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研究生: 方晨譽
Chen-Yu Fang
論文名稱: 批次發泡過程中的泡體成長現象模擬
Simulation of bubble growth phenomenon by Batch Foaming
指導教授: 葉樹開
Shu-Kai Yeh
口試委員: 林子仁
王鎮杰
學位類別: 碩士
Master
系所名稱: 工程學院 - 材料科學與工程系
Department of Materials Science and Engineering
論文出版年: 2023
畢業學年度: 111
語文別: 中文
論文頁數: 141
中文關鍵詞: 泡體成長流變參數有限差分法敏感性統御方程式
外文關鍵詞: Bubble growth, Rheological Parameters, Finite difference method, Sensitivity analysis, Governing equation
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泡孔尺寸對於高分子發泡材料的性質影響較深,控制泡孔尺寸的預測可以減少需要試誤法的實驗條件,本研究著重模擬精確的高分子發泡行為(PS/CO2)中的泡體成長現象,運用文獻精準量測到的物理參數(表面張力、零剪切黏度、擴散係數、亨利常數等),比對本實驗室利用高分子發泡視覺化設備拍攝之泡體成長數據,比較本研究的泡體成長模型的精確性,確認本研究在泡體成長行為模擬的可靠性,其結果建立了本實驗室對泡體成長模擬的初步研究。
也比對文獻中哪些物理參數對泡體成長現象是敏感的,代表那些物理參數對泡體成長行為是極重要的因子,驗證本研究對文獻結果有一定程度的還原性。


The size of the bubble is the critical factor that affects polymer foam's properties.
Controlling the prediction of bubble size can reduce the experimental conditions that
require a trial-and-error method. The present study focuses on the simulation of precise polymer foaming behaviors (PS/CO2) in the bubble growth phenomenon and the use of the literature to accurately measure physical parameters (surface tension, zero-shear viscosity, diffusion coefficient, Henry's constant, etc.), compared with this laboratory using polymer foaming visualization equipment to capture bubble growth data to confirm the accuracy of this study in the bubble growth model. We compare the accuracy of our bubble growth model with the bubble growth data captured by our laboratory using the polymer foaming visualization equipment and confirm the reliability of our study in simulating the bubble growth behavior. The results establish the preliminary study of bubble growth simulation in our laboratory.
The results also compare which physical parameters in the literature are sensitive to
the bubble growth phenomenon, and represent those physical parameters that are critical
factors for bubble growth behavior, verifying that the present study has a certain degree of reversibility to the literature results.

摘要 ................................... i Abstract ............................. ii 誌謝 ................................. iii 目錄 ................................. iv 圖目錄 ........................... viii 表目錄 ............................. xi 第一章 緒論 .................... 1 第二章 文獻回顧 .......................... 3 2.1 高分子發泡 .......................................... 3 2.2 高分子加工 ... ....................................... 4 2.2.1 批次發泡 ...................................5 2.2.2 押出發泡 ...................................5 2.2.3 射出發泡 ...................................6 2.3 古典成核理論 ....................7 2.3.1 古典同相成核 ........................................ 8 2.3.1.1 同相成核自由能 ......................... 8 2.3.1.2 同相成核成核率 ....................... 10 2.3.2 古典異相成核 ....................................... 11 2.3.2.1 異相成核自由能 ....................... 12 2.3.2.2 異相成核成核率 ....................... 12 2.3.3 擬穩態古典成核 .................................. 13 2.4 泡體成長模型 .................................... 14 2.4.1 單顆氣泡成長模型(Single Bubble Growth Model) ............................ 14 2.4.1.1 連續方程式(Equation of continuity)......................................... 15 2.4.1.2 動量平衡方程式(Equation of motion) ..................................... 16 2.4.1.3 質量守恆與擴散方程式 (Equation of Mass Balance) ............ 19 2.4.1.4 多項式濃度分布(Polynomial concentration profile) ................ 20 2.4.2 陣列同心球泡體模型(Cell Model) ....................... 29 2.4.2.1 連續方程式(Equation of continuity)......................................... 31 2.4.2.2 動量平衡方程式(Equation of motion) ..................................... 32 2.4.2.3 質量守恆與擴散方程式(Equation of Mass Balance and Diffusion) ......................................................33 2.4.2.4 完整濃度分布(Completely concentration profile) ................... 33 2.5 泡體成長重要物理參數 ................................. 38 2.5.1 零剪切黏度(Zero Shear Viscosity) ....................... 38 2.5.2 表面張力 (Surface tension) ................. 40 2.5.3 弛豫時間(Relaxation time) .................. 42 2.5.4 擴散係數(Diffusion coefficient) .......... 43 2.5.5 亨利常數(Henry’s constant) ................ 44 第三章 實驗步驟與計算方法 ...................................... 46 3.1 模擬流程 ................................... 46 3.2 計算流程 ................................... 46 3.2.1 網格建立 .............................................. 46 3.2.2 物理參數選用 ...................................... 47 3.2.3 起始條件 .............................................. 47 3.2.4 猜測泡體半徑 ...................................... 48 3.2.5 計算內應力 .......................................... 48 3.2.6 泡體壓力(1) ......................................... 49 3.2.7 濃度分布 .............................................. 49 3.2.8 泡體壓力(2) ......................................... 50 3.2.9 誤差 ... ...................................... 50 第四章 結果與討論 ..... .................................... 52 4.1 差分方程式之建立 ......................................... 52 4.2 差分方程式之穩定性分析 ............................. 57 4.2.1 猜測泡體半徑ODE 之穩定性 ............ 57 4.2.2 計算內應力ODE 之穩定性 ................ 58 4.2.3 擴散方程式穩定性 .............................. 58 4.3 與先前研究者之模型比較 ............................. 60 4.3.1 洩壓曲線 .............................................. 60 4.3.2 泡體成長曲線 ...................................... 61 4.3.3 敏感性分析 .......................................... 63 4.4 與發泡視覺化之實驗比較 ............................. 78 第五章 結論 .................. 87 參考文獻 ........................ 88 附錄 A 有限差分法的形式 ......................................... 97 A.1 有限差分法的推導 ........................................ 99 A.2 空間有限差分法形式(Spatial finite different method formula) .................. 102 A.3 時間有限差分法形式(Temporal finite different method formula) .............. 105 附錄 B 截斷誤差、一致性與數值穩定性 ............... 108 B.1 截斷誤差(truncation error) ........................... 108 B.2 一致性(Consistency) .................................... 109 B.3 穩定性(Stability) ........................................... 110 B.4 穩定性簡易判斷法 ....................................... 110 B.5 時間疊代法的穩定性推導 ........................... 112 B.6 馮諾依曼穩定性分析 ................................... 116 B.7 離散擴散方程式的穩定性推導 ................... 117 附錄 C Runge-Kutta Method ...................................... 120 附錄 D 均勻網格與非均勻網格之探討 ................... 124 附錄 E 辛普森積分法 ...................................126

Polymer Foam Market Research Report Information by Form (Flexible Foam
and Rigid Foam), Product [Polyurethane (PU), Polystyrene (PS), Polyethylene
(PE), Polypropylene (PP), Polyvinyl Chloride (PVC) and Others], Application
(Building and Construction, Automotive, Packaging, Furniture, Wind Turbine
Blade, and Others), And By Region (North America, Europe, Asia-Pacific,
And Rest Of The World) - Market Forecast Till 2030. Available from:
https://www.marketresearchfuture.com/reports/polymer-foam-market-4964.
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