本研究使用相位場方法模擬氣泡在固化之過程中於固液介面間之動態行為，本研究之模擬方式為使用二維兩相流模組，加入溫度控制變數及相位函數區分固相、液相、氣相三鄉之存在比例。本研究之制衡方程式為包含相百分比性質之能量、質量、動量以及濃度守恆方程式，考慮由於過飽和而形成氣泡，且之後氣泡並無質量傳遞，僅考慮溫度場對後續氣泡發展之影響。
結果顯示表面張力、固相變化潛熱值與上下邊界溫差大小影響氣泡形狀與固化速度，介面效應越大則越容易導致氣泡產生形變。

This study applies the phase-field method to simulate the behavior between bubble and liquid-solid front in the solidification. During the process, the two-phase flow module is used to match up with temperature and phase-field function to determine the percentage of- solid, liquid, and gas- in the domain. The governing equations for the relative percentage of phases contain energy , mass , momentum and concentration equations. Consider the formation of bubbles due to over saturation, and have no mass transfer for the bubble , only consider the effect of temperature field on the subsequent development of the bubble.
　　The results showed that the surface tension , solid phase change latent heat value , temperature difference between the upper and lower boundaries will influence the bubble’s shape and the velocity of solidification , the greater effect on the interface will easily influence the shape of bubble.

論文審定書 i
謝誌 ii
中文摘要 iii
Abstract iv
目錄 v
圖目錄 vii
符號說明 ix
下標符號說明 x
第一章 緒論 1
1-1 前言與研究背景 1
1-2相位場法（PFM）及二相流（Two Phase Flow） 2
1-3研究內容簡介與架構： 3
第二章系 統模型設定與理論之分析 4
2-1 模組之統御方程式 4
2-1-1 質量守恆方程式 & 相位場方程式 4
2-1-2動量方程式 5
2-1-3能量守恆方程式 9
2-1-4濃度方程式 9
2-2 模型設定與邊界假設 10
2-2-1 模型架構 10
2-2-2網格設置 11
2-2-3邊界與初始值設定 12
第三章 結果與討論 14
3-1模擬之基本性質 14
3-1-1流體性質 14
3-1-2上下邊界溫差198K 15
3-1-3 Delta 函數 21
3-2介面效應與遷移率微調參數 之影響 23
3-2-1介面厚度7.5×〖10〗^(-5)， =0.01 23
3-2-2介面厚度5×〖10〗^(-5)， =0.1 25
3-3結果討論 28
第四章 結論 29
參考文獻 30

[1] S.Kou, Welding Metallurgy. Wiley, New York, 1987
[2] M. C. Flemings, Solidification Processing, Mcgraw-Hill, New York, 1974
[3] Y. Sun and C. Beckermann, 2010,“Phase-field modeling of bubble growth and flow in a Hele-shaw cell”, J. Heat and Mass Transfer, 2969-2978.
[4]　David C. Venerus and Nadia Yala,1997,“Transport Analysis of Diffusion-Induced 　 Bubble Growth and Collapse in Viscous Liquids”, AIChE Journal,Vol.43,No.11.
[5] Tanai L. Marin,“Solidification of a Liquid Metal Droplet Impinging on a Cold Surface”,Excerpt from the Proceedings of the COMSOL Users Conference 2006 Boston.
[6] V.R.Voller and C.Prakash,“ A fixed grid numerical Modelling Methodology for convection-diffusion mushy region phase-change problem”,Jourmal of Heat and Mass Transfer,30(8),1709-1719(1987).
[7]　Ruben Scardovelli and Stephane Zaleski,1999,“Direct numerical simulation of
　　free-surface and interfacial flow”,Annu. Rev. Fluid Mech.,Vol.31,567-603.
[8] 　Y.Sun and C.Beckermann,2007, “Sharp interface tracking using the phase-field
equation”,Journal of Computational Physics 220,pp.626-653.

[9]　Naoki Takada, Masaki Misawa and Akio Tomiyama,2006,“A phase-field method
　 for nterface-tracking simulation of two-phase flows”,Mathematics and
Computersin Simulation 72,pp.220-226.
[10]　David Jacqmin,1999,“Calculation of Two-Phase Navier–Stokes Flows Using
Phase-Field Modeling”, Journal of Computational Physics 155,pp.96-127.
[11]　D. Jamet,O. Lebaigue, N. Coutris and J. M. Delhaye,2001,“The Second
Gradient Method for the Direct Numerical Simulation of Liquid–Vapor Flows
with Phase Change”, Journal of Computational Physics 169,pp.624-651.
[12]　J. A. WARREN and W. J. BOETTINGER,1995,“Prediction of dendritic growth
and microsegregation patterns in a binary alloy using the phase-field method”
Acta metal .mater,Vol.43,No.2,pp.689-703.
[13]　PENGTAO YUE, JAMES J. FENG, CHUN LIU and JIE SHEN,2004, “A
diffuse-interface method for simulating two-phase flows of complex fluids”, J.
Fluid Mech. , vol. 515, pp. 293–317.
[14]　F. Kong, H. Zhang and G. Wang,2008,“Numerical Simulation of Transient
Multiphase Field during Hybrid Plasma-Laser Deposition Manufacturing”J. Heat
Transfer, Vol.130, NO.112101, 1-7.