Bifurcation analyses of in-phase and out-of-phase oscillations in BWRs

Stability and bifurcation analysis of BWRs have been carried out using a reduced order two-channel model developed earlier by Karve et al. This model includes fuel rod heat conduction, single- and two-phase flow, and fundamental and first azimuthal mode modal neutron kinetics based on the ω-mode approach. The set of twenty-two ODEs in the reduced order model is analyzed using the bifurcation analysis code BIFDD. In addition to the stability boundary in the design and operating parameter spaces, nature of bifurcation along the entire stability boundaries is also determined. Effects of various bifurcation parameters on the nature of bifurcation are analyzed. Results are presented for the effects of flux asymmetry on bifurcation characteristics as parameterized by the azimuthal mode feedback coefficient. To parameterize different flux shapes, an amplification factor F is introduced to vary the azimuthal mode feedback coefficient. In-phase oscillations result for smaller values of F (F ~ 1; azimuthally symmetric flux). Depending upon the values of other parameters, both sub- and supercritical bifurcations are predicted. However, as F is increased, leading to azimuthally asymmetric flux shapes, the real part of a second pair of complex conjugate eigenvalues increases, and this pair of eigenvalues approaches the imaginary axis. This pair, for large F and small values of N_sub, actually becomes the eigenvalue with the largest real part, leading to out-of-phase oscillations when the stability boundary is crossed. In this case, for parameter values studied so far, only supercritical bifurcations are predicted. The eigenvectors corresponding to the two pairs of complex conjugate eigenvalues suggest that one is responsible for in-phase oscillations, while the second is responsible for the out-of-phase oscillations. Results of numerical integrations confirm the findings of the semi-analytical bifurcation analyses.