Numerical investigations of drop solidification on a cold plate in the presence of volume change

We present a front-tracking/finite difference method for simulation of drop solidification on a cold plate. The problem includes temporal evolution of three interfaces, i.e. solid–liquid, solid–gas, and liquid–gas, that are explicitly tracked under the assumption of axisymmetry. Method validation is carried out by comparing computational results with exact solutions for a two-dimensional Stefan problem, and with related experiments. We then use the method to investigate a drop solidifying on a cold plate in which there exists volume change due to density difference between the solid and liquid phases. Numerical results show that the shape of the solidified drop is profoundly different from the initial liquid one due to the effects of volume change and the tri-junction in terms of growth angles ϕ_gr on the solidification process. A decrease in the density ratio of solid to liquid ρ_sl or an increase in the growth angle results in an increase in the height of the solidified drop. The solidification process is also affected by the Stefan number St, the Bond number Bo, the Prandtl number Pr, the Weber number We, the ratios of the thermal properties of the solid to liquid phases k_sl and C_psl. Increasing St, Bo, Pr, We, or k_sl decreases the solidified drop height and the time to complete solidification. Moreover, the solidification growth rate is strongly affected by St, k_sl and C_psl. An increase in any of these parameters hastens the growth rate of the solidification interface. Contrarily, increasing ρ_sl decreases the growth rate. However, other parameters such as ϕ_gr, Bo, Pr and We have minor effects on the solidification growth rate.     

Numerical Approaches for microscopic modelling of solute redistribution during solidification of binary alloys

In the present study, two numerical approaches for single-domain modelling of microsegregation during solidification of binary alloys are presented. In the first approach, the concentration jump at the moving solid/liquid interface is formulated using a volumetric term and a Boolean function. The governing solute redistribution equation, valid for the whole domain comprising the solid and liquid regions, is derived in terms of the liquid phase composition. The effects of microstructure coarsening on microsegregation has been described and included in the model. In the second approach, the continuum mixture theory is utilized to derive a single domain solute redistribution equation in terms of the mixture composition. The solidification front motion and dendrite arm coarsening effects are accommodated by considering the representative elementary volume to consist of solid, interdendritic, and extradendritic liquid phases. Numerical solutions have been obtained using a control-volume based finite-difference method with a fixed grid. Good agreement has been observed between the predictions of the present fixed-domain models and the exact analytical and experimental results.     

Numerical study of solidification of a nano-enhanced phase change material (NEPCM) in a thermal storage system

The effects of nanoparticle dispersion on solidification of a Cu-n-hexadecane nanofluid inside a vertical enclosure are investigated numerically for different temperatures of the left vertical wall. An enthalpy porosity technique is used to trace the solid-liquid interface. The resulting nanoparticle-enhanced phase change materials (NEPCMs) exhibit enhanced thermal conductivity in comparison to the base material. The effect of the wall temperature and nanoparticle volume fraction are studied in terms of the solid fraction and the shape of the solid-liquid phase front. It has been found that a lower wall temperature and a higher nanoparticle volume fraction result in a larger solid fraction. The increase in the heat release rate of the NEPCM shows its great potential for diverse thermal energy storage applications.     

Numerical solution of solidification in a square prism using an algebraic grid generation technique

The solidification of an infinitely long square prism was analyzed numerically. A front fixing technique along with an algebraic grid generation scheme was used, where the finite difference form of the energy equation is solved for the temperature distribution in the solid phase and the solid–liquid interface energy balance is integrated for the new position of the moving solidification front. Results are given for the moving solidification boundary with a circular phase change interface. An algebraic grid generation scheme was developed for two-dimensional domains, which generates grid points separated by equal distances in the physical domain. The current scheme also allows the implementation of a finer grid structure at desired locations in the domain. The method is based on fitting a constant arc length mesh in the two computational directions in the physical domain. The resulting simultaneous, nonlinear algebraic equations for the grid locations are solved using the Newton-Raphson method for a system of equations. The approach is used in a two-dimensional solidification problem, in which the liquid phase is initially at the melting temperature, solved by using a front-fixing approach. The difference of the current study lies in the fact that front fixing is applied to problems, where the solid–liquid interface is curved such that the position of the interface, when expressed in terms of one of the coordinates is a double valued function. This requires a coordinate transformation in both coordinate directions to transform the complex physical solidification domain to a Cartesian, square computational domain. Due to the motion of the solid–liquid interface in time, the computational grid structure is regenerated at every time step.     

Numerical study of solidification of a drop with a growth angle difference

This study presents the direct numerical results of a drop solidifying on a plate, in which the difference between the growth angles is considered. The drop is two-dimensional with the presence of the left and right triple points, and the method used is a front-tracking technique. The growth angles at the right (ϕ_gr1) and left (ϕ_gr2) triple points are not equal, i.e. Δϕ_gr = ϕ_gr1–ϕ_gr2 ≠ 0°. Unlike the identical growth angles, the growth angle difference results in an asymmetric drop after complete solidification. In the presence of the solid-to-liquid density ratio ρ_sl < 1.0 (i.e. volume expansion), the tip of the solidified drop shifts more to the right as Δϕ_gr increases in the range of 0°–12°. In addition, the angle at the solidified drop top (i.e. tip angle) increases with Δϕ_gr. We also pay attention to the effects of some other parameters (such as the wetting angle ϕ₀, the growth angle ϕ_gr1 and ρ_sl) on the solidification process with the growth angle difference. The results reveal that the growth angle varied in the range of 6°–24° has a minor effect on the movement of the tip to the right while the tip shift increases with an increase in ϕ₀ in the range of 60°–130° or with a decrease in ρ_sl in the range of 0.8–1.1. The tip angle increases with an increase in ρ_sl or with a decrease in ϕ_gr1 or ϕ₀. We also investigate the solidification process under the influence of the Bond number.     

Numerical and experimental study of the traveling magnetic field effect on the horizontal solidification in a rectangular cavity part 2: Acting forces ratio and solidification parameters

In recent decades, many phase change processes in metals have been optimized using traveling magnetic fields due to a better understanding of their electromagnetic impact in such applications. In this paper, numerical and experimental study of the effect of traveling magnetic field on the solidification process was evaluated. A three-dimensional numerical model based on the multi-domain method was used to analyze the process of gallium horizontal solidification under the electromagnetic impact in a laboratory-size rectangular cavity. A linear inductor creating traveling magnetic field was designed and built for appropriate measurements and validation the calculations. The analysis was focused on the influence of the ratio between the applied electromagnetic forces and natural convective forces on the solidification front location and shape and on the velocity field. Since the overall electromagnetic force impact on the melt reduced during the solidification, when the melt area was converting into a solid, a new approach to control the solidification parameters was analyzed. In this approach, the value of electromagnetic force acting on the remaining melt during the process was maintained. The main result is the development and improvement of an effective tool for the analysis of direct solidification parameters.The experimental setup included an ultrasonic Doppler velocimeter (UDV) for noninvasive measurements of the velocities in the liquid part of the metal and the liquid-solid interface position, its profile and displacement. All important characteristics of the process were measured, and the results of computations agreed well enough with experimentally obtained data.     

Numerical modelling of a melting-solidification cycle of a phase-change material with complete or partial melting

A high accuracy numerical model is used to simulate an alternate melting and solidification cycle of a phase change material (PCM). We use a second order (in time and space) finite-element method with mesh adaptivity to solve a single-domain model based on the Navier-Stokes-Boussinesq equations. An enthalpy method is applied to the energy equation. A Carman-Kozeny type penalty term is introduced in the momentum equation to bring the velocity to zero inside the solid region. The mesh is dynamically adapted at each time step to accurately capture the interface between solid and liquid phases, the boundary-layer structure at the walls and the multi-cellular unsteady convection in the liquid. We consider the basic configuration of a differentially heated square cavity filled with an octadecane paraffin and use experimental and numerical results from the literature to validate our numerical system. The first study case considers the complete melting of the PCM (liquid fraction of 95%), followed by a complete solidification. For the second case, the solidification is triggered after a partial melting (liquid fraction of 50%). Both cases are analysed in detail by providing temporal evolution of the solid-liquid interface, liquid fraction, Nusselt number and accumulated heat input. Different regimes are identified during the melting-solidification process and explained using scaling correlation analysis. Practical consequences of these two operating modes are finally discussed.     

Inward solidification of a superheated liquid in a cooled horizontal tube

Inward solidification has been studied experimentally and analytically under conditions where the liquid phase is above the fusion temperature (i.e., superheated). The liquid was housed in a horizontal circular tube in which the surface was maintained at a uniform, time-invariant temperature during test runs. Three phase change materials (n-heptadecane,n-octadecane, and water) were used in the tests. Both analysis and experiments have established that for inward solidification, natural convection in a superheated liquid is not important in controlling the solidliquid interface motion for Stefan numbers less than unity. The interface velocity is determined primarily by the thermal resistance across the solid layer. Good agreement has been obtained between experimentally measured and analytically predicted solid-liquid interface positions when the density differences between the phases were accounted for.     

Effect of magnetic field on 3D flow and heat transfer during solidification from a melt

We present the effect of a magnetic field on three-dimensional fluid flow and heat transfer during solidification from a melt in a cubic enclosure. The walls of the enclosure are considered perfectly electrically conducting and the magnetic field is applied separately in three directions. The finite-volume method with enthalpy formulation is used to solve the mathematical model in the solid and liquid phases. The results obtained by our computer code are compared with the numerical and experimental data found in the literature. For Gr = 5 × 10⁵ and Ha = 0, 25, 50, 75, and 100 (where Gr and Ha are the Grashof and Hartmann numbers, respectively), the effects of magnetic field on flow and thermal fields, and on solid/liquid interface shape are presented and discussed. The interface is localized with and without magnetic field. The results show a strong dependence between the interface shape and the intensity and orientation of magnetic field. When the magnetic field is applied along the X-direction, the magnetic stability diagrams (V_max–Ha) and (Nu_avg–Ha) show the strongest stabilization of the flow field and heat transfer.     

Eutectic solidification patterns: Interest of microgravity environment

The solidification of binary eutectic alloys produces two-phase composite materials in which the microstructure, that is, the geometrical distribution of the two solid phases, results from complex pattern-formation processes at the moving solid–liquid interface. Since the volume fraction of the two solids depends on the local composition, solidification dynamics can be strongly influenced by thermosolutal convection in the liquid. In this contribution, we review our experimental and numerical work devoted to the understanding of eutectic solidification under purely diffusive conditions, which will soon be tested and extended during the microgravity experiment TRANSPARENT ALLOYS planned by the European Space Agency (ESA).     

Numerical Investigation on Marginal Stability and Convection with and without Magnetic Field in a Mushy Layer

In this article, an investigation is conducted to analyze the marginal stability with and without magnetic field in a mushy layer. During alloy solidification, such mushy layer, which is adjacent to the solidification front and composed of solid dendrites and liquid, is known to produce vertical chimneys. Here, we carry out numerical investigation for particular range of parameter values, which cover those of available experimental studies, to determine the convective flow at the onset of motion. The governing coupled non-linear partial differential equations are non-dimensionalised and solved to get the steady basic-state solution. The thickness of the mushy layer is determined as a part of the solution. Using multiple shooting technique, we determine the steady-state solutions in a range of critical Rayleigh number. We analyse the effect of several parameters, Chandrasekhar number Q, and Robert’s number τ on the problem. It was found that an increase in Q has a stabilizing effect on solidification and the critical Rayleigh number increases on increasing Q. It was also found that for moderate or small values of Robert’s number τ the critical Rayleigh number is mostly insensitive.     

Investigation of double-diffusive convection during the solidification of a binary mixture (NH4Cl–H2O) in a trapezoidal cavity

The solidification of binary mixture (NH₄Cl–H₂O) inside a trapezoidal cavity is investigated experimentally in this study. The effect of the initial concentration of ammonium chloride (0–19.8%) and boundary temperatures (−30 to 0°C) on the solidification process was investigated. Particle image velocimetry (PIV) technique was used for the visualization of the dynamic field in the melt. Thirty-two thermocouples were used to monitor the temperature distribution inside the cavity and on the cooling walls. The convective flow field, the temperature distribution, the frozen layer thickness and the moving solid/liquid interface were obtained for different initial concentrations of ammonium chloride and various boundary temperatures. The results obtained in the course of this study reveal that: (1) the process of solidification is slower with an increase in initial concentration levels of the binary solution: as the concentration increases, the time needed to get the same thickness of frozen layer increases; (2) an increase in the initial concentration of ammonium chloride solution reduce significantly the temperature in the melt; and (3) the initial concentration play a significant role in the evolution of convection flow patterns.     

A fixed domain model for microsegregation during solidification of binary alloys

A fixed-domain numerical model for microsegregation during alloy solidification is developed. The phenomena of solute partitioning at the moving solid/liquid interface and subsequent redistribution by diffusion in the solid and liquid phases have been formulated using volumetric terms. A solute balance equation valid for the whole domain comprising the solid and liquid phases has been obtained in terms of the liquid concentration. The effects of microstructure coarsening on microsegregation has been described and included in the present model. Numerical experiments and comparisons have been carried out between the present fixed-domain model, previous deforming-domain models, and the exact analytical solutions available in the literature. Good agreement has been observed between the predictions of the present fixed-domain model and the exact analytical solutions. Further extensions of the present model for the analysis of two-dimensional microsegregation have been also reported.     

On the transformation toughening of a crack along an interface between a shape memory alloy and an isotropic medium

In this study, a bilinear cohesive zone model is employed to describe the transformation toughening behavior of a slowly propagating crack along an interface between a shape memory alloy and a linear elastic or elasto-plastic isotropic material. Small scale transformation zones and plane strain conditions are assumed. The crack growth is numerically simulated within a finite element scheme and its transformation toughening is obtained by means of resistance curves. It is found that the choice of the cohesive strength t₀ and the stress intensity factor phase angle φ greatly influence the toughening behavior of the bimaterial. The presented methodology is generalized for the case of an interface crack between a fiber reinforced shape memory alloy composite and a linear elastic, isotropic material. The effect of the cohesive strength t₀, as well as the fiber volume fraction are examined.     

Sensitivity of mesh spacing on simulating macrosegregation during directional solidification of a superalloy

The sensitivity of mesh spacing on simulations of macrosegregation, particularly ‘freckles’, during vertical directional solidification of a superalloy in a rectangular mold was systematically analyzed to achieve accurate predictions in finite element calculations. It was observed that a coarser mesh spacing in the x‐direction horizontal tends to minimize the simulated macrosegregation, whereas a coarser mesh spacing in the y‐direction vertical artificially tends to make the system appear to have more macrosegregation. When solidification conditions either lead to a well‐established freckling case or to a well‐established non‐freckling case, the simulated results are not sensitive to the mesh spacing provided the elements are no larger than about 2d₁ by 2 D/V and 3d₁ by 4 D/V respectively, where d₁ is the primary dendrite arm spacing, D is the diffusivity of the alloy solute with the smallest diffusivity in the liquid, and V is the growth rate. However, when solidification conditions are very close to the transition between freckling and no freckling, the simulated results are sensitive to the mesh spacing, especially in the y‐direction. Based on the mesh sensitivity analysis from the two‐dimensional simulations of rectangular castings of René N5, the mesh with element dimensions no larger than 2d₁ in the x‐direction and 1.5 D/V in the y‐direction are recommended as the most stringent element size. Copyright © 2001 John Wiley & Sons, Ltd.     

3D macrosegregation simulation with anisotropic remeshing

The article presents a three-dimensional coupled numerical solution of momentum, mass, energy and solute conservation equations, for binary alloy solidification. The interdendritic flow in the mushy zone is assumed to obey the Darcy's law. Microsegregation is governed by the lever rule, assuming local equilibrium at phase interfaces. The resulting energy and solute advection–diffusion equations are solved using the Streamline-Upwind/Petrov–Galerkin (SUPG) finite element method. A SUPG-PSPG velocity-pressure formulation is applied for the momentum equation. The full algorithm was implemented in the 3D code THERCAST, together with an anisotropic remeshing method. Two applications have been considered: a small ingot of Pb-48wt%Sn alloy and a large steel ingot. The numerical results of these two cases are presented with the evolution of temperature, liquid velocity, and solute concentration fields during solidification. To cite this article: S. Gouttebroze et al., C. R. Mecanique 335 (2007).     

On the dispersion of a solute in oscillating flow of a non-Newtonian fluid in a channel

The paper presents an exact analysis of the dispersion of an immiscible solute in a non-Newtonian fluid (known as an incompressible second-order fluid which shows viscoelastic behaviour) flowing slowly in a parallel plate channel in the presence of a periodic pressure gradient. Using a generalized dispersion model which is valid for all times after the solute injection, the diffusion coefficients K _i (τ)(i=1,2,3,…) are obtained as functions of time τ in the case when the initial solute distribution is in the form of a slug of finite extent. The analysis leads to the novel result that K ₂(τ) (which is a measure of the longitudinal dispersion coefficient of the solute) has a steady part S in addition to a fluctuating part D ₂(τ) due to the pulsatility of the flow. It is found that S decreases with increase in the viscoelastic parameter M for given values of the amplitude λ and frequency ω of the pressure pulsation. On the other hand, it is found that at a fixed instant τ, the amplitude of D ₂(τ) increases with increase in M for given values of λ and ω. Further it is shown that at a given instant τ, the amplitude of D ₂(τ) decreases with increase in ω for given λ and M and the profile for D ₂(τ) becomes progressively flatter with increase in ω. Finally the axial distribution of the average concentration θ _m of the solute over the channel cross-section is determined at different instants after the solute injection for several values of M, λ and ω. The present study is likely to have important bearing on the problem of dispersion of tracers in blood flow through arteries.     

Asymptotic models of solidification in cooling thin-layer flows of a highly viscous fluid

Asymptotic models are constructed for the solidification process in a highly viscous film flow on the surface of a cone with a given mass supply at the cone apex. In the thin-layer approximation, the problem is reduced to two parabolic equations for the temperatures of the liquid and the solid coupled with an ordinary differential equation for the solidification front. For large Péclet numbers, an analytical steady-state solution for the solidification front is found. A nondimensional parameter which makes it possible to distinguish flows (i) without a solid crust, (ii) with a steady-state solid crust, and (iii) with complete solidification is determined. For finite Péclet numbers and large Stefan numbers, an analytical transient solution is found and the time of complete flow solidification is determined. In the general case, when all the governing parameters are of the order of unity, the original system of equations is studied numerically. The solutions obtained are qualitatively compared with the data of field observations for lava flows produced by extrusive volcanic eruptions.     

Series solutions for steady three-dimensional stagnation point flow of a nanofluid past a circular cylinder with sinusoidal radius variation

This article deals with the study of the steady three-dimensional stagnation point flow of a nanofluid past a circular cylinder that has a sinusoidal radius variation. By means of similarity transformation, the governing partial differential equations are reduced into highly non-linear ordinary differential equations. The resulting non-linear system has been solved analytically using an efficient technique namely homotopy analysis method (HAM). Expressions for velocity and temperature fields are developed in series form. In this study, three different types of nanoparticles are considered, namely alumina (Al₂O₃), titania (TiO₂), and copper (Cu) with water as the base fluid. For alumina–water nanofluid, graphical results are presented to describe the influence of the nanoparticle volume fraction φ and the ratio of the gradient of velocities c on the velocity and temperature fields. Moreover, the features of the flow and heat transfer characteristics are analyzed and discussed for foregoing nanofluids. It is found that the skin friction coefficient and the heat transfer rate at the surface are highest for copper–water nanofluid compared to the alumina–water and titania–water nanofluids.