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標題: 應用nodally exact差分法解具Soret效應傾斜封閉空間內填充多孔性介質之時變雙擴散自然對流問題
Application of the nodally exact difference method to transient double diffusive natural convection in an inclined porous cavity with Soret effect
作者: 謝文軒
Wen-Hsuan Hsieh
關鍵字: 克蘭克-尼科爾森法;對流擴散反應方程;熱質傳;Soret效應;Crank-Nicolson method;Convection-Diffusion-Reaction (CDR) equation;Heat and mass transfer;Soret effect.
引用: [1.] T. W. H. Sheu, S. K. Wang, and R. K. Lin,'An implicit scheme for solving the convection-diffusion-reaction equation in two dimensions' J.Comput. Phys. 164, 123-142,2000. [2.] A. Mansour,A. Amahmid and M. Hasnaoui,'Soret effect on thermosolutal convection developed in a horizontal shallow porous layer salted from below and subject to cross fluxes of heat' Int.J.Heat Fluid Flow,27,306-314,2008. [3.] M. Bouabid and A. B. Brahim,'Soret effect on entropy generation at magneto double-diffusive convection in an air filled square medium',IREC 6th,2015. [4.] B. M. Chen and L. Q. Zhang,'Numerically study on Soret effect and Dufour effect for double diffusive natural convection in porous enclosure',J. Hydrodyn,vol 6,82-85,2003. [5.] H.Darcy,'Les Fontaines Publiques de la Ville de Dijon', Paris,1856. [6.] T. L. Bergman and R. Srinivasan,'Numerical simulation of Soret induced double diffusion in an initially uniform concentration binary fluid',IJHMT 32,679-682,1990. [7.] N. Nithyadevi and R. J. Yang,'Magnetoconvection in an enclosure of water near its density maximum with Soret and Dufour effects',IJMF,1667-1676, 2009. [8.] N. Nithyadevi and R.J. Yang,'Double diffusive natural convection in a partially heated enclosure with Soret and Dufour effects',IJHF 30th,902-910,2009. [9.] R. Younsi,A. Harkati and D. Kalache,'Numerical simulation of double diffusive natural convection in porous cavity',AJSE 26th 145-155,2002. [10.] L. Leboucher, ' Monotone scheme and boundary conditions for finite volume simulation of magnetohynamic internal flows at high Hartmann number',J. Comput. Phys. 150,1991. [11.] Q. Zhang, C. Zhang,and L. Wangc,'The compact and Crank-Nicolson ADI schemes for two-dimensional semilinear multidelay parabolic equations',journal of Computational and Applied Mathematics,217-230,2015. [12.] M. Bourich,M. Hasnaoui,A. Amahmid,M. Mamou,'Soret convection in a shallow porous cavity submitted to uniform fluxes of heat and mass.'Int. Commun. Heat Mass Transfer 31,733-742,2004. [13.] T. Meis and U. Marcowitz,'Numerical solution of partial differential equations, A. M. Science Series,1981. [14.] X. Yu,'Finite difference methods for the reduced water wave equation', Computer methods in A. M.,265-270,1998. [15.] 廖百偉(2001)' 波狀微流管中電滲透流之熱效應數值研究'(未出版之博士論文)﹐國立中興大學﹐台中市。 [16.] 金展慶(1996)' 矩形空間內低溫水在飽和多孔性介質中之時變共軛熱傳分析'(未出版之碩士論文) 。國立中興大學﹐台中市。 [17.] 黃廷偉(2011)'Soret 效應和Dufour 效應對封閉空間內自然對流之影響'(未出版之碩士論文) 。逢甲大學� 台中市。 [18.] 陳冠霖(2013)' 微分法於熱傳問題之分析與應用'(未出版之碩士論文) 。中華大學﹐新竹市。
本文主要分為兩部分﹐第一部分是基於T. W. H. Sheu 等人所提出
的nodally exact 差分模型﹐結合Crank-Nicolson 數值方法﹐並將其用於解暫態二維Convection-Diffusion-Reaction Equation(CDR) 。從數值結果發現﹐在同樣數量格點下﹐此方法在一維或是二維的數值實驗都可以得到更好的精度。在數值計算的結果中﹐空間的精度可為O(h2)﹐時間的收斂精度則是可以維持在O(Δt2) 。
第二部分是將上述的數值方法﹐結合SOR 迭代法來處理一個暫態二維
質時熱質傳交互作用之自然問題。在此我們主要考慮旋轉角度ϕ 、Soret 係數M 、雷里數RT、路易斯數Le 、浮力比N 對平均紐賽數Nu 及平均雪耳伍
德數Sh 的影響。
結果顯示: 在本文問題中﹐旋轉角度ϕ 上升時﹐平均紐賽數Nu 上升。
當Soret 效應M 增加﹐平均紐賽數Nu 減少﹐平均雪耳伍德數Sh 減少。
雷里數RT 增加﹐流速增加而使對流占優﹐平均紐賽數Nu 及平均雪耳伍德
數Sh 皆上升﹐使熱傳及質傳增加。而路易斯數Le 增加時﹐質傳的效果會更
好使平均雪耳伍德數Sh 上升。浮力比為-1 時﹐平均紐賽數Nu 及平均雪
耳伍德數Sh 會最小。

This thesis consists of two main parts,in the first part,we presents a hybrid numerical method based on the nodally exact difference scheme developed by Sheu et al.and the Crank-Nicolson method for unsteady convection-diffusion-reaction(CDR) equation.By the numerical examples we find that with the same nodes,this hybrid method can have the better accuracy no matter on the 1D or 2D cases,and the accuracy in space is $ mathcal{O}(h^2) $,and in temporal coordinate is $ mathcal{O}(Delta t^2) $.
The problem under investigation is the transient double diffusive natural convection with Soret effect in an inclined enclosure filled with a fluid-saturated porous medium.Of interest are the effects of the angle of inclination($phi$),the Soret number($M$),the Rayleigh number($R_T$),the Lewis number($Le$) and the buoyancy ratio($N$) on the average Nusselt number $overline{Nu}$ and the average Sherwood number $overline{Sh}$.
The results indicated that the average Nusselt number $overline{Nu}$ and the average Sherwood number $overline{Sh}$ increase with an increase in the angle of in the Lewis number $Le$ and the angle of inclination $phi$ and the Rayleigh number,or a decrease in the Soret number.The $overline{Nu}$ and $overline{Sh}$ have the smallest values,respectively,when N = -1.
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