臺灣博碩士論文加值系統

傳統的管狀流動, 皆是探討邊界為無滑移條件的理想情況, 因此在流體力學當中才
能得到Hagen-Poiseuille 流動的經典例子。由於微機電系統製程的發展, 利用製程設
備在矽晶圓可製作出微小尺寸的流道, 而當流道的尺寸縮小且在相同的大氣環境下時,
Knudsen number 將相對增加, 使得傳統的無滑移條件及理論分析不再適用, 造成流動
進入滑移流動, 進而需討論流場中滑移流動行為, 並藉由巨觀的現象及微觀的行為來描
述流場的流動變化。
而滑移流動的邊界條件, 藉由Maxwell 所提出的一階滑移速度, 說明滑移速度是
利用機率的方式所推算出來的公式, 並需利用實驗的方式得到流場中的切線動量係數
(σm), 藉由文獻上所得到的滑移流動的質量流率公式反推切線動量係數, 以描述流道的
表面粗糙度關係; 並從質量流率公式中, 可得控制滑移流動的流場之重要的兩個參數, 其
為流場的壓縮性及稀薄程度, 此兩項主要決定流場中的流動現象。
本文利用數值模擬的方式, 模擬三維長矩形微型流道之滑移流動的行為,且設定切線
動量係數(σm) 皆為0.85, 探討出口絕對壓力操作皆低於一大氣壓時的流場, 並分別考
慮氬氣、氮氣及氦氣在不同的絕對壓力比下, 流場的流動行為及質量流率的比較。
就結果而言, 流場稀薄程度伴隨滑移流動的產生,而當流場處於較稀薄並且絕對壓力
比較小時, 流場中的滑移現象也具有重要性; 並且在低壓的操作模擬, 流場的Poiseuille
number 將隨著Knusden number 的增加而遞減。

Conventionally, no-slip boundary conditions are ideally imposed when analzing
fluid flowing through a duct. With the prominent development in the microfabrication
of microelectromechanical system (MEMS) devices, channels with micron
feature sizes can be fabricated on the silicon wafer. As the characteristic size of
the microchannels shrink down to a few microns, the Knudsen number becomes
non-negligible even under atmospheric pressure. This, in turn, invalidates the no
slip boundary conditions and results in the occruence of slip flow. The macroscopic
slip flow phenomena can be explained from the microscopic interaction between
the fluid and its confining walls.
Maxwell proposed a microscopic model to relate the slip velocity on the wall
with the gradient of the velocity profile on the wall. One of the critical parameters,
the tangential momentum accommodation coefficient (TMAC), can only be
obtained experimentally. Arkilic derived the mass flow rate of a slip flow driven by
pressure differential across the inlet and outlet ports and flowing through a long
2-D microchannel. Compressibility and rarefaction are the two most important
parameters affecting the mass flow rate.
In this work, the behaviors of the slip flow through a long 3-D rectangular
microchannel are simulated numerically. The TMACs of all simulations are set to
0.85. The slip flows of helium, nitrogen, and argon under various pressure ratios
are investigated. It is found that the slip flow phenomenon is most significant
when the pressure ratio is small and the gas is rarefied. It is also found that the
Poiseuille number increases with the increase of Knudsen number.

1 導論1
1.1 前言與研究目的. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
1.2 文獻回顧. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3
1.3 論文架構. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5
2 理論分析6
2.1 無滑移流動與滑移流動的差別. . . . . . . . . . . . . . . . . . . . . . 6
2.2 無滑移流動. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7
2.2.1 圓管流. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8
2.2.2 二維平板流動. . . . . . . . . . . . . . . . . . . . . . . . . . 9
2.2.3 矩形之穩態不可壓縮流. . . . . . . . . . . . . . . . . . . . . . 10
2.2.4 Fanno 流動. . . . . . . . . . . . . . . . . . . . . . . . . . . 11
2.3 滑移流動. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13
2.3.1 滑移速度之邊界條件. . . . . . . . . . . . . . . . . . . . . . . 13
2.3.2 滑移邊界之質量流率公式推導. . . . . . . . . . . . . . . . . . 13
2.4 流量與壓力關係. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20
3 模擬求解分析22
3.1 物理模型. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23
3.2 求解流程. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24
3.3 求解參數選定. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26
3.4 軟體之滑移速度. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27
3.5 流體性質. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28
3.5.1 氣體性質與溫度及壓力關係. . . . . . . . . . . . . . . . . . . 28
3.5.2 切線動量係數(σm) 之選定. . . . . . . . . . . . . . . . . . . 30
3.6 邊界條件設定. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31
3.7 殘餘值收斂條件. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32
3.8 求解之殘餘值選定. . . . . . . . . . . . . . . . . . . . . . . . . . . . 33
4 網格獨立性35
4.1 圓管流動模擬分析. . . . . . . . . . . . . . . . . . . . . . . . . . . . 35
4.2 矩形之微型流道模擬分析. . . . . . . . . . . . . . . . . . . . . . . . . 40
5 流場模擬分析47
5.1 選用氣體的重要性. . . . . . . . . . . . . . . . . . . . . . . . . . . . 47
5.2 可壓縮流之無滑移. . . . . . . . . . . . . . . . . . . . . . . . . . . . 48
5.3 滑移流動之流量公式討論. . . . . . . . . . . . . . . . . . . . . . . . . 52
5.4 流量與絕對壓力比. . . . . . . . . . . . . . . . . . . . . . . . . . . . 54
5.5 氣體分子與滑移現象. . . . . . . . . . . . . . . . . . . . . . . . . . . 61
5.6 近大氣壓時之變化. . . . . . . . . . . . . . . . . . . . . . . . . . . . 72
5.7 壁面剪應力與Poiseuille number . . . . . . . . . . . . . . . . . . . . 76
6 結論及未來發展84
6.1 結論. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 84
6.2 未來發展. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 84
參考文獻. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 86

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