Analysis of pressure fluctuations in fluidized beds. II. Reconstruction of the data by the Fokker–Planck and Langevin equations

Abstract

In this part of the sequel we develop a continuum representation of the pressure fluctuation time series p( t) for a fluidized bed (FB), analyzed in part I, by using stochastic methods based on the Markov processes. It is shown that the data may be represented by Markov series with a Markov time scale t M that is estimated based on the data. Using the relation between the Markov processes and the Kramers–Moyal (KM) expansion that is a continuum equation that involves, in principle, an infinite number of coefficients, we represent the pressure fluctuation time series by a KM expansion. However, since the third and higher-order coefficients of the expansion are very small, the KM expansion reduces to a Fokker–Planck (FP) equation that represents p( t) in terms of a drift and a diffusion coefficients that are computed based on the data. The FP equation is, in turn, equivalent to a Langevin equation, which is used to reconstruct the data, i.e. generate the time series that mimic, in a statistical sense, the original data. Thus, the Langevin equation may also be used to make probabilistic predictions for the future values of the pressure over time scales that are of the order of the Markov time scale t M . The accuracy of the reconstructed series and, hence, their continuum representation, is demonstrated. We also compute the frequency ν α + of level-crossing at a given level α , i.e. the relative frequency (probability) of occurrence of the event defined, for two times t i−1 and t i , by, ν α + = P [ p ( t i ) > α , p ( t i − 1 ) < α ] , where P( x) is the probability of the event. Thus, ν α + yields the frequency that a given pressure fluctuation level may be expected. The average time that one should wait in order to observe the pressure at a given level again is also computed. The two quantities may be particularly important for modeling of the FBs. A relation is presented between the drift and diffusion coefficients of the FP equation and the Hurst exponent that has previously been used to describe the pressure fluctuation time series in terms of self-affine stochastic distributions.