Statistical accuracy of scattered points filters and application to the dynamics of bubbles in gas-fluidized beds

Abstract

AbstractA novel analytical equation for the assessment of the accuracy of filters used for the interpolation and differentiation of scattered experimental data is presented. The equation takes into account the statistical nature of the filter output resulting from both the arbitrary positions of the data points and the randomness and noise present in the experimental data. Numerical estimation of the accuracy of the filter, using a Monte Carlo procedure, shows good agreement with the deduced analytical equation. This numerical procedure was also used to determine the accuracy of variance filters aimed at calculating the mean-square fluctuation of experimental data. The combination of the numerical results and analytical equations reveals the exact sources of inaccuracy arising in scattered point filters, namely: (i) the spectral inaccuracy of the weighting function; (ii) the noise or stochastic signal amplification; and (iii) the error arising from the random collocation of points within the filter window. The results also demonstrate that the use of the local mean in the calculation of the quadratic fluctuation leads to smaller estimation errors than the central mean. Finally, all these filters are used and critically evaluated in the framework of the stochastic position, diameter, and velocity of bubbles in a gas-fluidized bed. It is shown that the empirical coefficient of bubble coalescence in the two-dimensional bed tested, $\bar {\lambda } $, is in the range 2.0–2.4 when incorporating only the visible flow of bubbles. Here, the vertical distance over which a bubble survives without coalescing is $\bar {\lambda } {L}_{c} $, where ${L}_{c} $ is the characteristic separation between neighbouring bubbles in the horizontal direction prior to coalescence. It was also seen that the relative mean-square-root fluctuation of both bubble diameter and velocity is more than 50 % at the centre of the bed and remains nearly constant along the height of the bed.