OBJECIVES . . . The objectives of this paper are to give an overview of chaotic time series analysis and to survey studies of application of chaos theory in gas-solids fluidization in order to identify the potentials of chaos theory in improving fluidized bed design and operation.
OVERVIEW . . . The review begins with a brief introduction to chaos theory and gas-solids fluidization and follows with a discussion of the gamut of practical methods for the analysis of nonlinear, chaotic dynamical systems. It is described that chaos theory is a relatively new branch of science that is being developed as the nonlinear counterpart of classical linear signal processing.
Practical aspects of chaotic time series analysis are then discussed and recommendations are made as to the requirements for the measurement system, the optimum data-acquisition settings and standardized methods of data analysis.
Chaos studies in the field of bubbling and slugging fluidization were initiated about five years ago. These studies are considered next and it is shown that dynamical invariants such as correlation dimension and Kolmogorov entropy are able to quantitatively characterize the fluidized bed’s dynamics. Other analysis techniques such as principal component analysis and chaotic trajectory decomposition are then described and shown to be able to uniquely characterize different modes of dynamical bed behavior. It is found that these studies are merely prelimiiry in the sense that only specific examples are shown while so far only one study gives a more detailed impression about the dependency of chaotic invariants on varying fluidization conditions.
It is shown further that the dynamics of bubbling and slugging beds are spatially extended in the sense that the number of degrees of freedom (dimension) and the level of predictability (entropy) vary with the location in the bed. Novel methods are discussed that quantify this spatio-temporal behavior of fluidized beds in terms of the degree of dynamical coupling between different locations in the bed and of the level of predictability per unit of length.
Since two years also chaos studies in circulating fluidized beds (CFBs) operated at high gas velocity have been carried out and these have shown that Kolmogorov entropy can be used, for example, to quantify the turbulence level of flowing gas-solids suspensions as a function of fluidizing conditions in CFBs. These chaos studies of CFBs have been performed at a wide range of riser diameter and fluidizing conditions.
Two simple one-dimensional, ‘learning’ fluidized bed models that exhibit chaotic behavior are then examined. The first model describes single bubbles and bubble-to-bubble interactions and the second model is based on single particles and particle-to-particle interactions. The latter model is compared with experimental data about the dependence of entropy and dimension on gas velocity and is very well able to qualitatively reproduce the chaotic dependencies that were observed in the measurements.
OUTLOOK AND CONCLUSIONS ... An outlook is presented on two practical applications of chaos theory in the areas of design and operation of gas-solids fluidized beds.
First, it is proposed to include the chaoticness of fluidized beds in reactor scale-up. Two different routes are considered: the first route, proposes to include chaotic similarity as an additional requirement in dimensionless scaling laws for fluidized beds, and the second route is based on establishing (empirical) scaling correlations that relate the chaoticness of the fluidized bed to bed size, fluidizing conditions and particle properties.
Secondly, it is suggested that the concepts of chaos control that have recently been developed in the literature may also be applied to interactively control the dynamics of fluidization in order to, for example, enhance conversion and selectivity by influencing gas flow and bubble size distributions.
Conclusions are drawn that focus on general observations and suggestions for further work.
SUGGESTED PROJECTS.. Finally outlines are presented for four projects that may be considered for funding by IFPRI. In these outlines it is proposed to support chaos research in the following areas:
(A) dimensionless scaling of bubbling fluidized beds using the concept of chaotic similarity;
(B) scaling of bubbling fluidized beds based on empirical correlations that relate chaotic dynamics to bed size and operating conditions;
(C) chaos analysis of flow regimes in high velocity, circulating fluidized bed;
(D) development of a chaos control technique for bubbling fluidized beds.