Analysis of pressure fluctuations in fluidized beds. III. The significance of the cross correlations

In previous parts of this series we analyzed experimental times series that represent pressure fluctuation in a fluidized bed. Part I focused on the existence of long-range correlations in the series, and demonstrated that the multiscale probability density function of the successive increments of the data may be well-approximated by the Castaing equation that has been proposed for modeling velocity fluctuations in turbulent flows. Part II reconstructed the time series and demonstrated that they can be well-represented by the Fokker-Planck and Langevin equations, which provide probabilistic predictions for the future trends of the pressure. In the present paper we study cross correlations between the time series, measured at three locations in the fluidized bed. Past analyses had presumed that the time series are stationary, or that they can be transformed to one. The pressure time series are not, however, always stationary. Three distinct, but complementary methods that are suitable for non-stationary series are used to analyze the cross correlations. They are the multifractal detrended cross-correlation analysis (MF-DXA), the so-called Qcc(m) test in conjunction with the statistical test - the χ2(m) distribution - and the cross wavelet transform (XWT) function. The Qcc(m) test provides qualitative evidence for the presence of the cross correlations, whereas the MF-DXA identifies and quantifies the strength of long-range cross correlations between simultaneously recorded time series. The analysis by the MF-DXA also indicates that the pressure time series and the cross correlations between them are multifractal, hence confirming the long-range nature of the correlations identified in Part I. The results are confirmed further by the XWT analysis. The effect of time scales on the correlations and cross correlations is studied in detail. It is shown that such correlations exist at all the time scales. The cross correlations are then used for characterization of bubble sizes, bubbling frequency, and the sources of the pressure fluctuations.