One-Dimensional Models for Time-Dependent Transport in Solid Cylinders

Abstract

We present a set of 1-D transport models for solid cylinders of material irradiated with particles on the axial ends. The models are based on 1-D models originally developed for evacuated ducts with reflecting walls. The goal of this work is to show that 1-D models can be used for time-dependent transport in solid cylinders. A future use of this approach will apply these models to the high-energy density physics problem of a Marshak wave propagating down a cylindrical foam or other experiments. The models we present use a Galerkin procedure to project the radial dependence of the 3-D transport equation onto an expansion in terms of polynomials. Results demonstrate that for steady state problems with low scattering ratios, a three-basis function expansion can adequately capture the 3-D solution as computed via Monte Carlo. Smaller number of basis functions did not result in adequate solutions. As the radius of the cylinder increases, the 1-D model is more effective. Results from time dependent problems indicate that the 1-D models move particles too fast down the cylinder at early times, but are accurate on the order of 10 or more mean-free times. Our results indicate that 1-D models may be effective in modeling 2-D Marshak waves, but further work is necessary to answer this question.

Keywords:

• Particle transport
• mathematical modeling

Acknowledgments

The authors would like to thank the anonymous referees for their suggestions to improve this work, including pointing out some important references.