Perturbation and sensitivity theory for burnup analysis

Perturbation theory is developed for the nonlinear burnup equations describing the time-dependent behavior of the neutron and nuclide fields in a reactor core. General aspects of adjoint equations for nonlinear systems are first discussed and then various approximations to the burnup equations are rigorously derived and their areas for application presented. I In particular, the concept of coupled neutron/nuclide fields (in which perturbations in either the neutron or nuclide field are allowed to influence the behavior of the other field) is contrasted to the uncoupled approximation (in which the fields may be perturbed independently).

Adjoint equations are derived for each formulation of the burnup equations, with special attention given to the quasi-static approximation, the method employed by most space- and energy-dependent burnup codes. It is shown that, based on this formulation, three adjoint equations (for the flux shape, the flux normalization, and the nuclide densities) are required to account for coupled variations in the neutron and nuclide fields. The adjoint equations are derived in detail using a variational principle. The relation between coupled and uncoupled depletion perturbation theory is illustrated.

Solution algorithms are given for numerically solving the adjoint burnup equations, and the implementation of these procedures into existing computer codes is discussed. A physical interpretation is given for the burnup adjoint functions, which leads to a generalization of the principle of "conservation of importance" for coupled fields. Analytic example problems are solved to illustrate properties of the adjoint functions.

Perturbation theory is used to define sensitivity coefficients for burnup-dependent responses. Specific sensitivity coefficients are written for different types of nuclear data and for the initial condition of the nuclide field. Equations are presented for uncertainty analysis of burnup calculations.

Uncoupled depletion sensitivity theory is applied to the analysis of an irradiation experiment being used to evaluate new actinide crosssection data. The computed sensitivity coefficients are used to determine the sensitivity of various nuclide concentrations in the irradiated sample to actinide cross sections. Uncertainty analysis is used to calculate the standard deviation in the computed values for the plutonium isotopics.

Coupled depletion sensitivity theory is used to analyze a 300o Mw. th denatured LMFBR model (2 region, sphere). The changes in the final inventories of ²³²U, ²³³U, and ²³⁹Pu due to changes in concentrations of several nuclides at the beginning of cycle are predicted using depletion perturbation theory and are compared with direct calculation. In all cases the perturbation results show excellent agreement with the direct changes.