Asymptotic Numerical Method applied to stability and bifurcation analysis of a boiling flow at low pressure in a pipe
Résumé
In the framework of the RetD on GEN IV Sodium-cooled Fast Reactors (SFRs) at CEA, a hypothetical Unprotected Loss of Flow (ULOF) is investigated and could lead to sodium boiling if one assumes the non-activation of passive complementary safety devices. As a qualification basis, the CEA is currently developing and coupling reference codes at different scales. Supposing a complete integrity of hydraulic channels geometry, reactor case studies have shown the possibility of a periodic boiling regime the density wave mechanism, which is believed to be enhanced by low void worth core design distinctive of a GEN IV SFR, is interestingly characterized by pins claddings cooling possibility at sodium saturation temperature (instead of dry-out). In that scientific framework, this paper presents the ongoing development of an innovative semi-analytical methodology to perform stability and bifurcation analyses of boiling flows. While equations for a 1D multiphase model based on a drift model are classically used, the Asymptotic Numerical Method (ANM) is implemented to solve steady-state equations whose non linearities drive the observed dynamic phenomena. Inline, elements of the Theory of Dynamical Systems are first recalled, such as the Hopf bifurcation linked to the appearance of a periodic solution and its stability, to provide the necessary mathematical background. The added-value for this specific study of the ANM over a zero-order continuation method is illustrated on a test case. Some simulation results are then reported to investigate flow phenomenology that was addressed along boiling stability experiments led by Saha in 1970s. On one hand, the runs show our model's numerical ability to efficiently handle a non-linear set of equations while imposing a numerically stringent pressure difference for flow boundary conditions; on the other hand, no qualitative evidence of unstability onset has been observed while such a phenomenon is the central point of Saha experimental work. Even if only eigenvalues analysis, to be next carried out, could assert a miss of the model, this point could reveal some limitations regarding the set of closure laws or geometrical description that is up to now considered in our model. To support this statement, some additional simulation results on an academic case are given which exhibit how a non-linearity of the model, not engaged in the Saha experimental conditions, could lead to an oscillating response in the bifurcation diagram.
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