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.     

A LAGRANGIAN-EULERIAN FINITE-ELEMENT FORMULATION FOR STEADY-STATE SOLIDIFICATION PROBLEMS

Abstract

An accurate finite-element methodology is developed to solve a steady-state solidification problem for pure materials. Generally, this type of problem is governed by a set of conduction-advection differential equations. We consider a solidifying body moving at a constant velocity. At a steady state, the solid/liquid interface is fixed to an observer in a Eulerian frame, and the movement of the interface to an observer in a Lagrangian frame is determined by an energy balance equation at the interface. To determine the interface position in the Eulerian frame, we use the composition of the steady-state velocity of the moving body and the solid/liquid interface velocity in the Lagrangian frame through an iterative process. Meanwhile, the finite-element mesh is updated with a transfinite mapping scheme. In this new methodology, a weak formulation is applied to the interface energy balance equation to calculate the interface velocity in the Lagrangian frame. Numerical results are compared with analytical solutions for one-dimensional steady-state solidification problems, and excellent agreement is achieved. Several two-dimensional examples are provided to demonstrate the capability of the new methodology.     

A Fast Numerical Simulation for Modeling Simultaneous Metal Flow and Solidification in Thin Cavities Using the Lubrication Approximation

A numerical algorithm for modelling steady flow of liquid metal accompanied by solidification in a thin cavity is presented. The problem is closely related to a die cast process and in particular to the metal flow phenomenon observed in thin ventilation channels. Using the fact that the rate of metal flow in the channel is much higher than the rate of solidification, a numerical algorithm is developed by treating the metal flow as steady in a given time-step while treating the heat transfer in the thickness direction as transient. The flow in the thin cavity is treated as two dimensional after integrating the momentum and continuity equations over the thickness of the channel, while the heat transfer is modelled as a one-dimensional phenomenon in the thickness direction. The presence of a moving solid-liquid interface introduces non-linearity in the resulting set of equations, and which are solved iteratively. The location and shape of the solid-liquid interface are found as a part of the solution. The staggered grid arrangement is used to discretize the flow governing equations and the resulting set of partial differential equations is solved using the SIMPLE algorithm. The thickness direction heat-transfer problem accompanied by phase change is solved using a control volume formulation. The results are compared with the predictions of the commercial software FLOW3D® which solves the full three-dimensional set of flow and heat transfer equations accompanied with solidification. The Reynolds's lubrication equations accompanied by the through-the-thickness heat loss and solidification model can be successfully implemented to analyze flow and solidification of liquid metals in thin channel during the die cast process. The results were obtained with significant savings in CPU time.     

Exact solution of the problem of gas segregation in the process of crystallization

In the present work, we theoretically investigate the problem of gas segregation in the process of crystallization. The self-similar solutions of the problem for a flat and spherical solidification fronts moving with the velocity inversely proportional to the square root of time are found. The criteria of the elimination of gassing due to segregation are proposed. Using the Laplace transformation method, an analytical solution of the problem for a flat solidification front moving with a constant velocity is also obtained.     

Study of double-diffusive velocity during the solidification process using particle image velocimetry

This paper reports the velocity distribution of double-diffusive convection of a binary mixture in a rectangular enclosure during the solidification process. The mixture was a NH₄Cl–H₂O solution. The advanced technique, particle image velocimetry (PIV), was used to measure the velocity distribution in the liquid region during solidification. For the purpose of comparison, the solidification of pure water was studied with the same technique. The temperature of the cooling walls in the test chamber and the temperature of the test solution during solidification were also measured. The double-diffusive flow was found to be stronger at the beginning of solidification; the flow decays as solidification proceeds. The velocity distribution of the hypereutectic solution of NH₄Cl–H₂O has evident difference in comparison with hypoeutectic solution.     

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.     

BOUNDARY ELEMENT SOLUTIONS FOR FREE BOUNDARY CONVECTION-DIFFUSION PROBLEMS

The boundary element technique is used to solve the steady state convection-diffusion problem with constant velocity in a two-dimensional domain with a free interface. These problems arise in a number of important heat transfer applications involving melting or solidification, such as bulk crystal growth in Bridgman furnaces. The boundary element approach reduces the dimension of the problem, thereby improving the computational efficiency, and is particularly well suited to free-surface problems in which the position and shape of the solid-liquid interface are of primary importance. Results are presented for a case study problem representing solidification in a two-dimensional, rectangular configuration.     

An integrated model for interaction between melt flow and non-equilibrium solidification in thermal spraying

In this paper, a micro/macro-integrated model based on the VOF scheme is presented that accounts for free surface movement, thermal contact resistance, and fluid instability. A sub-model is developed to include the non-equilibrium solidification phenomena at the solid/liquid interface. The melt flow is incorporated into the microscopic model through prescribing a velocity profile that is obtained from the interpolation of melt velocities on the macroscopic grids near the interface. To improve the efficiency of the integration between the melt flow and the microscopic model, a relational database is developed and applied to the integrated micro/macro model. Three velocity profiles, e.g., linear, parabolic, and cubic velocity profiles, are considered and the results are compared with those obtained from the diffusion model.     

In-situ measurements of concentration and temperature during transient solidification of aqueous solution of ammonium chloride using laser interferometry

The variation in temperature and concentration plays a crucial role in predicting the final microstructure during solidification of a binary alloy. Most of the experimental techniques used to measure concentration and temperature are intrusive in nature and affect the flow field. In this paper, the main focus is laid on in-situ, non-intrusive, transient measurement of concentration and temperature during the solidification of a binary mixture of aqueous ammonium chloride solution (a metal-analog system) in a top cooled cavity using laser based Mach–Zehnder Interferometric technique. It was found from the interferogram, that the angular deviation of fringe pattern and the total number of fringes exhibit significant sensitivity to refractive index and hence are functions of the local temperature and concentration of the NH₄Cl solution inside the cavity. Using the fringe characteristics, calibration curves were established for the range of temperature and concentration levels expected during the solidification process. In the actual solidification experiment, two hypoeutectic solutions (5% and 15% NH₄Cl) were chosen. The calibration curves were used to determine the temperature and concentration of the solution inside the cavity during solidification of 5% and 15% NH₄Cl solution at different instants of time. The measurement was carried out at a fixed point in the cavity, and the concentration variation with time was recorded as the solid–liquid interface approached the measurement point. The measurement exhibited distinct zones of concentration distribution caused by solute rejection and Rayleigh Benard convection. Further studies involving flow visualization with laser scattering confirmed the Rayleigh Benard convection. Computational modeling was also performed, which corroborated the experimental findings.     

A comparison of hyperbolic and parabolic models of phase change of a pure metal

The solidification history of individual thermal spray particles has been the subject of many experimental and theoretical studies. Yet it is customary to assume that solidification occurs at the equilibrium temperature, and that heat propagates according to Fourier’s Law. To account for a finite thermal diffusion speed, a hyperbolic heat conduction equation is usually adopted to analyze heat transfer. However, under certain circumstances, this equation can violate the second law of thermodynamics, and so others have modified the original hyperbolic equation via theories of extended irreversible thermodynamics. In this work, we study non-equilibrium effects of rapid solidification of a pure metal particle, and compare the so-called parabolic, hyperbolic and modified hyperbolic equations for heat transfer, to predict the interface undercooling due to thermal effects and velocity as a function of time, for different relaxation times. Results indicate that differences are limited to the early part of the solidification process, when undercooling is most significant, the interface velocity is highest, and non-equilibrium effects are most evident. As solidification progresses, the non-equilibrium effects wane and solidification can then be properly modeled as an equilibrium process.     

A fixed-grid method for chemical etching

This paper presents a fixed-grid method for simulating chemical etching. This method is analogous to the enthalpy method used in the modeling of melting/solidification process. The total concentration of the etchant consists of two components: the unreacted etchant concentration and the reacted etchant concentration. The unreacted etchant concentration is solved in both the solution and the substrate (with zero unreacted etchant concentration). The reacted etchant concentration is used to capture the moving etch front of the solution-substrate interface. As the etch front is computed directly, there is no necessity to compute mesh velocity as for a moving grid approach, which changes a diffusion problem into a convective-diffusion problem. With the proposed approach, a diffusion problem remains a simple diffusion problem. In addition, by using a fixed grid, in contrast with a moving grid, Cartesian grid can be used to capture complicated etchfront in multi-dimensional problems. For illustration of the procedure, the proposed procedure is used to predict the etched front and etchant concentration in a one-dimensional diffusion-controlled etching problem.     

PIV study on the development of double-diffusive convection during the solidification effected by lateral cooling for a super-eutectic binary solution

The solidification of a binary solution occurs in a variety of industrial applications. The fundamental mechanism of the development of “double-diffusive convection” during solidification was one of the main subjects of study in the past. This study focused on the effect of the initial concentration of a super-eutectic aqueous ammonium chloride solution on the development of double-diffusive convection during the solidification process. Particle image velocimetry (PIV) technique was employed to measure the flow velocity and observe the morphological conditions associated with the solidification effected by cooling from the side wall. The transient temperature distribution within a test cell was also measured by using the designed experimental system. PIV measurement revealed that the flow structure was composed of a major circulation flow within the test cell for the binary solution with low-concentration and multiple layers of circulation flows developed in sequence in the melt of the test cell for the high-concentration binary solution. Test-cell images captured by CCD camera indicated the presence of more A-segregates within the mushy zone as the initial concentration of the binary solution increased. For the low-concentration binary solution, under circumstances of without or with very weak double-diffusive convection, transient temperature distribution presented thermal stratification within the test cell. However, strong double-diffusive convection resulted in intersection on temperature curves for the high-concentration binary solution.     

The study on polymer melt front,gas front and solid layer in filling stage of gas-assisted injection molding

The algorithms are developed to predict the polymer melt front, gas front and solid layer in gas-assisted injection molding. The simulation of two-dimensional, transient, non-isothermal and high viscous flow between two parallel plates with the generalized Newtonian fluid is presented in detail. During solidification while an injection mold fills, a solid-liquid interface moves and a two-phase zone exists; an enthalpy model is used to predict this interface in the two-phase flow problem. The model takes into account the three-phase flow including the effects of the gas front, solid layer and polymer melt front.     

Prediction of the phase change front using the optimization technique

A new method for tracking the solid-liquid interface is proposed. Experiments on solidification of a flowing fluid in a vertical tube were carried out to obtain the wall temperature of the tube as a function of time. Using the experimental data for the tube wall temperature, the governing equation, initial and boundary conditions for the solidified layer are transformed into an equivalent optimization problem. This problem is then discretized using the space-time finite element so that the continuously moving solid-liquid interface can be easily incorporated. This discretized problem is solved by Fibonacci search method to find the position of the transient solid-liquid interface. Solidified-layer thickness versus time curves are obtained from the present theory and they are compared with the curves obtained from the steady-state formula for solidification.     

Performance indicators for solar pipes with phase change storage

Performance indicators for a solar pipe system in which solar radiation is stored as latent heat of a phase changing material are proposed. These performance indicators are aimed at serving as a yardstick by which such multivariables systems are evaluated. The indicators are the melt and solidification times obtained for standardized systems and conditions. These indicator enable the comparison between the suitability of systems with different materials and configurations to store and release thermal energy. The indicators are obtained from numerical solutions of the nonlinear heat conduction problem of the axisymmetric liquid-solid interface motion within the solar pipe. Longitudinal dimension required for the determination of the solidification time is added by an axial superposition of axisymmetric sections. This simplified approach enables a simple numerical solution for an otherwise complicated problem. Estimates of performance characteristics that are based on a simplified model and realistic materials point to the practical potential of solar pipe utilization.     

Analysis of solidification in a finite PCM storage with internal fins by employing heat balance integral method

Here, a simplified analytical model has been proposed to predict solid fraction, solid–liquid interface, solidification time, and temperature distribution during solidification of phase change material (PCM) in a two‐dimensional latent heat thermal energy storage system (LHTES) with horizontal internal plate fins. Host of boundary conditions such as imposed constant heat flux, end‐wall temperature, and convective air environment on the vertical walls are considered for the analysis. Heat balance integral method was used to obtain the solution. Present model yields closed‐form solution for temperature variation and solid fraction as a function of various modeling parameters. Also, solidification time of PCM, which is useful in optimum design of PCM‐based thermal energy storages, has been evaluated during the analysis. The solidification time was found to be reduced by 93% by reducing the aspect ratio from 8 to 0.125 for constant heat flux boundary condition. While, for constant wall temperature boundary condition, the solidification time reduces by 99% by changing the aspect ratio from 5 to 0.05. In case of convective air boundary surrounding, the solidification time is found to reduce by 88% by reducing the aspect ratio from 8 to 0.125. Based on the analytical solution, correlations have been proposed to predict solidification time in terms of aspect ratio and end‐wall boundary condition.     

Heat transfer in high-temperature liquid jets

This study focuses on the hydrodynamics and heat transfer of a very high temperature liquid jet moving through air. The purpose was to determine the velocity and temperature fields in a jet of molten materials flowing from a furnace into casting devices. Understanding hydrodynamic and heat transfer properties of the jet is essential in controlling the flow and the solidification of molten products. The non-linear equations that govern this physical problem were solved numerically using a finite-difference method applied to a laminar and axisymmetric flow with no fluctuation of the interface between the liquid jet and the continuous phase. The exit velocity profile was analyzed in terms of its effect on liquid jet hydrodynamics and cooling properties; and the Peclet number and jet emissivity in terms of their influence on the thermal exchange. In addition to the theoretical approach, experimental values were provided to validate the numerical model. Jet diameter and surface temperature profile values were measured and compared. The numerically analyzed jet diameter contraction and surface temperature values were consistent with those obtained experimentally.     

ADJOINT VARIABLE METHOD FOR THE THERMAL DESIGN OF EUTECTIC DIRECTIONAL SOLIDIFICATION PROCESSES IN AN OPEN-BOAT CONFIGURATION

A computational method for the inverse design of a directional solidification process of a near-eutectic binary alloy driven by the coupled action of buoyancy, thermocapillary, and electromagnetic convection is presented. The objective is to calculate the mold cooling/heating conditions such that a stable desired interface growth with growth velocity v_f and thermal gradient G is achieved. The interface velocity v_f and thermal gradient G are chosen such that a diffusion-based growth is obtained in the presence of melt convection. Morphological stability is enforced by imposing an appropriate magnitude of G/|v_f|, which is determined a-priori based on the constitutional stability criterion. The design problem is posed as a functional optimization problem. The cost functional is defined so as to represent the deviation of the freezing interface thermal conditions from thermodynamic equilibrium. An appropriate continuum adjoint problem is defined such that an analytical expression for the gradient of the objective function is obtained. The conjugate gradient method coupled with the finite element solutions of the continuum direct, sensitivity, and adjoint problems is employed for solving the inverse problem. The method is demonstrated with an example of calculating the boundary thermal fluxes for the directional growth of an Sb-8.6% Ge melt in an open-boat configuration under the influence of an external horizontal magnetic field such that a stable vertical interface advances from left to right with a desired growth velocity.     

Numerical Modeling of the Solidification Phase Change in a Pipe and Evaluation of the Effect of Boundary Conditions

A numerical solution to a two-dimensional model of flow and transient heat transfer involving solidi- fication in a pipe has been established. Where the temperature of pipe wall is below the freezing point of fluid, phase change of flowing fluid and the influence of different boundary condition, such as pipe wall temperature, initial temperature and inlet velocity has been taken into account. Also it has been investigated to elicit proper non-dimensional numbers to show the solidification proceeding results. Additionally comparing the two acceptable inlet conditions, show distinctions between velocity inlet and pressure inlet boundary condition in such problems, which affect the whole freezing process.     

Solidification of aqueous ammonium chloride in a rectangular cell under constant wall temperature - an experimental study

The effect of a binary alloy solidification process was investigated for four different initial ammonium chloride concentrations (5%, 10%, 15% and 25%) and for four different constant wall temperature conditions (−13°C, −20°C, −25°C and −30°C). The solidification process inside the test cell was interrupted at various time steps to capture the phase front profile. Results show that a higher initial concentration and a lower constant wall temperature tend to increase the strength of thermal-driven buoyancy flow and solute-driven buoyancy flow respectively. The shape of both the solid and mushy phase front profiles are affected by the strength of these two flows. Isotherm plots and superimposed images are used to determine the location of solid-must interface and must-liquid interface respectively, inside the test cell at any time interval.