Sharp-interface simulation of dendritic growth with convection: benchmarks

We present and validate a numerical technique for computing dendritic growth of crystals from pure melts in the presence of forced convection. The Navier-Stokes equations are solved on a fixed Cartesian mesh and a mixed Eulerian-Lagrangian framework is used to treat the immersed phase boundary as a sharp solid-fluid interface. A conservative finite-volume discretization is employed which allows the boundary conditions to be applied exactly at the moving surface. Results are presented for a range of the growth parameters, namely crystalline anisotropy, flow Reynolds number and Prandtl number. Direct comparisons are made between the present results and those obtained with phase-field methods and excellent agreement is obtained. Values for the tip characteristics are tabulated to serve as benchmarks for computations of two-dimensional dendritic growth with convection.     

Numerical solutions for mixed controlled solidification of phase change materials

Kinetics of solidification of phase change materials (PCM) was analyzed for combined heat conduction in the PCM and container wall, and convection in the cold fluid. Stable and convergent numerical methods were derived after transformation to normalized size scale, and corresponding immobilization of the moving boundary. The accuracy was confirmed by comparing numerical solutions and corresponding analytical solutions for control by heat conduction in solidifying layers. The proposed method was used to assess when solidification is controlled solely by conduction in solidified layer, and to analyze relative roles of conduction through the container and convection in the cold fluid.     

Numerical simulation of filling and solidification of permanent mold castings

Filling of a mold is an essential part of the permanent mold casting process and affects significantly the heat transfer and solidification of the melt. For this reason, accurate prediction of the temperature field in permanent mold castings can be achieved only by including simulation of filling in the analysis. In this work we model filling and solidification of a casting of an automotive piston produced from an aluminum alloy. Filling of the three-dimensional mold is modeled by using the volume-of-fluid method. Fluid mechanics and heat transfer equations are solved by a finite element method. Comparisons of numerical results to available experimental data show that the formulated model provides a solution of acceptable accuracy despite some uncertainty in material properties and boundary and initial conditions. This implies that the model can be a viable tool to design permanent molds.     

A remedy for the temperature recovering method of solidification simulation

One of the methods of phase change simulation is the “temperature recovering method.” It has two main difficulties in practical application. The first one is the explicit nature of the method. The second one is the slow convergence of the solidification ratio. In this study, a method has been proposed to improve these difficulties. The method consists of two procedures. First, the solidification range is clustered into a discrete variable. A solidification ratio is sorted within a cluster as an integer variable. Second, the source term related to the change of the latent heat is reformulated into an implicit form by the “numerical linearization method” as previously proposed by the author. The benchmark test cases show that: (1) The convergence is faster, even for large latent heat cases, than the existing method. (2) The stability is independent of the time increment. © 2000 Scripta Technica, Heat Trans Asian Res, 29(5): 400–411, 2000     

Microscopic study on the solidification of aqueous solutions using laser interferometry

Solidification of an aqueous solution was studied using a laser interferometry technique combined with an optical microscope. In order to measure the concentration distribution in an aqueous NaCl solution near an ice crystal and the three-dimensional shape of the ice crystal in a known temperature field, a directional solidification stage was used. This was composed of low- and high-temperature blocks and a moving bed. In our experiment, ice crystals showed four shapes: flat, treelike, swordlike, and needlelike. As a result of the interferometry experiment, it was observed that the concentration increases in the bulk solution for thick samples, swordlike ice crystals have an asymmetric triangular cross section, and treelike crystals have a flat and low top. We also studied ice formation response to injection of a high-concentration solution. © 1998 Scripta Technica, Heat Trans Jpn Res, 27(5): 353–364, 1998     

Analytical and numerical solutions of radially symmetric inward solidification problems in spherical geometry

A short-time analytical solution is constructed by using a new technique which assumes fictitious initial temperatures in some fictitious extensions of the actual regions. Later, this short-time solution is compared with the numerical solution obtained by the finite difference scheme in which the space grid points change with the freezing front position. Even a small error in the initial values of the solid temperature and freezing front position, which are required for starting the numerical scheme, can, for a short time, give rise to considerable error in the freezing front position. However, the analytical and numerical solutions were found to be in close agreement if the numerical scheme is started with the analytical values of the solid temperature and freezing front.     

Numerical simulation of air solidification process in liquid hydrogen with LBM-CA coupled method

The solidification process of solid-air is a significant risk in liquid hydrogen system. To predict the solidification of air in liquid hydrogen, the coupled model of Lattice Boltzmann Method (LBM) and Cellular Automation (CA) has been proposed. Results shows that the local oxygen mass ratio can reach up to 30% at 0.1s when the air-proportioned nitrogen and oxygen mixture is solidified. The LBM-CA method is further used to study the effects of forced convection on the solid-air dendrite. It is shown that the upstream latent heat and the rejected solutes accumulate in the downstream with the flow field, leading to higher oxygen concentration in the downstream as the flow velocity increases. In addition, the solid-phase fraction of solid-air increases faster with the increasing flow velocity which indicates an increasing safety risk for the liquid hydrogen with solid-air system.     

Effect of micro mass transfer through phase interface on numerical simulation of solidification process

Existing models for the solute redistribution during solidification have been reviewed. Some typical models are applied for the numerical simulation of heat and mass transfer with phase change under experimental condition of inverse casting. The results show that the effect of micro mass transfer models on the formation of the new phase cannot be omitted. © 2004 Wiley Periodicals, Inc. Heat Trans Asian Res, 33(6): 393–401, 2004; Published online in Wiley InterScience ( www.interscience.wiley.com ). DOI 10.1002/htj.20024     

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.     

Growth of multi-crystalline silicon ingot by improved directional solidification process based on numerical simulation

A multi-crystalline silicon (Si) ingot was simulated and grown using the improved directional solidification (DS) process. Numerical simulations were performed with two different cooling paths and two different coolant flow rates in order to demonstrate the thermal characteristics in the improved DS furnace during the crystal growth. The temperature distributions in the furnace and locally (at the silicon ingot) were predicted as a function of time. From these result, a multi-crystalline Si ingot weighing 300 kg was grown within 40 h using the improved DS process. The Si ingot had a grain size that was larger than 5 mm, and the structure of the ingot was in the form of vertical columns. From the analysis results, the Si ingot exhibited a resistivity below 2 Ω cm and a life time above 3 μs.     

Approximate analytical solutions for the solidification of PCMs in fin geometries using effective thermophysical properties

Heat transfer enhancement of phase change materials (PCMs) is essential in order to achieve high charge and discharge powers of latent heat storage systems. The utilisation of highly conductive fins is an effective method to improve heat transfer. In the presented paper, solidification times of two fundamental geometries are examined by analytical modelling and numerical computation. These geometries are the finned plane wall and a single tube which is radial-finned on the outside. The paper describes approximate analytical solutions based on the effective medium approach which include the boundary conditions, as well as material and geometric parameters.     

A three-dimensional simulation of coupled turbulent flow and macroscopic solidification heat transfer for continuous slab casters

A numerical investigation was conducted for exploring the steady state transport phenomena of turbulent flow, heat transfer and macroscopic solidification in a continuous stainless steel slab caster. The numerical model is based on a generalized transport equation applicable to all the three regions, namely liquid, mushy and solid, which exist in a slab caster. The turbulence effects on the transport equations were taken into account using a low-Reynolds number k- turbulence model. The solidification of molten steel was modeled through the implementation of the popular enthalpy-porosity technique. A control volume based finite-difference scheme was used to solve the modeled equations on a staggered grid arrangement. A series of simulations was carried out to investigate the effects of the casting speed, the delivered superheat and the immersion depth of the twin-ported submerged entry nozzle (SEN) on the velocity and temperature distributions and on the extent of the solidified and mushy regions on the narrow and broad faces of the caster. In the absence of any known experimental data related to velocity profiles in a slab caster, the numerical predictions of the solidified profile on a caster's narrow face were compared with limited experimental data and a good agreement was found.     

Sharp interface numerical simulation of directional solidification of binary alloy in the presence of a ceramic particle

A sharp interface technique is employed to study the interaction of a solid–liquid interface in a solidifying binary alloy with a ceramic particle in the melt. The application targeted is solidification of a metal–matrix composite. A level-set based sharp interface numerical method is used to study the directional solidification process in the presence of the particle. The transport of solute and heat are computed. The directional solidification calculations are first validated against stability theory. The Mullins–Sekerka stability spectrum is reproduced with good agreement with the theory. The interaction of the cellular interface with a ceramic particle in the melt is then computed. It is shown that, in contrast to the case of a pure material, the ratio of thermal conductivity of the particle to the melt plays no role in determining the front morphology and the result of the particle–front interaction. The diffusion of species controls the evolution of the phase front around the particle. The implications of the results for particle–front interactions in a binary alloy are discussed.     

Irreversibility of solidification and of a cyclic solidification and melting process

The specific irreversibility of solidification and of cyclic solidification–melting process was considered on a general level. The variables affecting the irreversibility most are the Biot number, the dimensionless interfacial speed during recalescense and the extent of supercooling. The process path involving supercooling can become less irreversible than the path where phase transition completely happens in an equilibrium state if a thermal conductivity ratio between solid and liquid phases is sufficiently below unity. A finite maximum and a non-zero minimum value of irreversibility can be detected. These extreme values are independent of the geometrical shape of the substance.     

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.     

Crystallisation of undercooled aqueous solutions: Experimental study of free dendritic growth in cylindrical geometry

This paper reports a study of crystallisation kinetics in small volumes of undercooled water–MPG (monopropylene glycol) mixture. The experimental cell is a vertical cylinder (height 5 mm, diameter 2r_e = 7.5 mm); its bottom section is closed by a Plexiglas disc that transmits light from the lower part of the cylinder to a high-speed digital camera. Photographic recordings allow the determination of the crystal growth rate. When the antifreeze mass fraction is below 25 wt%, crystallisation is clearly divided into two stages: the growth of dendritic crystals in the undercooled solution followed by the passage of the interdendritic solidification front. Dendrite growth induces a sudden temperature increase in the mixture, while the passage of the interdendritic solidification front determines the time at which sensible heat effects again predominate. The results show that the dendrite growth rate is an increasing function of the degree of undercooling and a decreasing function of the antifreeze mass fraction.     

Thermal energy storage of phase change materials in the solidification process inside the quasi-square heat exchanger: CFD simulation

The high latent heat of phase change materials (PCMs) makes them one of the most important sources of heat energy storage systems. However, due to the slow rate of heat transfer in these materials, using conductive materials such as fins and nanoparticles could improve the thermal efficiency of these energy storage systems. So in this article, cross-shaped fins and Copper(II) oxide nanoparticles with different synthesized forms and various volume fractions have been employed to increase the thermal efficiency of paraffin PCMs. In this simulation, three fin models based on the installed size, the shape of the synthesized nanoparticles in brick, cylindrical, and platelet forms, and the nanoparticle volume fraction of the Copper(II) oxide is 1%–4% are studied. Increasing the volumetric ratio of nanoparticle and shape coefficient decrease the time of solidification, while increasing the length of the cross-shaped fins raises the solidification rate and improves heat transfer. Finally, it was found that when the inner and outer walls play a role in the solidification process at the same time, the solidification rate will increase by more than 66% as more zone of the surface is exposed to cold.