Parametric model for the analytical determination of the solidification and cooling times of semi-crystalline polymers

This work presents an efficient method to quickly calculate with good accuracy (to 5%) the solidification time of an injected semi-crystalline polymer slab. Under some hypotheses this polymer can be considered as a phase change material with a constant phase change temperature. We use a noteworthy property established as the ratio between the thickness of a solidifying phase change finite medium and the solidified thickness in a semi-infinite medium. The knowledge of this ratio enables to predict analytically the solidification time in a 1D finite medium. This ratio can be parameterized as a function of characteristic numbers in phase change problems: Stefan numbers and the ratio of thermal diffusivities of both phases. The results are compared with those given by a complete model integrating the physics of the coupling between heat transfer and crystallization kinetics. The solidification times computed from both models are very close, demonstrating the relevance of the simplified model. Finally, we also get a very good accuracy in calculating the total cooling time, from injection to ejection.     

Melt cavitation at its volumetric crystallization

It is shown that at volumetric crystallization of the undercooled melt the high negative pressures are generated in this melt; they are caused by matter shrinkage at solidification, what leads to intensive cavitation of uncrystallized melt. The mechanisms are studied and the kinetic model of this process is presented here. Considerable dependence of cavitation intensity on the cooling rate is shown. Numerical solutions to the problem are found at the example of cavitation of crystallizing metal melts. The sizes of crystalline grains and formed cavitation inclusions in the solidified material are determined.     

快速冷却条件下凝固潜热处理模型的研究

深入分析了有限差分法模拟凝固过程时，采用等效比热法处理相变潜热的原理；从理论和实际计算角度分析了快速冷却条件下常规的等效比热法会造成热量“虚增”，模拟计算结果失真的原因和影响程度；并进一步建立了快速冷却条件下凝固潜热处理的模型。计算结果表明，该模型可以充分保证模拟计算的精度和效率。     

The mathematical model of solidification latent heat under high cooling rate

Treatment of solidification latent heat is a key point in solidification simulation by the finite difference method. When latent heat is dealt with in a traditional method of equivalent latent heat, it was found that heat was increased when casting with a high cooling rate, and then the simulation result was distorted. In this paper, a new method is proposed to deal with solidification latent heat. Moreover, a mathematical model was suggested, in which the latent heat can be dealt with accurately under high or normal cooling rates. By contrasting the simulation results from this new method with the traditional one, it was indicated that this new model can obtain more accurate simulation results than the traditional model under high or normal cooling rates. © 2006 Wiley Periodicals, Inc. Heat Trans Asian Res, 35(2): 115–121, 2006; Published online in Wiley InterScience ( www.interscience.wiley.com ). DOI 10.1002/htj.20104     

Mechanics of transport phenomena in multi-component sessile drops with solidification

An experimental study is performed to determine the mechanics of transport phenomena in multicomponent sessile drops with internal solidification. These drops are cooled at the center of the drop base or over the enter base of the drop. Both the interfacial and internal flow structures are examined by means of laser shadowgraphy, while the microstructures are investigated using a microscope-video system. Three different degrees of cooling rate are imposed on the drops: low, medium and high cooling. It is disclosed that the center cooling at a low rate produces a radial flow induced by component separation but without solidification and Marangoni type (surface-tension induced) flow. In contrast, a high cooling rate results in Marangoni convection accompanied by solidification but no component separation. The effect of Prandtl number on the Marangoni flow velocity is determined. The interfacial disturbance is suppressed by center cooling but not by bulk cooling which induces the formation of Bernard cells. The impact of flow patterns on the mechanical property of melt solidification is discussed.     

Multi-scale modeling and simulation of crystallization during cooling in short fiber reinforced composites

To explore the crystallization during the cooling stage in short fiber reinforced composites, a multi-scale model in which macroscopic temperature and microscopic crystal morphology are related with each other is built up. According to this multi-scale model, the algorithm of coupling finite volume method with pixel coloring method is proposed. The finite volume method is used on the coarse grid to calculate the macroscopic temperature and the pixel coloring method is employed on the fine grid to capture the morphology of crystallization. Roles of cooling rate, initial temperature as well as the nucleation rate per unit fiber surface are investigated. To our knowledge this is the first time that a fully coupled multi-scale model and a multi-scale algorithm have been applied to the crystallization of short fiber reinforced composites.     

Effects of mould filling on evolution of the solid–liquid interface during solidification

This work presents a numerical analysis of simultaneous mould filling and phase change for solidification in a two-dimensional rectangular cavity. The role of residual flow strength and temperature gradients within the solidifying domain, caused by the filling process, on the evolution of solidification interface are investigated. An implicit volume of fluid (VOF)-based algorithm has been employed for simulating the free surface flows during the filling process, while the model for solidification is based on a fixed-grid enthalpy-based control volume approach. Solidification modeling is coupled with VOF through User Defined Functions developed in the commercial computational fluid dynamics (CFD) code FLUENT 6.3.26. Comparison between results of the conventional analysis without filling effect and those of the present analysis shows that the residual flow resulting from the filling process significantly influences the progress of the solidification interface. A parametric study is also performed with variables such as cooling rate, filling velocity and filling configuration, in order to investigate the coupled effects of the buoyancy-driven flow and the residual flow on the solidification behavior.     

Solidification of PCM around a curved tube

This paper presents the results of an experimental and numerical investigation on the solidification of PCM around a curved cold tube to determine the effects of the Dean number, cooling fluid flow rate and its temperature on the interface velocity, the time for complete solidification and the solidified mass. To formulate the solidification process around a curved tube a conduction model was used together with the immobilization technique and the Landau transform. The energy equation and the associated boundary conditions were discretized by the finite control volumes method. The computational program was optimized by numerical experiments and the optimized form was used to validate the model. Comparisons of the numerical predictions and experiments to investigate the effects the Dean number on the solidified mass showed agreement within 1% while the interface velocity and the time for complete solidification showed agreements of about 8% and less than 6%, respectively. The effects of the flow rate of the working fluid could be predicted within less than 8% for the solidified mass and to less than 4% for both the interface velocity and the time for complete solidification. The effects of the temperature of the working fluid are predictable to within less than 8% for the time for complete solidification and the interface velocity.     

Dynamic discharging characteristics simulation on solar heat storage system with spherical capsules using paraffin as heat storage material

The dynamic characteristics of solar heat storage system with spherical capsules packed bed during discharging process are studied. According to the energy balance of solar heat storage system, the dynamic discharging processes model of packed bed with spherical capsules is presented. Paraffin is taken as phase change material (PCM) and water is used as heat transfer fluid (HTF). The temperatures of PCM and HTF, solid fraction and heat released rate are simulated. The effects of inlet temperature of HTF, flow rate of HTF and porosity of packed bed on the time for discharging and heat released rate are also discussed. The following conclusion can be drawn: (1) the heat released rate is very high and decreases rapidly with time during the liquid cooling stage, it is stable at the solidification cooling stage, then it decreases to zero at the solid cooling stage. (2) The time for complete solidification decreases when the HTF flow rate increases, but the effect is not so obvious when the HTF flow rate is higher than 13 kg/min; (3) compared to the HTF inlet temperature and flow rate, the influence of porosity of packed bed on the time for complete solidification is not so significant.     

Analytical solution for solidification of close-celled metal foams

This paper introduces an analytical model capable of predicting the location of solidification front as well as the full solidification time for heterogeneous materials such as close-celled metallic foams. Full numerical simulations with the method of finite difference are separately conducted to validate the analytical model. The model predicts that an increase in porosity causes significant retardation of full solidification as a result of decreased effective thermal conductivity and diffusivity of the porous medium. Effects of pore shape and cooling temperature on overall solidification behavior were also studied.     

Thermodynamic optimization for crystallization process of gas hydrate

In this paper, thermodynamic optimization is applied to analyze the crystallization process of the gas hydrate related to the gas hydrate cool storage system. Thermodynamic optimization model of the gas hydrate crystallization process is established. By taking the entropy generation minimization as the optimization objective, both the optimal control strategy and the optimal cooling rate of the gas hydrate crystallization process are determined. The minimum entropy generation corresponding to the optimal cooling rate decreases by 7.8% compared with normal situation. The results presented in this paper can provide important guidelines for optimal design and operation of the gas hydrate crystallization process. Copyright © 2010 John Wiley & Sons, Ltd.     

Modeling of transport phenomena in hybrid laser-MIG keyhole welding

Mathematical models and associated numerical techniques have been developed to investigate the complicated transport phenomena in spot hybrid laser-MIG keyhole welding. A continuum formulation is used to handle solid phase, liquid phase, and the mushy zone during the melting and solidification processes. The volume of fluid (VOF) method is employed to handle free surfaces, and the enthalpy method is used for latent heat. Dynamics of weld pool fluid flow, energy transfer in keyhole plasma and weld pool, and interactions between droplets and weld pool are calculated as a function of time. The effect of droplet size on mixing and solidification is investigated. It is found that weld pool dynamics, cooling rate, and final weld bead geometry are strongly affected by the impingement process of the droplets in hybrid laser-MIG welding. Also, compositional homogeneity of the weld pool is determined by the competition between the rate of mixing and the rate of solidification.     

Design of hydrogen permeable Nb–Ni–Ti alloys by correlating the microstructures,solidification paths and hydrogen permeability

The compositional window in Nb–Ni–Ti alloys leading to the crystallization of primary α-Nb phase and the eutectic (α-Nb + NiTi) phase is of high technical relevance due to its favorable properties with respect to hydrogen permeation. The solidification behavior of Nb–Ni–Ti alloys in the primary α-Nb phase region is investigated to reveal the potential solidification paths. The study is based on the characterization of as-cast microstructures in combination with numerical calculations of solidification paths using the CALPHAD method coupled with a microsegregation model. Four different kinds of solidification paths depending on initial composition and cooling rate are found. Correspondingly, a new compositional window appropriate for hydrogen permeation is established in the primary α-Nb phase region. The variation of the hydrogen permeability of the alloys in this window is surprisingly high. High Nb content and Ni/Ti ratio lead to a high permeability. Nb₅₅Ni₂₀Ti₂₅ shows the highest permeability at 673 K, particularly 2.9 × 10⁻⁸ mol H₂ m⁻¹ s⁻¹ Pa^−1/2. This is about 1.8 times higher than that of pure Pd.     

A 3D computational model of heat transfer coupled to phase change in multilayer materials with random thermal contact resistance

An improved understanding of the heat transfer in materials consisting of two layers (splat and substrate) is essential for many industrial applications. We are interested in the deposition, rapid cooling and solidification of metal droplets (known as splats) brought into contact with a cold substrate. We therefore need to understand the temperature history in both the splat and the substrate, including phase change phenomena. A new model of the thermal contact resistance based on a random distribution of contact points, rather than the uniform distribution commonly used, is presented in this paper.Phase change has been also considered, using the enthalpy-porosity formulation. Simulations have been conducted with the commercial package CFX-4. The computational results for the cooling rate of the splat obtained using the random contact distribution model are in good agreement with available experimental results. In addition, results obtained from the random model provide information on the inhomogeneity affecting the temperature at the interface between the splat and the substrate.     

Influence of volumetric-flow rate in the crystallizer on the gas-hydrate cool-storage process in a new gas-hydrate cool-storage system

Experimental results of the gas-hydrate R141b cool-storage process are used to study the performance of a new type gas-hydrate cool-storage system. The relations among the cooling rate of the cool-storage medium, the degree of subcooling of crystallization, the formation rate of gas-hydrate, the cold energy stored and the volumetric flow rate in the crystallizer are provided. The experimental results indicate that the cool storage effect of the system is better when the volumetric flow rate in the crystallizer is between 150 and 450 l/h, the corresponding cooling rate of the cool-storage medium is between 0.055 and 0.105 °C/min, the degree of subcooling of crystallization is between 1.97 and 3.50 °C, the formation rate of the gas-hydrate is between 0.30 and 0.4 g/s, and the cold energy stored is between 2.9 and 4.0 MJ. The results presented provide guidance for the actual efficient operation of a new type of gas-hydrate cool-storage system.     

Numerical Analysis of Phase Change Material Characteristics Used in a Thermal Energy Storage Device

In this study, a numerical analysis is performed to investigate the freezing process of phase change materials (PCM) in a predesigned thermal energy storage (TES) device. This TES device is integrated with a milk storage cooling cycle operating under predefined practical conditions. Using this cooling unit, 100 litres of milk is kept cool at 4°C for 48 hours before it is collected. A 2-D model of the TES device is developed in COMSOL Multiphysics to analyze the phase change performance of water-based PCMs. The variations of thermal properties with temperature during the phase change are considered in the analysis. The model is used for exploring the solidification process of PCMs inside the TES device. Temperature variations with time, ice formation, and the impacts of boundary conditions are investigated in detail. Water PCM shows better characteristics in the solidification process in comparison to eutectic PCMs, which is mainly due to the differences between phase change temperatures of the PCMs.     

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.     

Modeling on transient microstructure evolution of solid-air solidification process under continuous cooling in liquid hydrogen

The safety hazard brought by the oxygen-rich solid formed by the solidification of air in liquid hydrogen cannot be ignored. A numerical model describing the evolution of dendrite during solidification of nitrogen-oxygen binary solutions in liquid hydrogen is developed. By introducing the reduction factor, the growth anisotropy of the six-fold symmetric dendrite simulated by the Cartesian grid is effectively reduced. The reliability of the model is verified by comparing the tip growth rate of dendrites with six-fold symmetry with analytical models. On this basis, the microstructure evolution and solute segregation of solid-air dendrites are investigated. The results show that with the increase of the cooling rate, the dendrite growth rate accelerates while aggravating the solute segregation, and the outer edge of the dendrite is more likely to reach the oxygen-rich state. The improvement in solute diffusibility enables dendrites to reach an oxygen-rich state at larger sizes, but also accelerates dendrite growth. The initial composition has little effect on the microstructure evolution and growth rate of the solid-air dendrite. However, in the presence of forced convection, the solid-air dendrite morphology loses its symmetry and makes the upstream dendrite arms reach an oxygen-rich state at a smaller size. This study helps to understand the air solidification process in liquid hydrogen and provides theoretical guidance for the safety research of liquid hydrogen system.     

Characteristics of Thermal Stress Evolution During the Cooling Stage of Multicrystalline Silicon

Directional solidification is one of the most popular techniques for the massive production of multicrystalline silicon (mc-Si) for solar cell application due to its well-balanced high conversion efficiency and low production cost. The grown crystal suffers from several types of defects that significantly degrade the photovoltaic performance of solar cells, among which dislocation is the most critical caused by thermal stress. To examine the characteristics of thermal stress and associated fields, therefore, it is significant for the understanding and optimization of the cooling process. In the article, an integrated simulation tool has been developed and used to investigate heat transfer, fluid flow, and thermal stress during the cooling process of mc-silicon. The simulation results were further proved by experimental observations. According to the distortion energy theory, the total strain energy consists of the volumetric strain energy and the shear strain energy, and yield occurs when the shear component exceeds that at the yield point, which is the major cause of the dislocation. Therefore, by analyzing von Mises stress aligned in the direction that has to support the maximum shear load, the regions in the ingot with dislocation generation and multiplication can be evaluated. The displacement indicates the motion of the crystal ingot, and reveals the regions of deformation due to the existence of thermal stress from uneven cooling. Based on the complete investigation of the characteristics of thermal stress and associated displacement, the cooling process could be well comprehended and further optimized with the minimization of dislocation density.     

Numerical analysis of rapid solidification in a single roller process

The rapid solidification in a single roller process has been used to make amorphous ribbons. Because this process occurs rapidly, it is difficult to obtain useful experimental data. Therefore, a numerical analysis has been performed on the rapid solidification in a single roller cooling process. The VOF (volume of fluid) method was adopted as the numerical method used to simulate transient two-dimensional thermal and fluid flow with a liquid-solid phase change and free surfaces. We simulated the behavior of an aluminum alloy. The geometry of the amorphous ribbon, flow and temperature fields, temperature history of alloy particle whose initial location is at the center of the nozzle, and the cooling rate were obtained using as parameters the roll velocity, the nozzle slot breadth, and the gap between nozzle and roller. © 1999 Scripta Technica, Heat Trans Asian Res, 28(1): 34–49, 1999