An analytical solution of the critical interface velocity for the encapturing of insoluble particles by a moving solid/liquid interface

An analytical model for the particle pushing phenomenon that occurs between spherical particles and advancing curved solid/liquid interfaces during solidification of pure melts is presented. An expression for the critical interface velocity for encapturing particles by moving solid/liquid interfaces has been developed for the steady-state condition. As a first step, the actual shape of the interface behind the particle is computed in terms of the thermal conductivity ratio of the particle to that of the melt and the temperature gradient ahead of the interface; based on assumed subject, the critical interface velocity is calculated using the force balance between the attractive forces and repulsive forces acting on the particle. The critical interface velocity under steady-state conditions in aluminum containing SiC particle (10 μm) comes out to be 5800 μm/s according to the present model; this calculated velocity is much closer to the experimental observations of Wu et al., as compared to the predictions of the models proposed by earlier workers.     

Creep behavior of a rotating functionally graded composite disc operating under thermal gradient

Creep behavior of a rotating disc made of isotropic functionally graded material (FGM) has been investigated. The disc under investigation is made of a composite containing silicon carbide particles in a matrix of pure aluminum. The creep behavior has been described by Sherby’s law. The disc is considered as having a thermal gradient in the radial direction. The present analysis indicates that for the assumed linear particle distribution, the steady-state strain rates are significantly lower compared to that in an isotropic disc with uniform particle distribution. It is also found that the strain rates in composite discs operating under thermal gradient are reduced as compared to similar discs under a uniform average temperature.     

Behavior of ceramic particles at the solid- liquid metal interface in metal matrix composites

Directional solidification experiments have been conducted to document SiC particle behavior at the solid-liquid interface in Al-2 pct Mg (cellular interface) and Al-6.1 pct Ni (eutectic interface) alloys. Particle size ranged from 20 to 150 μm diameter. Although predictions based on the thermodynamic approach suggest that no engulfment is possible, it was demonstrated that particles can be entrapped in the solid if adequate solidification rates and temperature gradients are used. The main factors responsible for this behavior are considered to be the difference between the thermal conductivities of particles and metal, the buildup of volume fraction of particles at the interface, and the morphological instability of the interface induced by the particles. A model including the contribution of drag and thermal conductivity has been proposed. Calculation with this model produced numbers for the critical velocity slightly higher than those evaluated experimentally. Various factors which can account for this discrepancy are discussed.     

Liquid convection effects on the pushing-engulfment transition of insoluble particles by a solidifying interface: Part I. Analytical calculation of the lift forces

During the solidification of a liquid containing insoluble particles, the particles can be instantaneously engulfed, or continuously pushed, or pushed and subsequently engulfed. A critical velocity for the pushing-engulfment transition is observed experimentally. Most models proposed to date ignore the complications arising from the liquid convection ahead of the solid-liquid interface. They simply solve the balance between the attractive drag force exercised by the liquid on the particle and the repulsive interfacial force. This work is an effort to calculate analytically the lift forces (Saffman and Magnus forces) under certain assumptions regarding the nature of fluid flow ahead of the solid/liquid interface. This makes possible the quantitative evaluation of the three experimentally observed regimes occurring during particle-interface interaction: (1) at low convection—no effect on the critical velocity for the particle engulfment transition; (2) at intermediate convection—increased critical velocity; (3) at high convection—no particle-interface interaction. The model was applied to evaluate the gravity level required for microgravity experimental work on particle pushing where the effect of liquid convection during solidification is negligible. This is necessary to validate existing theoretical models that do not take into account fluid flow parallel to the solidification interface.     

Experimental Study of Particle Motion on a Smooth Bed under Shoaling Waves Using Particle Image Velocimetry

The motion of spherical particles (diameter 1.58 mm, specific gravity 2.5) on 2 and 3% plane slope was studied in a laboratory wave flume for shoaling wave conditions. The range of wave-height-to-water-depth ratio was 0.24相似文献   

Finite Element Method (FEM) Calculations of Stress-Strain behavior of alpha-beta Ti-Mn alloys: Part II. Stress and strain distributions

The stress and strain distributions of Ti-Mn alloys have been calculated as a function of strain, volume fraction of phases, and, in one instance, as a function of particle size. Near the interface, strains in β are higher than they are away from the interface. In α the strains are lower near the interface than they are away from the interface. Stresses have the same behavior as the strains. The average strain in α is higher than the average strain in β. The ratio of the strain in α, εα, to the strain in β, εβ is minimum, for a given strain, near 50 vol pct of β• εα/εβ increases with strain up to two pct the limit of total strain examined. Direct experimental measurements with 12 μm square Al grid have confirmed calculations for three alloys,εα/εβ decreases with decreasing particle size, and uniformity of strain increases with decreasing particle size.     

Determining the three-dimensional morphology of γ′-particles in γ-γ′ superalloys

We propose qualitative and quantitative methods for determining the three-dimensional morphology of second-phase particles in Ni-based superalloys in the late stages of coarsening. Qualitatively, we determine a way to identify the three-dimensional shape of a particle from its (111) cross section. Quantitatively, we derive a method that uses stereological analysis on a single (111) section to compute the average shape of a particle in three dimensions. For cases where the average shape does not necessarily reflect the particles’ true morphology, we derive another method based on sectioning probability to compute the shape of individual particles. We are also able to determine a particle’s orientation in three dimensions by examining its (111) cross section. The methods were tested by using computer-generated (111) sections of three-dimensional arrays of rectangular particles. We conclude that (111) sections can be used to provide an accurate calculation of the interfacial area per unit volume (S _V ) of the structure. Finally, we illustrate the efficacy of using (111) sections to determine particle morphology by examining (111) transmission electron microscopy (TEM) micrographs of particles undergoing splitting.     

Numerical studies of the motion of particles in current-carrying liquid metals flowing in a circular pipe

A mathematical model has been developed to describe the motion of particles in current-carrying liquid metals flowing through a cylindrical pipe. The fluid velocity field was obtained by solving the Navier-Stokes equations, and the trajectories of particles were calculated using equations of motion for particles. These incorporate the drag, added mass, history, electromagnetic, and fluid acceleration forces. The results show that particle trajectories are affected by the magnetic pressure number R _H, the Reynolds number Re, the blockage ratio k, and the particle-fluid density ratio γ according to the relative importance of associated force terms. In the axial direction, the particles follow the fluid velocity closely and will move further axially before reaching the wall as the fluid velocity (Re) increases. In the radial direction, the outwardly directed electromagnetic force on the particle increases with radial distance from the axis, with increasing electric current (R _H), and increasing size (k) of particle. The competition between the electromagnetic force and the radial fluid acceleration force in the entrance region results in particle movement toward the central axis before moving toward the wall for small electric current (low R _H) and directly toward the wall for large current (high R _H). The low inertia (γ) bubbles move faster toward the wall than heavier particles do. The radial velocity of the particle movement as it approaches the wall is predicted to decrease due to wall effects. This model has been applied to the movement of inclusions within the electric sensing zone (ESZ) of the liquid metal cleanliness analyzer (LiMCA) system in molten aluminum, and it was proved that LiMCA system could be used in aluminum industries.     

Elastic moduli of titanium-hydrogen alloys in the temperature range 20 °C to 1100 °C

The elastic properties of a series of polycrystalline titanium-hydrogen alloys (containing up to 25 at. pct H) were measured over the temperature range 20 °C to 1100 °C. The latter limits permitted investigation of adjacent parts of the α+δ, α, and β phase fields. A laser ultrasonic technique was employed to measure the temperature and hydrogen-concentration dependencies of the elastic constants. The room-temperature elastic properties of the alloys depended only slightly on hydrogen concentration, remaining almost independent of the volume fraction of the δ hydride phase. In the α phase field, the addition of hydrogen decreased the shear and Young’s moduli and increased the bulk modulus, Lamé constant, and Poisson’s ratio. The Young’s and shear moduli decreased more rapidly with increasing temperature than in cubic phases. By contrast, Poisson’s ratio increased with temperature. In the β phase field, the temperature dependence of the elastic constants was weak. However, alloying with hydrogen increased the shear and Young’s moduli, decreased Poisson’s ratio, but did not appreciably affect the bulk modulus and Lamé constant. The different effects of hydrogen on the elastic constants of alpha and beta titanium are interpreted in terms of the influence of dissolved hydrogen on the stability of the hexagonal close-packed (hcp) and body-centered cubic (bcc) lattices in the vicinity of the α-to-β transformation. The present results are also used to help account for the effect of hydrogen concentration on the mechanical properties of Ti-H alloys.     

Numerical studies of the motion of spheroidal particles flowing with liquid metals through an electric sensing zone

A mathematical model has been developed to describe the motion of variously shaped and oriented spheroids entrained in liquid metals passing through a cylindrical electric sensing zone (ESZ) developed for liquid metals cleanliness analyzer (LiMCA) systems. The fluid velocity field within the ESZ was obtained by solving the Navier-Stokes equations, while the trajectories of particles within the ESZ were calculated using the equations of motion for particles. These incorporate forces resulting from drag, added mass, history, and electromagnetic and fluid acceleration balanced against the time rate of changes in a particle momentum. The effects of particle or inclusion shape and orientation were taken into account by including correction factors for drag (R ^D), added mass (M ^A), history (B), and electromagnetic force (E ^M). The numerical results show that particle trajectories are affected by the magnetic pressure number (R _H), the Reynolds number (Re), the blockage ratio (k), and the particle-fluid density ratio (γ). In the axial direction, spheroidal particles travel further axially before hitting the wall as the fluid velocity (Re) increases. In the radial direction, the outwardly directed electromagnetic force on nonconducting spheroids increases with radial distance from the axis, with increasing electric current (R _H) and with increasing size (k) of the particle. At low electric currents (low R _H), the competition between the electromagnetic force and the radial fluid acceleration force in the entrance region is predicted to result in particle movements first toward the central axis, before outward motion toward the wall, but directly toward the wall at large currents (high R _H). Spheroidal particles with symmetric axes perpendicular to the transverse axis of the ESZ move faster toward the sidewall as the particle aspect ratio (E) increases. The dominating increase in the added mass force over the increase in the electromagnetic force with decreasing E makes this effect much stronger for oblates (E<1) than for prolates (E>1). The stronger drag force on a prolate with its symmetric axis parallel to the axis of the ESZ makes it move slower toward the wall than a prolate with its axis of symmetry perpendicular to the axis of the ESZ. Low-inertia (low-γ) spheroidal particles move faster toward the sidewall than do heavier particles. This effect of γ is stronger for prolates than for oblates traversing with their symmetric axes perpendicular to the axis of the ESZ, owing to the decreased added mass effect as E increases, while the effect of γ becomes much stronger for a prolate traversing with its symmetric axis perpendicular rather than parallel to the axis of the ESZ, owing to its smaller added mass. The radial particle velocity when approaching the wall is predicted to decrease due to the wall effects. This model has been applied to the movement of spheroidal inclusions within the ESZ of a LiMCA system in molten aluminum, and it was proven from the theoretical point of view that LiMCA systems could be used in aluminum industries.     

Simulation of ferrite growth in continuously cooled low-carbon iron alloys

The growth of a planar ferrite (α): austenite (γ) boundary in low-carbon iron and Fe-Mn alloys continuously cooled from austenite through the (α+γ) two-phase field and the α single-phase field was simulated by incorporating carbon diffusion in austenite, intrinsic boundary mobility, and the drag of an alloying element. At a very high cooling rate (≥ 10³ °C/s), the width of the carbon diffusion spike in austenite approaches the limit at which spikes are viable, so that the growth of ferrite in which carbon is not partitioned can occur even above the α solvus. In this context, the upper limiting temperature of partitionless growth of ferrite is the T ₀ temperature. In the presence of drag of an alloying element, e.g., Mn, both carbon-partitioned and partitionless growth of ferrite begins to occur at finite undercoolings from the Ae ₃, T ₀, or α-solvus temperature, at which the driving force for transformation exceeds the drag force. The intrinsic mobility of the α:γ boundary may play a significant role at an extremely high cooling rate (≥10⁵ °C/s). This article is based on a presentation made at the symposium entitled “The Mechanisms of the Massive Transformation,” a part of the Fall 2000 TMS Meeting held October 16–19, 2000, in St. Louis, Missouri, under the auspices of the ASM Phase Transformations Committee.     

Force acting on spheres adhered to a vessel wall

To evaluate the force and torque acting on leukocytes attached to the vessel wall, we numerically study the flow field around the leukocytes by using rigid spherical particles adhered to the wall of a circular cylindrical tube as a model of adherent leukocytes. The adherent particles are assumed to be placed regularly in the flow direction with equal spacings, in one row or two rows. The flow field of the suspending fluid is analyzed by a finite element method applied to the Stokes equations, and the drag force and torque acting on each particle, as well as the apparent viscosity, are evaluated as a function of the particle to tube diameter ratio and the particle arrangements. For two-row arrangements of adhered particles where neighboring particles are placed alternately on opposite sides of the vessel, the drag and the torque exerted on each particle are higher than those for single-row arrangements, for constant particle to tube diameter ratio and axial spacing between neighboring particles. This is enhanced for larger particles and smaller axial spacings. The apparent viscosity of the flow through vessels with adhered particles is found to be significantly higher than that without adhered particles or when the particles are freely floating through the vessels.     

Deformation of an alloy with a lamellar microstructure: experimental behavior of individual widmanstatten colonies of an α-β titanium alloy

The deformation behavior of individual Widmanstatten colonies comprised of aligned lamellae of ductile phases has been investigated. Based on the α β Ti alloy, Ti-8Al-lMo-lV, this study shows the existence of a large (>2X) variation in the critical resolved shear stress for yielding of individual colonies. Schmid’s Law is not obeyed except for prism slip parallel to the β lamellae. In addition, colonies with a high yield stress exhibit a high work hardening rate and fine, uniform slip. This behavior appears to be independent of the α-phase slip system (basal, prism, or pyramidal) and of the microstructure (α βvs α α′ (martensite)). The experimental behavior is correlated to several colony orientation parameters including the stress axis, the slip plane, the slip direction and the orientation of the α β interface. The yield stress of a colony is found to increase as the slip direction of the dominant macroscopic slip plane approaches normality to the α β interface. These results indicate that the macroscopic flow behavior of colonies comprised of ductile lamellae depends on the ability of a slip system, once activated in the softer phase, to shear through the harder phase. The data also indicate that the interaction stresses at the phase interfaces are not a principal factor controlling macroscopic yielding in α β Ti alloys. Finally, the alignment of a slip system in the α-phase with a potential slip system in the β-phase lamellae does not appear to affect the yield stress strongly. formerly Graduate Students, Department of Metallurgical Engineering, Michigan Technological University     

高炉回旋区运动现象的数值模拟

风口回旋区是高炉内产生还原气体及热量的主要区域,颗粒相和气相在风口回旋区内的相互作用非常剧烈。回旋区的形状和大小决定了炉内煤气流的一次分布,是炉况顺行的基础。本文为了从颗粒尺度来描述回旋区内部气体和颗粒的运动行为,建立和发展了离散单元法和流体计算力学耦合模型。耦合模型考虑了多个相间力的作用(包括曳力、虚假质量力、升力和压力梯度力),得到了不同时刻风口前焦炭颗粒的分布图、体积分数云图、速度矢量图等。这些图表明在鼓风速度130 m/s时,风口前有颗粒被吹开,形成了明显的空腔,风口前及其附近区域的颗粒在做回旋运动,可以断定回旋区已经形成,且在6 s时风口回旋区达到稳定。     

Predicting the stress-strain behavior of polycrystalline α-lron containing hard spherical particles

Recent interest in the work hardening of metal crystals containing a dispersion of hard particles has resulted in analytical expressions relating the work hardening to strain, particle diameter, and volume fraction as well as other material parameters. In this study, these models have been used to calculate the tensile stress-strain behavior of polycrystalline α iron containing dispersions of the intermetallic compound Fe₂Ta. The structural characteristics of the Fe-Ta alloys were thoroughly evaluated. The particle morphology was measured for randomness, mean particle diameter, standard deviation of the particle diameter, volume fraction, and planar interparticle spacing. Also, the matrix flow strength, composition, crystallographic randomness, dislocation morphology and grain size were evaluated. It was found that an Orowan type relationship as modified by Ashby satisfactorily described the yield strength as a function of the interparticle spacing and particle diameter. An experimental slope of 11.1 x 10^-5 kg-cm/mm² and a calculated slope of 9.75 x 10^-5 kg-cm/mm² were found. Both the Hart revised FHP work hardening model and Ashby’s model based on the generation of secondary dislocations were in good agreement with the experimental data. Hart’s revised FHP model required the use of empirically obtained values for the particle volume fraction which differed by a factor of 10 from the measured volume fraction and therefore is not suitable for predictive purposes. At tensile strains greater than 5 pct, the work hardening was characteristic of the matrix without particles; therefore, deviation between the experimental and calculated results based on Ashby’s model occurred at large strains. It is hoped that this study represents a step towards applying work hardening models to more complex polycrystalline alloys.     

Mobility of the β1-γ 1 ′ martensitic interface in Cu-Al-Ni: Part II. Model calculations

The interaction of the twinnedβ _-1-γ ₁ ^′ , martensitic interface with various experimentally observed obstacle particles is analyzed using a specific dislocation model for the interface. The strain interaction of particles with simple shears and tetragonal distortions and their modulus interactions are treated as functions of particle crystallographic orientation, particle position with respect to interfacial intersection, and particle size. Differences from previous predictions of a simple general interface model arise primarily from differences in the assumed interfacial trajectory relative to the particles. The finite-particle calculations indicate that the point-particle approximation is valid for a particle radius less than one-tenth the interfacial twin period. Overall agreement with the experimentally measured interfacial mobility behavior is greatly improved over the previous simple model prediction. The measured athermal component of the driving force for interfacial motion is consistent with the strain and modulus interaction with 2H-phase particles. The activation-energy /driving-force relations obtained from the thermally activated component are reasonably represented by the strain interaction with the fine-scale atomic displacements of the tweed structure.     

Tensile properties to 650° C and deformation structures in a precipitation-strengthened Ti-Al Alloy

Appreciable strength levels were retained up to 650°C in a Ti-10 Al-1 Si alloy aged in the (α + α₂) phase field to yield best room-temperature strength and ductility. The aging treatment precipitated such a uniform distribution of the α₂ particles that at room temperature, dislocations bypassed instead of shearing the particles at low strains. Specimens fractured at room temperature exhibited fine uniform dimples even for those aging conditions that imparted no macroscopic ductility. A two-step aging process produced a higher volume fraction of bimodally distributed α₂ particles that led to higher strength levels at elevated temperatures. Both for the single-size and the bimodal α₂ particle distributions, elevated-temperature deformation structures consisted mainly of planar slip bands that sheared through the α₂ particles. Formerly National Research Council, Resident Postdoctoral Research Associate, NASA Lewis Research Center, Cleveland, Ohio.     

Microstructural and thermal stability of a Ti-43AI alloy containing dispersoids of titanium di-boride

Ti-43Al (atomic percent) alloy containing a dispersion of 7 vol pct TiB₂ particles was exposed to various thermal treatments to determine the stability of TiB₂ in an ⇌₂ + β-phase matrix. No new phases were detected at the particle/matrix interfaces even after thermal exposure at 1473 K for 7 days. The absence of an Al peak in the energy dispersive X-ray analysis system (EDS) spectra from TiB₂ particles chemically extracted from the specimens aged at 1473 K for 7 days indicated no diffusion of Al from the matrix to the particles. These results indicate that TiB₂ is stable in an α₂+ β matrix at 1473 K. E. Clevenger, formerly Undergraduate Student, Department of Mechanical and Materials Engineering, Wright State University