HEAT TRANSFER AND MICROSTRUCTURE DURINGTHE EARLY STAGES OF SOLIDIFICATION OFMETALSByCornelius Anaedu MuojekwuB.Sc., The University of Ife, Nigeria, 1987M.Sc., The University of Lagos, Nigeria, 1990A THESIS SUBMITTED IN PARTIAL FULFILLMENT OFTHE REQUIREMENTS FOR THE DEGREE OFMASTER OF APPLIED SCIENCE (M.A.Sc.)inTHE FACULTY OF GRADUATE STUDIESDEPARTMENT OF METALS AND MATERIALSENGINEERINGWe accept this thesis as conformingto the required standardTHE UNIVERSITY OF BRITISH COLUMBIAJuly 1993© Cornelius Anaedu Muojekwu, 1993In presenting this thesis in partial fulfilment of the requirements for an advanceddegree at the University of British Columbia, I agree that the Library shall make itfreely available for reference and study. I further agree that permission for extensivecopying of this thesis for scholarly purposes may be granted by the head of mydepartment or by his or her representatives. It is understood that copying orpublication of this thesis for financial gain shall not be allowed without my writtenpermission.(Signature)Department of /W. The University of British ColumbiaVancouver, CanadaDate c DE-6 (2/88)ABSTRACTThe future of solidification processing clearly lies not only in elucidating the various aspectsof the subject, but also in synthesizing them into unique qualitative and quantitative models.Ultimately, such models must predict and control the cast structure, quality and properties ofthe cast product for a given set of conditions Linking heat transfer to cast structure is an invaluableaspect of a fully predictive model, which is of particular importance for near-net-shape castingwhere the product reliability and application are so dependent on the solidification phenomena.This study focused on the characterization of transient heat transfer at the early stages ofsolidification and the consequent evolution of the secondary dendrite arm spacing. Water-cooledchills instrumented with thermocouples were dipped into melts of known superheats such thatunidirectional solidification was achieved. An inverse heat transfer model based on the sequentialregularization technique was used to predict the interfacial heat flux and surface temperature ofthe chill from the thermocouple measurements. These were then used as boundary conditionsin a 1-D solidification model of the casting. The secondary dendrite arm spacing (SDAS) atvarious locations within the casting was computed with various semi-empirical SDAS models.The predictions were compared with experimental measurements of shell thickness and secondarydendrite arm spacing from this work as well as results reported in the literature. The effects ofsuperheat, alloy composition, chill material, surface roughness and surface film (oil) wereinvestigated.The results indicate that the transient nature of the interface heat transfer between the chilland casting exerts the greatest influence in the first few seconds of melt-mold contact. Theinterfacial heat flux and heat transfer coefficient exhibited the typical trend common tosolidification where the initial contact between mold and melt is followed by a steadily growinggap. Both parameters increase steeply upon contact up to a peak value at a short duration (< 10iis), decrease sharply for a few seconds and then gradually decline to a fairly steady value. Heattransfer at the interface increased with increasing mold diffusivity, increasing superheat,decreasing thermal resistance of the interfacial gap, increasing thermal expansion of the mold,decreasing shrinkage of the casting alloy, decreasing mold thickness and initial temperature,and decreasing mold surface roughness. The secondary dendrite arm spacing decreased withincreasing heat flux for the same alloy system and depended on the cooling rate and localsolidification time. The secondary dendrite arm spacing was also found to be a direct functionof the heat transfer coefficient at distances very near the casting/mold interface.A three stage empirical heat flux model based on the thermophysical properties of the moldand casting was proposed for the simulation of the mold/casting boundary condition duringsolidification. The applicability of the various models relating secondary dendrite arm spacingto heat transfer parameters was evaluated and the extension of these models to continuous castingprocesses was pursued.iiiTable of ContentsABSTRACT^ iiTable of Contents ivTable of Tables ^ viiTable of Figures viiiNomenclature ^ xiAcknowledgement xviiiChapter 1 INTRODUCTION ^ 11.1 Fundamentals Of Solidification Processing. ^ 11.2 Heat and Fluid Flow During Solidification 41.3 Microstructural Evolution during Solidification. ^ 71.3.1 Nucleation ^ 71.3.2 Dendritic Growth 8Chapter 2 LI1ERATURE SURVEY^ 122.1 Solidification Modeling 122.2 Heat Flow - Interface Resistance ^ 142.3 Heat Flow - Latent Heat Evolution 212.3.1 Temperature Recovery Method^ 242.3.2 Specific Heat Method^ 252.3.3 Enthalpy Methods 272.3.4 Latent Heat Method^ 28iv2.3.5 The Nature of Latent Heat Evolution ^ 292.3.5.1 Cooling Curve Analysis 302.4 Fluid Flow During Solidification^ 332.5 Microstructural Evolution 342.5.1 Nucleation ^ 352.5.2 Growth 372.5.2.1 Dendrite Arm Spacing and Coarsening ^ 422.6 Coupling Heat Transfer and Microstructural Evolution 502.6.1 Complete Mixing Models^ 512.6.2 Solute Diffusion Models 51Chapter 3 SCOPE AND OBJECTIVES^ 543.1 Objectives/Importance ^ 543.2 Methodology^ 55Chapter 4 EXPERIMENTAL PROCEDURE AND RESULTS ^ 584.1 Design ^ 584.2 Instrumentation and Data Acquisition ^ 624.3 Dipping Campaigns ^ 634.3.1 Thermal Response of the Thermocouples ^ 674.4 Metallographic Examination^ 72Chapter 5 MATHEMATICAL MODELING^ 805.1 Chill Heat Flow Model ^ 805.2 Casting Heat Flow Model ^ 835.2.1 Latent Heat Evolution and Fraction Solid ^ 885.3 Dendrite Arm Spacing (DAS) Models^ 905.4 Sensitivity Analysis And Model Validation 91Chapter 6 RESULTS AND DISCUSSION^ 966.1 Heat Flow^ 966.2 Microstructure Formation ^ 1016.3 Effect of Process Variables 1056.3.1 Effect of Surface Roughness ^ 1056.3.2 Effect of Chill Material 1096.3.3 Effect of Superheat^ 1136.3.4 Effect of Alloy Composition ^ 1186.3.5 Effect of Oil Film ^ 1236.3.6 Effect of Bath Height 1296.4 Proposed Empirical Model ^ 1326.5 Implications for Continuous and Near-Net-Shape Casting ^ 138Chapter 7 SUMMARY AND CONCLUSIONS/RECOMMENDATIONS^ 142REFERENCES ^ 146APPENDIX A 158viTable of TablesTable 2.1 Various expressions for heat transfer coefficient ^ 17Table 2.2 Various expressions for primary dendrite arm spacing 45Table 2.3 Various expressions for secondary dendrite arm spacing ^ 47Table 2.4 Complete mixing models for the evaluation of solid fraction ^ 52Table 4.1 Some details of the experimental design ^ 60Table 4.2 Thermocouple calibration in boiling water 63Table 4.3 Properties of the oils used in the experiments ^ 65Table 4.4 Measured thermocouple response ^ 67Table 4.5 Typical secondary dendrite arm spacing measurement ^ 75Table 5.1 Thermophysical Properties Used in the Chill Model 83Table 5.2 Thermophysical Properties Used in the Casting Model^ 86Table 5.3 Input and recalculated interfacial heat flux ^ 94Table 6.1 Measured and predicted secondary dendrite arm spacing ^ 103Table 6.2 Measured shell surface roughness for various chill surface microprofiles ^ 106Table 6.3 Measured shell surface roughness for various chill materials ^ 110Table 6.4 Measured shell surface roughness for various superheats ^ 118Table 6.5 Measured shell surface roughness for different alloy compositions ^ 120Table 6.6 Measured shell surface roughness for the four oils ^ 127Table 6.7 Measured shell surface roughness for different bath heights ^ 129viiTable of FiguresFig. 1.1 Inter-relationship between microstructure and other process variables ^ 4Fig. 1.2 Schematic diagram of the casting/mold interface ^ 6Fig. 1.3 Schematic representation of dendritic growth 11Fig. 2.1 Process understanding and improvement with the aid of modeling^ 13Fig. 2.2 Typical variation of heat transfer coefficient with time ^ 19Fig. 2.3 Variation of interfacial heat flux with time ^ 21Fig. 2.4 Handling phase change in solidification modeling 23Fig. 2.5 Variation of specific heat with temperature during solidification^ 27Fig. 2.6 Variation of enthalpy with temperature^ 29Fig. 2.7 Growth phenomena during solidification 39Fig. 2.8 Various dendrite coarsening models ^ 44Fig. 3.1 Schematic illustration of the project methodology^ 57Fig. 4.1 Schematic representation of the experimental set-up 59Fig. 4.2 Operating thermal resistances ^ 61Fig. 4.3 Variation of the thermal resistances with time^ 62Fig. 4.4 Schematic representation of surface microprofile 64Fig. 4.5 Typical temperature data during casting ^ 68Fig. 4.6 Effect of surface roughness on measured temperature^ 69Fig. 4.7 Effect of chill material on measured temperature 69Fig. 4.8 Effect of superheat on measured temperature ^ 70Fig. 4.9 Effect of alloy composition on measured temperature^ 70Fig. 4.10 Effect of oil film on measured temperature^ 71Fig. 4.11 Effect of bath height on measured temperature 71Fig. 4.12 Typical micrographs of A1-7%Si alloy ^ 73viiiFig. 4.13 Typical measured secondary dendrite arm spacing (SDAS)^ 76Fig. 4.14 Effect of surface roughness on measured SDAS^ 76Fig. 4.15 Effect of chill material on measured SDAS 77Fig. 4.16 Effect of super heat on measured SDAS^ 77Fig. 4.17 Effect of alloy composition on measured SDAS 78Fig. 4.18 Effect of oil film on measured SDAS ^ 78Fig. 4.19 Effect of bath height on measured SDAS 79Fig. 5.1 Discretization of chill and casting ^ 82Fig. 5.2 Computer implementation of the models 87Fig. 5.3 Zones in the effective specific heat method ^ 89Fig. 5.4 Calculated and measured temperature profiles in the chill ^ 92Fig. 5.5 Analytical and numerical solutions of infinite slab assumption^ 93Fig. 5.6 Calculated and measured temperature profiles in the casting 95Fig. 5.7 Calculated and measured shell thickness ^ 95Fig. 6.1 Model predictions - interfacial heat flux and heat transfer coefficient ^ 97Fig. 6.2 Model predictions - shell thickness and interfacial gap ^ 98Fig. 6.3 Calculated surface temperature profiles for chill and casting ^ 99Fig. 6.4 Calculated and measured secondary dendrite arm spacing 104Fig. 6.5 Effect of surface roughness on heat transfer ^ 107Fig. 6.6 Effect of surface roughness on solidification and microstructure ^ 108Fig. 6.7 Effect of chill material on heat transfer^ 111Fig. 6.8 Effect of chill material on solidification and microstructure ^ 112Fig. 6.9 Variation of the chill thermal resistance with time ^ 113Fig. 6.10 Effect of superheat on heat transfer^ 116Fig. 6.11 Effect of superheat on solidification and microstructure ^ 117ixFig. 6.12 Effect of alloy composition on heat transfer ^ 121Fig. 6.13 Effect of alloy composition on solidification and microstructure ^ 122Fig. 6.14 Effect of oil film on heat transfer ^ 124Fig. 6.15 Effect of oil film on solidification and microstructure ^ 125Fig. 6.16 Surface temperatures of the chill and casting with the oils 126Fig. 6.17 Effect of bath height on heat transfer ^ 130Fig. 6.18 Effect of bath height on solidification and microstructure ^ 131Fig. 6.19 Proposed empirical heat flux model ^ 133Fig. 6.20 Variation of peak heat flux with some process variables ^ 135Fig. 6.21 Variation of peak heat flux with other process variables 136Fig. 6.22 Casting simulation with the empirical model for A1-7%Si ^ 137Fig. 6.23 Casting simulation with the empirical model for A1-3%Cu-4.5%Si^ 137Fig. 6.24 Empirical model applied to twin-roll casting of 0.8%C steel ^ 140Fig. 1A Schematic illustration of IHCP methods ^ 159Fig. 2A Schematic illustration of future time assumption 161Fig. 3A Flow diagram of the sequential IHCP technique^ 163NOMENCLATUREa,ao , A^constants^a^half width (horizontal) of a V-groove in Table 2.1, mA^area in Eq. (2.18), m2b,b0, B^constantsBi^Biot number = hljkc, Co, C 1 ,....C 11^constantsCa, ce, ci, cr, co^Ceff^composition, %effective specific heat capacity, J/Kg.KCp^specific heat capacity, J/Kg.K^Cpseudo^pseudo specific heat capacity, J/Kg.Kd, do^constantsD^diffusion coefficient, m2/sDAS^dendrite arm spacing, pme, eo^constantsFo^Fourier number = at/x 2 for heat conduction or Dt/x2 for diffusion,m 2/sf1^liquid fractionfs^solid fractiong^gravitational acceleration, m 2/sG^thermal gradient, °C/mAGA^diffusional activation energy for growth, JGe^solute gradient, m -1energy of formation of the critical nucleus, Jlumped material parameterAGehaxi^h^heat transfer coefficient, W/m2.K^h'^empirical constant in Table 2.1H enthalpy, J/kgHv^Vickers hardness of the softer solid in a contacting interface in Table2.1k^thermal conductivity, W/m.Kkb^Boltzmann constant in Eq. (2.31) = 1.38054 x 10-23 J/Kkp^partition coefficientK solidification constant in the parabolic shell growth expression, mis ic2K 1 , K2, K3, K4^empirical constants in Eqs. (2.32) and (2.33)length, mcharacteristic length (Volume/Area), mL latent heat, J/KgLc^coefficient of thermal contraction for the casting, nun/mLm^coefficient of linear thermal expansion for the mold, nun/mLv^volumetric latent heat, J/m3m^liquidus slope from the phase diagram, K/%ns^number of temperature sensorsnumber of surface atoms of the substrate per unit volume of liquid,atoms/m3N nucleated particle density, particles/m3N number of nodes in Eq. (5.7)Ns^original heterogeneous substrate density, substrates/m 3Nu^Nusselt number = h1c/kfP pressure, N/m2Pe^Peclet Number = vr/Dxii^Pr^Prandtl Number = a/p.q^heat flux, W/m2^Q'^rate of latent heat release, W/m 2r^radius, mr^number of future time steps in Appendix A^r *^radius of critical nucleeus, mR^growth rate, m/sR^thermal resistance in Fig. 4.2, m 2 .K/WRa^surface roughness, p.mRe^Reynolds Number = p u 1,./p.S^shell thickness, m^Si, S2^temperature jump distance, mSC^sensitivity coefficientS L^least square functionT^temperature, °CTC^calculated temperature, °CTM^measured temperature, °CTp^pouring or teeming temperature, °C^T 1'^temperature of a point midway between a node and the succeedingnode, °C^T 1^temperature of a point midway between a node and the preceding node,°Ct^time, s^tf^local solidification time, su^velocity along the x-axis, m/s^v^velocity along the y-axis, m/sV^volume, Kg/m3Vf1, Vf2^volume fractionsw^velocity along the z-axis, m/sx^distance along the x-axis, mx r^distance between roughness peaks of the rougher surface at an interfacein Table 2.1X^chill thickness, my^distance along the y-axis, mz^distance along the z-axis, ma^thermal diffusivity, m2/s^i3^constanta^thermal emissivityinterfacial energy, N/mGibbs-Thomson coefficient, K.m^X,^primary dendrite arm spacing, pmX2^secondary dendrite arm spacing, pmdynamic viscosity, Kg/m.s^112^nucleation constant, m-3 .K-222/7^p^density, Kg/m3a^Stefan-Boltzmann constant = 5.669 x 10-8iw m2.Ka0^angle between the vertical and the gap region of a V-groove in contactwith a melt, °xivhA^e^inverse of the time constant (e-Tic.--), s -1constantSubscripts^c^relating to casting^c^relating to capillarity in Eq. (2.40)c^relating to the critical value in Table 2.2cc^relating to cooling curvech^relating to the chillch/c^relating to chill/casting interfacechs^relating chill surfacecs^relating to casting surfacee^relating to eutecticeff^relating to an effective valueeq^relating to an equivalent valuef^relating to cooling fluidg^relating to the interfacial gapi^generic index for item or pointk^relating to kinetics in Eq. (2.40)I^relating to the liquidL^relating to the liquidusmax^relating to the maximum valuen^relating to the nucleus0^relating to the initial or original valuer^relating to radiation in Eq. (2.3)xv^s^relating to solid^s^relating the solute in Eq. (2.40)^S^relating to the solidusshell^relating to the solidified shellsi^relating to solid/liquid interface^t^relating to the dendrite tipt^relating to thermal in Eq. (2.40)^x^relating to the x-axisy^relating to the y-axis^z^relating to the z-axiszc^relating to the zero curve in Eq. (2.19)^a^relating to ferriterelating to austeniterelating to the ambient or surroundingSupercripts^a^exponent in Eq. (6.4)b^exponent in Table 2.1 and Eq. (6.4)c^exponent in Eq. (6.4)d^exponent in Eq. (6.5)e^exponent in Eq. (6.5)f^exponent in Eq. (6.5)i^time index in Appendix A (i=1,2,^r)j^space index in Appendix A (j=1,2, ^ns)m^time index for estimating heat flux (m=1,2^ tT)n^relating to the n'th timexvin i , n2,^ , n8^exponents^x^exponent in Eq. (2.27)y^exponent in Eq. (2.27)xviiAcknowledgementI would like to dedicate this work to the late Dr. P.E. Anagbo whose encouragement andassistance were instrumental to my studies at U.B.C., and to my CREATOR who has been makingmy life a continuing miracle.My gratitude goes to my supervisors, Dr. I.V. Samarasekera and Dr. J.K. Brimacombe,for their invaluable counsel and for providing the research assistantship that supported this study.The CONTA IHCP code provided by Dr. J.V. Beck is gratefully acknowledged. I alsoacknowledge the assistance of Neil Walker, Peter Musil and other MMAT technical staff incarrying out this research. My appreciation goes to my family, my friends and colleagues fortheir priceless support and solidarity.Cornelius Anaedu MuojekwuJuly 1993xviiiChapter 1INTRODUCTION1.1 Fundamentals Of Solidification Processing.Solidification can be defined as the transformation from a liquid phase to a solid phaseor phases. The phenomena associated with the process of solidification are complex and varied.It is especially difficult to conceive of the initial stages of the process, when the first crystalsor center of crystallization appears. Genders' proposed his solidification theory in 1926 but itwas Chalmers et al. 2 that later attempted to offer a comprehensive qualitative and quantitativeunderstanding of this theory.Chalmers and co-workers considered the instantaneous structure of liquid near its meltingpoint as one in which each atom is part of a "crystal-like cluster or micro-volume", orientatedrandomly and with "free space" between it and its neighboring clusters. These clusters wouldform and disperse very quickly through the transfer of atoms from one to another by movementacross the intervening free space. With reference to the extensive thermodynamics work ofGibbs 3 , they conceived of the possibility of clusters of all possible structures existing in theliquid near its melting point, such that those of lowest free energies become more stable andare favoured during nucleation. While each atom in the liquid is at a free energy minimum,these minima are nonetheless higher than those of the solid during nucleation. This accountsfor the evolution of the latent heat of fusion. So long as the clusters are below a certain criticalsize (embryo) corresponding to the liquidus temperature, they cannot grow and no tangiblesolid is formed. However, if the thermal condition is such that the critical size is less than thelargest cluster size, then nucleation occurs and the supercritical clusters (now nuclei) grow intocrystals.1The above consideration formed the basis of the usual conception of solidification as adual process of nucleation and growth. Ever since, solidification phenomena have been studiedfrom three major perspectives:1.Atomic Level; usually dominated by the atomic processes by which nucleation and growthoccurs. Emphasis has been on atomic sites (crystal structure), nucleation type and rate,atomic defects etc.2. Microscopic Level; dominated by microstructural evolution and growth. Such topics asphases and microstructures, interfacial phenomena, growth pattern, grain size and density,microscopic defects, etc, have been studied.3. Macroscopic Level; where the flow of liquid metal and the extraction of heat from thesolidifying casting predominate. Emphasis has been on fluid flow and heat transfer,macrostructure, shell thickness and pool profile, surface characteristics, shape,macroscopic defects, stress distribution, etc.Solidification modeling based exclusively on any of these three levels is important, butan integrated approach that couples the different levels will be an invaluable tool for theoptimization of the solidification process. As far back as 1964, Chalmers 2 recognized this whenhe noted in the preface to his book; "Principles Of Solidification", that the rapid progress madein elucidating the various separate aspects of solidification has not been matched by applicationof this knowledge to the problems encountered in industry. Of course, substantial progress hassince been made in terms of application of solidification knowledge but the pool of knowledgeremains so distant from application.It is strongly believed, therefore, that the future of solidification processing and modelinglies not only in understanding the various aspects of the subject, but also in coupling most ofthe different approaches and models into some uniquely comprehensive, qualitative and2quantitative packages. Such packages must allow for extensive prediction and control ofstructure, quality and properties of the solidified product, once the solidification conditionsand parameters are known. It is envisaged that in the distant future, a casting operator shouldbe able to establish a production route through a systematic material/process selection data base,once the casting quality and service requirements are known. This could be achieved if eachof the routes in Fig. 1.1 could be replaced by quantitative models linking the various stages inthe production schedule.In addition, the present trend towards near-net-shape casting minimizes or eliminates theneed for mechanical working of manufactured components and, often, the separateheat-treatment procedures. Therefore, the principal and enormous task of creating the requiredmicrostructure which determines the product quality, rests squarely on the solidification process.Thus, the reliability of the product is now solely dependent on the solidification phenomena.It is then obvious that any successful development of near-net-shape casting will dependcritically on the understanding and application of fundamental knowledge of solidificationcarried into the rapid solidification range. It is envisaged that the usefulness of this kind ofknowledge will require some definite links between the separate processes that contribute tosolidification. Of particular importance in this regard is the link between the microstructureand hence product quality, and other aspects of solidification such as fluid flow, heat transfer,nucleation and growth processes, as well as defects.This work focuses on the coupling of heat transfer phenomena and the resultantmicrostructure at the early stages of solidification in low melting point alloys. Dendrite armspacing (DAS) is used as a measure of the degree of fineness of the microstructure. The evolutionof the desired microstructure is ultimately linked to all the processes that contribute tosolidification, and the microstructure can be predicted if the quantitative relationships betweenit and the other processes are known.3Alloy Selectionmicrostructure = f(composition & grade)Melting & Teeming Practicesmicrostructure = f(porosity, inclusions,temperature) Casting Techniquemicrostructure = f(heat flow, fluid flow andsolidification parameters)Microstructurecasting quality & properties = f(microstructural parameters)Castingservice requirement = f(microstructural parameters)Casting Applicationperformance rating = f(microstructural parameters)Material PerformanceFig. 1.1 Schematic illustration of the inter-relationship between themicrostructure and other process variables.1.2 Heat and Fluid Flow During SolidificationDuring solidification of metal on a substrate surface, the overall heat flow is a functionof three major thermal resistances: the mold resistance, the interface resistance and the castingresistance. These resistances reduce the overall heat flow during casting. In most castingprocesses, it is desirable to control these resistances in order to optimize the solidificationprocess. The casting resistance usually depends on the shell thickness which in turn depends4on the mold and interface resistances. The mold resistance can be controlled by adequate choiceof mold material, mold dimensions and cooling method. The characterization of the interfacialresistance has always been a major source of uncertainty in the modeling of any solidificationprocess. This resistance is a time-dependent variable particularly at the initial stages. Thetransient nature of interfacial resistance during casting is attributed to the dynamics of themetal/mold contact surface or surfaces.In general, the metal/mold interface may exist in three major forms 4 - (a) clearance gap,(b) conforming contact or, (c) non-conforming contact as illustrated in Fig. 1.2. There couldbe a combination of these states at each stage of the solidification process. Each of these statesaffects the interface resistance by a different amount.In the case of conforming contact, perfect contact could be assumed such that heat transferacross the interface becomes a classical heterogeneous thermal contact problem. The thermalconductance in the interface is expressed in terms of thermal conductivities of the media incontact, the real area in contact, number of contact spots per unit area, actual surface profiles,etc. For nonconforming contact, interfacial oxide films and mold coatings together with the -factors mentioned above are limiting factors to interfacial heat transfer.When the surfaces of the metal and mold are separated by a gap of finite thickness, theheat transfer across this gap most often limits the effectiveness of heat transfer between themetal and the mold. Surface interactions, geometric effects, transformations of metal and moldmaterials are some of the factors that contribute to gap formation. Once the gap is formed, heattransfer across the gap could occur in any of the three modes of heat transfer: conduction,convection and radiation. The extent of each mode is controlled by the gap width, thecomposition of the gap and the temperature of the two surfaces separated by the gap.5Fluid flow during casting results from either induced or natural forces. Within the bulkliquid region, the teeming mechanism and any form of stirring or vibration are the major sourcesof induced forces that affect fluid dynamics during casting. Natural forces which originate fromthermal gradients, solute gradients, surface tension and transformation can also createsignificant fluid flow within a casting.Convection induced by fluid flow influences solidification at both the macroscopic andmicroscopic levels. At the microscopic level, it can change the shape of the isotherms andreduce the thermal gradients within the liquid region. Even if this does not dramatically modifythe overall solidification, the local solidification conditions, macrosegregation, and themicrostructure itself can be greatly affected by convection s. Within the mushy zone, volumechanges during solidification can drag the fluid in (or out) of the interdendritic region andultimately lead to microporosity formation.Fig. 1.2 Casting/Mold interface'.61.3 Microstructural Evolution during Solidification.Microstructural evolution during casting has been a subject of great interest to researchersfor some time. The degree of fineness of the microstructure determines the quality and propertiesof the cast component. The goal of most practical casting processes is to obtain fine isotropiccrystals such that segregation, porosity, and other defects are substantially reduced.Microstructural evolution is dependent on the dual process of nucleation and growth.1.3.1 NucleationNucleation may be defined as the formation of new phase (solid in the case ofsolidification) in a distinct region separated by a discrete boundary or boundaries. With respectto kinetics, nucleation can be classified either as continuous or instantaneous. Continuousnucleation assumes that nucleation occurs continuously once the nucleation temperature isreached while instantaneous nucleation assumes that all nuclei are generated at the same timeat a given nucleation temperature. Based on the nucleation sites, two distinct types of nucleationare known: homogeneous and heterogeneous nucleation. Homogeneous nucleation occurswhen all locations have an equal chance of being nucleation sites. On the other hand,heterogeneous nucleation occurs when certain locations are preferred sites. In most practicalcastings, the nucleation process is invariably heterogeneous; points on the substrate surfaceand any inhomogeneities in the bulk liquid being preferred sites.Not all the physical features which determine the properties of a surface for heterogeneousnucleation of a phase are understood. In terms of surface matching, the concept of coherencyis important'. A coherent interface is one in which matching occurs between atoms on eitherside of the interface. If there is only partial matching, the interface is considered to besemi-coherent. The ratio of the lattice parameter of the crystal being nucleated to that of thesubstrate is used as a measure of surface matching. Coherent surfaces are characterized by asingle source of strain energy at the interface (strain due to misfit) and allow for good wettability.7On the other hand, semi-coherent interfaces are characterized by both misfit and dislocationstrains, and therefore reduce the wettability of the surface by molten metal. Furthermore,charge distribution which leads to some electrostatic effects can influence the choice ofnucleation sites'.1.3.2 Dendritic GrowthOnce solid nuclei have been formed, they will grow provided the thermodynamicconditions (mainly energy reduction) are fulfilled. In terms of the nature of transformation,two main types of growth morphology have been identified8 - eutectic and dendritic. Eutecticgrowth involves the transformation of liquid simultaneously into two solids while dendriticgrowth involves transformation into a single solid phase.With respect to the solid/liquid interface geometry, growth can be dendritic, planar,cellular, lamellar or even armophous. Dendritic growth is by far the most common growthmorphology in alloys 9 except for the case of eutectics where cellular growth predominates.A dendrite element is defined as that portion of a grain at the completion of solidificationwhich is surrounded largely by an isoconcentration surface. Depending on the nucleation andheat flow conditions, dendritic growth could be equiaxed or columnar. Columnar dendriticgrowth is mainly solute diffusion controlled while equiaxed dendritic growth is heat and/orsolute diffusion controlled. Therefore, dendritic growth can be heat flux, solute flux, or heatand solute flux controlled. The shape of the dendrites has been found to depend on the heatflow conditions, small undercooling resulting in cylindrical dendrites while large undercoolingproduces spherical dendritee.The first set of dendrites grows parallel to the direction of heat flow (more pronouncedin the case of columnar growth) and are termed 'primary dendrites'. These dendritessubsequently become preferred sites for further nucleation and growth, leading to branching.8Growth stops when the dendrite tip encounters a barrier in its path, usually other dendrites.The idealized final form of the dendrite elements consists of primary, secondary, tertiary,quaternary and more arms as illustrated in Fig. 1.3. It is to be noted that there is a strongcompetition among the different arms and only the relatively larger ones survive at the end ofsolidification; the others shrink and eventually disappear as a result of coarsening.The driving force for arm coarsening is the reduction in total surface energy in the systemwhich acts through the Gibbs-Thomson effect at curved surfaces. Thus, solid surfaces ofdifferent curvatures, both positive and negative, establish different liquid concentrations attheir interfaces and diffusion in the liquid from high to low solute regions results inmorphological changes. The coarsening effect is more pronounced in secondary and higherorder arms, than in the primary arms. This is because the primary dendrites are moregeometrically constrained, thereby reducing the effectiveness of coarsening phenomena. Ithas been established that for most solidification processes, the coarsening phenomenon ratherthan the initial dendrite arm, is the overiding factor that controls the final dendrite arm spacing".Hence, the secondary dendrite arm spacing is a better indication of local heat flux and soluteflux conditions during solidification.The dendrite arm spacing (DAS) is a fundamental characteristics of microstructure andhas been used over the years as a measure of fineness and, hence, quality of cast products.Both primary and secondary dendrite arm spacings have been employed to quantify the degreeof fineness of microstructure. Dendrite arm spacing has been linked empirically to othersolidification parameters such as dendrite tip velocity, cooling rate, temperature gradient insolidifying material, local solidification time and distance from the chill surface.9The importance of DAS and its suitability as an efficient structural parameter forprocess-structure and structure-property relations in cast products have been illustrated byvarious researchers 12-19 . It has been shown that the dendrite arm spacing can be related to thefollowing:(i) tensile properties of a casting12 ' 13 (ultimate tensile strength, yield strength, percentelongation, etc)(ii) fractography of unidirectionally solidified alloys l 1 ' 14 (crack length, percentageelongation to failure, micro-hardness and impact energy)(iii) defects 15-17 (segregation, porosity, inclusion)(iv) heat treatment characteristics of casting" (homogenization)(v) subsequent mechanical working of cast ing" (extrusion)(vi) corrosion behavior of casting 18,191 0Fig. 1.3 Schematic illustration of dendritic growth". An initial dendritearm spacing, do, is formed early during solidification (a).Subsequently, some of the arms disappear (b & c), so that thedendrite arm spacing increases to the final size, df (c). The possibledendritic structure at the end of solidification is represented in (d).1 1Chapter 2LITERATURE SURVEY2.1 Solidification ModelingThe understanding and control of complex processes such as solidification are oftenachieved through a rigorous application of analysis and synthesis, a procedure known as processmodeling20. Process modeling could be defined as a comprehensive elucidation of a process orits component part in both qualitative and quantitative terms such that the process or its partcould be better understood, controlled or improved. The basic steps in process modeling areillustrated in Fig. 2.1. It is a dual process of analysis and synthesis that involves a combinationof two main tools21 :(i) experimental procedures (observations and measurements in one or more of thefollowing: laboratory, existing process, pilot plant and physical model)(ii) mathematical modelingThe first step is to break down the problem into its component parts that are sufficientlydetailed to allow a comprehensive study of the fine details using the above modeling tools.Following this step of analysis of individual building blocks of the process, the process is thensynthesized by incorporating these blocks into a model of the entire process. Now, the processis better understood and its behavior in practice can be predicted. It is then possible to control,modify and improve the process. The process can equally be scaled to other sizes or the improvedunderstanding of the process can be applied to develop a wholly new one.Solidification phenomenon has benefitted from all the basic tools of process modeling 20 .Observations and measurements yield the fundamental understanding and knowledge, but12sv►I►IEStSMATHEIAATICAL••400Et.PROCESStor.focasTowooRGANALYSISXPf RIME NIA(u00ELmathematical models provide the framework to assemble and apply this knowledgequantitatively, for a deeper understanding, control and improvement of the process. Hence,mathematical modeling is a very powerful tool in quantitative process analysis and synthesis'.Fig. 2.1 Schematic illustration of_process understanding and improvementwith the aid of modeling'.Based on the above, it is now widely accepted that a complete model for the simulationof solidification22 should include both the macroscopic modeling (heat transfer, fluid flow,stress distribution, macrostructure and macro-defects) and microscopic modeling(microstructure, microsegregation, microporosity and other micro-defects).From a mathematical viewpoint, solidification modeling has been directed towards asearch for analytical and numerical solutions to the continuity equations in the presence of aphase change s. Analytical solutions have been applied to a limited number of simplified cases'(mainly lumped capacity approximation, semi-infinite and finite slab analyses). The numerical13solution techniques include the finite difference methods 23 '24 (FDM), the finite elementmethods25 (FEM), the boundary element method 26 (BEM), the control volume method 27 (CVM)and the direct finite difference method 28 (DFDM)Whatever numerical technique is chosen for any particular problem, the efficiency of thesolution is limited by three main factors 45 :(1) the characterization of the interfacial resistance between the casting and the mold orother external cooling device.(2) the treatment of the latent heat release and the subsequent evolution of themicrostructure.(3) the treatment of accompanying fluid flow during solidification.2.2 Heat Flow - Interface ResistanceAs mentioned in Chapter 1, the characterization of interfacial heat transfer resistanceduring solidification has always been a major source of uncertainty in modeling of anysolidification process. The study of metal/mold interfacial heat transfer is very important intwo respects29 :(i) for promoting the accuracy of numerical heat transfer simulation(ii) for improving casting quality through better control of metal/mold thermal resistance.The problem here is to obtain a solution to Newton's law of cooling;q^—^ (2.1)For most casting processes, the variables in Equation (2. 1 ) - the heat transfer coefficient(h), the temperature of casting (T c) and the mold temperature (T.) must be determined at theinterface. However, surface measurements have serious experimental impediments. Firstly,14the physical situation at the interface may be unsuitable for attaching a sensor. Secondly, theaccuracy of the measurement may be seriously impaired by the presence of a sensor. Therefore,it is preferable to measure accurately the temperature history at an interior location and toestimate the surface condition from this measurement. This technique has become known asthe inverse heat conduction problem (IHCP).IHCP techniques have been applied extensively in characterizing the interface resistancein solidification modeling 30 . The numerical techniques involve the use of either the heat flux(discrete values or specified functions) boundary condition or the heat transfer coefficient(discrete values or specified functions) boundary condition. The basic assumption in either caseis that heat transfer conditions on both sides of the interface are exactly the same. In other wordsa quasi-steady state exists at the interface, there is no heat source, heat sink or accumulationacross the interface. Pehlke et al..' suggested a criterion to estimate the degree to which aquasi-steady state assumption is valid across an interface of finite thickness yi. This criterionis the square root of the dimensionless Fourier number;FFO= (at) u2/y, > 1.0 (2.2)where a is the average thermal diffusivity across the interface and y, is the interface thickness.In many heat transfer problems that attain steady state equilibrium, it has becomecustomary simply to assign a constant heat transfer coefficient. However, it has since beenrealized that the interfacial heat transfer is a time-dependent variable. The transient nature ofinterfacial heat transfer is attributed to the dynamics of the casting/mold contact. As statedearlier, the casting/mold interface often assumes a complex combination of finite gap,conforming and non-conforming contacts during solidification. Although the flow of heat nearthe interface is microscopically 3-dimensional, the overall heat transfer coefficient across suchan interface from a macroscopic standpoint may be written as the sum of three components 4 :15h=1-0-hg+h, (2.3)where h, is the part due to solid conduction through the points in contact, while h g and h i. denotethe contributions of gas conduction and radiation across the void spacing surrounding the contactpoints.Using measured temperatures in both casting and mold together with analytical and/ornumerical solutions, several researchers have attempted to quantify the transient interfacial heattransfer coefficient629 '31-47 . Earlier, several workers have proposed the use of a constanttime-averaged h to account for the transient nature of the interfacial heat flow 31-33 . Others34-38derived more specific expressions for h as summarized in Table 2. 1 .A review of the early studies on the interfacial resistance6.29 '3147 shows that such factorsas casting and mold geometry, mold surface roughness, contact pressure, time after teeming,thermal characteristics of casting and mold, mold coatings and nature of contact between castingand mold are known to affect the interfacial resistance. Tiller 34 observed that h decreases withtime from a peak value attained at contact.Using a chill immersion technique, Sun35 observed that h increases linearly with timewhich is exactly opposite to Tiller's result. The immersion technique used by Sun enabled acontinuous rise in h with time due to increased contact pressure as the casting contracts towardsthe chill and the chill expands towards the casting. Levy et a!. 38 used a similar geometry toshow that improved thermal contact could be achieved between casting and mold by utilizinga forced fit technique where the contraction and expansion of the mold are used to prevent gapformation.16Sully39 found that the heat transfer coefficient during solidification of metals can exhibitfeatures of the Sun mode135 and the model due to Tiller34 . He found that in most cases, h risesrapidly to a peak value and then declines to a low steady value under conditions where thecasting contracts away from the mold.Table 2.1 The various expressions for evaluation of heat transfer coefficient.Reference Expression for h Remark34 hhreceding interface= ,2 -Nit35 h = a +bt increasing contact pressure36 km , only hs was consideredh = A --NI(P IH,)xr37 h increases with increasing surfacesmoothness of the moldh = CV Ra-b6 k^r sin 0^a — sin 61 k= Yeghi. neglected.-h^ - ^+^Yi-^/Ia (^y- kghonly hg is considered— (xg + s, + 52)_ a(Tc+Tm)(T,2 +T,;,) only hr is consideredh—^(. +^— l)Studies on the effect of contact pressure on the heat transfer coefficient have beenconducted by several investigators 36 '4°'41 . It has been found that interfacial heat transfercoefficient is proportional to the square root of contact pressure 36 .Studies have also been undertaken on the effect of surface microprofile mainly in termsof roughness and surface coatings 33 '35 '39 '42-44. It was found that the heat transfer coefficientincreases with increasing surface smoothness. In the case of surface coating, the heat transfer17coefficient depends on the thermal conductivity, thickness and surface smoothness of the coatingmateria139 '4244 . It increases with increasing conductivity of coating, decreasing thickness ofcoating and increasing surface smoothness of coating.Suzuki et al. 45 measured the heat transfer coefficient between melt and chill by droppingliquid tin on a cylindrical chill made of different materials (brass, stainless steel,chromium-plated brass and nickel-plated brass). They claimed that the heat transfer coefficientdoes not depend on the thermal properties of the chill materials but presumably on the wettabilitybetween melt and chill.In a recent work, Sharma et a1. 6 proposed that an actual mold could be conceived to be acombination of v-grooves having different groove parameters such that the overall heat transfercoefficient of the surface can be calculated as series/parallel combinations of the constituentv-grooves. They proposed that the variation of h with time generally exhibits three distinctregions as illustrated in Fig. 2.2. From the time of initial contact (stage I), h rises rapidly to apeak value and decreases rapidly in a fluctuating manner In stage II, h is constant or fluctuatesaround a mean value. Stage III depends on the extent of contact pressure; h remains fairlyconstant if the contact pressure is constant but increases if the contact pressure is increased anddecreases if the interface recedes.Most of the recent IHCP techniques utilize the heat flux boundary condition. Earlier,Jacobi46 has used a time dependent interfacial heat flux such that when this transient heat fluxis divided by the estimated temperature drop across the interface, the transient interfacial heattransfer coefficient is obtained.Pehlke et al.4 '29 '47 did a comprehensive study of the heat transfer and solidification ofaluminum and copper bronze using a water cooled copper chill. They successfully characterizedthe metal/chill contact phenomena and gap formation by using transducers. They also simulated18the effect of chill location on melt/chill contact and found that a sizeable gap forms when thechill is on top of the melt while the melt and chill exhibit some form of non-conforming contactin the case when the chill is located below the melt. Their numerical analysis involves anextensive use of the 1-D inverse heat conduction technique based on the nonlinear estimationmethod of Beck".Fig. 2.2 Schematic illustration of the typical variation of heat transfercoefficient (h) with time during solidification'.19In a recent study by Kumar and Prabhu49 using the same technique as Pehlke andco-workers, it was shown that the maximum interfacial heat flux between a chill and solidifyingmetal could be represented as a power function of the chill thickness and chill thermal diffusivity:qmax = C 1 (Xla)n1 (2.4)Furthermore, they found that the heat flux after the maximum value could also be expressedas a power function of the thermal diffusivity and time in the form:(q /qina 0a0.05 c2(12 (2.5)The constants, C 1 and C2 were found to be dependent on the casting composition for thealuminum and copper alloys studied 49. Therefore, the interfacial heat flux should depend notonly on the thermophysical properties of the mold material but also on the properties of thecasting alloy. Bamberger et al. 5° found that for the same mold and casting conditions, theinterfacial heat flux depends on the alloy composition for Al-Si alloys. A typical heat fluxprofile is shown in Fig. 2.3. It is observed that the heat flux profile follows the same trend asthe heat transfer coefficient (See Fig. 2.2).20IIIII^I 40 80 120 180 200 2400Time (sec)Fig. 2.3 Estimated heat flux profile for 50 x 50 x 50 mm copper chill withoutcoating and Al-13.2% Si alloy42.2.3 Heat Flow - Latent Heat EvolutionThe energy conservation equation for a solidifying material is given by:21where, p, Cp and k assume the values of the particular phase or phases prevailing at a giventemperature and location in the casting. The source term, Q', describes the rate of latent heatevolution during any liquid-solid transformation and may be written asf,Q, = pL at(2.7)The solution to Eq. (2.7) has been of great interest to researchers of solidification andother fields where phase change occurs. The problem is two fold; (a) how is the latent heatactually released in practice, and (b) how should the latent heat phenomena be accounted forin a mathematical model?In terms of continuity at the interface, two major techniques can be identified from theliteratures - (i) the 1-domain and (ii) 2-domain or front tracking technique. These are illustratedin Fig. 2.4. The 1-domain techniques assume that the solid and liquid phases constitute thesame medium, with average thermal properties defined at each node as a function of temperature.This method is computational simpler since the phase boundary is not explicitly defined. Thisis advantageous in handling problems where the phase change region is a volume (such as themushy zone) rather than a surface (such as isothermal transformation front). The most common1-domain methods include the temperature recovery methods, the specific heat methods, theenthalpy methods, the latent heat methods, and other hybrids of the three.The 2-domain or front tracking techniques assume that the solid and liquid phases are twoseparate media. Accordingly, continuity equations are applied separately to each mediumtogether with a specified set of equations for the interface between them. These techniques aremore complicated with respect to computing and are best suited for isothermal transformationor transformations involving isolated cells or dendrites. The common 2-domain methods includethe line tracking methods51 '52 , spatial transformation method 53 and spatial grid deformation22(K, grad TI =^3ratZo)div (K, <rad T) =dt-v l = - v,(L7 )an ,^,methods 54-".Fig. 2.4 Schematic representation of phase change handling in heat flowmodeling of solidification s; (a) front tracking or 2-domain method,(b) 1-domain method.Poirier and Salcudean56 reviewed the various numerical methods used in mathematicalmodeling of phase change in liquid metals based on :23(a) the ability to solve multidimensional problems(b) ease of implementation(c) ability to account adequately for the mushy regionThey concluded that the 1-domain techniques are simpler, easier to use and are better suitedfor handling transformations with an appreciably mushy region commonly encountered insolidifying metal alloys. This conclusion agrees with that of other researchers in the field ofsolidifications . Hence, emphasis here is on the 1-domain techniques.2.3.1 Temperature Recovery MethodSometimes referred to as a postiterative method, the temperature recovery techniquewas first reported by Dusinbere57 and later by Doherty58 who used it in a FDM solution ofisothermal transformation. It has since be applied to non-isothermal cases59 and alsoincorporated into FEM solutions 60. In this method, the temperature of the node at which phasechange is occurring, is set back to the phase change temperature and the equivalent amountof heat is added to the enthalpy budget for that node. Once the enthalpy budget equals thelatent heat for the volume associated with that node, the temperature is allowed to fall accordingto the heat diffusion. This could be represented mathematically byT node = T^T >71^ (2.8a )T node = TL -- OH ILAT^Ts.7' Ti,^(2.8b )T node = T^T <Ts^(2.8c)The main advantage of this method is that conservation of energy is always ensured.However, the technique is known to produce undesired 'wiggles' or 'false eutectic plateaus'that result from energy conservation since any finite volume has a constant temperature during24isothermal solidification s . The method has also been shown to be very sensitive to the size ofthe time step56. Furthermore, the errors in the approximation are more magnified in the vicinityof the mushy zone than in the single phase regions 56 .23.2 Specific Heat MethodProbably the most commonly used method in solidification modeling, the specific heattechnique is attributed to Hashemi and Sliepcevich61 , who introduced it in an implicit FDMcode. It was later adopted to FEM formulation 62 . The procedure is to assign a pseudo specificheat to the region where the phase change is occurring.Substituting Eq. (2.7) into Eq. (2.6) above and re-arranging, the following is obtained:a 42K:^alax k ax ay k ay + a a aTT kaz)^pCp aTt_ aa.f.St{^ (= aT^ofaT (2.9)If a pseudo specific heat is defined asofC pseudo = C p —then Eq. (2.9) becomesa {421-')_i_ " k—If al a^Eax k ax^,^aZ C z^pCpseudo at(2.10)(2.11)Eq. (2.11) is the mathematical expression of the specific heat method.The accuracy of the method depends on the technique of solving Eq. (2.7), that is, theevolution of the solid fraction. Furthermore, it is difficult to ensure energy conservation s using25the specific heat method since no condition is generally imposed on Eq.(2.11). The simplestprocedure is to assume a linear release of the latent heat which results in the followingexpressions:C„d = Cp^T > 71^ (2.12a)^C „„de = C p + LI AT^Ts_.T 5_TL, (2.12b)C „„de = C p^T <Ts^(2.12c)This is the well known "apparent specific heat method". The main disadvantage of thistechnique lies in the discontinuity in specific heat at both the liquidus and solidus temperatures.A different method referred to as the "effective specific heat method" was proposed by Poirerand Salcudean56. In the effective method, a temperature profile is assumed between nodesand instead of calculating an apparent capacity based on the nodal temperature, an effectivecapacity is calculated based on an integration through the nodal volume. For a lineartemperature distribution, the effective capacity method can be represented byCode = Cp^T >> 71^ (2.13a)^C„„de = Ceff = 1 1 CpdVIIV^Ts-4-5_T 5_TL +^(2.13b)Code = C p^T «Ts^(2.13c)where is a number that accounts for the effect of the surrounding nodes on the nodal volumeof interest and is dependent on the node size. This method allows a node with a volume coveringtwo regions (liquid and mushy, or mushy and solid) to balance the effect of each region. Ithas been shown that this particular ability reduces the possibility of either over estimating orunder estimating the effect of latent heat. It also eliminates the discontinuity at the liquidusand solidus temperatures associated with the apparent heat method 56. A typical variation ofspecific heat with temperature utilizing the apparent and effective methods is shown in Fig.2.5.26Fig. 2.5 Calculated variation of specific heat with temperature for apparentand effective specific heat methods for a Al-7%Si alloy.23.3 Enthalpy MethodsMost of the pioneering work on this method were based on finding solutions to nonlinearequations using the implicit FDM scheme65. To date, both explicit and implicit FDM andFEM solutions based on the enthalpy method have been obtained 6648 . A hybrid of the enthalpyand apparent heat capacity methods has also been proposed with a novel three-time level FDMscheme69.27The method is based on the formulation of the right hand side of Eq.(2.9) in terms ofenthalpy instead of specific heat and temperature. Recalling Eq.(2.9) and rearranging, thefollowing is obtainedaaT^a( -,-a--; aT^af;Y^az az =^at — PL atwhereorThusHTdHH(T)= C==p T —Lf„CpdT — LCpdT +L(1—at (CP T —Lf)fa(T^T)df,s(T = 0)^fs)^(f;= 1.0=ataH(2.14)(2.15a)(2.15b)(2.15c)p atT = 0)The enthalpy method has some obvious advantages over the specific heat method. First,it ensures energy conservation at all times since the enthalpy is a direct dependent variable inthe energy equation. Secondly, there is no discontinuity at either the liquidus or solidustemperatures since any solidification path is characterized strictly by a decreasing enthalpyeven with recalescence. However, the enthalpy method is more difficult to implement withexisting standard codes and in most cases has been known to produce 'wiggles' or 'false eutecticplateaus in the cooling curves just like the temperature recovery method'. The typical enthalpyprofile using this method is shown in Fig.2.62.3.4 Latent Heat MethodThis method involves the solution of Eq.(2.6) without any transformation; the latentheat is neither incorporated into the specific heat nor the enthalpy budget. Eq.(2.7) is solveddirectly at each time step and substituted into Eq.(2.6). The method ensures energy conservationand is sometimes referred to as the solidification kinetics method 22. A typical enthalpy profile28using this method is also depicted in Fig.2.6.Fig. 2.6 Calculated variation of enthalpy with temperature for enthalpy andlatent heat methods for a Al-7%Si alloy?23.5 The Nature of Latent Heat EvolutionThe four methods discussed above are merely the techniques of handling latent heatrelease during mathematical modeling of solidification. The larger question now is how thislatent heat is actually released during solidifcation.Two major procedures have been adopted in determining the actual nature of latent heatrelease:(1) cooling curve analysis(2) nucleation and growth laws.29Only the cooling curve analysis technique will be discussed while the nucleation andgrowth laws will be taken up under microstructural evolution.23.5.1 Cooling Curve AnalysisExperimental cooling curves can be used to obtain pertinent information on the actualnature of latent heat release during solidification. This has been done by performingexperiments on lumped parameter systems223° with minimal temperature gradients since theyallow for simplified analysis. A solidifying metal can be treated in this way if its Biot numberis less than 0.1Bi -h 4< 0.1^ (2.16)kFor any casting that satisfies the above criterion, the basic energy conservation can be writtenashA (T -T)+Q‘=pC —dTV -^P dt(2.17)In the single phase region ( either liquid or solid), there is no heat source term such that theabove equation becomesdT^hA dt^pVCp(T -T) = 0(T -T)- (2.18)where 0 is the inverse of the time constant. It is noted that dT/dt is simply the cooling ratewhich can be obtained by immersing a thermocouple into the liquid or solid phase.In one of the cooling curve analysis techniques sometimes referred to as the zero curvemethod22, the latent heat released up to a time, t, is approximated by30f 1(_dTt c_ dTL(t) =0 ^dt Z, (2.19)where (dT/dt)ze is known as the zero curve and is obtained by simply joining the dT/dt obtainedfor the liquid phase to that obtained for the solid phase. Once the L(t) is known, thedetermination of the cooling rate as a function of time becomes trivial. However, the aboveequation is unique to the particular cooling rate obtained in a given experiment. For a moregeneral application, the rate of evolution of the solid fraction for a particluar alloy systemshould be known. The solid fraction up to time, t, can be obtained as followsfs(t)=L(t)L (2.20)By interpolation of experimental data at various cooling rates, it was found that dfjdt variesnot only with time but also with the cooling rate 22 such thatdfs ___( dT b }^(dT )2 d ((IT— a + ° + c^+ "^+ edt^° dt^° dt^° dt^° (2.21)By performing a series of experiments in a lumped system with a given alloy, the constantsa0 to e0, can be evaluated. The expression for dfjdt given by Eq.(2.21) is valid for a givenalloy under all conditions including the practical non-lumped systems.A similar technique recently published°, utilizes an artificial variation of 0 across thesolidification regions (liquid, mushy and solid) to estimate the amount of latent heat releasedup to a given time. For a linear variation of 031eL,_. (dT \dt 1(T_—T)^T>TL^ (2.22a)L —0 =el.+ 7T^(05' OL)^ (2.22b)L — Tses4dTt7}(71° — 7)^T <Ts^(2.22c)Once 0 is known, the variation of latent heat with temperature can be evaluated fromLdf^ (T —T.1)=C 1 +O 1+0dT dT/dt(2.23)By solving Eq.(2.23) for various measured temperatures, an empirical relationship betweenlatent heat release and casting temperature can be established. This relationship could be ofthe form:Ldf,=a+br+cT2 +dT3 +....dT(2.24)Equation (2.24) is valid for a given alloy under all conditions including the practicalnon-lumped systems since the latent heat release is expressed only as a function of castingtemperature. Once Eq.(2.24) is evaluated, the heat source term can be calculated easily fromelf dTQ =PL dT dt(2.25)The evolution of the solid fraction given by either method can then be handled in amathematical model via the specific heat, enthalpy or latent heat methods.322.4 Fluid Flow During SolidificationThe handling of fluid flow in modeling non-stationary problems such as solidification israther difficult because the regions (liquid and mushy) within which fluid flow have to beconsidered, changes continuously with time. As discussed in section 1.2, fluid flow in castingcan originate from two main sources - induced and natural forces.Three major approaches have been adopted in modeling the effect of fluid flow insolidification simulation:(a) the effect of fluid flow is incorporated into the heat flow model by simply increasing thethermal conductivity by an artificial amount.keff = ak^ (2.26)(b) the fluid flow pattern is replaced by a liquid region where complete mixing is assumed,plus a boundary layer whose thickness is estimated by a dimensional analysis. For instance,the Nusselt number is often expressed in terms of flow parameters and is used to determinethe heat transfer coefficient between a fluid and a solid surface. Such an expression canbe in the form:hL,(a/p)'' = aRe xPrY^(2.27)Nu = , = a (puD/g)x(c) the fluid flow is more precisely calculated from the Navier Stokes equationsDu^ap-f a2u^a2u^a2u(2.28a )pDt =pgx ax ax 2+.—.± —ay 2^az 2Dv^ap a2v^a2v^a2v (2.28b)Dt^"Y^ay^r-\ ax 2^ay 2^az 2Dw^ap a2w^a2w^a2w ) (2.28c)p=pgDt^z ++—aZ 2ax 2^ay 2and the energy conservation equation for incompressible flow given by33aT aT^aT)_, 1 a2T a2T a2T (aP +v?..12^4Y.4,w--)pCif u yx-± + w -t- ay2 az2 ) u ax ay az +1.143 (2.29)In 2-domain methods, the Navier-Stokes equations are solved only within the liquid regionas the velocity field is set equal to zero at the moving solid/liquid interface. In the 1-domainmethods, a fixed grid is defined for the entire system and several procedures have been developedto solve the Navier-Stokes equations in solidifying metals and alloyss . Morgan" set the velocityfield to zero as soon as a certain solid fraction is reached. Gartling 72 progressively increasesthe viscosity g in the equation as solidification proceeds. More recently, methods thatprogressively decrease the velocity field within the mushy zone have been developed 73 '74 . Inone analysis74, the average velocity field within the mushy zone can be represented by:fsys + (1— fs)vi (2.30)2.5 Microstructural EvolutionMicrostructure formation during the solidification of alloys is of prime importance for thecontrol of the properties and quality of cast products. In order to predict the properties and thesoundness of a casting, empirical methods or trial-and-error approaches have been adoptedover the decades. However, due to the complex interactions occurring during solidification,these methods have limited use and hardly can be extended to other solidification conditions.Furthermore, they usually give very little insight into basic mechanisms of solidification. Thisis particularly the case in equiaxed microstructure formation where nucleation, growth kinetics,solute diffusion and grain interactions have to be considered simultaneously with heatdiffusion75 .As stated in the last chapter, the evolution of solidification microstructures is dependenton the operating nucleation and growth phenomena. The degree of fineness of the microstructuredetermines the quality and properties of the cast component. The goal of most practical casting34processes is to obtain fine isotropic crystals such that segregation, porosity, and other defectsare substantially reduced. Exceptions are precision materials with little or no grain boundariessuch as fine wires and extremely thin plates or ribbons/wafers, produced by such unidirectionalcasting techniques as the 0.C.C 76 . These materials are mainly used as lead frames and bondingwires in such appliances as acoustic equipments and memory disks of computers. For theseapplications, long unidirectionally solidified columnar crystals are preferable, and the fewerthe number of crystals, the higher the quality of the cast component.23.1 NucleationThe classical theory of nucleation is based on the extensive thermodynamic treatmentof Gibbs3 . Gibbs analyzed the transformation of a liquid to a solid phase, taking into accountthe change of free energy between phases and the free energy change created by the introductionof the new surfaces.Kinetically, nucleation is a statistical process. A nucleation event may require a longperiod of time, say one per day, or the nucleation rate may be several hundred per second.The undercooling required for nucleation to occur plays an important role in the nucleationprocess. Undercooling can originate from either a thermal or constitutional gradient.According to Volmer and Weber'', two statistical probabilities can be considered:(a) the probability that an embryo will grow to the critical dimension of a nucleus and(b) the statistical probability of an atom joining the critical nucleus by transfer from theliquid in a diffusion process.The nucleation rate is that at which nuclei of critical dimension r * are converted into stablenuclei of radius, r>r * , by atom attachment from the liquid. This rate is proportional to theproduct of the two probabilities, (a) and (b).The rate of heterogeneous nucleation can be written as35n.D1 N)aN—a7=n 'yexp( kbTAG:et exp(_ AGAkb T (2.31)Taking into consideration that the initial nucleation site density N S within and around the meltwill decrease as nucleation proceeds, Hunt 71 suggested an approximation to the above equationin the formaNat= Ar)Kiexp(_ K2 OT )(2.32)where K1 —(2^AGAand^) DI .-- dyexp kbTand K2 = 01.02 111(N1(1)Equation (2.32) is based on the assumption of instantaneous nucleation, that is, all nuclei aregenerated at the same time once the critical nucleation temperature is reached. It predicts thesame final grain density irrespective of the cooling rate. Experimental results 5 '22 have shownthat both undercooling and grain density increase with increasing cooling rate. It has beensuggested22 that although most of the variables in the Hunt Model affect nucleation rate, onlythe initial number of available sites Ns will determine the final grain size. Thus a directrelationship between Ns and the cooling rate is expected. It has been proposed that thisrelationship can be described by a parabolic equation of the form 22 :dTN ,= K3 + IC4( dt(2.33)There are two possible explanations for the increase of the number of sites with coolingrate. The first is that because higher undercoolings are reached at higher cooling rates, different36types of nuclei become active, thus increasing the overall number of sites. Secondly, higherundercoolings are associated with a smaller critical radius which in turn results in an increasein the number of active sites.Other investigators 75 '79 assume that nucleation occurs continuously once the nucleationtemperature is attained - continuous nucleation. Oldfield79 proposed a parabolic dependenceof the number of active nuclei on the undercooling:N = 112(T,, — T)2^(2.34)which gives a nucleation rate ofaN^aTat = —2112(Tn —T)—at (2.35)It may be noted that the above is an empirical relationship derived from experimental data oncast iron. It is seen that nucleation rate is a function of both undercooling and cooling rate.Thevoz et al.75 extended this approach by assuming that at a given undercooling, AT, the graindensity is given by the integral of the nucleation site distribution from zero to AT1AT=^f(AT)d(AT) (2.36)The new grain density is updated at each time step as a function of undercooling with the finalgrain density corresponding to the onset of recaslescence.2.5.2 GrowthIn metallic systems, growth of the solid from liquid at moderate cooling rates has beenobserved to occur in three main regimes 9: planar, dendritic and cellular as illustrated in Fig.372.7 (a). With respect to the resulting microstructure, growth can either be eutectic or dendritic 8as depicted in Fig. 2.7 (b). Dendritic growth is by far the most common growth mode in alloysthat are not pure eutectics".Depending upon the nucleation and heat flow conditions, the growth process could resultin either equiaxed crystals (freely growing into an undercooled melt) or columnar crystals(growing into a positive temperature gradient). In the case of equiaxed growth, latent heatcreated during growth at the solid/liquid interface flows from the interface into the melt (thetemperature gradient in the liquid, G 1 , is negative) while in the case of columnar growth, heatflows from the liquid into the solid 8 (G1 is positive).The theories of dendritic growth are based on the same continuity equations, in particularthose of heat and solute diffusion which control the macroscopic aspects of solidification.However, unlike macroscopic solidification, a stationary state is considered in this case andadditional phenomena such as capillarity, local equilibrium of the various phases and possiblekinetic effects may be taken into account. As stated earlier, dendritic growth can be controlledby heat flux, solute flux or both. Columnar dendritic growth is mainly solute diffusioncontrolled and is common in alloys. On the other hand, equiaxed dendritic growth is heatand/or solute diffusion controlled and can occur in both alloys and pure metals.Many experimental measurements on free dendritic growth in undercooled melts havebeen carried out9. The early experiments were performed on (a) pure metals 2 '86-84 (Sn, Ni, Co,Pb, Ge, Bi), (b) binary alloys 2 '83 (Pb-Sn, Ni-Cu) and (c) non-metals 2 '85 (P, ice). Also, a greatdeal of effort86-88 has been devoted to experimental observations and measurements on a lowmelting point transparent "plastic crystal", succinonitrile, NC-CH2-CN, which facilitatesdirect observation. Furthermore, some work has been done on constrained or unidirectionaldendritic growth in a number of alloy systems 89-96. All these efforts have provided bothmorphological and empirical details of dendritic growth, furnishing information on dendrite38tip radius, tip velocity, side branch formation , arm spacing, remelting of dendrite arms,coarsening kinetics, the influence of thermal gradient (G1), the cooling rate and localsolidification tiMe2.80"97.Fig. 2.7 Growth phenomena during solidification (a) growth regimes', (b)growth microstructures8.39Observations on succinonitrile dendrites 86-88 have shown that the tip regions of dendritesare bodies of revolution very closely approximating a parabola. The cross-section is nearlycircular and approximates a body of nearly perfect axi-symmetry. Small undulations appearnear the tip which rapidly lengthen into side branches. The point behind the tip, at which thefirst distinguishable branch appears, varies slowly with the amount of melt supercooling.Dendritic growth can therefore be considered to proceed by three separate growthprocesses':(a) the initial propagation of the primary dendrite stem(b) evolution of dendrite branches(c) coarsening and coalescence of dendrite stem and arms.The initial propagation of the primary dendrite stem is dependent on the stability of thedendrite tip. An early analysis of the growth of a phase by diffusion was made by Zener 98based on the solid state transformation of ferrite (a) growing from austenite (7) in the Fe-Csystem. Assuming the a phase to be a disc with a spherical edge maintained at a radius, r,during transformation, the interface growth velocity can be expressed as:v, DL122F (2.37)^— cy^ATwhere —^c.(1 ^(AT +mcy)(1 —kp )Caand^kp = -Cyso that^yr cc AT240Following this type of analysis, many studies involving the mathematical descriptionof a branchless geometrical form growing at a constant rate and shape have been carried out 99-104 .The results of these theories express the axial dendritic growth velocity as a power functionof undercoolingv, = OG *(AT)b^(2.38)Hence, the driving force for dendritic growth is the tip undercooling. Ivantsov 1°2 gave amathematical analysis of the relation of the dendritic growth rate to undercooling in the form:AT E2 = (AT + m c) (1 — kp) Pe exp(Pe )Ei (Pe) = f(Pe) (2.39)where the Peclet number,^Peand^E, (Pe)=^(e- a a)daPeEquation (2.39) permits the calculation of the Peclet number, or the product, v tr„ as afunction of undercooling. The tip undercooling is made up of four components" - thermal(ATt), solutal (AT.), capillarity (AT,) and kinetics (ATk) undercooling, such thatAT = ATt + AT, + AT, + ATk^(2.40)In the analysis of Burden and Hunt89, and later modified by Laxmanan9, the total tipundercooling, assuming a negligible kinetic effect, is given byAT = ^ Rmic0(1—kP)rt k G r + 2721 R D1^P 1 t pSLrt(2.41)Vtrt2D41The dendrite tip radius, r„ is estimated from one of the available stability models 99-1°1305 .For example, Mullins and Sekerkam proposed that a marginally stable state is achieved at thedendrite tip when the tip radius rt. is equal to the perturbation wavelength X such thatr, = 1"= 24(mGA,(Pe)—G) (2.42)Equations (2.39) - (2.42) could then be combined to obtain solutions that give values of thedendrite tip radius, r1 , the dendrite tip velocity, v„ and the tip undercooling, AT.A knowledge of the growth law of the dendrite tip, that is, the relationship between v„r, and AT is not in itself sufficient to predict the final microstructural features and the coolingcurves since one still needs to predict how the solid fraction behind the tip changes withtemperature. Dendritic growth involves the advance of the dendrite tip into the liquid, and assolidification proceeds, perturbations around the tip may lead to branching. The branchingis made more complicated by secondary processes of coarsening and coalescence whichdetermine the final spacing between the dendrite arms. This spacing determines ultimatelythe distribution of solute on a macro-scale.2.5.2.1 Dendrite Arm Spacing and CoarseningMany studies have been made of the "as-solidified" microstructures of binary alloys inorder to determine experimentally the interdependence of dendrite arm spacings andsolidification parameters 99"6-1°8 . Also, many analytical solutions have been developed torelate the dendrite arm spacing to solidification variables. In both the theoretical andexperimental efforts, a number of solidification parameters have been found to influence thedendrite arm spacing. These include: the temperature gradient near the solidification front,growth rate of the dendrite tip, dendrite tip radius, cooling rate, extent of the mushy zoneand the time spent in the mushy zone.42The fineness of the dendritic structure that forms originally has little or no influenceon the final dendrite arm spacing due to the coarsening and coalescence phenomena whichchange the secondary and higher order arms more than the primary arms due to geometricalconstraints. The dendrite arms respond to changes in the solidification parameters mentionedabove by enlarging their size, merging with others, remelting, shrinking or completelydetaching from the parent arm.Five different models have been proposed to underpin the mechanisms that influencethe final dendrite arm spacing as illustrated in Fig. 2.8. These include:(i) Radial Remelting (Model I): A fine dendritic arm is surrounded by two coarser,identical arms. The smaller arm becomes thinner with time due to lateral dissolution, andshrinks back while the other two arms slightly increase in radius.(ii) Neck Remelting (Model II): A neck forms at the root of a dendrite arm surroundedby two larger arms. By further remelting at the neck of the arm and freezing at lower curvaturesites, the arm becomes disconnected from the main dendrite and gradually spheroidizes.(iii) Axial Remelting (Model III): A finer dendrite arm surrounded by two coarserarms of equal radii gradually shrinks back at constant radius by dissolution at its tip whilethe radii of the two coarser arms increase.(iv) Tip Remelting and Coalescence (Model IV): The tip of two equal arms dissolveand shrinks at constant radii while the concavity between them gradually fills in.(v) Coalescence (Model V): During the later stages of solidification, juxtaposed armsthat have coarsened enough may come into contact with each other and coalesce. Thecoalescence occurs by preferential solidification in regions of low or negative curvature.43Fig. 2.8 Schematic representation of the different dendrite coarseningmodelsu.Most of the theoretical and experimental results have established that the primarydendrite arm spacing (X i) could be expressed empirically as:X1 = C3G -N -lj4^(2.43)Some of the various theoretical analyses and experimental results are summarized inTable 2.2. It is clear that none of these results have universal application. The form of the44final equation in the theoretical analysis is dependent on the mathematical approximationsemployed, and on the physical model or models assumed (Models I-V). It is further notedthat these analytical theories are based on isolated dendrites.Table 2.2 Various expressions for the constants used in the evaluation of primarydendrite arm spacing.Model n3 n4 constant C3Trivedi" 0.5 0.517.8_V DJ-rtHunt' l° 0.5 0.25 2-4-i[Diffm (1 — k p )c 0 + kp G p Iv-t-1)-1 1/4Kurz and Fisher"' 1.0 1.0 6(v4T0— G1D1)(Gtpl—kpvt4T0)v, <vcIkp(1 _ 4)2s, 0.5 0.254.3{ ArATo}"^vt > velkpk^,An and Liu"' 0 02.38,\Iwor —kp )GPI + m(1 — kp)vtco'^vt < vclkpti 0.5 0.25 1.341,(D1rk4AT0)1'4,^yr > vcIkpRhotagi et al. 93 0.5 0.5 q8DATExperimental m 0.72 0.24-0.26 29.0 - 34.0 {manganese steel (0.59-1.48%C,1.10-1.14%Mn)}Experimental' 0.5 0.36- 0.5 30.5 - 56.5 {Al-Cu alloys (2.4-10.1%Cu)}Experimenta192 0.45 0.75 {Pb-Sb alloys (5-10%Sb)}For n3=n4 in Eq. (2.43) above, the product of the tip growth rate (R) and the temperaturegradient (G) can be represented by one term - the cooling rate since45aT aT ax—cooling rate = —at =^at GR^(2.44)Then equation (2.43) becomesX I or =aTCo at (2.45)Eq. (2.45) has been used extensively to predict both primary and secondary dendrite armspacing by many researchers in the field 107,109. It has been observed that n 3 n4 for secondaryarm spacings moreso than for primary spacings 106 , thereby making Eq. (2.45) more valid forestimating k than for A 1 . However, it is noted that the equation does not explicitly incorporatecoarsening and coalescence which are the dominant factors in determining the final value ofA.2. Equation (2.45) indicates that the dendrite arm spacing decreases with increasing coolingrate at a given location.As far back as 1966, Bower et a1.94 have shown that the secondary dendrite arm spacing(X2) is proportional to a certain power of local solidification time based on curve-fitting witha variety of experimental results 113-115 ,X2 = C5 (0n6 (2.46)Many theoretical models have been developed to account for this result. As in the case ofprimary dendrites, these models are limited by their mathematical approximations and thephysical models used in their derivation. Some of these theoretical models and selectedexperimental results are summarized in Table 2.3.These expressions have since been used to predict DAS once the local solidificationtime is known. The local solidification time is defined as the time spent in the mushy zone,46that is, the time required for the temperature at a point in the liquid metal to pass from theliquidus to the solidus temperature. It is often measured but can also be calculated with greatdifficulty (the liquidus is difficult to pinpoint) from a heat transfer model of the casting process.Table 2.3 The various expressions for the constants used in the evaluation ofsecondary dendrite arm spacing.Model n6 constant C5Kattamis et al. 96 1/3 7s1DTL^1 1/3 {LmCi(1 -kp )4)(fs)2 1n(1 - f)fiFuerer andWunderlin1161/3 f 166rDiln(ceico) 1/3}rn(kp - 1) (ce - co)Kirkwood 117 1/3 1 128DrYsr TL, 1*e/co) 1 1/3Lrn(kp - 1) (ce - co)Mortensen 118 1/3 271-D,^1 1/3{ 4c/ (-m)(1 -kp )f,(1 _ Arf-,-.)Voorhees' 19 1/3{8Tapytf.^) 1/39L[DLICp -(1-kp )mcloci]Experimenta150 0.43 11.5-15.3 x 106 Al-Si alloy (3.8-9.7% Si)Experimenta194 0.39 7.5 x 106 A1-4.5% Cu alloyWolf and Kurz12° have shown that an equivalent expression can be obtained bycorrelating the macroscopic shell growth with DAS. Employing the square root time law forshell growth,S =1(4-t^(2.47)the relation between DAS and cooling rate {Eq. (2.45) above}, an analytical solution fromSzekely et al. 121 and other assumptions, they obtained the following expression for DAS:47a2 ^4t ja5^tf. It5= C4 BK2 —C4 TL —TS(2.48)where B =psi.lcBased on the same kind of derivations and assumptions, the secondary dendrite armspacing has also been related to the distance from the chill surface by the followingexpression l°73 222l2 = CO ; 7^(2.49)where^C6 = C5{2and^n7 = 2/3}(KL — Ks) (1/3)K,IellWith either of the models, it is possible to predict DAS by simply solving the heattransfer equations to obtain the solidification time or shell thickness at various distances fromthe chill. However, the characterization of various constants for each alloy system stillconstitute a major barrier to this laudable goal.Another fundamental work in coupling the microstructure with heat transfer and fluidflow is the recent one by Hills 123 . Employing Eq. (2.45) with n5 = 1/3, and the relationshipbetween the cooling rate at the critical region just behind the dendrite tips and the growthrate,dT pL ,2.-...dt^k,(2.50)48Hills obtained expressions that directly linked the dendrite arm spacing to known heat transferparameters. For castings in a mold without convective cooling, DAS could be expressed asa function of shell thickness, solidification time, and the ratio of heat absorbing powers ofsolidifying metal and mold material in the form:whereLH* —^Cs TLks ps CsP =^Ic„,p.C„,13 =ftP * ,11*)1.35C4H* *2L2 ^ (P t)133/4(^)3(2.51)In the case of convective cooling conditions, the temperature of the fluid can be usedas a zero reference temperature such that DAS can be expressed as a function of interfacialheat transfer coefficient, solidification time, and the ratio of heat absorbing powers ofsolidifying metal and mold material in the form:Lpsksj21.2 = C4 [^expY* 20.53( v-H;) (2.52)where^Y* = (H * +0.55H*(")and^E —h2tPsks Cp49Hills 123 further linked the dendrite arm spacing to segregation and interdendritic fluidflow. Hills model is unique in that it links DAS to distance from chill surface, solidificationtime, heat transfer coefficient and other characteristics of the mold and casting.2.6 Coupling Heat Transfer and Microstructural EvolutionSection 2.5.2.1 has presented the interdependence between macroscopic heat transferduring solidification and microstructural evolution. It is obvious that the phenomena (heat andmass transport) at the dendrite tip control the dynamics of the mushy zone which in turn affectevents in the single phase regions (liquid and solid). The final dendrite arm spacing has beenexpressed as a function of microscopic parameters (tip radius, temperature near the tip, tipvelocity, tip undercooling) as well as macroscopic parameters (fraction solid, local solidificationtime, shell thickness and heat transfer coefficient).Microscopic and macroscopic models can be coupled via an appropriate determination ofsolid fraction at any stage in the solidification process. This is often achieved by evoking asolute diffusion model. The distribution of solute between the solid and liquid is influenced bythe diffusion of solute in the liquid, diffusion in the solid, convection, turbulence and physicaleffects involving the incorporation of atoms into an advancing interface. The solute rejectedinto the liquid during solidification builds up a boundary layer near the solidification front. Theactual transfer of solute to the solid is related to the solute content in the liquid near the interface.Solute diffusion models are mainly of two types depending on the assumptions employedin the solution of mass transport during solidification;(a) complete mixing models(b) diffusion models502.6.1 Complete Mixing ModelsTwo basic assumptions are made in the formulation of these models:(i) dendrite tip undercooling is negligible such that the liquid to solid transformationbegins and ends at the equilibrium liquidus and solidus temperatures respectively.(ii) there is complete mixing in the liquid even in the vicinity of the dendrite tip suchthat no concentration gradient exists in the liquid.The expressions for the fraction solid obtained from some of the commonly used completemixing models are shown in Table 2.4. These models are at best first approximations and arevery useful for columnar growth in most solidification processes where the cooling rate is nothigh enough to produce any appreciable undercooling. It is worth noting that equilibriumtransformation is an uphill task since thermodynamic considerations require a temperaturegradient between the two phases for transformation to proceed.2.6.2 Solute Diffusion ModelsSolute diffusion models take into account the dendrite tip undercooling and soluteconcentration gradient in the liquid during solidification . The aim is to obtain the solid fractionas a function of not only the temperature but also the dendrite tip velocity. The models arevery useful for modeling equiaxed growth and have been used for columnar growth underconditions of rapid cooling5 .For columnar growth, the commonly used solute models 126-128 are mere derivations ofthe complete mixing models discussed above. This is due to the fact that the region ofnon-complete mixing is limited to a small liquid region near the dendrite tip. In one suchmode1 126, the Scheil relationship truncated at the dendrite tip was applied to studycolumnar-to-equiaxed transition. Although such an approach has the advantage of being easilyimplemented in heat flow calculations, it does not satisfy an overall solute flux balance s .51Another mode1 127 takes into account the solute layer close to the dendrite but only along thegrowth direction with an assumption that the fraction solid increases linearly with time. Kurzet al.128 proposed a model based partly on the Brody and Flemings 95 derivations. They assumedthat beyond a certain concentration and a corresponding solid fraction, there is complete mixingand incomplete mixing below this point.Table 2.4 The various complete mixing models for the evaluation of fraction solidduring solidification.Model Solid Fraction (L)Lever Rule TL11 –kp T,–Ts(^)Scheil's Mode1124IT –Ts )1- '1'117,^7,i ,– i sClyne and Kurz1251( T –Ts ){ I-2Fo 'kp )Ds tf=—–0.5 exp(-0.5Fo -)kpFo1 — 2Fo 'kp [lwhere^Fo .and the FourierTL – Ts=Fo[l – exp(–Fo -1)]Number,x2Brody and Flemings951where[^ ,(kp – 1)– acokp 111-k°L kpco{kp(1–aa)– 1}=DIG,jmivicoFor equiaxed growth, the solid fraction is simply the product of grain density and thegrain volume. If the grain is considered to be a sphere, then the basic equation for the solutemodel is given by52f,(x,t)=N(x,t)i4 7tr 3(x,t)^ (2.53)The above equation can be solved easily for equiaxed eutectics by evaluating N(x,t) and r(x,t)from nucleation and growth laws. However, the solution is complicated for equiaxed dendriticsolidification since a dendritic grain is not fully solid within the mushy zone. In this case, thesolid fraction expression becomes a function of the dendritic grain fraction (4) and an internalsolid fraction (f). For example, in the model developed by Rappaz and Thevoz 129 '130, the totalsolid fraction was assumed to be composed of two components:(i) the internal volume fraction which depends on the variation of temperature with time(ii) the grain volume fraction that depends on the kinetics of the dendrite tipUnder these conditions, the total volume fraction is given byAfs(x,t)=4:0(x,t)(74+&^+S2(x ,t)v,(6.74)At)^(2.54)where^413(x , t) = mc0(1 — kp )and^C1(x, t) —c (1 kp )3^1^1 and^f(Pe) =1 + 2Pe + Pe 2 4Pe 3 +N(x,t)lnr 3f(Pe)N(x , t)47Er 2f(Pe)(c — c0)53Chapter 3SCOPE AND OBJECTIVES3.1 Objectives/ImportanceIt is evident that the evolution of the desired microstructure during casting is ultimatelylinked to all processes that contribute to solidification, such as heat and mass transfer and fluidflow. Furthermore, the casting properties which determine the suitability of a cast product fora given application are strongly dependent on the microstructure. An efficient solidificationmodel that links the microstructure to known casting conditions, coupled with structure-propertymodels, will be invaluable in the following respects:(i) appropriate selection of casting parameters for a given set of properties and applicationsor conversely, prediction of expected microstructure and consequently, casting properties fora given set of casting conditions.(ii) contribution to the development of nondestructive testing techniques to evaluate thedendrite arm spacing variations within a casting and correlate with other property measurementsas a quality control tool.This work focuses on the coupling of heat transfer phenomena and the resultantmicrostructure at the early stages of solidification. There are three main objectives of thisendeavor:(1) To characterize the transient heat flux, heat transfer coefficient and shell growth atthe very early stages of solidification utilizing dip experiments;(2) To characterize the evolution of the dendritic structure in terms of the secondarydendrite arm spacing during this period;54(3) To establish models and/or validate existing models that directly link themicrostructure, particularly the dendrite arm spacing to known heat transfer parameters.The ability to define processing conditions to obtain an optimum microstructure in thesolidified material is the key aspect of the design of many technologically important processes,including casting, welding, single crystal preparation and rapid solidification techniques. Fordendritic solidification, the microstructure is often quantified in terms of the dendrite armspacing. The difficulties of modeling dendritic solidification are due mainly to the fact that heatand mass transport equations have to be solved simultaneously at three levels- the microscopicdendrite tip, mushy zone and the single phase domain (liquid or solid). At the dendrite tip,nucleation and growth phenomena influence the latent heat and solid fraction evolution. At themushy zone, heat and mass transport are influenced by the solid fraction and the coarseningphenomena. The single phase domain is mainly controlled by the local cooling rate which inturn depends on the events at the other zones and the boundary surfaces.3.2 MethodologyUnidirectional solidification was chosen for this study because of its simplicity and thefact that it promotes the growth of dendrites in a single direction of heat flow. Thecharacterization of the dendrite arm spacing is thus easier. A dipping mechanism was chosenover the usual teeming technique in an attempt to reduce the influence of convection that resultsfrom pouring. Furthermore, dipping is conceived to be a better simulation of some castingprocesses (such as melt drag in near-net-shape casting) where the mold moves to make the initialcontact with the molten metal.In dip experiments as in many dynamic heat transfer situations, the heat transfer parametersat the surface are easier to determine from transient measurements at one or more interiorlocations. Such problems are classified as inverse heat conduction problems (IHCP) which, as55mentioned earlier, alleviate the difficulties associated with surface measurements. The generalmethodology consists of four stages as illustrated in Fig. 3.1 - design and fabrication,mathematical modeling, experimental campaigns and process analysis.The first stage involved the design, fabrication and instrumentation of a practicalunidirectional solidification apparatus based on the dipping mechanism. A water cooled chillinstrumented with thermocouples and mobile to permit dipping into a melting chamber wasdesired. Two mathematical models were developed to simulate heat transfer in the chill andcasting respectively. A third model was applied to the prediction of secondary dendrite armspacing (SDAS) from known heat transfer parameters. Then experimental campaigns wereundertaken utilizing four different chill materials dipped into Al-Si alloy melt (3-7 %). Thefollowing variables were investigated - chill material, chill surface roughness, superheat, alloycomposition, bath height and oil film. Finally, the results of the experiments were analyzedand model predicted values were compared with measurements.56Project Formulation- Literature Survey, Objectives,- Title, Scope and ImportanceDesign and Instrumentation- Chill and cooling channel- Melting & Holding Devices- Dipping Technique- Data AcquisitionMathematical Modeling- Chill Model (IHCP)- Casting Model (1-D)- DAS ModelsiExperimental Campaigns- Temperature Measurements- Metallographic Examinations- DAS Measurements- Surface MicroprofilestProcess Analysis- Modifications- Validation of Models- Analysis of Results- New ModelsiConclusions and RecommendationsFig. 3.1 Schematic illustration of the project methodology.57Chapter 4EXPERIMENTAL PROCEDURE AND RESULTSThe experimental part of this work was carried out in three steps:(i) design, fabrication and instrumentation of mold and cooling systems(ii) dipping campaigns(iii) metallographic examination and DAS measurements4.1 DesignThe experimental apparatus is shown schematically in Fig. 4.1. The apparatus was madeup of four major parts - the melting chamber, the chill, the cooling and the data acquisitionsystems. The cooling system consisted of a rectangular water channel connected to two 13mmdiameter water pipes serving as the inlet and outlet for the cooling water. The top plate wasfabricated out of transparent plexiglas to allow direct observation of water flow and had six3.1mm diameter holes to permit the passage of thermocouples from the chill to the dataacquisition system. The bottom plate of the water channel had a threaded hole at its center suchthat the chill could be screwed into it. The chill was essentially an insulated cylindrical metallicrod with exposed ends. A small part of the upper half (6.35mm) was threaded to fit into thebottom plate of the cooling channel.Commercial purity copper rod was used as the chill for most of the experiments. Castiron, brass (70%Cu-30%Zn) and low carbon steel rods were used to study the effects of differentchill materials on heat transfer and cast structure. For each experiment, the chill was insulatedwith 2mm thick fiberfrax that was held in place with 5gin iron wires tied tightly around the chill(fiberfrax of original thickness 5mm compacts to a final thickness of 2mm after fastening). Inorder to ensure good adherence between the solidified shell and the chill, two small pieces of58steel plate (20mm x 2.5mm x 0.5mm) were tied around the chill with thin iron wires (10timthick) and the protruding ends (about 5mm) were bent into an L-shape to serve as hooks. Thesolidified shell adhered to these hooks and also to the thermocouple in the melt, such that theshell could be lifted with the chill assembly. It was confirmed from calculations andmeasurements that these hooks did not affect heat transfer and microstructure in any tangibleway.Fig. 4.1 Schematic representation of the experimental set-up.A Phillips high frequency coreless induction furnace (0-12KW power, 0-110KHz) wasutilized for melting. A graphite crucible was used both as the container for the casting and asa susceptor of the electromagnetic field. The graphite crucible was insulated by inserting it intoa MgO crucible of slightly larger dimensions and packing any available space between them59with fine particles of alumina. Furthermore, the outer MgO crucible was wrapped with 5mmthick layer of fiberfrax. Preliminary measurements indicated that the heat losses from theinsulated surfaces accounted for less than one percent of the total heat loss by the solidifyingmetal.Al-Si alloys were melted in a graphite crucible and the water-cooled chill was dipped intothe melt at a known superheat while the data acquisition system recorded the temperature profilesat various locations in the chill and solidifying metal. Details of the equipment are summarizedin Table. 4.1.Table 4.1 Some details of the experimental design.Component Material DimensionCooling Channel(Rectangular Box)Top Plate:plexiglasOther Sides:stainlesssteelOutside: 300mm x 65mm X 62mmWater Channel . 285mm x 52 mm x 12.7mmChill(Cylindrical Rod)Commercial puritycopper, cast iron,70%Cu-30%Zn brass,low carbon steel.Diameter: 28.6mmHeight: 28.6mmInduction Unit Coil: 9mm, Cu tube Internal Diameter: 95mmHeight: 165mmNo. of Turns: 10Crucible Graphite Internal Diameter: 35mmThickness: 10mmHeight: 15 mmThermocouples Chromel-Alumel TypeKDiameter: 20ginAverage Bead Diameter: lmmData Acquisition Notebook on Compaqor MetrabyteFrequency: 2HzCasting Alloys Al - 7% SiAl - 5% SiAl - 3% Si-60Consider the electrical analogue shown in Fig. 4.2 in which it is desired that R chk >> Rfkh+R,1,, i.e. Rfm, and Rch have to be minimized. This was achieved by a proper design of thecooling water channel since the chill resistance, R ch, was limited by the chill thermal conductivityand the chill height. The cooling water channel was designed to ensure the following:(i) enough water velocity such that h f >>(ii) fully developed flow near the chill surface(iii) no appreciable recirculation or stalling near the chill surface.These goals were attained with the dimensions given in Table 4.1 and a water velocity of 2.0m/s such that the thermal resistance at the chill/casting interface constituted more than 70% ofthe total resistance for the whole duration of an experiment. A typical variation of the thermalresistances during solidification is presented in Fig.4.3 for Al-7%Si and copper chill. Waterw,RUchRdi = ZdjkdiChillCastingFig. 4.2 Schematic illustration of the operating thermal resistances.61Fig. 4.3 Estimated variation in thermal resistances for Al-7% Si alloy and copperchill.4.2 Instrumentation and Data AcquisitionThe chill was instrumented with three thermocouples located at 1.6mm, 3.2mm and22.2mm respectively from the hot face. The thermocouples with the beads bent into a slightL-form, were tightly secured in place in 2mm holes drilled at the center of 6.4mm diameterscrews. The screws were made of the same material as the chill and were tightened until the.the thermocouple beads could no longer move. A fourth thermocouple (in stainless steel sheath)attached to the chill insulation monitored the casting temperature at a preset location (12mmfrom chill /casting interface).For each experiment, the thermal responses of the thermocouples were recorded witheither a COMPAQ Type 18 computer (with 32-channel data board) or a Metrabyte Computer62(128-channel board) using Notebook 3.1 software. Preliminary measurements were made todetermine the experimental time step and the frequency of data acquisition. An experimentaltime step of 0.5s (2Hz frequency) was finally adopted since no significant loss in the generaltrend of the thermal responses was observed within this time interval. The thermocouples werecalibrated by immersing them into two mediums of known temperature - melting ice/watermixture and boiling water. The thermal response and standard deviations from the boiling watertest is shown in Table 4.2. The approximate response time of the thermocouples was estimatedfrom this test as 0.13s. It is noted that chromel-alumel thermocouple (type K) used for theexperiments has a design specification of ±2.2°C error limit between 0°C and 1250°C, with theupper end of the error operating at lower temperatures 131Table 4.2. Thermal response of four thermocouples (TC1-TC4) during calibrationin boiling water (B.P. ,--- 100°C at 1 atm. pressure).Time (s) Temperature (°C)TC1 TC2 TC3 TC45 101.3 101.3 102.5 102.510 101.3 102.4 101.3 101.315 101.2 101.3 101.3 102.520 102.6 101.3 101.3 102.525 101.3 102.5 101.3 101.3Mean 101.6 101.8 101.6 102.1StandardDeviation0.5 0.6 0.5 0.6Error 1.6 1.8 1.6 2.14.3 Dipping CampaignsThe chill surface was polished with different grades of abrasive paper or was filed in orderto obtain various degrees of surface texture. The surface roughness and surface microprofiles63were measured with a digital Talysurf 5 System 132, a stylus-type device that is capable ofmeasuring both the roughness and waviness components of the surface texture. Although surfaceroughness can be measured in terms of amplitude parameters (measure of the verticaldisplacements of a given profile) or spacing parameters (measures of the irregularity spacingsalong the surface, irrespective of the amplitude of these irregularities), the arithmetic mean ofthe departures of the amplitude profile from the mean line (Ra in gm) is the universallyrecognized, and most used, international parameter of roughness 132. Hence, surface roughnesswas quantified in terms of the Ra values in this study. With reference to Fig. 4.4, Ra is definedas:Ra =^Y(x)dxI o(4.1)Fig. 4.4 Schematic representation of surface microprofile measurement.Before each dipping campaign, the chill surface was prepared to obtain the desiredroughness before dipping. In some of the experiments, a thin film of oil was applied uniformlyto the chill surface before dipping. The properties of the oils used are shown in Table 4.3. The64chill and cooling system was meticulously sealed with liquid silicone to prevent water leaks.The data acquisition was turned on after running the cooling water through the cooling channelfor a few minutes. Each thermocouple response was compared with a measurement made witha high-accuracy hand held digital thermometer 131 (Omega CL23 calibrator / thermometer).Faulty contacts were revealed at this stage and were corrected immediately.Table 4.3 Properties of the oils used in the experiments.Properties Canola HEAR Steelskin BlachfordViscosity @ 160 185 200 21438°C (SUS)Flash Point (°C) >315 >300 226 200Fire Point (°C) >360 >350 252 -Boiling Point(°C)Start 205 215 170 -20% 280 280 230 -50% 315 320 300 -90% 335 335 330 -Fatty Acid CarbonContent Chain"Palmitic (%) 5.3 3.2 2.9 4.2 C 16:0Oleic (%) 57.7 14.9 40.3 61.5 C18:1Linoleic (%) 23.6 15.1 27.1 19.6 C18:2Linolenic (%) 9.2 9.3 26.0 9.7 C18:3Eicosenoic (%) 0.7 8.2 0.3 0.7 C20:1Erucic (%) 0.3 45.2 0.1 0.6 C22:1Others (%) 3.2 4.1 3.3 3.7** C18:1 implies 18 carbon atoms with one double bond in the chain.Small pieces of the alloy to be cast were weighed (the weight depends on the desired bathheight) and placed inside the graphite crucible. Three Al-Si alloys were utilized: Al-7%Si,A1-5%Si and Al-3%Si. The induction machine was started and melting was achieved within afew minutes; further heating raised the temperature of the melt well above the desired superheat.65The induction machine was turned off while a thermocouple encased in a mullite tube andconnected to the high-accuracy hand held digital thermometer was immersed in the melt tomonitor the superheat. The oxide layer at surface of melt was scooped off manually beforedipping since attempts to reduce oxidation either by argon blowing or by addition of Na 2A1F3did not yield positive result due to the high oxidation rate of aluminum alloys.Once the desired superheat was attained, the chill/casting contact was achieved by dippingthe chill and cooling channel assembly into the melt to a predetermined depth. Different depthswere experimented with and it was observed that the thermal response was more stable withina range of 2mm - 8mm. At depths less than 2mm, it appears that the thin surface oxide wasnot displaced enough to allow for a good initial chill/melt contact. At depths greater than 8mm,excessive burning of the fiberfrax insulation on the chill was noticed, leading to an increase intransverse heat transfer. Hence, the chill depth inside the melt was maintained at approximately5mm throughout the experiments. Three different dipping methods were tested - placing thechill/cooling system assembly on preset beams, using clamps to hold the assembly in place,and the use of a laboratory jack. No significant difference was observed between the thermalresponse of the three methods. Hence, the first of the three methods was adopted.The casting and chill were left in contact for about 60 seconds for each experiment. Thesolidified shell thickness was measured after cooling to room temperature. The surfaceroughness of the solidified shell was also measured with the Talysurf 5 System 132. To ensurethe reproducibility of results, at least two trials were conducted for each set of conditions. Theresults were observed to be reasonably reproducible as illustrated by Table 4.4 with standarddeviations ranging from 0.7 to 4.9.66Table 4.4 The measured thermal response at one location for three trials under thesame condition.Time (s) Trial 1 Trial 2 Trial 3 Mean (°C) Standard(°C) (°C) (°C) Deviation0 15.3 17.7 16.5 16.5 1.05 66.5 70.0 58.3 64.93 4.910 120.5 121.7 119.3 120.5 1.015 130.2 133.7 136.2 133.4 2.520 142.2 150.7 143.6 145.4 3.725 147.1 151.9 147.1 148.7 2.330 148.5 148.3 143.6 146.8 2.235 147.1 148.4 142.2 145.9 2.640 143.4 145.9 141.0 143.4 2.045 135.0 139.9 135.0 136.6 2.350 118.1 127.9 123.2 123.1 4.055 115.7 113.3 112.9 114.0 1.260 114.6 113.3 112.9 113.6 0.74.3.1 Thermal Response of the ThermocouplesThe thermal responses obtained for the dipping campaigns are as shown in Figures 4.5to 4.11. It is apparent that the temperature at a given location in the chill rises steeply uponcontact with the melt up to a peak value and then decreases as solidification progresses untila fairly steady state value is attained. On the other hand, the temperature at given location inthe casting decreases continuously with time as indicated in Fig. 4.5. The temperature profilein the chill decreases with increasing roughness of the chill surface (Fig. 4.6) and decreasingthermal diffusivity of the chill material (Fig. 4.7). Increasing the superheat increases thetemperature at a given location in the chill as depicted in Fig. 4.8. The temperature profile inthe chill increases with decreasing silicon content (Fig. 4.9). The effect of the oils on thetemperature profile in the chill is depicted in Fig. 4.10. The four oils utilized in this studydecreases the chill temperature for a few seconds (< 10 s) but finally increase this temperature67 22mm from hot face- - - 3.2mm from hot face 1.6mm from hot face- 12mm inside casting_......1:: .. ............. ... .. _,....... ......,--^—,beyond the value attained with no oil. Canola and HEAR oils appear to influence the chilltemperature more than the other two. The chill temperature also decrases with decreasing bathheight as shown in Fig. 4.11. The reason for each of the observed results are discussed laterin Chapter 6.700600---.0 5000..__.a),._ 400D0u1._ 3000-0.) 200t—100I^I^I^1^i-----0^10 20 30 40 50 60Time (s)Fig. 43 Typical temperature data during the casting campaigns for Al-7%Si alloyand copper chill.68110100900 80700- 60(^5 0ll121_•urfaefjoughnem(ziO m)- 0.018-- 0.090---- 0.291- 6.61----10 50 • 40302010 ^I^I^1^1^1 0^10 20 30 40 50 60Time (s)Fig. 4.6 Effect of surface roughness on measured temperature response of the chill(copper) at 1.6mm from chill/casting interface for Al-7%Si alloy.250200O15000_<L)E100 5000^10 20 30 40 50 60Time (s)Fig. 4.7 Effect of chill material on measured temperature response of the chill at1.6mm from chill/casting interface for Al-7%Si alloy.69melt superheat— 0-C- — 90'C00•C— 00•C— 120•C25020004) 150-4-,a_a)- 100Ea)50Fig. 4.8Fig. 4.910 20 30 40Time (s)Effect of superheat on measured temperature response of the chill (copper)at 1.6mm from chill/casting interface for Al-7%Si alloy.•0140120100 0) 80060Q4020050 600^10 20 30 40 50 60Time (s)Effect of alloy composition on measured temperature response of the chill(copper) at 1.6mm from chill/casting interface.70- no oilIllaohlordataalakin- REARoonola10080604020140120(-)0 10080.4860E40200Fig. 4.100^10 20 30 40 50 60Time (s)Effect of oil film on measured temperature response of the chill (copper) at1.6mm from chill/casting interface.Fig. 4.11 - 14.0om- -I^I^I^I 10 20 30 40 50 60Time (s)Effect of bath height on measured temperature response of the chill(copper) at 1.6mm from chill/casting interface for Al-7%Si alloy.714.4 Metallographic ExaminationLongitudinal specimens were cut from the center of the solidified shells for metallographicexamination. The specimens were mounted, ground and polished to 0.06Am surface finish.The Al-Si alloys were etched in Kellers reagent (1%HF-1.5%HC1-2.5%HNO 3-95%H20) forabout 30 seconds to reveal the dendritic solidification structure. The specimens were thenexamined with a metallurgical microscope and micrographs taken at various distances from theshell/chill interface. Typical micrographs are depicted in Fig. 4.12.The secondary dendrite arm spacing were measured from magnified micrographs or metricmicroscope utilizing a combination of the two common DAS counting methods - the lineintercept and individual arm counting methods.In the line intercept method, a line parallel to the primary arms was drawn and the numberof secondary arms were counted along one side of this line for short intervals of length (/). Atleast five measurements were made at any given distance from the surface. The averagesecondary dendrite arm spacing (SDAS) at a given mean distance is then given by1^1.^SDAS () —i(No . of Secondary Arms x Magnification (4.2)In the individual arm counting method, the primary arms were identified and the distancebetween the centers of adjacent secondary arms was measured. At least five measurementswere made at a given distance from the surface and the average of these measurements wastaken as the secondary arm spacing at the given location. Results of a typical SDASmeasurement together with the associated standard deviation is shown in Table 4.5. Figure4.13 shows a typical variation of measured secondary dendrite arm spacing with distancefrom surface. The effects of various variables on SDAS are depicted in Figs. 4.14 to 4.19.72(b)Fig. 4.12 Typical micrographs of the castings (etched in Kellers reagent, each scalereading represents 0.02mm) - (a) Al-7%Si alloy and copper chill, (b)Al-7%Si alloy and cast iron chill.73It is apparent that the SDAS increases with increasing distance from the interface. SDASincreases with increasing roughness of the chill surface (Fig. 4.14) and decreasing thermaldiffusivity of the chill material (Fig. 4.15). Increasing the superheat decreases the SDAS at agiven location in the casting as depicted in Fig. 4.16. SDAS at a given location increases withdecreasing silicon content (Fig. 4.17). The effect of the oils on SDAS is depicted in Fig. 4.18.The four oils utilized in this study appear not to have considerable effect on the SDAS at distancesnear the surface (< 10 mm) but decreases the SDAS at distances further away from the interface.Changes in bath height have no tangible effect on the measured SDAS as shown in Fig. 4.19.The reason for each of the observed results are discussed later in Chapter 6.74Table 4.5 Typical secondary dendrite arm spacing measurements for copper chill.Measured SDAS (gm)Shell(mm)1 2 3 4 5 6 7 Mean Std.** % ErrorSurface Roughness = 0.018 gm4.0 20.00 16.70 18.75 21.40 16.00 20.00 19.00 18.84 0.18 20.08.0 41.7 38.90 33.30 35.00 33.30 37.50 35.00 36.39 0.29 17.812.0 42.90 41.60 42.90 43.30 45.00 45.00 43.30 43.43 0.11 10.016.0 41.80 46.00 54.00 50.00 50.00 51.00 53.30 49.44 0.40 20.020.0 61.00 65.00 62.50 63.00 62.50 63.30 62.50 62.83 0.11 8.2Surface Roughness = 0.0304 i.tm18.60 18.60 20.00 26.30 22.20 22.90 22.90 25.00 22.56 0.25 20.440.00 40.00 38.75 38.70 40.00 41.67 40.00 42.00 40.16 0.12 10.748.00 48.00 42.90 42.00 45.00 45.70 47.50 46.70 45.40 0.21 13.553.00 53.00 52.50 46.70 46.70 52.00 53.30 53.30 51.07 0.28 15.555.00 55.00 57.50 60.00 65.00 60.00 62.00 63.30 60.40 0.32 16.1Surface Roughness = 0.291 iim27.00 27.00 27.50 22.50 25.00 30.00 32.50 33.30 28.26 0.36 22.650.00 50.00 44.30 44.00 42.50 45.00 50.00 50.00 46.54 0.31 17.048.50 48.50 46.00 50.00 57.50 53.20 54.50 57.50 52.46 0.41 20.360.00 60.00 60.00 62.00 61.00 55.00 60.00 61.00 59.86 0.21 12.170.00 70.00 62.00 63.30 63.30 63.30 65.00 62.50 64.20 0.25 13.4** Std stands for standard deviation75surface roughaeu(x10.4171)0 0.0160 0.030V o.zsty 6.6100 10.66o10080205^10^20Distance from surface (mm)Fig. 4.13 Typical measured secondary dendrite arm spacing with corresponding errorbars for Al-7%Si alloy and copper chill.i^I2515100908070coE 60io,- 50x...--40U)-.‹o 30U)20100 ^0Fig. 4.140^5^10^15^20^25Distance from surface (mm)Effect of surface roughness on measured secondary dendrite arm spacing forAl-7%Si alloy and copper chill.7610080CD^6040(r)2000^5^10^15^20^25Distance from the surface (mm)Fig. 4.15 Effect of chill material on measured secondary dendrite arm spacing forAI-7%Si alloy.1008060< 40C(f)200^5^10^15^20^25Distance from surface (mm)Fig. 4.16 Effect of superheat measured secondary dendrite arm spacing for Al-7%Sialloy and copper chill.77I^I^1^I0 A1-7XS10 Al-6X31V A1-3X31100806040CD(f)200^5^10^15^20^25Distance from surface (mm)Fig. 4.17 Effect of alloy composition on measured secondary dendrite arm spacing forcopper chill.100801 60< 40200^5^10^15^20^25Distance from surface (mm)Fig. 4.18 Effect of oil film on measured secondary dendrite arm spacing for Al-7%Sialloy and copper chill.7810080E(0I^600X..,Q 40w200^5^10^15^20^25Distance from surface (mm)Fig. 4.19 Effect of bath height on measured secondary dendrite arm spacing forA1-7%Si alloy and copper chill.79Chapter 5MATHEMATICAL MODELING5.1 Chill Heat Flow ModelWith adequate insulation on the cylindrical surface of the chill, heat flow through thechill could be reasonably approximated to a one-dimensional heat transfer problem. Thermalanalysis of the chill involves the following steps:(i) heat transfer from solidifying shell to the chill hot face. The mode of heat transfer herecould be convection, radiation or conduction depending on the form of contact betweenthe solidifying shell and the chill which determines the existence of an air gap betweenthem.(ii) heat transfer through the chill thickness by conduction.(iii)^heat transfer from the chill cold face to the cooling fluid mainly by convection.The governing equation for the chill heat flow model is therefore(,azInitial and Boundary Conditions1. Initial temperature of the chill is specified (l ay, < z < lch, 0 < r < ra, t=0)T = To = constant2. At the chill hot face (z = Iwo 0 < r < ro, t)aT–k —az = qc (t) = hc (t)^T)3.^At the chill cold face (z = l ch, 0 < r < ro, t)(5.1)(5.2)(5.3)80T. 7'eh (t)^ (5.4)Equation (5.1) was solved numerically using implicit finite difference equations based onthe Crank-Nicolson method which has a second-order accuracy in both space and time. Toprovide this accuracy, the finite difference approximations were developed at the midpoint ofthe time increment (At/2). The chill was discretized into N nodal points of width Az as shownin Fig. 5.1, such that a typical interior node (i.e. 2 i N) yields the following expression fora time step At:ocAt { 1 001 7, ±1 ocAt^aAt 7,„^ocAt^aAt 7,z i 1 —^2 z i^zi+1^2zi-1 2^Li^21i+12Az 2 -^Az^2Az2 2Az ^2AzAt the chill hot face, Eq. (5.3) becomes1+ °iAtin +1 + 41At T; +1 = 2At n+1 { aAt^n OCAt, Tiq —^i}i +T2-Az 2^Az2^pcp z^Az2 Az2 (5.6)At the chill cold face, Eq. (5.4) remains unchanged. The unknown heat flux, q: +1 , is bestassociated with the midpoint of the time step from the present time, e, to the future time,The temperatures are all known at time e and are needed at time t°+ 1 .Equations (5.5) and (5.6) were solved using the non-linear inverse heat conductiontechnique based on Beck's method 133334. This technique employs the least square minimizationwith regularization approach to obtain values of qc(t) and nodal temperatures in the chill (SeeAPPENDIX A). The measured temperature at 22.2mm from the hot face was used as a boundarycondition while the thermocouple readings at 1.6mm and 3.2mm were used as interiortemperature profiles respectively. The model output at each time step includes the following:the interfacial heat flux, temperature distribution in the chill, the difference between calculated81and measured temperatures at the thermocouple locations, and the root mean square of thesedifferences. The thermophysical properties employed in this model are summarized in Table5.1.Fig. 5.1 Discretization of both chill and casting.82Table 5.1. Thermophysical properties of materials used in the chill mode145'75,128,135-137.Chill Copper Brass Carbon Steel Cast IronDensity(p,Kgm 3)8940 8522 7400 7100Specific Heat 384 + 0.0988T 385 456 + 0.376T 514 + 0.38T(Cp ,JKg-IK-1)Thermal 400 - 0.0614T 104.93 + 0.3 T - 6.2 x 59.4 - 0.0418T 43.9 - 0.0131TConductivity(k, Wm -'K- ')10-4 T2 + 3.6 x 10 -6 T3Thermal 1.16 x 10-4 - 4.42 x 3.20 x 10 -5 + 9.14 x 1.73 x 10 -5 - 2.10 x 1.19 x 10-5 - 1.00 xDiffusivity 10-8 T 10-8T - 1.89 x 10-1° T2 108 T 104 T(1,m 2s --I)Average 16.9 19.9 14.5 13.0Coefficient ofLinear ThermalExpansion(L„,,pm1m.K)Emmisivity ofoxidized surface0.57 0.63 0.31 0.61(e)5.2 Casting Heat Flow ModelAlthough the furnace was turned off before the chill was dipped into the metal, it wasfound from both measurements and calculations that with adequate insulation around the crucible(> 5mm insulation thickness), the heat flux through the crucible constitutes less than 2% oftotal heat loss by the casting. Hence, the heat transfer in the casting can also be treated as 1-Dcase. Heat is extracted from the casting in the following steps:(i) heat transfer in the liquid pool (convection and conduction)(ii) heat transfer through the mushy zone (convection and conduction)(iii)^heat transfer through the solidified shell thickness ( mainly conduction)83(iv) heat transfer from the casting surface to the chill hot face. Here, the mode of transfercould be conduction, convection or radiation depending on the form of contact betweenshell and chill(v) heat transfer from the casting to the crucible wall (could be convection, conduction orradiation)(vi)^heat transfer from the exposed surface of the casting to the atmosphere (mainlyconvection and radiation)As in the chill model, the governing equation for this model is similar to Eq. (5.1) exceptthat a heat source term exists in this case.a ( aT) af^aTaz k-j.Initial and Boundary Conditions1. Initial temperature of the casting is specified (0 < z < l chk, 0 < r < ro, t)T =7' = constant2. At the chill/casting interface (z = lchk , 0 < r < ro, t)aTk—=qc (t)= hc (t)(T –Tchs)3.^At the casting/crucible interface (z = 0, 0 < r < ro, t)k—ar = qcr (t) 0az (5.10)The discretization procedure is similar to that of the chill model. The values of qc(t)obtained from the chill model solution were used as input data in the implicit finite difference(5.7)(5.8)(5.9)84casting model to yield values of the nodal temperatures which were then utilized to computesuch parameters as shell thickness, heat transfer coefficient, the thermal resistances,solidification times, local solidification times and cooling rate. These parameters were usedin the DAS models. The thermophysical properties utilized in the casting and DAS models aresummarized in Table 5.2. A schematic illustration of the complete computer implementationof the models is depicted in Fig. 5.2.Assumptions :(1) The latent heat evolution during solidification was accounted for by varying thespecific heat capacity, C p .(2) There are no other sources of heat generation (i.e apart from the latent heat ofsolidification).(3) There is no net heat consumption across the chill/casting interface. Hence, the heatflux profile calculated for the chill above was the same as the heat flux extracted from the casting.(4) The specific heat capacity, thermal conductivity and density are all functions oftemperature.(5) Heat transfer in the liquid by convection is very negligible and is not accounted forin the model.85Table 5.2. Thermophysical properties of materials used in the casting model45,75,128,135-137.Casting^A1-7%Si Al-5%Si Al-3%SiDensity 2680(p,Kgm -3)2690 2695Liquidus^610 632 648Temperature(Tr , °C)Solidus^577 577 577Temperature(T,, °C)Latent Heat^389000 389000 389000(L,JKg -1 )Specific Heat^963 963 963(Cp ,JKg -1K-1)Average^90 104 121ThermalConductivity(k,Wm -IK-1)Thermal^3.49 x 10-5 4.01 x 10-5 4.66 x 10 -5Diffusivity(oc m 2.9 -1)Volume^3.5 4.8 6.0Shrinkage (%)Average^23.5 24.0 24.7Coefficient ofLinear ThermalExpansion(La ,gmlm.K)Emmisivity of^0.19oxidized surface(E)0.19 0.19Other Parametersco^(%) 7ce^(%) 12.3kp 0.13D,^(m2s-1) 3.10 x 10-9m^(Kpct-1) -6T (mK) 0.9 x 10-786I nput DataMeasured Temps.ThermophysicalProperties of ChillyInverse ConductionModel0 Output DataTemp. Distributionin the ChillTime-dependentInterfacial Heat FluxInput DataInterfacial Heat FluxThermophysicalProperties of CastingCasting SuperheatSolidification ModelOutput DataTemp. Distributionin the CastingShell ThicknessSolidification TimeLocal Solidifcation TimeLocal Cooling RateHeat Transfer Coefficient Input DataShell ThicknessSolidification TimeLocal Solidifcation TimeLocal Cooling RateHeat Transfer CoefficientVDAS MODELVOutput DataDASCompare withMeasured DASa)O-oa5OO-oy Fig. 5.2 Schematic illustration of the complete computer implementation.875.2.1 Latent Heat Evolution and Fraction SolidThe effective heat capacity technique discussed in section 2.3.2 was adopted to handlethe latent heat evolution. A linear temperature distribution between nodes was assumed andthe latent heat release at any node within the vicinity of the mushy zone (illustrated in Fig. 5.3)is evaluated as follows:TrET+1TT - 2TL— T+2T-1(5.11)(5.12)Case I. Liquidus Vicinity: Node is above the liquidus while a fraction of its volume lies withinthe mushy zone (T1 > TL, T4,<TL) or node is within the mushy zone while a fraction of itsvolume is above the liquidus (Ts < T1 < TL, TT > TL).where^Vfl —TL —T,.TT - T1,AL)Ceff=Vfl.Cp(T)+(l—Vf1).(Cp(T)—LE-1,- (5.13)In this case, Afs/AT is evaluated at the temperature T .0.5(TL +TI)Case II. Complete Mushy: The whole nodal volume lies completely within the mushy zone(Tr > Ts, TA, < TL )•AfsCif = Cp (T)— L AT(5.14)AL/AT is evaluated at the nodal temperature, T.88Case III. Solidus Vicinity: Node is above the solidus while a fraction of its volume lies belowthe solidus (Ts < T1 > TL, T1 < Ts) or node is below the solidus while a fraction of its volumeis above the solidus (T, <Ts, TT > Ts).whereTT — TsV' P — TT — Ti,AfsCeff = Vf2 .0 p (T) + (1 — Vf2).(C p (T) — L a- (5.15)In this case, APAT is evaluated at the temperature T = 0 .5(TT +Ts)Time (s)Fig. 5.3 Schematic illustration of the various zones in the effective specificheat method - Case I. liquidus vicinity, Case II. complete mushyand Case III. solidus vicinity.The solid fraction was initially calculated using the Clyne and Kurz model' 25 :89fs1 1 —2Fo'kp(I-2Fo kpf1 —TL—Ts(T —7; (5.16) where^Fe' =Fo[1— exp(—Fo -1 )] — 0.5 exp(-0.5Fo -4 )and the Fourier Number,^Fo= AteEquation (5.17) reduces to the lever rule as Fo approaches infinity and to Scheil modelas Fo approaches zeros . A value of 3 x 10 m 2/s has been reported75 for the liquid diffusioncoefficient of Si in liquid Al-Si alloys. Assuming that the solid diffusion coefficient is aboutone fifth of this value, the Fourier number was found to be so small that the result of Eq. (5.17)was not different from that of Scheil model. Hence, the simpler Sheil model was subsequentlyused to calculate the solid fraction such thatf = 1(T —T juu-kP )71—Ts (5.17)53 Dendrite Arm Spacing (DAS) ModelsThe secondary dendrite arm spacings were computed using equations (2.37), (2.38),(2.40) and (2.42). The constant C5 was evaluated using the rate coarsening model due toMortensen 118 .271-DiA2 = C5{44-70(1—kp).fs(1—e)1/3(5.18)The constant C4 was evaluated by fitting the calculated cooling rate to the measuredsecondary dendrite arm spacing. The values for the parameters used in these equations werealso included in Table 5.1.905.4 Sensitivity Analysis And Model ValidationSensitivity analysis was carried out for both the chill and the casting models. For the chillIHCP model, the effect of node size, time step and number of future time steps (a period oftime where the heat flux is assumed to be temporarily constant) were studied. TheCrank-Nicolson implicit finite difference procedure which was used in this case has an accuracythat varies as (Az )2 and (At)2 respectively for constant thermophysical properties. By testingthe IHCP model for different time steps it was established that the heat flux does not varyappreciably at time steps At .__ 0.5s . A calculational time step of 0.25s was chosen in accordancewith the suggestion' 12 that the calculational time step be one half or one third of the experimentaltime (data was recorded at 0.5s interval). A node size of 6x10 -4m was also found to be adequate.The selection of the number of future time steps (see Appendix A) to be used in the programis more complicated. Increasing the number of future time steps has the beneficial effect ofreducing the sensitivity of the IHCP algorithm to measurement errors by "smoothing or biasing"the measurements to ensure a more stable output. However, there are two deleterious effectsof increasing the number of future time steps. First, there is less agreement between the measuredand calculated temperatures at a particular time step. Secondly, sudden changes in heat fluxcould be missed by biasing. Therefore, a balance must be achieved between two opposingconditions of minimum sensitivity of heat flux to measurement errors and minimum error inheat flux for errorless data. As measurement error increases, the number of future time stepsvalue should increase and vice versa. A value of 2 was found to be adequate for smooththermocouple readings while a value of 4 was better for readings with appreciable fluctuation.The model is validated by comparing the predicted values at the thermocouple locations withthe measured values as shown in Fig. 5.4. A good agreement exists between the measured andmodel predicted temperatures.91^ measured (1.01nm)— calculated (l.thara)V measured (8.2mra)— calculated (a_Etniu)1009080706050403020100 10^20^30^40^50^60Time (s)Fig. 5.4 Typical calculated and measured temperature profiles in the chill.For the casting model, the choice of Az and At was made by comparing the numericalresult with an infinite slab analytical predictions for the same boundary conditions. An exampleof this comparison is shown in Fig. 5.5. Good agreement is seen to exist between the analyticaland numerical results. Based on this comparison, a node size of 7x 10 4m was chosen while atime step of 0.5s was adopted. Furthermore, the energy balance at each time step wascross-checked by recalculating the surface heat flux from the predicted temperatures andcomparing the calculated values with the input values. Table 5.3 depict a typical result of thisprocedure. Good agreement is also observed between the two values. The model was alsovalidated by comparing the predicted temperature profile at a given location with thermocoupleoutput within the casting as shown in Fig. 5.6 and the predicted results compares well with the92measured ones. Further validation was carried out by comparing the model predicted shellthickness with the measured values. A typical result of this comparison is shown in Fig. 5.7.The predicted shell thickness compares favorably with the measured values.1^3^5^7^9^11^13^15Node NumberFig. 5.5 Temperature profile at the same location for both analytical andnumerical solutions of the transient infinite slab problem.66064062060058056054052050048046044042093Table 5.3. Comparison between the input heat flux and recalculated heat flux.Time (s) SurfaceTemp. (°C)q input(MW/m2)q recal.(MW/m2)StandardDeviation2 635 1.14 1.14 0.034 627 1.05 1.05 0.016 622 1.00 1.00 0.008 618 0.97 0.97 0.0110 615 0.95 0.95 0.1512 613 0.95 0.95 0.1514 610 0.93 0.93 0.1716 609 0.93 0.93 0.4218 607 0.91 0.91 0.3520 605 0.87 0.87 0.4222 602 0.85 0.85 0.4124 598 0.81 0.81 0.3926 594 0.77 0.77 0.3328 589 0.71 0.71 0.3330 584 0.66 0.66 0.10650600 00<L) 5500Q)E 500 450 0^10^20^30Time (s)Fig. 5.6 Typical calculated and measured temperature profiles in the casting(10mm depth).40^50^60943 0----.EE2501(f)wC-_Y01--- 20=(n150^30^60^90 120 150Superheat (°C)Fig. 5.7 Measured and calculated shell thickness profiles at varioussuperheats.95Chapter 6RESULTS AND DISCUSSION6.1 Heat FlowAs indicated in the measured temperature-time profiles (Figs. 4.5-4.11), it is evident thatthe temperature at any location in the chill increases very rapidly once contact is establishedwith the liquid metal until a peak value is attained. The magnitude and time of attaining thispeak value at a given location depend on the casting conditions. As expected, the magnitudeof the temperature peak increases while the time of attaining the peak decreases with decreasingdistance from the interface due to the chill thermal resistance. From this peak value thetemperature drops initially with a steep gradient and finally to a fairly steady value. This resultis similar to that observed by Pehlke et al. 47 for the case when the chill is located above thecasting.Typical model predictions for the interfacial heat flux, heat transfer coefficient, shellthickness, interfacial gap and surface temperature profiles are depicted in Figs. 6.1, 6.2 and 6.3respectively. These profiles correspond to a copper chill dipped into Al-7%Si alloy at a superheatof 30°C and with a chill surface roughness value of 0.03 gm and represent the general trend inthe results. It is obvious that the interfacial heat flux and heat transfer coefficient follow thesame trend as the measured chill temperatures. The radiation components of the heat flux andheat transfer coefficient which were calculated based on parallel plate assumption are seen tobe negligible, contributing less than one percent in each case. Initially, the casting/chill contactis localized according to the asperity profiles in the chill and the wettability of the chill surfaceby the molten metal, but increases continuously as the liquid metal spreads on the chill surface.96— inverse solution-^ radiation only10006(a)— 800NE-...Y 600 grad_..11- .................. — 3CT4002500— inverse solution^ radiation only20002N15001000... ...... . - ---129The peak heat flux (qmax) occurs at the time when a nearly perfect and continuous contact isestablished between casting and chill. Thus, the peak heat flux corresponds to the onset of asteadily growing shell.1200 ^200^iiiii ^00^10 20 30 40 50 60Time (s)500 ^ 60^10 20 30 40Time (s)Fig. 6.1 Typical model predictions for Al-7% Si alloy and copper chill - (a)interfacial heat flux, (b) heat transfer coefficient.50 6097casting contraction- -- interfacial gap-- chill expansion//1 ------------------------------------50,--...EE^40a0a*._.^30wcE 2001_.-I:vo^10C0(/)0— 00^10^20^30^40^50^60Time (s)3530EE 25toi0 20x.._.-w 150C0'01 10a500 10^20^30Time (s)40^50^60Fig. 6.2 Typical model predictions for Al-7% Si alloy and copper chill - (a)shell thickness (b) interfacial gap.987006000 500L_ 4000300E200100— eastirug surface I- — chill surface1 10 20 30 40 50 60Time (s)Fig. 63 Calculated surface temperature profiles in both chill and casting forAl-7% Si alloy and copper chill.Within this period, the dominant factors controlling interface heat transfer 39,47,49,50,138 are:(1) Wettability of Chill Surface: The interfacial heat transfer increases with increasingwettability of the chill surface. This is because wettability is a controlling factor that determinesthe extent of initial contact between casting and chill.(2) Chill Surface Geometry: Interfacial heat transfer increases with increasing surfacesmoothness. The total area in actual contact increases with increasing smoothness of the chillsurface.(3) Chill Thermal Properties: Interfacial heat transfer increases with increasing thermaldiffusivity of the chill particularly at the surface.99(4) Initial Chill and Casting Temperatures: The interfacial heat transfer increases withincreasing temperature gradient across the interface. It is noted that this gradient provides thedriving force for heat flow across the interface. Therefore, interfacial heat transfer increaseswith increasing casting temperature and decreasing chill temperature5° .Once the solidifying shell becomes self-supporting, it contracts in accordance with theshrinkage properties of the casting while the chill surface may expand. The relative magnitudeof the casting shrinkage and the chill expansion, together with any other pressure acting at theinterface determines the type of contact between chill and casting. In most cases, an air gap isformed unless the contact pressure is increased. The extra thermal resistance introduced by theair gap accounts for the decrease in interfacial heat flux. Hence, the drop in heat flux from thepeak value corresponds to the onset of a steadily growing air gap. At this point also, both thechill and casting surface may begin to oxidize. This mechanism has been confirmed by Pehlkeet al.47 who used transducers to monitor the electrical continuity between chill and casting. Theyfound that the electrical circuit breaks down at the onset of a sudden drop in interfacial heattransfer. A further drop in interfacial heat flux occurs as the gap grows coupled with increasingthermal resistance of the solidified shell.From the onset of a steadily growing gap, the following factors become dominant ininterfacial heat flow:(1) shrinkage characteristics of the casting(2) thermal conductivity and expansion characteristics of the chill(3) thermal conductivity of the air gap(4) surface oxidation characteristics of the chill and casting.(5) thermal properties of surface oxide layers.100(6) temperature gradient across the chill/casting interface(7) thermal conductivity of the solidifying shell.It is observed from Fig. 4.3 that the interfacial thermal resistance is the dominant resistancethroughout the duration of the experiment. The total thermal resistance decreases sharply fromcontact as a result of increasing interfacial heat transfer coefficient but starts to rise as theinterfacial heat transfer coefficient decreases and shell thickness increases.6.2 Microstructure FormationThe solidification structure in this work is predominantly columnar dendritic as shown inthe microstructures of Fig. 4.11. The dendrite arm spacings were sufficiently distinct for fairlyaccurate measurements except at distances very near the chill surface where a thin chill zone(<1mm) exists. It is difficult to resolve dendrite arms in the chill zone. The measured dendritearm spacing ranges from 181.1m (at 4mm from the interface) to 801.tm (at 20mm from the interface)as shown in Figs. 4.13 to 4.19 and, in all cases, increases with increasing distance from thechill surface. This implies that the secondary dendrite arm spacing (SDAS) increases withdecreasing cooling rate since the cooling rate decreases with distance from the chill surface.Therefore, as expected, a higher heat extraction rate leads to a finer microstructure. Thepredominance of columnar dendritic growth observed in these experiments is an indication ofsolute diffusion effects during solidification". Hence, the incorporation of a solute diffusionmodel into the solidification model is necessary.In general, microstructure formation starts at the onset of heterogeneous nucleation at themold wall asperities. The initial stage is probably pre-dendritic and has been found to consistof solid discs of the same composition as the liquid 139. The dendritic substructure is establishedwhen crystallographic alignment is attained. Although dendritic growth proceeds in three stages101(propagation of primary stems, evolution of side branches or arms, and coarsening andcoalescence), the final dendritic structure is controlled mainly by the coarsening and coalescencephenomena.Table 6.1 compares the various SDAS model predictions with the measured values whileFig. 6.4 depicts this comparison for the particular case of Al-7%Si and copper chill This tablerepresents the range of SDAS values obtained in this study. It is evident that only three of thesemodels are in consistently good agreement with the measured values - the theoretical modeldue to Mortensen 118 , the empirical cooling rate model and another empirical model due toBamberger et al. 5° . The constant in the cooling rate was found to be 58.0, compared with avalue of 53.0 reported for aluminum copper alloys 123. The fraction solid just before completesolidification (fs < 1.0) was used to evaluate the coarsening rate parameter in the Mortensenmodel.The Hills mode1 123 underpredicts the SDAS values as the distance from the chill surfaceincreases. It is noted that the Hills model was developed with near-net-shape casting in mind.This implies solidification of relatively smaller casting dimensions which allows for a largeinfluence of interfacial heat transfer coefficient on the SDAS. Based on these results, the Hillsmodel was found to be consistently valid for distances less than 8mm from the chill interfaceand could therefore be useful for near-net-shape castings. The model due to Shiau et al. 122 wasnot reasonably consistent in its predictions. This is attributed to the assumption adopted in thedevelopment of this model, namely, that both the liquidus and solidus curves obey the squareroot law. It was observed that this was not the case in most of the results from this work. It hasbeen shown that for most cases where thermal contact resistance exists between casting andmold, the square root law should be modified33 ' 14"42 .102Table 6.1 Measured and typical model predicted values of SDAS.Distance(mm)Secondary Dendrite Arm Spacing, SDAS (gm)Measured CalculatedRef. 107X, = cOvatf3C=58.0Ref. 118A,2 = 0 3_7C fromEq.(5.18)Ref. 50X2 = C tfi"C=15.0Ref. 122x2 = cx2r3Eq.(2.40)Ref. 1232‘..2 =f(h,t)Eq. (2.43)Casting: Al-7%Si^Superheat = 30°C^Surface Roughness = 0.018 gmChill:^Copper4 18.84 25.5 19.02 26.33 36.79 18.458 36.39 43.11 39.45 41.4 41.44 31.3812 43.43 44.99 41.19 43.67 47.82 38.5916 49.44 46.73 47.25 45.93 51.78 40.5520 62.83 51.62 55.13 52.11 56.67 47.62Casting: Al-7%Si^Superheat = 30°C^Surface Roughness = 10.560 gmChill -^Copper4 32.9 39.62 36.27 37.07 44.82 28.528 46.56 43.83 41.13 42.23 50.31 33.9412 54.37 54.75 54.12 56.33 56.45 41.5716 59.57 65.76 60.2 71.57 72.95 59.62Casting: A1-7%Si^Superheat = 30°C^Surface Roughness = 0.030 gmChill:^Brass4 38.41 39.68 37.34 35.60 42.72 35.538 50.24 47.12 49.68 51.28 56.65 46.8612 54.89 50.45 55.24 57.87 61.57 49.7716 59.17 58.72 58.65 62.36 67.23 52.18Casting: Al-3%Si^Superheat = 30°C^Surface Roughness = 0.030 gmChill:^Copper8 48.74 45.49 46.60 52.60 40.90 40.4312 58.84 51.67 59.18 62.18 45.76 47.9816 71.18 62.07 73.82 77.82 51.50 52.92103100908070Ox 60(f)50(/)403020100 5 10 15 20 25SDAS models0 measuredcooling rate118- - - MortensenBamberger et al.60122^ Shiau et al.^123 HillsIn general, the SDAS increases with increasing distance from the chill surface, increasinglocal solidification time and decreasing cooling rate. The theoretical coarsening model due toMortensen was used for estimating further SDAS values presented here.Distance from Surface (mm)Fig. 6.4 Typical calculated and measured secondary dendrite arm spacingfor Al-7% Si and copper chill (superheat=30 °C).1046.3 Effect of Process Variables63.1 Effect of Surface RoughnessThe effect of surface roughness on heat flow and microstructure formation is shown inFigs. 6.5 and 6.6. It is evident that heat extraction increases with increasing surface smoothnessof the chill This is manifested as increasing shell thickness and decreasing secondary dendritearm spacing as surface roughness decreases. It is observed that at the lower range of roughnessvalues covered in this study, an increase of roughness from 0.01811m to 0.0311m (-1.67 times)decreases the heat flux by an average of 8.5% while the heat transfer coefficient decreases by14%. If the surface roughness is further increased to 0.291pm (-16 times), the heat fluxdecreases by an average of 11% while the heat transfer coefficient decreases by an average of17%. This result is in agreement with the findings of other researchers 33.35 '3942-44. Prates andBiloni33 found that an empirical relationship exists between the surface microprofile and theoverall constant heat transfer coefficient in the form:h 2 = CRa ' (6.1)Values of C=4.68 x 10 6 and n=-0.03 reveal a very good fit (R 2=0.96) for kn. in the particularcase of AI-7%Si and copper chill.The influence of surface microgeometry on heat transfer and microstructure can beexplained by considering the initial contact between the surface layer of a liquid metal and themold surface. The first grains nucleate at the peaks or cusps of surface microprofile in apredendritic mode, thereby leading to thermal supercooling of surrounding liquid. Thisthermal supercooling subsequently increases the liquid surface tension. The increase in surfacetension reduces fluidity and, coupled with volume contraction as more liquid solidifies, reducesthe chances of the liquid contacting the surface valleys. The implication of this is that increasingsurface roughness decreases the total casting/mold contact area, thereby leading to lower heat105transfer rate. As a result of the localized solidification at the cusps, the surface roughness ofthe solidified shell is expected to increase with increasing surface roughness of the chill surface.This was confirmed by measurements of the surface microprofile of the solidified shell andthe result is depicted in Table 6.2. It is also observed that when the surface roughness of thechill is low (< lgm), the solidified shell is rougher than chill surface. On the other hand, athigher chill surface roughness (> 1gm), the surface of the solidified shell is smoother thanthat of the chill.Table 6.2 Measured shell surface roughness for various chill surface microprofile.Chill SurfaceRoughness (gm)Shell Surface Roughness (gm)1 2 3 4 5 Mean Std**0.018 1.60 1.34 1.07 1.03 1.19 1.25 0.230.030 2.01 1.44 1.08 1.58 1.24 1.47 0.360.291 2.67 2.39 3.01 2.84 2.23 2.63 0.325.61 3.28 3.74 3.07 2.98 3.82 3.37 0.3810.56 3.53 3.18 3.76 4.03 3.45 3.59 0.32** Std stands for standard deviation10620001600(- 1200800surface roughness(x10 -6m)0.018--- 0.030 0.291- 8.51---- 10_564000(a)10^20^30Time (s)40^50^600300025002000(NI E1500_c 100050000^10^20^30^40^50^60Time (s)Fig. 6.5 Effect surface roughness on heat transfer for AI-7% Si and copperchill (superheat=30 °C) - (a) interfacial heat flux, (b) heat transfercoefficient.107300^10^20Time (s)80ca60Ox 4001Cit206050400 0.030 (measured)0(b)1 5040E(1)^3020(1)_c^100 0^5^10^15^20^25Distance from Surface (mm)Fig. 6.6 Effect surface roughness on solidification and microstructure forA1-7% Si and copper chill (superheat=30°C) - (a) shell thickness (b)secondary dendrite arm spacing.6.3.2 Effect of Chill MaterialThe effect of chill material on heat flow and microstructure formation is shown in Figs.6.7 and 6.8. The heat flux and heat transfer coefficient increase with increasing thermaldiffusivity of the chill material. This result is in agreement with others reported in theliterature33 '3537 '39'"'47 '49 . Kumar et al." found that the maximum interfacial peak heat flux (q max)can be expressed as a power function of the thermal diffusivity and thickness of chill as shownin Eq. (2.3). A similar expression was found to exist in the present studies. For Al-7%Si, thisexpression is of the form:qmax = 2520( a 0.107mXmKW Im 2^(R2 = 0.965)^(6.2)The secondary dendrite arm spacing decreases with increasing thermal diffusivity of thechill material. The effect of chill material on the heat extraction and microstructure can beexplained by recalling that the chill thermal properties (thermal diffusivity, expansioncoefficient, emissivity and absorptivity) and surface oxidation characteristics are importantfactors in each stage of the solidification process. The ability of the chill to absorb and transportheat is of paramount importance during the first stage of solidification when there iscasting/mold contact. The thermal diffusivity of each of the chills decreases with increasingtemperature. The variation of the chill thermal diffusivity with temperature manifests in thechill thermal resistance (Xm/km), increasing thermal diffusivity resulting in decreasing thermalresistance. The variation of the thermal resistance of each of the chills with time is depictedin Fig. 6.9. It is obvious that copper has the least thermal resistance among the four materialsand therefore, exhibited the highest heat extraction rate. At the onset of the air gap, theradiation properties of the chills could become important although the main mode of heattransfer is conduction of heat across the air gap.109The effect of chill material on the surface roughness of the solidified shell is shown inTable 6.3 for a chill surface roughness of 0.031.un. Unlike the chill surface roughness, it isapparent that the chill material has no pronounced effect on the surface quality of the shell.Table 6.3 Measured shell surface roughness for different chill materials of similarmicroprofiles.Chill Material Shell Surface Roughness (gm)1 2 3 4 5 Mean Std**copper (Ra=0.031.1m) 2.01 1.44 1.08 1.58 1.24 1.47 0.36brass (Ra=0.031.tm) 1.58 2.13 1.34 1.17 1.27 1.50 0.38steel (Ra=0.03gm) 1.23 1.79 1.49 1.36 1.35 1.44 0.21cast iron (Ra=0.031.1m) 1.71 1.58 2.03 1.16 1.07 1.51 0.40Std stands for standard deviation110ECT14001200100080060040020000 10 20^30 40 50 6020002500cv^1500E1000500•• •1 20^30^40Time (s)( b )10 50 60Time (s)Fig. 6.7 Effect chill material on heat transfer for Al-7% Si(superheat=30°C) - (a) interfacial heat flux, (b) heat transfercoefficient.111Fig. 6.8 Effect chill material on solidification and microstructure for A1-7%Si (superheat=30 °C) - (a) shell thickness (b) secondary dendritearm spacing.tI^I I^I^Icast Iron----- steel- — - brass— copper--------------------------------- -- .- - - - _ - - - - - - -t70a)0 60Cac') '50- 0)--a) `---cr____<, ' 400 EE',1) ``),^30_c o1— ---= ,...,' 201000 10 20 30 40 50 60 70Time (s)Fig. 6.9 Variation of the chill thermal resistance of with time.6.3.3 Effect of SuperheatFigs. 6.10 and 6.11 depict the effect of superheat on heat flow and microstructure. Theinterfacial heat flux and heat transfer coefficient increase with increasing superheat. Thishigher heat extraction results in greater shell thickness and smaller dendrite arm spacing inagreement with earlier work reported in literature 143-144 The influence of superheat on heatextraction can be attributed to an increase in the interfacial contact between melt and chill asthe superheat increases. This increase in the interfacial contact with increasing superheat couldarise from two sources. Firstly, at higher superheats, the first solid shell that forms is relativelythin and remelts quickly, thereby allowing the melt to spread more uniformly across the surfaceof the chill. At lower superheats, the initial shell is thick and does not remelt but contracts113away from the chill, creating an interfacial gap which drastically reduce the interfacial heattransfer. Secondly, the fluidity of aluminum alloys increases slightly with increasingsuperheat37 and this enhances the spread of the melt across the chill surface such that moresurface asperities are filled. Furthermore, the casting superheat determines the extent of theinitial driving force (Tc-T.) for heat transfer across the interface. Increased melt temperatureimplies an increase in the initial temperature gradient across the mold/metal interface, therebyproviding a larger driving force for heat extraction. The increase in interfacial contact areacoupled with increasing driving force accounts for the increase in heat transfer with increasingsuperheat.The fact that the increase in heat extraction with increasing superheat is due to the natureof the melt/chill interface was confirmed with measurements of the surface roughness of thesolidified shell as shown in Table 6.4. It was found that the surface roughness of the solidifiedshell becomes smoother as the superheat increases. It is noted that the same chill surfacemicroprofile was utilized in all the experiments. The mechanism proposed here is furtherconfirmed by a similar result in literatures which showed that the wetting of the mold by themelt is the dominant factor that controls the heat transfer coefficient when liquid tin at differentsuperheats was dropped and solidified on a cylindrical chill of brass, stainless steel, nickel orchromium plated brass.When solidification recommences after remelting at high superheats, part of the meltsuperheat would have been lost to the chill and coupled with increased contact at the interface,the interfacial heat extraction increases, leading to the observed increase in shell thicknessand the decrease in secondary dendrite arm spacing. The increase in shell thickness withincreasing superheat implies that the solidification rate increases as the superheat is increased.114This indicates that higher superheats resulted in higher thermal gradient at the solidificationfront. It has been shown that under conditions of semi-infinite unidirectional solidification,the motion of the solid/liquid interface can be described by the expression 33S(t) = A .etf(/312,)t - C(Tc — TL) 1/2 (6.3)which shows that the shell thickness is a function of both heat transfer coefficient (h i) andsuperheat (Tc-TL). Therefore, the shell thickness can increase with increasing superheat if theincrease in heat transfer coefficient supersedes the effect of the superheat and, this is the casein this study.The decrease in secondary dendrite arm spacing with increasing superheat confirms theexistence of higher thermal gradients at the solidification front at higher superheats. It hasbeen found that in unidirectional solidification, the presence of a positive temperature gradientin front of the dendrite tip generally causes the velocity of the tip to be retarded to a greaterextent than the root at the commencement of solidification, thereby decreasing the mushyzone lengthW . As solidification progresses, the tip exhibits a speed up effect. The combinationof the reduction in mushy zone length and the speed up effect of the dendrite tip reduces thelocal solidification time. A positive temperature gradient can be maintained in front of thedendrite tips by use of superheat in the absence of convection or by supplying a heat input tothe melt in the presence of convection l". Hence, the net effect of increasing superheat in theabsence of convection is the acceleration of solidification with a consequent refinement ofmicrostructure. The fact that this was the case in our experiments support the fact that thedipping mechanism substantially reduced the effect of liquid convection during solidification.115160014001200-----,cs,E 1000800superheat- --- 0°C--- 30°C^ 60°C^ 90°C- 120°CCT60040020025002000(NE1500100050000^10^20^30^40Time (s)60500^10 20 30 40 50 60Time (s)Fig. 6.10 Effect superheat on heat transfer for AI-7% Si and copper chill - (a)interfacial heat flux, (b) heat transfer coefficient.116superheat---- 0°C--- 30°C— 60°C— 90°C-- 120°C0 30°C (measured)30Es's" 20c()cnC0I-1 0a)(f)00^10^20^30^40^50^60Time (s)8060E<0Qx 40(I)D(1)2025I^I^I^I 0^5^10^15^20Distance from surface (mm)Fig. 6.11 Effect superheat on solidification and microstructure for Al-7% Siand copper chill - (a) shell thickness (b) secondary dendrite armspacing.117Table 6.4 Measured shell surface roughness for various superheats.Superheat (°C) Shell Surface Roughness (iim)1 2 3 4 5 Mean Std**0 2.35 2.47 1.98 2.15 2.08 2.21 0.2030 2.01 1.44 1.08 1.58 1.24 1.47 0.3660 1.18 1.60 1.51 1.09 1.12 1.30 0.2490 0.98 1.16 1.34 1.00 1.10 1.12 0.15120 0.83 1.15 1.29 0.90 0.98 1.03 0.19** Std stands for standard deviation6.3.4 Effect of Alloy CompositionFigures 6.12 and 6.13 depict the effect of alloy composition on heat flow andmicrostructure. With reference to the Al-Si alloys, the heat flux and heat transfer coefficientincrease with decreasing silicon content of the alloy. It is noted that the heat flux and heattransfer coefficient for Al-3%Si and Al-5%Si exhibited much steeper gradients at the first 15sthan the Al-7%Si. The secondary dendrite arm spacing also increases with decreasing siliconcontent while the shell thickness decreases. This result is similar to the findings of Bambergeret a1.5° who investigated heat flow and dendrite arm spacings of Al-Si alloys ranging from 3.8to 9.7%Si. Other results pertaining to secondary dendrite arm spacing in hypoeutectic alloysindicate a decrease in spacing with increasing solute content 108,145-147.The increase in interfacial heat flux and heat transfer coefficient with a decrease in siliconcontent resulted from two main factors. Firstly, the casting temperature increases as the siliconcontent decreases due to the increase in the liquidus temperature. Thus, at a superheat of 30 °C,the casting temperature for alloys, Al-7%Si, Al-5%Si and Al-3%Si, were 640°C, 662°C and678°C respectively. The increase in casting temperature translates to an equivalent increasein the initial driving force for solidification. Secondly, there is an increase in the thermal118diffusivity of the alloys as the silicon content decreases. Both factors allow for an increase inheat extraction as the silicon content decreases such that instant freezing of the first shell aroundthe cusps of surface microprofile is expected to be more prevalent for A1-3%Si than for AI-7%Si.When this happens, uniform spreading of the melt across the chill surface is hampered suchthat less surface asperities are filled and this reduces the interfacial contact area. The fillingof surface asperities is further aggravated by the decreasing fluidity since the fluidity ofhypoeutectic alloys is known to be inversely proportional to the freezing range 136. The initialhigh heat extraction rate for Al-3%Si and Al-5%Si leads to a relatively thick shell, whichsubsequently contracts away from the chill surface, creating an interfacial gap. This gap bringsabout the rapid reduction in interfacial heat transfer and consequently the solidification rate.Measurement of the surface roughness of the solidified shell (Table 6.5) revealed that the shellsurface smoothness increases with increasing silicon content and this agrees with the proposedmechanism.The decrease in the solidification rate with decreasing silicon content can also beattributed to the increase in the mushy zone of these alloys as the silicon content decreases. Ithas been found that the secondary dendrite arm spacing is directly proportional to the extentiof the mushy zone in hypoeutectic binary alloys 145 . This ndicates that the solidification rateis inversely proportional to the extent of the mushy zone since the secondary dendrite armspacing decreases with increasing solidification rate. The increase in mushy zone length asthe silicon content decreases implies that the latent heat is released over a longer period, therebyresulting in a decrease in solidification rate that is manifested as reduced shell thickness andcoarser microstructure.The increase in solidification rate and the consequent decrease in secondary dendritearm spacing have further been explained in terms of the kinetics of dendritic growth 147348 . Ithas been proposed that dendritic growth occurs in two different stages - an initial transient119stage followed by a quasi-stationary stage 148 . The initial transient stage starts just behind thedendrite tip and extends until the solute diffusion fields from adjacent dendrites overlap. Thisstage is characterized by a rapid and dynamic increase in the solid volume fraction and isaccelerated by high solute content and/or fast growth rate. The quasi-stationary stage ischaracterized by the dendritic coarsening and is accelerated by low solute content and/or lowgrowth rate. Therefore, the main effect of decreasing the silicon content in Al-Si alloys is theretardation of transient stage and an acceleration of the coarsening stage, which finally resultsin an increase in secondary dendrite arm spacing.Table 6.5 Measured shell surface roughness for different alloy compositions.Alloy Composition Shell Surface Roughness (gm)1 2 3 4 5 Mean Std**Al-7%Si 2.01 1.44 1.08 1.58 1.24 1.47 0.36Al-5%Si 2.88 2.45 1.70 2.40 1.98 2.28 0.46Al-3%Si 2.43 3.02 3.75 3.16 2.99 3.07 0.47** Std stands for standard deviation120•2500^ A1-7ZSi— — — A1-5%SiA1-32;Si2000 —1500 —ss,10005000r^(a)0^10^20^30Time (s)40^50^6050004000(NA - 3000E2000100000^10^20^30^40^50^60Time (s)Fig. 6.12 Effect of alloy composition on heat transfer for Al-Si alloys andcopper chill (superheat=30°C) - (a) interfacial heat flux, (b) heattransfer coefficient.121^ A1-7%Si- - - A1-5%Si-^- A1-3%Si0^10 20 30 40 50 6002010300 A1-7%Si (measured)(b)8020Time (s)0^5^10^15^20^25Distance from Surface (mm)Fig. 6.13 Effect of alloy composition on solidification and microstructure forAl-Si alloys and copper chill (superheat=30°C) - (a) shell thickness(b) secondary dendrite arm spacing.1226.3.5 Effect of Oil FilmThe effect of oil film on heat flow and microstructure is presented in Figs. 6.14 and 6.15.It is noted that these oils were developed for, and are being used in, the continuous castingof steel billets as lubricants. Oil films increase the interface thermal resistance at initial contact.This trend is clearly manifested in the first few seconds of solidification for the temperature,heat flux, heat transfer coefficient and shell thickness profiles for the various oils. Themaximum heat flux decreases with increasing flash point of oil (see Table 4.3 for the flashpoints of the oils) while the time to attain this maximum value increases. The heat flux increaseswith increasing flash point for subsequent times after the peak value. The heat transfercoefficient also increases with increasing flash point of oil. It is noted that flash point of theseoils increases with increasing boiling range and decreasing viscosity. The increase in bothheat flux and heat transfer coefficient results in increasing shell thickness and decreasingsecondary dendrite arm spacing at a given location.This effect can be explained as follows. The oil film acts as an additional thermal batherupon contact of the chill by the metal thereby increasing the thermal resistance at the interface.However, within a short time, the oil temperature reaches its flash point, pyrolysing orcombusting and, releasing gases into the growing gap at the interface. This was observed tohappen during the experiments in the form of oil smoke that appeared a few seconds afterdipping. Model predictions of the surface temperature profiles for the chill and solidifyingshell (Fig. 6.16) indicate that while the oil layer near the shell surface reaches its flash point,the layer adjacent to the chill does not reach this temperature. This was confirmed by the factthat an oil film somewhat thinner than when first applied, was observed to remain at the chillsurface in all cases.123no oil- - - Blachford^ steelskinHEARcanola180015001200E900Cr60030010^20^30^40^50^60Time (s)350030002500c■IE 2000150010005000 0^10^20^30^40^50^60Time (s)Fig. 6.14 Effect of oil film on heat transfer for Al-7% Si and copper chill(superheat=30 °C) - (a) interfacial heat flux, (b) heat transfercoefficient.124500^10^20^30^40Time (s)800 no oil (measured)6040(b )I^(^I^I 0^5^10^15^20Distance from Surface (mm)60201040„--,E 3°E025Fig. 6.15 Effect of oil film on solidification and microstructure for AI-7% Siand copper chill (superheat=30°C) - (a) shell thickness (b)secondary dendrite arm spacing.125— Blachfox-d- – steelekin- 041101acasting surfac---------chill surface07006005004000a.) 300E20010000^10 20 30 40 50 60Time (s)Fig. 6.16 Calculated surface temperature profiles of chill and solidifying shellfor the four oils.The surface roughness of the solidified shell increases slightly with decreasing flashpoint and boiling range of the oils as shown in Table 6.6. It is also apparent from Table 6.6.that there is a slight increase in the surface roughness of the solidified shell with the oils whencompared to the case of no oil. This could be attributed to the pressure exerted on the firstsemi-solid shell that form by the products of oil boiling and pyrolysis.In general, oils are esters of fatty acid and glycerol. Fatty acids are long straight chainedhydrocarbons having a carboxylic acid group (-COOH) attached at one end while glycerol isa trihydroxy alcohol (CH 2OH-CHOH-CH2OH). The properties of oils depend on the chainlength of the molecule, on the degree of saturation, on the geometric isomerism and on therelative positions of the double bonds with respect to the carboxyl group and each other 149. In126the presence of heat and absence of oxygen, oils undergo thermally induced homolysis, atype of pyrolysis that results in rupture of carbon bonds and fragmentation into smaller units.Once formed, these smaller units or free radicals can enter into typical propagation reactionsyielding new free radicals. These chains of reactions continue until a termination stage isattained. The termination steps in pyrolysis of a given free radical may be either the joiningtogether (coupling) of two free radicals or their disproportionate (one is oxidized while theother is reduced).Table 6.6 Measured shell surface roughness for the four oils.Oil Shell Surface Roughness (gm)1 2 3 4 5 Mean Std**No oil 2.01 1.44 1.08 1.58 1.24 1.47 0.36Blachford Oil 1.43 1.58 1.97 1.44 1.34 1.55 0.25Steelskin Oil 1.49 1.56 1.45 1.62 1.46 1.52 0.07HEAR Oil 1.18 1.71 1.32 1.48 1.78 1.49 0.25Canola Oil 1.48 1.33 1.23 1.83 1.53 1.48 0.24** Std stands for standard deviationIn the presence of heat and a limited amount of oxygen, oils oxidize and undergoincomplete combustion in a number of stages, producing mainly carbon monoxide, watervapour and carbon black or soot. If excess oxygen is available together with heat, oils undergocomplete combustion, yielding mainly carbon dioxide and water vapour in the final stage.Combustion of most organic compounds produce energy as a by-product. Thus in the presenceof heat and oxygen, oils undergo a complex set of reactions producing H2, CO, CO2, H2O,C, and hydrocarbons. Some of these gases (mainly H2 and hydrocarbons) have higherconductivity than air and hence, enhances the heat conduction across the air gap.127It is observed that Canola and HEAR oils enhance heat transfer more than the Steelskinand Blachford oils. It is established that the double bonds and the carboxyl groups in oils arethe main reactive sites during pyrolysis and combustion. Hence, the rate of reaction increaseswith increasing number of double bonds (increasing unsaturation) and increasing length ofcarbon chains. With reference to Table 4.3, all the components of the oils except palmiticacid contain double bonds. Linolenic acid has three double bonds, linoleic acid has two doublebonds while the three remaining components have one double bond each. Hence, althoughthe total percentage of unsaturated components are about the same for these four oils (91.5%for Canola, 92.7% for HEAR, 93.8% for Steelskin and 92.1% for Blachford), Steelskin andCanola oils will exhibit greater unsaturation behavior than HEAR and Blachford oils. In termsof the total length of carbon chains, HEAR oil has the longest chain due its high euricic acidcontent while the chain length for the remaining three are about equal. Therefore, the highheat extraction rate with Canola oil can be attributed to its relatively high unsaturation level.On the other hand, the high heat extraction with HEAR oil can be due to its relatively longcarbon chains. Although high heat extraction is expected for Steelskin oil due to its highunsaturation, it appears the length and arrangement of the carbon chains may have supersededthis unsaturation factor. Blachford oil has the lowest heat extraction due its modest unsaturationand carbon chain length. Furthermore, the differences in the heat extraction rates of these oilsare influenced by the unknown relative arrangement and positions of the reactive sites.The exact amount of the evolved gases could not be determined due to the complexityof the possible reactions. A rough estimate could however be made if it is assumed that radiationand gas conduction are the main modes of heat transfer when the gap forms. Calculationsbased on this assumption reveal that the average gap conductivity likely increased from 0.046W/m.K to 0.091 W/m.K (-97.8%) to account for the increase in heat extraction observed forthe Canola and HEAR oils. In the case of Steelskin and Blachford oils, the increase in the128average gap conductivity is about 40%. Therefore, it would appear that Canola and HEARoils release higher amounts of H2 and hydrocarbons into the air gap than Steelskin and Blachfordoils.6.3.6 Effect of Bath HeightFigures 6.17 and 6.18 depict the effect of bath height on heat transfer and microstructureand shows a negligible influence on heat extraction and secondary dendrite arm spacing at theearly stages of solidification. This is expected for a unidirectional solidification since thecasting/mold surface contact area and other boundary conditions remain the same.Furthermore, the casting thermal resistance is small in relation to interfacial thermal resistance.The major effect of increasing bath height under these conditions should simply be a longerduration of dipping. Bath height also has no significant effect on the shell surface roughnessas shown in Table 6.7.Table 6.7 Measured shell surface roughness for different bath heights.Bath Height (cm) Shell Surface Roughness (Ltm)1 2 3 4 5 Mean Std**14 2.01 1.44 1.08 1.58 1.24 1.47 0.368.5 1.62 1.10 1.23 1.41 1.95 1.46 0.346 1.47 1.56 1.09 2.10 1.24 1.49 0.39** Std stands for standard deviation129Fig. 6.17 Effect of bath height on heat transfer for A1-7% Si and copper chill(superheat=30°C) - (a) interfacial heat flux, (b) heat transfercoefficient.130N^,Fig. 6.18 Effect of bath height on solidification and microstructure for Al-7%Si and copper chill (superheat=30°C) - (a) shell thickness (b)secondary dendrite arm spacing.1316.4 Proposed Empirical ModelFrom these results and the above discussions, it is evident that the interfacial heat flux isdependent on a variety of factors. The individual effects of some of these factors (surfaceroughness, chill material, superheat, alloy composition, and gap composition) have beenhighlighted. Following the trend of an earlier analysis by Kumar et al: 49, an attempt was madeto develop an empirical interfacial heat flux model to simulate the factors studied. It is envisagedthat such a model could provide an estimate of the interfacial heat flux transients that could beutilized for modeling the boundary conditions on the casting side for solidification simulation.The advantage of such a model is that the transient heat flux is estimated purely from thethermophysical properties of the chill and casting.The model is based on the Al-Si alloys and is divided into three stages as illustratedschematically in Fig. 6.19:(I) The first stage exhibits a linear relationship between heat flux and time, and coversthe period from time zero to the time at which the maximum heat flux is attained. The maximumheat flux can be expressed as follows:T^albmax = Co(^ Rac^W/m2^(R2 -= 0.99)^(6.4)where^Co = 1224-1988 = 136.50 x (TL-Ts)o.6352a = 1.9344b= 0.0657c= -0.0104The flux from onset of contact to the peak value (2 - 5 s in all cases studied) could be estimatedby linearizing the flux values from about 0.25q max at time zero to q max .132q^d= C1 L( --r n i ae if^W I M 2ax^L-qmt.(R2=0.92) (6.5)(II) The flux in the intervening period between 10 s and q ma„ could be estimated bylinearizing the heat flux values between qmax and the q value obtained by Eq. (6.5) for t=l0s.(III) The flux at any time greater than 10 seconds after the q max could be expressed aswhere C1 = 9.673d = 0.158e = 0.115f = -0.60Fig. 6.19 Schematic illustration of the proposed empirical heat flux model.133Figures 6.20 and 6.21 compare the values of the maximum heat flux obtained from Eq.(6.4) with the inverse solution values. Good agreement is observed for all cases considered. Acomplete simulation based on Eqs. (6.4) and (6.5) is shown in Fig. 6.22 for Al-7% Si. It isobserved that the empirical model compares well with the inverse solution. To further test thevalidity of this model, it was applied to A1-3%Cu-4.5%Si alloy (T L=627°C, Ts=525°C) underthe casting conditions reported by Kumar et al. 49 (Tc=750°C). A value of 1µm was assumed forthe surface roughness of the uncoated copper chill. Figure 6.23 depicts the result of thissimulation. Good agreement also exists between this model and the results obtained by Kumaret al42 .Three major factors were not included in Eqs. (6.4) and (6.5) - the latent heat, the thermaldiffusivity of the air gap and surface coating/film. It would seem logical to expect the constants,Co and C 1 , to increase with increasing latent heat of the metal since the latent controls the amountof energy released as a result of solidification. This could not be quantified in the present studydue to the fact that all the Al-Si alloys have the same latent heat. The effect of the thermaldiffusivity of the gap was demonstrated with the introduction of continuous casting oil films atthe surface. The pyrolysis or partial combustion of the oils released gases that enhanced thethermal conductivity of the air gap with a consequent increase in the heat flux. This effect couldnot be quantified since the actual composition of the released gases is not known. The oil filmsalso demonstrate the effect of surface coating on the heat flux. This could be incorporated intothe above expressions in the form of effective diffusivity of the mold. Coatings having a thermaldiffusivity smaller than that of the mold material will reduce the effective mold diffusivity andtherefore reduce the heat flux value.1341300Ex 12000ECT0 inverse solutionemprical model150014001100100040T /TC^fT135 4525002000E1500x0E0- 100050030^45^60^75TL —TS (c)c)Fig. 6.20 Variation of q„„,„ with different process variables - (a) initialtemperature ratio of melt and chill, (b) mushy zone.135O 20 400 inverse solution^ emprical model10000 5 101200Diffusivity/X rn (x10 -4m/s)Roughness (x10 -6m)Fig. 6.21 Variation of q„,.„ with different process variables - (a) ratio of chillthermal diffusivity to its thickness, (b) chill surface roughness.1360 inverse solutionempirical model1400120010008 0 0X6 0 00 400a)20000^20^40^60Time (s)Fig. 6.22 Variation of heat flux with time for Al-7% Si and copper chill(superheat=30 °C).2500 — Kumar's Model- This Model20001500100050020^40^60Time (s)Fig. 6.23 Simulated variation of heat flux with time for Al-3% Cu-4.5% Si(initial casting temperature = 750°C) and a 2.86cm thick copper chill.1376.5 Implications for Continuous and Near-Net-Shape CastingThe interfacial heat flux presented here for Al-Si alloys ranges from 0.97 - 2.0 MW/m 2while the heat transfer coefficient ranges from 1.95 - 4.30 KW/m 2.K. The predicted coolingrate is of the order of 10-100 °C/s at the surface leading to a value of 12-221.im for the secondarydendrite arm spacing very near the surface. These values are comparable to published data oncontinuous and thin slab casting of aluminium alloys 150,151. Szczypiorski et al. 150 reported SDASvalues of 8-161.tm at the surface of a 19mm thick slab cast on a twin-belt Hazelett machine and18-301.tm at the center. The cooling rate and local solidification times calculated here are seento be in the upper ranges of conventional DC casting and in the lower range of belt/roll castingfor alumimium alloys. It is therefore evident that the results obtained here have someimplications for continuous and near-net-shape casting processes in general; and the resultscould be projected to these processes for similar alloy systems.The observed heat flux transient is typical of any process where initial mold/casting contactis followed by the formation of a clearance gap. It is therefore possible to develop empiricalheat extraction models for these processes following the procedure adopted in this study. Ofcourse, the effects of convection in the melt and the relative velocity of casting and mold willhave to be included.It is evident from this study that the biggest variation in interfacial heat transfer occurs inthe first few seconds of metal-mold contact (< 45 s). Therefore, the transient nature of interfaceheat transfer exerts the greatest influence for short dwell times. The implication of this for thinsection casting is that the interfacial heat transfer is dynamic throughout the duration of casting.A constant overall boundary condition (h or q) cannot be used in the simulation of thesolidification process in this case. This is very relevant for continuous and near-net-shape casting138since the hallmark of these processes is the reduction in section thickness and consequently theshorter dwell times in the heat extraction device. The interfacial heat transfer is controlled bya myriad of factors within this time.Owing to the lack of directly applicable transient heat transfer data, the heat transfermeasurements of Shah et al. 138 for high carbon steel solidifying on a stationary copper chill wasemployed to estimate the values of the constants in the empirical model in order to apply themodel to the prediction of the initial heat transfer transient during twin-roll casting. The resultsare shown in Fig. 6.24 for a copper roll with a cooling channel located at 50mm below thesurface and with the assumption that the steady state value of interfacial heat flux reported inliterature 142 corresponds to the attainment of a fairly stable flux after the initial transient. Thefact that the interfacial heat flux is inversely proportional to the dwell time 142 was taken intoaccount by adjusting the constant, C o, in Eq. (6.4). It is observed that the steady state valuesare about 2-3 times less than the peak flux and are attained after 30-40 seconds of operation.It has been shown that the mold wall microgeometry as influenced by machining, polishingand coating has a profound influence on the cast structure through heat transfer. Most of themolds used in continuous and near-net-shape casting have fine ground surfaces and in somecases, the surfaces are coated with wear resistance materials. These surfaces are known to haveroughness values of 0.2wn to lt.tm . This study has shown that further reduction in this rangeof roughness could be beneficial by increasing the heat extraction rate and further refining ofmicrostructure. It was observed that a reduction of roughness from 0.29 1 Lim to 0.018[tm couldresult in a 17% average increase of interfacial heat transfer coefficient. This increment mayseem little by itself but for short dwell times and thin sections, it could make the differencebetween complete solidification and incomplete solidification within the mold. Furthermore,smooth surface finish will be beneficial in reducing the incidence of sticking where this occurs,thereby translating into productivity increase.139In continuous and near-net-shape casting processes, the mold surface could becomecritical not only in terms of the magnitude but also the distribution across the mold area incontact with the solidifying metal. It has been observed that small local heterogeneity on themicroprofile of the roll surface of a single roll caster could lead to local reduction of thesolidifying strip thickness by reducing the local heat transfer I52. Also, complete tearing of thestrip surface has been encountered when the roll surface was scratched 152 • 153. Perfect replicationof the surface microprofile of the roll surface by the cast strip was enhanced by decreasingsurface roughness and increasing casting speed 152. At high values of roll surface roughness,the surface of the cast strip is rough but generally becomes smoother than the roll surface 152 andthis observation is similar to the results obtained from this study.Fig. 6.24 Simulated variation of heat flux with time for high carbon steel(0.8%C) solidifying in a stationary copper chill and in twin-rollcaster at various dwell times.140The net effect of surface coating is the decrease in effective thermal diffusivity of themold since most coating materials have lower conductivity than the mold. The four continuouscasting oils investigated in this study increased the heat extraction rate once they attained theirflash point by releasing high conductivity gases into the interfacial gap. It is however notedthat this effect can only be realized if the oil flow is such that the flash point is reached. Adisadvantage of this is the possible increase in gaseous impurities from the casting process. Ifthere is no flashing of oil, the effect of oil lubrication will be similar to surface coating - reductionin interfacial heat transfer.Four properties of the chill material were shown to affect the interfacial heat extraction -chill thermal diffusivity, thickness, initial temperature and thermal expansion coefficient. Heatextraction increases with increasing thermal diffusivity, decreasing chill thickness anddecreasing initial temperature of the chill. Although increasing thermal expansion coefficientmay be beneficial by decreasing the air gap size, it is detrimental for thin molds by increasingthe tendency towards mold distortion.The initial effect of cast metal superheat is an increase in heat extraction rate if the moldthermal resistance is small. However, the overall effect will depend on the extent of convectionin the melt. With minimal convection in the melt, increasing superheat can accelerate thesolidification process but might retard it with increasing convection.Finally, for thin section castings, the result of this study suggests that a direct relationshipcould exist between the secondary dendrite arm spacing and the interfacial heat transfercoefficient via the Hills 123 equations. This implies that the effect of the interface is felt at alllocations throughout the casting duration.141Chapter 7SUMMARY AND CONCLUSIONS/RECOMMENDATIONSA one dimensional implicit finite difference model has been successfully developed topredict heat flow parameters and secondary dendrite arm spacing (SDAS) during unidirectionalsolidification. The model utilizes the Scheil equation in conjunction with the effective specificheat method to handle the release of the latent heat. Various secondary dendrite arm spacingmodels were incorporated for the prediction of SDAS. In order to characterize the boundarycondition at the mold/casting interface, a dip test was designed and experimental campaignswere carried out by dipping water-cooled cylindrical chills of different materials instrumentedwith thermocouples into Al-Si melts (3-7%Si). The dip test provided two kinds of measurements:(a) the thermal histories at thermocouple locations in the chill for about 60 seconds duration(b) the solidified shell within these short periodsThe thermal histories were fed into an inverse heat conduction (IHCP) model for the chillto predict transient interfacial heat flux, chill thermal histories at all locations and also to providea measure of the variability of the thermocouple readings. The sequential regularization IHCPmodel developed by Beck 134 was adapted and used for this purpose. The solidified shells wereused for metallographic examination and measurement of secondary dendrite arm spacing.The transient heat flux and chill surface temperature profiles were used as boundaryconditions for the casting model. The casting model predicts the interfacial heat transfercoefficient, temperature histories in the casting, cooling rate, local solidification times andsecondary dendrite arm spacings. The model is validated with temperature measurements in thecasting and the measured secondary dendrite arm spacing. The effects of some process variables142such as chill surface roughness, chill material, superheat, alloy composition, surface film (oil)and bath height were studied. Finally, an attempt was made to provide an empirical transientflux model that will take these variables into account.From the results of this study the following conclusions can be drawn:(i) The biggest variation in interfacial heat transfer occurs in the first few seconds ofmetal-mold contact (< 45 s). The transient nature of interface heat transfer exerts the greatestinfluence for short dwell times. Therefore, for thin sections castings, the interfacial heat transferis very dynamic throughout the duration of casting. A constant overall boundary condition (h orq) cannot be used in the simulation of the solidification process in this case.(ii) The interfacial heat flux, heat transfer coefficient and temperature profiles in the chillexhibit the typical trend common to solidification where the initial contact between mold andmelt is followed by the formation of a steadily growing gap. These three parameters increasesteeply upon contact up to a certain peak at very short time duration (0-10 s), decrease steeplyfor a few seconds and then gradually decline to a fairly steady value.(iii) The peak of the heat flux was found to correspond to the onset of a steadily growingshell and subsequent decrease in heat flux is attributed to the formation of a gap formed as aresult of the shell contraction away from the mold which is not balanced by enough moldexpansion.(iv) In general the heat flux increases with increasing mold thermal diffusivity, increasingsuperheat, increasing thermal diffusivity of the interfacial gap, decreasing mold thickness andinitial temperature, and decreasing mold surface roughness. Mold coatings or other surface filmssuch as oil reduce the heat flux at the onset of solidification by acting as an additional thermalbarrier since they often have lower thermal diffusivity than the mold. However, any furthereffect on the heat flux will depend on the nature of the chemical or thermal transformation of143these materials in the presence of heat.(v) The peak heat flux can be expressed as a power function of the superheat, surfaceroughness, chill initial temperature and thermal diffusivity of the chill material in the form:T, a ( an,q.= Co(F-^Ra^WIm2^(R2 -= 0.99)(vi) It was also found that the heat flux from 10 seconds after the peak can be expressedas power function of time, the chill thermal diffusivity and the ratio of the thermal expansion ofthe chill to the casting contraction in the form:d= C ,^ jaelf^W 2^(R2 = 0.92)qmax^4,(vii) A three stage heat flux empirical model is proposed. This model could be used as anapproximate boundary condition and can easily be evaluated from the knowledge of thethermophysical properties of the mold and casting.(viii) The secondary dendrite arm spacing depends on the cooling rate and localsolidification times in most cases. However at distances very near the surface (<8mm), it couldbe expressed as a direct function of the heat transfer coefficient and time via the Hills equation.(ix) Of all the variables investigated, the chill material and alloy composition were foundto have the greatest effect on the secondary dendrite arm spacing.(x) The results of this study have some implications for continuous and near-net-shapecasting. It was shown that by allowing for the small thickness of the mold used in these processes,the results presented here fall between the upper range of conventional DC casting of aluminiumalloys and lower range of such near-net-shape processes as the twin-belt caster. In these processes,the variables studied here could become critical due to the short dwell times and thin sections144involved.Finally, it is believed that the objectives of this study have been met substantially. 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Of TMS Conference on Casting ofNear-Net-Shape Products, 1988, pp.3-24.143. D.J. Hurtuk and A.A. Tzavaras: "Aberrations Observed in Relationship of DendriteSize-Alloying Elements for Low-Alloy Steel", Proc. of International Conference onSolidifcation and Casting, The Metals Soceity, London, 1979, pp.21-28.144. T.W. Caldwell, A.J. Campagna, M.C. Flemings and R. Mehrabian: "Refinement ofDendrite Arm Spacing in Aluminum Ingots Through Heat Flow Control", Metall. Trans.B, 1977, Vol. 8B, pp.261-270.145. P.C. Dann, J.A. Eady and L.M. Hogan: "Dendrite Arm Spacing in Aluminium-RichAlloys", Journal of Australian Institute of Metals, 1974, Vol. 19, pp.140-147.146. B. Alexander and F. Rhines: Trans. AIME, 1950, Vol.188, p.1267.147. R.N. Grugel: "Secondary and Tertiary Dendrite Arm Spacing Relationships in DirectinallySolidified Al-Si Alloys", Journal of Materials Science, 1993, Vol. 28, pp.677-683.148. D.J. Allen and J.D. Hunt: "Diffusion in the Semi-Solid Region During Denclritic", Metall.Trans. A, 1979, Vol. 10A, pp.1389-1397.149. R.J. Fessenden and J.S. Fessenden: "Organic Chemistry", PWS Publishers, Boston, 1982,pp.230-234, 887-890.150. W. Szczypiorski and R. Szczypiorski: "The Mechanical an Metallurgical Charateristicsof Twin-Belt Cast Al-Strip Using Current Hazellet Technology", TMS Light Metals, 1991,pp.805-814.151. B. Frischknecht and K.P. Maiwald: "Roll Caster Applications and Developments", TMSLight Metals, 1988, pp.365-372.156152. J.P. Birat, P. Blin, J.L. Jacquot, P. Riboud and B. Thomas: "Near Net Shape ContinuousCasting of Flat Products at IRSID", La Revue de Metallurgie - CIT, 1989, pp.919-930.153. P. Cremer and J. Bigot: "An Infrared Thermographic Study of the Temperature Variationof an Armophous Ribbon During Production by Planar Flow Casting", Material Scienceand Engineering, 1988, Vol. 98, pp.95-97.157APPENDIX ASEQUENTIAL IHCP SOLUTIONIn chill casting as in many dynamic heat transfer situations, the heat transfer parametersat the surface (heat flux, heat transfer coefficient and temperature history) are easier to determinefrom transient measurements at one or more interior locations. Such problems are classified asinverse heat conduction problems (IHCP). When the thermophysical properties of the material(density, thermal conductivity and heat capacity) are dependent on temperature, the IHCP issaid to be non-linear. In most practical IHCP situations, the solution does not satisfy the conditionsof uniqueness and stability such that the IHCP is said to be ill-posed. This implies that there areinfinite number of possible solutions producing almost similar results, and a solution is onlyaccepted by setting limits or defining boundaries.The various solution techniques currently available for IHCP are shown in Fig. 1A. Thediscrete value specification method attempts to match predicted and measured temperatures asclose as possible with the assumption that all input variables are errorless. On the other hand,the solution stabilization techniques attempts to reduce excessive fluctuations in the solution,thereby stabilizing it.The method adopted here is the sequential regularization method of Beck 133 '134. It utilizesa regularizing operator referred to as the sensitivity coefficient (SC). This is defined as the firstderivative of the a dependent variable such as temperature, with respect to the unknOwn parameterof interest (heat flux or heat transfer coefficient).158 INVERSE HEAT CONDUCTION PROBLEM[IHCP]Usually Non-Linear and Ill -posedTemperature Matching TechniquesAssumes that all input are errorless Solution Stabilization TechniquesAssumes that all input exceptmeasured temperatures are errorlessUtilizes least square minimization procedureDiscrete Value Specification MethodfMollification Methoc Function Specification MethodRegularization MethodWhole Domain Approach Sequential ApproachFig. 1A Schematic illustration of IHCP methods.orSC dri' —dedr"SC"' =dhmApplying this definition to Eq. (5.1), one obtains:a asc asc=pCp atInitial and Boundary Conditions1. Initial temperature of the chill is specified (l chk < z < I ch , 0 < r < ro, t=0)Sc = 02. At the chill hot face (z = lchk, 0 < r < ro, t)-K, asc .az3.^At the chill cold face (z = l ch , 0 < r < ro, t)Sc = 0^(6A)Eqs. (3A) and (5A) were solved with the aid of implicit finite difference formulation similarto Eqs. (5.5) and (5.6).ocAt sC,+1 — { 1+ aAt }sC,4" 1 + 0aAt^sc,p1 =2Az 2^Az2^26,z2 1+1aAt aAt ocAt (7A)SC" 1 — — 1}SCu +^SC!"2Az 2 Az2 2Az2At the chill hot face, Eq. (5A) becomes(1A)(2A)(3A)(4A)(5A)160AAq2^434 11:3(At — 1}SCr +^Sq."{^CCAt }1 SC" +1+^ Sz 2 *^A C;" = 16`t AZ2^pCpAz Az2^(8A )AZEqs. (7A) and (8A) can be linearized by assuming that for sufficiently small time steps,the thermophysical properties (p, k, Cp ) do not exhibit any tangible changes from one time stepto the next, even though there may be a large variation in such properties from one end of thebody to the other. However, the use of small time steps frequently introduces instabilities in theIHCP solution and often require some form of restriction on the time dependence of the heat flux.In the sequential method used here, it is assumed that the heat flux is temporarily constantfor a carefully selected 'r' future time steps as illustrated in Fig. 2A.m^m +1^m +2=0^1^2^3^M-1^ti^m+1^M+2Index for t.Fig. 2A Schematic illustration of the use of future time step in heat fluxcalculation.(9A )4 i161r nsE E (Min +1 -1 — TCn +1 -2)SCfn +1-2qm =qm-1 i=1 j=1r ns (scr +i -2)2(14A )The use of this future time steps in conjunction with the least square criterion enhances thestability of solution. The least square criterion in this case can be expressed in the form:r nsSL = E E^- TCT i -1 )2= 1 j = 1(10A)The function SL is minimized with respect to heat flux e whendSL=0^ (11A)deso that^r s^ dTC"1+1-12 y, y, (TM +1 -1 —^+ 1^—^ = 0^(12A)^i=1 j=1 dqmFor TCm = f(e), the first two terms of the Taylor series expansion will yield{dTC71+i — 2 }TC.n+i = TC-n i + (qm — qm-1 ) ^ +de(13A )Substituting Eq. (13A) into (12A), the following expression results:since^dTC +1-2—SC +1 —2deEquation (11A) is used to evaluate the heat flux at any given time step. The procedure for theimplementation of the IHCP algorithm is shown in Fig. 3A. The sequence of solution involvesobtaining the sensitivity coefficients employing Eqs. (6A), (7A) and (8A). The sensitivitycoefficients are then utilized in computing the heat flux via Eq. (14A). The heat flux valueobtained is finally used to evaluate the nodal temperatures with Eqs. (5.5) and (5.6). The procedure162Nois repeated when the time is incremented.INPUT DATAInitial temperature distributionFUNCTION 1Evaluates thermophysicalpropertiesCALL 1 RID IATridiagonal Matrix SolverOUTPUTNodal SensitivityCoefficientsFUNCTION2Evaluates Heat Fluxbased on SC valuesand future Temperatures1CALL TRIDIA Fig. 3A Flow diagram of the sequential IHCP technique.163
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Heat transfer and microstructure during the early stages of solidification of metals Muojekwu, Cornelius A. 1993
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Title | Heat transfer and microstructure during the early stages of solidification of metals |
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Muojekwu, Cornelius A. |
Date Issued | 1993 |
Description | The future of solidification processing clearly lies not only in elucidating the various aspects of the subject, but also in synthesizing them into unique qualitative and quantitative models. Ultimately, such models must predict and control the cast structure, quality and properties of the cast product for a given set of conditions Linking heat transfer to cast structure is an invaluable aspect of a fully predictive model, which is of particular importance for near-net-shape casting where the product reliability and application are so dependent on the solidification phenomena. This study focused on the characterization of transient heat transfer at the early stages of solidification and the consequent evolution of the secondary dendrite arm spacing. Water-cooled chills instrumented with thermocouples were dipped into melts of known superheats such that unidirectional solidification was achieved. An inverse heat transfer model based on the sequential regularization technique was used to predict the interfacial heat flux and surface temperature of the chill from the thermocouple measurements. These were then used as boundary conditions in a 1-D solidification model of the casting. The secondary dendrite arm spacing (SDAS) at various locations within the casting was computed with various semi-empirical SDAS models. The predictions were compared with experimental measurements of shell thickness and secondary dendrite arm spacing from this work as well as results reported in the literature. The effects of superheat, alloy composition, chill material, surface roughness and surface film (oil) were investigated. The results indicate that the transient nature of the interface heat transfer between the chill and casting exerts the greatest influence in the first few seconds of melt-mold contact. The interfacial heat flux and heat transfer coefficient exhibited the typical trend common to solidification where the initial contact between mold and melt is followed by a steadily growing gap. Both parameters increase steeply upon contact up to a peak value at a short duration (< 10s), decrease sharply for a few seconds and then gradually decline to a fairly steady value. Heat transfer at the interface increased with increasing mold diffusivity, increasing superheat, decreasing thermal resistance of the interfacial gap, increasing thermal expansion of the mold, decreasing shrinkage of the casting alloy, decreasing mold thickness and initial temperature, and decreasing mold surface roughness. The secondary dendrite arm spacing decreased with increasing heat flux for the same alloy system and depended on the cooling rate and local solidification time. The secondary dendrite arm spacing was also found to be a direct function of the heat transfer coefficient at distances very near the casting/mold interface. A three stage empirical heat flux model based on the thermo physical properties of the mold and casting was proposed for the simulation of the mold/casting boundary condition during solidification. The applicability of the various models relating secondary dendrite arm spacing to heat transfer parameters was evaluated and the extension of these models to continuous casting processes was pursued. |
Extent | 7850225 bytes |
Genre |
Thesis/Dissertation |
Type |
Text |
FileFormat | application/pdf |
Language | eng |
Date Available | 2008-09-15 |
Provider | Vancouver : University of British Columbia Library |
Rights | For non-commercial purposes only, such as research, private study and education. Additional conditions apply, see Terms of Use https://open.library.ubc.ca/terms_of_use. |
IsShownAt | 10.14288/1.0078429 |
URI | http://hdl.handle.net/2429/1966 |
Degree |
Master of Applied Science - MASc |
Program |
Materials Engineering |
Affiliation |
Applied Science, Faculty of Materials Engineering, Department of |
Degree Grantor | University of British Columbia |
GraduationDate | 1993-11 |
Campus |
UBCV |
Scholarly Level | Graduate |
AggregatedSourceRepository | DSpace |
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