Evaluation of the Eigenvalues of the Graetz Problem in Slip-flow.
LOUISIANA TECH UNIV RUSTON
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The objective of this research was to develop a new technique for evaluation of the eigenvalues of the Graetz problem in slip-flow - a heat transfer problem for gases at low pressures or in extremely small geometries. In this investigation, the velocity distribution with slip-flow has been obtained, expressed simply in terms of Knudsen Kn numbers. The expression shows that the velocity always increases as the Knudsen number increases. The relationship of Kn and molecular mean free path for a gas shows that Kn may become large enough to significantly affect the velocity distribution and consequently affect the heat transfer properties. A mathematical model of temperature distribution was established by combining the energy and momentum equations. A series solution was obtained by the method of Frobenius. Also, expressions for the local and overall Nusselt numbers were derived. All these expressions can be taken as functions of Knudsen numbers and Graetz numbers. Anew technique for evaluation of eigenvalues for the solution of the Graetz problem in slip-flow was developed. This method was based on the construction of a matrix. The computational results show that it is an effective method, and the lowest five values were found for Kn from 0.02 to 0.12. For practical calculations, relationships between eigenvalues and Knudsen numbers were obtained.