Abstract

The conjugated temperature distributions of a microchannel fluid flow between two semi-infinite parallel plates are obtained analytically. The variables separation and transformation techniques are implemented to introduce the degenerate hypergeometric differential equation, the solution of which is given in terms of Kummer’s functions. The eigenvalues of the corresponding transcendental characteristic equation are obtained using a mathematical solver software package. Non-dimensional analysis of the governing equations introduced the parameter of “solid-fluid heat conduction ratio” \(k_k\). Values of this parameter are considered to present two limiting case solutions, namely, the adiabatic boundary solution, when \(k_k\approx 0\) and the isothermal boundary solution, when \(k_k> 100\). The Nusselt number \(Nu\) of the two limiting solutions is obtained and compared accurately with the corresponding values from the literature. The effect of the Knudsen number \(Kn\), the Biot number \(Bi\), and the conductivity ratio \(k_k\) on the temperature, temperature jump, and the Nusselt number is investigated. It is found that the temperature jump near the flow entrance becomes more significant with increase in \(Kn\), \(Bi\), or \(k_k\). On the other hand, the Nusselt number is found to increase with growing \(Kn\) and decrease with increasing \(Bi\) or \(k_k\).

中文翻译：

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两个平行板之间微通道流体流的共轭传热的分析研究

摘要

通过分析获得两个半无限平行板之间的微通道流体流动的共轭温度分布。实施变量分离和变换技术以引入退化的超几何微分方程，该方程的解以库默函数给出。使用数学求解器软件包可获得相应的先验特征方程的特征值。控制方程的无量纲分析引入了“固流热传导率” \（k_k \）的参数。该参数的值被认为提供了两个极限情况解，即\（k_k \ approx 0 \）时的绝热边界解 和等温边界时的等温边界解。 \（k_k> 100 \）。 获得了两个极限解的努塞尔数\（Nu \），并将其与文献中的相应值进行了精确比较。研究了Knudsen数\（Kn \），Biot数\（Bi \）和电导率\（k_k \） 对温度，温度跃变和Nusselt数的影响。已经发现，随着\（Kn \）， \（Bi \）或\（k_k \）的增加，流入口附近的温度跃变变得更加明显 。另一方面，发现努塞尔特数随着 \（Kn \）的 增加而增加，而随着\（Bi \）的增加而减少 或\（k_k \）。