In-phase and out-of-phase oscillations in BWRs: Impact of azimuthal asymmetry and second pair of eigenvalues

Stability and bifurcation analyses of boiling water reactors have been carried out using a reduced-order two-channel model developed earlier by Karve et al. To parameterize azimuthal asymmetry in core loading, an amplification factor F is introduced into the model to vary azimuthal mode feedback coefficients. Bifurcation analysis code BIFDD and numerical integration are used to analyze the reduced-order model composed of 22 modified ordinary differential equations. Results are presented for effects of azimuthal asymmetry (as parameterized by the amplification parameter F) on characteristics of oscillations. Analysis of eigenvectors corresponding to the two pairs of complex conjugate eigenvalues with the largest and second largest real parts suggests that one of these pairs is responsible for in-phase oscillations and the other for the out-of-phase oscillations. For a uniform core without azimuthal asymmetry (F = 1), as a bifurcation parameter (total pressure drop) is varied, the pair of eigenvalues corresponding to the fundamental mode first cross the imaginary axis, thus making the system unstable and leading to in-phase oscillations. However, for azimuthally asymmetric cores (corresponding to large values of F) and small inlet subcooling, the pair of eigenvalues corresponding to the first azimuthal mode, whose real part is the second largest for F = I case, approach the vertical axis faster (as a bifurcation parameter is varied) than those corresponding to the fundamental mode, thus becoming the dominant pair of eigenvalues. This leads to out-of-phase oscillations. Results of bifurcation analyses show that both sub- and supercritical bifurcation can occur for large as well as small azimuthal asymmetry, depending on values of other operating parameters. Changes in characteristics of oscillations (in-phase or out-of-phase; super- or subcritical bifurcation), therefore, result along the stability boundary. Numerical integrations confirm the results of stability and bifurcation analyses.