Non-intrusive determination of bubble and slug length scales in fluidized beds by decomposition of the power spectral density of pressure time series
Introduction
A gas–solid fluidized bed is a commonly encountered reactor type in chemical industries for, e.g., efficiently contacting gaseous reactants with catalyst particles. Because the performance of a fluidized bed is highly dependent on the hydrodynamic state of fluidization, fluidized bed hydrodynamics have been the subject of study for many years now.
In bubbling fluidized beds especially the gas bubble characteristics, i.e. gas bubble size and velocity, are important for fluidized bed performance. On the one hand, gas bubbles provide mixing of the particles, thus enhancing heat and mass transfer; on the other hand, the gaseous reactants in the gas bubbles are barely contacted with catalyst particles, decreasing the reactor efficiency. Knowledge of these gas bubble characteristics is thus needed in order to properly design fluidized bed reactors.
The gas bubble characteristics can be investigated with direct measurement methods which measure the voidage in a fluidized bed, viz. optical, capacitance, and radioactive methods. Optical and capacitance methods provide local information on voidage with the disadvantage that only a small portion of the gas bubble is detected. With radioactive methods more global information can be obtained, but these methods are more expensive and are generally restricted to small-scale fluidized beds.
Instead of determining the gas bubble characteristics directly through voidage measurements, pressure fluctuations caused by the rising gas bubbles can be taken as an indirect measure of bubble size and velocity. Measuring pressure fluctuations is attractive because it is a relatively simple, non-intrusive, and inexpensive technique, applicable to a wide range of experimental conditions. However, in addition to pressure fluctuations originating from rising gas bubbles, other sources generating pressure fluctuations are present in a fluidized bed too, e.g. bed mass oscillation, bubble coalescence, bubble eruption, and gas turbulence.
In this paper the power spectral density (PSD) of time series of pressure fluctuations is used to characterize the gas bubble characteristics in freely bubbling fluidized beds. In literature, the dominant frequency in the PSD of a fluidized bed is commonly reported and associated with a regular fluctuation of the bed height (Roy et al., 1990). A power-law fall-off is generally observed at higher frequencies in the PSD of voidage and pressure time series. This power-law fall-off can be used to characterize fluidized bed hydrodynamics and to validate fluidized bed models (Ding and Tam, 1994; Nowak et al., 1993). Though the PSD is a commonly applied analysis tool in fluidized bed research, confusion still exists about the physical origin of the dominant frequencies and about the observed power-law fall-off. Therefore, in this paper, a method is proposed to decompose the PSD of time series of pressure fluctuations into its different components. Furthermore, these components are identified with the different physical phenomena underlying the time series of pressure fluctuations in gas–solid fluidized beds.
The proposed method is also applied to gas–solid flow in the riser of a circulating fluidized bed. In the riser, generally two different regions can be discerned: (1) a dense region in the bottom part of the riser with a constant particle concentration and irregularly shaped voids of gas, and (2) a dilute region in the upper part of the riser in which clusters of particles are present. The characteristics of the gas voids and the clusters are important for fluidized bed performance and, here too, the PSD of pressure time series can be decomposed into components corresponding to the physical phenomena underlying the pressure time series.
Section snippets
Theory
Most definitions given here can be found in any textbook on spectral analysis, e.g. Priestley (1981), Randall (1987), and Newland (1993); the most important definitions will be discussed briefly here. The PSD, Φxx, of a pressure time series px(t) measured at position x in a fluidized bed is defined aswhere is the one-sided Fourier transform of px(t), determined using a fast Fourier transform algorithm, and denotes ensemble averaging. For two time series measured
Bubbling fluidized bed
Time series of pressure fluctuations were recorded in a 0.385 m ID bubbling fluidized bed filled with Geldart-B sand particles (dp=0.39 mm, ρs=2650 kg/m3, umf=0.14 m/s). The settled bed height was 0.30 m (solids inventory = 51.8 kg), and the superficial gas velocity ranged from 0.20 to 0.95 m/s. In each experiment pressure time series were recorded simultaneously along column height and in the plenum. The in-bed pressure probes were located at 0.04, 0.09, 0.14,0.19, 0.24, and 0.29 m above the
Bubbling fluidized bed
For the bubbling fluidized bed typical PSDs are given in Fig. 2 at different measurement heights and at a gas velocity of 0.70 m/s. Typically, the PSDs display a power-law fall-off at high frequencies with a slope of approximately −4. This power-law fall-off was observed for all gas velocities. The PSD decreases in power with increasing measurement height. If the PSDs were caused by pressure fluctuations of gas bubbles only, the opposite would be expected: higher in the bed larger bubbles with
Conclusions
By using the coherence between time series of pressure fluctuations measured simultaneously in a fluidized bed and in its plenum, the PSD of the time series can be split up into the coherent-output and incoherent-output PSDs. In bubbling fluidized beds, the power in the coherent-output PSD is generated by fast pressure fluctuations originating primarily from bubble coalescence and bubble formation. The power in the incoherent-output PSD corresponds to pressure fluctuations generated by passing
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