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Melting of solids in liquid titanium during electron beam processing Ou, Jun 2015

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Melting of Solids in Liquid Titanium DuringElectron Beam ProcessingbyJun OuB.A.Sc., Central South University, 2007M.A.Sc., Central South University, 2010A THESIS SUBMITTED IN PARTIAL FULFILLMENT OFTHE REQUIREMENTS FOR THE DEGREE OFDOCTOR OF PHILOSOPHYinThe Faculty of Graduate and Postdoctoral Studies(Materials Engineering)THE UNIVERSITY OF BRITISH COLUMBIA(Vancouver)December, 2015© Jun Ou 2015AbstractBoth experiments and numerical modeling work have been carried out to understand the phenomena con-tributing to the melting of solid condensate in liquid titanium alloys during Electron Beam Cold HearthRe-melting (EBCHR). To begin, ice/water and ethanol/water analogue physical models were adopted tostudy the melting of a low melting point solid introduced into liquid and to provide data suitable for de-veloping a comprehensive numerical-based modeling framework. The results revealed that thermal andcompositional driven buoyancy and surface tension (Marangoni) flows, when present, can have a significantimpact on solid melting in a system where forced convection is not significant.In work that followed, the melting behavior of Commercial Purity Titanium (CP-Ti) rods in liquid CP-Tiwas investigated with the aid of an Electron Beam Button Furnace (EBBF) to examine the melting kineticsin the titanium system in the absence of compositional effects. The results showed that the liquid titaniuminitially froze onto the cold rod when it was immersed, resulting in the formation of a solid/solid interfacethat acted to retard melting when present. Data collected from the experiments included the evolution in thesolid profile of the rod with time and the evolution in temperature obtained from a thermocouple embeddedin the rod. The numerical modeling framework developed for the ethanol/water system was modified andapplied to support analysis of the experimental results including the determination of an effective interfacialheat transfer coefficient (EIHTC). A similarity solution was also developed to assess the numerical modelderived EIHTC.In the final phase of the study, work was conducted on Ti-Al solid rods partially immersed in liquid CP-Ti and liquid Ti-6wt%Al-4wt%V (Ti64) as a means of approximating the behavior of condensate in industry.The melting behavior of Ti-Al was observed to differ significantly from that of CP-Ti rods. Despite havinga lower melting point, the Ti-Al rod was found to heat up and melt at a much slower rate. Metallographicexamination of partially melted rods and a sensitivity analysis conducted with the numerical model has beenable to partially, but not fully explain this difference.iiPrefaceA significant amount of the work from this Ph.D. research program has been published in two peer-reviewedjournal papers, which are referred to below. The experimental program, mathematical model developmentand results and discussion sections associated with the two papers have been reformatted and appear inChapters 4 (publication 1) and 5 (publication 2). The work that has not been published at this time isincluded in Chapter 6.1) Ou, J.; Chatterjee, A.; Cockcroft, S. L.; Maijer, D. M.; Reilly, C. & Yao, L. Study of melting mechanismof a solid material in a liquid. International Journal of Heat and Mass Transfer, Elsevier, 2015, 80, 386-3972) Ou, J.; Cockcroft, S. L.; Maijer, D. M.; Yao, L.; Reilly, C. & Akhtar, A. An examination of the factorsinfluencing the melting of solid titanium in liquid titanium. International Journal of Heat and Mass Transfer,Elsevier, 2015, 86, 221-233In publication 1, my contribution was to develop the framework for the numerical model, conduct theanalytical analysis and assist in interpreting the results. Aniruddha Chatterjee, the second author, carried outthe experimental work and ran the model to support analysis of the experimental results in the context ofhis M.A.Sc program. We worked together to develop or define the material properties, initial and boundaryconditions, and analyze results. Drs. Carl Reilly and Lu Yao provided advice at various times during thestudy in support of both the experimental program and numerical modeling activities.In publication 2 and Chapter 6, I performed the experiments, collected the data, interpreted the results,developed the models and conducted the analytical analysis. Dr. Carl Reilly helped with the design andfabrication of a mechanism for placing samples in the molten pool in a controlled and reproducible manner.Dr. Ainul Akhtar supported the program by producing Ti-Al samples that were used in the work that isdescribed in Chapter 6. In addition, Drs. Ainul Akhatar and Lu Yao provided suggestions and commentsduring periodic reviews of the research program.Throughout the course of this program, Professors Steve Cockcroft and Daan Maijer, my supervisors,iiiprovided on-going support in the form of advice on all aspects of my Ph.D. program. All of the secondaryauthors listed on the papers identified above, provided editorial feedback during the preparation of the twomanuscripts for publication.ivTable of ContentsAbstract . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . iiPreface . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . iiiTable of Contents . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . vList of Tables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . viiiList of Figures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ixList of Symbols . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xiiList of Abbreviations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xvAcknowledgments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xvi1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.1 Melting Technologies . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21.1.1 Vacuum Arc Remelting . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21.1.2 Cold Hearth Melting . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21.2 Melt Related Defects . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41.3 Melting/Dissolution of Inclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52 Literature Review . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72.1 Melting/Dissolution of Solids in Liquid Metals . . . . . . . . . . . . . . . . . . . . . . . . 72.1.1 Basic Thermophysical Phenomena . . . . . . . . . . . . . . . . . . . . . . . . . . 72.1.2 Mechanism of Melting/Dissolution of Solids in Liquid Metals . . . . . . . . . . . . 82.1.3 Factors Influencing the Melting/Dissolution Kinetics . . . . . . . . . . . . . . . . . 102.1.4 Mathematical Characterization of Melting/Dissolution . . . . . . . . . . . . . . . . 112.2 Melting/Dissolution of Inclusions in Liquid Titanium . . . . . . . . . . . . . . . . . . . . 142.2.1 Hard Alpha Inclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 152.2.2 High Density Inclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 162.2.3 Al-rich Inclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 162.3 Mathematical Modeling of Heat, Mass and Momentum Transport in an Electron Beam But-ton Furnace . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 172.4 Titanium-Aluminum Binary Phase Diagram . . . . . . . . . . . . . . . . . . . . . . . . . 203 Scope and Objectives . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 223.1 Scope of the Research Program . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 233.2 Objectives of the Research Program . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24v4 Study of Melting Mechanism of a Solid Material in a Liquid . . . . . . . . . . . . . . . . . . 254.1 Experimental Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 254.1.1 Experimental Setup . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 254.2 Numerical Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 284.2.1 Computational Domain . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 284.2.2 Model Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 294.2.3 Boundary Conditions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 304.2.4 Material Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 314.2.5 Initial Conditions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 344.2.6 Computational Procedure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 344.3 Results and Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 354.3.1 General Comparison . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 354.3.2 Quantitative Comparison of Melt Rate and Temperature Evolution . . . . . . . . . 404.3.3 Effective Heat and Mass Transfer Coefficients . . . . . . . . . . . . . . . . . . . . 434.4 Summary and Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 475 An Examination of the Factors Influencing the Melting of Solid Titanium in Liquid Titanium 495.1 Experimental Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 495.1.1 Experimental Apparatus . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 505.1.2 Experimental Procedure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 515.2 Model Description . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 525.2.1 Governing Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 535.2.2 Computational Domain and Mesh . . . . . . . . . . . . . . . . . . . . . . . . . . . 535.2.3 Boundary Conditions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 555.2.4 Initial Conditions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 575.2.5 Material Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 585.2.6 Computational Procedure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 595.3 Result and Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 595.3.1 Pool Profile and Temperature before Dipping . . . . . . . . . . . . . . . . . . . . 595.3.2 Rod Solid Profiles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 615.3.3 Melting Rate . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 635.3.4 Temperature Monitored within the Rod . . . . . . . . . . . . . . . . . . . . . . . . 635.3.5 Examination of the Influencing Factors . . . . . . . . . . . . . . . . . . . . . . . . 665.3.6 Effective Heat Transfer Coefficient . . . . . . . . . . . . . . . . . . . . . . . . . . 675.4 Summary and Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 736 Study of the Melting of Ti-Al Solid in CP Titanium and Titanium Alloy (Ti64) Liquid . . . . 746.1 Experimental Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 746.1.1 Experimental Material . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 756.1.2 Experimental Procedure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 786.1.3 Experimental Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 786.2 Model Description . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 846.2.1 Governing Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 856.2.2 Computational Domain and Mesh . . . . . . . . . . . . . . . . . . . . . . . . . . . 856.2.3 Boundary Conditions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 856.2.4 Material Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 886.2.5 Initial Conditions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 966.2.6 Modeling Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 96vi6.3 Summary and Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1057 Summary and Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1078 Future Work . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 111Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 112viiList of Tables1.1 Comparison of material properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.2 Comparison of costs (unit: US$/lb) [6] . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21.3 Melt related defects and their possible causes [5] . . . . . . . . . . . . . . . . . . . . . . . 52.1 Description of the routines in Fig. 2.1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 84.1 Experimental table and examining factors . . . . . . . . . . . . . . . . . . . . . . . . . . . 274.2 Physical properties applied in the model . . . . . . . . . . . . . . . . . . . . . . . . . . . . 325.1 Chemical composition of CP-Ti that is used as the experimental material (wt%) . . . . . . . 495.2 Sensivitity analysis on grid size . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 555.3 Material properties of CP-Ti and boundary condition related parameters adopted in themodel . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 565.4 Reference temperature in Eq. 5.3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 575.5 Temperature comparison between the experimental measurements and model predictions . . 616.1 Details of attempts and the identified problems . . . . . . . . . . . . . . . . . . . . . . . . 766.2 Composition analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 77viiiList of Figures1.1 Schematic of a VAR furnace[1] . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31.2 Schematic of the cold hearth melting process with electron beam melting [5] . . . . . . . . . 42.1 Thermophysical phenomena occur during immersion of a solid addition within a bath ofsteel melt (modified from Ref [14]) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92.2 Experimental method in Powell’s study . . . . . . . . . . . . . . . . . . . . . . . . . . . . 162.3 Schematic of EB gun . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 182.4 Phase diagrams given in Ref [53] (a) and [54] (b) . . . . . . . . . . . . . . . . . . . . . . . 213.1 Schematic for explaining the “drop-in” event . . . . . . . . . . . . . . . . . . . . . . . . . 224.1 Schematic diagram of the physical model setup and numerically modeled domain . . . . . . 264.2 Schematic diagram showing the two experimental configurations and the flow drivers present 274.3 Physical properties applied in the model . . . . . . . . . . . . . . . . . . . . . . . . . . . . 334.4 Schematic diagrams showing the drivers for fluid flow for the four experimental conditionsexamined in the study (a) and a series of images comparing the experimental results andmodel predictions at 10 s elapsed time (b) . . . . . . . . . . . . . . . . . . . . . . . . . . . 364.5 Comparison between experimental and modeling results for the fully immersed ice/water case 374.6 Comparison between experimental and modeling results for the partially immersed ice/watercase . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 384.7 Comparison between experimental and modeling results for the partially immersed ethanol/watercase . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 394.8 Comparison on melting kinetics between experimental and modeling results . . . . . . . . . 414.9 Temperature comparison between experimental and modeling results at the two thermocou-ple locations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 424.10 Effective heat transfer coefficient during the melting process . . . . . . . . . . . . . . . . . 464.11 Effective mass transfer coefficient during the melting process (Ethanol/water fully immersedcase) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 475.1 Schematic diagram of the lab-scale EB melting furnace . . . . . . . . . . . . . . . . . . . . 505.2 Details of the sample holder . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 515.3 Dimensions of the puck (a) and dipped rods (b) and locations of the inserted thermocouples.The 5◦ slice at the right-hand-side is the modeling domain (c) . . . . . . . . . . . . . . . . . 515.4 Screenshot showing the rod dipped into the molten pool . . . . . . . . . . . . . . . . . . . 525.5 Three configurations of the model to simulate the three stages of the experiment . . . . . . . 535.6 Meshing details . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 545.7 Boundary conditions of the model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 555.8 Temperature dependent material properties of CP-Ti (Melting temperature: 1668◦C) . . . . 585.9 Pool profile comparison . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 605.10 Solid/liquid interface profile of the rods with different dipping times . . . . . . . . . . . . . 62ix5.11 Interface profile comparison between model predictions and experimental observations (Thedashed line shows the interface profile at the right side of the sample with 3 s since the profileis apparently asymmetric as observed in Fig. 5.10) . . . . . . . . . . . . . . . . . . . . . . 635.12 Comparison of melted volume ratio between experimental and modeling results (the diame-ter of the circles indicate the inaccuracies of the size measurements - less than 5%) . . . . . 645.13 Solid/solid interface formed when liquid CP-Ti solidified on the cold dipped rod . . . . . . . 645.14 Comparison between measured and predicted temperatures in the immersed rod at TC R(the size of the circles indicate the experimental inaccuracy - less than 6%) . . . . . . . . . . 655.15 Modelling results showing the effects of buoyancy and Marangoni forces on the solid/liquidinterface profile (the labels, Buo and Mara, in the figure denote buoyancy and Marangoniforces, respectively) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 665.16 Mesh and temperature distribution in proximity to the solid at t =2 s. . . . . . . . . . . . . . 675.17 The reference line over which the bulk liquid temperature was calculated (t=2 s) . . . . . . . 685.18 Bulk liquid temperatures adopted to calculate EIHTCs . . . . . . . . . . . . . . . . . . . . 685.19 Similarity solution related schematics (a) simplified case description for similarity solution,(b) simplified geometries (truncated cone or cone) with time and (c) transformation fromtruncated cone or cone to a cylinder . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 695.20 Effective heat transfer coefficients . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 715.21 Effects of thermal conductivity and flow velocity on heat transfer coefficient . . . . . . . . 726.1 CP-Ti crucible dimensions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 776.2 Ti-Al ingot . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 786.3 Photo captured showing the rod was stuck to the pool and detached from the sample holder . 796.4 Sample sections . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 806.5 Cross section of the sample – Ti-Al rod dipped in CP-Ti following type-B procedures . . . . 816.6 EDX line scan at the interface on face 1 (see Fig. 6.5a) . . . . . . . . . . . . . . . . . . . . 826.7 Rod profile - Ti-Al/Ti64 (type-A) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 826.8 Cross section of the Ti64 (type-A) sample . . . . . . . . . . . . . . . . . . . . . . . . . . . 836.9 Temperature data collected from thermocouple embedded in the three rods tested. . . . . . . 846.10 Comparison of the photos captured from the top view port of the furnace . . . . . . . . . . 846.11 Meshing details . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 866.12 Boundary conditions of the model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 866.13 Composition dependent activity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 876.14 Compositional surface tension coefficient . . . . . . . . . . . . . . . . . . . . . . . . . . . 886.15 Density of the Ti-Al . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 896.16 Ti-Al binary phase diagram adopted in the numerical model for latent heat associated withsolidification/melting . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 906.17 Solid fraction calculated based on the phase diagram . . . . . . . . . . . . . . . . . . . . . 916.18 Solid fraction dependent viscosity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 916.19 Solid fraction dependent permeability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 926.20 Effective specific enthalpy (a) and effective specific heat capacity (b) adopted in the model . 936.21 Illustration of adapting thermal conductivity . . . . . . . . . . . . . . . . . . . . . . . . . . 946.22 Thermal conductivity adopted in the model . . . . . . . . . . . . . . . . . . . . . . . . . . 946.23 Estimated vapour pressures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 966.24 Modeling results at different immersion times (contour at left: temperature in ◦C and right:mass fraction of Al) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 986.25 Temperature comparison between the model and experiment (Ti-Al in CP-Ti) . . . . . . . . 1006.26 Temperature comparison . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101x6.27 Modeling results for the no heat of vaporization case at different immersion times . . . . . . 1026.28 Temperature comparison between the tested case and base-case . . . . . . . . . . . . . . . . 1026.29 Tested phase diagrams . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1036.30 Temperature comparison between the tested cases and base-case . . . . . . . . . . . . . . . 1036.31 Temperature contours with velocity vectors of the two tested cases at different immersiontimes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1046.32 Aluminum mass fraction contours with velocity vectors of the two tested cases at differentimmersion times . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 105xiList of SymbolsSymbol Description UnitβT and βC thermal and compositional expansioncoefficients, respectively1/K and m3/kgγ surface tension N/mε emissivity -η pseudo-similarity variable -θ dimensionless temperature -µ viscosity Pa ⋅ sµx and µy position of the center of the beam spot mξ transformed axial coordinate -ρ (ρe, ρw, ρAl andρTi)density (of ethanol, water, aluminum andtitanium, respectively)kg/m3σEB standard deviation of the Gaussiandistributed EB powermσM Marangoni stress N/m2σrad Stefan-Boltzmann constant W/(m2 ⋅K4)τ stress tensor PaA(As/l) area (of the solid/liquid interface) m2C, Cm, m and n fitting parameters in the empiricalcorrelations-CP (CP,e) specific heat capacity (effective) J/(kg ⋅K)D (De and DAl) diffusivity (of ethanol in water and aluminumin titanium, respectively)m2/sDSV a microstructure dependent constant 1/m2Gr Grashof number -H specific enthalpy J/kgKperm permeability m2L and b parameters for evaluation of thedimensionless numbers - characteristiclength and space between the plates,respectivelymLm, Lα/β and Lvap specific latent heat for melting, for α/βphase transformation, and for vaporization,respectivelyJ/kgM (MAl) molar mass (molar mass of aluminum) kg/molNu (Nux or Nu) Nusselt number (local and mean,respectively)-P pressure PaxiiSymbol Description UnitPv(P0Al) vapor pressure (vapor pressure of purealuminum)PaPr Prandtl number -PEB EB power WQs/l total amount of heat transported through thesolid/liquid interfaceJQrad heat transported by radiation at the rod’ssurfaceJRg ideal gas constants J/(K ⋅mol)Rex Local Nusselt number -Sc Schmit number -SM (SM,darcy) momentum source term (for Darcyattenuation)kg/(m2 ⋅ s2)SE (SE,vap) energy source term (for vaporization) J/(m3 ⋅ s)Sh Sherwood number -T temperature KTbulk, Tmold,side,Tmold,bottom, Tpool ,Tchamber,Tceramicsheath andTs/ltemperatures at various locations:temperature at bulk liquid, mold side, moldbottom, pool, chamber, ceramic sheath andsolid/liquid interface, respectivelyKTliq and Tso liquidus and solidus KTm melting temperature KTre f reference temperature KU (Ubulk) velocity (bulk velocity) m/sV (V0 and Vs) volume of (the original and reaming solid) m3aAl activity of aluminum -f reduced stream function -fabs absorption factor -fs, fl , fα and fv solid fraction, liquid fraction, α phasefraction and vaporized liquid fraction,respectively-fEB EB scanning frequency Hzhcnt contact heat transfer coefficient W/(m2 ⋅K)he f f and hx effective interfacial heat transfer coefficientand local interfacial heat transfer coefficientW/(m2 ⋅K)k thermal conductivity W/(m ⋅K)me, mw, mAl andmTimass fraction of ethanol, water, aluminumand titanium, respectively-m˙ (m˙Al) mass flux (mass flux of aluminum) kg/(m2 ⋅ s)qEB heat flux input for EB heating W/m2qcnt heat flux by contact heat transfer W/m2qs/l heat flux through the solid/liquid interface W/m2r and h radius and height mxiiiSymbol Description Unitriand hi initial radius and height of the equivalentcylindermr0, rt , rb, h0 and lc dimensions of the transformed geometry (seeFig. 5.19 for details)mrEB radius of the circular EB scanning pattern mrV melting ratio based on volume -t time sx and y distance in x and y direction,respectively m∆T temperature change at a given locationduring the time interval ∆tK∆t time interval s∆Vs volume change of solid during the timeinterval ∆tm3xivList of AbbreviationsAP Atmosphere pressureCP-Ti Commercial pure titaniumCFD Computational Fluid DynamicsCHM Cold Hearth MeltingEB Electron BeamEBBF Electron Beam Button FurnaceEBCHR Electron Beam Cold Hearth Re-meltingEDX Energy-Dispersive X-ray SpectroscopyEIHTC/EIMTC Effective interfacial heat/mass transfer coefficientHDIs High Density InclusionsHIDs High Interstitial DefectsODE Ordinary differential equationPACHR Plasma Arc Cold Heart Re-meltingPDE Partial differential equationSEM Scanning Electron MicroscopyTC ThermocoupleTi64 Ti-6wt%Al-4wt%VVAR Vacuum Arc RemeltingWP Working pressurexvAcknowledgmentsForemost, I wish to express my sincere thanks to my supervisors, Dr. Steve Cockcroft and Dr. DaanMaijer, who have been supporting and guiding my research throughout this program with great patience andthoughtful encouragement. I am very grateful for their steady and high quality supervision that helped mefinish this program smoothly.I would also like to express my gratitude to Dr. Carl Reilly, Dr. Lu Yao and Dr. Ainul Akhtar for theirassistance with my experiments and insightful comments and suggestions provided in the weekly meetings.A big thank you to all of my colleagues and officemates for the advice throughout my study at UBC and thejoyful times we spent together. I also wish to thank Titanium Metals Corporation for the financial support.I appreciate the support from the technicians in our department including Ross Mcleod, Carl Ng, DavidTorok, Wonsang Kim and Jacob Kabel who helped me with sample machining, instrument fabrication andSEM/EDX analysis. I also want to thank the administrative staff, Fiona Webster, Mary Jansepar, MichelleTierney and Norma Donald who provided support during my study.I am very thankful to my parents and parents-in-law for caring for my wife and me. Finally I owe muchto my wife Min Xu, who is always there to support me with tolerance, understanding and love.xviChapter 1IntroductionTitanium alloys are attractive for a variety of applications, including in the aerospace, medical, marine,chemical processing, food processing and automotive industries, owing to their excellent strength to densityratio and corrosion resistance. The aerospace industry is responsible for 70-80 percent of the total titaniumconsumption (in the United States) [1]. Table 1.1 compares the density, tensile strength and ratio of strengthto density of CP-Ti and Ti64 to a commonly used high-strength aluminum alloy (7050) and steel (A514)alloy.Table 1.1: Comparison of material propertiesMetal or alloy Density atroomtemperature[kg/m3]Tensilestrength,yield [MPa]Tensilestrength,ultimate[MPa]Ratio of yieldstrength to density[MPa]/ [kg/m3]CP Titanium (grade 1) [2] 4,510 170 240 0.04Ti64 (typical wrought β annealed) [3] 4,430 860 955 0.19Al alloy 7050 (T736 die forgings) [3] 2,830 420 490 0.15A514 steel [4] 7,800 690 760 - 895 0.09One drawback of titanium and its alloys is that they are comparatively expensive produce, as highlightedin Table 1.2. The high cost has been a significant barrier to broader use particularly in commercial/consumerproducts. Therefore, in the past several decades, the major thrust in titanium development has been aimed atachieving cost reduction rather than in developing alloys with enhanced properties. One consequence of thehigh cost is that there is a strong economic incentive to recycle titanium scrap (often called revert). This istypically done by blending it with virgin material in the feedstock that is used in the various primary consol-idation technologies [5]. Because the resulting ingots may be intended for critical applications, significanteffort has been made to eliminate the melt related defects associated with revert material so as not to degradequality.The following sections in this chapter provide some basic knowledge related to titanium processing with1Table 1.2: Comparison of costs (unit: US$/lb) [6]Item Steel Aluminum TitaniumOre 0.02 0.01 0.22 (rutile)Metal 0.10 1.10 5.44Ingot 0.15 1.15 9.07Sheet 0.30 - 0.60 1.00 - 5.00 15.00 - 50.00emphasis on liquid metal refining during the primary consolidation step.1.1 Melting TechnologiesUnalloyed titanium and the various titanium alloys require special technologies to melt and consolidate thematerial for ingot production. Challenges include high reactivity with nitrogen and oxygen and high meltingtemperatures – i.e. 1668 ◦C for pure titanium. Two melting technologies dominate production: Vacuum ArcRemelting (VAR); and, Cold Hearth Melting (CHM).1.1.1 Vacuum Arc RemeltingVAR has been the most commonly used process for titanium ingot production since the commercial intro-duction of alloys occurred in the 1950s. To illustrate the process, a schematic of a VAR furnace is shown inFig. 1.1. The electrode is made up of mechanically compacted titanium sponge and alloying elements andmay include scrap. It is melted by the electric arc that forms between the electrode and the melt pool, whichis direct current, low voltage and high amperage (normally several kilo amperes).Decades of industrial practice have demonstrated the capability of VAR to produce high quality ingotswith good reproducibility. However, gravity induced segregation can occur in the process since it is operatedin a vertical orientation [5]. Additionally, it has limitations in its ability to treat revert material, because theelectrode is situated directly above the melt pool and the residence time of material in the liquid pool islimited.1.1.2 Cold Hearth MeltingCHM is a newer technology developed in the 1990s, and is being increasingly used, often in conjunctionwith VAR, because of its superior ability to process scrap. There are two commercial variants of CHM,2(-)(+)Drive motorVacuumElectrodeFurnace bodyWater outWater inSolidified ingotIngot poolArcCrucibleDrive screwFigure 1.1: Schematic of a VAR furnace[1]Electron Beam Cold Hearth Re-melting (EBCHR) and Plasma Arc Cold Heath Re-melting (PACHR). Aschematic of a typical EBCHR process configuration is shown in Fig. 1.2.One significant distinguishing feature of these processes is that they utilize a water-cooled copper vessel(called the cold hearth), which enables improved defect removal relative to the VAR process. Firstly, the coldhearth solidifies a thin layer of titanium alloy on its inner surface during melting (called the skull), whichprevents the molten liquid from contacting the hearth. This eliminates the possibility of any contamina-tion from the hearth material (liquid titanium cannot be held in conventional oxide refractories). Secondly,the residence time of molten titanium in the hearth is comparatively long and can be controlled to allowmore opportunity for reducing or removing the inclusions that constitute potential defects. For example,high-density inclusions, such as tungsten carbide tool bits (potentially introduced with the revert) can sinkto the bottom, where they become trapped in the mushy zone of the skull. Other inclusions, like type IIdefects, which possess relatively low density, may be decomposed by a combination of mechanisms includ-ing dissolution, melting and/or volatilization (the latter occurring in the EBCHR process, which operates ina vacuum). A second distinct advantage of the CHM processes is that they can produce ingots of varyingcross-section such as for example slabs; intended for plate or sheet production, thereby simplify conversion3Electron beamRaw material feedingIngot molten poolIngot beingwithdrawnMeltinghearthRefininghearthFigure 1.2: Schematic of the cold hearth melting process with electron beam melting [5]operations.The two variants of the CHM process differ with respect to the heat source and furnace environment.The PACHR uses a plasma-arc for melting and operates in an inert gas atmosphere (He or Ar), whereas theEBCHR process uses an electron beam for melting and operates in a vacuum. Typically, in both processes,a final VAR step is required to improve chemical homogenization and surface quality in the ingot if itis intended for disc material in turbine engines. For example, Al-rich regions, which may form duringEBCHM, have to be homogenized by applying a final VAR step for rotor grade applications [5].1.2 Melt Related DefectsVarious types of defects have been identified by the titanium industry. Table 1.3 lists the five principle typesof the melt related defects and their possible causes. The melt related defects that remain in the final ingotcan significantly degrade the material’s mechanical performance and on occasion have caused an in-servicefailure. Understanding the mechanisms by which these defects form and can be eliminated is thereforecritically important for obtaining defect free material.This research program focuses on the type II defect (Al-rich regions) caused by “drop-ins” during EBM.The “drop-ins” are associated with Al-rich vapor that condenses on the furnace roof and exposed sectionsof the mold wall. Occasionally during normal operation of the process, the condensate can dislodge and fallinto the melt. An Al-rich region can be formed in the final ingot if this material is not melted and chemically4Table 1.3: Melt related defects and their possible causes [5]Defect type Possible causesType I ("Hard Alpha"),also called High InterstitialDefects (HIDs)Sponge Production-Fires during handling or shearingFirst melt electrode production-Fires during compaction-Improperly conditioned scrap-Contaminated master alloy-Contamination during weldingMelting and remelting-Small water leak-Air leak-Aggressive grinding during ingot conditioningHigh Density Inclusions (HDIs) Scrap addition-Tungsten welding electrodes-Tool bits mixed into turningsBeta Flecks Melting segregationConversion too close to transus (including adiabaticheating effects)Type II (Alpha Stabilized) Improper final melt phase (excessive pipe formation)Improper ingot top removalAl-rich "drop-ins" during EBMVoids Incorporation of shrinkage pipe during conversionImproper conversion practicehomogenized before the melt solidifies in the mold. Additional information on the mechanism for formationof this type of defect is explained in Chapter 3.1.3 Melting/Dissolution of InclusionsAs mentioned previously, the EBCHM process has advantages in terms of its ability to process revert ma-terial owing to superior inclusion removal capabilities compared with the VAR process. Depending on theinclusion chemistry, removal may be realized in the EBCHR process by settling, in the case of HDIs, dis-solution, in the case of HIDs, or melting, in the case of Type II defects associated with Al-rich “drop-ins”.Over the last two decades there have been relatively few studies appearing in the open literature that areconcerned with inclusion removal in titanium melt processing. Examples include Refs [7], [8] and [9].While these studies have improved our overall understanding of the mechanism(s) involved, further workis necessary. This thesis focuses on understanding the factors that influence the melting of the material5associated with Al-rich “drop-ins” that could lead to Type II defects in the EBCHR process. The drop-inspossess a lower density than the melt and thus will likely reside on the melt surface. On the surface of themelt, the Marangoni forces (both thermal and compositional) in combination with the relative velocity dif-ference between the bulk liquid and drop-in will influence the melting behaviour. These conditions cannotbe considered to be analogous to natural convection. To quantify the melting kinetics under these condi-tions, a combination of experimental measurements, using an Electron Beam Button Furnace (EBBF), andnumerical modeling have been used to examine the factors that contribute to the melting of these inclusionsin liquid titanium.6Chapter 2Literature ReviewGiven the focus of the proposed research program to understand the melting of Al-rich inclusions in liquidtitanium, the literature review will encompass a discussion on the melting/dissolution and the factors affect-ing the melting/dissolution of solids in liquid metals. This will draw on foundational material appearing intextbooks in addition to the open literature. The discussion will then shift to the open literature focusingon experimental/mathematical studies conducted on the melting/dissolution of solids in liquid metals andvarious inclusions in liquid titanium, before reviewing the published work on the mathematical modeling ofthe liquid pool in an EBBF. In the end the Ti-Al binary phase diagrams extracted from a variety of studiesare presented as they are significant data for investigating the melting behavior of Ti-Al solid.2.1 Melting/Dissolution of Solids in Liquid Metals2.1.1 Basic Thermophysical PhenomenaThe melting/dissolution of solids in liquid metals is important in a number of metallurgical processes in-cluding copper, nickel and platinum group metal matte converting operations [10], steel making[11, 12] andin the refining of titanium [13, 7, 8]. The important phenomena range from the melting or dissolution ofalloy additions to the removal of deleterious constituents in refining operations, and span situations in whichthe solid (solute) has a higher melting temperature and lower melting temperature than the liquid (solvent).Moreover, the fluid flow conditions prevailing in the liquid solvent range from conditions in which the meltis vigorously moving to stagnant. Without considering fluid flow in the liquid, Sismanis et al. [14] haveidentified 6 scenarios that describe the possible ways in which a solid addition can interact with a steel melt,which are shown schematically in Fig. 2.1 and explained in Table 2.1. In the first four routines (routines 1 to4), the solid possesses a melting temperature lower than the temperature of the liquid and in routines 5 and6 the solid has a higher melting point. In addition, routines 4 and 6 include a reaction, while the others do7not. It can be seen that at the beginning of all the various scenarios liquid solidifies onto the immersed solidto form a shell, encasing the original solid, since the solid is initially cold. Normally, the shell melts prior tothe encased solid. In some situations however the melting temperature of the encased solid is lower than theshell, as illustrated in routines 1, 2 and 3, and internal melting occurs first. An example of such a situationis the immersion of aluminum solid in steel liquid investigated in Mucciardi’s study [15]. If an exothermicreaction is involved, for example routine 4, the shell can melt before the occurrence of internal melting dueto the generation of heat even though the encased solid’s melting temperature is lower. In routines 5 and6, where the solid has a higher melting temperature, the solid dissolves continuously into liquid from theoutside inward.Table 2.1: Description of the routines in Fig. 2.1Routine # Description1 Internal melting of the solid material occurs (1C) and is completed before there-melting of the solidified enclosing shell2 Similar to Routine 1, but the internal melting is not completed as the encasingshell melts first (2C)3 Once the solid material is re-exposed (3C), another shell or series of shellsmay be formed (3D)4 An exothermic reaction starts at (4B) and the shell melts back very fast5 Dissolution following melting of the encasing solid shell6 The solid material, generally a powder compact, is surrounded by a shell (6B).The duration of the shell is very short due to the low thermal conductivity ofthe compact. However, a reaction starts at the compact/shell interface andproceeds very quickly into the interior of the compact (6C and 6D)2.1.2 Mechanism of Melting/Dissolution of Solids in Liquid MetalsAs discussed above, the mechanisms by which a solid decomposes in the liquid may be classified as eithermelting or dissolution, depending on the melting temperature of the solid in relation to the liquid tempera-ture. Furthermore, these processes can be complicated by the formation of a solid shell, encasing the originalsolid, and the potential for chemical reactions. For more straightforward situations in which melting dom-inates, the thermal energy provided by the liquid solvent is sufficient to raise the temperature of the solid(solute) to the point where the liquid becomes the thermodynamically stable phase for the solute. In thissituation, heat transfer at the solid/liquid interface controls melting. Dissolution, in contrast, is a more com-plicated process and may occur in several steps. Sismanis et al. [14] considered that the dissolution in his8A B C DRoutine 1Routine 2Routine 3Routine 4Routine 5Routine 6Legend (shown below):bath meltsolid ma-terialsolidifiedshellinternalmeltingexothermicreactionFigure 2.1: Thermophysical phenomena occur during immersion of a solid addition within a bath of steelmelt (modified from Ref [14])9study (niobium and zirconium in liquid steel) consists of two consecutive steps. The first step is the surfacereaction where the solid goes through a phase change to liquid. The phase change is the result of the ruptureof the metallic bond at the solid/liquid interface, for which a large amount of energy is required. The secondstep is the transport of the resulting solute atoms from the interface into the bulk liquid through a boundarylayer. He argues that either step could be rate controlling. Bellot et al. [16] and Ghazal et al. [7], who inves-tigated the dissolution of titanium nitride (TiN) in liquid titanium, argued that in the case of TiN, dissolutionis controlled by the diffusion of nitrogen in the solid (TiN) to the solid/liquid interface. In addition, theyobserved two solid phases, α- and β -Ti, with different nitrogen diffusivities in each. Dissolution was foundto occur at the β -Ti/liquid interface, once the nitrogen level dropped low enough to allow melting at theinterface to occur (the melting temperature of TiN decreases with the decreasing nitrogen concentration).2.1.3 Factors Influencing the Melting/Dissolution KineticsThe melting/dissolution kinetics of solids in liquid are affected by many factors including the melting tem-perature of the solid, the diffusivities of the various species involved and the temperature, composition andfluid flow fields in the boundary layer at the solid/liquid interface. Given the interest of this research pro-gram, the factors related to the boundary layer at the solid/liquid interface are reviewed here.For melting, the distribution of temperature within the boundary layer in the vicinity of the solid/liquidinterface plays a critical role to determine the melting kinetics. In the case of dissolution, it is the distributionof both temperature and composition. Therefore, it is important to understand those factors that impact thedevelopment of the boundary layer. Bulk flow, or so called forced convection, when present, can substan-tially increase the melting/dissolution rate due to its effect on reducing the thickness of the thermal and masstransfer boundary layers. This has been widely recognized in many studies. For example, the investigationof Niinomi et al. [17] on the dissolution of a ferrous alloy in molten aluminum showed that the dissolu-tion rate increased at a higher rotating speed of the solid. Similarly, Reddy’s study [18] on the dissolutionof TiN in liquid Ti64 revealed that the dissolution rate could be substantially increased when stirring theliquid. In the absence of forced convection, natural convection dominates in influencing the developmentof the thermal and mass transfer boundary layers. Both thermal gradients and compositional gradients inthe thermal and mass transfer boundary layers, respectively, can induce buoyancy forces leading to naturalconvection conditions at the interface. Szekeley and Chhabra [19] recognized the significance of buoyancy10in heat transfer at the solid/liquid interface for metals and alloys for both melting and solidification in the1970s. This has also been confirmed by many other studies such as [20, 21, 22]. Chatterjee [23] comparedthe melting of frozen ethanol and ice in water and found that compositional buoyancy is the dominant flowdriver and one of the most important factors affecting the melting behavior.In cases where the solid is buoyant in the liquid and resides at the surface, gradients in temperature andcomposition in the boundary layer present at the surface can give rise to Marangoni (surface tension) forces,which can act as an additional flow driver to buoyancy. Marangoni forces (thermal and compositional)were also examined in Chatterjee’s study [23] and were found to be important influencing factors whenthe solids were partially immersed in the liquid (in these cases thermal and compositional gradients werecreated at the liquid free surface adjacent to the solid/liquid interface). A similar conclusion with respect tothe influence of Marangoni forces was also obtained in Chen’s study [24] in which the effect of Marangoniflow on slag-line dissolution of a refractory was investigated.In EBM processing of titanium alloys, there is the additional complication associated with the highlylocalized and intensive electron beam (EB) heating of the surface of the melt in that both large temperatureand compositional gradients may be present (the latter present in alloy systems in which one component hasa substantially higher vapor pressure than the others). Depending on the EB spot size, power density andspot velocity, substantial gradients in buoyancy and surface tension can be present at the surface to driveflow [25].2.1.4 Mathematical Characterization of Melting/DissolutionMathematical modeling is an important technique for either evaluating the important parameters like effec-tive interfacial heat and mass transfer coefficients (EIHTCs and EIMTCs) or aiding in analyzing experi-mental work. Three types of models have been identified in the relevant studies. These include: empiricalcorrelations, similarity solutions and numerical models.2.1.4.1 Empirical CorrelationsSpecifically in the studies of melting/dissolution of solids in liquid metals, the method of empirical corre-lations is often adopted to calculate the EIHTC and EIMTC at the solid/liquid interface. Examples includeBellot et al.’s study for TiN in liquid steel [16], Kim and Pehlke for Ferrovac E and 1045 steel in liquid iron11[26], Shoji et al. for copper in liquid Tin-Lead alloy [27], Argyropoulos and Guthrie for Ti in liquid steel[28], Sismanis for Ni and Zn in liquid steel [29] and Ghazal for tungsten in liquid titanium [8]. The EIHTCand EIMTC are employed to approximate the resistance to heat and mass transfer, respectively, occurringin the boundary layer in the fluid, and therefore may be used to calculate the melting/dissolution rate of thesolid.Generally, the method of calculation of the EIHTC centers on the calculation of the dimensionlessNusselt number (Nu) and dimensionless Sherwood number (Sh) for the EIMTC. These two numbers can beevaluated based on other dimensionless numbers that can be conveniently measured from the experiments.For example, under free convection conditions, Nu is a function of the Grashof number (Gr) and the Prandtlnumber (Pr) – i.e. Nu = f (Gr,Pr). Sh is a function of the Grashof number and the Schmidt number (Sc)– i.e. Sh = f (Gr,Sc). Whereas, under forced convection, the Grashof number is replaced by the Reynoldsnumber (Re) – i.e. the Nu = f (Re,Pr) and Sh = f (Re,Sc). Numerous forms of the functions (correlations)have been proposed in various studies. Examples include Ref [30], [31] and [32]. Specifically in the studiesrelated to the melting/dissolution of solids in liquid metals, examples include Ref [7], [28] and [33]. Eq. 2.1and Eq. 2.2 (Ranz-Marshall correlation) are adopted in Ref [28] for calculating the heat transfer during thedissolution of a vertically placed titanium cylinder in liquid steel under conditions of natural convection, andRef [7] for calculating the mass transfer during the dissolution of titanium nitride in liquid titanium duringEBM, respectively. Once the dimensionless numbers, Nu or Sh, are obtained, the EIHTC and EIMTC canbe easily derived based on the definition of these dimensionless numbers.Nux,cylinder = 0.2672 ⋅Pr−0.42 2√2Gr0.25x(xr )0.503(GrxPr)0.25[1+ (0.492Pr ) 916 ] 49 +1.011 ⋅Pr−0.02 0.503(GrxPr)0.25[1+ (0.492Pr ) 916 ] 49 (2.1)Sh = 2.0+0.6Re1/2Sc1/3 (2.2)As discussed above, using empirical correlations to evaluate the heat and mass transfer and melt-ing/dissolution rate is relatively straightforward and quick. But application of the empirical correlationsis generally confined to the specific conditions used for evaluation of the key dimensionless numbers, andtherefore attention should be paid to conditions used to evaluate the correlation to ensure acceptable accu-12racy for a specific problem. For example, the Ranz-Marshall correlation used by Ref [7] to evaluate theEIMTC for the dissolution of TiN in liquid Ti was originally developed to describe droplets of water fallingin air and hence may be prone to significant error.2.1.4.2 Similarity solutionsThe full Navier-Stokes equations, a set of three partial differential equations (PDEs) used to describe trans-port of heat, mass and momentum, have no analytical solution. But for some problems that show certainphysical similarities, the equations can be simplified allowing for similarity transformations, which convertthe PDEs to a set of ordinary differential equations (ODEs). The ODEs can then be solved either analyti-cally or numerically subject to appropriate boundary conditions. The so-called similarity solution methodhas been shown to describe certain situations accurately and to serve as a means of validating full numericalsolutions of the Navier-Stokes equations[34].Using this method, the fluid flow, temperature and composition fields within the respective boundarylayers can be solved. Once solved, the EIHTC and EIMTC at the solid/liquid interface can be derived andthe interfacial heat and mass transfer rates for the melting/dissolution problems estimated. Blasius [35]first developed the similarity solution to solve the external flow over a plate. Thereafter, new solutions wereobtained for a variety of other problems. For example, Jeffrey [36] extended the similarity solution for radialflows, and Gorla [37, 38] and Wang [39] for stagnation flows. However, it should be noted that this methodhas strict requirements on the problem description - e.g. geometry and flow characteristics - to allow for thesimilarity transformations and thus the applicable problems are very limited.2.1.4.3 Numerical modelsNumerical methods can approximately solve the Navier-Stokes equations over a broad range of conditions,but can be computationally intensive. The range and complexity of the problems that can be tackled hasgreatly expanded over the last several decades in step with increases in the availability of significant com-putational resources. The various methods basically consists of the following steps: first, the calculationdomain (geometry) is defined and discretized into discrete nodes comprising cells or elements; next, thematerial properties, boundary conditions and initial conditions (if the solution is transient) are specified;finally, the simulation is started and the equations solved. Iteration is generally required as many of the13boundary conditions and material properties are nonlinear. Numerical integration in time is also appliedusing a variety of time-stepping schemes in transient problems. In both, steady state and transient solutions,the “solution” is obtained when a given error tolerance is achieved, again using a variety of schemes toassess convergence. With respect to the melting/dissolution of solids in liquid metals, examples include thenumerical models developed by Bellot et al. [40] and Ghazal et al. [7] for calculating the dissolution of TiN(trajectory and dissolution kinetics) in liquid Ti64 during EBM and VAR, respectively, based on calculationof the fluid flow and temperature fields in the molten pool.In the case of the condensate-melting problem, relevant phenomena are occurring at both the scale of thehearth (meters) and at the scale of the condensate that enters the liquid (millimeters). Adoption of a meshresolution for the hearth that is also suitable for the condensate would yield a problem that is computation-ally intractable with the resources available. One approach is to develop separate macro- and meso-scalemodels and to couple them weakly. In this approach, the temperature and flow conditions predicted from thenumerical solution of the macro-scale hearth model are used as the boundary conditions for a standalone,heat-transfer only, meso-scale model to predict the condensate melting time (a key assumption being that thepresence of the condensate does not significantly impact on the macro-scale thermal and flow fields). Thecommercial software package Ansys-CFX, currently used for the hearth model, has the ability to output thetemperature and slip velocity for massless particles as they transit the hearth. These data can then be usedto determine an effective interfacial heat transfer coefficient (EIHTC) suitable for the condensate-meltingmodel. As a further refinement in this work, the full Navier-Stokes equations are first solved for the meso-scale problem. The results are then used to develop a correlation to evaluate the EIHTCs applicable to thecondensate in Ti-system. This approach avoids the pitfalls associated with the adoption of EIHTCs fromgeneralized literature-based correlations.2.2 Melting/Dissolution of Inclusions in Liquid TitaniumAs discussed in the chapter of Introduction, the inclusions present in the melt that are not eliminated duringthe melt processing and remain in the final ingot can significantly degrade the performance of the productmaterial, particularly in fatigue. In the past several decades, there have been a number of studies conductedto understand the melting/dissolution of the inclusions in liquid titanium.142.2.1 Hard Alpha InclusionsStarting first with hard alpha inclusions, industrial practice has shown that they are most frequently a nitro-gen rich compound (TiN). Bewlay [41] measured a dissolution rate of 2.2 µm/s for TiN in liquid Ti6242 ata temperature of 1725 ◦C. A nitrogen-containing solid α-Ti layer and a nitrogen-containing β -Ti solidifiedlayer were found between the solid TiN and liquid Ti6242 during dissolution. Reddy [18] investigated thedissolution of TiN cylindrical bars in liquid Ti64, pointing out the significance of stirring in increasing thedissolution kinetics. Schwartz confirmed this in his investigation, in which the dissolution rates in caseswith intense stirring were 10 times faster than those generally reported in literature. Additionally Mitchell[42] demonstrated that the dissolution rate strongly depends on the liquid temperature. In his study, thedissolution rate doubled when the liquid temperature was increased by 100 ◦C. Ghazal et al. [7], who in-vestigated the dissolution of the TiN inclusions, confirmed that the dissolution kinetics is highly dependenton the liquid temperature and the fluid flow surrounding the inclusion. In addition, under the conditionsexamined, the inclusions with initial diameters larger than 0.3 mm were shown to survive after a single VARmelt (the trajectory, temperature and slip velocity of the particle was obtained from a computational fluiddynamics (CFD) simulation of the melt pool in a VAR process). It was also concluded that a triple VARmelt had the potential to remove up to a 1 mm diameter inclusion. Powell [9] also studied the dissolution ofTiN in a titanium melt. The method employed is illustrated in Fig. 2.2. In his experimental method, a TiNrod was embedded vertically in a CP titanium (CP-Ti) ingot and a molten pool was produced by EB heatingso that a portion of the TiN rod was exposed to liquid titanium. A dissolution rate of approximately 0.16mm/min was measured in this study. Bellot et al. [16] investigated the dissolution of synthetic fully denseTiN samples of up to 6 wt pct and nitrided sponge of up to 15 wt pct in titanium and titanium alloy liquidbaths. The results show that the dissolution process is always controlled by the outward diffusion of nitrogeninto the bath through an external layer of beta phase. The growth of this beta phase layer depends on thevelocity of fluid flow in the bath. Another investigation of Bellot et al. [40] focused on the dissolution ofhard alpha inclusions during EBCHM. It was found that particles with a density similar to that of the liquid(less than 3 pct difference) can follow the liquid stream lines. In contrast, the particles whose density is verydifferent from that of liquid either settle to the bottom of melt and are trapped by mushy zone, or rise to thesurface.15CP Timolten poolTiN sample rodCP Ti plughearth surfaceFigure 2.2: Experimental method in Powell’s study2.2.2 High Density InclusionsAnother principal type of inclusions that may be present in the titanium melt is HDIs, or high densityinclusions. Sources include tungsten welding electrodes and tool bits mixed into turnings. Yamanaka andIchihashi [43] investigated the dissolution of three high-density materials, Ta, Mo and V, in liquid titaniumduring VAR processing. Particles of the three investigated elements with various sizes were mixed andcompacted with titanium sponge to produce electrodes followed by a VAR process. The results show Tawith initial size under 74 µm, Mo under 149µm and V under 3360µm can be fully dissolved by a singleVAR step under the conditions applied. Ghazal et al. [8] conducted a more quantitative investigation on thedissolution of the HDIs in liquid titanium. Cylinders, made from tungsten and molybdenum, were dippedinto various titanium alloys melt (CP-Ti, Ti17 and Ti64) in two different ways. In one way, samples wereremoved from the melt after various controlled periods of time so that the dissolving profiles of the dippedsamples could be observed and the dissolution rate measured. In another series of tests, the samples werekept inside the melt after shutting down the beam power in order to perform composition analysis. It wasshown that the dissolution kinetics are highly dependent on the liquid metal agitation and temperature. Theresults also revealed that the dissolution rates of both tungsten and molybdenum were higher in CP-Ti thanother alloys used.2.2.3 Al-rich InclusionsAs mentioned in the introduction, Al-rich “drop-ins” in the EBCHR process can cause type II defects.This type of inclusion possesses a density lower than titanium as well as a melting temperature lower thantitanium, which differs from the hard alpha inclusions and HIDs. It is therefore expected that melting instead16of dissolution is the mechanism of mass transportation from the solid to liquid. However, research directlyrelevant to the melting of Al-rich “drop-ins” in liquid titanium is very limited in the open literature. Tothe author’s knowledge, the only available work is Chatterjee’s study [23] on an analogue system based onethanol/water. The study revealed that thermal and compositional buoyancy and surface tension driven flows(Marangoni flows), when present, can have a significant impact on the melting behavior of the solid, sincethese flow drivers influence the development of the interfacial boundary layer and impact the heat and masstransfer through it.2.3 Mathematical Modeling of Heat, Mass and Momentum Transport in anElectron Beam Button FurnaceMathematical models are important for developing an improved understanding of the transport of heat, massand momentum during EBM of titanium, owing to the difficulty in making direct measurements – i.e. hightemperatures, reactive nature of titanium and its alloys and the vacuum environment. This also extends tothe small laboratory-scale furnaces often used for fundamental investigations, as an accurate mathematicalmodel of these furnaces is essential to compliment physical measurements that are often very limited inscope and accuracy.Characteristics of EB Heating - Before reviewing the mathematical models available in the open liter-ature, it is necessary to discuss the characteristics of EB heating since it affects the transport phenomenasignificantly. The EB gun (see Fig. 2.3) generates and accelerates a continuous stream of electrons towardthe target to be heated. The heat is generated within a short distance of the target surface by converting thekinetic energy of the electrons that are traveling at high speeds to heat. Depending on the intended applica-tion – e.g. melting or welding - the energy from the beam spot can be deposited into an area on the orderof a cm2 [44] or less than a mm2, respectively. The energy deposition depth is generally shallow and onthe order of µm [45]. Typically in melting operations, the EB is programmed to scan the area to be meltedusing lines, ellipses and circles or combination thereof, using deflection coils within the gun. Depending onthe scan rate (rate of translation of the beam) and pattern, EB heating can result in the generation of largetemperature gradients on the target surface.Mathematical models of EBBFs - P.D. Lee [46] developed a mathematical model and conducted exper-17~ion collectorfilamentcathodefocuselectrodeanodebeamguidenceand de-flectiontargetU fUsUB(a) Details of an EB gunelectronbeambeamguidancelensbeammatchinglensdeflectionsystemtarget(b) Details of the beam guidance anddeflection systemFigure 2.3: Schematic of EB guniments to investigate the effects of Marangoni flow resulting from thermal gradients at the free surface ina nickel alloy while being melted in an EBBF. The results revealed that Marangoni forces were significantflow drivers. The work also revealed that small changes in the concentration of surface-active elementscould completely alter the flow direction. This work highlights the potential for surface tension driven flowto be important when processing metals using EB heating. Meng [47] developed a mathematical modelto describe the fluid flow and heat transfer in a Ti64 button sample during EBBF melting. Three factors,buoyancy driven flow, Marangoni flow and the flow attenuation (also called Darcy damping flow) withinthe semi-solid zone (mushy zone), were examined to assess their significance in contributing to the fluidflow field and pool profile. It was found that buoyancy driven flow and flow attenuation within the mushyzone were the prevailing factors under the conditions in his study, while Marangoni flow had a limitedcontribution. Additionally, the ability to apply a time averaged surface heat flux to approximate the heatinput associated with a circular beam pattern was also assessed. The results indicated that a time-averagedapproach was a good approximation providing the pattern frequency is above 10 Hz, for the conditionsexamined in the study. Zhang [25] extended Meng’s research to examine buoyancy and Marangoni flowsinduced by the compositional gradients associated with aluminum evaporation in addition to the thermal18buoyancy and Marangoni flows. Under the conditions examined in the study, the composition within theliquid pool was found to be fairly uniform due to the strong fluid flow conditions prevailing within the pool.Consequently, the compositional buoyancy and Marangoni flows were found to only slightly impact the fluidflow and pool profile. The investigation also compared the melt pool profile and temperatures at discretelocations in the button and found them to be in good agreement with model predictions.Evaporation – As highlighted above, EBM can lead to changes in melt composition due to the preferen-tial loss of volatile alloy constituents, such as aluminum in the Al-bearing alloys like Ti64. This in turn canfurther complicate the flow drivers present during EBBF melting. Isawa et al. [48] summarized five possiblesites for aluminum evaporation in his study of the EBCHM process for Ti64 and Ti811: 1) the feedstock, 2)metal drops falling from the feedstock, 3) the hearth pool, 4) metal drops from the hearth, and 5) the ingot.His study revealed that evaporation took place mainly in the hearth pool.Many studies have utilized mathematical models to aid in understanding the evaporation process in EBMby identifying the key factors and predicting the evaporation rate. Most evaporation models are developedon the basis of Langmuir equation [49], which is given below in Eq. 2.3:m˙ = Pv√M2piRgT(2.3)where m˙ [kg/(m2 ⋅ s)] is the mass flux of the evaporated element, Pv [Pa] is the vapor pressure, M [kg/mol]is the molar mass, Rg [J/(K ⋅mol)] is the gas constant and T [K] is the temperature.Nakamura et al. [50] developed a two dimensional unsteady state heat and mass transfer model andconducted complimentary experimental work to investigate the effect of EB oscillation rate on aluminumevaporation in a Ti64 molten pool produced by an EBBF with a net power of 30 kW. Four rates (0, 0.1,1 and 10 Hz) were tested in his study and the results showed that evaporation can be greatly suppressedif the beam oscillation rate is more than 1 Hz. This is because the overheating at the melt surface canbe substantially reduced at a high oscillation rate. Bellot et al. [40] also developed a numerical model toexamine the influence of casting rate and EB scanning frequency on the evaporation of aluminum. Theresults revealed that the influence of casting rate is weak but scanning frequency is relatively strong. Anincrease of the scanning frequency from 0.5 to 11 Hz results in a 10 pct decrease of aluminum loss in thecase of melting Ti64 under the experimental conditions used. Akhonin et al. [51] developed a model to19describe the kinetics of aluminum evaporation during the EBCHM process. The model was used to assessthe effects of the process parameters, melting rate and power input, on the evaporation rate. Increasing themelting rate or reducing the power input can lead to a decrease of aluminum loss by evaporation. Powell etal. [52] had a more comprehensive and detailed study on this topic through both modeling and experimentalapproaches. Three factors, beam power, scanning frequency and background (chamber) pressure, wereexamined. Increasing beam power and decreasing beam spot size enhanced the evaporation, as expected.Background pressure does not present a clear correlation as it imposes two competing effects, gas focusingof the beam and interference with evaporation transport. Increasing scanning frequency led to a smallerpattern-size due to the beam guidance system problem, which was found to increase the evaporation rate. Inaddition, the activity coefficient of aluminum was estimated at 0.063 for the melting of Ti64 to calculate themass transfer rate of aluminum during evaporation based on the Langmuir equation.2.4 Titanium-Aluminum Binary Phase DiagramTitanium-Aluminum (Ti-Al) is a complex system that can form a number of phases depending on tem-perature and composition. The Ti-Al phase diagram has been investigated in many studies mainly for thepurpose of developing Ti-Al alloys for high temperature applications. Examples include Ref [53], [54] and[55]. The Ti-Al phase diagram can also provide information necessary for investigating the melting behav-ior of Al-rich regions in the EBM process. As previously described, the “drop-ins” are rich in aluminum(typically contained approximately 70% Al and 30% Ti by weight) and can vary in composition dependingon the Al content in melt as well as the operating conditions during melt processing. According to the phasediagram, in the range of compositions expected, the liquidus and solidus temperatures are expected to bebelow typical liquid metal processing temperature. Thus solution of the condensate would be expected tooccur by a process of melting.Additionally, it is worth noting that there are significant differences in the published phase diagrams. Forinstance, Fig. 2.4a and 2.4b are the phase diagrams provided in Ref [53] and [54], respectively, which aredifferent to some degree. In particular, the liquidus in Fig. b increases when the alloy is dilute in aluminum,whereas in Fig. 2.4a does not.20(βTi)(αTi+βTi)(αTi)+(βTi)(βTi)+L(αTi)+LTiAl+L(αTi)B2(αTi)(αTi)+Ti3AlTi3AlTi3Al+TiAl(αTi)+TiAlTiAlTi1-xAl1+xTi5Al11TiAl3(HT)L+TiAl3(HT)L+TiAl3(LT)Ti3Al5TiAl+TiAl2TiAl3(LT)+(Al) (Al)TiAl3(LT)TiAl2+TiAl3(LT)L1445◦1460◦1416◦1387◦735◦665◦1215◦TiAl2995◦850◦1170◦810◦Atomic Percent Aluminum20 40 60 80 AlTiTemperature◦ C80010001200140016001800(a)Lβαγα2Ti AlTiAl3TiAl2Ti3Al 7xAlTemperature (oC)0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.050060070080090010001100120013001400150016001700180019002000(b)Figure 2.4: Phase diagrams given in Ref [53] (a) and [54] (b)21Chapter 3Scope and ObjectivesType II defects (Al-rich regions) can degrade titanium fatigue performance and are not acceptable in rotorgrade applications [5]. One of the main formation mechanisms is the “drop-in” event, mentioned briefly inChapter 1 and schematically explained in Fig. 3.1. When melting Al-bearing alloys during EBCHR (for ex-ample Ti64), aluminum evaporates relatively quickly due to a combination of the comparatively high vaporpressure of aluminum, the high vacuum and high melt surface temperature. Note: titanium also evaporatesbut at a much slower rate. The vapor (mostly aluminum) then condenses on to the roof, hearth and moldwalls in the furnace chamber, resulting in the formation of a layer of Al (rich)-Ti material. Occasionally,the Al(rich)-Ti material may fall into the refining hearth or casting mold, which could potentially lead tothe formation of Al-rich regions in the final ingot if the Al(rich)-Ti material does not melt and sufficientlyhomogenize with the bulk titanium liquid before solidification occurs.Cold hearthElectron beamTi-6Al-4VmeltCasting moldVacuumchamberIngotCondensation of Al-Ti~3Al+1TivaporAl-Ti solidFigure 3.1: Schematic for explaining the “drop-in” eventThe kinetics of melting of the Al (rich)-Ti material in liquid titanium have not been investigated inany detail and the understanding of potential rate limiting phenomena is very limited. In order to provide22insights into effective removal of the Al-rich “drop-ins” during melt processing, it is necessary to carry outan investigation to improve our understanding at a fundamental level.3.1 Scope of the Research ProgramThe main goal of this research program is to investigate the melting of small amounts of cold solid Al-Tiintroduced into liquid titanium. A laboratory-scale EBBF will be utilized for the experimental componentof the work, which will be supported by the development of a numerical CFD model. The factors to bestudied include the various flow drivers including thermal and compositional buoyancy and thermal andcompositional Marangoni forces, which are expected to drive fluid-flow and influence heat and mass-transferin proximity to the solid/liquid interface.Many challenges exist when conducting experiments in titanium melt processing, which include highmelting temperature (>2000 ◦C), enclosed operating environment (vacuum or protective gas) and high re-activity of molten titanium. As a prelude to investigating solid Al-Ti melting in liquid titanium, work wasstarted with a low-temperature and transparent analogue system based on both frozen water/liquid water andfrozen ethanol/liquid water. This system facilitated observation of the basic phenomena of melting a solid inliquid, with similar flow drivers – i.e. thermal and compositional buoyancy and thermal and compositionalMarangoni forces. These experiments help refine the procedure for the following experiments with titaniumsamples and provided data suitable for developing and validating a computational modeling framework.Experiments using CP-Ti for both the solid and the liquid bath followed the experimental work con-ducted with the low-temperature analogue system. CP-Ti was used to simplify the experiments and analysisby removing compositional related phenomena from the system. This work was conducted in a laboratory-scale EBBF at UBC. Once completed, focus was then shifted to using Al-Ti compound for the solid, whichwas melted in both CP-Ti and Ti64 liquid. The Al-Ti/Ti system included both thermally and composition-ally related phenomena, which covers all of the phenomena believed to be occurring in an actual industrial“drop-in” event scenario. This system is more complicated but allows for important insights into the prac-tical problem to be obtained. Numerical models were also developed for the high-temperature systemsstudied which were validated against the experimental observations and measurements to allow for a morecomprehensive and detailed understanding of the melting process. These models were also used to examine23the main factors influencing the melting of the solid.3.2 Objectives of the Research ProgramThe objectives can be summarized as follows:1. Develop a modeling framework based on the low-temperature ethanol/water analogue system to un-derstand the basic phenomena of melting a solid in liquid which is to include heat, mass and momen-tum transport in both a liquid and solid phase.2. Design and construct an experimental setup based on the EBBF at UBC to allow experiments to beconducted on melting a solid in liquid titanium. The setup is to allow for the collection of bothqualitative and quantitative data for analysis, such as solid/liquid interface profile as a function oftime, melting rate and temperature evolution within the solid. The resulting experimental setup is tobe used to investigate the melting of CP-Ti solid in liquid CP-Ti and Ti-Al solid in liquid CP-Ti andin liquid Ti64.3. Develop comprehensive numerical models for the high-temperature systems identified in the 2nd ob-jective that incorporate the important phenomena related to heat, mass and momentum transport insolid and liquid phases that are capable of approximating the experimental observations and to aid ininterpreting key rate controlling phenomena.4. Using the numerical models in conjunction with the experimental data, derive effective heat and/ormass transfer coefficients suitable for use in particle melting/dissolution models in conjunction withdata derived from full-scale hearth models.24Chapter 4Study of Melting Mechanism of a SolidMaterial in a LiquidAs a prelude to investigating Al-Ti solid melting in the liquid Ti system (which is an experimentally chal-lenging system), work has been undertaken on a low-temperature analogue system (ethanol/water) to enablethe observation of the basic phenomena when melting a solid in liquid and to provide data suitable for devel-opment and validation of a modeling framework. The ethanol/water system was selected for several reasons:firstly, the solute (ethanol) has a lower melting point than the solvent (water) allowing solid ethanol to beintroduced into a higher temperature liquid, which is analogous to solid Al-Ti material added into liquid Ti;secondly, ethanol has a lower density, which is analogous to Al-Ti in liquid Ti; thirdly, ethanol has a lowersurface tension, again analogous to Al-Ti material in liquid Ti; fourthly, both are in the appropriate physicalstates at reasonable temperatures; and finally both are transparent, making direct observation possible.In this study, a series of experiments were conducted to examine the effects of four flow drivers on massand heat transport occurring at the interface of a solid in a liquid while in the process of melting underconditions of natural convection: 1) thermal buoyancy; 2) compositional buoyancy; 3) thermal Marangoni;and 4) compositional Marangoni. The empirical data generated from the system has been used to supportdevelopment and validation of a CFD-based numerical model of the system. The model in turn, has beenused to gain further insights into the role of the various transport processes involved in melting.4.1 Experimental Methods4.1.1 Experimental SetupFig. 4.1a shows a schematic diagram of the experimental apparatus. It consists of a transparent cylindri-cal glass beaker 50 mm in radius containing water (the solvent) into which a solid sample (the solute) is25introduced and held at a fixed position. The experiments were conducted in ambient air at 20 ◦C. Tempera-tures at two locations were measured by Type-T thermocouples: one, within the solute at the center-line atmid-height (TC1); and a second, near the wall of the cell, 5 mm from the top surface (TC2) - see Fig. 4.1a.Glass beakerSolvent (liquid)Modeling domainSolute (solid)Wooden holderTC1TC2CameraUnit: mm(a) Physical model setup0.5mm(b) Numerical model domainand meshFigure 4.1: Schematic diagram of the physical model setup and numerically modeled domainTo allow for a broad range of conditions to be explored both frozen ethanol and frozen water (ice) wereused as the solute, facilitating measurements to be conducted with and without the compositional drivers offlow present, respectively. In addition, the solid solute was introduced into the solvent in two configurations,as shown in Fig. 4.2. In the fully immersed configuration, Fig. 4.2a, Marangoni forces are eliminatedfrom the solid/liquid interface, leaving predominately buoyancy forces, whereas in the partially immersedcondition, Fig. 4.2b, Marangoni forces are present in proximity to the solid/liquid interface at the junctionof the three phases, solid, liquid and air. Four sets of experiments were conducted, as listed in Table 4.1.Note: reference is also made in Table 4.1 to the state of the flow driver as being either on (present) or off(absent) in proximity to the solid/liquid interface in the experiment.Holding the samples at a pre-determined height allowed for reproducible conditions to be maintainedduring the experiments. The configurations used allows for buoyancy driven flows to fully develop resultingin natural convection conditions. Marangoni driven flows on the other hand are only present in proximity tothe solid interface for the period of time solid is present at the free surface of the liquid - e.g. in the case ofthe partially immersed ice experiments, rapid melting at the free surface of the liquid limited the time solid26BuoyancySolventSoluteBeakerLiquid surface(a) Solid solute fully immersedMarangoniBuoyancySolventSolute BeakerLiquid surface(b) Solid solute partially immersedFigure 4.2: Schematic diagram showing the two experimental configurations and the flow drivers presentTable 4.1: Experimental table and examining factorsConfiguration Solute SolventMarangoni BuoyancyThermal Compositional Thermal CompositionalFullyimmersedIce Water Off Off On OffFullyimmersedFrozenethanolWater Off Off On OnPartiallyimmersedIce Water On Off On OffPartiallyimmersedFrozenethanolWater On On On Onwas present at the free surface. Conditions in which a solid of lower density was free to float and remainat the surface would allow for Marangoni driven flow to remain present at the solid liquid interface for theduration of melting.Since both solvent and solute in the study are transparent, dyes were used to differentiate them. Anorganic-soluble blue dye (insoluble in water) was used to distinguish ethanol from water, whereas a water-soluble yellow dye was used in the solute when water was both the solute and solvent. The use of dyesallowed some aspects of the basic fluid flow pattern to be ascertained and also the solid/liquid interface tobe visually emphasized.The experimental methodology was as follows: the cylindrical beaker was filled to a height of 60 mmwith distilled water (solvent) at 45∼50 ◦C. The solute sample with an embedded thermocouple and support27(made from wood) was then prepared by solidifying in liquid nitrogen. Once solidified, the frozen solutesample was removed and inserted into the solvent. In the case of the ethanol experiments, the solute wasimmediately immersed whereas in the case of ice, the sample was allowed to warm until a temperature ofapproximately -10 ◦C was recorded on the solute thermocouple. Temperature and video data were thenrecorded using a data acquisition system.4.2 Numerical ModelTo support analysis of the experimental data and ultimately to develop a framework for modeling the melt-ing of solids in titanium refining, a CFD model describing the experiments was developed. In order tocomprehensively describe the melting process, the numerical model necessarily incorporates the significantphenomena affecting heat, mass and momentum transport including compositional and thermal buoyancy,compositional and thermal Marangoni forces, Darcy damping flow and melting (phase change). A MULTI-COMPONENT MODEL was employed as two components (solute and solvent) are involved. The modelingwas performed using a commercial CFD package - ANSYS CFX V12.1.4.2.1 Computational DomainGiven that the geometry of the physical model is circumferentially symmetric, a pie-shaped domain repre-senting a 5◦ slice has been adopted for the computational domain. Note: CFX does not permit the use of a2-D axisymmetric model, hence the need for a 3-D 5◦ slice. The domain is shown in Fig. 4.1b and includesthe water (solvent), shaded light grey, and solid (solute), shaded blue, as presented in Fig. 4.1a. The soluteis differentiated from the solvent by virtue of a different set of initial conditions and material properties. Inaddition, the free surface of the liquid (solvent) and the solid (solute) are assumed to be flat.To generate the mesh, a vertical face of the domain was first discretized into a 0.5×0.5 mm grid, whichwas then rotated 5◦ about the centerline of symmetry. A single layer of elements was used in the circumfer-ential direction. At the domain centerline, the resulting layer of triangular prism elements is of poor qualityand was removed yielding a mesh comprised of 11,662 hexahedra elements containing 23,760 nodes. Asensitivity analysis to mesh size in the axial and radial directions was conducted by running cases with gridsof 0.25×0.25 mm, 1.0×1.0 mm and 1.5×1.5 mm and comparing the results to the base-case 0.5×0.5 mm.28Note: the melting time was used for the assessment, as it is of primary interest in this research program.For the example problem run, the predicted melting times where 79, 82, 97 and 109 s, in order of increasinggrid size. Model execution times were approximately 250, 25, 9 and 4 hrs, also in order of increasing gridsize. The base case grid of 0.5×0.5 mm was found to be a reasonable compromise between accuracy andexecution time.4.2.2 Model EquationsThe governing equations solved are the standard ones describing continuity, momentum and energy conser-vation for an incompressible fluid and are given by Eqs. 4.1 - 4.3, respectively.∂ρ∂ t+∇ ⋅ (ρU) = 0 (4.1)∂ (ρU)∂ t+∇ ⋅ (ρU ⊗U) = −∇P+∇ ⋅ τ +ρg+SM (4.2)∂ (ρCPT )∂ t+∇ ⋅ (ρUCPT ) = ∇ ⋅ (k∇T ) (4.3)where ρ [kg/m3] is the density, t [s] is the time, U [m/s] is the velocity of fluid, P [Pa] is the pressure, τ[N/m2] is viscous shear stress, g [m/s2] is the gravitational acceleration, Cp [J/(kg⋅K)] is the specific heat atconstant pressure, T [K] is the temperature and k [W/(m⋅K)] is the thermal conductivity.In the cases where the model is applied to the experiments involving the introduction of solid ethanolinto water - i.e. two components, an additional mass conservation equation is added in order to be able tosolve for the mass fraction of the solute (ethanol), which is given in Eq. 4.4.∂ρme∂ t= −∇ ⋅ρmeU +De∇2 ⋅ρme (4.4)where me is the mass fraction of ethanol, and De [m/s2] is the kinematic diffusivity of ethanol in water. Themass fraction of the solvent (water), mw, is obtained by applying the constraint me+mw = 1.CFX solves Eqs. 4.1 - 4.4 for the liquid and the solid as well as for any two-phase, solid-liquid, regionsthat exist at a given time within the computational domain. Two strategies are used to deal with the transition29from liquid to solid. The first entails adoption of a Darcy-based force (momentum source term) and thesecond entails increasing the effective viscosity of the liquid by four orders of magnitude. The Darcy-basedforce is included in the model via a momentum source term, SM, appearing on the right-hand-side of Eq.4.2. The expression for the momentum source term is given in Eq. 4.5.SM,Darcy = −µKpermU (4.5)where Kperm [m2] is the permeability. Note: U in the equation is the superficial velocity [56].The Darcyterm acts to dampen the flow in the semi-solid and fully solid regions of the domain proportionally withthe viscosity and inversely proportional with the permeability. The permeability is assumed to vary withfraction solid according to the Carman−Kozeny expression given in Eq. 4.6.Kperm =(1− fs)3DSV f 2s(4.6)where fs is the solid fraction, and DSV [m−2] is a coefficient related to the specific area-to-volume ratioof the solid, which is assumed to be a constant and is set to a value of 1.67× 1010 m−2 [57]. In order tokeep SM bounded, a limiting value for fs of 0.99 is imposed on the calculation of Eq. 4.6. As both thewater and ethanol used in the experiments are pure, both would be expected to possess a unique meltingpoint. However, an arbitrary small value of 1 ◦C has been applied for the temperature range over whichmelting/solidification occurs for both. The value chosen seeks to strike a balance between accuracy andconvergence (values below 1 ◦C resulted in convergence issues). The fraction solid has been assumed tovary linearly over 1 ◦C and the various material properties affected by the transition from solid to liquidhave also been ramped over the same range - see section 4.2.4 for the details.4.2.3 Boundary ConditionsIn terms of the continuity equation, there is no mass input or output through the external boundaries ofthe domain and from a heat transfer standpoint the external boundaries of the domain are assumed to beadiabatic. A shear stress σM [N/m2] based on Eq. 4.7 is assumed to prevail at the top liquid-air interface(free surface) and a no-slip condition is assumed at the water-glass beaker interface (the shear conditionallows Marangoni forces to be applied). The first term on the right-hand-side of Eq. 4.7 represents the30thermal-based surface tension force and the second term the compositional-based surface tension force.σM =∂γ∂T∂T∂n+∂γ∂C∂C∂n(4.7)where γ [N/m] is the surface tension, n [m] is a directional vector and C is the composition (mass fraction ofethanol). ∂γ∂T is generally referred to as the thermal surface tension coefficient and∂γ∂C the compositional sur-face tension coefficient. The gradient in temperature, ∂T∂n , and gradient in concentration,∂C∂n , are accessiblefrom within CFX at a given boundary location and time and are used within a user programmable subroutineto evaluate Eq. 4.7 on an element-by-element basis for those elements with faces located at the free surfaceof the liquid.4.2.4 Material PropertiesIn the experiments, compositional variation is expected to occur within the boundary layer between thefrozen ethanol and liquid and within the bulk liquid in those cases involving frozen ethanol as the solute.The compositional dependence of density is based on the expression given in Eq. 4.8 [58].1ρm =meρe +mwρw (4.8)where ρm [kg/m3], ρe [kg/m3] and ρw [kg/m3] are the densities of the mixture, pure ethanol and pure wa-ter, respectively. For the other properties k [W/(m⋅K)] and CP [J/(kg⋅K)], an ideal mixture is assumed asexpressed in Eq. 4.9 [58].Pm = mePe+mwPw (4.9)where Pe and Pw are the relevant properties of pure ethanol and pure water, respectively. Note: Eq. 4.8is equivalent to the ideal mixture rule with respect to specific volume (1/density), which is consistent withEq. 4.9. Temperature dependencies in the density, specific heat, and thermal conductivity of the purequantities are accounted for in the model, where available in the literature, and are presented graphically inFig. 4.3. A dashed vertical line is added to each figure to delineate the separation between the liquid and solidphases. Note: in the model the transition of material properties between solid and liquid (represented by thedashed vertical line) occurs in 1 ◦C as previously described in last section. The thermal and compositional31coefficients of surface tension as a function of composition are plotted in Fig. 4.3g and 4.3h, respectively.As they exhibit only small temperature dependence they are assumed to be independent of temperature. Thereferences for the various material properties described above and the units are summarized in Table 4.2.Table 4.2: Physical properties applied in the modelProperty SymbolValueUnitWater EthanolDensity ρ f (T ) - Fig. 4.3a [59] f (T ) - Fig. 4.3b [60] [kg/m3]Specific heat CP f (T ) - Fig. 4.3c [23] f (T ) - Fig. 4.3d [23] [J/(kg⋅K)]Viscosity µ 10 (solid) / 3e-4 (liquid) [23] [Pa⋅s]Thermal conductivity k f (T ) - Fig. 4.3e [59] f (T ) - Fig. 4.3f [60] [W/(m⋅K)]Latent heat of fusion Lm 334,000 [61] 109,000 [61] [J/kg]Thermal surface tensioncoefficient∂γ∂T f (T ) - Fig. 4.3g [62] [N/(m⋅K)]Compositional surfacetension coefficient∂γ∂C f (C) - Fig. 4.3h [62] [N/m]The latent heat of melting is accounted for in the heat balance by a modified specific heat methodology[63]. The expression used for this is shown in Eq. 4.10.CP,e =CP−Lm∂ fs∂T(4.10)where CP,e [J/(kg⋅K)] is the effective specific heat, and Lm [J/kg] is the specific heat of melting or solidifi-cation. This methodology also requires a finite temperature range over which melting occurs in order thatthe second term on the RHS of Eq. 4.10 remains finite. In the context of evaluation of Eq. 4.10, d fsdT is as-sumed constant over the applied melting range. In the case of the ice/water system, a 5 ◦C temperature rangewas assumed over which d fsdT has been evaluated, whereas for frozen ethanol/water system, the temperaturerange was doubled to 10 ◦C temperature range, resulting in values of 1/5 and 1/10 for d fsdT for ice meltingand ethanol melting, respectively. A detailed sensitivity analysis was conducted on the melting temperaturerange. For the ice/water case, temperature ranges of 3, 4, 6 and 7 ◦C were also run in the model in additionto the base case of 5 ◦C. A temperature range of 3 ◦C yielded a substantial reduction in melt time as themodel failed to accurately capture the release of latent heat. Ranges of 4, 6 and 7 ◦C resulted in similar melttimes to the base case value of 5 ◦C, hence the base case value of 5 ◦C was adopted for ice/water. A similarexercise was conducted with ethanol-ethanol, with the result that a 10 ◦C range was required.32200 150 100 50 0 50Temperature [ ◦ C]9209409609801000Density [kg/m3](a) Density of water200 150 100 50 0 50Temperature [ ◦ C]760780800820840860880900Density [kg/m3](b) Density of ethanol200 150 100 50 0 50Temperature [ ◦ C]1500200025003000350040004500Specific Heat [J/(kg·K)](c) Specific heat of water200 150 100 50 0 50Temperature [ ◦ C]2500300035004000450050005500Specific Heat [J/(kg·K)](d) Specific heat of ethanol200 150 100 50 0 50Temperature [ ◦ C]0.00.51.01.52.02.53.0Thermal Conductivity [W/(m·K)](e) Thermal conductivity of water200 150 100 50 0 50Temperature [ ◦ C]0.00.20.40.60.81.01.21.41.6Thermal Conductivity [W/(m·K)](f) Thermal conductivity of ethanol0.0 0.2 0.4 0.6 0.8 1.0Mass Fraction of Ethanol0.180.160.140.120.100.080.06γ T[N/(mm·K)](g) Thermal surface tension coefficient0.0 0.2 0.4 0.6 0.8 1.0Mass Fraction of Ethanol0.50.40.30.20.10.0γ C[N/m](h) Compositional surface tension coefficientFigure 4.3: Physical properties applied in the model334.2.5 Initial ConditionsFor the ice/water and frozen ethanol/water models the initial conditions for the solute component of the do-main were based on the temperatures measured by the thermocouple located within the solute in each exper-iment. For the ice/water models, isothermal temperatures of -10 and -12 ◦C were assumed for the fully andpartially immersed cases, respectively. For the frozen ethanol/water cases, isothermal temperatures of -185and -191 ◦C were assumed for the fully and partially immersed cases, respectively. The initial conditions forthe solvent were also based on the measurements recorded by the solvent thermocouple. Isothermal temper-atures of 48, 45, 49 and 49 ◦C were assumed for the ice/water fully immersed, ice/water partially immersed,frozen ethanol/water fully immersed and frozen ethanol/water partially immersed models, respectively. Thevelocity fields were set to zero and the pressure was set to atmospheric pressure.4.2.6 Computational ProcedureThe model was run in parallel with 2, 2.33 GHz Intel Xeon quad-core CPUs with 4.00 GB RAM. A high-resolution advection scheme available in CFX is used to solve the governing equations. The time step sizewas set at 0.01 s. A sensitivity analysis to time step was conducted and time steps of 0.005 and 0.02 werealso run for an example problem in addition to the base-case time step of 0.01 s. The melt times predictedwere 82, 82 and 36 s in order of increasing time step. Thus the base-case time step of 0.01 s was deemed tobe satisfactory. The convergence target was set to 0.0001 for the RMS residual with maximum 10 iterationsper time step. The RMS residual for convergence was chosen based on what is recommended for engineeringapplications within the Ansys CFX literature [58]. The 0.0001 RMS residual target was met within the 10iteration maximum in all of the results presented. Decreasing the target generally required a larger numberof iterations to achieve convergence, resulting in an increase in computational time without a substantialchange in the results. The computational time is approximately 25 hours for both the ethanol/water andice/water cases.344.3 Results and Discussion4.3.1 General ComparisonA general high-level summary of the results for the four experiments is presented in Fig. 4.4. The topseries of schematic images (Fig. 4.4a), illustrate the various flow drivers present in each of the four exper-iments. BT , BC, MT and MC stand for Buoyancy Thermal, Buoyancy Compositional, Marangoni Thermaland Marangoni Compositional, respectively. In the cases where competing forces (opposing force direc-tions) are present, the force(s) (denoted as solid formatted lines in Fig. 4.4a) is(are) the dominant directionas determined based on the results of the experiments and modeling. The series of images appearing in thelower half of the figure (Fig. 4.4b) represent a comparison between the experimental and modeling results at10 s elapsed time in each of the four experiments. The experimental results appear on the left-hand-side ofeach image and the modeling results on the right. The model results are presented as a temperature contourwith flow vectors superimposed (magnitude proportional to length). The experimental images have beenannotated with blue and red lines to help delineate the approximate location of the ice/water interface andthe frozen ethanol/water interface, respectively. The results in terms of the interface profiles and the amountof solid melted show good agreement for the four cases at 10 s elapsed time indicating that the model is ableto capture the basic physics of the problem.Taking a closer look at the fully immersed, ice/water experiment, which represents the simplest casefrom the standpoint of the number of flow drivers present, the results for 0, 10, 20, 30, 45 and 65 s arepresented in Fig. 4.5. It is apparent that the model slightly over predicts the melt rate, which becomesincreasingly apparent at longer times - see the 45 and 65 s images. This discrepancy is most likely relatedto the fluid flow boundary layer that develops adjacent to the interface. The geometries are slightly different(related to the supporting stick that was present in the experiment but not the model), which may affect theboundary layer development in the early stage. As can be seen in the model predictions, by 10 s a layer ofcolder fluid forms adjacent to the ice/water interface that results in the development of a downward buoyancyforce BT , due to its relatively higher density, and a downward flow. This downward flow, which occurs overboth the top and side surfaces, continues throughout the entire melting process resulting in the accumulationof cooler denser fluid at the bottom of beaker, which is also observed in the experiments as evidenced by theaccumulation of yellow die at the bottom of the beaker. Note the enhanced melting observed at the top of35icestickambientwaterBT icestickambientwaterBTMTstickambientwaterBCBTsolid ethanolstickambientwaterMCBTsolid ethanolMTBCI: B(T) only II: B(T),M(T) III: B(T),B(C) IV: B(T),M(T),B(C),M(C) (a)(b)Figure 4.4: Schematic diagrams showing the drivers for fluid flow for the four experimental conditionsexamined in the study (a) and a series of images comparing the experimental results and model predictionsat 10 s elapsed time (b)the sample relative to the bottom, which is also observed in the model predictions. This is clearly seen bycomparing the distances between the top of the sample and the free surface and between the bottom of thesample and the bottom of the container in the 0 and 65 s images - see the white arrows. Another feature ofthe melting relates to the profile of the vertical interface that develops with increasing time. At times greaterthan 20 s, the model predicts the development of a narrow-end-down profile, whereas the experiment showsthe opposite to be true. Given that relatively warm liquid is drawn in at the top of the sample as a resultof the downward flow of the cooler denser liquid along the surfaces of the sample one would expect anarrow-end-up profile to develop as is observed in the experiments.A more detailed look at the results for the partially immersed ice/water experiment, which includes thepotential for thermally induced Marangoni driven flow, are presented in Fig. 4.6. In this case the ice producesa temperature gradient at the free surface of the liquid, creating a thermal Marangoni force that drives theliquid toward the ice from an area of low surface tension to and area of high surface tension. The enhanced360.05 [m/s]0 s 10 s20 s 30 s45 s 65 sFigure 4.5: Comparison between experimental and modeling results for the fully immersed ice/water caseflow at the free surface locally increases heat transfer resulting in a noticeably higher rate of melting of theice adjacent to the free surface, which is both predicted by the model and observed experimentally. Only the0 and 10 s images have been presented for comparison as the results for the balance of the experiment showa similar trend to that observed in the fully immersed experiments owing to complete melting of the sampleat the top surface.The results for the partially immersed frozen ethanol experiment are presented in Fig. 4.7 for 5, 10, 15,20 and 25 s elapsed time. Note that only the partially immersed experiment for ethanol is presented, as it370.05 [m/s]10 s 10 s Figure 4.6: Comparison between experimental and modeling results for the partially immersed ice/watercaserepresents the case with the largest number of flow drivers present. In Fig. 4.7 the model predictions arepresented in terms of a series of contours of the mass fraction of ethanol to better delineate the compositionaleffects. As in the case of the thermal contours presented for the ice/water experiments, velocity vectors havebeen superimposed on the mass fraction contours, with the magnitude of the velocity proportional to thelength of the vector.As can be seen, the addition of a compositional driver for flow has substantially altered the flow patternand the evolution in the solid/liquid interface profile. The flow observed in the experiments and predicted bythe model is now upward along the vertical face of the solid liquid interface and outward from the interfacein proximity to the free surface. This results in the development of a big-end-up profile during melting. Incomparison to the ice/water model, the ethanol/water model predicts the correct evolution of the interfaceprofile with time and also more accurately predicts the overall melt rate - see 25 s image.The compositionally related phenomena giving rise to the difference observed between the ice/water andethanol/water experiments are as follows: firstly, melting leads to the introduction of lighter ethanol into theliquid with the result that the density of the liquid adjacent to the solid-liquid interface will tend to be reducedfrom a compositional standpoint. In contrast, as previously described, the thermally induced buoyancy forceassociated with the presence of cooler more dense liquid at the interface is directed downward. Under theconditions examined in this study, the compositional buoyancy term dominates over the thermal buoyancy,thus there is a net reduction in density at the interface and a force driving the fluid flow upwards along380.12 [m/s]0 s 5 s10 s 15 s20 s 25 sFigure 4.7: Comparison between experimental and modeling results for the partially immersed ethanol/watercasethe vertical interface of the solid. This results in the accumulation of liquid rich in ethanol at the top ofthe beaker, which is observed experimentally and predicted with the model. Secondly, the locally increasedethanol concentration in proximity to the free surface reduces the surface tension resulting in a shear force atthe free surface directed away from the interface. This is opposite in direction to the shear force associatedwith the presence of a thermal gradient at the interface. Based on the experiments and predictions, thecompositionally driven shear force dominates over the thermally induced shear force in this particular case.As a result, the preferential erosion at the top surface does not occur in the ethanol/water case. Note: a39dotted line has been used to estimate the profile of the solid adjacent to the top surface, as it is difficult todelineate this shape due to the large concentration of die present in the area. The reason why the modelis more accurate when applied to the frozen ethanol/water problem, both in terms of the evolution in theprofile of the vertical face and in terms of the melt rate is unclear.4.3.2 Quantitative Comparison of Melt Rate and Temperature EvolutionFig. 4.8 shows the variation in the amount of solute melted with time presented as the ratio of volume of thesolid solute melted to the initial volume of solid solute. The data presented includes the variation measuredexperimentally and predicted by the model for both the ice/water (Fig. 4.8a) and frozen ethanol/water (Fig.4.8b) experiments in the two configurations. In addition, to examine the influence of thermal buoyancy as aflow driver, the case for the ice/water system (fully immersed case) has been rerun with buoyancy switchedoff with the results also shown in Fig. 4.8a.As can be seen there is generally good agreement between the model and the experimental data for thetwo systems, in the two configurations examined, with buoyancy activated. There is a gradual reduction inmelt rate with time associated with the reduction in solid/liquid interfacial surface area as would be expected.Comparing Fig. 4.8a and Fig. 4.8b, it can be seen that the predictions for the frozen ethanol/water modelare slightly more accurate than those for the ice/water model, which is consistent with the assessment madein the previous section. The maximum error in the ratio in the ice/water model is ∼15% and occurs in thefully immersed system at about ∼35 s, elapsed time. Over a broader range of time the error is consistently∼10%. In comparison, the maximum error in the ratio for the frozen ethanol/water experiments is ∼9% andoccurs at one point in the partially immersed experiment at ∼12 s. Over a broader range, the error is in the5% range or less.As observed in the Fig. 4.8a, the melt rate of the immersed solid solute with buoyancy off becomesmuch slower compared to the case with buoyancy activated. This indicates that the convective heat transferinduced by thermal buoyancy plays an important role in determining heat transfer and melt rate in thefully immersed ice/water case. Comparing the fully immersed and partially immersed experiments in thetwo systems reveals only a small effect of Maragoni flow in the ice/water system and a moderate effectin the ethanol/water system. Recall that in the ice/water system, aggressive melting of the free surfaceresults in Marangoni forces present for only a short period of time in the experiment, hence the relatively40small difference in the ice/water system between the two experimental configurations. The outward inducedMarangoni flow in the ethanol/water system results in a decrease in the melt rate at the free surface. Hence,Marangoni flow is present at the solid/liquid interface for the entire duration of the experiment, leading toa large difference between the two experimental configurations. Overall, Marangoni induced flow has asmaller effect than buoyancy induced flow as it acts over a much smaller area of the solid surface.0 10 20 30 40 50 60 70 80Time (s)0.00.20.40.60.81.0Melted volume/Initial volumeBuoyance switched offModel (fully immersed)Model (partially immersed)Experiment (fully immersed)Experiment (partially immersed)(a) Ice/water0 5 10 15 20 25 30Time (s)0.00.20.40.60.81.0Melted volume/Initial volumeModel (fully immersed)Model (partially immersed)Experiment (fully immersed)Experiment (partially immersed)(b) Frozen ethanol/waterFigure 4.8: Comparison on melting kinetics between experimental and modeling resultsThe evolution of temperature with time measured by the two thermocouples is compared with the nu-merical model predictions in Fig. 4.9. Figs. 4.9a and 4.9b present the results of the ice/water cases for thefully immersed and partially immersed conditions, respectively, and 4.9c and 4.9d present the results of thefrozen ethanol/water cases for the fully immersed and partially immersed conditions, respectively. Focus-ing primarily on TC1, which is the thermocouple located in the solute, it can be seen that the temperatureevolution predicted by the model generally shows good agreement with the experimental measurements inall cases except for the initial stage of the fully immersed ethanol/water case.Starting first with the two ice/water cases, the curves show a gradual increase in temperature over thefirst ∼40 s as the solute heats up and some melting occurs. From ∼40 to 70 s the temperature is invariant withtime associated with the sample reaching 0 ◦C and the release of the latent heat of melting. The temperature410 20 40 60 80 100Time [s]1001020304050Temperature [◦ C]Model TC1Model TC2Experiment TC1Experiment TC2(a) Fully immersed ice/water0 20 40 60 80 100Time [s]1001020304050Temperature [◦ C]Model TC1Model TC2Experiment TC1Experiment TC2(b) Partially immersed ice/water0 5 10 15 20 25 30 35 40Time [s]20015010050050Temperature [◦ C]Model TC1Model TC2Experiment TC1Experiment TC2(c) Fully immersed ethanol/water0 5 10 15 20 25 30 35 40Time [s]20015010050050Temperature [◦ C]Model TC1Model TC2Experiment TC1Experiment TC2(d) Partially immersed ethanol/waterFigure 4.9: Temperature comparison between experimental and modeling results at the two thermocouplelocationsthen rapidly climbs to the solvent (bath) temperature at ∼70 s as the ice surrounding the thermocouple isfully melted and the thermocouple is exposed to the solvent. As can be seen, there is excellent quantitativeagreement between the model predictions and the data collected from TC1 and TC2. The only small error isthat the model slightly under predicts the time at which the temperature rapidly rises in Fig. 4.9a - i.e. whenthe thermocouple is exposed to the solvent in the fully immersed experiment.The results for the frozen ethanol/water experiments are a little more challenging to interpret due toa difference in the initial temperature response measured in the two experiments. In particular a gradualincrease in solute temperature was observed in the early stages of the fully immersed experiment, which42was not observed in the partially immersed experiment. The model predicts an initial period of temperatureinvariance for the first ∼10 s, which is followed by a gradual increase in temperature prior to reachingthe point where there is a rapid increase in temperature associated with the thermocouple contacting thesolvent. The data from the partially immersed experiment is consistent with this whereas the data fromthe fully immersed experiment is not. One possible explanation is that the wooden support, which was incontact with the solvent in the fully immersed experiment, acted to provide a path for heat conduction tothe solute. This path for conduction is small and ignored in the model. The other notable difference relativeto the ice/water experiment is that the rate of temperature increase following exposure of the thermocoupleto the solvent is slower in the frozen ethanol/water experiments. Moreover, this difference is not reflectedin the model predictions, which predict a more rapid increase consistent with what was observed in theice/water experiments. The observation of a less rapid increase in temperature may support the formation ofa two-phase (liquid-solid) zone, which is currently not supported in the model formulation.4.3.3 Effective Heat and Mass Transfer CoefficientsIn many practical applications, it is more straightforward to employ an effective interfacial heat transfercoefficient or mass transfer coefficient in order to calculate the melting and/or dissolution kinetics, thusavoiding the need to solve the general fluid flow problem. In both cases, an effective coefficient can becalculated using the results from the full CFD analysis and compared with literature-based correlations.This comparison supports validation of the numerically based derivation.Starting first with heat transfer, the effective interfacial heat transfer coefficient (he f f ) is defined as:he f f = q/(Ts/l −Tbulk) (4.11)where q [W/m2] is the heat flux transported through the boundary layer interface, Ts/l [K] is the temperatureat the interface, which is the melting temperature of the solid solute, and Tbulk [K] is the temperature of thebulk fluid or solvent in the case of this study.In the present work, the heat flux q may be calculated by the expression:q =QA∆t (4.12)43where Q [J] is the total amount of heat transported through the solid/liquid interface, A [m2] is the interfacialsurface area and ∆t [s] is the time interval over which Q is accumulated. In theory, Q can be calculated by:Q =∰ ρCP∆T dV +ρLm∆Vs (4.13)in which, ∆T [K] is the temperature increase at a given position within the solid during the time interval ∆t,and ∆Vs [m3] is the volume change in solid solute occurring in ∆t. The first term at the right hand side inthe equation above represents the enthalpy increase in the solid solute, whereas the second term is the latentheat absorbed by the melt due to melting.To calculate the area, A, in Eq. 4.12 an assumption is made that the solid solute maintains a cylindricalgeometry during the whole melting process. Moreover, the cylinder retains the same height/radius ratiothroughout the melting process. Based on this assumption, the area A can be calculated by:A = 2pirh+2pir2 (4.14)where r [m] and h [m] are the radius and height of the equivalent cylinder, respectively. h equals to r hiri wherehi [m] and ri [m] are the initial height and radius of the solid, respectively. Since the cylinder maintains aconstant height/radius ratio. The equivalent radius r can be calculated from the volume of the remainingsolid solute Vs by:r = 3√Vsripihi(4.15)Fig. 4.10 presents the effective heat transfer coefficients calculated from the modeling results of thefully immersed cases for both ice/water and ethanol/water, in which Marangoni forces are excluded, usingthe methodology outlined above. The evolution in the heat transfer coefficients for the two experimentalconditions, computed from the literature-based correlations - Eq. 4.16, are also plotted in the correspondingfigures for comparison.Nu =C(GrPr)n(L/b)m (4.16)where Nu is the mean Nusselt number, L is the characteristic length and b is the plate spacing (here thespacing between the peripheral surface of the cylindrical solid and the beaker’s wall is used). C, m and n are44fitting constants applicable for the system on which the measurements have been made. In Eq. 4.16, Ref[64] recommends the values 0.28, 0.25 and 0.25 to be used for the fitting constants C, m and n, respectively.However, the use of these recommended values results in underestimation of the heat transfer coefficientsfor the ice/water case and overestimation for the ethanol/water case (error< 30% for both cases). To betterfit the numerically derived data, the fitting constant C has been modified to 0.37 and 0.22 for the ice/waterand ethanol/water case, respectively. The Grashof number in the correlation, which approximates the ratioof buoyancy (due to temperature gradient) to viscous force acting on the fluid, is expressed as:Gr =ρ2βT g(Ts/l −Tbulk)L3µ2(4.17)where βT [1/K] is the volumetric thermal expansion coefficient (defined as βT = − 1ρ∂ρ∂T ), and L [m] is thecharacteristic length which is the height of the immersed cylinder. The above expression for Gr is appli-cable to the ice/water cases in which only thermal buoyancy is present. However, for ethanol/water casesa buoyancy force is exerted not only by the temperature gradient but also by the composition gradient. Toaccount for the added effect of the gradient in composition, the expression of Gr has been modified to Eq.4.18[65], which includes the compositional and thermal components (note: the upward direction is selectedas positive). The compositional component (the first term in the square bracket) is about twice as large asthe thermal component (the second). Therefore, the compositional buoyancy is dominant.Gr =ρ2g[βC(Cs/l −Cbulk)−βT (Ts/l −Tbulk)]L3µ2(4.18)where βC [m3/kg] is the volumetric compositional expansion coefficient defined as βC = − 1ρ∂ρ∂C ∣T , Cs/l andCbulk are the composition of solute at the solid/liquid interface and in the bulk liquid, respectively. ThePrandtl number is given by the following expression:Pr =CPµk(4.19)Comparing the results in Fig. 4.10, it can be seen that both the correlation and numerically derivedheat transfer coefficients show a similar variation with time, or extent of melting, and dependence on theexperimental conditions. Focusing first on the time dependence, under the two experimental conditions45examined the correlation and numerically derived heat transfer coefficients both exhibit a relatively slow rateof increase in magnitude initially and then rapidly increase toward the end of melting. The dependence ontime is actually related to the evolution in the length of the vertical face of the cylinder and the developmentof the heat transfer boundary layer along the face. As melting proceeds - i.e. the length of the face decreases- the average thickness of the thermal boundary layer is reduced resulting in an increase in the heat transfercoefficient. In the correlation, this is captured by the inverse square root dependence on the characteristiclength, L−0.5, which results in the rapid increase in the heat transfer coefficient predicted as L approaches 0.Shifting focus to the dependence on experimental conditions, significantly higher heat transfer coeffi-cients are predicted in ice/water experiment compared with the ethanol/water experiment. For example, theaverage effective heat transfer coefficient over the melting time in the ice/water case is 1,090 W/(m2⋅K)compared with 663 W/(m2⋅K) in the ethanol/water case. This difference arises due to: 1) the higher verticalflow in the ice/water system, which occurs because of the absence of competing thermal and compositionalbuoyancy forces present in the ethanol/water experiment, and 2) the higher thermal conductivity of water.Finally, comparing the correlation and numerically derived heat transfer coefficients, it can be seen thatthe correlation derived coefficients reproduce the form of the numerically derived coefficients very well, inboth cases, when the modified fitting constant, C (0.37 and 0.22 for the ice/water and ethanol/water caserespectively), is used.0 10 20 30 40 50 60 70 80Time [s]05001000150020002500300035004000Effective HTC [W/(m2 K)] ModelCorrelation(a) Ice/water fully immersed case0 5 10 15 20 25 30Time [s]0200400600800100012001400Effective HTC [W/(m2 K)] ModelCorrelation(b) Frozen ethanol/water fully immersed caseFigure 4.10: Effective heat transfer coefficient during the melting processTurning to mass transfer, and using a similar approach, an effective mass transfer coefficient can be46calculated for the fully immersed ethanol/water case, which involved mass transport. Unfortunately, thecorrelations available in the literature to estimate the mass transfer coefficient are more limited and to ourknowledge there is no correlation that exactly describes our case (vertical cylinder in an enclosed space). Thecorrelation closest to experimental conditions examined is Sh ∼ Gr0.25Sc0.33 (applicable to axi-symmetricbodies under conditions of natural convection) [30]. Initial application revealed that some modification wasrequired to bring the results more in line with numerically based results, which resulted in a correlation ofthe form similar to the relationship used for heat transfer:Sh =CmGr0.25Sc0.33(L/b)0.25 (4.20)where Cm is a fitting constant. The results calculated using this expression with Cm=6.0 are shown in Fig.4.11. As can be seen, the variation in the empirically derived mass transfer coefficient with time, or extentof melting, reproduces the form of the numerically derived results. The average effective mass transfercoefficient over the total melting time is 4.56×10−4 m/s.0 5 10 15 20 25 30Time [s]0.00000.00050.00100.00150.0020Effective MTC [m/s]ModelCorrelationFigure 4.11: Effective mass transfer coefficient during the melting process (Ethanol/water fully immersedcase)4.4 Summary and ConclusionsExperiments utilizing ice/water and frozen ethanol/water have been conducted as an analogue physicalmodel to study the factors influencing the melting of solids in liquid metals. The results of the exper-47imental measurements reveal that thermal and compositional driven buoyancy flows and surface tension(Marangoni) flows, when present, can impact on solid melting behavior in a system where forced convec-tion is not significant. In the ice/water experiments, the buoyancy flows that develop are downward alongthe various faces of the solute tending to enhance melting at the top of the sample as the recirculating flowthat develops draws in warmer water (solvent) at the top. In addition, the thermal Marangoni-based flowwas observed to significantly enhance melting where the solute (ice) intersects the free surface of the sam-ple. In the case of the frozen ethanol/water experiments, the thermal buoyancy and compositional buoyancyforces act in opposite directions. The resultant force causes a net upward flow along the various faces ofthe sample. This in turn draws in warm water at the base of the sample accelerating melting at the base.The thermal and compositional Marangoni forces also oppose one another with the result being a net forceacting away from the solid interface at the free surface reducing melting in proximity to the free surface.The mathematical model developed to support the analysis of the experimental results in this workappears to be able to describe the various physical phenomena influencing heat and mass transfer underthe range of conditions examined in this study. It may be concluded therefore that the model provides areasonable framework for application to liquid metal systems that exhibit similar characteristics in termsof thermo-physical property differences between the solvent and solute. One potential shortcoming of themodel relates to its ability to handle the development of a two-phase boundary layer at the solid liquidinterface, which likely occurs in some of the metallurgical systems of interest.Finally, the numerical model-based results for the ice/water and ethanol/water experiments were used toestimate an effective heat transfer coefficient and similarly the results of the ethanol/water experiments aneffective mass transfer coefficient. The effective heat and mass transfer coefficients were then compared withvalues calculated from literature-based correlations and were found to behave similarly with time (extent ofmelting) and in terms of their dependence on the experimental conditions. In the case of the heat transfercoefficient estimation, the correlation derived values agree with the numerically derived ones when the fittingconstant C was modified relative to the values obtained from literature for the cases examined in this study.48Chapter 5An Examination of the Factors Influencingthe Melting of Solid Titanium in LiquidTitaniumAs part of a broader program to understand the melting of Al(rich)-Ti solids in liquid titanium, this chapterpresents the findings of a study in which solid CP-Ti rods were dipped into a CP-Ti melt in an EBBF. CP-Tifor both the solid and for the liquid was initially studied in order that compositionally related phenomenacould be excluded, prior to moving to a system in which compositional flow drivers are present. In additionto experimental work, a numerical model, which incorporates the significant transport phenomena, hasbeen developed and validated against the experimental observations and measurements to allow a morecomprehensive and detailed characterization of the melting process. The modeling methodology follows anapproach previously presented in last chapter on the ice/water and frozen ethanol/water systems [66].5.1 Experimental MethodsSolid CP-Ti samples (see Table 5.1 for the chemical composition) in the form of cylindrical rods wereimmersed into a molten titanium pool for various periods of time and removed to observe the extent towhich the solid was melted. The immersion depth and position in the molten pool were fixed to improveexperimental repeatability. In addition, a thermocouple was embedded in the immersed sample to monitorthe temperature during immersion and quantify the evolution of temperature with time.Table 5.1: Chemical composition of CP-Ti that is used as the experimental material (wt%)C Fe H N O Other Ti0.080 0.300 0.015 0.030 0.250 0.400 Balance495.1.1 Experimental ApparatusThe experiments were performed in a lab-scale EBBF shown schematically in Fig. 5.1. It primarily consistsof a furnace (vacuum) chamber containing a water-cooled copper mold, two, two-stage, vacuum pumps(one for the furnace chamber and the other for the EB gun chamber), an electron beam gun with a maximumpower of 30 kW and other auxiliary systems such as a water-cooling system and a high voltage power supply.timer triggerEB guntorsion springvacuum chambercopper molddipping samplesample holderTi puckelectron beamoperating barcameraFigure 5.1: Schematic diagram of the lab-scale EB melting furnaceAn automated dipping and retraction system was fabricated to ensure that the rods could be accuratelyplaced within the liquid pool and the immersion times could be controlled. Typical immersion times werein the range of 1-4 s ± 0.1 s. The system consists of a timer, a timer switch, an actuator and a torsion springas shown schematically in Fig. 5.1. The system also includes a rod holder and thermocouple support, asshown in Fig. 5.2. Prior to dipping a rod, a molten pool was created in a titanium puck. To maintain astable liquid pool, the beam was left on during the experiments and was programmed to scan in a circularpattern. As the sample rod was placed at the center of the circular EB pattern, the arm of the sample holderwas made from molybdenum to tolerate being heated periodically by the beam. Likewise the thermocouplesheath was also protected with additional molybdenum shielding. Under the experimental conditions used,the arm assembly was able to tolerate the EB for in excess of 12 s.The dimensions of both the titanium puck and the sample rod are shown in Fig. 5.3. Thermocouples50NO. Item Material/Note1 sample rod CP Ti2 thermocouple (TC) type C3 TC sheath alumina4 TC shield molybdenum5 sample holder arm molybdenum6 sample clamp stainless steel7 clip stainless steelside view123234567Figure 5.2: Details of the sample holder(type C, wire diameter: 0.5 mm) were embedded in the puck and the sample rod (two in the puck and onein the rod) - see TC A, TC B and TC R in Fig. 5.3. Note: according to the information provided by the TCsupplier, the error in the Type-C TC is 4.5°C to 425°C and 1.0% to 2320°C [67].r=48r=45TC ATC B30rzunit: mm(a)35Φ=8.4TC R5(b)5◦puckrod(c)Figure 5.3: Dimensions of the puck (a) and dipped rods (b) and locations of the inserted thermocouples.The 5◦ slice at the right-hand-side is the modeling domain (c)5.1.2 Experimental ProcedureThe experimental procedure was as follows: firstly, a CP-Ti puck was placed in the copper mold and asample rod was installed in the sample holder. The furnace chamber was then evacuated to the desired levelof vacuum (typically in the range of 10−2 to 10−3 Pa). The EB was then switched on and the power rampedto the desired level of 11 kW, where it was maintained until the end of the test. The pattern scanned withthe EB is a circle with a diameter of ∼34 mm concentric with the center of the puck at a frequency of 80Hz. After a stable pool was established, the CP-Ti rod was immersed into the molten pool to a prescribeddepth (6 mm). Fig. 5.4 shows an image of a sample rod being immersed in the molten pool captured from51a video taken of the experiment. Four immersion times, 1, 2, 3 and 4 s, were used in this study to allow forthe evolution of the solid/liquid interface profile with time to be observed. Note: these times represent thetime from when the rod reached its specified depth in the liquid pool until when it was retracted.molten poolsample holderrodscanning patternFigure 5.4: Screenshot showing the rod dipped into the molten pool5.2 Model DescriptionIn this study, a numerical model has been developed to analyze the experiments and aid in understandingmelting. The model incorporates the significant transport phenomena present including heat, mass andmomentum transport. The flow drivers include thermal buoyancy and thermal Marangoni forces. Themodel was developed using the commercial CFD package, ANSYS CFXTM version 14.0, which is based onthe finite volume method. An analysis of the Reynolds number, following the solution for the flow field,revealed that the flow conditions are laminar and hence turbulence has been ignored. In addition, the liquidsurface is assumed to be flat, which allows the modeling of free surface to be avoided.Three configurations were modeled to simulate the three corresponding stages in the experiments, asschematically shown in Fig. 5.5. The first configuration computes the steady state temperature, and whereappropriate, velocity and pressure fields within the titanium puck prior to dipping the rod into molten pool.The second configuration utilized the IMMERSED SOLID METHOD in CFX to simulate the dipping event,which is described in more detail in Section 5.2.2. The final configuration simulates the melting process of52the immersed CP-Ti rod after being placed in the molten pool.Config 1: compute temperature and velocity fields within the puck before dippingConfig 2: simulate the effects of the dip-in event on the velocity in the poolConfig 3: simulate the meltingof the dipped solid sampleFigure 5.5: Three configurations of the model to simulate the three stages of the experiment5.2.1 Governing EquationsThe governing equations that were solved to obtain the temperature, velocity and pressure fields in the modelconfigurations described above are identical to those solved in the ice/water case. They have been describedand discussed in Chapter 4 – see Eqs. 4.1 ∼ 4.3. In addition, the method to implement the Darcy attenuationflow in the semi-solid region and stop motion in solid is also the same – using a momentum source term asexpressed in Eq. 4.5 and Eq. 4.6. Note: that in application to CP-Ti, the temperature range (1 ◦C) applied toevaluate Eq. 4.6 is from 1667 to 1668 ◦C. This arbitrarily small range has to be applied to achieve numericalstability, although the experimental material used in this study, CP-Ti, is a relatively pure metal and thus intheory the phase change (solid/liquid) occurs at a specific temperature - i.e. the melting point which is 1668◦C .5.2.2 Computational Domain and MeshGiven that the geometries of both the puck and rod when present are circumferentially symmetric, a 5o slicehas been adopted as the computational domain for each configuration, as illustrated in Fig. 5.3, to improvecomputational efficiency. Note: CFX does not permit the use of a 2-D axisymmetric model, hence the needfor a 3-D 5o slice.To generate the mesh, the vertical face of the domain was first discretized into a grid (see Fig. 5.6 fordetails). In order to further reduce computational time without losing accuracy, a finer grid spacing wasused in the area where liquid was present for the bulk of the analysis. The grid was then rotated 5o about thecenterline of symmetry. A single layer of elements was used in the circumferential direction. At the domaincenterline, the resulting layer of triangular prism elements is of poor quality and was removed yielding a53mesh comprised of 5,434 hexahedra elements containing 11,228 nodes. Note: only the domain and meshof the third configuration of the model is presented here for the sake of brevity. The mesh used in the threecases to describe the puck is identical. In the third case, the mesh associated with the rod is introducedinto the computational domain for full solution of the conservation equations. In the second model, therod is meshed and placed above the melt. Using the IMMERSED SOLID METHOD, the rod is prescribed avelocity and lowered into the melt. Using a momentum source term, CFX forces the velocity of the fluidin the appropriate nodes within the puck to be equal to the velocity of the solid rod as it is lowered into thepuck domain. This approach only impacts on the fluid flow (momentum transfer) and not the heat and masstransfer. The resultant error should be small due to the very short duration of this stage, less than 0.2 s. Theresults of a sensitivity analysis to grid size are presented in Table 5.2. The results for predicted melting timeshow sensitivity to grid size. As discussed in Section 5.3.4, the model with the base-case mesh was able topredict melting times close to what was observed after some tuning of one of the empirical parameters in themodel. Based on these results, the base case grid size was found to be a reasonable compromise betweenaccuracy and execution time.5mmFigure 5.6: Meshing details54Table 5.2: Sensivitity analysis on grid sizeParameter Base-case Case 1 Case 2 Case 3Grid size - No. of elements 5,434 2,199 14,916 26,240Predicted melting time [s] 4.3 6.5 3.6 3.1Model execution time 6h42m 5h15m 10h23m 17h57m5.2.3 Boundary ConditionsThe boundary conditions applied in the numerical model, which are presented in detail below, are presentedschematically in Fig. 5.7 together with the main flow drivers. Table 5.3 lists the values of the relatedparameters as well as the material properties.q(EB)copper moldsolid zoneliquid poolq(rad)q(cnt)q(rad)b(T)q(rad)q(cnt): contact heat transferq(rad): radiative heat tranferM(T): thermal Marangoni flowb(T): thermal buoyancyrzrodq(rad)M(T)watercooling channelFigure 5.7: Boundary conditions of the model5.2.3.1 Thermal Boundary ConditionsElectron Beam Heating - An EB provides heat by virtue of the conversion of the kinetic energy of electronsto thermal energy within a short distance of the surface of the material being irradiated (in the order of µm[45]). In the numerical model, this heat is characterized by a heat flux boundary condition applied at the topsurface of the puck. For a single stationary spot, the applied heat flux may be represented by a 2-D Gaussiandistribution, as shown in Eq. 5.1.qEB(x,y) = fabsPEB 12piσ2EBe−(x−µx)2+(y−µy)22σ2EB (5.1)55Table 5.3: Material properties of CP-Ti and boundary condition related parameters adopted in the modelProperty or Parameter Value Unit Ref(s)ρ f (T ) kg/m3 [68]µ 0.003(liquid) / 3(solid) Pa ⋅ s [68]CP f (T ) J/(kg ⋅K) [69, 70, 71]k f (T ) W/(m ⋅K) [71]ε 0.4(liquid) / 0.5(solid) - [71]hcnt 500 W/(m2 ⋅K) [72]Lm 440,000 J/kg [2]Lα/β 85,200 J/kg [2]Tm 1,940-1,941 K [71]∂γ/∂T -0.00027 N/(m ⋅K) [71]PEB 11,000 W -σEB 0.008 m -rEB 0.017 m -fabs 0.65 - -where fabs is the absorption factor, µx [m] and µy [m] represent the position of the center of the beam spot ata given time, PEB [W] is the EB power and σEB [m] is the standard deviation. Owing to the fact that the beampattern is repeated at 80 Hz, a time averaged heat flux distribution is used. According to the assessment doneby Meng [47], time averaging has a negligible effect on the results for EB pattern frequencies above ∼10Hz. The resulting expression for the time averaged heat flux is given in Eq. 5.2.qEB(x,y) = fabsPEB 12piσ2EB∫ 1/ fEB0e−[x−rEBcos(2pi fEBt)]2+[y−rEBsin(2pi fEBt)]22σ2EB dt (5.2)where fEB [Hz] is EB scanning frequency and rEB [m] is radius of the circular pattern.Thermal Radiation and Contact Heat Transfer - Heat is lost from the top and bottom of the puck viaradiation – see Fig. 5.7. The radiative heat transfer may be described by the Stefan-Boltzmann relationship(Eq. 5.3). Note: as the EB heating and melting process is conducted within a relatively high vacuum, theconductive heat transport to air is very small and has been neglected in the numerical model.qrad = εσrad(T 4−T 4re f ) (5.3)where ε is the emissivity, σrad [W/(m2 ⋅K4)] is the Stefan-Boltzmann constant, and the reference tempera-tures, Tre f [K], for the different surfaces are listed in Table 5.4.On the side surface of the puck that is in contact with the water-cooled crucible, a contact conductive56Table 5.4: Reference temperature in Eq. 5.3Radiative surface Tre f Value [◦C]Top surface @ puck Tchamber 25Side surface @ puck Tmold,side 200Bottom surface @ puck Tmold,bottom 50Side surface @ rod Tchamber / Tpool 25/2,000Top surface @ rod Tchamber 25Inner surface @ rod Tceramicsheath 75heat transfer boundary condition has been applied as described by Eq. 5.4.qcnt = hcnt (T −Tmold,side) (5.4)The value of the heat transfer coefficient hcnt [W/(m2 ⋅K)] is given in Table 5.3, and Tmold,side [K] inTable 5.4.5.2.3.2 Fluid Flow Boundary ConditionsThe side and bottom surfaces of the puck and all surfaces of the dipped rod (when inserted) are set to a noslip wall condition. A shear stress (σM [N/m2]) based on Eq. 5.5 was applied on the top surface of the puckwhere liquid was present to account for Marangoni forces [73].σM =∂γ∂T∂T∂n(n = x,y) (5.5)where ∂γ∂T is the thermal coefficient of surface tension and its value is given in Table 5.3.5.2.4 Initial ConditionsThe steady state temperature field at the end of first stage of the 3-part computation is independent of theinitial temperature. To minimize the computational time to arrive at the steady state solution, an arbitraryuniform temperature of 800 ◦C was used as the initial condition to reduce the computational time. Theinitial velocity and pressure were set to 0. Steady state was assessed by monitoring the temperatures atseveral locations within the domain. To begin the 2nd part of the computation, the temperature, velocityand pressure fields from the first configuration of the model at steady state, were imported into the secondconfiguration as the initial conditions. Similarly, the results computed from the second configuration were57then used as the initial conditions for the third configuration. Additionally, a second set of initial conditionswere imposed on the immersed CP-Ti rod, which included setting the velocity and pressure to 0 and thetemperature to that measured from the thermocouple embedded in the rod at the beginning of immersion (75◦C).5.2.5 Material PropertiesThe material properties adopted in the numerical model have been summarized in Table 5.3. The temperaturedependent properties listed in the table are shown graphically in Fig. 5.8.400 600 800 1000 1200 1400 1600 1800 2000Temperature [◦C]4000410042004300440045004600Density[kg/m3](a) Density400 600 800 1000 1200 1400 1600 1800 2000Temperature [◦C]1820222426283032k[W/(mK)](b) Thermal conductivity0.0 0.2 0.4 0.6 0.8 1.00.00.20.40.60.81.0CP,e[J/(kgK)]8000085000900009500010000400 600 800 1000 1200 1400 160 1800 20 0Temperature [◦C]40600800100012001400(c) Effective specific heat capacityFigure 5.8: Temperature dependent material properties of CP-Ti (Melting temperature: 1668◦C)The latent heat of the phase changes expected to occur – i.e. solidification, melting and the solid stateα/β transformation, are accounted for in the heat balance by utilizing an effective specific heat, which isexpressed in Eq. 5.6. The original temperature dependent specific heat and the effective specific heat areshown in Fig. 5.8 - the red dash line represents the actual specific heat and the black line is the effectivespecific heat accounting for the release of latent heat.CP, e =CP−Lm∂ fs∂T−Lα/β ∂ fα∂T (5.6)where Lm [J/kg] and Lα/β [J/kg] are specific latent heat for melting/solidification and α/β phase transfor-58mation, respectively, and fα is the fraction of α phase.For the α/β phase transformation, a 300 ◦C temperature range has been used to evaluate ∂ fα∂T , which waschosen based on the Ti-Fe binary phase diagram as the temperature range is very sensitive to the content ofiron. In the case of liquid to solid transformation, three temperature ranges were tested in the model, 1, 3and 5 ◦C. Originally it was hoped to use a 1 ◦C interval consistent with what is employed in the model inthe context of Eq. 4.6. However, the sensitivity analysis revealed that the model failed to capture the latentheat when a temperature of 1 ◦C was applied, whereas the modeling results for 3 and 5 ◦C were similar andindicative of capturing the latent heat. Hence, 3 ◦C has been chosen as the base-case range. Additionally, ∂ fs∂Tand ∂ fα∂T in Eq. 5.6, are assumed constant over the corresponding phase transformation temperature range.Therefore, the resulting values for ∂ fs∂T and∂ fα∂T , applied in the model, are 1/3 and 1/300◦C−1, respectively.5.2.6 Computational ProcedureThe model was run in parallel on 6 cores from 2.33 GHz Intel Xeon quad-core CPUs with 8.00 GB RAM.A high-resolution advection scheme available in ANSYS CFX was used to solve the governing equations.The time step size was set at 0.001 s. A sensitivity analysis to time step was conducted and time steps of0.0005 s and 0.002 s were also run for an example problem in addition to the base-case time step of 0.001 s.Comparing the temperatures predicted at the thermocouple location (TC R) for the three different time stepcases revealed little difference between the 0.001 and 0.0005 s time steps. Thus, the base-case time step of0.001 s was felt to strike a reasonable balance between accuracy and execution time. The convergence targetwas set to 1×10−4 for the RMS residual with a minimum of 3 and maximum of 10 iterations per time step,which was chosen based on what is recommended for engineering applications within the ANSYS CFXliterature [74]. Reducing the RMS residual target to 5× 10−5 or 1× 10−5 was found to have no significantimpact on the results.5.3 Result and Discussion5.3.1 Pool Profile and Temperature before DippingFig. 5.9 compares the experimentally obtained steady state pool profile based on an etched cross section ofthe melted puck (left-hand-side) with the predict temperature distribution at steady state (right-hand-side),59before the rod is added. The liquid pool profile can be identified by the difference in microstructure betweenthe material that was melted and re-solidified and the unmelted material – e.g. the solid/liquid interface ofthe steady state molten pool. To aid in visualizing this, the image has been annotated with a red dotted linelabeled “pool profile”.3 421 5 986 70 cmpool profile8001000120014001600180020002200T [oC]ridgeFigure 5.9: Pool profile comparisonAnother phenomenon observed is that the pool’s surface has sunk ∼1.5 mm in the center. There appearsto be three contributions to this: firstly, the solid portion of the puck has become permanently deformedunder gravity during melting. The resulting ∼1 mm deformation can be seen on the bottom surface of thepuck at the center. Secondly, some evaporation has occurred, which has led to mass loss from the liquidpool, and finally, liquid titanium was driven out radially onto the pool’s edge and solidified there to form aridge at the pool’s edge, as marked by the blue dashed box. The strong radial flow causing this is predictedin the model and is linked to Marangoni driven flow, which is acting radially outward at the edge of the pool(the Marangoni flow is discussed further in Section 5.3.5).To ease in comparing the model predicted pool profile with the observed pool profile, a line to demarkthe measured solid/liquid interface of the steady state pool has also been added to the right-hand-side image,together with the predicted location of the solidus isotherm, which appears as a black solid line. As can beseen, the numerical model is able to reproduce the experimentally observed pool profile relatively accuratelyin terms of both its subsurface shape and top surface radial dimension. It should be noted that this level ofagreement requires consistency between the fluid flow conditions in the model and experiments, as there isa substantial advective contribution to heat flow in the liquid pool predicted by the model.Finally, for a more quantitative comparison, temperatures at the locations of the two thermocouplesplaced in the puck (TC A and TC B shown in Fig. 5.3) have been compared with the model predictions at60the equivalent positions at steady state. This comparison is presented in Table 5.5 (errors of ±60 and ±50 ◦Chave been determined based on extensive experimental measurements). The model can be seen to predictthe temperatures at the two locations to within an error of less than 5%. The error on the experimentalmeasurements reported in Table 5.5 is based on the repeatability observed over a number of experiments.Slight variations in the beam power, focus and radius of the circular EB scanning pattern from test to testlikely were the largest contributors to this variability.Table 5.5: Temperature comparison between the experimental measurements and model predictionsLocation Experiment [◦C] Model [◦C]TC A 1350±60 1405TC B 1030±50 1065To summarize, both qualitative and quantitative comparisons have been made between the model predic-tions and experimentally derived data, which show that the numerical model is able to accurately reproducethe thermal and flow conditions that prevailed in the pool at steady state prior to the rod immersion.5.3.2 Rod Solid Profiles5.3.2.1 Experimentally Observed ProfilesThe profiles of the four immersed rods following the extraction from the molten pool are presented in Fig.5.10. The dashed blue line represents the initial dimension of the rod and the dashed red line representsthe free surface of the liquid titanium pool during immersion. The rod’s length increases for the 1 s case,while the width shows negligible change. This indicates that liquid titanium solidified on the bottom of therod initially. Solidification occurred only on the bottom due to the lower temperature and the absence ofMarangoni forces to drive flow, and heat transfer, in the liquid at the bottom of the pool. The evolution ofthe profile of remaining solid can be seen in the images taken of the rods dipped for 2, 3 and 4 s. Generally,there is a higher rate of mass loss from the sides than from the bottom. Note: there is some melting of the rodradially above the free surface of the melt, which is first obvious in the 2 s sample and increases thereafter.By 4 s, the immersed section of the rod is virtually gone and there is significant removal of material abovethe free surface resulting in a cone shaped profile.61t=1, 2, 3, 4 (s)melt surfaceoriginal profileFigure 5.10: Solid/liquid interface profile of the rods with different dipping times5.3.2.2 Comparison between the Experimental Observations and Modeling ResultsFig. 5.11 compares the predicted profiles extracted from the model with the experimentally observed pro-files. The observed profiles are coloured red and appear on the left-hand-side of the figure and the predictedprofiles are coloured blue, appearing on the right-hand-side. Note: the red lines are based on the left-hand-side sample profiles shown in Fig. 5.10 for the 1, 2 and 4 s rods, which showed a fairly high degree ofsymmetry. In contrast, the 3 s rod showed a significant degree of asymmetry, and thus both left and right-hand-side profiles are shown and drawn as the solid and dashed lines, respectively. In general, the modelwas able to reproduce the experimental observations over the range of immersion times in terms of bothprofile shape and degree of material melted. Additionally, the model captured the initial freezing of liquidtitanium on the bottom of the immersed rods.Taking a closer look, the model predicted less solidified titanium at the bottom and a greater degree ofmelting around the side of the rod at 1 s. One possible explanation for the disagreement is the inaccuratecharacterization of the initial conditions at the beginning of the third stage of the analysis. As discussedpreviously, the initial conditions for the third configuration, in terms of the fluid flow and temperature inthe molten pool, are obtained from the flow and temperature fields at the end of the second stage in theanalysis. However, the second stage analysis was only able to capture the effect of immersion of the rodon fluid flow, but not heat transfer. Consequently, the temperature field calculated during the dipping islikely inaccurate resulting in a relatively large error in the simulation of the initial stage of immersion. Withincreasing elapsed time (2, 3 and 4 s), the calculated temperature and fluid flow fields have more time todevelop and the agreement improves.620s1s2s3s4stExperimentModelliquid surfaceFigure 5.11: Interface profile comparison between model predictions and experimental observations (Thedashed line shows the interface profile at the right side of the sample with 3 s since the profile is apparentlyasymmetric as observed in Fig. 5.10)5.3.3 Melting RateThe ratio of melted volume to initial volume, defined in Eq. 5.7, has been extracted from both the exper-imental and modeling results as a function of immersion time and plotted in Fig. 5.12. Note: in Eq. 5.7,V0 and V denotes the original and current volumes, respectively, of solid that is immersed within the liquid.Due to the initial solidification of liquid titanium on to the surface of the rod, it can be seen that both theexperimental and modeling results show negative ratios initially. Since the model predicts less freezing atthe bottom and more melting at the side (as shown in Fig. 5.11), the ratio derived from the experimentalresults is more negative at 1 s. For the balance of the immersion times, 2, 3 and 4 s, the agreement is better,which, as mentioned, is consistent with what was observed in Fig. 5.11.rV =V0−VV0(5.7)5.3.4 Temperature Monitored within the RodBefore presenting the temperature results, it is necessary to consider the effect that solidification of liquidtitanium on the rod has on diffusive heat transport and in particular the resistance that develops at thesolid/solid interface that forms, an example of which is shown in Fig. 5.13. This solid/solid interface630 1 2 3 4Time [s]0.40.20.00.20.40.60.81.01.2Melted Volume Ratio r VmodelexperimentFigure 5.12: Comparison of melted volume ratio between experimental and modeling results (the diameterof the circles indicate the inaccuracies of the size measurements - less than 5%)decreases heat transfer through the interface. This phenomenon has also been recognized in other studiesrelated to the melting/dissolution of solids in liquid steel - see Refs [28, 75, 15]. This resistance vanishesonce the initially solidified titanium melts.dipped rodfrozen Tiinterface500 µm  Figure 5.13: Solid/solid interface formed when liquid CP-Ti solidified on the cold dipped rodIn the numerical model, the solid/solid interface has been approximated in terms of its influence on heattransfer by reducing the thermal conductivity in the region of the interface. The approach adopted is toreduce the conductivity within a region ±0.4 mm from the interface (0.8 mm thickness total) for materialwith solid fraction equal to one. Given the mesh size, this represents the smallest spatial resolution thatcould be adopted to describe the reduction in heat transfer associated with the interface. The modification64utilized was to reduce the conductivity from between 18~31 W/(m ⋅K) (depending on the temperature) to 1W/(m ⋅K). Note: the value of 1 W/(m ⋅K) was determined by trial and error fitting of the model results tothe experimental measurements. The model predictions are sensitive to the magnitude of the “interfacial”conductivity.Fig. 5.14 presents a comparison between the model predicted evolution in temperature and the measuredevolution at the location of TC R in the rod (shown in Fig. 5.3) for the 4 s sample. The evolution intemperature can be described as having three stages: 1) an initial stage where the temperature increases at aslow rate, (consistent with the presence of the solid/solid interface limiting the diffusion of heat into the rod);2) a second stage, beginning at approximately ∼1.5 s, where temperature increases at a fast rate (consistentwith the disappearance of the solid/solid interface and the presence of a large driving force for heat transfer);and 3) a period of moderating temperature increase (consistent with a gradual reduction in the driving forcefor heat transfer as the temperature in the rod approaches the liquid pool temperature and the release of latentheat of melting occurs). Once the thermocouple was exposed to the liquid titanium, its accuracy was rapidlydegraded and the data was not plotted. In comparing the predictions with the measurements, it is apparentthat the model is able to accurately predict the evolution in temperature in all three stages of behavior.0.0 0.5 1.0 1.5 2.0 2.5 3.0Time (s)020040060080010001200140016001800Temperature(◦C)ExperimentModelFigure 5.14: Comparison between measured and predicted temperatures in the immersed rod at TC R (thesize of the circles indicate the experimental inaccuracy - less than 6%)655.3.5 Examination of the Influencing FactorsTo examine the effects of the flow drivers on melting, three variants of the model were run: one, in whichthermal buoyancy was switched off; a second, in which thermal Marangoni was switched off; and a third,in which both were switched off. Note: the flow drivers (both buoyancy and Marangoni forces) were stillactive in the first and second configurations of the model to establish the bulk pool thermal and flow fieldsthat serve as the initial conditions for the third configuration. The corresponding flow driver(s) was (were)then switched off in the third configuration of the model. As a result, during rod melting, the impact of thetwo flow drivers on the behaviour of the boundary layer between the solid and liquid could be examinedseparately and in combination. Additionally, it is worth noting the corresponding bulk flow(s) are alsoimpacted and would gradually subside over a period of time due to viscous dissipation, thus the effect of theflow drivers on a combination of the boundary layer and bulk flows has in practice been assessed.The results are presented in Fig. 5.15, together with the results for the base-case model in which both areon. The results at 3 and 4 s show that the melting rate has been substantially reduced if either the buoyancyor the Marangoni force is deactivated. Of the two, the Marangoni force can be seen to exert a greaterimpact on the results by comparing the red line (Marangoni off) and green line (buoyancy off). Therefore,as reinforced by the results presented, it is of great importance to incorporate the two flow drivers, thermalbuoyancy and in particular thermal Marangoni, into the numerical model to ensure accuracy.1 s 2 s 3 s 4 sBuo On/Mara OnBuo On/Mara OffBuo Off/Mara OnBuo Off/Mara Offoriginal profileFigure 5.15: Modelling results showing the effects of buoyancy and Marangoni forces on the solid/liquid in-terface profile (the labels, Buo and Mara, in the figure denote buoyancy and Marangoni forces, respectively)Fig. 5.16 shows the development of the thermal boundary layer in proximity to the remaining solid att=2 s, with both the flow drivers (buoyancy and Marangoni force) on. The thermal boundary is between 1and 2.5 mm and varies with location. The number of elements within the boundary layer is between 4 and666 depending on the location allowing the temperature gradient in proximity to the solid to be adequatelyresolved, which was also supported by the results of the sensitivity analysis.−0.006−0.004−0.0020.000Depth [m]16671715176218101857190519522000T [◦C]0.06[m/ s]Figure 5.16: Mesh and temperature distribution in proximity to the solid at t =2 s.5.3.6 Effective Heat Transfer CoefficientFor the convenience of use in macro-scale models and/or heat transfer only models, it is necessary to employan EIHTC, to predict the melting and/or dissolution kinetics. The EIHTC (he f f ) may be defined as follows:he f f =qs/l(Ts/l −Tbulk) (5.8)Two approaches have been used in the present work: the first is based on the results of the numericalanalysis of the experiments and the second is based on boundary layer theory (similarity solution method).In the first method, the heat flux (qs/l), may be expressed as:qs/l = Qs/lAs/l∆t (5.9)and the total amount of the heat (Qs/l) transported through the solid/liquid interface (with area As/l), in atime interval ∆t, as:Qs/l =∰ ρCP∆T dV +ρLm∆VS −Qrad (5.10)In Eq. 5.10, the first term on the right-hand-side is the enthalpy accumulation in the immersed rod,the second term is the latent heat absorbed by melting or released by freezing, and the third term is theradiative heat transport from the rod’s surface above the liquid pool. All three of these terms can be evaluated67incrementally in time using the results from the numerical model. In addition, in Eq. 5.8, Ts/l and Tbulk alsorequire evaluation. The interfacial temperature, Ts/l , is straightforward and is the melting temperature ofCP-Ti, 1668 ◦C. The bulk liquid temperature, Tbulk, is more of a challenge in the context of the currentanalysis. In this study, three approaches were used. All three were based on the vertical line shown in Fig.5.17, which was assumed to be at the approximate location of the velocity/thermal boundary layer for therod at the beginning of the melting process. The three temperatures evaluated were based on the minimumalong the line, average (taken over the length of the line) and the maximum and are plotted as a function ofimmersion time in Fig. 5.18. It was reasoned that the minimum and maximum would provide a reasonablerange for Tbulk to assess sensitivity and that the average would be a reasonable approximation to Tbulk.0.06[m/ s]0.000 0.005 0.010 0.015 0.020 0.025 0.030 0.035Radius [m]−0.014−0.012−0.010−0.008−0.006−0.004−0.0020.000Depth [m]16671715176218101857190519522000T [◦C]Figure 5.17: The reference line over which the bulk liquid temperature was calculated (t=2 s)0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0Time [s]180018501900195020002050Temperature [◦ C]Line MaxLine MinLine AverageFigure 5.18: Bulk liquid temperatures adopted to calculate EIHTCsIn the second EIHTC analysis method, the problem is simplified to that illustrated in Fig. 5.19a - anaxisymmetric vertical, downward, flow over an isothermal cylinder. This is consistent with the boundarythat develops and flow direction seen in Fig. 5.17. To accommodate the change in geometry associated withmelting of the solid rod, the geometry is assumed to transition from a series of truncated cones to cones68as shown in Fig. 5.19b, consistent with the mass loss observed experimentally. The truncated cones/conesare then converted to a series of equivalent cylinders with r0 = (rt + rb)/2, and h0 = lc (see Fig. 5.19c for adrawing of the equivalent cylinder).Ubulk Tbulkvelocity boundary layerthermal boundary layerUbulkTbulkTs/lr0h0rx(a)tliquid surface(b)rbrtlcrtrbr0lch0(c)Figure 5.19: Similarity solution related schematics (a) simplified case description for similarity solution,(b) simplified geometries (truncated cone or cone) with time and (c) transformation from truncated cone orcone to a cylinderThe mathematical calculation of EIHTC follows the method described in Ref [76]. For brevity, thedetails will not be introduced here and readers are referred to the original work for more information.Two variables, ξ and η , and two functions, f and θ , have been defined as follows:ξ = 4r0 ( vxUbulk )1/2 , η = r2− r20r20ξ (5.11)69f (ξ ,η) = ψ(x,r)r0(vUbulkx)1/2 , θ (ξ ,η) = T (x,r)−T∞Ts/l −T∞ (5.12)After the required mathematical transformations, which have been introduced in Ref [76], the governingequations, which are partial differential equations (PDEs), can be converted to ordinary differential equations(ODEs), Eq. 5.13 for fluid flow and Eq. 5.14 for temperature, respectively, which may be solved eitheranalytically or numerically. Note: the bulk liquid velocity (Ubulk) adopted in the similarity solution is 0.06m/s, which is estimated from the velocity at the boundary layer given by the numerical model.For fluid flow:(1+ξη) f ′′′+ ( f +ξ ) f ′′ = 0 (5.13)and for temperature (the Prandtl number Pr is 0.093 here):1Pr(1+ξη)θ ′′+ ( f + ξPr)θ ′ = 0 (5.14)subject to the boundary conditions:f (ξ ,0) = f ′(ξ ,0) = 0, f ′(ξ ,∞) = 2, θ (ξ ,0) = 1 and θ (ξ ,∞) = 0After solving the equations, the thermal characteristics in terms of the local heat transfer coefficient andlocal Nusselt number can be obtained as follows.hx =qs/lTs/l −T∞ = 1Ts/l −T∞ ⋅ −k∂T∂ r ∣r=r0 = − 2kr0ξ ∂θ∂η ∣η=0 = − 2kr0ξ θ ′(ξ ,0) = − 2k4(vx/Ubulk)1/2θ ′(ξ ,0) (5.15)Nux =hxxk= −12Re1/2x θ ′(ξ ,0) (5.16)Then the overall averaged EIHTC can be calculated from the expression:he f f =∫ lc0 hxdxlc(5.17)70The results for the EIHTC calculated from the numerical model and the similarity solution have beenplotted together in Fig. 5.20. The results from the model-based calculation using all three values for Tbulkhave been included in the plot. Focusing first on the numerical model derived EIHTCs, it can be seen thatthey contain a fair amount of scatter, especially at the beginning (immersion times less than ∼0.8 s). Theoverall scatter likely relates to the use of a fixed mesh for the analysis domain and the increased scatter at thebeginning is likely due to the dynamics associated with the development of the thermal and fluid boundarylayer shortly after the rod is introduced in the melt. For the bulk of the experiment, the range is predicted tobe between 26,000∼36,000 W/(m2 ⋅K) for the case where Tbulk is based on the “Line Average”.0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0Time [s]100002000030000400005000060000h [W/(m2 K)]Line MaxLine MinLine AverageSimilarityFigure 5.20: Effective heat transfer coefficientsThe EIHTCs obtained from the similarity solution generally agree better with the “Line Min” Tbulkresults for the numerical model-based approach. This suggests that the alternative values for Tbulk basedon “Line Max” and “Line Average” may over estimate Tbulk for the line location selected. It should benoted that the similarity solution assumes steady state flow conditions and hence less agreement betweenthe two methods would be expected shortly after immersion. The similarity solution predicts a gradual rateof increase in the EIHTC until approximately 2.7 s immersion time, at which point the EIHTC is calculatedto rapidly increase, whereas the numerical model-based approach predicts a more uniform rate of increaseover the entire immersion time (ignoring the initial transient). The first stage of gradual increase (<∼ 2.7 s)in the similarity solution is due to decreasing radii of the equivalent cylinders. The change in behavior inthe second stage is related to the shape transition from truncated cone to cone geometries. As the length ofthe vertical surface decreases the boundary becomes less developed and therefore the EIHTC increases.The range of EIHTCs predicted in this work is comparable to the values calculated in Ref [15] – e.g.7110,000 - 26,000 W/(m2 ⋅K) for dipping aluminum wire into liquid steel. However, they are substantiallylarger than values determined for the EIHTC for the ice/water system of 600 – 4,000 W/(m2 ⋅K) and forfrozen ethanol/water system of 500 – 1,300 W/(m2 ⋅K) (presented in the previous chapter) [66]. The largedifference may be attributed to three factors: firstly, liquid titanium possesses a much higher thermal conduc-tivity than that of either water or ethanol. Secondly, the work reported in this chapter was conducted underconditions in which an appreciable bulk flow was present, whereas the ice/water and frozen ethanol/waterwere conducted under natural convection conditions (no appreciable bulk flow); and thirdly, the presentwork has a shorter characteristic length (∼0.005 m) compared to the previous work (initial length ∼0.036m).To further explore the contribution of these three factors, the heat transfer coefficient as a function ofcharacteristic length has been calculated using the similarity solution method described above for conditionsin which: 1) the conductivity of the liquid has been reduced; and 2) the bulk velocity has been reduced. Fig.5.21 compares the results. The black symbols represent the results for the base-case using the parametersbased on the current work (r0=0.0042 m, Ubulk =0.06 m/s, k=31.4 W/(m ⋅K)), the red symbols represent theresults when the thermal conductivity of the liquid is reduced to 0.6 W/(m ⋅K), which is around the valuefor water, and the blue symbols, when Ubulk is reduced to 0.0 1m/s. The results clearly show that all threefactors will have a significant effect on heat transfer.0.00000.00050.00100.00150.00200.00250.00300.00350.00400.0045Characteristic length [m]020000400006000080000100000120000140000h [W/(m2K)]k=31.4/u=0.06k=0.6/u=0.06k=31.4/u=0.01Figure 5.21: Effects of thermal conductivity and flow velocity on heat transfer coefficient725.4 Summary and ConclusionsThe melting behavior of solid CP-Ti in molten CP-Ti was investigated using a combination of experimentaland modeling techniques. The evolution in the solid/liquid interface profile was observed experimentally bydipping CP-Ti rods into a CP-Ti melt for various periods of time, ranging from 1 to 4 s. The results fromthe 4 rods were used to measure the evolution in the rod profile with time and the rate of melting with time.The temperature evolution obtained from the 4 s experiment was used to assess the evolution in temperaturein the rod with time.When immersed into the molten pool, the cold rod quickly cools down the surrounding melt and, at somelocations, freezes a layer of titanium on its surface. Most of the freezing was found to occur on the rod’sbottom surface due to the relatively low temperature and weaker convective heat transfer at this locationin the molten pool. A solid/solid interface forms between the rod and frozen titanium which introduces athermal resistance, significantly reducing heat transfer to the dipped rod while present. Once the frozentitanium melts and the solid/solid interface vanishes, the dipped rod is heated up rapidly and then melts. Thenumerical model was shown to reproduce the experimental results satisfactorily, indicating that the modelis capable of approximating the main physical phenomena influencing the melting behavior of a CP-Ti rodintroduced into liquid titanium. With the aid of the numerical model, it has been recognized that the thermalbuoyancy and the thermal Marangoni forces have significant effects on the melting behavior. Moreover, forthe convenience of use in practical industrial-scale applications, EIHTCs have been calculated based on theresults from the numerical model. The EIHTCs calculated with the model have been compared to valuesdetermined using boundary layer theory. The two results show reasonable agreement if an appropriate bulktemperature is selected.73Chapter 6Study of the Melting of Ti-Al Solid in CPTitanium and Titanium Alloy (Ti64) LiquidThis chapter presents the findings of a study in which solid Ti-Al rods were dipped into two types of liquid- CP-Ti and Ti64. The use of Ti-Al solid enables compositionally related phenomena to be examined in ad-dition to the thermally related phenomena previously considered. The compositionally related phenomenainclude compositional buoyancy and Marangoni forces, evaporation of aluminum, composition-dependentmelting and solidification and the associated latent heat absorption/release. These phenomena may signifi-cantly impact both the development of bulk flow pattern and the establishment of interfacial heat and masstransfer in the vicinity to the solid/liquid interface. Moreover, because Ti and Al are also present in thematerial associated with the “drop-in” events that have been known to occur industrially, as described inChapter 1 [5], the study of this system can provide insight into an important industrial problem.Since the studies presented in this chapter and in Chapter 5 are similar to some degree in terms of themethods employed for both conducting the experiments and developing the associated numerical model, forbrevity only the differences will be described in the Experimental Methods and Model Description sections.Readers are referred to the previous chapter for more details.6.1 Experimental MethodsThe dipped Ti-Al rods have the identical geometry and dimensions as the CP-Ti rods (see Fig. 5.3 for thedetails).746.1.1 Experimental MaterialTo achieve satisfactory experimental repeatability, the Ti-Al samples need to have a dense structure andhomogeneous composition (tests with porous Ti-Al rods showed unacceptably poor repeatability). However,production of a Ti-Al ingot with these characteristics proved to be a significant challenge. Before the finalsuccessful procedure was developed (introduced later), many attempts were conducted (see Table 6.1) usingvarious types of crucibles, forms of raw material and melting methods. The final successful procedureconsists of two key strategies: 1) using CP-Ti as the crucible material; and 2) using compacted Ti and Alpowder mixture as the raw material. Development of this approach was based on the experience gained fromthe various approaches that ultimately failed. As this work represent a significant expenditure of time andeffort the various approaches are briefly described below.As listed in Table 6.1, the first attempts involved melting Ti and Al bars (~8 mm in diameter) in alumina(attempt-1) and yttria (attempt-2) crucibles using an induction furnace (yttira is reported to be more stablethan alumina for melting titanium in Ref [77]). However, both crucibles cracked during melting and werefound to last between 30 and 60 mins, which was insufficient time to allow the Ti bars to be fully melted.Cracking of the ceramic crucibles was thought to be due to a combination of erosion/chemical attack fromthe molten titanium (highly reactive). In order to shorten the time for melting Ti and accelerate mixing,compacts of mixed Ti and Al powder (100 mesh and 10 mesh, respectively) were used as the raw materialfor attempt-3 (alumina crucible used) and attempt-4 (yttria crucible used). Although the compacts were fullymelted, the crucibles still cracked in 30 to 60 mins, resulting in the power to the furnace being switched off(relatively quick non-directional solidification). The Ti-Al ingots obtained using this approach were foundto have a porous structure, which was reasoned to be due to: 1) void formation associated with rapid non-directional solidification; and/or 2) the release of gas resulting from reaction with the crucible material. AnEBBF was used in the 5th attempt. However, this approach failed due to the loss of EB operation (the beamcould not be maintained due to excessive volatilization of gas trapped in the powder alloy compact). Finally,it was decided to use a Ti and Al powder compact in the induction furnace, but held in a CP-Ti crucible.The use of a powder compact shortened the time for melting and homogenization, and the CP-Ti crucibleenabled the molten Ti-Al to be held for a sufficient time to allow for mixing and release of entrapped gas.In addition, the CP-Ti crucible allowed the melt to be slowly directionally solidified. This eliminated theformation of solidification-based voids. Another minor advantage of using CP-Ti crucible is that crucible75contamination can be avoided as Ti is one of the alloying components. However, it should be noted that CP-Ti crucible partially dissolved into the Ti-Al melt, which resulted in an appreciable deviation in compositionof the final ingot from that of the original compact.Table 6.1: Details of attempts and the identified problemsNo. Form of rawmaterialCruciblematerialFurnace Problem identified1 Bars of Ti and Al Alumina InductionfurnaceNot fully melted, crucible cracked, porousstructure, oxidation contamination2 Bars of Ti and Al Yttria InductionfurnaceNot fully melted, crucible cracked, porousstructure, oxidation contamination3 Compacts of mixedTi and Al powderAlumina Inductionfurnacecrucible cracked, porous structure, oxidationcontamination4 Compacts of mixedTi and Al powderYttria Inductionfurnacecrucible cracked, porous structure, oxidationcontamination5 Compacts of mixedTi and Al powderAlumina EBBF Failed to melt as the trapped gas in thecompacts causes unstable pressure whichinterrupts the EB operation6 Compacts of mixedTi and Al powderCP-Ti InductionfurnaceNoticeable composition deviationAfter some trials with this approach, a suitable Ti-Al ingot was obtained with a composition of 24±0.6wt% (~36 at%) aluminum with the balance titanium. Three rods were machined from the Ti-Al ingot. Thedetailed procedure was as follows:1. Powders of pure aluminum and titanium were mixed in the ratio 75 at% aluminum and 25 at%titanium. After thorough mixing, the powder was compacted at room temperature using a hydraulic pressand a compression die. The compacts were short cylinders that were approximately 2.54 cm (1 inch) high,having a diameter of 1.91 cm (0.75 inch). The load applied for compaction was about 6 tons.2. The compacts were placed in a CP-Ti crucible (see Fig. 6.1) that was heated in an induction furnace(Radyne - 25 kW/400 kHz). The compacts were heated and melted under a high purity argon atmosphere,which was maintained at one atmospheric pressure. The temperature was maintained between 1500-1600 ◦Cto avoid melting of the CP-Ti crucible. Note: accurate measurement of temperature is not possible duringinduction melting due to the impact of the induction field on voltage output from thermocouples. To removetrace amounts of oxygen and nitrogen, a hot titanium filament was employed as a getter prior to the start ofmelting.3. After the compacts were fully melted, the molten material was maintained at temperature for one76r=0.75r=0.44.04.5unit: inFigure 6.1: CP-Ti crucible dimensionshour to allow for homogenization of the mixture.4. As a final step, the ingot was directionally solidified by withdrawing the CP-Ti crucible verticallydownward from the induction coil at the rate of approximately 17.78 cm (7 inch) per hour.A typical cross section of a Ti-Al ingot produced using this procedure is shown in Fig. 6.2a. Thedissolving of some of the crucible material resulted in a noticeable reduction of thickness in the cruciblewall and in a change in the final melt composition to about 36 at% aluminum from the original (75 at%).EDX-based analysis at the radial locations shown in Fig. 6.2a, revealed the composition of the final ingot tobe homogeneous at the macro-scale – see Table 6.2. An EDX-based line scan was also conducted to examinethe composition in micro-scale (Fig. 6.2b), which indicated that the Ti and Al powders were fully meltedand alloyed. Moreover, Figs. 6.2a and 6.2b show the ingot to be free from significant voids, rendering itsuitable for the purpose of fabricating rods for this investigation.Table 6.2: Composition analysisSpot Al - wt% Al - at%Center 24.5 36.5Middle 24.0 35.9Edge 23.4 35.177C M EEDX spots:Center (C)Middle (M)Edge (E)IngotCrucible10 mm(a) Cross section of the Ti-Al ingotscan path(b) EDX line scan resultsFigure 6.2: Ti-Al ingot6.1.2 Experimental ProcedureTwo types of experiments (Type-A and -B) were performed, which differ in terms of the procedure followingimmersion of the rod in the melt. In the Type-A experiment, the procedure was identical to the experimentscarried out in the CP-Ti/CP-Ti case, in which the rod was removed from the pool after a prescribed im-mersion time in order to observe the solid profile (see Chapter 5 for the details). In these experiments, theelectron beam was left on and the pool was maintained throughout the experiment. In the Type-B experi-ments, the electron beam was instantly shut off after a prescribed time of immersion without removing therod. As a result, the rod was solidified in place within the original melt-pool. In the Type-B experimentsdissolution/melting was characterized by sectioning the puck along a plane that approximately bisected therod.6.1.3 Experimental Results6.1.3.1 Basic ObservationsA total of three rods were obtained from the Ti-Al ingot produced using the method described previously.The first of these rods was dipped into the CP-Ti liquid with the intent to follow the Type-A procedure.However, it was not possible to removed rod from the CP-Ti liquid pool at the prescribed time. Aftervarious attempts to remove the rod it became detached from the sample clamp at 9.8 s – see Fig. 6.3. It78was apparent that the rod had become welded to bottom of the liquid pool. The results of this test werenot further investigated as it was felt that the various attempts to free the rod would have compromised thestructure of the interface between the rod and melt pool.Figure 6.3: Photo captured showing the rod was stuck to the pool and detached from the sample holderTo investigate the reason for the behavior observed in the first test, it was decided to change the procedureand use a Type-B test for the second Ti-Al rod, in which the Ti-Al rod was immersed in the CP-Ti liquidpool for 7.6 s followed by shutting off the power to the EB gun with the rod remaining in the liquid pool.After cooling to room temperature, a section approximately 5 mm thick was removed from the puck forfurther analysis, as shown in Fig. 6.4.Face-1 of the section is shown in Fig. 6.5a after polishing and face-2 is shown in Fig. 6.5b. As canbe seen in the expanded view of face-1, there are a number of voids present in the rod approximately 1-2 mm in diameter. Note: for the most part, the rod/melt interface is sharp (see right and bottom faces).However, there is evidence of melting/dissolution of the rod near the melt surface on the left face. On face-2 of the section – see Fig. 6.5b - there is a large void ~4 mm in diameter, a smaller void and evidence ofmelting/dissolution on the left and right-hand-side faces. Note: there is greater degree of melting/dissolutionon the right-hand-side of face-2 consistent with the large amount of melting/dissolution occurring on left-hand-side seen on face-1. All of the voids are located in the region of the rod that was below the melt surfaceand are approximately circular in cross-section. The current thinking is that these voids are a result of Al-vapour bubble formation in material that was either fully molten or semi-solid. In addition, the presence of79face 1face 2Figure 6.4: Sample sectionssharp interfaces suggested a very stable solid shell and therefore that the bubbles formed in liquid internalto the shell. Evidence of a distinct and persistent interface can be seen from the aluminum concentrationline scan appearing in Fig. 6.6. The location of the interface approximately matches the original dimensionof the rod, indicating little or no dissolution of the rod at the location examined after 7.6 s of immersion (incomparison, the CP-Ti rod, that has a higher melting point, was fully melted in 4 s).A review of the literature has produced two versions of the phase diagram, which were reproduced inChapter 2 - see Fig. 2.4a and 2.4b, given by Ref [53] and Ref [54], respectively. In the version of thephase diagram shown in Fig. 2.4b, there is a notable increase in the liquidus temperature with increasingconcentration of Al for dilute solutions. According to this phase diagram, CP-Ti (assumed to be pure) has amelting point of ~1668 ◦C. The liquidus temperature increases with increasing Al content, reaching a peakof ~1700 ◦C at 10.0 at% (5.6 wt% Al), prior to decreasing continuously with additional amounts of Al insolution. Given this, it is possible that Al could diffuse into the CP-Ti shell following its formation to createa higher melting temperature material and hence a persistent shell.To further explore this theory, the third and final rod was dipped into Ti64 liquid following the type-Aprocedure. As Ti64 contains 6 wt% Al it is not possible to form a shell with under 6 wt% Al.In the test, it was found that the Ti-Al rod could be removed successfully after the prescribed immersiontime - 6.1 s. Its profile is presented in Fig. 6.7, in which the dashed blue line represents the initial dimensionof the rod prior to dipping and the dashed red line represents the estimated free surface of molten pool80voidsinterface10 mm(a) Face 110 mm(b) Face 2Figure 6.5: Cross section of the sample – Ti-Al rod dipped in CP-Ti following type-B proceduresduring immersion. It can be seen that the length of the rod increased slightly, whereas the sides (below themelt surface) melted to a great degree. This behavior is similar to what was observed in the CP-Ti/CP-Ticase, with some melting occurring above the pool surface. Thus it would appear that the persistence shelland welding of the rod is at least related in part to the formation of a dilute solution of Al in Ti and thatthe melting temperature of this material is higher than that of CP-Ti, consistent with one of the two phasediagrams (Fig. 2.4b) appearing in the literature.Comparing the rate of melting, it would appear that the dissolution/melting rate of the Ti-Al rod in Ti64is significantly slower than that observed for the CP-Ti rod in CP-Ti liquid - i.e. at 6.1s the amount of theTi-Al rod remaining is comparable to the amount of CP-Ti rod remaining after 3 s. Thus there was theadditional phenomena influencing the melting behavior of the Ti-Al rod.To explore this result further, the rod dipped in Ti64 liquid was sectioned along the centerline to observethe sub-surface structure. The resulting section, following polishing, is presented in Fig. 6.8. As in theearlier experiment conducted on a Ti-Al rod, it can be seen that there are voids present in the region ofthe rod that was located below the melt-pool surface, most of which appear close to the melt surface. This81Scan pathinterfaceFigure 6.6: EDX line scan at the interface on face 1 (see Fig. 6.5a)melt surfaceoriginal profileFigure 6.7: Rod profile - Ti-Al/Ti64 (type-A)finding is again consistent with boiling of Al in semi-solid material (in this case it is likely that the Ti-Alwas not fully liquid as there was not a shell encasing this material). Based on the difference in meltingrate between the Ti-Al rod in Ti64 and CP-Ti rod in CP-Ti, and the presence of voids in the Ti-Al rod, itwould appear that the heat of vaporization of aluminum in semi-solid material plays an important role indetermining the melting kinetics of Ti-Al material in Ti and its alloys.6.1.3.2 Temperature MeasurementTo better understand the melting kinetics of Ti-Al, the temperature data obtained within the rods duringimmersion can be examined. The data for Ti-Al in CP-Ti, Ti-Al in Ti64 and CP-Ti in CP-Ti are presentedin Fig. 6.9. Notes: 1) the square blue data points that occasionally appear above the major trend in the82Figure 6.8: Cross section of the Ti64 (type-A) sampleTi-Al/CP-Ti data is noise that occasionally occurred as a result of EB interference; 2) the light grey datafor the CP-Ti/CP-Ti case denotes data collected following exposure of the thermocouple to titanium liquid;3) the light grey data for the Ti-Al/CP-Ti case indicates data collected after the power was turned off; and4) the light grey data for the Ti-Al/Ti64 case indicates data collected after the rod was extracted from thepool. As can be seen, the Ti-Al rods in both CP-Ti and Ti64 show a similar temperature response in the first5 s, reaching approximately 200 ◦C by 3 s and between 800 and 900 ◦C in 6 s. In contrast, the CP-Ti rod inCP-Ti reached approximately 200 ◦C in 1 s and 1,400 ◦C by 3 s. This is a substantial difference in heat-uprate, which is not associated with differences in melt pool conditions as the EB power and beam pattern wereapproximately the same. This could be due in part to the endothermic heat of vaporization of Al. However,if this were the case one would expect a similar initial heat-up rate in both the Ti-Al and CP-Ti rods until thevaporization temperature is reached (estimated previously to be 1,000 ◦C), which is not what was observed.Rather there is a notable delay in the initial heat up in both Ti-Al rods. Thus, it would appear there is anadditional phenomenon acting to absorb heat in the Ti-Al rod at lower temperatures before the boiling pointof Al is reached. Some evidence in support of this theory can be obtained from Fig. 6.10, which showsimages captured of the melt pool shortly after immersion of the CP-Ti and Ti-Al rods in CP-Ti. As can beseen, the liquid in the vicinity of the Ti-Al rod appears to be cooler (as indicated by the larger region of grey)than in vicinity of CP-Ti rod, supporting the notion that there is an additional phenomenon acting to absorb83heat in the Ti-Al rod in the early stages of immersion, not present in the CP-Ti rod.0 2 4 6 8 10Dip-in time [s]02004006008001000120014001600T [◦C]CP Ti/CP TiTi-Al/CP TiTi-Al/Ti64Figure 6.9: Temperature data collected from thermocouple embedded in the three rods tested.(a) CP-Ti solid in CP-Ti liquid (b) Ti-Al solid in CP-Ti liquidFigure 6.10: Comparison of the photos captured from the top view port of the furnace6.2 Model DescriptionThe previously developed numerical model may be used to examine some of the phenomena contributingto the difference in behavior observed between the CP-Ti and the Ti-Al rod immersion tests that have beenpostulated in the previous section, while also accounting for the difference in fluid flow conditions that mayarise as a result of the compositional flow drivers present in this system.84Only the case applicable to using CP-Ti liquid has been modeled. Development of a model to describeTi-Al melting in CP-Ti liquid proved to be a challenge owing to the limited amount of data available toquantify the thermo physical properties over the applicable composition and temperature ranges. In orderto attempt to account for some of the differences observed between the immersion tests involving CP-Tiand Ti-Al rods, there have been a number of modifications/additions to the model, which are discussed inthe following sections. Note: the previously developed CP-Ti/CP-Ti model took into account the increasedthermal resistance associated with the interface between the rod and the initial shell that solidifies on thesample. In addition, the model for the Ti-Al/CP-Ti case was performed using ANSYS CFX version 15.0,which allows for the implementation of composition-dependent latent heat associated melting/solidification(present in the following sections).6.2.1 Governing EquationsThe governing equations solved to quantify the velocity, temperature and composition fields are identical tothose used in the ethanol/water case – see Eq. 4.1 to Eq. 4.4 in Chapter 4. Since in the present application, thesolute is aluminum and the solvent is titanium, the subscripts indicating conserved component in equation,Eq. 4.4, have been modified and are now expressed in terms of mAl and DAl . Likewise, the mass fraction oftitanium is now obtained by applying the constraint: mAl +mTi = 1.6.2.2 Computational Domain and MeshThe computational domain was identical to that used in the model for the CP-Ti/CP-Ti system. The meshingmethod was also the same but a finer mesh was used – 14,916 hexahedra elements containing 30,418 nodes(shown in Fig. 6.11). A finer mesh was used to be able to better resolve details related to freezing/meltingprocesses as the behavior of Ti-Al solid is more complicated in comparison to CP-Ti.6.2.3 Boundary ConditionsFig. 6.12 summarizes the boundary conditions applied in the model as well as the flow drivers. For brevity,only the compositionally related ones, denoted in red in the figure, are introduced here as the thermallyrelated ones, denoted in black, have been described in the previous chapter.855mm5mm5mm5mmFigure 6.11: Meshing detailsq(EB)copper moldsolid zoneliquid poolq(rad)jAl , q(evap), q(rad)b(T,C)q(rad)q(cnt): contact heat transfer (conduction)q(rad): radiative heat tranferM(T,C): Marangoni forces (thermal, compositional)b(T,C): buoyancy (thermal, compositional)jAl: evaporation of Alq(evap): latent heat by evaporationrzrodq(rad)M(T,C)watercooling channelq(cnt)Figure 6.12: Boundary conditions of the model6.2.3.1 Composition Boundary ConditionsEvaporation - As the Ti-Al solid melts, aluminum is released into the liquid pool,and at the pool surfaceit can evaporate into the furnace chamber. Evaporation of aluminum is a significant phenomenon duringEBBM and EBCHR processing, which can lead to appreciable preferential loss of aluminum owing toits comparatively high vapour pressure. In the model, the evaporation of aluminum was characterized byimposing a mass flux boundary condition, expressed by Eq. 6.1 [49], to the puck’s top surface where liquidis present.86m˙Al = aAlP0Al√MAl2piRgT(6.1)where, m˙Al [kg/(m2⋅s)] is the mass flux of Al, aAl is the activity of Al given in Fig. 6.13 [78], P0Al [Pa] is thevapour pressure of pure aluminum at the local temperature, MAl [kg/mol] is the molar mass of aluminumand Rg [J/(K⋅mol)]is the ideal gas constant. Titanium also evaporates during melting, but at a rate 2 to 3orders of magnitude lower compared to aluminum. Ti evaporation was, therefore, ignored in the model.0.0 0.2 0.4 0.6 0.8 1.0Mass fraction of Al0.00.20.40.60.81.0aAlFigure 6.13: Composition dependent activity6.2.3.2 Thermal Boundary ConditionsLatent Heat Absorbed by Evaporation of Aluminum - The heat absorption associated with the evaporationof aluminum was accounted for in the model by applying a heat flux output (qevap) linked to the evaporativemass loss on the puck’s top surface, as given in Eq. 6.2.qevap = Lvapm˙Al (6.2)where Lvap [J/kg] is the latent heat of vaporization of Al (1.08e4 kJ/kg [79]) and m˙Al is calculated via Eq.6.1.876.2.3.3 Fluid Flow Boundary ConditionsMarangoni Forces - In addition to a thermal Maragoni force, a compositional Marangoni force was alsopresent at the pool surface due to the compositional gradient resulting from a combination of the meltingof Ti-Al solid at the solid/liquid interface and evaporation of Al from the melt pool surface. In the model,similar with the method used in the ethanol/water case, a shear stress (σM) based on Eq. 4.7 was applied tocharacterize both the thermal and compositional Marangoni forces [73]. But for this case, the symbol C inthe equation represents the composition of Al. The variation of ∂γ∂T with temperature is given in Table 5.3and the variation in ∂γ∂C with composition in Fig. 6.14 [80].0.0 0.2 0.4 0.6 0.8 1.0Mass fraction of Al3.02.52.01.51.00.50.00.5γ/C [N/m]Figure 6.14: Compositional surface tension coefficient6.2.4 Material PropertiesSince the material is a mixture of two components, Al and Ti, the material properties are therefore notonly dependent on temperature but also composition. In the model the following approaches were usedto approximately quantify the behavior of the relevant material properties. Note: although the approachesintroduced later calculate the properties in the full composition range (from pure Ti to pure Al), the currentmodel only used a range from 0 – 24 wt% Al (~36 at%) since 24 wt% Al is the composition of the rod, is themaximum composition present in the computational domain.886.2.4.1 DensityThe density for the Ti-Al mixture was calculated based on the expression given in Eq. 6.3.1ρ =mAlρAl(T ) + mTiρTi(T ) (6.3)where, ρ , ρAl and ρTi are the densities of the mixture, pure aluminum and pure titanium, respectively, andmTi and mAl are the mass fractions of Ti and Al, respectively. As a result of it being a binary system,mTi = 1−mAl . Note: both ρAl and ρTi in Eq. 6.3 are a function of temperature [81]. The resulting densityvariation with temperature and composition is visualized in Fig. 6.15.0.0 0.2 0.4 0.6 0.8 1.0Mass fraction of Al200400600800100012001400160018002000Temperature [◦ C]200023002600290032003500380041004400Density [kg/m3]Figure 6.15: Density of the Ti-Al6.2.4.2 Phase ChangeBased on the results of the experiments conducted on the Ti-Al rods, the Ti-Al phase diagram shown in Fig.2.4b [54] has been adopted for approximating the melting/solidification behavior as it indicates an increasein the liquidus temperature for dilute solutions of Al in Ti. There are two areas in the formulation of themodel that require quantification of the evolution in liquid fraction, fl , or solid fraction, fs. Note: fl + fs = 1.The first relates to the manner in which the flow is dampened with increasing solid fraction, or alternatively,un-dampened with increasing liquid fraction – i.e. as the rod melts. The second is concerned with the heatreleased/consumed that is associated with solidification/melting. Note: both solidification and melting occur89during rod immersion, with solidification occurring first (increase in fs) followed by melting (increase infl).Flow Dampening – Recalling, the model does not physically simulate a second solid phase, but adoptsa method of approximating the transition to solid by both increasing the viscosity and adding a momentumsource to reflect a decrease in permeability – see Chapter 4 Section 4.2.2. As can be seen from the phasediagram, Ti and Al will melt over a range in temperature that varies with composition. In the model, thesolid fraction evolution with temperature within the semi-solid region (mushy zone) for a given compositionwas calculated on the basis of Eq. 6.4.fs =Tliq−TTliq−Tsol(6.4)where Tsol [K] and Tliq [K] are the solidus and liquidus temperatures, respectively, which are a function ofcomposition based on simplified interpretation of the Ti-Al phase diagram (Fig. 2.4b). As discussed in thefollowing section on latent heat, in the model, in order to avoid divergence a minimum 50 ◦C range has beenimposed between the liquidus and solidus. Fig. 6.16 shows the modified phase diagram obtained followingapplication of a minimum 50 ◦C liquidus/solidus temperature interval. Fig. 6.17 shows the correspondingvariation in fs implemented in the model.LTi AlxAlTemperature (oC)0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.050060070080090010001100120013001400150016001700180019002000Liquidus (Real) Solidus (Real)Liquid-Solid region in the model Figure 6.16: Ti-Al binary phase diagram adopted in the numerical model for latent heat associated withsolidification/meltingIn the model, an effective viscosity was applied to decrease flow associated with the presence of a solid900.0 0.2 0.4 0.6 0.8 1.0Mass fraction of Al200400600800100012001400160018002000Temperature [◦ C]0.000.120.240.360.480.600.720.840.96Solid fractionFigure 6.17: Solid fraction calculated based on the phase diagramphase, which is a function of solid fraction as presented in Fig. 6.18.0.0 0.2 0.4 0.6 0.8 1.0Solid fraction0.00.51.01.52.02.53.0Viscosity [Pa s]Figure 6.18: Solid fraction dependent viscosityThe permeability decrease, also to account for the presence of a solid phase, is based on Eq. 4.6, whichis dependent on solid fraction as given in Fig. 6.19. The permeability was subsequently used to evaluate themomentum source term on the basis of Eq. 4.5, which dampens the fluid flow gradually in the semi-solid tosolid transition.Latent Heat of Melting/Solidification - In CFX, the release of latent heat associated with melting/solidificationfor a multi-component model can be accounted using the enthalpy method with the expression for the en-910.0 0.2 0.4 0.6 0.8 1.0Solid fraction0123456Kperm [m2]1e 7Figure 6.19: Solid fraction dependent permeabilitythalpy given in Eq. 6.5.H = ∫ TTre fCP,edT (6.5)where H [J/kg] is the specific enthalpy and Tre f [K] is the reference temperature, which is set to 25◦C. Note:CP,e in the above equation is the effective specific heat capacity which accounts for latent heat by adding anadditional term to the specific heat capacity as expressed in Eq. 6.6 [82]:CP,e =CP−Lm∂ fs∂T(6.6)In the second term on the right-hand-side, ∂ fs∂T is set to 0, for T > Tliquidus and for T < Tsolidus. ForTsolidus ≤ T ≤ Tliquidus,∂ fs∂T is held constant for a given composition, but varies with composition – e.g. fs isassumed to vary linearly with temperature for a given composition. The liquidus and solidus temperaturesand the corresponding value for ∂ fs∂T in the semi-solid region is based on the Ti-Al binary phase diagram (seeSection 6.2.4.2). The specific latent heat Lm is set to 300 kJ/kg. This is a guessed value based on those ofpure Ti [2], pure Al [83] and titanium alloys with aluminum as the primary alloying element [84, 25] .The term, Lm∂ fs∂T , results in an abrupt change of CP,e in the transition into or out of the semi-solid region.This makes the model prone to divergence, in particular at compositions where the temperature differencebetween the liquidus and solidus is small - e.g. for large values of ∂ fs∂T . Based on a sensitivity analysis92conducted with the model a minimum of 50 ◦C (solid-liquid region) is needed in the current model configu-ration to avoid divergence. Note: the two-phase solid-liquid region adopted in the model for dampening/un-dampening flow is consistent with that applied for the enthalpy calculation, and it is illustrated in Fig 6.16.The resulting effective specific enthalpy implemented in the model is shown in Fig. 6.20a and the effectivespecific heat capacity in Fig. 6.20b. Note: the abrupt increase in CP,e within the semi-solid region associatedwith solidification/melting.0.0 0.2 0.4 0.6 0.8 1.0Mass fraction of Al200400600800100012001400160018002000Temperature [◦C]17000038960060920082880010484001268000148760017072001926800Specific Enthalpy [J/kg](a)0.0 0.2 0.4 0.6 0.8 1.0Mass fraction of Al200400600800100012001400160018002000Temperature [◦C]80015442288303237764520526460086752Specific heat capacity [J/(kg·K)](b)Figure 6.20: Effective specific enthalpy (a) and effective specific heat capacity (b) adopted in the model6.2.4.3 Thermal ConductivityA review of the open literature found the following relevant thermal conductivity data: 1) the thermal con-ductivity of Ti-50 at% Al (36 wt% Al) over a temperature range from room temperature to 600 ◦C; and 2) thethermal conductivity of Ti-97 at% Al (94.8 wt% Al) at 20 ◦C. In addition, the thermal conductivities of CP-Al and CP-Ti are available over a broad temperature range. Generally, the thermal conductivity of an alloy islower for dilute solutions than for the pure metals. Fig. 6.21 shows the variation in thermal conductivity overthe entire composition range at a given temperature, T , adopted in model formulation. In this figure, whenmAl = 0, the literature-based data kTi at T is adopted and at mAl = 1, kAl at T is adopted. For compositionsin the range 0<mAl<A and B<mAl<1, the thermal conductivity is linearly varied with composition from thevalue for the pure alloy constituent at T to the value obtained from the literature for k36wt%Al at T . Note fortemperatures above 600 ◦C it was assumed to vary linearly until reaching the liquidus where is was assumed93to be 25 W/(m⋅K) and thereafter constant. The slope applied for the variation in both cases (dilute solutionsof Al and for dilute solutions of Ti) was determined from the expression: m = ± kAl−k94.8wt%Al1−0.948 , evaluated at Tfor kAl and 20◦C for k94.8wt%Al . For compositions in the range A<mAl<B, the thermal conductivity is heldconstant at k36wt%Al at T . The resulting variation in thermal conductivity with respect to both compositionand temperature is given in Fig. 6.22.kAl = f (T )kTi = f (T )k mixk94.8wt%AlTi A AlBmAlk36wt%Al = f (T )Figure 6.21: Illustration of adapting thermal conductivity0.0 0.2 0.4 0.6 0.8 1.0Mass fraction of Al200400600800100012001400160018002000Temperature [◦ C]10.038.867.696.4125.2154.0182.8211.6240.4Thermal conductivity [W/(mK)]0.00 0.05 0.10 0.15 0.20 0.25 0.30Mass fraction of Al200400600800100012001400160018002000Temperature [◦ C]101316192225283134Thermal conductivity [W/(mK)]Figure 6.22: Thermal conductivity adopted in the model6.2.4.4 Latent Heat of VaporizationOne of the features observed in the Ti-Al rods following immersion was the presence of porosity, whichhad developed as a result of exposure to elevated temperatures – see Section 6.1.3 for additional details.94It is postulated that these voids are a results of Al vapour bubble formation in liquid or semi-solid Ti-Al.Consequently, in addition to the latent heat of melting the latent heat of boiling was implemented as a heatsink and added to the energy conservation equation. The expression used is given in Eq. 6.7.SE,vap = −ρLvap∂ fv∂ t(6.7)where Lvap is the specific latent heat of vaporization,∂ fv∂ t is the rate of liquid fraction vaporized. To applythis term some adjustments and assumptions have been made to simplify the implementation process andkeep the calculation stable. First, Eq. 6.7 is rewritten toSE,vap = −ρLvap∂ fv∂T∂T∂ t(6.8)It is assumed that vaporization occurs over a temperature range of 50 ◦C, in which, fv varies linearlywith temperature (in other words ∂ fv∂T is a constant). The maximum volume fraction of vaporization canbe determined based on the Al content and is to be ~0.36, using the ideal-mixture rule, since the elementvaporized is predominantly aluminum. Given the maximum volume fraction and temperature range ofvaporization, ∂ fv∂T , can be obtained as it is assumed constant – e.g.∂ fv∂T =0.3650 [K] = 0.0072 [K−1]. The actualboiling temperature under the conditions in the semi-solid material is unknown. The results of a theoreticalanalysis based on the Clausius-Clapeyron Equation are shown in Fig. 6.23. Note: in the figure the labels“Al”, “Ti”, “Al_mix”, “Ti_mix” and “mix” denote the vapour pressures of pure aluminum, pure titanium,aluminum in solution, titanium in solution and the total vapour pressure of both Ti and Al in solution,respectively. A bubble could potentially occur at a temperature where the vapour pressure exceeds the localpressure. Horizontal lines have been added to Fig. 6.23 to demark atmospheric pressure (AP) and theworking pressure (WP) in the furnace. According to Fig. 6.23, vaporization could occur at ~1000 ◦C ator near the surface of the melt, which is below the melting temperature of the Ti-Al solid according to thephase diagram. This indicates that Al vapour-based voids formation could occur in semi-solid Ti-Al. Toapply this in the model, the heat source given in Eq. 6.7 is applied at temperatures above the solidus plus35 ◦C (corresponding solid fraction is 0.3) within a temperature range of 50 ◦C. Note: model sensitivityto the heat of vaporization is presented in the sensitivity analysis. In addition, this release of the heat waslimited to the region of rod immersed in the liquid pool consistent with where porosity was observed (see95the Section 6.1.3 for the details).work pressure (WP)atmospheric pressure (AP)Tb of Ti at APTb of Al at APesitmated Tb at WPFigure 6.23: Estimated vapour pressures6.2.5 Initial ConditionsIn the first configuration of the model (without the rod present), a temperature of 800 ◦C was set within thepuck domain and the mass fraction of Al was set to 0 along with the velocity and the pressure. The modelwas then run until a steady state was reached. The temperature, fluid flow and pressure, obtained in thefirst configuration were then imported into the second configuration (rod lowered into the liquid pool) asthe initial conditions. Similarly, the results computed from the second configuration, once the rod was fullylowered, were then used for the initial conditions in the third configuration (rod held in pool for varioustimes) – see Section 5.2.4 for a more complete description of the three model configurations. The initialconditions applied to the rod included the mass fraction of Al (as measured), the temperature recorded fromthe thermocouple embedded in the rod and a 0 velocity and pressure.6.2.6 Modeling ResultsThe model results are examined qualitatively in terms of contours of selected field variables and then quan-titatively by comparison to the data obtained from the thermocouple embedded in the rod. The results forthe base-case analysis are presented first, which is followed by a presentation of a sensitivity analysis. Thebase-case model conditions with respect to the evolution in liquidus temperature with composition and heatof vaporization of Al were described in the previous section. The sensitivity analysis entailed both changing96how the liquidus temperature evolves with composition and the volume fraction of Al that is vaporized inthe rod.6.2.6.1 Base-Case AnalysisThe modeling results following rod immersion with respect to the fields of fluid flow, composition andtemperature at the times of 0, 1, 3, 5, 7 and 10 s are shown in Fig. 6.24. The contours on the left handside represent the temperature profiles and the contours on the right are the profiles of the mass fraction ofaluminum. Note: the color map of the mass fraction is logarithmic. The vectors illustrate the fluid flowand the red lines delineate an iso-solid fraction equal to 0.1. The gray areas indicate solid fractions > 0.65(viscosity = 3 Pa⋅s). The space between the red lines and the gray regions are the semi-solid regions (mushyzone).Firstly, focusing on the evolution in solid fraction, fs > 0.65, in the rod (gray contour), it can be seenthat the immersion of the Ti-Al solid initially resulted in the prediction of some freezing of liquid titaniumonto the rod, mostly underneath the rod due to the lower temperature of the fluid in the molten pool andweaker fluid flow – see 1 s contour. Note: there is some aluminum released into the melt. By 3 s there issome melting of the material between the frozen material that formed on the rod and the rod predicted by themodel resulting in the formation of a shell at the bottom of the rod and the side. Note: the shell on the side isnon-continuous and the amount is much less than that at the bottom. Internal melting is predicted to continuewith time with some non-continuous shell predicted to remain on the vertical surface and continuous shellon bottom surface through 10 s of simulation – see 10 s contour. Thus, the model appears to be predictingthe formation of a stable and persistent shell consistent with observation, although perhaps to a lesser extentthan observed.Turning to the fs = 0.1 line (red isochron), as can be seen the line moves downward from the bottom ofthe rod and upward from the bottom of the liquid pool (directly below the rod), with increasing immersiontime over the duration of the simulation. Between 5 and 7 s the upper and lower fs = 0.1 isochron linesbelow the rod join indicating the formation of semi-solid bridge between the rod and bottom of the liquidpool, consistent with “welding” of the rod to the bottom of the pool, which was observed.Turning finally to fluid flow, there are four driving forces acting on the fluid in the current situation.These are: 1) thermal buoyancy; 2) compositional buoyancy; 3) thermal Marangoni; and 4) compositional970.06 [m/s]0.000 0.005 0.010 0.015 0.020 0.025 0.030 0.035 0.040radius [m]0.0140.0120.0100.0080.0060.0040.0020.000depth [m]10001150130014501600175019000.06 [m/s]0.000 0.005 0.010 0.015 0.020 0.025 0.030 0.035 0.040radius [m]0.0140.0120.0100.0080.0060.0040.0020.000depth [m]1e-032e-034e-037e-031e-023e-025e-021e-012e-014e-018e-01(a) 0 s0.06 [m/s]0.000 0.005 0.010 0.015 0.020 0.025 0.030 0.035 0.040radius [m]0.0140.0120.0100.0080.0060.0040.0020.000depth [m]10001150130014501600175019000.06 [m/s]0.000 0.005 0.010 0.015 0.020 0.025 0.030 0.035 0.040radius [m]0.0140.0120.0100.0080.0060.0040.0020.000depth [m]1e-032e-034e-037e-031e-023e-025e-021e-012e-014e-018e-01(b) 1 s0.06 [m/s]0.000 0.005 0.010 0.015 0.020 0.025 0.030 0.035 0.040radius [m]0.0140.0120.0100.0080.0060.0040.0020.000depth [m]10001150130014501600175019000.06 [m/s]0.000 0.005 0.010 0.015 0.020 0.025 0.030 0.035 0.040radius [m]0.0140.0120.0100.0080.0060.0040.0020.000depth [m]1e-032e-034e-037e-031e-023e-025e-021e-012e-014e-018e-01(c) 3 s0.06 [m/s]0.000 0.005 0.010 0.015 0.020 0.025 0.030 0.035 0.040radius [m]0.0140.0120.0100.0080.0060.0040.0020.000depth [m]10001150130014501600175019000.06 [m/s]0.000 0.005 0.010 0.015 0.020 0.025 0.030 0.035 0.040radius [m]0.0140.0120.0100.0080.0060.0040.0020.000depth [m]1e-032e-034e-037e-031e-023e-025e-021e-012e-014e-018e-01(d) 5 s0.06 [m/s]0.000 0.005 0.010 0.015 0.020 0.025 0.030 0.035 0.040radius [m]0.0140.0120.0100.0080.0060.0040.0020.000depth [m]10001150130014501600175019000.06 [m/s]0.000 0.005 0.010 0.015 0.020 0.025 0.030 0.035 0.040radius [m]0.0140.0120.0100.0080.0060.0040.0020.000depth [m]1e-032e-034e-037e-031e-023e-025e-021e-012e-014e-018e-01(e) 7 s0.06 [m/s]0.000 0.005 0.010 0.015 0.020 0.025 0.030 0.035 0.040radius [m]0.0140.0120.0100.0080.0060.0040.0020.000depth [m]10001150130014501600175019000.06 [m/s]0.000 0.005 0.010 0.015 0.020 0.025 0.030 0.035 0.040radius [m]0.0140.0120.0100.0080.0060.0040.0020.000depth [m]1e-032e-034e-037e-031e-023e-025e-021e-012e-014e-018e-01(f) 10 sFigure 6.24: Modeling results at different immersion times (contour at left: temperature in ◦C andright: mass fraction of Al)98Marangoni. Beginning first with a look at the buoyancy flow drivers, the thermally induced density differ-ence associated with the presence of cooler and denser liquid adjacent to the cold rod will tend to drive thefluid downward adjacent to the rod. In contrast, the melting of Ti-Al leads to a reduction in density froma compositional standpoint, which would tend to drive the fluid adjacent to the solid upward. As for theMarangoni flow drivers, the existence of cold material adjacent to the rod at the pool’s surface will drivefluid toward the rod – i.e. from an area of high temperature (low surface tension) to the area of low temper-ature (high surface tension) and the presence of material enriched in aluminum adjacent to the rod will tendto drive fluid away from the interface – e.g. from Al-rich regions (low surface tension) to Al-deplete regions(high surface tension). Thus, there are a number of competing flow drivers present.In the present case, it would appear that the thermal flow drivers dominate the flow adjacent to the rod –i.e. thermal buoyancy, giving rise to a large downward flow adjacent to the rod, and the thermal Marangoniforce, giving rise to a surface flow directed toward the rod. As a result, a recirculation flow cell adjacent to therod begins to develop within approximately 1 s that then intensifies through 10 s of the simulation, extendingat its peak approximately 1/2 of the distance across the liquid pool. Note: the formation of the additionalcell, which forms as a result of a plume of liquid rich in aluminum moving upward at approximately 1/2 theradius of the liquid pool. The resulting compositional gradient at the surface causes surface tension forcesdirected away from the enriched zone, both radially inward toward the rod and outward away from the rod.These forces in turn: 1) reinforce the large recirculating flow; and 2) generate a second weaker recirculatingflow in the outer 1/2 of the liquid pool.One of the main reasons for the limited impact of the compositional flow drivers is that the materialthat is enriched in Al is predicted to remain largely encased by the shell (see compositional contours attimes from 3 to 10 s). As a result no large gradient in composition developed in the simulation and thermalbuoyancy forces dominate close to the rod interface.Discrete Temperature Comparison - The temperature evolution with immersion time, extracted from thenumerical model at the position where the thermocouple was embedded, is shown in Fig. 6.25 (red solidline) together with the experimental measurement (blue circles). Note: in the experiment for which the datais presented the EB was shut off instantly after 7.6 s and thus data after that is not plotted in the figure. Ascan be seen from this comparison, there are two major differences between the model predictions and theexperimental measurements: 1) the model predicts a rapid temperature rise that starts at approximately 0.7599s, whereas the data shows this rise to occur at approximately 2 s; and 2) the model predicts a more rapidinitial increase in temperature with time than is measured.Previously, it was argued that the delay in heat up present in the Ti-Al rod-based experiments was notdue to Al vaporization or the presence of a stable shell. The results of the comparison shown in Fig. 6.25,appear to support this as both Al vaporization and the means by which a stable shell forms are accounted forin the model, yet the model does not predict the delay in heat up measured experimentally under the base-case conditions. A final conclusion on this rests on completing a sensitivity analysis, which is undertaken inthe following section.0 1 2 3 4 5 6 7Time [s]020040060080010001200140016001800Temperature [◦ C]experimentmodelFigure 6.25: Temperature comparison between the model and experiment (Ti-Al in CP-Ti)6.2.6.2 Sensitivity AnalysisSensitivity analyses were conducted with respect to two factors: 1) the latent heat associated with Al vapor-ization; 2) reduced thermal conductivity due to Al vaporization; and 3) the change of melting temperaturewith composition.Latent Heat Absorption by Al Vaporization – the effect of latent heat associated with aluminum vaporiza-tion is examined by running a case with the latent heat removed. The temperature monitored at the locationwhere the thermocouple was embedded is presented for the base case and for the case where Al vaporizationis removed in Fig. 6.26. As can be seen from this comparison, the latent heat of vaporization does not delaythe heating up rate as was argued in the previous section. The difference in temperature between the two100cases appears after ~1.2 s when the thermocouple has reached 400 ◦C. The difference emerges after materialin the rod has reached the vaporization temperature. Once reached, the heat up rate in the no vaporiza-tion case becomes higher than that of the base-case, which is expected as vaporization is endothermic andabsorbs in the rod.0 2 4 6 8 10Time [s]020040060080010001200140016001800Temperature [◦C]base caseno boilingFigure 6.26: Temperature comparisonThe temperature contours with velocity vectors at 3 and 7 s extracted from the tested case are presentedin Fig. 6.27. By comparing these results to those of the base-case (Fig. 6.24), it can be concluded that thelatent heat of vaporization impacts the melting in two aspects. First, melting of the rod slows down – e.gat 7 s the rod fully melts in the no vaporization case whereas it is partially melted in the base-case (~40%).Second, the shell formed encasing the rod is more persistent when the heat of vaporization is included.The main reason of this impact is in the model vaporization occurs at a temperature close to the liquidusat the rod composition - i.e. ~1600 ◦C. This temperature is lower than the melting temperature of the shellcontaining a dilute amount of Al. As a result, vaporization, which absorbs a large amount of heat, is able tomaintain the shell for a relatively long time.Thermal Resistance Induced by Al-Vaporization – if Al vapour bubbles are formed at the solid/liquidinterface when the rod is first introduced into the melt, they have the potential to introduce a resistance at theinterface, resulting in the delay in heating observed. However, as the model is unable to physically simulatethe formation of this vapour layer, the thermal conductivity of a region surrounding the rod (1 mm thick) wasreduced to 1 W/(m⋅K) (1 magnitude lower than the actual conductivity), to approximate the effect. Note:1010.06 [m/s]0.000 0.005 0.010 0.015 0.020 0.025 0.030 0.035 0.040radius [m]0.0140.0120.0100.0080.0060.0040.0020.000depth [m]1000115013001450160017501900(a) 3 s0.06 [m/s]0.000 0.005 0.010 0.015 0.020 0.025 0.030 0.035 0.040radius [m]0.0140.0120.0100.0080.0060.0040.0020.000depth [m]1000115013001450160017501900(b) 7 sFigure 6.27: Modeling results for the no heat of vaporization case at different immersion timesthis method exaggerates the effect as suppression of heat transfer is imposed immediately on immersion ofthe rod without the period of time needed for the rod surface to heat up to Al vaporization temperature.The results output for the location of the thermocouple for the modified thermal conductivity case andfor the base-case are plotted in Fig. 6.28. It can be seen that the rate of heat-up is reduced appreciably bythe introduction of a boundary layer resistance, however the approach has little or no impact on the delay ofheating observed in the experiments.0 2 4 6 8 10Time [s]020040060080010001200140016001800Temperature [◦ C]base-casereduce-KFigure 6.28: Temperature comparison between the tested case and base-caseEffect of Small Additions of Al on Liquidus Temperature - The various cases run with the model to exploresensitivity to the liquidus and solidus temperature behavior is presented in Fig. 6.29. Two additional caseshave been run for comparison to the base-case results: one in which the temperature increase is magnifiedand one in which the temperature exhibits a monotonic decrease.The evolution in temperature at the location of the thermocouple, for all three cases, is presented in102Ti AlxAlTemperature (oC)0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.050060070080090010001100120013001400150016001700180019002000LS16001700180019001500Increase-case liquidusIncrease-case solidusBase-case liquidusBase-case solidusDecrease-case liquidusDecrease-case solidus Figure 6.29: Tested phase diagramsFig. 6.30. In general, the three temperatures are similar with only slight differences. The largest differenceappears at ~ 4s and is ~200 ◦C. This result shows that the initial delay in heat up is not affect by the assumedbehavior of the liquidus temperature for dilute solutions of Al in Ti.0 2 4 6 8 10Time [s]020040060080010001200140016001800Temperature [◦ C]base-caseincrease-casedecrease-caseFigure 6.30: Temperature comparison between the tested cases and base-caseHowever, it is interesting to note that the “increase-case” results in a generally higher temperature pre-diction after approximately 2 s than is predicted for the other two cases. To explain this, the temperaturecontours with flow field (velocity vectors) at 3 and 7 s are plotted for the increased and decreased casesin Fig. 6.31. The difference is caused by the stronger flow in the “increase-case”, which can be seen bycomparing the velocity vectors in both 3 and 7 s. Stronger flow present in the “increased-case” enhancesthe convective heat transfer and also brings more fluid from the surface that possesses relatively high tem-103perature towards the rod, resulting in higher temperatures adjacent to the rod. As a result, the melting in the“increase-case” is faster, and the shell behaviors in the two cases are different - i.e. “welding” is preventedat the bottom and less shell formation on the side in the “increase-case” due to the enhanced heat transfer.0.06 [m/s]0.000 0.005 0.010 0.015 0.020 0.025 0.030 0.035 0.040radius [m]0.0140.0120.0100.0080.0060.0040.0020.000depth [m]1000115013001450160017501900(a) increase-case 3 s0.06 [m/s]0.000 0.005 0.010 0.015 0.020 0.025 0.030 0.035 0.040radius [m]0.0140.0120.0100.0080.0060.0040.0020.000depth [m]1000115013001450160017501900(b) increase-case 7 s0.06 [m/s]0.000 0.005 0.010 0.015 0.020 0.025 0.030 0.035 0.040radius [m]0.0140.0120.0100.0080.0060.0040.0020.000depth [m]1000115013001450160017501900(c) decrease-case 3 s0.06 [m/s]0.000 0.005 0.010 0.015 0.020 0.025 0.030 0.035 0.040radius [m]0.0140.0120.0100.0080.0060.0040.0020.000depth [m]1000115013001450160017501900(d) decrease-case 7 sFigure 6.31: Temperature contours with velocity vectors of the two tested cases at different immersion timesTo further explore the reason for this, it is necessary to look at the composition fields in the two cases.The contours of mass fraction of aluminum at 3 and 7 s are presented in Fig. 6.32. By comparing the twocontours at 3 s, it can be seen that the “increase-case” has less aluminum released out of the shell to thebulk liquid. This is because the shell in the “increase-case” is more continuous (though less at the bottom),due to the increasing melting temperature with the concentration of aluminum at dilute solutions. Thisleads to a smaller composition gradient and consequently weaker compositional flow drivers - buoyancy andMarangoni forces. As discussed previously in Section 6.2.6.1, the thermal and compositional buoyancy flowdrivers adjacent to the interface oppose one another, as do the compositional and thermal Marangoni flowdrivers. Thus, the initially reduced flux of aluminum into the liquid at the interface results in a stronger flowand increased heat transfer.In summary, the results of the sensitivity analyses reveal that the melting process (melting kinetics, shellbehavior and fluid flow adjacent to the rod) is sensitive to both aluminum vaporization and the behavior ofthe liquidus temperature for dilute solutions of Al in Ti. The base case results appear to give the best overallagreement with measurements. It is also clear from the sensitivity analysis that neither Al vaporization or1040.06 [m/s]0.000 0.005 0.010 0.015 0.020 0.025 0.030 0.035 0.040radius [m]0.0140.0120.0100.0080.0060.0040.0020.000depth [m]1e-032e-034e-037e-031e-023e-025e-021e-012e-014e-018e-01(a) increase-case 3 s0.06 [m/s]0.000 0.005 0.010 0.015 0.020 0.025 0.030 0.035 0.040radius [m]0.0140.0120.0100.0080.0060.0040.0020.000depth [m]1e-032e-034e-037e-031e-023e-025e-021e-012e-014e-018e-01(b) increase-case 7 s0.06 [m/s]0.000 0.005 0.010 0.015 0.020 0.025 0.030 0.035 0.040radius [m]0.0140.0120.0100.0080.0060.0040.0020.000depth [m]1e-032e-034e-037e-031e-023e-025e-021e-012e-014e-018e-01(c) decrease-case 3 s0.06 [m/s]0.000 0.005 0.010 0.015 0.020 0.025 0.030 0.035 0.040radius [m]0.0140.0120.0100.0080.0060.0040.0020.000depth [m]1e-032e-034e-037e-031e-023e-025e-021e-012e-014e-018e-01(d) decrease-case 7 sFigure 6.32: Aluminum mass fraction contours with velocity vectors of the two tested cases at differentimmersion timesthe behavior of the liquidus temperature are responsible for the delay in heat up observed in the Ti-Al rodsin comparison to the CP-Ti rod and that therefore there is a sink of heat active early in the melting processthat is currently missing from the model.6.3 Summary and ConclusionsA work has been carried out to investigate the melting of a Ti-Al solid in titanium liquid, which involved boththermal and compositionally related phenomena. The melting behavior was observed to differ significantlyfrom the CP-Ti/CP-Ti and in a way that is counter intuitive. The formation of persistent Ti-Al shell andvaporization of aluminum in semi-solid Ti-Al, both appear to contribute to a reduction in the melting kineticsof Ti-Al rods in comparison to CP-Ti rods, even though at the bulk composition of the rod the rod has asignificantly lower melting point than CP-Ti. The discrete temperature measurements have also reveal asignificant initial delay in the heat-up rate of the Ti-Al rod, the reason for which is not obvious.A numerical model was developed to aid in understanding the experimental observations in terms of thevarious physical phenomena contributing to melting. Qualitatively the model is able to capture the importantphenomena observed in the experiments - e.g. shell formation, internal melting and rod “welding” with thebottom of the molten pool. The model indicates the thermal flow drivers dominate the fluid flow adjacent to105the rod as well as the in the bulk liquid. The impact of the compositional flow drivers is limited, which ismostly because the material that is enriched in Al remained largely encased by the shell and consequently nolarge gradient in composition developed in the simulation. The quantitative comparison between the model-ing and experimental results shows the delay of heating up observed in experiments is not characterized bythe model. This delay is not attributed to either the Al vaporization or the increasing melting temperatureof the Ti-Al compound for dilute concentrations of aluminum, according to the sensitivity analysis with themodel.Overall an improved understanding on the melting of Ti-Al solid in titanium liquid has been developedwith the aid of this study. However, there are still some phenomena including the delay of heating upobserved in the experiments cannot be thoroughly understood and more work is required in terms of bothexperimental analysis and modeling development.106Chapter 7Summary and ConclusionsMotivated by reducing the occurrence of so-called condensate “drop-in’ related defects during EBCHRprocessing of Al-bear titanium alloys, work has been undertaken to study the factors that influence themelting/dissolution of solids in liquid titanium.Ethanol/Water Study - The investigation focused initially on a low-temperature, transparent, analoguesystem to allow observation of the basic phenomena and the straightforward collection of data suitable fordevelopment and validation of a numerical modeling framework. The materials examined included cylindersof both solid water (ice) and solid ethanol in liquid water, and the configurations tested included conditions inwhich the solid cylinders were both partially and fully immersed in the liquid. The combination of materialsand configurations tested allowed experiments to be conducted in which the following flow drivers werepresent: 1) thermal buoyancy only; 2) thermal buoyancy and thermal Marangoni (surface tension drivenforce); 3) thermal and compositional buoyancy; and 4) thermal and composition buoyancy and Marangoni.The data collected included the variation in the solid/liquid profile with time and the evolution of temperaturewith time as obtained from thermocouples embedded in the solid cylinders.The results indicated that the various flow drivers influence the development of the interfacial boundarylayer and impact on heat and mass transfer. In the ice/water case, the thermal buoyancy force enhanced themelting due to the induced flows at the solid/liquid interface. The thermal Marangoni-based flow was alsoobserved to enhance melting at the free surface. In the frozen ethanol/water case, the thermal buoyancyand compositional buoyancy forces act in opposite directions. The resultant force causes a net upwardflow along the various faces of the sample. This in turn draws in warm water at the base of the sampleaccelerating melting at the base. The thermal and compositional Marangoni forces also oppose one anotherwith the result being a net force acting away from the solid interface at the free surface reducing melting inproximity to the free surface.A mathematical model based on the commercial code ANSYS-CFX, was developed for the various con-107figurations and validated using the data obtained from the various tests. The mathematical model developedwas shown to be able to reproduce the variation in solid/liquid interface profile with time and evolution intemperature with time within minimal error for most of the experimental conditions examined. The oneexception was for the evolution in the solid/liquid interface in the ice/water test where the model was foundto predict a narrow-end-down profile whereas the experiment shows a narrow-end-up profile. Overall, it wasshown that the model is capable of describing the various physical phenomena influencing heat and masstransfer under the range of conditions examined. It may be concluded therefore that the model provides areasonable framework for application to liquid metal systems that exhibit similar characteristics in terms ofthermo-physical property differences between the solvent and solute.The numerical model-based results for the ice/water and ethanol/water experiments were then used toestimate EIHTCs and the results for the ethanol/water experiments, an EIMTC. For the conditions examined,the EIHTC was found to lie in the range of 600 to 4,000 W/(m2⋅K) for ice/water and 500 to 1,300 W/(m2⋅K)for ethanol/water. The EIMTC for water/ethanol was found to lie in the range 10−4 to 10−3 m/s. Theeffective heat and mass transfer coefficients were then compared with values calculated from literature-based correlations and were found to behave qualitatively similar with time (extent of melting) and in termsof their dependence on the experimental conditions. However, it was found that is was necessary to modifythe constant term(s) in the correlations in order to achieve quantitative agreement. Thus, it was concludedthat application of empirical-based heat and mass transfer coefficients will be prone to error and may needto be modified depending on the particular aspects of the system under analysis. One example would besystems in which the solid is lighter than the liquid and Maragoni driven flow is present.Solid CP-Ti/Liquid CP-Ti Study - Following completion of the work on the low-temperature analoguesystem, work shifted to focus on the melting behavior of solid CP-Ti in molten CP-Ti, which allowedtests to be completed on the titanium system but in the absence of compositional effects. These tests wereundertaken by dipping CP-Ti rods into a molten pool of CP-Ti and were conducted using the Electron BeamButton Furnace at UBC. The CP-Ti rods were dipped into the CP-Ti liquid for various periods of time andthen extracted to determine the evolution in the solid/liquid interface profile with time. A total of 4 rodswere used with immersion times of 1, 2, 3 and 4 s. In addition, the temperature evolution within the 4 s rodwas measured with a thermocouple embedded in the rod.When immersed into the molten pool, the cold rod quickly cools down the surrounding melt and, at some108locations, freezes a layer of titanium on its surface. Most of the freezing was found to occur on the rod’sbottom surface due to the relatively low temperature and weaker convective heat transfer at this location inthe molten pool. The solid/solid interface that forms between the rod and frozen titanium shell was foundto introduce a thermal resistance, which significantly reduced heat transfer to the dipped rod while present.Once the frozen titanium melts and the solid/solid interface vanishes, the dipped rod is heated up rapidlyand then melts.Following the experimental work, a numerical model was developed for the EBBF experiments based onthe modeling framework established from the ethanol/water study with the appropriate modifications madeto the analysis domain, material properties and boundary conditions. The numerical model was shownto reproduce the experimental results with only a minimal error, indicating that the model is capable ofapproximating the main physical phenomena influencing the melting behavior of a CP-Ti rod introducedinto liquid CP-Ti. With the aid of the model, a sensitivity analysis was conducted to examine the relativeimportance of thermal buoyancy and thermal Marangoni flow drivers. Both were found to be importantunder the conditions examined in the study. To support application of the data derived from this study inindustrial-scale applications, EIHTCs have been calculated based on the results from the numerical model.The EIHTC was found to lie in the range of 26,000 to 36,000 W/(m2⋅K). To confirm the numerically basedEIHTC results, EIHTCs were also estimated using boundary layer theory. The methods produced results inreasonable agreement providing an appropriate bulk temperature was selected in method applied using thenumerical model-based results.Solid Ti-Al/Liquid CP-Ti and Solid Ti-Al/ Liquid Ti64 - Finally attention was turned to study the meltingof a Ti-Al solid in liquid titanium using methods that were similar to those applied in the CP-Ti solid/CP-Tiliquid study. Both CP-Ti and Ti64 buttons were used to generate the liquid pool. Of the cases examined, thisfinal series of experiments and numerical analysis best represents the industrial problem of solid condensateoccasionally entering the melt pool in the refining hearth or mold, as it is focused on material with a relativelyhigh Al content and therefore includes both thermal and compositional effects.The results of the experiments revealed the melting behavior of Ti-Al to be significantly different fromthe CP-Ti solid and in a way that is counter intuitive. In both the CP-Ti button experiments and Ti64 buttonexperiments the melting kinetics were found to be significantly slower in comparison to CP-Ti rods, eventhough the Ti-Al rods tested have a significantly lower melting point than CP-Ti at the bulk composition of109rod. In the CP-Ti button experiments, the formation of persistent Ti-Al shell and vaporization of aluminumin encased semi-solid Ti-Al, both appear to contribute to a significant reduction in the melting/dissolutionkinetics of Ti-Al rods. In the Ti64 buttons, the melting kinetics was also retarded, but to a less extent,which was attributed to Al vaporization only (no persistent solid shell was observed). Al vaporization wasconfirmed from a metallographic examination of the partially melted rods, which showed voids present inthe sample that formed in the portion of the rod located below the liquid metal level. One of the two Ti-Alphase diagrams identified in the literature review showed an increase in the liquidus temperature for dilutesolutions of Al in Ti. This increase in liquidus temperature appears to be responsible for the formation of thestable shell in the CP-Ti button experiments. A comparison of the discrete temperature evolution data forthe three cases - CP-Ti/CP-Ti, Ti-Al/CP-Ti and Ti-Al/Ti64 - also confirmed a significant difference in theheat up rate between the CP-Ti rod experiments and Ti-Al rod experiments (including a significant initialdelay in heating).To better understand and confirm the mechanisms contributing to the slower kinetics observed in theTi-Al rod experiments, the numerical model was updated to include compositional effects. These includedthe following: 1) addition of a compositional dependence to the liquidus and solidus temperature, basedon the phase diagram which includes the increase in liquidus temperature for dilute solutions of Al in Ti;and 2) the addition of the endothermic heat of vaporization of aluminum above the liquidus temperatureof the Ti-Al alloy rod. Application of the model showed that the model is able to qualitatively capture theimportant phenomena observed in the experiments - e.g. stable shell formation, internal melting and rod“welding” with the bottom of the molten pool. The model indicates the thermal flow drivers dominate thefluid flow adjacent to the rod as well as the in the bulk liquid. The impact of the compositional flow driversis limited, which is mostly because the material that is enriched in Al remained largely encased by the shelland consequently no large gradient in composition developed in the simulation. The quantitative comparisonbetween the modeling and experimental results shows the delay of heating up observed in experiments is notcharacterized by the model. A sensitivity analysis conducted with the model confirmed that the initial delayin heating is not attributed to either the Al vaporization or the increasing melting temperature of the Ti-Alcompound for dilute concentrations of aluminum. Therefore there appears to be an additional endothermicreaction that occurs in the early stages of immersion of the rod in the liquid pool.110Chapter 8Future WorkIt is acknowledged that this work has limitations in terms of both experimental methods and analyses usedand in terms of the numerical model used to support the experimental program. Further work is recom-mended in the following areas:1. The reason for the delay in initial heat-up in the melting of Ti-Al solids is currently not undertood andadditional work is needed to clarifying reason of this.2. The synthetic Ti-Al solid tested in this study is different from the actual condensate that forms inindustrial furnaces in terms of both morphology and composition. The composition of the real con-densate is ~72 wt% Al, whereas the synthetic solid is ~24 wt % Al. Moreover, the real condensateis porous, whereas the synthetic solid is dense. 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