Abstract

The analytical discrete ordinates (ADO) method is used to develop a solution to a one-dimensional model of particle transport in ducts that includes wall migration. Particle reemission from the wall is described by a nonlocal, exponential displacement kernel. Since the governing transport equation of the model is not directly amenable to a solution by the ADO method, an alternative transport equation is derived first. For an approximation based on a half-range quadrature of order $N$, the ADO solution of the alternative equation becomes available once techniques of linear algebra are used to solve a quadratic eigenproblem of order $N$ for the eigenvalues and eigenvectors. The solution is expressed as a superposition of 4N modes, which are constructed from 2N positive/negative pairs of separation constants (the reciprocals of the square roots of the eigenvalues) and associated eigenvectors. Compatibility conditions that the solution must satisfy in order to also solve the governing equation of the model result in a reduction of the number of relevant modes to 2N + 2, just two in excess of the number of modes in the solution of the problem without wall migration. Highly accurate numerical results for the reflection and transmission probabilities are reported for isotropic and monodirectional incidence.

摘要

分析离散坐标 (ADO) 方法用于开发管道中粒子传输的一维模型的解决方案，该模型包括壁迁移。来自壁的粒子再发射由非局部指数位移核描述。由于模型的控制传输方程不能直接通过 ADO 方法求解，因此首先导出了一个替代传输方程。对于基于半程求积的近似值$ñ$，一旦使用线性代数技术求解阶次的二次特征问题，替代方程的 ADO 解就变得可用$ñ$为特征值和特征向量。解决方案表示为 4N 模式的叠加，这些模式由 2N 正/负分离常数对（特征值平方根的倒数）和相关特征向量构成。为了求解模型的控制方程，解决方案必须满足的相容条件导致相关模式的数量减少到 2N + 2，仅比无壁问题的解决方案中的模式数量多出两个移民。报告了各向同性和单向入射的反射和透射概率的高精度数值结果。