A model of the interfacial heat-transfer coefficient during unidirectional solidification of an aluminum alloy

A model is presented for the prediction of the interfacial heat-transfer coefficient during the unidirectional solidification vertically upward of an Al-7 wt pct Si alloy cast onto a water cooled copper chill. It has been experimentally determined that the casting surfaces were convex toward the chill, probably due to the deformation of the initial solidified skin of the casting. The model was, therefore, based upon a determination of the (macroscopic) nominal contact area between the respective rough surfaces and, within this region, the actual (microscopic) contact between the casting and the chill surfaces. The model produced approximate agreement with both experimentally determined values of the heat-transfer coefficient and the measured curvature of the casting surface and showed a reasonable agreement with measured temperatures in the casting and the chill also. A common experimental technique for the experimental determination of the heat-transfer coeffcient involves the assumption of one-dimensional heat transfer only. An implication of the approach adopted in this model is that the heat transfer in the region of the casting-chill interface may be two-dimensional, and the subsequent error in the experimentally determined values is discussed.     

A model of the interfacial heat-transfer coefficient for the aluminum gravity die-casting process

Interfacial heat-transfer coefficients were measured during the solidification of Al-Si alloys against coated die steel chills with varying chill temperature, coating thickness and coating type. Two principal resistances to heat transfer across the casting-chill interface were identified, namely, (1) the resistance to heat transfer of the coating itself and (2) the resistance to heat transfer of a layer of gas, (assumed to be air), trapped between the coating and casting surfaces by virtue of their roughness. These thermal resistances were evaluated by measurement of the coating thermal conductivity and determination of the thickness of the applied coatings and the thickness of the layer of air between the coating and casting surfaces. This produced a simple equation to predict the interfacial heat-transfer coefficient during the solidification of Al alloy die castings, which produced values that were found to agree well with the experimentally determined results. This equation was used to interpret the experimentally measured heat-transfer coefficients and to explain their variation with the different experimental conditions employed. A simple modification of the equation can also take into account the formation of an air gap, where the casting locally retreats away from the die surface, leading to a local reduction in the heat-transfer coefficient.     

Heat transfer and microstructure during the early stages of metal solidification

Transient heat transfer in the early stages of solidification of an alloy on a water-cooled chill and the consequent evolution of microstructure, quantified in terms of secondary dendrite arm spacing (SDAS), have been studied. Based on dip tests of the chill, instrumented with thermocouples, into Al-Si alloys, the influence of process variables such as mold surface roughness, mold material, metal superheat, alloy composition, and lubricant on heat transfer and cast structure has been determined. The heat flux between the solidifying metal and substrate, computed from measurements of transient temperature in the chill by the inverse heat-transfer technique, ranged from low values of 0.3 to 0.4 MW/m² to peak values of 0.95 to 2.0 MW/m². A onedimensional, implicit, finite-difference model was applied to compute heat-transfer coefficients, which ranged from 0.45 to 4.0 kW/m² °C, and local cooling rates of 10 °C/s to 100 °C/s near the chill surface, as well as growth of the solidifying shell. Near the chill surface, the SDAS varied from 12 to 22 (μm while at 20 mm from the chill, values of up to 80/smm were measured. Although the SDAS depended on the cooling rate and local solidification time, it was also found to be a direct function of the heat-transfer coefficient at distances very near to the casting/chill interface. A three-stage empirical heat-flux model based on the thermophysical properties of the mold and casting has been proposed for the simulation of the mold/casting boundary condition during solidification. The applicability of the various models proposed in the literature relating the SDAS to heat-transfer parameters has been evaluated and the extension of these models to continuous casting processes pursued.     

The movement of the concave casting surface during mushy-type solidification and its effect on the heat-transfer coefficient

The air gap formation process at the casting/mold interface of a hollow cylinder casting was investigated for alloys solidifying in a mushy type by measuring the displacements of the casting and the mold surfaces during solidification. The formation process of the air gap between the convex casting surface and the outer mold and the heat-transfer coefficient through the gap have been well documented by previous publications. However, the air gap between the concave casting surface and the inner mold, or the core, was found to form differently during mushy solidification, in which the air gap formed during solidification, reached a maximum gap distance, and then decreased due to the contraction of the solidified casting on the expanding inner mold. The gap formation was caused by an inward collapse of the coherent dendrite networks at the concave interface because of low pressure inside of the casting due to solidification shrinkage. The coherent dendrite networks at the convex interface did not collapse inward. The heat-transfer coefficients estimated by measuring the air gap thickness showed a similar tendency to the calculated values obtained by the inverse heat-conduction analysis.     

Selection of Initial Mold–Metal Interface Heat Transfer Coefficient Values in Casting Simulations—a Sensitivity Analysis

Mold–metal interface heat transfer coefficient values need to be determined precisely to accurately predict thermal histories at different locations in automotive castings. Thermomechanical simulations were carried out for Al-Si alloy casting processes using a commercial code. The cooling curve results were validated with experimental data from the literature for a cylindrical-shaped casting. Our analysis indicates that the interface heat transfer coefficient (IHTC) initial value choice between chill–metal and the sand mold–metal interfaces has a marked effect on the cooling curves. In addition, after choosing an IHTC initial value, the solidification rates of the alloy near the chill–metal interfaces varied during subsequent cooling when the gap began to form. However, the gap formation, which results in an IHTC change from the initial value, does not affect the cooling curves within the vicinity of the sand–metal interface. Optimized initial IHTC values of 3000 and 7000 W m⁻²-K⁻¹ were determined for a sand–metal interface and a chill (steel or copper)–metal interfaces, respectively. The initial IHTC had a significant effect on the prediction of secondary dendrite arm spacing (SDAS) (varying between approximately 15 microns and 70 microns) and ultimate tensile strength (UTS) (varying between approximately 250 MPa and 370 MPa) for initial IHTC values that were less than the optimized value of 7000 W m⁻² K⁻¹ for the chill–metal interfaces.     

Heat flux transients at the casting/chill interface during solidification of aluminum base alloys

Heat flow at the metal/chill interface of bar-type castings of aluminum base alloys was modeled as a function of thermophysical properties of the chill material and its thickness. Experimental setup for casting square bars of Al-13.2 pct Si eutectic and Al-3 pet Cu-4.5 pct Si long freezing range alloys with chill at one end exposed to ambient conditions was fabricated. Experiments were carried out for different metal/chill combinations with and without coatings. The thermal history at nodal locations in the chill obtained during the experiments was used to estimate the interface heat flux by solving a one-dimensional Fourier heat conduction equation inversely. Using the data on transient heat flux q, the heat flow at the casting/chill interface was modeled in two steps: (1) The peak in the heat flux curve q_max was modeled as a power function of the ratio of the chill thickness d to its thermal diffusivity a, and (2) the factor (q/q_max) X α^0.05 was also modeled as a power function of the time after the solidification set in. The model was validated for Cu-10 pct Sn -2 pct Zn alloy chill and Al-13.2 pct Si and Al-3 pct Cu-4.5 pct Si as the casting alloys. The heat flux values estimated using the model were used as one of the boundary conditions for solidification simulation of the test casting. The experimental and simulated temperature distributions inside the casting were found to be in good agreement. Formerly Assistant Professor with Karnataka Regional Engineering College     

Assessment of Heat Transfer During Solidification of Al–22% Si Alloy by Inverse Analysis and Surface Roughness Based Predictive Model

Heat flux transients were estimated during unidirectional downward solidification of Al?C22% Si alloy against copper, die steel and stainless steel chills. The chill instrumented with thermocouples was brought into contact with the liquid metal so as to avoid the effect of convection associated with the pouring of liquid metal. Heat flux transients were estimated by solving the inverse heat conduction problem. Higher thermal conductivity of chill material resulted in increased peak heat flux at the metal/chill interface. Peak heat flux decreased when 100???m thick alumina coating was applied on the chill surface. The lower thermal conductivity of alumina based coating and the presence of additional thermal resistance decreases the interfacial heat transfer. For uncoated chills, the ratio of the surface roughness (R_a) of the casting to chill decreased from 6.5 to 0.5 with decrease in the thermal conductivity of the chill material. However when coating was applied on the chill, the surface roughness ratio was nearly constant at about 0.2 for all chill materials. The measured roughness data was used in a sum surface roughness model to estimate the heat transfer coefficient. The results of the model are in reasonable agreement with experimentally determined heat-transfer coefficients for coated chills.     

Heat transport and solidification in the electromagnetic casting of aluminum alloys: Part II. Development of a mathematical model and comparison with experimental results

In this second article of a two-part series, a mathematical model for heat transport and solidification of aluminum in electromagnetic casting is developed. The model is a three-dimensional one but involves a simplified treatment of convective heat transport in the liquid metal pool. Heat conduction in the solid was thought to play a dominant role in heat transport, and the thermal properties of the two alloys used in measurements reported in Part I (AA 5182 and 3104) were measured independently for input to the model. Heat transfer into the water sprays impacting the sides of the ingot was approximated using a heat-transfer coefficient from direct chill casting; because this heat-transfer step appears not to be rate determining for solidification and cooling of most of the ingot, there is little inaccuracy involved in this approximation. Joule heating was incorporated into some of the computations, which were carried out using the finite element software FIDAP. There was good agreement between the computed results and extensive thermocouple measurements (reported in Part I) made on a pilot-scale caster at Reynolds Metals Company (Richmond, VA).     

Modeling the heat flow to an operating sirosmelt lance

A mathematical model of the heat flow to a Sirosmelt lance is presented, which predicts lance wall and air temperatures and the thickness of the slag layer on the lance. By measuring the distribution of wall temperature and slag thickness on an operating Sirosmelt lance, the model was used to determine both the heat-transfer coefficient between the vessel contents and the lance and the thermal conductivity of the slag layer. The slag layer thermal conductivity was found to be within the range of 0.5 to 1.1 W m⁻¹ K⁻¹, while the outside heat-transfer coefficient varied from 80 to 150 W m⁻² K⁻¹, both of which are smaller than quoted in the literature for metal/slag systems. The discrepancy was attributed primarily to the large quantities of combustion gases that envelop the lance and reduce convection and conduction from the melt to the lance. Other factors causing low thermal conductivity and a low heat-transfer coefficient include the thermal resistance at the slag/lance interface and the mushy region on the outside of the slag layer.     

板坯连铸三维鼓肚变形仿真研究

通过建立板坯连铸凝固过程的传热模型,获得板坯冷却传热过程的坯壳生长情况,在此基础上利用有限单元法建立了坯壳三维鼓肚变形仿真分析模型。针对实际铸机的设备和工艺状况,计算了板坯凝固过程的鼓肚变形情况,并就三维仿真分析的特点进行了分析。     

Reanalysis of solidification behaviour from the microstructure in near-net-shape casting of steels

Numerical calculations revealing the relationship between the as-cast structure and cooling conditions in near-net-shape casting of steels are presented. The solidification behaviour of steels of different composition was investigated for different process conditions with a one-dimensional heat-flow model. The dependence of the secondary dendrite arm spacing (SDAS) on the distance from the chill surface was determined on the basis of the empirical relationship between local solidification time and SDAS. The numerical results were compared with experimental values of SDAS for a stainless steel, a carbon tool steel and a high-speed steel, resp. Reasonable agreement of model calculations with the experimentally determined SDAS was obtained utilizing time-dependent effective heat-transfer coefficients of the order of 2.5 to 4.5 · 10³ W/m²K for typical thin-slab dimensions and 6.0 · 10³ W/m²K for thin-strip casting.     

Thermal Analysis and Microstructure of ZA8 Alloy Solidifying Against Chills

Thermal analysis during solidification of ZA8 alloy against copper, hot die steel and stainless steel chills instrumented with thermocouples was carried out in the present work. The investigation showed that the chill material and coating had a significant effect on the cooling curve of the casting. When casting was solidified against chills, the liquidus and eutectic start temperature of the casting remained nearly the same whereas eutectoid transformation occurred at a higher temperature. Cooling rate curve of the casting solidified against coated chill indicated that formation of solid shell and subsequent re-melting in the case of high thermal conductivity coated chill whereas in lower thermal conductivity coated chill, the re-melting of solid shell was absent. It was found that chilling during solidification causes the morphology of dendrites transform to nearly rounded shape with refinement of lamellar eutectic.     

Two-phase modeling directed toward hot tearing formation in aluminum direct chill casting

The two-phase mass and momentum conservation equations governing shrinkage-driven melt flow and thermally induced deformation are formulated for the aluminum direct chill (DC) casting process. Two main mechanisms associated with hot tearing formation during solidification and subsequent cooling are thus addressed simultaneously in the same mathematical model. The approach unifies the two-phase mushy zone model outlined by Farup and Mo, the constitutive relations that treat the mushy zone as a viscoplastic porous medium saturated with liquid outlined by Martin et al., and the “classical” mechanics approach to thermally induced deformations in solid (one-phase) materials using the linear kinematics approximation. A temperature field and a unique solidification path are considered as input to the model. The governing equations are solved for a one-dimensional (1-D) situation with some relevance to the DC casting process. The importance of taking into account the transfer of momentum from the liquid phase to the solid phase is then demonstrated through modeling examples. Furthermore, the modeling results indicate that the constitutive law governing the viscoplastic behavior of the solid skeleton of the mushy zone should take into account that the solid skeleton can be compressed/dilated as well as stress space anisotropy. Calculated peak values for liquid pressure and solid stress turn out to correlate to the hot tearing susceptibility measured in casting trials in the sense that trials having the largest cracks are those for which the highest pressures and stresses are computed.     

Determination of the heat-transfer coefficient between the AK7ch (A356) alloy casting and no-bake mold

The heat-transfer coefficient h between a cylindrical cast made of AK7ch (A356) aluminum alloy and a no-bake mold based on a furan binder is determined via minimizing the error function, which reflects the difference between the experimental and calculated temperatures in the mold during pouring, solidification, and cooling. The heat-transfer coefficient is h _L = 900 W/(m² K) above the liquidus temperature (617°C) and h _S = 600 W/(m² K) below the alloy solidus temperature (556°C). The variation in the heat-transfer coefficient in ranges h _L = 900–1200 W/(m² K) (above the alloy liquidus temperature) and h _S = 500–900 W/(m² K) (below the solidus temperature) barely affects the error function, which remains at ~22°C. It is shown that it is admissible to use a simplified approach when constant h = 500 W/(m² K) is specified, which leads to an error of 23.8°C. By the example of cylindrical casting, it is experimentally confirmed that the heat-transfer coefficient varies over the casting height according to the difference in the metallostatic pressure, which affects the casting solid skin during its solidification; this leads to a closer contact of metal and mold at the casting bottom.     

Measurements, simulations, and analyses of instantaneous heat fluxes from solidifying steels to the surfaces of twin roll casters and of aluminum to plasma-coated metal substrates

Using implanted thermocouples, and an inverse heat-transfer technique, heat fluxes and associated heat-transfer coefficients during the solidification of steel in a pilot scale 0.6-m-diameter twin roll caster, whose copper contact surfaces had been treated with a propriety coating, were measured. It was found that heat fluxes during initial contact of liquid steel with the rolls were low, rising to maximum values of about 5 to 6 MW per square meter halfway down the sump of liquid steel, but then diminishing toward zero as the strip approached the roll nip. These results corresponded to roll speeds of some 7 m/min, strip thicknesses of 7 mm, and a roll separating force of 20 kN. For higher speeds and thinner strip, a secondary peak in the heat flux was observed. Associated microstructures revealed acicular ferrite, large prior austenite grains, and secondary dendrite arm spacings in keeping with measurements. In parallel experiments simulating a single belt horizontal caster, heat fluxes from strips of various aluminum alloys to coated and uncoated steel and copper substrates were measured. Under these conditions, peak heat fluxes were recorded during the period of initial contact, and depending on the coating characteristics, these reduced to a lower plateau before declining, or continuously decreased toward zero, corresponding to complete solidification of the strip. A theoretical analysis of the maximum heat-transfer rates that can be expected given perfect thermal contact of metal with the rolls, and its moderations by gas films, and substrate coatings illustrates the dominant role of the gas film and the need for dynamic heat flux measurements for quantitative modeling of fluid flows and solidification phenomena in thin strip casting operations. A model for air gap formation is proposed, based on viscous laminar flows within the gas films. Predicted thicknesses are in reasonable accord with those deduced on the basis of heat flux measurements. This article is based on a presentation made in the “Geoffrey Belton Memorial Symposium,” held in January 2000, in Sydney, Australia, under the joint sponsorship of ISS and TMS.     

Heat-Transfer Measurements in the Primary Cooling Phase of the Direct-Chill Casting Process

Thermal modeling of the direct-chill casting process requires accurate knowledge of (1) the different boundary conditions in the primary mold and secondary direct water-spray cooling regimes and (2) their variability with respect to process parameters. In this study, heat transfer in the primary cooling zone was investigated by using temperature measurements made with subsurface thermocouples in the mold as input to an inverse heat conduction algorithm. Laboratory-scale experiments were performed to investigate the primary cooling of AA3003 and AA4045 aluminum alloy ingots cast at speeds ranging between 1.58 and 2.10 mm/s. The average heat flux values were calculated for the steady-state phase of the casting process, and an effective heat-transfer coefficient for the global primary cooling process was derived that included convection at the mold surfaces and conduction through the mold wall. Effective heat-transfer coefficients were evaluated at different points along the mold height and compared with values from a previously derived computational fluid dynamics model of the direct-chill casting process that were based on predictions of the air gap thickness between the mold and ingot. The current experimental results closely matched the values previously predicted by the air gap models. The effective heat-transfer coefficient for primary cooling was also found to increase slightly with the casting speed and was higher near the mold top (up to 824 W/m²·K) where the molten aluminum first comes in contact with the mold than near the bottom (as low as 242 W/m²·K) where an air gap forms between the ingot and mold because of thermal contraction of the ingot. These results are consistent with previous studies.     

Experimental Determination of Heat Transfer Across the Metal/Mold Gap in a Direct Chill (DC) Casting Mold��Part I: Effect of Gap Size and Mold Gas Type

An experimental apparatus to determine the heat-transfer coefficient in the gap formed between the cast metal and the mold wall of a vertical direct chill (DC) casting mold is described. The apparatus simulates the conditions existing within the confines of the DC casting mold and measures the heat flux within the gap. Measurements were made under steady-state conditions, simulating the steady-state regime of the DC casting process. A range of casting parameters that may affect the heat transfer was tested using this apparatus. In the current article, the operation of the apparatus is described along with the results for the effect of gas type within the mold, and the size of the metal-mold gap formed during casting. The results show that the gas type and the gap size significantly affect the heat transfer within a DC casting mold. The measured heat fluxes for all the conditions tested were expressed as a linear correlation between the heat-transfer coefficient and the metal-mold gap size, and the fluxes can be used to estimate the heat transfer between the metal and the mold at any gap size. These results are compared to values reported in the literature and recommendations are made for the future reporting of the metal/mold heat-transfer coefficient for DC casting. The results for the effect of the other parameters tested are described in Part II of the article.     

A correlation to describe interfacial heat transfer during solidification simulation and its use in the optimal feeding design of castings

It is known from experimental data that for pure aluminum castings manufactured via the gravity die casting process, the interfacial heat-transfer coefficient can vary in the range 500 to 16,000 W/m² K. These coefficients are of significant importance for the numerical simulation of the solidification process. The experimentally determined variation of interfacial heat-transfer coefficients with respect to time has been recalculated to highlight the variation with respect to casting temperature at the interface. This variation was observed to be of an exponential nature. Also, the pattern of variation was found to be similar in all the experimental results. It has been found that all these patterns of interfacial heat-transfer coefficient variation can be matched by a unique equation that has been proposed as a correlation to model the metal-mold interfacial heat transfer. The benefit of this correlation is in its ability to approximate the combined effects of geometry variation, insulation, chills, die coatings, air gap formation, etc. during the numerical simulation and its use in the optimal design of heat transfer at the metal-mold interface.     

Analysis of bulging for high speed continuous casting

In order to research the temperature distribution and mechanical deformation of slab bulging during high speed continuous casting,mathematical models have been developed to analyze the thermal and mechanical behavior of the slab.The thermal history of the slab has been predicted by a two-dimensional transient finite element heat transfer model,whose results serve as the input to the stress model.The stress model has been formulated for a two-dimensional longitudinal plane.In this case,the maximum tensile strain during the bulging process is located at the solidification front just past the top of the upstream roll,which may contribute to crack formation.The maximum tensile stresses are located at the cold surface in the middle of the two back-up rolls,just at the point of the maximum bulging.Stresses near the solidification front are small because of the high temperatures which produce lower elastic modulus values.Finally,the effect of the casting speed on the bulging deformation is discussed.     

板坯传热二冷动态控制模型的有限体积法

阐述了时空有限体积法的基本思想，应用其方法，以钢水的凝固过程为多元系相变的实际为出发点，建立了连铸板坯凝固传热的二冷动态控制模型，求得表面温度的分布，表面温度、二冷区各段水量随拉速的变化关系。模型的模拟数据和现场实测数据吻合良好。算法对连铸过程中拉速的变化具有良好的适应能力，拉速引起的表面温度波动很小，能可靠地反映浇注条件频繁发生变化的实际连铸坯的凝固传热过程。