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Theoretical Development : Formulation of Inverse Problem Probability density functions : definitions

A new method is developed for the determination of bubble size distributions from experimental laser Doppler anemometry (LDA) data through simultaneous measurement of only two orthogonal velocity components and the intermittent time gaps arising out of bubbles passing through the measurement point. An experimentally verifiable assumption of axisymmetry allowed the calculation of the three-dimensional velocity distributions from the two-dimensional measurement through the elegant use of a mathematical transform called Abel’s transform. In conjunction with the use of probability theory, it allows calculation of the chord length distribution. The bubble size distribution is obtained by an inversion procedure, which facilitates the transformation of chord lengths to bubble sizes. The inversion procedure is successfully tested with Monte Carlo simulations, in which bubbles of prescribed size distribution were generated for obtaining synthetic data on subtended chord lengths at any given point in the column. It is unambiguously demonstrated that assumptions that do not account for correlation between bubble size and direction of motion can yield highly erroneous size distributions from chord length data. The procedure was then used on experimental chord length distributions obtained using LDA to estimate local bubble size distributions in our bubble column. It is particularly interesting to observe that the mean bubble size estimated from the calculated bubble size distributions were consistent with an estimation based on slip velocity information in accord with the Zuber–Findlay drift flux model. © 2004 American Institute of Chemical Engineers AIChE J, 50: 3068–3084, 2004