From the Hilbert expansion of the Boltzmann equation to reactor homogenization
Although the Hilbert expansion of the Boltzmann equation of the nonlinear molecular theory of gases to obtain the Euler equations of the continuum theory of fluid mechanics usually is presented as a formal expansion begun by {open_quotes}inserting {close_quotes} a 1/e in front of the collision term, it also can be developed using the more physically intuitive idea of an asymptotic expansion in the ratio of the small collision mean-free {lambda} to the large characteristic system dimension L (or in the common infinite-medium kinetic theory setting, the large characteristic macroscopic scale length f/{vert_bar} {triangledown}f{vert_bar}). Thus the basic ideas, used for the past 20 yr in the development of diffusion-type equations as asymptotic approximations to the neutron transport equation and in the asymptotic development of lattice-cell Homogenized diffusion equations as asymptotic approximations, had their origins more than 70 yr ago in Hilbert`s original work. In that work, the solvability condition for the first-order equation leads to a linear Fredholm integral equation, derived from the nonlinear scattering operator, that has a fivefold degenerate zero eigenvalue, corresponding to the five collision invariants - mass, three momenta, and energy. This degeneracy, which leads to the Euler equations, is not present in the corresponding equations related to neutron scattering operators because they conserve only mass (particles, i.e., neutrons), transferring momenta and energy to the {open_quotes}background gas{close_quotes} (moderator, etc.). Hence, only one equation, the one-group diffusion equation, normally results from small mean-free-path asymptotic expansions of the neutron transport equation, even when the starting point is the energy-dependent transport equation or the multigroup transport equations.
- OSTI ID:
- 89120
- Report Number(s):
- CONF-941102-; ISSN 0003-018X; TRN: 95:004215-0192
- Journal Information:
- Transactions of the American Nuclear Society, Vol. 71; Conference: Winter meeting of the American Nuclear Society (ANS), Washington, DC (United States), 13-18 Nov 1994; Other Information: PBD: 1994
- Country of Publication:
- United States
- Language:
- English
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