# Transport in Random Media With Inhomogeneous Mixing Statistics

## Abstract

The Levermore-Pomraning (LP) model has been investigated for use in implicit Monte Carlo (IMC) algorithms to describe thermal photon transport computations in stochastically mixed cells with material energy coupling. For purely absorbing non-emitting materials, the LP-model is known to predict the exact material averaged uxes when the mixing statistics are Markovian. In the presence of scattering or reemmission, or if the mixing is non-Markovian, the LP-model is approximate and has greatly diminished accuracy for optically thick, highly scattering materials. Nevertheless, the model enables a robust extension of the simple atomic mix approximation to accommodate spatial correlation effects through well characterized parameters such as mean chord lengths. In this work, we construct exact analytical solutions of the LP-equations for piecewise constant spatial variation of the material chord lengths. It is not possible in general to obtain closed form solutions when the chord lengths are continuous functions of space, although we briefly examine special conditions under which this is feasible. However, by allowing an arbitrarily fine spatial segmentation of the medium it is possible to construct highly refined solutions to fit desired spatial chord length profiles. Material volume fractions that are consistent with the chord lengths are first constructed by solving themore »

- Authors:

- Univ. of New Mexico, Albuquerque, NM (United States). Dept. of Nuclear Engineering

- Publication Date:

- Research Org.:
- Lawrence Livermore National Lab. (LLNL), Livermore, CA (United States); Univ. of New Mexico, Albuquerque, NM (United States)

- Sponsoring Org.:
- USDOE

- OSTI Identifier:
- 1438709

- Report Number(s):
- LLNL-SR-750901

- DOE Contract Number:
- AC52-07NA27344

- Resource Type:
- Technical Report

- Country of Publication:
- United States

- Language:
- English

### Citation Formats

```
Prinja, Anil K., and O'Rourke, Patrick.
```*Transport in Random Media With Inhomogeneous Mixing Statistics*. United States: N. p., 2018.
Web. doi:10.2172/1438709.

```
Prinja, Anil K., & O'Rourke, Patrick.
```*Transport in Random Media With Inhomogeneous Mixing Statistics*. United States. doi:10.2172/1438709.

```
Prinja, Anil K., and O'Rourke, Patrick. Mon .
"Transport in Random Media With Inhomogeneous Mixing Statistics". United States.
doi:10.2172/1438709. https://www.osti.gov/servlets/purl/1438709.
```

```
@article{osti_1438709,
```

title = {Transport in Random Media With Inhomogeneous Mixing Statistics},

author = {Prinja, Anil K. and O'Rourke, Patrick},

abstractNote = {The Levermore-Pomraning (LP) model has been investigated for use in implicit Monte Carlo (IMC) algorithms to describe thermal photon transport computations in stochastically mixed cells with material energy coupling. For purely absorbing non-emitting materials, the LP-model is known to predict the exact material averaged uxes when the mixing statistics are Markovian. In the presence of scattering or reemmission, or if the mixing is non-Markovian, the LP-model is approximate and has greatly diminished accuracy for optically thick, highly scattering materials. Nevertheless, the model enables a robust extension of the simple atomic mix approximation to accommodate spatial correlation effects through well characterized parameters such as mean chord lengths. In this work, we construct exact analytical solutions of the LP-equations for piecewise constant spatial variation of the material chord lengths. It is not possible in general to obtain closed form solutions when the chord lengths are continuous functions of space, although we briefly examine special conditions under which this is feasible. However, by allowing an arbitrarily fine spatial segmentation of the medium it is possible to construct highly refined solutions to fit desired spatial chord length profiles. Material volume fractions that are consistent with the chord lengths are first constructed by solving the associated Chapmann-Kolmogorov equations and applied to obtain the material averaged fluxes in two settings: when the materials are purely absorbing, and when scattering is present in rod model geometry. Illustrative numerical results for the volume fractions and material averaged fluxes are obtained by implementing the algorithms in MATLAB for spatially linear and quadratic variations of the chord lengths.},

doi = {10.2172/1438709},

journal = {},

number = ,

volume = ,

place = {United States},

year = {Mon May 07 00:00:00 EDT 2018},

month = {Mon May 07 00:00:00 EDT 2018}

}