To the theory of directional solidification with a mushy zone in the presence of convection and kinetics in the melt

The directional solidification of binary systems in a frontal mode and in the presence of a phase transition region (mushy zone) is theoretically studied with allowance for convective-kinetic heat-and-mass transfer mechanisms. Nonlinear mathematical models are developed for this process, and their analytical solutions corresponding to various values of the system parameters are obtained.     

An analytical model for optimal directional solidification using liquid metal cooling

In what follows, a model is developed that describes the optimal processing parameters for directional solidification using liquid metal cooling (LMC). The model considers a sample with a flat geometry and, as a first approximation, can be used to treat the flat sections of a turbine blade. The model predicts (1) the optimal withdrawal rate of the casting from the hot zone, (2) the temperature gradient in the liquid at the solidification interface, and (3) the temperature profile along the length of the casting. The model is then used to perform a sensitivity analysis of the LMC process. Cooling bath temperature, baffle thickness, shell thickness, and shell thermal conductivity are shown to have a strong influence on system performance.     

AH36钢两相区非平衡凝固路径的数值预测

应用多元合金两相区凝固显微偏析数学模型,对AH36钢在一定冷却条件下的非平衡凝固路径进行了数值计算,获得了相应的局部固相分数与局部温度之间的变化关系,为该钢种对应的连铸浇铸过程仿真及其它凝固过程仿真提供了必要而准确的耦合数据,具有重要的理论意义和广泛的适用性.     

Modeling of solute redistribution in the mushy zone during solidification of aluminum-copper alloys

A mathematical model has been established to predict the formation of macrosegregation for a unidirectional solidification of aluminum-copper alloys cooled from the bottom. The model, based on the continuum formulation, allows the calculation of transient distributions of temperature, velocity, and species in the solidifying alloy caused by thermosolutal convection and shrinkage-induced fluid flow. Positive segregation in the casting near the bottom (inverse segregation) is found, which is accompanied by a moving negative-segregated mushy zone. The effects of shrinkage-induced fluid flow and solute diffusion on the formation of macrosegregation are examined. It is found that the redistribution of solute in the solidifying alloy is caused by the flow of solute-rich liquid in the mushy zone due to solidification shrinkage. A higher heat-extraction rate at the bottom increases the solidification rate, decreasing the size of the mushy zone, reducing the flow of solute-rich liquid in the mushy zone and, as a result, lessening the severity of inverse segregation. Comparisons between the theoretical predictions from the present study and previous modeling results and available experimental data are made, and good agreements are obtained.     

Synchrotron microradiography of temperature gradient zone melting in directional solidification

Using synchrotron microradiography, temperature gradient zone melting (TGZM) was observed in Sn-13 wt pct Bi alloy in real time during directional solidification. A significant amount of remelting was measured on the cold sides of the dendrite arms, whereas added solidification on the hot sides of the dendrite arms was observed during dendrite growth. Kinetics of TGZM was measured based on the real-time observations. TGZM had a significant effect on dendrite morphology during continuous cooling and holding within the solidification range. The presence of tertiary dendrite arms enhanced the rate of TGZM. Remelting also led to the disintegration of some secondary dendrite arms.     

Synchrotron microradiography of temperature gradient zone melting in directional solidification

Using synchrotron microradiography, temperature gradient zone melting (TGZM) was observed in Sn-13 wt pct Bi alloy in real time during directional solidification. A significant amount of remelting was measured on the cold sides of the dendrite arms, whereas added solidification on the hot sides of the dendrite arms was observed during dendrite growth. Kinetics of TGZM was measured based on the real-time observations. TGZM had a significant effect on dendrite morphology during continuous cooling and holding within the solidification range. The presence of tertiary dendrite arms enhanced the rate of TGZM. Remelting also led to the disintegration of some secondary dendrite arms.     

Synchrotron microradiography of temperature gradient zone melting in directional solidification

Using synchrontron microradiography, temperature gradient zone melting (TGZM) was observed in Sn-13 wt pct Bi alloy in real time during directional solidification. A significant amount of remelting was measured on the cold sides of the dendrite arms, whereas added solidification on the hot sides of the dendrite arms was observed during dendrite growth. Kinetics of TGZM was measured based on the real-time observations. TGZM had a significant effect on dendrite morphology during continuous cooling and holding within the solidification range. The presence of tertiary dendrite arms enhanced the rate of TGZM. Remelting also led to the disintegration of some secondary dendrite arms.     

The influence of convection during solidification on fragmentation of the mushy zone of a model alloy

Experiments have been conducted to observe fragmentation events in a model alloy (succinonitrile and acetone) solidifying in the presence of forced convection in the superheated melt. Measurements of fragmentation rates have been made, and an attempt was made to relate the results to the controllable parameters of the system. A microscope-video system recorded the mushy zone-melt interface, and the fragmentation process and fragmentation rates could be determined from a frame-by-frame analysis of the video images. Experiments were conducted for varying cooling rates, overall temperature differences, melt flow rates, and for two different concentrations of acetone (1.3 and 6.1 wt pct). Significant dendritic fragmentation occurred for all runs. In addition, the influence of buoyancy forces is clearly evident from particle motion near the mushy zone-melt interface. Fragmentation rates appear to correlate well with the magnitude of particle velocities near the interface, with increasing fragmentation being associated with higher particle velocity magnitude (either in the same or the opposite direction to the mean flow) for the 1.3 wt pct acetone mixture. However, the correlation is quite different for the higher concentration. The relationship between these results and the possible mechanisms for fragmentation are discussed. Although it appears that either constitutional remelting or capillary pinching are likely of importance, hydrodynamic shear forces or some other mechanism as yet undiscovered cannot be completely discounted, although circumstantial evidence suggests that mechanical shearing is inconsistent with observations made both here and already in published literature. The results provide a step in the development of solidification models that incorporate fragmentation processes in the mushy zone as an important mechanism of grain refinement and a potential source of macrosegregation in ingots and large castings.     

The influence of convection during solidification on fragmentation of the mushy zone of a model alloy

Experiments have been conducted to observe fragmentation events in a model alloy (succinonitrile and acetone) solidifying in the presence of forced convection in the superheated melt. Measurements of fragmentation rates have been made, and an attempt was made to relate the results to the controllable parameters of the system. A microscope-video system recorded the mushy zone-melt interface, and the fragmentation process and fragmentation rates could be determined from a frame-by-frame analysis of the video images. Experiments were conducted for varying cooling rates, overall temperature differences, melt flow rates, and for two different concentrations of acetone (1.3 and 6.1 wt pct). Significant dendritic fragmentation occurred for all runs. In addition, the influence of buoyancy forces is clearly evident from particle motion near the mushy zone-melt interface. Fragmentation rates appear to correlate well with the magnitude of particle velocities near the interface, with increasing fragmentation being associated with higher particle velocity magnitude (either in the same or the opposite direction to the mean flow) for the 1.3 wt pct acetone mixture. However, the correlation is quite different for the higher concentration. The relationship between these results and the possible mechanisms for fragmentation are discussed. Although it appears that either constitutional remelting or capillary pinching are likely of importance, hydrodynamic shear forces or some other mechanism as yet undiscovered cannot be completely discounted, although circumstantial evidence suggests that mechanical shearing is inconsistent with observations made both here and already in published literature. The results provide a step in the development of solidification models that incorporate fragmentation processes in the mushy zone as an important mechanism of grain refinement and a potential source of macrosegregation in ingots and large castings. C. J. PARADIES, formerly Graduate Research Assistant, Rensselaer Polytechnic Institute     

连铸流动与凝固耦合模拟中糊状区系数的表征及影响

分析提出了连铸流动与凝固耦合数值模拟中, 钢液在两相区流动时的糊状区系数(A_mush)与渗透率的关系; 通过建立大方坯连铸结晶器三维耦合数值模型, 揭示了不同糊状区系数对钢液流动、传热与凝固进程的影响, 以及早期相关研究结果差异的源头.结果表明: 糊状区系数越大, 钢液在糊状区内的流动阻力越强, 凝固时钢液流动速度降低越快.采用较大的糊状区系数时, 糊状区呈较窄的"带状"分布在固液相之间; 当糊状区系数较小时, 糊状区范围变大, 钢液在结晶器内温降过快, 自由液面处出现过冷现象, 凝固坯壳局部发生重熔.结合实验数据验证与模型分析, 认为糊状区系数取值1×108~5×108 kg·m-3·s-1可以较可靠地揭示连铸结晶器内的实际凝固现象.      

Mushy zone morphology during directional solidification of Pb-5.8 wt pct Sb alloy

The Pb-5.8 wt pct Sb alloy was directionally solidified with a positive thermal gradient of 140 K cm⁻¹ at a growth speed ranging from 0.8 to 30 μm s⁻¹, and then it was quenched to retain the mushy zone morphology. The morphology of the mushy zone along its entire length has been characterized by using a serial sectioning and three-dimensional image reconstruction technique. Variation in the cellular/dendritic shape factor, hydraulic radius of the interdendritic region, and fraction solid along the mushy zone length has been studied. A comparison with predictions from theoretical models indicates that convection remarkably reduces the primary dendrite spacing while its influence on the dendrite tip radius is not as significant.     

Two-phase modeling of mushy zone parameters associated with hot tearing

A two-phase continuum model for an isotropic mushy zone is presented. The model is based upon the general volume-averaged conservation equations, and quantities associated with hot tearing are included, i.e., after-feeding of the liquid melt due to solidification shrinkage is taken into account as well as thermally induced deformation of the solid phase. The model is implemented numerically for a one-dimensional model problem with some similarities to the aluminium direct chill (DC) casting process. The variation of some key parameters that are known to influence the hot-tearing tendency is then studied. The results indicate that both liquid pressure drop due to feeding difficulties and tensile stress caused by thermal contraction of the solid phase are necessary for the formation of hot tears. Based upon results from the one-dimensional model, it is furthermore concluded that none of the hot-tearing criteria suggested in the literature are able to predict the variation in hot-tearing susceptibility resulting from a variation in all of the following parameters: solidification interval, cooling contraction of the solid phase, casting speed, and liquid fraction at coherency.     

Primary spacing in directional solidification

A new analytical model is developed to explain the variation in primary spacing λ with growth velocity V. In this model, dendrite growth is resolved into two parts: the growth of the center core and that of the side arms, which are separately treated. In contrast to the assumption in the current models, it is only the dendrite core, not the entire dendrite, whose curvature radius at the tip is directly related to dendrite tip radius R. The primary spacing is considered to be the sum of core diameter and twice the sidearm length. As long as the growth of side arms is suppressed, it becomes cellular growth. As a result, this model gives a reasonable dependence of cell and dendrite spacing on the process parameters. The proposed model has been applied to several alloys to compare its predictions both with experimental data and with the analytical expression of the Hunt-Lu model.     

Thermosolutal convection during directional solidification

During solidification of a binary alloy at constant velocity vertically upward, thermosolutal convection can occur if the solute rejected at the crystal-melt interface decreases the density of the melt. We assume that the crystal-melt interface remains planar and that the flow field is periodic in the horizontal direction. The time-dependent nonlinear differential equations for fluid flow, concentration, and temperature are solved numerically in two spatial dimensions for small Prandtl numbers and moderately large Schmidt numbers. For slow solidification velocities, the thermal field has an important stabilizing influence: near the onset of instability the flow is confined to the vicinity of the crystal-melt interface. Further, for slow velocities, as the concentration increases, the horizontal wavelength of the flow decreases rapidly — a phenomenon also indicated by linear stability analysis. The lateral in-homogeneity in solute concentration due to convection is obtained from the calculations. For a narrow range of solutal Rayleigh numbers and wavelengths, the flow is periodic in time. Formerly with the Mathematical Analysis Division, Center for Applied Mathematics, National Bureau of Standards, Washington, DC 20234. This paper is based on a presentation made at the symposium “Fluid Flow at Solid-Liquid Interfaces” held at the fall meeting of the TMS-AIME in Philadelphia, PA on October 5, 1983 under the TMS-AIME Solidification Committee.     

Undercooling-Induced macrosegregation in directional solidification

The accepted primary mechanism for causing macrosegregation in directional solidification (DS) is thermal and solutal convection in the liquid. This article demonstrates the effects of under-cooling and nucleation on macrosegregation and shows that undercooling, in some cases, can be the cause of end-to-end macrosegregation. Alloy ingots of Pb-Sn were directionally solidified upward and downward, with and without undercooling. A thermal gradient of about 5.1 K/cm and a cooling rate of 7.7 K/h were used. Crucibles of borosilicate glass, stainless steel with Cu bottoms, and fused silica were used. High undercoolings were achieved in the glass crucibles, and very low undercoolings were achieved in the steel/Cu crucible. During under-cooling, large, coarse Pb dendrites were found to be present. Large amounts of macrosegregation developed in the undercooled eutectic and hypoeutectic alloys. This segre-gation was found to be due to the nucleation and growth of primary Pb-rich dendrites, continued coarsening of Pb dendrites during undercooling of the interdendritic liquid, Sn enrichment of the liquid, and dendritic fragmentation and settling during and after recalescence. Eutectic ingots that solidified with no undercooling had no macrosegregation, because both Pb and Sn phases were effectively nucleated at the start of solidification, thus initiating the growth of solid of eutectic composition. It is thus shown that undercooling and single-phase nucleation can cause significant macrosegregation by increasing the amount of solute rejected into the liquid and by the movement of unattached dendrites and dendrite fragments, and that macrosegregation in excess of what would be expected due to diffusion transport is not necessarily caused by convection in the liquid.     

Primary spacing in directional solidification

A new analytical model is developed to explain the variation in primary spacing λ with growth velocity V. In this model, dendrite growth is resolved into two parts: the growth of the center core and that of the side arms, which are separately treated. In contrast to the assumption in the current models, it is only the dendrite core, not the entire dendrite, whose curvature radius at the tip is directly related to dendrite tip radius R. The primary spacing is considered to be the sum of core diameter and twice the sidearm length. As long as the growth of side arms is suppressed, it becomes cellular growth. As a result, this model gives a reasonable dependence of cell and dendrite spacing on the process parameters. The proposed model has been applied to several alloys to compare its predictions both with experimental data and with the analytical expression of the Hunt-Lu model.     

The directional solidification of Pb-Sn-Cd alloys

The growing interest in composite structures for new material applications makes it necessary to determine just how generally we can apply existing solidification theory to controlled three-phase ternary solidification. The Pb-Sn-Cd ternary eutectic system was used as a suitable model system to completely map the phase morphology as a function of G/R and compositions. By carefully controlling the freezing rate and the thermal gradient in the liquid ahead of the solid-liquid interface (in the range 400 to 500 C/cm) the following areas of interest were investigated: 1) the effect of growth velocity and composition on coupled structures, 2) ternary impurities and their effect on the minimum G/R for coupled growth in a binary system, 3) the effect of growth velocity and composition on the nonplanar interface structures, and 4) the adaptability of present theories (the constitutional supercooling criterion and Cline’s binary analysis) in predicting the region of coupled growth in a three-component eutectic system growing at steady-state. It was found that much of the one and two-phase directional solidification theory and terminology can be directly extended to a ternary eutectic system. This suggests a further extension to n-phase, m-component systems (m ≥ n) with at least a qualitative understanding of the solidification process. The Authors wish to acknowledge the support of the National Science Foundation which made this study possible.