Nonlinear dynamics of a gas–solid fluidized bed by the state space analysis M. Karimi, N. Mostoufi, R. Zarghami n , R. Sotudeh-Gharebagh Multiphase Systems Research Lab., Oil and Gas Center of Excellence, School of Chemical Engineering, College of Engineering, University of Tehran, P.O. Box 11155/4563, Tehran, Iran article info Article history: Received 29 May 2010 Received in revised form 20 April 2011 Accepted 14 June 2011 Available online 24 June 2011 Keywords: Nonlinearity test Surrogate data series Time series The S-statistic Null hypothesis Fluidized bed abstract The S-statistic based on the comparison of two reconstructed attractors in the phase space was utilized to study whether two dependent time series (namely, evaluation and reference time series) were originated from the same mechanism or not. Evaluation time series were measured during the bed operation in a gas–solid bubbling fluidized bed and reference time series were generated according to phase randomized surrogate data series from the evaluation series. The results indicated that nonlinearity can be observed where the energy contribution of the macro-structures is more than that of the finer structures. It was found that a minimum in the energy of the finer structures with an increase in velocity corresponds to the transition between macro-structures and finer structures of the bubbling fluidization system. This conclusion can eventually help choose proper control systems and evaluation methods of signals in fluidized beds. & 2011 Elsevier Ltd. All rights reserved. 1. Introduction The hydrodynamics of gas–solids fluidized bed is governed by a complex nonlinear dynamic relationship, which is mainly con- trolled by different dynamic phenomena that occur in the bed. Examples of these phenomena are bubble formation, bubble coalescence and splitting, bubble passage as well as particles behaviors (van der Schaaf et al., 1998). If the hydrodynamics of the fluidized system is modeled with a set of nonlinear governing equations, then a proper understanding of the state of the fluidized bed at a certain time can be determined. However, the governing equations of these systems are complex and unknown. In this case, a quantitative interpretation of the hydrodynamics of the fluidized bed can be achieved through time series evaluation of the measured signals, such as pressure. In order to use nonlinear techniques in fluidized beds, the first question to answer is whether patterns of the measured signals from the fluidized beds are nonlinear systems or linearly correlated stochastic systems. Any UPO (Unstable Periodic Orbit) identified from experimental noisy fluidized bed time series has not been reported that can identify unambiguously a chaotic deterministic dynamics. In con- trast, what has been reported previously in fluidized bed (Daw and Halow, 1991; Schouten and van den Bleek, 1998; van der Stappen et al., 1993; Skrzycke et al., 1993; Hay et al., 1995) are indications of some chaotic dynamical features appearing in some measured time series, and according to that, it has been extensively shown the fact that tools derived from the deterministic chaos theory can be extremely useful to characterize the dynamics carried in those signals. A number of traditional nonlinear measures (TNM) such as the correlation dimension (Grassberger and Procaccia, 1983; Ellner, 1988), entropy (Schouten et al., 1994a, 1994b) and Lyapunove exponents have been developed based on concepts from nonlinear dynamics and theory of deterministic methods with several problems such as large data requirements, stationarity, lacunarity and edge effects (Basu and oufoula-Georgiou, 2002), presence of noise due to the influence on the cut-off length or the nominal attractor diameter (Hively and Protopopescu, 2004). The TNM are adequate to discriminate between clear-cut regular and chaotic dynamics; their sensitivity is not sufficient to distinguish between slightly different chaotic regimes, especially when the data are noisy (Hively and Protopopescu, 2004). A few researchers found that the nonlinear behavior of a fluidized bed was resembled to the noise rather than being nonlinear low dimensional system (Jonsson et al., 2000). Tam and Devine (1986) observed no low-dimensional attractor when evaluated their data in the state space domain. Letaief et al. (1995) found noise characteristics in bubbling flui- dized bed, which were similar to fractal Brownian motion. Zarghami (2009) found that the nonlinearity was mostly observed at low correlation dimensions when pressure fluctuations were analyzed in the state space. Tests of nonlinearity in the state space are based on compar- ison between two reconstructed attractors of two time series. These tests are based on generating a surrogate data series by a stochastic process from the original data set, which has some of the attributes of the reference or original data series (Theiler et al., 1992a, 1992b; Schreiber and Schmitz., 1996; Barnard et al., 2001). Contents lists available at ScienceDirect journal homepage: www.elsevier.com/locate/ces Chemical Engineering Science 0009-2509/$ - see front matter & 2011 Elsevier Ltd. All rights reserved. doi:10.1016/j.ces.2011.06.022 n Corresponding author. Fax: þ98 21 6646 1024. E-mail address: rzarghami@ut.ac.ir (R. Zarghami). Chemical Engineering Science 66 (2011) 4645–4653