加熱圓管內完全發展層流之熱傳近似解

An Analytical Solution of Fully Developed Laminar Flow in a Heated Circular Pipe

樂俞志(Yu-Zhi Yue)

指導教授 ： 許政行

中原大學/工學院/機械工程學系/碩士(2007年)

https://doi.org/10.6840/CYCU.2007.00650

摘要

摘要
本論文的目的在於探討圓管內速度已達到完全發展之層流，固定壁溫以及在管壁面施予非均勻熱通量加熱情況下，求解其無因次化溫度之近似解以及討論相關的熱傳係數。本文探討高普朗特數的流體即葛雷茲問題的假設將統御方程式簡化，然後採用分離變數法將偏微分方程式轉換為兩個獨立變數之常微分方程式。即Sturm-Liouville方法。在軸向為標準一階常微分式，而徑向的常微分方程式其特徵解為合流超幾何函數形式，運用數學套裝軟體mathematica® 可獲得展開式係數以及級數型式的解析解，合併兩常微分方程解即為最後之近似解。其次討論邊界條件為指數型式熱通量此近似解的解析方式。最後，探討在不同熱傳參數下，平均溫度、紐賽數以及熱傳特性。

關鍵字

分離變數法 ； 管流 ； 合流超幾何函數 ； 紐賽數 ； 總體溫度

並列摘要

Abstract
The purpose of this thesis is to study the approximation solutions of the dimensionless temperature and the heat transfer coefficient for a fully developed laminar flow in a circular pipe with constant wall temperature or externally heated non-uniform heat flux.
With large Prandtl Number, the Graetz Problem is assumed in this study. The governing partial differential equation is transformed by the Sturm-Liouville method into two independent ordinary differential equations. A first-order ordinary differential equation is obtained in the x-direction and a confluent hypergeometric function is obtained in the y-direction. The eigenvalues and analytical solutions are obtained by using the computer software Mathmatica®. In addition, the problem with an exponential-type heat flux boundary condition is also solved and discussed. The results are expressed in terms of bulk temperatures and heat transfer coefficients with different parameters.

並列關鍵字

confluent hypergeometric function ； bulk temperature ； separation of variables ； pipe flow ； Nusselt number

參考文獻

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