Characterization of dynamic behaviour in gas–solid turbulent fluidized bed using chaos and wavelet analyses
Introduction
In spite of the prodigious research effort devoted to its study for over half a century, some aspects of fluidized bed hydrodynamics remain elusive. The turbulent fluidization flow regime, in particular, is subject to continuing uncertainty and controversy. A comprehensive review covering what is known about the turbulent regime was published recently [1].
One of the factors underlying the uncertainty with respect to the turbulent fluidization flow regime relates to the conceptual basis for investigation employed by different research groups. Whereas the bubbling, slug flow and fast fluidization flow regimes can all be viewed and modelled as consisting of two distinct phases (bubbles and surrounding dense phase emulsion in the first case, gas slugs and surrounding dense phase in the second, and clusters and surrounding gas in the latter), the turbulent regime presents a more complex structure. Several different conceptual bases are possible:
- (a)
A number of studies, e.g. [2], [3], [4], [5] have assumed that the basic structure of bubbling fluidized beds, i.e., gas bubbles immersed in a solids-in-gas emulsion, can be extended into the turbulent fluidization flow regime. Effective bubble properties (e.g. void diameter, velocity and frequency) are then ascribed to the system, and the bed is treated as if it were a bubbling bed, albeit one with smaller, faster-moving voids.
- (b)
It is possible [6], [7] to view at least some turbulent fluidized beds as being subject to intermittency at each location, where periods of bubbling and fast fluidization alternate with each other, and the fraction of time corresponding to the latter increases as the superficial gas velocity is increased through the range beginning at the onset of turbulent fluidization, usually designated Uc, and terminating at the velocity corresponding to the transition from turbulent to fast fluidization. A variant of this approach [8], [9] is to statistically hybridize the different fluidization flow regimes, i.e., to probabilistically average properties from three limiting regime-specific models, one each applying to bubbling, turbulent and fast fluidization.
- (c)
For cases where neither (a) nor (b) turns out to be appropriate, it is possible to utilize concepts and methodologies developed for other complex flows (e.g. single-phase turbulent flows, gas–liquid churn/turbulent flow regime) to characterize the hydrodynamic patterns observed in the turbulent fluidization flow regime. Deterministic chaos analysis and wavelet analysis provide methodologies that can be adapted to fluidized beds. The ultimate goal of such tools is to provide a fundamental framework for representing and predicting the behaviour of fluidized beds over the entire spectrum of flow regimes.
This paper addresses some of these issues, in particular with respect to approaches (a) and (c) above. Based on local voidage fluctuations and pressure fluctuations measured in a column of diameter 0.29 m, we first examine whether or not a two-phase representation (approach (a) above) is capable of capturing the essence of the hydrodynamic behaviour. It is shown that there is no unambiguous way of distinguishing two phases and that, in fact, there is a broad and continuous spectrum of local voidages and phenomena which render the two-phase representation inappropriate as a descriptor or basis for modelling the turbulent fluidization investigated experimentally for FCC particles.
In view of this finding, we then turn our attention to (c) above, in particular to two analysis methods that may be promising, i.e., chaos and wavelet analysis. A number of research groups, e.g. [10], [11], [12], [13], [14], [15] have demonstrated that chaotic analysis methods can provide useful insights and can distinguish among flow regimes with respect to fluidized bed and other multiphase systems. Given the highly non-linear behaviour, chaos analysis should be especially applicable to the turbulent fluidization flow regime. Over at least some range of conditions, chaotic analysis has suggested multifractal behaviour of gas-fluidized beds [16], indicating that the complex flow patterns can be disaggregated and considered as being composed of different scales of motion. Wavelet analysis provides another method of analysing multiphase systems including fluidized beds, with due consideration of different scales contributing to the complex overall waveforms, e.g. [15], [17], [18]. Through wavelet analysis, a one-dimensional time series is transformed into a two-dimensional region displaying wavelet coefficient amplitudes as a function of both time and frequency. This enables time localization of spectral components to be interpreted [19]. According to Ren et al. [20], local voidage measurements from a fluidized bed can be decomposed into three scales: micro-scale (particle and fluid), meso-scale (voids, two-phase structure), and macro-scale (equipment, global). Information from wavelet analysis should complement that obtained from other techniques.
Section snippets
Analysis method
Hurst [21] initiated a new method of analysis for time series of natural phenomena. This analysis involves calculating the average rescaled range (R/S)τH for various values of subperiod length τH. The analysis was conducted carefully using a procedure developed and extensively tested by Briens et al. [22]. The R/S analysis was only one of many methods used to examine the signals. The results of the various methods are provided in Ellis [23]. A Hurst exponent, H, is estimated [22] by:
Experimental technique
The 0.29 m diameter, 4.5 m tall Plexiglas vessel is equipped with 58 sampling ports. The distributor is an aluminium perforated plate containing 98 holes of 5.6 mm diameter arranged in an equilateral triangular configuration with a 32 mm pitch, resulting in an open area ratio of 3.7%. Solids circulation is not controlled, but rather determined through a pressure balance between the return leg and the column. There are two flapper valves installed in the return leg to prevent gas from escaping up
Binary analysis
As previously reported [35], frequency analysis of pressure signals recorded in fluidized beds reveals dominant frequencies which characterize the flow regime. Analysis of the dominant peaks can be related to the physical behaviour, particularly in the bubbling and slugging flow regimes where the pressure fluctuations are a strong function of the movement of bubbles and slugs and when there is a clear bimodal distribution of voidage between bubbles and the surrounding emulsion phase. In
Conclusion
From the experimental work involving measurements of gauge and differential pressure and local voidage in fluidized beds operated at conditions approaching and beyond the onset of turbulent fluidization flow regime, several approaches to the underlying idea understanding the complex flow have been explored. Treating data from a turbulent fluidized bed as a combination of co-existing discrete bubbles and dense phase was found to be inappropriate because the bed operating in this flow regime
Acknowledgements
The authors are grateful for financial support from the Natural Sciences and Engineering Research Council of Canada (NSERC). We are also grateful to Cor van den Bleek who helped to interest us in chaos analysis as a tool for investigating fluidized beds.
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