Theoretical studies on the design criteria of double-sensor probe for the measurement of bubble velocity

This paper presents a study of the design criteria and the theoretical calibration factor of a double-sensor probe for the measurement of bubble velocity in two-phase flows. Due to the finite probe spacing between the two probe tips, any lateral motion of a bubble can complicate the interpretation of the measured value. To quantify the measurement error, a rigorous derivation was carried out for spherical bubbles to bridge the measured value with the true bubble velocity. Afterwards, the mean measurable value was then obtained by use of proper probability density functions identified via a necessary coordinate transform. It was discovered analytically that the measurable velocity might approach infinity if the probe spacing is too small compared to the bubble size. For practical applications, the probe spacing should be greater than roughly one half of the bubble diameter in order to avoid such a singularity problem. Numerical results indicate that the calibration factor, defined as the ratio of the mean bubble velocity to the average measured value, depends only on the relative bubble velocity fluctuation, if the probe spacing is about 0.5–2 times the bubble diameter. The relative bubble velocity fluctuation was then correlated with the standard deviation of the inverse measurable speed that can be readily obtained in the experiment. These results are also applicable for moderate non-isotropic velocity fluctuations. Finally the appropriate sample size was provided using a Monte Carlo approach.