature gradient and cooling rate illustrate that the model captures important phenomena observed in porosity formation in aluminum alloys. The transition from gas to shrinkage dominated porosity and the effects of different solubilities of hydrogen in the eutectic solid, capillary pressures at pore nucleation, and pore number densities are investigated in detail. Comparisons between predicted porosity percentages and previous experimental measurements show good correspondence, although some uncertainties remain regarding the extent of impingement of solid on the pores.

Modeling of Porosity Formation in Aluminium Alloys

A new approach based on microsegregation of gas dissolved in the melt is used to model pore formation during the solidification of aluminum alloys. The model predicts the amount and size of the porosity in a solidified casting. Computation of the micro-/macro-scale gas species transport in the melt is coupled with the simulation of the feeding flow and calculation of the pressure field. The rate of pore growth is calculated based on the local level of gas supersaturation in the melt. The effect of the microstructure on pore formation is also taken into account. Parametric studies for one-dimensional solidification under an imposed temperature gradient and cooling rate illustrate that the model captures important phenomena observed in porosity formation in aluminum alloys. Comparisons between predicted porosity percentages and previous experimental measurements show good correspondence.