感應式熔爐對金屬材料的加熱效率高、速度快，且兼具環保、安靜與良好可控性等優點，泛用於現今工業的金屬熔煉；而基於感應加熱原理的設備，也常使用於熱處理、焊接以及熱成型等製程。鈦合金材料的特性包含了強度高、抗腐蝕性強、高溫與低溫的力學性質好等，在各領域佔有重要價值，包括航太產業與生醫產業等。然而，因為鈦金屬本身熔點較高，並且熔融後具有部分程度的侵蝕性，對於熔爐的部件都會造成物理性與化學性的破壞，並且減短熔爐材料的壽命，亦衍生為經濟成本的顧慮。本文使用有限元素數值模擬方法，建立一套二維軸對稱的無鐵心式感應電爐模型，進行鈦合金熔煉爐體之熱應力分析。感應電爐的原理係基於電磁感應，以交流電線圈產生的交變磁場，使爐內的金屬材料產生渦電流，並經由焦耳效應產生熱能，直至材料完成熔融。本文的分析模式包含了電磁場、溫度場以及力學場，進行多重物理量耦合分析模擬，探討輸入交流電功率與頻率、爐體坩堝的壁厚、熔池的半徑與高度等參數，在鈦合金熔融的過程中，其對於爐體所產生的溫度與熱應力的影響。模擬結果顯示，當壁厚增加時，可有效降低坩堝外側之溫度，然而內側的壓縮主應力則會升高。以坩堝壁厚300 mm相較於100 mm為例，當鈦合金完全熔融時，坩堝外側之最大溫度可從1339 K大幅降低至635 K，減少約704 K；內側最大壓縮應力值，則會由6.61 GPa增加為11.11 GPa，增加了4.5 GPa，為約增加68%。此外，為減少應力集中的影響，坩堝爐壁與底部之間的圓角尺寸，須隨壁厚的增厚而擴大，相關結果可作為感應電爐爐體設計的重要參考。

This thesis utilizes finite element analysis on induction furnace simulation of melting titanium alloy, coupling fields including magnetic-electric field, heat transfer and solid mechanics. While titanium in liquidus temperature is in the nature of thermophysical and chemical aggressiveness, certain crucible failure due to thermal stresses is expected. The 2-D axisymmetric model contains a beryllia crucible with yttria coating, melted material (Ti-6Al-4V), copper coils and cooling water channel, formed a coreless induction furnace. Several assumptions and approaches were adopted, as well as the effects of alternating current power and frequency inputs were discussed in the very first part of the results. The main section of the simulation results demonstrates the comparisons of temperature and stresses distributions with different furnace geometries, including melting pool radii, crucible wall thicknesses, melting pool heights and melted material capacities, etc. The temperature and stresses distribution mainly depend on the relations of power input and material capacity, that is, the duration of completely melted. Under the identical power input, the crucibles with larger melting pool and/or thicker wall tend to be induced less thermal stresses. On the other hand, crucibles with smaller melting pool and/or thinner wall will have more heat conducted to outside, resulting in safety concerns. Moreover, the amounts of principal stresses that exceed material strength of the crucible were further described.

[1] V. Rudnev, D. Loveless, and R. Cook, Handbook of induction heating, Second edition. ed. (Manufacturing engineering and materials processing, no. 61). Boca Raton, FL: CRC Press, Taylor & Francis Group, 2017, pp. xxi, 749 pages.
[2] C. L. a. M. Peters, Titanium and Titanium Alloys: Fundamentals and Applications. Weinhiem: Wiley-VCH, 2003, p. 513.
[3] I. V. Gorynin, "Titanium alloys for marine application," Mat Sci Eng a-Struct, vol. 263, no. 2, pp. 112-116, May 15 1999.
[4] A. Khataee, H. M. Flower, and D. R. F. West, "New Titanium-Aluminum-X Alloys for Aerospace Applications," J Mater Eng, vol. 10, no. 1, pp. 37-44, 1988.
[5] A. Morita, H. Fukui, H. Tadano, S. Hayashi, J. Hasegawa, and M. Niinomi, "Alloying titanium and tantalum by cold crucible levitation melting (CCLM) furnace," Mat Sci Eng a-Struct, vol. 280, no. 1, pp. 208-213, Mar 15 2000.
[6] H. J. Rack and J. I. Qazi, "Titanium alloys for biomedical applications," Mat Sci Eng C-Bio S, vol. 26, no. 8, pp. 1269-1277, Sep 2006.
[7] M. Z. Yu et al., "3D Printed Ti-6Al-4V Implant with a Micro/Nanostructured Surface and Its Cellular Responses," Acs Omega, vol. 5, no. 49, pp. 31738-31743, Dec 15 2020.
[8] 李勝隆, 工程材料科學：原理與應用. 新北市: 高立圖書有限公司, 2016, p. 458.
[9] 中國材料科學學會材料手冊編審委員會, 許樹恩, Ed. 材料手冊 II. 非鐵金屬材料. 高雄市小港: 中國材料科學學會, 中華民國七十二年.
[10] B. Dutta and F. H. Froes, "Additive Manufacturing of Titanium Alloys," Adv Mater Process, vol. 172, no. 2, pp. 18-23, Feb 2014.
[11] P. Sun, Z. Z. Fang, Y. Xia, Y. Zhang, and C. S. Zhou, "A novel method for production of spherical Ti-6Al-4V powder for additive manufacturing," Powder Technol, vol. 301, pp. 331-335, Nov 2016.
[12] H. Mehboob et al., "A novel design, analysis and 3D printing of Ti-6Al-4V alloy bio-inspired porous femoral stem," J Mater Sci-Mater M, vol. 31, no. 9, Aug 20 2020.
[13] C. Y. Liang et al., "Preparation and surface modification of 3D printed Ti-6Al-4V porous implant," Rare Metals, May 19 2020.
[14] N. Bara, "Review Paper on Numerical Analysis of Induction Furnace," International Journal of Latest Trends in Engineering and Technology, vol. 2, no. 3, pp. 178-184, May 2013.
[15] A. Hiorth, "Design of a 30-ton induction electric furnace," J Ind Eng Chem-Us, vol. 3, pp. 849-855, Nov 1911.
[16] A. Gemant and J. Sticher, "Crucible Failure in the Induction Melting Process," J Appl Phys, vol. 16, no. 11, pp. 661-667, 1945.
[17] G. O. Thacker and C. H. G. Hands, "Low Frequency Induction Heating as Applied to Small Scale Chemical Plant," J Soc Chem Ind-L, vol. 66, no. 7, pp. 211-215, 1947.
[18] J. C. Matthews, "A Small Vacuum Induction Furnace," J Sci Instrum, vol. 34, no. 2, pp. 62-63, 1957.
[19] E. Taberlet and Y. Fautrelle, "Turbulent Stirring in an Experimental Induction Furnace," J Fluid Mech, vol. 159, no. Oct, pp. 409-431, 1985.
[20] I. V. Shurygina, Y. A. Polonskii, I. F. Masover, B. P. Aleksandrov, V. I. Dobrovolskaya, and V. M. Byndin, "Technology of Producing High-Grade Electroengineering Periclase by Melting in an Induction Furnace with a Cold Crucible," Refractories-Ussr+, vol. 22, no. 11-1, pp. 622-627, 1981.
[21] T. Takasu, K. Sassa, and S. Asai, "Direct Induction Skull Melting of Electrical Conducting Materials Using Crucible with Insulating Coat and Its Heat-Conduction Analysis," Tetsu to Hagane, vol. 80, no. 12, pp. 878-883, Dec 1994.
[22] L. Smith, "The Role of Coreless Induction Furnaces for Recycling Aluminum Scrap," Resour Conserv Recy, vol. 10, no. 1-2, pp. 185-191, Apr 1994.
[23] C. Marchand and A. Foggia, "2d Finite-Element Program for Magnetic Induction-Heating," Ieee T Magn, vol. 19, no. 6, pp. 2647-2649, 1983.
[24] D. C. Ko, G. S. Min, B. M. Kim, and J. C. Choi, "Finite element analysis for the semi-solid state forming of aluminium alloy considering induction heating," J Mater Process Tech, vol. 100, no. 1-3, pp. 95-104, Apr 3 2000.
[25] A. Bermudez, D. Gomez, M. C. Muniz, and P. Salgado, "Transient numerical simulation of a thermoelectrical problem in cylindrical induction heating furnaces," Adv Comput Math, vol. 26, no. 1-3, pp. 39-62, Jan 2007.
[26] A. Bermudez, D. Gomez, M. C. Muniz, R. Salgado, and R. Vazquez, "Numerical simulation of a thermo-electromagneto-hydrodynamic problem in an induction heating furnace," Appl Numer Math, vol. 59, no. 9, pp. 2082-2104, Sep 2009.
[27] A. Bermudez, C. Reales, R. Rodriguez, and P. Salgado, "Numerical analysis of a finite-element method for the axisymmetric eddy current model of an induction furnace," Ima J Numer Anal, vol. 30, no. 3, pp. 654-676, Jul 2010.
[28] P. Bulinski et al., "Effect of Turbulence Modelling in Numerical Analysis of Melting Process in an Induction Furnace," Arch Metall Mater, vol. 60, no. 3, pp. 1575-1579, 2015.
[29] B. Patidar, M. M. Hussain, S. K. Jha, A. Sharma, and A. P. Tiwari, "Analytical, numerical and experimental analysis of induction heating of graphite crucible for melting of non-magnetic materials," Iet Electr Power App, vol. 11, no. 3, pp. 342-351, Mar 2017.
[30] T. Tanaka, A. Kuroda, and K. Kurita, "Continuous-Casting of Titanium-Alloy by an Induction Cold Crucible," Isij Int, vol. 32, no. 5, pp. 575-582, 1992.
[31] A. Karwinski, W. Lesniewski, P. Wieliczko, and M. Malysza, "Casting of Titanium Alloys in Centrifugal Induction Furnaces," Arch Metall Mater, vol. 59, no. 1, pp. 403-406, 2014.
[32] M. Palacz et al., "Experimental Analysis of the Aluminium Melting Process in Industrial Cold Crucible Furnaces," Met Mater Int, vol. 26, no. 5, pp. 695-707, May 2020.
[33] E.J. Chapin and H.W. Friske, "A Metallurgical Evaluation of Refractory Compounds for Containing Molten Titanium. Part II - Carbon, Graphite, and Carbides," United States Naval Res. Lab., Washinton, D.C., USA, Jan 1955.
[34] N. M. Griesenauer, Alexande.Ca, and S. R. Lyon, "Vacuum Induction Melting of Titanium," J Vac Sci Technol, vol. 9, no. 6, pp. 1351-1355, 1972.
[35] C. E. Holcombe and T. R. Serandos, "Consideration of Yttria for Vacuum Induction Melting of Titanium," Metall Trans B, vol. 14, no. 3, pp. 497-499, 1983.
[36] H. R. Zhang, X. X. Tang, L. Zhou, M. Gao, C. G. Zhou, and H. Zhang, "Interactions between Ni-44Ti-5Al-2Nb-Mo alloy and oxide ceramics during directional solidification process," J Mater Sci, vol. 47, no. 17, pp. 6451-6458, Sep 2012.
[37] H. R. Zhang, X. X. Tang, C. G. Zhou, H. Zhang, and S. W. Zhang, "Comparison of directional solidification of gamma-TiAl alloys in conventional Al2O3 and novel Y2O3-coated Al2O3 crucibles," J Eur Ceram Soc, vol. 33, no. 5, pp. 925-934, May 2013.
[38] J. P. Kuang, R. A. Harding, and J. Campbell, "Investigation into refractories as crucible and mould materials for melting and casting gamma-TiAl alloys," Mater Sci Tech-Lond, vol. 16, no. 9, pp. 1007-1016, Sep 2000.
[39] J. Barbosa, C. S. Ribeiro, and C. Monteiro, "Processing of gamma TiAl, by ceramic crucible induction melting, and pouring in ceramic shells," Mater Sci Forum, vol. 426-4, pp. 1933-1938, 2003.
[40] J. Barbosa, H. Puga, C. S. Ribeiro, O. M. N. D. Teodoro, and A. C. Monteiro, "Characterisation of metal/mould interface on investment casting of gamma-TiAl," Int J Cast Metal Res, vol. 19, no. 6, pp. 331-338, Dec 2006.
[41] F. Smeacetto, M. Salvo, and M. Ferraris, "Protective coatings for induction casting of titanium," Surf Coat Tech, vol. 201, no. 24, pp. 9541-9548, Oct 15 2007.
[42] A. V. Kartavykh, V. V. Tcherdyntsev, and J. Zollinger, "TiAl-Nb melt interaction with AlN refractory crucibles," Mater Chem Phys, vol. 116, no. 1, pp. 300-304, Jul 15 2009.
[43] F. Gomes, H. Puga, J. Barbosa, and C. S. Ribeiro, "Effect of melting pressure and superheating on chemical composition and contamination of yttria-coated ceramic crucible induction melted titanium alloys," J Mater Sci, vol. 46, no. 14, pp. 4922-4936, Jul 2011.
[44] F. Gomes, J. Barbosa, and C. S. Ribeiro, "Induction melting of gamma-TiAl in CaO crucibles," Intermetallics, vol. 16, no. 11-12, pp. 1292-1297, Nov-Dec 2008.
[45] K. Liu et al., "Single step centrifugal casting TiAl automotive valves," Intermetallics, vol. 13, no. 9, pp. 925-928, Sep 2005.
[46] J. Barbosa, C. S. Ribeiro, and A. C. Monteiro, "Influence of superheating on casting of gamma-TiAl," Intermetallics, vol. 15, no. 7, pp. 945-955, Jul 2007.
[47] C. L. J. Morscheiser, and B. Friedrich, "Potential of Ceramic Crucibles for Melting of Titanium-Alloys and Gamma-Titaniumaluminide," 2008.
[48] A. H. Liu, B. S. Li, H. Nan, Y. W. Sui, J. J. Guo, and H. Z. Fu, "Study of interfacial reaction between TiAl alloys and four ceramic molds," Rare Metal Mat Eng, vol. 37, no. 6, pp. 956-959, Jun 2008.
[49] R. J. Cui, M. Gao, H. Zhang, and S. K. Gong, "Interactions between TiAl alloys and yttria refractory material in casting process," J Mater Process Tech, vol. 210, no. 9, pp. 1190-1196, Jun 19 2010.
[50] J. W. Koger, C. E. Holcombe, and J. G. Banker, "Coatings on Graphite Crucibles Used in Melting Uranium," Thin Solid Films, vol. 39, no. Dec, pp. 297-303, 1976.
[51] J. B. Condon and C. E. J. Holcombe, "Crucible Materials to Contain Molten Uranium," Oak Ridge Y-12 Plant, Tenn., USA, Report. Y--2084, Sep 1977.
[52] N. Alangi et al., "Liquid uranium corrosion studies of protective yttria coatings on tantalum substrate," J Nucl Mater, vol. 410, no. 1-3, pp. 39-45, Mar 2011.
[53] J. Sure et al., "Microstructural characterization and chemical compatibility of pulsed laser deposited yttria coatings on high density graphite," Thin Solid Films, vol. 544, pp. 218-223, Oct 1 2013.
[54] T. Tetsui, T. Kobayashi, T. Ueno, and H. Harada, "Consideration of the influence of contamination from oxide crucibles on TiAl cast material, and the possibility of achieving low-purity TiAl precision cast turbine wheels," Intermetallics, vol. 31, pp. 274-281, Dec 2012.
[55] M. Masuda, G. Yagawa, and Y. Ando, "Thermal-Analysis in Induction-Heating Stress Improvement," J Atom Energ Soc Jpn, vol. 22, no. 7, pp. 482-487, 1980.
[56] W. B. Kim and S. J. Na, "A Study on Residual-Stresses in Surface Hardening by High-Frequency Induction-Heating," Surf Coat Tech, vol. 52, no. 3, pp. 281-288, May 20 1992.
[57] J. I. Ghojel, "Thermal analysis of twin-channel induction furnaces," Metall Mater Trans B, vol. 34, no. 5, pp. 679-684, Oct 2003.
[58] S. A. Halvorsen, "A Model for High Temperature Inductive Heating," presented at the Proceedings of the COMSOL Conference, Milan, 2009.
[59] S. K. Seshadri, R. M. Pillai, and P. K. Rohatgi, "Pollution and Productivity Control Measures for Iron Foundries Using Cupola Furnace," J Sci Ind Res India, vol. 42, no. 7, pp. 377-382, 1983.
[60] H. Patil, G. Patange, and M. P. Khond, "Investigation of Pollution Emits By Cupola Furnace in Gujarat Foundry," IJETT, vol. 4, no. 5, pp. 1573-1575, May 2013.
[61] W. D. Callister and D. G. Rethwisch, Fundamentals of materials science and engineering : an integrated approach, 4th ed. Hoboken, N.J.: Wiley, 2012, pp. xxv, 910 p.
[62] S. Fashu et al., "A review on crucibles for induction melting of titanium alloys," Mater Design, vol. 186, Jan 15 2020.
[63] T. L. Chow, Introduction to electromagnetic theory : a modern perspective. Boston: Jones and Bartlett Publishers, 2006, pp. xix, 523 p.
[64] H. Crew, General physics; an elementary text-book for colleges. New York,: The Macmillan company, 1908, pp. xi, 522 p.
[65] R. W. Erickson, Fundamentals of power electronics. New York: Chapman & Hall, 1997, pp. xviii, 773 p.
[66] W. J. Carr, "Energy Loss Resulting from Domain Wall Motion," J Appl Phys, vol. 30, no. 4, Apr 1959.
[67] W. H. Hayt, Engineering electromagnetics, 5th ed. (McGraw-Hill series in electrical engineering Electromagnetics). New York: McGraw-Hill, 1989, pp. xiii, 472 p.
[68] R. A. Serway, Physics for scientists & engineers, with modern physics, 4th ed. (Saunders golden sunburst series). Philadelphia: Saunders College Pub., 1996, p. 687.
[69] J. D. Jackson, Classical electrodynamics, 3rd ed. New York: Wiley, 1999, pp. xxi, 808 p.
[70] J. C. Maxwell, "A dynamical theory of the electromagnetic field," Philosophical Transactions of the Royal Society, vol. 155, pp. 459–512, Jun 1865.
[71] J.-M. Jin, The finite element method in electromagnetics, Third edition. ed. Hoboken. New Jersey: John Wiley & Sons Inc., 2014, pp. xxix, 846 pages.
[72] A. F. Mills, Heat transfer, 2nd ed. Upper Saddle River, N.J.: Prentice Hall, 1999, pp. xxii, 954 p.
[73] F. Fiorillo, Measurement and Characterization of Magnetic Materials. Elsevier Academic Press, 2004.
[74] M. H. Sadd, Elasticity: theory, applications, and numerics, Third edition. ed. Amsterdam ; Boston: Elsevier/AP, Academic Press is an imprint of Elsevier, 2014, pp. xvii, 582 pages.
[75] D. S. Chandrasekharaiah and L. Debnath, Continuum Mechanics. Academic Press, 1994.
[76] W. D. Kingery, "Factors Affecting Thermal Stress Resistance of Ceramic Materials," J Am Ceram Soc, vol. 38, no. 1, pp. 3-15, 1955.
[77] D. P. Hasselman, "Thermal Stress Resistance Parameters for Brittle Refractory Ceramics - a Compendium," Am Ceram Soc Bull, vol. 49, no. 12, pp. 1033-+, 1970.
[78] P. F. Becher, D. Lewis, K. R. Carman, and A. C. Gonzalez, "Thermal-Shock Resistance of Ceramics - Size and Geometry-Effects in Quench Tests," Am Ceram Soc Bull, vol. 59, no. 5, pp. 542-+, 1980.
[79] A. Smalcerz, B. Oleksiak, and G. Siwiec, "The Influence a Crucible Arrangement on the Electrical Efficiency of the Cold Crucible Induction Furnace," Arch Metall Mater, vol. 60, no. 3, pp. 1711-1715, 2015.
[80] C. A. Valagiannopoulos and A. Sihvola, "On Modeling Perfectly Conducting Sharp Corners With Magnetically Inert Dielectrics of Extreme Complex Permittivities," Ieee T Antenn Propag, vol. 60, no. 10, pp. 4777-4784, Oct 2012.
[81] M. Boivineau et al., "Thermophysical properties of solid and liquid Ti-6Al-4V (TA6V) alloy," Int J Thermophys, vol. 27, no. 2, pp. 507-529, Mar 2006.
[82] M. Mots, R. Hooper, and P. Frost, "The Engineering Properties of Commercial Titanium Alloys," Battelle Memorial Inst. Titanium Metallurgical Lab., Columbus, Ohio, TML-15, Jun 1958.
[83] G. Welsch, R. Boyer, and E. W. Collings, Materials properties handbook: titanium alloys. Materials Park, OH: ASM International, 1994, pp. xxii, 1176 p.
[84] A. Schmon, K. Aziz, and G. Pottlacher, "Density of liquid Ti-6Al-4V," presented at the 16th International Conference on Liquid and Amorphous Metals (LAM-16), Bonn, Germany, Aug, 2017.
[85] C. Maniere, G. Lee, T. Zahrah, and E. A. Olevsky, "Microwave flash sintering of metal powders: From experimental evidence to multiphysics simulation," Acta Mater, vol. 147, pp. 24-34, Apr 1 2018.
[86] E. Kaschnitz, P. Reiter, and J. L. McClure, "Thermophysical properties of solid and liquid 90Ti-6Al-4V in the temperature range from 1400 to 2300 K measured by millisecond and microsecond pulse-heating techniques," Int J Thermophys, vol. 23, no. 1, pp. 267-275, Jan 2002.
[87] G. E. Darwin and J. H. Buddery, Beryllium (Metallurgy of the rarer metals). London: Butterworths Scientific, 1960, pp. ix, 392.
[88] J. Lillie, "Some Properties of Beryllium Oxide," California. Univ., Livermore, USA, Technical Report. UCRL-6457, May 1961.
[89] R. Pampuch, Ceramic materials: an introduction to their properties. Amsterdam ; New York: Elsevier Scientific Pub. Co., 1976, pp. vii, 344 p.
[90] D. Munz and T. Fett, Ceramics: mechanical properties, failure behaviour, materials selection (Springer series in materials science, no. 36). Berlin ; New York: Springer, 1999, pp. x, 298 p.
[91] P. Boch and J. C. Niepce, Ceramic Materials: Processes, Properties and Applications. Newport Beach, CA, USA: ISTE USA, 2007, pp. xvii, 573 p.
[92] J. R. Davis and ASM International. Handbook Committee., Copper and copper alloys (ASM specialty handbook). Materials Park, OH: ASM International, 2001, pp. vii, 652 p.
[93] V. R. Gandhewar, S. V. Bansod, and A. B. Borade, "Induction Furnace - A Review," International Journal of Engineering and Technology, vol. 3, no. 4, pp. 277-284, Aug 2011.