Thermo-mechanical phase-shift determination in Coriolis mass-flowmeters with added masses

The aim of this paper is to develop an approximate analytical solution for phase-shift (and thus mass flow) prediction along the length of the measuring tube of a Coriolis mass-flowmeter. A single, straight measuring tube is considered; added masses at the sensor and excitation locations are included in the model, and thus in the equation of motion. The measuring tube is excited harmonically by an electromagnetic driver. Taking into account thermal effects, the equation of motion is derived through use of the extended Hamilton's principle and constitutive relations. The equation of motion is discretized into a set of ordinary differential equations via Galerkin's technique. The method of multiple timescales is applied to the set of resultant equations, and the equations of order one and epsilon are obtained analytically for the system at primary resonance. The solution of the equation of motion is obtained by satisfying the solvability condition (making the solution of order epsilon free of secular terms). The flow-related phase-shift in the driver-induced tube vibration is measured at two symmetrically located points on either side of the mid-length of the tube. The analytical results for the phase-shift are compared to those obtained numerically. The effect of system parameters on the measured phase-shift is discussed. It is shown that the measured phase-shift depends on the mass flow rate, of course, but it is also affected by the magnitude of the added sensor mass and location, and the temperature change; nevertheless, the factors investigated do not induce a zero phase-shift.