Mass flowmeters for fluids with density gradient

Quadrature integration of flow velocity along prescribed parallel paths has been one of the preferred solutions for accurate computation of volumetric flowrate Q for over thirty years. The present work extends the quadrature volumetric flowmetering method to determining from sound speed c₃ along the same quadrature paths, the fluid temperature T, density ρ and mass flowrate M_F in the presence of a density gradient. Quadrature integration of ρ_i × V_i products V_i obtained from c_3i is useful to the extent the fluid is sufficiently pure and defined so that the uncertainty in sound speed c_i along each path generates only a small uncertainty in density pi along those same paths. Instead of "useful to the extent" one could just as well say "limited to the extent." Recognizing this limitation, it is interesting to compare the density and mass flowrate determined from (a) ultrasonic measurements of propagation across the fluid, as indicated above, with density and mass flowrate determinations based on two other known methods. These two methods are: (b) reflection coefficient measurement of fluid characteristic impedance Z, which would seem capable of leading to ρ after dividing by c; and determining ρ based on (c) torsional wavespeed in a waveguide of noncircular cross section.