G. C. Pomraning

Linear transport theory in a random medium

The time‐independent linear transport problem in a purely absorbing (no scattering) random medium is considered. A formally exact equation for the ensemble averaged distribution function 〈Ψ〉 is derived. Under the assumption of a two‐fluid statistical mixture, with the transition from one fluid to the other assumed to be determined by a Markov process, an exact solution to this equation for 〈Ψ〉 is obtained. In the source‐free case, this solution is shown to agree with the result obtained by ensemble averaging simple exponential attenuation. Several approximations to the exact equation for 〈Ψ〉 are considered, and numerical results given to assess the accuracy of these approximations.

Benchmark results for particle transport in a binary Markov statistical medium

Abstract We give numerical benchmark results for particle transport in a randomly mixed binary medium, with the mixing statistics described as a homogeneous Markov process. A Monte Carlo procedure is used to generate a physical realization of the statistics, and a discrete ordinate numerical transport solution is generated for this realization. The ensemble averaged solution, as well as the variance, is obtained by averaging a large number of such calculations. Reflection and transmission results are given for several problems in both rod and planar geometry. In a separate development, two coupled transport equations are derived which formally described transport in a random binary mixture for arbitrary mixing statistics. Closing these equations by approximating their coupling terms in a low order and intuitive way leads to a model for stochastic transport previously obtained via the master equation. The present derivation, based upon approximating exact equations, allows in principle the opportunity to develop more accurate models by making higher order approximations in the coupling terms.

Asymptotic analysis of radiative transfer problems

The equations of radiative transfer are systematically analyzed by asymptotic methods. To lowest order, the classical equilibrium diffusion approximation is recovered. The next order analysis leads to the equilibrium diffusion differential equations and initial condition, but with a boundary condition containing a linear extrapolation distance α. This quantity is related tothe solution of a canonical halfspace problem and is computed by deriving an appropriate variational principle. For the case of no scattering, an exact Wiener-Hopf solution is available. The FN solution technique is also applied to the problem of obtaining α with good results. Higher order asymptotic radiative transfer descriptions are discussed and, while not immediately constituting practical calculational techniques, do have implications for computing the parameters in the multiband treatment of the frequency variable.

Renewal theory for transport processes in binary statistical mixtures

Renewal theory is used to analyze linear particle transport without scattering in a random mixture of two immiscible fluids, with the statistics described by arbitrary (non‐Markovian) fluid chord length distributions. One conclusion (for unimodal distributions) that is drawn is that the mean and variance of the chord length distributions through each fluid is sufficient knowledge of the statistics to give a reasonably accurate description of the ensemble averaged intensity. Expressions for effective cross sections and an effective source to be used in the usual deterministic transport equation are also obtained. The use of these effective quantities allows statistical information to be introduced very simply into a standard transport equation. An analysis is given which shows how the transport description, including scattering, in a Markovian mixture can be modified to yield an approximate description of transport in a non‐Markovian mixture. Numerical results are given to assess the accuracy of this model...

Asymptotic and variational derivations of the simplified PN equations

Abstract It is demonstrated that the simplified PN equations are a leading order asymptotic limit of the transport equation. This limit is one of locally nearly planar transport involving scattering which is highly peaked in the forward direction. This asymptotic equivalence is valid in full generality, in that it allows time-dependent transport in a heterogeneous medium. It is also shown that the simplified PN equations can be derived variationally, making use of certain trial functions in a variational characterization of the even-parity transport equation. In this case, however, there appears to be an inherent restriction to time-independent transport in a homogeneous medium.

Radiative transfer in random media with scattering

Abstract For time-independent transport in planar systems, a formalism is developed to treat radiative transfer in a statistical medium, including the scattering interaction, for a certain class of problems. In this development, the statistics of the mixture are allowed to be completely general. As a special case, the purely scattering rod problem described by homogeneous Markov statistics is treated in detail, and numerical results given. These results agree with those reported earlier based upon the method of invariant imbedding. In particular, the existence of a transmission window is confirmed for this problem.

A statistical analysis of the double heterogeneity problem

Abstract The collision probability treatment for regions comprised of a uniform background medium with a random dispersion of small heterogeneities is analyzed using recently-developed statistical techniques. By assuming that the chord length distributions in such regions follow a renewal process we obtain an exact expression for the equivalent homogeneous cross section. Numerical comparisons show that the collision probability value for this cross section can be significantly in error in most cases, with an error of ∼8% for typical PWR poisoned fuel consisting of a mixture of uranium oxide and gadolinia grains. Renewal theory is generalized to the case of materials with internal structure allowing us to justify the central assumption on which the collision probability treatment is based. Also, new formulas are proposed for the calculation of the collision probabilities between matrix and grains in terms of the collision probabilities for the homogenized regions.

Initial and boundary conditions for diffusive linear transport problems

The initial and boundary layer analyses of an asymptotic expansion that yields a diffusive description of linear particle transport are carried out in some generality. This yields initial and boundary conditions that apply to the diffusion equation previously reported in the literature. Effects treated include boundary curvature, the variation of the transport boundary condition along the bounding surface, and spatial and temporal variations of the interaction coefficients (cross sections) in the initial and boundary layers.

Statistics, renewal theory, and particle transport

Abstract A model involving integral renewal equations is derived which describes the ensemble-averaged intensity for time-independent particle transport in a nonscattering medium composed of two randomly mixed, immiscible fluids. It is shown that for a special class of mixing statistics these renewal equations can be recast into integro-differential equations resembling usual transport/kinetic equations. An extension of this model is proposed which includes time dependence as well as scattering. This generalizes an earlier description proposed for the special case of Markov mixing statistics. An Appendix is included which gives several statistical results useful in this line of research.

The P N Theory as an Asymptotic Limit of Transport Theory in Planar Geometry —I: Analysis

In this paper the P{sub N} theory is shown to be an asymptotic limit of transport theory for an optically thick planar-geometry system with small absorption and highly anisotropic scattering. The asymptotic analysis shows that the solution in the interior of the system is described by the standard P{sub N} equations for which initial, boundary, and interface conditions are determined by asymptotic initial, boundary layer, and interface layer calculations. The asymptotic initial, (reflecting) boundary, and interface conditions for the P{sub N} equations agree with conventional formulations. However, at a boundary having a prescribed incident flux, the asymptotic boundary layer analysis yields P{sub N} boundary conditions that differ from previous formulations. Numerical transport and P{sub N} results are presented to substantiate this asymptotic theory.