Measurement of velocity in rotational flows using ultrasonic anemometry: the flowmeter

In this paper a previously developed theoretical model of the measurement process performed by a transittime ultrasonic anemometer is applied to a fluid flowing through a circular section pipe. This model considers the influence of the shift of the acoustic pulse trajectory from straight propagation due to the flow on the measured speed. The aim of this work is to estimate the errors induced in the measured velocity by the shift of the acoustic pulse trajectory. Using different duct’s flow models, laminar and turbulent regimes have been analyzed. The results show that neglecting the effect of shift of the acoustic pulse trajectory leads to flow rate measurement underestimation. List of symbols A cross-sectional area of the pipe c sound speed csc(/) cosecant of / angle C = {O,n,f} pipe coordinate system. CS , CC S transformation matrices between C and S C2 coefficient of the measured velocity correction term C20 part of C2 due to asymptotic expansion C2s part of C2 due to taking into account the deviation of the ray path from a straight line dc duct cross-section diameter Kh hydraulic correction factor l length of measurement path M Mach number n exponent of Nikuradse velocity profile model Q mean volumetric flow rate Re Reynolds number S = {O,x,z} acoustic path coordinate system u, w components of the velocity profile inside the pipe expressed in S coordinate system U, W dimensionless components of the velocity profile inside the pipe expressed in S coordinate system uM measured velocity along the measurement path UM dimensionless measured velocity along the measurement path un component of the measured velocity along the n axis VR reference velocity X, Z dimensionless coordinates Z+, Z– deviation of the ray path from a straight line in forward and backward directions Z1 coefficient of M order term in the expressions of Z+, Z– Z2 coefficient of M 2 order term in the expressions of Z+, Z– m fluid’s cinematic viscosity / measurement path inclination angle t velocity profile inside the pipe expressed in C coordinate system t average velocity across the cross-sectional area of the pipe S. Franchini (&) A. Sanz-Andrés A. Cuerva Instituto Universitario ‘Ignacio Da Riva’ (IDR), Escuela Técnica Superior de Ingenieros Aeronáuticos, Universidad Politécnica de Madrid, Madrid, Spain e-mail: [email protected] URL: http://www.idr.upm.es 123 Exp Fluids (2007) 42:903–911 DOI 10.1007/s00348-007-0299-x