A Variational Procedure for Determining Spatially Dependent Thermal Spectra

Abstract

Knowledge of the spatially dependent thermal spectrum near an interface between different media is important for many reactor calculations. The Kantorovich variational method is utilized to solve the equation A Variational Procedure for Determining Spatially Dependent Thermal SpectraAll authorsG. P. Calame & F. D. Federighihttps://doi.org/10.13182/NSE61-A25958Published online:12 May 2017Display full size where the subscript n denotes the nth spatial region. A functional, J, of Φ(E, ) and of Φ+(E, ), is found such that the solutions to the equation and its adjoint make J stationary. Trial functions for Φ and Φ+ are employed which are linear combinations of the infinite medium spectra and adjoints, respectively, of a hard and a soft region. The constants of combination are undetermined functions of . These trial functions are inserted into J and the energy integrations performed. When the resulting expression is made stationary with respect to arbitrary variations of the adjoint constants of combination, there results in the nth region a set of two coupled differential equations for the flux constants of combination. The equations are solved simultaneously, yielding the energy spectrum as a function of position. The spectrum is used to obtain activation rates, and the rates are compared to experiments. The agreement is excellent. The method, that of overlapping groups, appears to be a promising one for the solution of the thermal space energy problem in more complex reactor calculations.