Chaotic Behavior of Fluidized Beds Based on Pressure and Voidage Fluctuations zy D. Bai, zyxwvu €I. T. Bi, and J. R. Grace zyxw Dept. of Chemical Engineering, University of British Columbia, Vancouver, BC, Canada V6T 124 Absolute and differential fluctuations and local voidage fluctuations are commonly used in characterizing gas-flui- dized beds. Pressure fluctuations can result from several dif- ferent causes including bubble formation and eruption, self- excited oscillations of fluidized particles, pressure oscillations in the plenum chamber due to piston-like motion of the bed, bubble coalescence, and bubble splitting (Bi et al., 1995). Pressure waves from these sources can be propagated through the emulsion phase, although the amplitudes of the signals are attenuated during their propagation (Bi et al., 1995). Therefore, an absolute pressure probe records phenomena on a macroscopic scale from other locations in addition to local variations. Differential pressure measurements can, to a limited extent, filter out pressure waves originating outside the interval between the two ports connected to the differen- tial pressure transducer, that is, measurements are on a meso-scale. Voidage probes reflect local (that is, microscale) voidage variations caused by bubble passage and the particle movement of the emulsion phase. The amplitude of local voidage fluctuations is independent of bubble size, unlike pressure fluctuations whose amplitude increases with bubble size. Previous studies (Bi et al., 1995; Bi and Grace, 1995) have shown that statistical characteristics (such as standard deviation, skewness, dominant frequency, probability distri- bution) calculated from these three types of signals differ sig- nificantly from one another. In recent years, the application of deterministic chaos the- zyxwvut ory to fluidized beds has been pioneered by several research groups (such as Daw and Halow, 1991; van den Bleek and Schouten, 1993; Skrzycke et al., 1993; Halow and Daw, 1994; van der Stappen et al., 1995). Analysis of pressure fluctuation signals (such as the phase-space portrait, Lyapunov exponent, correlation dimension, and Kolmogorov entropy) has pro- vided evidence that gas-solids fluidized beds are determinis- tic chaotic systems. The correlation dimension of the time series has been calculated based on time series from absolute (van der Stappen et al., 1993) and differential (Daw and Halow, 1991) pressure fluctuation measurements. In view of the different information contained in time traces of absolute and differential pressure and local voidage, this article exam- Correspondence concerning this article should be addressed to J. R. Grace. ines the chaotic characteristics of signals from these three measurement methods based on analysis of signals recorded simultaneously in gas-solids fluidized beds. Experiments The experiments were carried out in a 102-mm-dia. col- umn described previously (Bi and Grace, 1995). Both FCC particles of zp = 60 pm, zyxw pp = 1,580 kg/m3 and sand particles of Lip = 214 pm, zyxwv pp = 2,640 kg/m3 were used in the tests with the static bed height maintained at 0.6 m for all tests. (ap is the mean particle diameter and pp is the particle density.) Three gas velocities were used, corresponding to different flow regimes. Pressure fluctuations were measured from ports on the wall of the column using Omega PX162 pressure transducers. A low-pass filter eliminates signals with frequen- cies above 20 zyxw Hz. Absolute pressure fluctuations and local voidages were measured 0.28 m above the distributor, while differential pressure fluctuations were measured across an interval from 0.20 to 0.41 m above the distributor. The opti- cal probe (see Zhou et al., 1994 for details) was located with its 1.5 mm tip on the axis of the column. All signals were sampled by a digital data logging system (12 bit DAS 8 A/D board, Labtech Notebook data acquisition software) at a fre- quency of 100 Hz for intervals of 40 s. A low-pass filter set at 20 Hz was used in the pressure fluctuation measurements, and each data set was numerically smoothed to eliminate small amplitude high frequency noise components before data analysis. Results and Discussion Power spectra The power spectrum of a chaotic time series is quite differ- ent from those of periodic and quasi-periodic time series. The former has a continuous, broad band noise-like spectrum. S is the power spectrum intensity. Beyond a certain frequency (f, Hz) range the power spectrum intensity decays with fre- quency, obeying a power law AIChE Journal May zyxwvu 1997 Vol. 43, No. 5 1357