Effects of Solutal Undercooling on Three-Dimensional Double-Diffusive Convection and Macrosegregation during Solidification of a Binary Alloy

ABSTRACT

A three-dimensional transient numerical model is developed for simulation of double-diffusive convection during binary alloy solidification processes, taking into account nonequilibrium effects due to solutal undercooling. Such an effect arising from microscopic convection near the diffusion boundary layer adjacent to the mushy region is captured by devising a macroscopic model based on a fixed-grid, enthalpy-based, control-volume approach. Microscopic features pertaining to solutal undercooling are incorporated through a modification of the partition coefficient by means of a number of macroscopically observable parameters. Numerical simulations are performed for solidification of a metallic alloy system kept in a side-cooled cubic enclosure. Typical curvatures of the streamlines and their nonequidistant characteristics, as projected on various cross-sectional planes, show an element of three-dimensionality in the double-diffusive convection (originating from the solidification process itself) and its interaction with the progressing solidification front. The three-dimensional transport leads to a global macrosegregation, with significant composition variations across the longitudinal planes, as dictated by the modified partition coefficient and thermosolutal convection mechanisms.     

A FULL 3-DIMENSIONAL ADAPTIVE FINITE VOLUME SCHEME FOR TRANSPORT AND PHASE-CHANGE PROCESSES,PART I: FORMULATION AND VALIDATION

A full three-dimensional (3-D) numerical formulation for accurate simulation of transport and phase-change processes is presented. These processes are characterized by a variety of flow and heat transfer mechanisms in irregular domains with or without the movement of phase-change interfaces and free surfaces. A generalized 3-D nonorthogonal curvilinear finite volume formulation is developed in conjunction with a robust mesh generation scheme known as multizone adaptive grid generation (MAGG) to tackle such problems. The coupling between the interfacial dynamics and transport phenomena in the bulk of the phases is inherent in this formulation. A 3-D k-epsilon model is also incorporated to tackle the turbulent flows in these applications. The unified numerical model is validated against classical 3-D problems such as turbulent natural convection in a differentially heated cube, solidification in a cavity, and so on. In a companion paper, Part II (see this issue), application of this formulation to 3-D simulation of hydrothermal crystal growth and low and high pressure Czochralski (Cz) crystal growth is presented.     

TEMPERATURE-TRANSFORMING MODEL FOR BINARY SOLID-LIQUID PHASE-CHANGE PROBLEMS PART II: NUMERICAL SIMULATION

Abstract

The model and numerical scheme developed in Part I were first verified with upward freezing experiments of an NH₄Cl-H₂O solution on a cold isothermal surface. Then, two-dimensional convection problems with different buoyancy terms in binary solid-liquid phase-change systems were studied. Finally, the model was used to simulate the solidification of an aqueous ammonium chloride solution in a rectangular cavity. The comparison of the results obtained from the present studies with the experimental and numerical results from the literature revealed a good agreement.     

Numerical simulation model by volume averaging for the dissolution process of GaSb into InSb in a sandwich system

ABSTRACT

The volume-averaging continuum technique has been utilized to obtain numerical predictions for the transport phenomena occurring during the dissolution process of GaSb into InSb melt in a sandwich system. Dissolution and subsequent growth in this system are achieved by the application of a temperature gradient. The developed model was first verified for two test cases [(i) fluid/solid conjugate heat transfer and (ii) the solidification process of the binary system]. The code was then utilized to simulate the dissolution process of GaSb into InSb in the GaSb/InSb/GaSb sandwich system. The present results show that the developed volume-averaging model provides accurate predictions.     

Pore-scale simulation of coupled multiple physicochemical thermal processes in micro reactor for hydrogen production using lattice Boltzmann method

A general numerical scheme based on the lattice Boltzmann method (LBM) is established to investigate coupled multiple physicochemical thermal processes at the pore-scale, in which several sets of distribution functions are introduced to simulate fluid flow, mass transport, heat transfer and chemical reaction. Interactions among these processes are also considered. The scheme is then employed to study the reactive transport in a posted micro reactor. Specially, ammonia (NH₃) decomposition, which can generate hydrogen (H₂) for fuel of proton exchange membrane fuel cells (PEMFCs), is considered where the endothermic decomposition reaction takes place at the surface of posts covered with catalysts. Simulation results show that pore-scale phenomena are well captured and the coupled processes are clearly predicted. Effects of several operating and geometrical conditions including NH₃ flow rate, operating temperature, post size, post insert position, post orientation, post arrangement and post orientation on the coupled physicochemical thermal processes are assessed in terms of NH₃ conversion, temperature uniformity, H₂ flow rate and subsequent current density generated in PEMFC.     

Studies on turbulent momentum, heat and species transport during binary alloy solidification in a top-cooled rectangular cavity

A two-dimensional transient fixed-grid enthalpy-based numerical method is developed to analyze the effects of turbulent transport during a binary alloy solidification process. Turbulence effects are introduced through standard k-ε equations, where coefficients are appropriately modified to account for phase-change. Microscopically-consistent estimates are made regarding temperature-solute coupling in a non-equilibrium solidification situation. The model is tested against laboratory experiments performed using an NH₄Cl-H₂O system in a rectangular cavity cooled and solidified from the top. Particular emphasis is laid on studying the interaction between Rayleigh-Benard type convection and directional solidification in the presence of turbulent transport. Numerical predictions are subsequently compared with experimental results regarding flow patterns, interface growth and evolution of the temperature field, and the agreement is found to be good.     

A hybrid discontinuous spectral element method and filtered mass density function solver for turbulent reacting flows

Abstract

We present a novel hybrid scheme for the large eddy simulation (LES) of turbulent reacting flows. The scheme couples the discontinuous spectral element method (DSEM) solver for the unsteady compressible Navier-Stokes equations with a Monte Carlo particle filtered mass density function (FMDF) solver for the transport of reacting species. The method is capable of high-order simulations on unstructured grids. Mean particle estimate construction mimics the DSEM numerical procedure and utilizes variable basis functions. The scheme is tested on non-reacting and reacting Taylor-Green vortex flows. Studies of varying polynomial order, different basis functions for constructing particle estimates, and varying particle quantities are conducted. We demonstrate that a tent kernel, in conjunction with high polynomial order, produces the most accurate results. The chemically reacting simulations validate the hybrid scheme and demonstrate its applicability across a range of reaction regimes. The hybrid scheme's computational cost is 2.1 times the DSEM-LES solver.     

THERMAL STRESSES IN THE EARLY STAGE OF SOLIDIFICATION OF STEEL

Abstract

A numerical model for calculation of thermal stresses and strains during solidification of steel is presented. Creep deformations are included. Stresses and strains during the early stage of solidification are studied for different steels under different cooling conditions. The results obtained may explain some observed cracking in continuously cast steel.     

GDQ Method for Natural Convection in a Cubic Cavity Using Velocity–Vorticity Formulation

ABSTRACT

Natural convection in a differentially heated cubic enclosure is studied by solving the velocity–vorticity form of the Navier–Stokes equations by a generalized differential quadrature (GDQ) method. The governing equations in the form of velocity Poisson equations, vorticity transport equations, and energy equation are solved using a coupled numerical scheme via a single global matrix for velocities, vorticities, and temperature. Vorticity and velocity coupling at the solid boundaries is enforced through a higher-order approximation by the GDQ method, thus assuring accurate satisfaction of the continuity equation. Nusselt numbers computed for Ra = 10³, 10⁴, 10⁵, and 10⁶ show good agreement with the benchmark results. A mesh independence study indicates that the present numerical procedure requires much coarse mesh compared to other numerical schemes to produce the benchmark solutions of the flow and heat transfer problems.     

Call for contributions to a numerical benchmark problem for 2D columnar solidification of binary alloys

This call describes a numerical comparison exercise for the simulation of ingot solidification of binary metallic alloys. Two main steps are proposed, which may be treated independently: 1. The simulation of the full solidification process. First a specified ‘minimal’ solidification model is used and the contributors are provided with the corresponding sets of equations. The objective is to verify the agreement of the numerical solutions obtained by different contributors. Then different physical solidification models may be compared to check the features that allow for the best possible prediction of the physical phenomena. 2. A separate preliminary exercise is also proposed to the contributors, only concerned with the convective problem in the absence of solidification, in conditions close to those met in solidification processes. Two problems are considered for the case of laminar natural convection: transient thermal convection for a pure liquid metal with a Prandtl number on the order of 10^?2, and double-diffusive convection in an enclosure for a liquid binary metallic mixture with a Prandtl number on the order of 10^?2 and a Lewis number on the order of 10⁴.     

A Study on the Impact and Solidification of Liquid Metal Droplets during Thermal Spray Deposition onto a Substrate with Concentric Grooves or Ridges

In this study, a numerical investigation has been performed on the spreading and solidification of a coating material droplet onto the rigid substrate in the thermal spray process. The computational model is validated through the comparison of the predicted numerical result and the experimental data for flat substrate. An analysis of the deposition formation on a substrate with small concentric grooves or ridges was performed. To examine the characteristic of the impact and solidification of a liquid droplet on the substrate with concentric grooves or ridges, a parametric study was conducted with various shapes and sizes of concentric grooves or ridges.     

Fluid Dynamics and Transfer Processes in Bended Microchannels

Abstract

The understanding of the flow processes in microchannels and micromixers is essential for the design of microfluidic devices like microreactors or analytical equipment. We have performed a systematic numerical CFD-study of mixing and mass transfer in sharp 90° bends and heat transfer in T-joints to obtain a detailed insight into the flow patterns and corresponding transfer processes in a wide range of Reynolds numbers. With increasing flow velocity, the straight laminar flow starts to form symmetrical vortices in the bend, at the entrance of the mixing channel, and in T-joints. The vortices enhance the transport processes like heat and mass transfer in the channels significantly. The influence of the geometry and the flow conditions is shown by an analytical estimation of the relevant forces. The appearance of convective transport processes is used for the definition of microflows, which are controlled by viscous forces and diffusive transfer processes.     

Modeling of Solidification Process in a Rotary Electromagnetic Stirrer

A macroscopic model of the solidification process in a rotary electromagnetic stirrer is presented. The fluid flow, heat, and mass transfer inside a rotary stirrer are modeled using, 3-D swirl flow equations in which turbulent flow is simulated using a k ? ? model. A hybrid model is used to represent the mushy zone, which is considered to be divided into two regions: a coherent region and a noncoherent region. Each region is represented by a separate set of governing equations. An explicit time-stepping scheme is used for solving the coupled temperature and concentration fields, while an implicit scheme is used for solving equations of motion. The coupling relations also include eutectic solidification, which is an important feature in modeling solidification with electromagnetic stirring, especially in the context of the formation of semi-solid slurry. The results from the present numerical solution agree well with those corresponding to experiments reported in literature.     

A block-centered finite-difference method for the time-fractional diffusion equation on nonuniform grids

ABSTRACT

In this article, a block-centered finite-difference scheme is introduced to solve the time-fractional diffusion equation with a Caputo derivative of order α ∈ (0, 1) on nonuniform grids. The resulting scheme is second-order-accurate in space and (2 ? α)-order-accurate in time, and the unconditional stability and convergence are proved theoretically. Moreover, numerical solutions of the unknown variable along with its first derivatives are obtained. Finally, numerical experiments, including boundary-layer and high-gradient problems, are carried out to support our theoretical analysis and indicate the efficiency of this method.     

Heat, mass and momentum transport behaviors in directionally solidifying blade-like castings in different electromagnetic fields described using a continuum model

A previous continuum model proposed and recently modified by the authors for describing the heat, mass and momentum transport phenomena in dendrite solidification process of alloy castings was further extended to the solidification cases in an arbitrary electromagnetic (EM)-fields. The extended continuum model and a FEM/FDM joint solution technique were successfully applied to the numerical simulations of directional solidification transport processes in blade-like castings of Pseudo-binary In718 base-4.85 wt.%Nb and Al-4.5 wt.%Cu alloys under a static or harmonic EM-field of different strengths/frequencies. The computational results demonstrate the availability of the present continuum modeling to treat an EM-STP problem, and also reveal that the volume-contraction-driven liquid feeding flow is much more difficult to be suppressed than the buoyancy-induced by means of applying a static magnetic field.     

RAPID TRANSIENT HEAT CONDUCTION IN MULTILAYER MATERIALS WITH PULSED HEATING BOUNDARY

ABSTRACT

Rapid transient heat conduction in multilayer materials under pulsed heating is solved numerically based on a hyperbolic heat conduction equation and taking into consideration the non-Fourier heat conduction effects. An implicit difference scheme is presented and a stability analysis conducted, which shows that the implicit scheme for the hyperbolic equation is stable. The code is validated by comparing the numerical results with an existing exact solution, and the physically unrealistic conditions placed on the time and space increments are identified. Using the validated model, the numerical solution of thermal wave propagation in multilayer materials is presented. By analyzing the results, the necessary conditions for observing non-Fourier phenomena in the laboratory can be inferred. The results are also compared with the numerical results from the parabolic heat conduction equation. The difference between them is clearly apparent, and this comparison provides new insight for the management of thermal issues in high-energy equipment. The results also illustrate the time scale required for metal films to establish equilibrium in energy transport, which makes it possible to determine a priori the time response and the measurement accuracy of metal film, thermal-resistant thermometers.     

The Impact of Thermal Contact Conductance on the Spreading and Solidification of a Droplet on a Substrate

The interfacial thermal contact conductance between an impinging molten droplet and a cold substrate plays an important role in the droplet spreading and solidification. In this paper, a simple correlation for the thermal contact conductance during a rapid contact solidification process was obtained. By introducing this correlation into the numerical model, a non-constant thermal contact conductance that varies with time and position was adopted for the first time to simulate the spreading and solidification of a molten droplet on a substrate. It was found that the droplet spreading and final bump shape are sensitive to the thermal contact conductance. Experiments were also performed to observe the final bump shape of the droplet. Qualitative agreement between the numerical and the experimental results justified the present method. Because the thermal contact conductance is not required to be prescribed, the present method is applicable to different operation conditions.     

Development of a mass-preserving level set redistancing algorithm for simulation of rising bubble

Abstract

This article is aimed to simulate the gas-liquid flow of rising bubbles with a mass-preserving level set method. To resolve the topological changes of gas-liquid interface where the classic finite difference scheme may yield oscillation solutions, the spatial terms in the level set advection equation will be approximated by an optimized compact reconstruction weighted essentially non-oscillatory (OCRWENO) scheme. This scheme achieves high-order accuracy in smooth regions, and meanwhile avoid numerical oscillation near discontinuities. Two benchmark problems including vortex flow and deforming field are chosen to compare the present simulation with previous numerical researches. Several rising bubble problems are validated by the proposed level set method.     

Thermal Energy Storage Behavior of a Paraffin during Melting and Solidification

Abstract

In this study, experiments are conducted to investigate charging and discharging characteristics of a paraffin as a phase change material (PCM). A vertical tube-in-shell geometry is designed to store the PCM. The thermophysical properties of the paraffin examined are determined through the differential scanning calorimeter (DSC) analysis. A series of experiments are carried out to investigate the effect of increasing the inlet temperature and the mass flow rate of the heat transfer fluid (HTF) both on the charging and discharging processes (i.e., melting and solidification) of the PCM.