Higher order power reactor noise analysis: The multigroup diffusion model

Power reactor noise analysis is one of the most powerful tools in online monitoring and diagnostics of nuclear power reactors. Unfortunately, since such an analysis belongs to the non-linear “parametric excitation” realm, its theoretical aspects and relations have been mostly carried out after linearization. In this paper a general framework, i.e. the Ladder Expansion Method, is developed to convert such equations to a series of coupled linear equations, up to any desired accuracy. This method is then applied to the single mode random fluctuations of the absorption cross sections in a power reactor which is modelled by the multigroup diffusion equation with multiple delayed neutron groups. A system of coupled pseudo steady state diffusion equations has been derived as the result. The procedure of numerically solving such a system, using the Finite Element Method is described and the previously reported GFEM code, which is a Galerkin FEM based diffusion solver and Linear Power Reactor Noise analyzer for two dimensional geometries, has been generalized to accommodate the LEM to evaluate the higher order power reactor noise moments. Use of these moments in the Power Spectral Density of the flux and its derivatives, such as the detection rate, has shown that the value of the PSD at the main harmonic of the fluctuation deviates with the predictions of the conventional LPRN method. Moreover, the generation of super-harmonic modes in the PSD of the neutron flux distribution has been shown to follow naturally by using the developed LEM. Three numerical benchmarks show the correctness and accuracy of the developed method. Finally, the error introduced in the linearization process is quantified by comparing numerical results of the LPRN method with that of the LEM, and some of the general trends of this error have been identified. As a result, the expectation of validity of the linearization process in the “sufficiently small perturbation” region is confirmed.