An electrostatic interpretation of the weighted flow theorem of electromagnetic flowmeters

Abstract

The weighted flow theorem of electromagnetic flowmeters links the output voltage of an electromagnetic flowmeter with the spatial velocity distribution of a fluid and the weight function. The existing derivations of the theorem are based on a formal Green's function treatment of the underlying partial differential equation describing flowmeter operation. The main aim of this paper is to provide a physically transparent and mathematically rigorous picture of the origin of the flowmeter output voltage. We show that electrostatic charges build up both in the volume of the meter and on its surfaces as a weakly conductive fluid enters the magnetic field of the meter. The flowmeter output voltage is completely determined by these volume and surface charges. Departing from the physical conditions in the fluid and the fluid boundaries we establish relations satisfied by those charge densities and then derive the analytical expression for the output voltage by standard electrostatic methods. The resulting expression is transformed into a volume integral of the scalar product of fluid velocity and the weight vector which in turn is equal to the vector product of the magnetic induction and a virtual current, as introduced by Bevir for the first time. We further show that this latter quantity may be interpreted as that part of the normalized current density of the operating flowmeter the contours of which are closed outside of the fluid. We also give a simple interpretation of the results based on conservation of energy. We hope that the proposed electrostatic approach could provide a more intuitive insight into flowmeter operation to electrical engineers than the general Green's function formalism.