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Synergy between the solution based on the Transversal Method Of Lines and the Leveque solution in the platform of the Graetz/Nusselt problem
International Communications in Heat and Mass Transfer ( IF 7 ) Pub Date : 2020-07-01 , DOI: 10.1016/j.icheatmasstransfer.2020.104590
Antonio Campo , Müslüm Arıcı , Yunesky Masip Macía

Abstract The aim of the present study is to generate an approximate, semi–analytical solution of the Graetz/Nusselt problem for the demanding upstream sub-region of the tube employing the Transversal Method Of Lines (TMOL). In the heat convection literature, the approximate, semi–analytical solution of the Graetz/Nusselt problem in the upstream sub-region of the tube that is available refers to the traditional physics-based Leveque problem that possesses certain idealizations. TMOL is a mathematics-based transformation that discretizes the partial derivative in the axial direction of the two–dimensional energy equation in cylindrical coordinates, while leaving the partial derivatives in the radial direction continuous. As a consequence, the TMOL procedure provides an adjoint one–dimensional energy equation in the radial variable linked to a transversal line placed in the cross-section of the tube. The resulting ordinary differential equation of second order with variable coefficients and inhomogeneous is named the confluent hypergeometric differential equation. This equation can be readily solved in exact, analytical form with a symbolic algebra code in terms of the Kummer function of first kind. This gives way to the approximate, analytical temperature profile in the upstream sub-region of the tube near the entrance. From here, the obtained approximate, analytical mean-bulk temperature profile based on TMOL gently overestimates the exact, analytical Graetz mean-bulk temperature profile, whereas the traditional approximate, analytical mean-bulk temperature profile rendered by Leveque gently underestimates the exact, analytical Graetz mean-bulk temperature profile. Overall, the two dissimilar methodologies share reasonable quality levels in the specific upstream sub-region of the tube, X ≤ 0.01.

中文翻译:

Graetz/Nusselt问题平台中基于横向线法的解与Leveque解的协同作用

摘要 本研究的目的是使用横向线法 (TMOL) 为要求苛刻的管道上游子区域生成 Graetz/Nusselt 问题的近似半解析解。在热对流文献中,可用的管子上游子区域中 Graetz/Nusselt 问题的近似、半解析解是指具有某些理想化的传统的基于物理的 Leveque 问题。TMOL 是一种基于数学的变换,它在柱坐标系中将二维能量方程的轴向偏导数离散化,同时保持径向的偏导数连续。作为结果,TMOL 程序在与放置在管子横截面中的横向线相关联的径向变量中提供了一个伴随的一维能量方程。所得的变系数非齐次二阶常微分方程称为合流超几何微分方程。根据第一类库默函数,可以使用符号代数代码以精确的解析形式轻松求解该方程。这让位于入口附近管子上游子区域的近似分析温度分布。从这里开始,获得的基于 TMOL 的近似分析平均体积温度曲线略微高估了精确的分析 Graetz 平均体积温度曲线,而传统的近似值,Leveque 提供的分析平均体积温度曲线略微低估了精确的分析 Graetz 平均体积温度曲线。总体而言,两种不同的方法在管道的特定上游子区域共享合理的质量水平,X ≤ 0.01。
更新日期:2020-07-01
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