Pressure Fluctuations In Fluidized Beds Research Articles

Detecting densified zone formation in membrane-assisted fluidized bed reactors through pressure measurements

This work reports the results of an experimental investigation on densified zone formation in a membrane fluidized bed reactor using combined pressure fluctuation and PIV (Particle Image Velocimetry) measurements. A pseudo 2D experimental setup was used, where porous plates on the back plate of column mimicked gas extraction through flat vertically inserted membranes. The maximum in the standard deviation of pressure fluctuations, commonly employed to indicate the transition to turbulent fluidization, shifted to lower fluidization velocities with an increase in the fraction of fluidizing gas being extracted. Flow visualization showed that this result is connected to the onset of stable densified zone formation, which occurred at progressively lower fluidization velocities as the gas extraction fraction was increased. It has also been found that the extent of densified zones quantified using instantaneous particle velocity maps collected by PIV increases with increasing gas extraction rates. This effect became larger for smaller particles. Results have therefore shown that the peak in pressure fluctuations in fluidized beds with gas extraction through flat vertical membranes indicates the onset of densified zones formation rather than turbulent fluidization. Such densified zones can have substantial detrimental effects (such as induced mass transfer limitations, gas bypass etc.) on the reactor performance and can be identified via pressure measurements, as illustrated in this work.

Analysis of pressure fluctuations in fluidized beds. III. The significance of the cross correlations

In previous parts of this series we analyzed experimental times series that represent pressure fluctuation in a fluidized bed. Part I focused on the existence of long-range correlations in the series, and demonstrated that the multiscale probability density function of the successive increments of the data may be well-approximated by the Castaing equation that has been proposed for modeling velocity fluctuations in turbulent flows. Part II reconstructed the time series and demonstrated that they can be well-represented by the Fokker–Planck and Langevin equations, which provide probabilistic predictions for the future trends of the pressure. In the present paper we study cross correlations between the time series, measured at three locations in the fluidized bed. Past analyses had presumed that the time series are stationary, or that they can be transformed to one. The pressure time series are not, however, always stationary. Three distinct, but complementary methods that are suitable for non-stationary series are used to analyze the cross correlations. They are the multifractal detrended cross-correlation analysis (MF-DXA), the so-called Qcc(m) test in conjunction with the statistical test – the χ2(m) distribution – and the cross wavelet transform (XWT) function. The Qcc(m) test provides qualitative evidence for the presence of the cross correlations, whereas the MF-DXA identifies and quantifies the strength of long-range cross correlations between simultaneously recorded time series. The analysis by the MF-DXA also indicates that the pressure time series and the cross correlations between them are multifractal, hence confirming the long-range nature of the correlations identified in Part I. The results are confirmed further by the XWT analysis. The effect of time scales on the correlations and cross correlations is studied in detail. It is shown that such correlations exist at all the time scales. The cross correlations are then used for characterization of bubble sizes, bubbling frequency, and the sources of the pressure fluctuations.

Multiscale probability distribution of pressure fluctuations in fluidized beds

Analysis of flow in fluidized beds, a common chemical reactor, is of much current interest due to its fundamental as well as industrial importance. Experimental data for the successive increments of the pressure fluctuations time series in a fluidized bed are analyzed by computing a multiscale probability density function (PDF) of the increments. The results demonstrate the evolution of the shape of the PDF from the short to long time scales. The deformation of the PDF across time scales may be modeled by the log-normal cascade model. The results are also in contrast to the previously proposed PDFs for the pressure fluctuations that include a Gaussian distribution and a PDF with a power-law tail. To understand better the properties of the pressure fluctuations, we also construct the shuffled and surrogate time series for the data and analyze them with the same method. It turns out that long-range correlations play an important role in the structure of the time series that represent the pressure fluctuation.

Comparison of bubble growth obtained from pressure fluctuation measurements to optical probing and literature correlations

Pressure fluctuation measurements (PFM) along with optical probing (OP) are conducted in a 145mm ID glass column fluidized bed with Geldart B particles to evaluate and compare the measured bubble growth with commonly used literature correlations. Bubble growth from PFM is evaluated by spectral data decomposition, while OP is evaluated by a bubble linking algorithm. It is shown that bubble sizes obtained from literature correlations vary widely for the same fluidization condition due to different methods in their determination. Compared to optical probe measurements, the best agreement is found with the correlation of Horio and Nonaka [A generalized bubble diameter correlation for gas-solid fluidized beds. AIChE J. 33 (1987) 1865–1872]. This agreement is in particular good for gas velocities u0/umf>5 and bed heights below the influence of slugging. Since PFM yield a characteristic length scale only proportional to the actual bubble size, comparison with bubble sizes from OP and literature correlations is not straightforward. However, this proportionality can be assessed by a physical model of Puncochár and Drahos [Origin of pressure fluctuations in fluidized beds. Chem. Eng. Sci. 60 (2005) 1193–1197]. Their physical model yields a proportionality constant of 3 which matches well the PFM data for gas velocities u0/umf>5.

Analysis of pressure fluctuations in fluidized beds. I. Similarities with turbulent flow

Experimental data for pressure fluctuation time series in a fluidized bed are analyzed by several distinct, but complementary methods. First, we compute a multiscale probability density function (PDF) for the successive increments of the data. The results demonstrate the evolution of the PDF from the short to long time scales, and indicate striking similarity to the Castaing equation that has been proposed for modeling velocity fluctuations in turbulent flows. Next, to further check the results we compute the structure function of the successive increments of the data. We find that the fluctuations exhibit multifractal behavior, which is also prevalent in turbulence. The multifractality implies that the pressure fluctuates differently over distinct time scales. To understand the origin of the multifractality, we use a powerful method of analysis, namely, the multifractal detrended fluctuation analysis (MF-DFA) in order to analyze the data. The results confirm the multifractal property of the data. To better understand the similarities between the pressure fluctuations in fluidized beds and velocity fluctuations in turbulent flow, and whether the multifractality is due to extended correlations in the data or because the PDF of the successive increments is broad, we also construct the shuffled and surrogate series for the data and analyze them by the MF-DFA method. Comparison of results for the original data with the shuffled and surrogate time series indicates that the correlations in the pressure are responsible for the multifractality of the data, rather than the broadness of the PDF.

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

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.

Comparison of decoupling methods for analyzing pressure fluctuations in gas‐fluidized beds

AbstractTwo methods of decoupling pressure fluctuations in fluidized beds by using the incoherent part (IOP) of absolute pressure (AP) and differential pressure (DP) fluctuations are evaluated in this study. Analysis is conducted first to demonstrate their similarities, differences, and drawbacks. Then, amplitudes, power spectral densities, mean frequencies, coherence functions, and filtering indices of the IOP of AP and DP fluctuations are calculated and compared based on experimental data from a two‐dimensional fluidized column of FCC particles. Derived bubble sizes are also compared with the sizes of bubbles viewed in the two‐dimensional bed. The results demonstrate the similarity of these two methods in filtering out global compression wave components from absolute pressure fluctuations, especially those generated from oscillations of fluidized particles and gas flow rate fluctuations. However, both methods are imperfect. Neither can filter out all the compression wave components and retain all the useful bubble‐related wave components. Their amplitudes can be used to characterize global bubble property and quality of gas–solids contacting in bed, but they do not give accurate measurement of bubble sizes. © 2009 American Institute of Chemical Engineers AIChE J, 2010

Standard deviation of absolute and differential pressure fluctuations in fluidized beds of group B particles

This work describes the behaviour of the standard deviation of pressure fluctuations, σ, in fluidized beds for group B particles in the bubbling regime. An empirical–theoretical function, which depends on the gas velocity, is proposed for predicting the pressure signal fluctuations, and the corresponding values of σ are calculated. The differences in the standard deviation of pressure fluctuations obtained for absolute or differential sensors are analyzed and compared to experimental values corresponding to different bed dimensions, pressure probe positions and particle properties.

Origin of pressure fluctuations in fluidized beds

The present paper shows a novel approach in interpreting the standard deviation of pressure fluctuations in a fluidized bed with respect to the spectral analysis. Based on several realistic assumptions for a freely bubbling bed, the physical model was proposed for the standard deviation of incoherent part of pressure fluctuations. Using the concept of energy dissipated in a fluidized bed the plausible explanation of linear dependence of standard deviation calculated from the total pressure signal on the excess gas velocity was given and verified experimentally.

Similarity between chaos analysis and frequency analysis of pressure fluctuations in fluidized beds

In literature the dynamic behavior of fluidized beds is frequently characterized by spectral analysis and chaos analysis of the pressure fluctuations that are caused by the gas–solids flow. In case of spectral analysis, most often the power spectral density (PSD) function is quantified, for example, by the frequency at the largest power and/or by the power-law fall-off at the higher frequencies. In case of chaos analysis, most often the correlation entropy (or Kolmogorov entropy) is used to characterize the fluidized bed dynamics. In this paper, it is shown theoretically that a relationship exists between the correlation entropy and the PSD function. This relationship is experimentally verified by a large set of experimental pressure fluctuation data from slugging, bubbling, and circulating fluidized beds. It is shown that the correlation entropy is linearly proportional to the average frequency that is obtained from the PSD function. As the average frequency can be directly interpreted in terms of the physical phenomena underlying the pressure fluctuations, the average frequency is suggested as a first, simple, characteristic of fluidized bed dynamics.

Statistical and deterministic chaos analysis of pressure fluctuations in a fluidized bed of polymer powders

The absolute and differential pressure fluctuations in gas-solid fluidized beds have been analyzed by statistical and deterministic chaos methods. Linear low-density polyethylene (LLDPE) particles with a mean diameter of 1.23 mm were used as a fluidizing material. The statistical methods are composed of the mean, standard deviation, skewness and kurtosis, and the deterministic methods are composed of autocorrelation, mutual information function, pseudo-phase space and correlation dimension. The minimum slug velocity of LLDPE particles is found to be 0.34 m/s by using the statistical and deterministic methods. As slugs appear and grow with increasing gas velocity, pressure fluctuations in the fluidized bed of LLDPE are oscillated and more periodic.

Multi-Fractal Signal Simulation of Local Instantaneous Pressure in a Bubbling Gas-Fluidized Bed—Part 1: Signal Simulation

The purpose of this paper, in two parts, is to (1) determine the fractal characteristics of time-series pressure data from gas-fluidized bed, (2) evaluate the possibility of using the Weierstrass-Mandelbrot (WM) function to simulate the fractal and dynamic characteristics of such pressure data, (3) refine the method of signal simulation introduced by Humphrey et al. (1992) for turbulence data, and (4) to investigate the nonlinear, fractal, and physical phenomena of pressure in a fluidized bed as a function of angular position around a cylinder, location in the bed, and the fluidization ratio. Part 1, (1) provides evidence that pressure signals from a bubbling fluidized bed are fractal, and delineates the range of scales over which these signals are fractal, (2) demonstrates that simulated signals have statistical measures essentially identical to experimental signals, and (3) relates parameters of the WM function to the physics of pressure fluctuations in fluidized beds. In addition, a refinement to the method of simulation employed by Humphrey et al. (1992) is proposed that employs the method presented by Higuchi (1988) to identify the fractal dimension and upper scaling frequency. Also, a direct means of specifying the amplitude of the fractal component of the signal results from using the peak-to-peak pressure of the experimental signal.

Pressure fluctuations in a fluidized bed

AbstractAn on‐line statistical study of the pressure fluctuations in fluidized beds was conducted by using pressure transducers, a correlation and probability analyzer and a Fourier transform analyzer. The causes of the fluctuations were explored, and the effects of the gas velocity, bed height, particle size and distributor design on the major frequency and amplitude of the fluctuations were investigated. The results indicate that the motion of bubbles appears to be the major cause of the pressure fluctuations in the upper portion of a fluidized bed. In the lower portion, the combined effects of the formation of large bubbles in the middle portion of the bed, the formation of small bubbles near the distributor, and the jet flow immediately above the distributor appear to be the major causes of pressure fluctuations.

Periodic pressure fluctuations in fluidized beds

In shallow gas fluidized beds harmonic oscillations of the pressure around its equilibrium value can be observed. Three aspects of these vibrations have been analysed: the frequency, the critical bed height and the damping. The frequency decreases with the inverse of the square root of the bed height for values below the critical height. For bed heights larger than the critical height the fluctuations cease to be harmonic, the bed breaks up and voids are formed leading to the formation of bubbles or slugs. The critical bed height can be calculated from the frequency and the wave velocity. The maximum value of the critical bed height is a few hundred particle diameters, thus most beds will fluidize heterogeneously. Damping of the oscillations is governed by the ratio of the fluid- to solids-density; the lower this values the higher the damping. The damping is liquid fluidized beds is such that oscillations are prevented.