气体分子以分子流态通过直圆管道时的位置分布计算

Calculation of Positional Distributions of Molecular Flux at Orifice of Cylindrical Pipe

针对气体分子以自由分子流态通过一段直圆管道的问题,将分子划分为从入口直达出口的直通分子和经过管壁反射后到达出口或入口的反射分子这两种类型,分别建立了流动数学模型。从余弦定律出发,推导出两类分子穿越出口与入口截面的分子总数和密度分布函数的积分公式,以及反射分子在管道内壁面的沿程分布规律。定义评价数ν为管口截面上某点的局部分子密度与整个截面平均分子密度之比,给出计算式。数值计算结果表明,按均匀分布进入管道入口的气体分子,在飞离管道出口截面时呈现出中心密集边缘稀疏的"位置束流效应",如管长L=0.5d时,中心处νdA(0)=1.1353,边缘处νdA(0.5)=0.8313;而返回入口截面的分子却呈现出中心稀疏边缘密集的分布。这种趋势随管长增大而减弱,当L>5d后趋于稳定。入口和出口截面的分子密度分布能够互补,两者叠加后恰好等于均匀分布。

The positional distributions at the orifice of the two kinds of gas molecules passing a cylindrical pipe-one fly freely and the other experience some reflection-were modeled and calculated,respectively.The integral formulas were derived in terms of cosine law,to describe the area density distributions at the inlet and outlet for both types of molecules.The possible positional distributions of the molecules reflected by the pipe walls were also calculated.A criterion,ν,defined as the ratio of the density at a given point to the average area density at the whole orifice section,was analytically evaluated.The calculated results show the ＂beaming effect＂,that is,on passing the outlet,the uniform area density of the beam through the inlet turns into non-uniform,decreasing in radial direction.In contrast,the area density of the molecule beam returning to the inlet increases in radial direction,depending on the pipe length.When L5d,the steady beam has a uniform area density.The positional distributions at the inlet and outlet were found to be complimentary,and their overlapping results in a uniform area density.