Random Paths in Random Arrays of Cylinders

Abstract

Distributions associated with random tracks in bodies are important to understanding the effects of radiation on objects and their response to it. Results of computer simulations are presented for the distribution densities, average lengths, and relative frequencies of the various types of random tracks resulting from the traversal of random arrays of cylinders by random lines. We consider arrays consisting of freely overlapping long circular cylinders distributed randomly with their axes parallel to a line or a plane, or oriented randomly in the three-dimensional space. The problem of random tracks lying outside the cylinders (external random tracks) is treated analytically, while the distributions of random tracks found within the cylinders (internal random tracks) are determined using a simulation procedure based on a discrete step-by-step random walk mechanism applied in finite samples of the arrays. The numerical results are used to validate various general analytical results for the distribution densities and average lengths of random tracks in structures of arbitrary shape. It is found that the distributions of internal random tracks exhibit strong dependence on the cylinder volume fraction, while the external random tracks are distributed independently of the cylinder volume fraction for all three cases of orientation distribution. At the limit of high cylinder volume fraction, the internal random track distributions approach the distributions of the external random tracks, while at the other extreme, they become, as expected, identical to those determined analytically for a single infinite cylinder.

Effective diffusion coefficients in square arrays of filament bundles

“…The effective diffusivities (or tortuosities) perpendicularly to the bundles (x and y direction) are equal to each other because of the isotropy of the fibrous structure on the ( x , y ) plane. It should be noted that i is equal to the mean length of the chords that are formed in the pore space when the porous medium is exposed to a field of random segments having mean length (Kellerer, 1971;Tomadakis and Sotirchos, 1993e), which in turn equals the average path between collisions of the molecule diffusing in the pore space.…”