Levitating spherical particle in a slightly tapered tube at low Reynolds numbers: Application to the low-flow rate rotameters

Abstract

is an open access repository that collects the work of Arts et Métiers ParisTech researchers and makes it freely available over the web where possible. In this study, a theoretical framework is developed to predict the equilibrium conditions of a nonneutrally buoyant sphere placed in a vertical conical tube as encountered in liquid rotameters. The analysis presented herein is applicable for a sphere heavier than the surrounding fluid, situated on the axis of a slightly tapered tube. The sphere is subject to the laminar flow conditions with the Reynolds numbers ranging between the Stokes type regimes up to values corresponding to slightly inertial regimes. In this work, we assume that the aperture angle of the tube is small and that the drag force is mainly due to the dissipation located in the gap between the tube and the sphere. Under these conditions, it is possible to consider the tube as locally cylindrical and we can use the results previously obtained for the correction factor of the Stokes force on a sphere subject to a Poiseuille flow in a tube of constant cross-section. We obtain an equation relating the flow rate to the vertical position of the sphere in the tube and the validity of this analysis is demonstrated by applying it to a commercially available rotameter. The present study provides a simple but sound theoretical method to calibrate such flowmeters.

Influência da Correção de Ladenburg para Movimentos Oscilatórios Amortecidos em Meios Viscosos por Meio de um Experimento de Baixo Custo

“…Como será visto adiante, esse acréscimo é substancial e, por esse motivo, faz-se necessário uma correção nessa força, usualmente, através de um coeficiente adimensional ()   . Dentre muitas propostas para esse coeficiente [Mitsugo 1979], será utilizado o limite assintótico da expressão de Haberman e Sayre [S. Champmartin 2012]:…”