New Methods for Extraction of Important Information from Various Time Series in Bubble Columns
Visiting Lecturer at Department of Chemical Engineering Tokyo Institute of Technology (Japan)
In this presentation will be demonstrated that useful information can be extracted from various time-dependent signals based on new methods employing the concept of reconstruction of the time series into numerous state vectors. This reconstruction technique is part of the nonlinear chaos theory (Schouten and Van den Bleek, Chem. Eng. J. 53, 75-87, 1993). The development of new methods for extraction of useful and hidden information from various time-dependent signals (differential pressure and gas holdup fluctuations as well as X-ray scans) is needed mainly for flow regime identification in various bubble column reactors. The topic is important since the degrees of mixing, mass and heat transfer are different in each hydrodynamic regime. In addition, both the design and scale-up of bubble columns depend strongly on the successful flow regime identification.
The first step in the time series reconstruction is the choice of the number of elements per state vector. This parameter is called embedding dimension and in all reconstructions in this presentation it has been fixed at 50. After the successful time series reconstruction, at least 289 vector pairs (without overlapping elements) have been generated. The elements of the second vector in the compared vector pair have been selected to be always with 100 positions later in the time series in order to avoid data overlapping. The distance between each two vectors has been estimated by means of the maximum norm. For this purpose, a cut-off length has been defined and set equal to three times the average absolute deviation (AAD) of the signal.
The vector reconstruction of the time series has been applied successfully to X-ray scans recorded by ultrafast X-ray tomography in an air-water bubble column (0.1 m in diameter, clear liquid height=0.66 m). The column has been equipped with a perforated plate distributor (55 holes, Ø 0.5×10-3 m). The pixel values (from the X-ray scans) in four regions of the column’s cross-section have been analyzed based on the information entropy (IE) algorithm (Nedeltchev and Shaikh, Chem. Eng. Sci. 100, 2-14, 2013). Only the vector pairs with distance smaller than the cut-off length have been taken into account. Every time-dependent signal (29,000 points) in the four regions has been reconstructed into 289 vector pairs and only the cases when the distance of the same vector pair is smaller than the cut-off length in only one region have been used in the IE algorithm. It has been assumed that these cases are associated with the degree of randomness in the signal. Based on the well-pronounced local minima in the IE profile as a function of superficial gas velocity UG two main transition velocities Utrans at 0.045 and 0.07 m/s have been identified. The first Utrans value has identified the end of the homogeneous flow regime. Its value has been confirmed by means of the overall gas holdup profile. The second Utrans value has distinguished the onset of the heterogeneous flow regime, which is characterized with gross liquid macrocirculation.
A new parameter called degree of randomness (DR) has been developed and tested on differential pressure fluctuations recorded in an air-water two-dimensional bubble column (width: 0.2 m, depth: 0.04 m, clear liquid height: 0.75 m). The column has been equipped with 6 needles (diameter: 1×10-3 m) and the distance between them has been set at 9×10-3 m. Again two well- pronounced local DR minima have been identified. The first one has appeared at Utrans=0.012 m/s (end of the homogeneous regime), whereas the second one has occurred at Utrans=0.039 m/s (onset of the heterogeneous regime). A comparison between the results in a two- and three- dimensional bubble column operated with an air-water system reveals that the transitions in a rectangular bubble column occur always earlier. The reason is the enhanced turbulence due to the presence of corners in the rectangular cross-section.
Finally, the main flow regime boundaries in a bubble column with internals have been investigated based on a novel statistical-chaotic method. The latter has been applied to gas holdup fluctuations recorded by means of a wire-mesh sensor (8 8 wires). The bubble column (0.1 m in ID) has been operated with an air-deionized water system at ambient conditions. The column has been equipped with a perforated plate distributor (55 holes, Ø 0.5×10-3 m). 37 vertical tubes (arranged in a square pitch with a diameter of 8 10-3 m) have been installed as internals. The time series (60,000 points) have been divided into 6 intervals and numerous vector pairs have been generated. Those with distance smaller than the cut-off length have been used to calculate the modified average absolute deviation (AAD) based on the six intervals.
As a result of an original combination of statistical and chaotic parameters it has been found that in the core of the bubble column with internals, the first transition velocity Utrans-1 (end of homogeneous regime) occurs at superficial gas velocity UG of 0.06 m/s, whereas the second transition velocity Utrans-2 (end of heterogeneous regime) appears at UG=0.13 m/s. At these critical velocities the modified AAD has exhibited well pronounced minima. In the column’s core the existence of transition flow regime has not been identified. In the annulus of the bubble column with internals, three transition velocities (at UG=0.03, 0.06 and 0.10 m/s) were identified. The first transition velocity identified the end of the gas maldistribution regime. The second and third critical velocities distinguished the ends of the homogeneous and heterogeneous regimes, respectively.