Variational Solutions of the Transport Equation

Francis, N.C.; Stewart, James J. P.; Bohl, L.S.; Krieger, T J

doi:10.2172/4305856

Cited by 8 publications

(6 citation statements)

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“…In the late 1950s, people derived explicit variational functionals for neutron transport [2]. Throughout the 1960s, 1970s, 1980s, and even the 1990s, much effort was poured into deriving improved variational techniques for diffusion and transport [3,6,14,18,21,24,25,30].…”

Section: Perturbation Theory and Variational Methods Applied To Matermentioning

confidence: 99%

“…. G. 2 The fuel pin axial profile in the FFTF benchmark. This is a sodium fast reactor (SFR) with mixed oxide (MOX) fuel [146].…”

Section: 3mentioning

confidence: 99%

“…Figure 1.2 illustrates this. Thus, when sodium voids and the spectrum shifts to higher energy, a reactor fueled primarily by 2 33 U will produce less reactivity than a reactor fueled primarily by 2 39 Pu. Furthermore, 2 33 U also has a higher delayed neutron fraction than 239 Pu, which further enhances safety.…”

Section: Sodium Void Worthmentioning

confidence: 99%

“…First order perturbation theory to compute reactivity coefficients is rapid and accurate, but it is already fully known. What isn't fully known, however, is (2). Boundary perturbation theory has still not been developed to the extent that we can accurately apply it to arbitrary core distortion scenarios.…”

Section: Why We Still Need Perturbation Theorymentioning

confidence: 99%

“…Since we already know the volumeaveraged radial flux gradients in cells 2 and 3, we can linearly interpolate these to estimate the gradient values on surface 3. If the axial heights of cells 2 and 3 are dz 2 and dz 3 , then we could interpolate like this:…”

Section: Estimation Of Boundary Flux Gradientsmentioning

confidence: 99%

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The “virtual density” theory of neutronics

Reed¹,

Smith

Forget

2018

Annals of Nuclear Energy

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Sustainable nuclear energy will likely require fast reactors to complement the current light water reactor paradigm. In particular, breed-and-burn sodium fast reactors (SFRs) offer a unique combination of fuel cycle and power density features. Unfortunately, large breed-and-burn SFRs are plagued by positive sodium void worth. In order to mitigate this drawback, one must quantify various sources of negative reactivty feedback, among which geometry distortions (bowing and flowering of fuel assemblies) are often dominant. These distortions arise mainly from three distinct physical phenomena: irradiation swelling, thermal swelling, and seismic events.Distortions are notoriously difficult to model, because they break symmetry and periodicity. Currently, no efficient and fully generic method exists for computing neutronic effects of distortions. Computing them directly via diffusion would require construction of exotic hyperfine meshes with continuous re-meshing. Many deterministic transport methods are geometrically flexible but would require tedious, intricate re-meshing or re-tracking to capture distortion effects. Monte Carlo offers the only high-fidelity approach to arbitrary geometry, but resolving minute reactivities and flux shift tallies within large heterogeneous cores requires CPU years per case and is thus prohibitively expensive. Currently, the most widely-used methods consist of various approximations involving weighting the uniform radial swelling reactivity coefficient by the power distribution. These approximations agree fairly well with experimental data for flowering in some cores, but they are not fully generic and cannot be trusted for arbitrary distortions. Boundary perturbation theory, developed in the 1980s, is fully general and mathematically rigorous, but it is inaccurate for coarse mesh diffusion and has apparently never been applied in industry.Our solution is the "virtual density" theory of neutronics, which alters material density (isotropically or anisotropically) instead of explicitly changing geometry. While geometry is discretized, material densities occupy a continuous domain; this allows density changes to obviate the greatest computational challenges of geometry changes. Although primitive forms of this theory exist in Soviet literature, they are only applicable to cases in which entire cores swell uniformly. Thus, we conceive a much more general and pragmatic form of "virtual density" theory to model non-uniform and localized geometry distortions via perturbation theory.In order to efficiently validate "virtual density" perturbation theory, we conceive the "virtual mesh" method for diffusion theory. This new method involves constructing a slightly perturbed "fake" mesh that produces correct first-order reactivity and flux shifts due to anisotropic swelling or expansion of individual mesh cells. First order reactivities computed on a "virtual mesh" agree with continuous energy Monte Carlo to within 1-uncertainty.We validate "virtual density" theory via the "virtual mesh" me...

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