MHD stability analysis of a liquid sodium flow at the annular gap of an EM pump

Abstract

A stability analysis of a viscous, incompressible, and electrically conducting liquid sodium flow in an annular linear induction electromagnetic pump for sodium coolant circulation of a Sodium Fast Reactor (SFR) was carried out when transverse magnetic fields permeate the sodium fluid across the narrow annular gap. Due to a negligible skin effect and the presence of a magnetic core outside the gap, radial magnetic field is assumed to be constant over the narrow channel gap, and the steady state solution of the axial velocity is obtained as a function of radius. Small perturbations for MHD fields were considered in sinusoidal form as a function of the angular frequency and wave number, and the resulting equations were linearized. The solutions of the perturbed equations were sought in the form of a linear combination of independent orthogonal functions in a non-dimensional radial interval (0, 1), and each orthogonal function was chosen to satisfy the boundary conditions of adhesion to the solid walls of the channel. Under the assumption that solutions of the equations were not oscillated rapidly according to the radial coordinate, finite numbers of orthogonal polynomials were considered. As a result, simultaneous equations with coefficients of steady-state solutions were arranged, and dispersion relations between angular frequency and wave number of perturbed state were sought. The imaginary part of the angular frequency was taken into consideration from the condition of existence of a nontrivial solution of the system, which leads to the relation between critical Reynolds number (Re-cr) and Hartmann number (H-a). In the present study, critical Reynolds number and wave numbers were plotted against the Hartmann number for a long wave perturbation; thus, it was shown that a magnetic field has a significant stabilizing effect on liquid sodium flow.