Stochastic, fractal and chaotic modeling of multiphase flow systems

Numerous process systems in general and multiphase flow systems in particular lend themselves to a stochastic description due to their inherent complexity and fluctuating characteristics. Bubble columns, boiling loops, and fluidized beds are notable examples of such systems wherein random generation and coalescence of bubbles lead to pressure and density fluctuations. Analysis of these systems utilizing short-memory models based on Markovian assumptions have been widely reported. Nevertheless, the results of the present work indicated that pressure fluctuations from various multiphase flow systems exhibit long-term correlation. These systems appear to be better described by the relatively novel concept of fractals, or more specifically "fractional Brownian motion" (fBm) comprising "fractional Gaussian noise" (fGn). In the present work, the pressure fluctuations in different multiphase flow systems have been analyzed and successfully modeled by resorting to the concept of fGn. The stochastic model developed for the pressure fluctuations comprised a fGn component and a wave-like component. A theoretical autocorrelation function has been developed based on the model. Comparison of the model with the experimental results indicated that fGn is indeed a pragmatic model for correlated time series.; A detailed simulation study of fGn has been conducted; analytic results concerning fGn and statistical properties of its characteristic parameter, the Hurst exponent, H, are very few in the available literature. Two methods of estimation of H, from the simulated data, namely, the rescaled range (R/S) and the maximum likelihood (ML) were compared. The mean squared errors of the rescaled range estimate, H{dollar}sb{lcub}rm R/S{rcub}{dollar}, and the maximum likelihood estimate, H{dollar}sb{lcub}rm ML{rcub}{dollar}, clearly indicated that H{dollar}sb{lcub}rm ML{rcub}{dollar} is a better estimator. Approximate acceptance regions for (1-C) level test for hypothesis H = h{dollar}sb{lcub}rm o{rcub}{dollar} are presented for both the methods of estimation. The confidence intervals based on the assumption of asymptotic normality of H{dollar}sb{lcub}rm ML{rcub}{dollar} were reasonable and improved with the sample size. The MSE of H{dollar}sb{lcub}rm ML{rcub}{dollar} also decreased with the increase in the series length.; Finally, to answer the often arising question as to the possible presence of chaotic dynamics in multiphase flow systems, pressure fluctuation signals for bubble column have been interpreted in terms of recently-developed methods for chaotic time-series analysis. The results confirmed that the pressure fluctuations are indeed governed by an underlying stochastic process.