Bifurcation analysis of density wave oscillations in natural circulation loop

Abstract

The present study explores the linear stability and bifurcation analysis of two-phase flow in the single heated channel for the natural circulation loop. The bifurcation analysis of this natural circulation system is limited to the detection of Hopf bifurcation, generalized Hopf bifurcation and turning point or limit point bifurcation of limit cycles. These natural circulation loops have several engineering applications and prone to thermal-hydraulic instability, which may lead to an operationally inefficient system. Stability boundary (linearly) obtained on a Npch- Nsub plane predicts the existence of limit cycles (a nonlinear phenomenon) for these loops and gives motivation for the further study on the detection of nonlinear phenomena. The subsequent results show a novel case of stability boundary (listed as Type A in the present work) emerging for higher subcooling numbers on Npch- Nsub parameter plane, which has not been noted in literature earlier and lies near to the Ledinegg stability boundary, which is a static instability. The characteristics of points nearby Type A stability boundary predicts that this boundary is different from the Ledinegg stability boundary. This Type A stability boundary divided by subcritical and supercritical Hopf bifurcation, which is separated by a generalized Hopf bifurcation (GH) point. Sustained oscillations in flow velocity can be observed near to the Hopf points on this boundary. These subcritical (hard and dangerous) and supercritical (soft and safe) Hopf bifurcations have been examined with time series graphs for all types (Type A-C) of stability boundaries present for the loops and occurring on the Npch- Nsub parameter plane. The two other type of boundaries known earlier is characterized here as Type B and C, respectively, which is basically Type I and Type II density wave oscillations. The transition of stability boundaries is discussed on the Npch- Nsub parameter plane, which shows that Type A stability boundary contains GH bifurcation coming from Type B stability boundary. The bifurcation of limit cycles originating from the Hopf points in the system has been examined and it is found that limit point bifurcation of limit cycles is occurring in the system, originating from the generalized Hopf points.