A model for obtaining the velocity vectors of spherical droplets in multiphase flows from measurements using an orthogonal four-sensor probe

This paper presents a mathematical model which enables the velocity vectors and diameters of spherical droplets (or bubbles) in bubbly two-phase flows to be determined from the outputs of a local four-sensor intrusive probe. Each of the four sensors functions by measuring the fluid conductivity at its tip and so use of the probe is relevant to flows where there is a contrast in the electrical conductivity of the continuous and dispersed phases. The motion of a non-conducting spherical droplet has been simulated as it moves across a four-sensor probe, which has a leading sensor and three rear sensors in an orthogonal arrangement. The technique described in this paper relies upon measuring the time intervals between the droplet surface first contacting the leading sensor and then coming into contact with each of the three rear sensors. Assuming that the surface of the droplet comes into contact twice with each of the rear sensors, as the droplet moves across the probe, there will be six such time intervals. It has been shown that in order to obtain the droplet velocity components in the x, y and z directions with an accuracy of ±2%, the six time delays must all be measured with an accuracy of ±10 µs. It has been shown that the probe dimensions are critical to the measurement technique and that the separation of the four sensors should be of the order of 1 mm. It has also been shown that if the droplets are oblate spheroids rather than spheres then, provided their aspect ratio is greater than 0.75, the magnitude of the additional error in the estimates of the droplet velocity components is less than 10.5%. No account has been taken of the influence of the probe on either the shape or the motion of the droplet