Numerical study of macrosegregation in Aluminum alloys solidifying on uneven surfaces

Solidification of Aluminum alloys is modeled on uneven surfaces characterized by sinusoidal curves. Wavelengths and amplitudes of these surfaces are varied to study the effect of changing surface topography on fluid flow, macrosegregation and inverse segregation in the solidifying alloy. Solidification is initiated by convective heat removal from the uneven surfaces and simulations are carried out in both vertical and horizontal configurations. Stabilized finite element methods, recently used for modeling solidification in the presence of shrinkage and buoyancy driven flows, are used to discretize and solve the governing transport equations derived by volume averaging. The effect of varying amplitudes and wavelengths is observed in heat transfer, fluid-flow, macrosegregation and inverse segregation processes. In vertical solidification, inverse segregation, that usually occurs at the bottom of the cavities, is studied for different sinusoidal topographies quantified by a particular wavelength and amplitude. The fluid flow here is driven by a combination of shrinkage and thermosolutal buoyancy. Shrinkage driven flow arises due to different densities of solid and liquid phases. During horizontal solidification of an Aluminum alloy from uneven surfaces, thermosolutal buoyancy plays a dominant role in fluid flow and the effect of shrinkage is neglected by assuming the individual phase densities to be equal. Convection in this case is much stronger than that in the vertical case and large scale redistribution of the solute element occurs. To measure variation in macrosegregation with changing surface topography, global extent of segregation and difference between maximum and minimum solute concentrations are calculated for different amplitudes and wavelengths. In both the cases, the main aim is to quantify changes in macrosegregation due to changing surface topography accomplished by varying amplitudes or wavelengths or both.     

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.     

Solidification of a supercooled liquid in stagnation-point flow

The solidification of a thermally supercooled liquid in stagnation-point flow is investigated. Due to the advancing solidifying front, both the temperature and flow fields are time dependent. A numerical solution to the problem using an interface tracking method is compared to analytical solutions obtained for instantaneous similarity (short time solution) and quasi-steady state (long time solution). The results show that the velocity of the solid-liquid interface eventually reaches a constant value and that the magnitude of the interface velocity increases with greater thermal supercooling. The solution to this problem provides insight into more complicated solidification problems relating to crystal growth.     

Control of macrosegregation during the solidification of alloys using magnetic fields

Numerical modeling of convection damping and macrosegregation suppression during solidification of alloys with prominent mushy zones through the use of tailored magnetic fields is demonstrated here. Macrosegregation leads to commonly observed defects such as freckles, channels and segregates in cast alloys that severely affect the performance and suitability of the alloy for further applications. The current work demonstrates the successful use of magnetic fields in suppressing thermosolutal convection and eliminating some of these defects in solidifying metallic alloys. The computational model presented utilizes volume-averaged governing transport equations and stabilized finite element techniques to discretize these equations. A finite-dimensional optimization problem, based on the continuum sensitivity method is considered to design the time history of the imposed magnetic field required to effectively damp convection. The coefficients that determine this time variation are the main design parameters of this optimization problem. Continuum sensitivity equations are derived by design-differentiating the governing equations of the direct problem. The cost functional here is given by the square of the L₂ norm of an expression representing the deviation of the volume-averaged velocity corresponding to conditions of convection less growth. The cost functional minimization process is realized through a non-linear conjugate gradient algorithm that utilizes finite element solutions of the continuum direct and sensitivity problems. Design of the time history of the imposed magnetic field is highlighted through different examples with the main objective being the suppression of convection and macrosegregation during alloy solidification.     

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.     

Solutocapillary convection in the float-zone process with a strong magnetic field

This paper treats the steady axisymmetric flow and mass transport in a cylindrical liquid bridge between the melting end of a feed rod and the solidifying end of an alloyed semiconductor crystal. There is a strong, uniform, steady, axial magnetic field. The surface tension depends on the temperature and the concentration of the species, while variations of the concentration occur because one species is rejected into the liquid during solidification. The thermocapillary and solutocapillary convections tend to cancel over part of the liquid bridge. For certain parameter ranges, there are two different stable solutions: one where the concentration gradient along the free surface leads to dominance by the solutocapillary convection and one where the mass transport due to the thermocapillary convection makes the concentration gradient along the free surface small, so that the thermocapillary convection is dominant.     

GRAVITY- AND SOLIDIFICATION-SHRINKAGE-INDUCED LIQUID FLOW IN A HORIZONTALLY SOLIDIFIED ALLOY INGOT

A basic-variable, explicit finite difference scheme is used to solve the unsteady and strongly pressure-coupled velocity problems of liquid flows induced both by thermosolutal buoyancies and solidification shrinkage in a binary alloy solidification process. A sample calculation, performed on an IBM personal computer, for a horizontally solidified Al-4.5%Cu alloy in a high-H/L ratio cavity with a top riser shows that the shrinkage established pressure gradient in the mushy region can be several hundred, even several thousand, times larger than that in the bulk liquid region.     

Multiscale modeling of particle–solidification front dynamics. Part II: Pushing–engulfment transition

The interaction between a particle and an advancing solidification front is studied using a multi-scale computational model developed in Part I. The flow and temperature fields are solved separately at two disparate scales, i.e. at the overall system scale (“outer region”) and in the thin melt layer (“inner region”) between the particle and the front. The solutions from the inner and outer regions are coupled at a matching region. The coupled dynamics of the particle and phase boundary motion, including lubrication and disjoining pressure effects in the premelted film between the particle and the front is captured in the simulations. Results show that particle pushing (as opposed to particle engulfment) can occur when the ratio of thermal conductivity of the particle to the melt, k_p/k_l < 1. The velocity of the solidification front at which the transition from particle pushing to particle engulfment occurs, i.e. the critical velocity for particle engulfment, is naturally obtained from the coupled dynamics. No ad hoc assumptions to identify the critical velocity need be made. The results also provide insights into the physics of particle–solidification front interactions.     

An investigation of the influence of natural convection on tin solidification using a quasi two-dimensional experimental benchmark

A well validated, quasi two-dimensional, unsteady solidification experimental benchmark is proposed to study the critical role of thermally driven natural convection using commercial pure tin. The experiment consists of solidifying a parallelipedic sample from two vertical sidewalls using two heat exchangers in a rectangular cavity. The mean temperature gradient G_T, and the mean cooling rate C_R, are set to control the experimental process. An array of 50 thermocouples allows us to measure the instantaneous temperature field and its evolution. While in the liquid state, the isotherms exhibit a plausible convective heat flow and its intensity increases as the Rayleigh number increases. In the solidification process, a novel recalescence phenomenon is observed by tracing the solidifying front in a relatively slow cooling rate case. By setting different mean temperature gradients, different patterns of the natural convection and the temperature field evolutions are obtained. A discussion is also presented regarding the crystallography.     

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     

Stability of buoyancy and surface tension driven convection in a horizontal double-diffusive fluid layer

Linear stability analysis has been applied to examine the stability of convection in a horizontal double-diffusive fluid layer driven by the combined effects of buoyancy and surface tension. Such a convective flow may serve as an idealized model of the horizontal Bridgman process for crystallization or solidification of liquid melts. Results show that salt-finger instability is excited over a wide range of thermal and solutal Grashof numbers. Travelling wave instabilities caused by surface tension effects are excited when the effective Marangoni number becomes large.     

Role of cellular wavelengths in self-acceleration of lean hydrogen-air expanding flames under turbulent conditions

An investigation is done upon the lean H₂-air flame under turbulent conditions to clarify the role of inherent instabilities and turbulence in self-acceleration of expanding flames. The result shows that, in weak turbulent flow fields, the two-stage self-acceleration feature still exists. As the turbulent intensity increases, the originally evident two-stage “transition-saturation” feature is weakened. When the Karlovitz number becomes greater than 1, the self-acceleration process no longer experiences the transition stage and performs a single-stage feature. These observations are correlated with the distribution and evolution of length scales in the cellular structure of the flamefront. A theoretical analysis is conducted to compare growth rates of disturbances induced by instability and turbulence, with consideration of their wavelength dependence. On this basis, interpretations are proposed on afore-mentioned phenomenal observations and response of quantitative parameters such as fractal excess and average cell wavelength.     

A numerical study of the combined effects of microsegregation, mushy zone permeability and fllow, caused by volume contraction and thermosolutal convection, on macrosegregation and eutectic formation in binary alloy solidification

Numerical simulations of the columnar dendritic solidification of a Pb-20 wt% Sn alloy in a square cavity cooled from one side and fed by a rectangular riser are reported. Overall macrosegregation patterns predicted using Scheil and lever-rule type microsegregation models are found to be similar, although the predicted eutectic fraction is significantly higher with the Scheil-type model. The choice of mushy zone permeability function significantly affects the predicted number, length and orientation of segregated channels. The inclusion of shrinkage-driven flow leads to the prediction of the well-known inverse macrosegregation pattern. However, macrosegregation caused by thermosolutal convection readily masks the inverse segregation. The microsegregation models predict different solid concentrations and eutectic fractions, leading to different solid density distributions which, in turn, cause differences in the extent of contraction-driven flow.     

Stimulation and flow processes in hot dry rock reservoirs

There was less agreement among the stimulation and flow processes group than in the other groups, reflecting the fact that this aspect remains the major uncertainty in the understanding of HDR systems. The differences arose from the different emphasis placed by the various research teams on the role of artificial as opposed to natural fractures. There was, however, general agreement that the flow in HDR reservoirs is predominantly laminar. It was also agreed that impedance can be reduced only by enlarging fractures apertures; the best way forward, however, depends on resolution of the “artificial” versus “natural” fracture controversy. This will require further theoretical work in close association with prolonged circulation in the major deep field experiments.     

Thermosolutal Convection During Melting in a Porous Medium Saturated with Aqueous Solution

The thermosolutal convection in a porous medium saturated with an aqueous solution near the temperature of the density maximum is studied. The fixed temperatures applied to vertical walls include the density maximum. The formulation of the problem is based on the Darcy-Brinkman model and the density variation is governed by a nonlinear approximation. The equations are solved by a finite-volume method. The numerical model is validated through experimental results. We show that the nonlinear variation of the density influences strongly the flow structure and the heat transfer. The structures of this flow show that the density maximum generates a complex flow structure of two contrarotating cells of unequal importance.     

Experimental analysis of the solidification of Sn–3 wt.%Pb alloy under natural convection

A quasi-two-dimensional solidification benchmark experiment with controlled thermal boundary conditions is proposed. The experiment consists in solidifying a rectangular ingot of Sn–3 wt.%Pb alloy using two lateral heat exchangers to extract the heat flux from one or two sides of the sample. The temperature difference between the two sides of the heat exchangers may vary from 0 to 40 K and the cooling rate from 0.02 to 0.04 K/s. This slow-cooling condition has been used to promote segregation formation. An array of fifty thermocouples placed on the corresponding sample walls is used to determine the instantaneous temperature distribution. During the solidification process, the temperature field is recorded versus time and analyzed. This makes it possible to estimate the change in temperature due to natural convection, the velocity field and the solidification macrostructure and segregation behavior. After each experiment, the segregation patterns are obtained by X-ray analysis and confirmed by eutectic fraction measurements. The local solute distribution is determined by means of induction coupled plasma analysis.     

Effect of air leak on long term performance of air heater

In this paper, the effect of an air leak on the air heater performance is investigated. “Air leaking in” and “air leaking out” systems are examined. The effect of various parameters like leakage rate, mass flow rate, solar insolation, plate length and ambient temperature has been studied. The possibility that air leakage can take place from any where along the length of the collector has been incorporated in the model. It is found that, for “air leaking in” systems, efficiency goes up, while for “air leaking out” systems, efficiency decreases from the no-leak situation, and it depends on the position of the leak.     

Heat transfer of gas–liquid mixture in micro-channel heat sink

The main objective of the present investigation is to study heat transfer in parallel micro-channels of 0.1 mm in size. Comparison of the results of this study to the ones obtained for two-phase flow in “conventional” size channels provides information on the complex phenomena associated with heat transfer in micro-channel heat sinks. Two-phase flow in parallel micro-channels, feeding from a common manifold shows that different flow patterns occur simultaneously in the different micro-channels: liquid alone (or single-phase flow), bubbly flow, slug flow, and annular flow (gas core with a thin liquid film, and a gas core with a thick liquid film). Although the gas core may occupy almost the entire cross-section of the triangular channel, making the side walls partially dry, the liquid phase always remained continuous due to the liquid, which is drawn into the triangular corners by surface tension. With increasing superficial gas velocity, a gas core with a thin liquid film is observed. The visual observation showed that as the air velocity increased, the liquid droplets entrained in the gas core disappeared such that the flow became annular. The probability of appearance of different flow patterns should be taken into account for developing flow pattern maps. The dependence of the Nusselt number, on liquid and gas Reynolds numbers, based on liquid and gas superficial velocity, respectively, was determined in the range of Re_LS = 4–56 and Re_GS = 4.7–270. It was shown that an increase in the superficial liquid velocity involves an increase in heat transfer (Nu_L). This effect is reduced with increasing superficial gas velocity, in contrast to the results reported on two-phase heat transfer in “conventional size” channels.     

The role of political connection on overinvestment of Chinese energy firms

This paper explores the role of political connection on the overinvestment of Chinese energy firms. We argue that political connection can act as a “helping hand” that enables energy firms to obtain more government support to invest and a “grabbing hand” that forces politically connected energy firms to heavily overinvest for the promotion benefit of local politicians. Our analysis shows that overinvestment of an energy firm is positively affected by its political connection. In terms of the “helping hand” effects, we find that politically connected energy firms are more likely overinvest when they receive more government subsidy. In terms of the “grabbing hand” effects, we show that local politician is more likely to force energy firms to overinvest if the politician is approaching the 65-year-old promotion line. Although firms are less likely to overinvest during the economic downturn, local politician shapes the investment decision of energy firms through the “grabbing hand” effects.     

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.